A penny-shaped cohesive crack model for material damage

304 G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316In principle, a more realistic micromechanicsmodel and a feasible homogenization schemeshould lead to a better constitutive law at macrolevel.For Gurson model, failure mechanism dueto void growth is supported by many experimentalobservations on failures of ductile materials (e.g.[8,10,20,24,29]). On the other hand, in most brittle,quasi-brittle, and even some ductile materials(such as concrete, rocks, ceramics and some metals),materialÕs failure mechanism may be attributedto propagation, nucleation and coalescenceof micro-cracks as well. Although several microcrackbased damage models have been proposedto describe elastic damage processes (e.g.[4,9,15,16,19] and others), few micro-crack damagemodels are available for inelastic damageprocesses.The motif of contemporary micromechanics isaimed at discovering unknown but importanteffective constitutive information by homogenizingmassive numbers of micro-objects with simplestructures. In this paper, new cohesive damagemodels are derived based on homogenization ofrandomly distributed penny-shaped cohesive cracks(Barenblatt–Dugdale type [1,2,7]) in an elasticRVE. At micro-level, the cohesive damage modelmimics realistic interactions among atomistic bondforces at crack tips, hence it may capture the overalldamage effects due to crack opening andpropagation.In general, damage means material degradationcaused by defects or deformation. Damage may bedefined as surface separations, permanent latticedistortion, various irreversible effects due to endochronicdissipation etc. In the context of thispaper, the term damage is strictly referred to thematerial degradation due to specific defect—permanentcrack opening, or volume fraction of permanentmicro-crack opening, which may beviewed as a second phase in a composite material.This definition of damage has been extensivelyused in engineering literature. The definition ofdamage used in Gurson model (e.g. void growth)belongs to this category as well.To distinguish the damage caused by deviatoricstress, such as dislocation, disclination, and surfacesliding, with the damage caused by hydrostaticstress, such as permanent crack opening,may simplify constitutive modeling. Of course inreality, material damage may be susceptible andrelated to both hydrostatic and deviatoric stressstates and it is sensitive to their combinations.However, it is a reasonable approximation to assumethat the damage due to permanent crackopening is only related to hydrostatic stress state,which renders a tractable homogenization solution.Throughout this paper, vectors and tensors aredenoted as bold-face letters, while their componentsare written as italic. The symbol ÔÆÕ denotesinner product of two vectors a Æ b = a i b i or contractionof adjacent indices of a vector and a tensor(e.g. n Æ r = n i r ij ). The symbol Ô:Õ denotes aninner product of two second-order tensors (e.g.c:d = c ij d ij ) or a double contraction of adjacentindices of tensors of rank two and higher (eg.C:e = C ijkl e kl ).2. Average theorem for an RVE with cohesivecracksOne of the fundamental concept in classicalmicromechanics is so called representative volumeelement (RVE). An RVE for a material point usuallycontains a very large number of microstructuresand it is statistical representative of thelocal continuum properties. Despite heterogeneityof microstructures, averaged stress or strain propertiesof an RVE can be derived under specificboundary conditions. For an RVE with randomlydistributed cohesive cracks inside, traditionalmicromechanics averaging theory for traction-freedefects [23] cannot be applied. In this section, averagingtheorem for RVE containing cohesive defectswith constant cohesive traction would bederived for our purpose.Define the volume average operator h Æ i, anddefine macro stress tensor R as the volume averageof micro stress tensor in an RVE,R ¼hri ¼ 1 ZrdVð1ÞV VFirst, consider the average stress in a three-dimensionalelastic RVE with a single penny-shapedBarenblatt–Dugdale crack at the center. Neglect-

