Conical defects in growing sheets

TL;DR

This paper investigates the equilibrium shapes, energy states, and stress distribution of conical defects in growing sheets, revealing multiple stable configurations and the conditions under which self-contact occurs.

Contribution

It constructs and analyzes the discrete stable conical states in the bending-dominated regime, providing insights into their energies and stability criteria.

Findings

Multiple stable conical states identified

Critical surplus angle for self-contact determined

Ground state exhibits two-fold symmetry

Abstract

A growing or shrinking disc will adopt a conical shape, its intrinsic geometry characterized by a surplus angle $se$ at the apex. If growth is slow, the cone will find its equilibrium. Whereas this is trivial if $se <= 0$, the disc can fold into one of a discrete infinite number of states if $se$ is positive. We construct these states in the regime where bending dominates, determine their energies and how stress is distributed in them. For each state a critical value of $se$ is identified beyond which the cone touches itself. Before this occurs, all states are stable; the ground state has two-fold symmetry.