# Simplified Calculation Model for Contact Resistance Based on Fractal Rough Surfaces Method
## 3 Results and Analysis
### Contact Resistance Analysis and Finite Element Verification  
Figure 11(a) illustrates that contact resistance tends to decrease with the increase in contact load, and the contact resistance reaches the local minimum with \(D=1.5\); this finding supports the established principle that a smaller real contact area leads to larger contact resistance. In Figure 11(b), we observe that the characteristic scale parameter \(G\) has a much greater influence on the contact resistance than the fractal dimension \(D\). This is attributed to \(G\) being positively correlated with the maximum height difference of the profile. When the height difference decreases, the number of contact asperities increases, leading to a greater number of parallel branches in the electric contact network. As a result, the resistance at the contact interface decreases.  
Figure 11: Comparison of model theoretical values and experimental data.  
Figure 13 shows the variation in contact resistance \(R\) with the resistivity \(\rho_{f}\) of the contaminated film layer for surface parameters \(D=1.53\), \(G=10^{-9}\) m, and dimensionless contact load \(F^{*}=1.29\). When the electrical contact surface is completely clean and ideal pure metal contact occurs, the contact resistance caused by the current line contraction is \(R=8.05\times 10^{-4}\)\(\Omega\). When the contaminated film meets \(d_{c}=1\)\(\mu\)m and \(\rho_{f}=10^{-4}\)\(\Omega\)m, the contact resistance \(R=29.1\times 10^{-4}\)\(\Omega\), which is of the same order of magnitude as the metal contact resistance. With the increase of film resistivity and thickness, the film resistance will be dominated by the total contact resistance.  
The 3D point cloud data is obtained based on the equivalent rough surface simulation method proposed in Section 2.1. These point cloud data are fitted to the surface, then patched and denoised to achieve surface model reconstruction, as shown in Figure 14. The rough body material is made of aluminum [28], with the following material parameters: \(E^{*}=69\) Gpa, \(v=0.33\), \(\sigma_{8}=276\)MPa. The contact model uses the Augmented Lagrangian method to enhance contact and limit penetration between surfaces. A smooth displacement along the \(Z\)-axis and a constant voltage are applied to the top surface of the rough body. Constant boundary conditions are applied to the bottom surface of the rigid body and do not consider small displacement slips in the x and y axes of the rough body. The contact surface consists of 150 by 100 nodes, resulting in 149 by 99 elements. The mesh is created using the moving node method, and it becomes increasingly coarser away from the surface.  
Figure 15 shows the variation in contact resistance with fractal parameters for different contact loads. It can be observed that the model proposed in this paper agrees well with the finite element simulation under small loads, with some deviations under large loads. The contact resistance is minimized at \(D=1.5\) and has a linear relationship with \(G\) on the double logarithmic axis. It is worth mentioning that the results of the finite element model may be inaccurate when the surface is relatively flat (i.e., when \(G\) is small).  
Figure 13: Variation in \(\rho_{f}\) versus \(R\).  
The reason is when the height difference between the point clouds is small and the surface is flat at the macro level, fitting the point clouds to generate solid surfaces results in the loss of a significant number of morphological features. Figure 16 demonstrates the reverse modeling approach employed in this study. While the accuracy of the method is improved by merging surfaces through fitting subregions, it still leads to the loss of some morphological features, which is more pronounced when dealing with flat surfaces.  
Figure 14: Reverse modeling to obtain computational domain, (**a**) solid model of the contacting bodies; (**b**) finite-element mesh.  
Figure 15: Comparison of contact resistance calculation and finite element analysis: (**a**) \(G=10^{-11}\) m; (**b**) \(D=1.53\).