Eur. Phys. J. E (2013) 36: 106
https://doi.org/10.1140/epje/i2013-13106-0

Regular Article

Dipoles in thin sheets

¹ Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apdo. Postal 70-543, 04510, México D.F., México
² Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Norris Hall (MC 0219), 495 Old Turner Street, 24061, Blacksburg, VA, USA
³ Equipe BioPhysStat, ICPMB-FR CNRS 2843, Université de Lorraine, 1 boulevard Arago, 57070, Metz, France
⁴ Institut Charles Sadron, CNRS, 23 rue du Loess, BP 84047, 67034, Strasbourg, France

^* e-mail: Martin-Michael.Mueller@univ-lorraine.fr

Received: 24 June 2013
Revised: 14 August 2013
Accepted: 19 August 2013
Published online: 26 September 2013

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.