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

TL;DR

This paper improves the conditions under which a family of isotropic strain energies based on the Hencky-logarithmic strain are polyconvex in 2D, enabling better mathematical guarantees for elastostatic problems.

Contribution

It extends the polyconvexity range for the exponentiated Hencky energies in planar elasticity from previous bounds, allowing for broader application in existence theorems.

Findings

Polyconvexity holds for k ≥ 1/4 and 𝑘̂ ≥ 1/8 in 2D.

Extension of existence results for elastostatic problems.

Improved mathematical properties of strain energy functions.

Abstract

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.