Nonlocal Qualitative and Quantitative Analysis of a Simple Shallow Shell Problem

Abstract

AbstractA nonlocal qualitative and quantitative analysis of a nonlinear problem is illustratively carried out for a shallow shell problem. The shell model is of “Marguerre” type and the problem is governed by a system of nonlinear partial differential equations. The complete nonlocal solution set is obtained equivalently by means of the boundary value problem and independently by means of each of the two genuinely dual variational problems, too. An identical morphological behaviour of the geometrically and of the statically approached complete solution set is proved. This implies a diffeomorphism between these sets and their identical stability behaviour. Thus the paradox connected with the different stability behaviour of the common solution points described by means of the two classical complementary variational problems is resolved for the considered problem. The solution set contains solution continua which are neither minimizers nor maximizers. Singular phenomena occur at any point of the involved complete degenerate solution set. This set is a natural extension of the notion of eigenvalues from linear to nonlocal nonlinear problems. The complete degenerate solution set introduced is indispensable in a nonlocal nonlinear morphology analysis. An exact numerical analysis is carried out in a space spanned by the first four active modes.