PHYSICAL REVIEW E 93, 033005 (2016)

Elastic theory of origami-based metamaterials

V. Brunck,1 F. Lechenault,1 A. Reid,1,2 and M. Adda-Bedia1

1Laboratoire de Physique Statistique, Ecole Normale Superieure, UPMC Universite Paris 06, Universite Paris-Diderot, CNRS,24 rue Lhomond 75005 Paris, France

2Department of Physics, North Carolina State University, North Carolina 27695, USA(Received 16 July 2015; revised manuscript received 8 January 2016; published 22 March 2016)

Origami offers the possibility for new metamaterials whose overall mechanical properties can be programedby acting locally on each crease. Starting from a thin plate and having knowledge about the properties of thematerial and the folding procedure, one would like to determine the shape taken by the structure at rest and itsmechanical response. In this article, we introduce a vector deformation field acting on the imprinted network ofcreases that allows us to express the geometrical constraints of rigid origami structures in a simple and systematicway. This formalism is then used to write a general covariant expression of the elastic energy of n-creases meetingat a single vertex. Computations of the equilibrium states are then carried out explicitly in two special cases:the generalized waterbomb base and the Miura-Ori. For the waterbomb, we show a generic bistability for anynumber of creases. For the Miura folding, however, we uncover a phase transition from monostable to bistablestates that explains the efficient deployability of this structure for a given range of geometrical and mechanicalparameters. Moreover, the analysis shows that geometric frustration induces residual stresses in origami structuresthat should be taken into account in determining their mechanical response. This formalism can be extended to ageneral crease network, ordered or otherwise, and so opens new perspectives for the mechanics and the physicsof origami-based metamaterials.

DOI: 10.1103/PhysRevE.93.033005

I. INTRODUCTION

Since the mid-1980s, the ancient art of paper foldinghas evolved into a fertile scientific field gathering distantdisciplines. Using the mathematics of flat foldability [1,2],computational origami has made possible the creation ofan enormous amount of new designs. The interest has alsoswitched from the depiction of concrete objects like thetraditional cranes of Japan’s Edo period to abstract structureslike tessellations, whose strange mechanical properties havearoused the curiosity of engineers and physicists alike. Origamimetamaterials display, for example, auxetic behavior [3,4]and multistability [5–7], the latter allowing reprogrammablereconfigurations [8].

Ubiquitous in nature where permanent creases can form asa result of growth in a constraining container (e.g., dragonflywings [9] and petal leaves [10,11]), origami structures arealso well represented in domains where strength or deployableproperties are attractive benefits, be it in fashion [12,13], ar-chitecture [14], medicine [15], or engineering, from airbags toconsumer electronics to deployable space structures [16–20].In all these areas, a question of paramount importance is howa particular crease affects the shape and mechanical propertiesof the overall structure.

To answer this question the folds are usually modeled byelastic hinges of specific stiffnesses and rest angles. In thisapproach, each crease lies at the intersection between twopanels, whose response is governed by a length scale L∗ thatcontrols whether the panels actually bend or fold [21,22].More precisely, this origami length L∗ = B/κ is the ratiobetween the bending modulus B of the faces and the torsionalstiffness κ of the crease. When the typical length of the panelsis small compared to L∗, the system behaves as a rigid origami,with faces remaining mostly undeformed and creases actuatingwhen submitted to stress. On the other hand, if the faces typical

size is larger than L∗, then the mechanical response will begoverned by the bending of the sheet, while the angles of thecreases will essentially keep their rest value [7]. The existenceof the length scale L∗ restricts the apparent scalability oforigami structures. Nevertheless, we look in the followingat the realm of rigid origami, which corresponds to takingsystems with infinitely rigid faces and flexible creases.

The process of making an origami structure consists inprinting a given planar graph on a flat sheet, which definesthe reference state, and then applying a plastic deformation tothis network that assigns to each edge a given stiffness andrest angle. The three-dimensional shape taken by the structureis a minimizer of the mechanical energy subject to kinematicconstraints. For this purpose, the first step is to consider asingle vertex with n creases coming out of it, which constitutesthe building block of any origami tessellation. In the caseof rigid origami, the standard approach is to find the energyminimum of such a vertex by summing over the creases, eachone having different stiffness κi and rest crease angle ψ0

i :Eh = ∑n

i=1 κi(ψi − ψ0i )2/2, where ψi is the opening angle

of the ith fold [4,5]. This energy must be supplemented bythe geometrical constraints that the faces remain rigid duringdeformation. However, in many practical applications theseconstraints are not explicitly given in terms of the foldsangles ψi , which forces us to adapt the energy minimizationprocedure for each study. This approach is thus not convenientfor building a general framework for the elasticity of origamistructures.

Here we provide a description of origami tessellations byexpressing the deformation in terms of the vectors carried bythe crease network instead of the folds angles. This approachoffers us the possibility to take into account the geometricfrustration using the same field of vectors, which rendersenergy minimization systematic and straightforward. Becauseorigami can be seen as a three-dimensional deformation of

2470-0045/2016/93(3)/033005(14) 033005-1 ©2016 American Physical Society

V. BRUNCK, F. LECHENAULT, A. REID, AND M. ADDA-BEDIA PHYSICAL REVIEW E 93, 033005 (2016)

FIG. 1. The geometry of a single straight crease can be describedby its length L and the three unit vectors (�u,�v, �w).

an initially flat planar graph, the vector field carried by thecrease network is a natural measure of the deformation of suchfolded structures that allows for studying complex origamidesigns [23] and models of crumpled paper [24].

After detailing our approach, we apply it to two specialcases that have the advantage of exhibiting a single degree offreedom: the generalized waterbomb base and the Miura-Ori.These are well studied systems allowing for comparison toexisting literature. By displaying applications of our descrip-tion, we hope to convey its ability to solve a vast array ofproblems in rigid origami by applying a simple and systematicprocedure.

II. EQUILIBRIUM STATE OF A SINGLE CREASE

We begin by considering a single straight crease of lengthL, that we simply model by the intersection of two rigid planarpanels. The geometry of the crease and the faces can then bedescribed by three unit vectors (Fig. 1): (�u, �w) for the first planeand (�v, �w) for the second one, such that �u · �w = �v · �w = 0 and�w lies along the crease.

Let us compute the elastic energy per unit length Estored in the crease. Being a scalar invariant quantity, E isa linear combination of the invariants built from the set ofvectors describing the geometry of the system. Here the onlynonvanishing quantities are the scalar product �u · �v and thetriple product (�u × �v) · �w. However, the latter term is a scalar if�w is a pseudovector (since �u × �v is a pseudovector). Therefore,�w should be written as

�w = �u × �v|�u × �v| . (1)

The leading order contributions to the elastic energy of thecrease are then given by:

E = L E(�u,�v, �w) = L (λ �u · �v + μ (�u × �v) · �w), (2)

where λ and μ are material real constants. Note that Eq. (1)ensures that the crease energy (2) is invariant under thetransformation �u ↔ �v. With the notations of Fig. 1, one has�u · �v = cos ψ and (�u × �v) · �w = | sin ψ |. We then introduce κ

and ψ0 ∈ [0,π ], such that

λ = −κ cos ψ0, μ = −κ sin ψ0, (3)

allowing us to rewrite Eq. (2) as

E ={−Lκ cos(ψ − ψ0) for 0 � ψ � π

−Lκ cos(ψ + ψ0) for π � ψ � 2π. (4)

For ψ ≈ ψ0, one recovers the usual energy of an elastichinge [4–7,22]

Eh = 12 Lκ (ψ − ψ0)2 + E0, (5)

where E0 is a constant, κ is the stiffness, and ψ0 is therest angle of the crease (the angle in the absence of externalloading [21,22]). The quadratic expression of the energy (5)sets the physical meaning of the constants λ and μ throughEq. (3) and shows that the covariant energy (2) does notintroduce additional material parameters.

Equation (4) shows that the energy landscape is even withrespect to ψ = π and is cusped at that location for all ψ0 �=0. Moreover, two equilibrium configurations are possible forψ ∈ [0,2π ],

ψ ≡ ψm = ψ0 and ψ ≡ ψv = 2π − ψ0. (6)

However, the result ψv = 2π − ψm indicates that these twosolutions correspond to the same state of the crease but“observed” from the two different sides of the folded sheet.Therefore, expressing the energy of the crease in terms ofthe vectors associated with the faces allows us to capture themirror-symmetry without enforcing it. The combination of thisresult with the cusp in the energy at ψ = π restricts the creaseequilibrium angle ψm (respectively, ψv) to vary in the interval[0,π ] (respectively, [π,2π ]). This physically based limitingangle variation is in agreement with the behavior of creasedsheets in real materials.

For a single fold, the crease energy as given by Eq. (5)is reasonable for several applications [22,25]. However, weshow in the following that for the case of interacting creasesthe parametrization of the crease energy in terms of thevectors (�u,�v, �w) becomes relevant. Finally, notice that suchparametrization is not unique, and, instead, one could usethe unit vectors (�nu,�nv, �w), where (�nu,�nv) are the normals tothe faces, and retrieve Eq. (4). Nevertheless, starting fromEq. (2) allows us to set a general framework for the elasticityof origami designs.

III. ENERGY OF A SINGLE VERTEX

We now extend our approach to determine the energy of avertex with n-creases of possibly different lengths Li (Fig. 2).Because every origami network, ordered or otherwise, can bedecomposed into a collection of such vertices, this structure

(a) (b)

FIG. 2. n-creases meeting at a single vertex. (a) The imprintedpattern of creases is described by their lengths Li and unit vectors �wi

separated by angles αi (i = 1, . . . ,n). (b) A subset of three adjacentcreases after deformation.

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(a) Mountain (b) Valley

FIG. 3. Definition of mountain and valley creases.

constitutes the building block of any rigid origami design. Foreach crease, we define the unit vector �wi , taken along theith crease and pointing outward from the vertex, and the unitvectors �ui , �vi (see Fig. 2). We also introduce for each creasethe material constants:

λi = −κi cos ψ0i , μi = −κi sin ψ0

i , (7)

with κi and ψ0i the stiffness and rest angle of the ith crease as

imposed by the operator when machining the structure. Let αi

be the angle between �wi and �wi+1, with∑n

i=1 αi = 2π , anduse the periodic convention �wi+n ≡ �wi . Because we considerthe rigid face limit, this angle remains constant during thedeformation of the sheet from its initial flat state to the foldedone. We then have n constraints given by:

Ci = �wi · �wi+1 − cos αi = 0, i = 1, . . . ,n. (8)

As there is no interaction between the folds apart from thegeometrical constraints, the energy of the vertex is obtainedby summing over all creases,

E =n∑

i=1

Li[λi �ui · �vi + μi (�ui × �vi) · �wi]. (9)

Note that the directions of �wi are fixed independently of thedirections of �ui and �vi (see Fig. 2). Thus one has (�ui × �vi) ·�wi = sin ψi , where the crease angle ψi is now defined as theoriented angle ( �ui,�vi). This writing does not modify the resultsof Sec. II and allows us to introduce the so-called mountainand valley creases (see Fig. 3).

Figure 2(b) shows that each panel contains four vectors. Forexample, the left panel in Fig. 2(b) contains ( �wi−1, �wi,�vi−1,�ui),whereas the right panel contains ( �wi+1, �wi,�vi,�ui+1). Since twovectors are sufficient to define each plane, one can write twovectors as a combination of the two others. Therefore thereexists a decomposition of �ui and �vi in terms of ( �wi−1, �wi, �wi+1),so we can rewrite Eq. (9) using the crease vectors �wi only.

A simple and visual way to find such a decomposition isto use the orthonormal base ( �ui, �wi ∧ �ui, �wi) (Fig. 4), in whichone has

�vi = (cos ψi, sin ψi,0), (10)

�wi−1 = (sin αi−1,0, cos αi−1), (11)

�wi+1 = (sin αi cos ψi, sin αi sin ψi, cos αi). (12)

FIG. 4. The projections of the vectors �vi , �wi−1, and �wi+1 in theorthonormal base ( �ui, �wi ∧ �ui, �wi).

Then one can show that

�ui · �vi = �wi−1 · �wi+1 − cos αi−1 cos αi

sin αi−1 sin αi

, (13)

(�ui × �vi) · �wi = ( �wi−1 × �wi+1) · �wi

sin αi−1 sin αi

. (14)

Using this result, one finds that Eq. (9) is transformed into:

E =n∑

i=1

Li[λi �wi−1 · �wi+1 + μi ( �wi−1 × �wi+1) · �wi], (15)

where a constant term has been discarded (the αi’s being fixed)and

Li = Li

sin αi sin αi−1(16)

is an effective length of the ith crease.The elastic energy of the vertex is expressed only in terms

of the field �wi that completely describes the deformation of thecrease network. Moreover, the geometric constraints (8) andthe conditions | �wi |2 = 1 are also given in terms of the samefield of vectors. Therefore, the equilibrium configuration of thevertex can be determined by minimizing the following energyfunctional:

E∗ = E −n∑

i=1

γiCi −n∑

i=1

δi(| �wi |2 − 1), (17)

where γi and δi are Lagrange multipliers. The minimizationwith respect to the components of the vectors �wi , γi , and δi

yields a closed system of 5n equations. The crease angles ψi

can be derived a posteriori from the vector field �wi using [seeEq. (13)]

cos ψi = �wi−1 · �wi+1 − cos αi−1 cos αi

sin αi−1 sin αi

. (18)

The main result of this section is that the vector field carriedby the crease network is the appropriate field that measuresthe deformation of a rigid origami structure from the planarreference state. While it is known that the geometric degrees offreedom of an origami metamaterial can be formulated in terms

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V. BRUNCK, F. LECHENAULT, A. REID, AND M. ADDA-BEDIA PHYSICAL REVIEW E 93, 033005 (2016)

of the unit vectors tangent to the folds and the faces [26–29],the reduction to a single vector field parametrization of thetotal energy, including both the elastic energy of the creasenetwork and the kinematic constraints, was lacking. Here weshow that starting from the mechanical response of a singlecrease given by Eq. (2), the total elastic energy of an origamivertex, as given by Eq. (17), can be reduced to a functional ofthe vector field �wi only.

