Evanescent Bloch waves and complex band structure in magnetoelectroelastic phononic crystals

We investigate the complex band structure of elastic waves in a magnetoelectroelastic phononic crystal, consisting of a polymeric matrix reinforced by BaTiO3-CoFe2O4 inclusions in a square lattice. We also study the influence of inclusion geometry - circular, square and rotated square on the band structure. The elastic wave propagation inside periodic structures consists of both propagating and evanescent modes. Accordingly, improved plane wave expansion (ω(k)) and extended plane wave expansion (k(ω)) methods are formulated to solve the governing equations of motion of a magnetoelectroelastic solid considering wave propagation in the xy plane in order to obtain the propagating and evanescent modes. Complete band gaps between the XY and Z modes are observed for all types of inclusion. Piezoelectricity and piezomagnetism influence significantly the band gaps and the attenuation. There are many modes of the magnetoelectroelastic phononic crystal that are not identified by the ω(k) method. However, they are well identified by the k(ω) method proposed. These complex modes are related to the evanescent nature of Bloch waves. The new findings of this study should be useful for vibration control using smart periodic structures with piezoelectricity and piezomagnetism.