2022-08-23 ジョージア工科大学
Researchers unfolded elegant equations to explain the enigma of expanding origami
研究者らは、幅広い種類の折り紙が応力に反応する方法についての一般法則を発表した。この法則は、薄い材料でできた平行四辺形（正方形、ひし形、長方形など）から作られる折り紙に適用される。
＜関連情報＞
平行四辺形折り紙における幾何学的力学を制御する離散的対称性 Discrete symmetries control geometric mechanics in parallelogram-based origami
James McInerney, Glaucio H. Paulino, and D. Zeb Rocklin
Proceedings of the National Academy of Sciences Published:August 3, 2022
DOI:https://doi.org/10.1073/pnas.2202777119
Significance
Origami-inspired metamaterials utilize crease geometries, often based on symmetries, to generate deployable and reconfigurable structures with unusual properties such as negative Poisson’s ratios and high stiffness-to-weight ratios. Mathematical treatments of such structures are often complex and ad hoc, thereby preventing far-reaching analytical solutions and obscuring the interplay between geometry and mechanical response. Here, we present a unifying analytical framework for the low-energy deformations (linear isometries) of quadrilateral-based origami, including mechanical metamaterials currently under intense study, that highlights the role of symmetry in a broad family of crease patterns exhibiting equal and opposite in-plane and out-of-plane Poisson’s ratios. This approach extends to nonuniform deformations and to other crease patterns where symmetry plays a fundamental role in the material properties.
Abstract
Geometric compatibility constraints dictate the mechanical response of soft systems that can be utilized for the design of mechanical metamaterials such as the negative Poisson’s ratio Miura-ori origami crease pattern. Here, we develop a formalism for linear compatibility that enables explicit investigation of the interplay between geometric symmetries and functionality in origami crease patterns. We apply this formalism to a particular class of periodic crease patterns with unit cells composed of four arbitrary parallelogram faces and establish that their mechanical response is characterized by an anticommuting symmetry. In particular, we show that the modes are eigenstates of this symmetry operator and that these modes are simultaneously diagonalizable with the symmetric strain operator and the antisymmetric curvature operator. This feature reveals that the anticommuting symmetry defines an equivalence class of crease pattern geometries that possess equal and opposite in-plane and out-of-plane Poisson’s ratios. Finally, we show that such Poisson’s ratios generically change sign as the crease pattern rigidly folds between degenerate ground states and we determine subfamilies that possess strictly negative in-plane or out-of-plane Poisson’s ratios throughout all configurations.