G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316 305ing body force, the equilibrium equation inside anRVE takes the formrr ¼ 0 8x 2 V ð2ÞAssume that the prescribed tractions on the remoteboundary of the RVE (oV 1 ) are generatedby a constant stress tensor r 1 . Let oV ec denotetraction-free part of a cohesive crack surface,and let oV pz denote the cohesive part of the cracksurface where constant traction force t is applied.Using divergence theorem and Eq. (2), it isstraightforward to show thathri ¼ 1 ZrdV ¼ 1 Zfr ðr xÞg T dVV V V V¼ 1 ZZr 1 dV fn ðr xÞg T dSV VoVZecfn ðr xÞg T dSoV pzZ¼ r 1 1fn ðr xÞg T dSV oV pzZ¼ r 1 1x tdSð3ÞV oV pzwhere t is the constant cohesive traction.Note that oV pz = oV pz+ [ oV pz and the surfaceareas joV pzþ j¼joV pz j¼ 1 joV 2 pzj, where subscriptÔ+Õ and Ô Õ are used to distinguish upperand lower part of the crack surfaces. So the lastterm in Eq. (3) becomesZ1x tdS ¼ 1 V oV pzVZZx t þ dS þoV pzþoV pz¼ 12V x ðtþ þ t ÞjoV pz j¼ 0x t dS!ð4Þwhere t + = t are the cohesive tractions actingon oV pz+ and oV pz respectively.Therefore, the average stress inside the RVEwill equal to remote stressR ¼hri ¼r 1ð5ÞBy superposition, it is straightforward to generatethis result to an RVE with N cohesive cracks randomlydistributed inside (see Fig. 1),hri ¼r 1 1VX Na¼1ZoV ðaÞpzx t ðaÞ dS ¼ r 1ð6ÞFig. 1. Randomly distributed micro-cracks within an RVE.Hence, the averaging theorem follows:Theorem. Suppose1. an elastic representative volume element containsN Barenblatt–Dugdale penny-shaped cracks withcohesive tractions in the cohesive zones;2. the tractions on the remote boundary of the RVEis generated by a constant stress tensor, i.e.,t 1 = n Æ r 1 and r 1 is constant.Then, macro stress of the RVE equals to the remoteconstant stress, i.e. R = hri = r 1 .3. Penny-shaped crack under uniform triaxialtensionBefore homogenization, the analytical solutionof three-dimensional (3D) penny-shaped crack inan RVE that is under uniform triaxial tension isoutlined in this section (see Fig. 2).Penny-shaped Dugdale crack problem has beenstudied by several authors. The early contributionwas made by Keer and Mura, who used the Trescayield criterion to link the cohesive strength tomicro yield stress [17]. In their study, only uniaxialtension loading was considered. More recently,

306 G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316where I (2) = d ij e i e j is the second-order identitytensor. By the averaging theorem, it is obvious thatmean macro stress, defined as R m ¼ 1 R 3 kk, equalsapplied remote stress,R m ¼ r 1ð8ÞIn cylindrical coordinate, the traction conditionson the remote boundary oV 1 and symmetric displacementboundary condition are expressed asr zz j oV 1¼ r hh j oV 1¼ r rr j oV 1¼ R m ð9ÞFig. 2. A penny-shaped cohesive crack in representativevolume element (the shaded region: cohesive zone—yieldedring).Chen and Keer re-examined the problem, and theyobtained general solutions for a penny-shapedcohesive crack under mixed-mode loading [5,6].On the other hand, however, the problem hasnot been thoroughly examined from micromechanicsperspective. For example, the connectionamong the onset value of cohesive strength, microyield stress in an RVE, and remote macro stresshas not been made. By examining a cohesivepenny-shaped crack in an RVE, the study providesa link among cohesive strength, yield stress of virginmaterial, and remote stresses on the boundaryof an RVE, which provides foundation for ensuinghomogenizations.Consider a three-dimensional penny-shapedDugdale crack of radius a with a ring-shaped cohesivezone with width b a in an RVE, which maybe viewed as an infinite isotropic space by ‘‘amicro-observer ’’ inside the RVE.Let the outward normal to crack surface parallelto Z(X 3 ) axis (see Fig. 2) and uniform triaxialtension stress is applied at the remote boundaryof the RVE, i.e.r 1 ¼ r 1 I ð2Þð7Þu z ðr; h; 0Þ ¼0 for b 6 r; 0 6 h 6 2p ð10ÞThe stress distribution on the crack surface andcohesive zone isr zz ðr; h; 0Þ ¼r 0 Hðr aÞfor 0 6 r 6 b; 0 6 h 6 2pð11Þwhere R m is the remote stress, H(r