Keywords: functionally graded, plates, vibration, dynamic stiffness, isotropic, Wittrick-Williams algorithm.

This paper considers the critical buckling and free vibration of functionally graded (FG) plates whose material properties vary continuously through the thickness. Such plates can be regarded as consisting of two phases, known as the matrix and reinforcement. The volume fraction of the reinforcement varies from zero to unity through the thickness of the plate, e.g. according to a power law, so that the composition is pure matrix material on the top surface and pure reinforcement material on the bottom surface. Applications include metal or ceramic components in aerospace and industrial applications which are required to withstand high temperature gradients.

It has previously been shown [1,2] that critical buckling loads and natural frequencies of FG plates are proportional to those for homogeneous isotropic plates. Moreover, coupling between in-plane and out-of-plane behaviour can be eliminated by an appropriate choice of the neutral surface [3,4].

These ideas are extended in this paper to obtain an equivalent isotropic model for a FG plate so that it can be analysed using existing methods based on classical plate theory for homogeneous plates. Analytical expressions are derived for the thickness, Young's modulus, Poisson's ratio and (for a vibration problem) the density of the equivalent isotropic plate. The position of its neutral surface is defined in terms of an offset above the geometric mid-surface. These expressions are shown to give an exact equivalence when the matrix and reinforcement materials have the same Poisson's ratio, and otherwise a small approximation is introduced.

The equivalent isotropic model is used to obtain critical buckling and undamped free vibration results for a simply supported square FG plate. The correctness and accuracy of the model are confirmed by comparing these results with well converged solutions from an approximate model in which the plate is divided into isotropic layers. Agreement is also demonstrated with theoretical predictions and results from the literature.

1

S. Abrate, "Free vibration, buckling, and static deflections of functionally graded plates", Composites Science and Technology, 66(14), 2382-2394, 2006. doi:10.1016/j.compscitech.2006.02.032

2

S. Abrate, "Functionally graded plates behave like homogeneous plates", Composites: Part B, 39(1), 151-158, 2008. doi:10.1016/j.compositesb.2007.02.026

3

D.G. Zhang, Y.H. Zhou, "A theoretical analysis of FGM thin plates based on physical neutral surface", Computational Materials Science, 44(2), 716-720, 2008. doi:10.1016/j.commatsci.2008.05.016

4

T. Prakash, M.T. Singha, M. Ganapathi, "Influence of neutral surface position on the nonlinear stability behavior of functionally graded plates", Computational Mechanics, 43(3), 341-350, 2009. doi:10.1007/s00466-008-0309-8

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