On a boundary value problem for conically deformed thin elastic sheets

Olbermann, Heiner

doi:10.2140/apde.2019.12.245

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On a boundary value problem for conically deformed thin elastic sheets

Heiner Olbermann

Vol. 12 (2019), No. 1, 245–258

DOI: 10.2140/apde.2019.12.245

Abstract

We consider a thin elastic sheet in the shape of a disk that is clamped at its boundary such that the displacement and the deformation gradient coincide with a conical deformation with no stretching there. These are the boundary conditions of a so-called “d-cone”. We define the free elastic energy as a variation of the von Kármán energy, which penalizes bending energy in $L^{p}$ with $p \in (2, \frac{8}{3})$ (instead of, as usual, $p = 2$ ). We prove ansatz-free upper and lower bounds for the elastic energy that scale like $h^{p ∕ (p - 1)}$ , where $h$ is the thickness of the sheet.

Keywords

thin elastic sheets, d-cone

Mathematical Subject Classification 2010

Primary: 49Q10, 74K20

Milestones

Received: 4 January 2018

Revised: 5 March 2018

Accepted: 10 April 2018

Published: 2 August 2018

Authors

Heiner Olbermann
Universität Leipzig Leipzig Germany

Vol. 12, No. 1, 2019