A penny-shaped cohesive crack model for material damage

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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
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