Research ArticleNew Two-Dimensional Polynomial Failure Criteria forComposite Materials

Shi Yang Zhao and Pu Xue

School of Aeronautics Northwestern Polytechnical University Xirsquoan 710072 China

Correspondence should be addressed to Shi Yang Zhao haiyangrongyangsinacom

Received 6 March 2014 Revised 27 April 2014 Accepted 28 April 2014 Published 28 May 2014

Academic Editor Gonzalo Martınez-Barrera

Copyright copy 2014 S Y Zhao and P XueThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The in-plane damage behavior and material properties of the composite material are very complex At present a large number oftwo-dimensional failure criteria such as Chang-Chang criteria have been proposed to predict the damage process of compositestructures under loading However there is still no good criterion to realize it with both enough accuracy and computationalperformance All these criteria cannot be adjusted by experimental data Therefore any special properties of composite materialcannot be considered by these criteria Here in order to solve the problem that the criteria cannot be adjusted by experiment newtwo-dimensional polynomial failure criteria with four internal parameters for composite laminates are proposed in the paper whichinclude four distinct failure modes fiber tensile failure fiber compressive failure matrix tensile failure and matrix compressivefailure In general the four internal parameters should be determined by experiments One example that identifies parameters ofthe new failure criteria is givenUsing the new criteria can reduce the artificialness of choosing the criteria for the damage simulationof the failure modes in composite laminates

1 Introduction

Fiber-reinforced polymer (FRP) composites are widelyapplied in aerospace marine and many other industries dueto their lightweight high stiffness strength and dampingproperties [1] In order to get strong and reliable structuresit is very important to study the mechanical behavior ofFRP In the studies the finite element method is proved tobe effective and successful which can predict the behaviorof composite structures under various loading conditionsFrom the literatures many researches can be found onthe behavior of composites such as buckling loads modalcharacteristics damage and failure Zhang and Yang [2]gave comprehensive reviews and some of the future researchon composite laminated plates Orifici et al [3] also gavea critical review to assess the state of the art in materialconstitutive modeling and composite failure theories Theysummarized the various theories and approaches within thecontext of the dissipated energy framework The dissipatedenergy function with units of energy per unit volume could

be determined from experimental testing and was postulatedto be a property of the material Based on the continuumdamage mechanics (CDM) Liu and Zheng [4] reviewedthe damage constitutive modeling the failure criteria andthe finite element implementation in the progressive failureanalysis which predicted the stiffness degradation and failurestrengths of composite laminates Specially the methodolo-gies to solve the numerical convergence problems due to theloss of element stiffness in the finite element analysis werediscussed

The failure behavior and damage analysis of the com-posite laminated structures by the finite element methodare paid the most attention by researchers Michopoulosrsquoswork [5] shows that there is a relationship among manycriteria Hinton et al [6ndash11] listed 19 theories of failure criteriaproposed for composite laminates and gave a comparisonof their predictive capabilities They showed similaritiesand differences between the predictions of the 19 theoriesby comparing the initial and final failure envelopes andrepresentative stress-strain curves They also explained the

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2014 Article ID 503483 7 pageshttpdxdoiorg1011552014503483

2 Advances in Materials Science and Engineering

differences between the predictions They did not intendto prove which theory is the best but just clarified theirdifferences between their predictions

Despite a large number of published papers on thedamage and failure behavior of the composite laminatesunder various loadings a very effective criterion to pre-dict the failure behavior of composites has not achievedyet Several widely used phenomenological failure criteriasuch as the maximum stress Hoffman Tsai-Wu and Tsai-Hill failure criteria [8 11ndash14] are proposed to describe thewhole ply failure of composite material structures Otherpopular phenomenological failure criteria such as Hashincriteria Hou criteria Chang-Chang criteria Linde criteriaLaRC04 criteria and Maimı criteria [15ndash21] can describethe damage process of composite structures in detail suchas fiber breakage fiber buckling matrix cracking thereforematrix compression All these criteria cannot be adjusted byexperiments Therefore any special properties of compositematerial cannot be considered by these criteria By study-ing the problems of dynamic bending failure of compositelaminated beams Santiuste et al [22] suggested that Hashincriteria may be suitable for ductile composite material andHou criteria may be suitable for brittle material in unidi-rectional laminated beams Therefore for different materialand different structures just one failure criterion whichcannot be adjusted by experiments could not satisfy variousrequirements

When we use some known criteria in our study directlyit has used a supposition that material properties used inyour own engineering are the same as that in the studywhich proposed the known criteria In fact the failurecriteria should be decided by experiments If one criterionis decided by experiments the criterion can consider specialproperties of any material used in engineering And alsoyou can use the new failure criteria in numerical simulationto get better numerical results Here we try to propose newtwo-dimensional polynomial failure criteria which can beadjusted by experiments

New two-dimensional polynomial failure criteria withfour internal parameters for composite laminates are pro-posed in the paper Four distinct failure modes includingfiber tensile failure fiber compressive failure matrix tensilefailure and matrix compressive failure are considered in theextended criteria Meanwhile when some special values aregiven to the four internal parameters the extended criteriawill become some classical failure criteria such as Chang-Chang criteria Hashin criteria and Hou criteria in twodimensional case Finally the features of the failure criteriaare discussed in detail And one example that chooses propervalues for the parameters in the criteria is given

2 New Two-Dimensional Failure Criteria

The damage of composite laminates under loading includesintraply failure and interply failure [23] Intraply failuremodes are fiber kinking fiber fracture matrix cracking andmatrix-fiber debonding Interply failure modes are interfacecracks between plies

New two-dimensional polynomial failure criteria forcomposite laminates are with four internal parameters Andeach failure mode has a parameter Therefore four failuremodes all can be adjusted by experiments And almost specialproperties of each failure mode can be considered Newtwo-dimensional polynomial failure criteria are obtained bycomparing common and characteristic parts of the presentfailure criteria such as Chang-Chang criteria Hou CriteriaHashin criteria and Linde criteria

21 Definition of Variables The related variables includestress components 120590

1 1205902 12059012 the tensile damage variable

of fiber 119889119891119905 the compressive damage variable of fiber 119889

119891119888

the tensile damage variable of matrix 119889119898119905 the compressive

damage variable of matrix 119889119898119888 some material properties

longitudinal failure stress strength119883119879 119883119862 transverse failure

stress strength 119884119879 119884119888 in-plane shear failure stress strength

11987812 the subscripts ldquo119879rdquo and ldquo119862rdquo refer to tension and compres-

sion

22 Fiber Tensile Failure Under tension loading fiber is themain portion to carry load In composites with high fibervolume fraction and those whose failure strain of the matrixis higher than that of the reinforcing fiber such as carbon-epoxy composites longitudinal failures start by isolated fiberfractures in weak zones [20] The localized fractures increasethe normal and interfacial shear stresses in adjoining fiberand the local stress concentration promotes matrix crackingand fiber and matrix debonding [24]

Fiber failure criteria in tension are the most simple dam-age form for composite laminates After the accumulationof fracture energy individual fiber will fail until that thewhole ply failure happens And the whole laminates will carryany loads In general the maximum stress strength or strainstrength is used to predict fiber failure

A new fiber failure criterion in tensile is

119889119891119905= 1198961lowast (

1205901

119883119879

)

2

+ (1 minus 1198961) lowast

1205901

119883119879

+ 1205731

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

ge 1 1205901ge 0

(1)

where 1198961is a parameter of the criteria of fiber failure 120573

1

is used to consider the interaction form involving in-planeshear which can take two values 0 or 1 120572 is a parameter forthe shear nonlinear effect which can take two values 0 or 1

23 Fiber Compressive Failure Under fiber compressionloads laminate failure modes are complex and depend onthe material properties The modes may be microbuckingkinking and fiber failure [25 26] The structures may be inthe state of collapse

The criteria for fiber compressive failure are almostthe same as the fiber tensile failure although the failurephenomenon of fiber compressive failure is more complexthan the fiber tensile failure

Advances in Materials Science and Engineering 3

A new fiber failure criterion in compression is

119889119891119888= 1198962lowast (

1205901

119883119888

)

2

+ 120574 lowast (1198962minus 1) lowast

1205901

119883119888

+ 1205732lowast (

12059012

119878

)

2

ge 1 1205901le 0

(2)

where 1198962is a parameter 120573

2is used to consider the interaction

form involving in-plane shear which can take two values 0or 1 120574 is a parameter which can take two values 1 and 0

24 Failure Criteria for Matrix Under transverse loadingthe behavior of the material is more complex than the fiberfailure A nonlinear stress-strain response will be observedif the laminate is loaded in the transverse direction or inthe shear directionThe famous Chang-Chang failure criteriaconsidered the nonlinear shear effects and introduced acoefficient to characterize the nonlinearity The nonlinearconstitutive relation proposed by Richard and Blacklock [27]introduced an exponential degradation form In general thefracture plane is perpendicular to the mid-plane of the plyunder transverse tensile load But the angle of the fractureplane will change if transverse stress increases under thecompressive load Generally for carbon-epoxy and glass-epoxy composites loaded in pure transverse compression thefracture plane is at an angle of 53∘ plusmn 3∘ with respect to thethickness direction [20 28]The LaRC04 criteria consider thealterative influence of the angle However the computationalefficiency decreases if the fracture angle changes frequentlyso Maimı et al assumed that the fracture angle can only takeone of the two discrete values 0∘ or 53∘ [20]

Here two-dimensional matrix failure criteria for com-posite materials are expressed by a quadratic polynomialThey will also not include the variable of the mutativefractural angle

241 Matrix Tensile Failure The criteria for matrix tensilefailure involve an interaction between the tensile normal andin-plane shear stresses

A new matrix failure criterion in tension is

119889119898119905= 1198963(

1205902

119884119879

)

2

+ (1 minus 1198963)

1205902

119884119879

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

1205902ge 0

(3)

where 1198963is a parameter 120572 is used to consider the nonlinear

interaction form involving in-plane shear

242 Matrix Compressive Failure The criteria for matrixtensile failure are complex in Hashin criteria Hou criteriaand Chang-Chang criteria The coefficients of quadraticpolynomials are different Therefore it is advisable to definethe matrix failure criterion in compression with parametersParameters should be chosen by experiments

A new matrix failure criterion in tension is as follows

119889119898119888= 1198964(

1205902

119884119888

)

2

+ (1198964minus 1)

1205902

119884119888

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

1205902le 0

(4)

where 1198964is a parameter 120572 is used to consider the nonlinear

interaction form involving in-plane shearSo far the two-dimensional failure criteria for composite

materials are given

3 Features of the New Criteria

31 Unified Expression of Fiber and Matrix Failure CriteriaThe new failure criteria for primary damage failure behaviorare given above In fact the new criteria can be rewritten as atensor form as follows

119889119894119897= 119896119894lowast (

120590119894

119883119894119897

)

2

+ 120574119894lowast (1 minus 119896) lowast

10038161003816100381610038161003816100381610038161003816

120590119894

119883119894119897

10038161003816100381610038161003816100381610038161003816

+ 120573119894lowast (

1205902

12211986612+ 34120572

1198941205904

12

1198782

12211986612+ 34120572

1198941198784

12

)

2

ge 1

(5)

where 119896119894is a parameter 120573

119894is used to consider the interaction

form involving in-plane shear 120574119894is a parameter which can

take two values 1 and 0 is the sign of the absolute value 119894is used to denote the damage category fiber or matrix 119897 is forthe stress direction tensile or compression

32 Some Special Forms of the New Criteria In fact Chang-Chang criteria Hou criteria and so forth are proposed fromsome special materials such as carbonepoxy and graphiteepoxy Therefore the validity of parameters in these criteriais very limited All these known criteria are just good atsimulating some special material If more accurate numericalresults are needed the failure criteria of everymaterial shouldconsider their own material properties and be given one setof special values of parameters And all these parametersare from experiments When some special values got byexperiments are given to four parameters of the new criteriait will get good numerical results Meanwhile when somespecial values are given to the four internal parameters thenew criteria will become these classical failure criteria suchas Chang-Chang criteria Linde criteria Hashin criteria andHou criteria in two dimensional case

When 1198961= 1 120573

1= 1 120572 = 0 119896

2= 1 120573

2= 0 120574 = 0 119896

3= 1

1198964= (119884119888211987823)2 the new criteria will becomeHashin criteria

in two-dimensional case as follows

119889119891119905= (

1205901

119883119879

)

2

+ (

12059012

119878

)

2

ge 1 1205901ge 0

119889119891119888= (

1205901

119883119888

)

2

ge 1 1205901le 0

4 Advances in Materials Science and Engineering

119889119898119905= (

12059022

119884119879

)

2

+ (

12059012

11987812

)

2

12059022ge 0

119889119898119888= (

119884119888

211987823

)

2

(

12059022

119884119888

)

2

+ [(

119884119888

211987823

)

2

minus 1]

12059022

119884119888

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(6)

When 1198961= 1 120573

1= 1 120572 = 0 119896

2= 1 120573

2= 1 120574 = 0 119896

3= 1

1198964= (119884119888211987812)2 the new criteria will becomeHou criteria in

two dimensional case as follows

119889119891119905= 1 lowast (

1205901

119883119879

)

2

+ (1 minus 1) lowast 1205901+ (

12059012

119878

)

2

ge 1 1205901ge 0

119889119891119888= (

1205901

119883119879

)

2

+ (

12059012

119878

)

2

ge 1 1205901le 0

119889119898119905= (

12059022

119884119879

)

2

+ (

12059012

11987812

)

2

+ (

12059023

11987823

)

2

12059022ge 0

119889119898119888=

1

119884119888

[(

119884119888

211987812

)

2

minus 1]12059022+

1205902

22

41198782

12

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(7)

When 1198961= 119883119879119883119888 1205731= 0 120572 = 0 119896

2= 119883119888119883119879 1205732= 0

120574 = 1 1198963= 119884119879119884119888 1198964= 119884119888119884119879 the new criteria will become

Linde criteria in two dimensional case as follows

119889119891= (

1

119883119879

minus

1

119883119888

)1205901+

1

119883119879119883119888

1205902

1

119889119898= (

1

119884119879

minus

1

119884119888

)1205902+

1

119884119879119884119888

1205902

2+ (

12059012

11987812

)

2

(8)

When 1198961= 1 120573

1= 1 120572 = 1 119896

2= 1 120573

2= 0 120574 = 0 119896

3= 1

1198964= 1 the new criteria will become Chang-Chang criteria as

follows

119889119891119905= (

1205901

119883119879

)

2

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

ge 1 1205901ge 0

119889119891119888= (

1205901

119883119888

)

