# Nonlinear waves in solids with slow dynamics: an internal-variable model

## TL;DR

This paper develops a thermodynamically consistent 3D internal-variable model for nonlinear wave softening in solids with slow dynamics, capturing hysteresis and strain-dependent sound speed in heterogeneous materials.

## Contribution

A new 3D thermodynamically admissible model incorporating internal variables for slow dynamics in solids is introduced, extending previous 1D phenomenological models.

## Key findings

- Model reproduces main experimental features of slow dynamics.
- The model is thermodynamically consistent and dissipative.
- Analytical results confirm qualitative agreement with experiments.

## Abstract

In heterogeneous solids such as rocks and concrete, the speed of sound diminishes with the strain amplitude of a dynamic loading (softening). This decrease known as "slow dynamics" occurs at time scales larger than the period of the forcing. Also, hysteresis is observed in the steady-state response. The phenomenological model by Vakhnenko et al. is based on a variable that describes the softening of the material [Phys. Rev. E 70-1, 2004]. However, this model is 1D and it is not thermodynamically admissible. In the present article, a 3D model is derived in the framework of the finite strain theory. An internal variable that describes the softening of the material is introduced, as well as an expression of the specific internal energy. A mechanical constitu-tive law is deduced from the Clausius-Duhem inequality. Moreover, a family of evolution equations for the internal variable is proposed. Here, an evolution equation with one relaxation time is chosen. By construction, this new model of continuum is thermodynamically admissible and dissipative (inelas-tic). In the case of small uniaxial deformations, it is shown analytically that the model reproduces qualitatively the main features of real experiments.