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↦W(F)=W^(logU) defined in terms of logarithmic strain logUlogU, where U=FTF, happens to be everywhere rank-one convex as a function of F  , the new function F↦W˜(F)=W^(logU−logUp) need not remain rank-one convex at some given plastic stretch Up (viz. Eplog≔logUp). This is in complete contrast to multiplicative plasticity (and infinitesimal plasticity) in which F↦W(FFp−1) remains rank-one convex at every plastic distortion Fp if F↦W(F)F↦W(F) is rank-one convex (∇u↦∥sym∇u−εp∥2 remains convex). We show this disturbing feature of the additive logarithmic plasticity model with the help of a recently introduced family of exponentiated Hencky energies.