Int J Fract (2006) 141:373–393DOI 10.1007/s10704-006-9000-2

ORIGINAL PAPER

Multiple cohesive crack growth in brittle materialsby the extended Voronoi cell finite element model

Shanhu Li · Somnath Ghosh

Received: 14 October 2005/Accepted: 6 July 2006 / Published online: 20 October 2006© Springer Science+Business Media B.V. 2006

Abstract This paper is aimed at modeling thepropagation of multiple cohesive cracks by theextended Voronoi cell finite element model orX-VCFEM. In addition to polynomial terms, thestress functions in X-VCFEM include branch func-tions in conjunction with level set methods andmulti-resolution wavelet functions in the vicinityof crack tips. The wavelet basis functions are adap-tively enriched to accurately capture crack-tipstress concentrations. Cracks are modeled by anextrinsic cohesive zone model in this paper. Theincremental crack propagation direction andlength are adaptively determined by a cohesivefracture energy based criterion. Numerical exam-ples are solved and compared with existing solu-tions in the literature to validate the effectivenessof X-VCFEM. The effect of cohesive zone param-eters on crack propagation is studied. Additionally,the effects of morphological distributions such aslength, orientation and dispersion on crack propa-gation are studied.

Keywords Extended Voronoi cell finite elementmodel · Multi-resolution wavelets · Cohesivezone model · Multiple crack propagation

S. Li · S. Ghosh (B)Department of Mechanical Engineering,The Ohio State University,650 Ackerman Road, Columbus, OH 43202USAe-mail: ghosh.5@osu.edu

1 Introduction

Numerical analysis and simulation of the growthand interaction of multiple cracks in materials isa challenging enterprise due to various kinematic,morphological and constitutive complexities thatgovern this process. Conventional finite elementapproaches suffer from very slow convergencesince the element formulation does not account forhigh gradients and singularities. Even a very highdensity mesh cannot overcome pathological meshdependence near the crack tips and avoid biasingthe direction of crack propagation. The difficultiesaggravate significantly in the presence of multiplecracks, due to their interaction with each other.Various methods have been proposed for improv-ing the effectiveness of computational methods inmodeling cracks. These include the singular ele-ment method using quarter-point elements(Barsoum 1976, 1977; Henshell and Shaw 1975;Hibbit 1977), the method of superposition thatintroduces singular terms to the finite elementinterpolations (Yagawa et al., 1980; Yamamoto andTokuda 1973), or the hybrid singular element meth-ods (Lin and Tong 1980; Piltner 1985; Tong et al.1973; Tong 1977), which augment interpolationfunctions using stress intensity factors from clas-sical elasticity theory. While most of these analysesare limited to stationary cracks, it is only in therecent years that effective numerical methods forsimulating crack propagation are being proposed.

374 S. Li, S. Ghosh

In this regard, the cohesive zone models or CZM(Camacho and Ortiz 1996; Foulk et al. 2000;Geubelle 1995; Needleman 1987, 1990, 1992; Ortizand Pandolfi 1999; Tvergaard 1990) have emergedas important tools for modeling crack propaga-tion in homogeneous and heterogeneous materi-als. A large number of studies e.g. Camacho andOrtiz (1996), Carpinteri (1989) and Xu andNeedleman (1994) have simulated crack propa-gation by inserting special cohesive elements be-tween continuum elements. The use of a highlyrefined computational mesh, especially near thecrack tip is still a requirement, even though theeffect is mitigated due to the finite crack tip stresswith the CZM. Alternatively, intra-element enrich-ment approaches, based on the incorporation ofembedded discontinuities in displacement or strainfields have been proposed (see Jirasek 2000 fora review), which eliminates mesh dependent pre-diction of the evolving crack path and hence theneed for remeshing. The extended FEM or X-FEM(Belytschko and Black 1999; Belytschko et al. 2001;Dolbow et al. 2000; Moës and Belytschko 2002;Moës et al. 1999) is a powerful recent additionto this family of intra-element enrichment. Cohe-sive crack propagation has been modeled in thiswork by using the partition of unity concept toincorporate local enrichment functions that allowsthe preservation of the general displacement basedFEM formalism.

The Voronoi cell finite element method(VCFEM), developed on the principles of theassumed stress hybrid FEM formulation (Tong et al.1973; Tong 1977), has had considerable success inthe micromechanical analysis of multi-phase het-erogeneous materials (Ghosh et al. 2000; Ghoshand Moorthy 1998, 2004; Ghosh and Mukhopad-hyay 1991; Li and Ghosh 2004, 2006; Moorthy andGhosh 1996, 2000). By introducing augmentationstress functions having forms that are motivatedfrom analytical micromechanics, the method is ableto overcome the need for high resolution meshin the proximity of morphological discontinuities,such as inclusions, voids, cracks, etc. VCFEM isbased on an unstructured mesh of closed polyg-onal cells as shown in Fig. 1a, where each cellcorresponds to the immediate neighborhood ofa heterogeneity like an inclusion or crack. Whilea typical mesh of Voronoi polygons evolves from

tessellating a heterogeneous domain (Ghosh andMukhopadhyay 1991; Ghosh and Moorthy 2004),an element in VCFEM need not adhere to this defi-nition and can be any closed polygonal cell consist-ing of a heterogeneity in a surrounding matrix. Forexample, in this paper, each Voronoi element in-volves a polygonal sub-domain that encompassesan initial crack. Conceptually, VCFEM has a strongresemblance to the mesh-free methods. Each cell inVCFEM represents the neighborhood and regionof immediate influence of a heterogeneity, whichmay be perceived of as the support domain for spe-cially constructed interpolation or kernel functionsof field variables and their evolution. A high levelof accuracy with significantly reduced degrees offreedom has been achieved with VCFEM in Ghoshet al. (2000); Ghosh and Moorthy (1998, 2004),Ghosh and Mukhopadhyay (1991), Li and Ghosh(2004, 2006) and Moorthy and Ghosh (1996, 2000).Computational efficiency of VCFEM is substan-tially higher than many conventional FE models.The cohesive crack propagation model has beenincorporated in VCFEM in Ghosh et al. (2000)and Li and Ghosh (2004) to model interfacial deb-onding in fiber reinforced composites. However,in these models, the debonding or crack evolutionpath is along the interface and hence the cohesivezone regions are known a-priori. In the event thatthe crack branches off into the matrix, the propaga-tion path is no longer pre-assessed and needs to bedetermined in each load increment, consistent withthe local state of stresses, strains and morphology.This task is considerably more formidable since aslight deviation from the actual path can lead tocompletely wrong prediction.