Equation (17) allows us to develop a unified elasticformalism of origami metametrials that can be applied toprobe various mechanical properties. Although, we haveconsidered the case of a single vertex of an arbitrary numberof creases, the generalization to complex origami tessellationsis straightforward (see Sec. VI). The equilibrium equations ofa general origami vertex can be determined by minimizingEq. (17) with respect to the vectors �wi and the Lagrangemultipliers γi and δi . Nevertheless, to show the effectivenessof this elastic model we study in the following two specialcases that allow for semianalytic resolution: the generalizedwaterbomb base and the Miura-Ori.

IV. GENERALIZED WATERBOMB BASE

We consider the origami vertex that consists in an alter-nation of n/2 mountain and n/2 valley folds of equal lengthL, separated by the same angle αi = α = 2π/n, where n isthe number of creases (see Fig. 5). The number of folds iseven and for the rigid generalized waterbomb to exist, thecondition n � 6 must be fulfilled. The case n = 8 correspondsto the usual waterbomb [30], a folding used as starting point ofmany origami constructions. We define a Cartesian coordinatessystem such that its origin is located at the vertex and thexy plane coincides with the reference planar state of thesheet before deformation. We consider configurations in whichall valleys (respectively, mountains) have the same materialparameters. Due to these symmetries, it is easy to show thatthe components of the crease vectors are given by:

�w2p = (sin θm cos 2pα, sin θm sin 2pα, cos θm),

�w2p+1 = (sin θv cos(2p+1)α, sin θv sin(2p+1)α, cos θv),

(19)

with p = 0, . . . ,(n/2 − 1). Here, θm (respectively, θv) denotesthe mountains (respectively, valleys) polar angles of the creasesand we use the periodicity convention �w0 ≡ �wn. Also, the

(a) (b)

FIG. 5. The generalized waterbomb base. (a) The imprinted net-work. (b) Experimental realization of a three-dimensional equilibriumstructure.

polar angles θm,v ∈ [0,π ] are chosen such that the referenceplane of the sheet before deformation corresponds to θ = π/2.Once these two angles are determined, the whole shape of thewaterbomb is achieved.

For the generalized waterbomb base with n-folds, the elasticenergy as given by Eq. (15) takes the following form:

EW (θm,θv)

= π

α

L

sin2 α[λm(cos2 α + sin2 α cos 2θv)

+λv(cos2 α+ sin2 α cos 2θm)

+μm(sin 2α sin2 θv cos θm− sin α sin 2θv sin θm)

+ μv(sin 2α sin2 θm cos θv− sin α sin 2θm sin θv)],

(20)

with:

λm = −κm cos ψ0m, μm = −κm sin ψ0

m,

λv = −κv cos ψ0v , μv = −κv sin ψ0

v , (21)

where (κm,ψ0m) and (κv,ψ

0v ) are, respectively, the stiffness

and rest angle of the mountains and valleys. Recall that ψi

is defined using the oriented angle from �ui to �vi which isequivalent to measuring crease angles on the same side ofthe folded sheet (Fig. 3). Therefore, we take the followingconvention for distinguishing mountains and valleys: ψ0

m ∈[0,π ] and ψ0

v ∈ [π,2π ]. In the following, we take κm = κv = κ

and scale the elastic energy such that κL = 1. The influenceof the ratio κm/κv adds extra adjustability that is not crucial tothe analysis [31].

Due to the symmetry of the waterbomb configuration, allthe geometric constraints Ci in Eq. (8) amount to one relationbetween θm and θv given by:

CW (θm,θv) = cos θm cos θv − cos α(1 − sin θm sin θv) = 0.

(22)Because the valleys are always lower than the mountains, onehas θv � θm and Eq. (22) yields:

θv(θm) =⎧⎨⎩arccos

(cos α cos θm

1+sin α sin θm

)0 � θm � π

2 ,

arccos(

cos α cos θm

1−sin α sin θm

)π2 < θm � π − α.

(23)

A. Equilibrium states of the waterbomb

We then minimize the elastic energy of the waterbomb,Eq. (20), subject to the geometric constraint (22) by introduc-ing the energy functional:

E∗W = EW (θm,θv) − γ CW (θm,θv), (24)

where γ is a Lagrange multiplier. The extrema of Eq. (24)are found graphically by determining the points where thecontour lines of the energy E(θm,θv) are tangent to the functionθv(θm) given by Eq. (23). Figure 6 shows the existence of twoequilibrium solutions: The system is always bistable. We labelthese solutions by (±), the (+) state corresponding to thesolution with θ+

m ∈ [0, π2 ] and the (−) state to the one with

θ−m ∈ [π

2 ,π ] (see Fig. 7). These two equilibrium states can beexperimentally obtained from one another through a snappingtransition.

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ELASTIC THEORY OF ORIGAMI-BASED METAMATERIALS PHYSICAL REVIEW E 93, 033005 (2016)

FIG. 6. Contour lines of the elastic energy, Eq. (20), of thewaterbomb and the parametric curve of the geometric constraint,Eq. (23) (solid line). The two points on the curve correspond tothe (±) equilibrium states. Here, the classical waterbomb (n = 8)is displayed with creases elastic parameters κm = κv , ψ0

m = π

8 , andψ0

v = 2π − π

2 .

For ψ0v = 2π − ψ0

m, the bistability of this system is ob-vious. The elastic energy of this completely symmetric casegiven by Eq. (20) satisfies the symmetry property EW (θm,θv) =EW (π − θv,π − θm), without imposing it a priori, and thegeometric constraint (23) is always symmetric with respect tothe line θv = π − θm: CW (θm,θv) = CW (π − θv,π − θm) (seeFig. 6). Therefore, in the complete symmetric case, one hasθ∓v = π − θ±

m . That means that one solution is obtained fromthe other by a first rotation (�ex,π ) and a second rotation (�ez,α)to exchange mountains and valleys. As shown in Fig. 7, thissymmetry property is suppressed when ψ0

m + ψ0v �= 2π and

the two equilibrium solutions are not superimposable.The minimization of the energy functional Eq. (24) can be

easily carried out numerically. Figure 8 displays the variationof the resulting equilibrium angles θ±

m and θ±v as function of

the number of creases. These angles determine completely theequilibrium shapes of the waterbomb. On the other hand, weperform in Appendix a series expansion for a large number ofcreases that corresponds to small angles α and find analyticalexpressions for θ±

m and θ±v at linear order in α. The results are

shown in Fig. 8. It is worth pointing out that when n → ∞,

(a) (+) solution (b) (−) solution

FIG. 7. The two equilibrium shapes of the waterbomb corre-sponding to the solutions of Fig. 6.

