Criterion of Hyperbolicity in Hyperelasticity in the Case of the Stored Energy in Separable Form

We consider the equations of hyperelasticity for isotropic solids in the Eulerian coordinates in a special case where the specific stored energy is a sum of two functions. The first one, the hydrodynamic part of the energy, depends only on the solid density and the entropy, and the second one, the shear energy, depends on the invariants of the Finger tensor in such a way that it is unaffected by the volume change. A new sufficient criterion of hyperbolicity for such a system is formulated: if the sound velocity is real and a symmetric 3×3 matrix determined in terms of the shear energy is positive definite on a one-parameter family of surfaces of the unit-determinant deformation gradient, the equations are hyperbolic.

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