An exact dimension-reduced dynamic theory for developable surfaces and curve-fold origami

TL;DR

This work develops an exact dimension-reduced dynamic theory for developable surfaces and curve-fold origami by exploiting the intrinsic 1D nature of isometric deformations. It constructs a dynamic rod-like description for the reference curve based on the Wunderlich energy and extends it to a nonlinear bi-rod model for curved folds, with a consistent Lagrangian framework. The theory is validated against finite element simulations across multiple scenarios, showing accurate predictions for wide panels where traditional narrow-rod models fail. The approach offers a powerful tool for designing and analyzing origami-based metamaterials and soft robotic systems with complex, wide-panel geometry. It also provides a foundation for future extensions to reinforceable developable structures and intersection-rich configurations.

Abstract

Curve-fold origami, composed of developable panels joined along a curved crease, exhibits rich dynamic behaviors relevant to metamaterials and soft robotic systems. Despite multiple approximated models, a comprehensive and exact dynamical theory for curve-fold origami remains absent, limiting the precise predictions of its dynamics, especially for those with wide panels. In this work, we develop an exact dimension-reduced theory that focuses on the dynamics of curve-fold origami, utilizing the intrinsic one-dimensional nature of developable surfaces. Starting from a single developable surface, we investigate the kinematics and kinetic energy of a moving developable surface. By overcoming the difficulty of describing the motion of local frames, we derive the exact velocity field of wide surfaces solely described by the motion of the reference curve, which leads to the kinetic energy of the entire surface. Owing to the one-dimensional feature, the Lagrangian of the system, composed of both kinetic and elastic energy, is a functional of the reference curve. Thus, we may variate the Lagrangian and derive a nonlinear dynamical theory for the reference curve, which comprises governing equations similar to the rod model but can precisely describe the motion of developable surfaces. The theory is validated consistently in both Lagrangian and Eulerian frameworks and is further extended to curved-fold origami modeled as a coupled bi-rod system. Utilizing our exact 1D model, we theoretically analyze the dynamical behaviors of various developable structures, revealing that the coupling of curvature and torsion along with the motion of local frames in our theory leads to the accurate modeling of arbitrarily deformed developable surfaces, which are validated by finite element analysis quantitatively.

Figures (9)

• Figure 1: Kinematics of a developable surface. (a) Frenet and Darboux frames. (b) Motion of a material point from $t_1$ to $t_2$ in the deformed configuration. (c) Motion of a material point from $t_1$ to $t_2$ in the reference configuration.
• Figure 2: (a) Correspondence between the local frames of a developable surface and those of the equivalent rod. (b) Force diagram for a differential rod element.
• Figure 3: (a) A rectangular wide strip is rolled and lifted to the initial position. (b) The dynamic process after releasing the load. The color bar denotes the rotation angle around $z$ axis. (c) The quantitative results (curvature) predicted by our model (left) and the Kirchhoff rod model (right). Our model achieves better agreement with the FEM results.
• Figure 4: (a) Correspondence between the local frames of the curved-fold origami and those of the equivalent bi-rod. (b) Force analysis of the corresponding differential rod elements.
• Figure 5: (a) FEM result of the dynamics of a curve-folded plate. (b) Comparison between the theoretical results and FEM results.