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The European Physical Journal Special Topics

, Volume 223, Issue 11, pp 2189–2204 | Cite as

Granular statistical mechanics – a personal perspective

  • R. Blumenfeld
  • S.F. Edwards
Review
Part of the following topical collections:
  1. Dynamic Systems: From Statistical Mechanics to Engineering Applications

Abstract

The science of granular matter has expanded from an activity for specialised engineering applications to a fundamental field in its own right. This has been accompanied by an explosion of research and literature, which cannot be reviewed in one paper. A key to progress in this field is the formulation of a statistical mechanical formalism that could help develop equations of state and constitutive relations. This paper aims at reviewing some milestones in this direction. An essential basic step toward the development of any static and quasi-static theory of granular matter is a systematic and useful method to quantify the grain-scale structure and we start with a review of such a method. We then review and discuss the ongoing attempt to construct a statistical mechanical theory of granular systems. Along the way, we will clarify a number of misconceptions in the field, as well as highlight several outstanding problems.

Keywords

Partition Function Granular Material European Physical Journal Special Topic Volume Function Mechanical Equilibrium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    W.O. Smith, P.D. Foote, P.F. Busang, Phys. Rev. 34, 1271 (1929)CrossRefADSGoogle Scholar
  2. 2.
    T. Kallstenius, W. Bergau, Proc. 5th Int. Conf. Soil Mech. Found. Eng. 1, 165 (1961)Google Scholar
  3. 3.
    T. Mogami, Soils Found. 5, 26 (1965)CrossRefGoogle Scholar
  4. 4.
    M. Oda, Soils Found. 14, 13 (1974)CrossRefGoogle Scholar
  5. 5.
    C.B. Brown, “The use of maximum entropy in the characterization of granular materials”, in Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials, edited by C.S. Cowin, M. Satake (Gakujutsu Bunken Fukyu-kai, 1978), p. 98Google Scholar
  6. 6.
    M. Oda, S. Nemat-Nasser, M.M. Mehrabadi, Int. J. Numer. Anal. Meth. Geomech. 6, 77 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    H. Herrmann, Physica A 263, 51 (1999)CrossRefADSGoogle Scholar
  8. 8.
    S.F. Edwards, R.B. Oakeshott, Physica A 157, 1080 (1989)MathSciNetCrossRefADSGoogle Scholar
  9. 9.
    S.F. Edwards, IMA Bull. 25, 94 (1989)Google Scholar
  10. 10.
    A. Mehta, S.F. Edwards, Physica A 157, 1091 (1989)MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    N. Xu, J. Blawzdziewicz, C.S. O’Hern, Phys. Rev. E 71, 061306 (2005)CrossRefADSGoogle Scholar
  12. 12.
    G.-J. Gao, J. Blawzdziewicz, C.S. O’Hern, Phys. Rev. E 74, 061304 (2006)CrossRefADSGoogle Scholar
  13. 13.
    G.-J. Gao, J. Blawzdziewicz, C.S. O’Hern, M. Shattuck, Phys. Rev. E 80, 061304 (2009)CrossRefADSGoogle Scholar
  14. 14.
    S. Henkes, C.S. O’Hern, B. Chakraborty, Phys. Rev. Lett. 99, 038002 (2007)CrossRefADSGoogle Scholar
  15. 15.
    L.A. Pugnaloni, I. Sánchez, P.A. Gago, J. Damas, I. Zuriguel, D. Maza, Phys. Rev. E 82, 050301(R) (2010)CrossRefADSGoogle Scholar
  16. 16.
    S.F. Edwards, R. Blumenfeld, in Powders and Grains, Stuttgart, edited by R. Garcia-Rojo, H.J. Herrmann, and S. McNamara (Balkema, Leiden, Netherlands, 2005), p. 3Google Scholar
  17. 17.
    R. Blumenfeld, J.F. Jordan, S.F. Edwards, Phys. Rev. Lett. 109, 238001 (2012)CrossRefADSGoogle Scholar
  18. 18.
    R.C. Ball, R. Blumenfeld, Phys. Rev. Lett. 88, 115505 (2002)CrossRefADSGoogle Scholar
  19. 19.
    R. Blumenfeld, S.F. Edwards, Phys. Rev. Lett. 90, 114303 (2003)CrossRefADSGoogle Scholar
  20. 20.
    R. Blumenfeld, S.F. Edwards, Eur. Phys. J. E 19, 23 (2006)CrossRefGoogle Scholar
  21. 21.
    V.A. Luchnikov, N.N. Medvedev, L. Oger, J.-P. Troadec, Phys. Rev. E. 59, 72057212 (1999)CrossRefGoogle Scholar
  22. 22.
    T. Aste, Phys. Rev. Lett. 96, 018002 (2006)CrossRefADSGoogle Scholar
  23. 23.
    C. Song, P. Wang, H.A. Makse, Nature 453, 629 (2008)CrossRefADSGoogle Scholar
  24. 24.
    M.P. Ciamarra, Phys. Rev. Lett. 99, 089401 (2007)CrossRefADSGoogle Scholar
  25. 25.
    R. Blumenfeld, S.F. Edwards, Phys. Rev. Lett. 99, 089402 (2007)CrossRefADSGoogle Scholar
  26. 26.
    G. Voronoi, J. Reine Ang. Math. 133, 97 (1908)zbMATHGoogle Scholar
  27. 27.
    B. Delaunay, Bulletin de l’Académie des Sci. de l’URSS, Classe des Sci. Mathématiques et Naturelles 6, 793 (1934)Google Scholar
  28. 28.
    A. Okabe, B. Boots, K. Sugihara, Spatial Tessellations. Concepts and Applications of Voronoi Diagrams (J. Wiley and Sons, Chichester, New York, Brisbane, Toronto and Singapore, 1992)Google Scholar
  29. 29.
    R. Blumenfeld, “Stress transmission and isostatic states of non-rigid particulate systems”, in IMA Volume in Mathematics and its Applications, Vol. 141, Modeling of Soft Matter, edited by M.-C.T. Calderer, E.M. Terentjev (Springer-Verlag, 2005), p. 235Google Scholar
  30. 30.
    T. Matsushima, R. Blumenfeld, Phys. Rev. Lett. 112, 098003 (2014)CrossRefADSGoogle Scholar
  31. 31.
    T. Matsushima, R. Blumenfeld, Powders and Grains (2013)Google Scholar
  32. 32.
    J.F. Jordan, R. Blumenfeld (in preparation) (2014)Google Scholar
  33. 33.
    S.F. Edwards, R. Blumenfeld, “Basic laws of granular materials”, in Physics of Granular Materials, edited by A. Mehta (Cambridge University Press, Cambridge, 2007)Google Scholar
  34. 34.
    R. Blumenfeld, S.F. Edwards, J. Phys. Chem. B 113, 3981 (2009)CrossRefGoogle Scholar
  35. 35.
    J.F. Jordan, R. Hihinashvili, R. Blumenfeld (in preparation) (2014)Google Scholar
  36. 36.
    R. Blumenfeld, J.F. Jordan, S.F. Edwards, Powders and Grains (2013)Google Scholar

Copyright information

© EDP Sciences and Springer 2014

Authors and Affiliations

  1. 1.ESE, Imperial College LondonLondonUK
  2. 2.Gonville and Caius CollegeCambridgeUK
  3. 3.College of Science, National Univ. of Defence TechnologyChangshaChina
  4. 4.Cavendish LaboratoryCambridgeUK

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