title: Wrinkling of a thin circular sheet bonded to a spherical substrate.
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 sheethas a small parameter, we determine the leading-order behaviour of the energy ashtends to zero, and we give (almost matching) upper and lower bounds for the next-order 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 distancerfrom the centre of the sheet (so that the number of wrinkles must increase withr). Our results on the next-order correction provide insight about how the wrinkling pattern should vary withrRoughly speaking, they suggest that the length scale of wrinkling shouldnotbe exactly constant-rather, it should vary slightly, so that the number of wrinkles at radiusrcan be approximately piecewise constant in its dependence onr, taking values that are integer multiples ofh-awith [Formula: see text]This article is part of the themed issue 'Patterning through instabilities in complex media: theory and applications'.
authors.affiliation: Institute of Mathematics, Leipzig University, Augustusplatz 10, 04109 Leipzig, Germany bella@math.uni-leipzig.de.
authors.forename: Peter
authors.identifier: http://orcid.org/0000-0002-1660-1711
authors.initials: P
authors.lastname: Bella
authors.affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA.
authors.forename: Robert V
authors.initials: RV
authors.lastname: Kohn
keywords: compressed thin elastic sheets; energy scaling laws; wrinkling