Research Papers

General Relations of Indentations on Solids With Surface Tension

[+] Author and Article Information
Jianmin Long

Institute of Soft Matter Mechanics,
College of Mechanics and Materials,
Hohai University,
Nanjing 210098, China;
Department of Engineering Mechanics,
Xi'an Jiaotong University,
Xi'an 710049, China

Yue Ding, Weike Yuan

Department of Engineering Mechanics,
Xi'an Jiaotong University,
Xi'an 710049, China

Wen Chen

Institute of Soft Matter Mechanics,
College of Mechanics and Materials,
Hohai University,
Nanjing 210098, China

Gangfeng Wang

Department of Engineering Mechanics,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: wanggf@mail.xjtu.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 2, 2017; final manuscript received March 4, 2017; published online April 5, 2017. Assoc. Editor: Harold S. Park.

J. Appl. Mech 84(5), 051007 (Apr 05, 2017) (8 pages) Paper No: JAM-17-1002; doi: 10.1115/1.4036214 History: Received January 02, 2017; Revised March 04, 2017

The conventional contact mechanics does not account for surface tension; however, it is important for micro- or nanosized contacts. In the present paper, the influences of surface tension on the indentations of an elastic half-space by a rigid sphere, cone, and flat-ended cylinder are investigated, and the corresponding singular integral equations are formulated. Due to the complicated structure of the integral kernel, it is difficult to obtain their analytical solutions. By using the Gauss–Chebyshev quadrature formula, the integral equations are solved numerically first. Then, for each indenter, the analytical solutions of two limit cases considering only the bulk elasticity or surface tension are presented. It is interesting to find that, through a simple combination of the solutions of two limit cases and fitting the direct numerical results, the dependence of load on contact radius or indent depth for general case can be given explicitly. The results incorporate the contribution of surface tension in contact mechanics and are helpful to understand contact phenomena at micro- and nanoscale.

