Curving Origami with Mechanical Frustration
CCurving Origami with Mechanical Frustration
T. Jules , ∗ F. Lechenault , and M. Adda-Bedia Universit´e de Lyon, Ecole Normale Sup´erieure de Lyon, Universit´e Claude Bernard,CNRS, Laboratoire de Physique, F-69342 Lyon, France and Laboratoire de Physique de l’Ecole Normale Sup´erieure,ENS, PSL Research University, CNRS, Sorbonne University,Universit´e Paris Diderot, Sorbonne Paris Cit´e, 75005 Paris, France (Dated: February 9, 2021)We study the three-dimensional equilibrium shape of a shell formed by a deployed accordion-likeorigami, made from an elastic sheet decorated by a series of parallel creases crossed by a centrallongitudinal crease. Surprisingly, while the imprinted crease network does not exhibit a geodesiccurvature, the emergent structure is characterized by an effective curvature produced by the de-formed central fold. Moreover, both finite element analysis and manually made mylar origamisshow a robust empirical relation between the imprinted crease network’s dimensions and the ap-parent curvature. A detailed examination of this geometrical relation shows the existence of threetypical elastic deformations, which in turn induce three distinct types of morphogenesis. We char-acterize the corresponding kinematics of crease network deformations and determine their phasediagram. Taking advantage of the frustration caused by the competition between crease stiffnessand kinematics of crease network deformations, we provide a novel tool for designing curved origamistructures constrained by strong geometrical properties.
INTRODUCTION
Origamis are the three-dimensional structures ob-tained by folding a thin sheet following a specific im-printed pattern of creases. Through their apparent scal-ability and the infinite number of crease network com-binations, origamis offer new methods to produce me-chanical metamaterials [1–3]. Their innovation potentialis only limited to our understanding of their kinemat-ics of deformation and mechanical properties. Interest-ingly, even a “simple” mathematical model such as rigid-foldable origamis [4], where the geometry imposes heavy,kinetic constraints through infinitely stiff faces, alreadyyields exciting behavior such as non-zero Gaussian cur-vature, saddle-shaped configuration, and auxetic defor-mation [5–7]. This construction leads to inverse origamidesign where a programmed tessellation pattern producesa target deployed shape [8–10].However, both initial and target structures are onlya subset of a larger configurational space, taking intoaccount the elastic deformation of the faces and thecreases’ stiffness. Both properties are crucial in under-standing folded elastic sheets’ behavior. Indeed, the free-dom granted by the bending and stretching of the facesenables continuous deformations between stable config-urations in multistable structures, otherwise forbiddenwith rigid faces constraints [11, 12]. Still, these consider-ations rely on fine-tuning the pattern of creases to obtain,at rest, a predefined deployed shape with flat faces.In contrast, even crumpled paper produces a complexthree-dimensional structure through a random pattern ofcreases [13, 14] and the local competition between differ-ent modes of deformation [15]. The internal frustrationgenerated by folding imposes local mechanical equilib-rium that shapes the origami. This behavior is evident for a simple vertex formed by intersecting creases [16, 17]or in the saddle-like shape obtained when the imprintedcrease displays geodesic curvature [18–20].In this letter, we explore the possibility of produc-ing curvature, an essential element in shaping origamistructures, without starting from curved-crease sculp-ture [18, 19] nor intricate crease network as in Reschpatterns [21, 22]. To illustrate our proposal, we study afolding pattern, coined the curved-accordion, and exploitthe frustration-induced deployment to display an appar-ent curvature along a single long crease. We describe thetypical deformations of the resulting three-dimensionalstructure. We show that both its local and global shapeare controlled by the crease network’s geometry and elas-ticity. This intriguing result confirms that, despite theimportance of the elastic response of the underlying ma-terial, the crease network structure acts as a backbonefor the origami, similarly to elastic gridshells [23].
