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Towards a constitutive law for the unsteady contact stress in granular media

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Abstract

In order to build a unified modelling for granular media by means of Eulerian averaged equations, it is necessary to study two contributions in the effective Cauchy stress tensor: the first one concerns solid and fluid matter, including contact and collisions between grains; the second one focuses on the random movements of grains and fluid, similar to Reynolds stress for turbulent flows. It is shown that the point of view of piecewise continuous media already used for two phase flows allows one to derive a constitutive equation for the first contribution, under the form of a partial differential equation. Similarly as for the Reynolds stress in turbulent flows, this equation cannot be written only in terms of averaged quantities without adequate approximations. The structure of the closed equation is discussed with respect to irreversible thermodynamics, and in connection with some already existing models. It is emphasised that numerical simulations by the discrete elements method can be used in order to validate these approximations, through numerical experiments statistically considered. Finally an extension of this approach to the second contribution of the effective Cauchy stress tensor is briefly discussed, showing how the modelling of both contributions have to be coupled.

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References

  1. Hutter K. and Rajagopal K.R. (1994). On flows of granular materials. Contin. Mech. Thermodyn. 6: 81–139

    Article  MATH  MathSciNet  Google Scholar 

  2. Luca I., Fang C. and Hutter K. (2004). A thermodynamic model of turbulent motions in granular materials. Contin. Mech. Thermodyn. 16: 363–390

    Article  MATH  MathSciNet  Google Scholar 

  3. Kataoka I. (1986). Local instant formulation of two phase flow. Int. J. Multiphase Flow 12(5): 745–758

    Article  MATH  Google Scholar 

  4. Nigmatulin, R.I.: Dynamics of multiphase media, vol. 1, 2. Hemisphere Pub. Corp. (1991)

  5. Johnson P.C. and Jackson R. (1987). Frictional-collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid. Mech. 176: 67–93

    Article  Google Scholar 

  6. Frenette, R., Zimmermann, T., Eyheramendy, D.: Unified modeling of a fluid or granular flow on dam-break case. J. Hydraul. Eng. 299 (2002)

  7. Josserand C., Lagrée P.-Y. and Lhuillier D. (2006). Stationary shear flow of dense granular materials. EuroPhys. Lett. 73(3): 363–369

    Article  MathSciNet  Google Scholar 

  8. Cundall P.A. and Strack O.D.L. (1979). A discrete numerical model for granular assemblies. Géotechnique 29: 47–65

    Article  Google Scholar 

  9. Radjai F. and Roux S. (2002). Turbulent-like fluctuations in quasistatic flow of granular media. Phys. Rev. Lett. 89(6): 064302

    Article  Google Scholar 

  10. Cambou B. (1998). Behaviour of Granular Materials. Springer, Heidelberg

    MATH  Google Scholar 

  11. Emeriault F. and Cambou B. (1996). Micromechanical modelling of anisotropic non-linear elasticity of granular medium. Int. J. Solids Struct. 33(18): 2591–2607

    Article  MATH  Google Scholar 

  12. Dafalias Y.F. (1998). Plastic spin: necessity or redundancy?. Int. J. Plast. 14(9): 909–931

    Article  MATH  Google Scholar 

  13. Liao C.-L. and Chang T.-P. (1997). Stress-strain relationship for granular materials based on the hypothesis of best fit. Int. J. Solids Struct. 34: 4087–4100

    Article  MATH  Google Scholar 

  14. Cambou B., Chaze M. and Dedecker F. (2000). Change of scale in granular materials. Eur. J. Mech. A/Solids 19: 999–1014

    Article  MATH  Google Scholar 

  15. Kuhl E., D’Adetta G.A., Herrmann H.J. and Ramm E. (2000). A comparison of discrete granular materail model with continuous microplane formulations. Granul. Matter 2: 113–121

    Article  Google Scholar 

  16. Jean M. (1999). The non-smooth contact dynamics method, Comput. Methods Appl. Mech. Eng. 177: 235–257

    Article  MATH  MathSciNet  Google Scholar 

  17. Kruyt N.P. (2003). Contact forces in anisotropic frictional granular materials. Int. J. Solids Struct. 40: 3537–3556

    Article  MATH  Google Scholar 

  18. MiDi G.D.R. (2004). On dense granular flows. Eur. Phys. J. E 14: 341–365

    Article  Google Scholar 

  19. Trentadue F. (2001). A micromechanical model for a non-linear elastic granular material base on local equilibrium conditions. Int. J. Solids Struct. 38: 7319–7342

    Article  MATH  Google Scholar 

  20. Jou D., Casas-vàzquez J. and Lebon G. (2001). Extended Irreversible Thermodynamics. Springer, Heidelberg

    MATH  Google Scholar 

  21. Simonin, O.: Continuum modelling of dispersed two-phase flows. In: Combustion and Turbulence in Two-Phase Flows, Von Karmann Institute of Fluid Dynamics Lecture Series, 2 (1996)

  22. Demoulin F.-X., Beau P.-A., Blokkeel G., Mura A. and Borghi R. (2007). A new model for turbulent flows with large density fluctuations: application to liquid atomization. Atomization Sprays 17: 1–31

    Article  Google Scholar 

  23. Fang C., Wang Y. and Hutter K. (2006). A thermomechanical continuum theory with internal length for chesionless granular material, PartI. Contin. Mech. Thermodyn. 17(8): 545–576

    Article  MATH  MathSciNet  Google Scholar 

  24. Savage S.B. (1998). Analyses of slow high-concentration flows of granular materials. J. Fluid Mech. 377: 1–26

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Ecole Centrale de Marseille, Technopôle de Château-Gombert, 38 rue Joliot Curie, 13451, Marseille, France

    Roland Borghi

  2. Laboratoire de Mécanique et d’Acoustique (UPR-CNRS 7051), 31 chemin Joseph Aiguier, 13402, Marseille, France

    Roland Borghi

  3. Cemagref, 3275 Route de Cézanne, CS 40061, 13182, Aix-en-Provence, France

    Stéphane Bonelli

Authors
  1. Roland Borghi
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  2. Stéphane Bonelli
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Correspondence to Roland Borghi.

Additional information

Communicated by S. Roux.

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Borghi, R., Bonelli, S. Towards a constitutive law for the unsteady contact stress in granular media. Continuum Mech. Thermodyn. 19, 329–345 (2007). https://doi.org/10.1007/s00161-007-0058-5

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  • DOI: https://doi.org/10.1007/s00161-007-0058-5

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