On the Initial Speed of Elastic-Plastic Boundaries in Longitudinal Wave Propagation in a Rod

Abstract

A study is given of elastic-plastic boundaries which start at the end x = 0 of a rod in one-dimensional wave propagation. The initial speed of the elastic-plastic boundaries at x = 0 and at any time, say t = t0, is determined analytically for all possible combinations of the time derivative σt of the stress σ(0, t) before and after t = t0. If σt at x = 0 is continuous and vanishes at t = t0, all possible combinations of σtt before and after t = t0 are considered. The analysis also gives the number of regions involved, the derivatives in each region, and distinguishes elastic regions from plastic regions. These are useful guides for a numerical solution of general initial and boundary-value problems.