Asymptotic Theory for the Vibrations of Elastic Plates

Abstract

Differential equations governing the vibrations of plates are established in a systematic and consistent manner without making any of the usual a priori assumptions regarding the distribution of stresses and displacements over the thickness of the plate. The analysis consists of the application of the boundary-layer technique to the equations of the three-dimensional theory of orthotropic elasticity. After suitably nondimensionalizing these equations, an asymptotic expansion of both the stress and displacement variables and length scales in terms of a small plate parameter is performed. By equating corresponding powers of the parameter, successive systems of differential equations are obtained. The lowest-order system in each case yields the different classical thin-plate equations. The higher-order systems yield the corresponding higher-order terms in the expansions and constitute thickness corrections associated with the effects of transverse shear, normal stress, and rotatory inertia.