## 4 Comparative Analysis of the Boundary Spatial Morphology of the TLEZ  
### Analytical Characterization of the Boundary of the TLEZ  
In SDCS, the front abutment pressure is the projection of each principal stress in the normal direction of the coal seam plane. Drawing on the analysis of elastic mechanics, the stress state of an arbitrary point within an object can be analogized to a small tetrahedron. In this tetrahedron, the other three surfaces are, respectively, parallel to the three coordinate planes, as shown in Figure 16. As the tetrahedron WABC approaches W infinitely closely, the stress of the tetrahedron evolves into the stress on the inclined plane. Let the outer normal of triangle ABC be \(N\), and the cosines of the normal direction are  
\[\left\{ \begin{array}{l}\cos(N,x)=l\\ \cos(N,y)=m\\ \cos(N,z)=n\end{array} \right. \tag{2}\]  
Suppose the volume of the tetrahedron WABC is \(\Delta\)V, and the area of triangle ABC is \(\Delta\)S. Then the areas of the other three faces are \(l\Delta\)S, \(m\Delta\)S, \(n\Delta\)S, respectively, which can be obtained according to the tetrahedron equilibrium condition:  
\[\left\{ \begin{array}{l}W_{x}=l\sigma_{x}+m\tau_{yx}+n\tau_{xx}\\ W_{y}=m\sigma_{y}+n\tau_{xy}+l\tau_{xy}\\ W_{z}=n\sigma_{z}+l\tau_{xz}+n\tau_{yz}\end{array} \right. \tag{3}\]  
Therefore, the normal stress \(\sigma_{N}\) on the triangle ABC is  
\[\sigma_{N}=l^{2}\sigma_{x}+m^{2}\sigma_{y}+n^{2}\sigma_{z}+2 mn\tau_{yz}+2 nl \tau_{xx}+2 lm\tau_{xy} \tag{4}\]  
In the fully mechanized top-coal caving slope with SDCS, as shown in Figure 4, the coordinate system is consistent with the three principal stress directions, and the dip angle  
Figure 16: Stress component of a point in the tetrahedron. of the coal seam is \(\alpha\). Assuming that the rock mass is an isotropic elastic body, because the unit body is constrained by the displacement of the adjacent rock mass in the horizontal direction, it is impossible to produce lateral deformation. According to the generalized [PERSON]'s law, we know that  
\[\sigma_{x}=\sigma_{y}=A\sigma_{z} \tag{5}\]  
Therefore, the front abutment pressure \(\sigma_{m}\) in front of the working face is  
\[\sigma_{m}=\cos^{2}\alpha\sigma_{1}+\sin^{2}\alpha A\sigma_{1} \tag{6}\]  
where \(\alpha\) is the dip angle of coal seam, \(\sigma_{1}\) is the maximum principal stress, and \(A\) is the pressure measuring coefficient. According to the relationship shown in Figure 17 in the TLEZ, the limit equilibrium equation established with the resultant force in the horizontal direction as zero is as follows [26]:  
\[h(\sigma_{3}+d\sigma_{3})-h\sigma_{3}-(f_{1}+f_{2})\sigma_{m}dx=0 \tag{7}\]  
where \(h\) is the top-coal thickness,\(f_{1}\) is the friction coefficient between the coal seams with an empirical value of 0.2,\(f_{2}\) is the friction coefficient between the coal seams and rock strata with an empirical value of 0.3, \(\sigma_{3}\) is the minimum principal stress of the top coal, and \(\sigma_{m}\) is the front abutment pressure of the top coal. When the top coal is in the limit equilibrium state, the coal obeys the Mohr-Coulomb strength criterion, and the following can be obtained:  
\[\sigma_{1}=\frac{2c\cos\varphi}{1-\sin\varphi}+\frac{1+\sin\varphi}{1-\sin \varphi}\sigma_{3} \tag{8}\]  
\[\frac{d\sigma_{1}}{d\sigma_{3}}=\frac{1+\sin\varphi}{1-\sin\varphi} \tag{9}\]  
where \(\varphi\) is the internal friction angle of the coal. Through the solution of Equations (7) and (9), the following can be obtained:  
\[\ln\sigma_{1}=\frac{(f_{1}+f_{2})\big{(}\cos^{2}\alpha+A\sin^{2}\alpha\big{)} x}{h}\bigg{(}\frac{1+\sin\varphi}{1-\sin\varphi}\bigg{)}+C \tag{10}\]  
Substituting \(x=0\) and \(\sigma_{m}=N\) into Equation (10) yields  
\[C=\ln\frac{N}{\cos^{2}\alpha+A\sin^{2}\alpha} \tag{11}\]  
Figure 17: Stress state of the TLEZ. where \(N\) is the supporting capacity of the wall.