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

Yue Ding, Weike Yuan

SVL,
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

SVL,
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

## Abstract

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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## Figures

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

Fig. 2

Dependence of indent depth on load for spherical indentation

Fig. 3

Fig. 4

Dependence of load on indent depth for spherical indentation

Fig. 5

Dependence of indent depth on load for conical indentation

Fig. 6

Fig. 7

Dependence of load on indent depth for conical indentation

Fig. 8

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

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