irregular lattice model for quasistatic crack...

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Irregular lattice model for quasistatic crack propagation

J. E. Bolander* and N. Sukumar†

Department of Civil and Environmental Engineering, University of California, Davis CA, 95616sReceived 13 June 2004; revised manuscript received 16 December 2004; published 24 March 2005d

An irregular lattice model is proposed for simulating quasistatic fracture in softening materials. Latticeelements are defined on the edges of a Delaunay tessellation of the medium. The dualsVoronoid tessellation isused to scale the elemental stiffness terms in a manner that renders the lattice elastically homogeneous. Thisproperty enables the accurate modeling of heterogeneity, as demonstrated through the elastic stress analyses offiber composites. A cohesive description of fracture is used to model crack initiation and propagation. Numeri-cal simulations, which demonstrate energy-conserving and grid-insensitive descriptions of cracking, are pre-sented. The model provides a framework for the failure analysis of quasibrittle materials and fiber-reinforcedbrittle-matrix composites.

DOI: 10.1103/PhysRevB.71.094106 PACS numberssd: 61.43.Bn, 62.20.Mk, 62.20.Dc

I. INTRODUCTION

The use of discrete, one-dimensional elements to repre-sent structural continua dates back to the work ofHrennikoff.1 The modern counterparts of such discrete sys-tems are lattice models, which are composed of simple, one-dimensional mechanical elements connected on a dense setof nodal sites that are either regularly or irregularlydistributed.2 These models have their primary justification inthe physical structure of matter at a very small scale, wherematerial can be seen as a collection of particles in equilib-rium with their interaction forces. Lattice models have alsobeen used to study the behavior of a variety of materials atlarger scales, with particular interest in their disorder andbreakdown under loading.2,3

When subjected to loading, the lattice network has elasticstrain energyHsud, whereu=hu1, . . . ,uNj are the general-ized displacements of theN nodal sites. On minimizingHwith respect tou, a system of equations is obtained, fromwhich u is determined. The breaking of an element in alattice network is based on criteria in terms of element strain,generalized force, or energy, as determined from the dis-placement solution. In the classical approach, an element isremoved from the network if it meets the breaking criterion.The procedure is repeated, where only the most critical ele-ment is removed for each solution of the equation set. Load-ing on the network is incremented only after all elements arewithin the limits set by the breaking criterion.

Lattice models differ mainly in the manner in whichneighboring nodal sites interact via the lattice elements. Withrespect to modeling material fracture, the simplest and one ofthe most popular forms of interaction is through central forcesor axiald springs.4–6 A more general form of interaction isprovided by the Born model,7 which includes both axial andtransverse stiffnesses, although this model in not rotationallyinvariant. With the introduction of rotational degrees of free-dom at the element nodes, bond-bending,8 granular, andEuler-Bernoulli beam9,10 models overcome the deficiency ofthe Born model and allow for a more general interactionbetween the neighboring sites that, in the limit, relate to agradient continuum, such as Mindlin-Toupin or Cosserat

continua. A gradient continuum includes higher-order termsand an internal length scale, which are absent in the con-tinuum theory of linear elasticity. By virtue of nodal sitesymmetry, lattices formed from regularly positioned nodalsites can model uniform straining. However, regular latticestend to provide low energy pathways for element breakingand, therefore, can strongly bias the cracking direction. Ir-regular lattices11,12exhibit less bias on cracking direction andoffer freedom in domain discretization, yet generally do notprovide an elastically uniform description of the material.

In this paper, we utilize beam-type elements in an irregu-lar lattice model of fracture for quasibrittle materials such asconcrete composites. The scaling of element stiffness termsis based on a Voronoi discretization of the material domainand provides an elastically uniform description of the mate-rial under uniform modes of straining. This Voronoi scalingwas introduced by Christet al.,13 who investigated the pos-sibility of carrying out quantum field theory computationsusing a random lattice. The elastically uniform lattice servesas a basis for the explicit modeling of heterogeneous fea-tures, such as inclusions and fibers. Fracture is modeled us-ing a cohesive description of cracking. Closing pressures as-sociated with the cohesive crack law blunt the singularitythat would exist at the crack tip in linear elastic fracturemechanics. This reduces nodal rotations that might otherwiseaccentuate differences between theories based on gradientcontinua and classical linear elasticity. In contrast to mostlattice models, element breaking is gradual and governed byrules that provide an energy conserving, objective represen-tation of fracture through the irregular lattice.