2

ge 1 1205901le 0

119889119898119905= (

12059022

119884119879

)

2

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

12059022ge 0

119889119898119888= (

12059022

119884119888

)

2

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

12059022le 0

(9)

Table 1 Failure strengths of T300914C

119883119879Mpa 119883

119862Mpa 119884

119879Mpa 119884

119862Mpa 119878

12Mpa

1500 900 27 200 80

Table 2 Failure strengths of E-glassLy556HT907DY063 epoxy

119883119879Mpa 119883

119862Mpa 119884

119879Mpa 119884

119862Mpa 119878

12Mpa 119866

119868Jm2

1500 900 27 200 80 165

When 1198963= 119866119868119888119866119868119868119888 120572 = 0 (3) will be one part of the

failure criteria LaR04 shown in (10)119866119868119888 119866119868119868119888

are the fracturetoughness of Mode I and Mode II respectively

119889119898119905=

119866119868119888

119866119868119868119888

(

1205902

119884119879

)

2

+ (1 minus

119866119868119888

119866119868119868119888

)

1205902

119884119905

+ (

12059012

11987812

)

2

1205902ge 0

(10)

From (5)ndash(9) we find that the new criteria with fourparameters have enough capability to match the failurecriteria which are used widely in the damage analysis ofcomposite materials

The parameters of the new failure criterion can be definedby experimental data to get better numerical results

4 Identifying Parameters ofthe New Failure Criteria

The process of identifying parameters of the new failurecriteria by experiments is shown here

Biaxial tests of E-glass and carbon fiber reinforced com-posite laminates were performed by Soden et al [29] Inbiaxial tests just 120590

1 1205902 and 120590

12can be obtained There-

fore just parameters 1198961 1198962 1198963 and 119896

4can be identified by

experimental data The other parameters 1205731 1205732 120572 120574 in the

new criteria are not the main items In fact 120574 is used toconsider the influence of linear item 120590

1 to the fiber tensile

damage 1205731is used to consider the influence of shear stress

12059012 to the fiber tensile damage 120573

2is used to consider the

influence of shear stress12059012 to the fiber compressive damage

120572 is used to consider the influence of nonlinear factor ofmaterial The corresponding real values of the parameters of1205731 1205732 120572 120574 in Chang-Chang criteria Linde criteria Hashin

criteria and Hou criteria in two dimensional case are almostthe same Therefore based on the four failure criteria abovethe parameters 120573

1= 1 120573

2= 1 120572 = 0 120574 = 1 are assumed

In order to reduce the complexity of identifying parameters1198961 1198962 1198963 1198964 we just choose the corresponding coefficients

of those parameters from Hashin Hou Linde and Chang-Chang criteria such as 119896

1= 1 119883

119879119883119862 1198962= 1 (119883

119888119883119879)2

and 119883119888119883119879 1198963= 1 119884

119879119884119888and 119866

119868119888119866119868119868119888 1198964= (119884119888211987812)2 1 or

119884119888119884119879

41 Experiments E-glass and carbon fiber reinforced com-posite laminates were used in biaxial tests [29]Theirmaterialproperties are listed in Tables 1 and 2 respectively

Biaxial failure for unidirectional T300914C carbonep-oxy lamina is expressed under combined longitudinal and

Advances in Materials Science and Engineering 5

0

20

40

60

80

100

120

140

0 500 1000 1500 2000

1205901 (Mpa)

12059012(M

pa)

Experimentk1 = 1

k1 = XTXC

Figure 1 Fiber tensile failure envelope

shear loading (1205901versus 120590

12)The specimens were in the form

of axially wound tubes made from prepreg T300BSL914Ccarbonepoxy The tubes were tested under combined axialtension or compression and torsion All the tubes were endreinforced and grips were used to transmit the torque to thetubes [29]

Biaxial failure for unidirectional E-glassLy556HT907DY063 epoxy lamina is expressed under combined trans-verse and shear loading (120590

2versus 120590

12) The tubes with

filament wound by circumferentially wound were 60mmin internal diameter and 2mm in thickness and wereconstructed from 62 by volume Vetrotes 21lowastK43 E-glassfiber (Gevetex) rovings and a Ciba-Geigy epoxy resin sys-tem Ly556HT907DY063 mixed in weight proportions of100 85 4 The tubes were cured at 100∘C for 2 h andpostcurved at 150∘C for 2 h [29]

42 Determination of Parameters The parameters 1205731= 1

1205732= 1 120572 = 0 120574 = 1 are assumed which are based

on Chang-Chang criteria Linde criteria Hashin criteria andHou criteria in two dimensional case And (1)ndash(4) of the newcriteria should be rewritten as follows

421 For Fiber Tensile Failure Consider

119889119891119905= 1198961lowast (

1205901

119883119879

)

2

+ (1 minus 1198961) lowast

1205901

119883119879

+ (

12059012

11987812

) ge 1 1205901ge 0

(11)

where 1198961is a parameter which can take two values 1 and

119883119879119883119862The curves of (11) are shown in Figure 1The curve of

1198961= 1 and the curve of 119896

1= 119883119879119883119862are both lying between

the experimental data which are both effective Howeverthe curve of 119896

1= 119883119879119883119862is close to a large number of

experimental data above in Figure 1 Therefore the curveof 1198961= 119883119879119883119862is used for failure analysis of T300914C

carbonepoxy lamina

422 For Fiber Compressive Failure Consider

119889119891119888= 1198962lowast (

1205901

119883119888

)

2

+ (1198962minus 1) lowast

1205901

119883119888

+ (

12059012

11987812

) ge 1 1205901le 0

(12)

0

20

40

60

80

100

120

minus1000 minus800 minus600 minus400 minus200 0

1205901 (Mpa)

12059012(M

pa)

k2 = 1

Experimentk2 = (XCXT)

2

k2 = XCXT

Figure 2 Fiber compressive failure envelope

0

10

20

30

40

50

60

70

80

0 10 20 30 40

1205902 (Mpa)

12059012(M

pa)

k3 = 1

k3 = YTYC

k3 = GIGII

Experiment

Figure 3 Matrix tensile failure envelope

where 1198962is a parameter which can take three values 1

(119883119888119883119879)2 and 119883

119888119883119879 The curves of (12) are shown in

Figure 2 The curve of 1198962= 1 is outmost among all curves

in Figure 2 and the experimental data is almost out ofthe envelope of all curves The difference of the curve of1198962= 1 with the data is minimal Therefore the value

1198962= 1 should be used for failure analysis of T300914C

carbonepoxy lamina

423 For Matrix Tensile Failure Consider

119889119898119905= 1198963lowast (

12059022

119884119879

)

2

+ (1 minus 1198963) lowast 12059022+ (

12059012

11987812

)

2

12059022ge 0

(13)

where 1198963is a parameter which can take three values 1 119884

119879119884119888

and 119866119868119888119866119868119868119888 The curves of (13) are shown in Figure 3 The

number of the experimental data is just fiveThe curve of 1198963=

119884119879119884119862is nethermost Three data are just in the curve of 119896

3=

119884119879119884119862 and the difference of the curve 119896

3= 119884119879119884119862isminimal

Therefore the value 1198963= 119884119879119884119862should be used for failure

analysis of E-glassLy556HT907DY063 epoxy lamina

6 Advances in Materials Science and Engineering

110

90

70

50

30

10

minus10minus50minus100minus150

1205902 (Mpa)

12059012(M

pa)

k4 = 1

Experimentk4 = YCYT

k4 = (YCS122)2

Figure 4 Matrix compressive failure envelope

424 For Matrix Compressive Failure Consider

119889119898119888= 1198964(

12059022

119884119888

)

2

+ (1198964minus 1)

12059022

119884119888

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(14)

where 1198964is a parameter which can take three values

(119884119888211987812)2 1 or 119884

119888119884119879 The curves of (14) are shown in

Figure 4 Because of the difference between the strengthobtained through experiments and that used in materialproperty three curves are all not perfect From Figure 4 thecurve of 119896

4= 119884119888119884119879is close to experimental data Therefore

the value of 1198964= 119884119888119884119879should be used in failure analysis of

E-glassLy556HT907DY063 epoxy laminaFrom the discussion above it is found that it is not

enough for the damage criterion that has only one set ofthe parameters The failure criteria should be adjusted byexperimental data for the materials to meet higher accuraterequirement

5 Conclusions

In this work new two-dimensional failure criteria with fourparameters for composite materials are given The new crite-ria include fiber failure criteria in tension and compressionand matrix failure criteria in tension and compression

The new criteria can be adjusted with four parametersto consider different material properties in engineering Ingeneral when we use some known criteria in our study ithas to assume that our material properties are the same asthat in the study which proposed the known criteria Thoseproblems above can be solved by the new criteria proposed inthis paper Meanwhile when some special values are given tothe four internal parameters the new criteria have the sameexpression as some classical failure criteria such as Chang-Chang criteria Hashin criteria Linde criteria and Houcriteria in two dimensional case which are used frequently inthe research of damage process for the composite structures

In general the four internal parameters should be chosenby experiments One example that chooses proper values forthe parameters is given which also validates that there is noone and only set of value for the parameters in any criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is financially supported by the National NaturalScience Foundation of China under Grant 11072202

References

[1] S Agrawal KalyanKumar Singh andPK Sarkar ldquoImpact dam-age on fibre-reinforced polymer matrix compositemdasha reviewrdquoJournal of Composite Materials vol 48 no 3 pp 317ndash332 2014

[2] Y X Zhang and C H Yang ldquoRecent developments in finiteelement analysis for laminated composite platesrdquo CompositeStructures vol 88 no 1 pp 147ndash157 2009

[3] A C Orifici I Herszberg and R S Thomson ldquoReview ofmethodologies for composite material modelling incorporatingfailurerdquoComposite Structures vol 86 no 1ndash3 pp 194ndash210 2008

[4] P F Liu and J Y Zheng ldquoRecent developments on damagemodeling and finite element analysis for composite laminatesa reviewrdquo Materials and Design vol 31 no 8 pp 3825ndash38342010

[5] J G Michopoulos ldquoOn the reducibility of failure theories forcomposite materialsrdquo Composite Structures vol 86 no 1ndash3 pp165ndash176 2008

[6] M J Hinton and P D Soden ldquoFailure criteria for compositelaminatesrdquoComposites Science and Technology vol 58 no 7 pp1001ndash1010 1998

[7] P D Soden M J Hinton and A S Kaddour ldquoLaminaproperties and lay-up configurations and loanding conditionsof a range fiber reinforced composite laminatesrdquo CompositesScience and Technology vol 58 pp 1137ndash1150 1998

[8] P D Soden M J Hinton and A S Kaddour ldquoA comparisonof the predictive capabilities of current failure theories forcomposite laminatesrdquo Composites Science and Technology vol58 no 7 pp 1225ndash1254 1998

[9] M J Hinton P D Soden and A S Kaddour ldquoEvaluation offailure prediction in composite laminates background to PartB of the exerciserdquo Composites Science and Technology vol 62pp 1481ndash1488 2002

[10] M J Hinton A S Kaddour and P D Soden ldquoA further assess-ment of the predictive capabilities of current failure theories forcomposite laminates comparison with experimental evidencerdquoComposites Science andTechnology vol 64 no 3-4 pp 549ndash5882004

[11] A S Kaddour M J Hinton and P D Soden ldquoA comparisonof the predictive capabilities of current failure theories for com-posite laminates additional contributionsrdquo Composites Scienceand Technology vol 64 no 3-4 pp 449ndash476 2004

[12] O Hoffman ldquoThe brittle strength of orthotropic materialrdquoJournal of Composite Materials vol 9 no 2 pp 200ndash205 1975

[13] R Narayanaswami and H M Adelman ldquoEvaluation of the ten-sor polynominal and Hoffman strength theories for compositematerialsrdquo Journal of CompositeMaterials vol 11 no 4 pp 366ndash377 1977

[14] S W Tsai and E M Wu ldquoA general theory for anisotropicmaterialsrdquo Journal of Composite Materials vol 5 pp 58ndash801971

Advances in Materials Science and Engineering 7

[15] Z Haszin ldquoFailure criteria for unidirectional fiber compositesrdquoJournal of Applied Mechanics Transactions ASME vol 47 no 2pp 329ndash334 1980

[16] J P Hou N Petrinic C Ruiz and S R Hallett ldquoPredictionof impact damage in composite platesrdquo Composites Science andTechnology vol 60 no 2 pp 273ndash281 2000

[17] F-K Chang and K-Y Chang ldquoA progressive damage modelfor laminated composites containing stress concentrationsrdquoJournal of Composite Materials vol 21 no 9 pp 834ndash855 1987

[18] K-Y Chang S Liu and F-K Chang ldquoDamage tolerance oflaminated composites containing an open hole and subjectedto tensile loadingsrdquo Journal of Composite Materials vol 25 no3 pp 274ndash301 1991

[19] P Maimi J A Mayugo and P P Camanho ldquoA three-dimensional damage model for transversely isotropic compos-ite laminatesrdquo Journal of CompositeMaterials vol 42 no 25 pp2717ndash2745 2008

[20] P Maimı P P Camanho J A Mayugo and C G Davila ldquoAcontinuum damage model for composite laminatesmdashpart Imdashconstitutive modelrdquo Mechanics of Materials vol 39 no 10 pp897ndash908 2007

[21] P Linde J Pleitner H D Boer and C Carmore ldquoModellingand simulation of fiber metal laminatesrdquo in Proceedings of theABAQUS Userrsquos Conference pp 421ndash439 Boston Mass USA2004

[22] C Santiuste S Sanchez-Saez and E Barbero ldquoA comparison ofprogressive failure of composite laminated beamsrdquo CompositeStructures vol 92 pp 2406ndash2414 2010

[23] T Anderson Fracture MechanicsmdashFundamentals and Applica-tions 2nd edition 1995

[24] I M Daniel and O Ishai Engineering Mechanics of CompositeMaterials Oxford University 1994

[25] S T Pinho Modelling failure of laminated composites usingphysically based failure models [PhD thesis] Imperial College2005

[26] M l Ribeiro Volnei Tita and D Vandepitte ldquoA new damagemodel for composite laminatesrdquo Composite Structures vol 94pp 635ndash642 2012

[27] R M Richard and J R Blacklock ldquoFinite element analysis ofinelastic structuresrdquo AIAA Journal vol 7 pp 432ndash438 1969

[28] A Puck and H Schurmann ldquoFailure analysis of FRP laminatesby means of physically based phenomenological modelsrdquo Com-posites Science and Technology vol 62 no 12-13 pp 1633ndash16622002

