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Front. Appl. Math. Stat. | doi: 10.3389/fams.2019.00034

Happy Catastrophe: recent progress in analysis and exploitation of elastic instability

  • 1University of Bath, United Kingdom
  • 2University of Bristol, United Kingdom
  • 3University of Exeter, United Kingdom
  • 4University College, Bristol, United Kingdom
  • 5Imperial College London, United Kingdom

A synthesis of recent progress is presented on a topic that lies at the
heart of both structural engineering and nonlinear 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 emphasised, where energy of the
unbuckled and fully-developed buckle patterns are equal, as is the
energetic preference of localised 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
localised 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.

Keywords: instability, Elastic, buckling, Sub-critical, localisation, path-following, Mountain-pass

Received: 13 Apr 2019; Accepted: 03 Jul 2019.

Edited by:

Federico Guarracino, University of Naples Federico II, Italy

Reviewed by:

Otti D'Huys, Aston University, United Kingdom
Luis A. Godoy, National Council for Scientific and Technical Research (CONICET), Argentina  

Copyright: © 2019 Hunt, Champneys, Dodwell, Groh, Neville, Pirrera, Sakhaei, Schenk and Wadee. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Mx. Giles Hunt, University of Bath, Bath, United Kingdom, g.w.hunt@bath.ac.uk