An ellipticity domain for the distortional Hencky-logarithmic strain energy

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 $C=F^{T}F$ for $F\in {\mathrm{G}\mathrm{L}}^{+}(n)$. 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.