[29] P D Soden M J Hinton and A S Kaddour ldquoBiaxial testresults for strength and deformation of a range of E-glass andcarbon fibre reinforced composite laminates failure exercisebenchmark datardquo Composites Science and Technology vol 62no 12-13 pp 1489ndash1514 2002

Submit your manuscripts athttpwwwhindawicom

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Nano

materials

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Journal ofNanomaterials

2 Advances in Materials Science and Engineering

differences between the predictions They did not intendto prove which theory is the best but just clarified theirdifferences between their predictions

Despite a large number of published papers on thedamage and failure behavior of the composite laminatesunder various loadings a very effective criterion to pre-dict the failure behavior of composites has not achievedyet Several widely used phenomenological failure criteriasuch as the maximum stress Hoffman Tsai-Wu and Tsai-Hill failure criteria [8 11ndash14] are proposed to describe thewhole ply failure of composite material structures Otherpopular phenomenological failure criteria such as Hashincriteria Hou criteria Chang-Chang criteria Linde criteriaLaRC04 criteria and Maimı criteria [15ndash21] can describethe damage process of composite structures in detail suchas fiber breakage fiber buckling matrix cracking thereforematrix compression All these criteria cannot be adjusted byexperiments Therefore any special properties of compositematerial cannot be considered by these criteria By study-ing the problems of dynamic bending failure of compositelaminated beams Santiuste et al [22] suggested that Hashincriteria may be suitable for ductile composite material andHou criteria may be suitable for brittle material in unidi-rectional laminated beams Therefore for different materialand different structures just one failure criterion whichcannot be adjusted by experiments could not satisfy variousrequirements

When we use some known criteria in our study directlyit has used a supposition that material properties used inyour own engineering are the same as that in the studywhich proposed the known criteria In fact the failurecriteria should be decided by experiments If one criterionis decided by experiments the criterion can consider specialproperties of any material used in engineering And alsoyou can use the new failure criteria in numerical simulationto get better numerical results Here we try to propose newtwo-dimensional polynomial failure criteria which can beadjusted by experiments

New two-dimensional polynomial failure criteria withfour internal parameters for composite laminates are pro-posed in the paper Four distinct failure modes includingfiber tensile failure fiber compressive failure matrix tensilefailure and matrix compressive failure are considered in theextended criteria Meanwhile when some special values aregiven to the four internal parameters the extended criteriawill become some classical failure criteria such as Chang-Chang criteria Hashin criteria and Hou criteria in twodimensional case Finally the features of the failure criteriaare discussed in detail And one example that chooses propervalues for the parameters in the criteria is given

2 New Two-Dimensional Failure Criteria

The damage of composite laminates under loading includesintraply failure and interply failure [23] Intraply failuremodes are fiber kinking fiber fracture matrix cracking andmatrix-fiber debonding Interply failure modes are interfacecracks between plies

New two-dimensional polynomial failure criteria forcomposite laminates are with four internal parameters Andeach failure mode has a parameter Therefore four failuremodes all can be adjusted by experiments And almost specialproperties of each failure mode can be considered Newtwo-dimensional polynomial failure criteria are obtained bycomparing common and characteristic parts of the presentfailure criteria such as Chang-Chang criteria Hou CriteriaHashin criteria and Linde criteria

21 Definition of Variables The related variables includestress components 120590

1 1205902 12059012 the tensile damage variable

of fiber 119889119891119905 the compressive damage variable of fiber 119889

119891119888

the tensile damage variable of matrix 119889119898119905 the compressive

damage variable of matrix 119889119898119888 some material properties

longitudinal failure stress strength119883119879 119883119862 transverse failure

stress strength 119884119879 119884119888 in-plane shear failure stress strength

11987812 the subscripts ldquo119879rdquo and ldquo119862rdquo refer to tension and compres-

sion

22 Fiber Tensile Failure Under tension loading fiber is themain portion to carry load In composites with high fibervolume fraction and those whose failure strain of the matrixis higher than that of the reinforcing fiber such as carbon-epoxy composites longitudinal failures start by isolated fiberfractures in weak zones [20] The localized fractures increasethe normal and interfacial shear stresses in adjoining fiberand the local stress concentration promotes matrix crackingand fiber and matrix debonding [24]

Fiber failure criteria in tension are the most simple dam-age form for composite laminates After the accumulationof fracture energy individual fiber will fail until that thewhole ply failure happens And the whole laminates will carryany loads In general the maximum stress strength or strainstrength is used to predict fiber failure

A new fiber failure criterion in tensile is

119889119891119905= 1198961lowast (

1205901

119883119879

)

2

+ (1 minus 1198961) lowast

1205901

119883119879

+ 1205731

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

ge 1 1205901ge 0

(1)

where 1198961is a parameter of the criteria of fiber failure 120573

1

is used to consider the interaction form involving in-planeshear which can take two values 0 or 1 120572 is a parameter forthe shear nonlinear effect which can take two values 0 or 1

23 Fiber Compressive Failure Under fiber compressionloads laminate failure modes are complex and depend onthe material properties The modes may be microbuckingkinking and fiber failure [25 26] The structures may be inthe state of collapse

The criteria for fiber compressive failure are almostthe same as the fiber tensile failure although the failurephenomenon of fiber compressive failure is more complexthan the fiber tensile failure

Advances in Materials Science and Engineering 3

A new fiber failure criterion in compression is

119889119891119888= 1198962lowast (

1205901

119883119888

)

2

+ 120574 lowast (1198962minus 1) lowast

1205901

119883119888

+ 1205732lowast (

12059012

119878

)

2

ge 1 1205901le 0

(2)

where 1198962is a parameter 120573

2is used to consider the interaction

form involving in-plane shear which can take two values 0or 1 120574 is a parameter which can take two values 1 and 0

24 Failure Criteria for Matrix Under transverse loadingthe behavior of the material is more complex than the fiberfailure A nonlinear stress-strain response will be observedif the laminate is loaded in the transverse direction or inthe shear directionThe famous Chang-Chang failure criteriaconsidered the nonlinear shear effects and introduced acoefficient to characterize the nonlinearity The nonlinearconstitutive relation proposed by Richard and Blacklock [27]introduced an exponential degradation form In general thefracture plane is perpendicular to the mid-plane of the plyunder transverse tensile load But the angle of the fractureplane will change if transverse stress increases under thecompressive load Generally for carbon-epoxy and glass-epoxy composites loaded in pure transverse compression thefracture plane is at an angle of 53∘ plusmn 3∘ with respect to thethickness direction [20 28]The LaRC04 criteria consider thealterative influence of the angle However the computationalefficiency decreases if the fracture angle changes frequentlyso Maimı et al assumed that the fracture angle can only takeone of the two discrete values 0∘ or 53∘ [20]

Here two-dimensional matrix failure criteria for com-posite materials are expressed by a quadratic polynomialThey will also not include the variable of the mutativefractural angle

241 Matrix Tensile Failure The criteria for matrix tensilefailure involve an interaction between the tensile normal andin-plane shear stresses

A new matrix failure criterion in tension is

119889119898119905= 1198963(

1205902

119884119879

)

2

+ (1 minus 1198963)

1205902

119884119879

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

1205902ge 0

(3)

where 1198963is a parameter 120572 is used to consider the nonlinear

interaction form involving in-plane shear

242 Matrix Compressive Failure The criteria for matrixtensile failure are complex in Hashin criteria Hou criteriaand Chang-Chang criteria The coefficients of quadraticpolynomials are different Therefore it is advisable to definethe matrix failure criterion in compression with parametersParameters should be chosen by experiments

A new matrix failure criterion in tension is as follows

119889119898119888= 1198964(

1205902

119884119888

)

2

+ (1198964minus 1)

1205902

119884119888

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

1205902le 0

(4)

where 1198964is a parameter 120572 is used to consider the nonlinear

interaction form involving in-plane shearSo far the two-dimensional failure criteria for composite

materials are given

3 Features of the New Criteria

31 Unified Expression of Fiber and Matrix Failure CriteriaThe new failure criteria for primary damage failure behaviorare given above In fact the new criteria can be rewritten as atensor form as follows

119889119894119897= 119896119894lowast (

120590119894

119883119894119897

)

2

+ 120574119894lowast (1 minus 119896) lowast

10038161003816100381610038161003816100381610038161003816

120590119894

119883119894119897

10038161003816100381610038161003816100381610038161003816

+ 120573119894lowast (

1205902

12211986612+ 34120572

1198941205904

12

1198782

12211986612+ 34120572

1198941198784

12

)

2

ge 1

(5)

where 119896119894is a parameter 120573

119894is used to consider the interaction

form involving in-plane shear 120574119894is a parameter which can

take two values 1 and 0 is the sign of the absolute value 119894is used to denote the damage category fiber or matrix 119897 is forthe stress direction tensile or compression

32 Some Special Forms of the New Criteria In fact Chang-Chang criteria Hou criteria and so forth are proposed fromsome special materials such as carbonepoxy and graphiteepoxy Therefore the validity of parameters in these criteriais very limited All these known criteria are just good atsimulating some special material If more accurate numericalresults are needed the failure criteria of everymaterial shouldconsider their own material properties and be given one setof special values of parameters And all these parametersare from experiments When some special values got byexperiments are given to four parameters of the new criteriait will get good numerical results Meanwhile when somespecial values are given to the four internal parameters thenew criteria will become these classical failure criteria suchas Chang-Chang criteria Linde criteria Hashin criteria andHou criteria in two dimensional case

When 1198961= 1 120573

1= 1 120572 = 0 119896

2= 1 120573

2= 0 120574 = 0 119896

3= 1

1198964= (119884119888211987823)2 the new criteria will becomeHashin criteria

in two-dimensional case as follows

119889119891119905= (

1205901

119883119879

)

2

+ (

12059012

119878

)

2

ge 1 1205901ge 0

119889119891119888= (

1205901

119883119888

)

2

ge 1 1205901le 0

4 Advances in Materials Science and Engineering

119889119898119905= (

12059022

119884119879

)

2

+ (

12059012

11987812

)

2

12059022ge 0

119889119898119888= (

119884119888

211987823

)

2

(

12059022

119884119888

)

2

+ [(

119884119888

211987823

)

2

minus 1]

12059022

119884119888

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(6)

When 1198961= 1 120573

1= 1 120572 = 0 119896

2= 1 120573

2= 1 120574 = 0 119896

3= 1

1198964= (119884119888211987812)2 the new criteria will becomeHou criteria in

two dimensional case as follows

119889119891119905= 1 lowast (

1205901

119883119879

)

2

+ (1 minus 1) lowast 1205901+ (

12059012

119878

)

2

ge 1 1205901ge 0

119889119891119888= (

1205901

119883119879

)

2

+ (

12059012

119878

)

2

ge 1 1205901le 0

119889119898119905= (

12059022

119884119879

)

2

+ (

12059012

11987812

)

2

+ (

12059023

11987823

)

2

12059022ge 0

119889119898119888=

1

119884119888

[(

119884119888

211987812

)

2

minus 1]12059022+

1205902

22

41198782

12

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(7)

When 1198961= 119883119879119883119888 1205731= 0 120572 = 0 119896

2= 119883119888119883119879 1205732= 0

120574 = 1 1198963= 119884119879119884119888 1198964= 119884119888119884119879 the new criteria will become

Linde criteria in two dimensional case as follows

119889119891= (

1

119883119879

minus

1

119883119888

)1205901+

1

119883119879119883119888

1205902

1

119889119898= (

1

119884119879

minus

1

119884119888

)1205902+

1

119884119879119884119888

1205902

2+ (

12059012

11987812

)

2

(8)

When 1198961= 1 120573

1= 1 120572 = 1 119896

2= 1 120573

2= 0 120574 = 0 119896

3= 1

1198964= 1 the new criteria will become Chang-Chang criteria as

follows

119889119891119905= (

1205901

119883119879

)

2

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

ge 1 1205901ge 0

119889119891119888= (

1205901

119883119888

)

2

ge 1 1205901le 0

119889119898119905= (

12059022

119884119879

)

2

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

12059022ge 0

119889119898119888= (

12059022

119884119888

)

2

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

12059022le 0

(9)

Table 1 Failure strengths of T300914C

119883119879Mpa 119883

119862Mpa 119884

119879Mpa 119884

119862Mpa 119878

12Mpa

1500 900 27 200 80

Table 2 Failure strengths of E-glassLy556HT907DY063 epoxy

119883119879Mpa 119883

119862Mpa 119884

119879Mpa 119884

119862Mpa 119878

12Mpa 119866

119868Jm2

1500 900 27 200 80 165

When 1198963= 119866119868119888119866119868119868119888 120572 = 0 (3) will be one part of the

failure criteria LaR04 shown in (10)119866119868119888 119866119868119868119888

are the fracturetoughness of Mode I and Mode II respectively

119889119898119905=

119866119868119888

119866119868119868119888

(

1205902

119884119879

)

2

+ (1 minus

119866119868119888

119866119868119868119888

)

1205902

119884119905

+ (

12059012

11987812

)

2

1205902ge 0

(10)

From (5)ndash(9) we find that the new criteria with fourparameters have enough capability to match the failurecriteria which are used widely in the damage analysis ofcomposite materials

The parameters of the new failure criterion can be definedby experimental data to get better numerical results

4 Identifying Parameters ofthe New Failure Criteria

The process of identifying parameters of the new failurecriteria by experiments is shown here

Biaxial tests of E-glass and carbon fiber reinforced com-posite laminates were performed by Soden et al [29] Inbiaxial tests just 120590

1 1205902 and 120590

12can be obtained There-

fore just parameters 1198961 1198962 1198963 and 119896

4can be identified by

experimental data The other parameters 1205731 1205732 120572 120574 in the

new criteria are not the main items In fact 120574 is used toconsider the influence of linear item 120590

1 to the fiber tensile

damage 1205731is used to consider the influence of shear stress

12059012 to the fiber tensile damage 120573

2is used to consider the

influence of shear stress12059012 to the fiber compressive damage

120572 is used to consider the influence of nonlinear factor ofmaterial The corresponding real values of the parameters of1205731 1205732 120572 120574 in Chang-Chang criteria Linde criteria Hashin

criteria and Hou criteria in two dimensional case are almostthe same Therefore based on the four failure criteria abovethe parameters 120573

1= 1 120573

2= 1 120572 = 0 120574 = 1 are assumed

In order to reduce the complexity of identifying parameters1198961 1198962 1198963 1198964 we just choose the corresponding coefficients

of those parameters from Hashin Hou Linde and Chang-Chang criteria such as 119896

