Geometric Mechanics of Periodic Pleated Origami

Abstract

Origami structures are mechanical metamaterials with properties that arise almost exclusively from the geometry of the constituent folds and the constraint of piecewise isometric deformations. Here we characterize the geometry and planar and nonplanar effective elastic response of a simple periodically folded Miura-ori structure, which is composed of identical unit cells of mountain and valley folds with four-coordinated ridges, defined completely by two angles and two lengths. We show that the in-plane and out-of-plane Poisson's ratios are equal in magnitude, but opposite in sign, independent of material properties. Furthermore, we show that effective bending stiffness of the unit cell is singular, allowing us to characterize the two-dimensional deformation of a plate in terms of a one-dimensional theory. Finally, we solve the inverse design problem of determining the geometric parameters for the optimal geometric and mechanical response of these extreme structures.

Origami tubes assembled into stiff, yet reconfigurable structures and metamaterials

“…For example, rigid origami systems are defined as those having a kinematic deformation mode in which movement is concentrated along the fold lines, whereas the panels remain flat (20,21). Among various rigid folding patterns, the Miura-ori has attracted attention for its folding characteristics (22,23), elastic stiffness properties beyond rigid folding (24,25), geometric versatility (26,27), and intrinsic material-like characteristics (28,29).…”

Geometrically controlled snapping transitions in shells with curved creases

“…The consequences of this coupling are seen in both naturally occurring scenarios, such as intestinal villi and pollen grains (4,5), and find use in man-made structures such as programmable metamaterials (6)(7)(8)(9). When these shells have multistable configurations, the transition between them is opposed by geometrically enhanced rigidity resulting from the dominant stretching energy.…”

Mechanical Response of a Creased Sheet

“…In this limit, the panels can be considered as rigid. Thus, the unfolding mechanism corresponds to what would be expected for usual origami whose deformation along the creases can be deduced from kinematics [10]. In the opposite limit, when the linear size of the elementary panels is large compared to L * , the extension of the structure is governed by the bending of these panels while the creases do not open up a lot.…”

“…Careful observations have revealed that decoration of a thin plate by a crease network dramatically modifies the global mechanical response of the plate resulting, for instance, in aging of the structure [9], negative Poisson ratio and unusual response to bending and stretching [10]. Clearly, the opening and closing of folded objects critically depends on the mechanical characteristics and on the geometrical network of the creases [11][12][13].…”

“…At the first level of description, the pattern of folds can be modeled by elastic hinges of specific stiffness where each crease lies at the intersection between two rigid, non-deformable panels. Hopefully, the whole mechanism could then be understood by studying the kinematics of geometrically interdependent rigid panels coupled to each other via the spatial distribution of the fold skeleton [2,10]. However, this general "geometrical" description dismisses the role played by the intrinsic elastic response of the panels during the folding and deployment of the structure.…”