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Continuum Mechanics and Thermodynamics

, Volume 17, Issue 8, pp 577–607 | Cite as

A thermo-mechanical continuum theory with internal length for cohesionless granular materials

Part II. Non-equilibrium postulates and numerical simulations of simple shear, plane Poiseuille and gravity driven problems
  • Chung FangEmail author
  • Yongqi Wang
  • Kolumban Hutter
Original Article

Abstract

This article continues Part I. Here the non-equilibrium responses of the constitutive variables t (Cauchy stress tensor), q (heat flux vector), h (equilibrated stress vector), Γ (flux term associated with the internal length ℓ), Π (production term associated with ℓ) and f (equilibrated intrinsic body force) as well as the Helmholtz free energy Ψ are postulated by use of a quasi-linear theory for three of four models deduced in Part I. In so doing, together with the equilibrium responses gained in Part I, a complete set of constitutive equations for the constitutive quantities for each model is obtained. The implemented models are applied to investigate typical isothermal steady granular shearing flows with incompressible grains, namely, simple plane shear flow, inclined gravity-driven flow and vertical channel-flow. The emphasis is on the models in which ℓ is considered a material constant (Model I) and an independent dynamic field quantity (Model III). Numerical results show that Model III is more appropriate than Model I since in the former model the effect of the motion of an individual grain can better be taken into account. Such a result is in particular significant for avalanches, since it verifies the existence of a thin layer immediately above the base of an avalanche, in which the grains are colliding strongly with one another, and provides a quantitative means to measure such a thin layer.

