Dispersive acoustic waves in elastic periodic media: a non-local dynamic homogenization approach

The paper is addressed to the analysis of low-frequency wave propagation in materials with periodic microstructure. A second-gradient continuum model is derived, which provides a sufficiently accurate simulation of for a wide range of wavelengths and a useful tool for the wave propagation analysis in bounded domains. The multi-scale kinematics is described through micro-fluctuation functions of the displacement field, which are derived by the solution of a recurrent sequence of cell BVPs and obtained as the superposition of a static and dynamic contribution. The latters are proportional to the even powers of the phase velocity and consequently the micro-fluctuation functions also depend on the direction of propagation. Therefore, both the higher order elastic moduli and the inertial terms results to depend on dynamic correctors. This approach is applied to the study of wave propagation in layered bimaterials with orthotropic phases, having an axis of orthotropy parallel to the direction of layering, in which case, the overall elastic and inertial constants can be determined analytically. The reliability of the proposed procedure is analysed by comparing the obtained dispersion functions with those derived by the Floquet-Bloch theory.