Engineering Fracture Mechanics 99 (2013) 118–131

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Engineering Fracture Mechanics

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Inverse estimation of cohesive zone laws from experimentallymeasured displacements for the quasi-static mode I fractureof PMMA

0013-7944/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engfracmech.2012.11.002

⇑ Corresponding author. Tel.: +82 2 9706309; fax: +82 2 9491458.E-mail address: khg@seoultech.ac.kr (H.-G. Kim).

Jae-Chul Oh, Hyun-Gyu Kim ⇑Department of Mechanical Engineering, Seoul National University of Science and Technology, 172 Gongneung-2dong, Nowon-gu, Seoul 139-743, South Korea

a r t i c l e i n f o

Article history:Received 6 April 2012Received in revised form 26 September 2012Accepted 5 November 2012

Keywords:Cohesive zone lawInverse analysisDigital image correlationField projection methodFinite element method

a b s t r a c t

In this paper, an efficient hybrid procedure combining experimental measurements and aninverse algorithm called the field projection method (FPM) is presented to estimate cohe-sive zone laws for the quasi-static mode I fracture of PMMA. The procedure is made up ofthe measurement of displacements in a region far away from the crack tip using the digitalimage correlation technique, its transfer to finite element (FE) nodes from measurementlocations using the moving least square approximation, and inverse analyses using numer-ical auxiliary fields and the conservation nature of the interaction J- and M-integrals. Sinceit is very difficult to measure the tractions and separations ahead of the crack tip where ahighly nonlinear behavior of materials emerges at very small scales, the present methodcan be a very successful approach to extract the cohesive zone laws from experimentallymeasured displacements in a far-field region. In particular, the FPM using numerical aux-iliary fields facilitates its use for determining the mesh-dependent cohesive zone laws in FEcomputations.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The cohesive zone model treats each potential crack path as two internal surfaces connected by cohesive tractions, anduses a traction–separation curve to describe the separation process between two surfaces. The cohesive energy and the cohe-sive strength, i.e. the area under the traction–separation curve and the peak traction on the traction–separation curve, rep-resent the local fracture toughness and local strength, respectively [1,2]. Hence, the shape of cohesive zone laws is offundamental importance and can greatly influence the fracture behavior of materials. The cohesive zone model has beenwidely applied to simulate delamination, crack propagation and failure process by pre-defining functional forms of cohesivezone laws [3–6]. These previous studies showed successful results for simulating fracture behavior efficiently in computa-tional fracture mechanics. However, the pre-defined cohesive zone laws may fail to correspond with the actual one whichincludes a micro-mechanism governing the fracture process of materials. Nevertheless, experimental measurements of cohe-sive zone laws near the crack tip are still highly non-trivial because the process zone size is very small and stresses cannot bemeasured directly, which requires measurements of the tractions and separations with high resolution and accuracy [7,8].Therefore, an efficient method, experimentally extracting the exact form of cohesive zone laws, is needed to understand themicro-mechanisms of the fracture process.

The extraction of cohesive zone laws is an inverse problem for finding unknown tractions and separations along cracksurfaces from measured displacements or strains. The problem is highly challenging since inverse problems are generally

Nomenclature

rij stress components in measured real fieldrij stress components in auxiliary fieldsui displacement components in measured real fieldui displacement components in auxiliary fieldsni unit normal vector components on the integration contourS[rij, ui,j] measured real field in the interaction integralsbS½rij; ui;j� auxiliary fields in the interaction integrals

JintC ½S; bS� interaction J-integral

MintC ½S; bS� interaction M-integral

tj cohesive traction componentsuþj displacement components on the upper crack surfaceu�j displacement components on the lower crack surfaceDuj separation components across the cohesive zoneAm integration domain of the interaction integralsCc cohesive crack surfacesU integral path around the crack tipNI(x) shape functions in finite element method