Following the trends set in Belytschko et al.(2001), Dolbow et al. (2000), Moës and Belytschko(2002) and Moës et al. (1999), an extended VCFEMor X-VCFEM has been recently developed in (Liand Ghosh 2006) for modeling the growth of mul-tiple cohesive cracks in brittle materials. The modelaccounts for interaction between cracks andinvokes an adaptive crack growth formulation torepresent the continuously changing direction ofevolving cracks. X-VCFEM augments the conven-tional VCFEM model by adding multi-resolutionwavelet stress functions (Glowinski et al. 1990;Jaffard 1992; Qian and Weiss 1993) in the vicinityof the crack tip, level set based branched functions

Multiple cohesive crack growth in brittle materials 375

Fig. 1 (a) A mesh of Voronoi cell elements, each containing a single pre-existing crack, (b) a typical Voronoi cell elementshowing different topological features and loads

for stress discontinuity across the crack. The incre-mental crack propagation direction and length areadaptively determined by a cohesive fractureenergy based criterion. No remeshing is neededin X-VCFEM for simulating crack growth, andthis adds to its desirability and effectiveness. Thepresent paper is an extended the developmentsin Li and Ghosh (2006) by adding crack mergingto its growth mechanism and investigates the pre-dictions of the model for different geometric andconstitutive manifestations. It begins with a reviewof the X-VCFEM formulation and demonstratesits convergence through a numerical example.X-VCFEM is then used to understand the influ-ence of cohesive parameters like peak stress andcritical separation, on crack growth in monolithicbrittle materials. Subsequently, the effect of mor-phological distributions including crack interac-tion, clustering, alignment, etc. on the growth andcoalescence are studied as factors critical to thefailure process.

2 Summary of X-VCFEM formulation forcohesive cracks

A comprehensive account of the development ofthe extended Voronoi cell finite element model(X-VCFEM) is given in Li and Ghosh (2006).Figure 1(a) shows a pre-cracked microstructuralregion � with a dispersion of cracks that is tes-sellated into an unstructured mesh of N polygonalVoronoi cell elements. The topology andconstituents of a typical Voronoi cell (VC) element�e are depicted in Fig. 1(b). Although each VCelement contains a single crack initially, the adap-

tive X-VCFEM formulation allows cracks to crossboundaries during the course of their evolution.The fracture process zone for each crack is rep-resented by a cohesive zone model. The elementboundary ∂�e with outward normal nE encom-passes segments of prescribed traction �te, pre-scribed displacement �ue and inter-elementboundaries �me, i.e. ∂�E

e = �te⋃

�ue⋃

�me. Theincompatible displacement field across a crack �crwith a normal ncr is facilitated through a set of con-nected node-pairs along the crack length. In theassumed stress hybrid finite element formulation,stationary conditions of the element energy func-tional in the variational principle yield weak formsof the kinematic equation and traction reciproc-ity conditions on the boundary and crack face. Inan incremental formulation for small deformationelasticity with evolving cracks, the element energyfunctional �e is expressed as:

�e(σij, �σij, uEi , �uE

i , ucri , �ucr

i )

= −∫

�e

�B(σij, �σij)d� −∫

�e

εij�σijd�

+∫

∂�e

(σij + �σij)nEj (uE

i + �uEi )d∂�

−∫

�te

(t̄i + �t̄i)(uEi + �uE

i)d�te

+∫

1�cr

(σij + �σij)ncrj (

1ucr

i + �1

ucri )d�cr

−∫

2�cr

(σij + �σij)ncrj (

2ucr

i + �2

ucri )d�cr

−∫

�cr

∫ 1ucr

i +�1

ucri −

2ucr

i −�2

ucri

1ucr

i −2

ucri

tcohi d(

1ucr

i −2

ucri )d�cr

(1)

376 S. Li, S. Ghosh

where σ , ε and B = ( 12σ : S : σ ) are the equili-

brated stress field, the corresponding strain fields,and the complimentary energy, respectively in anelement interior. uE is a kinematically admissibledisplacement field on the element boundary ∂�E

eand ucr represents displacements on the internalcohesive-crack surfaces �cr. Note that body forcesare neglected in this formulation. Variables withsuperscript E are on the element boundary whilethose with superscripts cr correspond to the cracksurface. The prefix � denote increments of respec-

tive variables. The notations1

(•) and2

(•) representtwo sides of the cohesive crack surface in a Voronoicell element. The last term in Eq. (1) correspondsto work done by cohesive tractions tcoh

i due to cracksurface separation. The element kinematic equa-tion is satisfied in a weak sense from the stationarycondition, obtained by setting the first variationof �e in Eq. (1) with respect to stress incrementsto zero. Weak satisfaction of the traction reciproc-ity conditions on (i) the inter-element boundary�me, (ii) the domain traction boundary �te and (iii)

the crack surfaces1

�cr and2

�cr (two sides of thecrack) are obtained by setting the first variationof the total energy functional � = ∑N

e=1 �e with

respect to the displacement fields �uE, �1

ucr and

�2

ucr, respectively to zero.The cohesive tractions are given by a rate inde-

pendent extrinsic linear cohesive zone model(Camacho and Ortiz 1996; Ortiz and Pandolfi 1999),which has infinite stiffness at low separations. Theeffective normal traction-separation response ofthis model is depicted in Fig. 2 (a, b). The tractionacross the cohesive surface is expressed in terms offree cohesive fracture energy potential φ as:

tcoh = ∂φ

∂δnn + ∂φ

∂δtt = t

δ(δnn + β2δtt) (2)

where δn and δt correspond to the normal and tan-gential components of the opening displacementsover the cohesive surface in the normal (n) and tan-gential (t) directions, respectively. The magnitude

of traction t = ∂φ∂δ

=√

tcohn

2 + β−2tcoht

2and the

effective opening displacement δ =√

δ2n + β2δ2

t .

Here tcohn and tcoh

t are the normal and tangentialcomponents of surface tractions and β is a coupling

coefficient to allow assignment of different weightsto normal and tangential opening displacements.For increasing δ, t is expressed as

t ={

σmax(δe−δ)δe

∀δ < δe

0 ∀δ ≥ δe(3)

The displacement δe is the separation at whicht goes to zero and σmax is the peak value of t.Unloading from any point on the traction-sepa-ration curve, proceeds along a linear path from thecurrent position to the origin following the relation

t = σmax

δe

δe − δmax

δmaxδ ∀δ ≤ δmax ≤ δe (4)

Reloading occurs with a reduced stiffness in com-parison with the original stiffness. For negativenormal displacement (compression), stiff penaltysprings with high stiffness are introduced betweenthe node-pairs on the crack face.

