# Mechanism of Stratum Instability and Dynamic Deformation under Discontinuous Boundary Conditions
## 2 Characteristics of Rock Movement in the Fault Zone
### Fault Mechanics Model  
A micro-element fault plane mechanical model based on the Anderson fracture mechanism has been established for the mechanical instability model of fault plane slip [27]. The micro-element mechanical model of the fault is shown in Figure 2. Take a micro-element in a two-dimensional fault section with the fault line as the diagonal; the fault plane where the fault line is located is the weak structural plane inside the micro-element. The normal stress on this weak structural plane (\(\sigma_{n}\)) and shear stress (\(\tau_{n}\)) can be expressed as follows:  
\[\left\{ \begin{array}{l}\sigma_{n}=\frac{1}{2}(\sigma_{h}+\sigma_{v})-\frac{ 1}{2}(\sigma_{h}-\sigma_{v})\cos 2\alpha\\ \tau_{n}=-\frac{1}{2}(\sigma_{h}-\sigma_{v})\sin 2\alpha\end{array} \right. \tag{1}\]  
where \(\alpha\) is the fault dip angle, in \({}^{\circ}\); \(\sigma_{h}\) is the horizontal stress received by the micro-element, in MPa; and \(\sigma_{v}\) is the vertical stress received by the micro-element, in MPa. If the micro-element in the fault plane slips along the weak structural plane, the shear stress on the weak structural plane should be greater than the sum of the maximum static friction force generated by the normal stress on the structural plane and the cohesion of the structural plane itself. According to the Mohr-Coulomb strength theory, the ultimate shear stress of the weak structural plane in the fault plane micro-element is as follows:  
\[\tau_{f}=\sigma_{n}\tan\varphi_{f}+c_{f} \tag{2}\]  
where \(\tau_{f}\) is the limit shear stress under the normal stress of the fault plane, in MPa; \(\tau_{n}\) is the internal friction angle of the fault plane, in \({}^{\circ}\); and \(c_{f}\) is the fault plane cohesion, in MPa. Meanwhile, a macroscopic mechanical instability model of the fault was obtained; the mechanical model of the fault is shown in Figure 2. The support stress \(F_{N}\) and horizontal stress \(T\) on the fault plane are decomposed along the fault:  
\[\left\{ \begin{array}{l}F_{x}=T\sin\alpha-F_{N}\cos\alpha\\ F_{y}=T\cos\alpha+F_{N}\sin\alpha\end{array} \right. \tag{3}\]  
where \(F_{N}\) is the support force of the football of the fault to the fault plane, in kN; \(t\) is the horizontal extrusion force of the hanging wall of the fault on the fault surface, in kN; \(\alpha\) is the fault dip angle, in \({}^{\circ}\); \(F_{x}\) is the force in the \(x\) direction of the fault plane, in N; and \(F_{y}\) is the force in the y direction at the fault plane, in N.  
The instability criterion of fault slip is as follows:  
\[F_{x}\tan\varphi=F_{y} \tag{4}\]  
where \(\varphi\) is the internal friction angle of the rock mass in the fault zone, in \({}^{\circ}\). In the process of coal mining, due to the existence of goaf and the movement of overlying strata, the abovementioned fault slip instability model can no longer meet and accurately describe the failure instability mode of stope faults. The difference in the layered structure and lithology of the overlying strata in goaf determines the diversity of the vertical  
Figure 2: Fault micro-element and macro-mechanical model. stratum movement mode of overlying strata in the stope. Therefore, the mechanisms and mechanical models of fault instability in different stratum movement models are different. Aiming at the stratum movement model of the stope without the surface subsidence area, considering that the movement state of the key stratum plays a leading role in the overlying stratum activity of the stope, the fault slip model and mechanical instability criterion after the key stratum is broken are given [28]. Based on the fault mechanics model, the model takes the key layer as the research object and considers the horizontal force \(T\) and vertical force \(R\)[30] between the broken rock blocks in the structure of the'masonry beam' after the key layer is broken. At the same time, the advanced abutment pressure \(F_{1}(x)\) and the original rock abutment pressure function \(F_{2}(x)\)[31] are introduced.