OptimizedPolynomialVirtualFieldsMethodforConstitutive...

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Research Article Optimized Polynomial Virtual Fields Method for Constitutive Parameters Identification of Orthotropic Bimaterials Xueyi Ma , Yu Wang, and Jian Zhao School of Technology, Beijing Forestry University, Beijing 100083, China Correspondence should be addressed to Jian Zhao; [email protected] Received 2 July 2020; Accepted 13 July 2020; Published 28 August 2020 Academic Editor: Jinyang Xu Copyright © 2020 Xueyi Ma et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Heterogeneous materials are widely applied in many fields. Owing to the spatial variation of its constitutive parameters, the mechanical characterization of heterogeneous materials is very important. e virtual fields method has been used to identify the constitutive parameters of materials. However, there is a limitation: constitutive parameters of one material have to be a priori; then, constitutive parameters of the other one can be identified. Aiming at this limitation, this article presents a method to identify the constitutive parameters of heterogeneous orthotropic bimaterials under the condition that constitutive parameters of both materials are all unknown from a single test. A constitutive parameter identification method of orthotropic bimaterials based on optimized virtual field and digital image correlation is proposed. e feasibility of this method is verified by simulating the deformation fields of a two-layer material under three-point bending load. e results of numerical experiments with FEM simulations show that the weighted relative error of the constitutive parameter is less than 1%. e results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of a confidence interval on the identified stiffness components. e results of numerical experiments with DIC simulations show that the weighted relative error is 1.44%, which is due to the noise in the strain data calculated by DIC method. 1. Introduction Heterogeneous materials, such as functionally gradient materials, wood [1, 2], polymeric foams [3], biological materials [4], and composites [5], have been widely applied in engineering. Constitutive parameters identification of heterogeneous materials is not only the important subject in experimental mechanics but also has attracted extensive notice in the fields of solid mechanics, structure health monitoring, and medical diagnosis. e traditional method of identifying constitutive parameters can directly deter- mine all constitutive parameters of constitutive models through standard experiments (e.g., uniaxial tensile test). However, the investing of experimental involved in such standard test methods may increase when using anisotropic models and inhomogeneous materials, as well as bimaterial [6]. With the development of the photomechanics, full-field measurement methods (e.g., digital image correlation and moir´ e interferometry) combined with inverse methods have been developed so that the multiple constitutive parameters can be identified by a heterogeneous strain field [7]. e inverse methods belong to the updating methods (e.g., finite element updating method) and nonupdating methods (e.g., virtual fields method). e finite element updating method (FEUM) extract constitutive parameters through minimizing the cost function between the re- sponse of the finite element model and the real behavior iteratively; its simulation must be close to experimental conditions and its initial guess choice will affect the identification result [8]. e virtual fields method (VFM), which was put forward by Perrion and Gr´ ediac [9, 10], is a typical nonupdating method used in a linear material model. Compared with the updating methods, the Hindawi Advances in Materials Science and Engineering Volume 2020, Article ID 2974723, 27 pages https://doi.org/10.1155/2020/2974723

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Research ArticleOptimized Polynomial Virtual Fields Method for ConstitutiveParameters Identification of Orthotropic Bimaterials

Xueyi Ma Yu Wang and Jian Zhao

School of Technology Beijing Forestry University Beijing 100083 China

Correspondence should be addressed to Jian Zhao zhaojian1987bjfueducn

Received 2 July 2020 Accepted 13 July 2020 Published 28 August 2020

Academic Editor Jinyang Xu

Copyright copy 2020 Xueyi Ma et al +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Heterogeneous materials are widely applied in many fields Owing to the spatial variation of its constitutive parameters themechanical characterization of heterogeneous materials is very important +e virtual fields method has been used to identify theconstitutive parameters of materials However there is a limitation constitutive parameters of one material have to be a priorithen constitutive parameters of the other one can be identified Aiming at this limitation this article presents a method to identifythe constitutive parameters of heterogeneous orthotropic bimaterials under the condition that constitutive parameters of bothmaterials are all unknown from a single test A constitutive parameter identification method of orthotropic bimaterials based onoptimized virtual field and digital image correlation is proposed +e feasibility of this method is verified by simulating thedeformation fields of a two-layer material under three-point bending load +e results of numerical experiments with FEMsimulations show that the weighted relative error of the constitutive parameter is less than 1 +e results suggest that thevariation coefficient-to-noise ratio can perform a priori evaluation of a confidence interval on the identified stiffness components+e results of numerical experiments with DIC simulations show that the weighted relative error is 144 which is due to thenoise in the strain data calculated by DIC method

1 Introduction

Heterogeneous materials such as functionally gradientmaterials wood [1 2] polymeric foams [3] biologicalmaterials [4] and composites [5] have been widely appliedin engineering Constitutive parameters identification ofheterogeneous materials is not only the important subjectin experimental mechanics but also has attracted extensivenotice in the fields of solid mechanics structure healthmonitoring and medical diagnosis +e traditional methodof identifying constitutive parameters can directly deter-mine all constitutive parameters of constitutive modelsthrough standard experiments (eg uniaxial tensile test)However the investing of experimental involved in suchstandard test methods may increase when using anisotropicmodels and inhomogeneous materials as well as bimaterial[6]

With the development of the photomechanics full-fieldmeasurement methods (eg digital image correlation andmoire interferometry) combined with inverse methodshave been developed so that the multiple constitutiveparameters can be identified by a heterogeneous strain field[7] +e inverse methods belong to the updating methods(eg finite element updating method) and nonupdatingmethods (eg virtual fields method) +e finite elementupdating method (FEUM) extract constitutive parametersthrough minimizing the cost function between the re-sponse of the finite element model and the real behavioriteratively its simulation must be close to experimentalconditions and its initial guess choice will affect theidentification result [8] +e virtual fields method (VFM)which was put forward by Perrion and Grediac [9 10] is atypical nonupdating method used in a linear materialmodel Compared with the updating methods the

HindawiAdvances in Materials Science and EngineeringVolume 2020 Article ID 2974723 27 pageshttpsdoiorg10115520202974723

advantage of VFM is that it requires less specimenboundary conditions and geometric configurations[11 12]

A volume of research has focused on full-field mea-surement and VFM to identify the constitutive parameters ofmaterials +e VFM within peridynamic framework wasproposed to identify material properties in linear elasticity[13] +e special virtual fields method was adopted to resolvethe difficulty in identifying constitutive properties of in-compressible and nonhomogeneous solids [14] +e VFMwas combined with an elaborately designed test configu-ration in order to realize the identification of the anisotropicyield constitutive parameters [15] Several inverse methodsbased on the VFM have been developed including theFourier series-based VFM [16 17] the sensitivity-basedVFM [18] the eigenfunction VFM [19 20] and severaloptimizations to improve the accuracy of identificationresults [21 22]

+e heterogeneity of strain field caused by multipleconstitutive parameters of the complex constitutive modelhas been solved by the traditional VFM For heterogeneousmaterials the heterogeneity of strain field caused by thespatial variation of constitutive parameters requires furtherimprovement of the virtual field method According to theavailable literature studies several methods and applicationshave been studied to identify constitutive parametersidentification of heterogeneous material using VFM[6 23 24] For instance a bimaterial constitutive parameteridentification method based on the combination of VFMandMoire interference was proposed [25] and the VFMwasalso extended to identify the parameters [26] In additionseveral methods have been researched to lower the effects ofnoise in strain fields [26ndash29] +e optimized fields methodsuch as polynomial optimized virtual field [30] and piece-wise optimized virtual field [31] was developed to improveconvenience and accuracy Although the above researchesmake identification of constitutive parameters more rapidand convenient there is still a limitation for bimaterials theconstitutive parameters of one material must be a priori inorder to determine the constitutive parameters of the othermaterial +e purpose of this work is to propose an opti-mized virtual field that can not only extract the constitutiveparameters of the heterogeneous orthotropic bimaterialswithout knowing any material constitutive parameters butalso require only one single test

In this study to extract constitutive parameters of thebimaterials under the condition that constitutive pa-rameters of the both materials are unknown specialoptimized polynomial virtual fields combined with three-point bending test was proposed In Section 2 the basicprinciple of optimized polynomial VFM and its im-provement applied in bimaterials without knowing anymaterial constitutive parameters are introduced In Sec-tion 3 the FEM simulated experiments and the simulatedDIC experiments are conducted to verify the feasibility ofthe abovementioned method Finally Section 4 is theconclusions

2 Methodology

21 Basic Principle of Optimized Polynomial Virtual FieldsMethod +e integral form of the mechanical equilibriumequation of the VFM based on virtual work for a continuoussolid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowast denotes the strain tensor corre-sponding to ulowast T represents the external force associatedwith the stress tensor σ by the boundary S is a vector whichis the volume force applied over the volume V of thespecimen and a denotes the distribution of accelerationSuch a distribution will cause an additional volumetricforce distribution equal to minusρawith DrsquoAlembertrsquos principlethe virtual displacement field being kinematically admis-sible (KA)

For an orthotropic material in a plane-stress state theprinciple of virtual work is suitable for every KA virtualfield +is equation (see equation 2) uses four indepen-dent KA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively (see Appendix A1for the detailed equation derivations)

AQ B (2)

where

2 Advances in Materials Science and Engineering

A

1113946Sε1εlowast (1)11113872 1113873dS 1113946

Sε2εlowast (1)21113872 1113873dS 1113946

Sε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 1113946

Sε6εlowast (1)61113872 1113873dS

1113946Sε1εlowast (2)11113872 1113873dS 1113946

Sε2εlowast (2)21113872 1113873dS 1113946

Sε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 1113946

Sε6εlowast (2)61113872 1113873dS

1113946Sε1εlowast (3)11113872 1113873dS 1113946

Sε2εlowast (3)21113872 1113873dS 1113946

Sε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 1113946

Sε6εlowast (3)61113872 1113873dS

1113946Sε1εlowast (4)11113872 1113873dS 1113946

Sε2εlowast (4)21113872 1113873dS 1113946

Sε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 1113946

Sε6εlowast (4)61113872 1113873dS

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q

Q11

Q22

Q12

Q66

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B

1113946Lf

Tiulowast (1)i1113872 1113873dl

1113946Lf

Tiulowast (2)i1113872 1113873dl

1113946Lf

Tiulowast (3)i1113872 1113873dl

1113946Lf

Tiulowast (4)i1113872 1113873dl

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents

+e constraints of KA virtual fields ulowast(1) ulowast(2) ulowast(3) andulowast(4) are as follows

1113946Sε1εlowast (1)11113872 1113873dS 1 1113946

Sε2εlowast (1)21113872 1113873dS 0 1113946

Sε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 0 1113946

Sε6εlowast (1)61113872 1113873dS 0

1113946Sε1εlowast (2)11113872 1113873dS 0 1113946

Sε2εlowast (2)21113872 1113873dS 1 1113946

Sε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 0 1113946

Sε6εlowast (2)61113872 1113873dS 0

1113946Sε1εlowast (3)11113872 1113873dS 0 1113946

Sε2εlowast (3)21113872 1113873dS 0 1113946

Sε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 1 1113946

Sε6εlowast (3)61113872 1113873dS 0

1113946Sε1εlowast (4)11113872 1113873dS 0 1113946

Sε2εlowast (4)21113872 1113873dS 0 1113946

Sε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 0 1113946

Sε6εlowast (4)61113872 1113873dS 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

+e exact values and white noise constitute the measuredstrain component Assume that the noise components areuncorrelated to each other and presume that the noise

between points is also uncorrelated +erefore the principleof virtual work is as equation (5) (see Appendix A2 for thedetailed equation derivations)

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(5)

whereN1N2 andN6 denote the processes of scalar zero-mean stationary Gaussian generalized by the corre-sponding strain components ε1 ε2 and ε6 respectivelyand c denotes the measured amplitude of the randomvariable strain

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by equation(6) So directly identify Q11 Q22 Q12 and Q66 using thesefour special virtual displacement fields as equation (7) (seeAppendix A3 for the detailed equation derivations)

Advances in Materials Science and Engineering 3

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (2)6 N6dS11138771113876 1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (3)6 N6dS11138771113876 1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q)the vector containing the variances ofQ11Q22Q12 and Q66 one can write as equation (8) (seeAppendix A4 for the detailed equation derivations)

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

and (η(i))2 have two kinds of unknowns the constitutiveparameters and the unknown coefficients of the virtualfields So (η(i))2 can be expressed as

η(i)1113872 1113873

212Ylowast(i)HYlowast(i)

Sc

nc

1113888 1113889

2

QappG(i)Qapp (9)

where Ylowast is the vector related to the coefficients of virtualstrain fields H is the Hessian matrix of all monomials in εlowastand G(j) j 1 2 3 4 is the following square matrix

1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113944n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113944n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)

and the (η(i))2 coefficients i 1 2 3 4 depend on theselected virtual fields Also the virtual fields are selectedfor the Qij parameters extraction +us select the ap-propriate virtual field that can reduce the influence of

strain In this article the virtual fields were expandedby the polynomials +e displacement fields and thevirtual strain are as shown in equations (11) and (12)respectively

4 Advances in Materials Science and Engineering

ulowast1 1113944

m

i01113944

n

j0Aij

x1

L1113874 1113875

i x2

w1113874 1113875

j

ulowast2 1113944

m

i01113944

n

j0Bij

x1

L1113874 1113875

i x2

w1113874 1113875

j

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

εlowast1 1113944m

i11113944

n

j0Aij

x1

L1113874 1113875

i minus1 x2

w1113874 1113875

j

εlowast2 1113944m

i01113944

n

j1Bij

x1

L1113874 1113875

i x2

w1113874 1113875

j minus1

εlowast6 1113944

m

i01113944

n

j1Aij

i

L

x1

L1113874 1113875

i x2

w1113874 1113875

j minus1+ 1113944

m

i11113944

n

j0Bij

i

L

x1

L1113874 1113875

i minus 1 x2

w1113874 1113875

j

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

where Aij and Bij are the coefficients of the monomials m

and n are the polynomial degrees that define that themaximum number of monomials has used and L and w arethe typical dimensions of the x1 and x2 directionsrespectively

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (13) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (13)

where λ(i) is the vector containing Lagrange multipliers+e KA condition can produce several linear equations

and the number of this equations is depending on thenumber of supports +e special virtual field condition alsoleads to several linear equations the number of which de-pends on the type of constitutive model of specimen ma-terial +e two types of conditions can produce the followinglinear system as equation (14) (see Appendix A5 for thedetailed equation derivations)

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (14)

Attention that (η(i))2 coefficients also correspond on theunknown Qij parameters +us the optimization problem issolved for a given set of Qij parameters and the process is

iterative the first guess of theQij parameters is used to find thefirst group of four special virtual fields+en these four specialvirtual fields are used to find a new set of Qij parameters andrepeat abovementioned steps until the optimal Qij parameteris found In practice no matter what the parameter Qij isinitially selected this iterative process will converge quicklyUsually only two loops are sufficient to converge

22 Optimized Polynomial Virtual Fields Applied to theBimaterials without Knowing Any Material ConstitutiveParameters As shown in Figure 1 for the plane-stressspecimen with two different materials Part A and Part B andboth of them are set with the elastic orthotropic material Butthe constitutive parameters of A and B are unknown Asmentioned above the sum of exact values and white noise isthe measured strain Assume that the noise components areuncorrelated to each other and presume that the noisebetween points is also uncorrelated +erefore the gov-erning equation of the virtual fields method for the totalspecimen is as equation (15) (see Appendix B1 for the de-tailed equation derivations)

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS

+ Q66a1113946Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS

+ Q66b1113946Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS1113876

+ Q12a1113946Sa

εlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946Saεlowast6 N6dS

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS

+ Q12b1113946Sb

εlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946Sbεlowast6 N6dS1113877

1113946Lf

Tiulowasti( 1113857dl

(15)

+e constraints of the eight special virtual fields denotedthat ulowast(1) ulowast(2) ulowast(3) ulowast(4) ulowast(5) ulowast(6) ulowast(7) and ulowast(8) mustsatisfy the following equation

Advances in Materials Science and Engineering 5

1113946Sa

ε1εlowast (1)11113872 1113873dS 0 1113946

Sa

ε2εlowast (1)21113872 1113873dS 0 1113946

Sa

ε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 1 1113946

Sa

ε6εlowast (1)61113872 1113873dS 0

1113946Sb

ε1εlowast (1)11113872 1113873dS 0 1113946

Sb

ε2εlowast (1)21113872 1113873dS 0 1113946

Sb

ε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 0 1113946

Sb

ε6εlowast (1)61113872 1113873dS 0

1113946Sa

ε1εlowast (2)11113872 1113873dS 0 1113946

Sa

ε2εlowast (2)21113872 1113873dS 0 1113946

Sa

ε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 1 1113946

Sa

ε6εlowast (2)61113872 1113873dS 0

1113946Sb

ε1εlowast (2)11113872 1113873dS 0 1113946

Sb

ε2εlowast (2)21113872 1113873dS 0 1113946

Sb

ε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 0 1113946

Sb

ε6εlowast (2)61113872 1113873dS 0

1113946Sa

ε1εlowast (3)11113872 1113873dS 0 1113946

Sa

ε2εlowast (3)21113872 1113873dS 0 1113946

Sa

ε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 1 1113946

Sa

ε6εlowast (3)61113872 1113873dS 0

1113946Sb

ε1εlowast (3)11113872 1113873dS 0 1113946

Sb

ε2εlowast (3)21113872 1113873dS 0 1113946

Sb

ε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 0 1113946

Sb

ε6εlowast (3)61113872 1113873dS 0

1113946Sa

ε1εlowast (4)11113872 1113873dS 0 1113946

Sa

ε2εlowast (4)21113872 1113873dS 0 1113946

Sa

ε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 1 1113946

Sa

ε6εlowast (4)61113872 1113873dS 0

1113946Sb

ε1εlowast (4)11113872 1113873dS 0 1113946

Sb

ε2εlowast (4)21113872 1113873dS 0 1113946

Sb

ε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 0 1113946

Sb

ε6εlowast (4)61113872 1113873dS 0

1113946Sa

ε1εlowast (5)11113872 1113873dS 0 1113946

Sa

ε2εlowast (5)21113872 1113873dS 0 1113946

Sa

ε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS 1 1113946

Sa

ε6εlowast (5)61113872 1113873dS 0

1113946Sb

ε1εlowast (5)11113872 1113873dS 0 1113946

Sb

ε2εlowast (5)21113872 1113873dS 0 1113946

Sb

ε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS 0 1113946

Sb

ε6εlowast (5)61113872 1113873dS 0

1113946Sa

ε1εlowast (6)11113872 1113873dS 0 1113946

Sa

ε2εlowast (6)21113872 1113873dS 0 1113946

Sa

ε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS 1 1113946

Sa

ε6εlowast (6)61113872 1113873dS 0

1113946Sb

ε1εlowast (6)11113872 1113873dS 0 1113946

Sb

ε2εlowast (6)21113872 1113873dS 0 1113946

Sb

ε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS 0 1113946

Sb

ε6εlowast (6)61113872 1113873dS 0

1113946Sa

ε1εlowast (7)11113872 1113873dS 0 1113946

Sa

ε2εlowast (7)21113872 1113873dS 0 1113946

Sa

ε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS 1 1113946

Sa

ε6εlowast (7)61113872 1113873dS 0

1113946Sb

ε1εlowast (7)11113872 1113873dS 0 1113946

Sb

ε2εlowast (7)21113872 1113873dS 0 1113946

Sb

ε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS 0 1113946

Sb

ε6εlowast (7)61113872 1113873dS 0

1113946Sa

ε1εlowast (8)11113872 1113873dS 0 1113946

Sa

ε2εlowast (8)21113872 1113873dS 0 1113946

Sa

ε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS 1 1113946

Sa

ε6εlowast (8)61113872 1113873dS 0

1113946Sb

ε1εlowast (8)11113872 1113873dS 0 1113946

Sb

ε2εlowast (8)21113872 1113873dS 0 1113946

Sb

ε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS 0 1113946

Sb

ε6εlowast (8)61113872 1113873dS 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

So directly identify Q11a Q11b Q12a

Q12b Q22a Q22b Q66a and Q66b using the abovementionedfour special virtual displacement fields If the noise is

ignored approximate parameters expressed as Qapp areidentified +e components of which are defined as equation(17) (see Appendix B2 for the detailed equation derivations)

6 Advances in Materials Science and Engineering

Qapp11a 1113946

Lf

Tiulowast (1)i dl minus Q22a1113946

Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS

minus Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS

Qapp22a 1113946

Lf

Tiulowast (2)i dl minus Q11a1113946

Saε1εlowast (2)1 dS minus Q12a1113946

Saε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66a1113946

Saε6εlowast (2)6 dS

minus Q11b1113946Sbε1εlowast (2)1 dS minus Q22b1113946

Sbε2εlowast (2)2 dS minus Q12b1113946

Sbε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (2)6 dS

Qapp12a 1113946

Lf

Tiulowast (3)i dl minus Q11a1113946

Saε1εlowast (3)1 dS minus Q22a1113946

Saε2εlowast (3)2 dS minus Q66a1113946

Saε6εlowast (3)6 dS

minus Q11b1113946Sbε1εlowast (3)1 dS minus Q22b1113946

Sbε2εlowast (3)2 dS minus Q12b1113946

Sbε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (3)6 dS

Qapp66a 1113946

Lf

Tiulowast (4)i dl minus Q11a1113946

Saε1εlowast (4)1 dS minus Q22a1113946

Saε2εlowast (4)2 dS minus Q12a1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS

minus Q11b1113946Sbε1εlowast (4)1 dS minus Q22b1113946

Sbε2εlowast (4)2 dS minus Q12b1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (4)6 dS

Qapp11b 1113946

Lf

Tiulowast (5)i dl minus Q22b1113946

Sbε2εlowast (5)2 dS minus Q12b1113946

Sbε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (5)6 dS

minus Q11a1113946Saε1εlowast (5)1 dS minus Q22a1113946

Saε2εlowast (5)2 dS minus Q12a1113946

Saε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66a1113946

Saε6εlowast (5)6 dS

Qapp22b 1113946

Lf

Tiulowast (6)i dl minus Q11b1113946

Sbε1εlowast (6)1 dS minus Q12b1113946

Sbε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (6)6 dS

minus Q11a1113946Saε1εlowast (6)1 dS minus Q22a1113946

Saε2εlowast (6)2 dS minus Q12a1113946

Saε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66a1113946

Saε6εlowast (6)6 dS

Qapp12b 1113946

Lf

Tiulowast (7)i dl minus Q11b1113946

Sbε1εlowast (7)1 dS minus Q22b1113946

Sbε2εlowast (7)2 dS minus Q66b1113946

Sbε6εlowast (7)6 dS minus Q11a1113946

Saε1εlowast (7)1 dS

minus Q22a1113946Saε2εlowast (7)2 dS minus Q12a1113946

Saε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS minus Q66a1113946

Saε6εlowast (7)6 dS

Qapp66b 1113946

Lf

Tiulowast (8)i dl minus Q11b1113946

Sbε1εlowast (8)1 dS minus Q22b1113946

Sbε2εlowast (8)2 dS minus Q12b1113946

Sbε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS

minus Q11a1113946Saε1εlowast (8)1 dS minus Q22a1113946

Saε2εlowast (8)2 dS minus Q12a1113946

Saε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS minus Q66a1113946

Saε6εlowast (8)6 dS

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

To minimize the effect of noise taking Q11a as an ex-ample the variance of Q11a is as follows

V Q11a( 1113857 c2E Q

app11a1113946

Saεlowast (1)1 N1dS + Q

app22a1113946

Saεlowast (1)2 N2dS + Q

app12a 1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66a 1113946

Saεlowast (1)6 N6dS1113876 1113877

21113888

+ Qapp11b1113946

Sblowast (1)N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113889

+ c2E Q

app11b1113946

Sbεlowast (1)1 N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app22b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113888 1113889

VA Q11a( 1113857 + VB Q11a( 1113857

(18)

Advances in Materials Science and Engineering 7

As described above the virtual fields are expanded bypolynomial basis functions based on equation (11) +eunknown coefficients Aij and Bij are included in vector Y asfollows

Y

A00

⋮A30

⋮

A03

⋮

A33

B00

⋮

B30

⋮

B03

⋮

B33

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)

+e integrals of equation (15) are calculated usingvectors B11B22B12 and B66 through discrete sum ap-proximation so that

1113946Sa

ε1εlowast1( 1113857dS asymp Lwaε1aε

lowastia B11a middot Y

1113946Sa

ε2εlowast2( 1113857dS asymp Lwaε2aε

lowast2a B22a middot Y

1113946Sa

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwa ε1aε

lowast2a + ε2aε

lowast1a1113872 1113873 B12a middot Y

1113946Sa

ε6εlowast6( 1113857dS asymp Lwaε6aε

lowast6a B66a middot Y

1113946Sb

ε1εlowast1( 1113857dS asymp Lwaε1bε

lowast1b B11b middot Y

1113946Sb

ε2εlowast2( 1113857dS asymp Lwbε2bε

lowast2b B22b middot Y

1113946Sb

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwb ε1bε

lowast2b + ε2bε

lowast1b1113872 1113873 B12b middot Y

1113946Sb

ε6εlowast6( 1113857dS asymp Lwbε6bε

lowast6b B66b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

+e others that the definition from the virtual fields(vectorsH11H22H12 andH66) are related to the terms ofequation (18)

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

ε1εlowast2( 1113857dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast6( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

ε1εlowast2( 1113857dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast6( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

+e first KA virtual field conditions of three-pointbending are ulowast2 0 when x1 0 and x2 0 which implies

B00 0 (22)

+e second KA virtual field conditions of three-pointbending re ulowast1 0 when x1 L which means that

1113944

m

i0Aij 0 j 0 middot middot middot n (23)

+e optimized virtual fields are totally defined +efollowing equations can identify the constitutive parameters

8 Advances in Materials Science and Engineering

Q11a F

tulowast (a)2 x1 L x2 w( 1113857

Q22a F

tulowast (a)2 x1 L x2 w( 1113857

Q12a F

tulowast (a)2 x1 L x2 w( 1113857

Q66a F

tulowast (a)2 x1 L x2 w( 1113857

Q11b F

tulowast (a)2 x1 L x2 w( 1113857

Q22b F

tulowast (a)2 x1 L x2 w( 1113857

Q12b F

tulowast (a)2 x1 L x2 w( 1113857

Q66b F

tulowast (a)2 x1 L x2 w( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

3 Numerical Verification and Discussion

To evaluate the accuracy of the abovementioned optimizedpolynomial VFM numerical experiments with FEM simu-lations and DIC simulations are employed in this section

31 Numerical Verification of Optimized Polynomial VirtualFieldswithFEMSimulations In FEM simulation in order to

facilitate the simulation of bimaterials configuration a two-dimensional rectangular beam specimen was adopted andapplied a three-point bending load to get inhomogeneousin-plane deformation fields Based on the principle of thesymmetry of three-point bending only the left of thespecimen was modelled as shown in Figure 1

+e length width and thickness of the bimaterial beamspecimen were set as L 30mm w 20mm andt 23mm respectively and the width of the two materialregions was the same wa wb 10mm+e force was set asF 2544N +e four-node plane-stress element withthickness (Solid-Quad 4 nodes 182Plane strs wthk) wasemployed +e displacement constraint uy 0 is added inthe lower left corner of the model and the line constraintux 0 is added on the right side of the model

+e beam specimen was set as orthotropic materials +eengineering elastic constants in the FEM simulation areshown in Table 1 and the converted constitutive parametersin the elastic matrix are shown in Table 2

+e displacement fields and the strain fields of the three-point bending test of the orthotropic bimaterial obtainedthrough FEM simulations using the commercial softwareANSYS are shown in Figures 2 and 3

Table 3 shows the identification results of the hetero-geneous orthotropic bimaterials by using the optimizedpolynomial VFM as shown in Section 22 To study theidentification accuracy of the optimized polynomial VFMthe relative error of every constitutive parameter componentand the weighted relative error Wre were calculated Owingto there were 8 unknown constitutive parameters theweighted relative error employed here can be expressed asfollows

Wre

Qi de11a minus Q

rfe11a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22a minus Q

re22a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12a minus Q

re12a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66a minus Q

rfe66a

11138681113868111386811138681113868

11138681113868111386811138681113868

+ Qi de11b minus Q

rfe

11b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22b minus Q

rfe

22b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12b minus Q

rfe

12b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66b minus Q

rfe

66b

11138681113868111386811138681113868

11138681113868111386811138681113868

Qrfe11a + Q

rfe22a + Q

rfe12a + Q

rfe66a + Q

rfe

11b + Qrfe

22b + Qrfe

12b + Qrfe

66b

(25)

It can be seen from Table 3 that the weighted relativeerror is lower than 1 +e relative error of Q66 of the twomaterials is the smallest Although the relative errors of Q11andQ22 of the twomaterials are slightly larger than Q66 bothof them are less than 2 +e relative error of Q12a and Q12b

is significantly higher than others because σ2 must be in-cluded in the virtual fields to identify Q12 while the stressdata density of σ2 is low in the three-point bending test +estress data density can be expressed by the followingequation

Advances in Materials Science and Engineering 9

ρ σi( 1113857 1113944

bpointsj1 σi Mj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868

npoints i 1 2 6 (26)

According to equation (7) the standard deviations of Qij

are equal to η(i)c If c is the standard deviation of the strainnoise the coefficients of variations (CV(Qij)) of eachstiffness component are given by

CV Q11( 1113857 η11Q11

c

CV Q22( 1113857 η22Q22

c

CV Q12( 1113857 η12Q12

c

CV Q66( 1113857 η66Q66

c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

+e ratio of coefficient of variation to the standarddeviation of the strain noise can show the sensitivity to thenoise It indicates the rate at which the coefficient of vari-ation increases with the increase of noise which can becalled the variation coefficient-to-noise ratio and can beexpressed as ηijQij +e lower variation coefficient-to-noiseratio indicates that the corresponding identification con-stitutive parameters are less susceptible to noise +e vari-ation coefficient-to-noise ratios ηijQij of eight identificationresults are shown in Table 3 Significant differences on thevariation coefficient-to-noise ratio and the coefficient-to-noise ratio of Q66 is the smallest the next is Q11 and then isthe Q22 and the biggest is Q12 +e quantitative results showthat if Q11 Q22 and Q66 are stable Q12 is less robustlyidentified +is result shows that this configuration is un-suitable for the transverse stiffness

32 Comparison with Results of Piecewise Virtual Fields andPolynomial Virtual Fields +is section will implement thesame numerical experiment as Section 31 to compare theidentified constitutive parameters of the piecewise virtual

L

w

wa

wb

F

Part A

Part B

Figure 1 Two-dimensional bimaterial model of three-point bending established in FEM simulation

Table 1 Engineering elastic constants of orthotropic bimaterialExa (Mpa) Eya (Mpa) Eza (Mpa) PRxya (Mpa) PRyza (Mpa) PRxza (Mpa) Gxya (Mpa) Gyza (Mpa) Gxza (Mpa)180000 30000 8300 017 025 031 5700 12000 12000Exb (Mpa) Eyb (Mpa) Ezb (Mpa) PRxyb (Mpa) PRyzb (Mpa) PRxzb (Mpa) Gxyb (Mpa) Gyzb (Mpa) Gxzb (Mpa)175000 32000 8300 025 025 031 5700 12000 12000

Table 2 Reference constitutive parameters of orthotropic bimaterial

Q11a (Mpa) Q22a (Mpa) Q12a (Mpa) Q66a (Mpa) Q11b (Mpa) Q22b (Mpa) Q12b (Mpa) Q66b (Mpa)

18193710 3071968 590720 5700 17822916 3297651 894780 5700

10 Advances in Materials Science and Engineering

field and polynomial virtual field At this situation the bi-linear shape function is used to interpolate the four-nodeelement to represent the virtual field instead of the poly-nomial virtual field

+e piecewise function expansion of the virtual dis-placement field is similar to the interpolation of the real

displacement in the FEM Derived from the relationshipbetween strain and displacement

ε Sulowast (28)

where

ndash133477

ndash113034

ndash092591

ndash072149

ndash051706

ndash031263

ndash010821

009622

030065

050507

(a)

ndash956093

ndash849861

ndash743628

ndash637396

ndash531163

ndash42493

ndash318698

ndash212465

ndash106233

0

(b)

Figure 2 Displacement distributions of heterogeneous orthotropic bimaterials (a) u1 (b) u2 (unit mm)

Advances in Materials Science and Engineering 11

ndash0235856

ndash018911

ndash014237

ndash009562

ndash004888

ndash000021

041162

009136

013811

018485

(a)

ndash213206

ndash186779

ndash160351

ndash133924

ndash107496

ndash081069

ndash054641

ndash028214

ndash001786

0024641

(b)

ndash362375

ndash32025

ndash278125

ndash236

ndash193874

ndash151749

ndash109624

ndash067499

ndash025373

016752

(c)

Figure 3 Strain distributions of heterogeneous orthotropic bimaterials (a) ε1 (b) ε2 (c) ε3

