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Edited by: Federico Guarracino, University of Naples Federico II, Italy

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

This article was submitted to Dynamical Systems, a section of the journal Frontiers in Applied Mathematics and Statistics

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.

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.

Bernard Budiansky famously used to say “everybody loves a buckling problem” [

The purpose here is to review modern developments in the theory and analysis of buckling instabilities, both in the work of the present authors and by others. In the process, we draw particular attention to new techniques of analysis—often applied to classical thorny buckliphobic problems—and highlight potential areas of buckliphilic exploitation. We place particular emphasis on the interplay between analytical, numerical and experimental techniques, showing how we pick our way through a plethora of unstable post-buckling equilibrium states, to focus on practically relevant solutions.

With its origins in singularity, or catastrophe, theory [

Subcritical bifurcations, exemplified by the classic responses of thin elastic shells, are known to carry distinctive features, such as the likelihood of extreme imperfection-sensitivity and wide experimental scatter, and certainly merit a general overview. The canonical example is that of the axially-loaded cylindrical shell, of interest to rocket designers, aircraft and storage tank manufacturers, as well as in the construction of buffers to absorb mechanical energy. Here, instability under realistic conditions occurs significantly below the critical load of the system as determined from linear stability analysis of the perfect problem, absent from imperfections.

One approach to deal with such imperfection-sensitivity is through stochastic methods. Eliashakoff, Arbocz, and others pioneered developments, such as the international databank of imperfections (see [

Methods based on sensitivity to perturbations have received a recent resurgence of interest inspired by theories of critical transitions in fluid dynamics, see for example [

Fundamentally, this paper takes a deterministic rather than stochastic point of view. Starting with the perturbation methods introduced by the Dutch engineer [

A key idea is that the perfect problem, devoid of imperfections or shocks, can give theoretical and practical insight into how structures buckle subcritically. We shall emphasize the significance of the Maxwell load, the level at the fundamental and periodic buckle patterns have the same energy (see e.g., [

The rest of this paper is outlined as follows. Section 2 gives a brief overview of non-linear post-buckling analysis of subcritical problems, starting from the pivotal work of Koiter, and including some general comments on analytical perturbation methods. A motivating simple pin-jointed “knee” model is presented as well as the classical problem of the axially loaded cylindrical shell. Section 3 surveys recent progress in computational path-following methods applied directly to a finite element representation to compute stable and unstable paths, with illustrations for a simple snap-through structure as well as the more complex cylindrical shell. Section 4 then considers computational energy-based methods that are able to identify Maxwell loads and mountain-pass solutions, again with reference to the cylindrical shell. Section 5 considers emerging experimental ideas to implement the numerical methods from the previous sections, via carefully controlled laboratory procedures. Section 6 surveys three examples, at different length-scales and from distinct engineering domains, that attempt to exploit subcritical buckling instabilities: prestressed stayed columns, adaptive aeroelastic structures, and a structural model for auxetic materials. Finally, section 7 draws conclusions and suggests avenues for future work.

Before the advent of modern computer-based methods, non-linear post-buckling of elastic structures was largely dealt with by systematic asymptotic analysis; i.e., perturbation procedures based on Taylor expansions about the critical bifurcation point [

“In the problem of the spherical shell under external pressure the systematic perturbation procedure is only valid in a range of load factors within a fraction of the order ^{1/2} times the shell thickness. It follows that the systematic perturbation procedure at the critical point has little, if any, practical significance for the present problem.”

Here

“A far more powerful method of achieving a second approximation to the post-buckling behavior was also developed already in our earlier work [

The reference is of course to his thesis [

In more recent years Koiter's ideas have been supplemented by other asymptotic techniques, such as expansions at the so-called Maxwell load [

Consider the simple mechanical model of

The system has two degrees-of-freedom, with associated generalized coordinates Δ and θ that respectively describe in-line displacement and rotation of the rigid link elements. It has three distinct possibilities for equilibrium. First, we have the simplest state in which the rotational spring does no work and the in-line spring simply squashes to give a fundamental equilibrium path describing the pre-buckling state:

Second, under the condition that the knee rotation

when the rotational spring is elastic with rotational stiffness

if it has passed its limiting plastic moment _{p}. In each case, the first two terms are the strain energies in the rotational and in-line springs, respectively; the final subtracted term is the work done by the dead load

The fundamental and post-buckling paths are plotted in

Responses of the knee models of _{p} = 1. (Left) Load vs. θ. (Right) Load vs. end-shortening.

