Conical dislocations in crumpling

A crumpled piece of paper is made up of cylindrically curved or nearly plan ar regions folded along line-like ridges, which themselves pivot about poin t-like peaks; most of the deformation and energy is focused into these loca lized objects. Localization of deformation in thin sheets is a diverse phen omenon(1-6), and is a consequence of the fact(7) that bending a thin sheet is energetically more favourable than stretching it. Previous studies(8-11) considered the weakly nonlinear response of peaks and ridges to deformatio n. Here we report a quantitative description of the shape, response and sta bility 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 deriv e scaling laws for the size of the core, characterize the geometry of the d islocation away from the core, and analyse the interaction between two coni cal dislocations in a simple geometry. Our results show that the initial st ages of crumpling (characterized by the large deformation of a few folds) a re 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 t wo interacting dislocations, suggesting that a cascade of related instabili ties is responsible for the focusing of energy to progressively smaller sca les during crumpling.