J.-C. Oh, H.-G. Kim / Engineering Fracture Mechanics 99 (2013) 118–131 119

ill-conditioned [9–11], so that a small amount of errors in the input data can cause large errors in the inverse solution. Afunctional form of the traction–separation relationship is often assumed a priori, with the associated shape parameters aswell as the cohesive energy, the cohesive strength and the separation length fitted to the measurement data [8,12–16]. Whilesuch approaches have been demonstrated convincingly, the accuracy of the experimentally fitted parameters is highly sen-sitive to the assumed traction–separation profile and an object function to match experimental results with simulations. Thisalso means that the uniqueness of cohesive zone laws obtained from these semi-empirical methods may not be guaranteed.Although the shape optimization approach minimizing an object function with a number of control points on the traction–separation curve [15,16] can be an affordable way to extract the cohesive zone law without a pre-defined form, sophisticatedregularization techniques and experimental measurements in a region near the crack tip may be required to prevent fromobtaining unphysical solutions due to the ill-posedness of inverse problems [11].

One promising method for reconstructing traction–separation relations from measured displacements without any pre-assumption on the shape of cohesive zone laws and comparing measured behaviors with model predictions for variousparameters is the field projection method (FPM) [17]. The FPM constructs an inverse system by combining the interactionJ-integral with analytical auxiliary fields along the contours in the cohesive zone and far-field region. The FPM was demon-strated to provide a systematic way of uncovering the shape of cohesive zone laws governed by the micro-mechanisms of thefracture process [18–20]. However, it is difficult to apply to the analysis of general models involving non-isotropic materialsand arbitrary-shaped cracks. In addition, there is a convergence issue in the FPM using eigenfunction expansions in a com-plex variable representation of analytical auxiliary fields. For that reason, the universality and its ease of numerical imple-mentation of the FPM has been extended by using numerical auxiliary fields [21]. In this augmented FPM, discrete uniformtractions are systematically applied on each element face of the cohesive crack surfaces in turn to construct independentauxiliary fields. The use of both interaction J- and M-integrals in this method reduces the number of required auxiliary fields,compared to the previous method [17]. The algorithm has proven its usefulness for inversely extracting cohesive tractionand separation values from displacements in a far-field region through numerical experiments [21].

Here, we experimentally measured the displacement field of polymethly methacrylate (PMMA) double cantilever beam(DCB) specimens with an end-notched crack by using the digital image correlation (DIC) technique. The DIC is a non-contactoptical measurement technique for obtaining full-field deformation by correlating digital images of material surfaces. Thistechnique can be used to determine the fracture characteristics of various materials. In this way, the DIC can also efficientlymeasure displacements in a region far away from the crack tip during the incremental crack growth in PMMA. However, acommon experimental technique such as the DIC to measure the displacement field inadvertently introduces unwantednoise in the measured data, and measurement locations do not match with the nodal positions in a finite element (FE) mesh.To resolve this problem, the moving least square (MLS) approximation is used to transfer displacements from measurementlocations to the nodes of FE meshes which are used for the inverse analysis. Since the MLS method involves local weightedleast square approximations as a smoothing effect on randomly distributed data, it can be a filtering process which reduces asharp noise in raw measurement displacements. There are many possible methods to transfer measured displacements frommeasurement locations to FE nodal positions. Among them, the MLS approximation can be a good candidate for the datatransfer with smoothing and filtering effects by minimizing a weighted error norm. In this study, we investigate the effectof the MLS approximation on extracted cohesive zone laws with varying the size of influence domains. The augmented FPMusing numerical auxiliary fields is then applied to inversely determine the cohesive tractions and separations along the

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cohesive crack surfaces of PMMA specimens. Since the measured and the auxiliary fields in the FPM use the same mesh,numerical computations of the interaction J- and M-integrals in a far-field region can be performed effectively. The cohesivezone laws extracted from experimentally measured deformation field are dependent on the FE mesh, and one needs to usemesh-dependent cohesive zone laws between crack surfaces in FE models to simulate a crack growth properly. Experimentaland numerical results show that the present hybrid procedure combining experimental measurements and the augmentedFPM using numerical auxiliary fields can be a very efficient way to estimate the cohesive zone laws for a crack growth inmaterials.

