Radially Symmetric Steady States of Nonlinearly Elastic Plates and Shells

Abstract

This paper treats the subject of its title, showing that despite its apparent simplicity, it is very rich: The existence, multiplicity, and qualitative behavior of solutions for problems accounting for the live loads due to hydrostatic pressure and centrifugal force depend critically on the material properties of the bodies, physically reasonable refined descriptions of which are given and examined here with great care, and on the nature of boundary conditions. The treatment here, giving new and sharp results, employs several different mathematical tools, ranging from phase-plane analysis to the mathematically more sophisticated direct methods of the calculus of variations, fixed-point theorems, and global continuation methods, each of which has different strengths and weaknesses for handling intrinsic difficulties in the mechanics. The practical importance of these results is that for each problem involving live loads they furnish thresholds in material response delimiting materials for which solutions are ill behaved. A mathematical or numerical study limited to a particular class of materials may dangerously indicate well-behaved solutions when there are other realistic materials for which solutions are ill behaved.