# Modeling of the Loading-Unloading Contact of Two Cylindrical Rough Surfaces with Friction
## 3 Fractal Contact Model in the Unloading Process with Regard of Friction
### Load-Unload Cycle Analysis  
Figure 2 shows the relation between dimensionless real contact area and dimensionless contact load of two cylindrical surfaces in different deformation stages. Figure 2a shows the load-unload cycle in the elastic stage and no difference can be seen between loading and unloading process as the plastic deformation does not happen in this stage. Figure 2b,c show the load-unload cycle in the first and second elastoplastic stage with the unloading points are 1.0 and 0.6 times of critical contact area. In the elastoplastic stage, the dimensionless real contact area in the unloading process is higher than that in the loading process with the same dimensionless contact load; the dimensionless real contact area of the unloading process, as well as the differences between the loading and unloading processes, increases with the loading interference. The reason is that asperities are in the elastoplastic deformation cannot fully recover in the unloading process and their curvature radius increases resulting in the adding of the contact area; the interference and quantity of asperities increase with the loading interference, leading to the further increase of the unrecoverable contact area. Figure 2: _Cont._The second elastoplastic stage is chosen to discuss the impacts of parameters and contact types in the model of the loading-unloading process of the two cylindrical surfaces. Figures 3 and 4 show the dimensionless contact area-load relations of two cylindrical surfaces in the second elastoplastic stage with effects of fractal dimension \(D\) and profile scaling parameter \(G\). In each figure, (a) shows the unloading at the dimensionless critical contact load and (b) shows the unloading at a given dimensionless contact load. The dimensionless critical real contact area and the dimensionless critical contact load increase with decrease of \(G\) or increase of \(D\); for a given dimensionless contact load, the dimensionless real contact area, as well as the differences of the dimensionless real contact area between the loading and unloading processes, are positively related to \(D\), and negatively correlated with \(G\). The reason is that the roughness of surface decreases as \(G\) decreases [46], as well as \(D\) increases [29; 47]. So it can be inferred that in the load-unload cycle in the elastoplastic stage, as the roughness of surface decreases, although the overall interference of asperities decreases, the quantity of asperities deformed increase rapidly which has a greater influence leading to the phenomenon shown in Figures 3 and 4. Figure 3: The relation between dimensionless real contact area and dimensionless contact load of two cylindrical surfaces in the second elastoplastic stage with different fractal dimensions with \(G=10^{-10}\) mm, \(R_{\text{c1}}=100\) mm and \(R_{\text{c2}}=60\) mm. (**a**) Unloading at the dimensionless critical contact load; (**b**) unloading at a given dimensionless contact load. Figure 2: The relation between dimensionless real contact area and dimensionless contact load of two cylindrical surfaces in different deformation stages with \(D=1.5\), \(G=10^{-10}\) mm, \(R_{\text{c1}}=100\) mm and \(R_{\text{c2}}=60\) mm. (**a**) The load–unload cycle in the elastic stage; (**b**) the load–unload cycle in the first elastoplastic stage; (**c**) the load–unload cycle in the second elastoplastic stage. Figures 5 and 6 show the load-unload cycle of two cylindrical surfaces in the second elastoplastic stage with different curvature radii \(R_{\text{c2}}\) and contact types. In each figure, (a) shows the unloading at the dimensional critical contact load, (b) shows the unloading at a dimensional given contact load, (c) shows the unloading at the dimensionless critical contact load and (d) shows the unloading at a given dimensionless contact load.