A non-ellipticity result, or the impossible taming of the logarithmic strain measure

Abstract

The logarithmic strain measures $\Vert \log U{\Vert }^2$, where $\log U$ is the principal matrix logarithm of the stretch tensor $U=\sqrt{F^{T}F}$ corresponding to the deformation gradient $F$ and $\Vert .\Vert$ denotes the Frobenius matrix norm, arises naturally via the geodesic distance of $F$ to the special orthogonal group $\mathrm{SO}(n)$. This purely geometric characterization of this strain measure suggests that a viable constitutive law of nonlinear elasticity may be derived from an elastic energy potential which depends solely on this intrinsic property of the deformation, i.e. that an energy function $W:\mathrm{G}{\mathrm{L}}^{+}(n)\to \mathbb{R}$ of the form \begin{equation} W(F)=Ψ(\lVert\log U\rVert^2) \tag{1} \end{equation} with a suitable function $\Psi :[0,\infty )\to \mathbb{R}$ should be used to describe finite elastic deformations. However, while such energy functions enjoy a number of favorable properties, we show that it is not possible to find a strictly monotone function $\Psi$ such that $W$ of the form (1) is Legendre-Hadamard elliptic. Similarly, we consider the related isochoric strain measure $\Vert {\mathrm{dev}}_n\log U{\Vert }^2$, where ${\mathrm{dev}}_n\log U$ is the deviatoric part of $\log U$. Although a polyconvex energy function in terms of this strain measure has recently been constructed in the planar case $n=2$, we show that for $n\ge 3$, no strictly monotone function $\Psi :[0,\infty )\to \mathbb{R}$ exists such that $F\mapsto \Psi (\Vert {\mathrm{dev}}_n\log U{\Vert }^2)$ is polyconvex or even rank-one convex. Moreover, a volumetric-isochorically decoupled energy of the form $F\mapsto \Psi (\Vert {\mathrm{dev}}_n\log U{\Vert }^2)+W_{\mathrm{vol}}(\det F)$ cannot be rank-one convex for any function $W_{\mathrm{vol}}:(0,\infty )\to \mathbb{R}$ if $\Psi$ is strictly monotone.

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