1= 1 119883

119879119883119862 1198962= 1 (119883

119888119883119879)2

and 119883119888119883119879 1198963= 1 119884

119879119884119888and 119866

119868119888119866119868119868119888 1198964= (119884119888211987812)2 1 or

119884119888119884119879

41 Experiments E-glass and carbon fiber reinforced com-posite laminates were used in biaxial tests [29]Theirmaterialproperties are listed in Tables 1 and 2 respectively

Biaxial failure for unidirectional T300914C carbonep-oxy lamina is expressed under combined longitudinal and

Advances in Materials Science and Engineering 5

0

20

40

60

80

100

120

140

0 500 1000 1500 2000

1205901 (Mpa)

12059012(M

pa)

Experimentk1 = 1

k1 = XTXC

Figure 1 Fiber tensile failure envelope

shear loading (1205901versus 120590

12)The specimens were in the form

of axially wound tubes made from prepreg T300BSL914Ccarbonepoxy The tubes were tested under combined axialtension or compression and torsion All the tubes were endreinforced and grips were used to transmit the torque to thetubes [29]

Biaxial failure for unidirectional E-glassLy556HT907DY063 epoxy lamina is expressed under combined trans-verse and shear loading (120590

2versus 120590

12) The tubes with

filament wound by circumferentially wound were 60mmin internal diameter and 2mm in thickness and wereconstructed from 62 by volume Vetrotes 21lowastK43 E-glassfiber (Gevetex) rovings and a Ciba-Geigy epoxy resin sys-tem Ly556HT907DY063 mixed in weight proportions of100 85 4 The tubes were cured at 100∘C for 2 h andpostcurved at 150∘C for 2 h [29]

42 Determination of Parameters The parameters 1205731= 1

1205732= 1 120572 = 0 120574 = 1 are assumed which are based

on Chang-Chang criteria Linde criteria Hashin criteria andHou criteria in two dimensional case And (1)ndash(4) of the newcriteria should be rewritten as follows

421 For Fiber Tensile Failure Consider

119889119891119905= 1198961lowast (

1205901

119883119879

)

2

+ (1 minus 1198961) lowast

1205901

119883119879

+ (

12059012

11987812

) ge 1 1205901ge 0

(11)

where 1198961is a parameter which can take two values 1 and

119883119879119883119862The curves of (11) are shown in Figure 1The curve of

1198961= 1 and the curve of 119896

1= 119883119879119883119862are both lying between

the experimental data which are both effective Howeverthe curve of 119896

1= 119883119879119883119862is close to a large number of

experimental data above in Figure 1 Therefore the curveof 1198961= 119883119879119883119862is used for failure analysis of T300914C

carbonepoxy lamina

422 For Fiber Compressive Failure Consider

119889119891119888= 1198962lowast (

1205901

119883119888

)

2

+ (1198962minus 1) lowast

1205901

119883119888

+ (

12059012

11987812

) ge 1 1205901le 0

(12)

0

20

40

60

80

100

120

minus1000 minus800 minus600 minus400 minus200 0

1205901 (Mpa)

12059012(M

pa)

k2 = 1

Experimentk2 = (XCXT)

2

k2 = XCXT

Figure 2 Fiber compressive failure envelope

0

10

20

30

40

50

60

70

80

0 10 20 30 40

1205902 (Mpa)

12059012(M

pa)

k3 = 1

k3 = YTYC

k3 = GIGII

Experiment

Figure 3 Matrix tensile failure envelope

where 1198962is a parameter which can take three values 1

(119883119888119883119879)2 and 119883

119888119883119879 The curves of (12) are shown in

Figure 2 The curve of 1198962= 1 is outmost among all curves

in Figure 2 and the experimental data is almost out ofthe envelope of all curves The difference of the curve of1198962= 1 with the data is minimal Therefore the value

1198962= 1 should be used for failure analysis of T300914C

carbonepoxy lamina

423 For Matrix Tensile Failure Consider

119889119898119905= 1198963lowast (

12059022

119884119879

)

2

+ (1 minus 1198963) lowast 12059022+ (

12059012

11987812

)

2

12059022ge 0

(13)

where 1198963is a parameter which can take three values 1 119884

119879119884119888

and 119866119868119888119866119868119868119888 The curves of (13) are shown in Figure 3 The

number of the experimental data is just fiveThe curve of 1198963=

119884119879119884119862is nethermost Three data are just in the curve of 119896

3=

119884119879119884119862 and the difference of the curve 119896

3= 119884119879119884119862isminimal

Therefore the value 1198963= 119884119879119884119862should be used for failure

analysis of E-glassLy556HT907DY063 epoxy lamina

6 Advances in Materials Science and Engineering

110

90

70

50

30

10

minus10minus50minus100minus150

1205902 (Mpa)

12059012(M

pa)

k4 = 1

Experimentk4 = YCYT

k4 = (YCS122)2

Figure 4 Matrix compressive failure envelope

424 For Matrix Compressive Failure Consider

119889119898119888= 1198964(

12059022

119884119888

)

2

+ (1198964minus 1)

12059022

119884119888

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(14)

where 1198964is a parameter which can take three values

(119884119888211987812)2 1 or 119884

119888119884119879 The curves of (14) are shown in

Figure 4 Because of the difference between the strengthobtained through experiments and that used in materialproperty three curves are all not perfect From Figure 4 thecurve of 119896

4= 119884119888119884119879is close to experimental data Therefore

the value of 1198964= 119884119888119884119879should be used in failure analysis of

E-glassLy556HT907DY063 epoxy laminaFrom the discussion above it is found that it is not

enough for the damage criterion that has only one set ofthe parameters The failure criteria should be adjusted byexperimental data for the materials to meet higher accuraterequirement

5 Conclusions

In this work new two-dimensional failure criteria with fourparameters for composite materials are given The new crite-ria include fiber failure criteria in tension and compressionand matrix failure criteria in tension and compression

The new criteria can be adjusted with four parametersto consider different material properties in engineering Ingeneral when we use some known criteria in our study ithas to assume that our material properties are the same asthat in the study which proposed the known criteria Thoseproblems above can be solved by the new criteria proposed inthis paper Meanwhile when some special values are given tothe four internal parameters the new criteria have the sameexpression as some classical failure criteria such as Chang-Chang criteria Hashin criteria Linde criteria and Houcriteria in two dimensional case which are used frequently inthe research of damage process for the composite structures

In general the four internal parameters should be chosenby experiments One example that chooses proper values forthe parameters is given which also validates that there is noone and only set of value for the parameters in any criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is financially supported by the National NaturalScience Foundation of China under Grant 11072202

References

[1] S Agrawal KalyanKumar Singh andPK Sarkar ldquoImpact dam-age on fibre-reinforced polymer matrix compositemdasha reviewrdquoJournal of Composite Materials vol 48 no 3 pp 317ndash332 2014

[2] Y X Zhang and C H Yang ldquoRecent developments in finiteelement analysis for laminated composite platesrdquo CompositeStructures vol 88 no 1 pp 147ndash157 2009

[3] A C Orifici I Herszberg and R S Thomson ldquoReview ofmethodologies for composite material modelling incorporatingfailurerdquoComposite Structures vol 86 no 1ndash3 pp 194ndash210 2008

[4] P F Liu and J Y Zheng ldquoRecent developments on damagemodeling and finite element analysis for composite laminatesa reviewrdquo Materials and Design vol 31 no 8 pp 3825ndash38342010

[5] J G Michopoulos ldquoOn the reducibility of failure theories forcomposite materialsrdquo Composite Structures vol 86 no 1ndash3 pp165ndash176 2008

[6] M J Hinton and P D Soden ldquoFailure criteria for compositelaminatesrdquoComposites Science and Technology vol 58 no 7 pp1001ndash1010 1998

[7] P D Soden M J Hinton and A S Kaddour ldquoLaminaproperties and lay-up configurations and loanding conditionsof a range fiber reinforced composite laminatesrdquo CompositesScience and Technology vol 58 pp 1137ndash1150 1998

[8] P D Soden M J Hinton and A S Kaddour ldquoA comparisonof the predictive capabilities of current failure theories forcomposite laminatesrdquo Composites Science and Technology vol58 no 7 pp 1225ndash1254 1998

[9] M J Hinton P D Soden and A S Kaddour ldquoEvaluation offailure prediction in composite laminates background to PartB of the exerciserdquo Composites Science and Technology vol 62pp 1481ndash1488 2002

[10] M J Hinton A S Kaddour and P D Soden ldquoA further assess-ment of the predictive capabilities of current failure theories forcomposite laminates comparison with experimental evidencerdquoComposites Science andTechnology vol 64 no 3-4 pp 549ndash5882004

[11] A S Kaddour M J Hinton and P D Soden ldquoA comparisonof the predictive capabilities of current failure theories for com-posite laminates additional contributionsrdquo Composites Scienceand Technology vol 64 no 3-4 pp 449ndash476 2004

[12] O Hoffman ldquoThe brittle strength of orthotropic materialrdquoJournal of Composite Materials vol 9 no 2 pp 200ndash205 1975

[13] R Narayanaswami and H M Adelman ldquoEvaluation of the ten-sor polynominal and Hoffman strength theories for compositematerialsrdquo Journal of CompositeMaterials vol 11 no 4 pp 366ndash377 1977

[14] S W Tsai and E M Wu ldquoA general theory for anisotropicmaterialsrdquo Journal of Composite Materials vol 5 pp 58ndash801971

Advances in Materials Science and Engineering 7

[15] Z Haszin ldquoFailure criteria for unidirectional fiber compositesrdquoJournal of Applied Mechanics Transactions ASME vol 47 no 2pp 329ndash334 1980

[16] J P Hou N Petrinic C Ruiz and S R Hallett ldquoPredictionof impact damage in composite platesrdquo Composites Science andTechnology vol 60 no 2 pp 273ndash281 2000

[17] F-K Chang and K-Y Chang ldquoA progressive damage modelfor laminated composites containing stress concentrationsrdquoJournal of Composite Materials vol 21 no 9 pp 834ndash855 1987

[18] K-Y Chang S Liu and F-K Chang ldquoDamage tolerance oflaminated composites containing an open hole and subjectedto tensile loadingsrdquo Journal of Composite Materials vol 25 no3 pp 274ndash301 1991

[19] P Maimi J A Mayugo and P P Camanho ldquoA three-dimensional damage model for transversely isotropic compos-ite laminatesrdquo Journal of CompositeMaterials vol 42 no 25 pp2717ndash2745 2008

[20] P Maimı P P Camanho J A Mayugo and C G Davila ldquoAcontinuum damage model for composite laminatesmdashpart Imdashconstitutive modelrdquo Mechanics of Materials vol 39 no 10 pp897ndash908 2007

[21] P Linde J Pleitner H D Boer and C Carmore ldquoModellingand simulation of fiber metal laminatesrdquo in Proceedings of theABAQUS Userrsquos Conference pp 421ndash439 Boston Mass USA2004

[22] C Santiuste S Sanchez-Saez and E Barbero ldquoA comparison ofprogressive failure of composite laminated beamsrdquo CompositeStructures vol 92 pp 2406ndash2414 2010

[23] T Anderson Fracture MechanicsmdashFundamentals and Applica-tions 2nd edition 1995

[24] I M Daniel and O Ishai Engineering Mechanics of CompositeMaterials Oxford University 1994

[25] S T Pinho Modelling failure of laminated composites usingphysically based failure models [PhD thesis] Imperial College2005

[26] M l Ribeiro Volnei Tita and D Vandepitte ldquoA new damagemodel for composite laminatesrdquo Composite Structures vol 94pp 635ndash642 2012

[27] R M Richard and J R Blacklock ldquoFinite element analysis ofinelastic structuresrdquo AIAA Journal vol 7 pp 432ndash438 1969

[28] A Puck and H Schurmann ldquoFailure analysis of FRP laminatesby means of physically based phenomenological modelsrdquo Com-posites Science and Technology vol 62 no 12-13 pp 1633ndash16622002

[29] P D Soden M J Hinton and A S Kaddour ldquoBiaxial testresults for strength and deformation of a range of E-glass andcarbon fibre reinforced composite laminates failure exercisebenchmark datardquo Composites Science and Technology vol 62no 12-13 pp 1489ndash1514 2002

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 3

A new fiber failure criterion in compression is

119889119891119888= 1198962lowast (

1205901

119883119888

)

2

+ 120574 lowast (1198962minus 1) lowast

1205901

119883119888

+ 1205732lowast (

12059012

119878

)

2

ge 1 1205901le 0

(2)

where 1198962is a parameter 120573

2is used to consider the interaction

form involving in-plane shear which can take two values 0or 1 120574 is a parameter which can take two values 1 and 0

24 Failure Criteria for Matrix Under transverse loadingthe behavior of the material is more complex than the fiberfailure A nonlinear stress-strain response will be observedif the laminate is loaded in the transverse direction or inthe shear directionThe famous Chang-Chang failure criteriaconsidered the nonlinear shear effects and introduced acoefficient to characterize the nonlinearity The nonlinearconstitutive relation proposed by Richard and Blacklock [27]introduced an exponential degradation form In general thefracture plane is perpendicular to the mid-plane of the plyunder transverse tensile load But the angle of the fractureplane will change if transverse stress increases under thecompressive load Generally for carbon-epoxy and glass-epoxy composites loaded in pure transverse compression thefracture plane is at an angle of 53∘ plusmn 3∘ with respect to thethickness direction [20 28]The LaRC04 criteria consider thealterative influence of the angle However the computationalefficiency decreases if the fracture angle changes frequentlyso Maimı et al assumed that the fracture angle can only takeone of the two discrete values 0∘ or 53∘ [20]

Here two-dimensional matrix failure criteria for com-posite materials are expressed by a quadratic polynomialThey will also not include the variable of the mutativefractural angle

241 Matrix Tensile Failure The criteria for matrix tensilefailure involve an interaction between the tensile normal andin-plane shear stresses

A new matrix failure criterion in tension is

119889119898119905= 1198963(

1205902

119884119879

)

2

+ (1 minus 1198963)

1205902

119884119879

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

1205902ge 0

(3)

where 1198963is a parameter 120572 is used to consider the nonlinear

interaction form involving in-plane shear

242 Matrix Compressive Failure The criteria for matrixtensile failure are complex in Hashin criteria Hou criteriaand Chang-Chang criteria The coefficients of quadraticpolynomials are different Therefore it is advisable to definethe matrix failure criterion in compression with parametersParameters should be chosen by experiments

A new matrix failure criterion in tension is as follows

119889119898119888= 1198964(

1205902

119884119888

)

2

+ (1198964minus 1)