(a) (+) solution. (b) (−) solution.

FIG. 8. Evolution of the two equilibrium solutions as a functionof the number of creases for the same material parameters as inFig. 6. Mountains angles are in black; valleys angles are in orange.The dashed curves correspond to the analytic results derived from theasymptotic expansions at first order in α (see Appendix). Note thatdespite the fact that continuous curves are used, only those pointscorresponding to integer even n are physically relevant.

the equilibrium angles for mountains and valleys converge tothe same limit θ∞ given by:

θ∞ ={

π2 − ψ0

v −ψ0m

4 (+) solution,

π2 + ψ0

v −ψ0m

4 (−) solution.(25)

Moreover, Fig. 8 shows that the equilibrium solutions for n �16 are well described by the analytical results of first-orderexpansions in α, suggesting that the complete behavior forall n can be achieved by taking into account a few additionalorders in the expansion.

Finally, we label ψm and ψv as the mountain and valleyangles of the creases. Using Eqs (18) and (19), one can showthat these angles are explicitly given by:

cos ψm = cos 2θv and 0 � ψm � π, (26)

cos ψv = cos 2θm and π � ψv � 2π. (27)

For a waterbomb structure, the behavior of the angles of thecreases can then be directly extracted from their orientations.

By determining the equilibrium configurations of thegeneralized waterbomb, we have quantified the geometricfrustration responsible for the deviation of the crease anglesfrom their rest values and shown that the system is genericallyin a prestressed state even in the absence of external loading.

B. Mechanical response of the waterbomb

The reference state of the waterbomb is set by theequilibrium solutions. To probe the mechanical response of thestructure, a protocol for applying an external loading shouldbe defined. The symmetry of the waterbomb configurationsuggests using the same setup as the one for the developablecones [32]. Figure 9 shows a schematic of the experimentin which a vertical force F is applied at the vertex of thewaterbomb base put on a cylinder of radius R < L. Dependingon the solution considered, the waterbomb is in contact withthe cylinder at either mountains or valleys creases. For the(+) solution [respectively, (−) solution], the applied forceprobes the response of the angle θv (respectively, θm) from itsequilibrium value.

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V. BRUNCK, F. LECHENAULT, A. REID, AND M. ADDA-BEDIA PHYSICAL REVIEW E 93, 033005 (2016)

Fv

m

v

m

R

x

z

(a) (+) solution

F

v

m

v

m

R

x

z

(b) (−) solution

FIG. 9. Schematic of an experiment to study the mechanicalresponse of a waterbomb depending on its equilibrium state.

Let us focus on the configuration of Fig. 9(a); the case ofFig. 9(b) follows the same steps. To calculate the stiffness ofthe structure, we note that the potential energy of a waterbombdeformed by a uniaxial force F+ in the z direction reads U =EW − ∫ z

z+vF+(z)dz. The external force F+(z) at equilibrium

state is obtained by solving the equation δU/δθv = 0:

F+(z) = − sin2 θv

R

dEW

dθv

. (28)

We look for the linear response of the waterbomb nearits equilibrium state: F+(z) k+(z − z+

v ), where k+ is thestiffness of the vertex in the experimental situation of Fig. 9(a)and is given by

k+ = dF+

dz

∣∣∣∣z=z+

v

= sin4 θ+v

R2

d2EW

dθ2v

∣∣∣∣θv=θ+

v

. (29)

Figure 10 shows the variation of the normalized stiffnessesk± with the number of creases. For both cases, we note that k±is not monotonic in the number of creases. The stiffness of thegeneralized waterbomb is very large for n = 6, decreases withthe number of creases for n > 6, and increases again linearlyfor large n, so there is an optimal waterbomb configurationfor which the stiffness is minimum. For the cases depictedin Fig. 10, the minimum of k+ (respectively, k−) is reachedfor n+

c = 20 (respectively, n−c = 22). In the general case, the

critical number of creases n±c depends on the material constants

κm/κv , ψ0m, and ψ0

v .

(a) (+) solution (b) (−) solution

FIG. 10. The stiffnesses k±, normalized by κL/R2, as function ofthe number of creases for the same material parameters as in Fig. 6.Asymptotic expansions are represented in red dashed curves. Integereven n are the only physically relevant points.

(a) (b)

(c) (d)

FIG. 11. Miura-Ori tessellation. (a) The imprinted network ofcreases. (b) Experimental realization of a three-dimensional equi-librium structure. (c) In the reference configuration, the unit cell isdefined by the red polygonal contour. (d) The folded structure of theunit cell and its crease vectors deformation field �wi .

V. MIURA-ORI

The Miura-Ori (Fig. 11) is a periodic tessellation built fromfour equal parallelograms [16] characterized by two creaselengths (L1,L2) and one plane angle α ∈ [0,π/2] [Fig. 11(c)].The vertex structure, with its three mountain and one valleyfolds [Fig. 11(d)], is similar to herringbone patterns that occurin leaves [10,11], fetal intestinal tissues [33], and biaxiallycompressed thin sheets supported on a substrate [34]. Thanksto its efficient packing-to-deployment ratio, it has been usedto design solar sails for satellites [17] and incidentally foldedmaps [18]. Because the Miura-Ori includes basic elements ofmany origami structures, it has been proposed as a frameworkfor origami metamaterials [3,4]. Although the approachesadopted in Refs. [3,4] are mainly geometrical, those workswill serve as a guideline to this section in which we quantifythe geometrical and mechanical properties of the Miura foldingpattern.

In Fig. 11(c), we define the unit cell of the tessellationwhose periphery does not include the crease network. Thischoice ensures that the elastic energy of all creases are takeninto account and that all cells equally contribute, independentlyof their orientation with respect to the unit cell. In the Cartesiancoordinates system of Fig. 11(d), the components of the creaseunit vectors �wi can be written as:

�w1 = (sin θ,0, cos θ ), �w3 = (− sin θ,0, cos θ ),(30)

�w2 = (cos φ, sin φ,0), �w4 = (cos φ, − sin φ,0),

where θ denotes the angle of �w1 with respect to the z axis andφ the angle of �w2 with respect to the x axis. The periodicityof the Miura tessellation imposes symmetry conditions to thefolded configuration. Vectors �w1 and �w3 lie in the xz planewhile �w2 and �w4 lie in the xy plane. Using Eq. (18), one finds

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ELASTIC THEORY OF ORIGAMI-BASED METAMATERIALS PHYSICAL REVIEW E 93, 033005 (2016)

that the crease angles are given by:

cos ψ1 = cos ψ3 = 1 − 2

(sin φ

sin α

)2

, (31)

cos ψ2 = cos ψ4 = 2

(cos θ

sin α

)2

− 1. (32)

The geometrical constraints Ci in Eq. (8) amount to onerelation between θ and φ given by:

CM (θ,φ) = sin θ cos φ + cos α = 0. (33)

The geometry of the Miura-Ori is then entirely described bythe four parameters (L1,L2,α,θ ), θ being the only degreeof freedom of the system. Note that Eq. (33) reproduces ina simple and condensed expression the same geometricalconditions given in Refs. [3,4]. Last, taking 0 < α < π/2enforces the conditions π/2 � θ � (π/2 + α) and (π − α) �φ � π .

