# Modeling of the Loading-Unloading Contact of Two Cylindrical Rough Surfaces with Friction
## 1 Introduction
[PERSON] [27; 28; 29] established the fractal model of the loading contact stiffness of the plane rough surfaces, cylindrical rough surfaces and spheroidal rough surfaces considering the friction factor successively by introducing the specific contact coefficient to the loading contact stiffness formula. It revealed the influence of parameters such as the friction coefficient and the fractal dimension on the loading contact stiffness. [PERSON] [30] developed an approach for contact mechanics of fractal surfaces by spline assisted discretization and revealed the significant effect of surface fractality on contact mechanics. [PERSON] [31] established a sliding frictional fractal contact model of two involute arc cylindrical gears. The model showed that the load-area relation was not only affected by the fractal parameters, but also by the contact radius and tooth line radius. [PERSON] [32] proposed a fractal contact stiffness model considering the effect of asperity interaction by introducing the concept of fractal smoothness which affected the contact stiffness and the effect could be offset by the asperity interaction. The effect of asperity interaction on contact stiffness was studied as well. [PERSON] [33; 34] established a normal contact stiffness fractal prediction model of dry-friction rough surface considering the friction factor based on three-dimensional fractal function, and the validity of the established model was verified by experiment. [PERSON] [35] proposed a contact stiffness model in view of the influence of domain extension factor between two spherical rough surfaces and analyzed evolution of the elastic-plastic contact involving three distinct stages ranged from complete elastic through elastic-plastic to fully plastic deformation. [PERSON] [36] used the results of reference [29] and investigated the conformal contact between piston and cylinder by employing the fractal theory. In this model, the contact coefficient was proposed, which was affected by the structural parameters of piston radius and radial clearance as well as the material properties. [PERSON] [37] proposed a novel contact stiffness model of a concrete-steel joint based on the fractal theory. Here, an iteration model was proposed to describe the contact state of the concrete-steel joint and revealed the nonlinear relationships between contact stiffness and load, fractal parameters. [PERSON] [38] proposed a revised MB model, and deduced the transition from elastic, elastoplastic to full plastic deformation for a single asperity whose contact area and load is related to asperity level. Further, [PERSON] [39] extended this model to develop a loading-unloading contact model between three-dimensional rough plane surfaces based on fractal theory, and the relation between the total contact load and the total real contact area was obtained in the loading and unloading processes. [PERSON] [40] developed the revised size distribution functions for the two-dimensional and three-dimensional fractal rough surfaces respectively, and the validity of the model was verified by comparing with the experimental data. As can be seen from the above literature review, a substantial effort has been made so far to investigate the contact between rough surfaces both statistical methods and fractal theory. However,the current research of the fractal model involved in the contact problem between two cylindrical surfaces mainly focused on the loading process, rather than the unloading process. Therefore, a loading-unloading contact analytical model between two cylindrical rough surfaces with friction, based on fractal theory is presented in this paper. The relation between the real contact area and the contact load in different deformation stages are given to investigate how the parameters in the model (such as fractal dimension, profile scaling parameter, curvature radius and contact type) affect the loading-unloading contact performance.