MATERIALS AND METHODSFolding pattern and experiments
The folding pattern for the curved-accordion origami isdepicted in Fig. 1(a). A rectangular sheet of thickness t isdecorated by a single central mountain crease crossed by N equally spaced perpendicular transverse creases. Thelatter have alternating directions that switch when cross-ing the central crease. The length l and width w of everyresulting rectangular face are identical. In other words,the proposed pattern corresponds to the extreme caseof Miura folding [5] with a single straight all-mountaincrease and right angles between folds.To study the relation between the shape of the origami a r X i v : . [ c ond - m a t . s o f t ] F e b FIG. 1. Schematics of the curved accordion origami. (a) Theimprinted crease network. Each face is a rectangle of dimen-sions l × w . The red (resp. blue) dashed (resp. dash-dotted)lines represent the mountain (resp. valley) folds. (b) Result-ing structure after folding and deployment. δθ i is the anglebetween successive segments of the central crease intersectingat the i th vertex. R is the apparent radius of curvature alongthe central crease. (c) Curved accordion origami made fromA4–paper sheet with N = 27. and the pattern of folds, we manually folded curved-accordions from 100 µ m thick polyethylene terephthalate(mylar) flat sheets with a various number of creases (9to 22), length l (10 mm to 140 mm), and width w (9 mmto 20 mm). Contrary to paper sheets, where the dam-age produced during the folding process yields unknowncrease properties [24], mylar sheets generate folds witha reproducible stiffness based on the elastic propertiesof the material [25]. After initial folding, we deployedeach origami by pulling on the faces manually to set therest angle Ψ at approximately 90 ° . As we do not havea precise method to measure it, each rest angle’s exactvalue is unknown. We obtained a three-dimensional shellstructure at equilibrium (see Fig. 1(c)) that is stable andrigid in the direction perpendicular to the central crease,similar to the rigidity displayed by an elastic sphericalshell [26]. To investigate its shape, we let the origami layon the free edges and took photos of the side such thatthe central crease is in the camera’s focal plane. Alter-nately, we used structured light scanning to analyze thethree-dimensional structures (See Supplemntary Materi-als). Numerical simulations
In parallel, we simulated the folding of an initially flatsheet following the pattern described in Fig. 1(a) usingfinite element methods (FEM) with the software COM-SOL. To do so, we followed the phase-field blueprint usedin [17]. We exploited the software’s multi-physics capa-
FIG. 2. Evolution of normalized radius
R/l for both simula-tions (full circles) and experiments (cross) with respect to thefaces’ aspect ratio. Each color corresponds to a given numberof transverse creases N . For the experiments, we measured R with N = 9, 15, 19, and 22. bilities to reproduce the change of reference configurationduring folding with local thermal expansions. This pro-tocol was experimentally tested to produce self-foldingsheets [27]. We chose unique elastic properties for thewhole sheet, with a Young modulus E = 4 GPa and aPoisson ratio ν = 0 .
