0
Research Papers

# Micro/Nanocontact Between a Rigid Ellipsoid and an Elastic Substrate With Surface Tension

[+] Author and Article Information
W. K. Yuan, J. M. Long, Y. Ding

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

G. F. Wang

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

1Corresponding author.

Manuscript received September 18, 2016; final manuscript received October 15, 2016; published online November 7, 2016. Assoc. Editor: Harold S. Park.

J. Appl. Mech 84(1), 011012 (Nov 07, 2016) (7 pages) Paper No: JAM-16-1460; doi: 10.1115/1.4035032 History: Received September 18, 2016; Revised October 15, 2016

## Abstract

For micro/nanosized contact problems, the influence of surface tension becomes prominent. Based on the solution of a point force acting on an elastic half space with surface tension, we formulate the contact between a rigid ellipsoid and an elastic substrate. The corresponding singular integral equation is solved numerically by using the Gauss–Chebyshev quadrature formula. When the size of contact region is comparable with the elastocapillary length, surface tension significantly alters the distribution of contact pressure and decreases the contact area and indent depth, compared to the classical Hertzian prediction. We generalize the explicit expression of the equivalent contact radius, the indent depth, and the eccentricity of contact ellipse with respect to the external load, which provides the fundament for analyzing nanoindentation tests and contact of rough surfaces.

<>
Your Session has timed out. Please sign back in to continue.

## References

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, p. 2728. [PubMed]
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(1–4), pp. 151–163.
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.
Andreotti, B. , Bäumchen, O. , Boulogne, F. , Daniels, K. E. , Dufresne, E. R. , Perrin, H. , Salez, T. , Snoeijer, J. H. , and Style, R. W. , 2016, “ Solid Capillarity: When and How Does Surface Tension Deform Soft Solids?,” Soft Matter., 12(12), pp. 2993–2996. [PubMed]
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.
Grierson, D. S. , Liu, J. J. , Carpick, R. W. , and Turner, K. T. , 2013, “ Adhesion of Nanoscale Asperities With Power-Law Profiles,” J. Mech. Phys. Solids, 61(2), pp. 597–610.
Greenwood, J. A. , and Williamson, J. B. P. , 1966, “ Contact of Nominally Flat Surfaces,” Proc. R. Soc. London Ser. A, 295(1442), pp. 300–319.
Bush, A. W. , Gibson, R. D. , and Thomas, T. R. , 1975, “ The Elastic Contact of a Rough Surface,” Wear, 35(1), pp. 87–111.
Majumdar, A. , and Bhushan, B. , 1991, “ Fractal Model of Elastic-Plastic Contact Between Rough Surfaces,” ASME J. Tribol., 113(1), pp. 1–11.
Hajji, M. A. , 1978, “ Indentation of a Membrane on an Elastic Half Space,” ASME J. Appl. Mech., 45(2), pp. 320–324.
Gurtin, M. E. , and Murdoch, A. I. , 1975, “ A Continuum Theory of Elastic Material Surfaces,” Arch. Ration. Mech. Anal., 57(4), pp. 291–323.
Gurtin, M. E. , Weissmüller, J. , and Larché, F. , 1998, “ A General Theory of Curved Deformable Interfaces in Solids at Equilibrium,” Philos. Mag. A, 78(5), pp. 1093–1109.
Huang, G. Y. , and Yu, S. W. , 2006, “ Effect of Surface Elasticity on the Interaction Between Steps,” ASME J. Appl. Mech., 74(4), pp. 821–823.
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.
Wang, G. F. , and Feng, X. Q. , 2009, “ Effects of Surface Stresses on Contact Problems at Nanoscale,” J. Appl. Phys., 101(1), p. 013510.
Koguchi, H. , 2008, “ Surface Green Function With Surface Stresses and Surface Elasticity Using Stroh's Formalism,” ASME J. Appl. Mech., 75(6), p. 061014.
Hayashi, T. , Koguchi, H. , and Nishi, N. , 2013, “ Contact Analysis for Anisotropic Elastic Materials Considering Surface Stress and Surface Elasticity,” J. Mech. Phys. Solids., 61(8), pp. 1753–1767.
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.
Long, J. M. , and Wang, G. F. , 2013, “ Effects of Surface Tension on Axisymmetric Hertzian Contact Problem,” Mech. Mater., 56, pp. 65–70.
Gao, X. , Hao, F. , Fang, D. , 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), pp. 2620–2630.
Liu, T. , Jagota, A. , and Hui, C. Y. , 2015, “ Adhesive Contact of a Rigid Circular Cylinder to a Soft Elastic Substrate—The Role of Surface Tension,” Soft Mater., 11(19), pp. 3844–3851.
Hui, C. Y. , Liu, T. , 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.
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.
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.
Shenoy, V. B. , 2005, “ Atomistic Calculations of Elastic Properties of Metallic FCC Crystal Surfaces,” Phys. Rev. B, 71(9), p. 094104.
Johnson, K. L. , 1985, Contact Mechanics, Cambridge University Press, London.
Erdogan, F. , and Gupta, G. D. , 1972, “ On the Numerical Solution of Singular Integral Equations,” Q. Appl. Math., 29, pp. 525–534.
Shenoy, V. V. , and Sharma, A. , 2001, “ Pattern Formation in a Thin Solid Film With Interactions,” Phys. Rev. Lett., 86(1), pp. 119–122. [PubMed]
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 Mater., 10(26), pp. 4625–4632.

## Figures

Fig. 1

Indentation of a rigid ellipsoid on an elastic half space

Fig. 2

The schematic of the elliptical contact region

Fig. 4

The pressure distribution within the contact region

Fig. 3

Variation of semimajor a with respect to η for different values of θ

Fig. 5

The surface normal displacement along the x-axis

Fig. 6

The surface normal bulk stress along the x-axis

Fig. 7

Variation of normalized load with respect to (ABδ/C)1/2/s

Fig. 8

Variation of normalized load with respect to the ratio of re to s

Fig. 9

Variation of normalized eccentricity with respect to the ratio of χ to s

## Errata

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections