Triangular buckling patterns of twisted inextensible strips

Korte, AP; Starostin, E. L.; Heijden, G. H. M. van der

doi:10.1098/rspa.2010.0200

Cited by 55 publications

(78 citation statements)

References 24 publications

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“…Our approach, of course, applies to open strips as well as to closed strips. In [28] we used a technique similar to the Möbius surgery discussed here to construct triangular buckling patterns of twisted strip in good agreement with experimental observations.…”

Section: Discussionsupporting

confidence: 50%

Equilibrium Shapes with Stress Localisation for Inextensible Elastic Möbius and Other Strips

Starostin

Heijden

2014

J Elast

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We formulate the problem of finding equilibrium shapes of a thin inextensible elastic strip, developing further our previous work on the Möbius strip. By using the isometric nature of the deformation we reduce the variational problem to a second-order onedimensional problem posed on the centreline of the strip. We derive Euler-Lagrange equations for this problem in Euler-Poincaré form and formulate boundary-value problems for closed symmetric one-and two-sided strips. Numerical solutions for the Möbius strip show a singular point of stress localisation on the edge of the strip, a generic response of inextensible elastic sheets under torsional strain. By cutting and pasting operations on the Möbius strip solution, followed by parameter continuation, we construct equilibrium solutions for strips with different linking numbers and with multiple points of stress localisation. Solutions reveal how strips fold into planar or self-contacting shapes as the length-to-width ratio of the strip is decreased. Our results may be relevant for curvature effects on physical properties of extremely thin two-dimensional structures as for instance produced in nanostructured origami.

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Section: Discussionsupporting

confidence: 50%

Equilibrium Shapes with Stress Localisation for Inextensible Elastic Möbius and Other Strips

Starostin

Heijden

2014

J Elast

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“…However, such spontaneous fold lines can, in fact, be observed in paper models, and indeed, mathematical models of origami folding show that developable deformations of facets bounded by straight lines must remain piecewise planar (19). Their formation is also motivated by the physics of stress concentration in thin-walled shells (20,21). To first-order approximation, the choice of diagonal for the additional fold line does not matter (SI Appendix, section S1).…”

Section: Folded Shell Structuresmentioning

confidence: 99%

Geometry of Miura-folded metamaterials

Schenk

Guest

2013

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

767

642

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This paper describes two folded metamaterials based on the Miura-ori fold pattern. The structural mechanics of these metamaterials are dominated by the kinematics of the folding, which only depends on the geometry and therefore is scale-independent. First, a folded shell structure is introduced, where the fold pattern provides a negative Poisson's ratio for in-plane deformations and a positive Poisson's ratio for out-of-plane bending. Second, a cellular metamaterial is described based on a stacking of individual folded layers, where the folding kinematics are compatible between layers. Additional freedom in the design of the metamaterial can be achieved by varying the fold pattern within each layer.I n this paper, we describe the use of origami for mechanical metamaterials, where the fold patterns introduce kinematic deformation modes that dominate the overall structural response. The geometry and kinematics of two types of folded metamaterial are described: a folded shell structure and a folded cellular metamaterial. The examples presented here are both based on a particular fold geometry: the classic Miura-ori pattern. This pattern has previously been considered for applications, such as deployable solar panels (1), and was observed in the biaxial compression of stiff thin membranes on a soft elastic substrate (2, 3).In recent years, origami has seen a surge in research interest from engineers and physicists. Developments include folded sandwich panel cores (4, 5), origami-inspired stents (6), selffolding membranes (7), and cellular materials made from folded cylinders (8). An important concept is rigid origami, where the fold pattern is modeled as rigid panels connected through frictionless hinges. These assumptions make the study of origami folding a matter of kinematics. Of particular interest here are fold patterns where four fold lines meet at each vertex (so-called degree-4 vertices). Each such vertex has one degree of freedom, a tessellated fold pattern is overconstrained, and folding is only possible under strict geometric conditions. In a landmark paper, Huffman (9) studied rigid folding using spherical geometry; recent work includes the modeling of crease patterns using quaternions (10) and an increased understanding of the foldability conditions for partly folded quadrilateral surfaces (11,12).In describing the properties of the folded metamaterials, we are here primarily concerned with the deformation kinematics. If required, these models can straightforwardly be extended to include simple constitutive behavior at the fold lines [for instance, elastic (13) or plastic (14) behavior].The paper is structured as follows. First, the Miura-ori unit cell is introduced, because its geometry plays a key role in the mechanical properties of the folded metamaterials. The first such metamaterial is based on a single planar Miura-ori sheet: a folded shell structure. Of particular interest are the shell's outof-plane kinematics. Second, a bulk metamaterial is proposed based on the stacking of individual Miura-or...

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“…Some of them are energetically costly and are blocked whereas others are activated and guide the global deformation of the structure. Often, however, facets remain rigid [10] or at least rigid by parts [6,11,12]. This means that when a facet deforms, new sub-creases emerge.…”

Section: Introductionmentioning

confidence: 99%

Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern

2017

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Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape, curvature and elongation on a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-trivial surfaces thus inspiring applications across a wide range of domains including structural engineering, architectural design and aerospace engineering. The present paper suggests a homogenization-type two-scale asymptotic method which, combined with standard tools from differential geometry of surfaces, yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin-jointed trusses. The outcome of the method is a set of nonlinear differential equations governing the parametrization, metric and curvature of surfaces that the initially discrete structure can fit. The theory is presented through a case study of a fairly generic example: the eggbox pattern. The proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated.

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Triangular buckling patterns of twisted inextensible strips

Cited by 55 publications

References 24 publications

Equilibrium Shapes with Stress Localisation for Inextensible Elastic Möbius and Other Strips

Equilibrium Shapes with Stress Localisation for Inextensible Elastic Möbius and Other Strips

Geometry of Miura-folded metamaterials

Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern

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