Numerical homogenisation of an elasto-plastic model-material with large elastic strains: macroscopic yield surfaces and the Eulerian normality rule

Abstract

This article presents the details of a numerical technique for computing the macroscopic response of a material with a given micro-structure to arbitrary prescribed loading histories. The method uses classical concepts of homogenisation theory in combination with the finite element method and focusses on the computation of macroscopic yield surfaces and inelastic strain rates. It places no restrictions on the magnitude of deformation and allows arbitrary combinations of stress- or strain control including the prescription of histories of the Cauchy stress. The method is illustrated by analysing a model material, consisting of a non-linearly elastic matrix with stiff elasto-plastic inclusions, which exhibits macroscopically associative elasto-plastic material behaviour with finite elastic strains. Yield surfaces and the directions of plastic flow after a prior finite simple shear deformation are computed for this material and are shown to be consistent with an additive decomposition of the Eulerian strain rate into elastic and plastic parts and a suitable formulation of the normality rule in Cauchy stress space; a novel version of the latter is derived which is valid for an arbitrary reference configuration.