Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Conical dislocations in crumpling

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.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

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.

References

  1. Calladine,C. Theory of Shell Structures (Cambridge Univ. Press, Cambridge, 1983).

    Book  Google Scholar 

  2. Wierzbicki,T. & Jones,N. (eds) Structural Failure (Wiley Interscience, New York, 1988).

    Google Scholar 

  3. Connelly,R. Rigidity and energy. Invent. Math. 66, 11–33 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  4. Nelson,D. R., Piran,T. & Weinberg,S. Statistical Mechanics of Membranes and Surfaces (World Scientific, Singapore, 1988).

    MATH  Google Scholar 

  5. Amirbayat,J. & Hearle,J. W. S. The complex buckling of flexible sheet materials. Int. J. Mech. Sci. 28, 339–370 (1986).

    Article  Google Scholar 

  6. da Vinci, Leonardo Notebooks Vol. I., Studies of Drapery (Dover Reprint, New York, 1984).

    Google Scholar 

  7. Rayleigh, Lord, Theory of Sound Vol. I, Ch. X a (Dover, New York, 1945).

    MATH  Google Scholar 

  8. Lobkovsky,A., Gentges,S., Li,H., Morse,D. & Witten,T. Stretched ridges in crumpling. Science 270, 1482–1485 (1995).

    Article  ADS  CAS  Google Scholar 

  9. Lobkovsky,A. & Witten,T. A. Properties of ridges in elastic membranes. Phys. Rev. E 55, 1577–1589 (1997).

    Article  ADS  CAS  Google Scholar 

  10. Chaieb,S. & Melo,F. Experimental study of developable cones. Phys. Rev. Lett. 80, 2354–2357 (1998).

    Article  ADS  CAS  Google Scholar 

  11. Cerda,E. & Mahadevan,L. Conical surfaces and crescent singularities in crumpled sheets. Phys. Rev. Lett. 80, 2358–2361 (1998).

    Article  ADS  CAS  Google Scholar 

  12. Ben Amar,M. & Pomeau,Y. Crumpled paper. Proc. R. Soc. Lond. A 453, 729–755 (1997).

    Article  MathSciNet  Google Scholar 

  13. Nabarro,F. R. N. Theory of Crystal Dislocations (Dover, New York, 1993).

    Google Scholar 

  14. Struik,D. J. Lectures on Classical Differential Geometry (Dover, New York, 1988).

    MATH  Google Scholar 

  15. Love,A. E. H. A Treatise on the Mathematical Theory of Elasticity (Dover, New York, 1944).

    MATH  Google Scholar 

  16. Timoshenko,S. & Gere,J. Theory of Elastic Stability (McGraw-Hill, New York, 1961).

    Google Scholar 

  17. Kramer,E. & Lobkovsky,A. Universal power law in the noise from a crumpled elastic sheet. Phys. Rev. E 53, 1465–1468 (1996).

    Article  ADS  CAS  Google Scholar 

  18. Houle,P. & Sethna,J. Acoustic emission from crumpling paper. Phys. Rev. E 54, 278–283 (1996).

    Article  ADS  CAS  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to L. Mahadevan.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/43395

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing