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

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.