Volume conservation during finite plastic deformation

TL;DR

This paper establishes a rigorous condition for volume conservation in finite strain elastoplasticity, proposing universal strategies applicable regardless of the use of unloading stress free configurations, supported by theoretical and numerical validation.

Contribution

It introduces a universal approach to ensure volume conservation in elastoplastic theories, regardless of the configuration assumptions, by linking it to the trace of the plastic stress rate.

Findings

Volume conservation is achieved when the plastic stress rate has zero trace.

Revising the stiffness tensor or choosing appropriate stress/strain measures can satisfy volume conservation.

The proposed methods are validated through theoretical proofs and numerical examples.

Abstract

An elastoplastic theory is not volume conserved if it improperly sets an arbitrary plastic strain rate tensor to be deviatoric. This paper discusses how to rigorously realize volume conservation in finite strain regime, especially when the unloading stress free configuration is not adopted or unique in the elastoplastic theories. An accurate condition of volume conservation is clarified and used in this paper that the density of a volume element after the applied loads are completely removed should be identical to that of the initial stress free states. For the elastoplastic theories that adopt the unloading stress free configuration (i.e. the intermediate configuration), the accurate condition of volume conservation is satisfied only if specific definitions of the plastic strain rate are used among many other different definitions. For the elastoplastic theories that do not adopt the unloading stress free configuration, it is even more difficult to realize volume conservation as the information of the stress free state lacks. To find a universal approach of realizing volume conservation for elastoplastic theories whether or not adopt the unloading stress free configuration, we propose a single assumption that the density of material only depends on the trace of the Cauchy stress, and interestingly find that the zero trace of the plastic stress rate is equivalent to the accurate condition of volume conservation. Two strategies are further proposed to satisfy the accurate condition of volume conservation: directly and slightly revising the tangential stiffness tensor or using a properly chosen stress/strain measure and elastic compliance tensor. They are implemented into existing elastoplastic theories, and the volume conservation is demonstrated by both theoretical proof and numerical examples.