Indentation load-depth relation for an elastic layer with surface tension

TL;DR

This paper develops a model for nanoindentation of elastic layers that includes surface tension effects, deriving explicit formulas and numerical solutions to understand how surface tension influences load-depth behavior.

Contribution

It introduces a surface tension-inclusive Green function and formulates a new integral equation for nanoindentation, providing explicit expressions for load-depth relations.

Findings

Surface tension significantly affects load-depth relations at small contact sizes.

Derived explicit formulas for load-depth and load-contact radius relations.

Numerical results show the impact of surface tension on contact pressure and surface deformation.

Abstract

Load-depth relation is the fundamental requisite in nanoindentation tests for thin layers, however, the effects of surface tension are seldom included. This paper concerns micro-/nano-sized indentation of a bonded elastic layer by a rigid sphere. The surface Green function with the incorporation of surface tension is first derived by applying the Hankel integral transform, and subsequently used to formulate the governing integral equation for the axisymmetric contact problem. By using a numerical method based on Gauss-Chebyshev quadrature formula, the singular integral equation is solved efficiently. Several numerical results are presented to investigate the influences of surface tension and layer thickness on contact pressure, surface deformation and bulk stress, respectively. It is found that when the size of contact is comparable to the ratio of surface tension to elastic modulus, the contribution of surface tension to the load-depth relation becomes quite prominent. With the help of a parametric study, explicit general expressions for the indentation load-depth relation as well as the load-contact radius relation are summarized, which provide the groundwork for practical applications.