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The stress response of a semi-infinite micropolar granular material subject to a concentrated force normal to the boundary

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Abstract

We consider the problem of a two dimensional semi-infinite granular material subject to a concentrated or point force normal to the boundary. This boundary value problem was originally solved for a classical elastic material by Flamant in 1892 and, hence, is also known as the ‘‘Flamant problem’’ (Johnson [8]). In this paper, the granular material is considered as an elastic micropolar or Cosserat continuum and is represented by a particular form of the general constitutive law derived in Walsh and Tordesillas [29]. The stress distribution predicted by the model is in good agreement with experimental data for small strains. In particular, two important features that are captured by the proposed model are: (i) the presence of tensile stress response regions, and (ii) the dependence of the stresses on the microstructural properties, i.e. the particles’ normal, tangential and rotational stiffness constants. The proposed analysis utilizes two new stress functions, similar to Airy’s stress functions in classical elastic theory.

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References

  1. M. Abramowitz & I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, Inc., New York, 1972)

  2. R. P. Behringer, Junfei Geng, D. Howell, E. Longhi, G. Reydellet, L. Vanel, E. Clément & S. Luding, Fluctuations in granular materials, Powders and Grains 2001, Kishino (ed.), (Swets & Zitlinger, Lisse, 2001), p. 347–354

  3. C. S. Chang & L. Ma, A micromechanically-based micropolar theory for deformation of granular solids, Int. J. Solids and Structures 28 (1) (1991), p. 67–86

    Google Scholar 

  4. C. S. Chang & J. Gao, Kinematic and static hypothesis for constitutive modelling of granulates considering particle rotation, Acta Mech. 115 (1996), p. 213–229

    Google Scholar 

  5. A. C. Eringen, Theory of micropolar elasticity, In: H. Liebowitz (ed.), Fracture - An advanced treatise, Vol. II, Chapter 7, (Academic Press, New York, 1968), p. 621– 693

  6. Y. C. Fung, Foundations of Solid Mechanics, Prentice Hall, inc., (Englewood Cliffs, New Jersey 1965)

  7. J. Hill & A. Tordesillas, The symmetrical adhesive contact problem for circular elastic cylinders, J. Elasticity 27 (1992), p. 1–36

    Google Scholar 

  8. K. L. Johnson, Contact Mechanics, (Cambridge University Press, Cambridge, 1985)

  9. B.S. Gardiner & A. Tordesillas, Micromechanical constitutive modelling of granular media: A focus on the evolution and loss of contacts in particle clusters. Submitted to J. Eng. Math. (2003)

  10. C. Gay & R. da Silveira, Continuum theory of frictional granular matter, cond-mat/0208155

  11. Junfei Geng, D. Howell, E. Longhi, R. P. Behringer, G. Reydellet, L. Vanel, E. Clément & S. Luding, Footprints in sand: The response of a granular material to local perturbations, Phys. Rev. Let. 87 (3) (2001), p. 035506

  12. Junfei Geng, G. Reydellet, E. Clément & R. P. Behringer, Green’s function Measurements of Force Transmission in 2D granular materials. Preprint submitted to Elsevier Science, 4 April 2003

  13. C. Goldenberg & I. Goldhirsch, Force chains, microelasticity, and macroelasticity, Phys. Rev. Let. 89 (8) (2002), p. 084302

    Google Scholar 

  14. I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series and Products, Sixth Edition, (Academic Press, San Diego, 2000)

  15. V. T. Granik & M. Ferrari, Microstructural mechanics of granular materials, Mechanics of Materials 15 (1993) p. 301–322

  16. D. A. Horner, J. F. Peters & A. Carrillo, Large scale discrete element modelling of vehicle-soil interaction, J. Eng. Mech. 127 (2001), p. 1027–1032

    Google Scholar 

  17. N. W. Mueggenburg, H. M. Jaeger & S. R. Nagel, Stress transmission through three-dimensional ordered granular arrays, Phys. Rev. E 66 (2002), p. 031304

    Google Scholar 

  18. D. M. Mueth, H. M. Jaeger & S. R. Nagel, Force distribution in a granular medium, Phys. Rev. E 57 (1998), p. 3164

    Google Scholar 

  19. M. Otto, J.-P. Bouchaud, P. Claudin & J. E. S. Socolar, Anisotropy in granular media: classical elasticity and directed-force chain network, Phys. Rev. E 67 (2003), p. 031202

    Google Scholar 

  20. D. R. J. Owen, Y. T. Feng, M. G. Cottrel & J. Yu, Discrete/finite element modelling of industrial applications with multi-fracturing and particulate phenomena, in Discrete Element Methods Numerical Modeling of Discontinua, B. J. Cook, R. P. Jensen, (Eds), (American Society of Civil Engineers, 2002), p. 11–16

  21. J. Rajchenbach, Is the classical elasto-plastic modelling relevant to describe the mechanical behaviour of cohesionless packings?, Powders and Grains 2001, Kishino (ed.), (Swets & Zitlinger, Lisse, 2001), p. 203–206

  22. L. Rothenburg, R. J. Bathurst & M. B. Dusseault, Micromechanical ideas in constitutive modelling of granular materials. In: J. Biarez, R. Gourves (Eds), (Powders and Grains, Balkema, Rotterdam, 1989), p. 355–363

  23. D. Serero, G. Reydellet, P. Claudin & E. Clement, Stress response function of a granular layer: quantitative comparison between experiments and isotropic elasticity, Eur. Phys. J. E 6 (2001), p. 169–179

    Google Scholar 

  24. S. A. Shoop, Finite Element Modelling of Tire-Terrain Interaction, Phd. Thesis, (The University of Michigan, 2001)

  25. A. Tordesillas ,& J. Shi, Frictional indentation of dilatant granular materials, Proc. Roy. Soc. Lond. A, 455 (1981) (1999), p. 261-283

  26. A. Tordesillas & J. Shi, Stresses, flow and deformation of soils in contact with metallic and/or rubber-like bodies, Proc. 13th International Conference of the ISTVS 1, p. 201-208

  27. A. Tordesillas & S. D. C. Walsh, Incorporating rolling resistance and contact anisotropy in micromechanical models of granular media, Powder Technology 124 (2002), p. 106–111

    Google Scholar 

  28. K. C. Valanis, A gradient theory of internal variables, Acta Mech. 116 (1996), p. 1–14

  29. S. D. C. Walsh & A. Tordesillas, A thermomechanical approach to the development of micropolar constitutive models of granular media, Acta Mech., Accepted for publication, September 2003

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Melbourne, Victoria, 3010, Australia

    Stuart D. C. Walsh & Antoinette Tordesillas

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  1. Stuart D. C. Walsh
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  2. Antoinette Tordesillas
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Correspondence to Antoinette Tordesillas.

Additional information

The support of the US Army Research Office through a grant to AT (Grant No. DAAD19-02-1-0216) and the Melbourne Research and Development Grant scheme is gratefully acknowledged. We thank our reviewers for their useful suggestions and insightful comments.

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Walsh, S., Tordesillas, A. The stress response of a semi-infinite micropolar granular material subject to a concentrated force normal to the boundary. GM 6, 27–37 (2004). https://doi.org/10.1007/s10035-004-0155-0

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  • DOI: https://doi.org/10.1007/s10035-004-0155-0

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