title: Elastic theory of origami-based metamaterials.
abstract: 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.
authors.affiliation: Laboratoire de Physique Statistique, Ecole Normale Supérieure, UPMC Université Paris 06, Université Paris-Diderot, CNRS, 24 rue Lhomond 75005 Paris, France.
authors.forename: V
authors.initials: V
authors.lastname: Brunck
authors.affiliation: Laboratoire de Physique Statistique, Ecole Normale Supérieure, UPMC Université Paris 06, Université Paris-Diderot, CNRS, 24 rue Lhomond 75005 Paris, France.
authors.forename: F
authors.initials: F
authors.lastname: Lechenault
authors.affiliation: Laboratoire de Physique Statistique, Ecole Normale Supérieure, UPMC Université Paris 06, Université Paris-Diderot, CNRS, 24 rue Lhomond 75005 Paris, France.|Department of Physics, North Carolina State University, North Carolina 27695, USA.
authors.forename: A
authors.initials: A
authors.lastname: Reid
authors.affiliation: Laboratoire de Physique Statistique, Ecole Normale Supérieure, UPMC Université Paris 06, Université Paris-Diderot, CNRS, 24 rue Lhomond 75005 Paris, France.
authors.forename: M
authors.initials: M
authors.lastname: Adda-Bedia