2. Field projection method using numerical auxiliary fields

2.1. Interaction integrals between real and auxiliary fields

In the fracture mechanics, one of important progresses is the use of conservative integrals such as J- and M-integrals, i.e.energy release rate represents the characteristic of a cracked body [22]. Acting under this idea, the FPM uses measured dis-placement information in the far-field region to inversely extract the parameters characterizing the deformation and failuremechanisms near the crack tip. Fig. 1 illustrates an edge crack with a cohesive zone behind the crack tip. Since there is nosingularity at the cohesive crack tip, J- and M-integrals at the cohesive crack surfaces Cc in Fig. 1 are equal to the values eval-uated along arbitrary integral path C around the crack tip. In order to extract unknown tractions and separations in the cohe-sive zone, Kim et al. [21] used the interaction J- and M-integrals between the measured real field S ¼ S½rij; ui;j� and the

auxiliary fields bS ¼ bS½rij; ui;j�. By using the domain integral method [23] and the path independency of the interaction inte-

grals such that JintC ½S; bS� ¼ Jint

Cc½S; bS� and Mint

C ½S; bS� ¼ MintCc½S; bS�, the relationship between the tractions and separations at the cohe-

sive crack surfaces can be evaluated by

Fig. 1.calcula

Z 0

�ctj@Duj

@x1þ tj

@Duj

@x1

� �dx1 ¼ �

ZAmðrijuj;1 þ rijuj;1 � rkluk;ld1iÞ

@q@xi

dA ð1aÞ

Z 0

�ctj@Duj

@x1þ tj

@Duj

@x1

� �x1 dx1 ¼ �

ZAmðrijui;jxk þ rijuj;kxk � rkluk;lxiÞ

@q@xi

dA ð1bÞ

where tj are the cohesive traction components on the lower crack surface, Duj ¼ uþj � u�j are the separation componentsalong the cohesive crack surfaces, c is the length of the cohesive zone with the origin (x1 = x2 = 0) located at the crack tip,q is an arbitrary continuous function with unit value on inner contour and zero value on outer contour, Am is the integrationdomain as shown in Fig. 1, and summation over repeated indices is implemented. A detailed derivation of the inverse for-mulation in Eq. (1) for extracting the cohesive tractions and separations is presented by Kim et al. [21].

Since the J- and M-integrals can be used for a cracked body with nonlinear deformation in a small region at the crack tip,these integrals evaluated along a contour far from the crack tip give us characteristic values associated with nonlinear defor-mation and failure mechanisms in the materials. Hence, the cohesive zone laws extracted from measured displacements in aregion far from the crack tip include both nonlinear material behavior near the crack tip and crack-bridging failure mecha-nisms, because the FPM in this study is based on a linear projection of far-fields data onto a local cohesive zone by using theconservation integrals in linear elastic materials.

Schematics of a cohesive zone of size c behind the cohesive crack tip. The interaction J- and M-integrals are taken over a remote path U, and areted using the domain integral method over the shaded region Am.

Fig. 2. Uniform (a) shear and (b) normal tractions imposed on the crack surfaces of the Kth element faces in the cohesive zone; the pseudo nodes aredenoted by open circles.

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2.2. Numerical auxiliary fields

In order to estimate the tractions and separations in the cohesive zone with a finite set of functions, a set of independentauxiliary fields should be provided for the interaction J- and M-integrals. Here, we use numerical auxiliary fields by applyingdiscrete uniform tractions on the FE faces along the cohesive crack surfaces [21]. As illustrated in Fig. 2, uniform normal andshear tractions are applied separately on the faces of pair elements on the cohesive crack surfaces to establish independentauxiliary fields. Let bSK;i denote the numerical auxiliary fields generated by imposing uniform tractions �tK in xi direction onKth elements in the cohesive zone as described in Fig. 2. Since the numerical auxiliary fields generated by different (K, i) com-binations are mutually exclusive, we obtain independent auxiliary fields by applying discrete uniform tractions sequentiallyon cohesive crack surfaces in the FE model.

