## 4 Shear Instability Analysis
### Shear Instability Model and Criterion  
Based on the shear instability mechanism of rock joints, the shear instability model of rock joints, as shown in Figure 12, is established. The model consists of a spring and a wedge in series. The spring characterizes the elastic deformation accumulated in the rockmass on both sides of the joint shear zone, and its characteristic parameter is the initial shear modulus (_E_); the wedge represents the weakening behavior of the joint shear zone, and the characteristic parameter is the post-peak soften modulus or shear instability parameter (_S_\({}_{p}\)). The spring compresses and accumulates elastic energy with the increase in shear displacement, and the wedge slips after reaching the peak shear strength. When the wedge is weakened, the spring is unloaded, and the elastic deformation is transformed into plastic deformation. Therefore, the characteristic parameters of the spring and the wedge together determine the occurrence of the shear instability of rock joints. The mechanical model of the shear instability of rock joints reflects the physical essence of shear instability, that is, the elastic rebound and energy release of rockmass on both sides of the joint shear zone. At present, there are few mechanical models that consider the influence of the elastic rebound of rockmass on the shear instability of rock joints. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**No. ** & **CNL/kN** & _d_/**mm** & _S\({}_{p}\)/MPa/mm** & _S\({}_{p}^{\prime}\)/MPa/mm** & _E_/MPa/mm** & **Experimental** & **Theoretical** & **Shear Instability** \\  & & & & & & & & & \\ C5 & 25 & 0.75 & 7.65 & 9.11 & 7.5 & \(E\) \(<\)_S\({}_{p}\)_ & \(E\) \(<\)_S\({}_{p}^{\prime}\)_ & Yes \\ D5 & 25 & 1.5 & 6.36 & 6.64 & 7.1 & \(E\)  
The necessary condition for the shear instability of rock joints is that the elastic rebound on both sides of the joint shear zone is greater than the weakening deformation of the joint shear zone [27; 28]. In this respect, the shear failure of rock joints can be accomplished only by the elastic rebound and energy release of the specimen itself. Therefore, the necessary conditions for the shear instability of rock joints (i.e., the shear instability criterion) are shown as follow:  
\[E<S_{p} \tag{2}\]  
Table 4 lists the values of _CNL_ and \(d\), which are influencing factors of shear instability, and the shear instability parameter \(S_{p}\) and elastic modules \(E\), respectively. According to the shear instability criterion of Equation (2), it can be determined that B4, C4, B5, C5, B6, and C6 occur shear instability by comparing the experimental values of \(S_{p}\) and \(E\). According to the experimental results from Figure 11, the relationship between the shear instability parameter (\(S_{p}\)) and its influencing factors _CNL_ (\(N\)) and grain size (\(d\)) can be obtained by fitting, as shown by the empirical formula of Equation (3) and its 3D surface in Figure 13. The empirical relationship (Equation (3)) between the shear instability parameter (\(S_{p}\)) and its influencing factors _CNL_ (\(N\)) and grain size (\(d\)) shows a moderate fit (e.g., \(R^{2}=0.8115\)). This may be due to the following factors: (1) there exists a slight dislocation between the upper and lower joint surface during the encapsulation process of the joint specimen; (2) the local damage of the joint surface is caused by the Brazilian-splitting method in the process of making joints; and (3) the discreteness of the specimens may affect the fitting result. In this respect, in view of the three potential factors affecting the test results, the repeated test is used to reduce the effect. \[S_{p}^{\prime}=-0.0085N^{2}+0.5155N-5.0183d^{2}+8.0014d-1.8621\,\left(R^{2}=0.8 115\right) \tag{3}\]  
Figure 12: Mechanical model of shear instability of rock joints. Figure 13: Three-dimensional surface for shear instability parameter and its influencing factors.