Elasticity and Stability of Shape Changing Structures

Douglas P. Holmesa

aDepartment of Mechanical Engineering, Boston University, Boston, MA, 02215 USA

Abstract

As we enter the age of designer matter – where objects can morph and change shape on command – what tools do we need to createshape–shifting structures? At the heart of an elastic deformation is the combination of dilation and distortion, or stretching andbending. The competition between the latter can cause elastic instabilities, and over the last fifteen years these instabilities haveprovided a multitude of ways to prescribe and control shape change. Buckling, wrinkling, folding, creasing, snapping have becomemechanisms that when harmoniously combined enable mechanical metamaterials, self–folding origami, ultralight and ultrathinkirigami, and structures that appear to grow from one shape to another. In this review, I aim to connect the fundamentals of elasticinstabilities to the advanced functionality currently found within mechanical metamaterials.

Keywords: Elasticity, Instability, Buckling, Snapping, Mechanical Metamaterials, Shape–Shifting, Programmable Matter,Origami & Kirigami, Swelling

1. Introduction

How do objects change shape? This question has formed thebasis of entire branches of mechanics and physics dating backcenturies, and so it is reasonable to wonder what new questionsand challenges remain. To drastically change an object’s shape,it should possess some degree of softness – either in a mate-rial sense, e.g. having a low elastic modulus, or in a geometricsense, e.g. having slenderness. This softness comes at a price –large deformations introduce nonlinear responses and instabili-ties. Material nonlinearities are common with traditional engi-neering materials, i.e. metals, wood, ceramics, which, whetherthey are brittle or ductile will tend to irreversibly deform in re-sponse to small strains, either by fracturing or flowing plasti-cally. Softer materials, like rubbers, gels, and biological tissuescan often withstand moderate amounts of strain without reach-ing a material limit, and so they can reversibly withstand elas-tic instabilities without permanent deformation, exhibiting geo-metric nonlinearities by bending, buckling, wrinkling, creasing,and crumpling. It is perhaps no surprise then, that a resurgencein studying elastic instabilities coincided with the emergence ofnew, simple, and inexpensive ways to prepare soft elastomers inany desired shape and size, thus enabling researchers to studyhow instabilities could perform useful functions. Now the studyof how to utilize elastic instabilities for mechanical functional-ity brings together the disciplines of soft matter physics, me-chanics, applied mathematics, biology, and materials sciencewith the aim to extend our understanding of structural stabilityfor both generating form and function.

Email address: dpholmes@bu.edu (Douglas P. Holmes)URL: www.bu.edu/moss (Douglas P. Holmes)

2. Elasticity of Slender Structures

2.1. Stretching and Bending

Slenderness, embodied by the canonical forms: rods, plates,and shells, provides the most direct way to deform a structure,as the reduced dimensionality enables large deformations whilethe material stress remains low, i.e. σ/E � 1, where σ is themaximum principal stress and E is Young’s elastic modulus.Thin structures are highly susceptible to instability, and this isdue in large part to their tendency to deform by bending1. Thefact that these structures are by definition thinner in one dimen-sion than the other two motivated the development of models ofelastic deformation of lower spatial dimension, i.e. reduced or-der models, to describe slender structures, such as rods, plates,and shells. With a shell being perhaps the most generic of theseforms, we can develop some intuition for how thin structuresdeform by looking at the strain energy of a thin elastic shell.Shell theories have a common structure: a stretching energyUs, which accounts for extension or compression of the middlesurface of the shell and is linear in the shell thickness, h, and abending energyUb, which accounts for the curvature change ofthe deformed shell, and is dependent on h3. The strain energyof a thin shell will scale as [1]

U ∼ Y∫

ε2 dω︸ ︷︷ ︸Us

+ B∫

κ2 dω︸ ︷︷ ︸Ub

(1)

where Y = Eh/(1 − ν2) is the stretching rigidity, B =

Eh3/[12(1 − ν2)] is the bending rigidity, ν is Poisson’s ratio,and dω is the area element. To gain some physical intuition

1Readers interested in a more thorough understanding of the ideas presentedthroughout this review should consult the Supplementary Information [1].

Preprint submitted to Current Opinion in Colloid and Interface Science September 14, 2018

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Figure 1: Stretching: a. Highly stretchable gel using double–network gels [2], b. Miura–ori fold [3], c. extremely stretchable thin sheets via kirigami [4], and d.negative swelling gels [5]. Bending: e. capillary origami [6], f . tetherless microgrippers [7], and g. programming curvature with origami [8]. Snapping: h. snappingpoppers [9], i. snapping elements that absorb elastic strain energy [10], j. bistable water bomb fold [3], k. bistable kirigami unit cells [11], and l. swelling–inducedsnapping shells [12]. Buckling: m. buckled silicon membranes [13] n. negative Poisson ratio [14], o. wrinkle patterns [33], p. buckling–induced kirigami [16]

about a particular problem it is often adequate to simply con-sider how the relevant energies scale. For instance, by sim-ply comparing the stretching and bending energies, we see thatUb/Us ∼ h2(κ/ε)2, where κ is the average curvature of inducedby bending the shell (units of reciprocal length), and ε is the av-erage strain induced when stretching the shell (unitless). Sincethe shell is thin, this quantity must be very, very small, indi-cating that it is far easier to bend a thin structure than it is tostretch it. This insight helps explain why thin structures areprone to instability: if you try to shorten the length of a thinrod by compressing it, the rod would much rather bend than becompressed, and to bend it must buckle.

The remainder of this review deals with the challenge ofcontrolling the shape change of a structure while overcomingthe constraints on bending and stretching it. An overview ofhow researchers have overcome these constraints is presented inFig 1, which highlights how tailoring materials, geometry, andtopology can enable structures to stretch and bend in uncon-ventional ways, and how elastic instabilities enable structuralmorphing and metamaterial behaviors such as negative Poissonratio and negative swelling. By building upon the fundamen-tals of elasticity (Section 2), and harnessing elastic instabilitiesfor enhanced functionality (Section 3), we are now ushering inan age of programmable matter (Section 4). The multitude ofapproaches for changing an object’s shape share similar tech-niques – mechanical metamaterials are built around bucklingand snapping mechanisms, origami and kirigami create localregions that stretch easily, shape–shifting structures swell moreor less locally – and the purpose of this review is to provide thereader the insight to see the underlying principles that governshape change.

2.2. ScalingUsing the scaling relations of a thin structure’s stretching and

bending energy, we were able to quickly see why thin object

bend rather than stretch. This approach can be useful for under-standing the relevant physics in a whole range of phenomena,and we will review some of the most relevant examples here.

2.2.1. Elastogravity LengthLet’s consider a simple question: if I slide a sheet of paper

over the edge of my desk, at what length will this sheet begin tobend under its own weight? Gravity is the relevant force here,and we know that the gravitational potential energy scales as

Ug ∼ ρg∫

δ2 dω, (2)

where ρ is the volumetric density of the paper, g is the accel-eration due to gravity, and δ is the distance over which gravityis acting. To determine when gravity will bend a thin sheet,we need to consider the term κ in equation 1 more carefully.Here, bending represents a vertical deflection δ occurring rela-tive to our unknown length `. Since κ is a curvature, for smalldeflections we may write κ ≈ ∂2δ/∂x2, such that from dimen-sional analysis we may write κ ∼ δ/`2. We expect the sheetof paper will bend when the energetic potential from gravity ison the same order as the energetic resistance to bending, andso by equating equation 2 and Ub we arrive at a characteristicelastogravity length scale2[17]

`eg ∼

(Bρg

)1/4

. (3)

Typical office paper has a bending rigidity of B = 40 mN·m, anddensity of ρ = 800 kg/m3, meaning that an overhang of about

2Depending on the choice of B (which often takes the form of B ∼ Er4 fora rod) and the choice of ρ (which may be taken to be an areal density), thisexponent may change accordingly.

2

47 mm (1.85 in) will cause the paper to start to sag (Fig. 2a).Beyond being a useful back–of–the–envelope calculation, theelastogravity length has been shown to play an important rolein determining the wavelength of wrinkling soap films [18],the ability of thin elastic sheets to “grab” water [19], the curl-ing [20] and twisting [21] of an elastic rod, the viscous peelingof plates [22], and the buckling [23] and the wrinkling [24] ofelastic sheets on water.

