Thermomechanical coupled waves in a nonlinear medium

Abstract

Several technological situations at moderate and high temperatures as well as many physical experiments at low temperatures show the necessity of taking into account the wave structure of heat transport. Motivated by experimental evidence a thermo-mechanical framework for a deformable heat-conducting continuum has been developed. In this framework, a history effect is present, it is introduced via a proportionality law between the heat flux vector and the gradient of a scalar thermal variable. The theory leads to a modified model of thermoelasticity with an extra thermal stress effect and wave-type heat conduction. The model is governed by a system of quasi-linear hyperbolic equations. All essential material functions are examined, and the impact of their nonlinearity on the solution to initial-boundary value problems is studied. A numerical scheme is developed to solve initial-boundary value problems on finite domains by a modification of Lax–Wendroff’s scheme. It turns out that the previously studied case of NaF crystals considered as rigid heat conductors was quantitatively quite acceptable: while it does not show any elastic waves, it still gives correct speeds and amplitudes of the pure thermal waves. However, this feature of upward compatibility of theories is lost for materials with higher thermal expansion coefficients. Hence, for new applications, it is of the essence to develop numerical methods for the fully coupled theory in 2D and 3D cases.