1205902

119884119888

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

1205902le 0

(4)

where 1198964is a parameter 120572 is used to consider the nonlinear

interaction form involving in-plane shearSo far the two-dimensional failure criteria for composite

materials are given

3 Features of the New Criteria

31 Unified Expression of Fiber and Matrix Failure CriteriaThe new failure criteria for primary damage failure behaviorare given above In fact the new criteria can be rewritten as atensor form as follows

119889119894119897= 119896119894lowast (

120590119894

119883119894119897

)

2

+ 120574119894lowast (1 minus 119896) lowast

10038161003816100381610038161003816100381610038161003816

120590119894

119883119894119897

10038161003816100381610038161003816100381610038161003816

+ 120573119894lowast (

1205902

12211986612+ 34120572

1198941205904

12

1198782

12211986612+ 34120572

1198941198784

12

)

2

ge 1

(5)

where 119896119894is a parameter 120573

119894is used to consider the interaction

form involving in-plane shear 120574119894is a parameter which can

take two values 1 and 0 is the sign of the absolute value 119894is used to denote the damage category fiber or matrix 119897 is forthe stress direction tensile or compression

32 Some Special Forms of the New Criteria In fact Chang-Chang criteria Hou criteria and so forth are proposed fromsome special materials such as carbonepoxy and graphiteepoxy Therefore the validity of parameters in these criteriais very limited All these known criteria are just good atsimulating some special material If more accurate numericalresults are needed the failure criteria of everymaterial shouldconsider their own material properties and be given one setof special values of parameters And all these parametersare from experiments When some special values got byexperiments are given to four parameters of the new criteriait will get good numerical results Meanwhile when somespecial values are given to the four internal parameters thenew criteria will become these classical failure criteria suchas Chang-Chang criteria Linde criteria Hashin criteria andHou criteria in two dimensional case

When 1198961= 1 120573

1= 1 120572 = 0 119896

2= 1 120573

2= 0 120574 = 0 119896

3= 1

1198964= (119884119888211987823)2 the new criteria will becomeHashin criteria

in two-dimensional case as follows

119889119891119905= (

1205901

119883119879

)

2

+ (

12059012

119878

)

2

ge 1 1205901ge 0

119889119891119888= (

1205901

119883119888

)

2

ge 1 1205901le 0

4 Advances in Materials Science and Engineering

119889119898119905= (

12059022

119884119879

)

2

+ (

12059012

11987812

)

2

12059022ge 0

119889119898119888= (

119884119888

211987823

)

2

(

12059022

119884119888

)

2

+ [(

119884119888

211987823

)

2

minus 1]

12059022

119884119888

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(6)

When 1198961= 1 120573

1= 1 120572 = 0 119896

2= 1 120573

2= 1 120574 = 0 119896

3= 1

1198964= (119884119888211987812)2 the new criteria will becomeHou criteria in

two dimensional case as follows

119889119891119905= 1 lowast (

1205901

119883119879

)

2

+ (1 minus 1) lowast 1205901+ (

12059012

119878

)

2

ge 1 1205901ge 0

119889119891119888= (

1205901

119883119879

)

2

+ (

12059012

119878

)

2

ge 1 1205901le 0

119889119898119905= (

12059022

119884119879

)

2

+ (

12059012

11987812

)

2

+ (

12059023

11987823

)

2

12059022ge 0

119889119898119888=

1

119884119888

[(

119884119888

211987812

)

2

minus 1]12059022+

1205902

22

41198782

12

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(7)

When 1198961= 119883119879119883119888 1205731= 0 120572 = 0 119896

2= 119883119888119883119879 1205732= 0

120574 = 1 1198963= 119884119879119884119888 1198964= 119884119888119884119879 the new criteria will become

Linde criteria in two dimensional case as follows

119889119891= (

1

119883119879

minus

1

119883119888

)1205901+

1

119883119879119883119888

1205902

1

119889119898= (

1

119884119879

minus

1

119884119888

)1205902+

1

119884119879119884119888

1205902

2+ (

12059012

11987812

)

2

(8)

When 1198961= 1 120573

1= 1 120572 = 1 119896

2= 1 120573

2= 0 120574 = 0 119896

3= 1

1198964= 1 the new criteria will become Chang-Chang criteria as

follows

119889119891119905= (

1205901

119883119879

)

2

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

ge 1 1205901ge 0

119889119891119888= (

1205901

119883119888

)

2

ge 1 1205901le 0

119889119898119905= (

12059022

119884119879

)

2

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

12059022ge 0

119889119898119888= (

12059022

119884119888

)

2

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

12059022le 0

(9)

Table 1 Failure strengths of T300914C

119883119879Mpa 119883

119862Mpa 119884

119879Mpa 119884

119862Mpa 119878

12Mpa

1500 900 27 200 80

Table 2 Failure strengths of E-glassLy556HT907DY063 epoxy

119883119879Mpa 119883

119862Mpa 119884

119879Mpa 119884

119862Mpa 119878

12Mpa 119866

119868Jm2

1500 900 27 200 80 165

When 1198963= 119866119868119888119866119868119868119888 120572 = 0 (3) will be one part of the

failure criteria LaR04 shown in (10)119866119868119888 119866119868119868119888

are the fracturetoughness of Mode I and Mode II respectively

119889119898119905=

119866119868119888

119866119868119868119888

(

1205902

119884119879

)

2

+ (1 minus

119866119868119888

119866119868119868119888

)

1205902

119884119905

+ (

12059012

11987812

)

2

1205902ge 0

(10)

From (5)ndash(9) we find that the new criteria with fourparameters have enough capability to match the failurecriteria which are used widely in the damage analysis ofcomposite materials

The parameters of the new failure criterion can be definedby experimental data to get better numerical results

4 Identifying Parameters ofthe New Failure Criteria

The process of identifying parameters of the new failurecriteria by experiments is shown here

Biaxial tests of E-glass and carbon fiber reinforced com-posite laminates were performed by Soden et al [29] Inbiaxial tests just 120590

1 1205902 and 120590

12can be obtained There-

fore just parameters 1198961 1198962 1198963 and 119896

4can be identified by

experimental data The other parameters 1205731 1205732 120572 120574 in the

new criteria are not the main items In fact 120574 is used toconsider the influence of linear item 120590

1 to the fiber tensile

damage 1205731is used to consider the influence of shear stress

12059012 to the fiber tensile damage 120573

2is used to consider the

influence of shear stress12059012 to the fiber compressive damage

120572 is used to consider the influence of nonlinear factor ofmaterial The corresponding real values of the parameters of1205731 1205732 120572 120574 in Chang-Chang criteria Linde criteria Hashin

criteria and Hou criteria in two dimensional case are almostthe same Therefore based on the four failure criteria abovethe parameters 120573

1= 1 120573

2= 1 120572 = 0 120574 = 1 are assumed

In order to reduce the complexity of identifying parameters1198961 1198962 1198963 1198964 we just choose the corresponding coefficients

of those parameters from Hashin Hou Linde and Chang-Chang criteria such as 119896

1= 1 119883

119879119883119862 1198962= 1 (119883

119888119883119879)2

and 119883119888119883119879 1198963= 1 119884

119879119884119888and 119866

119868119888119866119868119868119888 1198964= (119884119888211987812)2 1 or

119884119888119884119879

41 Experiments E-glass and carbon fiber reinforced com-posite laminates were used in biaxial tests [29]Theirmaterialproperties are listed in Tables 1 and 2 respectively

Biaxial failure for unidirectional T300914C carbonep-oxy lamina is expressed under combined longitudinal and

Advances in Materials Science and Engineering 5

0

20

40

60

80

100

120

140

0 500 1000 1500 2000

1205901 (Mpa)

12059012(M

pa)

Experimentk1 = 1

k1 = XTXC

Figure 1 Fiber tensile failure envelope

shear loading (1205901versus 120590

12)The specimens were in the form

of axially wound tubes made from prepreg T300BSL914Ccarbonepoxy The tubes were tested under combined axialtension or compression and torsion All the tubes were endreinforced and grips were used to transmit the torque to thetubes [29]

Biaxial failure for unidirectional E-glassLy556HT907DY063 epoxy lamina is expressed under combined trans-verse and shear loading (120590

2versus 120590

12) The tubes with

filament wound by circumferentially wound were 60mmin internal diameter and 2mm in thickness and wereconstructed from 62 by volume Vetrotes 21lowastK43 E-glassfiber (Gevetex) rovings and a Ciba-Geigy epoxy resin sys-tem Ly556HT907DY063 mixed in weight proportions of100 85 4 The tubes were cured at 100∘C for 2 h andpostcurved at 150∘C for 2 h [29]

42 Determination of Parameters The parameters 1205731= 1

1205732= 1 120572 = 0 120574 = 1 are assumed which are based

on Chang-Chang criteria Linde criteria Hashin criteria andHou criteria in two dimensional case And (1)ndash(4) of the newcriteria should be rewritten as follows

421 For Fiber Tensile Failure Consider

119889119891119905= 1198961lowast (

1205901

119883119879

)

2

+ (1 minus 1198961) lowast

1205901

119883119879

+ (

12059012

11987812

) ge 1 1205901ge 0

(11)

where 1198961is a parameter which can take two values 1 and

119883119879119883119862The curves of (11) are shown in Figure 1The curve of

1198961= 1 and the curve of 119896

1= 119883119879119883119862are both lying between

the experimental data which are both effective Howeverthe curve of 119896

1= 119883119879119883119862is close to a large number of

experimental data above in Figure 1 Therefore the curveof 1198961= 119883119879119883119862is used for failure analysis of T300914C

carbonepoxy lamina

422 For Fiber Compressive Failure Consider

119889119891119888= 1198962lowast (

1205901

119883119888

)

2

+ (1198962minus 1) lowast

1205901

119883119888

+ (

12059012

11987812

) ge 1 1205901le 0

(12)

0

20

40

60

80

100

120

minus1000 minus800 minus600 minus400 minus200 0

1205901 (Mpa)

12059012(M

pa)

k2 = 1

Experimentk2 = (XCXT)

2

k2 = XCXT

Figure 2 Fiber compressive failure envelope

0

10

20

30

40

50

60

70

80

0 10 20 30 40

1205902 (Mpa)

12059012(M

pa)

k3 = 1

k3 = YTYC

k3 = GIGII

Experiment

Figure 3 Matrix tensile failure envelope

where 1198962is a parameter which can take three values 1

(119883119888119883119879)2 and 119883

119888119883119879 The curves of (12) are shown in

Figure 2 The curve of 1198962= 1 is outmost among all curves

in Figure 2 and the experimental data is almost out ofthe envelope of all curves The difference of the curve of1198962= 1 with the data is minimal Therefore the value

1198962= 1 should be used for failure analysis of T300914C

carbonepoxy lamina

423 For Matrix Tensile Failure Consider

119889119898119905= 1198963lowast (

12059022

119884119879

)

2

+ (1 minus 1198963) lowast 12059022+ (

12059012

11987812

)

2

12059022ge 0

(13)

where 1198963is a parameter which can take three values 1 119884

119879119884119888

and 119866119868119888119866119868119868119888 The curves of (13) are shown in Figure 3 The

number of the experimental data is just fiveThe curve of 1198963=

119884119879119884119862is nethermost Three data are just in the curve of 119896

3=

119884119879119884119862 and the difference of the curve 119896

3= 119884119879119884119862isminimal

Therefore the value 1198963= 119884119879119884119862should be used for failure

analysis of E-glassLy556HT907DY063 epoxy lamina

6 Advances in Materials Science and Engineering

110

90

70

50

30

10

minus10minus50minus100minus150

1205902 (Mpa)

12059012(M

pa)

k4 = 1

Experimentk4 = YCYT

k4 = (YCS122)2

Figure 4 Matrix compressive failure envelope

424 For Matrix Compressive Failure Consider

119889119898119888= 1198964(

12059022

119884119888

)

2

+ (1198964minus 1)

12059022

119884119888

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(14)

where 1198964is a parameter which can take three values

(119884119888211987812)2 1 or 119884

119888119884119879 The curves of (14) are shown in

Figure 4 Because of the difference between the strengthobtained through experiments and that used in materialproperty three curves are all not perfect From Figure 4 thecurve of 119896

4= 119884119888119884119879is close to experimental data Therefore

the value of 1198964= 119884119888119884119879should be used in failure analysis of

E-glassLy556HT907DY063 epoxy laminaFrom the discussion above it is found that it is not

enough for the damage criterion that has only one set ofthe parameters The failure criteria should be adjusted byexperimental data for the materials to meet higher accuraterequirement

5 Conclusions

In this work new two-dimensional failure criteria with fourparameters for composite materials are given The new crite-ria include fiber failure criteria in tension and compressionand matrix failure criteria in tension and compression

The new criteria can be adjusted with four parametersto consider different material properties in engineering Ingeneral when we use some known criteria in our study ithas to assume that our material properties are the same asthat in the study which proposed the known criteria Thoseproblems above can be solved by the new criteria proposed inthis paper Meanwhile when some special values are given tothe four internal parameters the new criteria have the sameexpression as some classical failure criteria such as Chang-Chang criteria Hashin criteria Linde criteria and Houcriteria in two dimensional case which are used frequently inthe research of damage process for the composite structures

In general the four internal parameters should be chosenby experiments One example that chooses proper values forthe parameters is given which also validates that there is noone and only set of value for the parameters in any criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is financially supported by the National NaturalScience Foundation of China under Grant 11072202

References

[1] S Agrawal KalyanKumar Singh andPK Sarkar ldquoImpact dam-age on fibre-reinforced polymer matrix compositemdasha reviewrdquoJournal of Composite Materials vol 48 no 3 pp 317ndash332 2014

[2] Y X Zhang and C H Yang ldquoRecent developments in finiteelement analysis for laminated composite platesrdquo CompositeStructures vol 88 no 1 pp 147ndash157 2009

[3] A C Orifici I Herszberg and R S Thomson ldquoReview ofmethodologies for composite material modelling incorporatingfailurerdquoComposite Structures vol 86 no 1ndash3 pp 194ndash210 2008

[4] P F Liu and J Y Zheng ldquoRecent developments on damagemodeling and finite element analysis for composite laminatesa reviewrdquo Materials and Design vol 31 no 8 pp 3825ndash38342010

[5] J G Michopoulos ldquoOn the reducibility of failure theories forcomposite materialsrdquo Composite Structures vol 86 no 1ndash3 pp165ndash176 2008

[6] M J Hinton and P D Soden ldquoFailure criteria for compositelaminatesrdquoComposites Science and Technology vol 58 no 7 pp1001ndash1010 1998