This balance of bending and gravity is also what determinesthe length of curly hair [25, 26]. If the curl is treated as a circu-lar spring that is opening under the force of gravity, the bendingstiffness of the hair will be kcurl ∼ EIκ3

n, where κn is the cur-vature of the curl, or the intrinsic, natural curvature of the hair.Gravity acts upon each curl by ρgAL, where ρ is the density(mass/volume), A is the cross sectional area of the hair, and Lis the total length of the hair. Each individual curl, or circularspring, will open by an amount zcurl ∼ ρgAL/kcurl, and if thereare n curls, such that n ∼ Lκn, the vertical displacement of thefree end of the hair should be [25]

z ∼ρgAL2

EIκ2n. (4)

A similar argument can be used to design a Slinky. A Slinkyis little more than a very floppy spring, and one of the mosticonic features of this toy is that the can be bent into an arch.The bending energy for this discrete structure is slightly differ-ent than the continuous form given by Ub [27], however theconcept is the same. Balancing the spring’s effective bendingrigidity against the gravitational potential yields indicates thatthis Slinky would need 71 rings to form a stable arch [27]3.Since the Slinky typically has about 82 rings, cut off about 11or more and it won’t be as fun to play with. [27]

2.2.2. Elastocapillary LengthWe are familiar with fluids deform thin structures through

inertia, for example, consider the flapping of a flag in thewind [28]. In contrast, capillary forces are typically negligi-ble at macroscopic scales. However, at micro and millimetricscales, surface tension can cause thin structures to bend. So, anatural question to ask is: at what length scale should I start toworry that capillary forces will deform my structure? Here, weare concerned with surface energy, which scales as

Uγ ∼ γ

∫dω, (5)

where γ is the surface tension of the liquid. The most elegantscenario to imagine was outlined by Roman and Bico in theirreview papers on this topic [29, 30] in which they consider athin strip spontaneously wrapping around a wet cylinder of ra-dius R. For the strip to wrap around the cylinder, it must adopta curvature of κ ∼ R−1. Here again we can balance bending en-ergyUb with surface energy (eq. 5) to arrive at a characteristic

3Using nr ∼ EI/(mgRh), with radius R = 34.18mm, thickness h = 0.67mm,mass per ring m = 2.49g, and an experimentally measured effective bendingrigidity of EI = 40 × 10−6 N · m2.

elastocapillary length [31, 6]

`ec ∼

(Bγ

)1/2

. (6)

With this simple example, the prefactor can be worked out tobe exactly 1/2 [29], indicating that a sheet of office paper willspontaneously wrap around a cylinder wet with water (γ = 72mN·m) as long as its radius is greater than 372 mm (Fig. 2b).Intuitively, this makes sense – paper is intrinsically flat, and so itshould conform to a cylinder regardless of γ as R→ ∞, but be-low some critical R bending will be too costly. The role of sur-face stresses is a highly active research area at the moment [32],and those interested in this brief primer would be better servedreading the work of Bico and Roman [29, 30], along with re-cent reviews on deforming soft solids [33], bundling fiber ar-rays [34], and approaches for using fluids to assemble struc-tures [35].

2.2.3. Warping WafersIt is fair to say that one of the most classical examples of

a shape changing structure is the bending of a bimetallic stripwhen it is heated [36]. While thin bimetallic strips and beamsbend uniformly when subjected to a change in temperature, thinplates do not. This is perhaps familiar to those who have cookedin the oven with a metallic baking sheet, as it may buckle andwarp when heated above a certain temperature. The bowingand warping of thin plates was a particularly important prob-lem relating to the deposition of thin metallic films onto siliconwafers [37, 38]. Homogenous heating of a bimetal plate will en-dow the plate with a homogenous natural curvature κn, causingit to bend into the segment of a spherical cap, which has a pos-itive Gaussian curvature. Gauss’s famous Theorema Egregiumstates that you cannot change the intrinsic curvature of a surfacewithout stretching it, and so this deformation comes at the costof stretching the plate’s middle surface. Eventually, the ener-getic cost for the plate to bend into a cylinder becomes lowerthan the cost to continue bending into a spherical cap, and sothe bowing wafer will buckle into a cylindrical shape [1, 24].

The cost of stretching the plate’s middle surface into a spher-ical cap is quantified by ε in equation 1, and this will scale asthe square of the natural curvature κn times the plate radius r,such that ε ∼ (`κn)2 [40]. Alternatively, to avoid stretching thedisk will need to maintain its zero Gaussian curvature, which itcan accomplish by bending into a cylinder. If it deforms into acylindrical shape, the stretching energy is zero while the sheethas to suppress the curvature along one direction, such thatthe bending energy scales as it is written in equation 1, withκ ∼ κn [40]. By balancing the bending and stretching energiesfrom equation 1, we find that the plate should buckle when [40]

κn ∼h`2 , (7)

where, for thermal problems the natural curvature can be relatedto the temperature by calculating the curvature of a beam ofgiven modulus and thickness ratios using Timoshenko’s well

3

Figure 2: a. Demonstration of the elastogravity length with a sheet of paper.b. Elastocapillary encapsulation adapted from [6], c. Snap–through eversion ofa spherical cap, demonstrated with a tennis ball, adapted from [43], d. Wrin-kles on a soft substrate adapted from [44], and e. a demonstration of geometricrigidity [45].

known result [36]. In addition, the prefactor of this equation hasbeen worked out exactly for most plate geometries [40]. Thisscaling analysis also gives rise to a characteristic length scalethat is relevant to the anticlastic curvature of bent plates [41,42], and boundary layers in thin shells. By simply inverting therelation, we find

`bl ∼

√hkn∼√

hR, (8)

where R is the radius of curvature of a given shell. This char-acteristic length is easy to identify in thin shells. Take a tennisball, cut it in half, and turn it inside out (Fig 2c). The flat lipthat forms along the boundary of the everted shell has a lengthon the order of `bl.

2.2.4. Snapping ShellsIn the previous section, I encouraged an experiment that in-

volving cutting a tennis ball and turning it inside out. A naturalquestion may be: beginning at the ball’s apex, how far down,i.e. at what latitude from the north pole, should you cut throughit to ensure that it can be turned inside out? Turning a shellinside out requires the middle surface of the shell to stretch asit is everted. Pushing on a shell of radius R, like a ping pongball or water bottle, to a depth δ cause a dent with a charac-teristic length of `bl ∼

√δR – the same scaling that appears

in equation 8 – and this dent is equivalent to a locally evertedsegment of the shell. This gives us a measure of the stretch-ing strain, with ε ∼ δ2/`2

bl ∼ δ/R [43]. Deflection of the apexof the shell can be estimated in terms of the geometry of theshell, with δ = R(1 − cosα) ∼ α2R, where α is the planar anglesubtended between the pole and the free edge of the shell [43].Now the stretching energy of the shell can be found from equa-tion 1 using ε ∼ α2. To turn a shell inside out, it will adopt anew radius of curvature that is quite close to its original radius

of curvature, and the main distinction is that material pointsthat were originally on the outer (inner) surface of the shell arenow being compressed (stretched). This comes at the cost ofbending the shell, which according to equation 1 can be foundusing κ ∼ 1/R. Balancing these two energies, and, for histori-cal reasons, taking the fourth root of the result, gives rise to adimensionless parameter that characterizes the shell [46, 43]

Λ ∼ [12(1 − ν)]1/4(R

h

)1/2

α. (9)

Shells with Λ & 5.75 can be turned inside out, or everted, andremain that way – i.e. they are bistable [43]. For a typicaltennis ball, with R = 33.15mm, h = 3.3mm, and Poisson ratioof ν = 1/2, that means cutting the ball at a height of 15mmfrom its north pole will yield a bistable shell.

2.2.5. WrinklesProbably no instability is more responsible for the surge in

research interest on the topic of this review over the last fif-teen years than wrinkling [36, 26, 49, 38]. Like the others wehave encountered so far, this problem also has both a long his-tory [27] and a myriad of potential utility [52]. It is quite easy tosee the pattern characteristic of wrinkling by simply compress-ing the skin on the underside of your forearm between yourthumb and index finger. What is immediately apparent is theformation of ridges the are all equally spaced by some distanceλ and all seem to have approximately the same amplitude A(Fig. 2d). So, a natural question is: what sets the spacing ofthese wrinkles? The physics at play here don’t include grav-ity or a fluid, and balancing bending and stretching alone isnot enough. This is a buckling problem, but one where a stifffilm (skin) is resting on a softer substrate which is resisting de-formation – i.e. we can consider the mechanics of a beam onan elastic foundation [25]. Consider the outer portion of yourskin to be a stiff plate resting on a substrate that behaves like aWinkler foundation [25]. We can write the strain energy of thefoundation as

U f ∼ K∫

δ2 dω, (10)

where K is the stiffness of the foundation, and δ is its deflec-tion. A wrinkle will adopt a vertical deflection δ relative tosome unknown characteristic spacing λ. Therefore, the curva-ture of these wrinkles which costs bending energy will scale asκ ∼ δ/λ2. Balancing Ub with equation 10, we expect that thewrinkles will have a wavelength of [26]

λ ∼( B

K

)1/4

. (11)

The physical interpretation of this scaling is that the stiff skinwould like to buckle like an Euler column, with the lowesteigenmode. However, the substrate below would have to bestretched unevenly – more at the center of the skin than nearits edges. Therefore, the skin and the substrate reach a com-promise set by equation 11. The substrate does not need to besolid, indeed a fluid will impart an effective elastic stiffness of

4

K ∼ ρg, where ρ is the density of water [38, 40]. For instance,if you consider placing a droplet of water on a thin film float-ing on a bath of water, wrinkles will form on the surface due tosurface tension [38, 43, 42, 44, 48]. This simple scaling breaksdown when the loading is increased [40], as the wrinkles can nolonger be treated as a small perturbation from the initially flatstate, i.e. as the deformation transitions from the post–bucklingnear–threshold regime to the far–from–threshold regime [53]4.

