A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes

Abstract

We investigate a finite element formulation of the exponentiated Hencky-logarithmic model whose strain energy function is given by \[ W_\mathrm{eH}(\boldsymbol{F}) = \dfracμ{k}\, e^{\displaystyle k \left\lVert\mbox{dev}_n \log\boldsymbol{U}\right\rVert^2} + \dfracκ{2 \hat{k}}\, e^{\displaystyle \hat{k} [\mbox{tr} (\log\boldsymbol{U})]^2 }\,, \] where $μ>0$ is the (infinitesimal) shear modulus, $κ>0$ is the (infinitesimal) bulk modulus, $k$ and $\hat{k}$ are additional dimensionless material parameters, $\boldsymbol{U}=\sqrt{\boldsymbol{F}^T\boldsymbol{F}}$ and $\boldsymbol{V}=\sqrt{\boldsymbol{F}\boldsymbol{F}^T}$ are the right and left stretch tensor corresponding to the deformation gradient $\boldsymbol{F}$, $\log$ denotes the principal matrix logarithm on the set of positive definite symmetric matrices, $\mbox{dev}_n \boldsymbol{X} = \boldsymbol{X}-\frac{\mbox{tr} \boldsymbol{X}}{n}\boldsymbol{1}$ and $\lVert \boldsymbol{X} \rVert = \sqrt{\mbox{tr}\boldsymbol{X}^T\boldsymbol{X}}$ are the deviatoric part and the Frobenius matrix norm of an $n\times n$-matrix $\boldsymbol{X}$, respectively, and $\mbox{tr}$ denotes the trace operator. To do so, the equivalent different forms of the constitutive equation are recast in terms of the principal logarithmic stretches by use of the spectral decomposition together with the undergoing properties. We show the capability of our approach with a number of relevant examples, including the challenging "eversion of elastic tubes" problem.