Cellular buckling in long structures

A long structural system with an unstable (subcritical) post-buckling response that subsequently restabilizes typically deforms in a cellular manner, with localized buckles first forming and then locking up in sequence. As buckling continues over a growing number of cells, the response can be described by a set of lengthening homoclinic connections from the fundamental equilibrium state to itself. In the limit, this leads to a heteroclinic connection from the fundamental unbuckled state to a post-buckled state that is periodic. Under such progressive displacement the load tends to oscillate between two distinct values. The paper is both a review and a pointer to future research. The response is described via a typical system, a simple but ubiquitous model of a strut on a foundation which includes initially-destabilizing and finally-restabilizing nonlinear terms. A number of different structural forms, including the axially-compressed cylindrical shell, a typical sandwich structure, a model of geological folding and a simple link model are shown to display such behaviour. A mathematical variational argument is outlined for determining the global minimum postbuckling state under controlled end displacement (rigid loading). Finally, the paper stresses the practical significance of a Maxwell-load instability criterion for such systems. This criterion, defined under dead loading to be where the pre-buckled and post-buckled state have the same energy, is shown to have significance in the present setting under rigid loading also. Specifically, the Maxwell load is argued to be the limit of minimum energy localized solutions as end-shortening tends to infinity.