Almost conical deformations of thin sheets with rotational symmetry
Abstract
It has been found in numerical experiments that when one removes a sector from an elastic sheet and glues the edges of the sector back together, the resulting configuration is radially symmetric and nearly conical. We make a rigorous analysis of this setting under two simplyfying assumptions: Firstly, we only consider radially symmetric configurations. Secondly, we consider the so-called von-K\'arm\'an limit, where the size of the removed region as well as the deformations are small. We choose free boundary conditions for a sheet of infinite size. We show existence of minimizers of the suitably renormalized free energy functional. As a by-product, we obtain a lower bound for the elastic energy that has been conjectured in the related context of d-cones. Moreover, we determine the shape of minimizers at infinity up to exponentially decaying terms.
Keywords
Cite
@article{arxiv.1212.0705,
title = {Almost conical deformations of thin sheets with rotational symmetry},
author = {Stefan Müller and Heiner Olbermann},
journal= {arXiv preprint arXiv:1212.0705},
year = {2013}
}
Comments
18 pages. Revised and streamlined version. Section 1: Comment to relation to d-cones added. Section 2: Boundary conditions at 0 no longer imposed, but as consequence of energy estimates (Proposition 1). Section 3: Reorganized and shortened. Explanation of the strategy of proof added. All interpolation estimates now based on Lemma 2. Section 4: Shortened. Weighted energy estimates no longer needed