Numerical examples are presented to demonstrate the ac-curacy and applicability of the lattice model in terms of itselasticity and fracture properties. These capabilities providean effective framework for the modeling of fracture inheterogeneous materials. For instance, the lattice model pro-posed herein can serve as a basis for modeling fracturein fiber-reinforced brittle-matrix composites and in othermultiphase materials when each phase is assumed to behomogeneous.

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II. DOMAIN DISCRETIZATION

The elastic properties of the material are discretized usinga Voronoi diagram on an irregular set of points.14 By defini-tion, the Voronoi cell associated with siteI is the set of pointscloser to siteI than all other sites in the domainfFig. 1sadg.The Voronoi partitioning of the material domain is robust andfacilitates a high degree of preprocessing automation. Ad-vantages of this approach include the ability to do the fol-lowing:

s1d Explicitly model material structure, such as theboundaries between two phasesswhich generally do not runalong the grid lines produced by a regular triangular orsquare latticed. This can be done by strategically placingsemiregularly spaced points prior to random filling of thedomainfFig. 1sbdg.

s2d Grade average cell sizesgrid point densityd, whichcan be advantageous for improving resolution in critical re-gions of the domain, while reducing computational expenseelsewhere.

s3d Perform adaptive mesh refinement in nonlinear ortime-dependent problems.In the following, the generator points of the Voronoi diagramare the lattice sitessnodesd, whereas the edges of the corre-sponding Delaunay triangulation define the lattice elementconnectivitiesfFig. 1sadg. For the planar analyses considered

here, two translational and one rotational degrees of freedomare defined at each lattice node.

III. LATTICE ELEMENT DESCRIPTION

A. Elasticity model

The lattice element, shown in Fig. 2, consists of a zero-size spring set that is connected to the lattice nodes via rigidarms. The spring set is positioned at the midpoint of theVoronoi edge common to the two nodes. This general ap-proach has evolved from the rigid-body-spring concept ofKawai.15,16 Each spring set consists of normal, tangential,and rotational springs that are defined as local to the com-mon edge and assigned stiffnesseskn, kt, and kf, respec-tively. These stiffnesses are simple functions of the distance,hIJ, between the lattice nodesI and J and the length of thecommon Voronoi edge,sIJ,

kn = EAIJ/hIJ,

kt = EAIJ/hIJ, s1d

kf = knsIJ2 /12,

where AIJ=sIJt, with t being the thickness of the planarmodel andE being the elastic modulus of the continuummaterial. The systematic scaling of the spring stiffnessesgiven in Eq.s1d provides an elastically homogeneous repre-sentation of the continuum,15,17 which is necessary for theobjective modeling of fracture described later in this work.The samesIJ /hIJ scaling relation has been used for randomwalks on arbitrary sets18 and for solving two-dimensionaland three-dimensional diffusion problems on irregulargrids.19,20 Lattice models are limited in their ability to repre-sent Poisson effects in linear elasticity, due to the unidirec-tional structure of the lattice elements. A limited range ofmacroscopic Poisson ratios can be modeled by the regulararrangement of lattice elements or by the adjustment of ele-ment stiffness coefficients, but a local realization of the Pois-son effect is not obtained. The proposed model can represent

FIG. 1. sad Dual tessellations of a two-dimensional set of pointsand sbd Voronoi diagram partitioning of a multiphase material.

FIG. 2. Basic element of the spring network model.

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a range of macroscopic Poisson ratios by settingkt=akn,where 0øaø1, but for ktÞkn the material model is notelastically homogeneous.17 Schlangen and Garboczi10 pro-vide an alternative approach to obtaining an elastically uni-form random lattice, which involves the iterative refinementof the lattice element properties.

Due to the arrangement of its six nodal degrees of free-dom, the element shown in Fig. 2 is similar to a beam-springelement with axial stiffness, such as those used by Schlangenand van Mier21 to model elasticity and brittle fracture at thematerial scale. For the special case of a square lattice, thetwo approaches provide the same element stiffnessmatrices,15 provided thesIJ /hIJ scaling is also used for thebeam element properties. In general, however, the stiffnessformulations are different, partly because the spring set islocated eccentrically to the element axissDelaunay edged sothat elemental axial and rotational stiffness terms arecoupled.

