Analytical Solution for the Three-Dimensional Vibration of a Rectangular Functionally Graded Material Plate with Two Simply Supported Opposite Faces

Abstract

Analytical solutions based on three-dimensional (3D) elasticity for the vibrations of functionally graded material (FGM) plates are valuable for assessing the validity and accuracy of various plate theories and numerical approaches. Few benchmark 3D analytical solutions for the vibrations of FGM plates are available in the literature. In this study, analytical solutions based on Fourier series and 3D elasticity were developed for the first time for the vibrations of FGM rectangular plates with two simply supported opposite edge faces. The distributions of the properties of FGMs through the thickness follow a simple power law. The proposed solutions were validated by conducting comprehensive convergence studies on the vibration frequencies of square plates with different thickness-to-side ratios and boundary conditions as well as comparisons with published results. The benchmark nondimensional frequencies were tabulated for plates with free boundary conditions on the top and bottom faces and six combinations of boundary conditions on the other two faces. Moreover, the effects of aspect ratio and gradient index on the vibration frequencies of FGM plates were investigated. The influence of the thickness ratio of the FGM layer to the homogenous layer on the vibration frequencies of sandwich plates with FGM face sheets and a homogeneous core was also studied.