We investigate an immediate application in finite strain multiplicativeplasticity of the family of isotropic volumetric-isochoric decoupled strainenergies \begin{align*} F\mapsto W_{_{\rm eH}}(F):=\hat{W}_{_{\rmeH}}(U):=\{\begin{array}{lll} \frac{\mu}{k}\,e^{k\,\|{\rm dev}_n\log{U}\|^2}+\frac{\kappa}{{\text{}}{2\, {\hat{k}}}}\,e^{\hat{k}\,[{\rm tr}(\logU)]^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, $k,\hat{k}$ are dimensionless fittingparameters, $F=\nabla \varphi$ is the gradient of deformation, $U=\sqrt{F^T F}$is the right stretch tensor and ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n}\,{\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part of the strain tensor$\log U$. Based on the multiplicative decomposition $F=F_e\, F_p$, we couplethese energies with some isotropic elasto-plastic flow rules $F_p\,\frac{\rmd}{{\rm d} t}[F_p^{-1}]\in-\partial \chi({\rm dev}_3 \Sigma_{e})$ defined inthe plastic distortion $F_p$, where $\partial \chi$ is the subdifferential ofthe indicator function $\chi$ of the convex elastic domain $\mathcal{E}_{\rme}(W_{\rm iso},{\Sigma_{e}},\frac{1}{3}{\boldsymbol{\sigma}}_{\!\mathbf{y}}^2)$in the mixed-variant $\Sigma_{e}$-stress space and $\Sigma_{e}=F_e^T D_{F_e}W_{\rm iso}(F_e)$. While $W_{_{\rm eH}}$ may loose ellipticity, we show thatloss of ellipticity is effectively prevented by the coupling with plasticity,since the ellipticity domain of $W_{_{\rm eH}}$ on the one hand, and theelastic domain in $\Sigma_{e}$-stress space on the other hand, are closelyrelated.
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