An ellipticity domain for the distortional Hencky-logarithmic strain energy

Abstract

We describe ellipticity domains for the isochoric elastic energy $F\mapsto \Vert {\mathrm{dev}}_n\log U{\Vert }^2=\Vert \log \frac{\sqrt{F^{T}F}}{(\det F{)}^{1/n}}{\Vert }^2=\frac{1}{4}\Vert \log \frac{C}{(\det C{)}^{1/n}}{\Vert }^2$ for $n=2,3$, where $C=F^{T}F$ for $F\in {\mathrm{GL}}^{+}(n)$. Here, ${\mathrm{dev}}_n\log U=\log U-\frac{1}{n}\mathrm{tr}(\log U)\cdot 11$ 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 ${\mathcal{E}}_3(W_{{}_{\mathrm{H}}}^{\mathrm{iso}},\mathrm{LH},U,\frac{2}{3}):=\{U\in \mathrm{PSym}(3)|\Vert {\mathrm{dev}}_3\log U{\Vert }^2\le \frac{2}{3}\}$, 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 $W_{{}_{\mathrm{H}}}(F)=\mu \Vert {\mathrm{dev}}_3\log U{\Vert }^2+\frac{\kappa }{2}[\mathrm{tr}(\log U){]}^2$, $U=\sqrt{F^{T}F}$ with $\mu >0$ and $\kappa >\frac{2}{3}\mu$, previously obtained by Bruhns et al.

References in corpus (4)

• Geometry of logarithmic strain measures in solid mechanics
• The exponentiated Hencky-logarithmic strain energy. Part II: Coercivity, planar polyconvexity and existence of minimizers
• The exponentiated Hencky-logarithmic strain energy. Improvement of planar polyconvexity
• The exponentiated Hencky strain energy in modelling tire derived material for moderately large deformations

Cited by in corpus (6)

• A non-ellipticity result, or the impossible taming of the logarithmic strain measure
• The exponentiated Hencky energy: Anisotropic extension and case studies
• The exponentiated Hencky strain energy in modelling tire derived material for moderately large deformations
• Rank-one convexity vs. ellipticity for isotropic functions
• Morrey's conjecture for the planar volumetric-isochoric split. Part I: least convex energy functions
• A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes