# The Elasto-Plastic Contact Models of Cylinder-Based and Sphere-Based Fractal Rough Surfaces
## 5 Results and Discussion
spherical surface to a sphere-based fractal rough surface. For the same radius \(r\), the half contact width of the present model is approximately \(\sqrt{2}\) times that of Hertzian model under a given interference. The half contact width of the present model with \(r=0.1\) m equals approximately that of the Hertzian model with \(r=0.2\) m.  
In Figure 10a, when a cylinder-based fractal rough surface with asperities for which the frequency indexes range from 19 to 63 comes into contact with a rigid flat surface, the ratio of the real contact area of asperities with the lowest frequency index to the real contact area is maximum. As the frequency index increases, the ratio decreases. For \(\omega=0.3\delta_{n_{\min}}\) and \(\omega=0.9\delta_{n_{\min}}\), the sum of the ratio of the real contact area of asperities for which the frequency indexes range from \(n_{\min}\) to \(n_{\min}+5\) to the total real contact area can be obtained as \(\sum\limits_{i=1}^{N}\sum\limits_{19}^{24}A_{i,n}/A_{r}=0.9923\). Therefore, these asperities of the first six frequency indexes make major contributions to the cylinder-based fractal rough surface. Figure 10b shows the ratio of the contact area of asperities for which the frequency index is \(n\) to the total real contact area for the sphere-based fractal rough surface under different interferences. The results are similar to the cylinder-based fractal rough surface, and these asperities of the first six frequency indexes make major contributions to the sphere-based fractal rough surface because of \(\sum\limits_{i=1}^{N}\sum\limits_{19}^{24}S_{i,n}/S_{r}=0.9922\). Figure 11 shows the relation between the total real contact area and the total contact load. For the cylinder-based fractal rough surface, the asperity frequency indexes range from 19 to 66 (Figure 11a). We can obtain the elastic critical frequency index \(n_{ec}=32\) and the elasto-plastic critical frequency index \(n_{pc}=46\). Only elastic deformation takes place in these asperities for which the frequency indexes range from 19 to 32 during complete contact process (\(0<\omega\leq 0.99\delta_{n_{\min}}\)). The elastic deformation and elasto-plastic deformation can take place in these asperities for which the frequency indexes range from 33 to 46. The elastic deformation, elasto-plastic deformation, and full plastic deformation can take place in these asperities for which the frequency indexes range from 47 to 66. Because these asperities of the first six frequency indexes make major contributions to the cylinder-based fractal rough surface, the cylinder-based fractal rough surface appears to have an elastic property. The relation between the total real contact area and total contact load is approximately \(F_{r}\sim A_{r}^{3/2}\). For the sphere-based fractal rough surface, the asperity  
Figure 9: Comparison between the present fractal model and the Hertzian model on the half contact width under different interferences. (**a**) The cylinder-based fractal rough surface; (**b**) the sphere-based fractal rough surface. frequency indexes range from 19 to 67 (Figure 11b). The elastic critical frequency index is \(n_{cc}=33\), and the elasto-plastic critical frequency index is \(n_{pc}=46\). The sphere-based fractal rough surface appears to have an elastic property in the complete contact process (\(0<\omega\leq 0.99\delta_{n_{\min}}\)), and the relation between the total real contact area and total contact load is approximately \(P_{r}\sim S_{r}^{3/2}\). The comparison between the present fractal model and the Hertzian model based on pressure distributions has been obtained in Figure 12a. For the cylinder-based fractal rough surface with asperities for which the frequency indexes range from 19 to 66, it appears to have an elastic property. The smooth reference cylindrical surface with radius \(\rho=0.1\) m is only in elastic deformation. The values of pressure distributions increase with an increase in the interference.