Dipoles in thin sheets

TL;DR

This paper explores the complex shapes formed by dipole-like defects in thin elastic sheets, revealing multiple stable configurations and matching experimental observations.

Contribution

It introduces a nonlinear field theory approach to analyze dipoles in thin sheets, identifying multiple stable minima and unstable equilibria.

Findings

Two stable minima for dipole configurations.

Good agreement with experimental shapes.

Energy evaluations in the thin-sheet limit.

Abstract

A flat elastic sheet may contain pointlike conical singularities that carry a metrical "charge" of Gaussian curvature. Adding such elementary defects to a sheet allows one to make many shapes, in a manner broadly analogous to the familiar multipole construction in electrostatics. However, here the underlying field theory is non-linear, and superposition of intrinsic defects is non-trivial as it must respect the immersion of the resulting surface in three dimensions. We consider a "charge-neutral" dipole composed of two conical singularities of opposite sign. Unlike the relatively simple electrostatic case, here there are two distinct stable minima and an infinity of unstable equilibria. We determine the shapes of the minima and evaluate their energies in the thin-sheet regime where bending dominates over stretching. Our predictions are in surprisingly good agreement with experiments on paper sheets.