## 2 Multi Rigid Block Failure Mechanism
### Establishment of the Permissible Velocity Field
\[v_{i}=\frac{\sin\left(\frac{\pi}{2}\!+\!\varphi_{1}-\theta_{i-1}^{v}\!+\! \theta\!+\!\sum_{i=1}^{i-1}\alpha_{i}\right)}{\sin\left(\frac{\pi}{2}-\varphi _{1}\!+\!\theta_{i}^{v}-\theta-\sum_{i=1}^{i-1}\alpha_{i}\right)}v_{i-1} \tag{12}\]  
\[v_{i-1,i}=\frac{\sin\left(\theta_{i-1}^{v}-\theta_{i}^{v}\right)}{\sin\left( \frac{\pi}{2}-\varphi_{1}\!+\!\theta_{i}^{v}-\theta-\sum_{i=1}^According to the relationship of permissible velocity between rigid block \(i-1^{u}\) and \(i^{d}\), the direction angle of the velocity vector of rigid block \(i^{d}\) can be obtained as Equation (15).  
\[\theta_{i^{d}}^{v}=\frac{\pi}{2}-\beta_{i^{d}}+\varphi_{2}+\theta+\sum_{i=1}^{i -1}\alpha_{i}^{u} \tag{15}\]  
As shown in Figure 7, there are two cases of permissible velocity relationship between rigid block \(i-1^{u}\) and \(i^{d}\):  
According to the compatible velocity vector triangle in two cases:  
Case: III-\(i^{d}\) rigid block \(-1\) (Figure 7a)  
For Figure 7a, the relative velocity \(v_{i-1^{u},i^{d}}\) between the rigid block \(i-1^{u}\) and \(i^{d}\) face upwards along the velocity discontinuity and forms an angle \(\varphi_{2}\). \[v_{i^{d}}=\frac{\sin(\frac{\pi}{2}+\varphi_{2}+\theta_{i-1^{u}}^{v}-\theta- \sum_{i=1}^{i-1}\alpha_{i}^{u})}{\sin(\frac{\pi}{2}-\varphi_{2}-\theta_{i^{d}} ^{v}+\theta+\sum_{i=1}^{i-1}\alpha_{i}^{u})}v_{i-1^{u}} \tag{16}\]  
\[v_{i-1^{u},i^{d}}=\frac{\sin(\theta_{i^{d}}^{v}-\theta_{i-1^{u}}^{v})}{\sin( \frac{\pi}{2}-\varphi_{2}-\theta_{i^{d}}^{v}+\theta+\sum_{i=1}^{i-1}\alpha_{i} ^{u})}v_{i-1^{u}} \tag{17}\]  
Requirements of Equations (16) and (17):  
\[ \begin{array}{c}\theta_{i^{d}}^{v}>\theta_{i-1^{u}}^{v_{1}u}\\ -\frac{\pi}{2}<\varphi_{2}+\theta_{i-1^{u}}^{v}-\theta-\sum_{i=1}^{i-1}\alpha_ {i}^{u}<\frac{\pi}{2}\\ -\frac{\pi}{2}<-\varphi_{2}-\theta_{i-1^{u}}^{v}+\theta+\sum_{i=1}^{i-1}\alpha _{i}^{u}<\frac{\pi}{2}\end{array}  \tag{18}\]  
Case: III-\(i^{d}\) rigid block \(-2\) (Figure 7b)  
The relative velocity \(v_{i-1^{u},i^{d}}\) between the rigid block \(i-1^{u}\) and \(i^{d}\) in Figure 7b face downward along the velocity discontinuity and forms an angle \(\varphi_{2}\). \[v_{i^{d}}=\frac{\sin(\frac{\pi}{2}+\varphi_{2}-\theta_{i-1^{u}}^{v}+\theta+ \sum_{i=1}^{i-1}\alpha_{i}^{u})}{\sin(\frac{\pi}{2}-\varphi_{2}+\theta_{i^{d}} ^{v}-\theta-\sum_{i=1}^{i-1}\alpha_{i}^{u})}v_{i-1^{u}} \tag{19}\]  
Figure 7: Two different cases of permissible velocity fields between rigid block \(i-1^{u}\) and \(i^{d}\): (**a**) The relative velocity \(v_{i-1^{u},i^{d}}\) between the rigid block \(i-1^{u}\) and \(i^{d}\) face upwards along the velocity discontinuity; (**b**) The relative velocity \(v_{i-1^{u},i^{d}}\) between the rigid block \(i-1^{u}\) and \(i^{d}\) face downward along the velocity discontinuity. \[v_{i-1^{u},i^{u}}=\frac{\sin(\theta_{i-1^{u}}^{v}-\theta_{i^{d}}^{v})}{\sin(\frac{ \pi}{2}-\varphi_{2}+\theta_{i^{d}}^{v}-\theta-\sum_{i=1}^{i-1}\alpha_{i}^{u})}v_ {i-1^{u}} \tag{20}\]  
Requirements of Equations (19) and (20)  
\[ \begin{array}{c}\theta_{i^{d}}^{v}<\theta_{i-1^{u}}^{v}\\ -\frac{\pi}{2}<\varphi_{2}-\theta_{i-1^{u}}^{v}+\theta+\sum\limits_{i=1}^{i-1} \alpha_{i}^{u}<\frac{\pi}{2}\\ -\frac{\pi}{2}<-\varphi_{2}+\theta_{i^{d}}^{v}-\theta-\sum\limits_{i=1}^{i} \alpha_{i}^{u}<\frac{\pi}{2}\\ \end{array}  \tag{21}\]  
Since the direction angle \(\theta_{i^{d}}^{v}\) of the rigid block is known, it is possible to calculate the absolute and relative velocity vectors of the rigid block \(i^{d}\) directly for the identified multi-rigid block failure mechanism. Type III-\(i^{u}\):  
For rigid block \(i^{u}\), since the direction angle \(\theta_{i^{u}}^{v}\) of its absolute velocity vector cannot be determined directly, it is necessary to analyze the compatible velocity relationship between rigid block \(i-1^{u}\) and \(i^{u}\), rigid block \(i^{d}\) and \(i^{u}\), and perform simultaneous solution.