In the present paper, axisymmetric frictionless adhesive contact between a rigid punch and a power-law graded elastic half-space is analytically investigated with use of Betti's reciprocity theorem and the generalized Abel transformation, a set of general closed-form solutions are derived to the Hertzian contact and Johnson–Kendall–Roberts (JKR)-type adhesive contact problems for an arbitrary punch profile within a circular contact region. These solutions provide analytical expressions of the surface stress, deformation fields, and equilibrium relations among the applied load, indentation depth, and contact radius. Based on these results, we then examine the combined effects of material inhomogeneities and punch surface morphologies on the adhesion behaviors of the considered contact system. The analytical results obtained in this paper include the corresponding solutions for homogeneous isotropic materials and the Gibson soil as special cases and, therefore, can also serve as the benchmarks for checking the validity of the numerical solution methods.

References

References
1.
Hertz
,
H.
,
1882
, “
On the Contact of Elastic Solids
,”
J. Reine Angew. Math.
,
92
, pp.
156
171
.
2.
Johnson
,
K. L.
,
Kendall
,
K.
, and
Roberts
,
A. D.
,
1971
, “
Surface Energy and the Contact of Elastic Solids
,”
Proc. R. Soc. London, Ser. A
,
324
, pp.
301
313
.10.1098/rspa.1971.0141
3.
Maugis
,
D.
,
1995
, “
Extension of the Johnson-Kendall-Roberts Theory of the Elastic Contact of Spheres to Large Contact Radii
,”
Langmuir
11
, pp.
679
682
.10.1021/la00002a055
4.
Spolenak
,
R.
,
Gorb
,
S.
,
Gao
,
H.
, and
Arzt
,
E.
,
2005
, “
Effects of Contact Shape on the Scaling on Biological Attachments
,”
Proc. R. Soc. London, Ser. A
,
461
, pp.
305
319
.10.1098/rspa.2004.1326
5.
del Campo
,
A.
,
Greiner
,
C.
, and
Arzt
,
E.
,
2007
, “
Contact Shape Controls Adhesion of Bioinspired Fibrillar Surfaces
,”
Langmuir
23
, pp.
10235
10243
.10.1021/la7010502
6.
Waters
,
J.
,
Gao
,
H.
, and
Guduru
,
P.
,
2011
, “
On Adhesion Enhancement Due to Concave Surface Geometries
,”
J. Adhes.
87
, pp.
194
213
.10.1080/00218464.2011.557325
7.
Gao
,
H.
, and
Yao
,
H.
,
2004
, “
Shape Insensitive Optimal Adhesion of Nanoscale Fibrillar Structures
,”
Proc. Natl. Acad. Sci. U.S.A.
,
101
, pp.
7851
7856
.10.1073/pnas.0400757101
8.
Yao
,
H.
, and
Gao
,
H.
,
2006
, “
Optimal Shapes for Adhesive Binding Between Two Elastic Bodies
,”
J. Colloid Interface Sci.
,
298
, pp.
567
572
.10.1016/j.jcis.2005.12.059
9.
Guduru
,
P. R.
,
2007
, “
Detachment of a Rigid Solid From an Elastic Wavy Surface: Theory
,”
J. Mech. Phys. Solids
,
55
, pp.
445
472
.10.1016/j.jmps.2006.09.004
10.
Peng
,
Z.
, and
Chen
,
S.
,
2012
, “
The Effect of Geometry on the Adhesive Behavior of Bio-Inspired Fibrils
,”
Soft Matter
8
, pp.
9864
9869
.10.1039/c2sm26390d
11.
Sundaram
,
N.
, and
Chandrasekar
,
S.
,
2011
, “
Shape and Eccentricity Effects in Adhesive Contacts of Rodlike Particles
,”
Langmuir
27
, pp.
12405
12410
.10.1021/la202740b
12.
Hill
,
R.
, and
Strorakers
,
B.
,
1990
, “
A Concise Treatment of Axisymmetric Indentation in Elasticity
,”
Elasticity: Mathematical Methods and Application
(Sneddon 70th Anniversary Volume),
G.
Eason
,
R. W.
Ogden
, eds.,
Ellis Horwood
,
Chichester, UK
, pp.
199
209
.
13.
Sneddon
, I
. N.
,
1965
, “
The Relation Between Load and Penetration in the Axisymmetric Boussinesq Problem for a Punch of Arbitrary Profile
,”
Int. J. Eng. Sci.
,
3
, pp.
47
56
.10.1016/0020-7225(65)90019-4
14.
Barber
,
J. R.
,
2002
,
Elasticity
,
Kluwer Academic
,
Dordrecht, The Netherlands
.
15.
Maugis
,
D.
,
2000
,
Contact, Adhesion and Rupture of Elastic Solids
,
Springer-Verlag
,
Berlin
.
16.
Zhou
,
S.-S.
,
Gao
,
X.-L.
, and
He
,
Q.-C.
,
2011
, “
A Unified Treatment of Axisymmetric Adhesive Contact Problems Using the Harmonic Potential Function Method
,”
J. Mech. Phys. Solids
,
59
, p.
145
159
.10.1016/j.jmps.2010.11.006
17.
Kesari
,
H.
, and
Lew
,
A.
,
2011
, “
Adhesive Frictionless Contact Between an Elastic Isotropic Half-Space and a Rigid Axisymmetric Punch
,”
J. Elast.
,
106
, pp.
203
224
.10.1007/s10659-011-9323-8
18.
Suresh
,
S.
,
2001
, “
Graded Materials for Resistance to Contact Deformation and Damage
,”
Science
,
292
, pp.
2447
2451
.10.1126/science.1059716
19.
Sherge
,
M.
, and
Gorb
,
S.
,
2001
,
Biological Micro- and Nano-Tribology—Nature's Solutions
,
Springer
,
Berlin
.
20.
Chen
,
S.
,
Yan
,
C.
, and
Soh
,
A.
,
2009
, “
Adhesive Behavior of Two-Dimensional Power-Law Graded Materials
,.
Int. J. Solids Struct.
,
46
, pp.
3398
3404
.10.1016/j.ijsolstr.2009.05.006
21.
Chen
,
S.
,
Yan
,
C.
,
Zhang
,
P.
, and
Gao
,
H.
,
2009
, “
Mechanics of Adhesive Contact on a Power-Law Graded Elastic Half-Space
,”
J. Mech. Phys. Solids
,
57
, pp.
1437
1448
.10.1016/j.jmps.2009.06.006
22.
Jin
,
F.
, and
Guo
,
X.
,
2010
, “
Non-Slipping Adhesive Contact of a Rigid Cylinder on an Elastic Power-Law Graded Half-Space
,”
Int. J. Solids Struct.
,
47
, pp.
1508
1521
.10.1016/j.ijsolstr.2010.02.010
23.
Guo
,
X.
,
Jin
,
F.
, and
Gao
,
H.
,
2011
, “
Mechanics of Non-Slipping Adhesive Contact on a Power-Law Graded Elastic Half-Space
,”
Int. J. Solids Struct.
,
48
, pp.
2565
2575
.10.1016/j.ijsolstr.2011.05.008
24.
Jin
,
F.
, and
Guo
,
X.
,
2012
, “
Mode-Mixity-Dependent Adhesion of Power-Law Graded Elastic Solids Under Normal and Substrate Stretch-Induced Mismatch Strain
,”
Int. J. Solids Struct.
,
49
, pp.
2349
2357
.10.1016/j.ijsolstr.2012.05.003
25.
Yao
,
H.
, and
Gao
,
H.
,
2010
, “
Gibson-Soil-Like Materials Achieve Flaw-Tolerant Adhesion
,”
J. Comput. Theor. Nanosci.
,
7
, pp.
1299
1305
.10.1166/jctn.2010.1484
26.
Jin
,
F.
, and
Guo
,
X.
,
2013
, “
Mechanics of Axisymmetric Adhesive Contact of Rough Surfaces Involving Power-Law Graded Materials
,” Int. J. Solids Struct. (submitted).
27.
Booker
,
J. R.
,
Balaam
,
N. P.
, and
Davis
,
E. H.
,
1985
, “
The Behavior of an Elastic Non-Homogeneous Half-Space—Part II: Circular and Strip Footings
,”
Int. J. Numer. Anal. Methods Geomech.
,
9
, pp.
369
381
.10.1002/nag.1610090406
28.
Giannakopoulos
,
A. E.
and
Suresh
,
S.
,
1997
, “
Indentation of Solids With Gradients in Elastic Properties—Part II: Axisymmetric Indentors
,”
Int. J. Solids Struct.
,
34
, pp.
2393
2428
10.1016/S0020-7683(96)00172-2
29.
Gorenflo
,
R.
, and
Vessella
,
S.
,
1991
,
Abel Integral Equations: Analysis and Applications
,
Springer-Verlag
,
Berlin
.
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