  We investigate a finite element formulation of the exponentiated
Hencky-logarithmic model whose strain energy function is given by \[
  W_\mathrm{eH}(\boldsymbol{F}) =
  \dfrac{\mu}{k}\, e^{\displaystyle k \left\lVert\mbox{dev}_n
\log\boldsymbol{U}\right\rVert^2}
  + \dfrac{\kappa}{2 \hat{k}}\, e^{\displaystyle \hat{k} [\mbox{tr}
(\log\boldsymbol{U})]^2 }\,, \] where $\mu>0$ is the (infinitesimal) shear
modulus, $\kappa>0$ 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.
