The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issues and rank-one convexity

Abstract

We investigate 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 of the strain tensor $\log U$. For small elastic strains, $W_{{}_{\mathrm{eH}}}$ approximates the classical quadratic Hencky strain 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 not everywhere rank-one convex. In plane elastostatics, i.e. $n=2$, we prove the everywhere rank-one convexity of the proposed family $W_{{}_{\mathrm{eH}}}$, for $k\ge \frac{1}{4}$ and $\hat{k}\ge \frac{1}{8}$. Moreover, we show that the corresponding Cauchy (true)-stress-true-strain relation is invertible for $n=2,3$ and we show the monotonicity of the Cauchy (true) stress tensor as a function of the true strain tensor in a domain of bounded distortions. We also prove that the rank-one convexity of the energies belonging to the family $W_{{}_{\mathrm{eH}}}$ is not preserved in dimension $n=3$.