Equivalent spatial and material descriptions of finite deformation elastoplasticity in principal axes

Abstract

A spatial description of the theory of rate-independent finite deformation elastoplasticity, in which the stress tensor is defined through the strain energy function, is discussed. The main assumption of isotropic elastic response and invariance requirements under superposed rigid body motion restrict the acceptable forms of the strain energy function to those given in terms of principal values of the strain measure of elastic distortion. The formulation is developed on a manifold and the corresponding material description is obtained simply by pull-back of the derived spatial form, by appealing to the notion of covariance. The method of principal axes is systematically exploited to derive the explicit expression for the stress tensor computation for an arbitrary form of the strain energy function and the explicit form of the evolution equation for an arbitrary form of the yield function. A model problem of volume-preserving plastic flow is discussed in the closure.