Abstract
Owing to the significant effects of adhesive force and surface/membrane tension, the classical contact models often fail to describe the indentation responses of soft materials and biological systems. This work addresses the axisymmetric indentation of an elastic substrate with constant surface/membrane tension by a spherical, conical, or cylindrical flat indenter in the Johnson–Kendall–Roberts adhesive approximation. On the basis of non-adhesive contact solutions accounting for the surface/membrane tension effect, explicit expressions for the external load and depth with respect to the contact radius are derived for the adhesive contact cases, which act as the theoretical fundamental for the accurate analysis of indentation tests. Despite using different correction functions, the results for spherical indentation are consistent with the solution of previous studies. It is found that the role of surface/membrane tension in the adhesive contact behavior is controlled by a dimensionless parameter. As the parameter gets larger, the pull-off force and the contact size at zero-external load for spherical and conical indentations are smaller, whereas the pull-off force for cylindrical flat indentation is higher.
Issue Section:
Research Papers
References
1.
Zhang
, M. G.
, Cao
, Y. P.
, Li
, G. Y.
, and Feng
, X. Q.
, 2014
, “Spherical Indentation Method for Determining the Constitutive Parameters of Hyperelastic Soft Materials
,” Biomech. Model. Mechanobiol.
, 13
(1
), pp. 1
–11
. 2.
Niu
, T.
, and Cao
, G.
, 2014
, “Finite Size Effect Does not Depend on the Loading History in Soft Matter Indentation
,” J. Phys. D Appl. Phys.
, 47
(38
), p. 385303
. 3.
Dimitriadis
, E. K.
, Horkay
, F.
, Maresca
, J.
, Kachar
, B.
, and Chadwick
, R. S.
, 2002
, “Determination of Elastic Moduli of Thin Layers of Soft Material Using the Atomic Force Microscope
,” Biophys. J.
, 82
(5
), pp. 2798
–2810
. 4.
Ebenstein
, D. M.
, and Pruitt
, L. A.
, 2006
, “Nanoindentation of Biological Materials
,” Nano Today
, 1
(3
), pp. 26
–33
. 5.
Nguyen
, T. D.
, and Gu
, Y.
, 2014
, “Exploration of Mechanisms Underlying the Strain-Rate-Dependent Mechanical Property of Single Chondrocytes
,” Appl. Phys. Lett.
, 104
(18
), p. 183701
. 6.
Sen
, S.
, Subramanian
, S.
, and Discher
, D. E.
, 2005
, “Indentation and Adhesive Probing of a Cell Membrane With AFM: Theoretical Model and Experiments
,” Biophys. J.
, 89
(5
), pp. 3203
–3213
. 7.
Zamir
, E. A.
, and Taber
, L. A.
, 2004
, “On the Effects of Residual Stress in Microindentation Tests of Soft Tissue Structures
,” ASME J. Biomech. Eng.
, 126
(2
), pp. 276
–283
. 8.
Hajji
, M. A.
, 1978
, “Indentation of a Membrane on an Elastic Half Space
,” ASME J. Appl. Mech.
, 45
(2
), pp. 320
–324
. 9.
Hertz
, H.
, 1882
, “On the Contact Between Elastic Bodies
,” J. Reine Angew. Math.
, 92
, pp. 156
–171
. 10.
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
(1
), pp. 47
–57
. 11.
Johnson
, K. L.
, Kendall
, K.
, and Roberts
, A. D.
, 1971
, “Surface Energy and the Contact of Elastic Solids
,” Proc. R. Soc. A: Math. Phys. Eng. Sci.
, 324
(1558
), pp. 301
–313
. 12.
Maugis
, D.
, 1992
, “Adhesion of Spheres: the JKR-DMT Transition Using a Dugdale Model
,” J. Colloid Interface Sci.
, 150
(1
), pp. 243
–269
. 13.
Gerberich
, W. W.
, Tymiak
, N. I.
, Grunlan
, J. C.
, Horstemeyer
, M. F.
, and Baskes
, M. I.
, 2002
, “Interpretations of Indentation Size Effects
,” ASME J. Appl. Mech.
, 69
(4
), pp. 433
–442
. 14.
Zhang
, T. Y.
, and Xu
, W. H.
, 2002
, “Surface Effects on Nanoindentation
,” J. Mater. Res.
, 17
(7
), pp. 1715
–1720
. 15.
Maugis
, D.
, and Barquins
, M.
, 1981
, “Adhesive Contact of a Conical Punch on an Elastic Half-Space
,” J. Phys. Lett.
, 42
(5
), pp. 95
–97
. 16.
Borodich
, F. M.
, Galanov
, B. A.
