Refined theory for vibration of thick plates with the lateral and tangential loads

Based on the three-dimensional elastodynamics, without using the classical assumption, an operator method is established to refine the dynamic theory of an infinite homogenous isotropic plate by using the spectral decomposition of operators. By this method, the governing equations of the bending and stretching vibrations of plates with the lateral and tangential loads on the surface are derived from the Boussinesq–Galerkin solution of the three-dimensional elasticity, respectively. To effectively deduce the governing equations, a complex differential operator is introduced. Dispersion relations based on the refined equations and the three-dimensional elastodynamics are compared to verify the refined theory of plates. It is shown that the dispersion relation of the refined theory of plates agrees more with the result based on the three-dimensional elastodynamics than Mindlin’s theory. Therefore, the refined equations are accurate that can be used to solve the vibration of thick plates and determine high-order vibration modes of plates. The applicable conditions of the refined plate theory are analyzed and discussed.