# Modeling of the Loading-Unloading Contact of Two Cylindrical Rough Surfaces with Friction
## 1 Introduction  
The surface of the mechanical structure is rugged on a microscopic view, and consists of numerous asperities of different sizes. The contact between rough surfaces is characterized by the interaction of asperities, which leads to the real contact area being a small fraction of the apparent contact area. The contact characterization has a significant impact on physical phenomena such as friction, wear, lubrication, etc. The contact between two cylindrical rough surfaces are very common in the engineering field, such as gear and cam mechanism. It is of great significance to make an extensive research on mechanical properties of the contact form. Statistical methods are commonly used to analyze the contact problems of rough surfaces. There are many statistical models proposed recently, such as GW ([PERSON] and [PERSON]) model [1; 2; 3], CEB ([PERSON], [PERSON] and [PERSON]) model [4], KE ([PERSON] and [PERSON]) model [5; 6; 7], the ellipse model [8; 9], the size-dependent plasticity model [10] and so on. In particular, [PERSON] and [PERSON] [11; 12] proposed a model that predicts the static friction for elastic-plastic contact of rough surfaces (KE friction model); [PERSON] et al. [13] developed a numerical model for the elastic-plastic spherical contact under combined normal and tangential loading in full stick based on their single asperity models in full stick condition [14; 15]; [PERSON] et al. [16; 17] proposed statistical models for flat contacting rough surfaces and for spherical contacting rough surfaces under combined normal and tangential loading with the plasticity index \(\psi\leq 8\) respectively; [PERSON] et al. [18] extended the spherical model of [PERSON] up to \(\psi=32\); [PERSON] et al. [19] developed a model to predict the static friction coefficient considering the multi-scale nature of roughness; [PERSON] et al. [20] proposed an improved static friction model for elastic-plastic contacting surfaces based on KE frictionmodel. From the above statistical models and [PERSON]'s comment [21], the static friction coefficient is related to the normal load and decreases with an increasing normal load. However, the statistical parameters in the above statistical models depend on the sample length and the resolution of the instrument, so that it is not unique in determining the characterization and analysis of the rough surface. In order to overcome the shortcomings of the statistical model, the necessity for scale independent contact models motivated the growing use of fractal description of the multiscale nature of contacting rough surfaces. [PERSON] and [PERSON] [22] proposed the first fractal model for two-dimensional rough surface using the Weierstrass-Mandelbrot function (MB model). [PERSON] [23] extended the MB model to create the fractal model of three-dimensional rough surface (YK model). These two models revealed the relationship between the real contact area and the contact load through the numerical results, and the result of two models that the small contact spots were in plastic contact whereas large spots were in elastic contact was contrary to classical contact mechanics. [PERSON] [24] analyzed contact between a rigid cylindrical indenter and an elastic, perfectly plastic solid with fractal surface roughness. The conclusion was that idealizing surface roughness as a perfect fractal led to unphysical predictions for the contact size and number of contact spots. [PERSON] and [PERSON] [25] developed the fractal surface contact model (ME model) which agreed with classical contact mechanics by suggesting a revision to the MB model, in which the critical area of a single asperity was scale-dependent. [PERSON] [26] developed a complete contact model of a fractal surface by extending the ME model, and the total load, area and stiffness of a fractal rough surface were studied.