Geometric Mechanics of Origami Patterns Exhibiting Poisson’s Ratio Switchby Breaking Mountain and Valley Assignment

Phanisri P. Pratapa,* Ke Liu,* and Glaucio H. Paulino†

School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA

(Received 14 September 2018; published 19 April 2019)

Exploring the configurational space of specific origami patterns [e.g., Miura-ori (flat surface withparallelogram crease patterns), eggbox] has led to notable advances in science and technology. To augmentthe origami design space, we present a pattern, named “Morph,” which combines the features of its parentpatterns. We introduce a four-vertex origami cell that morphs continuously between a Miura mode and aneggbox mode, forming an homotopy class of configurations. This is achieved by changing the mountainand valley assignment of one of the creases, leading to a smooth switch through a wide range of negativeand positive Poisson’s ratios. We present elegant analytical expressions of Poisson’s ratios for both in-planestretching and out-of-plane bending and find that they are equal in magnitude and opposite in sign. Further,we show that by combining compatible unit cells in each of the aforementioned modes through kinematicbifurcation, we can create hybrid origami patterns that display unique properties, such as topological modelocking and tunable switching of Poisson’s ratio.

DOI: 10.1103/PhysRevLett.122.155501

Origami-inspired geometries have been used to designmetamaterials with unusual properties [1–7]. The aestheti-cally pleasant patterns and shapes typically start from 2Dsheets to construct 3D structures according to mountain andvalley assignments encoded in the crease patterns. In thisLetter, we present a new periodic pattern, named Morph,with a nondevelopable degree-4 unit cell that allows acertain crease to switch its mountain and valley assignment,leading to properties such as an arbitrarily tunablePoisson’s ratio that spans from positive to negative andtopological mode locking.Owing to their special geometries, origami metamate-

rials usually display interesting behavior [8–11]. Forinstance, the Miura-ori exhibits a negative Poisson’s ratiounder in-plane deformations [9], while the standard eggboxpattern has a positive Poisson’s ratio [12,13]. In compari-son, our proposed pattern morphs continuously between aMiura mode and an eggbox mode (see Fig. 1), thusbehaving as a single material possessing both positiveand negative Poisson’s ratio. Poisson’s ratio switching is anenticing phenomenon that has only been found recently forselected mechanical metamaterial designs, including nano-plates [14], reentrant origami tube assemblages [15],bistable auxetics [16], kirigami structures [17], and softnetworks [18]. Compared to other designs, the Morphexcels on having a wider tunable range of Poisson’s ratio,theoretically from negative infinity to positive infinity. Inaddition, the Morph unit cells can be assembled to form 2Dtessellations in which the unit cell can stay either in theMiura or eggbox mode, which allows the formation of

hybrid patterns—achieved by harnessing kinematicbifurcation.In their most general form, the panel angles α and β of

the Morph pattern are two independent geometric param-eters (see Fig. 1), thereby enriching the origami designspace, unlike the standard cases such as eggbox (β ¼ α) orMiura-ori (β ¼ π − α) whose vertex geometry is dictatedby just a single parameter α. Additionally, for α ≠ β, thedegree-4 nondevelopability feature that the Morph shareswith the standard eggbox makes it a generalization of thebasic pattern. Theoretically, Poisson’s ratio of the Morphsweeps the whole spectrum of real numbers as it morphsfrom one flat-folded state to the eggbox mode, to the Miuramode, and to another flat-folded state, as shown in Fig. 1.The red crease in Fig. 1 changes its mountain and valleyassignment as it transitions from eggbox mode to the Miuramode, which is made possible owing to the fact that theangle β is smaller than the angle α of the other two panels.By contrast, the standard eggbox or Miura-ori patterns donot allow any crease to switch its mountain and valleyassignment.To parametrize the rigid origami behavior of the

Morph unit cell, we define angles ϕ, ψ as the anglesbetween opposing crease lines and denote the dihedralangles between the panels as γ1, γ2, γ3, γ4, as shown inFig. 2(a). The unit cell has a degree-4 vertex, and thus it is asingle degree of freedom system. The dihedral angles arerelated to one another and to ϕ, ψ . We derive that γ2 ¼ γ4,as 0 ≤ γ2, γ4 ≤ π, indicating the existence of a plane ofsymmetry passing through the vertices denoted by O4, O5,