4, based on previous work with Mylarsheets [25] and tabulated values, while setting a thicknessidentical to the experiments. The creases characteristicwidth S = 3 t is chosen as a typical size for a manu-ally folded crease [25, 28], while the heating temperatureand thermal expansion coefficient are selected to imposea unique rest angle Ψ = π/ N totake advantage of the symmetry at the crease ( N + 1) / l from 5 mm to200 mm, for w from 3 . RESULTSCurvature of the origami
For both experimental and simulated origamis, thecentral crease naturally curves, as evidenced in Fig. 1(c).In fact, the transverse creases divide the central one intomultiple straight segments delimited by the vertices. Themechanical equilibrium resulting from the competitionbetween rigid creases connected by flexible faces requiresthe i th vertex to exhibit an angle δθ i between the i thand ( i + 1)th segment [29]. The contribution of all the δθ i yields a discrete curvature κ = 1 R = P Ni =1 δθ i ( N + 1) w . (1)To recover the corresponding radius of curvature R , wemeasured the radius of the best fitting arc to the pro-jection of the central crease on a plane, as shown inFig. 1(b). For the experimental structures, the planeof projection is defined by the image. We used a singularvalue decomposition algorithm to find the closest plane tothe three-dimensional coordinates of all the mesh pointsfrom the central crease for the simulated origamis.Fig. 2 shows slightly different behavior between theexperiments and the simulations. For moderately largenumber of creases ( N ≥ R and the dimensionsof the faces R simu ∝ w / l / . (2)In contrast, the experimental data seem to follow a linearrelation R exp ∝ w , (3)which is a direct consequence of Eq. (1) when assumingthe angles δθ i do not depend on the facet size. Thispoints to a decoupling of the vertices in the exploredexperimental regime, which might not be achieved in thesimulations.Overall, the simulations overestimate the curvature ofthe experimental accordions. It showcases two signifi-cant differences between our naive experimental protocoland FEM: the precise control of every rest angle and non-elastic mechanical behavior. On the one hand, the angles δθ i highly depend on the value of the rest angle at eachvertex [17]. Indeed, we observed a non-linear increase of R with respect to the rest angle during the folding of theaccordion in the simulations (See Supplemntary Materi-als). On the other hand, the folding pattern requires astress concentration near the vertices, where the geomet-ric incompatibility is the strongest. Consequently, therest angle of the transverse crease is larger in this regionthan near the free edges. Deformation modes
Even with these discrepancies, both simulations andexperiments display three typical deformation modes. Acloser inspection of the numerical data in Fig. (2) revealsadditional structures underlying the evolution of the ra-dius R with respect to l and w . These structures areconfirmed when analyzing the dependency of R on a sin-gle geometric parameter, for instance, the faces’ length l .Fig. 3 shows the existence of three regions I , II , and III , FIG. 3. Non monotonous evolution of R with respect to theface length l for w = 14 . N = 9 and 13. The dot-ted vertical lines indicate the approximate transition betweenthree typical types of elastic deformation given by l ≈ . w and l ≈ w . The point in the grey region displays localdeformations typical of both region II and region III . which correspond to very distinct regimes of deformation,as shown in Fig. 4. The region I corresponds to a sin-gle facet deformation for which each face is separated intothree areas. A central triangular flat facet is delimited bythe central crease and two bent regions accommodatingthe stiffness of the side crease in a deformation very sim-ilar to the shape of elastic ridges [30]. Above a first crit-ical length l , some faces begin to deform in two or morefacets. This behavior points to the start of the region II , characterized by the tiling of the faces with triangu-lar facets delimited by stretching ridges. The describeddeformation is similar to the one observed in twisted rib-bons [31]. A curving of the transverse creases themselvesnear the vertices accompany the faceting. Finally, abovea second critical length l , the free edges initially paral-lel to the central crease buckle out of symmetry throughEuler-like instability that defines the boundary of the re-gion III .We drew ideas from mechanical systems with compara-ble geometry to characterize the transition between thedeformation modes with the proper parameters. First,the faceting observed in region II is reminiscent of theshape displayed by twisted ribbons [31–33], so we focusedon quantifying the local twist in the faces. To that end,we extracted from the simulation the twist η along a facenext to the middle transverse crease i = ( N + 1) / η ( s ) ∝ w δφδs , (4)where s is the distance to the vertex along the trans-verse crease (see Fig. 1) and δφ is the angle after heat-ing between two initially parallel lines at s and s + δs .The twisting is constant along the whole face for singlefacet deformation. Strikingly, we notice from Fig. 5(a)that l not only indicates the transition to faceting but FIG. 4. Different elastic deformation regimes for the curvedaccordion origami. For all figures, the color represents thelocal mean curvature in m − . Typical deformations for thesimulation (left/top) and the experiment (right/bottom) for(a) a single facet with N = 9, l = 20 mm and w = 20 mm,(b) faceting with N = 9, l = 100 mm and w = 20 mm, and(c) buckling with N = 19, l = 140 mm and w = 10 mm.The bottom image is a raw experimental image with N = 20, l = 148 . w = 10 mm. The global shape is differentfrom the simulation, however a buckling behavior is clearlyobserve with both methods. For buckled origami, 3D scanningis hard to perform due to the complex structure. also defines the characteristic size of a boundary layerfor high twist near the central crease. In this buffer re-gion, the local deformation is different from the rest ofthe face. Beyond, the faces deform as twisted ribbons un-der slight tension [31, 34] with a characteristic triangularfaceting. The range over which the central crease affectsthe faces’ deformations is of the order of w . The localinfluence of boundary conditions on the deformation ofelastic origamis agrees with observations on their localactuation [35]. FIG. 5. (a) Local twist η of a face next to the middle trans-verse crease, with N = 11, w = 25 mm, and different lengths l . s = 0 at the central crease. (b) Squared angular defor-mation (Ψ edge − Ψ ) averaged over all transverse creases for N = 13 and w = 14 . l ≈ w between faceting andbuckling. The point in the grey region display buckling onlyon one side of the central crease. Since the transverse creases remain mostly straightwhile faceting, the arc length at the free boundary is smaller than the arc of the central crease. As a result, theaccordion is compressed near the free edge and store elas-tic crease energy, as shown in Fig. 5 (b). Subsequently,for l > l , this energy becomes large, and the systemprefers to adopt a new configuration that relaxes it bybuckling similarly to compressed elastic rods. The buck-ling requires the faces and the transverse creases to bendand twist accordingly. Each side of the central crease hasa different buckling threshold due to their different num-ber of mountains and valleys folds. In our simulations, itresults in a configuration for which only one side is buck-ling at the onset of the transition. This configurationdisplays a unique structure as highlighted in Fig. 5 (b).With the three distinct modes of deformation we de-scribed, we draw in Fig. 6 a phase diagram for the mor-phologies of the origami using the data from simulationsfor multiple patterns and dimensions. Just as we ob-served with the curvature, and even though the differ-ent elastic deformations rely on the origami’s complexstructures, the transitions between deformations only de-pend on the aspect ratio of the faces, with l ≈ . w and l ≈ w . FIG. 6. Phase space of the regimes of deformations. Thelines separating the different behaviors are given by l ≈ . w (dash-dotted) and l ≈ w (dashed). DISCUSSION
Clearly, the geometry of the pattern governs the detailsof the shell-like shape of the accordion. This prominenteffect comes from the stiffness of the creases. Since everyaccordion is stable, each shape corresponds to a mini-mal global elastic energy configuration that is the sum ofcontributions from the folds and the faces. Both dependon the elastic properties of the original sheet. However,in the limit of thin folds ( l, w (cid:29) t ), the energy cost toopen the crease ( E crease ∝ Elt [25]) is high. Much morethan the energy needed to deform the faces by bending( E bending ∝ Et [36]) or, when the boundary conditionsrequire non-isometric deformations, to produce stretch-ing ridges ( E ridge ∝ El / t / [37]). Consequently, thefaces’ deformation is subordinate to the network of rigidcreases, which acts as a backbone for the structure andimposes local boundary conditions. This characteriza-tion is evocative of elastic grid shells, where regular pla-nar grids of thin elastic rods are actuated into three-dimensional shell-like structures by the loading of theirextremities along a pre-determined path [23]. As a re-sult, the accordion’s effective curvature, characterized by R , is mainly governed by the pattern of folds and thevalue of the rest angles, while only marginally affectedby the elasticity of the sheet. We verified this claim byvarying the sheet’s thickness by one order of magnitude(from 20 µ m to 200 µ m) in simulations and obtained verysimilar curvature (See Supplemntary Materials). CONCLUSION
Our study investigates a poorly developed branch ofmechanical properties for origami, the frustrated shap-ing. Akin to shapes obtained with curved creases [20, 38],we manage to reach effective curvature for a deployedorigami at equilibrium from a straightforward pattern ofcreases, a Miura folding with right angles. The inter-actions between stiff folds yield three global geometricalshapes for the deployed origami, resulting from the com-petition between various deformation modes of the facessuch as bending, stretching, or twisting.Whereas we only considered the elasticity based onthe material mechanical properties, the local nature ofthe competition shows a clear path to design by control-ling the pattern of creases and the rest angle and stiffnessof the folds. Contrary to rigid-faces origami design, thecreases’ network’s choice does not limit the origami to asingle configuration. The shape and the resulting prop-erties are finely tunable after folding not only with localbistability [2], but also through the precise changes of therest angles [24]. However, quantitative applications willrequire a more in-depth elastic and geometric modelingof the complex competition between the creases and thefaces’ resulting deformations.
