314 G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316Fig. 7. Cohesive <strong>model</strong> prediction of axial stress and strain <strong>for</strong>different PoissonÕs ratios.ratio, it is still able to capture softening behavior atthe nearly incompressible limit.To account <strong>for</strong> the rapid void coalescence atfailure, Tvergaard and Needleman extended originalGurson <strong>model</strong> by introducing parameters q 1 ,q 2 and f*. As a modification of Eq. (63), the yieldingpotential function of Gurson–Tvergaard–Needleman(GTN) <strong>model</strong> [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 via8f<strong>for</strong> f 6 f c>:1=q 1 <strong>for</strong> f f < fð74ÞNote that f c and f f are volume fractions at onset ofvoid coalescence and total failure respectively.They are regarded as <strong>material</strong> 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 <strong>for</strong> macro stresses shrinks toa point, so <strong>material</strong> is ‘‘preprogrammed’’ to fail atf = f f . So in GTN <strong>model</strong>, effects of void coalescenceand total failure are addressed by scaling and interpolation,while failure prediction is embedded inthe <strong>cohesive</strong> <strong>crack</strong>ed <strong>model</strong>.Fig. 8. Gurson <strong>model</strong> prediction of axial stress and strain <strong>for</strong>different PoissonÕs ratios.Table 1Predicted critical volume fraction <strong>for</strong> 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). Gurson<strong>model</strong> reduces to J 2 plasticity and fail to predictany <strong>damage</strong> effect. While <strong>for</strong> the present <strong>model</strong>,because yielding potentials also depend on PoissonÕsFig. 9. Comparison of <strong>model</strong> predictions (v = 0.3).
G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316 315Predictions of all three <strong>model</strong>s are plotted inFig. 9 <strong>for</strong> PoissonÕs ratio v = 0.3. For simply illustrativepurpose, parameters <strong>for</strong> GTN <strong>model</strong> 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 by<strong>cohesive</strong> <strong>crack</strong> <strong>model</strong>. Although GTN matches<strong>cohesive</strong> <strong>model</strong> 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 based<strong>damage</strong> <strong>model</strong> is proposed <strong>for</strong> addressing failuremechanism of defected solids with randomly distributed<strong>penny</strong>-<strong>shaped</strong> <strong>cohesive</strong> micro-<strong>crack</strong>s(Barenblatt–Dugdale type). The distinguished featuresof the present <strong>cohesive</strong> <strong>crack</strong> <strong>damage</strong> <strong>model</strong>sare:• 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 <strong>for</strong>m of pressuresensitive plasticity versus the Huber–von Misesplasticity or <strong>cohesive</strong> laws at micro-level.• In <strong>cohesive</strong> <strong>damage</strong> <strong>model</strong>s, the effective yieldsurfaces depend on <strong>material</strong>s PoissonÕs ratio;whereas in the Gurson <strong>model</strong>, no such dependencecan be predicted, because of its assumptionof RVE is perfectly plastic. Different asymptoticyield surfaces are also observed <strong>for</strong> the case ofinfinitesimal <strong>damage</strong>.• Comparison of the present <strong>model</strong> with Gurson<strong>model</strong> and Gurson–Tvergaard–Needleman(GTN) <strong>model</strong> <strong>for</strong> a simple loading shows that,the present <strong>model</strong> can predict a critical volumefraction at which complete failure of <strong>material</strong>would occur. It can <strong>model</strong> and predict post elastic<strong>material</strong> degradation and catastrophic failuredue to <strong>cohesive</strong> micro-<strong>crack</strong> growth andcoalescence.The key step in the <strong>model</strong> development is howto accurately determine the energy release contributionto the <strong>material</strong> <strong>damage</strong> 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 trans<strong>for</strong>mation, etc. In fact,both Kfouri [18] and Wnuk [31] have studiedenergy release caused by the extension of Dugdale-BCS <strong>crack</strong>s in a two-dimensional space. To incorporatethose available results into the current<strong>for</strong>mulation, an in-depth study may be needed torefine the <strong>damage</strong> <strong>model</strong>s proposed here. A detailedcompanied mathematical exposition on<strong>cohesive</strong> <strong>damage</strong> <strong>model</strong> is reported in Li andWang [21].AcknowledgmentThis work is supported by a research fund toProfessor Shaofan Li by the committee on researchin University of Cali<strong>for</strong>nia at Berkeley,which is appreciated.References[1] G.I. Barenblatt, The <strong>for</strong>mation of equilibrium <strong>crack</strong>sduring 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 <strong>crack</strong>edsolid, International Journal of Solids and Structures 12(1976) 81–97.[5] W.R. Chen, L.M. Keer, Fatigue <strong>crack</strong> growth in mixedmode loading, ASME Journal of Engineering Materialsand Technology 113 (1991) 222–227.[6] W.R. Chen, L.M. Keer, Mixed-mode fatigue <strong>crack</strong> propagationof <strong>penny</strong>-<strong>shaped</strong> <strong>crack</strong>s, 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.