## 4 Establishment of Contact Mechanics Model  
### Analysis of Deformation State  
With the increment of applied load, the deformation of asperity will increase, resulting in elastic, elastic-plastic and purely plastic deformation. When analyzing the contact mechanics of asperities on rough surface, the assumptions are as follows:  
1 Deformations only occur on asperities on the rough surface, while the macro base does not deform. The volume of asperities remains unchanged before and after deformation. 2 In the process of surface contact, there is no interaction between asperities, and neither surface material strengthens. (1) Elastic deformation  
According to [PERSON] theory, the contact force in the direction of \(F_{\rm a}\) during elastic deformation is \(F_{\rm a}=\frac{4 ER^{0.5}\varphi_{\rm p}\cdot 15}{3}\). \(E\) represents the equivalent elastic modulus, which can be expressed as \(E=1/[(1-{v_{1}}^{2})/E_{1}+(1-{v_{2}}^{2})/E_{2}]\), \(\
u_{1}\), \(\
u_{2}\) and \(E_{1}\), \(E_{2}\) denotes the Poisson's ratio and elastic modulus of surface 1 and 2, respectively. \(R\) is the equivalent curvature radius and \(w\) is the deformation in \(F_{\rm a}\) direction. The contact area perpendicular to the direction of \(F_{\rm a}\), namely the area of inclined plane is \(a_{\rm a}=\pi Rw\), it is same with the intersection area \(S\) in Equation (10). The contact force in \(F_{\rm n}\) direction is \(F_{\rm n}=F_{\rm a}\cdot\cos\theta=\frac{4 ER^{0.5}\varphi_{\rm p}\cdot 15}{3}\cos\theta\). The contact area perpendicular to \(F_{\rm n}\) direction is \(a_{\rm n}=\pi Rw\mu\theta\). When elastic contact occurs, the average contact pressure in \(F_{\rm n}\) direction is  
\[p_{\rm n}=\frac{F_{\rm n}}{a_{\rm n}}=\frac{4E{a_{\rm n}}^{0.5}}{3\pi^{1.5}R}\]  
When the average contact pressure on the contact surface is less than the yield limit \(\sigma_{y}\) of the material, the asperities are in an elastic state. Under this condition, the elastic contact pressure can be expressed by \(p_{\rm n}\). When \(p_{\rm n}\) equals to \(\sigma_{y}\), the inception of yield occurs and the critical area \(a\)ec can be given by  
\[a_{\rm ec}=\left(\frac{3\pi^{1.5}R\sigma_{y}}{4E}\right)^{2} \tag{11}\]  
Purely plastic deformation  
The asperity yields fully when the average pressure is equal to \(3\sigma_{y}\)[42], in this condition  
\[\frac{p_{\rm n}}{\sigma_{y}}=\frac{2}{3}\left\{1+\ln\left[\frac{1}{3}\left( \frac{E}{\sigma_{y}}\right)\left(\frac{\pi}{R}\right)\right]\right\}\quad \mathrm{namely}\quad 3=\frac{2}{3}\left\{1+\ln\left[\frac{1}{3}\left(\frac{E}{ \sigma_{y}}\right)\left(\frac{a_{\rm pc}{}^{0.5}}{\pi^{0.5}R}\right)\right]\right\}\]  
After simplification, the critical plastic deformation area \(a_{pc}\) can be expressed as  
\[a_{\rm pc}=\left(\frac{3\mathrm{e}^{3.5}\sigma_{y}\pi^{0.5}R}{E}\right)^{2} \tag{12}\](3) Elastic-plastic deformation  
When \(a_{\rm ec}\leq a\leq a_{\rm pc}\), the asperity has elastic-plastic deformation. In this contact state, the relationship between the contact area and contact pressure becomes extremely complex. Considering that variations of the contact area and contact pressure should be continuous and smooth at the critical point of initial yield and be fully plastic, a template function \(f(a)\)[8] can be constructed. The template function is only used to distribute the weight of elastic area and plastic area in the elastic-plastic interval and does not involve the derivation conditions and process of the model, so it does not affect the compatibility of the model. \[f(a)=-2\bigg{(}\frac{a-a_{\rm ec}}{a_{\rm pc}-a_{\rm ec}}\bigg{)}^{3}+3\bigg{(} \frac{a-a_{\rm ec}}{a_{\rm pc}-a_{\rm ec}}\bigg{)}^{2} \tag{13}\]  
Then the average contact pressure in the elastic-plastic stage is given by  
\[