# The Elasto-Plastic Contact Models of Cylinder-Based and Sphere-Based Fractal Rough Surfaces
## 5 Results and Discussion
The half contact width of the present model and that of the Hertzian model are approximately identical with the same interference. For a given interference, the values of pressure distribution of the present model are greater than that of the Hertzian  
Figure 11: The relation between the real contact area and the total contact load. (**a**) The cylinder-based fractal rough surface with \(\rho=0.1\) m; (**b**) the sphere-based fractal rough surface with \(r=0.1\) m.  
Figure 10: The ratios of contact areas of asperities for which the frequency index is \(n\) to the total real contact area. (**a**) The cylinder-based fractal rough surface with \(\rho=0.1\) m; (**b**) the sphere-based fractal rough surface with \(r=0.1\) m.  
model. In Figure 12b, when the interference \(\omega\) is from \(0.001\delta_{n_{\min}}\) to \(0.99\delta_{n_{\min}}\), the ratio of the peak pressure of the present fractal model to that of the Hertzian model is approximately constant and equals 1.5675 in the complete contact process. In Figure 12c, the sphere-based fractal rough surface with asperities for which the frequency indexes range from 19 to 67 appears to have an elastic property under the contact load. The smooth reference spherical surface with radius \(r=0.1\) m is only in elastic deformation. The half contact width of the present model is approximately \(\sqrt{2}\) times that of Hertzian model under a given interference. The ratio of the peak pressure of the present fractal model to that of the Hertzian model is 1.3655 in the complete contact process. Figure 13a shows the relation between total real contact area and total contact load for a cylinder-based fractal surface with asperities for which the frequency indexes range from 38 to 85. The elastic and elasto-plastic deformation can take place in asperities for which the frequency indexes range from 38 to 46. These asperities for which the frequency indexes range from 38 to 43 make major contributions to the cylinder-based rough surface. When the right contact zone with the \(z\) axis is divided into \(N\) (\(N=100\)) subdivisions, the first subdivision nearest to the \(z\) axis appears to have an elasto-plastic property and the  
Figure 12: Comparison between the present fractal model and the Hertzian model based on pressure distributions. (**a**) The cylinder-based fractal rough surface; (**b**) the cylinder-based fractal rough surface; (**c**) the sphere-based fractal rough surface; (**d**) the sphere-based fractal rough surface. other \(N-1\) subdivisions appear to have an elastic property under the total contact load \(F_{r}\approx 3.164\times 10^{-8}\) N. With an increase in the total contact load, more and more subdivisions appear to have an elasto-plastic property. When the total contact load \(F_{r}<9.21\times 10^{-8}\) N, the ratio of the elastic deformation contact load to the total contact load is greater than 55%, and the load-area relation is approximately \(F_{r}\sim A_{r}^{3/2}\). With the total contact load continuously increasing, the ratio of the elasto-plastic deformation contact load to the total contact load increases. Therefore, the load-area relation gradually approaches \(F_{r}\sim A_{r}^{1.19}\), the cylinder-based rough surface appears to have an elasto-plastic property. In Figure 13b, the sphere-based fractal rough surface includes these asperities for which the frequency indexes range from 39 to 87. When the total contact load \(P_{r}<1.83\times 10^{-7}\) N, the ratio of the elastic deformation contact load to the total contact load is greater than 55%, so that the load-area relation approximately approaches \(P_{r}\sim S_{r}^{3/2}\). With an increase in the contact load, the load-area relation gradually approaches \(P_{r}\sim S_{r}^{1.19}\). Figure 14 shows the comparison between the present fractal model and the Hertzian model based on pressure distributions for the cylinder-based fractal rough surface.