# Simplified Calculation Model for Contact Resistance Based on Fractal Rough Surfaces Method
## Modeling  
### Rough Surface Morphology Structure  
The contact between two rough surfaces is usually simplified to a rigid ideal surface and an equivalent rough surface. The equivalent rough surface has the morphological characteristics and material properties of the two surfaces. From the two-dimensional W-M function, a cosine wave-like asperity is a basic unit of the rough surface model. It is assumed that the coupling effect between adjacent asperity and the strain-hardening effect of the material during the contact is neglected, and the tangential cross-section profile of the asperity is circular. Then, the profile height curve of the asperity before deformation can be described as  
\[z(x)=G^{D-1}l^{2-D}\cos\frac{\pi x}{l},-\frac{l}{2}<x<\frac{l}{2} \tag{1}\]  
where: \(D\) is the fractal dimension (1 < \(D\) < 2); \(G\) is the characteristic scale parameter reflecting the profile curve; \(l\) is the asperity base diameter; \(\delta\) is the asperity height under the base diameter of \(l\), satisfying \(\delta=G^{D-1}l^{2-D}\). Aserities of different scales with base diameter \(l\) satisfy the island area distribution law above sea level and the Monte Carlo method [23]  
\[N(L\geq l)=(\frac{l_{\max}}{l})^{D} \tag{2}\]  
The base diameter \(l_{i}\) and the corresponding height \(\delta_{i}\) of the asperity at different scales can be expressed by the following equation  
\[l_{i}=\frac{l_{\min}}{(1-R_{i})^{1/D}}=(\frac{l_{\min}}{l_{\max}})\frac{l_{ \max}}{(1-R_{i})^{1/D}} \tag{3}\]  
\[\delta_{i}=G^{D-1}[(\frac{l_{\min}}{l_{\max}})\frac{l_{\max}}{(1-R_{i})^{1/D} }]^{2-D} \tag{4}\]  
where \(l_{max}\) and \(l_{min}\) correspond to the maximum and minimum asperity base diameters, respectively, \(R_{i}\) is the random number, \(i\) = 1, 2, 3,..., \(N\), and \(N\) is the total number of random numbers generated, satisfying \(N\) = [(\(l_{max}/l_{min})^{D}\)]. The asperity corresponding to \(l_{max}\) is called the first level asperity, and the rest of the asperities are arranged sequentially according to the size of the base diameter. The range of the random number \(R_{i}\) is  
\[0\leq R_{i}\leq 1-\left(\frac{l_{\min}}{l_{\max}}\right)^{D} \tag{5}\]  
An asperity set is formed by stacking asperities of different scales with cosine functions of contours. The asperity corresponding to \(l_{max}\) is stacked as the base of the asperity set, and the asperity of the second level is stacked on top of the asperity of the first level until a total of \(N\) asperity sets of different levels are stacked. Figure 1 illustrates the stacked structures of the asperity set obtained for the specific number of layers. The height of the \(N\) level asperity set obtained through stacking is represented by \(l_{N}\) and can be expressed as  
\[h_{N}=G^{(D-1)}l_{\max}^{(2-D)}+\sum_{m=1}^{N-1}\delta_{i} \tag{6}\]It is worth noting that \(h_{N}\), as the highest point of the simulated surface, should be equal to the maximum height difference \(Sz\) (maximum height of the surface, \(Sz\)) of the measured surface. Based on Equation (6), the \(l_{max}\) is determined by iterative solution. The asperity sets are reconstructed concerning the contour height of the actual surface. The surface and the asperity sets satisfy the same height distribution law, as shown in Figure 2. Theoretically, surfaces of any height distribution can be simulated by simply adjusting the replication multiplication factor. The total number of asperity sets on the reconstructed simulated surface is \(N_{1}\), as distinguished from the number of asperity sets at different scales \(N\). The equivalent rough surface consists of several asperity sets with different stacked layers in the plane according to the Monte Carlo method. In this model, a square-shaped macroscopic contact base surface with a side length of \(L\) is used. The asperity set is randomly distributed on the plane and does not intersect with the boundary. The coordinates \((x(i),\)\(y(i))\) of the circle center of the base of the asperity set can be expressed as  
\[\left\{