Nonlinear dispersive wave equations for microstructured solids

Abstract

Dispersion is a characteristic feature for wave propagation in microstructured solids. In the case of linear elasticity, dispersion effects are modelled by higher-order derivatives included into the wave equation. Nonlinear effects are also well known in wave propagation in solids. In principle, such effects may appear at the macroscale as well as at the microscale. The microstructural influence is often taken into account by the introduction of internal variables. This suggests that internal variables may behave nonlinearly. It is shown that the nonlinear behaviour of internal variables may lead at the macroscale to the Benjamin– Bona–Mahoney equation or the Camassa–Holm equation.