2.3. Geometric Rigidity

Before we learn about how to control and direct the defor-mations of slender structures, it is useful to understand wherethese structures derive their rigidity from. It has long beenknown that geometry alone can provide functionality, such asenhanced structural integrity – arches have been used in archi-tecture to bear loads for over four thousand years, and thesestructures exhibit their rigidity due to their intrinsic curvature.Shell structures, such as domes, are no different. Research byVella et al. [60, 61, 62] and Reis et al. [45] has shed light onthe intimate connections between a shell’s geometry and it’smechanical behavior, demonstrating that tuning a shell’s shape,rather than the materials, provides a straightforward way to en-hance its ability to sustain a load. Consider a positively curved(e.g. spherical, ellipsoidal), pressurized shell loaded at its apexby a point force. The rigidity of the shell depends on both thedepth of indentation relative to shell thickness, and the degreeof shell pressurization. In the limit of weakly pressurized shells,the classical results from Reissner are recovered [60, 61, 45],which find that the stiffness is dependent on the shell’s Gaus-sian curvature K – the shell’s rigidity is correlated to the in–plane stretching of the shell [61], an energetically costly defor-mation. Consider an egg shell, which is effectively ellipsoidal.If we take the Gaussian curvature at any point on the shell tobe the product of the two principle curvatures, then K will belargest at the north pole of the shell, and lowest at the equator– therefore the enhanced stiffness observed with compressinga chicken egg at its poles compared to its equator is a conse-quence of geometry–induced rigidity [45] (Fig. 2e). In the limitof high pressure, the mean curvature H , not the Gaussian cur-vature, governs the shell stiffness [62]. The absence of a depen-dence on K in the large deflections of highly pressurized shellsimplies that the internal pressure negates the effect of geometricrigidity.

Understanding how a shell’s stiffness depends on both thedegree of internal pressure and the extent of deformation maybe an important tool for biomechanical characterization. Animportant question for quantifying the morphogenesis of cel-lular and multicellular structures is how to deconvolute mea-surements of the cell wall mechanics from measurements of thepressure caused by osmotic fluid flow through the cell wall [63],i.e. turgor pressure. Vella et al. [61] demonstrated that theirresults on the indentation of pressurized elastic shells couldcharacterize the turgor pressure within yeast cells, viz. Sac-charomyces cerevisiae. Using the stiffness from the Reissner

4This topic is discussed a bit more in the Supplemental Information

limit, a turgor pressure within the yeast cells was estimated thatwas consistent with those measured using other experimentaltechniques [64]. Preliminary work has begun on extending thisanalysis to the measurement of the elastic properties of tomatofruit cells [65] and plant tissues [66, 67], and further develop-ment of this model to include different loading types may pro-vide insight into the large deformations observed within arti-ficial biological microcapsules, for example see [68] and thereferences therein.

3. Elastic Instability Phenomena

The scaling from equation 1 tells us that thin structures preferto bend rather than stretch, and in doing so will often exhibit anelastic instability. The efficient design of thin and lightweightstructures out of high–strength materials leads to a conundrumat the heart of structural engineering: an optimum design isby its very nature prone to instability. With ample evidencethat structural instability plays an integral role in the morpho-genesis of biological materials [69] – from fingerprint forma-tion [34, 71], to the folds in the cerebral cortex [72, 73, 74],to the tendril perversion in climbing plants [75] – it is perhapsmore important than ever that scientists and engineers studyingsolid materials have a strong understanding of the mechanics ofelastic stability.

Stability requires us to consider the internal exchange of en-ergy within a structure, with the total potential energy consist-ing of the internal strain energy and the potential from the exter-nal loads, i.e. V = Us+Ub+P. We recall that the first variationof the total potential energy must be equal to zero for a struc-ture to be in equilibrium, δV = 0. This statement is equivalentto Newton’s second law. Equilibrium does not ensure stability,however. It is possible to balance a ball resting on the apex ofa steep hill, such that it is in equilibrium, but this equilibriumis unstable because any slight perturbation will cause the ballto role far away and not return. A ball resting at the bottom ofa hill is of course stable, because any perturbation to it, for ex-ample rolling it slightly up the hill will cause it to return to itsoriginal position once the perturbation is removed. We can alsospeak of neutral equilibrium, which would refer to the ball rest-ing on a flat surface – any perturbation will not change the ball’spotential energy. With this simple analogy, we may recognizethat these hills represent a potential energy landscape, and thatwe need to investigate the convexity or concavity of this land-scape at an equilibrium point to determine if this equilibrium isstable or unstable, respectively. This requires us to investigatethe character of the second variation of the total potential en-ergy – a structure is stable if δ2V > 0, unstable if δ2V < 0, andneutral if δ2V = 0 [1]. It is tempting to relate these ideas tofinding the extrema and curvature of a function in ordinary cal-culus, however energy is not a function, but rather a functional– a function of a function of a variable. Evaluating the characterof a functional requires the tools from variational calculus, andperhaps the most enjoyable primer on the calculus of variationscan be found in Feynman’s lectures [76].

For an elastic structure under a conservative load the criticalpoint at which an instability occurs will always correspond to

5

either a bifurcation (“buckling”) or a limit point (“snapping”).Imperfections in the structure’s shape, the eccentricity of theload, or within the material can cause bifurcation critical pointsto change in character to a limit point, suggesting that bucklingis perhaps more the exception than the rule [6]. On the surface,then it seems surprising that buckling problems are abound inthe literature, while limit point problems appear far less fre-quently. There are, in general, two reasons for this: (1.) thedistinction between bifurcations and limit points is not merelysemantic: one mathematical tool that allow us to study bifurca-tions – namely, linear stability analysis – is impossible to useto study a limit point instability [7, 8], making buckling prob-lems far easier to analyze, and (2.) slender structures are mostefficient if they carry their load in a primarily membrane stateof stress, and such configurations will typically fail by bucklingrather than snapping [6]. The primary mathematical differencebetween a bifurcation and a limit point instability is this: a limitpoint occurs when the equilibrium position becomes unstable,and the structure moves to the closest, stable point on the sameequilibrium path, while a bifurcation occurs when two equilib-rium pathways intersect, and an exchange of stability occursas the structure follows the stable equilibrium path. Concreteexamples of these two generic instability phenomena are givenin the Supplementary Information. For the chemist or physi-cist, these definitions may bring to mind a connection betweenthese instabilities and phase transitions [80, 81, 82], whereinone could draw an analogy between a limit point instability anda first–order phase transitions, and a bifurcation with a second–order phase transition. This is perhaps more useful conceptuallythan practically, but several similarities are shared.

3.1. Snapping

The snap–through instability presents an important mecha-nism for directed shape change in the design of shape–shiftingstructures. A recent review on this topic by Hu et al. willhopefully provide a nice compliment to the following discus-sion [83]. There have been demonstrations of actuating snap–through instabilities for just about every conceivable mechani-cal and non–mechanical stimulus, including temperature [84],light [85], acoustic excitation [86, 87], elastomer or gel swelling[88, 89, 90, 12], magnetic fields [91, 92], fluid flow [93], sur-face tension or elastocapillarity [94], and electrical current withmaterials that include from ceramic (piezoelectric) [95, 96],metallic (electrostatic) [97, 98], and rubber (dielectric elas-tomers) [99, 100, 23, 101]. Laminated composites of epoxy andcarbon fiber or fiber glass may exhibit bistability or multistabil-ity while thermally curing [102, 103, 104, 105]. Depending onthe fiber orientation, the presence of the fiber embedded in theepoxy matrix may give these materials an orthotropic response,thus providing design criteria for the orientation of the stimu-lus chosen to induce snapping [24]. Such structures may enablemorphing components for wind turbines [106], mechanisms forswimming or flying [107], ventricular assist devices [108], andfor lightweight, deployable structures [109, 110]. In a simi-lar manner, the orientation of the residual stress or prestressresulting from fabrication can alter the geometric criteria for

bistability [111, 112, 113]. In addition, rapid prototyping tech-niques, such as 3D printing and laser cutting, have made itmuch easier to generate bistable structures on a wide range ofscales. Of note is the elastomer coating technique pioneeredby Brun et al., which has made the preparation of thin, spheri-cal caps with nearly uniform thickness simple, fast, and afford-able [114]. With all of these ways to induce snapping, it is nosurprise that it has found utility in a wide range of engineeringfields.

