On the dissipation at a shock wave in an elastic bar

TL;DR

This study investigates the energy dissipation at shock waves in a nonlinear elastic bar by comparing it with oscillations in related conservative systems, using analytical, numerical, and approximate solutions.

Contribution

It introduces a quantitative relation between shock dissipation and oscillatory energy in three related elastic impact problems, including a new analytical and numerical approach.

Findings

Oscillatory energy increase correlates with shock dissipation.

Analytical solution for nonlinear elastic bar shock problem.

Numerical and approximate solutions for related systems.

Abstract

This paper aims to quantitatively relate the energy dissipated at a shock wave in a nonlinearly elastic bar to the energy in the oscillations in two related dissipationless, dispersive systems. In contrast to a phase boundary, there is no kinetic relation associated with a shock wave. Three one-dimensional dynamic impact problems are studied: Problem 1 concerns a nonlinearly elastic bar, Problem 2 a discrete chain of particles, and Problem 3 a continuum with a strain gradient term in the constitutive relation. In the impact problem considered, the free boundary of each initially quiescent body is subjected to a sudden velocity that is then held constant for all subsequent time. There is energy dissipation at the shock in Problem 1 but Problems 2 and 3 are conservative. Problem 1 is solved analytically, Problem 2 numerically and an approximate solution to Problem 3 is constructed using modulation theory. The rate of increase of the oscillatory energy in Problems 2 and 3 are calculated and compared with the dissipation rate at the shock in Problem 1. The results indicate that the former is a good measure of the latter.