Exact and numerical solutions to a Mindlin microcontinuum model

TL;DR

This paper develops exact and numerical solutions for a one-dimensional Mindlin microcontinuum model, demonstrating the effectiveness of the numerical scheme through comparison with exact solutions and applying it to inhomogeneous materials.

Contribution

It introduces a class of exact solutions for the Mindlin model and validates a robust numerical scheme for simulating micro-structural effects in elastic materials.

Findings

Exact solutions exhibit wave behavior due to potential energy properties.

Numerical scheme is accurate and robust, matching exact solutions.

Numerical example demonstrates applicability to inhomogeneous materials.

Abstract

In this paper we consider a one-dimensional Mindlin model describing linear elastic behaviour of isotropic materials with micro-structural effects. After introducing the kinetic and the potential energy, we derive a system of equations of motion by means of the Euler-Lagrange equations. A class of exact solutions is obtained. They have a wave behaviour due to a good property of the potential energy. Numerical solutions are obtained by using a weighted essentially non-oscillatory finite difference scheme coupled by a total variation diminishing Runge-Kutta method. A comparison between exact and numerical solutions shows the robustness and the accuracy of the numerical scheme. A numerical example of solutions for an inhomogeneous material is also shown.