The symmetry properties of the Miura tessellation alsoimpose conditions on the material constants λi and μi : Thestiffnesses and rest angles of the creases should satisfy

κ1 = κ3, ψ01 = 2π − ψ0

3 , 0 < ψ01 < π,

κ2 = κ4, ψ02 = ψ0

4 , 0 < ψ02 < π. (34)

Using Eq. (15), the elastic energy EM of the unit cell is thengiven by:

EM (θ,φ) = sin2 α

L1κ1EM (θ,φ)

= − cos ψ01 cos 2φ − β cos ψ0

2 cos 2θ

− sin ψ01 cos θ sin 2φ + β sin ψ0

2 sin 2θ sin φ,

(35)

where EM is a dimensionless elastic energy that is independentof the static angle α, and

β = κ2L2

κ1L1. (36)

The elastic energy does not depend explicitly on L2/L1

but rather on β, which embodies a mechanical-geometricalcoupling. Finally, by applying adequate rotations of the unitcell, one can check that the energy of all cells in the Miuratessellation are also given by Eq. (35).

In the following, we present results for parameters valuesβ = 1 and ψ0

1 = ψ02 = ψ0. The variation of these parameters

adds extra adjustability that is not crucial to the analysis [31].For this case, the geometric constraint (33) and the elasticenergy (35) are invariant under the transformation (θ,φ,ψ0) →(3π/2 − φ,3π/2 − θ,π − ψ0). This symmetry property al-lows us to explore the interval 0 � ψ0 � π/2 and deduce thebehavior for π/2 � ψ0 � π .

A. Equilibrium states of the Miura-Ori

The equilibrium shape of the Miura-Ori is found byminimizing the elastic energy EM given by Eq. (35) subject tothe geometric constraint (33). This can be done by introducingthe energy functional:

E∗M (θ,φ,γ ) = EM (θ,φ) − γ CM (θ,φ), (37)

FIG. 12. Contour lines of the energy EM (θ,φ) for ψ0 = π/2. Thetwo parametric curves correspond to the constraint CM (θ,φ) = 0 fortwo values of the plane angle α. The points on the curves correspondto the different equilibrium states.

where γ is a Lagrange multiplier. The extrema of Eq. (37) arefound graphically by determining the points where the contourlines of the energy EM (θ,φ) are tangent to the solution φ(θ )of Eq. (33). The fact that EM (θ,φ) is independent of the planeangle α allows us to determine the equilibrium configurationsfor all values of α using the same contour plot of EM (θ,φ).

Figure 12 shows that depending on the value of theplane angle, there is either a single equilibrium state or twometastable states. For a large range of values of α, the systemhas one equilibrium configuration given by the minimum ofEM along the parametric curve CM = 0. However for valuesof α close to π/2, one finds two possible stable configurationsseparated by an unstable solution for which EM is maximumalong the parametric curve CM = 0. This behavior is a genericsignature of a phase transition controlled by the parameter α.As will be shown below, the order of the transition dependson the rest angle of the creases ψ0. Figure 13 shows the threedimensional shapes of the Miura folding corresponding to theequilibrium solutions of the cases depicted in Fig. 12. Whilethe monostable state [Fig. 13(a)] reproduces a typical Miura-Ori configuration [3,4], the corresponding shape of the twometastable states differ considerably: The first configurationis closely packed [Fig. 13(b)] and the second one is widelydeployed [Fig. 13(c)].

For a systematic study as function of the rest angle ψ0 andthe static angle α, we perform numerically the minimizationof E∗

M given by Eq. (37) and determine the equilibriumangles θs and φs . The equilibrium crease angles ψ1s and ψ2s

are then calculated a posteriori using Eqs. (31) and (32).Figures 14(a)–14(d) display the variation of θs and ψ1s asfunction of α for two different rest angles ψ0. In addition tothe main continuous branch of solutions, a second equilibriumstate develops when the static angle approaches π/2. Thetransition from a monostable to two metastable states isgenerically of first order. However, the case of a rest angleψ0 = π/2 is peculiar: The two branches merge at αc 0.446, undergoing a second-order transition characterized by apitchfork bifurcation. Moreover, one notes that for ψ0 < π/2(respectively, ψ0 > π/2) the continuous branch of solutions

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V. BRUNCK, F. LECHENAULT, A. REID, AND M. ADDA-BEDIA PHYSICAL REVIEW E 93, 033005 (2016)

(a) α = π/5

(b) α = 0.48π (c) α = 0.45π

FIG. 13. Equilibrium configurations of the Miura-Ori for thesame parameters as in Fig. 12 and L1 = L2. (a) For α = π/5, thereis one equilibrium state. [(b) and (c)] For α = 0.48π , the system hastwo metastable states with one configuration that is approximatelyflat-folded and the second one that is widely deployed.

θs(α) correspond to the folded (respectively, deployed) state,while the second equilibrium solution that occurs for α >

αc(ψ0) corresponds to the deployed (respectively, folded) stateof the Miura-ori.

(a) ψ0 = π/4 (b) ψ0 = π/4

(c) ψ0 = π/2 (d) ψ0 = π/2

FIG. 14. Evolution of the equilibrium angles θs [(a) and (c)] andtheir corresponding crease angles ψ1s [(b) and (d)] as function of thestatic angle α for two values of ψ0. Red dashed curves correspond toasymptotic analytical solutions for α → π/2 (see Appendix). Notethat insets in (a) and (b) correspond to the second equilibrium statethat exists for α > αc(ψ0).

0. 0.2 0.4 0.6 0.8 1.0.44

0.46

0.48

0.5

0/

/

METASTABLE

MONO-STABLE

MONO-STABLE

FIG. 15. Phase diagram (α,ψ0) of the different equilibriumconfigurations of Miura folding for material parameters satisfyingψ0

1 = ψ02 = ψ0 and β = 1.

The phase diagram of the Miura-Ori equilibrium statesis determined by computing the threshold αc(ψ0) abovewhich the system becomes bistable. Figure 15 shows thatthe domain of existence of bistable solutions is alwaysnarrow and corresponds to quasirectangular building-blockparallelograms. The phase diagram provides a quantitative toolfor designing on request monostable or bistable structures andrationalizes the observations that optimal Miura-Ori reversibledeployable structures, such as solar sails or folded maps,should be designed with a plane angle α close to π/2 [17,18].An experimental realization of a bistable state and images ofa Miura-Ori folded map are shown in Fig. 16.

Finally, to study the relative stability of the folded anddeployed states, the energy landscape EM along the curve

FIG. 16. [(a) and (b)] Experimental realization of a bistableequilibrium configuration of Miura-Ori with a plane angle α = 85◦.The material used is a printable transparency film. [(c) and (d)] Thebistable equilibrium states of a Miura-Ori folded map with a planeangle α ≈ 88◦ (from Miura-ori Lab).

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ELASTIC THEORY OF ORIGAMI-BASED METAMATERIALS PHYSICAL REVIEW E 93, 033005 (2016)

0.6 0.7 0.8 0.9 1.0�1.5

�1

�0.5

0

FIG. 17. The energy landscape of a Miura-ori EM (θ,φ(θ )) withφ(θ ) given by the geometric constraint (33). The different curvescorrespond to α = 0.49π and to rest angles ψ0 = 0.4π (red), π/2(black), and 0.6π (blue).

satisfying the geometric constraint (33) is computed forvalues of static and rest angles allowing for two equilibriumstates. Figure 17 shows that for ψ0 < π/2 (respectively,ψ0 > π/2), the global energy minimum corresponds to thefolded (respectively, deployed) configuration. For ψ0 �= π/2,the second equilibrium state that appears for α > αc(ψ0) isa local minimum. For ψ0 = π/2, the two equilibrium statesare degenerate and the stability of the folded and deployedconfigurations is thus maximized. Creases in real origamistructures exhibit viscous response and relaxation [21] andcould smear out the secondary minima when deployment orfolding is attempted at finite speed due to transient stresses. Insuch situations, the corresponding metastable configurationswould appear unstable and difficult to reach. However, theintrinsic bistability of Miura-Ori can be enhanced by fine-tuning the crease stiffnesses and rest angles.

B. Mechanical response of the Miura-Ori

To study the mechanical response of the Miura-Ori, oneshould consider the equilibrium solutions as reference statesand define a protocol for applying an external loading. Herewe choose to apply equally on each vertex of the tessellationa force Fz along the z axis [see Fig. 11(d)]. The external forceFz(z) at equilibrium state is given by:

Fz(z) = dEM

dz= κ1

sin2 α sin θ

dEM

dθ, (38)

where dz = L1 sin θdθ . We look for the linear response of thesystem near its equilibrium state: Fz(z) kz(z − zs), where kz

is the stiffness of the Miura fold in the z direction given by:

kz = κ1

L1 sin2 α sin2 θs

d2EM

d2θ

∣∣∣∣θ=θs

. (39)

Note that the stiffnesses kx and ky can be determinedfollowing similar probing procedures [4]. Figure 18(a) showsthe stiffness of the different equilibrium configurations as afunction of the static angle α. For the main branch, kz isnot monotonic and there exists an optimal α for which thestiffness is minimal. In the bistable phase, we find that thestiffness of the deployed state is three orders of magnitude

0. 0.2 0.4 0.6 0.8 1.0

2004006008001000

0�

k z

0. 0.1 0.2 0.3 0.4 0.50

2004006008001000

�

k z

(a) ψ0 = π/4 (b) α = π/5

FIG. 18. The stiffness kz, normalized by κ1/L1 for two parametercases: (a) kz(α,ψ0 = π/4,β = 1). (b) kz(α = π/5,ψ0,β = 1). In (a)the Inset corresponds to the stiffness of the second stable branch andthe red dashed curves to asymptotic expansions for α → π/2.

softer than the stiffness of its corresponding folded state.This property is indeed suitable for deployable structures.In addition, Fig. 18(b) shows that for fixed static angle α

in the monostable phase, the stiffness kz is a monotonicallydecreasing function of ψ0.

We also study the coupling of deformations of the equilib-rium solutions. Following Refs. [3,4], we define the linearizedin-plane Poisson’s ratio ν as the ratio between the transversestrain (in the y direction) and the axial strain (in the x direction)of the Miura fold. This yields:

ν = 1 − 1

sin2 α sin2(ψ1/2). (40)

Figures 19 shows the expected result that ν is alwaysnegative [3,4], so the Miura-Ori is an auxetic material withrespect to deformations in the xy plane. The general behavioralso qualitatively follows the results of Refs. [3,4]. In addition,for α π/2, we show that the Poisson’s ratio of the deployedsolution scales as (π/2 − α)2 while it tends to a finite constantfor the folded state [see Appendix and Fig. 19(a)].

VI. DISCUSSION

We have developed an elastic theory of rigid origamistructures in which deformations are mediated by a creasenetwork that acts as a series of hinges connecting infinitelyrigid panels. Starting from a vectorial description of a singlecrease, we introduce a covariant elastic energy that naturallyembodies a nonlinear behavior without adding additionalmaterial constants besides crease stiffness and rest angle. Fora single vertex with arbitrary number of creases coming fromit, we performed simple geometric manipulations expressingrigid-face conditions to write both the elastic energy of thevertex and the geometric constraints in terms of the crease

0. 0.1 0.2 0.3 0.4 0.5�100�80�60�40�200

�

(a) ψ0 = π/4

0. 0.2 0.4 0.6 0.8 1.�100

�50

0

0�

(b) α = π/5

FIG. 19. Poisson’s ratio ν for the same two parameter cases as inFig. 18.

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V. BRUNCK, F. LECHENAULT, A. REID, AND M. ADDA-BEDIA PHYSICAL REVIEW E 93, 033005 (2016)

vectors only. This result allows us to define the vector fieldcarried by the crease network as the appropriate field forquantifying elastic deformations of origami structures withrespect to the planar reference configuration.

We then applied this model to the generalized waterbomband to Miura’s periodic tessellation. These two cases werechosen because of the single degree of freedom they exhibit,making a systematic analytical analysis possible. For thewaterbomb, we show a generic bistabilty of this system forany number of creases. For the Miura folding, we uncovera phase transition from monostable to bistable states thatexplains the efficient deployability of this structure for a givenrange of geometrical and mechanical parameters. Our studyprovides a quantitative and mechanical basis for the propertyof large volume ratio between the deployed and folded statesof Miura-Ori designs used for map folding and solar sails.Moreover, we show that geometric frustration forces us todefine the equilibrium configuration as a reference state forthe study of mechanical response of folded structures. Theanalysis of the waterbomb and Miura-Ori cases demonstratesthe power of using the vectors supported by the crease networkas a parametrization of the elastic deformation.

One could argue that our results, especially the bistablebehavior of Miura-ori, depend on the specific modeling of thecrease energy. Effectively, Eqs. (2) and (4) can be regarded asthe first-order terms in a Fourier expansion of the experimen-tally established quadratic form (5) [22]. However, a carefulinspection of Fig. 15 shows that the extent of the bistable regionin the (ψ0,α) parameter space is maximum for ψ0 = π/2, arest angle for which the difference between the covariant (2)and quadratic (5) forms of the crease energy is minimum. Ourresults show indeed that Miura-Ori fourfold vertices share thesame behavior as general four-degree vertices which are knownto be generically multibranched and at least bistable [5].

Until now, we considered rigid origami tessellations thatcan be reduced to a unit cell described by a single n-degreevertex. However, our approach can be extended to arbitrarynetworks for which the deformation of the crease vector fieldis a natural description. More precisely, consider a generalorigami tessellation built on an imprinted two-dimensionalundirected planar graph G of size (number of vertices) Nv . Theassociated network is defined by an adjacency matrix aij [35],the degree ki � 4 of each vertex [5], and the number of facesNf . For each edge (i,j ) connecting two adjacent vertices,we define the symmetric matrix elements (Lij ,λij ,μij ) as thelength and material constants of the corresponding crease. Theangles α±

ij between the edge (i,j ) and its two adjacent edgesat the vertex i are also prescribed. Finally, we introduce thevector deformation field of the network �wij running out fromvertex i to its adjacent one j . This definition of the vectordeformation field allows us to decouple the elastic energies

of the vertices i and j . The condition �wij = − �wji should beimposed as a constraint in the energy functional. However, thiscondition is in general not sufficient to reconstruct the creasenetwork. One must also impose Nf additional constraints thatensure a closed path for any oriented loop along the creasenetwork. These conditions, noted for simplicity Fi = 0, aresatisfied when the sum of the adjacent fold vectors Lij �wij

around each face vanishes. The Lagrangian E∗G of the crease

network in its folded state can then be written as:

E∗G = 1

2

Nv∑i=1

E∗i ({aij �wij ; j = 1, . . . ,N})

+Nv∑i=1

Nv∑j=1

ρij aij | �wij + �wji |2 +Nf∑i=1

�iFi , (41)

where the elastic energy E∗i of each vertex is given by Eq. (17),

ρij and �i are Lagrange multipliers, and the coefficient 1/2takes into account the double counting of each edge in thesum.

Therefore, the elastic energy of rigid origami can beformally written for any arbitrary crease network. This makesthis approach relevant for studying complex structures, be itperiodic tessellations built upon unit cells of more than onevertex such as Resch patterns [23], or disordered networkssuch as crumpled paper [24]. Moreover, Eq. (41) is suitablefor performing local mechanical perturbations that allow forstudying vibrational modes and elastic interactions with andbetween topological defects [29].

Finally, this formalism can also be extended to the situationin which the condition of infinitely rigid faces is relaxed.Indeed, when elastic deformations of the faces are taken intoaccount, the shape of the crease is generally curved and must bedetermined as the solution of a minimization problem [25,36].For the case of a single curved crease joining two flexiblepanels, the total energy of this structure is the sum of elasticenergies of the faces, induced by bending deformations, andthe crease elastic energy. The energy E in Eq. (2) is localalong the crease and thus can be generalized to creases witharbitrary shapes. Figure 1 shows that the vector fields ( �w,�u)and ( �w,�v) are the local tangent and normal unit vectors of theFrenet-Serret frames of each panel along the crease. Therefore,the covariant formulation given by Eq. (2) allows us to expressthe total elastic energy in terms of the same deformation fieldand thus to perform energy minimization properly.

ACKNOWLEDGMENTS

We thank S. Waitukaitis for discussions on origami model-ing. This research was supported by the ANR-14-CE07-0031METAMAT.

APPENDIX: ASYMPTOTIC EXPANSIONS

1. Generalised waterbomb

In the following, we look for equilibrium solutions of the waterbomb structure in the limit of large number of creases (α � 1).Let us write the angles θm and θv as:

θm = θ∞ + f1α + O(α2), (A1)

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ELASTIC THEORY OF ORIGAMI-BASED METAMATERIALS PHYSICAL REVIEW E 93, 033005 (2016)

θv = θ∞ + g1α + O(α2). (A2)

Using these expansions, the perturbation of the geometric constraint Eq. (22) yields12 (cos2 θ∞ − (f1 − g1)2)α2 + O(α3) = 0. (A3)

This gives:

f1 = g1 ± cos θ∞. (A4)

These two cases correspond to the (±) solutions. Here we detail only the (+) solution and give the final results of the (−) solution.For the (+) solution, one has θ∞ ∈ [0, π

2 ] and the condition θm < θv gives:

f1 = g1 − cos θ∞. (A5)

We are then left with two unknowns, θ∞ and g1. For the case κm = κv , the series expansion of the elastic energy Eq. (20) yields:

EW (θ∞,g1) = −(cos ψ0

m + cos ψ0v

)α + 2 sin θ∞

[sin

(θ∞ − ψ0

m

) + sin(θ∞ + ψ0

v

)]α2

+ 2[g1 sin

(2θ∞ − ψ0

m

) + (g1 − cos θ∞

)sin

(2θ∞ + ψ0

v

)]α3 + O(α4). (A6)

Energy minimization is carried out by differentiating with respect to θ∞ and g1. This gives:

∂EW

∂g1= 0 = 2α3

[sin

(2θ∞ − ψ0

m

) + sin(2θ∞ + ψ0

v

)] + O(α4), (A7)

∂EW

∂θ∞= 0 = 2

[sin

(2θ∞ − ψ0

m

) + sin(2θ∞ + ψ0

v

)]α2 + 2

[2g1 cos

(2θ∞ − ψ0

m

) + 2(g1 − cos θ∞

)cos

(2θ∞ + ψ0

v

)+ sin θ∞ sin

(2θ∞ + ψ0

v

)]α3 + O(α4). (A8)

Then, at this order in the perturbation one is left with two equations:

sin(2θ∞ − ψ0

m

) + sin(2θ∞ + ψ0

v

) = 0, (A9)

2g1 cos(2θ∞ − ψ0

m

) + 2(g1 − cos θ∞

)cos

(2θ∞ + ψ0

v

) + sin θ∞ sin(2θ∞ + ψ0

v

) = 0. (A10)

Using the definitions ψ0m ∈ [0,π ] and ψ0

v ∈ [π,2π ], one can show that the solutions for θ∞ ∈ [0, π2 ] and g1 are given by:

θ∞ = π

2+ ψ0

m − ψ0v

4, (A11)

g+1 = 1

8 cos ψ0m+ψ0

v

2

(sin

ψ0m + 3ψ0

v

4− 3 sin

3ψ0m + ψ0

v

4

). (A12)

Therefore the asymptotic expansions at first order in α of the equilibrium angles follow:

θ+m = π

2+ ψ0

m − ψ0v

4+ 1

8 cos ψ0m+ψ0

v

2

(sin

3ψ0m + ψ0

v

4− 3 sin

ψ0m + 3ψ0

v

4

)α + O(α2), (A13)

θ+v = π

2+ ψ0

m − ψ0v

4+ 1

8 cos ψ0m+ψ0

v

2

(sin

ψ0m + 3ψ0

v

4− 3 sin

3ψ0m + ψ0

v

4

)α + O(α2). (A14)

