Harm Askes, Andrei V. Metrikine, Aleksey V. Pichugin, Terry Bennett

 

Four simplified gradient elasticity theories for the simulation of dispersive wave propagation

                                                                     

Philosophical Magazine  

 

Summary. Gradient elasticity theories can be used to simulate dispersive wave propagation as it occurs in heterogeneous materials. Compared to the second-order partial differential equations of classicaly elasticity, in its most general format gradient elasticity also contains fourth-order spatial, temporal as well as mixed spatial-temporal derivatives. The inclusion of the various higher-order terms has been motivated through arguments of causality and asymptotic accuracy, but for numerical implementations it is also important that standard discretisation tools can be used for the interpolation in space and the integration in time. In this paper, we will formulate four different simplifications of the general gradient elasticity theory. We will study the dispersive properties of the models, their causality (or lack thereof) according to Einstein and their behaviour in simple initial/boundary value problems.