The piston problem in hyperelasticity with the stored energy in separable form

Abstract

The piston problem for a hyperelastic hyperbolic conservative model where the stored energy is given in separable form is studied. The eigenfields corresponding to the hyperbolic system are of three types: linearly degenerate fields (corresponding to the contact characteristics), the fields which are genuinely nonlinear in the sense of Lax (corresponding to longitudinal waves), and, finally, nonlinear fields which are not genuinely nonlinear (corresponding to transverse waves). Taking the initial state free of stresses, we presented possible auto-similar solutions to the piston problem. In particular, we have shown that the equations admit transverse shock waves having a remarkable property: the solid density is decreasing through such a shock, it is thus a ‘rarefaction’ shock.