When constant tractions and linear separations on element faces in the cohesive zone are assumed to determine thecohesive tractions and separations [21], Eq. (1) can be recast by

XM

E¼1

ZCE

@DuðK; iÞj

@x1dx1

" #tE

j þXKþ1

I¼K

tKj

ZCK

@NI

@x1dx1

" #DuI

j ¼ �JintC ðS; bSðK; iÞÞ ð2aÞ

XM

E¼1

ZCE

@DuðK;iÞj

@x1x1 dx1

" #tE

j þXKþ1

I¼K

tKj

ZCK

@NI

@x1x1 dx1

" #DuI

j ¼ �MintC ðS; bSðK;iÞÞ ð2bÞ

where tEj are unknown tractions,E = (1, . . . , M) are the element face numbers, NI(x1) are the shape functions of nodes I, and M

is the total number of element pairs in the cohesive zone. It should be noted that the solutions of inverse equations in Eq. (2)are not affected by the magnitude of uniform tractions in the auxiliary fields. Finally, by solving the above equations, we canobtain the cohesive zone law in terms of tractions tE

j and separations DuIj on cohesive crack surfaces.

3. Measurement of displacements in a far-field region

Digital image correlation (DIC) technique is a computer based image process as a non-contact optical technique to obtainfull-field information by recording the motion of speckle patterns on a specimen surface before and after the deformation ofa body. Although the DIC system cannot capture the fracture process in detail at micro- and nano-scale, it provides us themacro-scale measurement of full-field displacements in a region far from the crack tip. The DIC tracks the position of thesame physical points shown in a reference image and a deformed image. To achieve this, a square subset of pixels is iden-tified on the speckle patterns around the region of interest on a reference image and their corresponding locations are deter-mined on the deformed image. The displacements for subset centers can be estimated to minimize a correlation whichmeasures how well subsets match between the two digital images [24]. In order to obtain accurate DIC results, some param-eters including speckle size, subset size, subset overlap and gray level should be optimized through a series of experiments.Since the measurement locations of the DIC system do not match with FE nodes, the MLS approximation [25] is used totransfer measured displacements from subset centers to FE nodes. We also use the DIC system to obtain mechanical prop-erties of PMMA specimens in tensile test.

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3.1. PMMA double cantilever beam specimens

PMMA double cantilever beam (DCB) specimens of 2 mm thickness are loaded by the tensile test machine (INSTRON5848). Displacement controlled loading is applied incrementally to the DCB specimens to archive a stable quasi-static modeI crack growth. Initial sharp cracks are created by applying repeated loads on the DCB specimens with an initial notch madeby a razor knife. The size of the DCB specimens is shown in Fig. 3. Random speckle patterns of block dots are sprayed onwhite background painted on the specimens for the DIC measurement. The DIC system with two CCD cameras of full 0.8mega pixels is set in the front of the specimens, and the distance between the cameras and the specimens is adjusted onbasis of the object area to be measured. The subset size of DIC computations is related to the smoothness and accuracy ofevaluated data. If the subset size is comparatively small, the accuracy of the measurement data decreases but a local effectcan be captured better than those with large subsets. Since the inverse algorithm can be sensitive to local noise in the mea-sured displacements, the subset size should be as large as possible while preserving the traction–separation relationshipsbetween the cohesive crack surfaces. We take the subset size of 40 by 40 pixels centered at every computational point withdistance 10 pixels. The full-field of displacements measured by the DIC system is shown in Fig. 4. More high-resolution cam-eras and more elaborated spray patterns on the specimens can improve the quality of measurement data, but a smoothingprocess should be followed for filtering noisy data as a regularization procedure in solving inverse problems. We use the MLSapproximation for a smoothing and filtering process of data transfer from measurement locations to FE nodes, which min-imizes a weighted error norm with a monomial basis. Unlike the previous studies [8,22] which have tried to measure directlythe cohesive zone laws near the crack tip, the present method does not need the information near the crack tip or in thecohesive zone. It should be noted that displacements in the layers above and below the crack surfaces are not taken becausethe DIC system cannot compute displacements accurately in these layers due to insufficient and discontinuous subsetsacross the crack surfaces. Sequential images are taken during the deformation of the DCB specimens, and the DIC systemapplied the correlation algorithm to evaluate displacements as shown in Fig. 4. Since the crack is not growing continuouslyfor sequential impulses of displacement loadings, we chose the image just before it grows when the crack growth is suitablystable with a long straight crack. The displacement field at this moment is then corresponding to a stable quasi-static mode Icrack growth, which can be used to extract the cohesive zone laws for the simulation of a crack growth in PMMA.

