On the dissipation at a shock wave in an elastic bar

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.