## 3 The Mechanical Contribution of Waviness and Unevenness
### Contribution of Waviness and Unvevereness to the Shear Strength
#### 3.2.2 Contribution of Waviness and Unevenness to the Shear Strength  
In order to quantify the mechanical contribution of waviness and uneveness, the parameter \(K_{\mathit{s}}\)_ asperity was defined as a coefficient to represent the mechanical contribution of the asperities to the joint's shear strength (Equation (5)).  
\[K_{\mathit{s}}\mathit{s}_{\mathit{s}}=\frac{\tau_{p\_Wa+\mathit{Ue}}-\tau_{ p\_Flat}}{\tau_{p\_Wa+\mathit{Ue}}} \tag{5}\]  
where \(\tau_{p\_Wa+\mathit{Ue}}\) and \(\tau_{p\_Flat}\) are equal to the shear strengths of the joint specimen with two-order asperity and the specimen with the flat surface, respectively.  
The values of \(K_{\mathit{s}}\mathit{s}_{\mathit{s}}\) under different normal stresses with different roughness were calculated and are plotted in Figure 12. The parameter \(K_{\mathit{s}}\mathit{s}_{\mathit{s}}\) values gradually decreased with increasing normal stress (as shown in Figure 12), which indicated that the mechanical contribution of asperity to the joint specimen's strength decreased with the normal stress increase. Under the normal stress, the parameter \(K_{\mathit{s}}\mathit{s}_{\mathit{s}}\) increased with the roughness increase. The above results showed that there was a relationship between the contribution degree \(K_{\mathit{s}}\mathit{s}_{\mathit{s}}\) and the roughness coefficient of the joint surface.  
In order to further investigate the contribution of waviness and uneveness to the joint's shear strength, \(K_{\mathit{s}}\mathit{s}_{\mathit{s}}\mathit{s}_{\mathit{s}}\) and \(K_{\mathit{s}}\mathit{s}_{\mathit{s}}\mathit{s}_{\mathit{s}}\) were defined as coefficients representing the mechanical contribution of waviness and uneveness to the joint's shear strength (Equations (7) and (8)), respectively.  
\[\tau_{p\_Ue}=\tau_{p\_Wa+\mathit{Ue}}-\tau_{p\_Wa} \tag{6}\]  
\[K_{\mathit{s}}\mathit{s}_{\mathit{s}}\mathit{s}_{\mathit{s}}=\frac{\tau_{p\_Wa }-\tau_{p\_Flat}}{\tau_{p\_Wa+\mathit{Ue}}-\tau_{p\_Flat}}\times K_{\mathit{s} }\mathit{s}_{\mathit{s}}\mathit{s}_{\mathit{s}} \tag{7}\]  
\[K_{\mathit{s}}\mathit{s}_{\mathit{s}}\mathit{s}_{\mathit{s}}\mathit{e}=\frac{ \tau_{p\_Le}-\tau_{p\_Flat}}{\tau_{p\_Wa+\mathit{Ue}}-\tau_{p\_Flat}}\times K_ {\mathit{s}}\mathit{s}_{\mathit{s}}\mathit{s}_{\mathit{s}} \tag{8}\]  
where \(\tau_{p\_Le}\) equals the shear strength of the joint specimen containing only waviness.  
Figure 12: Mechanical contribution of asperity to the shear strength of the joint specimen.  
The values of \(K_{\textit{asperity\_Wa}}\) and \(K_{\textit{asperity\_Le}}\) under different normal stresses with different roughness were calculated and are shown in Figure 13a,b, respectively. The values of \(K_{\textit{asperity\_Wa}}\) and \(K_{\textit{asperity\_Le}}\) decreased with the increase in normal stress. The values of \(K_{\textit{asperity\_Wa}}\) and \(K_{\textit{asperity\_Le}}\) increased with the increase in joint roughness, as shown in Figure 13a,b. This result was different from [PERSON]'s research [64], which showed that the mechanical contribution of the unevenness was independent of the statistical parameters of the unevenness. Thus, it was necessary to carry out further research on the correlation between the mechanical contribution and the statistical parameter.