RESEARCH | REPORTS APPLIED ORIGAMI Hookean energy expression for all creases and calculate the dimensionless energy u as a func- tion of the folding configuration (Fig. 2E and fig. S5) (see supplementary materials for details). Using origami design principles We observe that the energy landscape has two distinct regions, each with its own local min- to fold reprogrammable imum (Fig. 2E, points I and II), that are con- nected at a single point corresponding to a fully mechanical metamaterials unfolded unit cell where f = e =180°.Formate- rials with a finite stretching modulus, this con- necting point widens along the f axis and allows Jesse L. Silverberg,1* Arthur A. Evans,2 Lauren McLeod,1 Ryan C. Hayward,3 for otherwise forbidden stretching-enabled tran- Thomas Hull,4 Christian D. Santangelo,2 Itai Cohen1 sitions. Because the stretching modulus is pro- portional to sheet thickness (23), the energy Although broadly admired for its aesthetic qualities, the art of origami is now being required to transition between states is a tunable recognized also as a framework for mechanical metamaterial design. Working with the design feature. Thus, while PTDs are introduced Miura-ori tessellation, we find that each unit cell of this crease pattern is mechanically reversibly, their stability to mechanical perturba- bistable, and by switching between states, the compressive modulus of the overall tions is controlled by the competition between the structure can be rationally and reversibly tuned. By virtue of their interactions, these material’s bending and stretching energy scales. mechanically stable lattice defects also lead to emergent crystallographic structures The mechanical consequences of a PTD were such as vacancies, dislocations, and grain boundaries. Each of these structures comes determined by folding laser-cut structures from from an arrangement of reversible folds, highlighting a connection between mechanical 24-lb paper with s2 unit cells, where s was varied metamaterials and programmable matter. Given origami’s scale-free geometric character, from 3 to 8 and a was fixed at 60°. A centrally this framework for metamaterial design can be directly transferred to milli-, micro-, and located defect was introduced by hand, and the nanometer-size systems. defect compressive modulus Ky ðeÞ was measured as a function of defect density n =1/s2 (see sup- etamaterials are rapidly emerging at the required (Fig. 1C); when e =180°– 2a,the plementary materials for details). Normalizing on December 8, 2016 frontier of scientific and technological in- structureismaximallycontractedintoafolded these data by the modulus of a defect-free lattice, normal novation due to their exotic and tunable state, and when e = 180°, the structure is an Ky ðeÞ, revealed a divergence in stiffness as a M material properties, which arise from ar- unfolded flat sheet. function of compression limited only by tearing rangements of smaller units within the If folded from an ideal material with infinite of the sheet at the highest strains (Fig. 2F and fig. bulk system (1–5). Once fabricated, structural mod- stretching modulus, the Miura-ori would have S4B).Thisdivergencearisesfromaviolationof ifications are generally difficult, making it rare only one degree of freedom described by e (14). Maekawa’s theorem, which requires the number to find metamaterials that can be reconfigured However, generating samples from laser-cut sheets of mountain folds minus the number of valley beyond their original design. Origami-inspired of paper and mylar (Fig. 1C) (see supplementary folds at every vertex to be T2 for a structure to mechanical metamaterials offer enhanced flex- materials for details), we find additional degrees fold flat (24). By linking enhanced mechanical ibility in performance because their properties of freedom, as evidenced by soft bending modes stiffness to violations of a canonical paper- are coupled to a dynamically alterable folding pat- (14, 15). As an extreme example, the Miura-ori folding theorem, these data experimentally demon- tern (6–16). Hence, multiple stable configurations supports highly localized heterogeneity intro- strate how geometric constraints in origami-based can manifest from a single structure yielding pro- duced after folding by applying force to a vertex design can dominate bulk material properties. http://science.sciencemag.org/ grammable metamaterials. in the normal direction and popping it into a We studied the Miura-ori tessellated folding categorically different mechanically stable state pattern, which has recently been proposed as an (Fig. 2A, fig. S4A, and movie S2). This pop- origami metamaterial (Fig. 1, A to C, figs. S1 to S3, through defect (PTD) changes lattice topology by and movie S1) (14, 15). Historically, this