On the exact in-plane and out-of-plane free vibration analysis of thick functionally graded rectangular plates: Explicit 3-D elasticity solutions

In this paper exact closed-form solutions of 3-D elasticity theory are presented to study both in-plane and out-of-plane free vibrations for thick functionally graded simply supported rectangular plates. The solution procedure of the transverse vibration utilizes Levinson’s representation form to describe the displacement; in this way, the 3-D elasto-dynamic equations are written in terms of some suitable independent functions satisfying ordinary differential equations. A similar procedure is presented for in-plane vibration by introducing an appropriate displacement field. In each case, the obtained ordinary differential equations are analytically solved and boundary conditions are satisfied. The proposed solutions are validated by comparing some of the present results with corresponding results known in the literature as well as with 3-D Finite Element Method. Finally, the influence of inhomogeneity on the natural frequencies for a thick functionally graded rectangular plate is discussed.