Exact Free Vibration Analysis of Lévy FGM Plates with Higher-Order Shear and Normal Deformation Theories

First-known exact solutions are derived for free vibration of thick and moderately thick functionally graded rectangular plates with at least one pair of opposite edges simply-supported on the basis of a family of two-dimensional shear and normal deformation theories with variable order. The boundary-value problem is first expressed in a compact unified form which is invariant with respect to the order of the kinematic theory. The Lévy method applied to this compact form yields a set of governing equations written in terms of invariant matrices, which are then appropriately expanded according to the order of the plate model. The resulting equations are put into a state-space representation and the frequency values are finally obtained by substituting the general solution of the state equation into the set of boundary conditions and solving the related homogeneous system. After discussing the way of recovering the through-the-thickness modal displacement and stress distribution at any point of the plate and how the effective elastic properties of the graded plate are computed, some numerical results are presented using various higher-order theories. Comparisons with exact three-dimensional and other two-dimensional approaches are provided for two-constituents metal-ceramic plates. New exact results for functionally graded plates with six combinations of boundary conditions are also obtained. They can be useful as valuable sources for validating other approaches and approximate methods.