Experimental results on the adhesion of 2D materials likegraphene and MoS2 have indicated that there is a snap intoadhesive contact between the film and a substrate containingroughness [115, 116], which has enabled researchers to demon-strate that this instability may be a useful metrology tool tomeasure the material properties of 2D materials [117]. Thesnap–through instability has also been utilized to tune a ma-terial’s properties in response to an applied load by alteringthe material’s lattice structure to generate dramatic, dynamicincreases in a material’s stiffness [118]. On a slightly largerscale, there is interest in the micro- and nanoelectromechani-cal (MEMS and NEMS, respectively) communities to use therapid snap–through of arches and shells in electromechani-cal systems [119] for accelerometers [120], or as a means torapidly change a surface’s texture or optical properties [88].The precise placement of folds in thin sheets can generate awide range of multistable structures, with the most fundamen-tal being the waterbomb [121, 122, 123, 3], which has genericbistability for any number of creases [3]. In addition to tra-ditional folding and cutting techniques, programming creasesinto a material through spatial variations in its thickness canenable bistability in folded shells – cylinders, spheres, or sad-dles [124]. One function that has drawn particular attention isthe design of bistable structures as a means to capture, trap, orharness elastic strain energy [125, 10, 126]. This process maybe amenable for use with soft elastomers [10], which can un-dergo very large snap–through instabilities without exhibitingmaterial failure [99, 127, 100, 101]. If these soft materials areused as actuators, rather than energy harvesters, the instabilitycan be used to trigger large, nonlinear changes in shape, pres-sure, and extension within soft, fluidic actuators [128].

3.2. BucklingThe elastic buckling of a beam or plate provides a straight-

forward way to get large, reversible, out–of–plane deforma-tions which can be utilized for generating advanced function-ality. For instance, embedding a flexible plate within a mi-crofluidic channel provides a means for mechanically actu-ated valving [129, 130]. Buckled plates have been utilizedto fabricate semiconductor nanoribbons for stretchable elec-tronics [131, 132, 133]. In recent years [134, 135, 136], re-searchers have utilized this principle to fabricate single–wallcarbon nanotube arches [137], single-crystalline silicon archesthat were used as metal-oxide semiconductor field–effect tran-sistors (MOSFETs) [138], and buckled lead zirconate titanate(PZT) ribbons for ceramic piezoelectric energy harvesting de-vices [139]. Buckling bilayer plates can be utilized to gen-erate shape–shifting structures [40] that may be used as soft

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grippers [140]. Buckling of shells has provided an intriguingway to control global shape [141, 142, 12] and local pattern-ing [143, 144, 145, 146], reduce aerodynamic drag [147, 148],generate lock–and–key colloids that can selectively aggre-gate [149], form liquid crystal shell actuators [150], and pavethe way for buckling microswimmers [151].

Wrinkles are what appear when a thin structure supportedby a soft substrate buckles [27, 36, 26, 49, 38, 52]. There area variety of ways to fabricate wrinkled surfaces, but the un-derlying principle is simple: bond a stiff film containing resid-ual compressive stress onto a compliant substrate. Some of thefirst experiments in this manner involved the deposition of thinmetallic film onto a PDMS substrate by electron beam evap-oration [152, 153]. This deposition locally heats the PDMSsurface, expanding it equibiaxially. Upon cooling, this com-pressive stress causes the metallic film to buckle with wrin-kles in a herringbone pattern that cover a large surface area.Wrinkles have long added functionality to structural materials,such as with their ability to damp the structural vibrations oc-curring on composite plates [154]. Similar to the buckled rib-bons used for flexible electronics, these wrinkled plates enabledthe fabrication of stretchable gold conductors [155], flexible-electronic devices using wrinkled ribbons [156, 136, 157] andwrinkled single-walled carbon nanotubes [158]. Sinusoidalwrinkles have been used to alter and tune friction [159, 160],to fabricate tunable phase gratings [161], and to improve themetrology of thin films via the strain–induced elastic buck-ling instability for mechanical measurement (SIEBIMM) tech-nique [162]. Additionally, there is significant evidence withinbiological systems to suggest that patterned surfaces enhanceadhesion [163, 164, 165, 166]. Control of the pattern topog-raphy is a crucial component for utilizing these structured sur-faces (Fig. 1o). Linear stability of the cylindrical pattern revealsthe emergence of hexagonal, triangular, and square modes, andthe commonly observed herringbone pattern [29, 30, 33]. Fi-nally, as the amount of overstress, or confinement, of the com-pressed plate increases, the bending energy along the plate goesfrom being broadly distributed among wrinkles to being lo-calized within sharper features [39, 40, 1], as the deformationtransitions from the post–buckling near–threshold regime to thefar–from–threshold regime [43, 44, 53].

4. Programmable Matter

4.1. Mechanical MetamaterialsMetamaterials are rationally designed structures composed

of building blocks to enable functionality that cannot be foundin natural materials [174]. While this term has been tradition-ally associated with electromagnetism and optics, the field hasrecently broadened to include elastostatic and elastodynamicsmetamaterials, collectively called mechanical metamaterials.There is an excellent review that will cover the details of thistopic far better than I can will do here, so I encourage the cu-rious reader to seek out Bertoldi et al. [175]. In general, theunderlying concept of a mechanical metamaterial is to use hi-erarchical structures, such as microscale or mesoscale geomet-ric features, to alter the macroscopic deformation of an object.

Figure 3: a. Spherical encapsulation with a “buckliball”, adapted from [170],b. static non–reciprocity in mechanical metamaterials [171], c. programmaticshape change with a pixelated cube, adapted from [172], and d. programmaticstructural polarity through external constraints, adapted from [173].

Foam has natural microscale features, and its atypical mechan-ical response exhibits a negative Poisson ratio [176]. This aux-etic behavior [177] has been explored in a variety of contexts,many of which utilize the buckling of micro/mesoscale beamswithin the macroscale structure [178, 14]. With the inclusionof additional constraints, this auxetic behavior can be finelytuned to provide a programmatic structural deformation witha tailored force–displacement response [173, 179] (Fig. 3d).Buckling of microstructural features to generate auxetic behav-ior can be extended from structured plates to structured shellsto potentially encapsulate material [170] (Fig. 3a). Instabilityis at the heart of mechanical metamaterials that exhibit negativecompressibility, in which they contract while being loaded intension, potentially enabling force amplification [180]. Buck-ling of subscale features becomes truly “meta” when one con-siders the Euler buckling of metabeams [181], wherein themicrostructure has a significant affect on the postbuckling re-sponse. More generally, precise control of pixelated structuresprovides a way to programmatically dictate a structure’s defor-mation across multiple length scales [182] (Fig. 3c). For ex-ample, the combinatorial design of specific building blocks canenable a cube to produce a smiling face in response to uniaxialcompression [172]. Mechanical metamaterials are now veeringcloser to classical metamaterials by examining the role of topol-ogy in addition to geometry to transform shape and mechanicalproperties [183], with notable examples including structures tobreak the static reciprocity formalized by the Maxwell–Bettireciprocity theorem [171] (Fig. 3b).

4.2. Origami & Kirigami

Recall that Gauss’s famous Theorema Egregium states thatyou cannot change the intrinsic curvature of a surface without

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stretching it, and we know from equation 1 that it is far easierto bend a thin object than stretch one. So, if you want to stretcha thin object or wrap a sheet of paper around a soccer ball,what is one to do? One answer is to design regions that enablestretching, such as with the inclusion of folds and cuts. A re-view by Witten [54], which in my opinion did much to catalyzethis entire discipline, describes at great lengths the way stresswithin thin sheets will localize into sharp creases and folds, es-sentially acting as sacrificial regions in which a material be-grudgingly stretches. Understanding the mechanical nature ofthese folds guides their strategic incorporation into engineeringsystems. If these folds are precisely placed and sequentially ac-tuated, they represent a means to develop advanced engineeringstructures. Taking inspiration from the Japanese art of origami,sequential folding has long been utilized in structured sys-tems [185]. Deployable structures, utilized in space technolo-gies, typically require damage–free actuation, reliable deforma-tion, and autonomous or automated conformational change, andhave been utilized within small satellite deployable structures,deploy booms, and array panels [186, 110]. Recent researchhas examined the mechanics of these foldable structures, witha focus on origami–inspired design [187, 188], and thus it maybe useful to the reader to seek out the review article by Peraza–Hernandez et al. on this specific topic [189]. The fundamen-tal mechanics of a fold [190], or more broadly a conical de-fect or singularity within a thin plate [191, 192, 70, 194] are atthe heart of understanding and designing origami mechanisms.While the geometric properties of a fold have been studied ingreat detail, the role of a material’s properties in these systemshas been largely overlooked to date. A notable exception isthe work by Croll et al. on the role of adhesion in crumpledstructures [195]. Unit cells of a repeating fold pattern provideorigami building blocks [196], and enable the programmaticdesign of structural deformations [197, 198, 199], 3D shape–shifting [200], and folding–induced curvatures [8]. Such sys-tems may exhibit multistability, as noted briefly in section onsnapping [201, 202].