In X-VCFEM formulation, the equilibrium con-ditions and constitutive relations in the elementinterior, as well as the compatibility conditions onthe element boundary and crack surface are satis-fied a-priori in a strong sense. Equilibrated stressfields in the interior of the element and compati-ble displacement fields on the element boundariesand crack faces are assumed in this method. Inthe incremental formulation, equilibriated stressincrements are obtained from the stress functionsincrements ��(x, y), and inter-element compati-ble displacement fields across the element bound-ary ∂�E

e and continuous displacement fields alonga crack face �

1/2cr (superscript 1/2 correspond to two

sides of the crack) in an element are generatedby interpolation of nodal displacements (Ghoshet al. 2000; Li and Ghosh 2006; Moorthy and Ghosh1996, 2000). Stress functions are constructed by thesuperposition of three different components, i.e.� = �poly +�branch +�wvlt. The stress incrementsin an element are computed by adding contribu-tions from each individual function to yield⎧⎨

⎩

�σxx

�σyy

�σxy

⎫⎬

⎭e

=[[P]poly [P]branch [P]wvlt

]

e

⎧⎨

⎩

�βpq

�βst

�βm,n,k,l

⎫⎬

⎭e

(5)

Multiple cohesive crack growth in brittle materials 377

Fig. 2 Normal and tangential traction-separation laws for the extrinsic linear cohesive zone model

{�β_e is the column of unknown stress coefficientincrements associated with the stress interpolationmatrix [P(x, y)]e. The pure polynomial component�poly represents far field stresses away from thecrack tip. The branch function �branch facilitatesdiscontinuity in stresses across the cracksurfaces without affecting the solution in the con-tinuous region beyond the crack. For the functionalrepresentation of the crack surface, a vector levelset method (Belytschko et al. 2001; Sethian 2001;Ventura et al. 2002, 2003) is used. An approxima-tion to the crack surface �

1/2cr in Fig. 1 is constructed

to describe the discontinuous stress fields acrosscrack paths in terms of a signed distance func-tion. The branch stress function increment is con-structed in terms of the radial distance and angularfunctions, based on this function. The multi-reso-lution wavelet stress function �wvlt is constructedfrom the family of Gaussian functions, i.e.

��a,b,c,d(ξ , η) = e−(ξ−b

a )2/2e−(η−d

c )2/2�βa,b,c,d (6)

The dilation and translation parameters (a, c)and (b, d) respectively, can vary in a continuousfashion. By changing the translation parameters,the multi-levels of wavelet bases can be made toclosely follow a moving crack tip. The dilationparameter with compact adjustable window sup-port can be used to provide high refinement andresolution near the crack tip. The wavelet basedstress function is constructed in a local orthogo-nal coordinate system (ξ , η) centered at the cracktip, with the ξ direction corresponding to the localtangent at the crack surface. For implementationin multi-resolution analysis involving discrete lev-

els, the translation and dilation parameters shouldbe expressed as discrete multiples of some startingvalues. Figure 3 shows the distribution of multiplelevels of wavelet bases in an extended Voronoi cellelement. This region of influence of the waveletbases is positioned symmetrically about the crackin the vicinity of an evolving crack tips. The crosses(×) corresponds to the position of each waveletbasis function bn, dn at a lower (m) level. As shownin Fig. 3, the squares (�) correspond to additionallocations at a higher level (m + 1) in the multi-resolution algorithm. The region covered by themulti-level wavelet functions reduce successivelywith levels, but provide higher resolution. Thus theregion of influence of the (m + 1)-th level wave-lets is smaller than that for m-th level. The waveletenriched incremental stress function in X-VCFEMis thus written as

��wvlt(ξ , η)

=mn,nn,kn,ln∑

m=1,n=−nn,k=1,l=−ln

��m,n,k,l(ξ , η) (7)

where the translation parameters n and l rangefrom −nn and −ln to nn and ln. This allows thewavelets to be positioned symmetrically about thecrack (Fig. 3). The addition of higher level waveletbases to the stress function, marked by squares (�)in Fig. 3, is done adaptively in accordance with astrain energy based element error measure that hasbeen derived in Li and Ghosh (2006) and Moorthyand Ghosh (2000). The necessary condition for sta-bility, that guarantees non-zero stress parametersβ for all non-rigid body element boundary/crack

378 S. Li, S. Ghosh

Fig. 3 Schematic diagram of an extended VC element,exhibiting the crack tip region and the surrounding re-gion of multi-level wavelet representation. The crosses (×)

corresponds to the position of lower level wavelets and thelocations of the adapted higher level wavelet bases are indi-cated by squares (�)

face displacement fields, is stated as

nβ > nEq + ncr

q − 3 (8)

where nβ is the number of β parameters, and nEq

and ncrq are the number of displacement degrees of

freedom on the element boundary and crack facerespectively.

Solution of the resulting nonlinear equations bya pure Newton–Raphson solver exhibits a discon-tinuous drop with the evolution of cracks, for caseswhere the loading process is monotonically con-trolled by incremental deformation or load condi-tions. To avert this, the arc-length solver is usedthrough the introduction of an unknown loadingparameter to govern the load increments (Crisfield1981, 1983; Li and Ghosh 2006; Schweizerhof andWriggers 1986).

3 Cohesive crack propagation and coalescence

Various criteria and conditions govern the evo-lution of multiple cohesive cracks in an elasticmedium. These are discussed next.

