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Propagation of elastic waves in damaged media (concrete, rocks) is studiedtheoretically and numerically. Such materials exhibit a nonlinear behavior,with long-time softening and recovery processes (slow dynamics). A constitutivemodel combining Murnaghan hyperelasticity with the slow dynamics is considered,where the softening is represented by the evolution of a scalar variable. Theequations of motion in the Lagrangian framework are detailed. These equationsare rewritten as a nonlinear hyperbolic system of balance laws, which is solvednumerically using a finite-volume method with flux limiters. Numerical examplesillustrate specific features of nonlinear elastic waves, as well as the effectof the material's softening. In particular, the generation of solitary waves ina periodic layered medium is illustrated numerically.

Plane-strain waves in nonlinear elastic solids with softening