# Modeling of the Loading-Unloading Contact of Two Cylindrical Rough Surfaces with Friction
## 3 Fractal Contact Model in the Unloading Process with Regard of Friction  
The contact model between two cylindrical surfaces in unloading process relies on the loading process. The corresponding unloading contact model varies with the final deformation stage of the loading process. Asperities on the surface still satisfy the fractal characteristic at the end of the unloading process [39; 44]. The size distribution in contact of two cylindrical surfaces in the unloading process can be written as:  
\[n^{\prime u}\Big{(}a^{\prime u}\Big{)}=\lambda_{C}\frac{D}{2}\Big{(}a^{\prime u }\Big{)}^{\frac{D}{2}}\Big{(}a^{\prime u}\Big{)}^{-\frac{D+2}{2}}, \tag{29}\]  
where \(a^{\prime u}_{\ l}\) is the maximum contact area for all asperities in the unloading process; and \(a^{\prime u}\) is the contact area of single asperity in the unloading process. Because the real contact area and contact load of asperities at the end of the loading process equal those at the beginning of the unloading process respectively, \(a^{\prime u}_{\ l}=a^{\prime}_{\ l}=a^{\prime}_{\max}\), and \(a^{\prime}_{\max}\) is the maximum contact area corresponding to the maximum interference \(a^{\prime}_{\max}\) for all asperities in the loading process. Thus, the unloading contact model between two cylindrical surfaces with regard of friction can be given in each deformation stage. 1. Elastic stage: when \(0\leq a^{\prime}_{\max}\leq a^{\prime}_{\ ex}\), only the elastic deformation occurs in asperities on the surface and the unloading process is an inverse process of the loading process in this case. So, the unloading contact model of the elastic stage can be written as: \[A^{u}_{ce}=\int_{0}^{a^{\prime u}_{\ l}}n^{\prime u}\Big{(}a^{\prime u}\Big{)} a^{\prime u}da^{\prime u}=\frac{\lambda_{C}D}{2-D}a^{\prime u}_{\ l}\ \text{and}\] (30) \[F^{u}_{ce}=\int_{0}^{a^{\prime u}_{\ l}}F^{u}_{\ e}n^{\prime u}\Big{(}a^{ \prime u}\Big{)}da^{\prime u}=\frac{1.1\lambda_{C}Dk_{\mu ac}\sigma_{Y}}{3-D}a ^{\prime-1/2}\Big{(}a^{\prime u}_{\ l}\Big{)}^{3/2},\] (31) where \(A^{u}_{ce}\) and \(F^{u}_{ce}\) are the real contact area and the contact load between two cylindrical surfaces in the unloading process in the elastic stage with regard of friction, respectively. 2. Elastoplastic stage: When \(a^{\prime}_{\alpha}<a^{\prime}_{\max}\leq a^{\prime}_{pc}\), the residual interference of asperities on the surface is the unrecoverable plastic portion of \(a^{\prime}_{\max}\) at the completion of the asperity unloading, which results in the surface topography after unloading different from the original one before loading process. The relation between the radius of curvature, \(R^{u}\), of the asperity after unloading and the radius before loading, \(R\), is [6]:  
\[\frac{R^{u}}{R}=1+1.275\Bigg{(}\frac{E}{\sigma_{y}}\Bigg{)}^{-0.216}\Bigg{(} \frac{a^{\prime}_{\max}}{a^{\prime}_{ce}}-1\Bigg{)}. \tag{32}\]  
The critical elastic contact area is proportional to the square of the radius of asperity curvature according to Hertzian theory. By combining Equation (32), the critical elastic contact area \(a^{\prime u}_{\ ex}\) of single asperity with regard of friction after unloading can be given as:  
\[a^{\prime u}_{\ ex}=\Bigg{[}1+1.275\Bigg{(}\frac{E}{\sigma_{y}}\Bigg{)}^{-0. 216}\Bigg{(}\frac{a^{\prime}_{\max}}{a^{\prime}_{\ ex}}-1\Bigg{)}\Bigg{]}^{2} a^{\prime}_{\ ex}. \tag{33}\]The relation between the maximum contact area and the maximum interference can be given by Equations (34) and (35) [5] when single asperity is in the first elastoplastic stage and the second elastoplastic stage, respectively.