[7] P D Soden M J Hinton and A S Kaddour ldquoLaminaproperties and lay-up configurations and loanding conditionsof a range fiber reinforced composite laminatesrdquo CompositesScience and Technology vol 58 pp 1137ndash1150 1998

[8] P D Soden M J Hinton and A S Kaddour ldquoA comparisonof the predictive capabilities of current failure theories forcomposite laminatesrdquo Composites Science and Technology vol58 no 7 pp 1225ndash1254 1998

[9] M J Hinton P D Soden and A S Kaddour ldquoEvaluation offailure prediction in composite laminates background to PartB of the exerciserdquo Composites Science and Technology vol 62pp 1481ndash1488 2002

[10] M J Hinton A S Kaddour and P D Soden ldquoA further assess-ment of the predictive capabilities of current failure theories forcomposite laminates comparison with experimental evidencerdquoComposites Science andTechnology vol 64 no 3-4 pp 549ndash5882004

[11] A S Kaddour M J Hinton and P D Soden ldquoA comparisonof the predictive capabilities of current failure theories for com-posite laminates additional contributionsrdquo Composites Scienceand Technology vol 64 no 3-4 pp 449ndash476 2004

[12] O Hoffman ldquoThe brittle strength of orthotropic materialrdquoJournal of Composite Materials vol 9 no 2 pp 200ndash205 1975

[13] R Narayanaswami and H M Adelman ldquoEvaluation of the ten-sor polynominal and Hoffman strength theories for compositematerialsrdquo Journal of CompositeMaterials vol 11 no 4 pp 366ndash377 1977

[14] S W Tsai and E M Wu ldquoA general theory for anisotropicmaterialsrdquo Journal of Composite Materials vol 5 pp 58ndash801971

Advances in Materials Science and Engineering 7

[15] Z Haszin ldquoFailure criteria for unidirectional fiber compositesrdquoJournal of Applied Mechanics Transactions ASME vol 47 no 2pp 329ndash334 1980

[16] J P Hou N Petrinic C Ruiz and S R Hallett ldquoPredictionof impact damage in composite platesrdquo Composites Science andTechnology vol 60 no 2 pp 273ndash281 2000

[17] F-K Chang and K-Y Chang ldquoA progressive damage modelfor laminated composites containing stress concentrationsrdquoJournal of Composite Materials vol 21 no 9 pp 834ndash855 1987

[18] K-Y Chang S Liu and F-K Chang ldquoDamage tolerance oflaminated composites containing an open hole and subjectedto tensile loadingsrdquo Journal of Composite Materials vol 25 no3 pp 274ndash301 1991

[19] P Maimi J A Mayugo and P P Camanho ldquoA three-dimensional damage model for transversely isotropic compos-ite laminatesrdquo Journal of CompositeMaterials vol 42 no 25 pp2717ndash2745 2008

[20] P Maimı P P Camanho J A Mayugo and C G Davila ldquoAcontinuum damage model for composite laminatesmdashpart Imdashconstitutive modelrdquo Mechanics of Materials vol 39 no 10 pp897ndash908 2007

[21] P Linde J Pleitner H D Boer and C Carmore ldquoModellingand simulation of fiber metal laminatesrdquo in Proceedings of theABAQUS Userrsquos Conference pp 421ndash439 Boston Mass USA2004

[22] C Santiuste S Sanchez-Saez and E Barbero ldquoA comparison ofprogressive failure of composite laminated beamsrdquo CompositeStructures vol 92 pp 2406ndash2414 2010

[23] T Anderson Fracture MechanicsmdashFundamentals and Applica-tions 2nd edition 1995

[24] I M Daniel and O Ishai Engineering Mechanics of CompositeMaterials Oxford University 1994

[25] S T Pinho Modelling failure of laminated composites usingphysically based failure models [PhD thesis] Imperial College2005

[26] M l Ribeiro Volnei Tita and D Vandepitte ldquoA new damagemodel for composite laminatesrdquo Composite Structures vol 94pp 635ndash642 2012

[27] R M Richard and J R Blacklock ldquoFinite element analysis ofinelastic structuresrdquo AIAA Journal vol 7 pp 432ndash438 1969

[28] A Puck and H Schurmann ldquoFailure analysis of FRP laminatesby means of physically based phenomenological modelsrdquo Com-posites Science and Technology vol 62 no 12-13 pp 1633ndash16622002

[29] P D Soden M J Hinton and A S Kaddour ldquoBiaxial testresults for strength and deformation of a range of E-glass andcarbon fibre reinforced composite laminates failure exercisebenchmark datardquo Composites Science and Technology vol 62no 12-13 pp 1489ndash1514 2002

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

4 Advances in Materials Science and Engineering

119889119898119905= (

12059022

119884119879

)

2

+ (

12059012

11987812

)

2

12059022ge 0

119889119898119888= (

119884119888

211987823

)

2

(

12059022

119884119888

)

2

+ [(

119884119888

211987823

)

2

minus 1]

12059022

119884119888

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(6)

When 1198961= 1 120573

1= 1 120572 = 0 119896

2= 1 120573

2= 1 120574 = 0 119896

3= 1

1198964= (119884119888211987812)2 the new criteria will becomeHou criteria in

two dimensional case as follows

119889119891119905= 1 lowast (

1205901

119883119879

)

2

+ (1 minus 1) lowast 1205901+ (

12059012

119878

)

2

ge 1 1205901ge 0

119889119891119888= (

1205901

119883119879

)

2

+ (

12059012

119878

)

2

ge 1 1205901le 0

119889119898119905= (

12059022

119884119879

)

2

+ (

12059012

11987812

)

2

+ (

12059023

11987823

)

2

12059022ge 0

119889119898119888=

1

119884119888

[(

119884119888

211987812

)

2

minus 1]12059022+

1205902

22

41198782

12

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(7)

When 1198961= 119883119879119883119888 1205731= 0 120572 = 0 119896

2= 119883119888119883119879 1205732= 0

120574 = 1 1198963= 119884119879119884119888 1198964= 119884119888119884119879 the new criteria will become

Linde criteria in two dimensional case as follows

119889119891= (

1

119883119879

minus

1

119883119888

)1205901+

1

119883119879119883119888

1205902

1

119889119898= (

1

119884119879

minus

1

119884119888

)1205902+

1

119884119879119884119888

1205902

2+ (

12059012

11987812

)

2

(8)

When 1198961= 1 120573

1= 1 120572 = 1 119896

2= 1 120573

2= 0 120574 = 0 119896

3= 1

1198964= 1 the new criteria will become Chang-Chang criteria as

follows

119889119891119905= (

1205901

119883119879

)

2

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

ge 1 1205901ge 0

119889119891119888= (

1205901

119883119888

)

2

ge 1 1205901le 0

119889119898119905= (

12059022

119884119879

)

2

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

12059022ge 0

119889119898119888= (

12059022

119884119888

)

2

+

1205902

12211986612+ 34120572120590

4

12

1198782

12211986612+ 34120572119878

4

12

12059022le 0

(9)

Table 1 Failure strengths of T300914C

119883119879Mpa 119883

119862Mpa 119884

119879Mpa 119884

119862Mpa 119878

12Mpa

1500 900 27 200 80

Table 2 Failure strengths of E-glassLy556HT907DY063 epoxy

119883119879Mpa 119883

119862Mpa 119884

119879Mpa 119884

119862Mpa 119878

12Mpa 119866

119868Jm2

1500 900 27 200 80 165

When 1198963= 119866119868119888119866119868119868119888 120572 = 0 (3) will be one part of the

failure criteria LaR04 shown in (10)119866119868119888 119866119868119868119888

are the fracturetoughness of Mode I and Mode II respectively

119889119898119905=

119866119868119888

119866119868119868119888

(

1205902

119884119879

)

2

+ (1 minus

119866119868119888

119866119868119868119888

)

1205902

119884119905

+ (

12059012

11987812

)

2

1205902ge 0

(10)

From (5)ndash(9) we find that the new criteria with fourparameters have enough capability to match the failurecriteria which are used widely in the damage analysis ofcomposite materials

The parameters of the new failure criterion can be definedby experimental data to get better numerical results

4 Identifying Parameters ofthe New Failure Criteria

The process of identifying parameters of the new failurecriteria by experiments is shown here

Biaxial tests of E-glass and carbon fiber reinforced com-posite laminates were performed by Soden et al [29] Inbiaxial tests just 120590

1 1205902 and 120590

12can be obtained There-

fore just parameters 1198961 1198962 1198963 and 119896

4can be identified by

experimental data The other parameters 1205731 1205732 120572 120574 in the

new criteria are not the main items In fact 120574 is used toconsider the influence of linear item 120590

1 to the fiber tensile

damage 1205731is used to consider the influence of shear stress

12059012 to the fiber tensile damage 120573

2is used to consider the

influence of shear stress12059012 to the fiber compressive damage

120572 is used to consider the influence of nonlinear factor ofmaterial The corresponding real values of the parameters of1205731 1205732 120572 120574 in Chang-Chang criteria Linde criteria Hashin

criteria and Hou criteria in two dimensional case are almostthe same Therefore based on the four failure criteria abovethe parameters 120573

1= 1 120573

2= 1 120572 = 0 120574 = 1 are assumed

In order to reduce the complexity of identifying parameters1198961 1198962 1198963 1198964 we just choose the corresponding coefficients

of those parameters from Hashin Hou Linde and Chang-Chang criteria such as 119896

1= 1 119883

119879119883119862 1198962= 1 (119883

119888119883119879)2

and 119883119888119883119879 1198963= 1 119884

119879119884119888and 119866

119868119888119866119868119868119888 1198964= (119884119888211987812)2 1 or

119884119888119884119879

41 Experiments E-glass and carbon fiber reinforced com-posite laminates were used in biaxial tests [29]Theirmaterialproperties are listed in Tables 1 and 2 respectively

Biaxial failure for unidirectional T300914C carbonep-oxy lamina is expressed under combined longitudinal and

Advances in Materials Science and Engineering 5

0

20

40

60

80

100

120

140

0 500 1000 1500 2000

1205901 (Mpa)

12059012(M

pa)

Experimentk1 = 1

k1 = XTXC

Figure 1 Fiber tensile failure envelope

shear loading (1205901versus 120590

12)The specimens were in the form

of axially wound tubes made from prepreg T300BSL914Ccarbonepoxy The tubes were tested under combined axialtension or compression and torsion All the tubes were endreinforced and grips were used to transmit the torque to thetubes [29]

Biaxial failure for unidirectional E-glassLy556HT907DY063 epoxy lamina is expressed under combined trans-verse and shear loading (120590

2versus 120590

12) The tubes with

filament wound by circumferentially wound were 60mmin internal diameter and 2mm in thickness and wereconstructed from 62 by volume Vetrotes 21lowastK43 E-glassfiber (Gevetex) rovings and a Ciba-Geigy epoxy resin sys-tem Ly556HT907DY063 mixed in weight proportions of100 85 4 The tubes were cured at 100∘C for 2 h andpostcurved at 150∘C for 2 h [29]

42 Determination of Parameters The parameters 1205731= 1

1205732= 1 120572 = 0 120574 = 1 are assumed which are based

on Chang-Chang criteria Linde criteria Hashin criteria andHou criteria in two dimensional case And (1)ndash(4) of the newcriteria should be rewritten as follows

421 For Fiber Tensile Failure Consider

119889119891119905= 1198961lowast (

1205901

119883119879

)

2

+ (1 minus 1198961) lowast

1205901

119883119879

+ (

12059012

11987812

) ge 1 1205901ge 0

(11)

where 1198961is a parameter which can take two values 1 and

119883119879119883119862The curves of (11) are shown in Figure 1The curve of

1198961= 1 and the curve of 119896

1= 119883119879119883119862are both lying between

the experimental data which are both effective Howeverthe curve of 119896

1= 119883119879119883119862is close to a large number of

experimental data above in Figure 1 Therefore the curveof 1198961= 119883119879119883119862is used for failure analysis of T300914C

carbonepoxy lamina

422 For Fiber Compressive Failure Consider

119889119891119888= 1198962lowast (

1205901

119883119888

)

2

+ (1198962minus 1) lowast

1205901

119883119888

+ (

12059012

11987812

) ge 1 1205901le 0

(12)

0

20

40

60

80

100

120

minus1000 minus800 minus600 minus400 minus200 0

1205901 (Mpa)

12059012(M

pa)

k2 = 1

Experimentk2 = (XCXT)

2

k2 = XCXT

Figure 2 Fiber compressive failure envelope

0

10

20

30

40

50

60

70

80

0 10 20 30 40

1205902 (Mpa)

12059012(M

pa)

k3 = 1

k3 = YTYC

k3 = GIGII

Experiment

Figure 3 Matrix tensile failure envelope

where 1198962is a parameter which can take three values 1

(119883119888119883119879)2 and 119883

119888119883119879 The curves of (12) are shown in

Figure 2 The curve of 1198962= 1 is outmost among all curves

in Figure 2 and the experimental data is almost out ofthe envelope of all curves The difference of the curve of1198962= 1 with the data is minimal Therefore the value

1198962= 1 should be used for failure analysis of T300914C

carbonepoxy lamina

423 For Matrix Tensile Failure Consider

119889119898119905= 1198963lowast (

12059022

119884119879

)

2

+ (1 minus 1198963) lowast 12059022+ (

12059012

11987812

)

2

12059022ge 0

(13)

where 1198963is a parameter which can take three values 1 119884

119879119884119888

and 119866119868119888119866119868119868119888 The curves of (13) are shown in Figure 3 The

number of the experimental data is just fiveThe curve of 1198963=

119884119879119884119862is nethermost Three data are just in the curve of 119896

3=

119884119879119884119862 and the difference of the curve 119896

3= 119884119879119884119862isminimal

Therefore the value 1198963= 119884119879119884119862should be used for failure

analysis of E-glassLy556HT907DY063 epoxy lamina

6 Advances in Materials Science and Engineering

110

90

70

50

30

10

minus10minus50minus100minus150

1205902 (Mpa)

12059012(M

pa)

k4 = 1

Experimentk4 = YCYT

k4 = (YCS122)2

Figure 4 Matrix compressive failure envelope

424 For Matrix Compressive Failure Consider

119889119898119888= 1198964(

12059022

119884119888

)

2

+ (1198964minus 1)

12059022

119884119888

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(14)

where 1198964is a parameter which can take three values

(119884119888211987812)2 1 or 119884

119888119884119879 The curves of (14) are shown in

Figure 4 Because of the difference between the strengthobtained through experiments and that used in materialproperty three curves are all not perfect From Figure 4 thecurve of 119896

