The exponentiated Hencky-logarithmic strain energy. Improvement of planar polyconvexity

In this paper we improve the result about the polyconvexity of the energies from the family of isotropic volumetric-isochoric decoupled strain exponentiated Hencky energies defined in the first part of this series, i.e. $$ W_{_{\rm eH}}(F)= \left\{\begin{array}{lll} \frac{\mu}{k}\,e^{k\,\|{\rm dev}_n\log U\|^2}+\frac{\kappa}{2\,\widehat{k}}\,e^{\widehat{k}\,[(\log {\rm det} U)]^2}&\text{if}& {\rm det}\, F>0,\\ +\infty &\text{if} &{\rm det} F\leq 0\,, \end{array}\right. $$ where $F=\nabla \varphi$ is the gradient of deformation, $U=\sqrt{F^T F}$ is the right stretch tensor and ${\rm dev}_n\log {U}$ is the deviatoric part of the strain tensor $\log U$. The main result in this paper is that in plane elastostatics, i.e. for $n=2$, the energies of this family are polyconvex for $k\geq \frac{1}{4}$, $\widehat{k}\geq \frac{1}{8}$, extending a previous result which proves polyconvexity for $k\geq \frac{1}{3}$, $\widehat{k}\geq \frac{1}{8}$. This leads immediately to an extension of the existence result.