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We consider a family of isotropic volumetric-isochoric decoupled strainenergies $$ F\mapsto W_{\rm eH}(F):=\widehat{W}_{\rmeH}(U):=\left\{\begin{array}{lll} \frac{\mu}{k}\,e^{k\,\|{\rm dev}_n\log{U}\|^2}+\frac{\kappa}{2\hat{k}}\,e^{\hat{k}\,[{\rm tr}(\log U)]^2}&\text{if}&{\rm det}\, F>0,\\ +\infty &\text{if} &{\rm det} F\leq 0,\end{array}\right.\quad $$ 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 Lam\'{e} constant, $k,\hat{k}$ are dimensionlessparameters, $F=\nabla \varphi$ is the gradient of deformation, $U=\sqrt{F^T F}$is the right stretch tensor and ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n} {\rmtr}(\log {U})\cdot 1\!\!1$ is the deviatoric part (the projection onto thetraceless tensors) of the strain tensor $\log U$. For small elastic strains theenergies reduce to first order to the classical quadratic Hencky energy $$  F\mapsto W{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U):={\mu}\,\|{\rm dev}_n\logU\|^2+\frac{\kappa}{2}\,[{\rm tr}(\log U)]^2, $$  which is known to be not rank-one convex.  The main result in this paper is that in plane elastostatics the energies ofthe 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-oneconvexity. Our method uses a judicious application of Steigmann's polyconvexitycriteria based on the representation of the energy in terms of the principalinvariants of the stretch tensor $U$. These energies also satisfy suitablegrowth and coercivity conditions. We formulate the equilibrium equations and weprove the existence of minimizers by the direct methods of the calculus ofvariations.

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