Singular global bifurcation problems for the buckling of anisotropic plates

Abstract

This paper treats a variety of unexpected pathologies that arise in the global bifurcation analysis of axisymmetrie buckled states of anisotropic plates. The geometrically exact plate theory used accounts for flexure, extension and shear. The nonlinear constitutive functions have very general form. As a consequence of the anisotropy the trivial solution may depend discontinuously on the load parameter. Accordingly, the equations for the bifurcation problem have the same character, so that bifurcating branches of solutions become disconnected as the load parameter crosses values at which discontinuities occur. The anisotropy furthermore implies that the governing equations have a singular behaviour much worse than that for isotropic plates. Consequently, a variety of novel constructions are required to demonstrate the validity of the essential results upon which global bifurcation theory stands. (These results include the compactness of certain operators and the uniqueness of solutions of initial value problems for singular ordinary differential equations.) It is shown that in regions of solution-parameter space in which the equations depend continuously on the load parameter there exist connected global branches of solution pairs that have detailed nodal properties inherited from eigenfunctions of the linearized problem. Moreover, these nodal properties are preserved across gaps occurring where discontinuities occur. The methodology used to show this result actually supports constructive methods for finding disconnected branches.