Loss of ellipticity for non-coaxial plastic deformations in additive logarithmic finite strain plasticity

TL;DR

This paper investigates the loss of ellipticity in additive logarithmic finite strain plasticity, revealing that rank-one convexity may not be preserved after plastic deformation, contrasting with multiplicative plasticity.

Contribution

It demonstrates that additive logarithmic plasticity can lose ellipticity after plastic deformation, unlike multiplicative plasticity where convexity is preserved.

Findings

Additive logarithmic plasticity may lose rank-one convexity after plastic deformation.

Contrast established between additive and multiplicative plasticity regarding convexity preservation.

Uses exponentiated Hencky energies to illustrate the loss of convexity.

Abstract

In this paper we consider the additive logarithmic finite strain plasticity formulation from the view point of loss of ellipticity in elastic unloading. We prove that even if an elastic energy $F\mapsto W(F)=\hat{W}(\log U)$ defined in terms of logarithmic strain $\log U$, where $U=\sqrt{F^T\, F}$, is everywhere rank-one convex as a function of $F$, the new function $F\mapsto \widetilde{W}(F)=\hat{W}(\log U-\log U_p)$ need not remain rank-one convex at some given plastic stretch $U_p$ (viz. $E_p^{\log}:=\log U_p$). This is in complete contrast to multiplicative plasticity in which $F\mapsto W(F\, F_p^{-1})$ remains rank-one convex at every plastic distortion $F_p$ if $F\mapsto W(F)$ is rank-one convex. We show this disturbing feature with the help of a recently considered family of exponentiated Hencky energies.