The rigid-plastic system on the other hand has no bifurcation point, and this highlights one of the key issues to be addressed in this paper. Over much of the range, the fundamental and post-buckling paths, although being relatively well-separated in the

The bifurcation point is restored for the elasto-plastic system as shown in the bottom row of panels in

Much has been written on the classical problem of the buckling of an axially loaded cylindrical shell (see e.g., [

It might seem strange in a forward-looking review paper to focus on such a problem, but with the modern impetus toward ever stronger and lighter structures and new materials, understanding the cylinder response remains a fundamental issue of continuing research. We will therefore use axially-compressed cylindrical shell buckling as the exemplar problem on which to illustrate the methodology reviewed in this paper.

When the fundamental deformation mode of a long, slender structure loses stability, it can either transition into a periodic buckling mode, spread equally over the domain, or a localized mode that is concentrated only over a portion of the domain. Such different kinds of buckle patterns were illustrated in the beautiful experimental and computational work of Yamaki [

The work by Horák et al. [

We shall continually return to the squashed cylinder problem throughout this study.

Another canonical shell buckling problem that has received a resurgence of interest due to the recent work of Hutchinson [

“An important result of Beaty's analysis [

He thus puts the poor performance of the perturbation method down to ever increasing influence of higher and higher-order terms—quartics larger than cubics, quintics more than quartics, and so on. This significant observation seems odd from the viewpoint of perturbation theory; using von Kármán–Donnell equations for the cylindrical shell [

The need for higher-order terms can be explained through the process of elimination of passive coordinates, as espoused in the book by Thompson and Hunt [_{i}, Λ), where {_{i}} describes a set of _{i} are deemed _{α} = _{α}(_{j}, Λ), where now 1 ≤ _{j}, Λ), equal in value to the

Differentiation using the chain rule then gives derivatives of _{ij} = 0 for

for a significant quartic term (see [

The derivative _{αα} appearing in the denominator of (3) is the so-called stability coefficient for the passive coordinate _{α}, and would have equated to zero had the coordinate been active and directly involved in the buckling process, If critical loads tend to bunch together on the fundamental path, as occurs for both the axially-compressed cylinder and pressurized sphere discussed above, then contamination from higher-modes close to the critical point of interest can clearly be extreme.

Modal analysis in the form of spectral or pseudo-spectral numerical methods made a resurgence in the 1990s and 2000s, allowing numerical continuation (path-following) methods to scale to models with hundreds of degrees-of-freedom (see e.g., [

In applied mathematics, methods for multi-parameter analysis, branch-switching and bifurcation tracking are well-established theoretically using the language of catastrophe (singularity) theory [

Our formulation considers a discretized model of a quasi-statically evolving, conservative and elastic structure, where the internal forces, _{T}(

Equilibrium is defined as a balance between internal and external forces acting on the structure. In a displacement-based finite element setting, this balance is written in terms of

The vectors ^{1}

The system (4) of

To turn this into a well-posed system of equations, one needs to add an additional scalar constraint, the most natural of which is the arclength constraint

Hence

where _{u} and _{λ} take different forms depending on the nature of the arclength constraint. By linearizing about the current equilibrium state,

we can find a set of solution points describing a continuous equilibrium curve. Note that the partial derivative of the residual with respect to the displacement vector, _{, u}_{, u}(_{T}(

More generally, Equation (4) can adapted to incorporate any number of additional parameters:

where

is a vector containing _{1} corresponds to parameters that influence the internal forces (e.g., material properties, geometric dimensions, temperature and moisture fields) and _{2} relates to externally applied mechanical loads (e.g., forces, moments, tractions).

