Nanoindentation of a half-space due to a rigid cylindrical roller based on Steigmann–Ogden surface mechanical model

In this article, a semianalytical method of solution is developed for the nanocontact problem of elastic half-space indented by a rigid cylindrical roller. The mechanical formulation is based on the complete version of Steigmann–Ogden surface elasticity theory. Surface tension, surface tensile stiffness and surface flexural rigidity of the half-space boundary are all taken into consideration. Fourier integral transform method converts the governing equations and displacement boundary conditions of the nanocontact problem into a singular integral equation. Gauss–Chebyshev quadrature and an iterative algorithm numerically solve this integral equation and the force equilibrium condition. The developed semianalytical solution is general in the sense that it can reduce to a few simplified theories. These include classical solution, considering only a single surface material parameter, and Gurtin–Murdoch surface elasticity theory, for which analytical kernel functions of the singular integral equation are presented. Dimension analysis demonstrates that the effects of Steigmann–Ogden surface elasticity on the two-dimensional Hertzian nanocontact properties are up to three dimensionless ratios among surface material parameters, shear modulus and the size of nanocontact. Moreover, least-squares regression analysis suggests that, in the presence of surface effects, an elliptic arc less than a half can represent the nanocontact pressure. When compared with their classical counterparts, lower maximum contact pressure and nonzero minimum pressure are found. Parametric experiments further show that surface tension and surface flexural rigidity significantly affect contact length, contact pressure, contact stiffness as well as displacements and stresses near the half-space boundary. In contrast, the effects of surface membrane stiffness are of secondary importance. In general, smaller indenters and larger surface constants lead to higher load-carrying capabilities of half-space and thus better mechanical responses.