Two-dimensional discrete granular phononic crystal for shear wave control

The phononic properties of a two-dimensional discrete phononic crystal, made of circular cross-section, infinitely long contacting elastic cylinders arranged on a simple square lattice, are described analytically. The significant interaction between shear and rotational waves is demonstrated in such granular phononic crystals. Controlling these interactions provides an opportunity for shear wave band design in metamaterials. The phononic band structure presents typical features of a square elementary cell of discrete phononic materials and also of continuous (composite) two-dimensional phononic materials with identical elementary cells. The theoretical analysis provides a clear physical explanation for the existence of a zero-group velocity point of the lowest-energy acoustic mode in particular directions of the phononic crystal and demonstrates the birefraction phenomenon.