Finite element modelling of wave dispersion with dynamically consistent gradient elasticity

Abstract

The discretisation aspects are investigated of a recently developed gradient elasticity model that is capable of describing wave dispersion and microstructural effects. The model includes higher-order stiffness terms alongside higher-order inertia terms. To avoid the need for ${\fancyscript{C}^1}$ -continuous interpolations in a numerical implementation, the field equations are rewritten using a recently developed operator split. The equations are discretised in space using finite elements and discretised in time using the Newmark time integrator. Firstly, the critical time step is derived for use with conditionally stable time integrators. Next, the dispersion behaviour of the continuum is compared with the dispersion behaviour of the discretised medium, which allows the formulation of rules on how to select the element size and the time step size. These rules are then verified in a one-dimensional bar example and in a two-dimensional example of wave propagation. It follows that the element size must be chosen roughly equal to the smallest of the intrinsic length scales, while the optimal time step follows from the ratio of element size and wave velocity and from the ratio of the length scales.