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0031-9007=19=122(15)=155501(6) 155501-1 © 2019 American Physical Society

andO6. While 0 ≤ γ1 < π, the ability of the creaseO5O6 toswitch between mountain and valley allows γ3 to vary from0 to 2π. In the flat-folded state I, ϕ ¼ ϕmax ¼ αþ β andγ3 ¼ 0. For 0 < γ3 < π, the unit cell is in eggbox mode andO5O6 is a mountain crease. As γ3 passes through π, O5O6

transitions from a mountain to a valley crease andthe panels on either side of O5O6 become coplanar. Inthe transition state, angle ψ also reaches its maximumψmax ¼ 2β. For π < γ3 < 2π, the unit cell is in Miura modeand O5O6 is a valley crease. Finally, as γ3 → 2π, the unitcell approaches the flat-folded II state with ϕ ¼ ϕmin ¼α − β. Let us define two intermediate variables

ξ ¼ cos β − cos α cosϕ ¼ sin α sinϕ cosðγ1=2Þ; ð1Þ

ζ ¼ cos α − cos β cosϕ ¼ sin β sinϕ cosðγ3=2Þ: ð2Þ

The configurational space of the Morph unit cell is thenfully described by ϕ (0 ≤ ψ ≤ 2β < π) and ψ (0 < α − β ≤ϕ ≤ αþ β < π) through the following equation:

cosψ ¼ cos 2αþ 2ξ2 csc2 ϕ; ð3Þ

which is presented for various choices of panel angles inFig. 2(b). We can observe in Fig. 2(b) that, as α → β, theMiura mode vanishes.We define Poisson’s ratio for in-plane stretching as the

tangential ratio of the orthogonal strains measured by thechange of width W and length L of a unit cell [8,9], whichare given by

W ¼ 2c sinðψ=2Þ; L ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2 − 2ab cosϕ

q: ð4Þ

To assure the bounding box of a unit cell being ortho-rhombic, which requires, for example, ðO1 O4 O7Þ⊥ðO1 O7 O9O3Þ and ðO1O2O3Þ⊥ðO1 O7O9O3Þ, the paneldimensions a and b are constrained by b ¼ aj cos α= cos βj(see Sec. I of the Supplemental Material [20]). Theanalytical expression for the in-plane Poisson’s ratio whenstretching in the L direction is

νsWL ¼ −dW=WdL=L

¼ 4c2L2

a2W2

���� cos βcos α

���� ξζ

sin4 ϕ: ð5Þ

As plotted in Fig. 2(c), it is clear that the stretchingPoisson’s ratio is negative in the Miura mode and positivein the eggbox mode, with a smooth transition near zero.

20 40 60 80 1000

20

40

60

80

100

0 20 40 60 80 1000

20

40

60

80

100

0 20 40 60 80 100-3

-2

-1

0

1

2

3

(a) (b)

(c)(d)

FIG. 2. Geometric configuration and in-plane mechanics of theMorph pattern. (a) Schematic of the unit cell with the descriptionof geometric parameters and vertices. (b),(c) The configurationspace and Poisson’s ratio in stretch, respectively, for differentchoices of α considering αþ β ¼ 100°. The solid and dashedlines represent the eggbox and Miura modes, respectively.(d) Stretching stiffness in W and L directions for α ¼ 60°,β ¼ 40°. The markers represent numerical results from origamistructural analyses using the bar-and-hinge reduced-order model[19]. We assume that a ¼ c ¼ 1.

X

YZ

Y

Z

X

Y

X Y

Z

X Y

Z

Eggbox Miura-ori

Morph(This Letter)

FIG. 1. (Top) Expanded design space of the Morph pattern(yellow shading) with standard eggbox (red line) and Miura-ori(blue line) as particular cases. (Middle) Fundamental modes ofthe Morph pattern: eggbox mode (left) and Miura mode (right).(Bottom) Configuration space showing transition of the Morphunit cell from one flat-folded state to another (see SupplementalVideo 1). The crease line shown in red morphs from a mountainfold in the eggbox mode to a valley fold in the Miura mode.

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Theoretically, νsWL approaches −∞ or þ∞ in the two flat-folded limits, thereby leading to a wide range of tunability.We note that, since W2=c2 and L2=a2 do not depend onthe length dimensions of the unit cell, Poisson’s ratio of theunit cell depends only on α, β, and ϕ, making it a purelygeometric quantity that is also independent of the lengthscale of the pattern.Accordingly, assuming that the energy of the unit cells is

composed of deformation from linear elastic rotationalhinges along the crease lines, we can derive the linear in-plane stretching stiffness of the pattern. Denoting kf as therotational spring modulus, the stored energy of the systemis given by

Us ¼kf2½aðγ1 − γ1;0Þ2 þ bðγ3 − γ3;0Þ2 þ 2cðγ2 − γ2;0Þ2�;

ð6Þ

where γ1;0, γ2;0, and γ3;0 are the neutral dihedral angles (inthe undeformed state). Expressing γ2 and γ3 in terms of γ1,the stiffnesses along the L direction is derived as

KL ¼ d2Us

dL2

����L¼L0

¼ d2Us

dγ21

�dLdϕ

dϕdγ1

�−2����γ1¼γ1;0

: ð7Þ

Similarly, we can get the stiffness along the W direction(see Sec. II of the Supplemental Material [20]). As shownin Fig. 2(d), the in-plane stiffness in the W direction(denoted by KW) is minimum at flat-folded states andreaches maximum at the transition state. Interestingly,while KL is maximal at flat-folded states, it is only closeto minimum at the transition but slightly away towards theeggbox mode.As revealed in previous research [8,9,12], a kinematically

single degree of freedom (d.o.f.) origami pattern mayexperience out-of-plane deformation, other than pure (in-plane) folding, if compliance of panels is taken intoconsideration. Accordingly, we define Poisson’s ratio inbending as the ratio of principal curvatures (νbWL ¼ −κW=κL)and find that the Morph pattern features a saddle-shapedgeometry in theMiura mode and a dome-shaped geometry inthe eggbox mode [see Figs. 3(a) and 3(b)]. It is intriguingthat the Morph pattern exhibits distinct Poisson’s ratio instretching and bending, similar to what have been found,separately, with the standard Miura-ori and the standardeggbox patterns. Here we show that, just like its two extremecases [9,12], the Morph pattern displays Poisson’s ratio withopposite sign but equal magnitude in stretching and bending.We can analytically calculate the principal bending curva-tures by allowing each panel of the origami pattern to bendalong one of its diagonals [9], under the assumption ofinfinitesimal deformation.We add infinitesimal rotations δ1, δ2, δ3, δ4 as shown in

Fig. 2(a) to provide further degrees of freedom to the

system in order to simulate bending of panels. Hence, thereare in total four extra degrees of freedom being added, yetisometric deformation is still ensured. Bending of the unitcell shall preserve the orthogonality between the two sidesof a unit cell (i.e., L and W). Thus, we enforce that thenormals of the side triangles of a unit cell after bending[e.g., ΔðO0

1 O02 O

03Þ, ΔðO0

7O08O

09Þ, ΔðO0

1O04O

07Þ,

ΔðO03 O

06O

09)] remain orthogonal to their respective side

directions, which leads to three independent constraints,

aξδ1l1

¼ bζδ2l2

; bζδ3l3

¼ aξδ4l4

;δ1l1

¼ δ4l4

; ð8Þ

where l1, l2, l3, and l4 are the lengths of the diagonalsO2O4, O2O6, O6O8, and O4O8, respectively. Thus, thebending is uniquely defined up to a single d.o.f. Theseconstraints automatically ensure that the deformed unit cellcan be periodically tessellated in the two principal direc-tions (i.e.,L andW), that is,∠ðO0

1O02O

03Þ ¼ ∠ðO0

7O08O

09Þ

and ∠ðO01O

04 O

07Þ ¼ ∠ðO0

3 O06 O

09Þ.

The curvatures in L andW directions are determined by

κL ¼ −jθ147j � jθ369j

L; ð9Þ

κW ¼ −jθ789j þ jθ123j

W; ð10Þ

where the þ or − in Eq. (9) for κL depends on whether thesystem is in the eggbox mode or the Miura mode,respectively, and θ147, θ369, θ789, and θ123 are the tilt angles

(c)

(a) (b)

(d)

FIG. 3. Out-of-plane bending of the Morph pattern. (a),(b) Bentshapes of the pattern in eggbox and Miura modes, respectively,obtained using the bar-and-hinge origami model. (c),(d) Triangu-lar face tilts creating a net angle change across length L ofthe Morph pattern in eggbox and Miura modes, respectively. Thenew coordinates of vertices O7 andO9 after bending (i.e., O0

7 andO0

9, respectively) can be calculated using the Rodrigues rotationformula [21].

