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-Bedia 1 1 Laboratoire de Physique Statistique, Ecole Normale Sup´ erieure, UPMC Universit´ e Paris 06, Universit´ e Paris-Diderot, CNRS, 24 rue Lhomond 75005 Paris, France 2 Department 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 programed by acting locally on each crease. Starting from a thin plate and having knowledge about the properties of the material and the folding procedure, one would like to determine the shape taken by the structure at rest and its mechanical response. In this article, we introduce a vector deformation field acting on the imprinted network of creases that allows us to express the geometrical constraints of rigid origami structures in a simple and systematic way. This formalism is then used to write a general covariant expression of the elastic energy of n-creases meeting at 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 any number of creases. For the Miura folding, however, we uncover a phase transition from monostable to bistable states that explains the efficient deployability of this structure for a given range of geometrical and mechanical parameters. Moreover, the analysis shows that geometric frustration induces residual stresses in origami structures that should be taken into account in determining their mechanical response. This formalism can be extended to a general crease network, ordered or otherwise, and so opens new perspectives for the mechanics and the physics of origami-based metamaterials. DOI: 10.1103/PhysRevE.93.033005 I. INTRODUCTION Since the mid-1980s, the ancient art of paper folding has evolved into a fertile scientific field gathering distant disciplines. Using the mathematics of flat foldability [1,2], computational origami has made possible the creation of an enormous amount of new designs. The interest has also switched from the depiction of concrete objects like the traditional cranes of Japan’s Edo period to abstract structures like tessellations, whose strange mechanical properties have aroused the curiosity of engineers and physicists alike. Origami metamaterials display, for example, auxetic behavior [3,4] and multistability [5–7], the latter allowing reprogrammable reconfigurations [8]. Ubiquitous in nature where permanent creases can form as a result of growth in a constraining container (e.g., dragonfly wings [9] and petal leaves [10,11]), origami structures are also well represented in domains where strength or deployable properties are attractive benefits, be it in fashion [12,13], ar- chitecture [14], medicine [15], or engineering, from airbags to consumer electronics to deployable space structures [16–20]. In all these areas, a question of paramount importance is how a particular crease affects the shape and mechanical properties of the overall structure. To answer this question the folds are usually modeled by elastic hinges of specific stiffnesses and rest angles. In this approach, each crease lies at the intersection between two panels, whose response is governed by a length scale L ∗ that controls whether the panels actually bend or fold [21,22]. More precisely, this origami length L ∗ = B/κ is the ratio between the bending modulus B of the faces and the torsional stiffness κ of the crease. When the typical length of the panels is small compared to L ∗ , the system behaves as a rigid origami, with faces remaining mostly undeformed and creases actuating when submitted to stress. On the other hand, if the faces typical size is larger than L ∗ , then the mechanical response will be governed by the bending of the sheet, while the angles of the creases will essentially keep their rest value [7]. The existence of the length scale L ∗ restricts the apparent scalability of origami structures. Nevertheless, we look in the following at the realm of rigid origami, which corresponds to taking systems with infinitely rigid faces and flexible creases. The process of making an origami structure consists in printing a given planar graph on a flat sheet, which defines the reference state, and then applying a plastic deformation to this network that assigns to each edge a given stiffness and rest angle. The three-dimensional shape taken by the structure is a minimizer of the mechanical energy subject to kinematic constraints. For this purpose, the first step is to consider a single vertex with n creases coming out of it, which constitutes the building block of any origami tessellation. In the case of rigid origami, the standard approach is to find the energy minimum of such a vertex by summing over the creases, each one having different stiffness κ i and rest crease angle ψ 0 i : E h = ∑ n i =1 κ i (ψ i − ψ 0 i ) 2 /2, where ψ i is the opening angle of the i th fold [4,5]. This energy must be supplemented by the geometrical constraints that the faces remain rigid during deformation. However, in many practical applications these constraints are not explicitly given in terms of the folds angles ψ i , which forces us to adapt the energy minimization procedure for each study. This approach is thus not convenient for building a general framework for the elasticity of origami structures. Here we provide a description of origami tessellations by expressing the deformation in terms of the vectors carried by the crease network instead of the folds angles. This approach offers us the possibility to take into account the geometric frustration using the same field of vectors, which renders energy minimization systematic and straightforward. Because origami can be seen as a three-dimensional deformation of 2470-0045/2016/93(3)/033005(14) 033005-1 ©2016 American Physical Society