The compensation of Gaussian curvature in developable cones is local

TL;DR

This study uses an angular deficit scheme to analyze Gaussian curvature distribution in developable cones, revealing local compensation phenomena near the rim that challenge global curvature constraints.

Contribution

It introduces a numerical method to analyze local Gaussian curvature distribution in d-cones, highlighting local compensation effects contrary to global Gauss-Bonnet expectations.

Findings

Negative Gaussian curvature at the rim where the sheet contacts the container.

Adjacent bands with positive Gaussian curvature compensate the negative curvature.

Local curvature compensation increases as the d-cone thickness decreases.

Abstract

In this paper we use the angular deficit scheme [V. Borrelli, F. Cazals, and J.-M. Morvan, {\sl Computer Aided Geometric Design} {\bf 20}, 319 (2003)] to determine the distribution of Gaussian curvature in developable cones (d-cones) [E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan, {\sl Nature} {\bf 401}, 46 (1999)] numerically. These d-cones are formed by pushing a thin elastic sheet into a circular container. Negative Gaussian curvatures are identified at the rim where the sheet touches the container. Around the rim there are two narrow bands with positive Gaussian curvatures. The integral of the (negative) Gaussian curvature near the rim is almost completely compensated by that of the two adjacent bands. This suggests that the Gauss-Bonnet theorem which constrains the integral of Gaussian curvature globally does not explain the spontaneous curvature cancellation phenomenon [T. Liang and T. A. Witten, {\sl Phys. Rev. E} {\bf 73}, 046604 (2006)]. The locality of the compensation seems to increase for decreasing d-cone thickness. The angular deficit scheme also provides a new way to confirm the curvature cancellation phenomenon.