On the relationship between the Eulerian and Lagrangian descriptions of finite rigid plasticity

Abstract

The motivation for the present paper is to clarify certain unresolved issues pertaining to the relation between the Lagrangian (or referential) and Eulerian (or spatial) strain-space formulations of finite plasticity. For conceptual simplicity, attention is confined to rigid-plastic materials. It is shown first that for constitutive equations in which the hardening parameter is a scalar, the Lagrangian and Eulerian descriptions are equivalent; and that, additionally, the choice of objective stress rate is immaterial. In the light of these developments, the role of objective rates is further explored in connection with more general (“anisotropic”) hardening laws which contain a shift tensor. A form of the constitutive equation for the rate of the shift tensor is motivated in which the choice of objective rate is arbitrary. It is then demonstrated that the structure of the constitutive equations of the theory — in both the Eulerian and Lagrangian descriptions — is form-invariant under arbitrary transformations of objective rate. The approach taken here contrasts with that adopted in a number of recent papers in which preference is given to one particular objective rate or another.