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A penny-shaped cohesive crack model for material damage

A penny-shaped cohesive crack model for material damage

A penny-shaped cohesive crack model for material damage

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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).

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