Free vibration of functionally graded thick circular plates: An exact and three-dimensional solution

This paper presents a simple and effective analytical method based on displacement potential functions for solving the 3D free vibration problem of the functionally graded circular plates with surface boundary conditions consisting of a transversely isotropic linearly elastic material. Material properties vary in the thickness direction according to exponential law and a special power law. Using the potential function method, the coupled governing equations are decomposed into two linear partial differential equations of fourth and second-order, and each equation solved independently by using the separation of variables with exact satisfaction of the two special boundary conditions in a pointwise manner. In addition, the analytical solutions are degenerated for two particular cases, including FG isotropic material and axisymmetric vibration. Also, the in-plane shear vibration has been discussed separately without considering the coupling of transverse and plane displacements. As an advantage of the proposed method, contrary to plate theories, all boundary conditions have been applied in a three-dimensional manner and surface-constraint form. Moreover, obtained solutions are applicable to any thickness ratio. In-plane/out of plane frequencies and mode shapes with nodal patterns are presented for different material properties and different thickness ratios. The results are compared with other analytical works as well as a 3D finite element (FE) solution obtained from the ABAQUS software, which demonstrates the effectiveness and high accuracy of the present method.