## 2 Contact Analysis of Rough Surface
### The Initial Porosity of Rough Sealing Surface  
[PERSON] [40] used fractal parameters to describe rough surfaces and found they can comprehensively characterize the overall, local and internal structure of the surface. For the fractal machined rough surface, W-M function can be used to describe the contour curve of any asperity before deformation  
\[z(x)=G^{D-1}l^{2-D}\cos\Bigl{(}\frac{\pi x}{l}\Bigr{)},-\frac{l}{2}<x<\frac{l}{2} \tag{1}\]  
Figure 4: Contact of two rough asperities.  
Figure 5: MATLAB simulation of the contact of two rough surfaces.  
Figure 3: Contact of smooth plane and rough asperity.  
where \(D\) is the fractal dimension of the surface, \(G\) is the fractal roughness, \(l\) is the base diameter of the asperity which can be given by \(l=1/\gamma^{n}\), \(\gamma^{n}\) is the spatial frequency of the asperity. When \(n=n_{\rm min}\), the base diameter of the largest asperity is \(l=1/\gamma^{n{\rm min}}\), when \(n=n_{\rm min}+1\), the base diameter of the corresponding asperity is \(l=1/\gamma^{n{\rm min}+1}\), etc.  
For the asperity satisfying Equation (1), its maximum height \(h=G^{D-1}l^{2-D}\) corresponds to the maximum surface roughness, and the corresponding \(l\) is the base diameter of the maximum asperity.  
The number of asperities on the sealing surface is related to the maximum contact area [23], which satisfies  
\[n(a)_{1,2}=\frac{D_{1,2}}{2}\psi^{\frac{2-D_{1,2}}{2}}a_{\rm L}^{\frac{D_{1,2} }{2}}a^{-\frac{D_{1,2}}{2}-1} \tag{2}\]  
where \(n(a)\) is the distribution density function for the contact area, subscript 1 and 2 represents two rough surfaces, respectively, \(n(a)\) takes the smaller value of \(n(a)_{1}\) and \(n(a)_{2}\), \(a\) is the contact area of asperity, \(a_{\rm L}\) is the maximum area of micro contact, \(\psi\) is the correction factor of \(A_{\rm r}/a_{\rm L}\), which is the ratio of the real contact area to the maximum area of micro contact. This can be obtained from the transcendental equation [41]:  
\[\frac{\psi^{(2-D)/2}-(1+\psi^{-D/2})^{(D-2)/D}}{(2-D)/D}=1 \tag{3}\]  
For the sealing surface with known surface topography parameters, the nominal contact area \(A_{0}\) is equal to the sum of the cross-sectional area of the largest asperity and all other smaller asperities on the surface:  
\[A_{01,2}=\int_{0}^{a_{\rm L1,2}}an(a)_{1,2}\mathrm{d}a=\frac{D_{1,2}}{2-D_{1,2 }}\psi^{\frac{2-D_{1,2}}{2}}a_{\rm L1,2} \tag{4}\]  
where \(a_{\rm L1,2}\) is the largest area of micro contact, \(a_{\rm L1,2}=\pi l_{1,2}{}^{2}/4\), m\({}^{2}\); \(A_{0}\) takes the larger value of \(A_{01}\) and \(A_{02}\).  
So far, the numerical expression can be used to calculate the porosity for sealing surface under certain topography parameters. Equation (5) is the initial porosity when a plane contacts a rough surface, and Equation (6) is the initial porosity when two rough surfaces contact.  
\[\phi_{\rm r-f}=1-\frac{\int_{0}^{h}n(a)a\cdot a\mathrm{dz}(x)}{A_{0}h}=1- \frac{Da_{\rm L}^{\frac{D}{2}}\pi^{0.5}\int_{0}^{a_{\rm L}}a^{\frac{1-D}{2}} \cdot\sin\frac{\sqrt{\pi n}}{l}\mathrm{d}a}{4A_{0}l}\] (  
When the fractal parameters of the two rough surfaces are the same, the calculated value based on the rough-rough contact is closer to the simulated value; the maximum relative error is less than 5%, while the relative error between the calculated and simulated value based on the rough-flat contact is 10.5-15.5%. Therefore, when the topography parameters of two surfaces are the same or similar, the contact should be considered shoulder-shoulder rather than top-top. Shoulder-shoulder contact should be used for the modeling and mechanical analysis of the rough surface.