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[see Figs. 3(c) and 3(d)]. The bending Poisson’s ratio isthen obtained as (see Secs. III and IV of the SupplementalMaterial [20])

νbWL ¼ −κWκL

¼ −4c2L2

a2W2

���� cos βcos α

���� ξζ

sin4 ϕ: ð11Þ

Comparing Eq. (5) with Eq. (11), we obtain the elegantresult νbWL ¼ −νsWL for the Morph pattern. The aboveexpression reduces to standard Miura-ori [8,9] and eggbox[12] cases as two particular cases for appropriate choicesof panel angles α and β (see Secs. V and VI of theSupplemental Material [20]).The aforementioned bending mode allows us to analyti-

cally derive the bending stiffness of the Morph pattern,which has similar characteristics to the in-plane stretchingstiffness (see Fig. S8 of the Supplemental Material [20]).By performing numerical simulation using the reduced-order bar-and-hinge model [19], we find that the analyticalmodel agrees well with the numerical simulations withvery small discrepancies, which further strengthens theassumption that infinitesimal rotations about panel diago-nals are sufficient to characterize first-order bendingresponse of the Morph pattern.Owing to its mode switching feature, the Morph pattern

unit cells do not have to be tessellated with uniformconfiguration. It is kinematically admissible to couplethe Morph unit cells into a hybrid pattern, such that thereare both Miura mode cells and eggbox mode cells in asingle tessellation, as demonstrated in Figs. 4(a) and 4(b).The feasibility of such a system can be understood bynoting that, in Fig. 2(b), a given ψ can correspond to theangle ϕ in either the eggbox mode or the Miura mode,which we denote as ϕe or ϕm, respectively. These angles aregiven by ϕe ¼ ϕ1 þ ϕ2 and ϕm ¼ ϕ1 − ϕ2, where ϕ1 andϕ2 are as shown in Fig. 4(c) and are given by cosϕ1 ¼cos α= cosðψ=2Þ and cosϕ2 ¼ cos β= cosðψ=2Þ (seeSec. VII of the Supplemental Material [20]).In Fig. 4(c), we show that one can smoothly deform a

homogeneous Morph pattern to a hybrid pattern using rigidorigami motion (no panel bending). By compatibility, allthe unit cells have the same ψ , i.e., ψm ¼ ψe ¼ ψ . Also,when ϕm ¼ ϕe, all the unit cells of the pattern are either inthe Miura mode or the eggbox mode, depending onwhether ϕ ¼ ϕm ¼ ϕ1 − ϕ2 or ϕ ¼ ϕe ¼ ϕ1 þ ϕ2, respec-tively. In the figure, these configurations are representedby the straight line in blue and red colors, respectively. Aswe move up the blue line, the ϕm increases and reachesthe transition point [which is ϕT ¼ cos−1ðcos α= cos βÞ]between Miura and eggbox modes. At this point, we notethat there is kinematic bifurcation in the configurationspace, which could either move all the unit cells into theeggbox mode by uniformly increasing the angle ϕ furtheracross all cells or switch some of the strips back into Miuramode and therefore generate hybrid patterns represented by

the green curve in the figure. This process is also demon-strated through animations and partly through physicaltesting in Videos 2 and 4, respectively, of the Supplemental

FIG. 4. Hybrid origami assemblages associated with the Morphpattern. (a) Alternating strips of Miura (M) and eggbox (E)modes. (b) Half pattern with strips in Miura (M) mode and theother half in eggbox (E) mode. (c) Creation of hybrid patternsfrom the Morph through kinematic bifurcation. (d) Change ofPoisson’s ratio with respect to varying number of Miura modestrips (nm) in a hybrid mode with 100 × 100 unit cells. Thenotation νsWL;h denotes Poisson’s ratio under stretch for the hybridpattern. (c),(d) We assume α ¼ 60°, β ¼ 40°. (e),(f) Mode lockingdue to extension in L direction when νsWL;h > 0. The positiveglobal Poisson’s ratio implies contraction in the W direction,resulting in decrease of ψe and ψm. The oppositely signed unitcell Poisson’s ratios of the two modes indicates that while ϕeincreases, ϕm decreases, meaning the Miura mode cells areaxially contracting, opposite to the global axial deformation. TheMiura mode cells with decreasing ϕm are locked because suchcells can no longer smoothly transition to their eggbox mode in arigid origami motion. (f) Contrasting global and local deforma-tions that occur in hybrid patterns leading to mode-lockingbehavior. (e),(f) The green lines represent the panel diagonals,whose projections provide a clean way of sketching the motions.

