Conical dislocations in crumpling

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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 phenomenon1,2,3,4,5,6, and is a consequence of the fact7 that bending a thin sheet is energetically more favourable than stretching it. Previous studies8,9,10,11 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.

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Figure 1: Geometry of an ideal conical dislocation.
Figure 2: Geometry of a real conical dislocation.
Figure 3: Mechanical response of a conical dislocation.
Figure 4: Geometry and mechanical response of two interacting conical dislocations.

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Acknowledgements

E.C. was supported by the Chilean Presidente de la República postdoctoral fellowship during the course of this work at MIT in 1997–98. S.C. was supported by a postdoctoral fellowship at Universidad de Santiago de Chile in 1997–98 during the course of this work. Additional support was provided by the Chilean Cátedra Presidencial en Ciencias (F.M.), the Karl van Tassel career development chair and the Sloan fund (L.M.) at MIT.

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Correspondence to L. Mahadevan.

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Cerda, E., Chaieb, S., Melo, F. et al. Conical dislocations in crumpling. Nature 401, 46–49 (1999) doi:10.1038/43395

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