Hidden (A)Symmetries of Elastic and Plastic Bifurcation

Abstract

Buckling at a distinct bifurcation point is more often than not symmetric, in the sense that equal and opposite amplitudes of the single buckling mode give identical physical shapes and hence energy levels. It is also commonly the case that combinations of modes break such symmetries, so that when critical loads are close (and sometimes when they are not), the preferred buckling involves a strong interaction between more than one mode. A single cubic cross-term of potential energy, readily identifiable from certain symmetry tests, then dominates the early (unstable) post-buckling, which later may be restabilized by positive quartic terms. In such instances the equilibrium paths over large deflections frequently exhibit a specific looping form, with an accompanying remote bifurcation, which was first predicted from a specialized application of the theorems underlying catastrophe theory. A number of well-known, and quite different, structural examples are given. A simple spring and link model due to Budiansky and Hutchinson, with a close similarity to the well-known Shanley model, shows that the latter can be interpreted in this manner, despite its being a problem of plastic bifurcation. Interactive buckling in sandwich panels, and in the long axially-loaded cylinder, provide two practical examples, others also being identified. A pilot scheme of perturbation analysis specifically designed to handle such problems is described, and used with considerable accuracy on the two link models, as demonstrated by comparison with exact solutions.