Homogenization framework for rigid and non-rigid foldable origami metamaterials

TL;DR

This work introduces a homogenization framework that unifies rigid and non-rigid origami metamaterials by representing origami faces as Kirchhoff–Love plates and deriving an equivalent continuum. It provides two complementary methods—asymptotic and energy-based homogenization—to extract the full set of effective stiffness coefficients $A^H$, $B^H$, and $D^H$ for a representative unit cell, demonstrated on Miura origami. Validation against detailed FE models and experiments shows the approach achieves up to 12.9% error on effective properties, outperforming rigid-panel and bar-and-hinge models by a wide margin, especially for bending and twisting responses. The framework also reveals how initial fold angle $ heta_0$ and crease stiffness $K_{ ext{cr}}$ nonlinearly influence stiffness, enabling precise design of origami metamaterials with tailored elastic properties for applications in soft robotics, deployable structures, and antennas.

Abstract

Origami metamaterials typically consist of folded sheets with periodic patterns, conferring them with remarkable mechanical properties. In the context of Continuum Mechanics, the majority of existing predictive methods are mechanism analogs which favor rigid folding and panel bending. While effective in predicting primary deformation modes, existing methods fall short in capturing the full spectrum of deformation of non-rigid foldable origami, such as the emergence of curvature along straight creases, local strain at vertices and warpage in panels. To fully capture the entire deformation spectrum and enhance the accuracy of existing methods, this paper introduces a homogenization framework for origami metamaterials where the faces are modeled as plate elements. Both asymptotic and energy-based homogenization methods are formulated and implemented. As a representative crease pattern, we examine the Miura origami sheet homogenized as an equivalent Kirchhoff-Love plate. The results reveal that certain effective elastic properties are nonlinearly related to both the initial fold angle and the crease stiffness. When benchmarked with results from fully resolved simulations, our framework yields errors up to 12.9\%, while existing models, including the bar-and-hinge model and the rigid-panel model, show up to 161\% error. The differences in errors are associated with the complex modes of crease and panel deformation in non-rigid origami, unexplored by the existing models. This work demonstrates a precise and efficient continuum framework for origami metamaterials as an effective strategy for predicting their elastic properties, understanding their mechanics, and designing their functionalities.

Figures (26)

• Figure 1: Origami metamaterials. Left is a Miura origami sheet made of tape and Mylar. Right is a waterbomb origami sheet made of cardboard.
• Figure 2: Homogenization scheme of a single-layered, periodic origami sheet. (a) Detailed Miura sheet $\Omega$. (b) Homogeneous Kirchhoff-Love plate. (c) Mid-surface of homogenized plate $\Omega^0$. (d) Zoom-in of origami sheet showing the location of the neutral surface (superimposed orange plane) with respect to the detailed origami sheet.
• Figure 3: Coordinate systems used in asymptotic homogenization. (a) A zoomed-in section of the origami sheet showing the macroscopic coordinates $\mathbf{x}$ and domain boundaries $\Gamma_h^+,\Gamma_h^-,\Gamma_h^{\textup{lat}}$. (b) Domain $\Omega_h$ of the origami sheet. (c) A unit cell $Y$ scaled up by $1/\varepsilon$ and the microscopic coordinates $\mathbf{y}$.
• Figure 4: Origami unit cell $Y$ consisting of thin plates and their mid-surface $Y^0$.
• Figure 5: Miura unit cells and geometric parameters. (a) A Miura tessellation. (b) A common choice of Miura unit cell. (c) Equivalent unit cell adopted in this work.