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

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. $$W_{{}_{\mathrm{eH}}}(F)=\{\begin{array}{lll}\frac{\mu }{k}{e}^{k\Vert {\mathrm{dev}}_n\log U{\Vert }^2}+\frac{\kappa }{2\hat{k}}{e}^{\hat{k}[(\log \det U){]}^2}& \text{if}& \det F>0,\\ +\infty & \text{if}& \det F\le 0,\end{array}$$ where $F=\nabla \phi$ is the gradient of deformation, $U=\sqrt{F^{T}F}$ is the right stretch tensor and ${\mathrm{dev}}_n\log U$ 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 $k\ge \frac{1}{4}$, $\hat{k}\ge \frac{1}{8}$, extending a previous result which proves polyconvexity for $k\ge \frac{1}{3}$, $\hat{k}\ge \frac{1}{8}$. This leads immediately to an extension of the existence result.

arXiv commentarXiv admin note: substantial text overlap with arXiv:1408.4430

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• The exponentiated Hencky-logarithmic strain energy. Part II: Coercivity, planar polyconvexity and existence of minimizers
• A Riemannian approach to strain measures in nonlinear elasticity
• A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm

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