Propagation and localization of elastic waves in highly anisotropic periodic composites via two-scale homogenization

Wave propagation in periodic elastic composites whose phases may have not only highly contrasting but possibly also (in particular) highly anisotropic stiffnesses and moderately contrasting densities is considered. A possibly inter-connected (i.e. not necessarily isolated) "inclusion" phase is assumed generally much softer than that in the connected matrix, although some components of its stiffness tensor may be of the same order as in the matrix. For a critical scaling, generalizing that of a "double porosity"-type for the highly anisotropic elastic case, we use the tools of "non-classical" (high contrast) homogenization to derive, in a generic setting, two-scale limiting elastodynamic equations. The partially high-contrast results in a constrained microscopic kinematics described by appropriate projectors in the limit equations. The effective equations are then uncoupled and explicitly analyzed for their band-gap structure. Their macroscopic component describes plane waves with a dispersion relation which is generally highly non-linear both in the frequency and in the wave vector. While it is possible in this way to construct band-gap materials without the high anisotropy, the number of propagating modes for a given frequency (including none, i.e. in the band-gap case) is independent of the direction of propagation. However the introduction of a high anisotropy does allow variation in the number of propagating modes with direction if the inclusion phase is inter-connected, including achieving propagation in some directions and no propagation in the others. This effect is explicitly illustrated for a particular example of a highly anisotropic fibrous composite