Shape selection in non-Euclidean plates

Gemmer, John; Venkataramani, Shankar C.

doi:10.1016/j.physd.2011.07.002

Cited by 37 publications

(74 citation statements)

References 39 publications

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“…We remark that the mentioned papers do not address the dimension reduction, but rather analyze the chosen actual configuration of the prestrained sheet. Closely related is also the literature on shape selection in non-Euclidean plates, exhibiting hierarchical buckling patterns in zero-strain plates (β = 2), where the complex morphology is due to the non-smooth energy minimization [19,20,21].…”

Section: Other Related Workmentioning

confidence: 87%

Dimension Reduction for Thin Films with Transversally Varying Prestrain: Oscillatory and Nonoscillatory Cases

Lewicka

Lučić

2019

Comm Pure Appl Math

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We study the non-Euclidean (incompatible) elastic energy functionals in the description of prestressed thin films, at their singular limits (Γ-limits) as h → 0 in the film's thickness h. Firstly, we extend the prior results [39,12,40] to arbitrary incompatibility metrics that depend on both the midplate and the transversal variables (the "non-oscillatory" case). Secondly, we analyze a more general class of incompatibilities, where the transversal dependence of the lower order terms is not necessarily linear (the "oscillatory" case), extending the results of [3,47] to arbitrary metrics and higher order scalings. We exhibit connections between the two cases via projections of appropriate curvature forms on the polynomial tensor spaces. We also show the effective energy quantisation in terms of scalings as a power of h and discuss the scaling regimes h 2 (Kirchhoff), h 4 (von Kármán) in the general case, as well as all possible (even powers) regimes for conformal metrics, thus paving the way to the subsequent complete analysis of the non-oscillatory setting in [34]. Thirdly, we prove the coercivity inequalities for the singular limits at h 2 -and h 4 -scaling orders, while disproving the full coercivity of the classical von Kármán energy functional at scaling h 4 .

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Section: Other Related Workmentioning

confidence: 87%

Dimension Reduction for Thin Films with Transversally Varying Prestrain: Oscillatory and Nonoscillatory Cases

Lewicka

Lučić

2019

Comm Pure Appl Math

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“…Our aim is to derive a self-consistent stability criterion, similar to Eqs. (56), for isometric immersions with constant negative Gaussian curvature [57]. Following Sec.…”

Section: Appendix B: Boundary Layer In a Sheet With Elliptic Referencmentioning

confidence: 99%

Strain tensor selection and the elastic theory of incompatible thin sheets

Oshri

Diamant

2017

Phys. Rev. E

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The existing theory of incompatible elastic sheets uses the deviation of the surface metric from a reference metric to define the strain tensor [Efrati et al., J. Mech. Phys. Solids 57, 762 (2009)]. For a class of simple axisymmetric problems we examine an alternative formulation, defining the strain based on deviations of distances (rather than distances squared) from their rest values. While the two formulations converge in the limit of small slopes and in the limit of an incompressible sheet, for other cases they are found not to be equivalent. The alternative formulation offers several features which are absent in the existing theory. (a) In the case of planar deformations of flat incompatible sheets, it yields linear, exactly solvable, equations of equilibrium. (b) When reduced to uniaxial (one-dimensional) deformations, it coincides with the theory of extensible elastica; in particular, for a uniaxially bent sheet it yields an unstrained cylindrical configuration. (c) It gives a simple criterion determining whether an isometric immersion of an incompatible sheet is at mechanical equilibrium with respect to normal forces. For a reference metric of constant positive Gaussian curvature, a spherical cap is found to satisfy this criterion except in an arbitrarily narrow boundary layer.

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“…Mathematically, however, that problem seems quite different from ours. Indeed, for a constant-negative-curvature disk there exist deformations with vanishing membrane energy and finite bending energy [12]. As a result, the minimum energy is of order h 2 and minimizers do not depend strongly on h. The increasing complexity seen experimentally [17] as h → 0 remains a puzzle.…”

Section: The Connection With Leaves Flowers and Torn Plasticmentioning

confidence: 99%

Metric-Induced Wrinkling of a Thin Elastic Sheet

Bella

Kohn

2014

J Nonlinear Sci

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We study the wrinkling of a thin elastic sheet caused by a prescribed non-Euclidean metric. This is a model problem for the patterns seen, for example, in torn plastic sheets and the leaves of plants. Following the lead of other authors we adopt a variational viewpoint, according to which the wrinkling is driven by minimization of an elastic energy subject to appropriate constraints and boundary conditions. We begin with a broad introduction, including a discussion of key examples (some well-known, others apparently new) that demonstrate the overall character of the problem. We then focus on how the minimum energy scales with respect to the sheet thickness h, for certain classes of displacements. Our main result is that when the deformations are subject to certain hypotheses, the minimum energy is of order h 4/3 . We also show that when the deformations are subject to more restrictive hypotheses, the minimum energy is strictly larger -of order h; it follows that energy minimization in the more restricted class is not a good model for the applications that motivate this work. Our results do not explain the cascade of wrinkles seen in some experimental and numerical studies; and they leave open the possibility that an energy scaling law better than h 4/3 could be obtained by considering a larger class of deformations.

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Shape selection in non-Euclidean plates

Cited by 37 publications

References 39 publications

Dimension Reduction for Thin Films with Transversally Varying Prestrain: Oscillatory and Nonoscillatory Cases

Dimension Reduction for Thin Films with Transversally Varying Prestrain: Oscillatory and Nonoscillatory Cases

Strain tensor selection and the elastic theory of incompatible thin sheets

Metric-Induced Wrinkling of a Thin Elastic Sheet

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