4= 119884119888119884119879is close to experimental data Therefore

the value of 1198964= 119884119888119884119879should be used in failure analysis of

E-glassLy556HT907DY063 epoxy laminaFrom the discussion above it is found that it is not

enough for the damage criterion that has only one set ofthe parameters The failure criteria should be adjusted byexperimental data for the materials to meet higher accuraterequirement

5 Conclusions

In this work new two-dimensional failure criteria with fourparameters for composite materials are given The new crite-ria include fiber failure criteria in tension and compressionand matrix failure criteria in tension and compression

The new criteria can be adjusted with four parametersto consider different material properties in engineering Ingeneral when we use some known criteria in our study ithas to assume that our material properties are the same asthat in the study which proposed the known criteria Thoseproblems above can be solved by the new criteria proposed inthis paper Meanwhile when some special values are given tothe four internal parameters the new criteria have the sameexpression as some classical failure criteria such as Chang-Chang criteria Hashin criteria Linde criteria and Houcriteria in two dimensional case which are used frequently inthe research of damage process for the composite structures

In general the four internal parameters should be chosenby experiments One example that chooses proper values forthe parameters is given which also validates that there is noone and only set of value for the parameters in any criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is financially supported by the National NaturalScience Foundation of China under Grant 11072202

References

[1] S Agrawal KalyanKumar Singh andPK Sarkar ldquoImpact dam-age on fibre-reinforced polymer matrix compositemdasha reviewrdquoJournal of Composite Materials vol 48 no 3 pp 317ndash332 2014

[2] Y X Zhang and C H Yang ldquoRecent developments in finiteelement analysis for laminated composite platesrdquo CompositeStructures vol 88 no 1 pp 147ndash157 2009

[3] A C Orifici I Herszberg and R S Thomson ldquoReview ofmethodologies for composite material modelling incorporatingfailurerdquoComposite Structures vol 86 no 1ndash3 pp 194ndash210 2008

[4] P F Liu and J Y Zheng ldquoRecent developments on damagemodeling and finite element analysis for composite laminatesa reviewrdquo Materials and Design vol 31 no 8 pp 3825ndash38342010

[5] J G Michopoulos ldquoOn the reducibility of failure theories forcomposite materialsrdquo Composite Structures vol 86 no 1ndash3 pp165ndash176 2008

[6] M J Hinton and P D Soden ldquoFailure criteria for compositelaminatesrdquoComposites Science and Technology vol 58 no 7 pp1001ndash1010 1998

[7] P D Soden M J Hinton and A S Kaddour ldquoLaminaproperties and lay-up configurations and loanding conditionsof a range fiber reinforced composite laminatesrdquo CompositesScience and Technology vol 58 pp 1137ndash1150 1998

[8] P D Soden M J Hinton and A S Kaddour ldquoA comparisonof the predictive capabilities of current failure theories forcomposite laminatesrdquo Composites Science and Technology vol58 no 7 pp 1225ndash1254 1998

[9] M J Hinton P D Soden and A S Kaddour ldquoEvaluation offailure prediction in composite laminates background to PartB of the exerciserdquo Composites Science and Technology vol 62pp 1481ndash1488 2002

[10] M J Hinton A S Kaddour and P D Soden ldquoA further assess-ment of the predictive capabilities of current failure theories forcomposite laminates comparison with experimental evidencerdquoComposites Science andTechnology vol 64 no 3-4 pp 549ndash5882004

[11] A S Kaddour M J Hinton and P D Soden ldquoA comparisonof the predictive capabilities of current failure theories for com-posite laminates additional contributionsrdquo Composites Scienceand Technology vol 64 no 3-4 pp 449ndash476 2004

[12] O Hoffman ldquoThe brittle strength of orthotropic materialrdquoJournal of Composite Materials vol 9 no 2 pp 200ndash205 1975

[13] R Narayanaswami and H M Adelman ldquoEvaluation of the ten-sor polynominal and Hoffman strength theories for compositematerialsrdquo Journal of CompositeMaterials vol 11 no 4 pp 366ndash377 1977

[14] S W Tsai and E M Wu ldquoA general theory for anisotropicmaterialsrdquo Journal of Composite Materials vol 5 pp 58ndash801971

Advances in Materials Science and Engineering 7

[15] Z Haszin ldquoFailure criteria for unidirectional fiber compositesrdquoJournal of Applied Mechanics Transactions ASME vol 47 no 2pp 329ndash334 1980

[16] J P Hou N Petrinic C Ruiz and S R Hallett ldquoPredictionof impact damage in composite platesrdquo Composites Science andTechnology vol 60 no 2 pp 273ndash281 2000

[17] F-K Chang and K-Y Chang ldquoA progressive damage modelfor laminated composites containing stress concentrationsrdquoJournal of Composite Materials vol 21 no 9 pp 834ndash855 1987

[18] K-Y Chang S Liu and F-K Chang ldquoDamage tolerance oflaminated composites containing an open hole and subjectedto tensile loadingsrdquo Journal of Composite Materials vol 25 no3 pp 274ndash301 1991

[19] P Maimi J A Mayugo and P P Camanho ldquoA three-dimensional damage model for transversely isotropic compos-ite laminatesrdquo Journal of CompositeMaterials vol 42 no 25 pp2717ndash2745 2008

[20] P Maimı P P Camanho J A Mayugo and C G Davila ldquoAcontinuum damage model for composite laminatesmdashpart Imdashconstitutive modelrdquo Mechanics of Materials vol 39 no 10 pp897ndash908 2007

[21] P Linde J Pleitner H D Boer and C Carmore ldquoModellingand simulation of fiber metal laminatesrdquo in Proceedings of theABAQUS Userrsquos Conference pp 421ndash439 Boston Mass USA2004

[22] C Santiuste S Sanchez-Saez and E Barbero ldquoA comparison ofprogressive failure of composite laminated beamsrdquo CompositeStructures vol 92 pp 2406ndash2414 2010

[23] T Anderson Fracture MechanicsmdashFundamentals and Applica-tions 2nd edition 1995

[24] I M Daniel and O Ishai Engineering Mechanics of CompositeMaterials Oxford University 1994

[25] S T Pinho Modelling failure of laminated composites usingphysically based failure models [PhD thesis] Imperial College2005

[26] M l Ribeiro Volnei Tita and D Vandepitte ldquoA new damagemodel for composite laminatesrdquo Composite Structures vol 94pp 635ndash642 2012

[27] R M Richard and J R Blacklock ldquoFinite element analysis ofinelastic structuresrdquo AIAA Journal vol 7 pp 432ndash438 1969

[28] A Puck and H Schurmann ldquoFailure analysis of FRP laminatesby means of physically based phenomenological modelsrdquo Com-posites Science and Technology vol 62 no 12-13 pp 1633ndash16622002

[29] P D Soden M J Hinton and A S Kaddour ldquoBiaxial testresults for strength and deformation of a range of E-glass andcarbon fibre reinforced composite laminates failure exercisebenchmark datardquo Composites Science and Technology vol 62no 12-13 pp 1489ndash1514 2002

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 5

0

20

40

60

80

100

120

140

0 500 1000 1500 2000

1205901 (Mpa)

12059012(M

pa)

Experimentk1 = 1

k1 = XTXC

Figure 1 Fiber tensile failure envelope

shear loading (1205901versus 120590

12)The specimens were in the form

of axially wound tubes made from prepreg T300BSL914Ccarbonepoxy The tubes were tested under combined axialtension or compression and torsion All the tubes were endreinforced and grips were used to transmit the torque to thetubes [29]

Biaxial failure for unidirectional E-glassLy556HT907DY063 epoxy lamina is expressed under combined trans-verse and shear loading (120590

2versus 120590

12) The tubes with

filament wound by circumferentially wound were 60mmin internal diameter and 2mm in thickness and wereconstructed from 62 by volume Vetrotes 21lowastK43 E-glassfiber (Gevetex) rovings and a Ciba-Geigy epoxy resin sys-tem Ly556HT907DY063 mixed in weight proportions of100 85 4 The tubes were cured at 100∘C for 2 h andpostcurved at 150∘C for 2 h [29]

42 Determination of Parameters The parameters 1205731= 1

1205732= 1 120572 = 0 120574 = 1 are assumed which are based

on Chang-Chang criteria Linde criteria Hashin criteria andHou criteria in two dimensional case And (1)ndash(4) of the newcriteria should be rewritten as follows

421 For Fiber Tensile Failure Consider

119889119891119905= 1198961lowast (

1205901

119883119879

)

2

+ (1 minus 1198961) lowast

1205901

119883119879

+ (

12059012

11987812

) ge 1 1205901ge 0

(11)

where 1198961is a parameter which can take two values 1 and

119883119879119883119862The curves of (11) are shown in Figure 1The curve of

1198961= 1 and the curve of 119896

1= 119883119879119883119862are both lying between

the experimental data which are both effective Howeverthe curve of 119896

1= 119883119879119883119862is close to a large number of

experimental data above in Figure 1 Therefore the curveof 1198961= 119883119879119883119862is used for failure analysis of T300914C

carbonepoxy lamina

422 For Fiber Compressive Failure Consider

119889119891119888= 1198962lowast (

1205901

119883119888

)

2

+ (1198962minus 1) lowast

1205901

119883119888

+ (

12059012

11987812

) ge 1 1205901le 0

(12)

0

20

40

60

80

100

120

minus1000 minus800 minus600 minus400 minus200 0

1205901 (Mpa)

12059012(M

pa)

k2 = 1

Experimentk2 = (XCXT)

2

k2 = XCXT

Figure 2 Fiber compressive failure envelope

0

10

20

30

40

50

60

70

80

0 10 20 30 40

1205902 (Mpa)

12059012(M

pa)

k3 = 1

k3 = YTYC

k3 = GIGII

Experiment

Figure 3 Matrix tensile failure envelope

where 1198962is a parameter which can take three values 1

(119883119888119883119879)2 and 119883

119888119883119879 The curves of (12) are shown in

Figure 2 The curve of 1198962= 1 is outmost among all curves

in Figure 2 and the experimental data is almost out ofthe envelope of all curves The difference of the curve of1198962= 1 with the data is minimal Therefore the value

1198962= 1 should be used for failure analysis of T300914C

carbonepoxy lamina

423 For Matrix Tensile Failure Consider

119889119898119905= 1198963lowast (

12059022

119884119879

)

2

+ (1 minus 1198963) lowast 12059022+ (

12059012

11987812

)

2

12059022ge 0

(13)

where 1198963is a parameter which can take three values 1 119884

119879119884119888

and 119866119868119888119866119868119868119888 The curves of (13) are shown in Figure 3 The

number of the experimental data is just fiveThe curve of 1198963=

119884119879119884119862is nethermost Three data are just in the curve of 119896

3=

119884119879119884119862 and the difference of the curve 119896

3= 119884119879119884119862isminimal

Therefore the value 1198963= 119884119879119884119862should be used for failure

analysis of E-glassLy556HT907DY063 epoxy lamina

6 Advances in Materials Science and Engineering

110

90

70

50

30

10

minus10minus50minus100minus150

1205902 (Mpa)

12059012(M

pa)

k4 = 1

Experimentk4 = YCYT

k4 = (YCS122)2

Figure 4 Matrix compressive failure envelope

424 For Matrix Compressive Failure Consider

119889119898119888= 1198964(

12059022

119884119888

)

2

+ (1198964minus 1)

12059022

119884119888

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(14)

where 1198964is a parameter which can take three values

(119884119888211987812)2 1 or 119884

119888119884119879 The curves of (14) are shown in

Figure 4 Because of the difference between the strengthobtained through experiments and that used in materialproperty three curves are all not perfect From Figure 4 thecurve of 119896

4= 119884119888119884119879is close to experimental data Therefore

the value of 1198964= 119884119888119884119879should be used in failure analysis of

E-glassLy556HT907DY063 epoxy laminaFrom the discussion above it is found that it is not

enough for the damage criterion that has only one set ofthe parameters The failure criteria should be adjusted byexperimental data for the materials to meet higher accuraterequirement

5 Conclusions

In this work new two-dimensional failure criteria with fourparameters for composite materials are given The new crite-ria include fiber failure criteria in tension and compressionand matrix failure criteria in tension and compression

The new criteria can be adjusted with four parametersto consider different material properties in engineering Ingeneral when we use some known criteria in our study ithas to assume that our material properties are the same asthat in the study which proposed the known criteria Thoseproblems above can be solved by the new criteria proposed inthis paper Meanwhile when some special values are given tothe four internal parameters the new criteria have the sameexpression as some classical failure criteria such as Chang-Chang criteria Hashin criteria Linde criteria and Houcriteria in two dimensional case which are used frequently inthe research of damage process for the composite structures

In general the four internal parameters should be chosenby experiments One example that chooses proper values forthe parameters is given which also validates that there is noone and only set of value for the parameters in any criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is financially supported by the National NaturalScience Foundation of China under Grant 11072202

References

[1] S Agrawal KalyanKumar Singh andPK Sarkar ldquoImpact dam-age on fibre-reinforced polymer matrix compositemdasha reviewrdquoJournal of Composite Materials vol 48 no 3 pp 317ndash332 2014

[2] Y X Zhang and C H Yang ldquoRecent developments in finiteelement analysis for laminated composite platesrdquo CompositeStructures vol 88 no 1 pp 147ndash157 2009

[3] A C Orifici I Herszberg and R S Thomson ldquoReview ofmethodologies for composite material modelling incorporatingfailurerdquoComposite Structures vol 86 no 1ndash3 pp 194ndash210 2008

[4] P F Liu and J Y Zheng ldquoRecent developments on damagemodeling and finite element analysis for composite laminatesa reviewrdquo Materials and Design vol 31 no 8 pp 3825ndash38342010

[5] J G Michopoulos ldquoOn the reducibility of failure theories forcomposite materialsrdquo Composite Structures vol 86 no 1ndash3 pp165ndash176 2008

[6] M J Hinton and P D Soden ldquoFailure criteria for compositelaminatesrdquoComposites Science and Technology vol 58 no 7 pp1001ndash1010 1998

[7] P D Soden M J Hinton and A S Kaddour ldquoLaminaproperties and lay-up configurations and loanding conditionsof a range fiber reinforced composite laminatesrdquo CompositesScience and Technology vol 58 pp 1137ndash1150 1998

[8] P D Soden M J Hinton and A S Kaddour ldquoA comparisonof the predictive capabilities of current failure theories forcomposite laminatesrdquo Composites Science and Technology vol58 no 7 pp 1225ndash1254 1998