The number ^{(n+p)} will be computed—the so-called

When ^{(n+p)}.

Posing the problem in this general manner allows the structural response to be viewed not only as a function of a varying load but also as a function of other parameters that define the structure. By treating these additional parameters as “forcing” variables in an arc-length solver, their effect on the structural response is readily obtained.

This general treatment naturally lends itself to the tracing of loci of singular points in parameter space. To constrain the system of

i.e., at least one eigenvector _{T}

Equation (9) describes

where the scalar equation restricts the magnitude of the eigenvector.

When evaluating one-dimensional curves (

on the curve described by

where

A solution to Equation (10) can be obtained through a consistent linearization coupled with Newton's method,

where the superscript denotes the

The above framework is quite general and can be adapted to find many different kinds of curves on an equilibrium surface; see [

Classic equilibrium paths in load-displacement space (a loading parameter is varied).

Parametric paths in parameter-displacement space (a geometric, constitutive or secondary loading parameter is varied).

Pinpointing singular points (bifurcation and limit points) on either of the two paths mentioned above.

Bifurcated branches emanating from a bifurcation point.

Singular paths that describe a locus of bifurcation and/or limit points in load-parameter-displacement space.

Branch-connecting paths that connect points on distinct equilibrium curves, e.g., a fundamental and a bifurcated path.

As an example, consider the snap-through behavior of the centrally loaded toggle frame with clamped ends, shown in

Schematic diagram of a toggle frame under transverse load

The toggle frame initially deforms symmetrically on the fundamental equilibrium path. This deformation mode becomes unstable at a symmetry-breaking bifurcation just before the maximum limit point on the curve. Because the connected non-symmetric path branching from the bifurcation point is unstable, the toggle frame snaps dynamically into the inverted stable shape. In

By imposing a singularity condition in the generalized path-following algorithm, the locus of limit and bifurcation points can be traced, illustrating how changes in the height of the frame affect the load-displacement solution of these singular points. There are multiple benefits of tracing such fold lines. First, they can be used to identify interesting points, such as the coincidence of limit and bifurcation points—the hilltop-branching points at

Consider a thin-walled isotropic cylindrical shell of thickness

The cylinder is modeled using isoparametric, geometrically non-linear finite elements based on a total Lagrangian formulation. The finite elements used are so-called “degenerated shell elements” [

_{cl}). The classical buckling load is given by

Path-following in the direction of decreasing displacement leads to a snaking sequence. The reason behind snaking has been established in a number of related contexts as the behavior of homoclinic orbits in the unfolding of a heteroclinic connection between flat and periodic states (see [

Starting from limit point 0 in _{cl} = 0.479). This critical point corresponds to the smallest possible compression to allow a single dimple as an equilibrium solution. Tracing the equilibrium path further, a series of destabilizations and restabilizations add further buckles to the left and right of the original single dimple. Proceeding along the snaking path, the single dimple thus grows in a sequence of 1, 3, 5, 7, and 9 waves until an entire ring around the cylinder exists. The mode shapes corresponding to limit points I–V in

The equilibrium path in

An additional snaking sequence starting from two dimples and representing growth of an even number of waves (2, 4, 6, 8, and 10) also exists. The even snaking sequence mirrors the behavior of the odd snaking sequence in its pattern formation and in the connection to another equilibrium path at a pitchfork bifurcation. In systems featuring spatial localization, snaking of both even and odd number of localizations is typical [

The snaking solution of even buckles also ends at a pitchfork bifurcation (point PB in

In closing this section we remark that the snaking results for the present paper were obtained with a mesh 4× denser than those in reference [

While continuation methods are an integral part of unraveling the often complex behavior associated with subcritical instabilities, they do not tell us which state (energetically) the system would prefer, nor quantify the sensitivity of a locally metastable equilibrium state, so-called

To address the problem of an infinite pre-buckling critical load (as in

A long structural system loaded axially typically prefers a localized to a distributed post-buckled response [