3.2. Material properties of PMMA

Since the FPM presented in the previous section is formulated for a linear elastic media, the domain Am in Eq. (1) shouldbe chosen by an elastically-deforming region. Hence, we need to find a linear elastic deformation region of the mechanicallyloaded specimens to evaluate the interaction J- and M-integrals correctly. It should be noted that nonlinear deformation nearthe crack tip is implicitly implemented in the traction–separation relationships between the cohesive crack surfaces becausethe entire domain except the cohesive zone is assumed that the deformation is linear elastic. As a result, the cohesive zonelaws should represent both the nonlinear zone near the crack tip and the actual cohesive zone ahead of the crack tip.

Two-dimensional strains of the tensile specimen shown in Fig. 5 are evaluated by the DIC system to determine the mate-rial properties of PMMA. The tensile specimen satisfying the ISO standard is deformed by the same test machine used for acrack growth of the DCB specimens. The deformation images are gradually taken by the DIC system, and stresses and strainsin the object area are fitted by the least square method to define the elastic modulus of PMMA. Furthermore, the Poisson’s

Fig. 3. PMMA double cantilever beam specimen with an end-notched crack and pin-holes for grips; the magnified image on the right is the initial crackshape observed by a microscope; all units are in mm.

Fig. 4. Displacement controlled loading is applied to the DCB specimens for a stable crack growth; distribution of displacements measured by DIC in (a) x-direction and (b) y-direction.

J.-C. Oh, H.-G. Kim / Engineering Fracture Mechanics 99 (2013) 118–131 123

ratio is evaluated by averaging strain values in x- and y-directions at the middle point of the specimen (see Fig. 5). Fig. 6shows the stress–strain curves of PMMA in elastic and nonlinear deformation ranges, respectively. The elastic modulus Eand the Poisson’s ratio m are determined as 2.692 GPa and 0.295, respectively. We find that linear elastic deformation is validin a region of which the equivalent stress is less than roughly 3 MPa, as indicated in Fig. 6b.

Fig. 5. Tensile test for obtaining the mechanical properties of PMMA; the closed red quadrangle denotes a middle point of the specimen.

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3.3. Transfer of the measured displacements from subset centers to FE nodes

Since the subset centers in the DIC system do not match with the nodes of FE meshes for extracting the cohesive zonelaws in the FPM, we need to transfer measured displacements suitably from subset centers to FE nodes. Additionally, themeasured displacements contain noise and fluctuations, which cause a deterioration of inverse solutions due to an inherentill-conditioning of the inverse system even though a large subset size is set to reduce the measurement noise. In our study,the measured displacements at subset centers are transferred to FE nodes by using the MLS approximation with the linearbasis, and then stresses and strains are obtained by imposing nodal displacements on the FE mesh. The MLS can approximaterandomly distributed data by taking space-varying coefficients in the least square method with a basis and weight functionsdefined on influence domains. Since the MLS approximation can reproduce the basis completely, the measured displace-ments can be transferred into mismatched FE nodes with a reasonable accuracy. This process can reduce experimental noisein the measured displacements, and makes the inverse process numerically stable. As a result, data smoothing and filteringcan be performed by a least square approximation with a basis in a local domain which is moving for a global approximation.The characteristics of data transfer from subset centers to FE nodes may depend on the size of influence domains in the MLSapproximation. In order to preserve a well-defined MLS approximation, the influence domains should be overlapped by more

Fig. 6. Stress–strain curves of PMMA specimens: (a) linear deformation range and (b) nonlinear deformation range measured by tensile test machine; rawstress–strain data are denoted by open circles in (a).

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than the number of a basis. As the size of influence domains increases, local fluctuations decrease because more measure-ment data are involved in the approximation. However, the original measurement data can be distorted when the size ofinfluence domains is comparatively large. In this study, we select the size of influence domains to satisfy the path indepen-dency of inverse solutions as best as possible.

