General Relations of Indentations on Solids With Surface Tension

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.

Boundary Element Calculations for Normal Contact of Soft Materials With Tensed Surface Membrane

“…The variations of indentation load and indentation depth with respect to the ratio s/a are plotted in Figures 5A,B, respectively. It is seen that our numerical results obtained from BEM calculations coincide well with existing semi-analytical solutions (Kim and Gouldstone, 2008;Long et al, 2017). The size dependent contact behavior is reproduced.…”

“…The constant membrane tension in Hajji (1978) and Kim and Gouldstone (2008) is corresponding to the residual surface tension, and the superficial layer modeled in Argatov and Sabina (2012) and Argatov and Mishuris (2016) can be considered as a solid surface with surface elasticity. Recently, problems of surface loadings and contacts have been widely studied based on the surface elasticity theory (He and Lim, 2006;Wang and Feng, 2007;Zhou and Gao, 2013;Long et al, 2017;. It is found that the deformation behavior would be distinctly different from classical models when the size of loading is comparable or even smaller than a critical length, usually at nano/microscale.…”

“…When the energy difference w = γ 1 +γ 2 -γ 12 (called work of adhesion) is appreciable and the effect of surface energy outside the contact can be neglected, adhesive contact model with specific work of adhesion would be appropriate (Johnson et al, 1971). On the other hand, if the energy difference equals to zero (w = 0) but the surface energy of one contact body is much larger than the other (i.e., γ 1 >> γ 2 and γ 1 ≈ γ 12 ), the contact problem can be addressed in the framework of surface elasticity theory (He and Lim, 2006;Wang and Feng, 2007;Zhou and Gao, 2013;Long et al, 2017;. In this work, we consider the latter circumstance with constant surface tension or, equivalently, the non-adhesive contact of solids with tensed surface membrane.…”

Effect of surface tension on the behavior of adhesive contact based on Maugis‒Dugdale model

“…Herewith, we skip all the details and only present the final solution. Based on the dimension analysis and the numerical results, we can parameterize the load P and the contact radius a in terms of the indent depth δ as (Long et al, 2017) respectively. Thus, in the ST-DMT model, the dependence of load P on the indent depth δ is given by…”

Size-dependent yield hardness induced by surface energy

“…In recent decades, attentions on the influence of surface tension or surface energy on contact problems of elastic materials has grown considerably. Based on the surface Green's functions with surface tension [17,18], Long et al [19,20] analysed the plane strain and axisymmetric contact problems respectively, and suggested that the existence of surface tension could lead to evident size-dependent contact responses.…”