Reflection of nonlinear shock waves in a lattice

Abstract

An analysis is made of the longitudinal shock response of a finite one-dimensional elastic lattice with a step jump in velocity applied at one end and the remaining end free. For linearly elastic interactions between nearest-neighbor particles, an exact solution is obtained using transform theory. For nonlinear elastic interactions, an asymptotic solution is obtained for the head of the pulse response near the free end, in terms of incident compression wave and reflected tension wave contributions. Nonlinear solutions for particle velocity and strain response near the free end are presented and discussed when the incident shock wave that has evolved in a lattice is of oscillatory form or is a sequence of solitary waves. On reflection of an incident shock wave at a free end, the strain response of a lattice behind the reflected wave rapidly alternates between compression and tension rather than behaving as an elastic continuum with a strain-free response. The cumulative effect of nonlinear elastic interaction is to accentuate these changes near the free end and noticeably increase the difference between lattice and continuum responses.