4. Cohesive zone laws for the quasi-static mode I fracture of PMMA

In this section, the inverse extraction of cohesive zone laws is carried out by using experimentally measured displace-ments and the FPM using numerical auxiliary fields. In the previous study [21], the numerical data of measured field, whichare obtained by solving FE model with cohesive elements of polynomial forms of cohesive zone laws, are used to verify thefield projection method using numerical auxiliary fields. In this study, experimentally measured displacements are imposedon three different meshes, as shown in Fig. 7, by using the MLS approximation. We use the general purpose FE code ABAQUS/Standard v6.10 to evaluate stresses and strains of measured field and auxiliary fields. In order to construct the auxiliary fieldsbS, we apply uniform tractions of 1:0� 103 N=mm2 element-by-element along the crack surfaces. It is convenient to use thesame mesh for the measurement and identification of the cohesive zone laws, because we need to compute the interaction J-

Fig. 7. Double cantilever beam models with an edge crack: (a) mesh A, (b) mesh B and (c) mesh C. The integration domains XA, XB and XC are used for thecalculation of the interaction J- and M-integrals.

Fig. 8. Schematic illustration of the weighted center of a cohesive zone.

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and M-integrals between the measured real field and the auxiliary fields in the same domain. As shown in Fig. 7, three dif-ferent mesh densities denoted by mesh A, mesh B and mesh C are used, and three integral domains XA, XB and XC of increas-

Fig. 9. Von-Mises stress distribution of the PMMA double cantilever beam; element edges of the mesh are removed to avoid over-shading of the stresscontours.

J.-C. Oh, H.-G. Kim / Engineering Fracture Mechanics 99 (2013) 118–131 127

ing distance from the crack tip are taken to investigate the effect of the integral contour to the inverse solution. In addition,the size of influence domains in the MLS approximation and mesh dependency to the inverse solution are also investigated inthis study.

To find the location and the size of cohesive zone, we can estimate the weighted center of the cohesive zone with respectto tjoDuj/ox1, denoted by xc

1 ¼ MCc ðSÞ=JCcðSÞ, which should lie within the cohesive zone as illustrated in Fig. 8. Note that the

weighted center is not the mid-point of the cohesive zone, but represents a first order estimate of the cohesive zone location[21]. In addition, in the absence of contact between the cohesive crack surfaces, the normal tractions within the cohesivezone should always be positive. In this regard, we perform iterative inverse analysis under the condition of positive normaltractions and no penetration of crack surfaces in the cohesive zone around the weighted center. As a result, the location ofcohesive crack tip and the size of cohesive zone can be determined to estimate the cohesive zone laws properly.

4.1. The size of influence domains in the MLS approximation

First, we investigate the effect of the size of influence domains in the MLS approximation, which alters the real-fielddeformation transferred from measurement locations to FE nodes. As a result, extracted cohesive zone laws can be variedby the size of influence domains due to a sensitive nature of the inverse system. Since the equivalent stress in some regionsof the integration domain A is greater than the elastic limit indicated in Fig. 6b (see Fig. 9), the interaction integrals in theintegration domain A can be different from those in the integration domains B and C. Hence, we do not consider the inte-gration domain A to investigate the effect of the size of influence domains. The inverse solutions using the mesh B are com-pared by changing the sizes of influence domains. As shown in Fig. 10, the mismatch between the cohesive zone laws usingthe integration domains B and C increases when the size of influence domains is larger or smaller than 14 mm. Hence, wetake 14 mm as the size of influence domains in the MLS approximation for extracting cohesive zone laws with the mesh B.Similarly, the sizes of influence domains for the meshes A and C are taken by 14 mm and 17 mm to be path independent asbest as possible for the integration domains B and C. Note that the variation of inverse solutions from displacement fieldbased on the MLS approximation is not significant when the size of influence domains is in a certain range at these selectedvalues. The dependency of inverse solutions to the size of influence domains is caused by a small change of stresses andstrains in the integral domain due to different number of data involved in the MLS approximation.

