EigenValue Problems in Shell Theory

Abstract

This thesis studies the justification of eigenvalue problems for classical lower dimensional models of linear elastic shells and rods. It is to show that the eigen solutions of the lower dimensional problem is the limit, in some suitable topology of the eignsolutions of the three dimensional problem when the thickness of the shell goes to zero. The different cases for shallow shells, rods, flexural shells, membrane shells, are discussed in different chapters respectively, using the techniques to prove the convergence rely on those used by Ciarlet and Lods. An eigen value problem in three dimensional elasticity posed over a shell, with different thicknesses E or 2E, is considered and clamped on different positions of its surface. By suitable scalings on the domain, eigen solutions etc., transforming this problem into a domain which is independent of E, and by some geometric assumptions, the main results are proved in each case. The important conclusions obtained and open problems raised by this study of eigenvalue problems of thin elastic shells are summarized in the concluding Remarks part of the thesis.