Energy scaling law for the regular cone

Abstract

We consider a thin elastic sheet in the shape of a disk whose reference metric is that of a singular cone. I.e., the reference metric is flat away from the center and has a defect there. We define a geometrically fully nonlinear free elastic energy, and investigate the scaling behavior of this energy as the thickness $h$ tends to 0. We work with two simplifying assumptions: Firstly, we think of the deformed sheet as an immersed 2-dimensional Riemannian manifold in Euclidean 3-space and assume that the exponential map at the origin (the center of the sheet) supplies a coordinate chart for the whole manifold. Secondly, the energy functional penalizes the difference between the induced metric and the reference metric in $L^{\infty }$ (instead of, as is usual, in $L^2$). Under these assumptions, we show that the elastic energy per unit thickness of the regular cone in the leading order of $h$ is given by $C^{\ast }{h}^2|\log h|$, where the value of $C^{\ast }$ is given explicitly.

26 pages, Sections 1 and 2 rearranged, introduction streamlined

References in corpus (1)

• Virus shapes and buckling transitions in spherical shells

Cited by in corpus (2)

• The shape of low energy configurations of a thin elastic sheet with a single disclination
• On a boundary value problem for conically deformed thin elastic sheets