The Miura-ori folding pattern is a well–known example offunctional origami – a parallelogram of folds defined by twofold angles and two lengths, which enables the compact fold-ing of a flat plate [203, 204, 205]. This functional arrayof pleated folds has been utilized for collapsable maps, de-ployable satellite arrays [206], and to increase the areal en-ergy density of folded lithium–ion batteries [207]. Mathemati-cians and mechanicians alike have borrowed concepts fromorigami to create biomedical stents [208], nanostructured fold-able electrodes [209], ultrathin high–resolution, folded opticallenses [210], photovoltaics [211], materials with tunable coef-ficients of thermal expansion [212], and for folding rigid, thin–walled structures around hinges [213]. The techniques for gen-erating folds include actuation by shape–memory alloys [214],light [215, 216], microfluidic flow [217], and direct–printedwet–origami [218].

A natural extension of using folds inspired by origami to en-able elaborate shape changes is to use cuts inspired by kirigami.Here, the fundamental behavior is governed by the nonlinearmechanics of thin frames [219]. With kirigami, the roles of

Figure 4: a. Morphing with halftone lithography adapted from [240], b. non–Euclidean plates adapted from [241], c. a monkey saddle, adapted from [242],d. pollen grain isometry of a spherical cap, related to the work in [12], e. 4Dprinting using shape memory polymer joints, adapted from [243], and f . activeprinted meshes, adapted from [244].

topology [220] and geometry are once again at the heart of de-sign rules that enable targeted shape changes [221, 222, 223].Mechanics enters the picture by enabling these flat sheets tobuckle into 3D structures when stretched or compressed be-yond a critical point [16, 224]. While lattice cuts have pri-marily been studied to date, with the work of Dias et al. be-ing a notable exception [224], additional functionality wasdemonstrated with multiscale cuts, i.e. kiri–kirigami [225],randomly oriented [226] and fractal cuts [227, 228], and pat-terns that naturally interlock [229]. One of the primary demon-strations of functional shape change via kirigami is the ex-treme stretchability of thin sheets [230, 231, 232], which eventranslates to 2D materials, such as graphene [4]. Dias et al.showed that to leading order, the out–of–plane deformationof a single cut is independent of the thickness of the sheet,providing mechanical insight into why this shape changingmechanism can scale down to the thinnest materials [224].Kirigami–inspired cut patterns have enabled the fabrication ofphotonic metamaterials [233], metamaterials with tunable ma-terial properties [234, 11], smart adhesives [235], solar trackingdevices [236], stretchable lithium ion batteries [237], opticalbeam steering [238], shape–shifting structures [239], and ultra-lightweight linear actuators [224].

4.3. Shape–Shifting Structures

The control of geometry and elasticity to create adaptive,morphing structures paves the way for an era of designer ma-terials [245]. We have reviewed the energetic limitations onbending and stretching structures, and we have seen how care-fully chosen stretching regions – through, for example, origamiand kirigami – enable a much wider range of shape changes. Analternative approach for shape–shifting structures is through theprogrammatic design of the volumetric strain within a material.This approach draws the closest analogy to synthetic structuralgrowth. Swelling presents a simple and effective technique to

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spatially tune volumetric strain, while falling short of growthby not permanently adding mass to the reference elastomer, orreference configuration. Swelling is the infiltration of an elasticnetwork with a favorable solvent, such that the characteristicsize of the polymer chains, i.e. their radius of gyration, dra-matically increases. The balance of fluid movement and elasticresponse dictates the time scale over which swelling–inducedphenomena will occur, and the ability for the fluid to swellthe polymer network dictates the magnitude of stress imparted.Fluid-flow within natural structures plays a crucial role in themorphology of growing tissues [246, 247, 248], the opening ofseed pods [249], the shrinkage of mud [250] and moss [251],and the curling of cartilage [252], leaves [253, 254], andpine cones [255]. Porous thin films, such as fuel cell mem-branes [256], are highly susceptible to swelling-induced delam-ination and buckling, which cripple their functionality.

Swelling–induced deformations provide a means for shapingtwo-dimensional sheets into three-dimensional structures [241,257, 240], with features spanning multiple length scales [258,259]. Differential swelling – where portions of the materiallocally swell more than others – have been used to createhelical ribbons [260], cylinders [40], saddles [261], pinchedspheres [12], and wavy strips and discs [262, 263, 264]. Theshape selection process is nontrivial, and many efforts havefocused on predicting what shape will emerge from program-ming the volumetric strain, often tailored using the metric ten-sor of the middle surface of a thin plate or shell [265, 266,267, 268, 269, 12]. The alternative, inverse problem – predict-ing what metric to program to achieve a specific 3D shape –may be even more desirable [270]. In addition to the contri-butions from the sheet’s elastic energy, understanding the dy-namic morphing of a swelling structure, such as the curlingof paper [271, 272], gels [273], and rubber [257, 274], posesadditional challenges [275, 276]. An intricate photolithogra-phy technique, e.g. halftone lithography, has been developed toscale this dynamic process down to create responsive, morph-ing structures on the micron scale [240]. The mechanics behindthis structural morphing combines dynamic aspects of swelling,structure, stability, and material properties, thereby creating arich environment for experimental and theoretical insights. Itis likely that building on the concept of programmable mat-ter, will inspire novel rapid–prototyping technologies, such as3D “inkjet” printing that utilizes small amounts of polymeriz-able solvents to create complex structures. Such ideas are al-ready emerging in a technique known as 4D printing, wherethe fourth dimension represents the timescale corresponding toswelling [277, 243, 244, 278, 279, 280, 281, 282].

5. Summary & Outlook

Shape changing materials require working with and aroundthe constraints of elasticity, and utilizing nonlinearity to gen-erate functionality. Most of what I have covered in this re-view does exactly that – overcoming the difficulty of stretchingthin sheets by folding or cutting them, using the buckling andsnapping of beams and plates and shells to generate metama-terial behavior, and playing with swelling to tune spatial distri-

butions of strain. So, where do we go from here? Certainly,important research will continue in the areas we have coveredat length, though new opportunities are beginning to emergein a few specific areas. I think one of the most significant ar-eas of research going forward will focus on woven fabrics andknits for tailoring shape and properties [283, 284], and for ex-ample, such ideas are already enabling the form finding of gridshells [285]. Another interesting area may lie at the interfacebetween elastic structures and granular or colloidal materials,so–called elastogranular interactions. Research in this regard isbeginning to lay out the fundamental behavior of bending andpacking of elastic rods within grains [286, 287, 288], whichis clearly relevant in the form and function of growing plantroots [289, 290, 291], and is beginning to show promise forprogrammable, reversible architecture [292, 293, 294]. Finally,in my opinion the biggest limitation in achieving the next gen-eration of shape–shifting structures is the absence of simple tofabricate and robust materials that are highly responsive to stim-uli – i.e. we need help from chemists and materials scientists.Dielectric elastomers offer a lot of promise, yet require extremevoltages and fail often. Swelling elastomers and gels is slow, re-quires a bath of fluid, and usually involves brittle materials thatfail when they either deform too much or dry out. We need ad-dressable and programmable materials to take full advantage ofthe recent advances in mechanical metamaterials. Too often theresearch covered in this review utilizes ordinary office paper,dental polymers, and traditional acrylic plastics and urethanerubbers. This approach is fine, and should even be applauded,when the purpose is to show how an appropriate understand-ing of mechanics, geometry, and topology can make profoundpredictions with run–of–the–mill materials, but additional ma-terials science advances are necessary to fully realize the poten-tial opportunities for technological insertion of shape–shiftingmaterials. Among the leaders of this effort to connect materi-als and mechanics is the work of Hayward et al., see for ex-ample [295, 296, 297]. Another recent interesting effort to tunethe material response of traditional thermal bimorphs highlightsthe ability to use encapsulated phase change materials to spon-taneously induce thermal bending at critical temperatures [298].There is much work to be done to better blend the insights frommechanics with the advances in materials.

