## 4 Establishment of Contact Mechanics Model
### Analysis of Deformation State
#### 4.1.1 Real Contact Area and Contact Load of Rough Surface  
For two rough surfaces in contact when the number of asperities on the interface satisfies Equation (2), its nominal contact area \(A_{0}\) can be expressed by Equation (4). When \(a_{\rm L}<a_{\rm ec}\), all the contact asperities are in elastic deformation, therefore, the real contact area \(A_{\rm r}\) is the elastic contact area \(A_{\rm re}\) and the contact force \(F_{\rm n}\) is the elastic contact force \(F_{\rm ne}\), which can be given by  
\[A_{\rm r1,2}=A_{\rm re1,2}=\int_{0}^{a_{\rm L}}an(a)_{1,2}\mathrm{d}a=\frac{D_ {1,2}}{2-D_{1,2}}k_{1,2}a_{\rm L} \tag{15}\]  
\[F_{\rm n1,2}=F_{\rm ne1,2}=\int_{0}^{a_{\rm L,1,2}}p_{\rm ne}an(a)_{1,2} \mathrm{d}a=\frac{4 kED_{1,2}a^{1.5}}{3(3-D_{1,2})\pi^{15}R} \tag{16}\]  
where \(k_{1,2}=\psi^{\frac{2-D_{1,2}}{2}}\), \(a_{\rm L}\) is the contact area of the largest asperity; \(A_{\rm r}\) is the larger value between \(A_{\rm r1}\) and \(A_{\rm r2}\); \(F_{\rm n}\) takes the larger value of \(F_{\rm n1}\) and \(F_{\rm n2}\). When \(a_{\rm ec}\leq a_{\rm L}\leq a_{\rm pc}\), the contact asperities are in elastic-plastic deformation. In this scenario, the total contact area \(A_{\rm r}\) is the sum of elastic contact area \(A_{\rm re}\) and elastic-plastic contact area \(A_{\rm rep}\), and the total force \(F_{\rm n}\) is the sum of elastic contact force \(F_{\rm ne}\) and elastic-plastic contact force \(F_{\rm ne}\)  
\[ \begin{array}{ll}A_{\rm r1,2}&=A_{\rm re1,2}+A_{\rm rep1,2}+A_{\rm rp1,2}\\ &=\int_{0}^{a_{\rm ec}}an(a)_{1,2}\mathrm{d}a+\int_{a_{\rm ec}}^{a_{\rm L}}an( a)_{1,2}\mathrm{d}a=\frac{D_{1,2}}{2-D_{1,2}}k_{1,2}a_{\rm L}\end{array}  \tag{17}\]  
\[ \begin{array}{ll}F_{\rm n1,2}&=F_{\rm ne1,2}+F_{\rm ne1,2}\\ &=\int_{0}^{a_{\rm ec}}p_{\rm ne}an(a)_{1,2}\mathrm{d}a+\int_{a_{\rm ec}}^{a_{ \rm L}}p_{\rm np}an(a)_{1,2}\mathrm{d}a\\ &=\int_{0}^{a_{\rm ec}}\frac{4E_{a}a^{0.5}}{3\pi^{15}R}an(a)_{1,2}\mathrm{d}a+ \int_{a_{\rm ec}}^{a_{\rm L}}\bigg{(}\frac{4E_{a}a^{0.5}}{3\pi^{15}R}+\Big{(}3 \sigma_{\rm y}-\frac{4E_{a}a^{0.5}}{3\pi^{15}R}\Big{)}\bigg{(}-2\Big{(}\frac {a-a_{\rm ec}}{a_{\rm pc}-a_{\rm ec}}\Big{)}^{3}+3\Big{(}\frac{a-a_{\rm ec} }{a_{\rm pc}-a_{\rm ec}}\Big{)}^{2}\bigg{)}\bigg{)}an(a)_{1,2}\mathrm{d}a\\ &=\frac{4 kED_{1,2}a_{\rm ec}-3D_{1,2}}{3(3-D_{1,2})\pi^{15}R}a_{\rm L}^{\frac{ D_{1,2}}{2}}+\frac{kD_{1,2}}{2}a_{\rm L}\frac{D_{1,2}}{2}\int_{a_{\rm ec}}^{a_{ \rm L}}\bigg{(}\frac{4E_{a}a^{0.5}}{3\pi^{15}R}+\Big{(}3\sigma_{\rm y}-\frac{4 E_{a}a^{0.5}}{3\pi^{15}R}\Big{)}\bigg{(}-2\Big{(}\frac{a-a_{\rm ec}}{a_{\rm pc}-a_{ \rm ec}}\Big{)}^{3}+3\Big{(}\frac{a-a_{\rm ec}}{a_{\rm pc}-a_{\rm ec}}\Big{)}^ {2}\bigg{)}\bigg{)}a^{-\frac{D_{1,2}}{2}}\mathrm{d}a\end{array}  \tag{18}\]  
When \(a_{\rm L}>a_{\rm pc}\), the contact asperities are in plastic deformation. In this condition, the total contact area \(A_{\rm r}\) is the sum of elastic contact area \(A\)re, elastic-plastic contact area \(A\)repand plastic contact area \(A\)rp, and the total force \(F_{\rm n}\) is the sum of elastic contact force \(F_{\rm ne}\), elastic-plastic contact force \(F_{\rm ne}\) and plastic contact force \(F_{\rm np}\). \[