We investigate a family of isotropic volumetric-isochoric decoupled strain energies $$\begin{aligned} F\mapsto W_{\mathrm{eH}}(F):=\widehat{W}_{\mathrm{eH}}(U):=\left \{ \begin{array}{l@{\quad}l} \frac{\mu}{k} e^{k\|\operatorname {dev}_n\log{U}\|^2}+\frac{\kappa}{{2 {\widehat {k}}}} e^{\widehat{k}[\operatorname {tr}(\log U)]^2}&\text{if}\ \det F>0,\\ +\infty&\text{if}\ \det F\leq0, \end{array} \right . \end{aligned}$$ based on the Hencky-logarithmic (true, natural) strain tensor logU, where μ>0 is the infinitesimal shear modulus, $\kappa=\frac{2\mu+3\lambda}{3}>0$
 is the infinitesimal bulk modulus with λ the first Lame constant, $k,\widehat{k}$
 are additional dimensionless material parameters, F=∇φ is the gradient of deformation, $U=\sqrt{F^{T} F}$
 is the right stretch tensor and is the n-dimensional deviatoric part of the strain tensor logU. For small elastic strains, W
 eH approximates the classical quadratic Hencky strain energy $$\begin{aligned} F\mapsto W_{\mathrm{H}}(F):=\widehat{W}_{\mathrm{H}}(U)&:={\mu}\| \operatorname {dev}_n\log U\|^2+\frac{\kappa}{2}\bigl[\operatorname{tr}( \log U)\bigr]^2, \end{aligned}$$ 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
 eH, for $k\geq\frac{1}{4}$
 and $\widehat{k}\geq\frac{1}{8}$
 . 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
 eH is not preserved in dimension n=3 and that the energies $$\begin{aligned} F\mapsto\frac{\mu}{k}e^{k\|\log U\|^2},\qquad F\mapsto\frac{\mu }{k}e^{\frac{k}{\mu} \bigl(\mu\|\operatorname {dev}_n\log U\|^2+\frac{\kappa}{2}[\operatorname {tr}(\log U)]^2 \bigr)}, \quad F \in\mathrm{GL}^+(n), n\in \mathbb {N}, n\geq2 \end{aligned}$$ are also not rank-one convex.

https://www.math.uaic.ro/~ghiba/lucrari/NeffGhibaLankeit-Arxiv-Hencky-Part-I.pdf

The Exponentiated Hencky-Logarithmic Strain Energy. Part I: Constitutive Issues and Rank-One Convexity

Gao Xin-Lin

Papers

Finite deformation elasto-plastic solution for the pure bending problem of a wide plate of elastic linear-hardening material