Plasticity of three-dimensional foams

Abstract

This chapter gives in the first part a summary of some important elements in continuum mechanics, i.e. the decomposition of the stress tensor in its spherical and the deviatoric part and the use of stress invariants to describe the physical content of the stress tensor. In the next part, the elastic behaviour of isotropic materials based on generalised Hooke’s law is summarised and a notation appropriate for computer implementation is introduced. The constitutive description is then extended to plastic material behaviour and the description based on a yield condition, flow rule and hardening law is introduced. The concept of invariants is consistently applied and explained for the characterisation of yield conditions. A classical simple cubic cell model based on beams (Gibson/Ashby model) is investigated in the next chapter in order to highlight the assumptions and the derivation of the macroscopic material properties (elastic constants and yield stress). In the following, a strategy to determine the influence of the hydrostatic stress on the yield behaviour is proposed and conceptionally realised by a state of plane strain and a state of uniaxial strain. In addition, alternative ways to determine the complete set of elastic constants are shown. The last part covers the implementation of yield conditions into finite element codes. The understanding of the predictor-corrector concept is required to provide new constitutive equations in commercial computational codes.