# The Elasto-Plastic Contact Models of Cylinder-Based and Sphere-Based Fractal Rough Surfaces
## 5 Results and Discussion
For the cylinder-based fractal rough surface with asperities for which the frequency indexes range from 38 to 85, the values of the pressure distribution of the present model are greater than that of the Hertzian model for a given interference in Figure 14a,b. The half contact width of the present model and that of the Hertzian model are approximately identical for a given interference. In Figure 14c, when the interference \(0<\omega<0.067\delta_{n_{\rm min}}\), these asperities for which the frequency index is 38 to 41 are in elastic deformation in the first subdivision nearest to the \(z\) axis, and the ratio of the peak pressure of the present fractal model to that of the Hertzian model equals approximately 344.2. When the interference \(0.067\delta_{n_{\rm min}}<\omega<0.129\delta_{n_{\rm min}}\), the ratio decreases from 344.2 to 341.3. When the interference \(0.129\delta_{n_{\rm min}}<\omega<0.1784\delta_{n_{\rm min}}\), these asperities of frequency index 35 are in elastic deformation in the first subdivision nearest to the \(z\) axis, and the ratio decreases from 341.3 to 338.8. When the interference \(0.1784\delta_{n_{\rm min}}<\omega<0.99\delta_{n_{\rm min}}\), the inelastic deformation gradually takes place in more and more subdivisions, and the ratio decreases from 338.8 to 279.3. Figure 13: The relation between the real contact area and the total contact load. (**a**) The cylinder-based rough surface with \(\rho\) = 0.1 m; (**b**) the sphere-based rough surface with \(r\) = 0.1 m.  
For the sphere-based fractal rough surface with asperities for which the frequency indexes range from 39 to 87, the values of pressure distribution of the present model are greater than that of the Hertzian model for a given interference in Figure 15a,b. And the half contact width of the present model is approximately \(\sqrt{2}\) times that of Hertzian model for a given interference. In Figure 15c, when the interference \(0<\omega<0.0643\delta_{n_{\rm min}}\), these asperities for which the frequency index is 39 to 42 are in elastic deformation in the first subdivision nearest to the \(z\) axis. The ratio of the peak pressure of the present fractal model to that of the Hertzian model equals approximately 397.9. When the interference \(0.0643\delta_{n_{\rm min}}<\omega<0.1702\delta_{n_{\rm min}}\), ratio of inelastic to elastic deformation increases in the first subdivision nearest to the \(z\) axis and the ratio decreases from 397.9 to 391.4. When the interference \(0.1702\delta_{n_{\rm min}}<\omega<0.99\delta_{n_{\rm min}}\), more and more subdivisions appear to have an elasto-plastic property, and the ratio decreases from 391.4 to 320.7. Figure 14: Comparison between the present fractal model and the Hertzian model based on pressure distributions for the cylinder-based fractal rough surface. (**a**) The pressure distribution with \(\omega=0.3\delta_{n_{\rm min}}\), (**b**) the pressure distribution with \(\omega=0.9\delta_{n_{\rm min}}\), (**c**) the ratio of the peak pressure of the present fractal model to the peak pressure of the Hertzian model. Figure 16a shows the relation between the total real contact area and total contact load for a cylinder-based fractal surface with asperities for which the frequency indexes range from 50 to 97. In the complete contact process, the ratio of elastic deformation to inelastic deformation is so small that the relation between the total real contact area and total contact load appears to have an inelastic property. For the interference \(\omega\approx 0.279\delta_{n_{\rm min}}\) or the total contact \(F_{r}\approx 1.76\times 10^{-11}\) N, the first subdivision nearest to the \(z\) axis appears to have a full plastic property, and the other subdivisions appear to have an elasto-plastic or elastic property.