A Gurson-type criterion for plastically anisotropic solids containing arbitrary ellipsoidal voids

The aim of this paper is to provide a homogenized criterion for porous ductile materials incorporating both void shape and plastic anisotropy effects. This is done by extending recent criteria of Madou and Leblond (2012a,b) for general ellipsoidal cavities in plastically isotropic matrices, and Monchiet et al. (2008), Keralavarma and Benzerga (2010) for spheroidal cavities in plastically anisotropic matrices, to general ellipsoidal cavities in plastically anisotropic matrices. A limit-analysis is performed of an ellipsoidal representative volume made of some rigid-ideal-plastic Hill material, containing a confocal ellipsoidal void and loaded under conditions of homogeneous boundary strain rate. Use is made in this analysis of some trial incompressible velocity fields discovered by Leblond and Gologanu (2008), satisfying such conditions on an arbitrary family of confocal ellipsoids. Approximations resulting from asymptotic studies of the microscopic plastic dissipation near the void and at infinity lead to an analytic yield function, the coefficients of which are not fully determined at this stage. Complete determination of these coefficients is done using finite element simulations for hydrostatic loadings, on the one hand, and a rigorous bound of Ponte-Castaneda (1991), Willis (1991) and Michel and Suquet (1992) for nonlinear composites for deviatoric loadings, on the other hand.