2D-to-3D transformation of ring origami via snap-folding instabilities

Abstract

Ring origami, consisting of closed-loop rods, is a class of shape-morphing structures that undergo shape transformation through folding enabled by snap-buckling instabilities, referred to as snap-folding instabilities. Previous studies have shown that 2D ring origami composed of rod segments with in-plane natural curvature (i.e., the stress-free curved state lies in the plane of the planar ring) can achieve diverse and intriguing 2D-to-2D shape transformations. Here, we propose a 2D-to-3D shape transformation strategy for ring origami by introducing out-of-plane natural curvature (i.e., the stress-free curved state lies in a plane perpendicular to the planar ring) into the rod segments. Due to natural curvature-induced out-of-plane bending moments, a 2D elastic ring spontaneously snaps out-of-plane and reaches equilibrium in a 3D configuration. These snapping-induced out-of-plane shape transitions not only enable self-guided, spontaneous shape morphing, but also allow the construction of complex structures from simple geometries, making them promising for the design of functional deployable and foldable structures. By combining a multi-segment Kirchhoff rod model with finite element simulations and experiments, we systematically investigate the 3D equilibrium states and transition behavior of these systems. Using square and hexagonal rings as representative examples, we demonstrate that by rationally designing the out-of-plane natural curvature of rod segments, 2D rings can exhibit a range of functional behaviors, including spontaneous 2D-to-3D shape transformation (e.g., planar square to sphere) via snap-folding, multistability with various 3D configurations, and monostability with a compact zero-energy 3D configuration.