This snaking sequence with localizations developing over the length of a structure has now been recognized in a number of different circumstances (see for example [

First introduced by Ambrosetti and Rabinowitz [

a suitable (energy) functional

a stationary point _{1}, which is a local minimum

a second point _{2}, for which

We note that a suitable function is normally available in the form of total potential energy, with local minima appearing on a stable fundamental equilibrium path [

The theorem states that, over the set of all continuous paths connecting _{1} and _{2}, i.e.,

one can find the infimum of the maxima of the energy functional

The physical significance of the mountain pass is that it represents the connecting point in solution space with the smallest energy hump,

required to escape the local minimum at _{1} and transition to a lower energy state at _{2}. Therefore, the Maxwell load/displacement (depending on the loading regime), at which

The application of the Theorem provides a computable energy hump to assess shock sensitivity; the mountain pass state _{c} itself is significant, since at this point the system has just one negative eigenvalue for which the system is unstable. This eigenvector marks a direction in solution space _{1} to _{2}. This eigenvalue at the mountain pass point therefore indicates the imperfection or probing modes that a subcritical system could be most sensitive to.

The literature gives a variety of algorithms for finding mountain pass solutions e.g., the nudged elastic band method [

Conjugate peak refinement is an iterative scheme performing alternating line search maximization and minimization steps to find the mountain pass solution _{c}. The approach generates a sequence of piecewise-linear approximations to a path γ^{⋆} which passes through _{c}. For the ^{th} iteration, we denote this approximation γ^{(k)} characterized by a set of points ^{(0)} as the straight line connecting _{1}_{2}, so that

^{(k−1)}, to obtain line maximization point

^{(k)}^{(k)}, so that points defining the line are

At any iteration of the state

We now demonstrate the mountain pass procedure geometrically with a generic, two degree-of-freedom energy landscape given by a modified Müller-Brown potential [_{1} and _{2}, with a non-trivial mountain pass connecting them. The approach provides a good approximation to the saddle point in just two iterations. In the first iteration, we see the algorithm starts by approximating the mountain path with a straight line between the _{1} and _{2}. A maximum is located along this line; see the ° in the far left panel of ^{(0)} is updated to γ^{(1)}, characterized by three points connected by the pair of straight lines. For iteration 2 the procedure continues in the same way, first a maximization step over the path to produce the second ° in the third panel of

and the algorithm terminates.

Symbols denote: (×) Local minima, (−−−) potential “mountain path,” (→) search direction ○ line maximum, Δ line minimum.

Classically, stability of equilibrium is governed entirely by the local Hessian of the total potential energy; wells with respect to all degrees-of-freedom denote stability, whereas saddles or maxima denotes instability. This framework fails in the case of infinite critical loads presented in

As is seen in the insert of _{u} = _{M}/_{cl} = 0.486 (_{M}

The Maxwell displacement could serve as a lower-bound estimate for the cylinder's first instability load, by marking the onset of “shock sensitivity” [

Rather than apply a computationally expensive infinite degree-of-freedom mountain-pass algorithm as in Horák et al. [

To implement such an analysis numerically, consider applying such a poker at right angles to the cylinder mid-surface, half-way along its length. Such a process involves two fundamental parameters, applied end-compression

The stability landscape of an axially compressed cylinder with a probing side force. ^{3}) vs. normalized probe displacement (Δ

For increased levels of end-shortening, the equilibrium manifold traces S-shaped curves; as the dimple develops, lateral resistance reduces, until limit points are traversed leading to regions of negative stiffness (paths 2–3 in

While numerical methods for the analysis of non-linear structures are well-developed, experimental methods tailored to such structures, in particular shell buckling, have received comparatively little attention (see e.g., [

Conventional test methods fail to capture all but the simplest non-linear behavior, and consequently researchers lack reliable methods to validate their ideas experimentally. The main reason traditional test methods fail is the difficulty in measuring unstable parts of the response. Any structure whose equilibrium curve features limit points can snap under force- or displacement-controlled test methods, as illustrated in

Numerical analysis succeeds where experiments fail because in a numerical setting the force and displacement at a control point can be controlled independently and simultaneously. This freedom allows the solver to set combined limits on force and displacement, and prevents jumping to other solutions when an equilibrium becomes unstable (see

There are several interesting published approaches to work around this problem. Wiebe and Virgin [

Consider the centrally-loaded shallow arch studied by Neville et al. [_{1} and _{2}.