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Material [20]. It can be seen that, along the green curve, ϕmreduces and ϕe increases, as to be expected from Fig. 2(b),for a compatible ψ , across the two types of unit cells inthe system.Depending on the coupling mode of the hybrid pattern,

the tessellated sheet exhibits a different Poisson’s ratio, νsWL;h

[see Eq. (S98) of the Supplemental Material [20] ]. Thereexists a transition point when νsWL;h varies from positiveextremum (all unit cells in eggbox mode) to negativeextremum (all unit cells in Miura mode), which, however,does not happen when the number of Miura mode andeggbox mode strips are the same, due to unequal contribu-tions from both modes. We consider a system with100 × 100 cells and increase the number of Miura modecells (in strips) along the L direction (denoted as nm) from 0to 100 [see Fig. 4(d)]. For a given pattern, the switching ofPoisson’s ratio can be tuned to occur at different fold anglesby smoothly modifying the number of Miura mode strips inthe system (see Sec. VII B of the Supplemental Material[20]), which renders the Morph pattern reprogrammable.The hybrid patterns also exhibit interesting behavior in

bending due to the combined action from Miura andeggbox mode cells (see Video 3 of the SupplementalMaterial [20]). For example, a hybrid pattern with alter-nating Miura and eggbox mode strips bends into a domeshape [Fig. 4(a)], whereas that with a set of Miura modestrips adjacent to one another bends into a complexgeometry that has both saddle and dome shapes [Fig. 4(b)].The interplay between the contrasting Poisson’s ratios of

the eggbox and Miura mode unit cells coupled with theglobal Poisson’s ratio of the hybrid pattern leads to mode-locking behavior. The most obvious mode locking is thetensile mode locking (demonstrated in Video 1 of theSupplemental Material [20]). For certain types of hybridmodes, if we stretch the hybrid pattern along the L (axial)direction, the Miura mode cells, which normally wouldsmoothly transition into eggbox mode under stretching,would rather lock themselves in Miura mode and foldtoward flat-folded state II. Tensile mode locking happenswhen a hybrid pattern displays a positive Poisson’s ratioglobally, such that it shrinks in the lateral direction understretching. For a Miura mode unit cell, this means that itmust contract in the axial direction (as well as the lateraldirection), despite the fact that the global pattern isexpanding in the axial direction in which it is stretched,as illustrated in Fig. 4(f). Similarly, compressive modelocking happens to eggbox unit cells when a hybrid patternwith globally negative Poisson’s ratio is contracted (seeSec. VII C of the Supplemental Material [20]). We remarkthat the mode locking of a hybrid Morph pattern istopological. It locks the mountain and valley assignmentof certain unit cells, but still allows the pattern to foldsmoothly as a rigid origami to the flat-folded states. Thisis different from motion locking [22] where the panelscome into contact with each other, hindering the rigid

foldability and preventing the pattern from reaching theflat-folded state.The Morph pattern exhibits morphing characteristics by

breaking the mountain and valley assignment, which leadsto smooth switching of Poisson’s ratio across a very widerange of negative to positive values and topological modelocking as a consequence of kinematic bifurcation. Ouranalysis reveals that the Morph pattern exhibits Poisson’sratio with equal magnitude but opposite sign when subjectto in- and out-of-plane deformations. Moreover, we discusshybrid patterns that can be created by coupling Morph unitcells in distinct modes, creating a tessellation with reprog-rammable Poisson’s ratio and topological mode locking.The locking feature of the hybrid patterns can be useful increating structures with multistability [3]. We envision thathybrid patterns can also have many applications in topo-logical mechanics due their ability to transform the sym-metry of the system under in-plane deformations [23,24].

We acknowledge useful discussions with ProfessorDavid Zeb Rocklin. We thank the support from the U.S.National Science Foundation (NSF) through GrantNo. 1538830, the China Scholarship Council (CSC), andthe Raymond Allen Jones Chair at Georgia Tech.

*P. P. P. and K. L. contributed equally to this work.†Corresponding author.paulino@gatech.edu

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