3.1 Criterion for direction of incremental crackadvance

In X-VCFEM, the incremental crack advancedirection is determined from the cohesive fractureenergy at the crack tip. From the equivalence ofcohesive fracture energy φ for complete decohe-sion and the energy release rate Gc (see Ortizand Pandolfi 1999), the crack growth direction isestimated as that, along which the cohesive frac-ture energy φ is maximized for a given crack tipstate of stress. The cohesive fracture energy φAat the crack tip A along any direction α can beexpressed in terms of the separation δ(α) in thatdirection as:

φA(α) =(∫ δ(α)

0t(α)dδ

)

A

=(∫ t(α)

σmax

√

(tcohn )2 + β−2(tcoh

t )2 · ∂δ

∂tdt

)

A

(9)

The effective cohesive traction for direction α canbe deduced to be

t(α) =√

(σxxsin2α − σxysin 2α + σyycos2α)2 + β−2(−12σxxsin 2α + σxycos 2α + 1

2σyysin 2α)2 (10)

Multiple cohesive crack growth in brittle materials 379

The incremental direction of crack propagation isassumed as that which maximizes the cohesive frac-ture energy at A according to the criteria:

∂φA(α)

∂α= 0 and

∂2φA(α)

∂α2 < 0 (11)

From Eqs. (9) and (10), the criteria (11) is explicitlyexpressed as

φA(α) = δe

2σmax

(σ 2

max − t(α)2) (a)

∂φA

∂α= − δe

σmaxt∂t∂α

= 0 �⇒∂t∂α

= 1√(

σxxsin2α − σxysin 2α + σyycos2α)2 + β−2

(− 12σxxsin 2α + σxycos 2α + 1

2σyysin 2α)2

[

(σxxsin2α − σxysin 2α + σyycos2α)(σxxsin 2α − 2σxycos 2α − σyysin 2α)

+β−2(−1

2σxxsin 2α + σxycos 2α + 1

2σxxsin 2α

)(−σxxcos 2α − 2σxysin 2α + σyycos 2α)]

= 0 (b)

∂2φA

∂α2 = δe

σmax

[(σxysin 2α − σxxsin2α − σyycos2α

)

×(σxxsin 2α − 2σxycos 2α − σyysin 2α)

+β−2(1

2σxxsin 2α − σxycos 2α − 1

2σyysin 2α

)2( − σxxcos 2α − 2σxysin 2α + σyycos 2α)]

< 0 (c) (12)

The direction of crack propagation αc is obtained as the solution of Eq. (12)b as

αXVCFEMc =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

arctan

(−σxx+σyy±

√(σxx−σyy)2+4σ 2

xy

2σxy

)

arctan

(2(β2−1)σxy±

√(4β4−8β2+4)σ 2

xy−(4β4−4β2+2)σxxσyy−(2β2−1)(σ 2xx+σ 2

yy)

2β2σxx−σxx+σyy

) (13)

The optimal angle αXVCFEMc is chosen as the one that satisfies the condition in Eq. (12)c.

3.2 Length of the incremental cohesive crackadvance

The length of cohesive zone advance (�l̄) in thecrack evolution scheme is estimated using a crite-rion that the cohesive fracture energy goes to zeroat the end of the new segment as shown in Fig.4(a). Hence the directional gradient of the cohe-sive fracture energy plays a role in this criterion.

The cohesive fracture energy at two points A (pres-ent crack tip) and B (close to A in the direction ofcrack propagation) are evaluated by substitutingthe value of the stresses in Eq. (12a). The tip of thecohesive zone is obtained from the linear extrapo-lation of this line that corresponds to zero cohesivefracture energy. From Fig. 4(a), the increment ofcohesive zone for probable crack growth is givenby the dimension:

�l̄ = φA

φA − φB|AB| (14)

where |AB| is the distance between the two pointsin the incremental crack direction.

3.3 Cracks crossing interelement boundaries

Crack advance from one Voronoi cell element tothe next occurs in X-VCFEM following an algo-rithm depicted in Fig. 4(b). A continuous tracking

380 S. Li, S. Ghosh

Fig. 4 Algorithms forincremental propagationof cohesive cracks: (a) fordetermining direction andincremental length of thecohesive zone, (b) forcrossing elementboundaries and (c) formerging with other cracks

method is implemented to monitor if a cohesivecrack surface has reached an element boundary orgone past it. The intersection of the crack surfacewith an element boundary is first checked by solv-ing the equation system

x − xi

xi+1 − xi= y − yi

yi+1 − yi,

x − xn

xn+1 − xn= y − yn

yn+1 − yn

(15)

where (xi, yi) represents the tip of the cohesivecrack line for the ith increment, and (xn, yn) is theposition of the nth node on the element boundary.If the intersection point is outside of the cohesiveline or the element boundary, it is assumed thatthere is no intersection. Once a cohesive crack hasreached its intersection with the boundary, a newnode pair (n1, n2) is introduced on the elementboundary at this point as shown in Fig. 4(b). The

node pair belongs to the intersection of the ele-ment boundary and the cohesive crack, i.e. n1n2 ∈∂�E

e ∩ �cr. The crack is subsequently advanced tothe next element following the intra-element crackadvance procedure discussed above.

3.4 Crack coalescence

The coalescence or merging of multiple cracks isalso considered is this paper. Figure 4(c) showsthe merger of two cohesive cracks. The algorithmfor crack merging, outlined below, is an extensionof the element boundary intersection algorithm inSect. 3.3.

• Record all the cracks that have propagated inan increment.

Multiple cohesive crack growth in brittle materials 381

• Check for the intersection of the last incremen-tal segment of the cohesive crack with those ofall neighboring cracks that belong to either thesame element or neighboring elements, usingEq. (15).

• Insert a three-node junction (m1, m2, m3) at thepoint of intersection, once intersection of twocrack segments is ascertained. This junction isshown in Fig. 4(c).

• Apply cohesive zone models to all node pairsof the three nodes at this junction.

• Contribution of the junctions nodes, e.g. (m1,m2), to the load vector in the assembled ma-trix equation requires special treatment. Foreach of these nodes, contributions of integralsfrom adjoining crack segments belonging totwo different cohesive cracks are summed.

• As soon as crack tip merges with other cracks,the stress concentration at the crack tip disap-pears. Consequently, remove the wavelet func-tions corresponding to this crack tip from thestress functions for the element.

4 X-VCFEM validation for accuracy andefficiency

A set of numerical examples are conducted for val-idating the accuracy and efficiency of X-VCFEM,prior to its application for parametric studies onmultiple crack propagation in brittle media. In thissection, three sets of numerical examples are solvedand compared with other codes or established re-sults in the literature, for examining the accuracyand efficiency of X-VCFEM. These convergencestudies for static and quasi-static cracks augmentstudies that have been conducted in Li and Ghosh(2006).

