We study the non-Euclidean (incompatible) elastic energy functionals in the description of prestressed thin films, at their singular limits ( @math -limits) as @math in the film's thickness @math . Firstly, we extend the prior results [Lewicka-Pakzad, Bhattacharya-Lewicka-Schaffner, Lewicka-Raoult-Ricciotti] 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 [Agostiniani-Lucic-Lucantonio, Schmidt] 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 @math and discuss the scaling regimes @math (Kirchhoff), @math (von Karman) 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 [Lewicka]. Thirdly, we prove the coercivity inequalities for the singular limits at @math - and @math - scaling orders, while disproving the full coercivity of the classical von Karman energy functional at scaling @math .
Alternative geometrically nonlinear thin plate theories can be used to analyze the self-similar structures with metric asymptotically flat at infinity @cite_51 , a disk with edge-localized growth @cite_27 , the shape of a long leaf @cite_19 , or torn plastic sheets @cite_4 . In @cite_7 @cite_24 a variant of the F "oppl-von K 'arm 'an equilibrium equations has been formally derived from finite incompressible elasticity, via the multiplicative decomposition of deformation gradient @cite_17 similar to ours. See also models related to cockling of paper, grass blades, sympatelous flowers and movement of euglenids DCB, BMT, D . The forward and inverse problems in the study of self-folding in thin sheets of patterned hydrogel bilayers are discussed in the forthcoming paper @cite_44 .
@cite_4 @cite_7 @cite_24 @cite_19 @cite_27 @cite_44 @cite_51 @cite_17
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We have measured the multiscale wrinkling that occurs along the edge of torn plastic sheets. The plastic deformations produced by tearing define a new metric on the sheet, which then relaxes elastically. The resultant patterns of wrinkles correspond to a superposition of waves of different wavelengths. Measurements of the variation of the pattern as a function of the distance from the edge reveal a set of transitions, each of which adds a new mode to the cascade. The wavelengths in the cascade depend on both a geometrical length scale Lgeo given by the metric near the sheet’s edge, and the sheet thickness t: t 0.3 Lgeo . This scaling implies vanishingly short wavelengths in the limit t → 0. A possible geometrical origin of this behavior is discussed. Finally, we show that our measurement and analysis techniques are applicable to the study of some wavy patterns of leaves. These measurements reveal that the intrinsic geometry of a wavy leaf resembles that of the torn plastic sheets. This supports the possibility that some leaves form waves through a spontaneous wrinkling, rather than through an explicit three-dimensional construction. Recently, much attention has been given to a noteworthy property of some soft tissues: their ability to grow. Many attempts have been made to model this behavior in biology, chemistry, and physics. Using the theory of finite elasticity, Rodriguez has postulated a multiplicative decomposition of the geometric deformation gradient into a growth-induced part and an elastic one needed to ensure compatibility of the body. In order to fully explore the consequences of this hypothesis, the equations describing thin elastic objects under finite growth are derived. Under appropriate scaling assumptions for the growth rates, the proposed model is of the Foppl-von Karman type. As an illustration, the circumferential growth of a free hyperelastic disk is studied. Long leaves in terrestrial plants and their submarine counterparts, algal blades, have a typical, saddle-like midsurface and rippled edges. To understand the origin of these morphologies, we dissect leaves and differentially stretch foam ribbons to show that these shapes arise from a simple cause, the elastic relaxation via bending that follows either differential growth (in leaves) or differential stretching past the yield point (in ribbons). We quantify these different modalities in terms of a mathematical model for the shape of an initially flat elastic sheet with lateral gradients in longitudinal growth. By using a combination of scaling concepts, stability analysis, and numerical simulations, we map out the shape space for these growing ribbons and find that as the relative growth strain is increased, a long flat lamina deforms to a saddle shape and or develops undulations that may lead to strongly localized ripples as the growth strain is localized to the edge of the leaf. Our theory delineates the geometric and growth control parameters that determine the shape space of finite laminae and thus allows for a comparative study of elongated leaf morphology. Non-Euclidean plates are a subset of the class of elastic bodies having no stress-free configuration. Such bodies exhibit residual stress when relaxed from all external constraints, and may assume complicated equilibrium shapes even in the absence of external forces. In this work we present a mathematical framework for such bodies in terms of a covariant theory of linear elasticity, valid for large displacements. We propose the concept of non-Euclidean plates to approximate many naturally formed thin elastic structures. We derive a thin plate theory, which is a generalization of existing linear plate theories, valid for large displacements but small strains, and arbitrary intrinsic geometry. We study a particular example of a hemispherical plate. We show the occurrence of a spontaneous buckling transition from a stretching dominated configuration to bending dominated configurations, under variation of the plate thickness. We discuss self-folding of a thin sheet by using patterned hydrogel bilayers, which act as hinges connecting flat faces. Folding is actuated by heterogeneous swelling due to different cross-linking densities of the polymer network in the two layers. Our analysis is based on a dimensionally reduced plate model, obtained by applying a recently developed theory [1], which provides us with an explicit connection between (three-dimensional) material properties and the curvatures induced at the hinges. This connection offers a recipe for the fabrication and design of the bilayers, by providing the values of the cross-linking density of each layer that need to be imprinted during polymerization in order to produce a desired folded shape upon swelling. We study the buckling of thin elastic plates caused by residual strains concentrated near a free edge. This is a model for plant leaves and torn plastic sheet morphologies. We derive new governing equations explaining self-similar patterns reported earlier in experiments. We reveal the cascade mechanism, determine the bounds for its wavelengths, and predict a similarity factor of 3 in agreement with experiments. This is confirmed by numerical solutions with up to five generations of wrinkles.