# Modeling of the Loading-Unloading Contact of Two Cylindrical Rough Surfaces with Friction
## 2 Contact Model in the Loading Process with Regard of Friction
### Fractal Contact Model of Two Cylindrical Surfaces in Loading Process with Regard of Friction  
In the fractal theory, the contact spots are of different sizes and spread randomly over the contact interface. The size distribution of contact spots is \(n(a)\), such that \(n(a)da\) is equal to the number of contact spots of area between \(a\) and \(a+da\). According to the MB model, the size distribution of contact spots \(n(a)\) is:  
\[n(a)=\left|\frac{dN(A>a)}{da}\right|=\frac{D}{2}\frac{a^{D/2}_{\max}}{a^{(D+2)/ 2}}, \tag{20}\]  
where \(N\) is the total number of contact spots with area larger than \(a\); \(a_{\max}\) is the area of the largest contact spot. However, Equation (20) cannot apply to the rough curved surface since the deformation of curved surface is different from that of the planar surface, and results in the disparity for the calculation of variable \(N\) in Equation (20). With support of the cylindrical contact coefficient, \(\lambda_{C}\), the asperity size distribution, \(n^{\prime}(a^{\prime})\), in the contact of two cylindrical surfaces [27], can be given as:  
\[n^{\prime}(a^{\prime})=\lambda_{C}\frac{D}{2}a^{\prime}\frac{D}{2}a^{\prime} \frac{D+2}{2}, \tag{21}\]where \(a^{\prime}_{l}\) is the maximum contact area for all asperities with a given interference and \(\lambda_{C}\) is the cylindrical contact coefficient, defined as:  
\[\lambda_{C}=\left[\frac{\left(\frac{4\beta}{\pi E}\frac{R_{c1}R_{c2}}{R_{c1}+R_{ c2}}\right)^{\frac{1}{2}}}{\pi(R_{c1}\pm R_{c2})}\right]^{\frac{1}{R_{c2}}\pm \frac{1}{R_{c1}}}, \tag{22}\]  
where \(R_{c1}\) and \(R_{c2}\) are the curvature radius of two cylinders, respectively; \(B\) is the contact length of two cylinders; \"+\" is for the external contact; \"-\" is for the internal contact and \(R_{c1}>R_{c2}\). Thus, the contact model between two cylindrical surfaces in loading process can be given in each deformation stage with regard of friction. 1. Elastic stage:  
The elastic deformation only happens in asperities on the surface when \(0\leq a^{\prime}_{l}\leq a^{\prime}_{cc}\). The contact model in the elastic stage can be written as:  
\[A_{cc}=\int_{0}^{a^{\prime}_{l}}n^{\prime}(a^{\prime})a^{\prime}da^{\prime}= \frac{\lambda_{C}D}{2-D}a^{\prime}_{l}\ \text{and} \tag{23}\]  
\[F_{cc}=\int_{0}^{a^{\prime}_{l}}{{}^{I}F}_{c}n^{\prime}(a^{\prime})da^{\prime}= \frac{1.1\lambda_{C}Dk_{\mu cc}\sigma_{Y}}{3-D}a^{\prime-1/2}_{cc}a^{\prime 3 /2}_{l}, \tag{24}\]  
where \(A_{cc}\) and \(F_{cc}\) are the real contact area and the contact load between two cylindrical surfaces in the elastic stage with regard of friction, respectively. 2. Elastoplastic stage:  
When \(a^{\prime}_{cc}<a^{\prime}_{l}\leq a^{\prime}_{cc}\), both the elastic deformation and the first elastoplastic deformation happen in asperities on the surface. The contact model in the first elastoplastic stage can be written as:  
\[A_{ccp1}=\int_{0}^{a^{\prime}_{cc}}n^{\prime}(a^{\prime})a^{\prime}da^{\prime}+ \int_{a^{\prime}_{cc}}^{a^{\prime}_{l}}n^{\prime}(a^{\prime})a^{\prime}da^{ \prime}=\frac{\lambda_{C}D}{2-D}a^{\prime}_{l}\ \text{and} \tag{25}\]  
\[ \begin{array}{ll}F_{ccp1}&=\int_{0}^{a^{\prime}_{cc}}F_{c^{\prime}}n^{\prime }(a^{\prime})da^{\prime}+\int_{a^{\prime}_{cc}}^{a^{\prime}_{l}}F_{c^{\prime} \mu 1}n^{\prime}(a^{\prime})da^{\prime}\\ &=\frac{1.1\lambda_{C}Dk_{\mu cc}\sigma_{Y}}{3-D}a^{\prime-2}_{c}a^{\prime-2}_{ c}a^{\prime-2}_{c}a^{\prime-2}_{c}a^{\prime-2}_{c}a^{\prime-2}_{c}a^{\prime}_{l} \cdot\frac{0.6621a^{\prime-2}_{c}a^{\prime-2}_{c}a^{\prime}_{l}}{1.2544-0.5D} \Big{(}a^{\prime}_{l}a^{\prime 1.2544-0.5D}-a^{\prime 1.2544-0.5D}\Big{)}a^{ \prime D/2}_{l}\,\end{array}  \tag{26}\]  
where \(A_{ccp1}\) and \(F_{ccp1}\) are the real contact area and the contact load between two cylindrical surfaces in the first elastoplastic stage with regard of friction, respectively. When \(a^{\prime}_{cc}<a^{\prime}_{l}\leq a^{\prime}_{pc}\), the deformation of asperities on the surface includes the elastic deformation, the first elastoplastic deformation and the second elastoplastic deformation.