ACKNOWLEDGEMENTS
T.-J. thanks Marcelo Dias and Dominic Vella for fruit-ful discussions and for providing access to COMSOL.
Supplementary Materials
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Santangelo, The shape and mechan-ics of curved-fold origami structures, EPL (EurophysicsLetters) 100 (5) (2012) 54005. doi:10.1209/0295-5075/100/54005 . urving Origami with Mechanical FrustrationSupplementary Materials
3D SCANNING AND ANALYSIS.
Structured light scanning is a technique to measure an object’s three-dimensional shape using a projected lightpattern and a camera. For our experiment, we use the DAVID 3D software coupled with a webcam Logitech C920Pro HD and a projector ACER H6517ST. After a calibration step achieved by the software on a specifically patternedright angle, we analyze the shape of the origamis under multiple orientations thanks to a rotating platform. To ensurethe origami is visible, we fold the transparent elastic sheet and then spray opaque white paint on top of the externalsurface. Then, we fuse the multiple meshes we obtain into a single one and analyze it.To get the local curvature from the mesh, we use the software MeshLab [1], and more precisely, the filter Colorizecurvature (APSS) [2, 3] and changing the MLS - Filter scale to 10 in order to reduce the noise of the scan. The otherparameters are kept to their default values.
FOLDING SIMULATIONFolding simulation
To simulate the folding of the curved accordion, we begin by setting the geometry of the pattern. To do that, westart by modelling a rectangular surface of dimensions 2 l × ( N + 1) w . Then, we delimit the pattern of creases on thesurface. Every crease is centered on the pattern and is 2 S wide, with S = 3 t , where t is the thickness of the sheet.We puncture a hole of radius √ S at every vertex to avoid a too large stress concentration. Finally, we only choosean odd number of creases to have a symmetric crease pattern and thus to simulate only half of the system. Thanksto each side of this crease’s symmetry, we increase the computing speed. {{ { FIG. 1. (a) Schematics of the folding through thermal dilatation of a crease of size 2 S . (b) Typical mesh around the crease. For the meshing, we differentiate between the faces and the creases. For the former, we choose free triangularmeshes with a size fine to extremely fine . For the latter, we choose rectangular meshes that ensure a line of nodesalong the center line of the creases while connecting the vertex of triangles from the two connected faces with astraight line. A schematics of it is shown in Fig. 1 (b). a r X i v : . [ c ond - m a t . s o f t ] F e b To incorporate the physics, we choose to model our system with the shell model from COMSOL, suited for analyzing3D structures created from thin sheets. With this model, we only need to produce a 2D surface, the thickness beingadded by the equations of the model afterward. Then, we impose a thermal dilatation coefficient α T to each creasedepending on whether it is a mountain ( α T = +1 K − ) or a valley ( α T = − − ). At this step, we also attribute tothe sheet its mechanical properties, its Young modulus E = 4 GPa, and its Poisson ratio ν = 0 . T . We also chooseto separate the heating of the central crease and the transverse creases. The heating protocol is as follow: We start byheating the central crease until the gradient reaches 0 . T . Then, we heat the transverse creases until the gradientalso reaches 0 . T . Finally, we repeat the operation by 0 . T increment until all the creases reach a gradient of ∆ T .We set three kinematic constraints for the accordion, all at the ( N + 1) / T such that a target rest angle Ψ is reached. The dilation of the top part of thecrease increases its spatial width up to L t = (1 + α T ∆ T )2 S . (1)Through simple geometrical relations, L t is linked to the angular opening of the crease at rest by π − Ψ = L h − S o t (2)= 2 α T ∆ T S o t . (3)We verified Eq. (3) by simulating a single fold for multiple thickness t and size of crease S . We found a similarrelation π − Ψ = Aα T ∆ T S o t , (4)with A = 2 .