12 Advances in Materials Science and Engineering

ε

ε1

ε2

ε6

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

ulowast ulowast1

ulowast2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

S

z

zx10

0z

zx2

z

zx2

z

zx1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(29)

For the quadrilateral element the virtual displacementcan be expressed by the following formula

ulowast ξ1 ξ2( 1113857≃ulowast ξ1 ξ2( 1113857 11139444

a1N(a) ξ1 ξ2( 1113857ulowast(a)

(30)

where ulowast is the vector containing the virtual displacement ofany point (ξ1 ξ2) in the element and 1113957ulowast(a) is the vectorcontaining the virtual displacement of the nodes in the el-ement +e expression of N(a) is defined as

N(a) ξ1 ξ2( 1113857 N(a) ξ1 ξ2( 1113857I (31)

where I is the identity matrix +e shape function expressionof N(a) can be defined as follows

N(1)

14

1 minus ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 + ξ2( 1113857

N(2)

14

1 minus ξ1( 1113857 1 + ξ2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

So the virtual displacement can be expressed by thefollowing formula

ulowast1 asymp N

(1)1113957ulowast(1)1 + N

(2)1113957ulowast(2)1 + N

(3)1113957ulowast(3)1 + N

(4)1113957ulowast(4)1

ulowast2 asymp N

(1)1113957ulowast(1)2 + N

(2)1113957ulowast(2)2 + N

(3)1113957ulowast(3)2 + N

(4)1113957ulowast(4)2

⎧⎨

⎩

(33)

+erefore the unknown coefficient vector Y (equation(17)) in the polynomial virtual fields can be denoted on eachnode

Y

1113957ulowast(1)1

1113957ulowast(1)2

⋮

1113957ulowast(n)1

1113957ulowast(n)2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

where n is the total number of nodesUse the piecewise virtual field instead of the polynomial

virtual field to perform the same experiment (Section 31)Table 4 shows the values of relative and weighted relativeerror of A and B It can be seen from Table 4 that theidentification results of piecewise virtual fields are similar tothat of polynomial virtual fields +e weighted relative erroris 097 which is slightly larger than that of polynomialvirtual fields +e relative error of Q66 of the two materials isthe smallest Although the relative errors of Q11 and Q22ofthe two materials are slightly larger than Q66 both of themare less than 2 Similar to the results of polynomial virtualfields the relative error of Q12a and Q12b is significantlyhigher than others because σ2 must be included in thevirtual fields to identify Q12 while the data density of σ2 islow in the three-point bending test

+e variation coefficient-to-noise ratios of eight iden-tification results are also shown in Table 4 Significant dif-ferences can be seen on the variation coefficient-to-noiseratios and the quantitative results show that if Q11 Q22 andQ66 are stable Q12 is less robustly identified Compared withthe polynomial virtual field the variation coefficient-to-noise ratio of the eight identification parameters calculatedby the optimized piecewise virtual field is larger

33 Influence of Noise on Identification Results +e noise isinevitable in the experiments To investigate the influence ofnoise on identification results zero-mean Gaussian white

Table 3 Identification results of constitutive parameters of orthotropic bimaterial using the optimized polynomial virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18159713 280642 minus019

083

Q22a 3071968 3035043 106390 minus120Q12a 590720 479127 2080530 minus1889Q66a 5700 570102 53999 002Q11b 17822916 17735394 234711 minus049Q22b 3297651 3259546 91601 minus116Q12b 894780 827581 1018827 minus751Q66b 5700 569879 59903 minus002

Advances in Materials Science and Engineering 13

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 2: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

advantage of VFM is that it requires less specimenboundary conditions and geometric configurations[11 12]

A volume of research has focused on full-field mea-surement and VFM to identify the constitutive parameters ofmaterials +e VFM within peridynamic framework wasproposed to identify material properties in linear elasticity[13] +e special virtual fields method was adopted to resolvethe difficulty in identifying constitutive properties of in-compressible and nonhomogeneous solids [14] +e VFMwas combined with an elaborately designed test configu-ration in order to realize the identification of the anisotropicyield constitutive parameters [15] Several inverse methodsbased on the VFM have been developed including theFourier series-based VFM [16 17] the sensitivity-basedVFM [18] the eigenfunction VFM [19 20] and severaloptimizations to improve the accuracy of identificationresults [21 22]

+e heterogeneity of strain field caused by multipleconstitutive parameters of the complex constitutive modelhas been solved by the traditional VFM For heterogeneousmaterials the heterogeneity of strain field caused by thespatial variation of constitutive parameters requires furtherimprovement of the virtual field method According to theavailable literature studies several methods and applicationshave been studied to identify constitutive parametersidentification of heterogeneous material using VFM[6 23 24] For instance a bimaterial constitutive parameteridentification method based on the combination of VFMandMoire interference was proposed [25] and the VFMwasalso extended to identify the parameters [26] In additionseveral methods have been researched to lower the effects ofnoise in strain fields [26ndash29] +e optimized fields methodsuch as polynomial optimized virtual field [30] and piece-wise optimized virtual field [31] was developed to improveconvenience and accuracy Although the above researchesmake identification of constitutive parameters more rapidand convenient there is still a limitation for bimaterials theconstitutive parameters of one material must be a priori inorder to determine the constitutive parameters of the othermaterial +e purpose of this work is to propose an opti-mized virtual field that can not only extract the constitutiveparameters of the heterogeneous orthotropic bimaterialswithout knowing any material constitutive parameters butalso require only one single test

In this study to extract constitutive parameters of thebimaterials under the condition that constitutive pa-rameters of the both materials are unknown specialoptimized polynomial virtual fields combined with three-point bending test was proposed In Section 2 the basicprinciple of optimized polynomial VFM and its im-provement applied in bimaterials without knowing anymaterial constitutive parameters are introduced In Sec-tion 3 the FEM simulated experiments and the simulatedDIC experiments are conducted to verify the feasibility ofthe abovementioned method Finally Section 4 is theconclusions

2 Methodology

21 Basic Principle of Optimized Polynomial Virtual FieldsMethod +e integral form of the mechanical equilibriumequation of the VFM based on virtual work for a continuoussolid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowast denotes the strain tensor corre-sponding to ulowast T represents the external force associatedwith the stress tensor σ by the boundary S is a vector whichis the volume force applied over the volume V of thespecimen and a denotes the distribution of accelerationSuch a distribution will cause an additional volumetricforce distribution equal to minusρawith DrsquoAlembertrsquos principlethe virtual displacement field being kinematically admis-sible (KA)

For an orthotropic material in a plane-stress state theprinciple of virtual work is suitable for every KA virtualfield +is equation (see equation 2) uses four indepen-dent KA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively (see Appendix A1for the detailed equation derivations)

AQ B (2)

where

2 Advances in Materials Science and Engineering

A

1113946Sε1εlowast (1)11113872 1113873dS 1113946

Sε2εlowast (1)21113872 1113873dS 1113946

Sε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 1113946

Sε6εlowast (1)61113872 1113873dS

1113946Sε1εlowast (2)11113872 1113873dS 1113946

Sε2εlowast (2)21113872 1113873dS 1113946

Sε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 1113946

Sε6εlowast (2)61113872 1113873dS

1113946Sε1εlowast (3)11113872 1113873dS 1113946

Sε2εlowast (3)21113872 1113873dS 1113946

Sε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 1113946

Sε6εlowast (3)61113872 1113873dS

1113946Sε1εlowast (4)11113872 1113873dS 1113946

Sε2εlowast (4)21113872 1113873dS 1113946

Sε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 1113946

Sε6εlowast (4)61113872 1113873dS

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q

Q11

Q22

Q12

Q66

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B

1113946Lf

Tiulowast (1)i1113872 1113873dl

1113946Lf

Tiulowast (2)i1113872 1113873dl

1113946Lf

Tiulowast (3)i1113872 1113873dl

1113946Lf

Tiulowast (4)i1113872 1113873dl

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents

+e constraints of KA virtual fields ulowast(1) ulowast(2) ulowast(3) andulowast(4) are as follows

1113946Sε1εlowast (1)11113872 1113873dS 1 1113946

Sε2εlowast (1)21113872 1113873dS 0 1113946

Sε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 0 1113946

Sε6εlowast (1)61113872 1113873dS 0

1113946Sε1εlowast (2)11113872 1113873dS 0 1113946

Sε2εlowast (2)21113872 1113873dS 1 1113946

Sε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 0 1113946

Sε6εlowast (2)61113872 1113873dS 0

1113946Sε1εlowast (3)11113872 1113873dS 0 1113946

Sε2εlowast (3)21113872 1113873dS 0 1113946

Sε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 1 1113946

Sε6εlowast (3)61113872 1113873dS 0

1113946Sε1εlowast (4)11113872 1113873dS 0 1113946

Sε2εlowast (4)21113872 1113873dS 0 1113946

Sε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 0 1113946

Sε6εlowast (4)61113872 1113873dS 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

+e exact values and white noise constitute the measuredstrain component Assume that the noise components areuncorrelated to each other and presume that the noise

between points is also uncorrelated +erefore the principleof virtual work is as equation (5) (see Appendix A2 for thedetailed equation derivations)

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(5)

whereN1N2 andN6 denote the processes of scalar zero-mean stationary Gaussian generalized by the corre-sponding strain components ε1 ε2 and ε6 respectivelyand c denotes the measured amplitude of the randomvariable strain

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by equation(6) So directly identify Q11 Q22 Q12 and Q66 using thesefour special virtual displacement fields as equation (7) (seeAppendix A3 for the detailed equation derivations)

Advances in Materials Science and Engineering 3

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (2)6 N6dS11138771113876 1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (3)6 N6dS11138771113876 1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q)the vector containing the variances ofQ11Q22Q12 and Q66 one can write as equation (8) (seeAppendix A4 for the detailed equation derivations)

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

and (η(i))2 have two kinds of unknowns the constitutiveparameters and the unknown coefficients of the virtualfields So (η(i))2 can be expressed as

η(i)1113872 1113873

212Ylowast(i)HYlowast(i)

Sc

nc

1113888 1113889

2

QappG(i)Qapp (9)

where Ylowast is the vector related to the coefficients of virtualstrain fields H is the Hessian matrix of all monomials in εlowastand G(j) j 1 2 3 4 is the following square matrix

1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113944n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113944n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)

and the (η(i))2 coefficients i 1 2 3 4 depend on theselected virtual fields Also the virtual fields are selectedfor the Qij parameters extraction +us select the ap-propriate virtual field that can reduce the influence of

strain In this article the virtual fields were expandedby the polynomials +e displacement fields and thevirtual strain are as shown in equations (11) and (12)respectively

4 Advances in Materials Science and Engineering

ulowast1 1113944

m

i01113944

n

j0Aij

x1

L1113874 1113875

i x2

w1113874 1113875

j

ulowast2 1113944

m

i01113944

n

j0Bij

x1

L1113874 1113875

i x2

w1113874 1113875

j

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

εlowast1 1113944m

i11113944

n

j0Aij

x1

L1113874 1113875

i minus1 x2

w1113874 1113875

j

εlowast2 1113944m

i01113944

n

j1Bij

x1

L1113874 1113875

i x2

w1113874 1113875

j minus1

εlowast6 1113944

m

i01113944

n

j1Aij

i

L

x1

L1113874 1113875

i x2

w1113874 1113875

j minus1+ 1113944

m

i11113944

n

j0Bij

i

L

x1

L1113874 1113875

i minus 1 x2

w1113874 1113875

j

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

where Aij and Bij are the coefficients of the monomials m

and n are the polynomial degrees that define that themaximum number of monomials has used and L and w arethe typical dimensions of the x1 and x2 directionsrespectively

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (13) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (13)

where λ(i) is the vector containing Lagrange multipliers+e KA condition can produce several linear equations

and the number of this equations is depending on thenumber of supports +e special virtual field condition alsoleads to several linear equations the number of which de-pends on the type of constitutive model of specimen ma-terial +e two types of conditions can produce the followinglinear system as equation (14) (see Appendix A5 for thedetailed equation derivations)

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (14)

Attention that (η(i))2 coefficients also correspond on theunknown Qij parameters +us the optimization problem issolved for a given set of Qij parameters and the process is

iterative the first guess of theQij parameters is used to find thefirst group of four special virtual fields+en these four specialvirtual fields are used to find a new set of Qij parameters andrepeat abovementioned steps until the optimal Qij parameteris found In practice no matter what the parameter Qij isinitially selected this iterative process will converge quicklyUsually only two loops are sufficient to converge

22 Optimized Polynomial Virtual Fields Applied to theBimaterials without Knowing Any Material ConstitutiveParameters As shown in Figure 1 for the plane-stressspecimen with two different materials Part A and Part B andboth of them are set with the elastic orthotropic material Butthe constitutive parameters of A and B are unknown Asmentioned above the sum of exact values and white noise isthe measured strain Assume that the noise components areuncorrelated to each other and presume that the noisebetween points is also uncorrelated +erefore the gov-erning equation of the virtual fields method for the totalspecimen is as equation (15) (see Appendix B1 for the de-tailed equation derivations)

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS

+ Q66a1113946Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS

+ Q66b1113946Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS1113876

+ Q12a1113946Sa

εlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946Saεlowast6 N6dS

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS

+ Q12b1113946Sb

εlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946Sbεlowast6 N6dS1113877

1113946Lf

Tiulowasti( 1113857dl

(15)

+e constraints of the eight special virtual fields denotedthat ulowast(1) ulowast(2) ulowast(3) ulowast(4) ulowast(5) ulowast(6) ulowast(7) and ulowast(8) mustsatisfy the following equation

Advances in Materials Science and Engineering 5

1113946Sa

ε1εlowast (1)11113872 1113873dS 0 1113946

Sa

ε2εlowast (1)21113872 1113873dS 0 1113946

Sa

ε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 1 1113946

Sa

ε6εlowast (1)61113872 1113873dS 0

1113946Sb

ε1εlowast (1)11113872 1113873dS 0 1113946

Sb

ε2εlowast (1)21113872 1113873dS 0 1113946

Sb

ε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 0 1113946

Sb

ε6εlowast (1)61113872 1113873dS 0

1113946Sa

ε1εlowast (2)11113872 1113873dS 0 1113946

Sa

ε2εlowast (2)21113872 1113873dS 0 1113946

Sa

ε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 1 1113946

Sa

ε6εlowast (2)61113872 1113873dS 0

1113946Sb

ε1εlowast (2)11113872 1113873dS 0 1113946

Sb

ε2εlowast (2)21113872 1113873dS 0 1113946

Sb

ε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 0 1113946

Sb

ε6εlowast (2)61113872 1113873dS 0

1113946Sa

ε1εlowast (3)11113872 1113873dS 0 1113946

Sa

ε2εlowast (3)21113872 1113873dS 0 1113946

Sa

ε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 1 1113946

Sa

ε6εlowast (3)61113872 1113873dS 0

1113946Sb

ε1εlowast (3)11113872 1113873dS 0 1113946

Sb

ε2εlowast (3)21113872 1113873dS 0 1113946

Sb

ε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 0 1113946

Sb

ε6εlowast (3)61113872 1113873dS 0

1113946Sa

ε1εlowast (4)11113872 1113873dS 0 1113946

Sa

ε2εlowast (4)21113872 1113873dS 0 1113946

Sa

ε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 1 1113946

Sa

ε6εlowast (4)61113872 1113873dS 0

1113946Sb

ε1εlowast (4)11113872 1113873dS 0 1113946

Sb

ε2εlowast (4)21113872 1113873dS 0 1113946

Sb

ε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 0 1113946

Sb

ε6εlowast (4)61113872 1113873dS 0

1113946Sa

ε1εlowast (5)11113872 1113873dS 0 1113946

Sa

ε2εlowast (5)21113872 1113873dS 0 1113946

Sa

ε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS 1 1113946

Sa

ε6εlowast (5)61113872 1113873dS 0

1113946Sb

ε1εlowast (5)11113872 1113873dS 0 1113946

Sb

ε2εlowast (5)21113872 1113873dS 0 1113946

Sb

ε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS 0 1113946

Sb

ε6εlowast (5)61113872 1113873dS 0

1113946Sa

ε1εlowast (6)11113872 1113873dS 0 1113946

Sa

ε2εlowast (6)21113872 1113873dS 0 1113946

Sa

ε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS 1 1113946

Sa

ε6εlowast (6)61113872 1113873dS 0

1113946Sb

ε1εlowast (6)11113872 1113873dS 0 1113946

Sb

ε2εlowast (6)21113872 1113873dS 0 1113946

Sb

ε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS 0 1113946

Sb

ε6εlowast (6)61113872 1113873dS 0

1113946Sa

ε1εlowast (7)11113872 1113873dS 0 1113946

Sa

ε2εlowast (7)21113872 1113873dS 0 1113946

Sa

ε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS 1 1113946

Sa

ε6εlowast (7)61113872 1113873dS 0

1113946Sb

ε1εlowast (7)11113872 1113873dS 0 1113946

Sb

ε2εlowast (7)21113872 1113873dS 0 1113946

Sb

ε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS 0 1113946

Sb

ε6εlowast (7)61113872 1113873dS 0

1113946Sa

ε1εlowast (8)11113872 1113873dS 0 1113946

Sa

ε2εlowast (8)21113872 1113873dS 0 1113946

Sa

ε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS 1 1113946

Sa

ε6εlowast (8)61113872 1113873dS 0

1113946Sb

ε1εlowast (8)11113872 1113873dS 0 1113946

Sb

ε2εlowast (8)21113872 1113873dS 0 1113946

Sb

ε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS 0 1113946

Sb

ε6εlowast (8)61113872 1113873dS 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

So directly identify Q11a Q11b Q12a

Q12b Q22a Q22b Q66a and Q66b using the abovementionedfour special virtual displacement fields If the noise is

ignored approximate parameters expressed as Qapp areidentified +e components of which are defined as equation(17) (see Appendix B2 for the detailed equation derivations)

6 Advances in Materials Science and Engineering

Qapp11a 1113946

Lf

Tiulowast (1)i dl minus Q22a1113946

Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS

minus Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS

Qapp22a 1113946

Lf

Tiulowast (2)i dl minus Q11a1113946

Saε1εlowast (2)1 dS minus Q12a1113946

Saε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66a1113946

Saε6εlowast (2)6 dS

minus Q11b1113946Sbε1εlowast (2)1 dS minus Q22b1113946

Sbε2εlowast (2)2 dS minus Q12b1113946

Sbε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (2)6 dS

Qapp12a 1113946

Lf

Tiulowast (3)i dl minus Q11a1113946

Saε1εlowast (3)1 dS minus Q22a1113946

Saε2εlowast (3)2 dS minus Q66a1113946

Saε6εlowast (3)6 dS

minus Q11b1113946Sbε1εlowast (3)1 dS minus Q22b1113946

Sbε2εlowast (3)2 dS minus Q12b1113946

Sbε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (3)6 dS

Qapp66a 1113946

Lf

Tiulowast (4)i dl minus Q11a1113946

Saε1εlowast (4)1 dS minus Q22a1113946

Saε2εlowast (4)2 dS minus Q12a1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS

minus Q11b1113946Sbε1εlowast (4)1 dS minus Q22b1113946

Sbε2εlowast (4)2 dS minus Q12b1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (4)6 dS

Qapp11b 1113946

Lf

Tiulowast (5)i dl minus Q22b1113946

Sbε2εlowast (5)2 dS minus Q12b1113946

Sbε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (5)6 dS

minus Q11a1113946Saε1εlowast (5)1 dS minus Q22a1113946

Saε2εlowast (5)2 dS minus Q12a1113946

Saε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66a1113946

Saε6εlowast (5)6 dS

Qapp22b 1113946

Lf

Tiulowast (6)i dl minus Q11b1113946

Sbε1εlowast (6)1 dS minus Q12b1113946

Sbε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (6)6 dS

minus Q11a1113946Saε1εlowast (6)1 dS minus Q22a1113946

Saε2εlowast (6)2 dS minus Q12a1113946

Saε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66a1113946

Saε6εlowast (6)6 dS

Qapp12b 1113946

Lf

Tiulowast (7)i dl minus Q11b1113946

Sbε1εlowast (7)1 dS minus Q22b1113946

Sbε2εlowast (7)2 dS minus Q66b1113946

Sbε6εlowast (7)6 dS minus Q11a1113946

Saε1εlowast (7)1 dS

minus Q22a1113946Saε2εlowast (7)2 dS minus Q12a1113946

Saε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS minus Q66a1113946

Saε6εlowast (7)6 dS

Qapp66b 1113946

Lf

Tiulowast (8)i dl minus Q11b1113946

Sbε1εlowast (8)1 dS minus Q22b1113946

Sbε2εlowast (8)2 dS minus Q12b1113946

Sbε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS

minus Q11a1113946Saε1εlowast (8)1 dS minus Q22a1113946

Saε2εlowast (8)2 dS minus Q12a1113946

Saε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS minus Q66a1113946

Saε6εlowast (8)6 dS

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

To minimize the effect of noise taking Q11a as an ex-ample the variance of Q11a is as follows

V Q11a( 1113857 c2E Q

app11a1113946

Saεlowast (1)1 N1dS + Q

app22a1113946

Saεlowast (1)2 N2dS + Q

app12a 1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66a 1113946

Saεlowast (1)6 N6dS1113876 1113877

21113888

+ Qapp11b1113946

Sblowast (1)N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113889

+ c2E Q

app11b1113946

Sbεlowast (1)1 N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app22b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113888 1113889

VA Q11a( 1113857 + VB Q11a( 1113857

(18)

Advances in Materials Science and Engineering 7

As described above the virtual fields are expanded bypolynomial basis functions based on equation (11) +eunknown coefficients Aij and Bij are included in vector Y asfollows

Y

A00

⋮A30

⋮

A03

⋮

A33

B00

⋮

B30

⋮

B03

⋮

B33

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)

+e integrals of equation (15) are calculated usingvectors B11B22B12 and B66 through discrete sum ap-proximation so that

1113946Sa

ε1εlowast1( 1113857dS asymp Lwaε1aε

lowastia B11a middot Y

1113946Sa

ε2εlowast2( 1113857dS asymp Lwaε2aε

lowast2a B22a middot Y

1113946Sa

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwa ε1aε

lowast2a + ε2aε

lowast1a1113872 1113873 B12a middot Y

1113946Sa

ε6εlowast6( 1113857dS asymp Lwaε6aε

lowast6a B66a middot Y

1113946Sb

ε1εlowast1( 1113857dS asymp Lwaε1bε

lowast1b B11b middot Y

1113946Sb

ε2εlowast2( 1113857dS asymp Lwbε2bε

lowast2b B22b middot Y

1113946Sb

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwb ε1bε

lowast2b + ε2bε

lowast1b1113872 1113873 B12b middot Y

1113946Sb

ε6εlowast6( 1113857dS asymp Lwbε6bε

lowast6b B66b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

+e others that the definition from the virtual fields(vectorsH11H22H12 andH66) are related to the terms ofequation (18)

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

ε1εlowast2( 1113857dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast6( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

ε1εlowast2( 1113857dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast6( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

+e first KA virtual field conditions of three-pointbending are ulowast2 0 when x1 0 and x2 0 which implies

B00 0 (22)

+e second KA virtual field conditions of three-pointbending re ulowast1 0 when x1 L which means that

1113944

m

i0Aij 0 j 0 middot middot middot n (23)

+e optimized virtual fields are totally defined +efollowing equations can identify the constitutive parameters

8 Advances in Materials Science and Engineering

Q11a F

tulowast (a)2 x1 L x2 w( 1113857

Q22a F

tulowast (a)2 x1 L x2 w( 1113857

Q12a F

tulowast (a)2 x1 L x2 w( 1113857

Q66a F

tulowast (a)2 x1 L x2 w( 1113857

Q11b F

tulowast (a)2 x1 L x2 w( 1113857

Q22b F

tulowast (a)2 x1 L x2 w( 1113857

Q12b F

tulowast (a)2 x1 L x2 w( 1113857

Q66b F

tulowast (a)2 x1 L x2 w( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

3 Numerical Verification and Discussion

To evaluate the accuracy of the abovementioned optimizedpolynomial VFM numerical experiments with FEM simu-lations and DIC simulations are employed in this section

31 Numerical Verification of Optimized Polynomial VirtualFieldswithFEMSimulations In FEM simulation in order to

facilitate the simulation of bimaterials configuration a two-dimensional rectangular beam specimen was adopted andapplied a three-point bending load to get inhomogeneousin-plane deformation fields Based on the principle of thesymmetry of three-point bending only the left of thespecimen was modelled as shown in Figure 1

+e length width and thickness of the bimaterial beamspecimen were set as L 30mm w 20mm andt 23mm respectively and the width of the two materialregions was the same wa wb 10mm+e force was set asF 2544N +e four-node plane-stress element withthickness (Solid-Quad 4 nodes 182Plane strs wthk) wasemployed +e displacement constraint uy 0 is added inthe lower left corner of the model and the line constraintux 0 is added on the right side of the model

+e beam specimen was set as orthotropic materials +eengineering elastic constants in the FEM simulation areshown in Table 1 and the converted constitutive parametersin the elastic matrix are shown in Table 2

+e displacement fields and the strain fields of the three-point bending test of the orthotropic bimaterial obtainedthrough FEM simulations using the commercial softwareANSYS are shown in Figures 2 and 3

Table 3 shows the identification results of the hetero-geneous orthotropic bimaterials by using the optimizedpolynomial VFM as shown in Section 22 To study theidentification accuracy of the optimized polynomial VFMthe relative error of every constitutive parameter componentand the weighted relative error Wre were calculated Owingto there were 8 unknown constitutive parameters theweighted relative error employed here can be expressed asfollows

Wre

Qi de11a minus Q

rfe11a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22a minus Q

re22a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12a minus Q

re12a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66a minus Q

rfe66a

11138681113868111386811138681113868

11138681113868111386811138681113868

+ Qi de11b minus Q

rfe

11b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22b minus Q

rfe

22b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12b minus Q

rfe

12b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66b minus Q

rfe

66b

11138681113868111386811138681113868

11138681113868111386811138681113868

Qrfe11a + Q

rfe22a + Q

rfe12a + Q

rfe66a + Q

rfe

11b + Qrfe

22b + Qrfe

12b + Qrfe

66b

(25)

It can be seen from Table 3 that the weighted relativeerror is lower than 1 +e relative error of Q66 of the twomaterials is the smallest Although the relative errors of Q11andQ22 of the twomaterials are slightly larger than Q66 bothof them are less than 2 +e relative error of Q12a and Q12b

is significantly higher than others because σ2 must be in-cluded in the virtual fields to identify Q12 while the stressdata density of σ2 is low in the three-point bending test +estress data density can be expressed by the followingequation

Advances in Materials Science and Engineering 9

ρ σi( 1113857 1113944

bpointsj1 σi Mj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868

npoints i 1 2 6 (26)

According to equation (7) the standard deviations of Qij

are equal to η(i)c If c is the standard deviation of the strainnoise the coefficients of variations (CV(Qij)) of eachstiffness component are given by

CV Q11( 1113857 η11Q11

c

CV Q22( 1113857 η22Q22

c

CV Q12( 1113857 η12Q12

c

CV Q66( 1113857 η66Q66

c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

+e ratio of coefficient of variation to the standarddeviation of the strain noise can show the sensitivity to thenoise It indicates the rate at which the coefficient of vari-ation increases with the increase of noise which can becalled the variation coefficient-to-noise ratio and can beexpressed as ηijQij +e lower variation coefficient-to-noiseratio indicates that the corresponding identification con-stitutive parameters are less susceptible to noise +e vari-ation coefficient-to-noise ratios ηijQij of eight identificationresults are shown in Table 3 Significant differences on thevariation coefficient-to-noise ratio and the coefficient-to-noise ratio of Q66 is the smallest the next is Q11 and then isthe Q22 and the biggest is Q12 +e quantitative results showthat if Q11 Q22 and Q66 are stable Q12 is less robustlyidentified +is result shows that this configuration is un-suitable for the transverse stiffness

32 Comparison with Results of Piecewise Virtual Fields andPolynomial Virtual Fields +is section will implement thesame numerical experiment as Section 31 to compare theidentified constitutive parameters of the piecewise virtual

L

w

wa

wb

F

Part A

Part B

Figure 1 Two-dimensional bimaterial model of three-point bending established in FEM simulation

Table 1 Engineering elastic constants of orthotropic bimaterialExa (Mpa) Eya (Mpa) Eza (Mpa) PRxya (Mpa) PRyza (Mpa) PRxza (Mpa) Gxya (Mpa) Gyza (Mpa) Gxza (Mpa)180000 30000 8300 017 025 031 5700 12000 12000Exb (Mpa) Eyb (Mpa) Ezb (Mpa) PRxyb (Mpa) PRyzb (Mpa) PRxzb (Mpa) Gxyb (Mpa) Gyzb (Mpa) Gxzb (Mpa)175000 32000 8300 025 025 031 5700 12000 12000

Table 2 Reference constitutive parameters of orthotropic bimaterial

Q11a (Mpa) Q22a (Mpa) Q12a (Mpa) Q66a (Mpa) Q11b (Mpa) Q22b (Mpa) Q12b (Mpa) Q66b (Mpa)

18193710 3071968 590720 5700 17822916 3297651 894780 5700

10 Advances in Materials Science and Engineering

field and polynomial virtual field At this situation the bi-linear shape function is used to interpolate the four-nodeelement to represent the virtual field instead of the poly-nomial virtual field

+e piecewise function expansion of the virtual dis-placement field is similar to the interpolation of the real

displacement in the FEM Derived from the relationshipbetween strain and displacement

ε Sulowast (28)

where

ndash133477

ndash113034

ndash092591

ndash072149

ndash051706

ndash031263

ndash010821

009622

030065

050507

(a)

ndash956093

ndash849861

ndash743628

ndash637396

ndash531163

ndash42493

ndash318698

ndash212465

ndash106233

0

(b)

Figure 2 Displacement distributions of heterogeneous orthotropic bimaterials (a) u1 (b) u2 (unit mm)

Advances in Materials Science and Engineering 11

ndash0235856

ndash018911

ndash014237

ndash009562

ndash004888

ndash000021

041162

009136

013811

018485

(a)

ndash213206

ndash186779

ndash160351

ndash133924

ndash107496

ndash081069

ndash054641

ndash028214

ndash001786

0024641

(b)

ndash362375

ndash32025

ndash278125

ndash236

ndash193874

ndash151749

ndash109624

ndash067499

ndash025373

016752

(c)

Figure 3 Strain distributions of heterogeneous orthotropic bimaterials (a) ε1 (b) ε2 (c) ε3

12 Advances in Materials Science and Engineering

ε

ε1

ε2

ε6

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

ulowast ulowast1

ulowast2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

S

z

zx10

0z

zx2

z

zx2

z

zx1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(29)

For the quadrilateral element the virtual displacementcan be expressed by the following formula

ulowast ξ1 ξ2( 1113857≃ulowast ξ1 ξ2( 1113857 11139444

a1N(a) ξ1 ξ2( 1113857ulowast(a)

(30)

where ulowast is the vector containing the virtual displacement ofany point (ξ1 ξ2) in the element and 1113957ulowast(a) is the vectorcontaining the virtual displacement of the nodes in the el-ement +e expression of N(a) is defined as

N(a) ξ1 ξ2( 1113857 N(a) ξ1 ξ2( 1113857I (31)

where I is the identity matrix +e shape function expressionof N(a) can be defined as follows

N(1)

14

1 minus ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 + ξ2( 1113857

N(2)

14

1 minus ξ1( 1113857 1 + ξ2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

So the virtual displacement can be expressed by thefollowing formula

ulowast1 asymp N

(1)1113957ulowast(1)1 + N

(2)1113957ulowast(2)1 + N

(3)1113957ulowast(3)1 + N

(4)1113957ulowast(4)1

ulowast2 asymp N

(1)1113957ulowast(1)2 + N

(2)1113957ulowast(2)2 + N

(3)1113957ulowast(3)2 + N

(4)1113957ulowast(4)2

⎧⎨

⎩

(33)

+erefore the unknown coefficient vector Y (equation(17)) in the polynomial virtual fields can be denoted on eachnode

Y

1113957ulowast(1)1

1113957ulowast(1)2

⋮

1113957ulowast(n)1

1113957ulowast(n)2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

where n is the total number of nodesUse the piecewise virtual field instead of the polynomial

virtual field to perform the same experiment (Section 31)Table 4 shows the values of relative and weighted relativeerror of A and B It can be seen from Table 4 that theidentification results of piecewise virtual fields are similar tothat of polynomial virtual fields +e weighted relative erroris 097 which is slightly larger than that of polynomialvirtual fields +e relative error of Q66 of the two materials isthe smallest Although the relative errors of Q11 and Q22ofthe two materials are slightly larger than Q66 both of themare less than 2 Similar to the results of polynomial virtualfields the relative error of Q12a and Q12b is significantlyhigher than others because σ2 must be included in thevirtual fields to identify Q12 while the data density of σ2 islow in the three-point bending test