6. Acknowledgements

I gratefully acknowledge the financial support from NSFCMMI – CAREER through Mechanics of Materials and Struc-tures (#1454153). I am grateful to L. Stein–Montalvo and X.Jiang for photos, and I would like to thank M.A. Dias for manydiscussions on this topic over the years. I also thank the orga-nizers of the Kavli Institute for Theoretical Physics (NSF PHY-1748958) program on Geometry, Elasticity, Fluctuations, andOrder in 2D Soft Matter for providing me an opportunity tothink through some of the ideas detailed within this article. Iapologize if I did not cite your work in this review, it was anaccident and an oversight on my part, don’t @ me.

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15

Supplementary Information:Elasticity and Stability of Shape Changing Structures

A. A bit more on stretching & bending

Shell theories have a common structure: a stretching energyUs, which accounts for in–plane deformations, that is linear inthe shell thickness, h, and a bending energyUb, which accountsfor the curvature change of the deformed shell, and is dependenton h3 [1, 2, 3, 4]. The dimensionless strain energy of the Koitershell equations are written in covariant form as [5]

Us =Y2

∫ [(1 − ν)εαβεαβ + ν(εαα)2

]dω ∼ Y

∫ε2dω, (1a)

Ub =B2

∫ [(1 − ν)καβκαβ + ν(καα)2

]dω ∼B

∫κ2dω, (1b)

where Y = Eh/(1 − ν2) is the stretching rigidity, B =

Eh3/[12(1 − ν2)] is the bending rigidity, E is Young’s elasticmodulus, ν is Poisson’s ratio, dω is the area element, and εand κ are the tensors that describe the stretching and bending ofthe middle surface of the shell, respectively. The Greek indicesα ∈ [1, 2], and raised indices refer to contravariant componentsof a tensor, while lowered indices refer to covariant compo-nents. The intricacies of this notation are not important upon afirst introduction to this topic, but for the curious reader I wouldrecommend the instructive text by Niordson [5]. The energieseach contain the square of two invariants, since the shell is two–dimensional, for example with εαα being a measure of how thearea changes and εαβεαβ being a measure of how the shell isdistorted.

B. A bit more on elastic stability

Unfortunately, most student’s first and only exposure to aproblem of elastic stability is a rather deceptive one – the Eulerbuckling of a column. In studying the Euler buckling of a col-umn, it is customary to make a one particularly significant as-sumption – that the column is incompressible. This assumptionis advantageous because it is both reasonable, as most columnsdo not shorten significantly before buckling, and it simplifiesthe calculation. However, it is quite misleading, and I will letone of the forefathers of stability theory, W.T. Koiter explainwhy (emphasis mine):

“Here it appears that a negative second variation ofthe potential energy of the external loads is the causeof a loss of stability, but this state of affairs is due to oursimplifying assumption of an incompressible rod, andwe may not generalize this experience to other cases,as still happens only too frequently. In fact, in the caseof elastic structures under dead loads the potential en-ergy of the external loads does not enter at all into thestability condition. Loss of stability of elastic struc-tures is always due to an internal exchange of en-ergy.” [6]

−ncα

0

ncα

P/EA

a.

−α − α√3

0 α√3α

−α2

0

αUnstable

q

∂2 qqV

b.

0 α 2α

−ncα

0

ncα

i.

ii.iii.

iv.v.

Snap

(α− q)

P/EA

c.

Figure 1: a. Normalized force P/EA plotted as a function of angle q showingextrema at ncα, where nc = − 2π3

75√

3. b. The second variation of the total potential

energy is plotted as a function of angle q. The equilibrium configuration isunstable when ∂2

qq ≤ 0. c. Normalized force is plotted as a function of rotation,showing the loading path from i. the initial configuration to ii. the limit point atncα, at which point stability is lost and the structure snaps to iii.Upon unloadingfrom iii. to the limit point at iv. the structure snaps to v. Removal of the loadrestores the initial configuration i.

What that means physically is that it is the energy, and morespecifically its second variation, in the deformed or fundamen-tal state that needs to be evaluated for stability, and not the po-tential acting on the structure in its original or reference state.For a more fundamental and at times conceptual discussion ofelastic stability, I recommend the two textbooks by Thompsonand Hunt [7, 8].

C. The simplest snapping structure

The simplest structure that exhibits snapping is the bistabletruss first analyzed by von Mises, in which two elastic barsof elastic modulus E, cross sectional area A, and an axialstiffness EA/(L/ cosα) are pinned together with an initial an-gle α, and pinned at supports separated by a distance 2L. Aforce P applied to the apex of the truss will change the an-gle from α to q, and induce a compressive strain in the barsof ε = cosα/ cos q − 1, which causes a vertical displacementw = L(tanα − tan q). In this single degree of freedom exam-ple, the total potential energy is simply the strain energy in thespring minus the potential energy of the load,

V(q) =EALcosα

(cosαcos q

− 1)2

− PL(tanα − tan q). (2)

Equilibrium is found by setting the first variation of the poten-tial energy to zero,

∂V∂q

=L

cos2 q[2EA(cosα tan q − sin q) + P

]= 0, (3)

from which we can find the equilibrium path

PEA

= 2(sin q − cosα tan q), (4)

which is plotted in figure 1a. Here we note an important dis-tinction between snapping, which is a limit point instability, andbuckling, which is a bifurcation – there is only on equilibriumpath, meaning the truss will deflection under any infinitesimal

16

load P. A system that exhibits a limit point instability has onlyone equilibrium path, and loss of stability will cause a discon-tinuous jump in a given parameter, i.c. a finite change in q inresponse to an infinitesimal change in P. Stability of this trusswill be lost when the second variation of its potential energy,δ2V = 1

2!∂2V∂q2 δq2, ceases to be positive definite. We will eval-

uate the stability of the equilibrium state at a constant load byinserting the load from equation 4 into equation 3, and calculat-ing the second variation of V as

∂2V∂q2 = 2

EALcos4 q

(cosα − cos3 q

), (5)

which is plotted in which is plotted in figure 1b. It is clearfrom the graph that a region of this second variation is belowzero, specifically when −α/

√3 ≤ q ≤ α/

√3. This region of

the equilibrium path given by equation 4 is unstable. Since inthis example we have considered a load–controlled experiment,meaning the magnitude of the low is being increased, the trusshas no choice but to jump (horizontally) from point ii. to pointiii. on Fig. 1c., since this is the closest, stable part of the equi-librium curve at the same fixed value of P/EA.

In general, finding the critical point of a snap–through insta-bility is a challenge, because when you linearize the equationsyou lose all information about the instability. This can be im-mediately seen with our simple example of the von Mises trussfrom equation 2 – a Taylor series expansion of q for any α leavesyou with a linear equation, and thus the critical point is lost –its second variation will never be negative, so one would erro-neously think the system is always stable. Numerically, thereare multiple approaches for finding, following, and continu-ing through instability points [9], including utilizing arc–lengthmethods or dynamic simulations.

D. The simplest buckling structure

To see the buckling and postbuckling behavior of a structure,it is useful to consider a nonlinear, finite–deflection theory –something that is straightforward to examine in a discrete, onedegree of freedom system. Consider a rigid bar of length L thatis held vertical by a rotational spring of stiffness kr, and loadedat the free end of the bar by a a load P. If it is perfectly vertical,there is no initial inclination angle to the bar5, i.e. α = 0. Wecan again use q to measure the the inclination angle of the barafter loading, so that the total potential energy may be writtenas

V(q) =12

krq2 − PL(1 − cos q), (6)

where the first term on the right hand side is the strain energystored in the rotational spring, and the second term is the workof the load. Its first variation yields

∂V∂q

= krq − PL sin q, (7)

5To consider the role of imperfections, or to analyze the stability of a wholearray of structures and materials, I recommend pouring over the invaluable tomeby Bazant and Cedolin [10], especially chapter 4.

