# Simplified Calculation Model for Contact Resistance Based on Fractal Rough Surfaces Method
## Modeling
### Contact Properties  
The contact properties of the asperity set have a direct impact on the mechanical properties of the entire surface. To investigate the relationship between the total contact load and the actual contact area between rough surfaces, it is essential to study the contact properties of a single asperity set. The asperity set, which is a multilayer stacked structure, can be approximated as a sinusoidal asperity structure with the same height and base length, as shown in Figure 4.  
Based on finite elements, the contact characteristics of a three-dimensional sinusoidal contact asperity can be described by a fitting equation for the contact area versus contact load [24]  
\[A=\begin{cases}A_{IGH1}(1-\left[\frac{\overline{p}}{p*}\right]^{1.51})+A_{IGH2 }\left[\frac{\overline{p}}{p*}\right]^{1.04}&\frac{\overline{p}}{p*}<0.8\\ A_{IGH2}&\frac{\overline{p}}{p*}\geq 0.8\end{cases} \tag{9}\]  
where: \(A\) is the real contact area of a single asperity; \(\overline{p}\) is the average contact pressure of a sinusoidal asperity, which satisfies \(\overline{p}=F/l_{max}^{2}\), \(F\) is the contact load of this sinusoidal asperity; \(p*\) is the average contact pressure of asperity fully deformed, which satisfies \(p*=\sqrt{2\pi}E*h_{N}/l_{max}\). \(E*\) is the equivalent sinusoidal asperity Young's modulus, which can be obtained from the [PERSON]'s ratio and elastic modulus \(v_{l}\), \(v_{2}\), and \(E_{1}\), \(E_{2}\) of the two original contact surfaces by the formula \(E*=E_{1}E_{2}/(E_{2}(1-v_{1}^{2})+(E2(1-v_{2}^{2}))\). \(A_{IGH1}\) and  
Figure 4: Equivalent schematic diagram of the asperity set.  
Figure 3: Calculation flowchart for simulated surfaces.  
\(A_{jCH2}\) are the first contact area and the second contact area of the sinusoidal asperity contact model, which satisfy  
\[\left\{\begin{array}{l}A_{jGH1}=2\pi(l_{\max})^{2}\Big{[}\frac{3}{8\pi}\frac{ \overline{p}}{p^{\ast}}\Big{]}^{2/3}\\ A_{jGH2}=(l_{\max})^{2}\Big{(}1-\frac{3}{2\pi}\Big{[}1-\frac{\overline{p}}{p^{ \ast}}\Big{]}\Big{)}\end{array}\right. \tag{10}\]  
Figure 5 depicts a single sinusoidal asperity in contact with a rigid plane. The dashed line represents the original contour of the asperity, while the solid line represents the deformed contour of the asperity. The contact area between the asperity set and the rigid plane is an ideal circle, with \(r_{i}\) is the radius of the asperity contact spot, and \(d\) is the distance of the rigid plane from the base plane of the asperity, the real contact area of the asperity is \(A_{i}=\pi r_{i}^{2}\).  
For a sinusoidal asperity contact structure, the contact spot radius \(r_{i}\) can be calculated from the following equation [25]  
\[r_{i}=\left(\sqrt{\frac{h_{i}}{d}-0.75}-0.5\right)\frac{l_{\max}}{2}\ \ \ 0\leq d\leq\frac{1}{3}h_{i} \tag{11}\]  
The radius \(r_{i}\) and area \(A_{i}\) of the contact spot of the sinusoidal asperity in the plane can be calculated by specifying the distance \(d\) between the rigid plane and the base plane. The contact load \(F_{i}\) can be obtained from Equation (9). Extending to the entire contact plane, all sinusoidal asperities in the plane with height greater than \(d\) are in contact with the rigid plane, and their contact areas and contact loads can be calculated from Equations (9) and (11). The real contact area \(A_{r}\) and the total contact load \(F\) of the plane are the sums of the real contact area and the contact load of the contacting asperities, respectively, denoted as  
\[\left\{\begin{array}{l}A_{r}=\sum\limits_{i=1}^{n^{\prime}}\pi r_{i}^{2}\\ F=\sum\limits_{i=1}^{n^{\prime}}F_{i}\end{array}\right. \tag{12}\]  
where: \(n^{\prime}\) is the number of contacting asperities.  
Figure 5: Single sinusoidal asperity contact model.