title: Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern.
abstract: Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape, curvature and elongation on a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-trivial surfaces thus inspiring applications across a wide range of domains including structural engineering, architectural design and aerospace engineering. The present paper suggests a homogenization-type two-scale asymptotic method which, combined with standard tools from differential geometry of surfaces, yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin-jointed trusses. The outcome of the method is a set of nonlinear differential equations governing the parametrization, metric and curvature of surfaces that the initially discrete structure can fit. The theory is presented through a case study of a fairly generic example: the eggbox pattern. The proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated.
authors.affiliation: Laboratoire Navier, Ecole des Ponts-ParisTech, IFSTTAR, CNRS, UPE, 6 et 8 Avenue Blaise Pascal, 77455 Marne-la-Vallée cedex 2, France; CERMICS, Ecole des Ponts-ParisTech, Université Paris-Est, 6 et 8 Avenue Blaise Pascal, 77455 Marne-la-Vallée, France.
authors.forename: H
authors.identifier: 0000-0001-7235-4498
authors.initials: H
authors.lastname: Nassar
authors.affiliation: Laboratoire Navier , Ecole des Ponts-ParisTech , IFSTTAR, CNRS, UPE, 6 et 8 Avenue Blaise Pascal, 77455 Marne-la-Vallée cedex 2, France.
authors.forename: A
authors.identifier: 0000-0002-3202-5558
authors.initials: A
authors.lastname: Lebée
authors.affiliation: CERMICS, Ecole des Ponts-ParisTech , Université Paris-Est , 6 et 8 Avenue Blaise Pascal, 77455 Marne-la-Vallée, France.
authors.forename: L
authors.identifier: 0000-0001-8090-9107
authors.initials: L
authors.lastname: Monasse
keywords: eggbox; floppy modes; form finding; metasurface; origami