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  1. Origami — the ancient art of paper folding — has been widely adopted as a design and fabrication framework for many engineering applications, including multi-functional structures, deployable spacecraft, and architected materials. These applications typically involve complex and dynamic deformations in the origami facets, necessitating high-fidelity models to better simulate folding-induced mechanics and dynamics. This paper presents the formulation and validation of such a new model based on the Absolute Nodal Coordinate Formulation (ANCF), which exploits the tessellated nature of origami and describes it as an assembly of flexible panels rotating around springy creases. To estimate the crease folding, we mathematically formulate a “torsional spring connector” in the framework of ANCF and apply it to the crease nodes, where the facets meshed by ANCF plate elements are interconnected. We simulate the dynamic folding of a Miura-ori unit cell and compare the results with commercial finite element software (ABAQUS) to validate the modeling accuracy. The ANCF model can converge using significantly fewer elements than ABAQUS without sacrificing accuracy. Therefore, this high-fidelity model can help deepen our knowledge of folding-induced mechanics and dynamics, broadening the appeals of origami in science and engineering. 
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  2. Multi-stable structures have gathered extensive interest because they can provide a broad spectrum of adaptive functions for many engineering systems. Especially, origami sheets with a translational periodicity can be stacked and assembled to form a multi-stable cellular solid, which has emerged as a promising platform to design functional structures. This paper investigates the multi-stability characteristics of a non-rigid stacked Miura-origami mechanism consisting of Miura-ori sheets and accordion-shaped connecting sheets, focusing on the elemental unit cell. A nonlinear mechanical model based on the barhinge approach is established to quantitatively study the unit cell’s multi-stability with intentionally relaxed rigid-folding conditions. Results show that only two stable states are achievable in the unit cell with enforced rigid-folding kinematics. However, if one relaxes the rigid-folding conditions and allows the facet to deform (i.e. non-rigid folding), four stable states are reachable in the unit cell if the crease torsional stiffness of the connecting sheets becomes sufficiently larger than that of the Miura-ori sheets, or the stress-free folding angle deviates away from 0°. A close examination of the potential energy composition of the non-rigid unit cell provides a detailed principle underpinning the multi-stability. By showing the benefits of exploiting facet compliance, this study can become the building blocks for origami-based structures and material systems with a wider variety of novel functionalities. 
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  3. Origami-inspired structures and material systems have been used in many engineering applications because of their unique kinematic and mechanical properties induced by folding. However, accurately modeling and analyzing origami folding and the associated mechanical properties are challenging, especially when large deformation and dynamic responses need to be considered. In this paper, we formulate a high-fidelity model — based on the iso-parametric Absolute Nodal Coordinate Formulation (ANCF) — for simulating the dynamic folding behaviors of origami involving large deformation. The centerpiece of this new model is the characterization of crease deformation. To this end, we model the crease using rotational spring at the nodes. The corresponding folding angle is calculated based on the local surface normal vectors. Compared to the currently popular analytical methods for analyzing origami, such as the rigid-facet and equivalent bar-hinge approach, this new model is more accurate in that it can describe the large crease and facet deformation without imposing many assumptions. Meanwhile, the ANCF based origami model can be more efficient computationally compared to the traditional finite element simulations. Therefore, this new model can lay down the foundation for high-fidelity origami analysis and design that involves mechanics and dynamics. 
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