Skip to main content
SpringerLink
Log in

Contact forces distribution for a granular material from a Monte Carlo study on a single grain

  • Original Paper
  • Published:
Granular Matter Aims and scope Submit manuscript

Abstract

The force network ensemble is one of the most promising statistical descriptions of granular media, with an entropy accounting for all force configurations at mechanical equilibrium consistent with some external stress. It is possible to define a temperature-like parameter, the angoricity \(\alpha ^{-1}\), which under isotropic compression is a scalar variable. This ensemble is frequently studied on whole packings of grains; ho wever, previous works have shown that spatial correlations can be neglected in many cases, opening the door to studies on a single grain. Our work develops a Monte Carlo method to sample the force ensemble on a single grain at constant angoricity on two and three-dimensional mono-disperse granular systems, both with or without static friction. The results show that, despite the steric exclusions and the constrictions of Coulomb’s limit and repulsive normal forces, the pressure per grain always show a k-gamma distribution with scale parameter \(\nu =\alpha ^{-1}\) and shape parameter k close to \(k'\), the number of degrees of freedom in the system. Moreover, the average pressure per grain fulfills an equipartition theorem \(\langle p \rangle = k' {\alpha }^{-1} \) in all cases (in close parallelism with the one for an ideal gas). Those results are in good agreement with the analysis of previous experimental and numerical results on many-grain systems, and with our own molecular dynamics simulations. These results suggest the existence of \(k'\) independent random variables (i.e. elementary forces) with identical exponential distributions as the basic elements for describing the force network ensemble at low angoricities under isotropic compression, in analogy with the volume ensemble of granular materials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Agnolin, I., Roux, J.: Internal states of model isotropic granular packings. I. assembling process, geometry, and contact networks. Phys. Rev. E 76(6), 061302 (2007). https://doi.org/10.1103/PhysRevE.76.061302

    Article  ADS  MathSciNet  Google Scholar 

  2. Anikeenko, A., Medvedev, N., Aste, T.: Structural and entropic insights into the nature of the random-close-packing limit. Phys. Rev. E 77, 031101 (2008). https://doi.org/10.1103/PhysRevE.77.031101

    Article  ADS  Google Scholar 

  3. Aste, T., Di Matteo, T.: Emergence of gamma distributions in granular materials and packing models. Phys. Rev. E (2008). https://doi.org/10.1103/PhysRevE.77.021309

    Google Scholar 

  4. Aste, T., Di Matteo, T., Saadalfar, M., Senden, T., Swinney, H.: An invariant distribution in static granular media. Euro. Phys. Lett. 79(2), 24003 (2007). https://doi.org/10.1209/0295-5075/79/24003

    Article  ADS  Google Scholar 

  5. Blumenfeld, R., Jordan, J., Edwards, S.: Interdependence of the volume and stress ensembles and equipartition in statistical mechanics of granular systems. Phys. Rev. Lett. 109(23), 238001 (2012). https://doi.org/10.1103/PhysRevLett.109.238001

    Article  ADS  Google Scholar 

  6. Clusel, M., Corwin, E., Siemens, A., Brujić, J.: A granocentric model for random packing of jammed emulsions. Nature 460(7255), 611 (2009). https://doi.org/10.1038/nature08158

    Article  ADS  Google Scholar 

  7. Corwin, E., Clusel, M., Siemens, A., Brujić, J.: Model for random packing of polydisperse frictionless spheres. Soft Matter 6(13), 2949 (2010). https://doi.org/10.1039/c000984a

    Article  ADS  Google Scholar 

  8. Cowin, S., Satake, M.: Continuum mechanical and statistical approaches in the mechanics of granular materials. J. Rheol. 23(23), 243 (1979). https://doi.org/10.1122/1.549526

    Article  ADS  Google Scholar 

  9. Delaney, G., Di Matteo, T., Aste, T.: Combining tomographic imaging and dem simulations to investigate the structure of experimental sphere packings. Soft Matter 6(1), 2992 (2010). https://doi.org/10.1039/B927490A

    Article  ADS  Google Scholar 

  10. Devroye, L.: Non-Uniform Random Variable Generation, 1st edn. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  11. Duran, J.: An Introduction to the Physics of Granular Materials, 2nd edn. Springer, Berlin (2000)

    MATH  Google Scholar 

  12. Edwards, S., Grinev, D.: Statistical mechanics of vibration-induced compaction of powders. Phys. Rev. E 58(4), 4758 (1998). https://doi.org/10.1103/PhysRevE.77.021309

    Article  ADS  Google Scholar 

  13. Edwards, S., Oakeshott, R.: Theory of powders. Phys. A Stat. Mech. Appl. 157(3), 1080 (1989). https://doi.org/10.1016/0378-4371(89)90034-4

    Article  MathSciNet  Google Scholar 

  14. Henkes, S., Chakraborty, B.: Statistical mechanics framework for static granular matter. Phys. Rev. E 79(6), 061301 (2009). https://doi.org/10.1103/PhysRevE.79.061301

    Article  ADS  MathSciNet  Google Scholar 

  15. Makse, H., Kurchan, J.: Testing the thermodynamic approach to granular matter with a numerical model of a decisive experiment. Nature 415(1), 614 (2002). https://doi.org/10.1038/415614a

    Article  ADS  Google Scholar 

  16. Matsushima, T., Blumenfeld, R.: Universal structural characteristics of planar granular packs. Phys. Rev. Lett. 112(9), 098003 (2014). https://doi.org/10.1103/PhysRevLett.112.098003

    Article  ADS  Google Scholar 

  17. Maxwell, J.: L. on the calculation of the equilibrium and stiffness of frames. Philos. Mag. 27(182), 294 (1864)

    Article  Google Scholar 

  18. Metzger, P., Donahue, C.: Elegance of disordered granular packings: a validation of edwards hypothesis. Phys. Rev. Lett. 94(1), 148001 (2005). https://doi.org/10.1103/PhysRevLett.94.148001

    Article  ADS  Google Scholar 

  19. Oquendo, W., Muñoz, J., Radjai, F.: An equation of state for granular media at the limit state of isotropic compression. EPL 114(1), 14004 (2016). https://doi.org/10.1209/0295-5075/114/14004

    Article  ADS  Google Scholar 

  20. Puckett, J., Lechenault, F., Daniels, K.: Local origins of volume fraction fluctuations in dense granular materials. Phys. Rev. E 83(4), 041301 (2011). https://doi.org/10.1103/PhysRevE.83.041301

    Article  ADS  Google Scholar 

  21. Radjai, F., Dubois, F. (eds.): Discrete-Element Modeling of Granular Materials, 1st edn. Wiley, New York (2011)

    Google Scholar 

  22. Radjai, F., Roux, S., Moreau, J.: Contact forces in a granular packing. Chaos 9(1), 544 (1999). https://doi.org/10.1063/1.166428

    Article  ADS  MATH  Google Scholar 

  23. Saadatfar, M., Sheppard, A., Senden, T., Kabla, A.: Mapping forces in a 3d elastic assembly of grains. J. Mech. Phys. Solids 60(1), 55 (2012). https://doi.org/10.1016/j.jmps.2011.10.001

    Article  ADS  MATH  Google Scholar 

  24. Snoeijer, J., Vlugt, T., Ellenbroek, W., van Hecke, M., van Leeuwen, J.: Ensemble theory for force networks in hyperstatic granular matter. Phys. Rev. E 70(6), 061306 (2004). https://doi.org/10.1103/PhysRevLett.92.054302

    Article  ADS  Google Scholar 

  25. Snoeijer, J., Vlugt, T., van Hecke, M., van Saarloos, W.: Force network ensemble: A new approach to static granular matter. Phys. Rev. Let. 92(5), 054302 (2004). https://doi.org/10.1103/PhysRevLett.92.054302

    Article  ADS  Google Scholar 

  26. Tighe, B., van Eerd, A., Vlugt, T.: Entropy maximization in the force network ensemble for granular solids. Phys. Rev. Lett. 100(23), 238001 (2008). https://doi.org/10.1103/PhysRevLett.100.238001

    Article  ADS  Google Scholar 

  27. Tighe, B., Snoeijer, J., Vlugt, T., van Hecke, M.: The force network ensemble for granular packings. Soft Matter 6(13), 2908 (2010). https://doi.org/10.1039/B926592A

    Article  ADS  Google Scholar 

  28. Tighe, B., Socolar, J., Schaeffer, D., Mitchener, G., Huber, M.: Force distributions in a triangular lattice of rigid bars. Phys. Rev. E 72(3), 031306 (2005). https://doi.org/10.1103/PhysRevE.72.031306

    Article  ADS  Google Scholar 

  29. Tighe, B., Vlugt, T.: Force balance in canonical ensembles of static granular packings. J. Stat. Mech. Theory Exp. 2010(1), P01015 (2010). https://doi.org/10.1088/1742-5468/2010/01/P01015

    Article  Google Scholar 

  30. Vlugt, T., Tighe, B.: Stress fluctuations in granular force networks. J.Stat. Mech. 2011(04), 04002 (2011). https://doi.org/10.1088/1742-5468/2011/04/P04002

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Simulation of Physical Systems Group, Department of Physics, Universidad Nacional de Colombia, Carrera 30 No. 45-03, Ed. 404, Of. 348, Bogotá D.C., Colombia

    Manuel A. Cárdenas-Barrantes, Jose Daniel Muñoz & William F. Oquendo

  2. Department of Mathematics, Physics, and Statistics, Faculty of Engineering, Universidad de la Sabana, Km 7 Autopista Norte de Bogota, Chía, Colombia

    William F. Oquendo

Authors
  1. Manuel A. Cárdenas-Barrantes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jose Daniel Muñoz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. William F. Oquendo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manuel A. Cárdenas-Barrantes.

Ethics declarations

Conflict of interest

We declare that we have no conflict of interest with the work submitted.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cárdenas-Barrantes, M.A., Muñoz, J.D. & Oquendo, W.F. Contact forces distribution for a granular material from a Monte Carlo study on a single grain. Granular Matter 20, 1 (2018). https://doi.org/10.1007/s10035-017-0773-y

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1007/s10035-017-0773-y

Keywords

Search

Navigation