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Granular Matter

, Volume 12, Issue 3, pp 239–252 | Cite as

Stress, stress asymmetry and couple stress: from discrete particles to continuous fields

  • Isaac Goldhirsch
Article

Abstract

Explicit and closed expressions for the stress and couple-stress fields for discrete (classical) mechanical systems in terms of the constituents’ degrees of freedom and interactions are derived and compared to previous results. This is done by using an exact and general coarse graining formulation, which allows one to predetermine the resolution of the continuum fields. Since the full dynamics of the pertinent fields is considered, the results are not restricted to static states or quasi-static deformations; the latter comprise mere limiting cases, which are discussed as well. The fields automatically satisfy the equations of continuum mechanics. An explicit expression for the antisymmetric part of the stress field is presented; the question whether the latter vanishes, much like its nature when it does not, have been debated in the literature. Physical explanations of some of the obtained results are offered; in particular, an interpretation of the expression for the stress field provides an argument in favor of its uniqueness, yet another topic of debate in the literature. The formulation and results are valid for single realizations, and can of course be used in conjunction with ensemble averaging. Part of the paper is devoted to a biased discussion of the notion of coarse graining in general, in order to set the presented results in a certain perspective. Although the results can be applied to molecular (nanoscale included) and granular systems alike, the presentation and some simplifying assumptions (which can be easily relaxed) target granular systems. The results should be useful for the analysis of experimental and numerical findings as well as the development of constitutive relations.

Keywords

Coarse graining Averaging Homogenization Stress Stress asymmetry Couple Stress Cosserat Continuum mechanics Discrete mechanical systems Granular systems Nanoscale systems 