+e variation coefficient-to-noise ratios of eight iden-tification results are also shown in Table 4 Significant dif-ferences can be seen on the variation coefficient-to-noiseratios and the quantitative results show that if Q11 Q22 andQ66 are stable Q12 is less robustly identified Compared withthe polynomial virtual field the variation coefficient-to-noise ratio of the eight identification parameters calculatedby the optimized piecewise virtual field is larger

33 Influence of Noise on Identification Results +e noise isinevitable in the experiments To investigate the influence ofnoise on identification results zero-mean Gaussian white

Table 3 Identification results of constitutive parameters of orthotropic bimaterial using the optimized polynomial virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18159713 280642 minus019

083

Q22a 3071968 3035043 106390 minus120Q12a 590720 479127 2080530 minus1889Q66a 5700 570102 53999 002Q11b 17822916 17735394 234711 minus049Q22b 3297651 3259546 91601 minus116Q12b 894780 827581 1018827 minus751Q66b 5700 569879 59903 minus002

Advances in Materials Science and Engineering 13

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 3: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

A

1113946Sε1εlowast (1)11113872 1113873dS 1113946

Sε2εlowast (1)21113872 1113873dS 1113946

Sε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 1113946

Sε6εlowast (1)61113872 1113873dS

1113946Sε1εlowast (2)11113872 1113873dS 1113946

Sε2εlowast (2)21113872 1113873dS 1113946

Sε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 1113946

Sε6εlowast (2)61113872 1113873dS

1113946Sε1εlowast (3)11113872 1113873dS 1113946

Sε2εlowast (3)21113872 1113873dS 1113946

Sε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 1113946

Sε6εlowast (3)61113872 1113873dS

1113946Sε1εlowast (4)11113872 1113873dS 1113946

Sε2εlowast (4)21113872 1113873dS 1113946

Sε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 1113946

Sε6εlowast (4)61113872 1113873dS

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q

Q11

Q22

Q12

Q66

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B

1113946Lf

Tiulowast (1)i1113872 1113873dl

1113946Lf

Tiulowast (2)i1113872 1113873dl

1113946Lf

Tiulowast (3)i1113872 1113873dl

1113946Lf

Tiulowast (4)i1113872 1113873dl

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents

+e constraints of KA virtual fields ulowast(1) ulowast(2) ulowast(3) andulowast(4) are as follows

1113946Sε1εlowast (1)11113872 1113873dS 1 1113946

Sε2εlowast (1)21113872 1113873dS 0 1113946

Sε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 0 1113946

Sε6εlowast (1)61113872 1113873dS 0

1113946Sε1εlowast (2)11113872 1113873dS 0 1113946

Sε2εlowast (2)21113872 1113873dS 1 1113946

Sε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 0 1113946

Sε6εlowast (2)61113872 1113873dS 0

1113946Sε1εlowast (3)11113872 1113873dS 0 1113946

Sε2εlowast (3)21113872 1113873dS 0 1113946

Sε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 1 1113946

Sε6εlowast (3)61113872 1113873dS 0

1113946Sε1εlowast (4)11113872 1113873dS 0 1113946

Sε2εlowast (4)21113872 1113873dS 0 1113946

Sε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 0 1113946

Sε6εlowast (4)61113872 1113873dS 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

+e exact values and white noise constitute the measuredstrain component Assume that the noise components areuncorrelated to each other and presume that the noise

between points is also uncorrelated +erefore the principleof virtual work is as equation (5) (see Appendix A2 for thedetailed equation derivations)

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(5)

whereN1N2 andN6 denote the processes of scalar zero-mean stationary Gaussian generalized by the corre-sponding strain components ε1 ε2 and ε6 respectivelyand c denotes the measured amplitude of the randomvariable strain

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by equation(6) So directly identify Q11 Q22 Q12 and Q66 using thesefour special virtual displacement fields as equation (7) (seeAppendix A3 for the detailed equation derivations)

Advances in Materials Science and Engineering 3

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (2)6 N6dS11138771113876 1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (3)6 N6dS11138771113876 1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q)the vector containing the variances ofQ11Q22Q12 and Q66 one can write as equation (8) (seeAppendix A4 for the detailed equation derivations)

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

and (η(i))2 have two kinds of unknowns the constitutiveparameters and the unknown coefficients of the virtualfields So (η(i))2 can be expressed as

η(i)1113872 1113873

212Ylowast(i)HYlowast(i)

Sc

nc

1113888 1113889

2

QappG(i)Qapp (9)

where Ylowast is the vector related to the coefficients of virtualstrain fields H is the Hessian matrix of all monomials in εlowastand G(j) j 1 2 3 4 is the following square matrix

1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113944n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113944n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)

and the (η(i))2 coefficients i 1 2 3 4 depend on theselected virtual fields Also the virtual fields are selectedfor the Qij parameters extraction +us select the ap-propriate virtual field that can reduce the influence of

strain In this article the virtual fields were expandedby the polynomials +e displacement fields and thevirtual strain are as shown in equations (11) and (12)respectively

4 Advances in Materials Science and Engineering

ulowast1 1113944

m

i01113944

n

j0Aij

x1

L1113874 1113875

i x2

w1113874 1113875

j

ulowast2 1113944

m

i01113944

n

j0Bij

x1

L1113874 1113875

i x2

w1113874 1113875

j

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

εlowast1 1113944m

i11113944

n

j0Aij

x1

L1113874 1113875

i minus1 x2

w1113874 1113875

j

εlowast2 1113944m

i01113944

n

j1Bij

x1

L1113874 1113875

i x2

w1113874 1113875

j minus1

εlowast6 1113944

m

i01113944

n

j1Aij

i

L

x1

L1113874 1113875

i x2

w1113874 1113875

j minus1+ 1113944

m

i11113944

n

j0Bij

i

L

x1

L1113874 1113875

i minus 1 x2

w1113874 1113875

j

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

where Aij and Bij are the coefficients of the monomials m

and n are the polynomial degrees that define that themaximum number of monomials has used and L and w arethe typical dimensions of the x1 and x2 directionsrespectively

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (13) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (13)

where λ(i) is the vector containing Lagrange multipliers+e KA condition can produce several linear equations

and the number of this equations is depending on thenumber of supports +e special virtual field condition alsoleads to several linear equations the number of which de-pends on the type of constitutive model of specimen ma-terial +e two types of conditions can produce the followinglinear system as equation (14) (see Appendix A5 for thedetailed equation derivations)

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (14)

Attention that (η(i))2 coefficients also correspond on theunknown Qij parameters +us the optimization problem issolved for a given set of Qij parameters and the process is

iterative the first guess of theQij parameters is used to find thefirst group of four special virtual fields+en these four specialvirtual fields are used to find a new set of Qij parameters andrepeat abovementioned steps until the optimal Qij parameteris found In practice no matter what the parameter Qij isinitially selected this iterative process will converge quicklyUsually only two loops are sufficient to converge

22 Optimized Polynomial Virtual Fields Applied to theBimaterials without Knowing Any Material ConstitutiveParameters As shown in Figure 1 for the plane-stressspecimen with two different materials Part A and Part B andboth of them are set with the elastic orthotropic material Butthe constitutive parameters of A and B are unknown Asmentioned above the sum of exact values and white noise isthe measured strain Assume that the noise components areuncorrelated to each other and presume that the noisebetween points is also uncorrelated +erefore the gov-erning equation of the virtual fields method for the totalspecimen is as equation (15) (see Appendix B1 for the de-tailed equation derivations)

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS

+ Q66a1113946Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS

+ Q66b1113946Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS1113876

+ Q12a1113946Sa

εlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946Saεlowast6 N6dS

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS

+ Q12b1113946Sb

εlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946Sbεlowast6 N6dS1113877

1113946Lf

Tiulowasti( 1113857dl

(15)

+e constraints of the eight special virtual fields denotedthat ulowast(1) ulowast(2) ulowast(3) ulowast(4) ulowast(5) ulowast(6) ulowast(7) and ulowast(8) mustsatisfy the following equation

Advances in Materials Science and Engineering 5

1113946Sa

ε1εlowast (1)11113872 1113873dS 0 1113946

Sa

ε2εlowast (1)21113872 1113873dS 0 1113946

Sa

ε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 1 1113946

Sa

ε6εlowast (1)61113872 1113873dS 0

1113946Sb

ε1εlowast (1)11113872 1113873dS 0 1113946

Sb

ε2εlowast (1)21113872 1113873dS 0 1113946

Sb

ε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 0 1113946

Sb

ε6εlowast (1)61113872 1113873dS 0

1113946Sa

ε1εlowast (2)11113872 1113873dS 0 1113946

Sa

ε2εlowast (2)21113872 1113873dS 0 1113946

Sa

ε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 1 1113946

Sa

ε6εlowast (2)61113872 1113873dS 0

1113946Sb

ε1εlowast (2)11113872 1113873dS 0 1113946

Sb

ε2εlowast (2)21113872 1113873dS 0 1113946

Sb

ε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 0 1113946

Sb

ε6εlowast (2)61113872 1113873dS 0

1113946Sa

ε1εlowast (3)11113872 1113873dS 0 1113946

Sa

ε2εlowast (3)21113872 1113873dS 0 1113946

Sa

ε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 1 1113946

Sa

ε6εlowast (3)61113872 1113873dS 0

1113946Sb

ε1εlowast (3)11113872 1113873dS 0 1113946

Sb

ε2εlowast (3)21113872 1113873dS 0 1113946

Sb

ε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 0 1113946

Sb

ε6εlowast (3)61113872 1113873dS 0

1113946Sa

ε1εlowast (4)11113872 1113873dS 0 1113946

Sa

ε2εlowast (4)21113872 1113873dS 0 1113946

Sa

ε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 1 1113946

Sa

ε6εlowast (4)61113872 1113873dS 0

1113946Sb

ε1εlowast (4)11113872 1113873dS 0 1113946

Sb

ε2εlowast (4)21113872 1113873dS 0 1113946

Sb

ε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 0 1113946

Sb

ε6εlowast (4)61113872 1113873dS 0

1113946Sa

ε1εlowast (5)11113872 1113873dS 0 1113946

Sa

ε2εlowast (5)21113872 1113873dS 0 1113946

Sa

ε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS 1 1113946

Sa

ε6εlowast (5)61113872 1113873dS 0

1113946Sb

ε1εlowast (5)11113872 1113873dS 0 1113946

Sb

ε2εlowast (5)21113872 1113873dS 0 1113946

Sb

ε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS 0 1113946

Sb

ε6εlowast (5)61113872 1113873dS 0

1113946Sa

ε1εlowast (6)11113872 1113873dS 0 1113946

Sa

ε2εlowast (6)21113872 1113873dS 0 1113946

Sa

ε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS 1 1113946

Sa

ε6εlowast (6)61113872 1113873dS 0

1113946Sb

ε1εlowast (6)11113872 1113873dS 0 1113946

Sb

ε2εlowast (6)21113872 1113873dS 0 1113946

Sb

ε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS 0 1113946

Sb

ε6εlowast (6)61113872 1113873dS 0

1113946Sa

ε1εlowast (7)11113872 1113873dS 0 1113946

Sa

ε2εlowast (7)21113872 1113873dS 0 1113946

Sa

ε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS 1 1113946

Sa

ε6εlowast (7)61113872 1113873dS 0

1113946Sb

ε1εlowast (7)11113872 1113873dS 0 1113946

Sb

ε2εlowast (7)21113872 1113873dS 0 1113946

Sb

ε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS 0 1113946

Sb

ε6εlowast (7)61113872 1113873dS 0

1113946Sa

ε1εlowast (8)11113872 1113873dS 0 1113946

Sa

ε2εlowast (8)21113872 1113873dS 0 1113946

Sa

ε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS 1 1113946

Sa

ε6εlowast (8)61113872 1113873dS 0

1113946Sb

ε1εlowast (8)11113872 1113873dS 0 1113946

Sb

ε2εlowast (8)21113872 1113873dS 0 1113946

Sb

ε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS 0 1113946

Sb

ε6εlowast (8)61113872 1113873dS 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

So directly identify Q11a Q11b Q12a

Q12b Q22a Q22b Q66a and Q66b using the abovementionedfour special virtual displacement fields If the noise is

ignored approximate parameters expressed as Qapp areidentified +e components of which are defined as equation(17) (see Appendix B2 for the detailed equation derivations)

6 Advances in Materials Science and Engineering

Qapp11a 1113946

Lf

Tiulowast (1)i dl minus Q22a1113946

Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS

minus Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS

Qapp22a 1113946

Lf

Tiulowast (2)i dl minus Q11a1113946

Saε1εlowast (2)1 dS minus Q12a1113946

Saε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66a1113946

Saε6εlowast (2)6 dS

minus Q11b1113946Sbε1εlowast (2)1 dS minus Q22b1113946

Sbε2εlowast (2)2 dS minus Q12b1113946

Sbε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (2)6 dS

Qapp12a 1113946

Lf

Tiulowast (3)i dl minus Q11a1113946

Saε1εlowast (3)1 dS minus Q22a1113946

Saε2εlowast (3)2 dS minus Q66a1113946

Saε6εlowast (3)6 dS

minus Q11b1113946Sbε1εlowast (3)1 dS minus Q22b1113946

Sbε2εlowast (3)2 dS minus Q12b1113946

Sbε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (3)6 dS

Qapp66a 1113946

Lf

Tiulowast (4)i dl minus Q11a1113946

Saε1εlowast (4)1 dS minus Q22a1113946

Saε2εlowast (4)2 dS minus Q12a1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS

minus Q11b1113946Sbε1εlowast (4)1 dS minus Q22b1113946

Sbε2εlowast (4)2 dS minus Q12b1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (4)6 dS

Qapp11b 1113946

Lf

Tiulowast (5)i dl minus Q22b1113946

Sbε2εlowast (5)2 dS minus Q12b1113946

Sbε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (5)6 dS

minus Q11a1113946Saε1εlowast (5)1 dS minus Q22a1113946

Saε2εlowast (5)2 dS minus Q12a1113946

Saε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66a1113946

Saε6εlowast (5)6 dS

Qapp22b 1113946

Lf

Tiulowast (6)i dl minus Q11b1113946

Sbε1εlowast (6)1 dS minus Q12b1113946

Sbε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (6)6 dS

minus Q11a1113946Saε1εlowast (6)1 dS minus Q22a1113946

Saε2εlowast (6)2 dS minus Q12a1113946

Saε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66a1113946

Saε6εlowast (6)6 dS

Qapp12b 1113946

Lf

Tiulowast (7)i dl minus Q11b1113946

Sbε1εlowast (7)1 dS minus Q22b1113946

Sbε2εlowast (7)2 dS minus Q66b1113946

Sbε6εlowast (7)6 dS minus Q11a1113946

Saε1εlowast (7)1 dS

minus Q22a1113946Saε2εlowast (7)2 dS minus Q12a1113946

Saε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS minus Q66a1113946

Saε6εlowast (7)6 dS

Qapp66b 1113946

Lf

Tiulowast (8)i dl minus Q11b1113946

Sbε1εlowast (8)1 dS minus Q22b1113946

Sbε2εlowast (8)2 dS minus Q12b1113946

Sbε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS

minus Q11a1113946Saε1εlowast (8)1 dS minus Q22a1113946

Saε2εlowast (8)2 dS minus Q12a1113946

Saε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS minus Q66a1113946

Saε6εlowast (8)6 dS

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

To minimize the effect of noise taking Q11a as an ex-ample the variance of Q11a is as follows

V Q11a( 1113857 c2E Q

app11a1113946

Saεlowast (1)1 N1dS + Q

app22a1113946

Saεlowast (1)2 N2dS + Q

app12a 1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66a 1113946

Saεlowast (1)6 N6dS1113876 1113877

21113888

+ Qapp11b1113946

Sblowast (1)N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113889

+ c2E Q

app11b1113946

Sbεlowast (1)1 N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app22b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113888 1113889

VA Q11a( 1113857 + VB Q11a( 1113857

(18)

Advances in Materials Science and Engineering 7

As described above the virtual fields are expanded bypolynomial basis functions based on equation (11) +eunknown coefficients Aij and Bij are included in vector Y asfollows

Y

A00

⋮A30

⋮

A03

⋮

A33

B00

⋮

B30

⋮

B03

⋮

B33

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)

+e integrals of equation (15) are calculated usingvectors B11B22B12 and B66 through discrete sum ap-proximation so that

1113946Sa

ε1εlowast1( 1113857dS asymp Lwaε1aε

lowastia B11a middot Y

1113946Sa

ε2εlowast2( 1113857dS asymp Lwaε2aε

lowast2a B22a middot Y

1113946Sa

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwa ε1aε

lowast2a + ε2aε

lowast1a1113872 1113873 B12a middot Y

1113946Sa

ε6εlowast6( 1113857dS asymp Lwaε6aε

lowast6a B66a middot Y

1113946Sb

ε1εlowast1( 1113857dS asymp Lwaε1bε

lowast1b B11b middot Y

1113946Sb

ε2εlowast2( 1113857dS asymp Lwbε2bε

lowast2b B22b middot Y

1113946Sb

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwb ε1bε

lowast2b + ε2bε

lowast1b1113872 1113873 B12b middot Y

1113946Sb

ε6εlowast6( 1113857dS asymp Lwbε6bε

lowast6b B66b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

+e others that the definition from the virtual fields(vectorsH11H22H12 andH66) are related to the terms ofequation (18)

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

ε1εlowast2( 1113857dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast6( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

ε1εlowast2( 1113857dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast6( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

+e first KA virtual field conditions of three-pointbending are ulowast2 0 when x1 0 and x2 0 which implies

B00 0 (22)

+e second KA virtual field conditions of three-pointbending re ulowast1 0 when x1 L which means that

1113944

m

i0Aij 0 j 0 middot middot middot n (23)

+e optimized virtual fields are totally defined +efollowing equations can identify the constitutive parameters

8 Advances in Materials Science and Engineering

Q11a F

tulowast (a)2 x1 L x2 w( 1113857

Q22a F

tulowast (a)2 x1 L x2 w( 1113857

Q12a F

tulowast (a)2 x1 L x2 w( 1113857

Q66a F

tulowast (a)2 x1 L x2 w( 1113857

Q11b F

tulowast (a)2 x1 L x2 w( 1113857

Q22b F

tulowast (a)2 x1 L x2 w( 1113857

Q12b F

tulowast (a)2 x1 L x2 w( 1113857

Q66b F

tulowast (a)2 x1 L x2 w( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

3 Numerical Verification and Discussion

To evaluate the accuracy of the abovementioned optimizedpolynomial VFM numerical experiments with FEM simu-lations and DIC simulations are employed in this section

31 Numerical Verification of Optimized Polynomial VirtualFieldswithFEMSimulations In FEM simulation in order to

facilitate the simulation of bimaterials configuration a two-dimensional rectangular beam specimen was adopted andapplied a three-point bending load to get inhomogeneousin-plane deformation fields Based on the principle of thesymmetry of three-point bending only the left of thespecimen was modelled as shown in Figure 1

+e length width and thickness of the bimaterial beamspecimen were set as L 30mm w 20mm andt 23mm respectively and the width of the two materialregions was the same wa wb 10mm+e force was set asF 2544N +e four-node plane-stress element withthickness (Solid-Quad 4 nodes 182Plane strs wthk) wasemployed +e displacement constraint uy 0 is added inthe lower left corner of the model and the line constraintux 0 is added on the right side of the model

+e beam specimen was set as orthotropic materials +eengineering elastic constants in the FEM simulation areshown in Table 1 and the converted constitutive parametersin the elastic matrix are shown in Table 2

+e displacement fields and the strain fields of the three-point bending test of the orthotropic bimaterial obtainedthrough FEM simulations using the commercial softwareANSYS are shown in Figures 2 and 3

Table 3 shows the identification results of the hetero-geneous orthotropic bimaterials by using the optimizedpolynomial VFM as shown in Section 22 To study theidentification accuracy of the optimized polynomial VFMthe relative error of every constitutive parameter componentand the weighted relative error Wre were calculated Owingto there were 8 unknown constitutive parameters theweighted relative error employed here can be expressed asfollows

Wre

Qi de11a minus Q

rfe11a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22a minus Q

re22a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12a minus Q

re12a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66a minus Q

rfe66a

11138681113868111386811138681113868

11138681113868111386811138681113868

+ Qi de11b minus Q

rfe

11b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22b minus Q

rfe

22b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12b minus Q

rfe

12b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66b minus Q

rfe

66b

11138681113868111386811138681113868

11138681113868111386811138681113868

Qrfe11a + Q

rfe22a + Q

rfe12a + Q

rfe66a + Q

rfe

11b + Qrfe

22b + Qrfe

12b + Qrfe

66b

(25)

It can be seen from Table 3 that the weighted relativeerror is lower than 1 +e relative error of Q66 of the twomaterials is the smallest Although the relative errors of Q11andQ22 of the twomaterials are slightly larger than Q66 bothof them are less than 2 +e relative error of Q12a and Q12b

is significantly higher than others because σ2 must be in-cluded in the virtual fields to identify Q12 while the stressdata density of σ2 is low in the three-point bending test +estress data density can be expressed by the followingequation

Advances in Materials Science and Engineering 9

ρ σi( 1113857 1113944

bpointsj1 σi Mj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868

npoints i 1 2 6 (26)

According to equation (7) the standard deviations of Qij

are equal to η(i)c If c is the standard deviation of the strainnoise the coefficients of variations (CV(Qij)) of eachstiffness component are given by

CV Q11( 1113857 η11Q11

c

CV Q22( 1113857 η22Q22

c

CV Q12( 1113857 η12Q12

c

CV Q66( 1113857 η66Q66

c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

+e ratio of coefficient of variation to the standarddeviation of the strain noise can show the sensitivity to thenoise It indicates the rate at which the coefficient of vari-ation increases with the increase of noise which can becalled the variation coefficient-to-noise ratio and can beexpressed as ηijQij +e lower variation coefficient-to-noiseratio indicates that the corresponding identification con-stitutive parameters are less susceptible to noise +e vari-ation coefficient-to-noise ratios ηijQij of eight identificationresults are shown in Table 3 Significant differences on thevariation coefficient-to-noise ratio and the coefficient-to-noise ratio of Q66 is the smallest the next is Q11 and then isthe Q22 and the biggest is Q12 +e quantitative results showthat if Q11 Q22 and Q66 are stable Q12 is less robustlyidentified +is result shows that this configuration is un-suitable for the transverse stiffness

32 Comparison with Results of Piecewise Virtual Fields andPolynomial Virtual Fields +is section will implement thesame numerical experiment as Section 31 to compare theidentified constitutive parameters of the piecewise virtual

L

w

wa

wb

F

Part A

Part B

Figure 1 Two-dimensional bimaterial model of three-point bending established in FEM simulation

Table 1 Engineering elastic constants of orthotropic bimaterialExa (Mpa) Eya (Mpa) Eza (Mpa) PRxya (Mpa) PRyza (Mpa) PRxza (Mpa) Gxya (Mpa) Gyza (Mpa) Gxza (Mpa)180000 30000 8300 017 025 031 5700 12000 12000Exb (Mpa) Eyb (Mpa) Ezb (Mpa) PRxyb (Mpa) PRyzb (Mpa) PRxzb (Mpa) Gxyb (Mpa) Gyzb (Mpa) Gxzb (Mpa)175000 32000 8300 025 025 031 5700 12000 12000

Table 2 Reference constitutive parameters of orthotropic bimaterial

Q11a (Mpa) Q22a (Mpa) Q12a (Mpa) Q66a (Mpa) Q11b (Mpa) Q22b (Mpa) Q12b (Mpa) Q66b (Mpa)

18193710 3071968 590720 5700 17822916 3297651 894780 5700

10 Advances in Materials Science and Engineering

field and polynomial virtual field At this situation the bi-linear shape function is used to interpolate the four-nodeelement to represent the virtual field instead of the poly-nomial virtual field

+e piecewise function expansion of the virtual dis-placement field is similar to the interpolation of the real

displacement in the FEM Derived from the relationshipbetween strain and displacement

ε Sulowast (28)

where

ndash133477

ndash113034

ndash092591

ndash072149

ndash051706

ndash031263

ndash010821

009622

030065

050507

(a)

ndash956093

ndash849861

ndash743628

ndash637396

ndash531163

ndash42493

ndash318698

ndash212465

ndash106233

0

(b)

Figure 2 Displacement distributions of heterogeneous orthotropic bimaterials (a) u1 (b) u2 (unit mm)

Advances in Materials Science and Engineering 11

ndash0235856

ndash018911

ndash014237

ndash009562

ndash004888

ndash000021

041162

009136

013811

018485

(a)

ndash213206

ndash186779

ndash160351

ndash133924

ndash107496

ndash081069

ndash054641

ndash028214

ndash001786

0024641

(b)

ndash362375

ndash32025

ndash278125

ndash236

ndash193874

ndash151749

ndash109624

ndash067499

ndash025373

016752

(c)

Figure 3 Strain distributions of heterogeneous orthotropic bimaterials (a) ε1 (b) ε2 (c) ε3

12 Advances in Materials Science and Engineering

ε

ε1

ε2

ε6

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

ulowast ulowast1

ulowast2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

S

z

zx10

0z

zx2

z

zx2

z

zx1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(29)

For the quadrilateral element the virtual displacementcan be expressed by the following formula

ulowast ξ1 ξ2( 1113857≃ulowast ξ1 ξ2( 1113857 11139444

a1N(a) ξ1 ξ2( 1113857ulowast(a)

(30)

where ulowast is the vector containing the virtual displacement ofany point (ξ1 ξ2) in the element and 1113957ulowast(a) is the vectorcontaining the virtual displacement of the nodes in the el-ement +e expression of N(a) is defined as

N(a) ξ1 ξ2( 1113857 N(a) ξ1 ξ2( 1113857I (31)

where I is the identity matrix +e shape function expressionof N(a) can be defined as follows

N(1)

14

1 minus ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 + ξ2( 1113857

N(2)

14

1 minus ξ1( 1113857 1 + ξ2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

So the virtual displacement can be expressed by thefollowing formula

ulowast1 asymp N

(1)1113957ulowast(1)1 + N

(2)1113957ulowast(2)1 + N

(3)1113957ulowast(3)1 + N

(4)1113957ulowast(4)1

ulowast2 asymp N

(1)1113957ulowast(1)2 + N

(2)1113957ulowast(2)2 + N

(3)1113957ulowast(3)2 + N

(4)1113957ulowast(4)2

⎧⎨

⎩

(33)

+erefore the unknown coefficient vector Y (equation(17)) in the polynomial virtual fields can be denoted on eachnode

Y

1113957ulowast(1)1

1113957ulowast(1)2

⋮

1113957ulowast(n)1

1113957ulowast(n)2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

where n is the total number of nodesUse the piecewise virtual field instead of the polynomial

virtual field to perform the same experiment (Section 31)Table 4 shows the values of relative and weighted relativeerror of A and B It can be seen from Table 4 that theidentification results of piecewise virtual fields are similar tothat of polynomial virtual fields +e weighted relative erroris 097 which is slightly larger than that of polynomialvirtual fields +e relative error of Q66 of the two materials isthe smallest Although the relative errors of Q11 and Q22ofthe two materials are slightly larger than Q66 both of themare less than 2 Similar to the results of polynomial virtualfields the relative error of Q12a and Q12b is significantlyhigher than others because σ2 must be included in thevirtual fields to identify Q12 while the data density of σ2 islow in the three-point bending test

+e variation coefficient-to-noise ratios of eight iden-tification results are also shown in Table 4 Significant dif-ferences can be seen on the variation coefficient-to-noiseratios and the quantitative results show that if Q11 Q22 andQ66 are stable Q12 is less robustly identified Compared withthe polynomial virtual field the variation coefficient-to-noise ratio of the eight identification parameters calculatedby the optimized piecewise virtual field is larger

33 Influence of Noise on Identification Results +e noise isinevitable in the experiments To investigate the influence ofnoise on identification results zero-mean Gaussian white

Table 3 Identification results of constitutive parameters of orthotropic bimaterial using the optimized polynomial virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18159713 280642 minus019

083

Q22a 3071968 3035043 106390 minus120Q12a 590720 479127 2080530 minus1889Q66a 5700 570102 53999 002Q11b 17822916 17735394 234711 minus049Q22b 3297651 3259546 91601 minus116Q12b 894780 827581 1018827 minus751Q66b 5700 569879 59903 minus002

Advances in Materials Science and Engineering 13

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 4: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (2)6 N6dS11138771113876 1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (3)6 N6dS11138771113876 1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS + Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q)the vector containing the variances ofQ11Q22Q12 and Q66 one can write as equation (8) (seeAppendix A4 for the detailed equation derivations)

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

and (η(i))2 have two kinds of unknowns the constitutiveparameters and the unknown coefficients of the virtualfields So (η(i))2 can be expressed as

η(i)1113872 1113873

212Ylowast(i)HYlowast(i)

Sc

nc

1113888 1113889

2

QappG(i)Qapp (9)

where Ylowast is the vector related to the coefficients of virtualstrain fields H is the Hessian matrix of all monomials in εlowastand G(j) j 1 2 3 4 is the following square matrix

1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113944n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113944n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)

and the (η(i))2 coefficients i 1 2 3 4 depend on theselected virtual fields Also the virtual fields are selectedfor the Qij parameters extraction +us select the ap-propriate virtual field that can reduce the influence of

strain In this article the virtual fields were expandedby the polynomials +e displacement fields and thevirtual strain are as shown in equations (11) and (12)respectively

4 Advances in Materials Science and Engineering

ulowast1 1113944

m

i01113944

n

j0Aij

x1

L1113874 1113875

i x2

w1113874 1113875

j

ulowast2 1113944

m

i01113944

n

j0Bij

x1

L1113874 1113875

i x2

w1113874 1113875

j

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

εlowast1 1113944m

i11113944

n

j0Aij

x1

L1113874 1113875

i minus1 x2

w1113874 1113875

j

εlowast2 1113944m

i01113944

n

j1Bij

x1

L1113874 1113875

i x2

w1113874 1113875

j minus1

εlowast6 1113944

m

i01113944

n

j1Aij

i

L

x1

L1113874 1113875

i x2

w1113874 1113875

j minus1+ 1113944

m

i11113944

n

j0Bij

i

L

x1

L1113874 1113875

i minus 1 x2

w1113874 1113875

j

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

where Aij and Bij are the coefficients of the monomials m

and n are the polynomial degrees that define that themaximum number of monomials has used and L and w arethe typical dimensions of the x1 and x2 directionsrespectively

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (13) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (13)

where λ(i) is the vector containing Lagrange multipliers+e KA condition can produce several linear equations

and the number of this equations is depending on thenumber of supports +e special virtual field condition alsoleads to several linear equations the number of which de-pends on the type of constitutive model of specimen ma-terial +e two types of conditions can produce the followinglinear system as equation (14) (see Appendix A5 for thedetailed equation derivations)

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (14)

Attention that (η(i))2 coefficients also correspond on theunknown Qij parameters +us the optimization problem issolved for a given set of Qij parameters and the process is

iterative the first guess of theQij parameters is used to find thefirst group of four special virtual fields+en these four specialvirtual fields are used to find a new set of Qij parameters andrepeat abovementioned steps until the optimal Qij parameteris found In practice no matter what the parameter Qij isinitially selected this iterative process will converge quicklyUsually only two loops are sufficient to converge

22 Optimized Polynomial Virtual Fields Applied to theBimaterials without Knowing Any Material ConstitutiveParameters As shown in Figure 1 for the plane-stressspecimen with two different materials Part A and Part B andboth of them are set with the elastic orthotropic material Butthe constitutive parameters of A and B are unknown Asmentioned above the sum of exact values and white noise isthe measured strain Assume that the noise components areuncorrelated to each other and presume that the noisebetween points is also uncorrelated +erefore the gov-erning equation of the virtual fields method for the totalspecimen is as equation (15) (see Appendix B1 for the de-tailed equation derivations)

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS

+ Q66a1113946Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS

+ Q66b1113946Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS1113876

+ Q12a1113946Sa

εlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946Saεlowast6 N6dS

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS

+ Q12b1113946Sb

εlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946Sbεlowast6 N6dS1113877

1113946Lf

Tiulowasti( 1113857dl

(15)

+e constraints of the eight special virtual fields denotedthat ulowast(1) ulowast(2) ulowast(3) ulowast(4) ulowast(5) ulowast(6) ulowast(7) and ulowast(8) mustsatisfy the following equation

