The propagation and growth of acceleration waves in heat-conducting elastic materials

Abstract

In the thermo-mechanical theory of continua an acceleration wave may be defined as a propagating singular surface on which the strain, velocity, temperature and entropy are continuous and on which discontinuities in acceleration, strain rate and strain gradient occur. We study here basic properties of such waves in homogeneous heat-conducting elastic bodies, placing no restriction on the symmetry of the material. In a paper which has strongly influenced the course of recent research in the theory of non-linear elastic waves TRUESDELL [1961] has discussed the propagation of acceleration waves through a finitely strained material according to the purely mechanical theory of elasticity. The central result in this analysis is the FRESNEL-HADAMARD theorem, requiring the acceleration amplitude of a wave travelling in a given direction to be a right proper vector of a certain tensor, called the acoustical tensor, which depends upon the direction of propagation and the state of deformation at the location of the wave. The wave speed is determined by the proper number of the acoustical tensor associated with the acceleration amplitude, and thus in a given direction there are up to three speeds with which an acceleration wave can propagate, different speeds corresponding to distinct amplitudes. The FRES~,mL-HADAMARO theorem determines the direction of the acceleration amplitude but places no restriction on its magnitude. For plane acceleration waves propagating through a homogeneously deformed isotropic elastic material GREEN [1964, 1965] has shown that the magnitude of the acceleration amplitude either increases without bound over a finite interval of the time t, decays towards zero as t ~ ~ , or remains constant. These results have been extended to acceleration waves of arbitrary form by CHEN [1968, 1, 2] and JtrNEJA & NARiBOLI [1970], and to general elastic materials by CrrEN [1970, 1, 2] (in respect of longitudinal and transverse waves*) and CHADWICK & OGDEN [1971, 1]. An analysis of the growth of acceleration waves in hyperelastic materials embracing both of these generalizations has been given by SUI-IVBI [1970], and BOWEN & WANG [1970] have studied.