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References

  1. 1.
    Irving J.H., Kirkwood J.G.: The statistical mechanical theory of transport properties IV. The equations of hydrodynamics. J. Chem. Phys. 18, 817–829 (1950)CrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Born, M., Huang, K.: Dynamical Theory of Crystal Lattices. Oxford (1954, reissue 2000)Google Scholar
  3. 3.
    Reichl, L.E.: A Modern Course in Statistical Physics. Wiley and numerous references therein (1998)Google Scholar
  4. 4.
    Weber J.: Recherches concernantes les contraintes intergranulaires dans les milieux pulvérantes. Bull. de Liais P. C. 20, 1–31 (1966)Google Scholar
  5. 5.
    Cambou B., Dubujet P., Emeriault F., Sidoroff F.: Homogenization for granular media. Eur. J. Mech. A Solids 14, 255–276 (1995)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Chang C.S., Gao J.: Kinematic and static hypotheses for constitutive modeling of granulates considering particle rotation. Acta Mech. 115, 213–229 (1996)zbMATHCrossRefGoogle Scholar
  7. 7.
    Bagi K.: Stress and strain in granular assemblies. Mech. Mat. 22(3), 165–177 (1996)CrossRefGoogle Scholar
  8. 8.
    Babic M.: Average balance equations for granular materials. Int. J. Eng. Sci. 35, 523–548 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Goddard J.D.: Continuum modeling of granular assemblies: quasi-static dilatancy and yield. In: Herrmann, H.J., Hovi, J.-P., Luding, S. (eds) Physics of Dry Granular Media, pp. 1–24. Kluwer, Dordrecht (1998)Google Scholar
  10. 10.
    Nemat-Nasser S.: A micromechanically-based constitutive model for frictional deformation of granular materials. J. Mech. Phys. Solids 48, 1541–1563 (2000)zbMATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Lätzel M., Luding S., Herrmann H.J.: From discontinuous models towards a continuum description. In: Vermeer, P.A., Diebels, S., Ehlers, W., Herrmann, H.J., Luding, S., Ramm, E. (eds) Continuous and Discontinuous Models of Cohesive-Frictional Materials, Lecture Notes in Physics 568, pp. 215–230. Springer, Berlin (2001)CrossRefGoogle Scholar
  12. 12.
    Bardet J.P., Vardoulakis I.: The asymmetry of stress in granular media. Int. J. Solids Struct. 38, 353–367 (2001)zbMATHCrossRefGoogle Scholar
  13. 13.
    Zhu H.P., Yu A.B.: Averaging method for granular materials. Phys. Rev. E 66, 021302 (2002)ADSGoogle Scholar
  14. 14.
    Ball R.C., Blumenfeld R.: Stress field in granular systems, loop forces and potential formulation. Phys. Rev. Lett. 88, 115505 (2002)CrossRefADSGoogle Scholar
  15. 15.
    Ehlers W., Ramm E., Diebels S., D’Addetta G.A.: From particle ensembles to Cosserat continua: homogenization of contact forces towards stresses and couple stresses. Int. J. Solids Struct. 40, 6681–6702 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kruyt N.P., Rothenburg L.: Kinematic and static assumptions for homogenization in micromechanics of granular media. Mech. Mater. 36(22), 1157–1173 (2004)CrossRefGoogle Scholar
  17. 17.
    Froiio F., Tomassetti G., Vardoulakis I.: Mechanics of granular materials: the discrete and continuum descriptions juxtaposed. Int. J. Solids Struct. 43, 7684–7720 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Goddard J.D.: From granular matter to generalized continuum. In: Capriz, G., Giovine, P., Mariano, P.M. (eds) Mathematical Models of Granular Matter, Lecture Notes in Applied Mathematics, vol. 1937, Ch. 1, pp. 1–20. Springer, Berlin (2008)Google Scholar
  19. 19.
    Goddard J.D.: Microstructural origins of continuum stress fields—a brief history and some unresolved issues. In: DeKee, D., Kaloni, P.N. (eds) Recent Developments in Structured Continua, Pitman Research Notes in Mathematics, No. 143, pp. 179–208. Longman/ Wiley, New York (1986)Google Scholar
  20. 20.
    Goldhirsch I.: Rapid granular flows. Ann. Rev. Fluid Mech. 35, 267–293 (2003)CrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Glasser B.J., Goldhirsch I.: Scale dependence, correlations and fluctuations of stresses in rapid granular flows. Phys. Fluids 13, 407–420 (2001)CrossRefADSGoogle Scholar
  22. 22.
    Love A.E.H.: Treatise of Mathematical Theory of Elasticity. Cambridge, UP, London (1927)zbMATHGoogle Scholar
  23. 23.
    Voigt W.: Theoretische Studien über die Elasticitätverhhältnisse der Krystalle. Abhandt. Ges. Wiss. Gött. 34, 3–52 (1887)Google Scholar
  24. 24.
    Goldenberg C., Goldhirsch I.: Continuum mechanics for small systems and fine resolutions. In: Rieth, M., Schommers, W. (eds) Handbook of Theoretical and Computational Nanotechnology, pp. 330–386. American Scientific, New York (2006)Google Scholar
  25. 25.
    Murdoch A.I.: On the microscopic interpretation of stress and couple stress. J. Elast. 71(1–3), 105–131 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Murdoch A.I.: A critique of atomistic definitions of the stress tensor. J. Elast. 88(2), 113–140 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Goldhirsch I., Goldenberg C.: On the microscopic foundations of elasticity. Eur. Phys. J. E 9(3), 245–251 (2002)Google Scholar
  28. 28.
    Goldenberg C., Goldhirsch I.: Small and large scale granular statics. Granul. Matter 6(2–3), 87–96 (2004)zbMATHCrossRefGoogle Scholar
  29. 29.
    Goldhirsch I., Goldenberg C.: Stress in dense granular materials. In: Wolf, D.E., Hinrichsen, H. (eds) The Physics of Granular Media, pp. 3–22. Wiley, Weinheim (2004)Google Scholar
  30. 30.
    Schofield P., Henderson J.R.: Statistical mechanics of inhomogeneous fluids. Prof. R. Soc. Lond. Ser. A 379(1776), 231–246 (1982)zbMATHCrossRefADSGoogle Scholar
  31. 31.
    Wajnryb E., Altenberger A.R., Dahler J.S.: Uniqueness of the microscopic stress tensor. J. Chem. Phys. 103, 9782–9787 (1995)CrossRefADSGoogle Scholar
  32. 32.
    Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge, UP, Cambridge (1967)zbMATHGoogle Scholar
  33. 33.
    Goldenberg C., Atman A.P.F., Claudin P., Combe G., Goldhirsch I.: Scale separation in granular packings: stress plateaus and fluctuations. Phys. Rev. Lett. 96(16), 168001 (2006)CrossRefADSGoogle Scholar
  34. 34.
    Goldenberg C., Goldhirsch I.: Effects of friction and disorder on the quasistatic response of granular solids to a localized force. Phys. Rev. E 77(4), 041303 (2008)ADSGoogle Scholar
  35. 35.
    Tan M.-L., Goldhirsch I.: Rapid granular flows as mesoscopic systems. Phys. Rev. Lett. 81(14), 3022–3025 (1998)CrossRefADSGoogle Scholar
  36. 36.
    Arslan H., Sture S.: Evaluation of a physical length scale for granular materials. Comput. Mater. Sci. 42, 525–530 (2008)CrossRefGoogle Scholar
  37. 37.
    Eringen A.C.: Theory of micropolar elasticity. In: Leibowitz, H. (eds) Fracture: An Advanced Treatise, vol. I, Mathematical Fundamentals, pp. 621–729. Academic Press, New York (1968)Google Scholar
  38. 38.
    Cercignani C.: The Boltzmann Equation and Its Applications, Ser. “Applied Mathematical Sciences”, No. 67. Springer, Berlin (1988)Google Scholar
  39. 39.
    Goldhirsch I.: Scales and kinetics of granular flows. CHAOS 9(3), 659–672 (1999)zbMATHCrossRefADSGoogle Scholar
  40. 40.
    Goldenberg C., Goldhirsch I.: Friction enhances elasticity in granular solids. Nature 435(7039), 188–191 (2005)CrossRefADSGoogle Scholar
  41. 41.
    Zhang, J., Behringer, R.P., Goldhirsch, I.: Coarse graining of a physical granular system. Prog. Theor. Phys. Suppl. (2010, in press)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.School of Mechanical Engineering, Faculty of EngineeringTel-Aviv UniversityTel-AvivIsrael

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