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Dynamics of singularities in a constrained elastic plate

Nature volume 407, pages 718–720 (2000)Cite this article

Abstract

Large deformations of thin elastic plates usually lead to the formation of singular structures which are either linear1,2,3,4 (ridges) or pointlike5,6,7,8 (developable cones). These structures are thought to be generic for crumpled plates3,5, although they have been investigated quantitatively only in simplified geometries1,2,3,4,6,7,8. Previous studies9,10,11 have also shown that a large number of singularities are generated by successive instabilities. Here we study, experimentally and numerically, a generic situation in which a plate is initially bent in one direction into a cylindrical arch, then deformed in the other direction by a load applied at its centre. This induces the generation of pairs of singularities; we study their position, their dynamics and the corresponding resistance of the plate to deformation. We solve numerically the equations describing large deformations of plates; developable cones are predicted, in quantitative agreement with the experiments. We use geometrical arguments to predict the observed patterns, assuming that the energy of the plate is given by the energy of the singularities.

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Figure 1: Observed patterns with a Mylar sheet of thickness h = 0.35 mm, Young's modulus E = 3.8 × 109 N m-2 and Poisson ratio ν = 0.4.
Figure 2: Two numerical situations with α = 20° and d = 16.5 cm.
Figure 3: Force F versus displacement of the centre Z with α = 20° and d = 16.5 cm.
Figure 4: Distance between the dominant d-cones and the centre with various α and d values.

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References

  1. Pogorelov, A. V. Bendings of Surfaces and Stability of Shells. (Translations of Mathematical Monographs No. 72, American Mathematical Society, 1988).

    Book  Google Scholar 

  2. Witten, T. A. & Li, H. Asymptotic shapes of a fullerene ball. Europhys. Lett. 23, 51– 55 (1993).

    Article  ADS  Google Scholar 

  3. Lobkovsky, A. E., Gentges, S., Li, H., Morse, D. & Witten, T. A. Scaling properties of stretching ridges in a crumpled elastic sheet. Science 270, 1482– 1485 (1995).

    Article  ADS  CAS  Google Scholar 

  4. Lobkovsky, A. E. Boundary layer analysis of the ridge singularity in a thin plate. Phys. Rev. E 53, 3750–3759 (1996).

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Chaı ¨eb, S. & Melo, F. Experimental study of developable cones. Phys. Rev. Lett. 80, 2354–2357 (1998).

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  8. Cerda, E., Chaı ¨eb, S., Melo, F. & Mahadevan, L. Conical dislocations in crumpling. Nature 401, 46–49 (1999).

    Article  ADS  CAS  Google Scholar 

  9. Chaı ¨eb, S. & Melo, F. Experimental study of crease formation in an axially compressed sheet. Phys. Rev. E 56, 4736–4745 ( 1997).

    Article  ADS  Google Scholar 

  10. Pomeau, Y. & Rica, S. Plaques très comprimées. C.R. Acad. Sci. Paris 325, Série IIb, 181–187 (1997).

    ADS  CAS  MATH  Google Scholar 

  11. Pauchard, L. & Rica, S. Contact and compression of elastic spherical shells. Phil. Mag. B 78, 225– 233 (1998).

    Article  ADS  CAS  Google Scholar 

  12. Pippard, A. B. The elastic arch and its modes of instability. Eur. J. Phys. 11, 359–365 (1990).

    Article  Google Scholar 

  13. Patrício, P., Adda-Bedia, M. & Ben Amar, M. An elastica problem: Instabilities of an elastic arch. Physica D 124, 181– 187 (1998).

    Article  MathSciNet  Google Scholar 

  14. Landau, L. D. & Lifshitz, E. M. Theory of Elasticity (Pergamon, Oxford, 1986).

    MATH  Google Scholar 

  15. Patrício, P. & Krauth, W. Numerical solutions of the Von Karman equations for a thin plate. Int. J. Mod. Phys. C 8, 427–434 ( 1997).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank L. Quartier and M. Haddad for their valuable help. This work was finished during the stay of M.B.A. and A.B. at the Department of Mechanical Engineering of the MIT; M.B.A. and A.B. are grateful for the hospitality they received.

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Authors and Affiliations

  1. Laboratoire de Physique Statistique de l'ENS (associated with CNRS and the universities Paris VI and Paris VII) , 24 rue Lhomond, Paris, F-75231 Cedex 05, France

    Arezki Boudaoud, Yves Couder & Martine Ben Amar

  2. Centro de Física da Matéria Condensada, Universidade de Lisboa Av. Prof. Gama Pinto 2, Lisbon, 1649-003, Portugal

    Pedro Patrício

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  1. Arezki Boudaoud
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  2. Pedro Patrício
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Correspondence to Arezki Boudaoud.

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Boudaoud, A., Patrício, P., Couder, Y. et al. Dynamics of singularities in a constrained elastic plate. Nature 407, 718–720 (2000). https://doi.org/10.1038/35037535

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