On Formation of Singularities in One-Dimensional Nonlinear Thermoelasticity

Abstract

It is well known that smooth motions of nonlinear elastic bodies generally will break down in finite time due to the formation of shock waves. On the other hand, for thermoelastic materials, the conduction of heat provides dissipation that competes with the destabilizing effects of nonlinearity in the elastic response. The work of Coleman & Gurtin [2] on the growth and decay of acceleration waves provides a great deal of insight concerning the interplay between dissipation and nonlinearity in one-dimensional nonlinear thermoelastic bodies. Assuming that the elastic modulus, specific heat, and thermal conductivity are strictly positive, the stress-temperature modulus is nonzero, and that the elastic response is genuinely nonlinear they show that acceleration waves of small initial amplitude decay but waves of large initial amplitude can explode in finite time. In other words, thermal diffusion manages to restrain waves of small amplitudes but nonlinearity in the elastic response is dominant for waves of large amplitudes.