4.1 Comparison of a single crack propagationmodel with ABAQUS

As a benchmark, a static problem of a plate with asingle cohesive edge crack is solved using the com-mercial FEM code ABAQUS and compared withresults by X-VCFEM. The problem, schematicallydepicted in Fig. 5(a), is solved for plane strain con-ditions under a remote tension load. The material

modeled has a Young’s modulus E = 70 × 103 MPaand Poisson ratio ν = 0.33. A special user ele-ment (UEL) subroutine is developed in ABAQUSfor incorporating the cohesive model at the crackface. It is difficult to incorporate the extrinsic cohe-sive model in the displacement based ABAQUS,since there is a stress jump at zero displacement.To circumvent this problem in ABAQUS, a bilin-ear cohesive zone model discussed in Li and Ghosh(2004) is used to describe crack growth in the plate.The cohesive parameters for this model are σmax =5 MPa, δc = 1 × 10−6 mm, δe = 5 × 10−3 mm andβ = 0.707. The uniform load in the y direction is in-creased from 0 − 3 MPa. A total of 12840 QUAD4elements with 13250 nodes and 77 cohesive ele-ments are used in ABAQUS. In X-VCFEM, theentire domain is represented by a single elementconsisting of 142 nodes on the boundary and crackfaces. Adaptive multi-level enrichment of the wave-let bases is performed by the strain energy crite-rion (Li and Ghosh 2006). The optimal parametersfor stress function representations in X-VCFEMhave been determined in Li and Ghosh (2006) tobe pn = qn = 13, sn = tn = 0, nn = 4, ln = 1,mn = kn = 4, a1 = c1 = b1 = d1 = 0.1 andtra = trc = 0.5. This corresponds to 102 termsin the polynomial stress function �poly, 1 term inthe branch function �branch and 128 terms in thewavelet function �wvlt for a total of 231 terms. Thisstructure of element stress functions is retained forall subsequent simulations. The crack is assumed topropagate horizontally due to problem symmetryand hence, the modules for determining incremen-tal crack direction in Sect. 3 are switched off in thisproblem. The automatic load stepping algorithmin AQAQUS requires a total of 37 load steps whilethe X-VCFEM simulation is conducted in 8 stepsto reach the desired maximum load. Figure 5(b)shows the load-vertical displacement (σ − uy) plotat the point A of the crack face. The load-displace-ment results by X-VCFEM are in excellent agree-ment with the high resolution ABAQUS model,thus exhibiting its accuracy. On the other hand, theX-VCFEM simulation takes only 1.6 minutes on asingle CPU in the Pentium 4 cluster with 2.4GhzIntel P4 Xeon processors, as opposed to 13.9 min-utes for ABAQUS on the same machine. Even forthis simple example, an excess of tenfold advantagein computing speed is achieved by X-VCFEM. This

382 S. Li, S. Ghosh

Fig. 5 (a) A plate with anedge crack under remotetension load, (b)comparison ofload-deformationresponse by X-VCFEMand ABAQUS

factor is expected to increase considerably withincreasing complexity, such as more cracks.

Figure 6 shows the contour plot of the load direc-tion stress σyy and the corresponding stress distri-butions due to various terms in the stress functionat the end of the first load step. The figure in theinset of Fig. 6(c) is a rescaled plot of the stressdue to the branch function, since this is orders ofmagnitude smaller (mode I crack in this problem)than those due to polynomial or wavelet represen-tations. The set of contour plots point to severalimportant observations summarized below.

• The stress distribution due to polynomial termsin Fig. 6(b) describes the stress field far fromthe crack tip well, but it fails to predict a highgradient around the crack tip.

• The stress distribution due to the branch func-tion in Fig. 6(c) clearly shows its capability todepict stress discontinuity along the crack sur-face. However, the magnitude of this stress isof the order of 10−11, which is almost trivial

compared with contributions from the polyno-mial and wavelet functions. This is because thestress discontinuity does not exist at the crackface for this mode I problem, as shown in thestress contour plot of Fig. 6(a).

• Figures 6(d, e and f) show stress distributionsdue to three levels of wavelet functions, respec-tively. High gradients in the stress componentare missed by the low level wavelet functions,but these are represented much better by higherlevels of wavelets in the adapted smallerregions. This demonstrates the advantage ofthe multi-resolution wavelet function repre-sentation.

4.2 Comparison with the results of a classicaldynamic crack propagation problem

This is a numerical experiments, based on the wellknown Kalthoff’s experiment on dynamic crack

Multiple cohesive crack growth in brittle materials 383

Fig. 6 Contour plot of the load direction stress σyyand the corresponding distributions due to differentterms in the stress functions: (a) total σyy, (b) contri-bution from �poly, (c) contribution from �branch (in-

set shows a re-scaled contour near the crack face), (d)contribution from level-1 �wvlt, (e) contribution fromlevel-2 �wvlt, and (f) contribution from level-3 �wvlt

propagation in an impact loaded prenotched plate.This problem has been the subject of many studies(Kalthoff 2000; Kalthoff and Winkler 1988;Rethore et al. 2005). The studies suggest that acrack, subjected to a tension-compression load asshown in Fig. 7(a), propagates at an angle of approx-imately 60–70o with respect to the initial notchin the plate. The present X-VCFEM developmentdoes not incorporate inertia terms, and hence aquasi-static crack propagation problem is simulatedinstead of the dynamic test. The configuration in

Fig. 7(a), shows that the experimental projectilemotion is replaced by the traction boundary con-ditions in the simulation under plane strain condi-tions. A small initial crack length (a = 0.02, a

b =0.11 in Fig. 7(a)) is chosen to mitigate the effectof the constrained right hand boundary on crackpropagation. Material properties for this problemare: Young’s modulus E = 207 GPa and Poissonratio ν = 0.3. Since the cohesive zone modelparameters corresponding to the experiments areessentially unknown to the authors, the X-VCFEM

384 S. Li, S. Ghosh

Fig. 7 (a) A plate with an edge crack for the classicalKalthoff experiment, (b) cohesive models with three differ-ent sets of parameters, (c) prediction of the crack path by

X-VCFEM, and (d)applied stress-horizontal displacementplots at point A for different cohesive zone models

simulation is conducted with three sets of cohe-sive parameters as shown in Fig. 7(b). They are:(i) CP-1: σmax = 40.0 MPa, δe = 0.6 e-4m, β = 1,(ii) CP-2: σmax = 20.0 MPa, δe = 1.2 e-4m, β =1, and (iii) CP-3: σmax = 10.0 MPa, δe=2.4 e-4m,β = 1. The sets correspond to identical cohesivefracture energy (area under the curve) but differ-ent peak stress and critical separation values. TheX-VCFEM simulations also investigate the sensi-tivity of the solutions to the cohesive parameters.The entire computational domain is representedby a single element in X-VCFEM with 132 nodaldegrees of freedom. The crack paths, predicted byX-VCFEM simulation for the three sets of cohe-sive parameters are shown in Fig. 7(c). The initialcrack growth angle for all cases is around 70o. Thisangle is corroborated by brittle failure experimentsat very low velocities in Erdogan and Sih (1963).Subsequently, the crack propagation takes placewithin the envelope between 60o and 70o. This is

in agreement with studies in Kalthoff (2000), Kalt-hoff and Winkler (1988), Rethore et al. (2005). Fig-ure 7(c) also shows that the crack growth path forthis problem is not sensitive to the cohesive zoneparameters. The dynamic conditions, as well asboundary constraints are responsible for the smalldifference between X-VCFEM results and thosein Rethore et al. (2005). Figure 7(d) shows the plotof the applied stress as a function of the horizon-tal displacement at the base of the notch at A. Alinear response is seen for all the models.

