An ellipticity domain for the distortional Hencky-logarithmic strain energy

Abstract: We describe ellipticity domains for the isochoric elastic energy $ F\mapsto |{\rm dev}n\log U|^2=\bigg|\log \frac{\sqrt{F^TF}}{(\det F)^{1/n}}\bigg|^2 =\frac{1}{4}\,\bigg|\log \frac{C}{({\rm det} C)^{1/n}}\bigg|^2 $ for $n=2,3$, where $C=F^TF$ for $F\in {\rm GL}^+(n)$. Here, ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n}\, {\rm tr}(\log {U})\cdot 1!!1$ is the deviatoric part of the logarithmic strain tensor $\log U$. For $n=2$ we identify the maximal ellipticity domain, while for $n=3$ we show that the energy is Legendre-Hadamard elliptic in the set $\mathcal{E}_3\bigg(W{{\rm H}}^{\rm iso}, {\rm LH}, U, \frac{2}{3}\bigg)\,:=\,\bigg{U\in{\rm PSym}(3) \;\Big|\, |{\rm dev}_3\log U|^2\leq \frac{2}{3}\bigg}$, which is similar to the von-Mises-Huber-Hencky maximum distortion strain energy criterion. Our results complement the characterization of ellipticity domains for the quadratic Hencky energy $ W{_{\rm H}}(F)=\mu \,|{\rm dev}_3\log U|^2+ \frac{\kappa}{2}\,[{\rm tr} (\log U)]^2 $, $U=\sqrt{F^TF}$ with $\mu>0$ and $\kappa>\frac{2}{3}\, \mu$, previously obtained by Bruhns et al.