Numerical solutions to a microcontinuum model using WENO schemes

In this paper, we consider a one-dimensional Mindlin model describing linear elastic behaviour of isotropic materials with microstructural 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. We transform the set of hyperbolic partial differential equations in a particular form, which makes clear how to impose boundary conditions correctly. Next 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.