1. Departamento de Física de la Universidad de Santiago de Chile, Avda Ecuador 3493, Casilla 307 Correo 2, Santiago, Chile
2. Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

Correspondence to: L. Mahadevan² Correspondence and requests for materials should be addressed to L.M. (Email: l_m@mit.edu).

Abstract

A crumpled piece of paper is made up of cylindrically curved or nearly planar regions folded along line-like ridges, which themselves pivot about point-like peaks; most of the deformation and energy is focused into these localized objects. Localization of deformation in thin sheets is a diverse phenomenon^{1, 2, 3, 4, 5, 6}, and is a consequence of the fact⁷ that bending a thin sheet is energetically more favourable than stretching it. Previous studies^{8, 9, 10, 11} considered the weakly nonlinear response of peaks and ridges to deformation. Here we report a quantitative description of the shape, response and stability of conical dislocations, the simplest type of topological crumpling deformation. The dislocation consists of a stretched core, in which some of the energy resides, and a peripheral region dominated by bending. We derive scaling laws for the size of the core, characterize the geometry of the dislocation away from the core, and analyse the interaction between two conical dislocations in a simple geometry. Our results show that the initial stages of crumpling (characterized by the large deformation of a few folds) are dominated by bending only. By considering the response of a transversely forced conical dislocation, we show that it is dynamically unstable above a critical load threshold. A similar instability is found for the case of two interacting dislocations, suggesting that a cascade of related instabilities is responsible for the focusing of energy to progressively smaller scales during crumpling.