# Plane-strain waves in nonlinear elastic solids with softening

## TL;DR

This paper investigates the propagation of nonlinear elastic waves in damaged, softening media like concrete and rocks, using a combined theoretical and numerical approach to understand wave behavior and solitary wave generation.

## Contribution

It introduces a novel constitutive model that integrates Murnaghan hyperelasticity with slow dynamics to analyze nonlinear wave propagation in damaged solids.

## Key findings

- Softening affects wave propagation characteristics.
- Numerical simulations show formation of solitary waves.
- Material softening influences wave speed and shape.

## Abstract

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