Axisymmetric Hertzian contact problem accounting for surface tension and strain gradient elasticity

In this paper, we investigate the axisymmetric Hertzian contact problem at micro-/nanoscale. The deformation of material bulk is described by a simplified theory of strain gradient elasticity, and the influence of surface tension is integrated based on the surface elasticity theory. Using the Mindlin's potential function method and double Fourier integral transform, the normal surface displacement induced by a concentrated force is derived in a closed form. Following this, the contact between a rigid sphere and an elastic half-space is formulated in terms of singular integral equation, which is numerically solved by applying the Gauss-Chebyshev method. The results indicate that the distribution of contact pressure is distinctly different from that in classical elasticity theory. The indented substrate tends to perform stiffer due to the effects of surface tension and strain gradient elasticity. When the contact radius is comparable with the material length parameter, the indentation force (depth) can be ten (three) times of that given by classical Hertz theory.