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

$$F \mapsto W_{\rm eH}(F):=\widehat{W}_{\rm eH}(U):=\left\{\begin{array}{lll}\frac{\mu}{k}\,e^{k\,\|{\rm dev}_n{\rm log} {U}\|^2}+\frac{\kappa}{2\hat{k}}\,e^{\hat{k}\,[{\rm tr}({\rm log} U)]^2}&\text{if}& \det\, F > 0,\\+\infty &\text{if} &\det F\leq 0,\end{array}\right.$$

based on the Hencky-logarithmic (true, natural) strain tensor log U, where μ > 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, ${F=\nabla \varphi}$ is the gradient of deformation, ${U=\sqrt{F^T F}}$ is the right stretch tensor and ${{\rm dev}_n{\rm log} {U} ={\rm log} {U}-\frac{1}{n}{\rm tr}({\rm log} {U})\cdot{1\!\!1}}$ 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

$$\begin{array}{ll}F\mapsto W{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U)&:={\mu}\,\|{\rm dev}_n{\rm log} U\|^2+\frac{\kappa}{2}\,[{\rm tr}({\rm log} U)]^2,\end{array}$$

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 ${W_{_{\rm eH}}}$ are polyconvex for ${k\geq \frac{1}{3},\,\widehat{k}\geq \frac{1}{8}}$ , 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.