Dispersion relations of elastic waves in one-dimensional piezoelectric phononic crystal with initial stresses

The effects of initial stresses on dispersion relations of elastic waves in a one-dimensional piezoelectric phononic crystal are studied in this paper. It is taken into account that initial normal and shear stresses acted on piezoelectric slabs and their influences on constitutive equations, governing equations and boundary conditions under in-plane and anti-plane strain cases. Based on the transfer matrices of ingredient piezoelectric slabs, the total transfer matrix of a single cell is derived. Furthermore, the Bloch theorem is used to obtain the dispersive equation of in-plane and anti-plane Bloch waves. The dispersive equations are solved numerically and the numerical results are shown graphically. In the case of normal propagation of elastic waves within piezoelectric slabs, the analytical expressions of the dispersion equation are derived and compared with other literatures. The influences of initial stresses on the dispersive relations are discussed based on the numerical results. It is found that the initial normal stresses have more evident influences than that of initial shear stresses, no matter that in-plane Bloch waves or anti-plane Bloch wave are concerned. Moreover, the influences of initial stress are more evident on the dispersion curves at high frequency than that at the low frequency.