## 2 Multi Rigid Block Failure Mechanism
### Establishment of the Permissible Velocity Field
(1) Type III-\(i^{u}\) rigid block and \(i-1^{u}\) rigid block:  
The compatible velocity relationship of rigid block \(i^{u}\) and \(i-1^{u}\), as shown in Figure 8:  
Case: 1 (Figure 8a)  
For Figure 8a, the relative velocity \(v_{i-1^{u},i^{u}}\) between the rigid block \(i-1^{u}\) and \(i^{u}\) face upwards along the velocity discontinuity and forms an angle \(\varphi_{1}\)  
\[v_{i^{u}}=\frac{\sin(\frac{\pi}{2}+\varphi_{1}+\theta_{i-1^{u}}^{v}-\theta- \sum_{i=1}^{i-1}\alpha_{i}^{u})}{\sin(\frac{\pi}{2}-\varphi_{1}-\theta_{i^{u} }^{v}+\theta+\sum_{i=1}^{i-1}\alpha_{i}^{u})}v_{i-1^{u}} \tag{22}\]  
\[v_{i-1^{u},i^{u}}=\frac{\sin(\theta_{i^{u}}^{v}-\theta_{i-1^{u}}^{v})}{\sin( \frac{\pi}{2}-\varphi_{1}-\theta_{i^{u}}^{v}+\theta+\sum_{i=1}^{i-1}\alpha_{i }^{u})}v_{i-1^{u}} \tag{23}\]  
Figure 8: Possible compatible velocity relationship between rigid block \(i-1^{u}\) and \(i^{u}\): (**a**) The relative velocity \(v_{i-1^{u},i^{u}}\) between the rigid block \(i-1^{u}\) and \(i^{u}\) face upwards along the velocity discontinuity; (**b**) The relative velocity \(v_{i-1^{u},i^{u}}\) between the rigid block in Figure 8b face downward along the velocity discontinuity. Requirements of Equations (22) and (23):  
\[ \begin{array}{c}\theta_{\vec{u}}^{v}>\theta_{i-1^{u}}^{v}\\ -\frac{\pi}{2}<\varphi_{1}+\theta_{i-1^{u}}^{v}-\theta-\sum\limits_{i=1}^{i-1} \alpha_{i}^{u}<\frac{\pi}{2}\\ -\frac{\pi}{2}<-\varphi_{1}-\theta_{i^{u}}^{v}+\theta+\sum\limits_{i=1}^{i-1} \alpha_{i}^{u}<\frac{\pi}{2}\end{array} \]  
Case: 2 (Figure 8b)  
The relative velocity \(v_{i-1^{u},\vec{u}}\) between the rigid block in Figure 8b face downward along the velocity discontinuity and forms an angle \(\varphi_{1}\). \[v_{\vec{u}}=\frac{\sin(\frac{\pi}{2}+\varphi_{1}-\theta_{i-1^{u}}^{v}+\theta+ \sum_{1}^{i-1}\alpha_{i}^{u})}{\sin(\frac{\pi}{2}-\varphi_{1}+\theta_{i^{u}}^{ v}-\theta-\sum_{1}^{i-1}\alpha_{i}^{u})}v_{i-1^{u}} \tag{24}\]  
\[v_{i-1^{u},\vec{u}}=\frac{\sin(\theta_{i-1^{u}}^{v}-\theta_{i^{u}}^{v})}{\sin (\frac{\pi}{2}-\varphi_{1}+\theta_{i^{u}}^{v}-\theta-\sum_{1}^{i-1}\alpha_{i }^{u})}v_{i-1^{u}} \tag{25}\]  
Requirements of Equations (24) and (25):  
\[ \begin{array}{c}\theta_{\vec{u}}^{v}<\theta_{i-1^{u}}^{v}\\ -\frac{\pi}{2}<\varphi_{1}-\theta_{i-1^{u}}^{v}+\theta+\sum\limits_{i=1}^{i-1} \alpha_{i}^{u}<\frac{\pi}{2}\\ -\frac{\pi}{2}<-\varphi_{1}+\theta_{i^{u}}^{v}-\theta-\sum\limits_{1}^{i-1} \alpha_{i}^{u}<\frac{\pi}{2}\end{array}  \tag{26}\]  
(2)  
Type III-\(\vec{u}\) and \(\vec{u}\):  
The possibility of two compatible velocity relationships between rigid block \(\vec{u}\) and \(\vec{u}\) is shown in Figure 9:  
Case: 3 (Figure 9a)  
Figure 9: Two possible compatible velocity relationships between rigid block \(\vec{u}\) and \(\vec{u}\): **(a)** The relative velocity \(\vec{v}_{\vec{u},\vec{u}}\) between the rigid block \(\vec{u}\) and \(\vec{u}\) face upwards along the velocity discontinuity; **(b)** The relative velocity \(\vec{v}_{\vec{u},\vec{u}}\) between the rigid block in Figure 9b faces downward along the velocity discontinuity. For Figure 9a, the relative velocity \(v_{\mu^{i},\mu^{u}}\) between the rigid block \(i^{d}\) and \(i^{u}\) face upwards along the velocity discontinuity and forms an angle \(\varphi_{2}\). \[v_{\mu^{i}}=\frac{\sin(-\frac{\pi}{2}+\varphi_{2}+\beta_{i}^{u}+\theta_{\mu}^{ \varphi}-\theta-\sum_{1}^{i-1}\alpha_{i}^{u})}{\sin(\frac{3\pi}{2}-\varphi_{2} -\beta_{i}^{u}-\theta_{i^{u}}^{v}+\theta+\sum_{1}^{i-1}\alpha_{i}^{u})}v_{\mu ^{d}} \tag{27}\]  
\[v_{\mu^{i},\mu^{u}}=\frac{\sin(\theta_{\mu}^{\varphi}-\theta_{\mu}^{\varphi})} {\sin(\frac{3\pi}{2}-\varphi_{2}-\beta_{i}^{u}-\theta_{i^{u}}^{v}+\theta+ \sum_{1}^{i-1}\alpha_{i}^{u})}v_{\mu^{d}} \tag{28}\]  
Requirements of Equations (27) and (28):  
\[