Hydroelastic wake on a thin elastic sheet floating on water

A curved elastic sheet is flattened on a rigid flat plate by vertical end forces. The problem is governed by a non-dimensional parameter, B, which signifies the relative importance of flexural rigidity to the applied force and the natural radius. The elastica equations are solved by elliptic functions, perturbation for small B, and numerical integration. Force-displacement characteristics and sheet configurations are found. The results may be applied to sandwiched leaf springs.

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Nonlinear deflection of a thin elastic sheet on a liquid foundation

Quarterly of Applied Mathematics ◽

10.1090/qam/724054 ◽

1984 ◽

Vol 41 (4) ◽

pp. 433-442 ◽

Cited By ~ 3

Author(s):

Chang Yi Wang

Keyword(s):

Elastic Sheet

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Elastohydrodynamics of a pre-stretched finite elastic sheet lubricated by a thin viscous film with application to microfluidic soft actuators

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.967 ◽

2019 ◽

Vol 862 ◽

pp. 732-752 ◽

Cited By ~ 5

Author(s):

Evgeniy Boyko ◽

Ran Eshel ◽

Khaled Gommed ◽

Amir D. Gat ◽

Moran Bercovici

Keyword(s):

Closed Form ◽

Closed Form Solution ◽

Finite Domain ◽

Viscous Film ◽

Elastic Sheet ◽

Von Karman ◽

Von Kármán Equations ◽

Wide Range ◽

Soft Actuators ◽

Von Kármán

The interaction of a thin viscous film with an elastic sheet results in coupling of pressure and deformation, which can be utilized as an actuation mechanism for surface deformations in a wide range of applications, including microfluidics, optics and soft robotics. Implementation of such configurations inherently takes place over finite domains and often requires some pre-stretching of the sheet. Under the assumptions of strong pre-stretching and small deformations of the lubricated elastic sheet, we use the linearized Reynolds and Föppl–von Kármán equations to derive closed-form analytical solutions describing the deformation in a finite domain due to external forces, accounting for both bending and tension effects. We provide a closed-form solution for the case of a square-shaped actuation region and present the effect of pre-stretching on the dynamics of the deformation. We further present the dependence of the deformation magnitude and time scale on the spatial wavenumber, as well as the transition between stretching- and bending-dominant regimes. We also demonstrate the effect of spatial discretization of the forcing (representing practical actuation elements) on the achievable resolution of the deformation. Extending the problem to an axisymmetric domain, we investigate the effects arising from nonlinearity of the Reynolds and Föppl–von Kármán equations and present the deformation behaviour as it becomes comparable to the initial film thickness and dependent on the induced tension. These results set the theoretical foundation for implementation of microfluidic soft actuators based on elastohydrodynanmics.

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The diffusion of load from a stiffener into an infinite elastic sheet

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1957.0040 ◽

1957 ◽

Vol 239 (1218) ◽

pp. 296-310 ◽

Cited By ~ 10

Keyword(s):

Mixed Boundary ◽

Functional Differential Equations ◽

Mixed Boundary Conditions ◽

Limited Range ◽

The Other ◽

Shear Stresses ◽

Variable Theory ◽

Elastic Sheet ◽

Complex Variable Theory ◽

In Series

The use of complex variable theory to express problems in generalized plane stress is well known, but methods of finding particular solutions are available for only a limited range of problems. This paper and its sequel will develop a new technique, reducing certain problems with mixed boundary conditions to second order functional differential equations, whose solutions can be found in series form. Exact solutions are given to three fundamental problems of the diffusion of load in an infinite two-dimensional elastic sheet to which a semi-infinite elastic stiffener is continuously attached throughout its length. The first problem has a load applied to the end of the stiffener, with its line of action along the stiffener and its reactions at infinity. In the other two problems the stiffener end is unloaded but a uniform tension is applied to the sheet at infinity, in one case parallel to the stiffener, in the other perpendicular to it. Expressions for the load in the stiffener and for the direct and shear stresses in the sheet are found and plotted in non-dimensional form.

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Effect of morphology on the large-amplitude flapping dynamics of an inverted flag in a uniform flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.474 ◽

2019 ◽

Vol 874 ◽

pp. 526-547 ◽

Cited By ~ 5

Author(s):

Boyu Fan ◽

Cecilia Huertas-Cerdeira ◽

Julia Cossé ◽

John E. Sader ◽

Morteza Gharib

Keyword(s):

Large Amplitude ◽

Free Edge ◽

Flow Speed ◽

Uniform Flow ◽

Analytical Theory ◽

Natural Occurrence ◽

Fluid Forces ◽

Elastic Sheet ◽

Divergence Instability ◽

Flow Speeds

The stability of a cantilevered elastic sheet in a uniform flow has been studied extensively due to its importance in engineering and its prevalence in natural structures. Varying the flow speed can give rise to a range of dynamics including limit cycle behaviour and chaotic motion of the cantilevered sheet. Recently, the ‘inverted flag’ configuration – a cantilevered elastic sheet aligned with the flow impinging on its free edge – has been observed to produce large-amplitude flapping over a finite band of flow speeds. This flapping phenomenon has been found to be a vortex-induced vibration, and only occurs at sufficiently large Reynolds numbers. In all cases studied, the inverted flag has been formed from a cantilevered sheet of rectangular morphology, i.e. the planform of its elastic sheet is a rectangle. Here, we investigate the effect of the inverted flag’s morphology on its resulting stability and dynamics. We choose a trapezoidal planform which is explored using experiment and an analytical theory for the divergence instability of an inverted flag of arbitrary morphology. Strikingly, for this planform we observe that the flow speed range over which flapping occurs scales approximately with the flow speed at which the divergence instability occurs. This provides a means by which to predict and control flapping. In a biological setting, leaves in a wind can also align themselves in an inverted flag configuration. Motivated by this natural occurrence we also study the effect of adding an artificial ‘petiole’ (a thin elastic stalk that connects the sheet to the clamp) on the inverted flag’s dynamics. We find that the petiole serves to partially decouple fluid forces from elastic forces, for which an analytical theory is also derived, in addition to increasing the freedom by which the flapping dynamics can be tuned. These results highlight the intricacies of the flapping instability and account for some of the varied dynamics of leaves in nature.

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The one-dimensional model for d-cones revisited

Advances in Calculus of Variations ◽

10.1515/acv-2014-0031 ◽

2016 ◽

Vol 9 (3) ◽

pp. 201-215 ◽

Cited By ~ 3

Author(s):

Heiner Olbermann

Keyword(s):

Necessary Conditions ◽

Energy Functional ◽

Developable Surface ◽

Dimensional Model ◽

Bending Energy ◽

Main Motivation ◽

One Dimensional ◽

Elastic Sheet ◽

Physics Literature ◽

The One

AbstractA d-cone is the shape one obtains when pushing an elastic sheet at its center into a hollow cylinder. In a simple model, one can treat the elastic sheet in the deformed configuration as a developable surface with a singularity at the “tip” of the cone. In this approximation, the renormalized elastic energy is given by the bending energy density integrated over some annulus in the reference configuration. The thus defined variational problem depends on the indentation ${{h}}$ of the sheet into the cylinder. This model has been investigated before in the physics literature; the main motivation for the present paper is to give a rigorous version of some of the results achieved there via formal arguments. We derive the Gamma-limit of the energy functional as ${{h}}$ is sent to 0. Furthermore, we analyze the minimizers of the limiting functional, and list a number of necessary conditions that they have to fulfill.

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