Advances in Materials Science and Engineering 5

1113946Sa

ε1εlowast (1)11113872 1113873dS 0 1113946

Sa

ε2εlowast (1)21113872 1113873dS 0 1113946

Sa

ε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 1 1113946

Sa

ε6εlowast (1)61113872 1113873dS 0

1113946Sb

ε1εlowast (1)11113872 1113873dS 0 1113946

Sb

ε2εlowast (1)21113872 1113873dS 0 1113946

Sb

ε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 0 1113946

Sb

ε6εlowast (1)61113872 1113873dS 0

1113946Sa

ε1εlowast (2)11113872 1113873dS 0 1113946

Sa

ε2εlowast (2)21113872 1113873dS 0 1113946

Sa

ε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 1 1113946

Sa

ε6εlowast (2)61113872 1113873dS 0

1113946Sb

ε1εlowast (2)11113872 1113873dS 0 1113946

Sb

ε2εlowast (2)21113872 1113873dS 0 1113946

Sb

ε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 0 1113946

Sb

ε6εlowast (2)61113872 1113873dS 0

1113946Sa

ε1εlowast (3)11113872 1113873dS 0 1113946

Sa

ε2εlowast (3)21113872 1113873dS 0 1113946

Sa

ε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 1 1113946

Sa

ε6εlowast (3)61113872 1113873dS 0

1113946Sb

ε1εlowast (3)11113872 1113873dS 0 1113946

Sb

ε2εlowast (3)21113872 1113873dS 0 1113946

Sb

ε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 0 1113946

Sb

ε6εlowast (3)61113872 1113873dS 0

1113946Sa

ε1εlowast (4)11113872 1113873dS 0 1113946

Sa

ε2εlowast (4)21113872 1113873dS 0 1113946

Sa

ε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 1 1113946

Sa

ε6εlowast (4)61113872 1113873dS 0

1113946Sb

ε1εlowast (4)11113872 1113873dS 0 1113946

Sb

ε2εlowast (4)21113872 1113873dS 0 1113946

Sb

ε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 0 1113946

Sb

ε6εlowast (4)61113872 1113873dS 0

1113946Sa

ε1εlowast (5)11113872 1113873dS 0 1113946

Sa

ε2εlowast (5)21113872 1113873dS 0 1113946

Sa

ε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS 1 1113946

Sa

ε6εlowast (5)61113872 1113873dS 0

1113946Sb

ε1εlowast (5)11113872 1113873dS 0 1113946

Sb

ε2εlowast (5)21113872 1113873dS 0 1113946

Sb

ε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS 0 1113946

Sb

ε6εlowast (5)61113872 1113873dS 0

1113946Sa

ε1εlowast (6)11113872 1113873dS 0 1113946

Sa

ε2εlowast (6)21113872 1113873dS 0 1113946

Sa

ε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS 1 1113946

Sa

ε6εlowast (6)61113872 1113873dS 0

1113946Sb

ε1εlowast (6)11113872 1113873dS 0 1113946

Sb

ε2εlowast (6)21113872 1113873dS 0 1113946

Sb

ε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS 0 1113946

Sb

ε6εlowast (6)61113872 1113873dS 0

1113946Sa

ε1εlowast (7)11113872 1113873dS 0 1113946

Sa

ε2εlowast (7)21113872 1113873dS 0 1113946

Sa

ε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS 1 1113946

Sa

ε6εlowast (7)61113872 1113873dS 0

1113946Sb

ε1εlowast (7)11113872 1113873dS 0 1113946

Sb

ε2εlowast (7)21113872 1113873dS 0 1113946

Sb

ε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS 0 1113946

Sb

ε6εlowast (7)61113872 1113873dS 0

1113946Sa

ε1εlowast (8)11113872 1113873dS 0 1113946

Sa

ε2εlowast (8)21113872 1113873dS 0 1113946

Sa

ε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS 1 1113946

Sa

ε6εlowast (8)61113872 1113873dS 0

1113946Sb

ε1εlowast (8)11113872 1113873dS 0 1113946

Sb

ε2εlowast (8)21113872 1113873dS 0 1113946

Sb

ε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS 0 1113946

Sb

ε6εlowast (8)61113872 1113873dS 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

So directly identify Q11a Q11b Q12a

Q12b Q22a Q22b Q66a and Q66b using the abovementionedfour special virtual displacement fields If the noise is

ignored approximate parameters expressed as Qapp areidentified +e components of which are defined as equation(17) (see Appendix B2 for the detailed equation derivations)

6 Advances in Materials Science and Engineering

Qapp11a 1113946

Lf

Tiulowast (1)i dl minus Q22a1113946

Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS

minus Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS

Qapp22a 1113946

Lf

Tiulowast (2)i dl minus Q11a1113946

Saε1εlowast (2)1 dS minus Q12a1113946

Saε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66a1113946

Saε6εlowast (2)6 dS

minus Q11b1113946Sbε1εlowast (2)1 dS minus Q22b1113946

Sbε2εlowast (2)2 dS minus Q12b1113946

Sbε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (2)6 dS

Qapp12a 1113946

Lf

Tiulowast (3)i dl minus Q11a1113946

Saε1εlowast (3)1 dS minus Q22a1113946

Saε2εlowast (3)2 dS minus Q66a1113946

Saε6εlowast (3)6 dS

minus Q11b1113946Sbε1εlowast (3)1 dS minus Q22b1113946

Sbε2εlowast (3)2 dS minus Q12b1113946

Sbε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (3)6 dS

Qapp66a 1113946

Lf

Tiulowast (4)i dl minus Q11a1113946

Saε1εlowast (4)1 dS minus Q22a1113946

Saε2εlowast (4)2 dS minus Q12a1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS

minus Q11b1113946Sbε1εlowast (4)1 dS minus Q22b1113946

Sbε2εlowast (4)2 dS minus Q12b1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (4)6 dS

Qapp11b 1113946

Lf

Tiulowast (5)i dl minus Q22b1113946

Sbε2εlowast (5)2 dS minus Q12b1113946

Sbε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (5)6 dS

minus Q11a1113946Saε1εlowast (5)1 dS minus Q22a1113946

Saε2εlowast (5)2 dS minus Q12a1113946

Saε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66a1113946

Saε6εlowast (5)6 dS

Qapp22b 1113946

Lf

Tiulowast (6)i dl minus Q11b1113946

Sbε1εlowast (6)1 dS minus Q12b1113946

Sbε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (6)6 dS

minus Q11a1113946Saε1εlowast (6)1 dS minus Q22a1113946

Saε2εlowast (6)2 dS minus Q12a1113946

Saε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66a1113946

Saε6εlowast (6)6 dS

Qapp12b 1113946

Lf

Tiulowast (7)i dl minus Q11b1113946

Sbε1εlowast (7)1 dS minus Q22b1113946

Sbε2εlowast (7)2 dS minus Q66b1113946

Sbε6εlowast (7)6 dS minus Q11a1113946

Saε1εlowast (7)1 dS

minus Q22a1113946Saε2εlowast (7)2 dS minus Q12a1113946

Saε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS minus Q66a1113946

Saε6εlowast (7)6 dS

Qapp66b 1113946

Lf

Tiulowast (8)i dl minus Q11b1113946

Sbε1εlowast (8)1 dS minus Q22b1113946

Sbε2εlowast (8)2 dS minus Q12b1113946

Sbε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS

minus Q11a1113946Saε1εlowast (8)1 dS minus Q22a1113946

Saε2εlowast (8)2 dS minus Q12a1113946

Saε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS minus Q66a1113946

Saε6εlowast (8)6 dS

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

To minimize the effect of noise taking Q11a as an ex-ample the variance of Q11a is as follows

V Q11a( 1113857 c2E Q

app11a1113946

Saεlowast (1)1 N1dS + Q

app22a1113946

Saεlowast (1)2 N2dS + Q

app12a 1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66a 1113946

Saεlowast (1)6 N6dS1113876 1113877

21113888

+ Qapp11b1113946

Sblowast (1)N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113889

+ c2E Q

app11b1113946

Sbεlowast (1)1 N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app22b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113888 1113889

VA Q11a( 1113857 + VB Q11a( 1113857

(18)

Advances in Materials Science and Engineering 7

As described above the virtual fields are expanded bypolynomial basis functions based on equation (11) +eunknown coefficients Aij and Bij are included in vector Y asfollows

Y

A00

⋮A30

⋮

A03

⋮

A33

B00

⋮

B30

⋮

B03

⋮

B33

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)

+e integrals of equation (15) are calculated usingvectors B11B22B12 and B66 through discrete sum ap-proximation so that

1113946Sa

ε1εlowast1( 1113857dS asymp Lwaε1aε

lowastia B11a middot Y

1113946Sa

ε2εlowast2( 1113857dS asymp Lwaε2aε

lowast2a B22a middot Y

1113946Sa

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwa ε1aε

lowast2a + ε2aε

lowast1a1113872 1113873 B12a middot Y

1113946Sa

ε6εlowast6( 1113857dS asymp Lwaε6aε

lowast6a B66a middot Y

1113946Sb

ε1εlowast1( 1113857dS asymp Lwaε1bε

lowast1b B11b middot Y

1113946Sb

ε2εlowast2( 1113857dS asymp Lwbε2bε

lowast2b B22b middot Y

1113946Sb

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwb ε1bε

lowast2b + ε2bε

lowast1b1113872 1113873 B12b middot Y

1113946Sb

ε6εlowast6( 1113857dS asymp Lwbε6bε

lowast6b B66b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

+e others that the definition from the virtual fields(vectorsH11H22H12 andH66) are related to the terms ofequation (18)

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

ε1εlowast2( 1113857dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast6( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

ε1εlowast2( 1113857dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast6( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

+e first KA virtual field conditions of three-pointbending are ulowast2 0 when x1 0 and x2 0 which implies

B00 0 (22)

+e second KA virtual field conditions of three-pointbending re ulowast1 0 when x1 L which means that

1113944

m

i0Aij 0 j 0 middot middot middot n (23)

+e optimized virtual fields are totally defined +efollowing equations can identify the constitutive parameters

8 Advances in Materials Science and Engineering

Q11a F

tulowast (a)2 x1 L x2 w( 1113857

Q22a F

tulowast (a)2 x1 L x2 w( 1113857

Q12a F

tulowast (a)2 x1 L x2 w( 1113857

Q66a F

tulowast (a)2 x1 L x2 w( 1113857

Q11b F

tulowast (a)2 x1 L x2 w( 1113857

Q22b F

tulowast (a)2 x1 L x2 w( 1113857

Q12b F

tulowast (a)2 x1 L x2 w( 1113857

Q66b F

tulowast (a)2 x1 L x2 w( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

3 Numerical Verification and Discussion

To evaluate the accuracy of the abovementioned optimizedpolynomial VFM numerical experiments with FEM simu-lations and DIC simulations are employed in this section

31 Numerical Verification of Optimized Polynomial VirtualFieldswithFEMSimulations In FEM simulation in order to

facilitate the simulation of bimaterials configuration a two-dimensional rectangular beam specimen was adopted andapplied a three-point bending load to get inhomogeneousin-plane deformation fields Based on the principle of thesymmetry of three-point bending only the left of thespecimen was modelled as shown in Figure 1

+e length width and thickness of the bimaterial beamspecimen were set as L 30mm w 20mm andt 23mm respectively and the width of the two materialregions was the same wa wb 10mm+e force was set asF 2544N +e four-node plane-stress element withthickness (Solid-Quad 4 nodes 182Plane strs wthk) wasemployed +e displacement constraint uy 0 is added inthe lower left corner of the model and the line constraintux 0 is added on the right side of the model

+e beam specimen was set as orthotropic materials +eengineering elastic constants in the FEM simulation areshown in Table 1 and the converted constitutive parametersin the elastic matrix are shown in Table 2

+e displacement fields and the strain fields of the three-point bending test of the orthotropic bimaterial obtainedthrough FEM simulations using the commercial softwareANSYS are shown in Figures 2 and 3

Table 3 shows the identification results of the hetero-geneous orthotropic bimaterials by using the optimizedpolynomial VFM as shown in Section 22 To study theidentification accuracy of the optimized polynomial VFMthe relative error of every constitutive parameter componentand the weighted relative error Wre were calculated Owingto there were 8 unknown constitutive parameters theweighted relative error employed here can be expressed asfollows

Wre

Qi de11a minus Q

rfe11a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22a minus Q

re22a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12a minus Q

re12a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66a minus Q

rfe66a

11138681113868111386811138681113868

11138681113868111386811138681113868

+ Qi de11b minus Q

rfe

11b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22b minus Q

rfe

22b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12b minus Q

rfe

12b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66b minus Q

rfe

66b

11138681113868111386811138681113868

11138681113868111386811138681113868

Qrfe11a + Q

rfe22a + Q

rfe12a + Q

rfe66a + Q

rfe

11b + Qrfe

22b + Qrfe

12b + Qrfe

66b

(25)

It can be seen from Table 3 that the weighted relativeerror is lower than 1 +e relative error of Q66 of the twomaterials is the smallest Although the relative errors of Q11andQ22 of the twomaterials are slightly larger than Q66 bothof them are less than 2 +e relative error of Q12a and Q12b

is significantly higher than others because σ2 must be in-cluded in the virtual fields to identify Q12 while the stressdata density of σ2 is low in the three-point bending test +estress data density can be expressed by the followingequation

Advances in Materials Science and Engineering 9

ρ σi( 1113857 1113944

bpointsj1 σi Mj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868

npoints i 1 2 6 (26)

According to equation (7) the standard deviations of Qij

are equal to η(i)c If c is the standard deviation of the strainnoise the coefficients of variations (CV(Qij)) of eachstiffness component are given by

CV Q11( 1113857 η11Q11

c

CV Q22( 1113857 η22Q22

c

CV Q12( 1113857 η12Q12

c

CV Q66( 1113857 η66Q66

c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

+e ratio of coefficient of variation to the standarddeviation of the strain noise can show the sensitivity to thenoise It indicates the rate at which the coefficient of vari-ation increases with the increase of noise which can becalled the variation coefficient-to-noise ratio and can beexpressed as ηijQij +e lower variation coefficient-to-noiseratio indicates that the corresponding identification con-stitutive parameters are less susceptible to noise +e vari-ation coefficient-to-noise ratios ηijQij of eight identificationresults are shown in Table 3 Significant differences on thevariation coefficient-to-noise ratio and the coefficient-to-noise ratio of Q66 is the smallest the next is Q11 and then isthe Q22 and the biggest is Q12 +e quantitative results showthat if Q11 Q22 and Q66 are stable Q12 is less robustlyidentified +is result shows that this configuration is un-suitable for the transverse stiffness

32 Comparison with Results of Piecewise Virtual Fields andPolynomial Virtual Fields +is section will implement thesame numerical experiment as Section 31 to compare theidentified constitutive parameters of the piecewise virtual

L

w

wa

wb

F

Part A

Part B

Figure 1 Two-dimensional bimaterial model of three-point bending established in FEM simulation

Table 1 Engineering elastic constants of orthotropic bimaterialExa (Mpa) Eya (Mpa) Eza (Mpa) PRxya (Mpa) PRyza (Mpa) PRxza (Mpa) Gxya (Mpa) Gyza (Mpa) Gxza (Mpa)180000 30000 8300 017 025 031 5700 12000 12000Exb (Mpa) Eyb (Mpa) Ezb (Mpa) PRxyb (Mpa) PRyzb (Mpa) PRxzb (Mpa) Gxyb (Mpa) Gyzb (Mpa) Gxzb (Mpa)175000 32000 8300 025 025 031 5700 12000 12000

Table 2 Reference constitutive parameters of orthotropic bimaterial

Q11a (Mpa) Q22a (Mpa) Q12a (Mpa) Q66a (Mpa) Q11b (Mpa) Q22b (Mpa) Q12b (Mpa) Q66b (Mpa)

18193710 3071968 590720 5700 17822916 3297651 894780 5700

10 Advances in Materials Science and Engineering

field and polynomial virtual field At this situation the bi-linear shape function is used to interpolate the four-nodeelement to represent the virtual field instead of the poly-nomial virtual field

+e piecewise function expansion of the virtual dis-placement field is similar to the interpolation of the real

displacement in the FEM Derived from the relationshipbetween strain and displacement

ε Sulowast (28)

where

ndash133477

ndash113034

ndash092591

ndash072149

ndash051706

ndash031263

ndash010821

009622

030065

050507

(a)

ndash956093

ndash849861

ndash743628

ndash637396

ndash531163

ndash42493

ndash318698

ndash212465

ndash106233

0

(b)

Figure 2 Displacement distributions of heterogeneous orthotropic bimaterials (a) u1 (b) u2 (unit mm)

Advances in Materials Science and Engineering 11

ndash0235856

ndash018911

ndash014237

ndash009562

ndash004888

ndash000021

041162

009136

013811

018485

(a)

ndash213206

ndash186779

ndash160351

ndash133924

ndash107496

ndash081069

ndash054641

ndash028214

ndash001786

0024641

(b)

ndash362375

ndash32025

ndash278125

ndash236

ndash193874

ndash151749

ndash109624

ndash067499

ndash025373

016752

(c)

Figure 3 Strain distributions of heterogeneous orthotropic bimaterials (a) ε1 (b) ε2 (c) ε3

12 Advances in Materials Science and Engineering

ε

ε1

ε2

ε6

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

ulowast ulowast1

ulowast2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

S

z

zx10

0z

zx2

z

zx2

z

zx1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(29)

For the quadrilateral element the virtual displacementcan be expressed by the following formula

ulowast ξ1 ξ2( 1113857≃ulowast ξ1 ξ2( 1113857 11139444

a1N(a) ξ1 ξ2( 1113857ulowast(a)

(30)

where ulowast is the vector containing the virtual displacement ofany point (ξ1 ξ2) in the element and 1113957ulowast(a) is the vectorcontaining the virtual displacement of the nodes in the el-ement +e expression of N(a) is defined as

N(a) ξ1 ξ2( 1113857 N(a) ξ1 ξ2( 1113857I (31)

where I is the identity matrix +e shape function expressionof N(a) can be defined as follows

N(1)

14

1 minus ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 + ξ2( 1113857

N(2)

14

1 minus ξ1( 1113857 1 + ξ2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

So the virtual displacement can be expressed by thefollowing formula

ulowast1 asymp N

(1)1113957ulowast(1)1 + N

(2)1113957ulowast(2)1 + N

(3)1113957ulowast(3)1 + N

(4)1113957ulowast(4)1

ulowast2 asymp N

(1)1113957ulowast(1)2 + N

(2)1113957ulowast(2)2 + N

(3)1113957ulowast(3)2 + N

(4)1113957ulowast(4)2

⎧⎨

⎩

(33)

+erefore the unknown coefficient vector Y (equation(17)) in the polynomial virtual fields can be denoted on eachnode

Y

1113957ulowast(1)1

1113957ulowast(1)2

⋮

1113957ulowast(n)1

1113957ulowast(n)2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

where n is the total number of nodesUse the piecewise virtual field instead of the polynomial

virtual field to perform the same experiment (Section 31)Table 4 shows the values of relative and weighted relativeerror of A and B It can be seen from Table 4 that theidentification results of piecewise virtual fields are similar tothat of polynomial virtual fields +e weighted relative erroris 097 which is slightly larger than that of polynomialvirtual fields +e relative error of Q66 of the two materials isthe smallest Although the relative errors of Q11 and Q22ofthe two materials are slightly larger than Q66 both of themare less than 2 Similar to the results of polynomial virtualfields the relative error of Q12a and Q12b is significantlyhigher than others because σ2 must be included in thevirtual fields to identify Q12 while the data density of σ2 islow in the three-point bending test

+e variation coefficient-to-noise ratios of eight iden-tification results are also shown in Table 4 Significant dif-ferences can be seen on the variation coefficient-to-noiseratios and the quantitative results show that if Q11 Q22 andQ66 are stable Q12 is less robustly identified Compared withthe polynomial virtual field the variation coefficient-to-noise ratio of the eight identification parameters calculatedby the optimized piecewise virtual field is larger

33 Influence of Noise on Identification Results +e noise isinevitable in the experiments To investigate the influence ofnoise on identification results zero-mean Gaussian white

Table 3 Identification results of constitutive parameters of orthotropic bimaterial using the optimized polynomial virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18159713 280642 minus019

083

Q22a 3071968 3035043 106390 minus120Q12a 590720 479127 2080530 minus1889Q66a 5700 570102 53999 002Q11b 17822916 17735394 234711 minus049Q22b 3297651 3259546 91601 minus116Q12b 894780 827581 1018827 minus751Q66b 5700 569879 59903 minus002

Advances in Materials Science and Engineering 13

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 5: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

ulowast1 1113944

m

i01113944

n

j0Aij

x1

L1113874 1113875

i x2

w1113874 1113875

j

ulowast2 1113944

m

i01113944

n

j0Bij

x1

L1113874 1113875

i x2

w1113874 1113875

j

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

εlowast1 1113944m

i11113944

n

j0Aij

x1

L1113874 1113875

i minus1 x2

w1113874 1113875

j

εlowast2 1113944m

i01113944

n

j1Bij

x1

L1113874 1113875

i x2

w1113874 1113875

j minus1

εlowast6 1113944

m

i01113944

n

j1Aij

i

L

x1

L1113874 1113875

i x2

w1113874 1113875

j minus1+ 1113944

m

i11113944

n

j0Bij

i

L

x1

L1113874 1113875

i minus 1 x2

w1113874 1113875

j

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

where Aij and Bij are the coefficients of the monomials m

and n are the polynomial degrees that define that themaximum number of monomials has used and L and w arethe typical dimensions of the x1 and x2 directionsrespectively

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (13) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (13)

where λ(i) is the vector containing Lagrange multipliers+e KA condition can produce several linear equations

and the number of this equations is depending on thenumber of supports +e special virtual field condition alsoleads to several linear equations the number of which de-pends on the type of constitutive model of specimen ma-terial +e two types of conditions can produce the followinglinear system as equation (14) (see Appendix A5 for thedetailed equation derivations)

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (14)

Attention that (η(i))2 coefficients also correspond on theunknown Qij parameters +us the optimization problem issolved for a given set of Qij parameters and the process is

iterative the first guess of theQij parameters is used to find thefirst group of four special virtual fields+en these four specialvirtual fields are used to find a new set of Qij parameters andrepeat abovementioned steps until the optimal Qij parameteris found In practice no matter what the parameter Qij isinitially selected this iterative process will converge quicklyUsually only two loops are sufficient to converge

22 Optimized Polynomial Virtual Fields Applied to theBimaterials without Knowing Any Material ConstitutiveParameters As shown in Figure 1 for the plane-stressspecimen with two different materials Part A and Part B andboth of them are set with the elastic orthotropic material Butthe constitutive parameters of A and B are unknown Asmentioned above the sum of exact values and white noise isthe measured strain Assume that the noise components areuncorrelated to each other and presume that the noisebetween points is also uncorrelated +erefore the gov-erning equation of the virtual fields method for the totalspecimen is as equation (15) (see Appendix B1 for the de-tailed equation derivations)

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS

+ Q66a1113946Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS

+ Q66b1113946Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS1113876

+ Q12a1113946Sa

εlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946Saεlowast6 N6dS

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS

+ Q12b1113946Sb

εlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946Sbεlowast6 N6dS1113877

1113946Lf

Tiulowasti( 1113857dl

(15)

+e constraints of the eight special virtual fields denotedthat ulowast(1) ulowast(2) ulowast(3) ulowast(4) ulowast(5) ulowast(6) ulowast(7) and ulowast(8) mustsatisfy the following equation

Advances in Materials Science and Engineering 5

1113946Sa

ε1εlowast (1)11113872 1113873dS 0 1113946

Sa

ε2εlowast (1)21113872 1113873dS 0 1113946

Sa

ε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 1 1113946

Sa

ε6εlowast (1)61113872 1113873dS 0

1113946Sb

ε1εlowast (1)11113872 1113873dS 0 1113946

Sb

ε2εlowast (1)21113872 1113873dS 0 1113946

Sb

ε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 0 1113946

Sb

ε6εlowast (1)61113872 1113873dS 0

1113946Sa

ε1εlowast (2)11113872 1113873dS 0 1113946

Sa

ε2εlowast (2)21113872 1113873dS 0 1113946

Sa

ε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 1 1113946

Sa

ε6εlowast (2)61113872 1113873dS 0

1113946Sb

ε1εlowast (2)11113872 1113873dS 0 1113946

Sb

ε2εlowast (2)21113872 1113873dS 0 1113946

Sb

ε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 0 1113946

Sb

ε6εlowast (2)61113872 1113873dS 0

1113946Sa

ε1εlowast (3)11113872 1113873dS 0 1113946

Sa

ε2εlowast (3)21113872 1113873dS 0 1113946

Sa

ε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 1 1113946

Sa

ε6εlowast (3)61113872 1113873dS 0

1113946Sb

ε1εlowast (3)11113872 1113873dS 0 1113946

Sb

ε2εlowast (3)21113872 1113873dS 0 1113946

Sb

ε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 0 1113946

Sb

ε6εlowast (3)61113872 1113873dS 0

1113946Sa

ε1εlowast (4)11113872 1113873dS 0 1113946

Sa

ε2εlowast (4)21113872 1113873dS 0 1113946

Sa

ε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 1 1113946

Sa

ε6εlowast (4)61113872 1113873dS 0

1113946Sb

ε1εlowast (4)11113872 1113873dS 0 1113946

Sb

ε2εlowast (4)21113872 1113873dS 0 1113946

Sb

ε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 0 1113946

Sb

ε6εlowast (4)61113872 1113873dS 0

1113946Sa

ε1εlowast (5)11113872 1113873dS 0 1113946

Sa

ε2εlowast (5)21113872 1113873dS 0 1113946

Sa

ε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS 1 1113946

Sa

ε6εlowast (5)61113872 1113873dS 0

1113946Sb

ε1εlowast (5)11113872 1113873dS 0 1113946

Sb

ε2εlowast (5)21113872 1113873dS 0 1113946

Sb

ε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS 0 1113946

Sb

ε6εlowast (5)61113872 1113873dS 0

1113946Sa

ε1εlowast (6)11113872 1113873dS 0 1113946

Sa

ε2εlowast (6)21113872 1113873dS 0 1113946

Sa

ε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS 1 1113946

Sa

ε6εlowast (6)61113872 1113873dS 0

1113946Sb

ε1εlowast (6)11113872 1113873dS 0 1113946

Sb

ε2εlowast (6)21113872 1113873dS 0 1113946

Sb

ε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS 0 1113946

Sb

ε6εlowast (6)61113872 1113873dS 0

1113946Sa

ε1εlowast (7)11113872 1113873dS 0 1113946

Sa

ε2εlowast (7)21113872 1113873dS 0 1113946

Sa

ε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS 1 1113946

Sa

ε6εlowast (7)61113872 1113873dS 0

1113946Sb

ε1εlowast (7)11113872 1113873dS 0 1113946

Sb

ε2εlowast (7)21113872 1113873dS 0 1113946

Sb

ε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS 0 1113946

Sb

ε6εlowast (7)61113872 1113873dS 0

1113946Sa

ε1εlowast (8)11113872 1113873dS 0 1113946

Sa

ε2εlowast (8)21113872 1113873dS 0 1113946

Sa

ε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS 1 1113946

Sa

ε6εlowast (8)61113872 1113873dS 0

1113946Sb

ε1εlowast (8)11113872 1113873dS 0 1113946

Sb

ε2εlowast (8)21113872 1113873dS 0 1113946

Sb

ε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS 0 1113946

Sb

ε6εlowast (8)61113872 1113873dS 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

So directly identify Q11a Q11b Q12a

Q12b Q22a Q22b Q66a and Q66b using the abovementionedfour special virtual displacement fields If the noise is

ignored approximate parameters expressed as Qapp areidentified +e components of which are defined as equation(17) (see Appendix B2 for the detailed equation derivations)

6 Advances in Materials Science and Engineering

Qapp11a 1113946

Lf

Tiulowast (1)i dl minus Q22a1113946

Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS

minus Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS

Qapp22a 1113946

Lf

Tiulowast (2)i dl minus Q11a1113946

Saε1εlowast (2)1 dS minus Q12a1113946

Saε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66a1113946

Saε6εlowast (2)6 dS

minus Q11b1113946Sbε1εlowast (2)1 dS minus Q22b1113946

Sbε2εlowast (2)2 dS minus Q12b1113946

Sbε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (2)6 dS

Qapp12a 1113946

Lf

Tiulowast (3)i dl minus Q11a1113946

Saε1εlowast (3)1 dS minus Q22a1113946

Saε2εlowast (3)2 dS minus Q66a1113946

Saε6εlowast (3)6 dS

minus Q11b1113946Sbε1εlowast (3)1 dS minus Q22b1113946

Sbε2εlowast (3)2 dS minus Q12b1113946

Sbε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (3)6 dS

Qapp66a 1113946

Lf

Tiulowast (4)i dl minus Q11a1113946

Saε1εlowast (4)1 dS minus Q22a1113946

Saε2εlowast (4)2 dS minus Q12a1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS

minus Q11b1113946Sbε1εlowast (4)1 dS minus Q22b1113946

Sbε2εlowast (4)2 dS minus Q12b1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (4)6 dS

Qapp11b 1113946

Lf

Tiulowast (5)i dl minus Q22b1113946

Sbε2εlowast (5)2 dS minus Q12b1113946

Sbε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (5)6 dS

minus Q11a1113946Saε1εlowast (5)1 dS minus Q22a1113946

Saε2εlowast (5)2 dS minus Q12a1113946

Saε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66a1113946

Saε6εlowast (5)6 dS

Qapp22b 1113946

Lf

Tiulowast (6)i dl minus Q11b1113946

Sbε1εlowast (6)1 dS minus Q12b1113946

Sbε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (6)6 dS

minus Q11a1113946Saε1εlowast (6)1 dS minus Q22a1113946

Saε2εlowast (6)2 dS minus Q12a1113946

Saε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66a1113946

Saε6εlowast (6)6 dS

Qapp12b 1113946

Lf

Tiulowast (7)i dl minus Q11b1113946

Sbε1εlowast (7)1 dS minus Q22b1113946

Sbε2εlowast (7)2 dS minus Q66b1113946

Sbε6εlowast (7)6 dS minus Q11a1113946

Saε1εlowast (7)1 dS

minus Q22a1113946Saε2εlowast (7)2 dS minus Q12a1113946

Saε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS minus Q66a1113946

Saε6εlowast (7)6 dS

Qapp66b 1113946

Lf

Tiulowast (8)i dl minus Q11b1113946

Sbε1εlowast (8)1 dS minus Q22b1113946

Sbε2εlowast (8)2 dS minus Q12b1113946

Sbε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS

minus Q11a1113946Saε1εlowast (8)1 dS minus Q22a1113946

Saε2εlowast (8)2 dS minus Q12a1113946

Saε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS minus Q66a1113946

Saε6εlowast (8)6 dS

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

To minimize the effect of noise taking Q11a as an ex-ample the variance of Q11a is as follows

V Q11a( 1113857 c2E Q

app11a1113946

Saεlowast (1)1 N1dS + Q

app22a1113946

Saεlowast (1)2 N2dS + Q

app12a 1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66a 1113946

Saεlowast (1)6 N6dS1113876 1113877

21113888

+ Qapp11b1113946

Sblowast (1)N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113889

+ c2E Q

app11b1113946

Sbεlowast (1)1 N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app22b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113888 1113889

VA Q11a( 1113857 + VB Q11a( 1113857

(18)

Advances in Materials Science and Engineering 7

As described above the virtual fields are expanded bypolynomial basis functions based on equation (11) +eunknown coefficients Aij and Bij are included in vector Y asfollows

Y

A00

⋮A30

⋮

A03

⋮

A33

B00

⋮

B30

⋮

B03

⋮

B33

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)

+e integrals of equation (15) are calculated usingvectors B11B22B12 and B66 through discrete sum ap-proximation so that

1113946Sa

ε1εlowast1( 1113857dS asymp Lwaε1aε

lowastia B11a middot Y

1113946Sa

ε2εlowast2( 1113857dS asymp Lwaε2aε

lowast2a B22a middot Y

1113946Sa

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwa ε1aε

lowast2a + ε2aε

lowast1a1113872 1113873 B12a middot Y

1113946Sa

ε6εlowast6( 1113857dS asymp Lwaε6aε

lowast6a B66a middot Y

1113946Sb

ε1εlowast1( 1113857dS asymp Lwaε1bε

lowast1b B11b middot Y

1113946Sb

ε2εlowast2( 1113857dS asymp Lwbε2bε

lowast2b B22b middot Y

1113946Sb

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwb ε1bε

lowast2b + ε2bε

lowast1b1113872 1113873 B12b middot Y

1113946Sb

ε6εlowast6( 1113857dS asymp Lwbε6bε

lowast6b B66b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

+e others that the definition from the virtual fields(vectorsH11H22H12 andH66) are related to the terms ofequation (18)

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

ε1εlowast2( 1113857dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast6( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

ε1εlowast2( 1113857dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast6( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

+e first KA virtual field conditions of three-pointbending are ulowast2 0 when x1 0 and x2 0 which implies

B00 0 (22)

+e second KA virtual field conditions of three-pointbending re ulowast1 0 when x1 L which means that

1113944

m

i0Aij 0 j 0 middot middot middot n (23)

+e optimized virtual fields are totally defined +efollowing equations can identify the constitutive parameters