B. Fracture model

One motivation for utilizing lattice models is the explicitrepresentation of discontinuous material structure at finescales, including the simulation of bond breaking in atomicstructures. At coarser resolutions, however, the model repre-sents the material as a continuum. Heterogeneity is intro-duced into the network either explicitly or through statisticalvariations in the element properties.2 As a fundamental re-quirement for either case, the lattice formulation must beable to provide an unbiased representation of fracturethrough a homogeneous medium. Consider the discretizationof a two-phase composite shown in Fig. 1sbd. At this scale,heterogeneity is apparent at the junction of different phases,whereas each phase by itself is often regarded as a homoge-neous material. From a modeling perspective, the lattice el-ements eithers1d correspond to specific features of the ma-terial structure, such as an interface between different phasesor s2d are not related to any material feature, such as whenrepresenting homogeneous properties within a single phase.The proposed lattice model is applied to the latter of thesetwo cases, although the approach can be tailored to treat bothcases.

In this study, modeI cracking is simulated by degradingthe strengths and stiffnesses of the lattice elements, accord-ing to the crack-band approach.22 The crack-band model isbased on the observation that microcracking, crack branch-ing, and other toughening mechanisms associated withcracking occur within a fracture process zone, the width ofwhich is typically related to the maximum size of the hetero-geneities. These various crack openings are assumed to beuniformly distributed over the crack-band width. In the nu-merical implementation of the model, cracks form andpropagate through the element interiors and the band widthgenerally conforms to the element size, rather than to theactual width of the fracture process zonesi.e., the fracturelocalizes into the smallest width permitted by the elementdiscretization of the material domain, provided there is noartificial locking of the crack opening that might otherwisecause the crack band to widend. In this sense, the crack-band

model is functionally equivalent to the cohesive crack model,as considered by Dugdale23 and Barenblatt24 and later ap-plied to softening materials by Hillerborget al.25 Accordingto this type of fracture model, separation takes place across acohesive zone and is resisted by cohesive tractions. Withinthis framework, the physically unrealisticÎr singularity thatarises in linear elastic fracture mechanics26 is avoided. Forbrittle materials, the interface traction law can be linked withthe gradual breaking of atomic bonds toward the formationof a new traction-free surface. The area under the traction-displacement curve is the specific fracture energy,GF, andthe maximum effective stress,st, the maximum effectiveseparation,wc, and the shape of the traction-displacementcurve are material parameters. When cohesive tractions di-minish with increasing separation, the traction-displacementcurve can be referred to as a softening relation.

An important feature of the fracture model is that thecrack band can form at an angleuR to the element axis andthe dimensions of the crack band are determined accordingto the local geometry of the Voronoi diagram17 fFig. 3sadg.The forces carried by the spring sets are known at any stageof the loading history. An average value of tensile stress canbe calculated from the resultant of this force pair, divided bythe projected area,

sR =FR

AIJ cosuR. s2d

To obtain proper fracture energy consumption for differentmeshing strategies, strain values characterizing the softeningresponse are22

FIG. 3. sad Crack-band geometry within an element of the springnetwork model andsbd associated softening relation.

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«cr =w

hIJ cosuR, s3d

wherew is the crackopening displacement, which is assumedto be uniformly distributed over the crack-band width ofhIJ cosuR fFig. 3sadg. In essence, modeI cracking is con-trolled by the shape of the tension-softening diagram and thefracture energy,GF, which are assumed to be material prop-erties and thus do not depend on the domain discretization.The softening diagram, shown in Fig. 3sbd, can be defined bystress and crack-opening values determined through inverseanalysis of fracture test results.17 After each load increment,the resultant tensile stress in each spring set is checkedagainst the softening relation. For a critical spring set, frac-ture involves an isotropic reduction of the spring stiffnessesand an associated release of spring forces, so that the result-ant stress lies on the corresponding softening relation. Therelease of spring forces causes an imbalance between theexternal and internal nodal force vectors, which often pro-motes additional fracture within the load step. The process ofpartially breaking the single most critical element and thensolving the associated system of linear equations is repeatedwithin each load step until the fracture criterion is satisfiedthroughout the problem domain. Although this process iscomputationally demanding, negative stiffness terms areavoided and zero-energy modes of deformation are not pro-duced.

