Eigenmode Asymptotics in Thin Elastic Plates

Abstract

In this paper, we investigate the behavior of the vibration modes (eigenvalues) of an isotropic homogeneous plate as its thickness tends to zero. As lateral boundary conditions, we consider clamped or free edge. We prove distinct asymptotics for bending and membrane modes: the smallest bending eigenvalues behave as the square of the thickness whereas the membrane eigenvalues tend to non-zero limits. Moreover, we prove that all these eigenvalues have an expansion in power series with respect to the thickness regardless of their multiplicities or of the multiplicities of the limit in-plane problems.