A Note on Nonbreaking Waves in Hyperelastic Materials

Abstract

This note treats the evolution of waves on hyperelastic materials, due to initial jump discontinuities in the gradient of strain. In general, these discontinuities become unbounded in finite time, leading to discontinuous strain. There are, however, certain cases in which the gradient jump remains finite for all times. We show here that the class of materials admitting such exceptional waves is fairly large, including Hadamard materials and generalized Hooke materials. An earlier example of Jeffrey and Teymur [6] is also discussed and reset in a more general framework.

Some solutions for a compressible isotropic elastic material

“…Earlier, the same material model was used in numerical studies of azimuthal shear (see [16] and [17]), anti-plane shear [35], combined torsion and anti-plane shear [36] and finite torsion [32]. Lustman [34] showed that from all possible forms of the function h(J) in the Hadamard material (8) its Levinson-Burgess form (29) is the only one for which all the generated waves (possibly three-dimensional) are non-breaking waves, i.e. for initial data with weak discontinuities (jumps in the n th or higher order derivatives for n th order equations) lower-order derivatives (and therefore the strain) do not become discontinuous.…”