The propagation of plane irrotational waves through an elastoplastic medium

Abstract

This paper is an attempt at a systematic investigation of wave propagation in a metal, treating interactions between elastic and plastic waves, and the formation and propagation of shock waves, in the general case of motions with unidirectional strain arising from an initial smooth loading­ unloading pulse. A stress-strain relation with linear elastic paths and concave-upward plastic paths (where compression is measured as positive) is derived and used so that the elastic wave velocity is uniform, and the plastic wave velocity an increasing function of stress. The analysis is in terms of engineering stress and strain with a Lagrangian co-ordinate system. Analytic solutions to the interactions between different types of continuous waves are developed incorporating an expression for the motion of the elastic-plastic boundary. An analysis of the breakdown of a smooth plastic compression wave into a shock wave is presented, and the propaga­ tion conditions derived. It is shown that the heat dissipated is proportional to the cube of the strain jump, its low value for moderate shock strength suggests that the shock does not appreciably affect the stress-strain relation, an assumption from which a solution for the unloading of a plastic compression front by an overtaking elastic wave, while shock formation is taking place, is derived. A numerical illustration of this solution for a particular pulse in aluminum is given