Element size and time step selection procedures for the numerical analysis of elasticity with higher-order inertia

Continuum theories with additional higher-order inertia terms have been suggested in the literature in order to be able to describe dispersive wave propagation and in order to include microstructural information in the macroscopic model. In this short note, we investigate two numerical discretisation aspects of such a model. Firstly, the critical time step is derived which is relevant for conditionally stable time integrators. Secondly, a discrete dispersion analysis is carried out by which the accuracy of the discretised model can be assessed a priori. Recommendations for the finite element size and the time step size can thus be made.