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

Abstract

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, $\mu >0$ is the infinitesimal shear modulus, $\kappa =\frac{2\mu +3\lambda }{3}>0$ is the infinitesimal bulk modulus with $\lambda$ the first Lamé constant, $k,\hat{k}$ are dimensionless fitting parameters, $F=\nabla \phi$ is the gradient of deformation, $U=\sqrt{F^{T}F}$ is the right stretch tensor and ${\mathrm{dev}}_n\log U=\log U-\frac{1}{n}\mathrm{tr}(\log U)\cdot 11$ 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 $F_p\frac{\mathrm{d}}{\mathrm{d}t}[F_p^{-1}]\in -\partial \chi ({\mathrm{dev}}_3{\Sigma }_e)$ defined in the plastic distortion $F_p$, where $\partial \chi$ is the subdifferential of the indicator function $\chi$ of the convex elastic domain ${\mathcal{E}}_{\mathrm{e}}(W_{\mathrm{iso}},{\Sigma }_e,\frac{1}{3}{\mathbf{\sigma }}_{\mathbf{y}}^2)$ in the mixed-variant ${\Sigma }_e$-stress space and ${\Sigma }_e=F_e^T{D}_{F_e}{W}_{\mathrm{iso}}(F_e)$. While $W_{{}_{\mathrm{eH}}}$ may loose ellipticity, we show that loss of ellipticity is effectively prevented by the coupling with plasticity, since the ellipticity domain of $W_{{}_{\mathrm{eH}}}$ on the one hand, and the elastic domain in ${\Sigma }_e$-stress space on the other hand, are closely related.

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