IV. NUMERICAL SIMULATIONS

A. Elastic uniformity of irregular lattices

In Fig. 4, quasiuniform and graded discretizations of aunit square domain are illustrated. For uniform strain condi-tions imposed on the boundaries, the lattice sites should dis-place so that uniform strain occurs throughout the lattice.Regular lattice models, by virtue of site symmetry, exhibitsuch elastic uniformity, whereas irregular lattices are gener-ally not elastically uniform.10

The lattices indicated in Fig. 4 are subjected to both uni-form stretchingsu1=x1; u2=0d and combined stretching andshear su1=u2=x1+x2d. The numerical solution accuratelyrepresents the theoretical displacement field, as indicated bythe small relative errors indicated in Table I. The relativeerror is defined as

er =iu − uhi2

iui2, iui2 =Îo

I=1

N

usxId ·usxId, s4d

wherei ·i2 is theL2 norm of the indicated argument,u anduh

are the exact and numerical solutions, respectively, andN isthe total number of nodes. The networks do not exhibit spu-rious heterogeneity arising from either random mesh geom-etry or varying element size. If each of the lattice elements isassigned a constant area, equal to the average of the Voronoifacet areasAIJ, then the lattices are not elastically uniform, asevidenced by the large relative errors given in Table I. Inproducing the results for shear loading, rotations of the

Voronoi cells have been constrained for both sets of analyses.The placement of nodes along the boundaries is only to fa-cilitate the analyses and is not a requirement here.

B. Elastic analysis of fiber composites

To illustrate the importance of elastic uniformity, we firstconsider a short fiber embedded within an irregular latticemodel of a matrix material. A fiber contributes to the stiff-ness of a lattice element if it crosses the common boundaryof the two Voronoi cellssFig. 5d. To model this stiffnesscontribution, a zero-length spring connects the two cells atthe boundary crossing. The spring is aligned in the fiber di-rection and assigned axial stiffness,

FIG. 4. Irregular lattices.sad Quasiuniform nodal discretizationand sbd graded nodal discretization.


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kf =Afs fsxcd

shv/coscd«m, s5d

where Af is the fiber cross-section area,«m is the matrixstrain in the fiber direction,s fsxcd is the fiber axial stress atthe boundary crossing, andc is the angle between the fiberaxis and the direction of loading on the composite. Thespring stiffness is linked to the computational degrees offreedom, which are defined at the lattice nodes, by assumingthe cells to be rigid. The approach is general in that anyappropriate model relatings fsxcd and«m can be used to de-

terminekf through Eq.s5d. The relation used here, and plot-ted in Fig. 6sbd, is based on an elastic shear lag theory pro-

posed by Cox,27 in which perfect bonding is assumedbetween the fiber and matrix.

In general, fibers cross multiple Voronoi cell boundariesand therefore contribute to the stiffness and internal forcecalculations of multiple lattice elements. Figure 6sad shows asingle, inclined fiber of lengthl f embedded in a homoge-neous matrix subjected to uniaxial tension. The fiber elasticmodulus isEf =10Em, whereEm is the elastic modulus of thematrix. For comparisons with Cox’s theory,27 which assumesa uniform strain in the matrix, the fiber diameter is chosen tobe small relative to the thickness of the matrix. Axial stresslevels in the fiber at each cell boundary crossing are shownin Fig. 6sbd, where the axial stress has been normalized bys f =Ef« cos2 c, with « being the strain in the loading direc-tion. The computed axial stress values agree with theorywhen the Voronoi scaling of the lattice stiffness coefficientsis used. The constant-area scaling does not provide an elas-tically uniform representation of the matrix, and the associ-ated artificial heterogeneity of the lattice is manifested asnoise in the axial stress profile.

As a second example, we consider a fiber composite witha random distribution of short fibers. Fibers are randomlyinserted in aa3a square domain with thicknessa/10 fFig.7sadg. The matrix discretization is shown in Fig. 6sad. Thefiber lengthl f =a/8, fiber diameterf= l f /100,Ef =10Em, and6 519 fibers are used to achieve a fiber volume fraction of1%. The composite system is subjected to tensile loading inthe vertical direction and the fiber axial stress valuessforeach individual fiber at each boundary crossingd are plottedin Fig. 7sbd. Once again the numerical results for the axialstress in the fibers are in good agreement with Cox’stheory.27 Of the 11 497 data points shown in Fig. 7sbd, a fewstress values differ from the theoretical predictions, sincenonuniformity and the discrete nature of the fiber distributioncauses variations in the stiffness and therefore fluctuations instrain. Similarly, good agreement with theory is obtained forseveralfold increases in the fiber volume fraction, althoughfor much higher fiber contentssor for significantly largerfibers relative to the domain thicknessd, the heterogeneitymore strongly affects straining of the matrix and thereforethe axial strain profiles of the fibers. The modeling of localinteraction effects between the fibers is approximate, andtherefore the physical interpretation of the results also be-comes difficult for high volume fractions of fibers.

