Branching of Nonlinear Boundary Problem Solutions

Abstract

Branching, i.e., the splitting of solutions into two or more branches, is considered to be the essential feature of nonlinear boundary problems solutions behaviour for deformation of thin shells. The types of branching and branching (singular) points are distinguished as bifurcation points and limit points. Such special cases as multiple branching, symmetric bifurcation points, and isolated branches are considered as well. The static and energy criteria of stability and expected behaviour of shell structure are presented. The vector–matrix formulation of a bifurcation problem as the instrument to reveal the non-uniqueness of the solution is presented. The connection between the bifurcation problem for boundary and Cauchy problems is demonstrated. The properties of the correspondent Frechet matrix as the key parameters of branching are considered. The specifics of eigenvalues and eigenforms (obtained in the frameworks of linear theory) for most typical shell loading cases which demonstrate the buckling modes of behaviour—external pressure and axial compression—are presented.