On the dissipation associated with equilibrium shocks in finite elasticity

Equilibrium fields with discontinuous displacement gradients can occur in finite elasticity for certain materials. The presence of such equilibrium shocks affects the energy balance in the elastostatic field, and the present paper is concerned with a notion of dissipation associated with this energy balance. A dissipation inequality is proposed for three-dimensional equilibrium shocks for both compressible and incompressible materials. The consequences of this inequality are studied for weak shocks in plane strain for compressible materials and for shocks of arbitrary strength in anti-plane strain for a class of incompressible materials. A thermodynamic argument for the dissipation inequality is also given.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 between the California Institute of Technology and the Office of Naval Research.