C. Fracture of uniaxial tension specimen

The lattice model is used to simulate fracture in a rectan-gular panel of homogeneous material under uniaxial tensile

TABLE I. Relative errorser for uniform strain loading.

Boundaryconditions

Quasiuniform mesh Graded mesh

Voronoi scaling AIJ=AIJ Voronoi scaling AIJ=AIJ

u1=x1 1.25310−7 4.41310−2 5.46310−8 8.68310−2

u2=0

u1=x1+x2 1.602310−7 3.640310−2 1.191310−7 7.610310−2

u2=x1+x2

FIG. 5. sad Lattice element with fiber inclusion andsbd fibercontribution to element stiffness.


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loading. The panel has cross-section areaA and the softeningproperties of the material are shown in Fig. 3sbd. Figure 8shows a Voronoi discretization of the panel. For imposedrelative displacementd between the two ends of the speci-mensand reactive forcePd, stresssR is determined for eachlattice element per Eq.s2d; for each element,sR has magni-tude P/A and acts in the direction of the axial load. Withincreasing axial load,sR reaches the tensile strengthst in allelements simultaneously. Only a small reduction of thestvalue is needed to initiate fracture at any specific location.For a perturbation of 1310−7st in the tensile strength of theelement atA or B sFig. 8d, fracture initiates at those loca-tions. From the point of crack initiation, fracture propagatesthrough the cross section and the material separatesfFigs.9sad and 9sbdg. From these plots, it appears that cracking isconstrained to follow the intercell boundaries. However, asdiscussed in Sec. III B, fracture is distributed within the lat-tice elements and the direction of fracture is not constrainedby the discretization. In Fig. 10, average stresss=P/A andaxial displacementd have been normalized to better indicatethat the cohesive softening curvefFig. 3sbdg, used as input tothe model, is manifested at the structural scale. The separa-tion process involves only two nodes per element; this facili-tates the transition from continuous to discontinuous behav-ior and avoids significant stress locking.

If the crack band is constrained to form normal to theelement axis, as would be the case for a central force springlattice, the model response is not a direct reflection of thesoftening curve. There are two main sources of error:s1d themagnitude of the stress component aligned with the elementaxis, sn=Fn/AIJ, is less than the average axial stresss=P/A, unless the element is aligned with the direction oftensile loading, ands2d after fracture initiation, the compo-nent of crack opening in the direction of the element axis issmaller than that in the direction of loading. These two

sources of error combine to produce excess strength and en-ergy consumption, as indicated by the broken line in Fig. 10,which can be viewed as a form of stress locking. In this casefFig. 9scdg, fracture initiation does not occur at the point ofreduced strength, which happens to be the element at loca-tion B. Furthermore, elements at different locations partiallyfracture prior to localization and ultimate failure along thepath shown in Fig. 9scd. Even in most classical latticeapproaches,2 where, upon violation of the fracture criteria,the element is completely removed, the first of these sourcesof error would still be present. The sensitivity to fluctuationsin strength, provided by the elastically uniform materialmodel and thesR stress measure, is a prerequisite to analysesbased on the statistical assignment of strength values.

D. Fracture of three-point bend specimen

The lattice model is used to simulate the three-point bendtest illustrated in Fig. 11. The specimen has dimensionsL=300 mm,d=100 mm,,=d/2, and a uniform thickness of100 mm. For an actual concrete mix, the physical test28 pro-vided the load versus crack mouth opening displacementsCMODd response shown in Fig. 12. Using the regular,straight-line discretization of the potential crack path be-tween the prenotch tip and load application pointfFig.11sbdg, an inverse analysis based on a Levenburg-Marquardtminimization algorithm provided the softening relation pa-rameterssst=4.12 MPa, wc=0.154 mm, b=0.247, andh=0.118d. These softening parameters were then used in aforward analysis to obtain the corresponding load versusCMOD curve in Fig. 12. Differences between the experimen-tal result and the inverse fitting are due, in part, to the use ofa bilinear softening relation, which is assumed to be constantduring fracture through the ligament length. The dimensionsof the three-point bend specimen, boundary conditions, and

FIG. 6. sad Inclined fiber in a homogeneous matrix andsbd axial stress along fiber.


