Geometric Mechanics of Periodic Pleated Origami

TLDR

Origami structures are mechanical metamaterials whose properties arise almost exclusively from the geometry of their folds and the constraint of piecewise isometric deformations. The authors aim to characterize the geometry and elastic response of a simple periodically folded Miura‑ori structure composed of identical unit cells defined by two angles and two lengths. They analyze the planar and nonplanar effective elastic response and solve the inverse design problem to determine the geometric parameters that yield optimal mechanical behavior. They find that the in‑plane and out‑of‑plane Poisson's ratios are equal in magnitude but opposite in sign, independent of material properties, and that the unit cell’s bending stiffness is singular, enabling a two‑dimensional deformation to be described by a one‑dimensional theory.

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.