Band-gap calculations of anisotropic phononic crystals in a square lattice

A boundary element method is extended to calculate the band structures of anisotropic phononic crystals with different components forming the square lattice. The system may be composed of anisotropic inclusions embedded in the isotropic matrix or isotropic inclusions embedded in the anisotropic matrix. In a periodic unit cell, boundary integral equations of the matrix and the inclusion are given. Substituting the periodic boundary conditions and the interface conditions, a set of linear equations is formed. Then the relations between the wave number and the frequency are determined, which clearly exhibit the band gaps of the phononic system. Some numerical examples are used to illustrate the accuracy and efficiency of the boundary element method. Additionally, the results show that for anisotropic phononic crystal, band gaps not only depend on the periodic lattice but also the angle between the symmetry axis of orthotropic materials (different rotating angles).