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.

Friction enhances elasticity in granular solids

Nature volume 435, pages 188–191 (2005)Cite this article

Abstract

For years, engineers have used elastic and plastic models to describe the properties of granular solids, such as sand piles and grains in silos1,2,3. However, there are theoretical4,5,6 and experimental7,8,9,10,11,12,13,14 results that challenge this approach. Specifically, it has been claimed4,5,6 that stress in granular solids propagates in a manner described by wave-like (hyperbolic) equations, rather than the elliptic equations of static elasticity. Here we report numerical simulations of the response of a two-dimensional granular slab to an external load, revealing that both approaches are valid—albeit on different length scales. For small systems that can be considered mesoscopic on the scale of the grains, a hyperbolic-like, strongly anisotropic response is expected. However, in large systems (those typically considered by engineers), the response is closer to that predicted by traditional isotropic elasticity models. Static friction, often ignored in simple models, plays a key role: it increases the elastic range and renders the response more isotropic, even beyond this range.

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

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Get just this article for as long as you need it

$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Force distribution and linearity below a lattice of disks.
Figure 2: The changes (the teardrop) in the contact network induced by an applied force, for different values of Fext and µ.
Figure 3: Effects of friction.
Figure 4: Effects of disorder and a phase diagram.

References

  1. Nedderman, R. M. Statics and Kinematics of Granular Materials (Cambridge Univ. Press, Cambridge, 1992)

    Book  Google Scholar 

  2. Savage, S. B. in Physics of Dry Granular Media (eds Herrmann, H. J., Hovi, J. P. & Luding, S.) 25–95 (NATO ASI Series, Kluwer, Dordrecht, 1998)

    Book  Google Scholar 

  3. Jackson, R. in Theory of Dispersed Multiphase Flow (ed. Meyer, R. E.) 291–337 (Academic, New York, 1983)

    Book  Google Scholar 

  4. Wittmer, J. P., Claudin, P., Cates, M. E. & Bouchaud, J.-P. An explanation for the central stress minimum in sand piles. Nature 382, 336–338 (1996)

    Article  ADS  CAS  Google Scholar 

  5. Bouchaud, J.-P., Claudin, P., Levine, D. & Otto, M. Force chain splitting in granular materials: a mechanism for large-scale pseudo-elastic behaviour. Eur. Phys. J. E 4, 451–457 (2001)

    Article  CAS  Google Scholar 

  6. Tkachenko, A. V. & Witten, T. A. Stress propagation through frictionless granular material. Phys. Rev. E 60, 687–696 (1999)

    Article  ADS  CAS  Google Scholar 

  7. Drescher, A. & de Josselin de Jong, G. Photoelastic verification of a mechanical model for the flow of a granular material. J. Mech. Phys. Solids 20, 337–351 (1972)

    Article  ADS  Google Scholar 

  8. Geng, J. et al. Footprints in sand: the response of a granular material to local perturbations. Phys. Rev. Lett. 87, 035506 (2001)

    Article  ADS  CAS  Google Scholar 

  9. Geng, J., Reydellet, G., Clément, E. & Behringer, R. P. Green's function measurements of force transmission in 2D granular materials. Physica D 182, 274–303 (2003)

    Article  ADS  Google Scholar 

  10. Mueggenburg, N. W., Jaeger, H. M. & Nagel, S. R. Stress transmission through three-dimensional ordered granular arrays. Phys. Rev. E 66, 031304 (2002)

    Article  ADS  Google Scholar 

  11. Moukarzel, C. F., Pacheco-Martinez, H., Ruiz-Suarez, J. C. & Vidales, A. M. Static response in disk packings. Granular Matter 6, 61–66 (2004)

    Article  CAS  Google Scholar 

  12. da Silva, M. & Rajchenbach, J. Stress transmission through a model system of cohesionless elastic grains. Nature 406, 708–710 (2000)

    Article  ADS  CAS  PubMed  Google Scholar 

  13. Smid, J. & Novosad, J. Pressure distribution under heaped bulk solids. In Proc. 1981 Powtech Conf., Institution of Chemical Engineers Symp. 63, D3/V/1–12 (1981).

