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.

Geometric Mechanics of Origami Patterns Exhibiting Poisson’s Ratio Switch by Breaking Mountain and Valley Assignment

“…It is intriguing that the Morph pattern exhibits distinct Poisson's ratio in stretching and bending, similar to what have been found, separately, with the standard Miura-ori and the standard Eggbox patterns. Here we show that, just like its two extreme cases [9,12], the Morph pattern displays Poisson's ratio with opposite sign but equal magnitude in stretching and bending. We can analytically calculate the principal bending curvatures by allowing each panel of the origami pattern to bend along one of its diagonals [9], under the assumption of infinitesimal deformation.…”

“…As revealed in previous research [8,9,12], kinematically single Degree of Freedom (DOF) origami pattern may experience out-of-plane deformation, other than pure (in-plane) folding, if compliance of panels is taken into consideration. Accordingly, we define the Poisson's ratio in bending as the ratio of principal curvatures (ν b WL = −κ W /κ L ) and find that the Morph pattern features a saddle shaped geometry in the Miura mode, and a dome shape geometry in the Eggbox mode (see Figs.…”

“…Owing to their special geometries, origami metamaterials usually display interesting behavior [8][9][10][11]. For instance, the Miura-ori exhibits a negative Poisson's ratio under in-plane deformations [9], while the standard Eggbox pattern has a positive Poisson's ratio [12,13]. In comparison, our proposed pattern morphs continuously between a Miura mode and an Eggbox mode (see Fig.…”

Materials and Structures toward Soft Electronics

“…A wide variety of origami structures with distinct crease patterns have been demonstrated, including Miura‐ori, Yoshimura, diagonal, zigzag, square‐twist, diamond, waterbomb, and eggbox, among which the Miura‐ori pattern is the most commonly used for soft electronics due to its rigid foldability, simple geometry, and advanced theory . As shown in Figure a, the crease pattern of Miura‐ori can be represented as a quadrilateral mesh given by a set of vertices, lines, and facets, where the blue lines and red lines correspond to the valley and mountain folds, respectively, which are formed when the paper is folded toward or away from the viewer .…”

Design and Optimization of Origami-Inspired Orthopyramid-Like Core Panel for Load Damping

“…Inspired by the kirigami tessellation, Wang [32] proposed two design methods that could map the target three-dimensional geometries into two-dimensional patterns of cuts and creases. Nassar [33] presented a method, which output a set of nonlinear differential equations governing the parametrization, metric, and curvature of surfaces that the initially discrete structure could fit. Employing the Miura pattern, Zhang [34] studied the sandwich beam with the origami-inspired core and found it has unique mechanical properties that can be used as programmable materials to fit specified engineering demands.…”