Skip to main content

Reversibility of crumpling on compressed thin sheets

Reversibility of crumpling


Compressing thin sheets usually yields the formation of singularities which focus curvature and stretching on points or lines. In particular, following the common experience of crumpled paper where a paper sheet is crushed in a paper ball, one might guess that elastic singularities should be the rule beyond some compression level. In contrast, we show here that, somewhat surprisingly, compressing a sheet between cylinders make singularities spontaneously disappear at large compression. This “stress defocusing” phenomenon is qualitatively explained from scale-invariance and further linked to a criterion based on a balance between stretching and curvature energies on defocused states. This criterion is made quantitative using the scalings relevant to sheet elasticity and compared to experiment. These results are synthesized in a phase diagram completed with plastic transitions and buckling saturation. They provide a renewed vision of elastic singularities as a thermodynamic condensed phase where stress is focused, in competition with a regular diluted phase where stress is defocused. The physical differences between phases is emphasized by determining experimentally the mechanical response when stress is focused or defocused and by recovering the corresponding scaling laws. In this phase diagram, different compression routes may be followed by constraining differently the two principal curvatures of a sheet. As evidenced here, this may provide an efficient way of compressing a sheet that avoids the occurrence of plastic damages by inducing a spontaneous regularization of geometry and stress.

Graphical abstract

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


  1. V. Pereira, A. Castro Neto, H. Liang, L. Mahadevan, Phys. Rev. Lett. 105, 156603 (2010).

    ADS  Article  Google Scholar 

  2. J. Genzer, J. Groenewold, Soft Matter 2, 310 (2006).

    ADS  Article  Google Scholar 

  3. U. Seifert, Adv. Phys. 46, 13 (1997).

    ADS  Article  Google Scholar 

  4. M. Alava, K. Niskanen, Rep. Prog. Phys. 69, 699 (2006).

    ADS  Article  Google Scholar 

  5. M. Golombek, F.S. Anderson, M.T. Zuber, J. Geophys. Res. 106, 811 (2001).

    Google Scholar 

  6. B. Roman, A. Pocheau, Phys. Rev. Lett. 108, 074301 (2012).

    ADS  Article  Google Scholar 

  7. L. Landau, E. Lifshitz, Theory of Elasticity (Elsevier, Cambridge, UK, 1986).

  8. T. Witten, Rev. Mod. Phys. 79, 643 (2007).

    ADS  Article  MATH  MathSciNet  Google Scholar 

  9. B. Audoly, Y. Pomeau, Elasticity and Geometry: From Hair Curls to the non-linear response of Shells (Oxford University Press, Oxford, UK, 2010).

  10. A. Lobkovsky, S. Gentges, H. Li, D. Morse, T. Witten, Science 270, 1482 (1995).

    ADS  Article  Google Scholar 

  11. A. Lobkovsky, Phys. Rev. E 53, 3750 (1996).

    ADS  Article  MathSciNet  Google Scholar 

  12. A. Lobkovsky, T. Witten, Phys. Rev. E 55, 1577 (1997).

    ADS  Article  Google Scholar 

  13. M. Ben Amar, Y. Pomeau, Proc. R. Soc. London, Ser. A 453, 729 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  14. E. Cerda, L. Mahadevan, Phys. Rev. Lett. 80, 2358 (1998).

    ADS  Article  Google Scholar 

  15. S. Chaïeb, F. Melo, J.C. Géminard, Phys. Rev. Lett. 80, 2354 (1998).

    ADS  Article  Google Scholar 

  16. L. Euler, Methodus Inveniendi Lineas Curvas Maximi Minimivi Propreitate Gaudentes. Additamentum I: De Curvis Elasticas (Lausanne & Geneva, 1744).

  17. W.A. Oldfather, C.A. Ellis, D. Brown, Isis 20, 72 (1930).

    Article  Google Scholar 

  18. D.J. Struik, Lectures on Classical Differential Geometry (Addison-Wesley, Reading, MA, 1961).

  19. B. Roman, A. Pocheau, Europhys. Lett. 46, 602 (1999).

    ADS  Article  Google Scholar 

  20. B. Roman, A. Pocheau, J. Mech. Phys. Solid 50, 2379 (2002).

    ADS  Article  MATH  Google Scholar 

  21. A. Pocheau, B. Roman, Physica D 192, 161 (2004).

    ADS  Article  MATH  MathSciNet  Google Scholar 

  22. J. Hure, B. Roman, J. Bico, Phys. Rev. Lett. 109, 054302 (2012).

    ADS  Article  Google Scholar 

  23. O. Kruglova, F. Brau, D. Villers, P. Damman, Phys. Rev. Lett. 107, 164303 (2011).

    ADS  Article  Google Scholar 

  24. A. Pogorelov, Bendings of Surfaces and Stability of Shells (American Mathematical Society, Providence, 1988).

  25. C. Quilliet, Eur. Phys. J. E 35, 1 (2012).

    Article  Google Scholar 

  26. B. Du, O.C. Tsui, Q. Zhang, T. He, Langmuir 17, 3286 (2001).

    Article  Google Scholar 

  27. A. Boudaoud, P. Patricio, Y. Couder, M. Ben Amar, Nature 407, 718 (2000).

    ADS  Article  Google Scholar 

  28. B. Davidovitch, Phys. Rev. E 80, 025202 (2009).

    ADS  Article  Google Scholar 

  29. B. Davidovitch, R.D. Schroll, D. Vella, M. Adda-Bedia, E.A. Cerda, Proc. Natl. Acad. Sci. U.S.A. 108, 18227 (2011).

    ADS  Article  Google Scholar 

  30. J. Huang, M. Juszkiewicz, W. de Jeu, E. Cerda, T. Emrick, N. Menon, T. Russell, Science 317, 650 (2007).

    ADS  Article  Google Scholar 

  31. H. Vandeparre, M. Piñeirua, F. Brau, B. Roman, J. Bico, C. Gay, W. Bao, C. Lau, P. Reis, P. Damman, Phys. Rev. Lett. 106, 224301 (2011).

    ADS  Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Alain Pocheau.

Additional information

Contribution to the Topical Issue “Irreversible Dynamics: A topical issue dedicated to Paul Manneville” edited by Patrice Le Gal and Laurette S. Tuckerman.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pocheau, A., Roman, B. Reversibility of crumpling on compressed thin sheets. Eur. Phys. J. E 37, 28 (2014).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI:


  • Topical issue: Irreversible Dynamics: A topical issue dedicated to Paul Manneville