Buckling cascades in free sheets

Sharon, Eran; Roman, Benoît; Marder, Michael; Shin, Gyu Seung; Swinney, Harry L.

doi:10.1038/419579a

Cited by 229 publications

(196 citation statements)

References 6 publications

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“…The geometry of such surfaces will then serve as a model for the observed morphology in many non-Euclidean sheets including torn plastic [1] and lettuce leaves [10] which do not have a globally defined number of waves, but rather have local buckling behavior which increases the number of waves as we approach the boundary.…”

Section: Discussionmentioning

confidence: 99%

“…Free non-Euclidean thin elastic sheets arise in a variety of physical [1,3,8] and biological [2,11,33] systems. The morphology of these sheets is usually modeled as the equilibrium configurations for an appropriate elastic energy.…”

Section: Discussionmentioning

confidence: 99%

“…The differential growth of thin elastic sheets can generate highly nontrivial configurations such as the multiple generation of waves along the edge of a torn elastic sheet [1] and the wrinkled patterns on leaves [2]. Recently, environmentally responsive gel disks of uniform thickness have been created in the laboratory that mimic this type of growth by differentially shrinking in the radial direction when thermally activated in a hot water bath [3].…”

Section: Introductionmentioning

confidence: 99%

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Shape selection in non-Euclidean plates

Gemmer

Venkataramani

2011

Physica D: Nonlinear Phenomena

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We investigate isometric immersions of disks with constant negative curvature into R 3 , and the minimizers for the bending energy, i.e. the L 2 norm of the principal curvatures over the class of W 2,2 isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in H 2 into R 3 . In elucidating the connection between these immersions and the non-existence/singularity results of Hilbert and Amsler, we obtain a lower bound for the L ∞ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally W 2,2 , and numerically have lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grow approximately exponentially with the radius of the disc. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Shape selection in non-Euclidean plates

Gemmer

Venkataramani

2011

Physica D: Nonlinear Phenomena

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“…For instance, geometry alone cannot select a shape when multiple isometric embeddings exist, and then mechanics provides the selection principle, through bending energy minimization (Lewicka and Reza Pakzad, 2011). Furthermore, the target metric may not be realizable, leading to frustrated shapes that may relax in-plane stretching by curving into possibly very complex shapes (Sharon et al, 2002;Marder, 2003).…”

Section: Introductionmentioning

confidence: 99%

Shape control of active surfaces inspired by the movement of euglenids

Arroyo

DeSimone

2014

Journal of the Mechanics and Physics of Solids

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We examine a novel mechanism for active surface morphing inspired by the cell body deformations of euglenids. Actuation is accomplished through in-plane simple shear along prescribed slip lines decorating the surface. Under general non-uniform actuation, such local deformation produces Gaussian curvature, and therefore leads to shape changes. Geometrically, a deformation that realizes the prescribed local shear is an isometric embedding. We explore the possibilities and limitations of this bioinspired shape morphing mechanism, by first characterizing isometric embeddings under axisymmetry, understanding the limits of embeddability, and studying in detail the accessibility of surfaces of zero and constant curvature. Modeling mechanically the active surface as a non-Euclidean plate (NEP), we further examine the mechanism beyond the geometric singularities arising from embeddability, where mechanics and buckling play a decisive role. We also propose a non-axisymmetric actuation strategy to accomplish large amplitude bending and twisting motions of elongated cylindrical surfaces. Besides helping understand how euglenids delicately control their shape, our results may provide the background to engineer soft machines.

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“…However, a detailed understanding of the pre-buckled shapes is necessary since post-buckling is extremely sensitive to the initial morphology, even more so for confined systems such as end-supported nanoribbons. These become tractable only in the (linear) small amplitude limit and a naive approach would discount their utility as the postbuckling in these atomically thin sheets is expected to take place primarily through bending -the stretching is prohibitively expensive [18][19][20][21] . This is in stark contrast to recent computations on edge morphologies of semiinfinite graphene sheets that show that stretching plays crucial, if not decisive role 11 .…”

Section: Introductionmentioning

confidence: 99%

Rippling instabilities in suspended nanoribbons

Wang

Upmanyu

2012

Phys. Rev. B

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Morphology mediates the interplay between the structure and electronic transport in atomically thin nanoribbons such as graphene as the relaxation of edge stresses occurs preferentially via outof-plane deflections. In the case of end-supported suspended nanoribbons that we study here, past experiments and computations have identified a range of equilibrium morphologies, in particular for graphene flakes, yet a unified understanding of their relative stability remains elusive. Here, we employ atomic-scale simulations and a composite framework based on isotropic elastic plate theory to chart out the morphological stability space of suspended nanoribbons with respect to intrinsic (ribbon elasticity) and engineered (ribbon geometry) parameters, and the combination of edge and body actuation. The computations highlight a rich morphological shape space that can be naturally classified into two competing shapes, bending-like and twist-like, depending on the distribution of ripples across the interacting edges. The linearized elastic framework yields exact solutions for these rippled shapes. For compressive edge stresses, the body strain emerges as a key variable that controls their relative stability and in extreme cases stabilizes co-existing transverse ripples. Tensile edge stresses lead to dimples within the ribbon core that decay into the edges, a feature of obvious significance for stretchable nanoelectronics. The interplay between geometry and mechanics that we report should serve as a key input for quantifying the transport along these ribbons.

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Buckling cascades in free sheets

Cited by 229 publications

References 6 publications

Shape selection in non-Euclidean plates

Shape selection in non-Euclidean plates

Shape control of active surfaces inspired by the movement of euglenids

Rippling instabilities in suspended nanoribbons

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