a) is theHeaviside function, and r 0 is the materialÕs cohesivestrength, the onset value for crack opening,and it is different from the micro yielding stress.The problem can be solved via superposition oftwo sub-problems: a trivial problem—an intactRVE in uniform triaxial tension state, i.e. "x2V,r ð0Þzzr ð0Þrz¼ r ð0Þhh¼ r ð0Þrh¼ rð0Þ rr¼ R m ð12Þ¼ rð0Þ zh ¼ 0 ð13Þand a crack problem—an RVE with a center crackthat is subjected to the following boundary conditions(see Fig. 3):r ðcÞzz j oV 1¼ r ðcÞhh j oV 1¼ r ðcÞrr j oV 1¼ 0 ð14Þr ðcÞzz ðr; h; 0Þ ¼ R m þ r 0 Hðr aÞfor 0 6 r 6 b; 0 6 h 6 2pð15Þu ðcÞzðr; h; 0Þ ¼0 for b 6 r; 0 6 h 6 2p ð16ÞFor the crack problem, the crack opening displacementcould be solved as8>:2p2p1 v l 1 v l R mR mpffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2 r 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2 r 2R bpffiffiffiffiffiffiffiffiffiffiffiffiffipr 0 ta2 a 2 = ffiffiffiffiffiffiffiffiffiffiffiffiffi t 2 r 2 dt ; 0 < r < aR bpffiffiffiffiffiffiffiffiffiffiffiffiffipr 0 tr2 a 2 = ffiffiffiffiffiffiffiffiffiffiffiffiffi t 2 r 2 dt ; a < r < bð17Þ

G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316 307In the yield ring (z = 0 and a < r < b), stress distributionsare found asr ðcÞzz¼ r 0 R m ð18ÞFig. 3. Illustration of superposition of cohesive crack problem.rr¼ 1 þ 2v R m 2þ 1 2v2r ðcÞ1 þ a2r 2þ 2v r 0ð19Þr ðcÞhh ¼ 1 þ 2v 2þ 1 2v2r ðcÞrz¼ r ðcÞrhR m1a 2r 2þ 2v r 0ð20Þ¼ rðcÞ zh ¼ 0 ð21Þwhere PoissonÕs ratio v*, to be treated as eitheroverall or matrix material property, is unspecifiedat this moment, which may depend on the laterhomogenization procedures.To ensure the stresses at crack tip to be finite,the size of the cohesive zone a/b, macro stressR m , and the cohesive stress r 0 are related throughthe following expression:ab ¼sffiffiffiffiffiffiffiffiffiffiffiffiffiffiR 2 m1r 2 0orR mr 0¼sffiffiffiffiffiffiffiffiffiffiffiffiffi1a 2b 2ð22ÞDenote the projection of traction-free crack surfaceonto X 1 X 2 plane as X 1 , and the projectionof the cohesive zone (ring shape) as X 2 . The totalvolume of crack opening by a single cohesive crackis the integration of crack opening displacementFig. 4. Projection domain of crack surface and cohesive zone.over the entire projection area, X = X 1 [ X 2 (seeFig. 4). It is readily to show thatZ Z ZV c ¼ ½u z ŠdS ¼ ½u z ŠdS þ ½u z ŠdSXX 1 X 2¼ 8ð1 v Þhib 3 R3l m r 0 ð1 ða=bÞ 2 Þ 3=2¼ 8ð1 v Þa 3 R mqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3l 1 ðR m =r 0 Þ 2ð23Þwhere the crack opening displacement, COD, isdefined by½u z Š¼u þ zu z¼ 2u z ð24ÞIf one is mainly interested in inelastic deformationof quasi-brittle materials, it may be assumed that

308 G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316micro-scale yielding due to hydrostatic stress stateis small-scale yielding: a2 1. For a 6 r 6 b, a2 1.b 2 r 2The total stress distribution within cohesive zone(a 6 r 6 b) may be approximated asr ðtÞzzr ðtÞrr¼ r ð0Þzz¼ r ð0Þrrþ r ðcÞzz¼ r 0 ð25Þþ r ðcÞrr¼ 1 2v R m þ r 02r ðtÞhh ¼ rð0Þ hh þ rðcÞ hh ¼ 1r ðtÞrz¼ r ðtÞrh ¼ rðtÞ zh ¼ 02v R m þ 2v r 02ð26Þð27Þð28ÞIt is assumed that inside the cohesive zone microplastic yielding is controlled by the Huber–vonMises criterion. Therefore, one can link the cohesivestrength, r 0 , with the yield stress of the virginmaterial, r Y ,by12r ðtÞrrr ðtÞzz 2 2þ rðtÞ 2þ rðtÞhhr ðtÞzzrrr ðtÞhh¼ r 2 Yð29ÞSubstitute Eqs. (25)–(27) into (29) and solve for r 0 .The following quadratic equation may be obtained:4 r 2 02 r 202 r Yþ 1¼ 0R m R m 1 2v R mð30Þwhich has two roots. The positive root is chosen tolink the cohesive stress r 0 with the yield stress r Yof virgin material in uniaxial tension,1 þr 0¼R msffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 24 r Y31 2v R m4ð31Þ4. Effective elastic material properties of an RVEOne of the goals of homogenization is to find theeffective material properties of an RVE to characterizethe constitutive relation of stress and strainrelation at the macro-level. Conjugate to the macrostress R, the macro strain E is defined through theoverall complementary energy density W c of an RVEE ¼ oW coRð32ÞDefine the effective compliance D of the microcrackedRVE throughE ¼ D : R ¼ D : r 1ð33Þwhere the macro stress R = r 1 , according to averagetheorem, is regarded as prescribed. It is notedthat the macro strain of an RVE may not be thevolume average strain, i.e, E 5 hei. FurthermoreEq. (33) may not be a linear relationship, becauseD may depend on R in general, which will be substantializedin later sections.A common strategy for homogenization of randomlydistributed defects is to divide the macrostrain E into two parts,E ¼ e ð0Þ þ e ðaÞð34Þwhere e (0) = D:R is known, and D is the elasticcompliance of the corresponding virgin material.The second term e (a) is so-called additional straintensor representing the effect of defects. If the relationshipbetween additional strain and macrostress can be found, say e (a) = H:R, where H isthe added compliance due to micro-cracks, subsequentlythe effective elastic compliance moduli,D ¼ D þ H, can be deduced.Energy method is applied to find an additionalstrain formula for cohesive cracks. The essenceof energy methods is to find the energy release ina cohesive fracture process and hence to find theequivalent reduction of material properties. Forelastic cracks, the energy release rates were discussedin the classical works of Rice and hisco-worker [3,25,26]. Nevertheless, the energy dissipationprocess of cohesive fracture is much morecomplicated than a purely elastic fracture process.It includes energy dissipation from both surfaceseparation and plastic dissipation. A related discussioncan be also found in [18,22,31].Although accurate determination of energy lossduring a damage process requires an in-depthunderstanding of the physical process involved,an upper bound estimate may be made based onsimplified assumptions. It is assumed that the totalenergy release of a cohesive crack is completelyconsumed in surface separation, which may ormay not be true in cohesive fracture, because ofplastic dissipation in the cohesive zone.

G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316 309Subjected to uniform triaxial loading r 1 =R m I (2) , the total energy release rate of an RVE witha single penny-shaped cohesive micro-crack can beestimated asZZR ¼ R m ½u z ŠdS r 0 ½u z ŠdSð35ÞXX 2Carrying out the integration using crack displacementsolutions Eqs. (17), the energy release estimatecan be written as following expression:R ¼ 16ð1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv Þr 23l 0 a3 a1 1 ðR m =r 0 Þ 2 ð36ÞConsider that there are N penny-shaped cracks insidethe RVE. The density of energy release of theRVE is estimated as sum of each crack contribution,i.e. R ¼ P Na¼1 R a. Define the crack openingvolume fraction asf ¼ XN 4pa 3 a3V bð37Þa¼1where a a is the radius of the ath crack, and 4pa 3 a =3is the volume of a sphere with radius a a , and b isthe ratio between the volume of permanent crackopening and the volume of total crack openingof a cohesive crack. For simplicity, it is assumedthat this ratio is fixed for every crack inside anRVE. Obviously, 0 < b

310 G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316D ¼ 13K E1 þ 12l E2ð46ÞH ¼ðh 1 þ h 2 ÞE 1 þ h 2 E 2ð47Þwhere E 1 ¼ 1 3 Ið2Þ I ð2Þ , E 2 ¼ 1 3 Ið2Þ I ð2Þ þ I ð4sÞ .Since the traction stress state on the remoteboundary of the RVE is hydrostatic, H tensor inEq. (44) cannot be uniquely determined in thiscase. Information carried in Eq. (43) is only volumetricand admits only one scalar relation,D : R m I ð2Þ ¼ðD þ HÞ : R m I ð2Þð48ÞConsider Eq. (34) and identities E 1 : I (2) = I (2) andE 2 : I (2) = 0. By virtue of Eqs. (41) and (48), itcan be shown that13K ¼ 13K þðh 1 þ h 2 Þ¼ 13K þ 4ð1 vÞ fqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið49Þ3bpl1 ðR m =r 0 Þ 2There are two unknowns, K and l, or equivalentlyh 1 and h 2 in Eq. (49). Additional condition isneeded to uniquely determine D or H. Imposethe restriction that the relative reduction of theshear modulus is the same as that of the bulkmodulus,KK ¼ l ð50ÞlThis restriction further guarantees the positive definitenessof the overall strain energy.Consider the relations13K ¼ 1 2vE12l ¼ 1 þ vEandand13K ¼ 1 2vE12l ¼ 1 þ vEð51ÞA direct consequence of Eq. (50) is v ¼ v, so13K ¼ 1 1 2v¼ 1 1 2v2l 1 þ v 2l 1 þ v13K ¼ 1 ð52Þ1 2v2l 1 þ vKK ¼ l l ¼ 1 8ð1 v 2 Þ fqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3bpð1 2vÞ1 ðR m =r 0 Þ 25. Micro-cohesive-crack damage modelð53ÞHomogenization of nonlinear problems is oftendifficult. Without proper statistical closure, averagingalong may not be sufficient to provide sensibleresults. In this paper, it is postulated that thereis a limit for the amount of distortional energy thata given material ensemble can store. This reflectsin the following hypothesis on the condition ofmacro-yielding:The macroscopic yielding of an RVE begins whenthe distortional strain energy density of an RVEreaches to a threshold. In other words, the maximumelastic distortional energy of an RVE is a materialconstant,U d 6 U crdð54ÞIt is noted that above criterion is a reminiscenceof the HenckyÕs maximum distortional energyprinciple in traditional infinitesimal plasticity.The criterion can be calibrated using an uniaxialtension test of the virgin materialU crd ¼ 16l r2 Yð55ÞDefine the macro deviatoric stress tensor and itssecond invariant asR 0 ¼ R 1R 3 kkI ð2Þ ð56ÞJ 2 ¼ 1 2 R0 : R 0ð57Þand the equivalent macro stress R eq is defined aspR eq ¼ffiffiffiffiffiffiffi3J 2ð58ÞIn a real damage evolution process, the above criteriatake the following form:U d ¼ J 22l ¼ R2 eq6l 6 U crdð59ÞSo the criterion of the maximum distortional energydensity of an RVE becomeswhich, when substituted into Eq. (49), leads to theestimates of effective elastic moduliR 2 eqr 2 Y¼ l lð60Þ

G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316 311Using Eq. (53), one may derive the following effectiveyielding potential:WðR eq ; R m ; qÞ ¼ R2 eqþ 8ð1 v2 Þr 2 Y3bpð1 2vÞf qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ¼ 0 ð61Þ1 ðR m =r 0 Þ 2where R eq and R m are defined as the equivalentmacro stress and mean macro stress, and q representsthe other internal variables, which may beimplicitly embedded in r Y .In terms of the stress ratio R m /r Y , the effectiveyielding potential function of plastic flow W canbe recast asWðR eq ; R m ; qÞ¼ R2 eqþ 8ð1 v2 Þfr 2 Y3bpð1 2vÞrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 24r1 þYð1 2vÞR m3 0r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 212@4r1 þY3 16Að12vÞR m121 ¼ 0ð62ÞThe newly derived pressure-sensitive yieldingfunction W, is displayed in Fig. 5 with differentPoissonÕs ratios. It should be mentioned that theself-consistent scheme based damage model W willfail at v = 0.5, since for incompressible materials,uniform triaxial tension load will not be able toproduce dilatational strain energy.In Fig. 6, the cohesive damage model is juxtaposedwith the popular Gurson model [12], whoseeffective yield function takes the following form:U ¼ R2 eqþ 2f cosh 3 R mð1 þ f 2 Þ¼0 ð63Þr 2 Y2 r YBoth yield functions will reduce to classical J 2plasticity when f = 0. However, it is worth noticingthat for the limit case of infinitesimal amount ofdamage (i.e. f ! 0), the proposed cohesive damagemodel predicts yielding of material when the stressratio of hydrostatic stress and the cohesive strength,or equivalently the ratio of hydrostatic stress andtrue yield stress, approaches a finite value, i.e.R mR m 4! 1 or ! pffiffiffiffiffið64Þr 0 r Y 12 ð1 2vÞFor Gurson model, when the amount of damage isinfinitesimal the material will not yield unless thehydrostatic stress becomes infinite. The asymptoticyielding potentials for infinitesimal damage arehighlighted as dashed lines in Fig. 6(c) and (d).Obviously, the Gurson model is not realistic, becauseany material will fail when the remote loadexceeds the materialÕs theoretical strength.In the followings, features of the present modelwill be explored in the framework of continuummechanics. The macro response the reversible partof effective constitutive relation is characterized asa nonlinear elasticity, whereas the irreversible partof effective constitutive relation is a form of pressure-sensitiveplasticity. The rate form of constitutiverelation