Elastic waves and homogenization in oblique periodic structures

A mathematical model has been constructed to describe elastic waves propagating in a two-dimensional solid containing a doubly periodic parallelogram array of circular holes. A multipole expansion method is employed which takes into account a coupling between shear and dilatational waves via the traction boundary conditions and determines the structure of the propagating modes. It is important that the homogenized elastic structure be anisotropic; this follows from analysis presented here. The algorithm has been implemented into a computer code, which was used to construct the dispersion diagrams and analyse the filtering properties of the composite structure. It is of particular interest to study the hexagonal and rhombic types of parallelogram lattices, which can be shown to exhibit phononic band-gaps.