[9] M J Hinton P D Soden and A S Kaddour ldquoEvaluation offailure prediction in composite laminates background to PartB of the exerciserdquo Composites Science and Technology vol 62pp 1481ndash1488 2002

[10] M J Hinton A S Kaddour and P D Soden ldquoA further assess-ment of the predictive capabilities of current failure theories forcomposite laminates comparison with experimental evidencerdquoComposites Science andTechnology vol 64 no 3-4 pp 549ndash5882004

[11] A S Kaddour M J Hinton and P D Soden ldquoA comparisonof the predictive capabilities of current failure theories for com-posite laminates additional contributionsrdquo Composites Scienceand Technology vol 64 no 3-4 pp 449ndash476 2004

[12] O Hoffman ldquoThe brittle strength of orthotropic materialrdquoJournal of Composite Materials vol 9 no 2 pp 200ndash205 1975

[13] R Narayanaswami and H M Adelman ldquoEvaluation of the ten-sor polynominal and Hoffman strength theories for compositematerialsrdquo Journal of CompositeMaterials vol 11 no 4 pp 366ndash377 1977

[14] S W Tsai and E M Wu ldquoA general theory for anisotropicmaterialsrdquo Journal of Composite Materials vol 5 pp 58ndash801971

Advances in Materials Science and Engineering 7

[15] Z Haszin ldquoFailure criteria for unidirectional fiber compositesrdquoJournal of Applied Mechanics Transactions ASME vol 47 no 2pp 329ndash334 1980

[16] J P Hou N Petrinic C Ruiz and S R Hallett ldquoPredictionof impact damage in composite platesrdquo Composites Science andTechnology vol 60 no 2 pp 273ndash281 2000

[17] F-K Chang and K-Y Chang ldquoA progressive damage modelfor laminated composites containing stress concentrationsrdquoJournal of Composite Materials vol 21 no 9 pp 834ndash855 1987

[18] K-Y Chang S Liu and F-K Chang ldquoDamage tolerance oflaminated composites containing an open hole and subjectedto tensile loadingsrdquo Journal of Composite Materials vol 25 no3 pp 274ndash301 1991

[19] P Maimi J A Mayugo and P P Camanho ldquoA three-dimensional damage model for transversely isotropic compos-ite laminatesrdquo Journal of CompositeMaterials vol 42 no 25 pp2717ndash2745 2008

[20] P Maimı P P Camanho J A Mayugo and C G Davila ldquoAcontinuum damage model for composite laminatesmdashpart Imdashconstitutive modelrdquo Mechanics of Materials vol 39 no 10 pp897ndash908 2007

[21] P Linde J Pleitner H D Boer and C Carmore ldquoModellingand simulation of fiber metal laminatesrdquo in Proceedings of theABAQUS Userrsquos Conference pp 421ndash439 Boston Mass USA2004

[22] C Santiuste S Sanchez-Saez and E Barbero ldquoA comparison ofprogressive failure of composite laminated beamsrdquo CompositeStructures vol 92 pp 2406ndash2414 2010

[23] T Anderson Fracture MechanicsmdashFundamentals and Applica-tions 2nd edition 1995

[24] I M Daniel and O Ishai Engineering Mechanics of CompositeMaterials Oxford University 1994

[25] S T Pinho Modelling failure of laminated composites usingphysically based failure models [PhD thesis] Imperial College2005

[26] M l Ribeiro Volnei Tita and D Vandepitte ldquoA new damagemodel for composite laminatesrdquo Composite Structures vol 94pp 635ndash642 2012

[27] R M Richard and J R Blacklock ldquoFinite element analysis ofinelastic structuresrdquo AIAA Journal vol 7 pp 432ndash438 1969

[28] A Puck and H Schurmann ldquoFailure analysis of FRP laminatesby means of physically based phenomenological modelsrdquo Com-posites Science and Technology vol 62 no 12-13 pp 1633ndash16622002

[29] P D Soden M J Hinton and A S Kaddour ldquoBiaxial testresults for strength and deformation of a range of E-glass andcarbon fibre reinforced composite laminates failure exercisebenchmark datardquo Composites Science and Technology vol 62no 12-13 pp 1489ndash1514 2002

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

6 Advances in Materials Science and Engineering

110

90

70

50

30

10

minus10minus50minus100minus150

1205902 (Mpa)

12059012(M

pa)

k4 = 1

Experimentk4 = YCYT

k4 = (YCS122)2

Figure 4 Matrix compressive failure envelope

424 For Matrix Compressive Failure Consider

119889119898119888= 1198964(

12059022

119884119888

)

2

+ (1198964minus 1)

12059022

119884119888

+ (

12059012

11987812

)

2

ge 1 12059022lt 0

(14)

where 1198964is a parameter which can take three values

(119884119888211987812)2 1 or 119884

119888119884119879 The curves of (14) are shown in

Figure 4 Because of the difference between the strengthobtained through experiments and that used in materialproperty three curves are all not perfect From Figure 4 thecurve of 119896

4= 119884119888119884119879is close to experimental data Therefore

the value of 1198964= 119884119888119884119879should be used in failure analysis of

E-glassLy556HT907DY063 epoxy laminaFrom the discussion above it is found that it is not

enough for the damage criterion that has only one set ofthe parameters The failure criteria should be adjusted byexperimental data for the materials to meet higher accuraterequirement

5 Conclusions

In this work new two-dimensional failure criteria with fourparameters for composite materials are given The new crite-ria include fiber failure criteria in tension and compressionand matrix failure criteria in tension and compression

The new criteria can be adjusted with four parametersto consider different material properties in engineering Ingeneral when we use some known criteria in our study ithas to assume that our material properties are the same asthat in the study which proposed the known criteria Thoseproblems above can be solved by the new criteria proposed inthis paper Meanwhile when some special values are given tothe four internal parameters the new criteria have the sameexpression as some classical failure criteria such as Chang-Chang criteria Hashin criteria Linde criteria and Houcriteria in two dimensional case which are used frequently inthe research of damage process for the composite structures

In general the four internal parameters should be chosenby experiments One example that chooses proper values forthe parameters is given which also validates that there is noone and only set of value for the parameters in any criterion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is financially supported by the National NaturalScience Foundation of China under Grant 11072202

References

[1] S Agrawal KalyanKumar Singh andPK Sarkar ldquoImpact dam-age on fibre-reinforced polymer matrix compositemdasha reviewrdquoJournal of Composite Materials vol 48 no 3 pp 317ndash332 2014

[2] Y X Zhang and C H Yang ldquoRecent developments in finiteelement analysis for laminated composite platesrdquo CompositeStructures vol 88 no 1 pp 147ndash157 2009

[3] A C Orifici I Herszberg and R S Thomson ldquoReview ofmethodologies for composite material modelling incorporatingfailurerdquoComposite Structures vol 86 no 1ndash3 pp 194ndash210 2008

[4] P F Liu and J Y Zheng ldquoRecent developments on damagemodeling and finite element analysis for composite laminatesa reviewrdquo Materials and Design vol 31 no 8 pp 3825ndash38342010

[5] J G Michopoulos ldquoOn the reducibility of failure theories forcomposite materialsrdquo Composite Structures vol 86 no 1ndash3 pp165ndash176 2008

[6] M J Hinton and P D Soden ldquoFailure criteria for compositelaminatesrdquoComposites Science and Technology vol 58 no 7 pp1001ndash1010 1998

[7] P D Soden M J Hinton and A S Kaddour ldquoLaminaproperties and lay-up configurations and loanding conditionsof a range fiber reinforced composite laminatesrdquo CompositesScience and Technology vol 58 pp 1137ndash1150 1998

[8] P D Soden M J Hinton and A S Kaddour ldquoA comparisonof the predictive capabilities of current failure theories forcomposite laminatesrdquo Composites Science and Technology vol58 no 7 pp 1225ndash1254 1998

[9] M J Hinton P D Soden and A S Kaddour ldquoEvaluation offailure prediction in composite laminates background to PartB of the exerciserdquo Composites Science and Technology vol 62pp 1481ndash1488 2002

[10] M J Hinton A S Kaddour and P D Soden ldquoA further assess-ment of the predictive capabilities of current failure theories forcomposite laminates comparison with experimental evidencerdquoComposites Science andTechnology vol 64 no 3-4 pp 549ndash5882004

[11] A S Kaddour M J Hinton and P D Soden ldquoA comparisonof the predictive capabilities of current failure theories for com-posite laminates additional contributionsrdquo Composites Scienceand Technology vol 64 no 3-4 pp 449ndash476 2004

[12] O Hoffman ldquoThe brittle strength of orthotropic materialrdquoJournal of Composite Materials vol 9 no 2 pp 200ndash205 1975

[13] R Narayanaswami and H M Adelman ldquoEvaluation of the ten-sor polynominal and Hoffman strength theories for compositematerialsrdquo Journal of CompositeMaterials vol 11 no 4 pp 366ndash377 1977

[14] S W Tsai and E M Wu ldquoA general theory for anisotropicmaterialsrdquo Journal of Composite Materials vol 5 pp 58ndash801971

Advances in Materials Science and Engineering 7

[15] Z Haszin ldquoFailure criteria for unidirectional fiber compositesrdquoJournal of Applied Mechanics Transactions ASME vol 47 no 2pp 329ndash334 1980

[16] J P Hou N Petrinic C Ruiz and S R Hallett ldquoPredictionof impact damage in composite platesrdquo Composites Science andTechnology vol 60 no 2 pp 273ndash281 2000

[17] F-K Chang and K-Y Chang ldquoA progressive damage modelfor laminated composites containing stress concentrationsrdquoJournal of Composite Materials vol 21 no 9 pp 834ndash855 1987

[18] K-Y Chang S Liu and F-K Chang ldquoDamage tolerance oflaminated composites containing an open hole and subjectedto tensile loadingsrdquo Journal of Composite Materials vol 25 no3 pp 274ndash301 1991

[19] P Maimi J A Mayugo and P P Camanho ldquoA three-dimensional damage model for transversely isotropic compos-ite laminatesrdquo Journal of CompositeMaterials vol 42 no 25 pp2717ndash2745 2008

[20] P Maimı P P Camanho J A Mayugo and C G Davila ldquoAcontinuum damage model for composite laminatesmdashpart Imdashconstitutive modelrdquo Mechanics of Materials vol 39 no 10 pp897ndash908 2007

[21] P Linde J Pleitner H D Boer and C Carmore ldquoModellingand simulation of fiber metal laminatesrdquo in Proceedings of theABAQUS Userrsquos Conference pp 421ndash439 Boston Mass USA2004

[22] C Santiuste S Sanchez-Saez and E Barbero ldquoA comparison ofprogressive failure of composite laminated beamsrdquo CompositeStructures vol 92 pp 2406ndash2414 2010

[23] T Anderson Fracture MechanicsmdashFundamentals and Applica-tions 2nd edition 1995

[24] I M Daniel and O Ishai Engineering Mechanics of CompositeMaterials Oxford University 1994

[25] S T Pinho Modelling failure of laminated composites usingphysically based failure models [PhD thesis] Imperial College2005

[26] M l Ribeiro Volnei Tita and D Vandepitte ldquoA new damagemodel for composite laminatesrdquo Composite Structures vol 94pp 635ndash642 2012

[27] R M Richard and J R Blacklock ldquoFinite element analysis ofinelastic structuresrdquo AIAA Journal vol 7 pp 432ndash438 1969

[28] A Puck and H Schurmann ldquoFailure analysis of FRP laminatesby means of physically based phenomenological modelsrdquo Com-posites Science and Technology vol 62 no 12-13 pp 1633ndash16622002

[29] P D Soden M J Hinton and A S Kaddour ldquoBiaxial testresults for strength and deformation of a range of E-glass andcarbon fibre reinforced composite laminates failure exercisebenchmark datardquo Composites Science and Technology vol 62no 12-13 pp 1489ndash1514 2002

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 7

[15] Z Haszin ldquoFailure criteria for unidirectional fiber compositesrdquoJournal of Applied Mechanics Transactions ASME vol 47 no 2pp 329ndash334 1980

[16] J P Hou N Petrinic C Ruiz and S R Hallett ldquoPredictionof impact damage in composite platesrdquo Composites Science andTechnology vol 60 no 2 pp 273ndash281 2000

[17] F-K Chang and K-Y Chang ldquoA progressive damage modelfor laminated composites containing stress concentrationsrdquoJournal of Composite Materials vol 21 no 9 pp 834ndash855 1987

[18] K-Y Chang S Liu and F-K Chang ldquoDamage tolerance oflaminated composites containing an open hole and subjectedto tensile loadingsrdquo Journal of Composite Materials vol 25 no3 pp 274ndash301 1991

[19] P Maimi J A Mayugo and P P Camanho ldquoA three-dimensional damage model for transversely isotropic compos-ite laminatesrdquo Journal of CompositeMaterials vol 42 no 25 pp2717ndash2745 2008

[20] P Maimı P P Camanho J A Mayugo and C G Davila ldquoAcontinuum damage model for composite laminatesmdashpart Imdashconstitutive modelrdquo Mechanics of Materials vol 39 no 10 pp897ndash908 2007

[21] P Linde J Pleitner H D Boer and C Carmore ldquoModellingand simulation of fiber metal laminatesrdquo in Proceedings of theABAQUS Userrsquos Conference pp 421ndash439 Boston Mass USA2004

[22] C Santiuste S Sanchez-Saez and E Barbero ldquoA comparison ofprogressive failure of composite laminated beamsrdquo CompositeStructures vol 92 pp 2406ndash2414 2010

[23] T Anderson Fracture MechanicsmdashFundamentals and Applica-tions 2nd edition 1995

[24] I M Daniel and O Ishai Engineering Mechanics of CompositeMaterials Oxford University 1994

[25] S T Pinho Modelling failure of laminated composites usingphysically based failure models [PhD thesis] Imperial College2005

[26] M l Ribeiro Volnei Tita and D Vandepitte ldquoA new damagemodel for composite laminatesrdquo Composite Structures vol 94pp 635ndash642 2012

[27] R M Richard and J R Blacklock ldquoFinite element analysis ofinelastic structuresrdquo AIAA Journal vol 7 pp 432ndash438 1969

[28] A Puck and H Schurmann ldquoFailure analysis of FRP laminatesby means of physically based phenomenological modelsrdquo Com-posites Science and Technology vol 62 no 12-13 pp 1633ndash16622002

[29] P D Soden M J Hinton and A S Kaddour ldquoBiaxial testresults for strength and deformation of a range of E-glass andcarbon fibre reinforced composite laminates failure exercisebenchmark datardquo Composites Science and Technology vol 62no 12-13 pp 1489ndash1514 2002

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials