# Modeling of the Loading-Unloading Contact of Two Cylindrical Rough Surfaces with Friction
## 2 Contact Model in the Loading Process with Regard of Friction
### Contact Model of Single Asperity in Loading Process with Regard of Friction
Combining Equations (5) and (8), the relation between the critical elastic contact areas \(a_{ec}\) without friction and \(a^{\prime}_{ec}\) with friction can be given as:  
\[a^{\prime}_{ec}=\frac{2C}{3.3k_{\mu ac}}a_{ec} \tag{11}\]  
Thus, the contact model of single asperity in loading process can be given in elastic, elastoplastic and plastic stages respectively. 1. Elastic stage:  
The single asperity is in elastic deformation when \(0\leq a^{\prime}\leq a^{\prime}_{ec}\). The contact area and contact load is in the form of:  
\[F^{\prime}_{e}=F^{\prime}_{ec}\Bigg{(}\frac{\omega^{\prime}}{\omega^{\prime} _{ec}}\Bigg{)}^{3/2} \tag{12}\]  
\[a^{\prime}=a^{\prime}_{ec}\Bigg{(}\frac{\omega^{\prime}}{\omega^{\prime}_{ec}} \Bigg{)} \tag{13}\]  
where \(F^{\prime}_{e}\) is the contact load of single asperity in the elastic stage and \(\omega^{\prime}\) is the interference of single asperity. Combining Equations (8), (12) and (13), the relation between the contact area and the contact load of single asperity in the elastic deformation with regard of friction can be given as:  
\[F^{\prime}_{e}=1.1k_{\mu ec}\sigma_{\gamma}a^{\prime-1/2}_{ec}a^{\prime 3/2} \tag{14}\]  
2. Elastoplastic stage:When the interference is greater than critical elastic interference, the elastoplastic deformation occurs. When \({\omega^{\prime}}_{\alpha}<{\omega^{\prime}}\leq 6{\omega^{\prime}}_{\alpha}\)[5], the single asperity is in the first elastoplastic stage, where the relation between the contact area and the contact load can be given as:  
\[\frac{F^{\prime}}{F_{\alpha}}=1.03\bigg{(}\frac{a^{\prime}}{0.93a_{cc}}\bigg{)} ^{1.2544} \tag{15}\]  
When \(6{\omega^{\prime}}_{\alpha}<{\omega^{\prime}}\leq 110{\omega^{\prime}}_{\alpha}\), the single asperity is in the second elastoplastic stage, where the relation between the contact area and the contact load can be given as:  
\[\frac{F^{\prime}}{F_{\alpha}}=1.40\bigg{(}\frac{a^{\prime}}{0.94a_{cc}}\bigg{)} ^{1.1021} \tag{16}\]  
By substituting Equations (5), (8) and (11) into Equations (15) and (16) respectively, the relation between the contact area and the contact load of single asperity in the elastoplastic deformation with regard of friction can be expressed by Equations (17) and (18) respectively. \[{F^{\prime}}_{\alpha p1}=0.6621{\sigma_{Y}}{\rm C}^{1.2544}{k^{-0.2544}_{ \mu cc}}{a^{\prime}}_{\alpha}^{-0.2544}({a^{\prime}})^{1.2544}\left({a^{\prime }}_{cc}<a^{\prime}\leq{a^{\prime}}_{\alpha c}\right) \tag{17}\]  
\[{F^{\prime}}_{\alpha p2}=0.9494{\sigma_{Y}}{\rm C}^{1.1021}{k^{-0.1021}_{\mu cc }}{a^{\prime}}_{cc}^{-0.1021}({a^{\prime}})^{1.1021}\left({a^{\prime}}_{\alpha c }<a^{\prime}\leq{a^{\prime}}_{\alpha c}\right) \tag{18}\]  
where \({F^{\prime}}_{\alpha p1}\) is the contact load of single asperity in the first elastoplastic stage; \({a^{\prime}}_{\alpha c}\) is the critical contact area of the first elastoplastic stage with regard of friction, which is \({a^{\prime}}_{\alpha c}=11.7475{k_{\mu\alpha c}}{Ca^{\prime}}_{\alpha c}\) with \({a^{\prime}}=6{\omega^{\prime}}_{\alpha c}\); \({F^{\prime}}_{\alpha p2}\) is the contact load of single asperity in the second elastoplastic stage; \({a^{\prime}}_{\alpha c}\) is the critical contact area of the second elastoplastic stage with regard of friction, which is \({a^{\prime}}_{\alpha c}=338.8815{k_{\mu\alpha c}}{Ca^{\prime}}_{\alpha c}\) with \({a^{\prime}}=110{\omega^{\prime}}_{\alpha c}\). 3. Plastic stage:  
When \({a^{\prime}}>{a^{\prime}}_{\alpha c}\), the contact is in the plastic stage. The relation between the contact area and the contact load of single asperity can be expressed as:  
\[{F^{\prime}}_{p}=Ha^{\prime}, \tag{19}\]  
where \({F^{\prime}}_{p}\) is the contact load of single asperity in the plastic stage and \(H\) is the hardness of the softer material. The above results can be used in contact modeling of two cylindrical surfaces with regard of friction.