An asymptotic theory for dynamic response of anisotropic inhomogeneous and laminated plates

We develop an asymptotic theory for dynamic analysis of anisotropic inhomogeneous plates within the framework of three-dimensional elasticity. The inhomogeneities are considered to be in the thickness direction and the laminated plates belong to an important class of this type of inhomogeneous plates. Through nondimensionalization and introduction of the multiple time scales in the formulation, we obtain a uniform expansion of the field variables in even powers of a small plate parameter. The expansion yields an asymptotic solution valid regardless of the time span, whereas a straightforward expansion fails to produce convergent results. We show by successive integration that the equations for the asymptotic solution are of the same form as those in the classical laminated plate theory (CLT). While the asymptotic solution is no more difficult than the CLT solution, it is capable of yielding displacements and all the stress components in a consistent and systematic manner. Modifications to the lower-order solutions are made by eliminating the secular terms in the equations according to the method of multiple scales. The basic theory is illustrated by determining the free vibration characteristics of a symmetric cross-ply laminated plate.