A hyperbolic generalized Zener model for nonlinear viscoelastic waves

Abstract

A macroscopic model describing nonlinear viscoelastic waves is derived in Eulerian formulation, through the introduction of relaxation tensors. It accounts for both constitutive and geometrical nonlinearities. In the case of small deformations, the governing equations recover those of the linear generalized Zener model (GZM) with memory variables, which is widely used in acoustics and seismology. The structure of the relaxation terms implies that the model is dissipative. The chosen family of specific internal energies ensures also that the model is unconditionally hyperbolic. A Godunov-type scheme with relaxation is implemented. A procedure for maintaining isochoric transformations at the discrete level is introduced. Numerical examples are proposed to illustrate the properties of viscoelastic waves and nonlinear wave phenomena.