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} \fracμ{k}\,e^{k\,\|{\rm dev}_n\log {U}\|^2}+\fracκ{\text{}{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, $μ>0$ is the infinitesimal shear modulus, $κ=\frac{2μ+3λ}{3}>0$ is the infinitesimal bulk modulus with $λ$ the first Lamé constant, $k,\hat{k}$ are dimensionless fitting parameters, $F=\nabla φ$ 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 couple these energies with some isotropic elasto-plastic flow rules $F_p\,\frac{\rm d}{{\rm d} t}[F_p^{-1}]\in-\partial χ({\rm dev}_3 Σ_{e})$ defined in the plastic distortion $F_p$, where $\partial χ$ is the subdifferential of the indicator function $χ$ of the convex elastic domain $\mathcal{E}_{\rm e}(W_{\rm iso},{Σ_{e}},\frac{1}{3}{\boldsymbolσ}_{\!\mathbf{y}}^2)$ in the mixed-variant $Σ_{e}$-stress space and $Σ_{e}=F_e^T D_{F_e} W_{\rm iso}(F_e)$. 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 $W_{_{\rm eH}}$ on the one hand, and the elastic domain in $Σ_{e}$-stress space on the other hand, are closely related.

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