Evolution of shock waves in a one-dimensional lattice

Abstract

A normal impact problem for a lattice is considered. A solution is formulated for an arbitrary interaction potential between particles at the two impact ends and simple linear force interaction between particles within each lattice. Asymptotic response of a lattice to impact with truncated interaction and to different end accelerations applied to achieve a shock velocity are considered and related to the response to a step jump in velocity. The essential difference in the asymptotic head of the pulse response for different shock excitations is a constant phase shift in the arrival time of the dispersive oscillations relative to the step jump response. Then, weakly nonlinear interactions of quadratic form are considered within a lattice. A modification is made for the far-field evolution equation previously obtained [J. Appl. Phys. 44, 4569 (1973)] for the asymptotic first-order frequency dispersion and nonlinear response to a step jump in velocity. Comparison is made with numerical solutions of the lattice equations of motion and the differences are discussed. For different end accelerations applied to achieve a shock velocity, the asymptotic nonlinear dispersive oscillations are determined by a phase shift in response relative to the nonlinear response to a step jump in velocity. The amount of the phase shift is the same as in the case of linear interaction.