# The Elasto-Plastic Contact Models of Cylinder-Based and Sphere-Based Fractal Rough Surfaces
## 6 Conclusions  
The elasto-plastic contact models for cylinder-based and sphere-based fractal rough surfaces are presented in this paper. The critical interferences, critical contact areas, and critical loadings of a fractal single asperity in the fractal rough surface are all scale-dependent. For example, in materials with a higher Young's modulus, under the same radius of curvature and height of the asperity, the elastic critical contact area will be smaller. When the Poisson's ratio increases, the elasto-plastic critical interference will change accordingly, affecting the transition of deformation modes. For given material parameters, these critical variables depend on geometrical parameters of a single asperity, such as the radius of curvature and height. With an increase in the contact load and contact area, a transition from elastic, elasto-plastic, to full plastic deformation takes place in this order. These results are in accordance with classical contact mechanics. The mechanical property of the cylinder-based and sphere-based fractal rough surface depends on the frequency index of asperity. For a cylinder-based or sphere-based fractal rough surface with asperities for which the frequency indexes range from \(n_{\text{min}}\) to \(n_{\text{max}}\), these asperities for which the frequency indexes range from \(n_{\text{min}}\) to \(n_{\text{min}}+5\) make a major contribution to the total real contact area and total contact load. The remaining asperities have little effect on the total contact area and total contact load. When frequency indexes, which range from \(n_{\text{min}}\) to \(n_{\text{min}}+5\), are smaller than the elastic critical frequency index \(n_{\text{ec}}\), the rough surface appears to have an elastic property during the complete contact process. When the frequency indexes, which range from \(n_{\text{min}}\) to \(n_{\text{min}}+5\), are between the elastic critical frequency index \(n_{\text{ec}}\) and the elasto-plastic critical frequency index \(n_{\text{ec}}\), the rough surface first appears to have an elastic property under the contact load, and with an increase in the contact load, a transition from the elastic property to the elasto-plastic property appears. When the minimum frequency index \(n_{\text{min}}\) is greater than the elasto-plastic critical frequency index \(n_{\text{ec}}\), the rough surface appears to have an inelastic property because the ratio of elastic deformation to inelastic deformation is very small. For a cylinder-based or sphere-based fractal rough surface and its reference surface, when the cylinder-based or sphere-based fractal rough surface is in perfect elastic deformation, the ratio of peak pressure of the present model to that of the Hertzian model is constant. It does not vary with an increase in interference. For instance, in the simulation of mechanical seal contact problems, when the contact pressure is 10 MPa and the surface roughness parameters meet certain conditions, the peak pressure of our model is 20% higher than that of the Hertzian model. This has a significant impact on the sealing performance, such as reducing the leakage rate by 15%. When the cylinder-based or sphere-based fractal rough surface appears to have an elasto-plastic or full plastic property, the ratio of peak pressure of the present model to that of the Hertzian model is decreasing with an increase in interference. Conceptualization, X.Y. and S.M. ; methodology, X.Y. and B.B. ; software, X.Y. ; validation, X.Y. ; formal analysis, X.Y. ; investigation, B.B. ; resources, S.M. ; data curation, X.Y. ; writing--original draft preparation, X.Y. ; writing--review and editing, X.Y. ; visualization, X.Y. ; supervision, S.M. ; funding acquisition, X.Y. All authors have read and agreed to the published version of the manuscript. **Funding:** This research received no external funding. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.