Using these results, one can perform the series expansion of the stiffness. Equation (29) can be rewritten as:

k+ = sin4 θv

[∂2EW

∂θ2v

+ 2dθm

dθv

∂EW

∂θm∂θv

+(

dθm

dθv

)2∂2EW

∂θ2m

]∣∣∣∣θv=θ+

v

, (A15)

where k+ is scaled by κL/R2. After straightforward computations, one finds

k+ = − cos4 ψ0m − ψ0

v

4cos

ψ0m + ψ0

v

2

[7 + cos

(ψ0

m − ψ0v

)]π

α+ π

32cos3 ψ0

m − ψ0v

4

×[

55 cos ψ0m + 65 cos ψ0

v − 3 cos(ψ0

m − 2ψ0v

) + 11 cos(2ψ0

m − ψ0v

) − 48[7 + cos

(ψ0

m − ψ0v

)]cos

ψ0m + ψ0

v

2

]+ O(α).

(A16)

By carrying out similar computations for the (−) solution, one obtains:

θ−m = π

2+ ψ0

v − ψ0m

4+ 1

8 cos ψ0m+ψ0

v

2

(sin

3ψ0m + ψ0

v

4− 3 sin

ψ0m + 3ψ0

v

4

)α + O(α2), (A17)

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V. BRUNCK, F. LECHENAULT, A. REID, AND M. ADDA-BEDIA PHYSICAL REVIEW E 93, 033005 (2016)

θ−v = π

2+ ψ0

v − ψ0m

4+ 1

8 cos ψ0m+ψ0

v

2

(sin

ψ0m + 3ψ0

v

4− 3 sin

3ψ0m + ψ0

v

4

)α + O(α2), (A18)

and

k− = − cos4

(ψ0

m − ψ0v

4

)cos

ψ0m + ψ0

v

2

[7 + cos

(ψ0

m − ψ0v

)]π

α+ π

32cos3 ψ0

m − ψ0v

4

{55 cos ψ0

v + 65 cos ψ0m

− 3 cos(ψ0

v − 2ψ0m

) + 11 cos(2ψ0v − ψ0

m) − 48[7 + cos

(ψ0

m − ψ0v

)]cos

ψ0m + ψ0

v

2

}+ O(α). (A19)

2. Miura-Ori

In the following, we look for equilibrium solutions of the Miura-Ori tesselation when the static angle α is close to π/2. Wesolve this problem in the general case for any given material parameters β, ψ0

1 , and ψ02 . Define α = π/2 − ε, with 0 � ε � 1

and:

θ = θ0 + f1ε + O(ε2), (A20)

φ = φ0 + g1ε + O(ε2). (A21)

The series expansion of Eq. (33) to first order in ε yields:

CM (θ,φ) = sin θ0 cos φ0 + (1 + f1 cos θ0 cos φ0 − g1 sin θ0 sin φ0)ε + O(ε2). (A22)

The geometrical constraint CM = 0 gives:

sin θ0 cos φ0 = 0, (A23)

g1 sin θ0 sin φ0 − f1 cos θ0 cos φ0 = 1. (A24)

Equation (A23) admits two solutions: either θ0 = π or φ0 = π/2. Here we detail only the case θ0 = π and give the final resultsfor the case φ0 = π/2.

Using the solution θ0 = π , Eq. (A24) reduces to:

f1 = 1

cos φ0. (A25)

We are then left with two unknowns, φ0 and g1. The series expansion of the elastic energy as given by Eq. (35) gives:

EM (φ0,g1) = −[cos

(2φ0 + ψ0

1

) + β sin ψ02

] + 2[g1 sin

(2φ0 + ψ0

1

) + β sin ψ02 tan φ0

]ε + O(ε2). (A26)

Energy minimization is carried out by differentiating with respect to φ0 and g1. This gives:

∂EM

∂φ0= 0 = 2 sin

(2φ0 + ψ0

1

) + 2

[2g1 cos

(2φ0 + ψ0

1

) + βsin ψ0

2

cos2 φ0

]ε + O(ε2), (A27)

∂EM

∂g1= 0 = 2 sin

(2φ0 + ψ0

1

)ε + O(ε2). (A28)

Then, at this order in the perturbation, one is left with two equations:

sin(2φ0 + ψ0

1

) = 0, (A29)

2g1 cos(2φ0 + ψ0

1

) + βsin ψ0

2

cos2 φ0= 0. (A30)

Using the definitions φ0 ∈ [π2 ,π ] and ψ1 ∈ [0,π ], one can show that the solutions for φ0 and g1 are given by:

φ0 = π − ψ01

2, (A31)

g1 = − β sin ψ02

cos ψ01 + 1

. (A32)

Therefore the asymptotic expansions to first order in ε of the equilibrium angles follow as:

θs = π − 1

cos(ψ0

1 /2)ε + O(ε2), (A33)

φs = π − ψ01

2− β sin ψ0

2

cos ψ01 + 1

ε + O(ε2). (A34)

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Using these results, one can derive from Eqs. (31) and (32) the series expansion of the equilibrium crease angles ψ1s and ψ2s :

ψ1s = ψ01 + 2β sin ψ0

2

cos ψ01 + 1

ε + O(ε2), (A35)

ψ2s = 2ε tanψ0

1

2+ O(ε2), (A36)

and the series expansion of the Poisson’s ratio as given by Eq. (40) reads

ν = − 1

tan2(ψ0

1

/2) + 8β sin ψ0

2

2 sin ψ01 − sin 2ψ0

1

ε + O(ε2). (A37)

Finally, the series expansion of Eq. (39) that defines the stiffness kz yields

kz =(1 + cos ψ0

1

)3

4(1 − cos ψ0

1

)[1 + 3√

2+ 3(3 −

√2) cos ψ0

1 − 3√2

(√

2 − 1) cos 2ψ01

]ε−4 + O(ε−3), (A38)

where kz is is scaled by κ1/L1.By carrying out similar computations for the solution φ0 = π/2, one obtains:

θs = π − ψ02

2+ sin ψ0

1

β(1 − cos ψ0

2

)ε + O(ε2), φs = π

2+ 1

sin(ψ0

2

/2)ε + O(ε2),

ψ1s = π − 2

tan(ψ0

1

/2)ε + O(ε2), ψ2s = ψ0

2 − 2 sin ψ01

β(1 − cos ψ0

2

)ε + O(ε2),

ν = −2

1 − cos ψ02

ε2 + O(ε4), kz = 8β

1 − cos ψ02

+ O(ε2). (A39)

Note that the two equilibrium solutions exist for all values of 0 � ψ01 < π and 0 < ψ0

2 � π . For ψ01 = π (respectively, ψ0

2 = 0),the only solution that persists is the one that corresponds to the branch φ0 = π/2 (respectively, θ0 = π ).

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