design suppressing one fold and bending adjacent facets was invented to efficiently pack solar panels for through an angle f (Fig. 2, B and C). Quantify- space missions (17), but the morphology also nat- ing lattice distortion by the surface’s Monge Downloaded from urally occurs in leaves (18) and embryonic in- patch mean curvature calculated from three- testine (19, 20) and generally arises when thin dimensional (3D) digitized scans, we find that sheets tethered to a surface undergo biaxial com- displacements of vertices in the xy plane are pression. The geometry of a Miura-ori is a herring- negligible within a distance of one unit cell; bone pattern that emerges from a series of convex this compares favorably with theoretical pre- mountain and concave valley creases (Fig. 1C). dictions that estimate a distortion decay length Vertices are formed when four creases intersect, ≈1/2 the unit cell width (Fig. 2D) (see supplemen- and four adjacent vertices bound equal-area facets tary materials for details) (21, 22). Although highly arranged with inversion symmetry (Fig. 1B). This localized, PTDs are unlike conventional lattice fold pattern defines a lattice characterized by two defects because they involve only elastic facet static crease lengths ‘1;‘2,andonestaticplane bending and hence are reversibly removable. Fig. 1. Schematic and 3D reconstruction of the angle a.Toquantifyfolding,avertexanglee is To determine the robustness of PTDs to mechan- Miura-ori with definition of important geometric ical perturbations, we theoretically model a single parameters. (A) The Miura-ori crease pattern on a unit cell with one mountain and three valley flat sheet is depicted with alternating mountain 1Physics Department, Cornell University, Ithaca, NY 14853, folds in mechanical equilibrium (similar to the (red) and valley (blue) folds. (B) Each facet is USA. 2Department of Physics, University of Massachusetts, paper model in Fig. 2, B and C). To account for bounded by four vertices labeled 1 through 4. (C) 3 Amherst, MA 01003, USA. Department of Polymer Science experimentally observed facet bending, two ad- An experimentally recorded 3D image of a 4 by 4 and Engineering, University of Massachusetts, Amherst, MA 4 ditional symmetrically placed creases are included laser-cut Miura-ori with two in-plane projections 01003, USA. Department of Mathematics, Western New f England University, Springfield, MA 01119, USA. whose folding angles are equal and have equi- shows the folded structure at equilibrium. The *Corresponding author. E-mail: [email protected] librium values of 180° (14, 15). We assume a color bar gives height in mm. SCIENCE sciencemag.org 8 AUGUST 2014 • VOL 345 ISSUE 6197 647 RESEARCH | REPORTS PTD Nevertheless, such constraints are still malleable; introduced here, soft robotics-based technology trinsic PTD compressive modulus, Ky (see sup- defect the value of e where Ky increases is shifted (25), microelectromechanical systems actuator plementary materials for details). This was found simply by varying n. Thus, PTDs make attractive arrays (26), or even thermal fluctuations (27), by noting that the stress a PTD exerts is propor- elements for metamaterial design because the on- provide alternative avenues to introduce and tional to its extension relative to the neighboring defect set and strength of Ky arerationallytunable, remove PTDs. undefected unit cells. At low defect densities for and moreover, these “mechanical pixels” can be To understand both the density dependence a lattice of arbitrary size, the compressive modulus activated on demand to vary the bulk compres- and compression dependence of the Miura-ori ofalatticewithaPTDcanbeexpressedinadipole defect normal PTD PTD sive properties. Although PTDs were manually with a PTD, we analytically calculated the in- expansion, Ky = Ky + nKy ,whereKy Fig. 2. Experimental data showing spatial confinement of a PTD and combined experi- mental/theoretical data for PTD mechanics. (A) 3D reconstruction of a Miura-ori with a centrally located PTD (red facets). The color bar is height in mm. (B) Photograph of a single vertex shows that it is mechanically stable in isolation, as is a PTD (C). (D) Mean curvature of 3D scan from (A) shows a modified crease pattern in the presence of a PTD. The color bar is in units of mm–1. Overlapping vertex position of same lattice without a PTD (black dots) shows that the PTD is highly localized and causes minimal distortion in finite-sized lattices. (E) Theoretical energy diagram of a single vertex has two energy minima cor- responding to a (I) normal and (II) PTD state. Arrows indicate direct pathway to PTD formation. on December 8, 2016 The white background is geometrically forbidden. (F) Experimentally measured modulus as a function