_{a} is applied at the mid-span actuation point, generating reaction force _{a}. _{p} across both probes, while they are allowed to move horizontally. _{a} = 5 mm are numbered 1–5.

At _{a} = 5 there are several equilibria available; each with distinct values of _{a}. Each equilibrium is also associated with a unique deformation shape (

By moving the probes and actuation point in concert, a simple form of path-following can be performed [_{p} = 0). Small perturbations can be used to avoid large deviations from the equilibrium curve. If the actuation point steps past a limit point, the probes will not be able to find equilibrium and the actuation point direction is then reversed. This approach allows the equilibrium path of the shallow arch to be followed around a displacement limit point, as shown in

Results of the experimental path-following method. _{p}| > 0.1 N when searching for equilibria) are shown in green, and equilibria (_{p} < 0.1 N) are indicated by the black dots.

Experimental results are naturally affected by phenomena and imperfections not included in theoretical models. The shallow arch example, for instance, is sensitive to changes in geometry and probe location, as well as displaying complex behavior in response to the two input parameters (_{a} and _{p}). Virtual testing is a technique that can address these issues, and aid in experimental design and interpretation of results. A successful example of such a virtual testing environment coupled to the commercial FE solver A

Inspired by the theoretical work of Horák et al. [

Test for assessing the resilience of the cylinder to perturbations, proposed by Thompson [

In fact, the idea of poking axially-compressed cylinders from the side to assess resilience to buckling has a long history predating any mountain-pass considerations. By tapping axially-loaded cylinders with a finger, Eßlinger and Geier [

The poker force vs. displacement response of the cylinder for different levels of axial compression was shown previously using FE simulations in

Such a probing experiment was successfully implemented by Virot et al. [

Even though the idea is simple to implement, in practice the system can bifurcate by pivoting around the point load. To offset this symmetry-breaking effect, as highlighted by Thompson and Sieber [

As stated in the Introduction and indeed reflected in the title of this paper, instability need not solely be considered as something to be avoided or designed against; it is also possible to utilize instability in a positive manner [

Prestressed stayed columns are important elements of many modern large-scale structures; see

Prestressed stayed columns in practice. (Left to right) An example as a slender support for a façade in Chiswick Park, West London; a set of roof supports in the former Eurostar terminal in Waterloo Station in Central London; an example in a shopping mall in Dalian, China. Photographs courtesy of Dr. Daisuke Saito and Dr. Jialiang Yu.

Such columns tend to be slender and have intermediately placed cross-arms and associated pretensioned cables, thereby reducing the buckling effective length _{e}. The length _{e} provides a measure of the critical buckling eigenmode wavelength and the Euler strut buckling load is proportional to _{e} significantly and hence provide a commensurate increase in critical buckling load and ultimate capacity. Depending on the overall geometry, this change in critical buckling load can also be associated with quantitative and qualitative changes in the triggered buckling mode within the non-linear range. The behavior has been discussed at length in previous work, with the focus falling on qualitative critical [

Some of these works use conventional finite element modeling, where post-buckling shapes are initiated by introducing imperfections that are affine to linear buckling modes. A drawback is that the full picture of modal interaction only becomes available under a combination of symmetric and anti-symmetric imperfections. Other modes may also be drawn in, for example, should it be thin enough, localized buckling in the main tubular column, and the numerical methodology discussed in section 3.3, can be useful. Nevertheless, there has been a sequence of increasingly-sophisticated low-dimensional models, to capture mode interaction [