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specific fracture energy of the material preclude the appear-ance of instability in the global load versus CMOD response.The solution process can accommodate instabilityssuch assnap backd in the specimen response by advancing and/orretracting the load point displacements so that the materialfracture criterion is precisely followed.

Using the same set of softening parameters, forwardanalyses are repeated for two different semirandom, irregulardiscretizations of the ligament length. The deformed mesh ata near-final stage in the loading history for one of the analy-ses is shown in Fig. 13. The resulting load versus CMODplots is also given in Fig. 12. The main observation is thatmesh size and irregular geometry do not appreciably influ-ence cracking behavior and, therefore, the resulting load-displacement curves agree well.

To help illustrate the consequences of theAIJ /hIJ scalingof the stiffness and crack-band dimensions, two variationsare made.

s1d The lattice elements in the fracture ligament regionare assigned a constant area, equal to the average of theVoronoi facet areas over the same region. For this case, theinitial slope of the load-CMOD diagram is nearly the same,whereas the remainder of the curve is in fair agreement withthe previous resultssFig. 14d.

s2d As for the uniaxial tension test simulation, the crackband is constrained to form normal to the element axissi.e.,sn=Fn/AIJ is used to guide fractured. For this case, the glo-bal response curve is too toughsFig. 14d. Schlangen andGarboczi29 have noted a strong directional dependence of thefracture properties of regular lattices when employing thistype of normal force fracture criterion.

From these results, it appears that the scaling of the ele-ment stiffness terms and the condition of elastic uniformityare of secondary importance when compared to the influenceof the element-breaking rules. However, the performanceevaluation of the different models should not be based on theglobal load versus CMOD results alone. The following sec-tion focuses on the performance of each model in terms ofthe rate and variation of energy consumption along the cracktrajectory.

E. Energy conservation in crack growth processes

As is common for lattice models, only one lattice elementis modified per computational cycle. The energy associated

FIG. 7. Analysis of random fiber composite.sad Fiber distribu-tion andsbd axial stress along fibers.

FIG. 8. Voronoi discretization of uniaxial tension testspecimen.

FIG. 9. Failure patternssad CaseA, sbd caseB, and scd caseBusingsn fracture criterion.

FIG. 10. Normalized load versus displacement relations forsimulated uniaxial tension test.


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with the breaking of an element can be computed from thechanges in reactive force at the load points, whose displace-ments are controlled through a finite stiffness device. Thecumulative amount of energy consumed by each lattice ele-ment is obtained, so that the distribution of local energy con-sumption can be viewed at any load stage. The total amountof energy assigned to the breaking elements is equal to thedifference between the area under the global force versusdisplacement curve and the elastic strain energy stored in thesystem.

In Fig. 15, we depict the maps of specific fracture energyconsumption,gF, along the ligament length for the regularand two irregular discretizations of the ligament region. Theenergies correspond to a near-final stage in the loading his-tory and have been normalized byGF, the area under thebilinear softening diagramfFig. 3sbdg. As expected, the regu-lar discretization of the ligament produces uniform energyconsumption along the crack trajectory. The ratiogF /GF de-creases near the top of the ligament length, sincew becomesless thanwc and tends to zero when approaching the neutralaxis of bending. For the random geometry analysis, the en-ergy distributions are nearly uniform along the principalcrack trajectory, with gF /GF<1.0. Crack propagationthrough the random mesh produces nearly the same results ascrack propagation along a smooth, predefined pathway. Thisis desirable in that network random geometry does not rep-

resent any structural features within the material. The simu-lated uniaxial tension test discussed earlier also exhibits auniform distribution ofgF /GF<1.0 over the fracture surface.

