# Homogenization framework for rigid and non-rigid foldable origami metamaterials

## TL;DR

This paper develops a homogenization framework modeling origami faces as plates to accurately predict complex deformations in non-rigid foldable origami metamaterials, surpassing existing rigid models in accuracy.

## Contribution

It introduces a novel continuum homogenization approach using plate elements for origami, capturing non-rigid deformation modes and improving predictive accuracy.

## Key findings

- Effective elastic properties depend nonlinearly on fold angle and crease stiffness.
- The framework achieves up to 12.9% error compared to detailed simulations.
- Existing models show errors up to 161%, highlighting the new method's superiority.

## 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.