# Motion Characteristics of Collapse Body during the Process of Expanding a Rescue Channel
## 4 Analysis of the Movement Mechanism of the Collapse Body in the Expansion Process  
### Particle Mechanics Analysis  
Unexpanded collapsed accumulation bodies are in a quasi-static state [22], with high solid concentration and relatively stable structure, and their mechanical behavior and structural failure modes are similar to those of solid structures. However, due to the high discreteness of the collapse body, the collapse body above the expansion device will have displacements in different directions and sizes rather than form an overall unidirectional movement [23]. In view of the difference between the quasi-static state and the discrete state before and after the expansion of the collapse body, the mechanical state of the collapse body in the expansion process is analyzed in combination with the relevant theories of solid yielding and granular flow. When the lower part is expanded, the collapse particles are affected by their own gravity, vertical force, and lateral compressive stress. As shown in Figure 7a, the relative motion of the particles occurs as the expansion device rises, and the collapse body directly above the expansion device is lifted and compacted, while the collapse body at a higher position is loosened, and the horizontal compressive stress in the collapse body gradually decreases with the increase in the porosity between the particles. As shown in the lateral stress curve in Figure 10, when the expansion device is advanced to the position shown in Figure 7b, the relative slip of the collapse body occurs with the increase in the expansion height; the lateral compressive stress of the granular media is the minimum, and the displacement regions are divided into the direct and the indirect displacement regions. Figure 11 shows the stress state of the collapse body at the slip plane in the direct and the indirect displacement regions. Figure 10: Boundary stress curve. At this time, any point on the slip plane is:  
\[F_{h}=F_{g}\tan(\theta_{2}-\varphi) \tag{1}\]  
\[F_{g}=\frac{Y^{2}\rho g\tan(90^{\circ}-\theta_{2})}{2} \tag{2}\]  
where \(F_{h}\) is the lateral stress value in the collapse body, \(F_{g}\) is the gravity, \(\theta_{2}\) is the dip angle of the slip plane, \(\varphi\) is the internal friction angle of the granular particles, and \(\rho\) is the granular particle density. Let \(\tan\varphi=f\), and introducing Equation (2) into Equation (1), we obtain:  
\[F_{h}=\frac{Y^{2}\rho g\tan(90^{\circ}-\theta_{2})\tan(\theta_{2}-\varphi)}{2 }=\frac{Y^{2}\rho g(\tan\theta_{2}-f)}{2(1+f\tan\theta_{2})\tan\theta_{2}} \tag{3}\]  
\(\tan\theta_{2}\) is always substituted by \(x\); let \(dF_{h}/dx=0\) because the lateral stress value \(F_{h}\) of the collapse body at the slip plane is the minimum, and we obtain:  
\[\frac{x+fx^{2}-(x-f)(1+2 fx)}{(1+fx)^{2}x^{2}}=0 \tag{4}\]  
The calculation results are:  
\[x=\tan\theta_{2}=\tan\Bigl{(}45^{\circ}+\frac{\varphi}{2}\Bigr{)} \tag{5}\]  
The dip angle of slip plane 2 is:  
\[\theta_{2}=45^{\circ}+\frac{\varphi}{2} \tag{6}\]  
As shown by the red slip plane 1 in Figure 7b, the collapse bodies in the upper indirect displacement region and the lower stable region on both sides of the expansion device also show a relative slip phenomenon with the further expansion of the rescue channel. During the formation of the slip plane, the principal stress leading to the failure of the \"solid structure\" of the accumulation body is the thrust along the orthogonal direction of  
Figure 11: Force analysis of slip plane 2.  
the slip plane 2. Combined with the Mohr-Coulomb failure criterion in the solid structure and the envelope of the accumulation body, the angle between the slip plane 1 and the principal stress is \(45^{\circ}-\varphi/2\).