The exponentiated Hencky-logarithmic strain energy. Part III: Coupling with idealized isotropic finite strain plasticity

We investigate an immediate application in finite strain multiplicative plasticity of the family of isotropic volumetric-isochoric decoupled strain energies \begin{align*} F\mapsto W_{{\rm eH}}(F):=\hat{W}{{\rm eH}}(U):={\begin{array}{lll} \frac{\mu}{k},e^{k,|{\rm dev}n\log {U}|^2}+\frac{\kappa}{{$$}{2, {\hat{k}}}},e^{\hat{k},[{\rm tr}(\log U)]^2}&$\text{if}$& {\rm det}, F>0,\ +\infty &$\text{if}$ &{\rm det} F\leq 0, \end{array}.\quad \end{align*} based on the Hencky-logarithmic (true, natural) strain tensor $\log U$. Here, $\mu >0$ is the infinitesimal shear modulus, $\kappa =\frac{2\mu +3\lambda }{3}>0$ is the infinitesimal bulk modulus with $\lambda$ the first Lam'{e} constant, $k,\widehat{k}$ are dimensionless fitting parameters, $F=\mathrm{\nabla }\phi$ is the gradient of deformation, $U=\sqrt{F^{T}F}$ is the right stretch tensor and ${\rm dev}n\log {U} =\log {U}-\frac{1}{n}, {\rm tr}(\log {U})\cdot 1!!1$isthedeviatoricpartofthestraintensor$\log U$\mathrm{.}Basedonthemultiplicativedecomposition$F=F_e, F_p$, we couple these energies with some isotropic elasto-plastic flow rules $F_p,\frac{\rm d}{{\rm d} t}[F_p^{-1}]\in-\partial \chi({\rm dev}3 \Sigma{e})$ defined in the plastic distortion $F_p$, where $\mathrm{\partial }\chi$ is the subdifferential of the indicator function $\chi$ of the convex elastic domain $$\mathcal{E}${\rm e}(W{\rm iso},{\Sigma{e}},\frac{1}{3}{\boldsymbol{\sigma}}{!$\mathbf{y}$}^2)$inthemixed-variant$\Sigma{e}$-stressspaceand$\Sigma_{e}=F_e^T D_{F_e} W_{\rm iso}(F_e)$\mathrm{.}While$W_{_{\rm eH}}$ may loose ellipticity, we show that loss of ellipticity is effectively prevented by the coupling with plasticity, since the ellipticity domain of $W_{{}_{\mathrm{e}\mathrm{H}}}$ on the one hand, and the elastic domain in ${\mathrm{\Sigma }}_e$-stress space on the other hand, are closely related.