Smooth solutions for nonlinear elastic waves with softening

A new hyperbolic softening model has been proposed for wave propagation in damaged solids [2]. The linear elasticity becomes nonlinear through an additional internal variable. This thermodynamically relevant model yields a dissipative energy. The 3×3 nonlinear hyperbolic system so-obtained is totally linearly degenerate like the well-known Kerr-Debye system. Existence of global smooth solutions is studied here thanks to the Kawashima condition. Moreover, shocks never appear with smooth initial data.