Abstract
The elastic properties of a regular packing of spheres with different tolerances were evaluated using the discrete element method to elucidate the mechanisms behind the discrepancies between laboratory experiments and theoretical predictions of the classic Hertz-Mindlin contact law. The simulations indicate that the elastic modulus of the packing is highly dependent on the coordination number and the magnitude and distribution of contact normal forces, and this dependence is macroscopically reflected as the influence of confining pressure and void ratio. The increase of coordination number and the uniformity of contact normal forces distribution with increasing confining pressure results in the stress exponent \(n\) for elastic modulus being higher than 1/3 as predicted by the Hertz-Mindlin law. Furthermore, the simulations show that Poisson’s ratio of a granular packing is not a constant as commonly assumed, but rather it decreases as confining pressure increases. The variation of Poisson’s ratio appears to be a consequence of the increase of the coordination number rather than the increase of contact normal forces with confining pressure.
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References
Hardin, B.O., Richart, F.E.: Elastic wave velocities in granular soils. J. Soil Mech. Found. Eng. Div. 89(SM1), 39–56 (1963)
Richart, F.E., Hall, J.R., Woods, R.D.: Vibrations of Soils and Foundations. Prentice-Hall, Englewood Cliffs (1970)
Hardin, B.O., Drnevich, V.P.: Shear modulus and damping in soil: Measurement and parameter effects. J. Soil Mech. Found. Div. 98(7), 603–624 (1972)
Iwasaki, T., Tatsuoka, F.: Effect of grain size and grading on dynamic shear moduli of sand. Soils Found. 17(3), 19–35 (1977)
Kokusho, T.: Cyclic triaxial test of dynamic soil properties for wide strain range. Soils Found. 20(2), 45–60 (1980)
Stokoe, K.H.I.I., Hwang, S.K., Lee, J.N.-K.: Effects of various parameters on the stiffness and damping of soils at small to medium strains. Proc. First Int. Conf. Prefail. Deformat. Charact. Geomater. 2, 785–816 (1995)
Wichtmann, T., Triantafyllidis, T.: On the influence of the grain size distribution curve on P-wave velocity, constrained elastic modulus \(M_{\rm max}\) and Poisson’s ratio of quartz sands. Soil Dyn. Earthq. Eng. 30, 757–766 (2010)
Yang, J., Gu, X.Q.: Shear stiffness of granular material at small strain: does it depend on grain size? Géotechnique 63(2), 165–179 (2013)
Duffy, J., Mindlin, R.D.: Stress-strain relations and vibrations of a granular medium. J. Appl. Mech. 24, 585–593 (1956)
Petrakis, E., Dobry, R.: Micromechanical Modeling of Granular Soil at Small Strain by Arrays of Elastic Spheres. Report CE-87-02, Dept. Civil Eng., Rensselaer Polytechnic Institute, Troy, NY (1987)
Wichtmann, T., Triantafyllidis, Th: Influence of the grain-size distribution curve of quartz sand on small strain shear modulus \(G_{\rm max}\). J. Geotech. Geoenviron. Eng. 135(10), 1404–1418 (2009)
Yang, J., Gu, X.Q.: Dynamic shear modulus of dry sand: effect of test method. In: Proceedings of the 14th European Conference on Earthquake Engineering, Ohrid, Macedonia (2010)
Goddard, J.D.: Nonlinear elasticity and pressure-dependent wave speeds in granular media. Proc. R. Soc. Lond. 430, 105–131 (1990)
Chen, Y.-C., Ishibashi, I., Jenkins, J.T.: Dynamic shear modulus and fabric: Part I, depositional and induced anisotropy. Géotechnique 38(1), 25–32 (1988)
Santamarina, J.C., Cascante, G.: Effect of surface roughness on the wave propagation parameters. Géotechnique 48(1), 129–136 (1998)
Yimsiri, S., Soga, K.: Effect of surface roughness on small-strain modulus: Micromechanics view. Proc. 2nd Int. Symp. Prefail. Deform. Charact. Geomater. 1, 597–602 (1999)
Kumar, J., Madhusudhan, B.N.: Effect of relative density and confining pressure on Poisson ratio from bender-extender element tests. Géotechnique 60(7), 561–567 (2010)
Gu, X.Q.: Dynamic Properties of Granular Materials at the Macro and Micro Scales. PhD thesis, The University of Hong Kong, Hong Kong (2012)
McDowell, G.R., Bolton, M.D.: Micro mechanics of elastic soil. Soils Found. 41(6), 147–152 (2001)
Chang, C.S., Misra, A., Sundaram, S.S.: Properties of granular packing under low amplitude cyclic loading. Soil Dyn. Earthq. Eng. 10(4), 201–211 (1991)
Itasca: User’s manual for PFC\(^{2D}\). Itasca Consulting Group, Inc., Minneapolis (2005)
Hoque, E., Tatsuoka, F.: Effects of stress ratio on small-strain stiffness during triaxial shearing. Géotechnique 54(7), 429–439 (2004)
Thornton, C.: Numerical simulations of deviatoric shear deformation of granular media. Géotechnique 50(1), 43–53 (2000)
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The work presented in this paper was supported by the University of Hong Kong through the Seed Funding for Basic Research scheme (11159098) and the Outstanding Young Researcher Award scheme (2006–2007). This support is gratefully acknowledged.
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Gu, X.Q., Yang, J. A discrete element analysis of elastic properties of granular materials. Granular Matter 15, 139–147 (2013). https://doi.org/10.1007/s10035-013-0390-3
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DOI: https://doi.org/10.1007/s10035-013-0390-3