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

TL;DR

This paper explores coupling a specific exponential Hencky-logarithmic strain energy with isotropic finite strain plasticity, demonstrating that plasticity coupling can prevent loss of ellipticity in the energy model.

Contribution

It introduces a novel coupling of exponential Hencky strain energy with plasticity, showing how plasticity prevents ellipticity loss in finite strain models.

Findings

Plasticity coupling effectively prevents ellipticity loss.

The energy model is based on the logarithmic strain tensor.

Coupling relates the energy's ellipticity domain to the elastic domain.

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} \frac{\mu}{k}\,e^{k\,\|{\rm dev}_n\log {U}\|^2}+\frac{\kappa}{{\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 Lam\'{e} constant, $k,\hat{k}$ are dimensionless fitting parameters, $F=\nabla \varphi$ 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 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 $\partial \chi$ is the subdifferential of the indicator function $\chi$ of the convex elastic domain $\mathcal{E}_{\rm e}(W_{\rm iso},{\Sigma_{e}},\frac{1}{3}{\boldsymbol{\sigma}}_{\!\mathbf{y}}^2)$ 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 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 $\Sigma_{e}$-stress space on the other hand, are closely related.