Dispersion relations of elastic waves in two-dimensional tessellated piezoelectric phononic crystals

A type of two-dimensional tessellated piezoelectric phononic crystal is theoretically studied in this paper, formed by homogeneous piezoelectric and inhomogeneous functionally graded rectangular columns. Firstly, the propagation properties of in-plane and anti-plane Bloch waves in each piezoelectric rectangular column are systematically investigated. Furthermore, the total transfer matrices of Bloch waves are obtained based on transfer matrices of homogeneous piezoelectric and inhomogeneous functionally graded rectangular columns. Finally, the Bloch theorem is used to obtain the dispersion relations of in-plane and anti-plane Bloch waves. The influences of non-dimensional geometrical parameters and gradient profile functions on the dispersion relations are discussed based on the graphically numerical results. Under a special value of modal parameter, the dispersion surfaces and curves of in-plane Bloch waves approximatively have plane and axis symmetries respectively, and an approximate total band-gap of anti-plane Bloch waves arise. The theoretical models and numerical discussions will provide a direct guidance of multi-material 3D printing for inhomogeneous periodic structures with dispersion and band-gaps properties.