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Research Papers

Interaction Model for “Shelled Particles” and Small-Strain Modulus of Granular Materials

[+] Author and Article Information
Shun-hua Zhou

Professor
Key Laboratory of Road and Traffic Engineering
of the State Ministry of Education,
Tongji University,
Shanghai 200092, China
e-mail: zhoushh@tongji.edu.cn

Peijun Guo

Professor
Key Laboratory of Road and Traffic Engineering
of the State Ministry of Education,
Tongji University,
Shanghai 200092, China;
Department of Civil Engineering,
McMaster University,
Hamilton L8S 4L7, ON, Canada
e-mail: guop@mcmaster.ca

Dieter F. E. Stolle

Professor
Department of Civil Engineering,
McMaster University,
Hamilton L8S 4L7, ON, Canada
e-mail: stolle@mcmaster.ca

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 25, 2018; final manuscript received May 22, 2018; published online June 27, 2018. Assoc. Editor: Thomas Siegmund.

J. Appl. Mech 85(10), 101001 (Jun 27, 2018) (11 pages) Paper No: JAM-18-1170; doi: 10.1115/1.4040408 History: Received March 25, 2018; Revised May 22, 2018

The elastic modulus of a granular assembly composed of spherical particles in Hertzian contact usually has a scaling law with the mean effective pressure p as KGp1/3. Laboratory test results, however, reveal that the value of the exponent is generally around 1/2 for most sands and gravels, but it is much higher for reclaimed asphalt concrete composed of particles coated by a thin layer of asphalt binder and even approaching unity for aggregates consisting of crushed stone. By assuming that a particle is coated with a thin soft deteriorated layer, an energy-based simple approach is proposed for thin-coating contact problems. Based on the features of the surface layer, the normal contact stiffness between two spheres varies with the contact force following knFnm and m[1/3,1], with m=1/3 for Hertzian contact, m=1/2 soft thin-coating contact, m=2/3 for incompressible soft thin-coating, and m=1 for spheres with rough surfaces. Correspondingly, the elastic modulus of a random granular packing is proportional to pm with m[1/3,1].