The particular complication in stayed columns is produced by the cable stay, where there is the possibility of a sudden loss in elastic stiffness caused by cables slackening. The outcome is similar to that described in _{e} and _{m}, the exact details of which are presented in Yu and Wadee [

Prestressed stayed column. (Row 1: left to right) Geometric definitions; effect of prestressing and buckled shape showing deformations of main column, cross-arms and stays used in the Rayleigh–Ritz model. (Row 2: left to right) Equilibrium paths showing: distinct mode 1 buckling (_{1}); distinct mode 2 buckling (_{2}); interactive buckling with a secondary mode jumping path. (Rows 3–4) Illustration of mode jumping through different points on the secondary mode jumping path from the third case in row 2 now plotted as _{1} vs. _{2} in row 3 and the deformed structure presented in row 4.

The column subsequently restabilizes once it finds a configuration that restores equilibrium. Both the numerical continuation procedure for the analytical model, and the Riks algorithm used in ABAQUS, can capture this behavior. One advantage of the former is that it tends to crystallize the detailed mechanical response into a few distinctive characteristics; the main column buckling modes are discretized into a Rayleigh–Ritz type model, and the non-linear results provide straightforward output of the contributions of the linear buckling modes to the post-buckling profile, _{1} and _{2} being amplitudes of the first two main column buckling modes. This analysis allows the interpretation of the effects of symmetry-breaking, and the potential to trigger higher pure or interactive modes in the post-buckling range.

All the consequences for the post-critical strength, stiffness and potential to jump between different equilibrium states owing to the cable stay behavior, can be determined directly. This information can then be used to determine parametric spaces where practical geometric quantities, such as stay diameter, layout of the stayed column system and initial prestressing forces, can generate qualitatively different, yet predictable, responses [

The simplest configuration with a single-cross arm can also be considered as a single cell within a larger lattice material. The performance of metal lattices, for example with a criss-cross structure as in Queheillalt and Wadley [

Sandwich panel with prestressed lattice core, the unit cells of which can be represented as prestressed stayed columns, as highlighted in the lower diagram [

So-called adaptive structures are able to change shape and/or material properties in response to varying external stimuli [

Consider for simplicity the buckling response of a simply-supported Euler strut, illustrated in

Buckling of a simply-supported strut and corresponding equilibrium diagrams.

Compression levels required to produce any meaningful shape-change from δ to −δ are typically sufficient to cause snap-through at a symmetry-breaking bifurcation (see asymmetric red shape in

Specifically, the fold line tracks the two limit points with respect to changes in the compressive displacement,

For values of compression,

Reducing the compression,

By decreasing the level of compression,

The control of geometrical parameters, material properties and/or boundary conditions can be used to tailor the equilibrium manifolds and adapt the multistability of the system to specific working and environmental conditions. We now consider a practical example.

A passive adaptive air inlet can regulate the opening aperture of a connected duct by interacting with the fluid flow around it. As shown in

Adaptive air inlet demonstrator.

As the airflow streaming over the panel accelerates into the connected duct, the decreasing pressure field creates an upwards force on the panel causing it to snap shut at a critical airspeed. If the airspeed is lowered beneath another threshold, the inlet automatically opens again. Unlike traditional shape-changing systems, the inlet does not rely on auxiliary devices for actuation. By increasing the amount of compression beyond the limit point on the broken-away path, the inlet can be transformed into a bistable structure that remains closed once the airflow is reduced. The greater the applied compression, the higher the airspeed required to actuate snap-through; the system's parameters can be tailored to meet specific operating requirements (see [

This device has potential for engineering applications where cooling and drag reduction create competing design drivers. Examples include air inlets on cars or cooling ducts on jet engines, which use fresh-air cooling for reliable engine operation although this cooling induces a drag penalty. For additional engineering examples that use the non-linear taxonomy described above (see e.g., [

The fact that materials exist with negative Poisson's ratio is not only intriguing but also of practical significance. So-called auxetic materials typically exhibit high energy absorption and fracture resistance, and have a broad range of practical applications from blast curtains and shock absorbers to running trainers and the ability to control waves (see for example [

Experimental and numerical work led by Bertoldi et al. [

(

Most examples of auxetic behavior in the literature are based on re-entrant structures [_{1} and _{2}, linked by the linear spring of stiffness

A new model for an auxetic cell, depicted in the unloaded state where _{1} = _{2} =

Response of the cell of _{1} − _{2}, and the symmetry-breaking variable (_{1} − _{2}). _{1} − _{2}) plane.

