Constitutive relations and their time integration for anisotropic elasto-plastic porous materials

Finite deformation constitutive relations are developed for a class of plastically anisotropic porous solids with an underlying evolving microstructure. They are based on a model obtained by homogenization for rigid-perfectly plastic materials containing non-spherical voids. To facilitate numerical implementation, heuristic extensions are proposed to incorporate weak elasticity, strain hardening and accurate void shape evolution. A semi-implicit time integration scheme is used along with the Newton–Raphson method to solve the system of equations resulting from the discretization of the constitutive equations. The procedure to calculate the consistent tangent matrix, which is needed to solve the global force–displacement matrix equation, is summarized. The framework is used to illustrate the predictive capabilities of the model, first under conditions previously assessed against finite element cell model calculations, then under conditions heretofore not examined. The latter include situations of initial anisotropy as well as situations involving significant void distortions, not only in terms of void enlargement or shape change, but also in terms of void rotations. In particular, various combinations of stress triaxiality, initial void shape, void orientation, matrix orthotropy properties and loading directions are simulated. In addition, the finite element implementation of the model is addressed and illustrated for simple cases.