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

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.

Buckling and postbuckling of architectured materials: A review of methods for lattice structures and metal foams

“…33 3D printing is also regarded as a versatile tool for implementing nudges for tailored buckling configurations. 34 As emphasized in ref., 30 the interplay between analytical, numerical and experimental techniques is of utmost importance, such that the way through a plethora of unstable postbuckling equilibrium can be determined. Thus, hereafter the further important building blocks are presented.…”

Exploring Adaptive Behavior of Non-linear Hexagonal Frameworks

“…When transitioning between stable states it is often desirable to select the path that minimizes the actuation energy. The barrier to actuation can be considered analogous to the Maxwell energy (see e.g., Hunt et al, 2000b;Wadee and Edmunds, 2005;Champneys et al, 2019). For a system to transition between two minima, it must overcome an energetic barrier, the smallest possible barrier is characterized by one or a series of adjacent saddle points.…”

“…Harnessing structural instabilities is not without its own challenges, see e.g., Champneys et al (2019) who outline recent advances in the field. One prominent example is the buckling of an axially compressed cylindrical shell, which, beyond the classical periodic buckling waveforms distributed over the entire domain of the cylinder, has a strong proclivity to localize the buckling mode over only a small portion of the domain, thereby forming localized dimples (Groh and Pirrera, 2019a).…”

Effects of Disruptive Inclusions in Sandwich Core Lattices to Enhance Energy Absorbency and Structural Isolation Performance

“…The conventional wisdom in terms of the perception of structural instabilities has been that they are best avoided in practice. However, with the rapid emergence of nonlinear mathematics, computational power, alongside numerical, and manufacturing techniques, the exploitation of the geometrically nonlinear range is becoming increasingly feasible for a wider range of applications (Reis, 2015;Champneys et al, 2019). Of course, in the field of thin-walled structures, the naturally stable post-buckling of plates has been exploited since the 1940s due to the pioneering work of, amongst others, von Kármán et al (1932) and Winter (1947).…”