Asymptotic analysis of 3D dynamic equations in linear elasticity for a thin layer resting on a Winkler foundation

A 3D dynamic problem for a thin elastic layer resting on a Winkler foundation is considered. The stiffness of the layer is assumed to be much greater than that of the foundation in order to allow low-frequency bending motion. The leading long-wave approximation valid outside the vicinity of the cut-off frequency is derived. It is identical to the classical Kirchhoff plate theory. A novel near cut-off 2D approximation is also established. It involves both bending and extension motions which are not separated from each other due to the effect of the foundation. The associated dispersion relation appears to be non-uniform over the small wavenumber domain.