As aforementioned, the conservation nature of the J- and M-integrals should be preserved when the deformation in theintegral domains is within the linear elastic range. The cohesive zone law using the integration domain A is deviated fromthose using the integration domains B and C as shown in Fig. 10. The integration domain A may lead to somewhat differentcohesive zone laws in terms of fracture energy due to nonlinear deformation in this domain. Since the stresses in the inter-action J- and M-integrals are obtained from measured displacements assigned on FE nodes by using linear elastic constitutiveequations, the stresses near the cohesive crack tip are different from actual values. As a result, we cannot use integrationdomains near the crack tip when the linear deformation is not valid in this region, because the FPM in this study is formu-lated on the basis of the linear elasticity. However, the deformation in a far-field region is the same as the actual one, and thecrack growing behavior such as crack opening loads and the energy released to the domain can be obtained by using thecohesive zone laws. Consequently, the characteristics of nonlinear deformation near the crack tip and crack-bridging failuremechanisms are implicated in the cohesive zone laws extracted from experimentally measured displacements in a far-fieldregion.

Fig. 10. Numerically extracted cohesive zone laws in mesh B using integration domains A–C of which nodal displacements are transferred fromexperimentally measured data with different size of influence domains in the MLS approximation: the sizes of influence domains are (a) 13 mm, (b) 14 mmand (c) 15 mm.

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4.2. Mesh dependency of cohesive zone laws

Although mesh densities for the meshes A, B and C are different, deformations in a far-field region are close to each otherbecause the same measured displacements at the subset centers of the DIC system are transferred to the FE nodes of thesethree meshes. However, traction–separation relationships between the cohesive crack surfaces can be dependent on themesh density because of different resolutions of deformation fields near the cohesive zone. In other words, the cohesive zonelaws can be associated with the mesh design near the cohesive zone in order to represent the characteristics of a growingcrack correctly with different mesh densities. We obtain mesh-dependent inverse solutions as plotted in Fig. 11. As the meshis refined, a sharp increase of traction at the beginning of separation can be captured in the inverse analysis. In addition, thecohesive zone size for a coarse mesh is longer than that for a fine mesh. When the mesh is not fully refined in a region nearthe cohesive zone, the traction–separation relationships can be different because the resolution of uniform tractions and lin-ear separations on the faces of bilinear finite elements neighboring with cohesive crack surfaces is not enough to capture theshape of cohesive zone laws. Although the shape of cohesive zone laws is varied with mesh density, fracture energies asso-ciated with these cohesive zone laws are almost the same. The mesh dependency of inverse solutions of the augmented FPMwith numerical auxiliary fields is also explained by Kim et al. [21], in which FE results with pre-defined cohesive zone lawsare used instead of experimentally measured displacements. We can expect that the cohesive zone laws approach to a the-oretical traction–separation relationship as the mesh density increases. Although we do not study details about the conver-gence of cohesive zone laws with mesh refinement, a tendency to converge towards a cohesive zone law can be found inFig. 11.

The stress fields induced by tractions on the element crack faces cannot be accurately represented by a coarse FE mesh[21]. Tomer et al. [26] showed that very fine FE meshes are necessary to resolve details of nonlinear deformation fieldsaround the cohesive crack tip and the traction distributions within the cohesive zone. Turon et al. [27] showed that a mod-ified cohesive zone law with a reduced cohesive strength and an enlarged cohesive zone should be used for the simulation ofa crack growth with coarse meshes. Consequently, the cohesive zone laws representing the nonlinear material deformationand the crack-bridging mechanisms can be dependent on the FE mesh near the cohesive crack tip when the entire domain

Fig. 11. Cohesive zone laws extracted from measured displacements in the integration domain B for FE meshes with different mesh densities.

J.-C. Oh, H.-G. Kim / Engineering Fracture Mechanics 99 (2013) 118–131 129

except the cohesive zone is modeled by using a linear elastic material. Hence, we have to use mesh-dependent cohesive zonelaws in accordance with the mesh design when the mesh is not fully refined.

The feature of the cohesive zone laws shown in Fig. 11 is the non-zero stress at zero crack opening. When the cohesiveseparation initiates, the cohesive traction acting on the crack surfaces can be started from zero or a certain value. In general,the cohesive traction increases from zero value as the cohesive separation increases until a maximum value is reaches, andthe cohesive traction decreases as the cohesive separation increases after the cohesive traction reaches the maximumstrength. In this cohesive model, initially elastic cohesive law is implemented between the crack surfaces. The initially elasticcohesive-zone law has been widely used because it is easy to use numerically stable algorithm when pre-defined crack sur-faces are considered in the crack growth simulation. However, mesh-dependent solutions may be obtained when initiallyelastic cohesive-zone laws are used between all inter-element faces [28,29]. The extracted traction–separation relationshipsshow initially rigid cohesive zone laws, which can avoid the element detachment when the normal traction is below than acritical value.

