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Scale invariance and universality of force networks in static granular matter

Nature volume 439, pages 828–830 (2006)Cite this article

Abstract

Force networks form the skeleton of static granular matter1,2. They are the key factor that determines mechanical properties such as stability3, elasticity4,5 and sound transmission6,7, which are important for civil engineering and industrial processing. Previous studies have focused on investigations of the global structure of external forces8,9,10,11 (the boundary condition) and on the probability distribution of individual contact forces4,12. So far, however, precise knowledge of the disordered spatial structure of the force network has remained elusive. Here we report that molecular dynamics simulations of realistic granular packings reveal scale invariance of clusters of particles interacting by means of relatively strong forces. Despite visual variation, force networks for various values of the confining pressure and other parameters have identical scaling exponents and scaling function, thereby determining a universality class. Unexpectedly, the flat ensemble of force configurations13,14,15 (a simple generalization of equilibrium statistical mechanics) belongs to this universality class, whereas some widely studied simplified models16,17,18 do not. This implies that the elasticity of the grains and their geometrical disorder do not affect the universal mechanical properties.

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Figure 1: Dependence of the force network on pressure and polydispersity.
Figure 2: Force chains at different scales.
Figure 3: The universal scaling function.

References

  1. Garcia-Rojo, R., Herrmann, H. J. & McNamara, S. (eds) Powders and Grains 2005 (Balkema, Leiden, 2005)

  2. Jaeger, H. M., Nagel, S. R. & Behringer, R. P. Granular solids, liquids and gases. Rev. Mod. Phys. 68, 1259–1273 (1996)

    Article  ADS  Google Scholar 

  3. Daerr, A. & Douady, S. Two types of avalanche behaviour in granular media. Nature 399, 241–243 (1999)

    Article  ADS  CAS  Google Scholar 

  4. Makse, H. A., Gland, N., Johnson, D. L. & Schwartz, L. M. Why effective medium theory fails in granular materials. Phys. Rev. Lett. 83, 5070–5073 (1999)

    Article  ADS  CAS  Google Scholar 

  5. Goldenberg, C. & Goldhirsch, I. Friction enhances elasticity in granular solids. Nature 435, 188–191 (2005)

    Article  ADS  CAS  Google Scholar 

  6. Jia, X., Caroli, C. & Velicky, B. Ultrasound propagation in externally stressed granular media. Phys. Rev. Lett. 82, 1863–1866 (1999)

    Article  ADS  CAS  Google Scholar 

  7. Somfai, E., Roux, J.-N., Snoeijer, J. H., van Hecke, M. & van Saarloos, W. Wave propagation in confined granular systems. Phys. Rev. E 72, 021301 (2005)

    Article  ADS  Google Scholar 

  8. Vanel, L., Howell, D., Clark, D., Behringer, R. P. & Clement, 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 

  9. Wittmer, J. P., Claudin, P., Cates, M. E. & Bouchaud, J.-P. An explanation for the central stress minimum in sand piles. Nature 382, 336–339 (1996)

    Article  ADS  CAS  Google Scholar 

  10. 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 

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

    Article  ADS  CAS  Google Scholar 

  12. Majmudar, T. S. & Behringer, R. P. Contact force measurements and stress-induced anisotropy in granular materials. Nature 435, 1079–1082 (2005)

    Article  ADS  CAS  Google Scholar 

  13. Edwards, S. F. & Oakeshott, R. Theory of powders. Physica A 157, 1080–1090 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  14. Makse, H. A. & Kurchan, J. Testing the thermodynamics approach to granular matter with a numerical model of a decisive experiment. Nature 415, 614–617 (2002)

    Article  ADS  CAS  Google Scholar 

  15. Snoeijer, J. H., Vlugt, T. J. H., van Hecke, M. & van Saarloos, W. Force network ensemble: a new approach to static granular matter. Phys. Rev. Lett. 92, 054302 (2004)

    Article  ADS  Google Scholar 

  16. Liu, C.-h. et al. Force fluctuations in bead packs. Science 269, 513–515 (1995)

    Article  ADS  CAS  Google Scholar 

  17. Coppersmith, S. N., Liu, C.-h., Majumdar, S., Narayan, O. & Witten, T. A. Model for force fluctuations in bead packs. Phys. Rev. E 53, 4673–4685 (1996)

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  19. Radjai, F., Jean, M., Moreau, J.-J. & Roux, S. Force distributions in dense two-dimensional granular systems. Phys. Rev. Lett. 77, 274–277 (1996)

    Article  ADS  CAS  Google Scholar 

  20. Makse, H. A., Johnson, D. L. & Schwartz, L. M. Packing of compressible granular materials. Phys. Rev. Lett. 84, 4160–4163 (2000)

    Article  ADS  CAS  Google Scholar 

  21. Mueth, D. M., Jaeger, H. M. & Nagel, S. R. Force distribution in a granular medium. Phys. Rev. E 57, 3164–3169 (1998)

    Article  ADS  CAS  Google Scholar 

  22. Blair, D. L., Mueggenburg, N. W., Marshall, A. H., Jaeger, H. M. & Nagel, S. R. Force distributions in 3D granular assemblies: effects of packing order and inter-particle friction. Phys. Rev. E 63, 041304 (2001)

    Article  ADS  CAS  Google Scholar 

  23. Erikson, J. M., Mueggenburg, N. W., Jaeger, H. M. & Nagel, S. R. Force distributions in three-dimensional compressible granular packs. Phys. Rev. E 66, 040301 (2002)

    Article  ADS  Google Scholar 

  24. Fortuin, C. M. & Kasteleyn, P. W. On the random cluster model. Physica 57, 536–564 (1971)

    Article  ADS  Google Scholar 

  25. Stauffer, D. & Aharony, A. Introduction to Percolation Theory (Taylor & Francis, London, 1991)

    MATH  Google Scholar 

  26. Brujic, J. et al. 3D bulk measurements of the force distribution in a compressed emulsion system. Faraday Disc. 123, 207–220 (2003)

    Article  ADS  CAS  Google Scholar 

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

    Article  Google Scholar 

  28. Johnson, K. L. Contact mechanics (Cambridge Univ. Press, Cambridge, UK, 1985)

    Book  Google Scholar 

  29. Ostojic, S. & Panja, D. Response of a hexagonal granular packing under a localized external force. Europhys. Lett. 71, 70–77 (2004)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank B. Behringer, W. Ellenbroek, C. Goldenberg, M. van Hecke, W. van Saarloos and K. Shundyak for discussions. S.O. is financially supported by the Dutch research organization FOM (Fundamenteel Onderzoek der Materie). E.S. is supported by the PHYNECS training network of the European Commission.

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Author notes

  1. Ellák Somfai

    Present address: The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, UK

Authors and Affiliations

  1. Institute for Theoretical Physics, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE, Amsterdam, The Netherlands

    Srdjan Ostojic & Bernard Nienhuis

  2. Universiteit Leiden, Instituut-Lorentz, PO Box 9506, 2300 RA, Leiden, The Netherlands

    Ellák Somfai

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  1. Srdjan Ostojic
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Correspondence to Bernard Nienhuis.

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Supplementary Figure 1

This file contains a Supplementary Figure displaying probability distributions of force magnitudes in all the systems studied. (PDF 69 kb)

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Ostojic, S., Somfai, E. & Nienhuis, B. Scale invariance and universality of force networks in static granular matter. Nature 439, 828–830 (2006). https://doi.org/10.1038/nature04549

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