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Happy Catastrophe: Recent Progress in Analysis and Exploitation of Elastic Instability

Research output: Contribution to journalArticle

Original languageEnglish
Number of pages30
JournalFrontiers in Applied Mathematics and Statistics
Volume5
Issue number34
DOIs
DateAccepted/In press - 3 Jul 2019
DatePublished (current) - 30 Jul 2019

Abstract

A synthesis of recent progress is presented on a topic that lies at the heart of both structural engineering and non-linear science. The emphasis is on thin elastic structures that lose stability subcritically—without a nearby stable post-buckled state—a canonical example being a uniformly axially-loaded cylindrical shell. Such structures are hard to design and certify because imperfections or shocks trigger buckling at loads well below the threshold of linear stability. A resurgence of interest in structural instability phenomena suggests practical stability assessments require stochastic approaches and imperfection maps. This article surveys a different philosophy; the buckling process and ultimate post-buckled state are well-described by the perfect problem. The significance of the Maxwell load is emphasized, where energy of the unbuckled and fully-developed buckle patterns are equal, as is the energetic preference of localized states, stable, and unstable versions of which connect in a snaking load-deflection path. The state of the art is presented on analytical, numerical and experimental methods. Pseudo-arclength continuation (path-following) of a finite-element approximation computes families of complex localized states. Numerical implementation of a mountain-pass energy method then predicts the energy barrier through which the buckling process occurs. Recent developments also indicate how such procedures can be replicated experimentally; unstable states being accessed by careful control of constraints, and stability margins assessed by shock sensitivity experiments. Finally, the fact that subcritical instabilities can be robust, not being undone by reversal of the loading path, opens up potential for technological exploitation. Several examples at different length scales are discussed; a cable-stayed prestressed column, two examples of adaptive structures inspired by morphing aeroelastic surfaces, and a model for a functional auxetic material.

    Research areas

  • instability, elastic, buckling, sub-critical, localization, path-following, mountain-pass

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    Rights statement: This is the final published version of the article (version of record). It first appeared online via Frontiers Media S.A. at https://www.frontiersin.org/articles/10.3389/fams.2019.00034/full . Please refer to any applicable terms of use of the publisher.

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