4.3 Crack propagation in a three-point bendingspecimen

In this example, a mixed-mode cohesive crack prop-agation is modeled in a three-point bend test withan unsymmetrically positioned initial crack. The

Multiple cohesive crack growth in brittle materials 385

Fig. 8 (a) A three-point bending specimen with an un-symmetric initial crack, (b) comparison of load-deflec-tion response from results of X-VCFEM (Mariani and

Perego 2003) and, (c,d) comparison of the crack paths byX-VCFEM with that in Mariani and Perego (2003) forα = 0.25 and α = 0.5 respectively

problem has been studied by Mariani and Perego(2003) using XFEM under plane stress conditions.Figure 8(a) shows the geometrical dimensions, loadP at point A and boundary constrains. The ini-tial crack position is determined by the offset ra-tio α, defined as the ratio of the distance of theinitial crack from the mid-span cross-section tohalf of the beam span. The material has a Young’smodulus E = 31, 370 MPa and Poisson ratioν = 0.2. The cohesive parameters for this problemare CP1: σmax = 4.4 MPa, δe = 0.07719298 mm,β = 1, which have been used in Mariani and Per-ego (2003). Once again, the entire domain is repre-sented by a single Voronoi element in X-VCEFMwith 154 nodal degrees of freedom, and the stressinterpolation functions consist of 102 terms in the

polynomial stress function, 1 term in the branchfunction and 128 terms in the wavelet function fora total of 231 terms. The arc-length method ex-plained before is used to govern the loadincrements so as to describe the softening in load-deflection responses. Figure 8(b) shows the load-deflection curve (at the load point A) for twovalues of the offset parameter, i.e. α = 0.5 andα = 0.25. The initial elastic response in the loadP-deflection u curve is stiffer and also the peak loadis higher for higher values of α. The load-deflectionresponse exhibits significant softening in the laterstages of crack propagation due to the extent ofthe evolved crack. The path of crack propagationfor the two cases are shown in Fig. 8(c) and (d).The cracks move towards the point of applied load

386 S. Li, S. Ghosh

and align themselves perpendicular to the edge ofthe specimen. Excellent agreement is obtained be-tween the results by X-VCFEM and in Mariani andPerego (2003).

5 Effect of cohesive parameters on crackevolution

Cohesive zone model parameters, e.g. σmax and δe

in Eq. 3, can significantly affect crack propagationand the overall behavior of a cracking material.Issues related to the implementation of cohesivezone models for the process zone in a crack pathhave been discussed in detail in Elices et al. (2002)and Yang and Cox (2005). In Elices et al. (2002), ithas been argued that intrinsic models with a hard-ening branch followed by a softening branch ex-tend a cohesive crack to a zone with large numberof cracks, thus inhibiting localized cracking. Fur-thermore, the intrinsic models use a high initialhardening slope, which results in mesh dependenceof the solution and very small elements are neces-sary near the fracture surface. An extrinsic model,which consists of a monotonically decreasing trac-tion function, is consequently a desirable modelespecially in the absence of an explicitly definedcrack path. However, for displacement-based finiteelement analysis, the extrinsic models face a ma-jor problem arising from the stress jump at δ = 0.This requires regularization for stability, and con-sequently the intrinsic models are preferred withdisplacement based FEM. Stress intensity factors(KI and KII) and the J-integral are used as alterna-tives to traction at the crack tip. In comparison, theassumed stress based X-VCFEM can provide thestress explicitly at the crack tip without dependingon the displacements, and hence is an advanta-geous method with the extrinsic cohesive models.This section considers the effect of various param-eters in the extrinsic cohesive model, as well asthe morphology e.g. distribution and orientationof pre-existing cracks, on their evolution.

5.1 Crack propagation in a three-point bendingspecimen

This example revisits the three-point bend simula-tion in Sect. 4.3 with three different sets of cohesive

parameters, to examine their effect on crack prop-agation. The cohesive parameters are:

• CP1: σmax = 4.4 MPa, δe = 0.07719298 mm,(highest peak stress, lowest critical displace-ment)

• CP2: σmax = 2.2 MPa, δe = 0.15438596 mm• CP3: σmax = 1.1 MPa, δe = 0.30877192 mm

(lowest peak stress, highest critical displace-ment)

The coefficient β = 1 in Eq. 2 for all cases. Thesimulations are carried out till softening behav-ior is observed in the load-displacement response.Figure 9(a) is a plot of the applied load to thedeflection of the point of application of the load.Significant sensitivity to the cohesive parametersis seen in these plots. The load-deflection responseexhibits softening in the later stages of crack prop-agation due to the significantly evolved crack. Thestiffness of the P-u curve is higher for higher val-ues of the peak stress σmax. On the other hand,larger δe’s in CP2 and CP3 delay relatively theonset of softening for the cases with smaller peakstresses during the damage process. The path ofcrack propagation for the three cases are shown inthe Fig. 9(b). The cracks move towards the pointof applied load and align themselves perpendicu-lar to the edge of the specimen. The crack growthpath does not depend significantly on the cohesiveparameters.