When using a constant area,AIJ, for each lattice elementin the ligament region, it is clear that local energy con-sumption does not follow that prescribed by the softeningrelation fFig. 16sadg. For elements withAIJ.AIJ, the ratiogF /GF,1, whereas for elements withAIJ,AIJ, the converseis true. This tendency forgF /GF to be on either side of unityprovides reasonable global load-displacement results, as seenearlier in Fig. 14. When constraining fracture to form normalto the element axisssn criteriond, however,gF /GFù1 for allcracks withwùwc fFig. 16sbdg, resulting in overstrength andexcess energy consumption in the structural response. Thestress locking is more pronounced, relative to that seen forthe uniaxial tension specimen, since the fracture path fromthe prenotch tip includes elements that are significantly in-clined to the loading direction. For the uniaxial tension test,there are low energy pathways available, where the elementsare closely aligned with the loading direction. In these simu-lations, there is a temptation to associate the nonuniformdistribution of energy consumption with heterogeneous fea-tures present in the material. However, the effects of thisartificial heterogeneity can be severe and bear no relation tothe actual material features.

F. Fracture of fiber-reinforced brittle-matrix composites

The irregular lattice serves as a framework for modelingthe fracture of fiber-reinforced brittle-matrix composites.Discontinuous, short fibers are added to brittle matrix mate-rials to provide additional toughness after matrix fracture.The increase in toughness is due to debonding along thefiber-matrix interface, followed by frictional pullout of thefibers traversing an opening matrix crack.30 Various types offibers are used, depending on the application, with the fiber

FIG. 11. sad Three-point bend test specimen and dimensions andsbd Voronoi discretization for inverse analysis of softeningparameters.

FIG. 12. Experimental curve and forward analysis runs.

FIG. 13. Mesh for simulating fracture in three-point bendtest.

FIG. 14. Influence of discretization and fracture criterion onspecimen response.


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dosages ranging up to several percent by volume of the com-posite, which is a practical limit for achieving uniform dis-persions of fibers with high aspect ratios.

Prior to matrix fracture, fiber contributions to compositestiffness and strength are modeled using elastic-shear lagtheory, as described in Sec. IV B. At the onset of matrixfracture, fibers that cross the developing crack are identified,and the associated springs at the crossing locationsfFig.5sbdg are modified as follows. Considering equilibrium andcompatability conditions, the axial stiffness of a modifiedspring is based on a composite pulloutsi.e., axial force ver-sus pullout displacementd curve, which is derived from thepullout relations for the embedded fiber lengths to each sideof the spring. The crack opening is modeled as the separationbetween adjacent Voronoi cells at the fiber-crossing location.The spring aligns with the point of entry of the fiber intoeach Voronoi cell. With continued crack opening, the shorterembedded length eventually pulls out, whereas the other end

of the fiber unloads after peak fiber load. The pullout rela-tions for each embedded length are derived from the consti-tutive properties of the matrix-fiber interfacefshown in Fig.17sad, whereta andt f indicate adhesional and constant fric-tional bond strengths, respectivelyg. Slip hardening or fric-tional decay along the matrix-fiber interface can be consid-ered in the modeling approach. In addition, the springelement is augmented with a beam-spring component tomodel the flexural and shear properties of the fiber, whichbecome active after fracture initiation. This lumping of thefiber postcracking behavior into a nonlinear spring bridgingthe crack is valid, provided the fiber crosses only one crack,which is the case in most applications.

Assuming constantt f =ta for the fiber-matrix interface,the direct pullout of a single embedded length from the ir-regular lattice yields the plot of axial forcesat the load pointdversus pullout displacement shown in Fig. 17sbd. The peakload is equal toP0=pflet f, wherele is the embedded length.

FIG. 15. Crack trajectories andlocal energy consumption.sadRegular discretizationsbd irregu-lar discretizationscase Id and scdirregular discretizationscase IId ofthe ligament length.


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Within the figure inset, the numerical solution is comparedwith theory31 for the initial part of the curve that is primarilyaffected by fiber debonding, i.e., the loss of perfect bond.After peak load, the fiber is completely debonded and theaxial force declines linearly due to the loss of embeddedlength as the fiber is pulled out from the matrix.

To demonstrate and verify the modeling procedure formultiple fiber composites, short fibers are randomly posi-

tioned within an irregular lattice model of a uniaxial tensiontest specimen. The Voronoi discretization of thea35a rect-angular domainswith thicknessa/2d and the model bound-ary conditions are as previously shown in Fig. 8. The fiberlength l f =a/2, fiber diameterf= l f /100,Ef =2Em, and 2547fibers are used to achieve a fiber volume fraction of 1%. Thematrix-fiber interface is assigned constant bond strength,t=ta=t f, and t /st=1. Fiber axial strength is assumed to begreater than the maximum pullout strengths0.5pfl ft fd, sothat tensile rupture of the fibers is avoided, which is normallythe material design objective.

