TY - JOUR

T1 - Wrinkling of a thin circular sheet bonded to a spherical substrate

AU - Bella, Peter

AU - Kohn, Robert V.

N1 - Funding Information:
This project was initiated as a result of discussions with Benny Davidovitch and Gregory Grason, which we gratefully acknowledge. Those discussions took place at the Kavli Institute for Theoretical Physics, during a spring 2016 programme on Geometry, Elasticity, Fluctuations, and Order in 2D Soft Matter (supported by the NSF through grant no. PHY11-25915). We also gratefully acknowledge input from the anonymous referees, whose comments led to substantial improvements in the presentation of our results. P.B. was a postdoc at the Max Planck Institute for Mathematics in the Sciences (Leipzig) when this work was begun.
Publisher Copyright:
©2017 The Author(s) Published by the Royal Society. All rights reserved.

PY - 2017/5/13

Y1 - 2017/5/13

N2 - We consider a disc-shaped thin elastic sheet bonded to a compliant sphere. (Our sheet can slip along the sphere; the bonding controls only its normal displacement.) If the bonding is stiff (but not too stiff), the geometry of the sphere makes the sheet wrinkle to avoid azimuthal compression. The total energy of this system is the elastic energy of the sheet plus a (Winkler-Type) substrate energy. Treating the thickness of the sheet h as a small parameter, we determine the leading-order behaviour of the energy as h tends to zero, and we give (almost matching) upper and lower bounds for the nextorder correction. Our analysis of the leading-order behaviour determines the macroscopic deformation of the sheet; in particular, it determines the extent of the wrinkled region, and predicts the (non-Trivial) radial strain of the sheet. The leading-order behaviour also provides insight about the length scale of the wrinkling, showing that it must be approximately independent of the distance r from the centre of the sheet (so that the number of wrinkles must increase with r). Our results on the next-order correction provide insight about how the wrinkling pattern should vary with r. Roughly speaking, they suggest that the length scale of wrinkling should not be exactly constant-rather, it should vary slightly, so that the number of wrinkles at radius r can be approximately piecewise constant in its dependence on r, taking values that are integer multiples of h-A with a≈ 1/2. This article is part of the themed issue 'Patterning through instabilities in complex media: Theory and applications'.

AB - We consider a disc-shaped thin elastic sheet bonded to a compliant sphere. (Our sheet can slip along the sphere; the bonding controls only its normal displacement.) If the bonding is stiff (but not too stiff), the geometry of the sphere makes the sheet wrinkle to avoid azimuthal compression. The total energy of this system is the elastic energy of the sheet plus a (Winkler-Type) substrate energy. Treating the thickness of the sheet h as a small parameter, we determine the leading-order behaviour of the energy as h tends to zero, and we give (almost matching) upper and lower bounds for the nextorder correction. Our analysis of the leading-order behaviour determines the macroscopic deformation of the sheet; in particular, it determines the extent of the wrinkled region, and predicts the (non-Trivial) radial strain of the sheet. The leading-order behaviour also provides insight about the length scale of the wrinkling, showing that it must be approximately independent of the distance r from the centre of the sheet (so that the number of wrinkles must increase with r). Our results on the next-order correction provide insight about how the wrinkling pattern should vary with r. Roughly speaking, they suggest that the length scale of wrinkling should not be exactly constant-rather, it should vary slightly, so that the number of wrinkles at radius r can be approximately piecewise constant in its dependence on r, taking values that are integer multiples of h-A with a≈ 1/2. This article is part of the themed issue 'Patterning through instabilities in complex media: Theory and applications'.

KW - Compressed thin elastic sheets

KW - Energy scaling laws

KW - Wrinkling

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U2 - 10.1098/rsta.2016.0157

DO - 10.1098/rsta.2016.0157

M3 - Article

C2 - 28373380

AN - SCOPUS:85017514372

VL - 375

JO - Philosophical transactions. Series A, Mathematical, physical, and engineering sciences

JF - Philosophical transactions. Series A, Mathematical, physical, and engineering sciences

SN - 0962-8428

IS - 2093

M1 - 20160157

ER -