5.2 Crack growth in a sheared plate

Crack propagation in a plate with a central crackwas experimentally studied by Erdogan and Sih(1963), where the plate was subjected to a far fieldshear load. An optical micrograph of the crackedspecimen in Erdogan and Sih (1963) is shown inFig. 10(a). The material of the specimen in theexperiment was assumed to be homogeneous, iso-tropic and linearly elastic and the crack wasassumed to be brittle. This problem is simulatedby X-VCFEM with a single element of dimen-sion 10 × 8 cm as shown in Fig. 10(c). The initialcrack length is l0 = 1.6 cm. The material param-eters are: Young’s modulus E = 100 GPa, Pois-son ratio ν = 0.3. Since the cohesive parametershave not been assessed in the experiments, the

Multiple cohesive crack growth in brittle materials 387

Fig. 9 (a) Comparison ofload-deflection behaviorand (b) the crack paths byX-VCFEM for differentcohesive zone parameterswith α = 0.5

sensitivity of the solution to five different sets ofcohesive parameters is studied in the simulations.These sets, illustrated in Fig. 10(b) correspond tothe same cohesive fracture energy and are:

• CP1: σmax=3.0 MPa, δe = 3.0 × 10−3 mm• CP2: σmax=6.0 MPa, δe = 1.5 × 10−3 mm• CP3: σmax=3.0 MPa, δe = 6.0 × 10−3 mm• CP4: σmax=6.0 MPa, δe = 3.0 × 10−3 mm• CP5: σmax=1.5 MPa, δe = 6.0 × 10−3 mm

The coefficient β = 1 in Eq. 2 for all cases. In Fig.10(c), a uniform shear load per unit length τ0 isapplied on the top and bottom surfaces. The loadincrement is applied by controlling the incrementalcrack propagation (Li and Ghosh 2006). A total of8 load increments are applied to execute this prob-lem. In each increment, the applied load is scaledby the arc-length parameter (Li and Ghosh 2006)to yield an equilibrated applied load correspond-ing to a prescribed crack propagation length.

The crack paths predicted by X-VCFEM for allthe different cohesive parameters compare verywell with experimental observations in Erdoganand Sih (1963). The path shows very little sen-sitivity to the cohesive parameters. However thecohesive parameters have a significant influenceon the shear load response as a function of thecrack length. This is demonstrated in Fig. 10(d),where the normalized crack length is defined aslr = current crack length

initial crack length . Thus the crack propagationrate, but not its direction is dependent on the cohe-sive parameters. For the cases with larger peaktraction i.e. CP2 and CP4, higher applied loads areneeded for causing similar extent of crack growthas for CP1 and CP3 with lower peak stresses. Com-parison of the results for parameters CP2 and CP4show that a smaller δe results in quicker reductionof the local cohesive traction. This makes the over-all load with CP2 to increase slower than that withCP4 corresponding to a higher δe. The result with

388 S. Li, S. Ghosh

Fig. 10 (a) Optical micrograph showing the path of crack-ing in a plate with a central crack subjected to far-field shear(Erdogan and Sih 1963), (b) 5 different sets of cohesive

parameters for X-VCFEM simulations, (c) correspondingcrack paths generated by X-VCFEM, (d) comparison ofthe growth of cracks for the different parameters

parameters CP5 is consistent with the trends exhib-ited with the other parameters. Although the simu-lation results show that both σmax and δe affect thecrack growth, comparison of cases CP1, CP2, CP3with CP4 shows that the crack growth is more sen-sitive to σmax than to δe. The results also imply thatthe cohesive fracture energy or effectively the en-ergy release rate Gc does not alone determine thecrack propagation behavior. Individual parame-ters, affecting the shape of the cohesive law, play animportant role in predicting growth characteristics.

5.3 Propagation of multiple cracks in a plate

A plate with 28 randomly located and orientedcracks is simulated under a tensile loading in this

example. Figure 11(a) and (b) show the two micro-structures with different crack distributions. Forthe microstructure 1, cracks of equal length arerandomly dispersed but are oriented horizontally.The microstructure 2 has cracks of random lengthand orientation. In addition, it contains a clusterof 8 cracks in an otherwise random distributionas shown in Fig. 11(b). The plate is of dimen-sion 0.1×0.1 m, and the material parameters are:Young’s modulus E = 104 MPa and Poissonratio ν = 0.3. The Voronoi cell mesh in Fig. 11(a)consists of 694 nodal degrees of freedom on thecell boundaries and 1004 nodal degrees of free-dom on the cracks. On the other hand, the mesh inFig. 11(b) consists of 650 nodal degrees of freedomon the cell boundaries and 1008 nodal degrees offreedom on the cracks. The stress interpolation

Multiple cohesive crack growth in brittle materials 389

Fig. 11 Crack propagation in two square micro-regionscontaining 28 cracks, by X-VCFEM: (a) domain with hori-zontal cracks of equal length and random distribution, (b)domain with random orientation, length and distribution of

cracks but containing a cluster, (c,d) contour plots of σyy(MPa) with cohesive parameters CP1 for the domains in(a) and (b), (e,f) contour plots of σyy (MPa) with cohesiveparameters CP2 for the domains in (a) and (b)

functions for each element are the same as dis-cussed before. As a crack crosses the element boun-dary or merges with another crack, the waveletterms in the stress interpolations are adjusted fol-lowing the procedure described in Sect. 3.4. Tounderstand the effect of the cohesive parameterson crack propagation, two different sets of

cohesive parameters are considered:

• CP1: σmax=1.0 MPa, δe=1.0 e-5 m, β = 0.707• CP2: σmax=2.0 MPa, δe=0.5 e-5 mm, β = 0.707

A uniform tension load σ per unit length is appliedon the top and bottom surfaces as shown in Fig.11(a,b). The applied load is increased in a total of

390 S. Li, S. Ghosh

Fig. 12 Comparison ofthe growth of crack A inmicrostructure 1 withdifferent cohesiveparameters

9 increments and is scaled by the arc-length param-eter for load equilibrium corresponding to a pre-scribed crack opening.

Figure 11(c,d) and (e,f) show the stress σyy con-tour plots along with evolved position of the cracksfor the two sets of microstructures and cohesiveparameter, respectively. The plots are shown forthe final load increment. The growth pattern ofeach crack can be observed by comparing withits initial configuration in Fig. 11(a) and (b). Thecracks propagate across element boundaries, inter-act with each other and in some cases merge. Thedependence of the propagation of multiple crackson the morphology and cohesive parameters iscomplicated, in general. However, several obser-vations can be made based on the results of theX-VCFEM simulations.

• Larger stress concentrations develop at tipsof cracks that are nearly perpendicular to thedirection of loading. Consequently, this subsetof cracks grows more easily than others thatare more aligned with the loading direction.From Fig. 11(d) and (f), it can be seen thatsome cracks that are nearly parallel to the loaddirection never propagate.