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Hertz, H. , 1882, “On the Contact Between Elastic Bodies,” J. Reine Angew. Math., 92, pp. 156–171.
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. [CrossRef]
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(1558), pp. 301–313. [CrossRef]
Derjaguin, B. V. , Muller, V. M. , and Toporov, Y. P. , 1975, “Effect of Contact Deformations on the Adhesion of Particles,” J. Colloid Interface Sci., 53(2), pp. 314–325. [CrossRef]
Maugis, D. , 1992, “Adhesion of Spheres: The JKR-DMT Transition Using a Dugdale Model,” J. Colloid Interface Sci., 150(1), pp. 243–269. [CrossRef]
Muller, V. M. , Yushchenko, V. S. , and Derjaguin, B. V. , 1980, “On the Influence of Molecular Forces on the Deformation of an Elastic Sphere and Its Sticking to a Rigid Plane,” J. Colloid Interface Sci., 77(1), pp. 91–101. [CrossRef]
Greenwood, G. A. , 1997, “Adhesion of Elastic Spheres,” Proc. R. Soc. London, Ser. A, 453(1961), pp. 1277–1297. [CrossRef]
Oliver, W. C. , and Pharr, G. M. , 1992, “An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments,” J. Mater. Res., 7(6), pp. 1564–1583. [CrossRef]
Stelmashenko, N. A. , Walls, M. G. , Brown, L. M. , and Milman, Y. V. , 1993, “Microindentations on W and Mo Oriented Single Crystals: An STM Study,” Acta Metall. Mater., 41(10), pp. 2855–2865. [CrossRef]
Ma, Q. , and Clarke, D. R. , 1995, “Size Dependent Hardness of Silver Single Crystals,” J. Mater. Res., 10(4), pp. 853–863. [CrossRef]
Nix, W. D. , and Gao, H. , 1998, “Indentation Size Effects in Crystalline Materials: A Law for Strain Gradient Plasticity,” J. Mech. Phys. Solids, 46(3), pp. 411–425. [CrossRef]
Begley, M. R. , and Hutchinson, J. W. , 1998, “The Mechanics of Size Dependent Indentation,” J. Mech. Phys. Solids, 46(10), pp. 2049–2068. [CrossRef]
Huang, Y. G. , Zhang, F. , and Hwang, K. C. , 2006, “A Model of Size Effects in Nano-Indentation,” J. Mech. Phys. Solids, 54(8), pp. 1668–1686. [CrossRef]
Tymiak, N. I. , Kramer, D. E. , Bahr, D. F. , and Gerberich, W. W. , 2001, “Plastic Strain and Strain Gradients at Very Small Indentation Depths,” Acta Mater., 49(6), pp. 1021–1034. [CrossRef]
Horstemeyer, M. F. , and Baskes, M. I. , 1999, “Atomistic Finite Deformation Simulations: A Discussion on Length Scale Effects in Relation to Mechanical Stresses,” J. Eng. Mater. Technol., 121(2), pp. 114–119. [CrossRef]
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. [CrossRef]
Gurtin, M. E. , and Murdoch, A. I. , 1975, “A Continuum Theory of Elastic Material Surfaces,” Arch. Ration. Mech. Anal., 57(4), pp. 291–323. [CrossRef]
Gurtin, M. E. , Weissmuller, J. , and Larche, F. , 1998, “A General Theory of Curved Deformable Interfaces in Solids at Equilibrium,” Philos. Mag. A, 78(5), pp. 1093–1109. [CrossRef]
Huang, Z. P. , and Wang, J. , 2006, “A Theory of Hyperelasticity of Multi-Phase Media With Surface/Interface Energy Effect,” Acta Mech., 182(1), pp. 195–210. [CrossRef]
Huang, Z. P. , and Sun, L. , 2007, “ Size-Dependent Effective Properties of a Heterogeneous Material With Interface Energy Effect: From Finite Deformation Theory to Infinitesimal Strain Analysis,” Acta Mech., 190(1), pp. 151–163. [CrossRef]
Miller, R. E. , and Shenoy, V. B. , 2000, “ Size-Dependent Elastic Properties of Nanosized Structural Elements,” Nanotechnology, 11(3), pp. 139–147. [CrossRef]
Sapsathiarn, Y. , and Rajapakse, R. K. N. D. , 2012, “A Model for Large Deflections of Nanobeams and Experimental Comparison,” IEEE Trans. Nanotechnol., 11(2), pp. 247–254. [CrossRef]
Duan, H. L. , Wang, J. , Huang, Z. P. , and Karihaloo, B. L. , 2005, “ Size-Dependent Effective Elastic Constants of Solids Containing Nano-Inhomogeneities With Interface Stress,” J. Mech. Phys. Solids, 53(7), pp. 1574–1596. [CrossRef]
Mogilevskaya, S. G. , Crouch, S. L. , La Grotta, A. , and Stolarski, H. K. , 2010, “The Effects of Surface Elasticity and Surface Tension on the Overall Elastic Behavior of Unidirectional Nano-Composites,” Comput. Sci. Technol., 70(3), pp. 427–434. [CrossRef]
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. [CrossRef]
Wang, G. F. , and Feng, X. Q. , 2009, “Timoshenko Beam Model for Buckling and Vibration of Nanowires With Surface Effects,” J. Phys. D: Appl. Phys., 42(15), p. 155411. [CrossRef]
Olsson, P. A. T. , and Park, H. S. , 2012, “On the Importance of Surface Elastic Contributions to the Flexural Rigidity of Nanowires,” J. Mech. Phys. Solids, 60(12), pp. 2064–2083. [CrossRef]
Hajji, M. A. , 1978, “Indentation of a Membrane on an Elastic Half Space,” ASME J. Appl. Mech., 45(2), pp. 320–324. [CrossRef]
He, L. H. , and Lim, C. W. , 2006, “Surface Green Function for a Soft Elastic Half-Space: Influence of Surface Stress,” Int. J. Solids Struct., 43(1), pp. 132–143. [CrossRef]
Huang, G. Y. , and Yu, S. W. , 2007, “Effect of Surface Elasticity on the Interaction Between Steps,” ASME J. Appl. Mech., 74(4), pp. 821–823. [CrossRef]
Wang, G. F. , and Feng, X. Q. , 2007, “Effects of Surface Stresses on Contact Problems at Nanoscale,” J. Appl. Phys., 101(1), p. 013510. [CrossRef]
Koguchi, H. , 2008, “Surface Green Function With Surface Stresses and Surface Elasticity Using Stroh's Formalism,” ASME J. Appl. Mech., 75(6), p. 061014. [CrossRef]
Chen, W. Q. , and Zhang, C. , 2010, “ Anti-Plane Shear Green's Functions for an Isotropic Elastic Half-Space With a Material Surface,” Int. J. Solids Struct., 47(11–12), pp. 1641–1650. [CrossRef]
Gao, X. , Hao, F. , Fang, D. N. , and Huang, Z. P. , 2013, “Boussinesq Problem With the Surface Effect and Its Application to Contact Mechanics at the Nanoscale,” Int. J. Solids Struct., 50(16–17), pp. 2620–2630. [CrossRef]
Long, J. M. , Wang, G. F. , Feng, X. Q. , and Yu, S. W. , 2012, “ Two-Dimensional Hertzian Contact Problem With Surface Tension,” Int. J. Solids Struct., 49(13), pp. 1588–1594. [CrossRef]
Long, J. M. , and Wang, G. F. , 2013, “Effects of Surface Tension on Axisymmetric Hertzian Contact Problem,” Mech. Mater., 56(1), pp. 65–70. [CrossRef]
Wang, G. F. , and Niu, X. R. , 2015, “Nanoindentation of Soft Solids by a Flat Punch,” Acta Mech. Sin., 31(4), pp. 531–535. [CrossRef]
Rimai, D. , Quesnel, D. , and Busnaina, A. , 2000, “The Adhesion of Dry Particles in the Nanometer to Micrometer-Size Range,” Colloids Surf., A, 165(1–3), pp. 3–10. [CrossRef]
Chakrabarti, A. , and Chaudhury, M. K. , 2013, “Direct Measurement of the Surface Tension of a Soft Elastic Hydrogel: Exploration of Elastocapillary Instability in Adhesion,” Langmuir, 29(23), pp. 6926–6935. [CrossRef] [PubMed]
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(4), p. 2728.
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. [CrossRef] [PubMed]
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, 471(2175), p. 20140727. [CrossRef]
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. [CrossRef]
Chen, T. Y. , Chiu, M. S. , and Weng, C. N. , 2006, “Derivation of the Generalized Young–Laplace Equation of Curved Interfaces in Nanoscaled Solids,” J. Appl. Phys., 100(7), p. 074308. [CrossRef]
Ru, C. Q. , 2010, “Simple Geometrical Explanation of Gurtin–Murdoch Model of Surface Elasticity With Clarification of Its Related Versions,” Sci. China Phys. Mech., 53(3), pp. 536–544. [CrossRef]
Shenoy, V. B. , 2005, “Atomistic Calculations of Elastic Properties of Metallic FCC Crystal Surfaces,” Phys. Rev. B, 71(9), p. 094104. [CrossRef]
Erdogan, F. , and Gupta, G. D. , 1972, “On the Numerical Solution of Singular Integral Equations,” Q. Appl. Math., 29(4), pp. 525–534. [CrossRef]
Shenoy, V. , and Sharma, A. , 2001, “Pattern Formation in a Thin Solid Film With Interactions,” Phys. Rev. Lett., 86(1), pp. 119–122. [CrossRef] [PubMed]
Corless, R. M. , Gonnet, G. H. , Hare, D. E. G. , Jeffrey, D. J. , and Knuth, D. E. , 1996, “On the Lambert W Function,” Adv. Comput. Math., 5(1), pp. 329–359. [CrossRef]
Johnson, K. L. , 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK.
Sneddon, I. N. , 1948, “Boussinesq's Problem for a Rigid Cone,” Math. Proc. Camb. Philos. Soc., 44(4), pp. 492–507. [CrossRef]
Sneddon, I. N. , 1946, “Boussinesq's Problem for a Flat-Ended Cylinder,” Math. Proc. Camb. Philos. Soc., 42(1), pp. 29–39. [CrossRef]


Grahic Jump Location
Fig. 1

Axisymmetric indentations of an elastic half-space by (a) a rigid sphere, (b) a rigid cone, and (c) a flat-ended rigid cylinder

Grahic Jump Location
Fig. 2

Dependence of indent depth on load for spherical indentation

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Fig. 3

Dependence of load on contact radius for spherical indentation

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Fig. 4

Dependence of load on indent depth for spherical indentation

Grahic Jump Location
Fig. 5

Dependence of indent depth on load for conical indentation

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Fig. 6

Dependence of load on contact radius for conical indentation

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Fig. 7

Dependence of load on indent depth for conical indentation

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Fig. 8

Dependence of indent depth on base radius for flat-ended cylindrical indentation



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