, and Suarez-Alvarez
, M. M.
, 2014
, “The JKR-Type Adhesive Contact Problems for Power-Law Shaped Axisymmetric Punches
,” J. Mech. Phys. Solids
, 68
, pp. 14
–32
. 17.
Argatov
, I.
, Li
, Q.
, Pohrt
, R.
, and Popov
, V. L.
, 2016
, “Johnson-Kendall-Roberts Adhesive Contact for a Toroidal Indenter
,” Proc. R. Soc. A: Math. Phys. Eng. Sci.
, 472
(2191
), p. 20160218
. 18.
Style
, R. W.
, Hyland
, C.
, Boltyanskiy
, R.
, Wettlaufer
, J. S.
, and Dufresne
, E. R.
, 2013
, “Surface Tension and Contact With Soft Elastic Solids
,” Nat. Commun.
, 4
(1
), p. 2728
. 19.
Cao
, Z.
, Stevens
, M. J.
, and Dobrynin
, A. V.
, 2014
, “Adhesion and Wetting of Nanoparticles on Soft Surfaces
,” Macromolecules
, 47
(9
), pp. 3203
–3209
. 20.
Cao
, Z.
, and Dobrynin
, A. V.
, 2015
, “Contact Mechanics of Nanoparticles: Pulling Rigid Nanoparticles From Soft, Polymeric Surfaces
,” Langmuir
, 31
(45
), pp. 12520
–12529
. 21.
Gurtin
, M. E.
, and Murdoch
, A. I.
, 1975
, “A Continuum Theory of Elastic Material Surfaces
,” Arch. Ration. Mech. Anal.
, 57
(4
), pp. 291
–323
. 22.
Povstenko
, Y. Z.
, 1993
, “Theoretical Investigation of Phenomena Caused by Heterogeneous Surface Tension in Solids
,” J. Mech. Phys. Solids
, 41
(9
), pp. 1499
–1514
. 23.
Huang
, Z. P.
, and Wang
, J.
, 2006
, “A Theory of Hyperelasticity of Multi-Phase Media With Surface/Interface Energy Effect
,” Acta Mech.
, 182
(3–4
), pp. 195
–210
. 24.
Dingreville
, R.
, Qu
, J. M.
, and Cherkaoui
, M.
, 2005
, “Surface Free Energy and its Effect on the Elastic Behavior of Nano-Sized Particles, Wires and Films
,” J. Mech. Phys. Solids
, 53
(8
), pp. 1827
–1854
. 25.
He
, L. H.
, and Lim
, C. W.
, 2006
, “Surface Green Function for a Soft Elastic Half-Space: Influence of Surface Stress
,” Int. J. Solid Struct.
, 43
(1
), pp. 132
–143
. 26.
Wang
, G. F.
, and Feng
, X. Q.
, 2007
, “Effects of Surface Stresses on Contact Problems at Nanoscale
,” J. Appl. Phys.
, 101
(1
), p. 013510
. 27.
Zhou
, S.
, and Gao
, X. L.
, 2013
, “Solutions of Half-Space and Half-Plane Contact Problems Based on Surface Elasticity
,” Z. Angew. Math. Phys.
, 64
(1
), pp. 145
–166
. 28.
Long
, J.
, Ding
, Y.
, Yuan
, W.
, Chen
, W.
, and Wang
, G.
, 2017
, “General Relations of Indentations on Solids With Surface Tension
,” ASME J. Appl. Mech.
, 84
(5
), p. 051007
. 29.
Long
, J.
, Yuan
, W.
, Chen
, W.
, and Wang
, G.
, 2018
, “Analytic Relations for Two-Dimensional Indentations With Surface Tension
,” Mech. Mater.
, 119
, pp. 34
–41
. 30.
Pinyochotiwong
, Y.
, Rungamornrat
, J.
, and Senjuntichai
, T.
, 2013
, “Rigid Frictionless Indentation on Elastic Half Space With Influence of Surface Stresses
,” Int. J. Eng. Sci.
, 71
, pp. 15
–35
. 31.
Argatov
, I.
, 2022
, “The Surface Tension Effect Revealed via the Indentation Scaling Index
,” Int. J. Eng. Sci.
, 170
, p. 103593
. 32.
Zhou
, W.
, and Yang
, F.
, 2022
, “Effects of Surface Stress on the Indentation Response of an Elastic Half-Space
,” Int. J. Mech. Sci.
, 229
, p. 107512
. 33.
Xia
, K.
, Liu
, C. F.
, Leong
, W. H.
, Kwok
, M. H.
, Yang
, Z. Y.
, Feng
, X.
, Liu
, R. B.
, and Li
, Q.
, 2019
, “Nanometerprecision Non-Local Deformation Reconstruction Using Nanodiamond Sensing
,” Nat. Commun.
, 10
(1
), p. 3259
. 34.
Cheng
, Z.
, Shurer
, C. R.
, Schmidt
, S.
, Gupta
, V. K.
, Chuang
, G.
, Su
, J.
, Watkins
, A. R.
, et al, 2020
, “The Surface Stress of Biomedical Silicones Is a Stimulant of Cellular Response
,” Sci. Adv.
, 6
(15
), p. eaay0076
. 35.
Style
, R. W.
, Jagota
, A.
, Hui
, C. Y.
, and Dufresne
, E. R.
, 2017
, “Elastocapillarity: Surface Tension and the Mechanics of Soft Solids
,” Annu. Rev. Condens. Matter Phys.
, 8
(1
), pp. 99
–118
. 36.
Kim
, J. H.
, and Gouldstone
, A.
, 2008
, “Spherical Indentation of a Membrane on an Elastic Half-Space
,” J. Mater. Res.
, 23
(8
), pp. 2212
–2220
. 37.
Salez
, T.
, Benzaquen
, M.
, and Raphael
, E.
, 2013
, “From Adhesion to Wetting of a Soft Particle
,” Soft Matter
, 9
(45
), pp. 10699
–10704
. 38.
Long
, J. M.
, Wang
, G. F.
, Feng
, X. Q.
, and Yu
, S. W.
, 2016
, “Effects of Surface Tension on the Adhesive Contact Between a Hard Sphere and a Soft Substrate
,” Int. J. Solids Struct.
, 84
, pp. 133
–138
. 39.
Gao
, X.
, Hao
, F.
, Huang
, Z. P.
, and Fang
, D. N.
, 2014
, “Mechanics of Adhesive Contact at the Nanoscale: The Effect of Surface Stress
,” Int. J. Solids Struct.
, 51
(3–4
), pp. 566
–574
. 40.
Hui
, C. Y.
, Liu
, T. S.
, Salez
, T.
, Raphael
, E.
, and Jagota
, A.
, 2015
, “Indentation of a Rigid Sphere Into an Elastic Substrate With Surface Tension and Adhesion
,” Proc. R. Soc. A: Math. Phys. Eng. Sci.
, 471
(2175
), p. 20140727
. 41.
Zhang
, L.
, and Ru
, C. Q.
, 2019
, “A Refined JKR Model for Adhesion of a Rigid Sphere on a Soft Elastic Substrate
,” ASME J. Appl. Mech.
, 86
(5
), p. 051004
. 42.
Xu
, X. J.
, Jagota
, A.
, and Hui
, C. Y.
, 2014
, “Effects of Surface Tension on the Adhesive Contact of a Rigid Sphere to a Compliant Substrate
,” Soft Matter
, 10
(26
), pp. 4625
–4632
. 43.
Zhu
, X. Y.
, and Xu
, W.
, 2018
, “Effect of Surface Tension on the Behavior of Adhesive Contact Based on Lennard-Jones Potential law
,” J. Mech. Phys. Solids
, 111
, pp. 170
–183
. 44.
Jia
, N.
, Yao
, Y.
, Peng
, Z.
, and Chen
, S.
, 2021
, “Surface Effect in Nanoscale Adhesive Contact
,” J. Adhesion
, 97
(4
), pp. 380
–398
. 45.
Zhu
, X.
, Xu
, W.
, and Qin
, Y.
, 2020
, “Effect of Surface Tension on the Behavior of Adhesive Contact Based on Maugis‒Dugdale Model
,” Eur. J. Mech. A-Solid
, 81
, p. 103930
. 46.
Ciavarella
, M.
, 2018
, “An Approximate JKR Solution for a General Contact, Including Rough Contacts
,” J. Mech. Phys. Solids
, 114
, pp. 209
–218
. 47.
Popov
, V. L.
, 2018
, “Solution of Adhesive Contact Problem on the Basis of the Known Solution for Non-Adhesive One
,” Facta Universitatis
, 16
(1
), pp. 93
–98
. 48.
Kendall
, K.
, 1971
, “The Adhesion and Surface Energy of Elastic Solids
,” J. Phys. D Appl. Phys.
, 4
(8
), pp. 1186
–1195
. 49.
Liu
, T.
, Jagota
, A.
, and Hui
, C. Y.
, 2016
, “Effect of Surface Tension on the Adhesion Between a Rigid Flat Punch and a Semi-Infinite Neo-Hookean Half-Space
,” Extreme Mech. Lett.
, 9
, pp. 310
–316
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