8 Advances in Materials Science and Engineering

Q11a F

tulowast (a)2 x1 L x2 w( 1113857

Q22a F

tulowast (a)2 x1 L x2 w( 1113857

Q12a F

tulowast (a)2 x1 L x2 w( 1113857

Q66a F

tulowast (a)2 x1 L x2 w( 1113857

Q11b F

tulowast (a)2 x1 L x2 w( 1113857

Q22b F

tulowast (a)2 x1 L x2 w( 1113857

Q12b F

tulowast (a)2 x1 L x2 w( 1113857

Q66b F

tulowast (a)2 x1 L x2 w( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

3 Numerical Verification and Discussion

To evaluate the accuracy of the abovementioned optimizedpolynomial VFM numerical experiments with FEM simu-lations and DIC simulations are employed in this section

31 Numerical Verification of Optimized Polynomial VirtualFieldswithFEMSimulations In FEM simulation in order to

facilitate the simulation of bimaterials configuration a two-dimensional rectangular beam specimen was adopted andapplied a three-point bending load to get inhomogeneousin-plane deformation fields Based on the principle of thesymmetry of three-point bending only the left of thespecimen was modelled as shown in Figure 1

+e length width and thickness of the bimaterial beamspecimen were set as L 30mm w 20mm andt 23mm respectively and the width of the two materialregions was the same wa wb 10mm+e force was set asF 2544N +e four-node plane-stress element withthickness (Solid-Quad 4 nodes 182Plane strs wthk) wasemployed +e displacement constraint uy 0 is added inthe lower left corner of the model and the line constraintux 0 is added on the right side of the model

+e beam specimen was set as orthotropic materials +eengineering elastic constants in the FEM simulation areshown in Table 1 and the converted constitutive parametersin the elastic matrix are shown in Table 2

+e displacement fields and the strain fields of the three-point bending test of the orthotropic bimaterial obtainedthrough FEM simulations using the commercial softwareANSYS are shown in Figures 2 and 3

Table 3 shows the identification results of the hetero-geneous orthotropic bimaterials by using the optimizedpolynomial VFM as shown in Section 22 To study theidentification accuracy of the optimized polynomial VFMthe relative error of every constitutive parameter componentand the weighted relative error Wre were calculated Owingto there were 8 unknown constitutive parameters theweighted relative error employed here can be expressed asfollows

Wre

Qi de11a minus Q

rfe11a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22a minus Q

re22a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12a minus Q

re12a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66a minus Q

rfe66a

11138681113868111386811138681113868

11138681113868111386811138681113868

+ Qi de11b minus Q

rfe

11b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22b minus Q

rfe

22b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12b minus Q

rfe

12b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66b minus Q

rfe

66b

11138681113868111386811138681113868

11138681113868111386811138681113868

Qrfe11a + Q

rfe22a + Q

rfe12a + Q

rfe66a + Q

rfe

11b + Qrfe

22b + Qrfe

12b + Qrfe

66b

(25)

It can be seen from Table 3 that the weighted relativeerror is lower than 1 +e relative error of Q66 of the twomaterials is the smallest Although the relative errors of Q11andQ22 of the twomaterials are slightly larger than Q66 bothof them are less than 2 +e relative error of Q12a and Q12b

is significantly higher than others because σ2 must be in-cluded in the virtual fields to identify Q12 while the stressdata density of σ2 is low in the three-point bending test +estress data density can be expressed by the followingequation

Advances in Materials Science and Engineering 9

ρ σi( 1113857 1113944

bpointsj1 σi Mj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868

npoints i 1 2 6 (26)

According to equation (7) the standard deviations of Qij

are equal to η(i)c If c is the standard deviation of the strainnoise the coefficients of variations (CV(Qij)) of eachstiffness component are given by

CV Q11( 1113857 η11Q11

c

CV Q22( 1113857 η22Q22

c

CV Q12( 1113857 η12Q12

c

CV Q66( 1113857 η66Q66

c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

+e ratio of coefficient of variation to the standarddeviation of the strain noise can show the sensitivity to thenoise It indicates the rate at which the coefficient of vari-ation increases with the increase of noise which can becalled the variation coefficient-to-noise ratio and can beexpressed as ηijQij +e lower variation coefficient-to-noiseratio indicates that the corresponding identification con-stitutive parameters are less susceptible to noise +e vari-ation coefficient-to-noise ratios ηijQij of eight identificationresults are shown in Table 3 Significant differences on thevariation coefficient-to-noise ratio and the coefficient-to-noise ratio of Q66 is the smallest the next is Q11 and then isthe Q22 and the biggest is Q12 +e quantitative results showthat if Q11 Q22 and Q66 are stable Q12 is less robustlyidentified +is result shows that this configuration is un-suitable for the transverse stiffness

32 Comparison with Results of Piecewise Virtual Fields andPolynomial Virtual Fields +is section will implement thesame numerical experiment as Section 31 to compare theidentified constitutive parameters of the piecewise virtual

L

w

wa

wb

F

Part A

Part B

Figure 1 Two-dimensional bimaterial model of three-point bending established in FEM simulation

Table 1 Engineering elastic constants of orthotropic bimaterialExa (Mpa) Eya (Mpa) Eza (Mpa) PRxya (Mpa) PRyza (Mpa) PRxza (Mpa) Gxya (Mpa) Gyza (Mpa) Gxza (Mpa)180000 30000 8300 017 025 031 5700 12000 12000Exb (Mpa) Eyb (Mpa) Ezb (Mpa) PRxyb (Mpa) PRyzb (Mpa) PRxzb (Mpa) Gxyb (Mpa) Gyzb (Mpa) Gxzb (Mpa)175000 32000 8300 025 025 031 5700 12000 12000

Table 2 Reference constitutive parameters of orthotropic bimaterial

Q11a (Mpa) Q22a (Mpa) Q12a (Mpa) Q66a (Mpa) Q11b (Mpa) Q22b (Mpa) Q12b (Mpa) Q66b (Mpa)

18193710 3071968 590720 5700 17822916 3297651 894780 5700

10 Advances in Materials Science and Engineering

field and polynomial virtual field At this situation the bi-linear shape function is used to interpolate the four-nodeelement to represent the virtual field instead of the poly-nomial virtual field

+e piecewise function expansion of the virtual dis-placement field is similar to the interpolation of the real

displacement in the FEM Derived from the relationshipbetween strain and displacement

ε Sulowast (28)

where

ndash133477

ndash113034

ndash092591

ndash072149

ndash051706

ndash031263

ndash010821

009622

030065

050507

(a)

ndash956093

ndash849861

ndash743628

ndash637396

ndash531163

ndash42493

ndash318698

ndash212465

ndash106233

0

(b)

Figure 2 Displacement distributions of heterogeneous orthotropic bimaterials (a) u1 (b) u2 (unit mm)

Advances in Materials Science and Engineering 11

ndash0235856

ndash018911

ndash014237

ndash009562

ndash004888

ndash000021

041162

009136

013811

018485

(a)

ndash213206

ndash186779

ndash160351

ndash133924

ndash107496

ndash081069

ndash054641

ndash028214

ndash001786

0024641

(b)

ndash362375

ndash32025

ndash278125

ndash236

ndash193874

ndash151749

ndash109624

ndash067499

ndash025373

016752

(c)

Figure 3 Strain distributions of heterogeneous orthotropic bimaterials (a) ε1 (b) ε2 (c) ε3

12 Advances in Materials Science and Engineering

ε

ε1

ε2

ε6

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

ulowast ulowast1

ulowast2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

S

z

zx10

0z

zx2

z

zx2

z

zx1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(29)

For the quadrilateral element the virtual displacementcan be expressed by the following formula

ulowast ξ1 ξ2( 1113857≃ulowast ξ1 ξ2( 1113857 11139444

a1N(a) ξ1 ξ2( 1113857ulowast(a)

(30)

where ulowast is the vector containing the virtual displacement ofany point (ξ1 ξ2) in the element and 1113957ulowast(a) is the vectorcontaining the virtual displacement of the nodes in the el-ement +e expression of N(a) is defined as

N(a) ξ1 ξ2( 1113857 N(a) ξ1 ξ2( 1113857I (31)

where I is the identity matrix +e shape function expressionof N(a) can be defined as follows

N(1)

14

1 minus ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 + ξ2( 1113857

N(2)

14

1 minus ξ1( 1113857 1 + ξ2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

So the virtual displacement can be expressed by thefollowing formula

ulowast1 asymp N

(1)1113957ulowast(1)1 + N

(2)1113957ulowast(2)1 + N

(3)1113957ulowast(3)1 + N

(4)1113957ulowast(4)1

ulowast2 asymp N

(1)1113957ulowast(1)2 + N

(2)1113957ulowast(2)2 + N

(3)1113957ulowast(3)2 + N

(4)1113957ulowast(4)2

⎧⎨

⎩

(33)

+erefore the unknown coefficient vector Y (equation(17)) in the polynomial virtual fields can be denoted on eachnode

Y

1113957ulowast(1)1

1113957ulowast(1)2

⋮

1113957ulowast(n)1

1113957ulowast(n)2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

where n is the total number of nodesUse the piecewise virtual field instead of the polynomial

virtual field to perform the same experiment (Section 31)Table 4 shows the values of relative and weighted relativeerror of A and B It can be seen from Table 4 that theidentification results of piecewise virtual fields are similar tothat of polynomial virtual fields +e weighted relative erroris 097 which is slightly larger than that of polynomialvirtual fields +e relative error of Q66 of the two materials isthe smallest Although the relative errors of Q11 and Q22ofthe two materials are slightly larger than Q66 both of themare less than 2 Similar to the results of polynomial virtualfields the relative error of Q12a and Q12b is significantlyhigher than others because σ2 must be included in thevirtual fields to identify Q12 while the data density of σ2 islow in the three-point bending test

+e variation coefficient-to-noise ratios of eight iden-tification results are also shown in Table 4 Significant dif-ferences can be seen on the variation coefficient-to-noiseratios and the quantitative results show that if Q11 Q22 andQ66 are stable Q12 is less robustly identified Compared withthe polynomial virtual field the variation coefficient-to-noise ratio of the eight identification parameters calculatedby the optimized piecewise virtual field is larger

33 Influence of Noise on Identification Results +e noise isinevitable in the experiments To investigate the influence ofnoise on identification results zero-mean Gaussian white

Table 3 Identification results of constitutive parameters of orthotropic bimaterial using the optimized polynomial virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18159713 280642 minus019

083

Q22a 3071968 3035043 106390 minus120Q12a 590720 479127 2080530 minus1889Q66a 5700 570102 53999 002Q11b 17822916 17735394 234711 minus049Q22b 3297651 3259546 91601 minus116Q12b 894780 827581 1018827 minus751Q66b 5700 569879 59903 minus002

Advances in Materials Science and Engineering 13

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 6: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

1113946Sa

ε1εlowast (1)11113872 1113873dS 0 1113946

Sa

ε2εlowast (1)21113872 1113873dS 0 1113946

Sa

ε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 1 1113946

Sa

ε6εlowast (1)61113872 1113873dS 0

1113946Sb

ε1εlowast (1)11113872 1113873dS 0 1113946

Sb

ε2εlowast (1)21113872 1113873dS 0 1113946

Sb

ε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS 0 1113946

Sb

ε6εlowast (1)61113872 1113873dS 0

1113946Sa

ε1εlowast (2)11113872 1113873dS 0 1113946

Sa

ε2εlowast (2)21113872 1113873dS 0 1113946

Sa

ε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 1 1113946

Sa

ε6εlowast (2)61113872 1113873dS 0

1113946Sb

ε1εlowast (2)11113872 1113873dS 0 1113946

Sb

ε2εlowast (2)21113872 1113873dS 0 1113946

Sb

ε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS 0 1113946

Sb

ε6εlowast (2)61113872 1113873dS 0

1113946Sa

ε1εlowast (3)11113872 1113873dS 0 1113946

Sa

ε2εlowast (3)21113872 1113873dS 0 1113946

Sa

ε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 1 1113946

Sa

ε6εlowast (3)61113872 1113873dS 0

1113946Sb

ε1εlowast (3)11113872 1113873dS 0 1113946

Sb

ε2εlowast (3)21113872 1113873dS 0 1113946

Sb

ε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS 0 1113946

Sb

ε6εlowast (3)61113872 1113873dS 0

1113946Sa

ε1εlowast (4)11113872 1113873dS 0 1113946

Sa

ε2εlowast (4)21113872 1113873dS 0 1113946

Sa

ε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 1 1113946

Sa

ε6εlowast (4)61113872 1113873dS 0

1113946Sb

ε1εlowast (4)11113872 1113873dS 0 1113946

Sb

ε2εlowast (4)21113872 1113873dS 0 1113946

Sb

ε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS 0 1113946

Sb

ε6εlowast (4)61113872 1113873dS 0

1113946Sa

ε1εlowast (5)11113872 1113873dS 0 1113946

Sa

ε2εlowast (5)21113872 1113873dS 0 1113946

Sa

ε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS 1 1113946

Sa

ε6εlowast (5)61113872 1113873dS 0

1113946Sb

ε1εlowast (5)11113872 1113873dS 0 1113946

Sb

ε2εlowast (5)21113872 1113873dS 0 1113946

Sb

ε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS 0 1113946

Sb

ε6εlowast (5)61113872 1113873dS 0

1113946Sa

ε1εlowast (6)11113872 1113873dS 0 1113946

Sa

ε2εlowast (6)21113872 1113873dS 0 1113946

Sa

ε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS 1 1113946

Sa

ε6εlowast (6)61113872 1113873dS 0

1113946Sb

ε1εlowast (6)11113872 1113873dS 0 1113946

Sb

ε2εlowast (6)21113872 1113873dS 0 1113946

Sb

ε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS 0 1113946

Sb

ε6εlowast (6)61113872 1113873dS 0

1113946Sa

ε1εlowast (7)11113872 1113873dS 0 1113946

Sa

ε2εlowast (7)21113872 1113873dS 0 1113946

Sa

ε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS 1 1113946

Sa

ε6εlowast (7)61113872 1113873dS 0

1113946Sb

ε1εlowast (7)11113872 1113873dS 0 1113946

Sb

ε2εlowast (7)21113872 1113873dS 0 1113946

Sb

ε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS 0 1113946

Sb

ε6εlowast (7)61113872 1113873dS 0

1113946Sa

ε1εlowast (8)11113872 1113873dS 0 1113946

Sa

ε2εlowast (8)21113872 1113873dS 0 1113946

Sa

ε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS 1 1113946

Sa

ε6εlowast (8)61113872 1113873dS 0

1113946Sb

ε1εlowast (8)11113872 1113873dS 0 1113946

Sb

ε2εlowast (8)21113872 1113873dS 0 1113946

Sb

ε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS 0 1113946

Sb

ε6εlowast (8)61113872 1113873dS 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

So directly identify Q11a Q11b Q12a

Q12b Q22a Q22b Q66a and Q66b using the abovementionedfour special virtual displacement fields If the noise is

ignored approximate parameters expressed as Qapp areidentified +e components of which are defined as equation(17) (see Appendix B2 for the detailed equation derivations)

6 Advances in Materials Science and Engineering

Qapp11a 1113946

Lf

Tiulowast (1)i dl minus Q22a1113946

Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS

minus Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS

Qapp22a 1113946

Lf

Tiulowast (2)i dl minus Q11a1113946

Saε1εlowast (2)1 dS minus Q12a1113946

Saε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66a1113946

Saε6εlowast (2)6 dS

minus Q11b1113946Sbε1εlowast (2)1 dS minus Q22b1113946

Sbε2εlowast (2)2 dS minus Q12b1113946

Sbε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (2)6 dS

Qapp12a 1113946

Lf

Tiulowast (3)i dl minus Q11a1113946

Saε1εlowast (3)1 dS minus Q22a1113946

Saε2εlowast (3)2 dS minus Q66a1113946

Saε6εlowast (3)6 dS

minus Q11b1113946Sbε1εlowast (3)1 dS minus Q22b1113946

Sbε2εlowast (3)2 dS minus Q12b1113946

Sbε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (3)6 dS

Qapp66a 1113946

Lf

Tiulowast (4)i dl minus Q11a1113946

Saε1εlowast (4)1 dS minus Q22a1113946

Saε2εlowast (4)2 dS minus Q12a1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS

minus Q11b1113946Sbε1εlowast (4)1 dS minus Q22b1113946

Sbε2εlowast (4)2 dS minus Q12b1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (4)6 dS

Qapp11b 1113946

Lf

Tiulowast (5)i dl minus Q22b1113946

Sbε2εlowast (5)2 dS minus Q12b1113946

Sbε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (5)6 dS

minus Q11a1113946Saε1εlowast (5)1 dS minus Q22a1113946

Saε2εlowast (5)2 dS minus Q12a1113946

Saε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66a1113946

Saε6εlowast (5)6 dS

Qapp22b 1113946

Lf

Tiulowast (6)i dl minus Q11b1113946

Sbε1εlowast (6)1 dS minus Q12b1113946

Sbε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (6)6 dS

minus Q11a1113946Saε1εlowast (6)1 dS minus Q22a1113946

Saε2εlowast (6)2 dS minus Q12a1113946

Saε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66a1113946

Saε6εlowast (6)6 dS

Qapp12b 1113946

Lf

Tiulowast (7)i dl minus Q11b1113946

Sbε1εlowast (7)1 dS minus Q22b1113946

Sbε2εlowast (7)2 dS minus Q66b1113946

Sbε6εlowast (7)6 dS minus Q11a1113946

Saε1εlowast (7)1 dS

minus Q22a1113946Saε2εlowast (7)2 dS minus Q12a1113946

Saε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS minus Q66a1113946

Saε6εlowast (7)6 dS

Qapp66b 1113946

Lf

Tiulowast (8)i dl minus Q11b1113946

Sbε1εlowast (8)1 dS minus Q22b1113946

Sbε2εlowast (8)2 dS minus Q12b1113946

Sbε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS

minus Q11a1113946Saε1εlowast (8)1 dS minus Q22a1113946

Saε2εlowast (8)2 dS minus Q12a1113946

Saε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS minus Q66a1113946

Saε6εlowast (8)6 dS

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

To minimize the effect of noise taking Q11a as an ex-ample the variance of Q11a is as follows

V Q11a( 1113857 c2E Q

app11a1113946

Saεlowast (1)1 N1dS + Q

app22a1113946

Saεlowast (1)2 N2dS + Q

app12a 1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66a 1113946

Saεlowast (1)6 N6dS1113876 1113877

21113888

+ Qapp11b1113946

Sblowast (1)N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113889

+ c2E Q

app11b1113946

Sbεlowast (1)1 N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app22b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113888 1113889

VA Q11a( 1113857 + VB Q11a( 1113857

(18)

Advances in Materials Science and Engineering 7

As described above the virtual fields are expanded bypolynomial basis functions based on equation (11) +eunknown coefficients Aij and Bij are included in vector Y asfollows

Y

A00

⋮A30

⋮

A03

⋮

A33

B00

⋮

B30

⋮

B03

⋮

B33

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)

+e integrals of equation (15) are calculated usingvectors B11B22B12 and B66 through discrete sum ap-proximation so that

1113946Sa

ε1εlowast1( 1113857dS asymp Lwaε1aε

lowastia B11a middot Y

1113946Sa

ε2εlowast2( 1113857dS asymp Lwaε2aε

lowast2a B22a middot Y

1113946Sa

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwa ε1aε

lowast2a + ε2aε

lowast1a1113872 1113873 B12a middot Y

1113946Sa

ε6εlowast6( 1113857dS asymp Lwaε6aε

lowast6a B66a middot Y

1113946Sb

ε1εlowast1( 1113857dS asymp Lwaε1bε

lowast1b B11b middot Y

1113946Sb

ε2εlowast2( 1113857dS asymp Lwbε2bε

lowast2b B22b middot Y

1113946Sb

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwb ε1bε

lowast2b + ε2bε

lowast1b1113872 1113873 B12b middot Y

1113946Sb

ε6εlowast6( 1113857dS asymp Lwbε6bε

lowast6b B66b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

+e others that the definition from the virtual fields(vectorsH11H22H12 andH66) are related to the terms ofequation (18)

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

ε1εlowast2( 1113857dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast6( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

ε1εlowast2( 1113857dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast6( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

+e first KA virtual field conditions of three-pointbending are ulowast2 0 when x1 0 and x2 0 which implies

B00 0 (22)

+e second KA virtual field conditions of three-pointbending re ulowast1 0 when x1 L which means that

1113944

m

i0Aij 0 j 0 middot middot middot n (23)

+e optimized virtual fields are totally defined +efollowing equations can identify the constitutive parameters

8 Advances in Materials Science and Engineering

Q11a F

tulowast (a)2 x1 L x2 w( 1113857

Q22a F

tulowast (a)2 x1 L x2 w( 1113857

Q12a F

tulowast (a)2 x1 L x2 w( 1113857

Q66a F

tulowast (a)2 x1 L x2 w( 1113857

Q11b F

tulowast (a)2 x1 L x2 w( 1113857

Q22b F

tulowast (a)2 x1 L x2 w( 1113857

Q12b F

tulowast (a)2 x1 L x2 w( 1113857

Q66b F

tulowast (a)2 x1 L x2 w( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

3 Numerical Verification and Discussion

To evaluate the accuracy of the abovementioned optimizedpolynomial VFM numerical experiments with FEM simu-lations and DIC simulations are employed in this section

31 Numerical Verification of Optimized Polynomial VirtualFieldswithFEMSimulations In FEM simulation in order to

facilitate the simulation of bimaterials configuration a two-dimensional rectangular beam specimen was adopted andapplied a three-point bending load to get inhomogeneousin-plane deformation fields Based on the principle of thesymmetry of three-point bending only the left of thespecimen was modelled as shown in Figure 1

+e length width and thickness of the bimaterial beamspecimen were set as L 30mm w 20mm andt 23mm respectively and the width of the two materialregions was the same wa wb 10mm+e force was set asF 2544N +e four-node plane-stress element withthickness (Solid-Quad 4 nodes 182Plane strs wthk) wasemployed +e displacement constraint uy 0 is added inthe lower left corner of the model and the line constraintux 0 is added on the right side of the model

+e beam specimen was set as orthotropic materials +eengineering elastic constants in the FEM simulation areshown in Table 1 and the converted constitutive parametersin the elastic matrix are shown in Table 2

+e displacement fields and the strain fields of the three-point bending test of the orthotropic bimaterial obtainedthrough FEM simulations using the commercial softwareANSYS are shown in Figures 2 and 3

Table 3 shows the identification results of the hetero-geneous orthotropic bimaterials by using the optimizedpolynomial VFM as shown in Section 22 To study theidentification accuracy of the optimized polynomial VFMthe relative error of every constitutive parameter componentand the weighted relative error Wre were calculated Owingto there were 8 unknown constitutive parameters theweighted relative error employed here can be expressed asfollows

Wre

Qi de11a minus Q

rfe11a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22a minus Q

re22a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12a minus Q

re12a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66a minus Q

rfe66a

11138681113868111386811138681113868

11138681113868111386811138681113868

+ Qi de11b minus Q

rfe

11b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22b minus Q

rfe

22b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12b minus Q

rfe

12b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66b minus Q

rfe

66b

11138681113868111386811138681113868

11138681113868111386811138681113868

Qrfe11a + Q

rfe22a + Q

rfe12a + Q

rfe66a + Q

rfe

11b + Qrfe

22b + Qrfe

12b + Qrfe

66b

(25)

It can be seen from Table 3 that the weighted relativeerror is lower than 1 +e relative error of Q66 of the twomaterials is the smallest Although the relative errors of Q11andQ22 of the twomaterials are slightly larger than Q66 bothof them are less than 2 +e relative error of Q12a and Q12b

is significantly higher than others because σ2 must be in-cluded in the virtual fields to identify Q12 while the stressdata density of σ2 is low in the three-point bending test +estress data density can be expressed by the followingequation

Advances in Materials Science and Engineering 9

ρ σi( 1113857 1113944

bpointsj1 σi Mj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868

npoints i 1 2 6 (26)

According to equation (7) the standard deviations of Qij

are equal to η(i)c If c is the standard deviation of the strainnoise the coefficients of variations (CV(Qij)) of eachstiffness component are given by

CV Q11( 1113857 η11Q11

c

CV Q22( 1113857 η22Q22

c

CV Q12( 1113857 η12Q12

c

CV Q66( 1113857 η66Q66

c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

+e ratio of coefficient of variation to the standarddeviation of the strain noise can show the sensitivity to thenoise It indicates the rate at which the coefficient of vari-ation increases with the increase of noise which can becalled the variation coefficient-to-noise ratio and can beexpressed as ηijQij +e lower variation coefficient-to-noiseratio indicates that the corresponding identification con-stitutive parameters are less susceptible to noise +e vari-ation coefficient-to-noise ratios ηijQij of eight identificationresults are shown in Table 3 Significant differences on thevariation coefficient-to-noise ratio and the coefficient-to-noise ratio of Q66 is the smallest the next is Q11 and then isthe Q22 and the biggest is Q12 +e quantitative results showthat if Q11 Q22 and Q66 are stable Q12 is less robustlyidentified +is result shows that this configuration is un-suitable for the transverse stiffness

32 Comparison with Results of Piecewise Virtual Fields andPolynomial Virtual Fields +is section will implement thesame numerical experiment as Section 31 to compare theidentified constitutive parameters of the piecewise virtual

L

w

wa

wb

F

Part A

Part B

Figure 1 Two-dimensional bimaterial model of three-point bending established in FEM simulation

Table 1 Engineering elastic constants of orthotropic bimaterialExa (Mpa) Eya (Mpa) Eza (Mpa) PRxya (Mpa) PRyza (Mpa) PRxza (Mpa) Gxya (Mpa) Gyza (Mpa) Gxza (Mpa)180000 30000 8300 017 025 031 5700 12000 12000Exb (Mpa) Eyb (Mpa) Ezb (Mpa) PRxyb (Mpa) PRyzb (Mpa) PRxzb (Mpa) Gxyb (Mpa) Gyzb (Mpa) Gxzb (Mpa)175000 32000 8300 025 025 031 5700 12000 12000

Table 2 Reference constitutive parameters of orthotropic bimaterial

Q11a (Mpa) Q22a (Mpa) Q12a (Mpa) Q66a (Mpa) Q11b (Mpa) Q22b (Mpa) Q12b (Mpa) Q66b (Mpa)

18193710 3071968 590720 5700 17822916 3297651 894780 5700

10 Advances in Materials Science and Engineering

field and polynomial virtual field At this situation the bi-linear shape function is used to interpolate the four-nodeelement to represent the virtual field instead of the poly-nomial virtual field

+e piecewise function expansion of the virtual dis-placement field is similar to the interpolation of the real

displacement in the FEM Derived from the relationshipbetween strain and displacement

ε Sulowast (28)

where

ndash133477

ndash113034

ndash092591

ndash072149

ndash051706

ndash031263

ndash010821

009622

030065

050507

(a)

ndash956093

ndash849861

ndash743628

ndash637396

ndash531163

ndash42493

ndash318698

ndash212465

ndash106233

0

(b)

Figure 2 Displacement distributions of heterogeneous orthotropic bimaterials (a) u1 (b) u2 (unit mm)

Advances in Materials Science and Engineering 11

ndash0235856

ndash018911

ndash014237

ndash009562

ndash004888

ndash000021

041162

009136

013811

018485

(a)

ndash213206

ndash186779

ndash160351

ndash133924

ndash107496

ndash081069

ndash054641

ndash028214

ndash001786

0024641

(b)

ndash362375

ndash32025

ndash278125

ndash236

ndash193874

ndash151749

ndash109624

ndash067499

ndash025373

016752

(c)

Figure 3 Strain distributions of heterogeneous orthotropic bimaterials (a) ε1 (b) ε2 (c) ε3

12 Advances in Materials Science and Engineering

ε

ε1

ε2

ε6

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

ulowast ulowast1

ulowast2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

S

z

zx10

0z

zx2

z

zx2

z

zx1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(29)

For the quadrilateral element the virtual displacementcan be expressed by the following formula

ulowast ξ1 ξ2( 1113857≃ulowast ξ1 ξ2( 1113857 11139444

a1N(a) ξ1 ξ2( 1113857ulowast(a)

(30)

where ulowast is the vector containing the virtual displacement ofany point (ξ1 ξ2) in the element and 1113957ulowast(a) is the vectorcontaining the virtual displacement of the nodes in the el-ement +e expression of N(a) is defined as

N(a) ξ1 ξ2( 1113857 N(a) ξ1 ξ2( 1113857I (31)

where I is the identity matrix +e shape function expressionof N(a) can be defined as follows

N(1)

14

1 minus ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 + ξ2( 1113857

N(2)

14

1 minus ξ1( 1113857 1 + ξ2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

So the virtual displacement can be expressed by thefollowing formula

ulowast1 asymp N

(1)1113957ulowast(1)1 + N

(2)1113957ulowast(2)1 + N

(3)1113957ulowast(3)1 + N

(4)1113957ulowast(4)1

ulowast2 asymp N

(1)1113957ulowast(1)2 + N

(2)1113957ulowast(2)2 + N

(3)1113957ulowast(3)2 + N

(4)1113957ulowast(4)2

⎧⎨

⎩

(33)

+erefore the unknown coefficient vector Y (equation(17)) in the polynomial virtual fields can be denoted on eachnode

Y

1113957ulowast(1)1

1113957ulowast(1)2

⋮

1113957ulowast(n)1

1113957ulowast(n)2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

where n is the total number of nodesUse the piecewise virtual field instead of the polynomial

virtual field to perform the same experiment (Section 31)Table 4 shows the values of relative and weighted relativeerror of A and B It can be seen from Table 4 that theidentification results of piecewise virtual fields are similar tothat of polynomial virtual fields +e weighted relative erroris 097 which is slightly larger than that of polynomialvirtual fields +e relative error of Q66 of the two materials isthe smallest Although the relative errors of Q11 and Q22ofthe two materials are slightly larger than Q66 both of themare less than 2 Similar to the results of polynomial virtualfields the relative error of Q12a and Q12b is significantlyhigher than others because σ2 must be included in thevirtual fields to identify Q12 while the data density of σ2 islow in the three-point bending test

+e variation coefficient-to-noise ratios of eight iden-tification results are also shown in Table 4 Significant dif-ferences can be seen on the variation coefficient-to-noiseratios and the quantitative results show that if Q11 Q22 andQ66 are stable Q12 is less robustly identified Compared withthe polynomial virtual field the variation coefficient-to-noise ratio of the eight identification parameters calculatedby the optimized piecewise virtual field is larger

33 Influence of Noise on Identification Results +e noise isinevitable in the experiments To investigate the influence ofnoise on identification results zero-mean Gaussian white

Table 3 Identification results of constitutive parameters of orthotropic bimaterial using the optimized polynomial virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18159713 280642 minus019

083

Q22a 3071968 3035043 106390 minus120Q12a 590720 479127 2080530 minus1889Q66a 5700 570102 53999 002Q11b 17822916 17735394 234711 minus049Q22b 3297651 3259546 91601 minus116Q12b 894780 827581 1018827 minus751Q66b 5700 569879 59903 minus002

Advances in Materials Science and Engineering 13

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 7: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

Qapp11a 1113946

Lf

Tiulowast (1)i dl minus Q22a1113946

Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS

minus Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS

Qapp22a 1113946

Lf

Tiulowast (2)i dl minus Q11a1113946

Saε1εlowast (2)1 dS minus Q12a1113946

Saε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66a1113946

Saε6εlowast (2)6 dS

minus Q11b1113946Sbε1εlowast (2)1 dS minus Q22b1113946

Sbε2εlowast (2)2 dS minus Q12b1113946

Sbε1εlowast (2)2 + ε2ε

lowast (2)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (2)6 dS

Qapp12a 1113946

Lf

Tiulowast (3)i dl minus Q11a1113946

Saε1εlowast (3)1 dS minus Q22a1113946

Saε2εlowast (3)2 dS minus Q66a1113946

Saε6εlowast (3)6 dS

minus Q11b1113946Sbε1εlowast (3)1 dS minus Q22b1113946

Sbε2εlowast (3)2 dS minus Q12b1113946

Sbε1εlowast (3)2 + ε2ε

lowast (3)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (3)6 dS

Qapp66a 1113946

Lf

Tiulowast (4)i dl minus Q11a1113946

Saε1εlowast (4)1 dS minus Q22a1113946

Saε2εlowast (4)2 dS minus Q12a1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS

minus Q11b1113946Sbε1εlowast (4)1 dS minus Q22b1113946

Sbε2εlowast (4)2 dS minus Q12b1113946

Sbε1εlowast (4)2 + ε2ε

lowast (4)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (4)6 dS

Qapp11b 1113946

Lf

Tiulowast (5)i dl minus Q22b1113946

Sbε2εlowast (5)2 dS minus Q12b1113946

Sbε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (5)6 dS

minus Q11a1113946Saε1εlowast (5)1 dS minus Q22a1113946

Saε2εlowast (5)2 dS minus Q12a1113946

Saε1εlowast (5)2 + ε2ε

lowast (5)11113872 1113873dS minus Q66a1113946

Saε6εlowast (5)6 dS

Qapp22b 1113946

Lf

Tiulowast (6)i dl minus Q11b1113946

Sbε1εlowast (6)1 dS minus Q12b1113946

Sbε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (6)6 dS

minus Q11a1113946Saε1εlowast (6)1 dS minus Q22a1113946

Saε2εlowast (6)2 dS minus Q12a1113946

Saε1εlowast (6)2 + ε2ε

lowast (6)11113872 1113873dS minus Q66a1113946

Saε6εlowast (6)6 dS

Qapp12b 1113946

Lf

Tiulowast (7)i dl minus Q11b1113946

Sbε1εlowast (7)1 dS minus Q22b1113946

Sbε2εlowast (7)2 dS minus Q66b1113946

Sbε6εlowast (7)6 dS minus Q11a1113946

Saε1εlowast (7)1 dS

minus Q22a1113946Saε2εlowast (7)2 dS minus Q12a1113946

Saε1εlowast (7)2 + ε2ε

lowast (7)11113872 1113873dS minus Q66a1113946

Saε6εlowast (7)6 dS

Qapp66b 1113946

Lf

Tiulowast (8)i dl minus Q11b1113946

Sbε1εlowast (8)1 dS minus Q22b1113946

Sbε2εlowast (8)2 dS minus Q12b1113946

Sbε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS

minus Q11a1113946Saε1εlowast (8)1 dS minus Q22a1113946

Saε2εlowast (8)2 dS minus Q12a1113946

Saε1εlowast (8)2 + ε2ε

lowast (8)11113872 1113873dS minus Q66a1113946

Saε6εlowast (8)6 dS

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

To minimize the effect of noise taking Q11a as an ex-ample the variance of Q11a is as follows

V Q11a( 1113857 c2E Q

app11a1113946

Saεlowast (1)1 N1dS + Q

app22a1113946

Saεlowast (1)2 N2dS + Q

app12a 1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66a 1113946

Saεlowast (1)6 N6dS1113876 1113877

21113888

+ Qapp11b1113946

Sblowast (1)N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113889

+ c2E Q

app11b1113946

Sbεlowast (1)1 N1dS + Q

app22b1113946

Sbεlowast (1)2 N2dS + Q

app22b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp66b1113946

Sbεlowast (1)6 N6dS1113876 1113877

21113888 1113889

VA Q11a( 1113857 + VB Q11a( 1113857

(18)

Advances in Materials Science and Engineering 7

As described above the virtual fields are expanded bypolynomial basis functions based on equation (11) +eunknown coefficients Aij and Bij are included in vector Y asfollows

Y

A00

⋮A30

⋮

A03

⋮

A33

B00

⋮

B30

⋮

B03

⋮

B33

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)

+e integrals of equation (15) are calculated usingvectors B11B22B12 and B66 through discrete sum ap-proximation so that

1113946Sa

ε1εlowast1( 1113857dS asymp Lwaε1aε

lowastia B11a middot Y

1113946Sa

ε2εlowast2( 1113857dS asymp Lwaε2aε

lowast2a B22a middot Y

1113946Sa

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwa ε1aε

lowast2a + ε2aε

lowast1a1113872 1113873 B12a middot Y

1113946Sa

ε6εlowast6( 1113857dS asymp Lwaε6aε

lowast6a B66a middot Y

1113946Sb

ε1εlowast1( 1113857dS asymp Lwaε1bε

lowast1b B11b middot Y

1113946Sb

ε2εlowast2( 1113857dS asymp Lwbε2bε

lowast2b B22b middot Y

1113946Sb

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwb ε1bε

lowast2b + ε2bε

lowast1b1113872 1113873 B12b middot Y

1113946Sb

ε6εlowast6( 1113857dS asymp Lwbε6bε

lowast6b B66b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

+e others that the definition from the virtual fields(vectorsH11H22H12 andH66) are related to the terms ofequation (18)

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

ε1εlowast2( 1113857dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast6( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

ε1εlowast2( 1113857dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast6( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

+e first KA virtual field conditions of three-pointbending are ulowast2 0 when x1 0 and x2 0 which implies

B00 0 (22)

+e second KA virtual field conditions of three-pointbending re ulowast1 0 when x1 L which means that

1113944

m

i0Aij 0 j 0 middot middot middot n (23)

+e optimized virtual fields are totally defined +efollowing equations can identify the constitutive parameters

8 Advances in Materials Science and Engineering

Q11a F

tulowast (a)2 x1 L x2 w( 1113857

Q22a F

tulowast (a)2 x1 L x2 w( 1113857

Q12a F

tulowast (a)2 x1 L x2 w( 1113857

Q66a F

tulowast (a)2 x1 L x2 w( 1113857

Q11b F

tulowast (a)2 x1 L x2 w( 1113857

Q22b F

tulowast (a)2 x1 L x2 w( 1113857

Q12b F

tulowast (a)2 x1 L x2 w( 1113857

Q66b F

tulowast (a)2 x1 L x2 w( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

3 Numerical Verification and Discussion

To evaluate the accuracy of the abovementioned optimizedpolynomial VFM numerical experiments with FEM simu-lations and DIC simulations are employed in this section

31 Numerical Verification of Optimized Polynomial VirtualFieldswithFEMSimulations In FEM simulation in order to

facilitate the simulation of bimaterials configuration a two-dimensional rectangular beam specimen was adopted andapplied a three-point bending load to get inhomogeneousin-plane deformation fields Based on the principle of thesymmetry of three-point bending only the left of thespecimen was modelled as shown in Figure 1

+e length width and thickness of the bimaterial beamspecimen were set as L 30mm w 20mm andt 23mm respectively and the width of the two materialregions was the same wa wb 10mm+e force was set asF 2544N +e four-node plane-stress element withthickness (Solid-Quad 4 nodes 182Plane strs wthk) wasemployed +e displacement constraint uy 0 is added inthe lower left corner of the model and the line constraintux 0 is added on the right side of the model

+e beam specimen was set as orthotropic materials +eengineering elastic constants in the FEM simulation areshown in Table 1 and the converted constitutive parametersin the elastic matrix are shown in Table 2

+e displacement fields and the strain fields of the three-point bending test of the orthotropic bimaterial obtainedthrough FEM simulations using the commercial softwareANSYS are shown in Figures 2 and 3

Table 3 shows the identification results of the hetero-geneous orthotropic bimaterials by using the optimizedpolynomial VFM as shown in Section 22 To study theidentification accuracy of the optimized polynomial VFMthe relative error of every constitutive parameter componentand the weighted relative error Wre were calculated Owingto there were 8 unknown constitutive parameters theweighted relative error employed here can be expressed asfollows

Wre

Qi de11a minus Q

rfe11a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22a minus Q

re22a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12a minus Q

re12a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66a minus Q

rfe66a

11138681113868111386811138681113868

11138681113868111386811138681113868

+ Qi de11b minus Q

rfe

11b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22b minus Q

rfe

22b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12b minus Q

rfe

12b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66b minus Q

rfe

66b

11138681113868111386811138681113868

11138681113868111386811138681113868

Qrfe11a + Q

rfe22a + Q

rfe12a + Q

rfe66a + Q

rfe

11b + Qrfe

22b + Qrfe

12b + Qrfe

66b

(25)

It can be seen from Table 3 that the weighted relativeerror is lower than 1 +e relative error of Q66 of the twomaterials is the smallest Although the relative errors of Q11andQ22 of the twomaterials are slightly larger than Q66 bothof them are less than 2 +e relative error of Q12a and Q12b

is significantly higher than others because σ2 must be in-cluded in the virtual fields to identify Q12 while the stressdata density of σ2 is low in the three-point bending test +estress data density can be expressed by the followingequation

Advances in Materials Science and Engineering 9

ρ σi( 1113857 1113944

bpointsj1 σi Mj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868

npoints i 1 2 6 (26)

According to equation (7) the standard deviations of Qij

are equal to η(i)c If c is the standard deviation of the strainnoise the coefficients of variations (CV(Qij)) of eachstiffness component are given by

CV Q11( 1113857 η11Q11

c

CV Q22( 1113857 η22Q22

c

CV Q12( 1113857 η12Q12

c

CV Q66( 1113857 η66Q66

c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

+e ratio of coefficient of variation to the standarddeviation of the strain noise can show the sensitivity to thenoise It indicates the rate at which the coefficient of vari-ation increases with the increase of noise which can becalled the variation coefficient-to-noise ratio and can beexpressed as ηijQij +e lower variation coefficient-to-noiseratio indicates that the corresponding identification con-stitutive parameters are less susceptible to noise +e vari-ation coefficient-to-noise ratios ηijQij of eight identificationresults are shown in Table 3 Significant differences on thevariation coefficient-to-noise ratio and the coefficient-to-noise ratio of Q66 is the smallest the next is Q11 and then isthe Q22 and the biggest is Q12 +e quantitative results showthat if Q11 Q22 and Q66 are stable Q12 is less robustlyidentified +is result shows that this configuration is un-suitable for the transverse stiffness

32 Comparison with Results of Piecewise Virtual Fields andPolynomial Virtual Fields +is section will implement thesame numerical experiment as Section 31 to compare theidentified constitutive parameters of the piecewise virtual

L

w

wa

wb

F

Part A

Part B

Figure 1 Two-dimensional bimaterial model of three-point bending established in FEM simulation

Table 1 Engineering elastic constants of orthotropic bimaterialExa (Mpa) Eya (Mpa) Eza (Mpa) PRxya (Mpa) PRyza (Mpa) PRxza (Mpa) Gxya (Mpa) Gyza (Mpa) Gxza (Mpa)180000 30000 8300 017 025 031 5700 12000 12000Exb (Mpa) Eyb (Mpa) Ezb (Mpa) PRxyb (Mpa) PRyzb (Mpa) PRxzb (Mpa) Gxyb (Mpa) Gyzb (Mpa) Gxzb (Mpa)175000 32000 8300 025 025 031 5700 12000 12000

Table 2 Reference constitutive parameters of orthotropic bimaterial

Q11a (Mpa) Q22a (Mpa) Q12a (Mpa) Q66a (Mpa) Q11b (Mpa) Q22b (Mpa) Q12b (Mpa) Q66b (Mpa)

18193710 3071968 590720 5700 17822916 3297651 894780 5700

10 Advances in Materials Science and Engineering

field and polynomial virtual field At this situation the bi-linear shape function is used to interpolate the four-nodeelement to represent the virtual field instead of the poly-nomial virtual field

+e piecewise function expansion of the virtual dis-placement field is similar to the interpolation of the real

displacement in the FEM Derived from the relationshipbetween strain and displacement

ε Sulowast (28)

where

ndash133477

ndash113034

ndash092591

ndash072149

ndash051706

ndash031263

ndash010821

009622

030065

050507

(a)

ndash956093

ndash849861

ndash743628

ndash637396

ndash531163

ndash42493

ndash318698

ndash212465

ndash106233

0

(b)

Figure 2 Displacement distributions of heterogeneous orthotropic bimaterials (a) u1 (b) u2 (unit mm)

Advances in Materials Science and Engineering 11

ndash0235856

ndash018911

ndash014237

ndash009562

ndash004888

ndash000021

041162

009136

013811

018485

(a)

ndash213206

ndash186779

ndash160351

ndash133924

ndash107496

ndash081069

ndash054641

ndash028214

ndash001786

0024641

(b)

ndash362375

ndash32025

ndash278125

ndash236

ndash193874

ndash151749

ndash109624

ndash067499

ndash025373

016752

(c)

Figure 3 Strain distributions of heterogeneous orthotropic bimaterials (a) ε1 (b) ε2 (c) ε3

12 Advances in Materials Science and Engineering

ε

ε1

ε2

ε6

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

ulowast ulowast1

ulowast2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

S

z

zx10

0z

zx2

z

zx2

z

zx1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(29)

For the quadrilateral element the virtual displacementcan be expressed by the following formula

ulowast ξ1 ξ2( 1113857≃ulowast ξ1 ξ2( 1113857 11139444

a1N(a) ξ1 ξ2( 1113857ulowast(a)

(30)

where ulowast is the vector containing the virtual displacement ofany point (ξ1 ξ2) in the element and 1113957ulowast(a) is the vectorcontaining the virtual displacement of the nodes in the el-ement +e expression of N(a) is defined as

N(a) ξ1 ξ2( 1113857 N(a) ξ1 ξ2( 1113857I (31)

where I is the identity matrix +e shape function expressionof N(a) can be defined as follows

N(1)

14

1 minus ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 + ξ2( 1113857

N(2)

14

1 minus ξ1( 1113857 1 + ξ2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

So the virtual displacement can be expressed by thefollowing formula

ulowast1 asymp N

(1)1113957ulowast(1)1 + N

(2)1113957ulowast(2)1 + N

(3)1113957ulowast(3)1 + N

(4)1113957ulowast(4)1

ulowast2 asymp N

(1)1113957ulowast(1)2 + N

(2)1113957ulowast(2)2 + N

(3)1113957ulowast(3)2 + N

(4)1113957ulowast(4)2

⎧⎨

⎩

(33)

+erefore the unknown coefficient vector Y (equation(17)) in the polynomial virtual fields can be denoted on eachnode

Y

1113957ulowast(1)1

1113957ulowast(1)2

⋮

1113957ulowast(n)1

1113957ulowast(n)2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

where n is the total number of nodesUse the piecewise virtual field instead of the polynomial

virtual field to perform the same experiment (Section 31)Table 4 shows the values of relative and weighted relativeerror of A and B It can be seen from Table 4 that theidentification results of piecewise virtual fields are similar tothat of polynomial virtual fields +e weighted relative erroris 097 which is slightly larger than that of polynomialvirtual fields +e relative error of Q66 of the two materials isthe smallest Although the relative errors of Q11 and Q22ofthe two materials are slightly larger than Q66 both of themare less than 2 Similar to the results of polynomial virtualfields the relative error of Q12a and Q12b is significantlyhigher than others because σ2 must be included in thevirtual fields to identify Q12 while the data density of σ2 islow in the three-point bending test

+e variation coefficient-to-noise ratios of eight iden-tification results are also shown in Table 4 Significant dif-ferences can be seen on the variation coefficient-to-noiseratios and the quantitative results show that if Q11 Q22 andQ66 are stable Q12 is less robustly identified Compared withthe polynomial virtual field the variation coefficient-to-noise ratio of the eight identification parameters calculatedby the optimized piecewise virtual field is larger

33 Influence of Noise on Identification Results +e noise isinevitable in the experiments To investigate the influence ofnoise on identification results zero-mean Gaussian white

Table 3 Identification results of constitutive parameters of orthotropic bimaterial using the optimized polynomial virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18159713 280642 minus019

083

Q22a 3071968 3035043 106390 minus120Q12a 590720 479127 2080530 minus1889Q66a 5700 570102 53999 002Q11b 17822916 17735394 234711 minus049Q22b 3297651 3259546 91601 minus116Q12b 894780 827581 1018827 minus751Q66b 5700 569879 59903 minus002

Advances in Materials Science and Engineering 13

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 8: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

As described above the virtual fields are expanded bypolynomial basis functions based on equation (11) +eunknown coefficients Aij and Bij are included in vector Y asfollows

Y

A00

⋮A30

⋮

A03

⋮

A33

B00

⋮

B30

⋮

B03

⋮

B33

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)

+e integrals of equation (15) are calculated usingvectors B11B22B12 and B66 through discrete sum ap-proximation so that

1113946Sa

ε1εlowast1( 1113857dS asymp Lwaε1aε

lowastia B11a middot Y

1113946Sa

ε2εlowast2( 1113857dS asymp Lwaε2aε

lowast2a B22a middot Y

1113946Sa

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwa ε1aε

lowast2a + ε2aε

lowast1a1113872 1113873 B12a middot Y

1113946Sa

ε6εlowast6( 1113857dS asymp Lwaε6aε

lowast6a B66a middot Y

1113946Sb

ε1εlowast1( 1113857dS asymp Lwaε1bε

lowast1b B11b middot Y

1113946Sb

ε2εlowast2( 1113857dS asymp Lwbε2bε

lowast2b B22b middot Y

1113946Sb

ε1εlowast2 + ε2ε

lowast1( 1113857dS asymp Lwb ε1bε

lowast2b + ε2bε

lowast1b1113872 1113873 B12b middot Y

1113946Sb

ε6εlowast6( 1113857dS asymp Lwbε6bε

lowast6b B66b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

+e others that the definition from the virtual fields(vectorsH11H22H12 andH66) are related to the terms ofequation (18)

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast1( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

ε1εlowast2( 1113857dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sa

εlowast6( 11138572dS asymp

Lwa

npointsa1113888 1113889

2

Y middot H11a middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast1( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

ε1εlowast2( 1113857dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

1113946Sb

εlowast6( 11138572dS asymp

Lwb

npointsa1113888 1113889

2

Y middot H11b middot Y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

+e first KA virtual field conditions of three-pointbending are ulowast2 0 when x1 0 and x2 0 which implies

B00 0 (22)

+e second KA virtual field conditions of three-pointbending re ulowast1 0 when x1 L which means that

1113944

m

i0Aij 0 j 0 middot middot middot n (23)

+e optimized virtual fields are totally defined +efollowing equations can identify the constitutive parameters

8 Advances in Materials Science and Engineering

Q11a F

tulowast (a)2 x1 L x2 w( 1113857

Q22a F

tulowast (a)2 x1 L x2 w( 1113857

Q12a F

tulowast (a)2 x1 L x2 w( 1113857

Q66a F

tulowast (a)2 x1 L x2 w( 1113857

Q11b F

tulowast (a)2 x1 L x2 w( 1113857

Q22b F

tulowast (a)2 x1 L x2 w( 1113857

Q12b F

tulowast (a)2 x1 L x2 w( 1113857

Q66b F

tulowast (a)2 x1 L x2 w( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

3 Numerical Verification and Discussion

To evaluate the accuracy of the abovementioned optimizedpolynomial VFM numerical experiments with FEM simu-lations and DIC simulations are employed in this section

31 Numerical Verification of Optimized Polynomial VirtualFieldswithFEMSimulations In FEM simulation in order to

facilitate the simulation of bimaterials configuration a two-dimensional rectangular beam specimen was adopted andapplied a three-point bending load to get inhomogeneousin-plane deformation fields Based on the principle of thesymmetry of three-point bending only the left of thespecimen was modelled as shown in Figure 1

+e length width and thickness of the bimaterial beamspecimen were set as L 30mm w 20mm andt 23mm respectively and the width of the two materialregions was the same wa wb 10mm+e force was set asF 2544N +e four-node plane-stress element withthickness (Solid-Quad 4 nodes 182Plane strs wthk) wasemployed +e displacement constraint uy 0 is added inthe lower left corner of the model and the line constraintux 0 is added on the right side of the model

+e beam specimen was set as orthotropic materials +eengineering elastic constants in the FEM simulation areshown in Table 1 and the converted constitutive parametersin the elastic matrix are shown in Table 2

+e displacement fields and the strain fields of the three-point bending test of the orthotropic bimaterial obtainedthrough FEM simulations using the commercial softwareANSYS are shown in Figures 2 and 3

Table 3 shows the identification results of the hetero-geneous orthotropic bimaterials by using the optimizedpolynomial VFM as shown in Section 22 To study theidentification accuracy of the optimized polynomial VFMthe relative error of every constitutive parameter componentand the weighted relative error Wre were calculated Owingto there were 8 unknown constitutive parameters theweighted relative error employed here can be expressed asfollows

Wre

Qi de11a minus Q

rfe11a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22a minus Q

re22a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12a minus Q

re12a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66a minus Q

rfe66a

11138681113868111386811138681113868

11138681113868111386811138681113868

+ Qi de11b minus Q

rfe

11b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22b minus Q

rfe

22b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12b minus Q

rfe

12b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66b minus Q

rfe

66b

11138681113868111386811138681113868

11138681113868111386811138681113868

Qrfe11a + Q

rfe22a + Q

rfe12a + Q

rfe66a + Q

rfe

11b + Qrfe

22b + Qrfe

12b + Qrfe

66b

(25)

It can be seen from Table 3 that the weighted relativeerror is lower than 1 +e relative error of Q66 of the twomaterials is the smallest Although the relative errors of Q11andQ22 of the twomaterials are slightly larger than Q66 bothof them are less than 2 +e relative error of Q12a and Q12b

is significantly higher than others because σ2 must be in-cluded in the virtual fields to identify Q12 while the stressdata density of σ2 is low in the three-point bending test +estress data density can be expressed by the followingequation

Advances in Materials Science and Engineering 9

ρ σi( 1113857 1113944

bpointsj1 σi Mj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868

npoints i 1 2 6 (26)

According to equation (7) the standard deviations of Qij

are equal to η(i)c If c is the standard deviation of the strainnoise the coefficients of variations (CV(Qij)) of eachstiffness component are given by

CV Q11( 1113857 η11Q11

c

CV Q22( 1113857 η22Q22

c

CV Q12( 1113857 η12Q12

c

CV Q66( 1113857 η66Q66

c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

+e ratio of coefficient of variation to the standarddeviation of the strain noise can show the sensitivity to thenoise It indicates the rate at which the coefficient of vari-ation increases with the increase of noise which can becalled the variation coefficient-to-noise ratio and can beexpressed as ηijQij +e lower variation coefficient-to-noiseratio indicates that the corresponding identification con-stitutive parameters are less susceptible to noise +e vari-ation coefficient-to-noise ratios ηijQij of eight identificationresults are shown in Table 3 Significant differences on thevariation coefficient-to-noise ratio and the coefficient-to-noise ratio of Q66 is the smallest the next is Q11 and then isthe Q22 and the biggest is Q12 +e quantitative results showthat if Q11 Q22 and Q66 are stable Q12 is less robustlyidentified +is result shows that this configuration is un-suitable for the transverse stiffness

32 Comparison with Results of Piecewise Virtual Fields andPolynomial Virtual Fields +is section will implement thesame numerical experiment as Section 31 to compare theidentified constitutive parameters of the piecewise virtual

L

w

wa

wb

F

Part A

Part B

Figure 1 Two-dimensional bimaterial model of three-point bending established in FEM simulation

Table 1 Engineering elastic constants of orthotropic bimaterialExa (Mpa) Eya (Mpa) Eza (Mpa) PRxya (Mpa) PRyza (Mpa) PRxza (Mpa) Gxya (Mpa) Gyza (Mpa) Gxza (Mpa)180000 30000 8300 017 025 031 5700 12000 12000Exb (Mpa) Eyb (Mpa) Ezb (Mpa) PRxyb (Mpa) PRyzb (Mpa) PRxzb (Mpa) Gxyb (Mpa) Gyzb (Mpa) Gxzb (Mpa)175000 32000 8300 025 025 031 5700 12000 12000

Table 2 Reference constitutive parameters of orthotropic bimaterial

Q11a (Mpa) Q22a (Mpa) Q12a (Mpa) Q66a (Mpa) Q11b (Mpa) Q22b (Mpa) Q12b (Mpa) Q66b (Mpa)

18193710 3071968 590720 5700 17822916 3297651 894780 5700

10 Advances in Materials Science and Engineering

field and polynomial virtual field At this situation the bi-linear shape function is used to interpolate the four-nodeelement to represent the virtual field instead of the poly-nomial virtual field

+e piecewise function expansion of the virtual dis-placement field is similar to the interpolation of the real

displacement in the FEM Derived from the relationshipbetween strain and displacement

ε Sulowast (28)

where

ndash133477

ndash113034

ndash092591

ndash072149

ndash051706

ndash031263

ndash010821

009622

030065

050507

(a)

ndash956093

ndash849861

ndash743628

ndash637396

ndash531163

ndash42493

ndash318698

ndash212465

ndash106233

0

(b)

Figure 2 Displacement distributions of heterogeneous orthotropic bimaterials (a) u1 (b) u2 (unit mm)

Advances in Materials Science and Engineering 11

ndash0235856

ndash018911

ndash014237

ndash009562

ndash004888

ndash000021

041162

009136

013811

018485

(a)

ndash213206

ndash186779

ndash160351

ndash133924

ndash107496

ndash081069

ndash054641

ndash028214

ndash001786

0024641

(b)

ndash362375

ndash32025

ndash278125

ndash236

ndash193874

ndash151749

ndash109624

ndash067499

ndash025373

016752

(c)

Figure 3 Strain distributions of heterogeneous orthotropic bimaterials (a) ε1 (b) ε2 (c) ε3

12 Advances in Materials Science and Engineering

ε

ε1

ε2

ε6

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

ulowast ulowast1

ulowast2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

S

z

zx10

0z

zx2

z

zx2

z

zx1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(29)

For the quadrilateral element the virtual displacementcan be expressed by the following formula

ulowast ξ1 ξ2( 1113857≃ulowast ξ1 ξ2( 1113857 11139444

a1N(a) ξ1 ξ2( 1113857ulowast(a)

(30)

where ulowast is the vector containing the virtual displacement ofany point (ξ1 ξ2) in the element and 1113957ulowast(a) is the vectorcontaining the virtual displacement of the nodes in the el-ement +e expression of N(a) is defined as

N(a) ξ1 ξ2( 1113857 N(a) ξ1 ξ2( 1113857I (31)

where I is the identity matrix +e shape function expressionof N(a) can be defined as follows

N(1)

14

1 minus ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 + ξ2( 1113857

N(2)

14

1 minus ξ1( 1113857 1 + ξ2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

So the virtual displacement can be expressed by thefollowing formula

ulowast1 asymp N

(1)1113957ulowast(1)1 + N

(2)1113957ulowast(2)1 + N

(3)1113957ulowast(3)1 + N

(4)1113957ulowast(4)1

ulowast2 asymp N

(1)1113957ulowast(1)2 + N

(2)1113957ulowast(2)2 + N

(3)1113957ulowast(3)2 + N

(4)1113957ulowast(4)2

⎧⎨

⎩

(33)

+erefore the unknown coefficient vector Y (equation(17)) in the polynomial virtual fields can be denoted on eachnode

Y

1113957ulowast(1)1

1113957ulowast(1)2

⋮

1113957ulowast(n)1

1113957ulowast(n)2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

where n is the total number of nodesUse the piecewise virtual field instead of the polynomial

virtual field to perform the same experiment (Section 31)Table 4 shows the values of relative and weighted relativeerror of A and B It can be seen from Table 4 that theidentification results of piecewise virtual fields are similar tothat of polynomial virtual fields +e weighted relative erroris 097 which is slightly larger than that of polynomialvirtual fields +e relative error of Q66 of the two materials isthe smallest Although the relative errors of Q11 and Q22ofthe two materials are slightly larger than Q66 both of themare less than 2 Similar to the results of polynomial virtualfields the relative error of Q12a and Q12b is significantlyhigher than others because σ2 must be included in thevirtual fields to identify Q12 while the data density of σ2 islow in the three-point bending test

+e variation coefficient-to-noise ratios of eight iden-tification results are also shown in Table 4 Significant dif-ferences can be seen on the variation coefficient-to-noiseratios and the quantitative results show that if Q11 Q22 andQ66 are stable Q12 is less robustly identified Compared withthe polynomial virtual field the variation coefficient-to-noise ratio of the eight identification parameters calculatedby the optimized piecewise virtual field is larger

33 Influence of Noise on Identification Results +e noise isinevitable in the experiments To investigate the influence ofnoise on identification results zero-mean Gaussian white

Table 3 Identification results of constitutive parameters of orthotropic bimaterial using the optimized polynomial virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18159713 280642 minus019

083

Q22a 3071968 3035043 106390 minus120Q12a 590720 479127 2080530 minus1889Q66a 5700 570102 53999 002Q11b 17822916 17735394 234711 minus049Q22b 3297651 3259546 91601 minus116Q12b 894780 827581 1018827 minus751Q66b 5700 569879 59903 minus002

Advances in Materials Science and Engineering 13

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 9: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

Q11a F

tulowast (a)2 x1 L x2 w( 1113857

Q22a F

tulowast (a)2 x1 L x2 w( 1113857

Q12a F

tulowast (a)2 x1 L x2 w( 1113857

Q66a F

tulowast (a)2 x1 L x2 w( 1113857

Q11b F

tulowast (a)2 x1 L x2 w( 1113857

Q22b F

tulowast (a)2 x1 L x2 w( 1113857

Q12b F

tulowast (a)2 x1 L x2 w( 1113857

Q66b F

tulowast (a)2 x1 L x2 w( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

3 Numerical Verification and Discussion

To evaluate the accuracy of the abovementioned optimizedpolynomial VFM numerical experiments with FEM simu-lations and DIC simulations are employed in this section

31 Numerical Verification of Optimized Polynomial VirtualFieldswithFEMSimulations In FEM simulation in order to

facilitate the simulation of bimaterials configuration a two-dimensional rectangular beam specimen was adopted andapplied a three-point bending load to get inhomogeneousin-plane deformation fields Based on the principle of thesymmetry of three-point bending only the left of thespecimen was modelled as shown in Figure 1

+e length width and thickness of the bimaterial beamspecimen were set as L 30mm w 20mm andt 23mm respectively and the width of the two materialregions was the same wa wb 10mm+e force was set asF 2544N +e four-node plane-stress element withthickness (Solid-Quad 4 nodes 182Plane strs wthk) wasemployed +e displacement constraint uy 0 is added inthe lower left corner of the model and the line constraintux 0 is added on the right side of the model

+e beam specimen was set as orthotropic materials +eengineering elastic constants in the FEM simulation areshown in Table 1 and the converted constitutive parametersin the elastic matrix are shown in Table 2

+e displacement fields and the strain fields of the three-point bending test of the orthotropic bimaterial obtainedthrough FEM simulations using the commercial softwareANSYS are shown in Figures 2 and 3

Table 3 shows the identification results of the hetero-geneous orthotropic bimaterials by using the optimizedpolynomial VFM as shown in Section 22 To study theidentification accuracy of the optimized polynomial VFMthe relative error of every constitutive parameter componentand the weighted relative error Wre were calculated Owingto there were 8 unknown constitutive parameters theweighted relative error employed here can be expressed asfollows

Wre

Qi de11a minus Q

rfe11a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22a minus Q

re22a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12a minus Q

re12a

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66a minus Q

rfe66a

11138681113868111386811138681113868

11138681113868111386811138681113868

+ Qi de11b minus Q

rfe

11b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de22b minus Q

rfe

22b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de12b minus Q

rfe

12b

11138681113868111386811138681113868

11138681113868111386811138681113868 + Qi de66b minus Q

rfe

66b

11138681113868111386811138681113868

11138681113868111386811138681113868

Qrfe11a + Q

rfe22a + Q

rfe12a + Q

rfe66a + Q

rfe

11b + Qrfe

22b + Qrfe

12b + Qrfe

66b

(25)

It can be seen from Table 3 that the weighted relativeerror is lower than 1 +e relative error of Q66 of the twomaterials is the smallest Although the relative errors of Q11andQ22 of the twomaterials are slightly larger than Q66 bothof them are less than 2 +e relative error of Q12a and Q12b

is significantly higher than others because σ2 must be in-cluded in the virtual fields to identify Q12 while the stressdata density of σ2 is low in the three-point bending test +estress data density can be expressed by the followingequation

Advances in Materials Science and Engineering 9

ρ σi( 1113857 1113944

bpointsj1 σi Mj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868

npoints i 1 2 6 (26)

According to equation (7) the standard deviations of Qij

are equal to η(i)c If c is the standard deviation of the strainnoise the coefficients of variations (CV(Qij)) of eachstiffness component are given by

CV Q11( 1113857 η11Q11

c

CV Q22( 1113857 η22Q22

c

CV Q12( 1113857 η12Q12

c

CV Q66( 1113857 η66Q66

c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

+e ratio of coefficient of variation to the standarddeviation of the strain noise can show the sensitivity to thenoise It indicates the rate at which the coefficient of vari-ation increases with the increase of noise which can becalled the variation coefficient-to-noise ratio and can beexpressed as ηijQij +e lower variation coefficient-to-noiseratio indicates that the corresponding identification con-stitutive parameters are less susceptible to noise +e vari-ation coefficient-to-noise ratios ηijQij of eight identificationresults are shown in Table 3 Significant differences on thevariation coefficient-to-noise ratio and the coefficient-to-noise ratio of Q66 is the smallest the next is Q11 and then isthe Q22 and the biggest is Q12 +e quantitative results showthat if Q11 Q22 and Q66 are stable Q12 is less robustlyidentified +is result shows that this configuration is un-suitable for the transverse stiffness

32 Comparison with Results of Piecewise Virtual Fields andPolynomial Virtual Fields +is section will implement thesame numerical experiment as Section 31 to compare theidentified constitutive parameters of the piecewise virtual

L

w

wa

wb

F

Part A

Part B

Figure 1 Two-dimensional bimaterial model of three-point bending established in FEM simulation

Table 1 Engineering elastic constants of orthotropic bimaterialExa (Mpa) Eya (Mpa) Eza (Mpa) PRxya (Mpa) PRyza (Mpa) PRxza (Mpa) Gxya (Mpa) Gyza (Mpa) Gxza (Mpa)180000 30000 8300 017 025 031 5700 12000 12000Exb (Mpa) Eyb (Mpa) Ezb (Mpa) PRxyb (Mpa) PRyzb (Mpa) PRxzb (Mpa) Gxyb (Mpa) Gyzb (Mpa) Gxzb (Mpa)175000 32000 8300 025 025 031 5700 12000 12000

Table 2 Reference constitutive parameters of orthotropic bimaterial

Q11a (Mpa) Q22a (Mpa) Q12a (Mpa) Q66a (Mpa) Q11b (Mpa) Q22b (Mpa) Q12b (Mpa) Q66b (Mpa)

18193710 3071968 590720 5700 17822916 3297651 894780 5700

10 Advances in Materials Science and Engineering

field and polynomial virtual field At this situation the bi-linear shape function is used to interpolate the four-nodeelement to represent the virtual field instead of the poly-nomial virtual field

+e piecewise function expansion of the virtual dis-placement field is similar to the interpolation of the real

displacement in the FEM Derived from the relationshipbetween strain and displacement

ε Sulowast (28)

where

ndash133477

ndash113034

ndash092591

ndash072149

ndash051706

ndash031263

ndash010821

009622

030065

050507

(a)

ndash956093

ndash849861

ndash743628

ndash637396

ndash531163

ndash42493

ndash318698

ndash212465

ndash106233

0

(b)

Figure 2 Displacement distributions of heterogeneous orthotropic bimaterials (a) u1 (b) u2 (unit mm)

Advances in Materials Science and Engineering 11

ndash0235856

ndash018911

ndash014237

ndash009562

ndash004888

ndash000021

041162

009136

013811

018485

(a)

ndash213206

ndash186779

ndash160351

ndash133924

ndash107496

ndash081069

ndash054641

ndash028214

ndash001786

0024641

(b)

ndash362375

ndash32025

ndash278125

ndash236

ndash193874

ndash151749

ndash109624

ndash067499

ndash025373

016752

(c)

Figure 3 Strain distributions of heterogeneous orthotropic bimaterials (a) ε1 (b) ε2 (c) ε3

12 Advances in Materials Science and Engineering

ε

ε1

ε2

ε6

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

ulowast ulowast1

ulowast2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

S

z

zx10

0z

zx2

z

zx2

z

zx1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(29)

For the quadrilateral element the virtual displacementcan be expressed by the following formula

ulowast ξ1 ξ2( 1113857≃ulowast ξ1 ξ2( 1113857 11139444

a1N(a) ξ1 ξ2( 1113857ulowast(a)

(30)

where ulowast is the vector containing the virtual displacement ofany point (ξ1 ξ2) in the element and 1113957ulowast(a) is the vectorcontaining the virtual displacement of the nodes in the el-ement +e expression of N(a) is defined as

N(a) ξ1 ξ2( 1113857 N(a) ξ1 ξ2( 1113857I (31)

where I is the identity matrix +e shape function expressionof N(a) can be defined as follows

N(1)

14

1 minus ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 + ξ2( 1113857

N(2)

14

1 minus ξ1( 1113857 1 + ξ2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

So the virtual displacement can be expressed by thefollowing formula

ulowast1 asymp N

(1)1113957ulowast(1)1 + N

(2)1113957ulowast(2)1 + N

(3)1113957ulowast(3)1 + N

(4)1113957ulowast(4)1

ulowast2 asymp N

(1)1113957ulowast(1)2 + N

(2)1113957ulowast(2)2 + N

(3)1113957ulowast(3)2 + N

(4)1113957ulowast(4)2

⎧⎨

⎩

(33)

+erefore the unknown coefficient vector Y (equation(17)) in the polynomial virtual fields can be denoted on eachnode

Y

1113957ulowast(1)1

1113957ulowast(1)2

⋮

1113957ulowast(n)1

1113957ulowast(n)2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

where n is the total number of nodesUse the piecewise virtual field instead of the polynomial

virtual field to perform the same experiment (Section 31)Table 4 shows the values of relative and weighted relativeerror of A and B It can be seen from Table 4 that theidentification results of piecewise virtual fields are similar tothat of polynomial virtual fields +e weighted relative erroris 097 which is slightly larger than that of polynomialvirtual fields +e relative error of Q66 of the two materials isthe smallest Although the relative errors of Q11 and Q22ofthe two materials are slightly larger than Q66 both of themare less than 2 Similar to the results of polynomial virtualfields the relative error of Q12a and Q12b is significantlyhigher than others because σ2 must be included in thevirtual fields to identify Q12 while the data density of σ2 islow in the three-point bending test

+e variation coefficient-to-noise ratios of eight iden-tification results are also shown in Table 4 Significant dif-ferences can be seen on the variation coefficient-to-noiseratios and the quantitative results show that if Q11 Q22 andQ66 are stable Q12 is less robustly identified Compared withthe polynomial virtual field the variation coefficient-to-noise ratio of the eight identification parameters calculatedby the optimized piecewise virtual field is larger

33 Influence of Noise on Identification Results +e noise isinevitable in the experiments To investigate the influence ofnoise on identification results zero-mean Gaussian white

Table 3 Identification results of constitutive parameters of orthotropic bimaterial using the optimized polynomial virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18159713 280642 minus019

083

Q22a 3071968 3035043 106390 minus120Q12a 590720 479127 2080530 minus1889Q66a 5700 570102 53999 002Q11b 17822916 17735394 234711 minus049Q22b 3297651 3259546 91601 minus116Q12b 894780 827581 1018827 minus751Q66b 5700 569879 59903 minus002

Advances in Materials Science and Engineering 13

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 10: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

ρ σi( 1113857 1113944

bpointsj1 σi Mj1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868

npoints i 1 2 6 (26)

According to equation (7) the standard deviations of Qij

are equal to η(i)c If c is the standard deviation of the strainnoise the coefficients of variations (CV(Qij)) of eachstiffness component are given by

CV Q11( 1113857 η11Q11

c

CV Q22( 1113857 η22Q22

c

CV Q12( 1113857 η12Q12

c

CV Q66( 1113857 η66Q66

c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

+e ratio of coefficient of variation to the standarddeviation of the strain noise can show the sensitivity to thenoise It indicates the rate at which the coefficient of vari-ation increases with the increase of noise which can becalled the variation coefficient-to-noise ratio and can beexpressed as ηijQij +e lower variation coefficient-to-noiseratio indicates that the corresponding identification con-stitutive parameters are less susceptible to noise +e vari-ation coefficient-to-noise ratios ηijQij of eight identificationresults are shown in Table 3 Significant differences on thevariation coefficient-to-noise ratio and the coefficient-to-noise ratio of Q66 is the smallest the next is Q11 and then isthe Q22 and the biggest is Q12 +e quantitative results showthat if Q11 Q22 and Q66 are stable Q12 is less robustlyidentified +is result shows that this configuration is un-suitable for the transverse stiffness

32 Comparison with Results of Piecewise Virtual Fields andPolynomial Virtual Fields +is section will implement thesame numerical experiment as Section 31 to compare theidentified constitutive parameters of the piecewise virtual

L

w

wa

wb

F

Part A

Part B

Figure 1 Two-dimensional bimaterial model of three-point bending established in FEM simulation

Table 1 Engineering elastic constants of orthotropic bimaterialExa (Mpa) Eya (Mpa) Eza (Mpa) PRxya (Mpa) PRyza (Mpa) PRxza (Mpa) Gxya (Mpa) Gyza (Mpa) Gxza (Mpa)180000 30000 8300 017 025 031 5700 12000 12000Exb (Mpa) Eyb (Mpa) Ezb (Mpa) PRxyb (Mpa) PRyzb (Mpa) PRxzb (Mpa) Gxyb (Mpa) Gyzb (Mpa) Gxzb (Mpa)175000 32000 8300 025 025 031 5700 12000 12000

Table 2 Reference constitutive parameters of orthotropic bimaterial

Q11a (Mpa) Q22a (Mpa) Q12a (Mpa) Q66a (Mpa) Q11b (Mpa) Q22b (Mpa) Q12b (Mpa) Q66b (Mpa)

18193710 3071968 590720 5700 17822916 3297651 894780 5700

10 Advances in Materials Science and Engineering

field and polynomial virtual field At this situation the bi-linear shape function is used to interpolate the four-nodeelement to represent the virtual field instead of the poly-nomial virtual field

+e piecewise function expansion of the virtual dis-placement field is similar to the interpolation of the real

displacement in the FEM Derived from the relationshipbetween strain and displacement

ε Sulowast (28)

where

ndash133477

ndash113034

ndash092591

ndash072149

ndash051706

ndash031263

ndash010821

009622

030065

050507

(a)

ndash956093

ndash849861

ndash743628

ndash637396

ndash531163

ndash42493

ndash318698

ndash212465

ndash106233

0

(b)

Figure 2 Displacement distributions of heterogeneous orthotropic bimaterials (a) u1 (b) u2 (unit mm)

Advances in Materials Science and Engineering 11

ndash0235856

ndash018911

ndash014237

ndash009562

ndash004888

ndash000021

041162

009136

013811

018485

(a)

ndash213206

ndash186779

ndash160351

ndash133924

ndash107496

ndash081069

ndash054641

ndash028214

ndash001786

0024641

(b)

ndash362375

ndash32025

ndash278125

ndash236

ndash193874

ndash151749

ndash109624

ndash067499

ndash025373

016752

(c)

Figure 3 Strain distributions of heterogeneous orthotropic bimaterials (a) ε1 (b) ε2 (c) ε3

12 Advances in Materials Science and Engineering

ε

ε1

ε2

ε6

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

ulowast ulowast1

ulowast2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

S

z

zx10

0z

zx2

z

zx2

z

zx1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(29)

For the quadrilateral element the virtual displacementcan be expressed by the following formula

ulowast ξ1 ξ2( 1113857≃ulowast ξ1 ξ2( 1113857 11139444

a1N(a) ξ1 ξ2( 1113857ulowast(a)

(30)

where ulowast is the vector containing the virtual displacement ofany point (ξ1 ξ2) in the element and 1113957ulowast(a) is the vectorcontaining the virtual displacement of the nodes in the el-ement +e expression of N(a) is defined as

N(a) ξ1 ξ2( 1113857 N(a) ξ1 ξ2( 1113857I (31)

where I is the identity matrix +e shape function expressionof N(a) can be defined as follows

N(1)

14

1 minus ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 + ξ2( 1113857

N(2)

14

1 minus ξ1( 1113857 1 + ξ2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

So the virtual displacement can be expressed by thefollowing formula

ulowast1 asymp N

(1)1113957ulowast(1)1 + N

(2)1113957ulowast(2)1 + N

(3)1113957ulowast(3)1 + N

(4)1113957ulowast(4)1

ulowast2 asymp N

(1)1113957ulowast(1)2 + N

(2)1113957ulowast(2)2 + N

(3)1113957ulowast(3)2 + N

(4)1113957ulowast(4)2

⎧⎨

⎩

(33)

+erefore the unknown coefficient vector Y (equation(17)) in the polynomial virtual fields can be denoted on eachnode

Y

1113957ulowast(1)1

1113957ulowast(1)2

⋮

1113957ulowast(n)1

1113957ulowast(n)2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

where n is the total number of nodesUse the piecewise virtual field instead of the polynomial

virtual field to perform the same experiment (Section 31)Table 4 shows the values of relative and weighted relativeerror of A and B It can be seen from Table 4 that theidentification results of piecewise virtual fields are similar tothat of polynomial virtual fields +e weighted relative erroris 097 which is slightly larger than that of polynomialvirtual fields +e relative error of Q66 of the two materials isthe smallest Although the relative errors of Q11 and Q22ofthe two materials are slightly larger than Q66 both of themare less than 2 Similar to the results of polynomial virtualfields the relative error of Q12a and Q12b is significantlyhigher than others because σ2 must be included in thevirtual fields to identify Q12 while the data density of σ2 islow in the three-point bending test

+e variation coefficient-to-noise ratios of eight iden-tification results are also shown in Table 4 Significant dif-ferences can be seen on the variation coefficient-to-noiseratios and the quantitative results show that if Q11 Q22 andQ66 are stable Q12 is less robustly identified Compared withthe polynomial virtual field the variation coefficient-to-noise ratio of the eight identification parameters calculatedby the optimized piecewise virtual field is larger

33 Influence of Noise on Identification Results +e noise isinevitable in the experiments To investigate the influence ofnoise on identification results zero-mean Gaussian white

Table 3 Identification results of constitutive parameters of orthotropic bimaterial using the optimized polynomial virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18159713 280642 minus019

083

Q22a 3071968 3035043 106390 minus120Q12a 590720 479127 2080530 minus1889Q66a 5700 570102 53999 002Q11b 17822916 17735394 234711 minus049Q22b 3297651 3259546 91601 minus116Q12b 894780 827581 1018827 minus751Q66b 5700 569879 59903 minus002

Advances in Materials Science and Engineering 13

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 11: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

field and polynomial virtual field At this situation the bi-linear shape function is used to interpolate the four-nodeelement to represent the virtual field instead of the poly-nomial virtual field

+e piecewise function expansion of the virtual dis-placement field is similar to the interpolation of the real

displacement in the FEM Derived from the relationshipbetween strain and displacement

ε Sulowast (28)

where

ndash133477

ndash113034

ndash092591

ndash072149

ndash051706

ndash031263

ndash010821

009622

030065

050507

(a)

ndash956093

ndash849861

ndash743628

ndash637396

ndash531163

ndash42493

ndash318698

ndash212465

ndash106233

0

(b)

Figure 2 Displacement distributions of heterogeneous orthotropic bimaterials (a) u1 (b) u2 (unit mm)

Advances in Materials Science and Engineering 11

ndash0235856

ndash018911

ndash014237

ndash009562

ndash004888

ndash000021

041162

009136

013811

018485

(a)

ndash213206

ndash186779

ndash160351

ndash133924

ndash107496

ndash081069

ndash054641

ndash028214

ndash001786

0024641

(b)

ndash362375

ndash32025

ndash278125

ndash236

ndash193874

ndash151749

ndash109624

ndash067499

ndash025373

016752

(c)

Figure 3 Strain distributions of heterogeneous orthotropic bimaterials (a) ε1 (b) ε2 (c) ε3

12 Advances in Materials Science and Engineering

ε

ε1

ε2

ε6

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

ulowast ulowast1

ulowast2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

S

z

zx10

0z

zx2

z

zx2

z

zx1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(29)

For the quadrilateral element the virtual displacementcan be expressed by the following formula

ulowast ξ1 ξ2( 1113857≃ulowast ξ1 ξ2( 1113857 11139444

a1N(a) ξ1 ξ2( 1113857ulowast(a)

(30)

where ulowast is the vector containing the virtual displacement ofany point (ξ1 ξ2) in the element and 1113957ulowast(a) is the vectorcontaining the virtual displacement of the nodes in the el-ement +e expression of N(a) is defined as

N(a) ξ1 ξ2( 1113857 N(a) ξ1 ξ2( 1113857I (31)

where I is the identity matrix +e shape function expressionof N(a) can be defined as follows

N(1)

14

1 minus ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 + ξ2( 1113857

N(2)

14

1 minus ξ1( 1113857 1 + ξ2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

So the virtual displacement can be expressed by thefollowing formula

ulowast1 asymp N

(1)1113957ulowast(1)1 + N

(2)1113957ulowast(2)1 + N

(3)1113957ulowast(3)1 + N

(4)1113957ulowast(4)1

ulowast2 asymp N

(1)1113957ulowast(1)2 + N

(2)1113957ulowast(2)2 + N

(3)1113957ulowast(3)2 + N

(4)1113957ulowast(4)2

⎧⎨

⎩

(33)

+erefore the unknown coefficient vector Y (equation(17)) in the polynomial virtual fields can be denoted on eachnode

Y

1113957ulowast(1)1

1113957ulowast(1)2

⋮

1113957ulowast(n)1

1113957ulowast(n)2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

where n is the total number of nodesUse the piecewise virtual field instead of the polynomial

virtual field to perform the same experiment (Section 31)Table 4 shows the values of relative and weighted relativeerror of A and B It can be seen from Table 4 that theidentification results of piecewise virtual fields are similar tothat of polynomial virtual fields +e weighted relative erroris 097 which is slightly larger than that of polynomialvirtual fields +e relative error of Q66 of the two materials isthe smallest Although the relative errors of Q11 and Q22ofthe two materials are slightly larger than Q66 both of themare less than 2 Similar to the results of polynomial virtualfields the relative error of Q12a and Q12b is significantlyhigher than others because σ2 must be included in thevirtual fields to identify Q12 while the data density of σ2 islow in the three-point bending test

+e variation coefficient-to-noise ratios of eight iden-tification results are also shown in Table 4 Significant dif-ferences can be seen on the variation coefficient-to-noiseratios and the quantitative results show that if Q11 Q22 andQ66 are stable Q12 is less robustly identified Compared withthe polynomial virtual field the variation coefficient-to-noise ratio of the eight identification parameters calculatedby the optimized piecewise virtual field is larger

33 Influence of Noise on Identification Results +e noise isinevitable in the experiments To investigate the influence ofnoise on identification results zero-mean Gaussian white

Table 3 Identification results of constitutive parameters of orthotropic bimaterial using the optimized polynomial virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18159713 280642 minus019

083

Q22a 3071968 3035043 106390 minus120Q12a 590720 479127 2080530 minus1889Q66a 5700 570102 53999 002Q11b 17822916 17735394 234711 minus049Q22b 3297651 3259546 91601 minus116Q12b 894780 827581 1018827 minus751Q66b 5700 569879 59903 minus002

Advances in Materials Science and Engineering 13

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 12: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

ndash0235856

ndash018911

ndash014237

ndash009562

ndash004888

ndash000021

041162

009136

013811

018485

(a)

ndash213206

ndash186779

ndash160351

ndash133924

ndash107496

ndash081069

ndash054641

ndash028214

ndash001786

0024641

(b)

ndash362375

ndash32025

ndash278125

ndash236

ndash193874

ndash151749

ndash109624

ndash067499

ndash025373

016752

(c)

Figure 3 Strain distributions of heterogeneous orthotropic bimaterials (a) ε1 (b) ε2 (c) ε3

12 Advances in Materials Science and Engineering

ε

ε1

ε2

ε6

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

ulowast ulowast1

ulowast2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

S

z

zx10

0z

zx2

z

zx2

z

zx1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(29)

For the quadrilateral element the virtual displacementcan be expressed by the following formula

ulowast ξ1 ξ2( 1113857≃ulowast ξ1 ξ2( 1113857 11139444

a1N(a) ξ1 ξ2( 1113857ulowast(a)

(30)

where ulowast is the vector containing the virtual displacement ofany point (ξ1 ξ2) in the element and 1113957ulowast(a) is the vectorcontaining the virtual displacement of the nodes in the el-ement +e expression of N(a) is defined as

N(a) ξ1 ξ2( 1113857 N(a) ξ1 ξ2( 1113857I (31)

where I is the identity matrix +e shape function expressionof N(a) can be defined as follows

N(1)

14

1 minus ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 + ξ2( 1113857

N(2)

14

1 minus ξ1( 1113857 1 + ξ2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

So the virtual displacement can be expressed by thefollowing formula

ulowast1 asymp N

(1)1113957ulowast(1)1 + N

(2)1113957ulowast(2)1 + N

(3)1113957ulowast(3)1 + N

(4)1113957ulowast(4)1

ulowast2 asymp N

(1)1113957ulowast(1)2 + N

(2)1113957ulowast(2)2 + N

(3)1113957ulowast(3)2 + N

(4)1113957ulowast(4)2

⎧⎨

⎩

(33)

+erefore the unknown coefficient vector Y (equation(17)) in the polynomial virtual fields can be denoted on eachnode

Y

1113957ulowast(1)1

1113957ulowast(1)2

⋮

1113957ulowast(n)1

1113957ulowast(n)2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

where n is the total number of nodesUse the piecewise virtual field instead of the polynomial

virtual field to perform the same experiment (Section 31)Table 4 shows the values of relative and weighted relativeerror of A and B It can be seen from Table 4 that theidentification results of piecewise virtual fields are similar tothat of polynomial virtual fields +e weighted relative erroris 097 which is slightly larger than that of polynomialvirtual fields +e relative error of Q66 of the two materials isthe smallest Although the relative errors of Q11 and Q22ofthe two materials are slightly larger than Q66 both of themare less than 2 Similar to the results of polynomial virtualfields the relative error of Q12a and Q12b is significantlyhigher than others because σ2 must be included in thevirtual fields to identify Q12 while the data density of σ2 islow in the three-point bending test

+e variation coefficient-to-noise ratios of eight iden-tification results are also shown in Table 4 Significant dif-ferences can be seen on the variation coefficient-to-noiseratios and the quantitative results show that if Q11 Q22 andQ66 are stable Q12 is less robustly identified Compared withthe polynomial virtual field the variation coefficient-to-noise ratio of the eight identification parameters calculatedby the optimized piecewise virtual field is larger

33 Influence of Noise on Identification Results +e noise isinevitable in the experiments To investigate the influence ofnoise on identification results zero-mean Gaussian white

Table 3 Identification results of constitutive parameters of orthotropic bimaterial using the optimized polynomial virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18159713 280642 minus019

083

Q22a 3071968 3035043 106390 minus120Q12a 590720 479127 2080530 minus1889Q66a 5700 570102 53999 002Q11b 17822916 17735394 234711 minus049Q22b 3297651 3259546 91601 minus116Q12b 894780 827581 1018827 minus751Q66b 5700 569879 59903 minus002

Advances in Materials Science and Engineering 13

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 13: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

ε

ε1

ε2

ε6

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

ulowast ulowast1

ulowast2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

S

z

zx10

0z

zx2

z

zx2

z

zx1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(29)

For the quadrilateral element the virtual displacementcan be expressed by the following formula

ulowast ξ1 ξ2( 1113857≃ulowast ξ1 ξ2( 1113857 11139444

a1N(a) ξ1 ξ2( 1113857ulowast(a)

(30)

where ulowast is the vector containing the virtual displacement ofany point (ξ1 ξ2) in the element and 1113957ulowast(a) is the vectorcontaining the virtual displacement of the nodes in the el-ement +e expression of N(a) is defined as

N(a) ξ1 ξ2( 1113857 N(a) ξ1 ξ2( 1113857I (31)

where I is the identity matrix +e shape function expressionof N(a) can be defined as follows

N(1)

14

1 minus ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 minus ξ2( 1113857

N(2)

14

1 + ξ1( 1113857 1 + ξ2( 1113857

N(2)

14

1 minus ξ1( 1113857 1 + ξ2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

So the virtual displacement can be expressed by thefollowing formula

ulowast1 asymp N

(1)1113957ulowast(1)1 + N

(2)1113957ulowast(2)1 + N

(3)1113957ulowast(3)1 + N

(4)1113957ulowast(4)1

ulowast2 asymp N

(1)1113957ulowast(1)2 + N

(2)1113957ulowast(2)2 + N

(3)1113957ulowast(3)2 + N

(4)1113957ulowast(4)2

⎧⎨

⎩

(33)

+erefore the unknown coefficient vector Y (equation(17)) in the polynomial virtual fields can be denoted on eachnode

Y

1113957ulowast(1)1

1113957ulowast(1)2

⋮

1113957ulowast(n)1

1113957ulowast(n)2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

where n is the total number of nodesUse the piecewise virtual field instead of the polynomial

virtual field to perform the same experiment (Section 31)Table 4 shows the values of relative and weighted relativeerror of A and B It can be seen from Table 4 that theidentification results of piecewise virtual fields are similar tothat of polynomial virtual fields +e weighted relative erroris 097 which is slightly larger than that of polynomialvirtual fields +e relative error of Q66 of the two materials isthe smallest Although the relative errors of Q11 and Q22ofthe two materials are slightly larger than Q66 both of themare less than 2 Similar to the results of polynomial virtualfields the relative error of Q12a and Q12b is significantlyhigher than others because σ2 must be included in thevirtual fields to identify Q12 while the data density of σ2 islow in the three-point bending test

+e variation coefficient-to-noise ratios of eight iden-tification results are also shown in Table 4 Significant dif-ferences can be seen on the variation coefficient-to-noiseratios and the quantitative results show that if Q11 Q22 andQ66 are stable Q12 is less robustly identified Compared withthe polynomial virtual field the variation coefficient-to-noise ratio of the eight identification parameters calculatedby the optimized piecewise virtual field is larger

33 Influence of Noise on Identification Results +e noise isinevitable in the experiments To investigate the influence ofnoise on identification results zero-mean Gaussian white

Table 3 Identification results of constitutive parameters of orthotropic bimaterial using the optimized polynomial virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18159713 280642 minus019

083

Q22a 3071968 3035043 106390 minus120Q12a 590720 479127 2080530 minus1889Q66a 5700 570102 53999 002Q11b 17822916 17735394 234711 minus049Q22b 3297651 3259546 91601 minus116Q12b 894780 827581 1018827 minus751Q66b 5700 569879 59903 minus002

Advances in Materials Science and Engineering 13

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 14: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

noise with different amplitudes is added in the strain data ofthe orthotropic bimaterial specimen and the above-mentioned experiments are repeated to obtain the coeffi-cients of variation corresponding to the eight identificationresults with the increasing noise amplitudes In order toassess the influence of noise on the identified parameters itis necessary to discuss the standard deviation of strain noiseIt can be seen from equation (27) that the coefficients ofvariation (CV(Qij)a) are proportional to the uncertainty (c)of strain measurement Since the coefficient of variation canbe directly compared between different constitutive com-ponents this article chose to discuss the coefficient ofvariation rather than standard deviation +us coefficientsof variation were used to discuss the sensitivity to noise

+e coefficients of variation of the identified stiffnesscomponents of the two materials were calculated for a range ofstrain noise standard deviation values+e strain noise standarddeviation c varies from 5times10minus5 to 1times 10minus3 by steps of 5times10minus5For each value of c the identification is repeated 30 times usingthe optimized polynomial virtual fields +e coefficients ofvariation (CV(Qij)) were calculated by equation (34) Figure 4indicates the graph plotting the coefficients of variations for the

eight stiffness components of the two materials as a function ofc+e points are fitted by linear regression to calculate the slopeand the slope was compared to the theoretical value of thevariation coefficient-to-noise ratio ηijQij

CV Qij1113872 1113873

1113944Qij minus Qij)

2n minus 1

1113936 Qij1113872 1113873n⎛⎝

11139741113972

(35)

Table 4 Identification results of constitutive parameters of orthotropic bimaterial using the optimized piecewise virtual fields

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18123694 297551 minus038

097

Q22a 3071968 3022644 116049 minus161Q12a 590720 497948 2458494 minus5871Q66a 5700 570070 61965 001Q11b 17822916 17724989 250937 minus055Q22b 3297651 3245653 97825 minus158Q12b 894780 818467 1259329 minus853Q66b 5700 569921 68896 minus001

Table 5 +e fitted and the theoretical values of the variationcoefficient-to-noise ratios

Fitted value +eoretical valueη11aQ11a 2713 266534η22aQ22a 1034 96984η12aQ12a 16575 1721564η66aQ66a 529 50576η11bQ11b 2394 220878η22bQ22b 850 84713η12bQ12b 8804 847326η66bQ66b 594 56006

times10ndash3

0002004006008

01012014016018

02

CV(Q

ij)a

CV(Q11)aCV(Q22)a

CV(Q12)aCV(Q66)a

02 04 06 08 100γ

(a)

times10ndash3

CV(Q11)bCV(Q22)b

CV(Q12)bCV(Q66)b

02 04 06 08 100γ

0001002003004005006007008009

01

CV(Q

ij)b

(b)

Figure 4 +e coefficients of variation of the identified stiffness components (a) Part A (b) Part B

14 Advances in Materials Science and Engineering

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 15: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

n

m3 4 5

6

5

4

3

6

266265264263262261260259258257

(a)

n

6

5

4

3

m3 4 5 6

9894929088868482807876

(b)

n

6

5

4

3

m3 4 5 6

170

165

160

155

150

(c)

n

6

5

4

3

m3 4 5 6

50

48

46

44

42

40

38

(d)

n

6

5

4

3

m3 4 5 6

220

218

216

214

212

210

(e)

n

6

5

4

3

m3 4 5 6

84

82

80

78

76

74

72

70

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 15

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 16: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

For both materials A and B Figure 4 shows that thesmallest coefficient of variation value is CV(Q66) +eresults of Figure 4 are consistent with the variationcoefficient-to-noise ratio studies in Section 31 Q66 wasthe most stable of the identified stiffness componentsCV(Q22) was the second and CV(Q11) was the third andfinally CV(Q12) which is also consistent with the resultsin Section 31 +e slope of the fitted curve is the fittedvariation coefficient-to-noise ratio +e comparisons ofthe fitted and theoretical value of the variation coeffi-cient-to-noise ratio are listed in Table 5 Figure 4 andTable 5 not only show the linear relationship but alsoshow that the theoretical ηijQij values are consistentwith the fitted ones It suggests that the procedure canperform a priori evaluation of a confidence intervalon the identified stiffness components In additiondue to the hypothesis of equation (16) the straightline is unsuitable for the highest values of noise Equa-tion (16) shows the noise should be smaller than thesignal

34 Influence of the Polynomial Degrees According toequation (10) when the basic functions to expand the virtualfield is selected the only parameters the degrees of the x1and x2 monomials denoted as m and n need to be selectedm and nare the polynomial degrees that represent themaximum number of monomials +e total number ofpolynomial degrees is 2(m + 1)(n + 1) for the case of op-timized polynomial virtual fields applied to the bimaterialwithout knowing any material constitutive parameters thecondition of equation (35) needs to be satisfied

2(m + 1)(n + 1)gt n + 10 (36)

For the optimized polynomial virtual fields on the onehand m and n cannot be selected too small which will leadto too few polynomial degrees and thus cannot satisfyequation (35) On the other hand m and n should not beselected too large which will result in ill-conditioned matrixFigure 5 shows the evolution of the variation coefficient-to-noise ratio ηijQij as a function of m and n It can be seen

n

6

5

4

3

m3 4 5 6

84

82

80

78

74

76

(g)

n

6

5

4

3

m3 4 5 6

56

54

52

50

46

48

42

44

(h)

Figure 5 Values of ηijQij (a) η11aQ11a (b) η22aQ22a (c) η12aQ12a (d) η66aQ66a (e) η11bQ11b (f ) η22bQ22b (g) η12bQ12b and (h)η66bQ66b

(a) (b)

Figure 6 +e reference and the deformation speckle pattern images (a) Reference speckle pattern (b) Deformed speckle pattern

16 Advances in Materials Science and Engineering

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 17: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

from Figure 5 that there are slight variations in the ηijQij

with higher gradients for the values related to Q12a and Q12aDue to the three-point bending test configuration theseparameters are relatively less good identifiability It can alsoshow that the dependance on n is higher than that on m +isis because all the constraints affect the x2 monomials

35 Numerical Verification of Optimized Polynomial VirtualFields with DIC Simulations In practice to provide a morereliable strain input for the virtual fields method digitalimage correlation (DIC) was applied to obtain deformation

fields DIC is an effective optical technique for noncontactfull-field deformation measurement for various materialsand structures +e DIC computes the displacement of eachimage point by comparing the gray intensity of images of thetest specimen surface in different loading states +e cor-responding strain fields are calculated using a numericaldifferentiation method In addition to the systematic error ofDIC algorithm aberration distortion and out-of-planedisplacement will affect the accuracy of displacementmeasurement

For the purpose of evaluating the simulated results ofconstitutive parameters for bimaterials and the influence of

0 5 10 15 20 25 300

5

10

15

20

minus012

minus01

minus008

minus006

minus004

minus002

0

002

004

(a)

0 5 10 15 20 25 300

5

10

15

20

minus09

minus08

minus07

minus06

minus05

minus04

ndash03

ndash02

ndash01

(b)

Figure 7 Displacement fields of speckle images using DIC (a) u1 (b) u2(unit mm)

Advances in Materials Science and Engineering 17

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 18: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

5 10 15 20 25

2

4

6

8

10

12

14

16

18

times 10ndash3

minus10

minus8

minus6

minus4

minus2

0

2

4

6

(a)

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus08

minus1

minus06

minus04

minus02

0

02

04

06

08

(b)

Figure 8 Continued

18 Advances in Materials Science and Engineering

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 19: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

strain input calculated by DIC on the identification ofconstitutive parameters the simulated DIC results were usedas the input of the VFM to exclude the influence of lensdistortion and out-of-plane displacements +e displace-ment fields of the bimaterial beam specimen obtainedthrough abovementioned FEM simulations were exportedand converted to reference and deformed speckle patterns byGaussian speckle [32] as shown in Figure 6 +e dis-placement fields and strain fields obtained from DIC areshown in Figures 7 and 8 respectively

