# Modeling of the Loading-Unloading Contact of Two Cylindrical Rough Surfaces with Friction
## 5 Conclusions  
This paper developed the first fractal model of the loading-unloading process for cylindrical contact surface with friction in different deformation stages. The loading-unloading contact model for single asperity with friction were obtained firstly, and then deduced the loading-unloading contact model between two cylindrical surfaces with friction. Based on numerical results with MATLAB, the trend of the loading-unloading contact performance between two cylindrical surfaces with friction were obtained by changing the fractal dimension \(D\), the profile scaling parameter \(G\), the curvature radius \(R_{c2}\) and the contact type. The following conclusions can be drawn:  
* The dimensionless real contact area of the loading-unloading process increases with the dimensionless contact load. In the elastic stage, the unloading process is the same with the loading one. In the elastoplastic stage, the dimensionless real contact area of the unloading process is greater than that of the loading process for a given dimensionless contact load; the dimensionless real contact area of the unloading process, as well as the differences of the dimensionless real contact area between the loading and unloading processes, increase with the loading interference.
* The dimensionless critical real contact area and the dimensionless critical contact load increase with the increase of the fractal dimension, decrease of the profile scaling parameter and the curvature radius, or the substitution of external contact for internal contact.
* In the elastoplastic stage, for a given dimensionless contact load, the dimensionless real contact area of the loading-unloading process of cylindrical contact surface with friction, as well as the differences of the dimensionless real contact area between the loading and unloading processes, increase with the increase of the fractal dimension, decrease of the profile scaling parameter and the curvature radius, or the substitution of external contact for internal contact.  
Figure 8: The comparison between the present model and KE statistical model. (**a**) \(G=10^{-10}\) m, \(1.4\leq D\leq 1.5\); (**b**) \(D=1.45\), \(10^{-11}\) m \(\leq G\leq 10^{-9}\) m.  
**Author Contributions:** Conceptualization, H.W., P.J. and M.L.; methodology, [PERSON] and M.L.; writing--original draft preparation, [PERSON], P.J., [PERSON] and G.W.; writing--review and editing, [PERSON], F.Y., M.L. and X.W.; visualization, [PERSON] and X.W.; supervision, [PERSON] and L.W.; funding acquisition, P.J., L.W., F.Y. and G.W. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the National Key Research and Development Program of China, grant number 2018 YFC0310500; the High-Tech Ship Research Projects sponsored by the Ministry of Industry and Information Technology, grant number 2018 GXB01; Fundamental Research Funds for the Central Universities, grant number 3072019 CF0705; Yantai Special Fund for School Land Integration Development (development and test platform of subsea production system), grant number 2019-105-13; National Natural Science Foundation of China, grant number 51779064; the Science and Technology Projects Sponsored by the West Coast of Qingdao New District, grant number 2019-157-2018-1-1.  
**Conflicts of Interest:** The authors declare no conflicts of interest.