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].

References

References
1.
Mavko
,
G.
,
Mukerji
., and
Dvorkin
,
J.
,
2009
,
The Rock Physics Handbook
,
Cambridge University Press
,
Cambridge, UK
.
2.
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
.
3.
Hertz
,
H.
,
1881
, “
Über Die Beruhrüng Fester Elastischer Körper
,”
J. Für Die Reine Angew. Math.
,
92
, pp.
156
171
.
4.
Johnson
,
K. L.
,
1985
,
Contact Mechanics
,
Cambridge University Press
,
Cambridge, UK
.
5.
Mindlin
,
R. D.
, and
Deresiewicz
,
H.
,
1953
, “
Elastic Spheres in Contact Under Varying Oblique Force
,”
ASME J. Appl. Mech.
,
20
(3), pp.
327
344
.
6.
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
.
7.
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
.
8.
Yimsiri
,
S.
, and
Soga
,
K.
,
2000
, “
Micromechanics-Based Stress-Strain Behaviour of Soils at Small Strains
,”
Géotechnique
,
50
(
5
), pp.
559
571
.
9.
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
.
10.
Itasca Consulting Group, Inc.,
2006
, “
PFC3D User's Manual
,”
Itasca Consulting Group, Inc.
,
Minneapolis, MN
.
11.
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
.
12.
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
13.
Goddard
,
J. D.
,
1990
, “
Nonlinear Elasticity and Pressure-Dependent Wave Speeds in Granular Media
,”
Proc. R. Soc. London A
,
430
(
1878
), pp.
105
131
.
14.
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
.
15.
Mitchell
,
J. K.
, and
Soga
,
K.
,
2005
,
Fundamentals of Soil Behaviour
,
3rd ed.
,
Wiley
,
Hoboken, NJ
.
16.
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
.
17.
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
.
18.
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
.
19.
Sneddon
,
I. N.
,
1948
, “
Boussinesq's Problem for a Rigid Cone
,”
Proc. Cambridge Philos. Soc.
,
44
(
4
), pp.
492
507
.
20.
De Gennes
,
P.-G.
,
1996
, “
Static Compression of a Granular Medium: The ‘Soft Shell’ Model
,”
Europhys. Lett.
,
35
(
2
), pp.
145
149
.
21.
Alshibli
,
K. A.
, and
Alsaleh
,
M. I.
,
2004
, “
Characterizing Surface Roughness and Shape of Sands
,”
J. Comput. Civ. Eng.
,
18
(
1
), pp.
36
45
.
22.
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
.
23.
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
.
24.
Herrmann
,
H. J.
,
Stauffer
,
D.
, and
Roux
,
S.
,
1987
, “
Violation of Linear Elasticity Due to Randomness
,”
Europhys. Lett.
,
3
(
3
), pp.
265
267
.
25.
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
.
26.
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
.
27.
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
.
28.
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
.
29.
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
.
30.
Towhata
,
I.
,
2008
,
Geotechnical Earthquake Engineering
,
Springer
,
Berlin
.
31.
Popov
,
V. L.
,
2010
,
Contact Mechanics and Friction: Physical Principles and Applications
,
Springer-Verlag
,
Berlin
.
32.
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
.
33.
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
.
34.
Reedy
,
E. D.
, Jr.
,
2006
, “
Thin-Coating Contact Mechanics With Adhesion
,”
J. Mater. Res.
,
21
(
10
), pp.
2660
2668
.
35.
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
.
36.
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
.
37.
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
.
38.
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
.
39.
Gladwell
,
G. M. L.
,
1980
,
Contact Problems in the Classical Theory of Elasticity
,
Sijthoff & Noordhoff International Publishers
, Alphen aan den Rijn,
The Netherlands
.
40.
Meijers
,
P.
,
1968
, “
Rigid Cylinder on an Elastic Layer
,”
Sci. Res.
,
18
(
1
), pp.
353
358
.
41.
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
42.
Matthewson
,
M. J.
,
1981
, “
Axi-Symmetric Contact on Thin Compliant Coatings
,”
J. Mech. Phys. Solids
,
29
(
2
), pp.
89
113
.
43.
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
.
44.
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/
45.
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
.
46.
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
.
47.
Greenwood
,
J. A.
, and
Williamson
,
J. B. P.
,
1966
, “
Contact of Nominally Flat Surfaces
,”
Proc. R. Soc. London A
,
295
(
1442
), pp.
300
319
.
48.
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
.
49.
Pohrt
,
R.
, and
Popov
,
V. L.
,
2013
, “
Contact Stiffness of Randomly Rough Surfaces
,”
Sci. Rep.
,
3
(
3293
), pp.
1
6
.
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