for damaged materials is written as_R ¼ C : ð E _ pE _ Þ ð65Þwhere C is the effective elastic moduli andC ¼ D 1 p; E _ and E _ are the average rate of totaldeformation and the average rate of plastic deformationrespectively.The macro plastic flow direction may be givenby the associative rule_E p ¼ kN _ ð66Þwhere N is the normal of yield surface in stressspaceN ¼ oWð67ÞoRand the plastic multiplier k _ can be determined byconsistency condition as usual,hN : C : Ei_k ¼_ ð68ÞN : C : N W q Nwhere W q ¼ oW , h Æ iis the Macauley bracket, and qoqare internal variables whose evolutions are assumedto be governed by_q ¼ _ khðR; qÞð69ÞThe damage evolution law for micro-cracks in acohesive elastic environment may be significantlydifferent from that of voids in a perfectly viscoplasticenvironment. For this moment, conventional

312 G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316Fig. 5. Cohesive micro-crack damage model, W(R eq , R m , q), with different PoissonÕs ratios (b = 1/3).damage evolution law is adopted. Let the volumeof the RVE be denoted as V = V m + V c , whereV c is the crack opening volume and V m is the volumeof matrix. Assuming the total rate change ofthe crack volume is proportional to the volume ratechange in an RVE, i.e._V c ¼ c _Vð70Þwhere c is the proportionality constant. Then_f ¼ d dtV cV¼ðcf Þ _ VV ¼ðc f Þtraceð _ EÞ ð71ÞAssume a specimen is originally intact (i.e. f = 0),and it is subjected to an uniaxial tension test underdisplacement control. The onset of material damage,which is physically initiation and propagationof micro-cohesive-cracks, will occur after the specimenfirst reaches its yield limit (r Y , e Y ).Let r and e denote axial stress and axial strainof the specimen, it is obvious that R eq = r,R m = r/3 and traceð EÞ¼ð1 _ 2vÞ_e. Eq. (71) isused to describe post-elastic damage evolutionprocess, and for the purpose of illustration onecan simply choose c = 1. Integrate the rate form,volume fraction f could therefore be deduced froma given axial strain e,1f ¼ 1ð72Þexpðð1 2vÞðe e Y ÞÞThe corresponding axial stress during persistentplastic loading could be derived via solving yieldingpotential functions W(r, e) =0orU(r, e)=0.In Figs. 7 and 8, the predicted post-elastic materialbehaviors of the present model and Gursonmodel are plotted for various PoissonÕs ratios.For clarity, initial yield strain e Y is neglected in

314 G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316Fig. 7. Cohesive model prediction of axial stress and strain fordifferent PoissonÕs ratios.ratio, it is still able to capture softening behavior atthe nearly incompressible limit.To account for the rapid void coalescence atfailure, Tvergaard and Needleman extended originalGurson model by introducing parameters q 1 ,q 2 and f*. As a modification of Eq. (63), the yieldingpotential function of Gurson–Tvergaard–Needleman(GTN) model [30] is written asU ¼ R2 eqþ 2qr 2 1 f cosh 3q 2 R mð1 þ q 2 1Y2 r f 2 Þ¼0Yð73Þwhere f* is specified as a piecewise function of f via8ffor f 6 f c>:1=q 1 for f f < fð74ÞNote that f c and f f are volume fractions at onset ofvoid coalescence and total failure respectively.They are regarded as material constant and needto be specified. It is seen that as volume fraction fgrows towards its failure value f f and f* approaches1/q 1 the yield surface for macro stresses shrinks toa point, so material is ‘‘preprogrammed’’ to fail atf = f f . So in GTN model, effects of void coalescenceand total failure are addressed by scaling and interpolation,while failure prediction is embedded inthe cohesive cracked model.Fig. 8. Gurson model prediction of axial stress and strain fordifferent PoissonÕs ratios.Table 1Predicted critical volume fraction for different PoissonÕs ratiosv 0.100 0.200 0.300 0.400 0.499f f 0.632 0.489 0.343 0.186 0.002volume fraction change throughout the tensiontest is very tinny according to Eq. (72). Gursonmodel reduces to J 2 plasticity and fail to predictany damage effect. While for the present model,because yielding potentials also depend on PoissonÕsFig. 9. Comparison of model predictions (v = 0.3).

