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 phenomenon1±6, and is a
consequence of the fact7 that bending a thin sheet is energetically
more favourable than stretching it. Previous studies8±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.