The continuous smooth curve replicates the response of a single arch [

The absence of homogeneity in the natural loading path gives the potential for considerable complexity of response once cells are combined, as in

Combined cells of the form of _{i},

Response of the three-tier system of _{1} − _{4}). _{1} = _{4} and _{2} = _{3}.

This theme issue focuses on the notion of stability in a variety of different contexts, both mathematical and practical. It could be argued that there is no more classical context in which one thinks of stability than structural engineering. It is fundamentally the job of the structural engineer to avoid buckling, failure or collapse. This paper takes a slightly different point of view on the topic. We focus on emerging ideas of elastic stability and post-instability behavior of structures that fail subcritically, via irreversible jumps in energy. Such problems are of current interest for at least three reasons.

First and foremost, because of their sensitivity to shocks and imperfections, there is difficulty in certifying such structures for safety. We have argued that despite over 70 years since Koiter's pioneering work, a robust methodology for analyzing the stability of such structures has yet to emerge. We have promoted here a promising line of attack, based on the Maxwell equal energy criterion and the concept of the mountain pass, as well as emerging ideas on how such ideas might be applied experimentally. However, there remains much to be done before such ideas can provide a practical assessment and design tool. It is also interesting to note how the method relies on understanding the structure of unstable, localized post-buckling paths, which form the energy barriers or basin boundaries of the problem. In that sense, there is a strong connection to other active areas of stability-related research; tipping points in natural systems (see e.g., [

Second, the structural engineering domain is changing. Across numerous lengthscales, there is a quest for ever more lightweight structures. It could be said that the revolution in composites and other nano-structured materials has been threatening to revolutionize just about the whole of the built environment for almost 50 years. Yet, despite the huge investment within academia and industry, why are we not yet seeing carbon fiber motor cars come off the production line, wholly composite airplanes in our airports, or fiber-reinforced polymer buildings being constructed en masse? There are doubtless a range of reasons for this slow penetration of composite technologies, and as most disruptive technologies, the revolution may actually be just around the corner. Nevertheless, we would argue that one of the bottlenecks still to be overcome, is that we do not understand how such structures fail. Most lightweight structures are optimized for strength, but such optimization typically leads to subcritical failure modes (see e.g., [

Finally, there is the point of view that we have been also trying to promote in this article that instability is not necessarily a bad thing. We have highlighted three areas of possible engineering exploitation of non-reversible structural instability. More generally though, we are quite used to the notion of things that snap and pop into instability. These include the pressure required to depress the keys on a computer keyboard being controlled by dome buckling, to old-fashioned bi-metallic strips being used to control switches, as in a motor car indicator light. Crash barriers and crumple zones also exploit the idea that elastic deformation of a subcritical structure can lead to transfer of significant amounts of energy into permanent plastic deformation. Origami also provides an inspiration to engineers in how small energy barriers need to be overcome in order to fold (or unfold) a structure into a new shape (see e.g., [

Clearly, there remain many lessons that engineers and designers need to learn by taking inspiration from the natural world. Not least among such lessons, as we seek to build a more resilient world in the face of global change, must surely be that there need not necessarily be anything to fear from an instability. Not only are sudden irreversible instabilities not necessarily to be feared, they can in fact be designed to be exploited for the greater good. Happy catastrophes indeed!

All data generated for this study can be found in the manuscript and/or supplementary material.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors acknowledge useful conversations with D. Avitabile, G. J. Lord, J. Sieber, and J. M. T. Thompson.

^{1}The comma notation is used throughout to denote differentials with respect to subscripted variables