Homogenisation for elastic photonic crystals and dynamic anisotropy

We treat in-plane elasticity, with coupled shear and compressional waves and therefore a full vector problem, demonstrating that a two-scale asymptotic approach using a macroscale and microscale results in effective scalar continuum equations posed entirely upon the macroscale; the vector nature of the problem being incorporated on the microscale. This rather surprising result is comprehensively verified by comparing the resultant asymptotics to full numerical simulations for the Bloch problem of perfectly periodic media. The dispersion diagrams for this Bloch problem are found both numerically and asymptotically. Periodic media exhibit dynamic anisotropy, e.g. strongly directional fields at specific frequencies, and both finite element computations and the asymptotic theory predict this. Periodic media in elasticity can be related to the emergent fields of metamaterials and photonic crystals in electromagnetics and relevant analogies are drawn. As an illustration we consider the highly anisotropic cases and show how their existence can be predicted naturally from the homogenisation theory.