Abstract:
The equations of hyperelasticity are hyperbolic if the specific energy $e$ as a function of the deformation gradient $\mathbf F$ is rank-one convex. This condition is not easy to verify even in the case of isotropic solids, where the energy depends only on the invariants of the right or the left Cauchy-Green deformation tensor. However, in practice, it is important to guarantee the hyperbolicity in all domain of $\mathbf F$ having the positive determinant. The domain of large deformations occurs, in particular, in studing of rubber-like materials. Another application of large deformations hyperelasticity comes from the numerical treatment of mathematical models of elastic-plastic solids where one usually uses a splitting procedure : the “elastic” step is followed by the “plastic” relaxation step (Miller & Collela (2001), Godunov & Romenskii (2003), Godunov & Peshkov (2010), Barton et al. (2010), Favrie & Gavrilyuk (2011, 2012)). It is necessary to assure the hyperbolicity condition at each “elastic” step. Indeed, the hyperbolicity is a necessary condition for the wellposdness of the Cauchy problem and corresponding numerical Godunov-type methods.
We will consider the Eulerian formulation of the hyperelasticity for isotropic solids. These equations are invariant under rotation group $SO(3)$. The consequence of that are immediate: for hyperbolicity it is sufficient to consider only 1D case. Indeed, the normal characteristic direction can always be transformed by rotation to the one of Cartesian basis vectors (we have to use three composed rotations defined by Euler angles between the Cartesian basis and a natural local basis on characteristic surface). So, the problem to assure the hyperbolicity of one-dimensional sub-systems for arbitrary strains and shears becomes the basic one. Such a 1D problem stays complex because the number of unknowns involved in such a formulation is large (14 scalar partial differential equations).
We will concentrate on a particular class of elastic materials described by a stored energy $e$ taken in separable form:
\begin{equation*}
e=e^{h}(|\mathbf G|,\eta)+e^{e}(\mathbf g),
\end{equation*}
where $\mathbf G$ is the Finger tensor, $\mathbf g=\mathbf G\vert\mathbf G\vert^{-1/3}$, and
$\vert\mathbf G\vert$ is the determinant of $\mathbf G$. The choice of the Finger tensor is natural for the Eulerian description of isotropic solids. The energy $e^{h}(\vert\mathbf G\vert,\eta)$ is the hydrodynamic part of the energy, depending only on the determinant of $\mathbf G$ and the entropy $\eta$, and
$e^e(\mathbf g) $ is the shear elastic energy. This part of the energy is unaffected by the volume change. Under classical convexity hypothesis of $e^h$, we reduce the problem of hyperbolicity to a simpler one: show that a symmetric $3\times3$ matrix (determined in terms of the shear energy $e^e$ only) is positive definite on a one-parameter family of unit-determinant deformation gradient compact surfaces. Explicit expressions of the stored energy satisfying this criterium are proposed.
