Conical instabilities on paper

TL;DR

This paper analyzes the stability of conical defects in unstretchable flat sheets, revealing conditions for stability and instability of various defect modes through a detailed mathematical framework.

Contribution

It introduces a second-order expansion of bending energy and a global constraint to analyze defect stability, providing new insights into defect mode behavior.

Findings

2-fold ground state is stable

Excited states have multiple unstable modes

Critical surplus angle determines mode stability

Abstract

The stability of the fundamental defects of an unstretchable flat sheet is examined. This involves expanding the bending energy to second order in deformations about the defect. The modes of deformation occur as eigenstates of a fourth-order linear differential operator. Unstretchability places a global linear constraint on these modes. Conical defects with a surplus angle exhibit an infinite number of states. If this angle is below a critical value, these states possess an n-fold symmetry labeled by an integer, n \geq 2. A nonlinear stability analysis shows that the 2-fold ground state is stable, whereas excited states possess 2(n - 2) unstable modes which come in even and odd pairs.