Abstract

Propagating discontinuities in continua, such as, shocks and phase transition fronts often cause dissipation. The dissipated energy is said to manifest in high frequency oscillations behind the propagating front, yet there is no quantitative accounting of it. In this talk we relate the energy dissipated at a shock wave in a non-linearly elastic bar to the energy in the oscillations in two related dissipationless, dispersive systems. We study three one-dimensional dynamic impact problems. Problem 1 concerns a nonlinearlu 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 based on dispersive shock waves is constructed analytically. The rate of increase in 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 qualitative measure of the latter. The quantitative agreement is satisfactory at larger impact speeds but less so at smaller speeds, some reasons for which are discussed. Joint work with Rohan Abeyaratne, Massachusetts Institute of Technology, Cambridge, USA .