Time-Harmonic Waves in Pre-Stressed Dissipative Materials

Abstract

Dissipative materials are modeled as viscoelastic solids or fluids. Wave propagation is considered by letting time-harmonic waves propagate on an equilibrium configuration with a nonzero stress. Equations of motion for pre-stressed materials are reviewed. Next nonlinear constitutive equations are considered and linearized around the equilibrium state. The thermodynamic restrictions on the linearized equations are then derived explicitly. Two descriptions of wave propagation are applied and the thermodynamic restrictions are shown to play a crucial role. Inhomogeneous waves are proved to propagate in such a way that the amplitude decays in the direction of the energy flux. Rays are shown to belong to three types connected to the three eigenvalues of the acoustic tensor. For each type of rays, the amplitude evolution is established and the decay is found to be a consequence of the thermodynamic restrictions.