Topological transitions in the configuration space of non-Euclidean origami

Berry, M.; Lee-Trimble, M. E.; Santangelo, Christian

doi:10.1103/physreve.101.043003

Cited by 18 publications

(17 citation statements)

References 22 publications

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“…For instance, Kawasaki's condition can certainly be relaxed and still yield an origami unit cell. In addition, advances in 3-D/4-D printing and manufacturing make it sensible to also consider: 1) origami that is absent a flat configuration [51,52], and 2) simple building blocks of nonisometric origami [53,54] made of active materials. In all such cases, as long as the unit cell has four and only four corners, the design equations for the group parameters (i.e., the equations (14-16)) directly apply to construct any Helical Origami.…”

Section: Discussionmentioning

confidence: 99%

Helical Miura origami

2020

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Section: Discussionmentioning

confidence: 99%

Helical Miura origami

2020

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“…For models of this type, there are singular configurations where several branches of allowed configurations meet [8,21,23]. Each branch has a tangent space where it meets the singular configuration and many such branches meet at this point [8].…”

Section: A Folding Rigid Origamimentioning

confidence: 99%

Robust Folding of Elastic Origami

Lee-Trimble¹,

Kang²,

Hayward³

et al. 2021

Preprint

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Self-folding origami, structures that are engineered flat to fold into targeted, three-dimensional shapes, have many potential engineering applications. Though significant effort in recent years has been devoted to designing fold patterns that can achieve a variety of target shapes, recent work has also made clear that many origami structures exhibit multiple folding pathways, each of which can compete with the desired one. This can lead to structures that are programmed for one shape, yet fold incorrectly. To disentangle the features that lead to misfolding, we introduce a model of self-folding origami that accounts for the finite stretching rigidity of the origami faces and allows the computation of energy landscapes that lead to misfolding. We find that, in addition to the geometrical features of the origami, the finite elasticity of the nearly-flat origami configurations regulates the proliferation of potential misfolded states through a series of saddle-node bifurcations. We apply our model to one of the most common origami motifs, the symmetric "bird's foot," a single vertex with four folds. We show that though even a small error in programmed fold angles induces metastability in rigid origami, elasticity allows one to tune resilience to misfolding. In a more complex design, the "Randlett flapping bird," which has thousands of potential competing states, we further show that the number of actual observed minima and folding robustness is strongly determined by the structure's elasticity.

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“…6 furnishes only two constraints per vertex when all edges lie in a plane. These additional folding motions are paired with zero modes that correspond to vertices popping up or down out of the plane (19,20). Generally, this yields an extra Nv − 1 linear folding motions for developable origami in the flat state (the rigid-body translation in the direction normal to the sheet can be written as a linear combination of the Nv additional modes arising from developability) which do not all extend to rigid folding motions.…”

Section: Nonlinear Constraints Lead To Branching Between Cylindrical Configurationsmentioning

confidence: 99%

“…Origami principles are used to engineer deployable solar cells (7), stent grafts (8), flexible electronics (9, 10), impact mitigation devices (11), and tunable antennas (12) as well as to characterize patterns in biological systems (13). Yet determining whether a crease pattern can be rigidly folded into a particular shape is a nondeterministic in polynomial-time-hard problem (14) due to nonlinear geometric constraints (15) that can lead to disjoint (16) or branched (17)(18)(19)(20) configuration spaces with multiple energetic minima (21,22).…”

mentioning

confidence: 99%

Hidden symmetries generate rigid folding mechanisms in periodic origami

McInerney

Chen

Theran

et al. 2020

Proc. Natl. Acad. Sci. U.S.A.

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We consider the zero-energy deformations of periodic origami sheets with generic crease patterns. Using a mapping from the linear folding motions of such sheets to force-bearing modes in conjunction with the Maxwell–Calladine index theorem we derive a relation between the number of linear folding motions and the number of rigid body modes that depends only on the average coordination number of the origami’s vertices. This supports the recent result by Tachi [T. Tachi,Origami6, 97–108 (2015)] which shows periodic origami sheets with triangular faces exhibit two-dimensional spaces of rigidly foldable cylindrical configurations. We also find, through analytical calculation and numerical simulation, branching of this configuration space from the flat state due to geometric compatibility constraints that prohibit finite Gaussian curvature. The same counting argument leads to pairing of spatially varying modes at opposite wavenumber in triangulated origami, preventing topological polarization but permitting a family of zero-energy deformations in the bulk that may be used to reconfigure the origami sheet.

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Topological transitions in the configuration space of non-Euclidean origami

Cited by 18 publications

References 22 publications

Helical Miura origami

Helical Miura origami

Robust Folding of Elastic Origami

Hidden symmetries generate rigid folding mechanisms in periodic origami

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