Keywords

Internal length Granular materials Simple shear flows 

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References

  1. 1.
    Fang, C., Wang, Y., Hutter, K.: A thermo-mechanical continuum theory with internal length cohesionless granular materials. Part I. A class of constitutive models. Continuum Mech. Thermodyn. (in press) (2006)Google Scholar
  2. 2.
    Wang, Y., Hutter, K.: Shearing flows in a Goodman-Cowin type granular material-theory and numerical results. Particulate Science and Technology 17, 97–124 (1999)CrossRefGoogle Scholar
  3. 3.
    Wang, Y., Hutter, K.: A constitutive model of multiphase mixtures and its application in shearing flows of saturated solid-fluid mixtures. Granular Matter 1, 163–181 (1999)CrossRefGoogle Scholar
  4. 4.
    Wang, Y., Hutter, K.: A constitutive theory of fluid-saturated granular materials and its application in gravitational flows. Rheol. Acta. 38, 214–223 (1999)CrossRefGoogle Scholar
  5. 5.
    Aris, R.: Vectors, Tensors and the Basic Equations of Fluid Mechanics. Prentice-Hall (1962)Google Scholar
  6. 6.
    Serrin, J.J.: Math. Mech. 8, 459–469 (1959)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Bagnold, R.A.: Experiments on a gravity free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. London, Ser. A 225, 49–63 (1954)CrossRefGoogle Scholar
  8. 8.
    Goodman, M.A., Cowin, S.C.: Two problems in the gravity flow of granular materials. J. Fluid Mech. 45, 321–339 (1971)zbMATHCrossRefGoogle Scholar
  9. 9.
    Passman, S.L., Nunziato, J.W., Bailey, P.B.: Shearing motion of a fluid-saturated granular material. J. Rheology 30(1), 167–192 (1986)zbMATHCrossRefGoogle Scholar
  10. 10.
    Passman, S.L., Nunziato, J.W., Bailey, P.B., Thomas, J.P.: Shearing flow of granular materials. J. Eng. Mech. Division, ASCE 106, 773-783 (1980)Google Scholar
  11. 11.
    Nunziato, J.W., Passman, S.L.: Gravitational flows of granular materials with incompressible grains. J. Rheology 24, 395–420 (1980)zbMATHCrossRefGoogle Scholar
  12. 12.
    Savage, S.B.: Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92 53–96 (1979)Google Scholar
  13. 13.
    Cowin, S.C.: Constitutive relations that imply a generalized Mohr-Coulomb criterion. Acta Mechanica 20, 41–46 (1974)CrossRefzbMATHGoogle Scholar
  14. 14.
    Fang, C.: A thermo-mechanical continuum theory with internal length of cohesionless granular materials. Ph.D. thesis, Darmstadt University of Technology, Germany (2005)Google Scholar
  15. 15.
    Kirchner, N.: Thermodynamically consistent modelling of abrasive granular materials. I: Non-equilibrium theory. Proc. R. Soc. Lond. A 458, 2153–2176 (2002)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Kirchner, N., Teufel, A.: Thermodynamically consistent modelling of abrasive granular materials. II: Thermodynamic equilibrium and applications to steady shear flows. Proc. R. Soc. Lond. A 458, 3053–3077 (2002)zbMATHCrossRefGoogle Scholar
  17. 17.
    Haff, P.K.: Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 410–430 (1983)CrossRefGoogle Scholar
  18. 18.
    Hanes, D.M., Jenkins, J.T., Richman, M.W.: The thickness of steady plane shear flow of circular disks driven by identical boundaries. J. Applied Mech 55, 969–980 (1988)zbMATHCrossRefGoogle Scholar
  19. 19.
    Hui, K., Haff, P.K., Ungar, J.E.: Boundary conditions for high-shear grain flows. J. Fluid Mech, vol. 145, 223–233 (1984)zbMATHCrossRefGoogle Scholar
  20. 20.
    Jenkins, J.T., Richmann, M.W.: Boundary conditions for plane flows of smooth, nearly elastic, circular disks. J. Fluid Mech 171, 53–56 (1986)zbMATHCrossRefGoogle Scholar
  21. 21.
    Johnson, P.C., Jackson, R.J.: Frictional-collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech 176, 67–93 (1987)CrossRefGoogle Scholar
  22. 22.
    Johnson, P.C., Nott, P., Jackson, R.: Frictional-collisional equations of motion for particulate flows and their application to chutes. J. Fluid Mech. 210 501–535 (1990)Google Scholar
  23. 23.
    Richman, M.W.: Boundary conditions based upon a modified Maxwellian velocity distribution for flows of identical, smooth, nearly elastic spheres. Acta Mechanica 75, 227–240 (1988)CrossRefGoogle Scholar
  24. 24.
    Richman, M.W., Chou, C.S.: Boundary effects on granular shear flows of smooth disks. ZAMP 39, 885–901 (1988)zbMATHCrossRefGoogle Scholar
  25. 25.
    Hanes, D.M., Inman, D.: Observations of rapidly flowing granular-fluid materials. J. Fluid Mech. 150, 357–380 (1985)CrossRefGoogle Scholar
  26. 26.
    Savage, S.B., Sayed, M.: Stresses developed by dry cohesionless granular materials sheared in an annular shear cell. J. Fluid Mech. 142, 391–430 (1984)CrossRefGoogle Scholar
  27. 27.
    Hutter, K., Rajagopal, K.R.: On flows of granular materials. Continuum Mech. Thermodyn. 6, 81–139 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Savage, S.B.: Mechanics of Granular Flows. In: Hutter, K. (ed.) Continuum mechanics in environmental sciences and geophysics pp. 467–522. Springer Verlag, (1993)Google Scholar
  29. 29.
    Anderson, D.A., Tannehill, J.C., Pletcher, R.H.: Computational Fluid Mechanics and Heat Transfer. Hemisphere Publishing Corporation, New York (1984)Google Scholar
  30. 30.
    Savage, S.B., Lun, C.K.K.: Particle size segregation in inclined chute flow of dry cohesionless granular solids. J. Fluid Mech. 189, 311–335 (1988)CrossRefGoogle Scholar
  31. 31.
    Vallance, J.W., Savage, S.B.: Particle segregation in granular flows down chutes. In:Rosato, A.D., Blackmore, D.L. (eds.) IUTAM Symposium on Segregation in Granular Flows, pp. 31–52. Kluwer Academic Publishers, Netherlands (2000)Google Scholar
  32. 32.
    Hutter, K., Szidarovsky, F., Yakowitz, S.: Plane steady flow of a cohesionless granular material down an inclined plane: A model for flow avalanches, Part II: Numerical results. Acta Mechanica 65, 239–361 (1986)CrossRefGoogle Scholar
  33. 33.
    Duran, J.: Sands, Powders and Grains. Springer Verlag, Heidelberg (2000)zbMATHGoogle Scholar
  34. 34.
    Gray, J.M.N.T., Hutter, K.: Pattern formation in granular avalanches. Continuum Mech. Thermodyn. 9, 341–345 (1997)CrossRefGoogle Scholar
  35. 35.
    Ristow, G.H.: Pattern Formation in Granular Materials. Springer Verlag Berlin, Heidelberg (2000)Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Institut für MechanikTechnische Universität DarmstadtDarmstadtGermany

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