Spherical wave propagation in elastic-plastic work-hardening materials

Abstract Equations governing the propagation of finite amplitude waves with polar symmetry through an isotropic elastic-plastic work-hardening material are formulated. Formal characteristics solutions for elastically and plastically deforming regions are given, and the respective validity conditions obtained. Matching conditions across a moving interface between elastic and plastic regions at which the normal stress is continuous are determined from the balance laws and constitutive response. Analogous to the situation for waves in uni-axial strain, six types of interaction can occur corresponding to interface propagation speeds in the six ranges limited by the elastic and plastic wave propagation speeds. Given the initial motion, the required number of interface conditions to determine the interface path and wave motion in the two regions is obtained for each type. The role of vanishing plastic work rate is shown precisely. Linearized equations for infinitesimal deformation and a parabolic work-hardening law are recovered, and give rise to displacement potentials which satisfy a non-homogeneous wave equation in both regions. Global solutions can then be formally expressed in terms of wave functions.