For the same lattice geometry, three different random re-alizations of the fiber distribution are considered, providingthe three global response curves shown in Fig. 18, wheren isthe number of fibers bridging the crack andle is the corre-sponding average embedded length. The load-displacementresponse is linear until the onset of matrix fracture, afterwhich the toughening actions of the fibers are mobilized. Theresponse curves are normalized by the mathematically ex-pected postcracking strength,

Pe = snpfl f/4dt f , s6d

where n is the average ofn for the three simulations. Thescatter in the results is caused by the differences inn and thedifferences in the distribution of embedded lengths about theexpected average ofl f /4, both of which are due to randomvariation in the fiber positions. The load-free crack conditionoccurs at an axial displacement ofl f /2, which corresponds tothe maximum possible embedded length. As the post-cracking strength is significantly less than the first crackingstrength of the composite, a single crack forms, followed byfiber pullout, as shown in Fig. 19. Nonuniformity of the fiberdistribution causes variation in the load carried by the ma-trix; therefore the crack location differs for each of the threesimulations.

V. CONCLUSION

In this paper, we have described an irregular lattice modelthat is suitable for simulating fracture in quasibrittle materi-als, such as concrete, rock, and other geomaterials. Thismodel differs from previous lattice models in several re-spects.

FIG. 16. Crack trajectories and local energy consumption.sadConstant element areasAIJ=Ad andsbd axial stress fracture criterionssn=Fn/AIJd.

FIG. 17. sad Constitutive relations for the fiber-matrix interfaceand sbd axial load versus pullout displacement for a single fiber.

FIG. 18. Axial load versus displacement curves for three nomi-nally identical fiber-reinforced brittle matrix composites.


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s1d Lattice geometry is based on a Voronoi discretizationof the material domain, allowing effective gradations ofnodal point density. The Voronoi diagram provides scalingrules for the elemental stiffness relations, so that the irregularlattice is elastically homogeneous under uniform modes ofstraining. The elastically uniform lattice serves as a basis forthe explicit modeling of heterogeneous features, such asshort fibers.

s2d Fracture is modeled using a crack-band approach,where the crack band can form at arbitrary angles to the axesof the lattice elements. The dimensions of the crack band andthe softening relation are also defined by the local geometryof the Voronoi diagram. Fracture localizes into the narrowestband permitted by lattice discretization and, therefore, themodel can also be regarded as a cohesive zone representationof fracture.

s3d The crack-band representation of fracture involves anincremental softening of the lattice elements, according tothe prescribed traction-displacement relation. This is in con-trast to the conventional approach in which elements arecompletely removed from the lattice upon violating the frac-ture criterion.

s4d The fracture model is objective with respect to theirregular geometry of the lattice. During simulated fracturetesting of a uniaxial tension specimen and a notched concretebeam, uniform fracture energy is consumed along the crack

trajectory regardless of the mesh geometry. The common ap-proach of assigning equal areas to the lattice elements, andthe use of element axial forces to define the fracture criteria,leads to strongly nonuniform specific energy consumptionalong the crack path that, on the average, is much higher thanthat prescribed through the cohesive traction-displacementrelation. Although disordered materials exhibit fluctuationsin specific energy consumption as the fracture process ad-vances, the fluctuations exhibited by the common lattice ap-proach can be extreme and can bear no connection to thephysical processes. This form of stress locking leads to over-strength and excess energy consumption in the global load-displacement response.

The fracture simulations presented in this paper involvedstatistically homogeneous softening materials as well asfiber-reinforced brittle-matrix composites. In the former case,the various sources of energy consumption during fracturewere represented by a cohesive law. A similar approach canbe applied in the fracture analysis of multiphase materialswhere each phase is assumed to be homogeneous. In thelatter case, the stiffness, strength, and toughening mecha-nisms of individual fibers were explicitly represented withinthe material model. This enables the quantification and studyof the effects of nonuniform fiber distributions on compositeperformance measures, such as postcracking strength andtoughness. The explicit modeling of material structure, andits relations to system breakdown and failure, is a long-termobjective of this research.

ACKNOWLEDGMENT

The research support of the National Science Foundationthrough Contract No. CMS-0201590 to the University ofCalifornia, Davis, is gratefully acknowledged.

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FIG. 19. Fracture localization in the fiber-reinforced compositemodel.


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