Calculus of Variations and Geometric Measure Theory
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S. Müller - H. Olbermann

Almost conical deformations of thin sheets with rotational symmetry

created by olbermann on 23 Jul 2019



Inserted: 23 jul 2019

Year: 2012

ArXiv: 1212.0705 PDF


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.

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