Two-Dimensional Approximation of Eigenvalue Problems in Shell Theory: Flexural Shells

Abstract

The eigenvalue problem for a thin linearly elastic shell, of thickness 2ε, clamped along its lateral surface is considered. Under the geometric assumption on the middle surface of the shell that the space of inextensional displacements is non-trivial, the authors obtain, as ε → 0, the eigenvalue problem for the two-dimensional “flexural shell” model if the dimension of the space is infinite. If the space is finite dimensional, the limits of the eigenvalues could belong to the spectra of both flexural and membrane shells. The method consists of rescaling the variables and studying the problem over a fixed domain. The principal difficulty lies in obtaining suitable a priori estimates for the scaled eigenvalues.