Fig. 12. Comparison of von-Mises stresses in the integration domains B and C; (a) experimental results and (b) FE results by solving a forward problem withthe mesh C.

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4.3. Forward simulation using an extracted cohesive zone law

In order to verify the cohesive zone laws extracted from experimentally measured displacements, a forward problem ofquasi-static crack growth is solved by using cohesive elements placed between finite elements. The linear elastic propertiesobtained from tensile tests of PMMA are used for the FE model of DCB specimens. It should be noted that linear elastic mate-rials should be used in finite elements on the entire domain except the cohesive zone because the FPM in this study is basedon a linear projection of far-fields data onto the cohesive zone by using the conservation integrals in linear elastic materials.The mesh C is used in the forward analysis, and loads are applied at the left ends of top and bottom surfaces of the FE model.The cohesive elements using the extracted cohesive zone laws are implemented in the user-defined elements (UEL) subrou-tine in ABAQUS. A continuous cohesive zone law through the cubic spline interpolation of a set of discrete tractions and sep-arations along the cohesive crack surfaces is used to compute the tangent stiffness in the UEL. As shown in Fig. 12, von-Misesstress of FE results of the forward problem is compared with that computed from experimentally measured displacements inthe integration domains B and C. Stress distribution in experiments is not symmetric due to an inaccuracy in experimentalmeasurements. On the contrary, forward solutions are perfectly symmetric with respect to the crack surfaces. However,stress contours in Fig. 12 have a similar feature in terms of spatial patterns and levels of stresses. Note that forward solutionsusing the extracted cohesive zone law show a good agreement with measured displacements in terms of the J- and M-inte-grals. In general, an inverse analysis is performed to minimize errors between measured data and calculated results. Simi-larly, the forward solutions using inversely estimated cohesive zone laws match with experimentally measureddisplacements with respect to the conservation integrals.

5. Concluding remarks

The inverse extraction of crack-tip cohesive zone laws from far-field deformation information is a challenging but impor-tant task for quantitative understanding of the micro-mechanics involved in the fracture process. In this work, we have pro-posed a hybrid (experimental/numerical) method which makes use of both experimentally measured displacements andinverse numerical computations. The technique involves an efficient computational scheme of inverse analysis, opticalexperiments based on the DIC, and their approximation by using the MLS method. The size of influence domains in theMLS approximation is taken to satisfy the path independency of inverse solutions as best as possible. The numerical auxiliaryfields are generated by imposing uniform tractions element-by-element along the crack faces. The path-independent inter-action J- and M-integrals between these auxiliary fields and real measurement field are then used to extract the cohesivezone laws. Integration domains should be taken not to involve nonlinear deformation in the inverse formulations, whichcan be easily determined from stress contours computed from experimentally measured displacements assigned on FEnodes. In this study, the cohesive zone laws for the quasi-static mode I fracture of PMMA are estimated effectively by theinverse procedure. Our results show that path-independent cohesive zone laws can be obtained unless integration domainsare close to the crack tip. In addition, our numerical experiments, which have been performed to investigate the effect ofmesh density, show that the cohesive zone laws depend on the mesh design of which resolution is not sufficient to representdeformations near the cohesive crack tip. The proposed procedure is a useful approach to estimate the cohesive zone lawsfrom experimentally measured displacements in a far-field region. Improved results can be obtained by employing higherresolution DIC images and a more refined scheme for transferring measured displacements from subset centers to FE nodes.Cohesive zone laws in this study include not only tractions and separations in the crack-bridging region but also nonlineardeformation near the crack tip. The development of a nonlinear FPM is required for the isolation of nonlinear deformationeffects on the cohesive zone laws.

Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF)funded by the Ministry of Education, Science and Technology (2010-0004875).

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