The exponentiated Hencky-logarithmic strain energy. Part II: Coercivity, planar polyconvexity and existence of minimizers

Abstract

We consider a family of isotropic volumetric-isochoric decoupled strain energies $$F\mapsto W_{\mathrm{eH}}(F):={\hat{W}}_{\mathrm{eH}}(U):=\{\begin{array}{lll}\frac{\mu }{k}{e}^{k\Vert {\mathrm{dev}}_n\log U{\Vert }^2}+\frac{\kappa }{2\hat{k}}{e}^{\hat{k}[\mathrm{tr}(\log U){]}^2}& \text{if}& \det F>0,\\ +\infty & \text{if}& \det F\le 0,\end{array}$$ based on the Hencky-logarithmic (true, natural) strain tensor $\log U$, where $\mu >0$ is the infinitesimal shear modulus, $\kappa =\frac{2\mu +3\lambda }{3}>0$ is the infinitesimal bulk modulus with $\lambda$ the first Lamé constant, $k,\hat{k}$ are dimensionless parameters, $F=\nabla \phi$ is the gradient of deformation, $U=\sqrt{F^{T}F}$ is the right stretch tensor and ${\mathrm{dev}}_n\log U=\log U-\frac{1}{n}\mathrm{tr}(\log U)\cdot 11$ is the deviatoric part (the projection onto the traceless tensors) of the strain tensor $\log U$. For small elastic strains the energies reduce to first order to the classical quadratic Hencky energy $$F\mapsto W{}_{\mathrm{H}}(F):={\hat{W}}_{{}_{\mathrm{H}}}(U):=\mu \Vert {\mathrm{dev}}_n\log U{\Vert }^2+\frac{\kappa }{2}[\mathrm{tr}(\log U){]}^2,$$ which is known to be not rank-one convex. The main result in this paper is that in plane elastostatics the energies of the family $W_{{}_{\mathrm{eH}}}$ are polyconvex for $k\ge \frac{1}{3}$, $\hat{k}\ge \frac{1}{8}$, extending a previous finding on its rank-one convexity. Our method uses a judicious application of Steigmann's polyconvexity criteria based on the representation of the energy in terms of the principal invariants of the stretch tensor $U$. These energies also satisfy suitable growth and coercivity conditions. We formulate the equilibrium equations and we prove the existence of minimizers by the direct methods of the calculus of variations.

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