8. To reach a rest angle of 90 deg, we set the temperature to ∆ T = 0 .
187 K.A large part of the resolution method we used is set by default by COMSOL. We use the MUMPS algorithm witha pivot threshold of 0.1 and a relative tolerance of 0.001. Two parameters we change are the maximum number ofiterations that we set to 250 to let the algorithm more time to reach the solution, and we take into account thegeometrical nonlinearities.
Effect of the rest angle
As we detail in the previous section, we increase the heating of the central crease and of all the transverse creasesalternatively during the folding process. As a result, there are multiple solutions for which every crease’s rest angleis the same. In Fig. 2, we plot the evolution of the corresponding curvature of the accordion with respect to the restangle.We observe a clear nonlinear increase of R c when we open the rest angle. As expected, R c → ∞ when Ψ → ° since it get closer and closer to the flat sheet. We also notice that the experimental data hint at an effective rest anglemuch larger than 90 ° . A part of it may come from the experimental protocol since we do not control precisely therest angles, which could be higher than 90 ° . However, the angle 145 ° expected from the simulation seems too largefor the shape we observe experimentally. So the difference of curvature between experiments and simulations has alsoanother origin. Effect of the sheet thickness and the holes at the vertices
An essential assertion in this work is that the overall shape mainly depends on the pattern of creases and not thesheet’s elasticity, as long as the thickness is negligible compared to the origami’s characteristic lengths. To confirm
FIG. 2. Evolution of normalized radius of curvature R c /l with respect to the rest angle of every crease Ψ , for an accordion ofparameters N = 13, l = 150 mm and w = 14 . our claim, we simulate folding the curved accordion for identical patterns but multiple thicknesses of sheets, from 20to 200 µ m, one order of magnitude, for N = 13 and w = 14 . µ m in Fig. 3, we notice that the thickness only has a marginal impact on the global curvature of the accordion. FIG. 3. (a) Evolution of normalized radius R c /l with respect to the aspect ratio l/w . (b) Evolution of normalized radius R c /l with respect to the aspect ratio l/w with N = 13 and w = 14 . µ mthick sheet without hole at the vertices and the color of the circle correspond to different thickness from 20 to 200 µ m. A final element to analyze is the impact of the holes at the vertices used in our simulations to minimize the solvingissues caused by the stress concentration at the vertices. Since they represent a sizeable structural difference betweenexperiments and simulations, they might be responsible for the inadequacy between the curvature of both types ofdata. However, we see in Fig. 3 that the holes only have a minimal impact on the resulting curvature. [1] P. Cignoni, M. Callieri, M. Corsini, M. Dellepiane, F. Ganovelli, and G. Ranzuglia, Meshlab: an open-source mesh processingtool, in
Eurographics Italian Chapter Conference , edited by V. Scarano, R. D. Chiara, and U. Erra (The EurographicsAssociation, 2008).[2] G. Guennebaud and M. Gross, Algebraic point set surfaces, in
ACM SIGGRAPH 2007 papers on - SIGGRAPH ' (ACMPress, 2007). [3] G. Guennebaud, M. Germann, and M. Gross, Dynamic sampling and rendering of algebraic point set surfaces, ComputerGraphics Forum27