Asymptotic derivation of 2D dynamic equations of motion for transversely inhomogeneous elastic plates

Abstract

The 3D dynamic equations in elasticity for a thin transversely inhomogeneous plate are subject to asymptotic analysis over the low-frequency range. The leading and first order approximations are derived. The former is given by a biharmonic equation on the mid-plane generalizing the classical Kirchhoff equation for plate bending. A simple explicit formula for the effective bending stiffness is presented. The refined first order equation involves the same biharmonic operator, as the leading order one, along with corrections expressed through Laplacians. However, the constant coefficients at these corrections take the form of sophisticated repeated integrals across the plate thickness. The formulae for the transverse variations of the displacement and stress components, especially relevant for FGM structures, are also obtained. The scope for comparison of the developed asymptotic results and the existing ad hoc considerations on the subject seems to be limited, in contrast to the homogeneous setup, due to a more substantial deviation between the predictions offered by these two approaches.