#### Vol. 12, No. 1, 2019

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On a boundary value problem for conically deformed thin elastic sheets

### Heiner Olbermann

Vol. 12 (2019), No. 1, 245–258
##### 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 $p\in \left(2,\frac{8}{3}\right)$ (instead of, as usual, $p=2$). We prove ansatz-free upper and lower bounds for the elastic energy that scale like ${h}^{p∕\left(p-1\right)}$, 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