Three-dimensional free vibrations analysis of functionally graded rectangular plates by the meshless local Petrov–Galerkin (MLPG) method

In this paper, the meshless local Petrov–Galerkin (MLPG) method is used for three-dimensional (3D) elastodynamic analysis of functionally graded (FG) isotropic plates. The 3D linear elasticity theory and infinitesimal strains are the basic assumptions in this analysis. Unlike 2-D theories, in 3D theory, no restricting assumptions are made about the kinematics of deformation in plates. Therefore, a higher accuracy can be achieved by using 3D theory. Heaviside step functions are used as test functions on the local sub-domain of each node and the field variables are interpolated using the 3D moving least squares (MLS) approximation. To impose the essential boundary conditions, the penalty method is adopted and Young's modulus is assumed to be graded through the thickness of plates by the Mori–Tanaka estimation and Poisson's ratio is assumed to be constant. The natural frequencies of thick rectangular plates are obtained for different boundary conditions. Also, the effects of different parameters such as functionally graded power law index, thickness-to-length ratio, and the aspect ratio on the natural frequencies of plates are also studied in details. In order to validate this method, the results are compared with the available exact 3D solutions. The results show the accuracy and effectiveness of the present method.