## 4 Characterization of the Joint Surface Roughness Coefficient
### Roughness Characterization Formula Considering the Mechanical Contribution  
The mechanical contributions of different asperity orders are different. There is a relationship between the mechanical contribution and the statistical parameters. Therefore, it was necessary to establish a formula representing the joint roughness that accurately considered the mechanical contributions of different asperity orders. The shear strength can be divided into three parts [48]: the basic frictional resistance, the resistance provided by the waviness, and the resistance provided by the unevenness, as shown in Equation (9). \[\tau=\tau_{Wa}+\tau_{Le}+\tau_{b} \tag{9}\]  
[PERSON] [56] divided the joint surface into waviness and unevenness by different interval methods and established a formula representing the joint surface roughness, considering the statistical parameters of waviness and unevenness (Equation (10)). \[JRC=\mathsf{a}\times\left(P^{1\mathsf{st}}\right)^{m}+b\times\left(P^{2 \mathsf{nd}}\right)^{n}+c \tag{10}\]  
Inserting Equations (11) and (12) into [PERSON]'s formula, it is rewritten as Equation (11). \[\tau_{p}=\tau_{p\_Wa+Ue}=\sigma_{n}\tan\biggl{[}\left(a\times\left(P^{1 \mathsf{st}}\right)^{m}+b\times\left(P^{2\mathsf{nd}}\right)^{n}+c\right) \cdot\log_{10}\biggl{(}\frac{JCS}{\sigma_{n}}\biggr{)}+\phi_{b}\biggr{]} \tag{11}\]  
By decomposing \(\mathsf{c}\) in Equation (13) into two parts, \(\mathsf{c}_{1}\) and \(\mathsf{c}_{2}\), Equation (13) is rewritten as Equation (12). \[\tau_{p}=\tau_{p\_Wa+Ue}=\sigma_{n}\tan\biggl{[}\left(a\times\left(P^{1 \mathsf{st}}\right)^{m}+c_{1}+b\times\left(P^{2\mathsf{nd}}\right)^{n}+c_{2} \right)\cdot\log_{10}\biggl{(}\frac{JCS}{\sigma_{n}}\biggr{)}+\phi_{b}\biggr{]} \tag{12}\]  
where \(a\times\left(P^{1\mathsf{st}}\right)^{m}+c_{1}\) represents the waviness's roughness, and \(b\times\left(P^{2\mathsf{nd}}\right)^{n}+c_{2}\) represents the unevenness's roughness. Therefore, Equation (14) is transformed into Equation (13). \[\tau_{p}=\tau_{p\_Wa+Ue}=\sigma_{n}\tan\biggl{[}\left(JRC^{1\mathsf{st}}+JRC^ {2\mathsf{nd}}\right)\cdot\log_{10}\biggl{(}\frac{JCS}{\sigma_{n}}\biggr{)}+ \phi_{b}\biggr{]} \tag{13}\]For joint specimens that contain only waviness, the shear strength equation is expressed as Equation (14). \[\tau_{p}=\tau_{p\_Wa}=\sigma_{n}\tan\biggl{[}JRC^{\rm 1 st}\cdot\log_{10} \biggl{(}\frac{JCS}{\sigma_{n}}\biggr{)}+\phi_{b}\biggr{]} \tag{14}\]  
In order to calculate \(JRC^{\rm 1 st}\) and \(JRC^{\rm 2 nd}\), Equations (13) and (14) are deduced backward, and the results are shown in Equations (15) and (16). \[JRC^{\rm 1 st}+JRC^{\rm 2 nd}=\biggl{(}\arctan\biggl{(}\frac{\tau_{p\_Wa+Le}}{ \sigma_{n}}\biggr{)}-\phi_{b}\biggr{)}\biggl{/}\log_{10}\biggl{(}\frac{JCS}{ \sigma_{n}}\biggr{)} \tag{15}\]  
\[JRC^{\rm 1 st}=\biggl{(}\arctan\biggl{(}\frac{\tau_{p\_Wa}}{\sigma_{n}}\biggr{)}- \phi_{b}\biggr{)}\biggl{/}\log_{10}\biggl{(}\frac{JCS}{\sigma_{n}}\biggr{)}=a \times\left(P^{\rm 1 st}\right)^{m}+c_{1} \tag{16}\]  
The second-order roughness coefficient \(JRC^{\rm 2 nd}\) is obtained by subtracting Equation (16) from Equation (15), as shown in Equation (17). \[JRC^{\rm 2 nd}=\biggl{(}\arctan\biggl{(}\frac{\tau_{p\_Wa+Le}}{\sigma_{n}} \biggr{)}-\arctan\biggl{(}\frac{\tau_{p\_Wa}}{\sigma_{n}}\biggr{)}\biggr{)} \biggl{/}\log_{10}\biggl{(}\frac{JCS}{\sigma_{n}}\biggr{)}=b\times\left(P^{\rm 2 nd }\right)^{n}+c_{2} \tag{17}\]  
Equations (16) and (17) are fitted according to the statistical parameters given in Figure 12; the fitting results are shown in Figure 18.  
According to the fitting result, the formula considering the mechanical contribution of the waviness and unevenness was established, as shown in Equations (18) and (19).