# 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  
According to the von Mises yield criterion, the critical elastic interference \(\omega_{\alpha}\) and the critical elastic contact area \(a_{cc}\) of single asperity without friction can be expressed as [41]:  
\[\omega_{cc}=\bigg{(}\frac{\pi C\sigma_{Y}}{2E}\bigg{)}^{2}R \tag{1}\]  
\[a_{cc}=\pi R\omega_{cc}=\pi\bigg{(}\frac{\pi C\sigma_{Y}}{2E}\bigg{)}^{2}R^{2} \tag{2}\]  
where \(C\) is the critical yield stress coefficient, \(C=1.295\exp(0.736\
u)\); \(\
u\) is the Poisson ratio of the softer material; \(\sigma_{Y}\) is the yield strength of the softer material; \(R\) is the radius of asperity curvature; \(E\) is the equivalent elastic modulus, which is:  
\[\frac{1}{E}=\frac{1-\
u_{1}^{2}}{E_{1}}+\frac{1-\
u_{2}^{2}}{E_{2}}, \tag{3}\]  
and \(E_{1}\), \(E_{2}\) are the elastic modulus of two contact surfaces, respectively; \(\
u_{1}\) and \(\
u_{2}\) are the Poisson ratio of two contact surfaces, respectively. By Hertzian theory, the relation between the critical elastic contact load \(F_{cc}\) and the critical elastic interference \(\omega_{cc}\) of single asperity without friction is:  
\[F_{cc}=\frac{4}{3}ER^{\frac{1}{2}}\omega_{\alpha}^{\frac{3}{2}} \tag{4}\]  
By substituting Equations (1) and (2) into Equation (4), the relation between the critical elastic contact load \(F_{cc}\) and the critical elastic contact area \(a_{cc}\) can be given as:  
\[F_{cc}=\frac{2}{3}C\sigma_{Y}a_{cc} \tag{5}\]  
According to the fractal theory, the critical elastic interference \(\omega_{cc}\) and the critical elastic contact area \(a_{cc}\) can be expressed as [26]:  
\[\omega_{\alpha}=\frac{C^{2}\sigma_{Y}^{2}l^{D}}{4E^{2}G^{D-1}} \tag{6}\]  
\[a_{cc}=\frac{C^{2}\sigma_{Y}^{2}l^{2D}}{4E^{2}G^{2D-2}\pi} \tag{7}\]  
where \(G\) is the profile scaling parameter; \(D\) is the fractal dimension and \(l\) is the asperity base diameter. For a given rough surface, the value of the critical elastic interference and the critical elastic contact area are constant, which depend on the material properties, the fractal dimension and the profile scaling parameter, as well as the maximum asperity base diameter on the base plane [29]. If the slide happens between contact surfaces, the relation between the critical elastic contact load \(F^{\prime}_{ec}\) and the critical elastic contact area \(a^{\prime}_{ec}\) of single asperity with regard of friction can be given as by Equation (8) when single asperity begins to yield [42]. \[F^{\prime}_{ec}=1.1k_{\mu ec}\sigma_{\gamma}a^{\prime}_{ec} \tag{8}\]  
where, \(F^{\prime}_{ec}\) is the critical elastic contact load of single asperity with regard of friction; \(a^{\prime}_{ec}\) is the critical elastic contact area of single asperity with regard of friction; \(k_{\mu ec}\) is the value of the correction factor with regard of friction, \(k_{\mu}\), when single asperity begins to yield, and \(k_{\mu}\) can be expressed as:  
\[k_{\mu}= \begin{cases}1-0.228\mu&0\leq\mu\leq 0.3\\ 0.932e^{-1.58(\mu-0.3)}&0.3<\mu\leq 0.9\end{cases}  \tag{9}\]  
where the friction coefficient \(\mu\) is related to the normal load and decreases with an increasing normal load in the deformation process of single asperity [11; 12; 13; 14; 15; 16; 17; 21]. Due to the neglecting of the adhesion force, the relation between the friction coefficient and the normal load in the elastic regime can be expressed as [11]:  
\[\mu=0.516\Big{(}\frac{F}{F_{ec}}\Big{)}^{-0.345} \tag{10}\]  
The critical elastic contact load of the spherical surface are the same both with and without friction according to [PERSON]'s measurement [43] and the GW model [1]. The friction force acting on asperities is the tangential force which affects the normal contact by changing the contact area, and the load-area relation of single asperity in the elastic stage satisfies \(F\propto a^{\prime 3/2}\)[28; 29; 36], where \(F\)' is the contact load of single asperity and \(a\)' is the contact area.