Wrinkling of a thin circular sheet bonded to a spherical substrate

Peter Bella, Robert Kohn

Research output: Contribution to journalArticle

Abstract

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'.

Original languageEnglish (US)
Article number20160157
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume375
Issue number2093
DOIs
StatePublished - May 13 2017

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Wrinkling
wrinkling
Substrate
Substrates
Energy
Length Scale
Vary
Patterning
Small Parameter
Slip
elastic sheets
Upper and Lower Bounds
Compression
Geometry
Radius
Tend
energy
Predict
Integer
Zero

Keywords

  • Compressed thin elastic sheets
  • Energy scaling laws
  • Wrinkling

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

Cite this

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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'.

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