Computational aspects of a new multi-scale dispersive gradient elasticity model with micro-inertia

Computational aspects of a recently developed gradient elasticity model are discussed in this paper. This model includes the (Aifantis) strain gradient term along with two higher-order acceleration terms (micro-inertia contributions). It has been demonstrated that the presence of these three gradient terms enables one to capture the dispersive wave propagation with great accuracy. In this paper, the discretisation details of this model are thoroughly investigated, including both discretisation in time and in space. Firstly, the critical time step is derived that is relevant for conditionally stable time integrators. Secondly, recommendations on how to choose the numerical parameters, primarily the element size and time step, are given by comparing the dispersion behaviour of the original higher-order continuum with that of the discretised medium. In so doing, the accuracy of the discretised model can be assessed a priori depending on the selected discretisation parameters for given length-scales. A set of guidelines can therefore be established to select optimal discretisation parameters that balance computational efficiency and numerical accuracy. These guidelines are then verified numerically by examining the wave propagation in a one-dimensional bar as well as in a two-dimensional example.