Almost Conical Deformations of Thin Sheets with Rotational Symmetry
Stefan Müller, Heiner Olbermann · SIAM Journal on Mathematical Analysis · 2014
It has been found in numerical experiments [T. Liang and T. A. Witten, Phys. Rev. E (3), 73 (2006), 046604] 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: First, we only consider radially symmetric configurations. Second, we consider the so-called von Kármán 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 [S. Müller and H. Olbermann, Calc. Var. Partial Differential Equations, (2013), 10.1007/s00526-013-0616-6]. Moreover, we determine the shape of minimizers at infinity up to terms that decay like $\exp(-c\sqrt{r})$.