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References

Mavko, G. , Mukerji ., and Dvorkin, J. , 2009, The Rock Physics Handbook, Cambridge University Press, Cambridge, UK. [CrossRef] [PubMed] [PubMed]
Fleischmann, J. A. , Drugan, W. J. , and Plesha, M. E. , 2013, “ Direct Micromechanics Derivation and DEM Confirmation of the Elastic Moduli of Isotropic Particulate Materials—Part II: Particle Rotation,” J. Mech. Phys. Solids, 61(7), pp. 1585–1599. [CrossRef]
Hertz, H. , 1881, “ Über Die Beruhrüng Fester Elastischer Körper,” J. Für Die Reine Angew. Math., 92, pp. 156–171.
Johnson, K. L. , 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK. [CrossRef]
Mindlin, R. D. , and Deresiewicz, H. , 1953, “ Elastic Spheres in Contact Under Varying Oblique Force,” ASME J. Appl. Mech., 20(3), pp. 327–344.
Gao, Y. F. , Lucas, B. N. , Hay, J. C. , Oliverd, W. C. , and Pharra, G. M. , 2006, “ Nanoscale Incipient Asperity Sliding and Interface Micro-Slip Assessed by the Measurement of Tangential Contact Stiffness,” Scr. Mater., 55(7), pp. 653–656. [CrossRef]
Annett, J. , Gao, Y. , Cross, G. L. , Herbert, E. G. , and Lucas, B. N. , 2011, “ Mesoscale Friction Anisotropy Revealed by Slidingless Tests,” J. Mater. Res., 26(18), pp. 2373–2378. [CrossRef]
Yimsiri, S. , and Soga, K. , 2000, “ Micromechanics-Based Stress-Strain Behaviour of Soils at Small Strains,” Géotechnique, 50(5), pp. 559–571. [CrossRef]
Potyondy, D. O. , and Cundall, P. A. , 2004, “ A Bonded-Particle Model for Rock,” Int. J. Rock Mech. Min. Sci., 41(8), pp. 1329–1364. [CrossRef]
Itasca Consulting Group, Inc., 2006, “PFC3D User's Manual,” Itasca Consulting Group, Inc., Minneapolis, MN.
Plassiard, J.-P. , Belheine, N. , and Donzé, F.-V. , 2009, “ A Spherical Discrete Element Model: Calibration Procedure and Incremental Response,” Granular Mater., 11(5), pp. 293–306. [CrossRef]
Hardin, B. O. , and Richart, F. E., Jr. , 1963, “ Elastic Wave Velocities in Granular Soils,” J. Soil Mech. Found. Div., 89(1), pp. 33–65. http://cedb.asce.org/CEDBsearch/record.jsp?dockey=0013008
Goddard, J. D. , 1990, “ Nonlinear Elasticity and Pressure-Dependent Wave Speeds in Granular Media,” Proc. R. Soc. London A, 430(1878), pp. 105–131. [CrossRef]
Agnolin, I. , and Roux, J.-N. , 2005, “ Elasticity of Sphere Packings: Pressure and Initial State Dependence,” Powders & Grains, R. Garcia-Rojo , H. J. Hermann , and McNamara , eds., Balkema Publishers, Rotterdam, The Netherlands, pp. 87–91.
Mitchell, J. K. , and Soga, K. , 2005, Fundamentals of Soil Behaviour, 3rd ed., Wiley, Hoboken, NJ.
Wichtmann, T. , and Triantafyllidis, T. , 2009, “ On the Influence of the Grain Size Distribution Curve of Quartz Sand on the Small Strain Shear Modulus Gmax,” J. Geotech. Geoenviron. Eng., 135(10), pp. 1404–1418. [CrossRef]
Yang, J. , and Gu, X. Q. , 2013, “ Shear Stiffness of Granular Material at Small Strains: Does It Depend on Grain Size?,” Géotechnique, 63(2), pp. 165–179. [CrossRef]
Zimmer, M. A. , Prasad, M. , Mavko, G. , and Nur, A. , 2007, “ Seismic Velocities of Unconsolidated Sands—Part 1: Pressure Trends From 0.1 to 20 MPa,” Geophys., 72(1), pp. E1–E13. [CrossRef]
Sneddon, I. N. , 1948, “ Boussinesq's Problem for a Rigid Cone,” Proc. Cambridge Philos. Soc., 44(4), pp. 492–507. [CrossRef]
De Gennes, P.-G. , 1996, “ Static Compression of a Granular Medium: The ‘Soft Shell’ Model,” Europhys. Lett., 35(2), pp. 145–149. [CrossRef]
Alshibli, K. A. , and Alsaleh, M. I. , 2004, “ Characterizing Surface Roughness and Shape of Sands,” J. Comput. Civ. Eng., 18(1), pp. 36–45. [CrossRef]
Yang, H. , Baudet, B. A. , and Yao, T. , 2016, “ Characterization of the Surface Roughness of Sand Particles Using an Advanced Fractal Approach,” Proc. R. Soc. London A, 472(2194), pp. 20–21. [CrossRef]
Huang, B. , Li, G. , Vukosavljevic, D. , Shu, X. , and Egan, B. K. , 2005, “ Laboratory Investigation of Mixing Hot-Mix Asphalt With Reclaimed Asphalt Pavement,” Transp. Res. Rec., 1929, pp. 37–45. [CrossRef]
Herrmann, H. J. , Stauffer, D. , and Roux, S. , 1987, “ Violation of Linear Elasticity Due to Randomness,” Europhys. Lett., 3(3), pp. 265–267. [CrossRef]
Costea, C. , and Gilles, B. , 1999, “ On the Validity of Hertz Contact Law for Granular Material Acoustics,” Eur. Phys. J., B7(1), pp. 155–168. [CrossRef]
Payan, M. , Kostas Senetakis, K. , Khoshghalb, A. , and Khalili, N. , 2017, “ Effect of Gradation and Particle Shape on Small-Strain Young's Modulus and Poisson's Ratio of Sands,” Int. J. Geomech., 17(5), p. 04016120. [CrossRef]
Menq, F. Y. , and Stokoe, K. H. , 2003, “ Linear Dynamic Properties of Sandy and Gravelly Soils From Large-Scale Resonant Tests,” Third International Symposium on Deformation Characteristics of Geomaterial, Lyon, France, Sept. 22–24, pp. 63–71.
Kohata, Y. , Tatsuoka, F. , Wang, L. , Jiang, G. L. , Hoque, E. , and Kodaka, T. , 1997, “ Modelling the Non-Linear Deformation Properties of Stiff Geomaterials,” Géotechnique, 47(3), pp. 563–580. [CrossRef]
Jardine, R. J. , 1995, “ One Perspective of the Pre-Failure Deformation Characteristics of Some Geomaterials,” Pre-Failure Deformation of Geomaterials, S. , Shibuya , T. Mitachi , and S. Miura , eds., Vol. 2, Balkema, Rotterdam, The Netherlands pp. 855–885.
Towhata, I. , 2008, Geotechnical Earthquake Engineering, Springer, Berlin. [CrossRef]
Popov, V. L. , 2010, Contact Mechanics and Friction: Physical Principles and Applications, Springer-Verlag, Berlin. [CrossRef]
Johnson, K. L. , and Sridhar, I. , 2001, “ Adhesion Between a Spherical Indenter and an Elastic Solid With a Compliant Elastic Coating,” J. Phys. D: Appl. Phys., 34(5), pp. 683–689. [CrossRef]
Liu, S. B. , Peyronnel, A. P. , Wang, Q. J. , and Keer, L. M. , 2005, “ An Extension of the Hertz Theory for Three-Dimensional Coated Bodies,” Tribol. Lett., 18(3), pp. 303–314. [CrossRef]
Reedy, E. D., Jr. , 2006, “ Thin-Coating Contact Mechanics With Adhesion,” J. Mater. Res., 21(10), pp. 2660–2668. [CrossRef]
Greenwood, J. A. , and Barber, J. R. , 2012, “ Indentation of an Elastic Layer by a Rigid Cylinder,” Int. J. Solids Struct., 49(21), pp. 2962–2977. [CrossRef]
McGuiggan, P. M. , Wallace, J. S. , Smith, D. T. , Sridhar, I. , Zheng, Z. W. , and Johnson, K. L. , 2007, “ Contact Mechanics of Layered Elastic Materials: Experiment and Theory,” J. Phys. D: Appl. Phys., 40(19), pp. 5984–5994. [CrossRef]
Reedy, E. D., Jr. , 2007, “ Contact Mechanics for Coated Spheres That Includes the Transition From Weak to Strong Adhesion,” J. Mater. Res., 22(9), pp. 2617–2622. [CrossRef]
Sohn, D. , Won, H.-S. , Jang, B. , Kim, J.-H. , Lee, H.-J. , and Choi, S. T. , 2015, “ Extended JKR Theory on Adhesive Contact Between Elastic Coatings on Rigid Cylinders Under Plane Strain,” Int. J. Solids Struct., 71, pp. 244–254. [CrossRef]
Gladwell, G. M. L. , 1980, Contact Problems in the Classical Theory of Elasticity, Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, The Netherlands. [CrossRef]
Meijers, P. , 1968, “ Rigid Cylinder on an Elastic Layer,” Sci. Res., 18(1), pp. 353–358.
Hlaváčk, M. , 2008, “ Elliptical Contact on Elastic Incompressible Coating,” Eng. Mech., 15(4), pp. 249–261. http://www.engineeringmechanics.cz/pdf/15_4_249.pdf
Matthewson, M. J. , 1981, “ Axi-Symmetric Contact on Thin Compliant Coatings,” J. Mech. Phys. Solids, 29(2), pp. 89–113. [CrossRef]
Jaffar, M. J. , 1988, “ A Numerical Solution for Axisymmetric Contact Problems Involving Rigid Indenters on Elastic Layers,” J. Mech. Phys. Solids, 36(4), pp. 401–416. [CrossRef]
Hopkins, T. C. , Beckham, T. L. , and Sun, C. , 2007, “ Resilient Modulus of Compacted Crushed Stone Aggregate Bases,” Kentucky Transportation Centre, University of Kentucky, Lexington, KY, Report No. KTC-05-27/SPR-229-01-1F. https://uknowledge.uky.edu/ktc_researchreports/176/
Margolis, S. V. , and Krinsley, D. H. , 1974, “ Processes of Formation and Environmental Occurrence of Microfeatures on Detrital Quartz Grains,” Am. J. Sci., 274(5), pp. 449–464. [CrossRef]
Uthus, L. , Hoff, I. , and Horvli, I. , 2005, “ Evaluation of Grain Shape Characterization Methods for Unbound Aggregates,” Seventh International Conference on the Bearing Capacity of Road, Railways and Airfields, Trondheim, Norway, June 27–29, p. 12.
Greenwood, J. A. , and Williamson, J. B. P. , 1966, “ Contact of Nominally Flat Surfaces,” Proc. R. Soc. London A, 295(1442), pp. 300–319. [CrossRef]
Gao, Y.-F. , and Bower, A. F. , 2006, “ Elastic-Plastic Contact of a Rough Surface With Weierstrass Profile,” Proc. R. Soc. London A, 462(2065), pp. 319–348. [CrossRef]
Pohrt, R. , and Popov, V. L. , 2013, “ Contact Stiffness of Randomly Rough Surfaces,” Sci. Rep., 3(3293), pp. 1–6.

Figures

Grahic Jump Location
Fig. 1

Variation of exponent n with the coefficient of uniformity and mean particle size d50

Grahic Jump Location
Fig. 2

Contact between coated elastic bodies: (a) a rigid sphere and an elastic layer bonded on a rigid substrate and (b) two coated rigid spheres

Grahic Jump Location
Fig. 3

Contact of rigid spheres coated by incompressible elastic layers: (a) εr=εθ=0 and (b) εz+εr+εθ=0

Grahic Jump Location
Fig. 4

Contact of a rigid frictionless sphere with a thin incompressible layer bounded on a semi-infinite rigid substrate: (a) εr=εθ=0 and (b) εz+εr+εθ=0

Grahic Jump Location
Fig. 5

Particle size distribution of the virgin aggregate and select RAPs

Grahic Jump Location
Fig. 6

Influence of RAP content on the variation of resilient modulus with bulk stress for different aggregate-RAP blends: (a) virgin aggregate and aggregate-RAP 3 blends and (b) virgin aggregate and aggregate-RAP 1 blends

Grahic Jump Location
Fig. 7

Variation of K2 with RAP content

Grahic Jump Location
Fig. 8

Contact of rough surfaces (after Ref. [31]): (a) A random rough surface according to Greenwood and Williamson (1966) and (b) contact of a rough surface composed of cone-shaped asperities

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