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.

在本文中，开发了一种半解析解方法，用于解决由刚性圆柱滚子压入的弹性半空间的纳米接触问题。机械公式基于Steigmann–Ogden表面弹性理论的完整版本。半空间边界的表面张力，表面拉伸刚度和表面弯曲刚度都被考虑在内。傅里叶积分变换法将纳米接触问题的控制方程和位移边界条件转换为奇异积分方程。Gauss-Chebyshev求积和一个迭代算法在数值上求解该积分方程和力平衡条件。从可以简化为一些简化理论的意义上说，开发的半分析解决方案是通用的。其中包括经典解决方案，仅考虑单个表面材料参数，以及Gurtin-Murdoch表面弹性理论，为此提出了奇异积分方程的解析核函数。尺寸分析表明，Steigmann-Ogden表面弹性对二维Hertzian纳米接触特性的影响在表面材料参数，剪切模量和纳米接触尺寸之间高达三个无量纲的比率。此外，最小二乘回归分析表明，在存在表面效应的情况下，小于一半的椭圆弧可以表示纳米接触压力。与传统的同类产品相比，发现较低的最大接触压力和非零的最小压力。参数实验进一步表明，表面张力和表面抗弯刚度会显着影响接触长度，接触压力，接触刚度以及半空间边界附近的位移和应力。相反，表面膜刚度的影响是次要的。通常，较小的压头和较大的表面常数会导致较高的半空间承载能力，从而带来更好的机械响应。