## 4 Characterization of the Joint Surface Roughness Coefficient  
### Relationship between the Mechanical Contribution and Statistical Parameter  
#### 4.1.1 Statistical Parameters of Waviness and Unevenness  
In order to study the relationship between the asperity's mechanical contribution and its statistical parameters, the parameters \(Z_{2}\) and \(R_{\text{P}}\)-1 of the standard JRC profile and the waviness component (1 st order) and unevenness component (2 nd order) were calculated, and the results are shown in Figure 14.  
The statistical parameters (\(Z_{2}\) and \(R_{\text{P}}\)-1) increased with the increase in roughness (as shown in Figure 14). There were some profile's statistical parameters that did not obey this law, however. As shown in Figure 14a, the statistical parameter \(Z_{2}^{\text{2 nd}}\) showed a decreasing trend from profile \(4^{\#}\) to \(5^{\#}\), while the statistical parameter \(Z_{2}^{\text{1 st}}\) showed an increasing trend at this position; the statistical parameter \(Z_{2}^{\text{2 nd}}\) showed an increasing trend from profile \(8^{\#}\) to \(9^{\#}\), while the statistical parameter showed a decreasing trend at this position (as shown in Figure 14a). According to the analysis, compared with the \(5^{\#}\) profile, the \(4^{\#}\) profile  
Figure 14: Statistical parameters of the standard JRC profile at critical decomposition levels (**a**) Statistical parameter \(Z_{2}\), (**b**) Statistical parameter \(R_{\text{P}}\)-1.  
Figure 13: (**a**) Mechanical contribution of waviness, (**b**) Mechanical contribution of unevenness.  
contained more unevenness and less waviness. Compared with the 9\({}^{\#}\) profile, the 8\({}^{\#}\) profile contained more unevenness and waviness. [PERSON] [52] also found this trend.  
In order to analyze the mechanical contribution difference between the waviness component and the unevenness component, \(K_{W-U}\) was defined as the parameter representing the mechanical contribution ratio of the waviness and the unevenness. The value of \(K_{W-U}\) was calculated and is shown in Figure 15. \(K_{W-U}\) barely changed with the normal stress increase.  
In order to further establish the relationship between the mechanical contribution and the statistical parameters, the ratio of the statistical parameters of waviness to unevenness was calculated, and the calculated results are shown in Figure 16. Comparing Figures 15 and 16, we found that the mechanical contribution ratio of the waviness and unevenness and the statistical parameter ratio of the waviness and unevenness showed an opposite trend. KW-U and the statistical parameter ratio of the waviness and unevenness showed different change laws. The larger the KW-U, the smaller the statistical parameter ratio of the waviness and unevenness. In Figure 15, \(K_{W\text{-U-9.5}}>K_{W\text{-U-12.5}}>K_{W\text{-U-10.8}}>K_{W\text{-U-16.7}}>K_ {W\text{-U-18.7}}\); at the same time, \(({Z_{2}}^{\text{1 st}}/{Z_{2}}^{\text{2 nd}})_{9.5}<({Z_{2}}^{\text{1 st}}/{Z_{2}} ^{\text{2 nd}})_{12.5}<({Z_{2}}^{\text{1 st}}/{Z_{2}}^{\text{2 nd}})_{10.8}<({Z_{ 2}}^{\text{1 st}}/{Z_{2}}^{\text{2 nd}})_{16.7}<({Z_{2}}^{\text{1 st}}/{Z_{2}}^{ \text{2 nd}})_{18.7}\). [PERSON] [78] also found this trend in his research.  
Figure 16: Statistical parameter ratio of waviness and unevenness.  
Figure 15: Mechanical contribution ratio of waviness and unevenness.