−1.0 −0.5 0.0 0.5 1.0

0

1

q

PL/k

r

Figure 2: A plot of the two equilibrium curves of a buckling rigid bar, usingthe equilibrium paths given by equation 8. When PL/kr ≤ 1, the bar remainsvertical, and it buckles either to the left (q < 0) or right (q > 0) when PL/kr > 1.At the critical point, which is stable [7, 8], there is an exchange of stabilitybetween the gray equilibrium curve and the black curve as the structure buckles.

and since equilibrium requires the first variation to be station-ary, i.e. ∂V = 0, we find the equilibrium relation between forceand angle to be

P(q) =kr

L

(q

sin q

). (8)

Upon inspection of equation 8, we see that there are two solu-tions that satisfy this equation: q = 0 or q , 0 (Fig. 2). Thebar can stay perfectly vertical, i.e. q = 0 ∀ P, but that solutionis not always stable. We can check the stability of our systemby evaluating the second variation of V , and requiring it to bepositive definite. The second variation is

∂2V∂q2 = kr − PL cos q. (9)

By setting this second variation equal to zero, we find the criti-cal buckling force

Pc =kr

L cos q, (10)

and by inserting equation 8 into equation 9 we see that this post-buckling path is stable for all q. For P ≤ Pc, the bar will remainvertical and undeflected, and for P > Pc, the bar will buckleeither to the left or the right, breaking symmetry, and this buck-ling direction will be dictated by imperfections in the bar or theeccentricity of the load.

E. Buckling of continuous structures

E.1. Elastica

The history behind the mathematical treatment of a thin elas-tic is rich, and a detailed discussion of it is well beyond thescope of this review. I would encourage all interested to read a

17

history of it as detailed by Levien [11]. A nice recent article thatrevisited the mathematical intricacies of the problem was pre-pared by Singh et al. [12], along with a detailed derivation ofthe governing equations and various solutions in a recent bookby Bigoni [13]. There are typically three flavors of equilibriumequations for the elastica, and their utility very much dependson either what community the research is embedded within, orhow the particulars of a given problem lend themselves to astraightforward solution. In general, you will encounter theequations parameterized by the angle along the arc length ofthe elastic curve θ(s), the curvature along the arc length κ(s), orthrough a force and moment balance. All three are equivalentand can be recovered with a little effort and some geometry –an exercise left up to the reader. For instance, a moment andforce balance yields

m′(s) + t(s) × f (s) = 0, (11a)f ′(s) + n(s) = 0, (11b)

where m(s) and f (s) are the moments and forces acting on thecurve, respectively, and t(s) and n(s) are the tangent and nor-mal vectors along the curve, respectively. Here, we use an apos-trophe to denote an ordinary derivative with respect to the arclength. Using the Frenet–Serret frame, the equations can berewritten in terms of the curvature along the arc length as

2κ′′(s) + κ3 − µκ(s) + σ = 0, (12)

where µ and σ are constants corresponding to how the elasticais loaded. Finally, this equation can in turn be shown to beequivalent to the equilibrium equation of a planar elastica asparameterized by the angle along the arc length as

θ′′(s) + λ2p sin θ(s) = 0, (13)

where s is the arc length that parameterizes the curve, θ is theangle that the tangent vector at a given point s makes with thehorizon, and λ2

p = P/B is the ratio of the applied load P to thebending rigidity B of the beam.

The buckling of an elastica, or an Euler column, is a problemencountered by most engineers during their studies. Its solutionbegins by linearizing equation 13, which is nonlinear becauseof the term sin θ(s). Linearizing about the flat state allows us toconsider small angles, such that sin θ(s) ≈ θ(s), such that whatstarted as a nonlinear eigenvalue problem now becomes a lineareigenvalue problem. For example, for an elastic that is simplysupported at its ends, the lowest eigenvalue is gives a criticalbuckling load of

Pc = π2 BL2 , (14)

where L is the length of the elastica. For a clamped–clampedelastica, the buckling load is four times as large. Linearizing theequations comes at a cost, however – while we gain insight intothe force at which buckling will occur, along with the modeshape of the buckled structure, we lose the ability to quantifythe amplitude of the deflection.

Figure 3: (Top) Shapes of a simply supported elastica subjected to various endshortenings. Graphs were generated in Mathematica using equations 15a,b.(Bottom) Shapes of a clamped–clamped elastica subjected to various end short-enings. Graphs were generated in Mathematica using equations 16a,b using thecode block provided below these equations.

The post–buckling shape of a simply supported elastica isdetermined by the parametric equations [13]

x(s) =2

Λ(k)(E[am[sΛ(k) +K[k]|k]|k]

− E[am[K[k], k], k]) − s, (15a)

y(s) = −2

Λ(k)kcn[sΛ(k) +K[k]|k], (15b)

where k = sin α2 , α is the angle of rotation at the inflection

point at s = L/2, i.e. symmetry allows the analysis to focus ononly half of the rod length, Λ(k) = 2mK[k] which for a modeone deformation m = 1, K[·] is the complete elliptic integralof the first kind, E[·|·] is the incomplete elliptic integral of thesecond kind, am[·|·] is the amplitude for Jacobi elliptic func-tions, and cn[·|·] is the Jacobi cn elliptic function. Increasingα will increase the amplitude of the elastica while conservingthe elastica’s arc length. Similarly, the post–buckling shape ofclamped–clamped elastica is then determined by the parametricequations [13]

x(s) =2

Λ(k)E[am[sΛ(k)|k]|k] − s, (16a)

y(s) =2

Λ(k)k(1 − cn[sΛ(k)|k]), (16b)

where k = sin α2 , α is the angle of rotation at the inflection point

at s = L/4, i.e. symmetry allows the analysis to focus on onlyquarter of the rod length, and Λ(k) = 2(m + 1)K[k] which fora mode one deformation m = 1. Increasing α will increasethe amplitude of the elastica while conserving the elastica’s arclength. The following Mathematica code is provided to en-able the reader to visualize the deformed shape of the clamped–clamped elastica given by equations 16a

In[1]:= k[α_]:= Sin[α/2]

In[2]:= λc[k_, m_]:= 2 (m + 1) EllipticK[k]

In[3]:= xc[s_, α_, m_]:= (-s +

2/λc[k[α],

18

m] (EllipticE[JacobiAmplitude[

s λc[k[α], m], k[α]], k[α]]))

In[4]:= yc[s_, α_, m_]:= (2 k[α]/λc[k[α],m] (1 - JacobiCN[s λc[k[α], m],

k[α]]))

In[5]:= ParametricPlot[

{xc[s, 1.5, 1], yc[s, 1.5, 1]},

{s, 0, 1}]

where for the parametric plot, values of α = 1.5 and mode num-ber of m = 1 were chosen arbitrarily. Example elastica curvesare plotted in Fig. 3.

In both the simply supported and clamped–clamped case, theend shortening u(α)/L is given by

u(α)L

= 2(1 −E(k)K(k)

). (17)

E.2. Plates & Shells

We began this review with Koiter’s thin shell equations, i.e.equations 1a and 1b. There are various simplifications that canbe made to these equations to make them more mathemati-cally tractable [5], and we will begin with a commonly usedset of equations to describe the mechanics of thin plates: theFoppl-von Karman (FvK) plate equations. The FvK equationsare rooted in an approximation for the plate’s in–plane strainεαβ where the non–linear terms describing the plate’s out–of–plane deflection w are retained, while the non–linear terms cor-responding to the plate’s in–plane displacements uα are dis-carded, such that εαβ ≈ 1/2(uα,β + uβ,α + w,αw,β), where thecomma represents differentiation, i.e. f,α ≡ ∂ f /∂xα. Using theFvK approximation for strain, a plate’s equilibrium equationscan be arrived at from a variational approach that minimizesthe plate’s free energy [14].

B∇4w − hσαβw,αβ = 0, (18a)σαβ,β = 0, (18b)

where B = Eh3/12(1 − ν2) is the bending stiffness. Alterna-tively, by introduction of the Airy potential, σαβ = εαγεβµφ,λµ,where εαβ is the two–dimensional Levi–Civita symbol, and the“die” operator, which represents a symmetric contraction of twocomponents,i.e. ♦4[ f , g] ≡ f,ααg,ββ − 2 f,αβg,αβ + f,ββg,αα [15],the equilibrium equations can be written as

B∇4w − ♦4[φ,w] = 0, (19a)1Y∇4φ +

12♦4[w,w] = 0, (19b)

where Y = Eh is the stretching stiffness. These equations arenon–linear, with the term ♦4[φ,w] coupling the plate’s curva-ture with in–plane stress, and the term ♦4[w,w]/2 representingthe Gauss curvature K. The explicit appearance of the Gausscurvature illustrates the intimate connection between a plate’selasticity and its geometry [16], and this interplay will consis-tently appear in the various examples that follow.

A plate is thin structure that is flat in its stress–free con-figuration, while a shell has a non–zero initial curvature. Aninstructive approach to the complexities that arise with shellmechanics is to regard the shell as consisting of two distinctsurfaces: one which sustains stretching stress resultants andthe other which sustains the bending stress resultants. Calla-dine [17] covers this two–surface approach to the equilibriumequations for shells in a comprehensive manner, and we willonly highlight the key components here. In this two–surfaceapproach, the stretching surface is equivalent to the membranehypothesis of shells, and the bending surface is very similar tothe FvK analysis detailed above. The general interaction be-tween these two imaginary surfaces, i.e. the way an externallyapplied force is distributed between them, leads directly to theDonnell–Mushtari–Vlasov (DMV) equations [18].

B∇4w + ∇2kφ − ♦

4[φ,w] = 0, (20a)1Y∇4φ − ∇2

kw +12♦4[w,w] = 0. (20b)

These equilibrium equations contain the Vlasov operator∇2

k( f ) ≡ R−1β f,αα + R−1

α f,ββ, which incorporates the shell’s prin-cipal curvatures. It should be immediately apparent that in theabsence of any initial curvature, equations 20a and 20b revertdirectly to the FvK equations given by equations 19a and 19b.Since these equations are an extension of the FvK plate equa-tions, they retain the same variational structure, and are appli-cable under similar assumptions of small strains and moderaterotations [16].

Few exact solutions to the Foppl-von Karman equations areknown to exist, but geometrical simplifications make the FvKequations useful for describing the Euler buckling of a platein response to an in–plane stress. A uniaxially compressedplate that buckles out–of–plane will have cylindrical symme-try, which greatly simplifies the geometry, since a cylindri-cal shape is developable to a plane, such that in equation 19bK = ♦4[w,w]/2 = 0. Since there is only stress in one direction,say the x–direction, equation 19b reduces to φ,xxxx = 0. Consid-ering a plate with clamped boundary conditions, this equationfor φ can be integrated, allowing equation 19a to reduce to anordinary differential equation, which can be solved in terms ofthe plate’s deflection [16],

w(x, y) = ±h√

3

(σ

σEu− 1

)1/2 (1 + cos

πxL

), (21)

where L is the length of the plate, σ is the uniaxial stress, andthe Euler buckling stress is

σEu =π2E

3(1 − ν2)

(hL

)2

. (22)

As noted in the subsection on Warping Wafers, buckling ofthin plates can also occur from a mismatch of strains throughthe thickness, in particular with heated or differentially swollenplates. Some additional, relevant references to this topic in-clude [19, 20, 21, 22, 23, 24].

19

E.3. WrinklingWhen a thin plate is bound to a compliant substrate and com-

pressed, higher buckling modes emerge, and the pattern forma-tion of these ordered buckled structures, or wrinkles, have gar-nered significant recent interest. The wavelength of wrinklesis selected by a balance of plate’s bending energy and the en-ergy required to deform the underlying elastic substrate. Thebending resistance of the sheet penalizes short wavelengths,while deformation of the elastic foundation that is supportingthe sheet penalizes long wavelengths. An intermediate wave-length emerges when we consider that the reaction of the un-derlying layer K is proportional to the deflection of the platew. In the simplest case, for 1D wrinkles extending in the x–direction, equation 18a becomes

Bw,xxxx − hσxxw,xx + Kw = 0, (23)

when a Winkler foundation is included [25]. By linearizing thisequation, and discarding any stretching of the mid–plane due tocurvature, i.e. the second term is zero, a characteristic lengthscale emerges based on a balance of the two rigidities. Thescaling we found in equation 11 can be recovered here fromdimensional analysis. In the limit of wrinkles on a very deepsubstrate, i.e. h � λ � Hs, the stiffness of the elastic foun-dation scales as K ∼ Es/λ [26], and the wrinkle wavelengththerefore scales as

λ ∼ h(

EEs

)1/3

. (24)

We can explicitly determine the critical stress required forwrinkles to form and the resulting wavelength by lineariz-ing equations 18a and 18b about the flat, unbuckled state,i.e. uα = w = 0, and performing linear stability analy-sis [27, 28, 29, 30, 31, 32, 33]. Linearizing these equationsrequires the strain tensor to simplify to εαβ ≈ 1/2(uα,β + uβ,α).The plate will be considered infinite in the xα directions, andexposed to equibiaxial stress, such that the linearized equationsare B∇4w − hσ0∇

2w = −p and ∇4φ = 0. The stress compo-nent exerted by the substrate onto the plate is introduced as p.These ordinary differential equations allow periodic solutionsin the form w(x, y) = w cos (k1x), which result in a cylindricalpattern described by a critical threshold σc and wavelength λgiven by [34, 28, 29, 31]

σc = E∗(

3E∗s2E∗

)2/3

, (25a)

λ = πh(

2E∗

3E∗s

)1/3

. (25b)

The starred quantities E∗ and E∗s represent the effective, or re-duced, modulus of the plate and substrate, respectively. Thereduced modulus of the plate is E∗ = E[1 − ν2]−1, while thereduced modulus of the substrate includes the tangential trac-tion forces exerted by the substrate onto the plate, such thatE∗s = Es(1 − νs)[(1 + νs)(3 − 4νs)]−1 [30, 31, 32].

Wrinkles that form under compression are familiar to mosteveryone – they appear by simply compressing our skin. It

is less intuitive to observe wrinkles that form as a free elas-tic sheet is pulled in tension [35, 36, 26, 37]. This problemis no longer confined to 1D as there is a tension T in they–direction, and localized regions of compressive in the x–direction near the clamped boundaries. The equilibrium equa-tion that emerges from minimizing the free energy for this con-figuration is Bw,xxxx − hσxxw,xx − Tw,yy = 0 [26]. Cerda etal. [26] identified an analogy between the energy in an elasticfoundation supporting a thin sheet, U f ∼

∫AKw2 dA,and the

sheet’s stretching energy, Um ∼∫

AT(w,x

)2 dA. Since these en-ergies are of similar form, and with the in–plane strain scalingas w,x ∼ w/L, comparing the two leads to the emergence ofeffective stiffness of the elastic foundation, K ∼ T/L2. Thisconnection is convenient as it allows equation 23, and the scal-ing in equation 24, to hold for a variety of wrinkling problems,with each problem varying only in the actual form of the effec-tive stiffness of the elastic foundation, K.

E.4. FoldingAs the amount of overstress, or confinement, of the com-

pressed plate increases, the bending energy along the plate goesfrom being broadly distributed among wrinkles to being local-ized within sharper features [38, 39, 40, 41, 42, 43, 44, 45, 46,47, 48, 49, 50, 51, 52, 53]. When the compressive strain ex-ceeds a critical value of ε ' 0.2, a pitchfork bifurcation of thewrinkle morphology emerges as one wrinkle grows in ampli-tude while neighboring wrinkles decrease [39, 40, 41]. Thisfocusing of bending energy leads to a break in the up–downsymmetry of the wrinkled plate, and is analogous to period–doubling bifurcations in dynamical systems [41]. The sym-metry breaking occurs from the nonlinear contribution of thecompliant foundation that supports the stiff plate as the out–of–plane deflection and in–plane compression of the plate are nolonger equivalent [41, 49]. Further compression of the platebeyond a strain of ε ' 0.26 leads to a period-quadrupling bifur-cation [41], in which sharp folds appear, localizing the much ofthe stress within highly curved ridges [54, 39, 40, 41, 55, 49].This folding is a means for focusing the elastic energy withinthe plate. Qualitatively, a fold occurs when the radius of curva-ture of the deformed feature is on the same order as the thick-ness of the film.

When folds occur over a large scale, for instance with thecrumpling of a piece of paper, these stress-focused ridges cansignificantly alter the mechanical properties of the structure. Acrumpled piece of paper is characterized by sharp ridges thatterminate at point–like singularities. These singularities areconical dislocations, and they emerge as the sheet tries to re-sist stretching, thereby localizing the stretched region into thetip of the cone. Referred to as developable cones, or d–cones,due to their isometry to a flat plate, the shape is a particu-lar solution to the FvK equations in the limit of large deflec-tions [56, 57, 58, 59, 60]. A d–cone is a building block to acrumpled sheet, and two of these singularities are connected bya stretching ridge, which contains much of the deformation en-ergy within a strongly buckled sheet [61, 62, 63]. The width ofthese stretching ridges can be arrived at either by scaling con-siderations of the sheets elastic energy [62] or by a boundary

20

layer analysis of the FvK equations [64]. Upon formation ofa ridge, the bending and stretching energies are of the sameorder, and for a sharp fold on a plate with a given thickness,these energies will scale linearly with the crease length χ, i.e.Um ∼ Ub ∼ χ. In reality, the curvature of the fold is soft-ened by a small amount of stretching along the ridge, and asthe length of the fold increases these energies grow at a slowerrate, Um ∼ Ub ∼ χ

1/3 [61, 62, 64, 63]. Recent work has focusedon the dynamic deformations of d–cones and ridges [65, 66],the existence and annihilation of multiple singularities in a con-strained elastic plate to minimize stretching [67, 68], the me-chanics of curved–folds [69], and the nature of dipoles in thinsheets [70].

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