# The Elasto-Plastic Contact Models of Cylinder-Based and Sphere-Based Fractal Rough Surfaces
## 5 Results and Discussion  
The relations between the total real contact area and total contact load for cylinder-based and sphere-based fractal rough surfaces are presented in this section. These parameters of equivalent rough surfaces are shown in Table 1  
In order to obtain the relations between the total real contact area and the total contact loading, a simple loading method is adopted in this chapter. \(\omega_{i,n}\) is the maximum interference for asperities for which the frequency index is \(n\) in the \(i\)th subdivision. When a cylinder-based or sphere-based rough surface makes contact with a rigid flat surface, we assume that the interference \(\omega_{i,n}\) is proportional to full height of asperity for which the frequency index is \(n\), \(\delta_{i,n}(\delta_{i,n}=\delta_{n})\), namely \(\frac{\omega_{i,n}\omega_{i,n}}{\delta_{i,n}\omega_{i,n}}=\frac{\omega_{i,n} \omega_{i,n}+1}{\delta_{i,n}\omega_{i,n}+1}=\cdots\frac{\omega_{i,n}}{\delta_{ i,n}}=\cdots=\frac{\omega_{i,n}\omega_{i,n}}{\delta_{i,n}\omega_{n}}\). Then \(a^{\prime}_{i,nl}\) can be obtained by \(\omega_{i,n}\). According to \(D=2.4\), \(\rho=0.1\) m, \(G=1.36\times 10^{-11}\) m, and \(H=5.5\times 10^{9}\) N/m\({}^{2}\), the relations between the total real contact area and numbers of subdivisions under different interferences are shown in Figure 8. These interferences are \(\omega=0.3\delta_{n_{\min}}\), \(\omega=0.6\delta_{n_{\min}}\), and \(\omega=0.9\delta_{n_{\min}}\), respectively. These interferences in the \(i\)th subdivision can be obtained using Equations (36) and (75). The total real contact area is proportional to the interference. For the cylinder-based fractal rough surface (Figure 8a), when the value of subdivision number increases, the total real contact area first increases until \(N=12\) and then is approximately constant to \(N=100\). For the sphere-based fractal rough surface (Figure 8b), the total real contact area first increases until \(N=14\) and then is approximately constant to \(N=100\). The results show that the value of total real contact area is convergent with an increase in the subdivision number. In order to ensure that the calculation results are stable, the number of subdivision \(N\) equals 100. In order to obtain the relation between the rough surface and smooth surface in the contact process, a smooth cylindrical surface with radius \(\rho\) (base circle) can be regarded as the reference cylindrical surface to a cylinder-based fractal rough surface with a radius of base circle \(\rho\). For a given interference \(\omega\), \(\omega\ll\rho\), only elastic deformation takes place in the reference cylindrical surface. The half contact width of the reference surface can be obtained using Hertzian model [30]. Figure 9a shows comparison between the cylinder-based fractal rough surface and reference cylindrical surface on the half contact width with different radii of base circles. When the radius \(\rho\) increases, the half contact width increases under a given interference. The results of the present model and Hertzian model are approximately identical. In Figure 8b, a smooth spherical surface with radius \(r\) is regarded as the reference \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Fractal dimension \(D\) & 2.4 \\ Range of asperity frequency index \(n\) & 19–101 \\ Subdivisions \(N\) & 100 \\ Young’s modulus \(E\) [N/m\({}^{2}\)] & \(7.2\times 10^{10}\) \\ Poisson’s ratio \(v\) & 0.17 \\ Characteristic length scale of the surface \(G\) [m] & \(1.36\times 10^{-11}\) \\ Hardness \(H\) [N/m\({}^{2}\)] & \(5.5\times 10^{9}\) \\ The cut-off length \(L_{a}\) [m] & \(1.0\times 10^{-9}\) \\ The radius of base circle \(\rho/r\) [m] & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The calculation parameters. Figure 8: The relations between the total real contact area and numbers of subdivisions under different interferences. (**a**) The cylinder-based fractal rough surface; (**b**) the sphere-based fractal rough surface.