  14. Brockbank, R., Huntley, J. M. & Ball, R. C. Contact force distribution beneath a three-dimensional granular pile. J. Phys. II 7, 1521–1532 (1997)

    CAS  Google Scholar 

  15. Vanel, L., Howell, D., Clark, D., Behringer, R. P. & Clément, E. Memories in sand: experimental tests of construction history on stress distributions under sandpiles. Phys. Rev. E 60, R5040–R5043 (1999)

    Article  ADS  CAS  Google Scholar 

  16. Serero, D., Reydellet, G., Claudin, P., Clément, E. & Levine, D. Stress response function of a granular layer: quantitative comparison between experiments and isotropic elasticity. Eur. Phys. J. E 6, 169–179 (2001)

    Article  CAS  Google Scholar 

  17. Goldenberg, C. & Goldhirsch, I. Force chains, microelasticity, and macroelasticity. Phys. Rev. Lett. 89, 084302 (2002)

    Article  ADS  CAS  PubMed  Google Scholar 

  18. Reydellet, G. & Clément, E. Green's function probe of a static granular piling. Phys. Rev. Lett. 86, 3308–3311 (2001)

    Article  ADS  CAS  Google Scholar 

  19. Cundall, P. A. & Strack, O. D. L. A discrete numerical model for granular assemblies. Geotechnique 29, 47–65 (1979)

    Article  Google Scholar 

  20. Otto, M., Bouchaud, J.-P., Claudin, P. & Socolar, J. E. S. Anisotropy in granular media: classical elasticity and directed-force chain network. Phys. Rev. E 67, 031302 (2003)

    Article  ADS  CAS  Google Scholar 

  21. Duffy, J. & Mindlin, R. D. Stress–strain relations and vibrations of a granular medium. J. Appl. Mech. 24, 585–593 (1957)

    MathSciNet  Google Scholar 

  22. Bathurst, R. J. & Rothenburg, L. Micromechanical aspects of isotropic granular assemblies with linear contact interactions. J. Appl. Mech. 55, 17–23 (1988)

    Article  ADS  Google Scholar 

  23. Chang, C. S. & Ma, L. Elastic material constants for isotropic granular solids with particle rotation. Int. J. Solids Struct. 29, 1001–1018 (1992)

    Article  Google Scholar 

  24. Gay, C. & da Silveira, R. Anisotropic elastic theory of preloaded granular media. Europhys. Lett. 68, 51–57 (2004)

    Article  ADS  CAS  Google Scholar 

  25. da Silveira, R., Vidalenc, G. & Gay, C. Stress propagation in two dimensional frictional granular matter. Preprint at 〈http://arXiv.org/cond-mat/0208214〉 (2002).

  26. Johnson, K. L. Contact Mechanics 220 (Cambridge Univ. Press, Cambridge, 1985)

    Book  Google Scholar 

  27. Tournat, V. Probing weak forces in granular media through nonlinear dynamic dilatancy: clapping contacts and polarization anisotropy. Phys. Rev. Lett. 92, 085502 (2004)

    Article  ADS  CAS  PubMed  Google Scholar 

Download references

Acknowledgements

We thank A. P. F. Atman, R. P. Behringer, P. Claudin, E. Clément, J. Geng, N. Mueggenburg, M. van Hecke, W. van Saarloos and T. A. Witten for discussions. This work was supported by the Israel Science Foundation (ISF) and the US-Israel Binational Science Foundation (BSF).

Author information

Author notes

  1. C. Goldenberg

    Present address: Physique et Mécanique des Milieux Hétérogènes, ESPCI, 10 rue Vauquelin, 75231, Paris, Cedex 5, France

Authors and Affiliations

  1. School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences,

    C. Goldenberg

  2. Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering, Tel-Aviv University, 69978, Tel-Aviv, Israel

    I. Goldhirsch

Authors
  1. C. Goldenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Goldhirsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. Goldhirsch.

Ethics declarations

Competing interests

The authors declare that they have no competing financial interests.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Goldenberg, C., Goldhirsch, I. Friction enhances elasticity in granular solids. Nature 435, 188–191 (2005). https://doi.org/10.1038/nature03497

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature03497

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

Advanced 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