G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316 315Predictions of all three models are plotted inFig. 9 for PoissonÕs ratio v = 0.3. For simply illustrativepurpose, parameters for GTN model arechosen as q 1 = 1.5, q 2 =1, and f f = 0.343, f c =f f /3. It is noted that above f f value is chosen onpurposely to be the same as that predicted bycohesive crack model. Although GTN matchescohesive model at both ends (of course), itsundergoes an abrupt change once f exceeds f c .Damage process towards failure is piecewise quasi-linearand catastrophic failure effect is stillabsent.6. Concluding remarksIn this paper, a novel micromechanics baseddamage model is proposed for addressing failuremechanism of defected solids with randomly distributedpenny-shaped cohesive micro-cracks(Barenblatt–Dugdale type). The distinguished featuresof the present cohesive crack damage modelsare:• Homogenized macro-constitutive relations aredifferent from the micro-constitutive relations:the reversible part of macro-constitutive relationis nonlinear elastic versus the linear elasticbehaviors at micro-level; the irreversible part ofmacro-constitutive relation is a form of pressuresensitive plasticity versus the Huber–von Misesplasticity or cohesive laws at micro-level.• In cohesive damage models, the effective yieldsurfaces depend on materials PoissonÕs ratio;whereas in the Gurson model, no such dependencecan be predicted, because of its assumptionof RVE is perfectly plastic. Different asymptoticyield surfaces are also observed for the case ofinfinitesimal damage.• Comparison of the present model with Gursonmodel and Gurson–Tvergaard–Needleman(GTN) model for a simple loading shows that,the present model can predict a critical volumefraction at which complete failure of materialwould occur. It can model and predict post elasticmaterial degradation and catastrophic failuredue to cohesive micro-crack growth andcoalescence.The key step in the model development is howto accurately determine the energy release contributionto the material damage process. The energyrelease in nonlinear fracture mechanical process isconsumed in several different dissipation processes,e.g. surface separation, dislocation movement andhence plastic dissipation, heat conduction, andmay be even phase transformation, etc. In fact,both Kfouri [18] and Wnuk [31] have studiedenergy release caused by the extension of Dugdale-BCS cracks in a two-dimensional space. To incorporatethose available results into the currentformulation, an in-depth study may be needed torefine the damage models proposed here. A detailedcompanied mathematical exposition oncohesive damage model is reported in Li andWang [21].AcknowledgmentThis work is supported by a research fund toProfessor Shaofan Li by the committee on researchin University of California at Berkeley,which is appreciated.References[1] G.I. Barenblatt, The formation of equilibrium cracksduring brittle fracture, Journal of Applied Mathematics& Mechanics 23 (1959) 622–636, English translation fromPMM 23 (1959) 434–444.[2] G.I. Barenblatt, Mathematical Theory of EquilibriumCracks in Brittle Fracture, Advances in Applied Mechanics,vol. 7, Academic Press, 1962, pp. 55–129.[3] B. Budiansky, J.R. Rice, Conservation laws and energyrelease rates, Journal of Applied Mechanics 26 (1973) 201–203.[4] B. Budiansky, R.J. OÕConnell, Elastic moduli of a crackedsolid, International Journal of Solids and Structures 12(1976) 81–97.[5] W.R. Chen, L.M. Keer, Fatigue crack growth in mixedmode loading, ASME Journal of Engineering Materialsand Technology 113 (1991) 222–227.[6] W.R. Chen, L.M. Keer, Mixed-mode fatigue crack propagationof penny-shaped cracks, ASME Journal of EngineeringMaterials and Technology 115 (1993) 365–372.[7] D.S. Dugdale, Yielding of steel sheets containing slits,Journal of Mechanics and Physics of Solids 8 (1960) 100–104.

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