Formulation of the extended plane wave expansion for in‐plane and out‐of‐plane vibrations in viscoelastic phononic thin plates under a constant temperature field

Abstract In this study, we develop an extended plane wave expansion method to determine the complex elastic band gaps for both in‐plane and out‐of‐plane vibrations in a viscoelastic phononic crystal (PnC) thin plate model under a constant temperature field. The periodic structure of the thin plate comprises a unit cell (UC) consisting of an epoxy matrix and circular steel inclusions. In this configuration, we consider the viscoelastic effects described by the generalized Maxwell model arising from the epoxy matrix. Furthermore, we incorporate thermal effects in the UC, accounting for the linear dependence of the Young's modulus on the temperature of the epoxy matrix, assuming the absence of heat flow in the dimensions of the UC. We present a band structure illustrating the displacement of the real and imaginary parts of the complex band structure for evanescent waves arising from the viscoelastic PnC thin plate, considering both in‐plane and out‐of‐plane vibrations.