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

The dispersion relations of elastic waves in a one-dimensional phononic crystal formed by periodically repeating of a pre-stressed piezoelectric slab and a pre-stressed piezomagnetic slab are studied in this paper. The influences of initial stress on the dispersive relation are considered based on the incremental stress theory. First, the incremental stress theory of elastic solid is extended to the magneto-electro-elasto solid. The governing equations, constitutive equations, and boundary conditions of the incremental stresses in a magneto-electro-elasto solid are derived with consideration of the existence of initial stresses. Then, the transfer matrices of a pre-stressed piezoelectric slab and a pre-stressed piezomagnetic slab are formulated, respectively. The total transfer matrix of a single cell in the phononic crystal is obtained by the multiplication of two transfer matrixes related with two adjacent slabs. Furthermore, the Bloch theorem is used to obtain the dispersive equations of in-plane and anti-plane Bloch waves. The dispersive equations are solved numerically and the numerical results are shown graphically. The oblique propagation and the normal propagation situations are both considered. In the case of normal propagation of elastic waves, the analytical expressions of the dispersion equation are derived and compared with other literatures. The influences of initial stresses, including the normal initial stresses and shear initial stresses, on the dispersive relations are both discussed based on the numerical results.