An ellipticity domain for the distortional Hencky-logarithmic strain energy

TL;DR

This paper characterizes the ellipticity domains of the isochoric elastic energy based on the deviatoric part of the logarithmic strain tensor for 2D and 3D cases, identifying maximal domains and conditions for Legendre-Hadamard ellipticity.

Contribution

It provides a detailed analysis of ellipticity domains for the distortional Hencky-logarithmic strain energy, including the maximal domain in 2D and a specific elliptic set in 3D, extending prior work.

Findings

Identified the maximal ellipticity domain for 2D case.

Established the Legendre-Hadamard elliptic set for 3D case.

Complemented previous characterizations of the quadratic Hencky 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.