# The Elasto-Plastic Contact Models of Cylinder-Based and Sphere-Based Fractal Rough Surfaces
## 5 Results and Discussion
When the interference \(0.007\delta_{n_{\rm min}}<\omega<0.68\delta_{n_{\rm min}}\) or the total contact \(9.468\times 10^{-14}<F_{r}\leq 5.703\times 10^{-11}\) N, the ratio of the elasto-plastic deformation contact load to the total contact load is less than 55%, and the load-area relation is approximately \(F_{r}\sim A_{r}^{1.19}\). When \(F_{r}>5.703\times 10^{-11}\) N, the cylinder-based fractal rough surface gradually appears to have a full plastic property with an increase in the total contact load, and the load-area relation gradually approaches \(F_{r}\sim A_{r}\). In Figure 16b, the frequency indexes of asperities range from 53 to 101 for the sphere-based fractal rough surface. When the interference \(0.004\delta_{n_{\rm min}}<\omega\leq 0.4\delta_{n_{\rm min}}\) or the total contact load \(1.775\times 10^{-14}<F_{r}\leq 1.159\times 10^{-11}\) N, the ratio of the elasto-plastic deformation contact load to the total contact load is less than 55%, and the load-area relation is approximately \(F_{r}\sim A_{r}^{1.19}\). When \(F_{r}>1.159\times 10^{-11}\) N, the sphere-based fractal rough surface gradually appears to have a full plastic property  
Figure 15: Comparison between the present fractal model and the Hertzian model based on pressure distributions for the sphere-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 present fractal model to the peak pressure of the Hertzian model. with an increase in total contact load, and the load-area relation gradually approaches \(F_{r}\sim A_{r}\). In Figure 17a, for the cylinder-based fractal surface with asperities for which the frequency indexes range from 50 to 97, the values of pressure distribution of the present model are greater than those of the Hertzian model with interference \(\omega=0.9\delta_{n_{\min}}\), and the half contact width of the present model and that of the Hertzian model are approximately identical. Figure 17b shows the ratio of the peak pressure of the present fractal model to that of the Hertzian model with interference ranging from \(0.0001\delta_{n_{\min}}\) to \(0.99\delta_{n_{\min}}\). When the interference \(\omega<0.003\delta_{n_{\min}}\), the ratio is approximately constant. The ratio in elastic deformation is so small that it is invisible in Figure 17b. When the interference \(0.003\delta_{n_{\min}}<\omega<0.99\delta_{n_{\min}}\), the fractal rough surface appears to have an inelastic property, and the ratio decreases from 10,381.1 to 3147.5. In Figure 18a, the values of pressure distribution of the present model are greater than those of the Hertzian model with interference \(\omega=0.9\delta_{n_{\min}}\) for a sphere-based fractal surface with asperities for which the frequency indexes range from 53 to 101. When the  
Figure 16: 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 8:0 Comparison between the present fractal model and the elastic foundation model based on pressure distributions for the cylinder-based fractal rough surface. (**a**) The pressure distribution with \(\omega=0.9\delta_{n_{\min}}\); (**b**) the ratio of the peak pressure. interference \(0.0018\delta_{n_{\min}}<\omega<0.99\delta_{n_{\min}}\), the fractal rough surface appears to have an inelastic property, and the ratio of the peak pressure of the present fractal model to that of the Hertzian model decreases from 21,097.2 to 4854.2 in Figure 19b. The pressure distributions depend on the range of frequency indexes of asperities under a given interference for the fractal rough surface. Figure 19a shows the relation between the peak pressure and range of frequency indexes of asperities in the logarithmic coordinates system.