Large scale modulation of high frequency waves in periodic elastic composites

This paper addresses the modelling of large scale modulations of high frequency mechanical waves propagating in periodic elastic composite. By means of an asymptotic approach, we derive (i) the equations governing the large scale modulations, (ii) the specific features of the modulation propagation, and (iii) the domain of validity of the description. Using a multi-cell approach, the theory is implemented for 3D elastic composites with heterogeneous moduli and density. The physical parameters of the up-scaled description are determined from (mutli-)cell eigenmodes. This approach provides specific information on the physics of large scale modulations that are not directly accessible from the Bloch wave decomposition. In particular, the nature of the modulation is shown to differ significantly for simple or multiple degenerated eigenmodes. The theoretical formulation enables (i) to identify the frequency range for which wave motions are correlated over long distances, and (ii) simple calculations of high frequency wave field based on a two-step procedure separating the (multi-)cell scale and the large modulation scale. The links between the high frequency modulation approach and the usual low frequency homogenisation are discussed. Finally, an illustrating example is presented and the results and their applications are commented.