Fast and slow dynamics in a nonlinear elastic bar excited by longitudinal vibrations

TL;DR

This paper introduces a physical model and numerical method for simulating the complex, multi-scale dynamics of heterogeneous materials like rocks and concrete under longitudinal vibrations, capturing both fast elastic responses and slow defect evolution.

Contribution

It proposes a coupled hyperbolic system with relaxation to model nonlinear elasticity, viscoelasticity, and defect evolution, along with a numerical scheme for solving these equations.

Findings

Numerical simulations qualitatively match experimental observations.

The model captures both fast vibrational and slow defect processes.

A splitting-based numerical scheme effectively solves the evolution equations.

Abstract

Heterogeneous materials, such as rocks and concrete, have a complex dynamics including hysteresis, nonlinear elasticity and viscoelasticity. It is very sensitive to microstructural changes and damage. The goal of this paper is to propose a physical model describing the longitudinal vibrations of this class of material, and to develop a numerical strategy for solving the evolution equations. The theory relies on the coupling between two processes with radically-different time scales: a fast process at the frequency of the excitation, governed by nonlinear elasticity and viscoelasticity; a slow process, governed by the evolution of defects. The evolution equations are written as a nonlinear hyperbolic system with relaxation. A time-domain numerical scheme is developed, based on a splitting strategy. The numerical simulations show qualitative agreement with the features observed experimentally by Dynamic Acousto-Elastic Testing.