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

We investigate a family of isotropic volumetric-isochoric decoupled strain energies $$F\mapsto W_{_{\rm eH}}(F):=\widehat{W}_{_{\rm eH}}(U):=\left\{\begin{array}{lll} \frac{\mu}{k}\,e^{k\,\|{\rm dev}_n\log {U}\|^2}+\frac{\kappa}{{2\, {\widehat{k}}}}\,e^{\widehat{k}\,[{ \rm tr}(\log U)]^2}&\text{if}& { \rm det} F>0,\\ +\infty &\text{if} &{ \rm det} F\leq 0, \end{array}\right.\quad$$ 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, $k,\widehat{k}$ are dimensionless parameters, $F=\nabla \varphi$ is the gradient of deformation, $U=\sqrt{F^T F}$ is the right stretch tensor and ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n} {\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part of the strain tensor $\log U$. For small elastic strains, $W_{_{\rm eH}}$ approximates the classical quadratic Hencky strain energy $$F\mapsto W_{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U):={\mu}\,\|{\rm dev}_n\log U\|^2+\frac{\kappa}{2}\,[{\rm tr}(\log U)]^2,$$ 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 $W_{_{\rm eH}}$, 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 $W_{_{\rm eH}}$ is not preserved in dimension $n=3$

Linear constrained Cosserat-shell models including terms up to O(h5){O}(h^5). Conditional and unconditional existence and uniqueness

In this paper we linearise the recently introduced geometrically nonlinear constrained Cosserat-shell model. In the framework of the linear constrained Cosserat-shell model, we provide a comparison of our linear models with the classical linear Koiter shell model and the "best" first order shell model. For all proposed linear models we show existence and uniqueness based on a Korn's inequality for surfaces.Comment: arXiv admin note: text overlap with arXiv:2208.04574, arXiv:2010.1430

An ellipticity domain for the distortional Hencky-logarithmic strain energy

We describe ellipticity domains for the isochoric elastic energy $F\mapsto \|{\rm dev}_n\log U\|^2=\bigg\|\log \frac{\sqrt{F^TF}}{(\det F)^{1/n}}\bigg\|^2 =\frac{1}{4}\,\bigg\|\log \frac{C}{({\rm det} C)^{1/n}}\bigg\|^2$ for $n=2,3$, where $C=F^TF$ for $F\in {\rm GL}^+(n)$. Here, ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n}\, {\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part of the logarithmic strain tensor $\log U$. For $n=2$ we identify the maximal ellipticity domain, while for $n=3$ we show that the energy is Legendre-Hadamard elliptic in the set $\mathcal{E}_3\bigg(W_{_{\rm H}}^{\rm iso}, {\rm LH}, U, \frac{2}{3}\bigg)\,:=\,\bigg\{U\in{\rm PSym}(3) \;\Big|\, \|{\rm dev}_3\log U\|^2\leq \frac{2}{3}\bigg\}$, which is similar to the von-Mises-Huber-Hencky maximum distortion strain energy criterion. Our results complement the characterization of ellipticity domains for the quadratic Hencky energy $W_{_{\rm H}}(F)=\mu \,\|{\rm dev}_3\log U\|^2+ \frac{\kappa}{2}\,[{\rm tr} (\log U)]^2$, $U=\sqrt{F^TF}$ with $\mu>0$ and $\kappa>\frac{2}{3}\, \mu$, previously obtained by Bruhns et al