The stability of a plane longitudinal shock wave in an isotropic elastic medium

Abstract

The problem of the linear stability of fast plane longitudinal shock waves (SW) in an isotropic elastic body whose elastic potential is a given function of the deformation tensor invariants depending additively on entropy is considered. When the medium is in a state of uniaxial compression or extension, the resulting dispersion equation can be factorized. Assuming that ahead of the SW the medium is in a state of uniaxial compression (extension), sufficient conditions for the instability of longitudinal SWs are obtained. When the medium is in a state of uniaxial extension ahead of the SW and the velocity of the SW is such that the deformations behind the SW are close to zero and much smaller than those ahead of the SW, the problem of linear stability is solved completely, i.e. the necessary and sufficient conditions for stable, unstable and neutrally stable SWs to exist are stated. All the results obtained remain valid in the case of a medium with transverse anisotropy (the direction of the anisotropy axis coinciding with the direction of SW propagation), and also for an isotropic medium in a state of compression (extension) in the direction of two mutually perpendicular axes lying in the plane perpendicular to the direction of SW propagation, the deformations along these axes being equal.