# Mechanism of Stratum Instability and Dynamic Deformation under Discontinuous Boundary Conditions
## 4 Existence Analysis of Separation Space
### Mechanical Determination of Critical Layer Slip Instability  
The stable compression module, deviatoric stress transition module and deviatoric stress instability module defined in Figure 6 were divided into rock strata and analyzed using a mechanical model. Combined with the three-zone division of the stop, the key stratum theory and the basic lithology of the rock strata, the rock strata in the three module areas are divided. The stable compression module is basically composed of the caving-zone rock layer and the fracture rock layer. The deviatoric stress transition module can be the fracture zone or the bending subsidence zone. The deviatoric stress instability module is composed of the key layer and its weak upper rock layer. The above theoretical division can only infer the approximate positions of the three modules and cannot accurately determine them. Therefore, it is necessary to use the mechanical model to determine the positions of the modules. The slip instability model of the key-layer support module is shown in Figure 7. The model can be used to analyze the slip instability of the fault boundary module rock working face when the fault dip angle is consistent with the direction of the rock movement angle and the fault dip angle \(\alpha\) is less than the rock movement angle \(\beta\). Figure 5: Diagram of fault separation space. Figure 6: Non-boundary support module partition. \(p_{2}\) is the maximum tensile stress at the fracture limit of the rock stratum. When the rock stratum reaches the ultimate tensile strength and the rock mass does not deflect and displace, the rock stratum in this area breaks, and the fault surface does not produce separation space. On the contrary, the rock layer and the fault surface produce separation space. The horizontal force balance condition of the rock stratum is as follows: \(f_{1}+f_{2}=F\)  
In the Equation:  
\[\text{P}_{3}u(x_{2})=\text{P}_{1}u(x_{1})+G+\text{P}_{2}l\cos\beta \tag{15}\]  
\[\text{P}_{2}=\text{p}\sin\beta \tag{16}\]  
\[l=\frac{h}{\sin\beta} \tag{17}\]  
\[F(x)=f_{1}+f_{2} \tag{18}\]  
From the above equation:  
\[F(x)=f_{1}+f_{2}=\mu p_{1}u(x_{1})+\mu p_{1}u(x_{1})+G+\frac{p_{2}h}{\tan\beta} =F(x_{1}) \tag{19}\]  
By combining Equations (13) and (19):  
\[F(x_{1})=\left\{ \begin{array}{l}2\mu p_{1}\Big{(}1-\frac{K_{1}}{\tan\alpha} \Big{)}x_{1}+\mu\Big{(}\frac{2p_{1}K_{1}x_{0}}{\tan\alpha}+G+\frac{p_{2}h}{ \tan\beta}\Big{)}\Big{(}x_{0}\leq x_{1}\leq x_{0}+\frac{h_{1}}{K_{1}}\Big{)}\\ 2\mu p_{1}\Big{(}1-\frac{K_{2}}{\tan\alpha}\Big{)}x_{1}+\mu\Big{(}\frac{2p_{1} K_{2}x_{0}}{\tan\alpha}+G+\frac{p_{2}h}{\tan\beta}\Big{)}\Big{(}x_{0}+\frac{h_{1}}{K _{1}}\leq x_{1}\leq x_{0}+\frac{h_{1}+h_{2}}{K_{2}}\Big{)}\\ \ldots\\ 2\mu p_{1}\Big{(}1-\frac{K_{n}}{\tan\alpha}\Big{)}x_{1}+\mu\Big{(}\frac{2p_{1} K_{n}x_{0}}{\tan\alpha}+G+\frac{p_{2}h}{\tan\beta}\Big{)}\Big{(}x_{0}+\frac{h_{1}+h_{2}+ \cdots+h_{n-1}}{K_{n-1}}\leq x_{1}\leq x_{0}+\frac{h_{1}+h_{2}+\cdots+h_{n}}{K_ {n}}\Big{)}\\ \end{array} \right. \tag{20}\]  
where \(\mu\) is the friction coefficient of rock strata; \(h\) is the thickness of rock, in m; \(p_{1}\) and \(p_{3}\) are uniformly distributed loads on and below the end rock mass, in kN/m\({}^{2}\); \(p\) is the tensile strength of the rock stratum, in MPa; \(\beta\) is the rock movement angle, in \({}^{\circ}\); and \(u(x_{1})\) and \(u(x_{2})\) are the horizontal distance between the fault line and the rock movement line, in m.  
When the fault dip angle is less than the rock movement angle, \(2\mu p_{1}\Big{(}1-\frac{K_{n}}{\tan\alpha}\Big{)}\leq 2\mu p_{1}\Big{(}1- \frac{\tan\beta_{\text{min}}}{\tan\alpha}\Big{)}<0\). Therefore, as \(x_{1}\) increases, \(F(x_{1})\) decreases gradually. When \(F(x)=F(x_{1})<F=p_{2}h,u(x)\) corresponds to the horizontal displacement of the end of the rock layer, which forms a separation space with the fault plane.