The shape of the edge of a leaf

TL;DR

This paper investigates the shapes of thin sheets with non-flat metrics, analyzing when they buckle and how their energy-minimizing configurations can be determined both analytically and numerically.

Contribution

It provides new analytical conditions and numerical methods for understanding buckling and shape formation in stretched sheets with non-uniform metrics.

Findings

Buckling occurs when a stretched sheet loses axial symmetry.

Derived energy expressions for numerical and analytical studies.

Numerical solutions for shape configurations of strips with linear metric gradients.

Abstract

Leaves and flowers frequently have a characteristic rippling pattern at their edges. Recent experiments found similar patterns in torn plastic. These patterns can be reproduced by imposing metrics upon thin sheets. The goal of this paper is to discuss a collection of analytical and numerical results for the shape of a sheet with a non--flat metric. First, a simple condition is found to determine when a stretched sheet folded into a cylinder loses axial symmetry, and buckles like a flower. General expressions are next found for the energy of stretched sheet, both in forms suitable for numerical investigation, and for analytical studies in the continuum. The bulk of the paper focuses upon long thin strips of material with a linear gradient in metric. In some special cases, the energy--minimizing shapes of such strips can be determined analytically. Euler--Lagrange equations are found which determine the shapes in general. The paper closes with numerical investigations of these equations.