• Stress concentrations are higher at tips of thelonger cracks. For the same external load, sec-tions corresponding to longer cracks leave lessintact material, leading to higher stress con-centrations. As a consequence, it is seen in Fig.

11(d) and (f) that the longer cracks propagatemore than the shorter ones.

• Irrespective of the initial orientation, the evol-ved crack path tends to align in a direction per-pendicular to the applied load direction. Thiscorresponds to an optimal direction for releas-ing the cohesive fracture energy. This observa-tion is dominant, when the influence of nearbycracks on the local stress field is small. The localstress field in this case is mainly governed bythe influence of the applied load on this singlecrack.

• Cracks are attracted towards weak surfaces,such as other cracks or voids and prefer topropagate in those directions. This may be attri-buted to the fact that the cohesive fracture en-ergy in the direction of these weaker surfaceswith lower (or zero) tractions is naturally lowerin comparison with other directions. Highercrack tip hoop stress appears on the side closeto the weak surface, e.g. near another crack.This drives the crack propagation in that direc-tion towards the other crack, eventually result-ing in a merger. Once a crack merger occurs,the high concentrated stresses no longer existnear the merged regions, and hence the crackceases to propagate further.

• Cracks in the two configurations of Fig. 11(a)and (b) behave very differently due to the evo-lution of local stress and cohesive energy. Eventhough the two cracks at the upper right cor-ner are initially not that dissimilar, their final

Multiple cohesive crack growth in brittle materials 391

Fig. 13 Macroscopicstress–strain response fordifferent microstructuralmorphologies andcohesive parameters

configurations in the Fig. 11(e) and (f) are quitedifferent. Different crack mergers are observedfor these two cases. For Fig. 11(a), the crack onthe upper right hand corner propagates to thedomain boundary and this causes a big changein the stress state at the crack tip, affectingthe crack propagation direction. While the twocracks at the upper right corner of Fig. 11(a) donot merge, they do in Fig. 11(b) due to differentevolving stress states.

• Figure 11(d) and (f) show that the longest crackdoes not necessarily evolve from a cluster. Notall cracks in a cluster grow considerably. Thisis somewhat in contrast to observations madewith particle reinforced composites, wherealmost always clusters cause a local stress con-centration. The interaction between neighbor-ing cracks contributes to the enhancement ormitigation of stresses, depending on their ori-entations and length. This dictates their prop-agation and just being in a cluster does notguarantee significant growth.

• Different cohesive parameters show very lit-tle difference in the final configuration andhence the propagation direction. However, therate of crack growth varies considerably withthese parameters as seen in the crack length-macroscopic strain plot of Fig. 12.

Figure 13 shows the macroscopic stress–strainresponse for the two microstructures and cohe-sive parameters. Even before the cracks propagate(signaled by the change in slope), the stiffness of

the microstructure 2 is higher than that of micro-structure 1 due to a higher level of effective damagecaused by crack lengths and more importantly ori-entations. Orientations perpendicular to the loaddirection cause a larger reduction in stiffness incomparison with other directions. With additionalloading, the overall damage caused by the growthof cracks is also higher for the microstructure 1.This is seen by the lower values of the macro-scopic stress for this case. The effect of the cohesiveparameters on the stress–strain response is quitepronounced. The maximum macroscopic stress forboth microstructures increases significantly forhigher values of σmax, even though the cohesivefracture energy is the same for the two cohesivemodels. This is caused by a slowdown in the growthrate of the cracks with overall deformation.

6 Concluding remarks

In this paper, an extended Voronoi cell finite ele-ment model or X-VCFEM developed in Li andGhosh (2006) is further augmented to accommo-date crack coalescence. The resulting X-VCFEMmodel is used to predict multiple crack interactionand propagation in brittle materials. X-VCFEMincorporates special enhancements to the stressfunctions in a hybrid assume stress element for-mulation to the element to allow stress disconti-nuities across the cohesive crack and to accuratelydepict the crack tip stress concentrations. Thesefeatures are accommodated through the incorpo-

392 S. Li, S. Ghosh

ration of branch functions in conjunction with levelset methods across crack contours, and adaptivemulti-resolution wavelet functions in the vicinity ofthe crack tip. This results in a powerful numericalmodel that avoids the use of cumbersome reme-shing or mesh refinement with growing cracks. Thecracks are modeled by an extrinsic cohesive zonemodel, for which the stress jump at zero displace-ment can be easily modeled by the hybrid stress-based VCFEM. This is a special advantage ofVCFEM, which can use extrinsic models and thusavoid intrinsic models that show mesh dependence.A particular focus of this paper is on the effect ofcohesive parameters on crack growth. The incre-mental growth directions and lengths of thesecracks are adaptively determined in terms of cohe-sive fracture energy near the crack tip.

Various problems are solved with X-VCFEMand compared with existing solutions in the litera-ture to validate the model and show convergence.The X-VCFEM results show excellent accuracyand efficiency in comparison with other numer-ical solutions and experimental solutions in theliterature. Numerical simulations with X-VCFEMare then used to understand the effect of differentcohesive parameters on the crack propagation. Inaddition to the total cohesive fracture energy, indi-vidual cohesive parameters have significant effectespecially on the rate of crack growth. However,the path of growth for single cracks is relativelyunaffected by the change in cohesive parameters.Finally, a problem with multiple cracks is analyzedto study the effect of morphology, e.g. length, po-sition and orientations of cracks on their propa-gation and interaction. The simulations reveal thatlonger cracks that are nearly perpendicular to load-ing direction are more amenable to grow larger ata faster rate than other cracks. The crack prop-agation direction is also dependent on the localstress field, which is a function of both the externalload and local morphology, such as other cracksin the neighborhood. In the recent years X-FEMhas proven to be an effective tool modeling mov-ing discontinuities without the need for continuousremeshing. X-VCFEM is also a significant additionto this family. It has been proven to be a powerfultool for analyzing large regions of the microstruc-ture with multiple growing and interacting cracks.Research is now underway for extending the

evolving crack problem to heterogeneous micro-structures with inclusions.

Acknowledgements This work has been partially suppor-ted by the Air Force Office of Scientific Research throughgrant No.F49620-98-1-01-93 (Program Director: Dr. B. L.Lee) and by the Army Research Office through grant No.DAAD19-02-1-0428 (Program Director: Dr. B. Lamatti-na). This sponsorship is gratefully acknowledged. Computersupport by the Ohio Supercomputer Center through grantPAS813-2 is also gratefully acknowledged.

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