Table 6 shows the identification results and the cor-responding variation coefficient-to-noise ratio and rela-tive error and weighted relative error for both materialsSimilar to the results in Table 3 it shows significantdifferences on the variation coefficient-to-noise ratio andthe coefficient-to-noise ratio of Q66 which is the smallestthe next is Q11 and then is the Q22 and the biggest is Q12+e lowest value is Q66 +is result is consistent withstudies in Section 33 Q66was the most stable of the fouridentified stiffness components in Section 33 Q11 is thesecond Q22 is the third and Q12 is the last +is result is alsoconsistent with the studies in Section 33 As shown in Table 6the relative errors of Q12a and Q12b are more than 10 while

others are notmore than 4 and the weighted relative error is114 which is slightly larger than the results in Table 3 +isindicates that the strain data calculated by DIC contains noiseresulting in an increase in the error of the identificationresults +is comparison result reveals the feasibility of theoptimized polynomial virtual fields combined with DIC forextracting the constitutive parameters of the unknownbimaterial It should be pointed out that in the actual DICcalculation the displacement measurement errors caused byaberration and out-of-plane displacements will lead to ad-ditional strain noise so the relative errors and weightedrelative error of the identification results will be larger thanthe results in Table 6

4 Conclusions

In this article optimized polynomial virtual fields areproposed to extract the constitutive parameters of theheterogeneous orthotropic bimaterials without knowing anymaterial constitutive parameters Numerical experimentswith FEM simulations are employed to investigate the ac-curacy of the optimized polynomial virtual fields method+e main conclusions are as follows

5 10 15 20 25

2

4

6

8

10

12

14

16

18

minus02

minus015

ndash01

ndash005

0

(c)

Figure 8 Strain fields of simulated speckle images using DIC (a) ε1 (b) ε2 (c) ε6

Table 6 Identification results of constitutive parameters of orthotropic bimaterials with speckle images

Reference (MPa) Identification (MPa) ηijQij Relative error () Weighted relative error ()

Q11a 18193710 18055558 280682 minus076

114

Q22a 3071968 3044846 105995 minus088Q12a 590720 479989 2071620 minus1875Q66a 5700 567847 54031 minus038Q11b 17822916 17825847 234412 002Q22b 3297651 3173088 89408 minus378Q12b 894780 788406 1037439 minus1189Q66b 5700 572527 59455 044

Advances in Materials Science and Engineering 19

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 20: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

(1) +e optimized polynomial virtual fields were pro-posed to identify the constitutive parameters ofheterogeneous orthotropic bimaterials under thecondition that constitutive parameters of both ma-terials are all unknown from a single test +e resultsshow that the weighted relative error of the con-stitutive parameter is less than 1 +e stress datadensity was used to explain the difference in therelative error of the constitutive parameter Forheterogeneous orthotropic bimaterials withoutknowing any material constitutive parameters thismethod is suitable for extracting the constitutiveparameters

(2) To investigate the effect of the noise a series ofnumerical experiments with different strain noiseamplitudes were used in FEM simulations +evariation coefficient-to-noise ratio and coeffi-cients of variation are applied to quantify theinfluence of noise on the identification results andthe results suggest that the variation coefficient-to-noise ratio can perform a priori evaluation of aconfidence interval on the identified stiffnesscomponents

(3) For comparison piecewise virtual fields were used toextract the constitutive parameters of the hetero-geneous orthotropic bimaterials +e results denotedthat the optimized polynomial virtual fields have ahigher reliability than the optimized piecewise vir-tual fields

(4) To study the accuracy of the optimized polynomialVFM combined with DIC method the numericalexperiments with DIC simulations are employed+e comparison result shows the feasibility of theoptimized polynomial virtual fields combined withDIC for extracting the constitutive parameters of theunknown bimaterials +e result also indicates thatthe strain data calculated by DIC contains noiseresulting in an increase in the error of the identifi-cation results and the weighted relative error is114

Nomenclature

σ Cauchy stress tensorulowast Virtual displacement

vectorεlowast Strain tensorT External forceS Area of the specimenb Vector of volume forceV Specimen volumea Distribution of

accelerationKA Virtual displacement

field being kinematicallyadmissible

Q11 Q22 Q12 Q66 In-plane constitutiveparameters

ulowast(1) ulowast(2) ulowast(3) ulowast(4) Four independent KAvirtual fields

εlowast(1) εlowast(2) εlowast(3) εlowast(4) Four independent KAvirtual fields

N1N2N6 Processes of scalar zero-mean stationaryGaussian

ε1 ε2 ε6 Strain componentsc Measured amplitude of

the random variablestrain

Qapp Approximate parametercomponents

εi(i 1 2 6) Norm of straincomponent

V(Q11) V(Q22) V(Q12) V(Q66) Variances ofQ11 Q12 Q22 and Q66

Ylowast Vector concerning thecoefficients of virtualstrain fields

H Hessian matrixL(i) Lagrangian functionλ(i) Vector containing

Lagrange multipliersQ11a Q22a Q12a Q66a Constitutive parameters

of material aQ11b Q22b Q12b Q66b Constitutive parameters

of material bL Typical dimensions of

the x1w Typical dimensions of

the x2F External force of the

specimenWre Weighted relative error

of Q

N(i)(i 1 2 3 4) Shape function

Appendix

A Equation Derivation of Section 21

A1lte Principle of VirtualWork for an OrthotropicMaterialin a Plane-Stress State +e integral form of the mechanicalequilibrium equation of the VFM based on virtual work for acontinuous solid can be expressed as [31]

minus 1113946Sσ εlowastdS + 1113946

ST middot ulowastdS + 1113946

Vb middot ulowastdV

1113946Vρa middot ulowastdV forallulowastKA

(A1)

where σ is the Cauchy stress tensor ulowast is the virtual dis-placement vector εlowastdenotes the strain tensor correspondingto ulowast T represents the external force associated with thestress tensor σ by the boundary S is a vector which is thevolume force applied over the volume V of the specimenand a denotes the distribution of acceleration Such a dis-tribution will cause an additional volumetric force

20 Advances in Materials Science and Engineering

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 21: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

distribution equal to minusρa with DrsquoAlembertrsquos principle thevirtual displacement field being kinematically admissible(KA)

In the case of a quasistatic load acceleration acan beomitted +e volume force bcan usually also be omittedUnder this circumstance equation (A1) can be written asfollows

minus1113946Sσ εlowastdS 1113946

ST middot ulowastdS forallulowastKA (A2)

Equation (A2) denotes the plane-stress problem For anorthotropic material in a plane-stress state the principle ofvirtual work becomes

1113946SQ11 ε1ε

lowast1( 1113857dS + 1113946

SQ22 ε2ε

lowast2( 1113857dS + 1113946

SQ12 ε1ε

lowast2 + ε2ε

lowast1( 1113857dS

+ 1113946SQ66 ε6ε

lowast6( 1113857dS 1113946

Lf

Tiulowasti( 1113857dl

(A3)

where Q11 Q22 Q12 and Q66 are the in-plane stiffnesscomponents and equation (A3) is linear It is suitable forevery KA virtual field +is equation uses four independentKA virtual fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) instead ofulowast(1) ulowast(2) ulowast(3) and ulowast(4) respectively So the relationshipbetween AQ and B is as follows

AQ B (A4)

A2 lte Principle of Virtual Work with Uncorrelated Noise+e exact values and white noise constitute the measuredstrain component So equation (A3) in Appendix A1 can bereexpressed as follows

Q111113946Sε1 minus cN1( 1113857εlowast1 dS + Q221113946

Sε2 minus cN2( 1113857εlowast2 dS

+ Q121113946S

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+ Q661113946Sε6 minus cN6( 1113857εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(A5)

where N1N2 and N6 denote the processes of scalar zero-mean stationary Gaussian generalized by the correspondingstrain components ε1 ε2 and ε6 respectively and c denotesthe measured amplitude of the random variable strainAssume that the noise components are uncorrelated to eachother and presume that the noise between points is alsouncorrelated Equation (A5) can be rewritten as follows

Q11 1113946Sε1εlowast1( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139811

+ Q22 1113946Sε2εlowast2( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

+ Q12 1113946Sε1εlowast2 + ε2ε

lowast1( 1113857

1113980radicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradic11139810

dS + Q66 1113946Sε6εlowast6( 1113857dS

1113980radicradicradicradicradic11139791113978radicradicradicradicradic11139810

minus

c Q111113946Sεlowast1 N1dS + Q221113946

Sεlowast2 N2dS + Q121113946

Sεlowast2 N1 + εlowast1 N2( 1113857dS + Q661113946

Sεlowast6 N6dS1113876 1113877 1113946

Lf

Tiulowasti( 1113857dl

(A6)

A3 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11 Q22 Q12 and Q66 using these fourspecial virtual displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4)

Q11 c Q111113946Sεεlowast(1)

1 N1dS + Q221113946Sεlowast (1)2 N2dS + Q121113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowast (1)i1113872 1113873dl

(A7)

Q22 c Q111113946Sεlowast (2)1 N1dS + Q221113946

Sεlowast (2)2 N2dS + Q121113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (2)6 N6dS1113877 + 1113946

Lf

Tiulowast (2)i1113872 1113873dl

(A8)

Q12 c Q111113946Sεlowast (3)1 N1dS + Q22 1113946 εlowast (3)

2 N2dS + Q121113946Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (3)6 N6dS1113877 + 1113946

Lf

Tiulowast (3)i1113872 1113873dl

(A9)

Advances in Materials Science and Engineering 21

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 22: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

Q66 c Q11 1113946 εlowast (4)5 N1dS + Q221113946

Sεlowast (4)2 N2dS + Q121113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS1113876

+ Q661113946Sεlowast (4)6 N6dS1113877 + 1113946

Lf

Tiulowast (4)i1113872 1113873dl

(A10)

If the noise is ignored approximate parametersexpressed as Qapp these components are defined by

Qapp11 1113946

Lf

Tiulowast (1)i1113872 1113873dl

Qapp22 1113946

Lf

Tiulowast (2)i1113872 1113873dl

Qapp12 1113946

Lf

Tiulowast (3)i1113872 1113873dl

Qapp66 1113946

Lf

Tiulowast (4)i1113872 1113873dl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A11)

where c can be negligible compared to the ε1 ε2 and ε6respectively and the εi(i 1 2 6) is the norm of straincomponent L2 +us

c≪ min ε1

ε2

ε6

1113872 1113873 (A12)

Equations (A7)ndash(A10) mentioned above can be reex-pressed by the actual values of the Qij instead of the ap-proximate counterparts +us

Q11 c Qapp11 1113946

Sεlowast (1)1 N1dS1113876 + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (1)6 N6dS1113877 + Q

app1

Q22 c Qapp11 1113946

Sεlowast (2)1 N1dS + Q

app22 1113946

Sεlowast (2)2 N2dS + Q

app12 1113946

Sεlowast (2)2 N1 + εlowast (2)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (2)6 N6dS1113877 + Q

app22

Q12 c Qapp11 1113946

Sεlowast (3)1 N1dS + Q

app22 1113946

Sεlowast (3)2 N2dS + Q

app12 1113946

Sεlowast (3)2 N1 + εlowast (3)

1 N21113872 1113873dS1113876

+ Qapp66 1113946

Sεlowast (3)6 N6dS1113877 + Q

app12

Q66 c Qapp11 1113946

Sεlowast (4)1 N1dS1113876 + Q

app22 1113946

Sεlowast (4)2 N2dS + Q

app12 1113946

Sεlowast (4)2 N1 + εlowast (4)

1 N21113872 1113873dS

+ Qapp66 1113946

Sεlowast (4)6 N6dS1113877 + Q

app66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A13)

A4 lte Variance of Identified Constitutive ParametersTo lower the effect of noise the objective function can bedenoted as the difference between the real parametersQ and

approximate parametersQapp For instance the variance forQ11 can be written as follows

V Q11( 1113857 c2E Q

app11 1113946

Sεlowast (1)1 N1dS + Q

app22 1113946

Sεlowast (1)2 N2dS + Q

app12 1113946

Sεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Qapp60 1113946

Sεlowast (1)6 N6dS1113876 1113877

21113888 1113889

(A14)

Using the rectangle method to discretize the above-mentioned integrals

22 Advances in Materials Science and Engineering

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 23: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

V Q11( 1113857 c2 S

n1113874 1113875

2E Q

app11 1113944

n

i1ε(1)1 Mi( 1113857N1 Mi( 1113857 + Q

app22 1113944

n

i1εlowast (1)2 Mi( 1113857N2 Mi( 1113857+⎡⎣⎛⎝

Qapp12 1113944

n

i1εlowast (1)2 Mi( 1113857N1 Mi( 1113857 + εlowast (1)

1 Mi( 1113857N2 Mi( 11138571113872 1113873 + Qapp66 1113944

n

i1εlowast (1)6 Mi( 1113857N6 Mi( 1113857⎤⎦

2⎞⎠

(A15)

where S is the specimen area and n is the number ofrectangular elements by discretizing the specimen geometry

Derived from the abovementioned equations are the fol-lowing total six different types of Si i 1 2 6

S1 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E N1 Mi( 1113857Nl Mj1113872 11138731113960 1113961 ine jampk l 1 2 6

S2 1113944n

i1ε2k Mi( 1113857E N

2k Mi( 11138571113960 1113961 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857E N

2k Mi( 11138571113960 1113961 k l 1 2ampkne l

S4 1113944n

i11113944

n

j1εk Mi( 1113857εk Mj1113872 1113873E Nk Mi( 1113857Nk Mj1113872 11138731113960 1113961 ine jampk l 1 2

S5 1113944n

i11113944

n

j1εk Mi( 1113857εl Mj1113872 1113873E Np Mi( 1113857Nq Mj1113872 11138731113960 1113961 k l p q 1 2amppne q

S6 1113944n

i11113944

n

j1εk Mi( 1113857ε6 Mj1113872 1113873E N1 Mi( 1113857N6 Mj1113872 11138731113960 1113961 k l 1 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A16)

Due to the autocorrelation of functions Ni i 1 2 6the mathematical expectation in the abovementionedequation is equal to 0 or 1 then

S1 S4 S5 S6 0

S2 1113944

n

i1ε2k Mi( 1113857 k 1 2 6

S3 1113944n

i1εk Mi( 1113857εl Mi( 1113857 k l 1 2ampkne l

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A17)

Hence V(Q11) can be rewritten as follows

V Q11( 1113857 c2 S

n1113874 1113875

2Q

app11( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113944

n

i1εlowast (1)1 Mi( 11138571113872 1113873

2+⎡⎣

Qapp22( 1113857

2+ Q

app12( 1113857

21113872 1113873 1113936

n

i1εlowast (1)2 Mi( 11138571113872 1113873

2+ 2 Q

app11 + Q

app22( 1113857Q

app12 1113944

n

i1εlowast (1)1 Mi( 1113857εlowast (1)

2 Mi( 1113857

+ Qapp66( 1113857

21113944

n

i1εlowast (1)6 Mi( 11138571113872 1113873

2⎤⎦

(A18)

Advances in Materials Science and Engineering 23

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 24: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

Similar results are obtained for Q22 Q12 and Q66Denoting V(Q) the vector containing the variances ofQ11Q22Q12 and Q66 one can write

V Q11( 1113857 c2 S

n1113874 1113875

2QappG(1)Qapp

V Q22( 1113857 c2 S

n1113874 1113875

2QappG(2)Qapp

V Q12( 1113857 c2 S

n1113874 1113875

2QappG(3)Qapp

V Q66( 1113857 c2 S

n1113874 1113875

2QappG(4)Qapp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A19)

where G(j) j 1 2 3 4 is the following square matrix

1113936n

i1εlowast (j)1 Mi( 11138571113872 1113873

20 1113936

n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

0 1113936n

i1εlowast (j)2 Mi( 11138571113872 1113873

21113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 0

1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113936n

i1εlowast (j)1 Mi( 1113857εlowast (j)

2 Mi( 1113857 1113944

n

i1εlowast (j)1 Mi( 11138571113872 1113873

2+ 1113944

n

i1εlowast (j)2 Mi( 11138571113872 1113873

20

0 0 0 1113936n

i1εlowast (j)6 Mi( 11138571113872 1113873

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A20)

+ese variances are proportional to c2 expressing

η(i)1113872 1113873

2

S

n1113874 1113875

2QappG(i)Qapp

(A21)

Equation (A19) can be rewritten as

V Q11( 1113857 η(1)1113872 1113873

2c2

V Q22( 1113857 η(2)1113872 1113873

2c2

V Q12( 1113857 η(3)1113872 1113873

2c2

V Q66( 1113857 η(4)1113872 1113873

2c2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A22)

A5 lte Linear System of KA Virtual Fields +e KA con-dition can produce several linear equations and the numberof these equations is depending on the number of supports+e special virtual field condition also leads to several linearequations the number of which depends on the type ofconstitutive model of specimen material +e two types ofconditions can produce the following linear system

AYlowast(i) Z(i)

(A23)

Using the Lagrange multiplier approach the Lagrangianfunction L(i) can be constructed for each constitutive pa-rameter sought and its constraint is equation (23) and theobjective function is (η(i))2 +erefore the expression of theLagrange function is

24 Advances in Materials Science and Engineering

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 25: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

L(i)

12Ylowast(i)HYlowast(i)

+ λ(i) AYlowast(i)minus Z(i)

1113872 1113873 (A24)

where λ(i) is the vector containing Lagrange multipliersEquation (A23) can be re-expressed as the following linearrelationship

H A

A 01113890 1113891

Y(i)

λ(i)⎛⎝ ⎞⎠

0

Ζ(i)1113888 1113889 (A25)

B Equation Derivation of Section 22

B1 lte Governing Equation of the Virtual Fields Method forthe Total Specimen For the plane-stress specimen with twodifferent materials A and B and both of them are set with theelastic orthotropic material But the constitutive parametersof A and B are unknown +e governing equation of thevirtual fields method for the total specimen is as follows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS+

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS 1113946

Lf

Tiulowasti( 1113857dl

(B1)

As mentioned above the sum of exact values and whitenoise is the measured strain as equation (B1) can bereexpressed as follows

Q11a1113946Sa

ε1 minus cN1( 1113857εlowast1 dS + Q22a1113946Sa

ε2 minus cN2( 1113857εlowast2 dS + Q12a1113946Sa

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS

+Q66a1113946Sa

ε6 minus cN6( 1113857εlowast6 dS + Q11b1113946Sb

ε1 minus cN1( 1113857εlowast1 dS + Q22b1113946Sb

ε2 minus cN2( 1113857dS+

Q12b1113946Sb

ε1 minus cN1( 1113857εlowast2 + ε2 minus cN2( 1113857εlowast11113858 1113859dS + Q66b1113946Sb

ε6 minus cN6( 1113857εlowast6 dS 1113946Lf

Tiulowasti( 1113857dl

(B2)

Assume that the noise components are uncorrelated toeach other and presume that the noise between points is also

uncorrelated +erefore equation (B2) can be rewritten asfollows

Q11a1113946Saε1εlowast1 dS + Q22a1113946

Saε2εlowast2 dS + Q12a1113946

Saε1 _εlowast2 + ε2ε

lowast1( 1113857dS + Q66a1113946

Saε6εlowast6 dS +

Q11b1113946Sbε1εlowast1 dS + Q22b1113946

Sbε2εlowast2 dS + Q12b1113946

Sbε1εlowast2 + ε2ε

lowast1( 1113857dS + Q66b1113946

Sbε6εlowast6 dS minus

c Q11a1113946Saεlowast1 N1dS + Q22a1113946

Saεlowast2 N2dS + Q12a1113946

Saεlowast2 N1 + εlowast1 N2( 1113857dS + Q66a1113946

Saεlowast6 N6dS1113876

Q11b1113946Sbεlowast1 N1dS + Q22b1113946

Sbεlowast2 N2dS + Q12b1113946

Sbεlowast2 N1 + εlowast1 N2( 1113857dS + Q66b1113946

Sbεlowast6 N6dS1113877 1113946

Lf

Tiulowasti( 1113857dl

(B3)

Advances in Materials Science and Engineering 25

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 26: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

B2 lte Principle of Virtual Work with Ignored NoiseDirectly identify Q11a Q11b Q12a Q12b Q22a Q22b Q66a andQ66b using the abovementioned four special virtual

displacement fields εlowast(1) εlowast(2) εlowast(3) and εlowast(4) For instanceulowast(1) is a special virtual field that identifies Q11a which canbe written as follows

Q11a minus Q22a1113946Saε2εlowast (1)2 dS minus Q12a1113946

Saε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66a1113946

Saε6εlowast (1)6 dS minus

Q11b1113946Sbε1εlowast (1)1 dS minus Q22b1113946

Sbε2εlowast (1)2 dS minus Q12b1113946

Sbε1εlowast (1)2 + ε2ε

lowast (1)11113872 1113873dS minus Q66b1113946

Sbε6εlowast (1)6 dS +

c Q11a1113946Saεlowast (1)1 N1dS + Q22a1113946

Saεlowast (1)2 N2dS + Q12a1113946

Saεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66a1113946Saεlowast (1)6 N6dS1113876

Q11b1113946Sbεlowast (1)1 N1dS + Q22b1113946

Sbεlowast (1)2 N2dS + Q12b1113946

Sbεlowast (1)2 N1 + εlowast (1)

1 N21113872 1113873dS + Q66b1113946Sbεlowast (1)6 N6dS1113877 + 1113946

Lf

Tiulowasti( 1113857dl

(B4)

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is research was funded by the Fundamental ResearchFund for the Central Universities (Grant nos 2018ZY08 and2015ZCQ-GX-02) and the National Natural Science Foun-dation of China (Grant no 11502022)

References

[1] J Xavier S Avril F Pierron and J Morais ldquoNovel experi-mental approach for longitudinal-radial stiffness character-isation of clear wood by a single testrdquo Holzforschung vol 61no 5 pp 573ndash581 2007

[2] J Xavier and F Pierron ldquoMeasuring orthotropic bendingstiffness components of pinus pinaster by the virtual fieldsmethodrdquo lte Journal of Strain Analysis for Engineering De-sign vol 53 no 8 pp 556ndash565 2018

[3] P Wang F Pierron M Rossi P Lava and O T +omsenldquoOptimised experimental characterisation of polymeric foammaterial using DIC and the virtual fields methodrdquo Strainvol 52 no 1 pp 59ndash79 2015

[4] L Zhang S G +akku M R Beotra et al ldquoVerification of avirtual fields method to extract the mechanical properties ofhuman optic nerve head tissues in vivordquo Biomechanics andModeling in Mechanobiology vol 16 no 3 pp 871ndash887 2017

[5] X Dai and H Xie ldquoConstitutive parameter identification of3D printing material based on the virtual fields methodrdquoMeasurement vol 5 pp 38ndash43 2015

[6] M Zhou and H Xie ldquoAn identification method of mechanicalproperties of materials based on the full-field measurementmethod based on the fringe patternrdquo Strain vol 55 no 5Article ID 12326 2019

[7] S Avril M Bonnet A S Bretelle et al ldquoOverview of iden-tification methods of mechanical parameters based on full-

field measurementsrdquo Experimental Mechanics vol 48 no 4pp 381ndash402 2008

[8] E Markiewicz B Langrand and D Nottacuvier ldquoA review ofcharacterisation and parameters identification of materialsconstitutive and damage models from normalised directapproach to most advanced inverse problem resolutionrdquoInternational Journal of Impact Engineering vol 110pp 371ndash381 2017

[9] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields method 1-Principle and definitionrdquo Interna-tional Journal of Solids and Structures vol 39 no 10pp 2691ndash2705 2002

[10] M Grediac E Toussaint and F Pierron ldquoSpecial virtual fieldsfor the direct determination of material parameters with thevirtual fields methodrdquo 2-Application to in-plane propertiesrdquoInternational Journal of Solids and Structures vol 39 no 10pp 2707ndash2730 2002

[11] S Cooreman D Lecompte H Sol J Vantomme andD Debruyne ldquoIdentification of mechanical material behaviorthrough inverse modeling andDICrdquo Experimental Mechanicsvol 48 no 4 pp 421ndash433 2008

[12] L Bruno ldquoMechanical characterization of composite mate-rials by optical techniques a reviewrdquo Optics and Lasers inEngineering vol 104 pp 192ndash203 2017

[13] R Delorme P Diehl I Tabiai L L Lebel and M LevesqueldquoExtracting constitutive mechanical parameters in linearelasticity using the virtual fields method within the ordinarystate-based peridynamic frameworkrdquo Journal of Peridynamicsand Nonlocal Modeling vol 2 no 2 pp 111ndash135 2020

[14] M Yue and S Avril ldquoOn improving the accuracy of non-homogeneous shear modulus identification in incompressibleelasticity using the virtual fields methodrdquo InternationalJournal of Solids and Structures vol 178-179 pp 136ndash1442019

[15] J Fu W Xie J Zhou and L Qi ldquoA method for the si-multaneous identification of anisotropic yield and hardeningconstitutive parameters for sheet metal formingrdquo Interna-tional Journal of Mechanical Sciences vol 181 pp 1ndash43 2020

[16] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method forthe identification of three-dimensional stiffness distributionsand its application to incompressible materialsrdquo Strainvol 53 no 5 Article ID 12229 2017

[17] T T Nguyen J M Huntley I A Ashcroft P D Ruiz andF Pierron ldquoA Fourier-series-based virtual fields method for

26 Advances in Materials Science and Engineering

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27

Page 27: OptimizedPolynomialVirtualFieldsMethodforConstitutive ...downloads.hindawi.com/journals/amse/2020/2974723.pdfadvantage of VFM is that it requires less specimen boundary conditions

the identification of 2-d stiffness and traction distributionsrdquoStrain vol 50 no 5 pp 454ndash468 2014

[18] A Marek F M Davis and F Pierron ldquoSensitivity-basedvirtual fields for the non-linear virtual fields methodrdquoComputational Mechanics vol 60 no 3 pp 409ndash431 2017

[19] N Nigamaa and S J Subramanian ldquoIdentification oforthotropic elastic constants using the eigenfunction virtualfields methodrdquo International Journal of Solids and Structuresvol 51 no 2 pp 295ndash304 2014

[20] W Hao Y Zhang and Y Yuan ldquoEigenfunction virtual fieldsmethod for thermo-mechanical parameters identification ofcomposite materialsrdquo Polymer Testing vol 50 pp 224ndash2342016

[21] K Syedmuhammad E Toussaint M Grediac S Avril andJ Kim ldquoCharacterization of composite plates using the virtualfields method with optimized loading conditionsrdquo CompositeStructures vol 85 no 1 pp 70ndash82 2008

[22] J Kim F Barlat F Pierron and M Lee ldquoDetermination ofanisotropic plastic constitutive parameters using the virtualfields methodrdquo Experimental Mechanics vol 54 no 7pp 1189ndash1204 2014

[23] J Xavier S Avril F Pierron and J Morais ldquoldquoVariation oftransverse and shear stiffness properties of wood in a treerdquoComposites Part Ardquo Applied Science and Manufacturingvol 40 no 12 pp 1953ndash1960 2009

[24] E Toussaint G Michel and F Pierron ldquo+e virtual fieldsmethod with piecewise virtual fieldsrdquordquo International Journal ofMechanical Sciences vol 48 no 3 pp 256ndash264 2006

[25] Q Cao Y Li and H Xie ldquoOrientation-Identified virtual fieldsmethod combined with moire interferometry for mechanicalcharacterization of single crystal Ni-based superalloysrdquoOpticsand Lasers in Engineering vol 125 no 123 pp 1ndash9 2020

[26] K Syed-Muhammad E Toussaint and M Grediac ldquoOpti-mization of a mechanical test on composite plates with thevirtual fields methodrdquo Structural and Multidisciplinary Op-timization vol 38 no 1 pp 71ndash82 2009

[27] M Grediac and F Pierron ldquoNumerical issues in the virtualfields methodrdquo International Journal for Numerical Methodsin Engineering vol 59 no 10 pp 1287ndash1312 2004

[28] M Rossi and F Pierron ldquoOn the use of simulated experimentsin designing tests for material characterization from full-fieldmeasurementsrdquo International Journal of Solids and Structuresvol 49 no 3 pp 420ndash435 2012

[29] X Gu and F Pierron ldquoTowards the design of a new standardfor composite stiffness identificationrdquo Composites Part AApplied Science and Manufacturing vol 91 pp 448ndash4602016

[30] J Fu W Xie and L Qi ldquoAn identification method for an-isotropic plastic constitutive parameters of sheet metalsrdquoProcedia Manufacturing vol 47 pp 812ndash815 2020

[31] F Pierron and M Grediac ldquolte Virtual Fields MethodExtracting ConstitutiveMechanical Parameters from Full-FieldDeformation Measurementsrdquo Springer Verla New York NYUSA 2012

[32] P Zhou and K E Goodson ldquoSubpixel displacement anddeformation gradient measurement using digital imagespeckle correlation (DISC)rdquo Optical Engineering vol 40no 8 pp 1613ndash1620 2001

Advances in Materials Science and Engineering 27


Amse 1st announcment - mf.vu.lt

Amse 1st announcment - mf.vu.lt

Dfid Jo Abbot - VfM and Civil Society

Dfid Jo Abbot - VfM and Civil Society

VFM self-assessment workshop This presentation provides a VFM toolkit with a self-assessment template

VFM self-assessment workshop This presentation provides a VFM toolkit with a self-assessment template

Disraeli Bridges Project VFM Report Final

Disraeli Bridges Project VFM Report Final

Trinhbhmsb60440 Vfm

Trinhbhmsb60440 Vfm

Value For Money (VFM) Statement - Greatwell Homes...Value For Money (VFM) Statement Our commitment to Value For Money (VFM) as one of our key corporate strategies includes being transparent

Value For Money (VFM) Statement - Greatwell Homes...Value For Money (VFM) Statement Our commitment to Value For Money (VFM) as one of our key corporate strategies includes being transparent

Vortex Meter Forbes Marshall VFM 7700 Manual

Vortex Meter Forbes Marshall VFM 7700 Manual

Zeetex VFM Range

Zeetex VFM Range

VFM Value for Money

VFM Value for Money

VIETFUND MANAGEMENT (VFM · 2019. 12. 12. · MONTHLY UPDATE VFM August/ 2013 VIETFUND MANAGEMENT (VFM) HEAD OFFICE: Room 1701²1704, 17th thFloor, Melinh Point Tower, 02 Ngo Duc

VIETFUND MANAGEMENT (VFM · 2019. 12. 12. · MONTHLY UPDATE VFM August/ 2013 VIETFUND MANAGEMENT (VFM) HEAD OFFICE: Room 1701²1704, 17th thFloor, Melinh Point Tower, 02 Ngo Duc

Zeetex NEW TBR RANGE Global 2019 - preview · ZSH10 vfm ZOH10 vfm ZOH11 vfm ZOH12 vfm ZOH13 vfm ZOH14 vfm ZOH15 vfm ZOH16 vfm ZDR10 vfm ZDR11 vfm ZDR12 vfm fm fm fm ZDH10 vfm ZOM10

Zeetex NEW TBR RANGE Global 2019 - preview · ZSH10 vfm ZOH10 vfm ZOH11 vfm ZOH12 vfm ZOH13 vfm ZOH14 vfm ZOH15 vfm ZOH16 vfm ZDR10 vfm ZDR11 vfm ZDR12 vfm fm fm fm ZDH10 vfm ZOM10

Wr Pr ravail - AMSE

Wr Pr ravail - AMSE

TV Logic VFM-056W User Manual

TV Logic VFM-056W User Manual

VIETFUND MANAGEMENT (VFM...MONTHLY UPDATE VFM October 2013 VIETFUND MANAGEMENT (VFM) HEAD OFFICE: Room 1701²1704, 17th Floor, Melinh Point Tower, 02 Ngo Duc Ke Street, District 1,

VIETFUND MANAGEMENT (VFM...MONTHLY UPDATE VFM October 2013 VIETFUND MANAGEMENT (VFM) HEAD OFFICE: Room 1701²1704, 17th Floor, Melinh Point Tower, 02 Ngo Duc Ke Street, District 1,

Value for Money (VFM) Strategy - SHINE Supply · The VFM framework starts with an elaboration of the effectiveness dimension of VFM, which is the most important VFM factor, as it

Value for Money (VFM) Strategy - SHINE Supply · The VFM framework starts with an elaboration of the effectiveness dimension of VFM, which is the most important VFM factor, as it

Agenda 2009 AMSE

Agenda 2009 AMSE

Rodrigo Santoro - VFM#8

Rodrigo Santoro - VFM#8

VFM in assessment and care management

VFM in assessment and care management

Public-Private Partnerships: New VfM Guidance

Public-Private Partnerships: New VfM Guidance

CRIDF VfM Framework: Overviewcridf.net/RC/wp-content/uploads/2018/01/VfM-Framework.pdf · CRIDF VfM GN v1 Page 1 of 13 Part I: Value for Money (VfM) in DFID Programming Goal (of the

CRIDF VfM Framework: Overviewcridf.net/RC/wp-content/uploads/2018/01/VfM-Framework.pdf · CRIDF VfM GN v1 Page 1 of 13 Part I: Value for Money (VfM) in DFID Programming Goal (of the