# An FDM-Based Dynamic Zoning Method for Disturbed Rock Masses above a Longwall Mining Panel
## 2 The Dynamic Zoning Method
### Adaptive Adjustment of Mechanical Parameters of Disturbed Rock Masses  
In the process of underground mining, the deformation and failure characteristics of the overlying rock masses dynamically vary, and the mechanical properties correspondingly change. With the mining of the coal seam, at first, the overlying disturbed rock mass undergoes small deformation, and the rock mass is in the elastic deformation stage. After the rock mass yields, internal fissures occur, and the rock mass gradually loses continuity, but it can still maintain the layered form. At this time, the rock mass is in the post-peak softening stage, and the strength parameters (cohesion \(c\) and friction angle \(\varphi\)) are gradually reduced. The cohesion \(c\) and friction angle \(\varphi\) tend to stabilize until the rock masses reach the residual strength stage. When the coal-seam roof begins to cave and fill into the mined-out area, the rock masses are in the caved material deformation stage, and the cohesion \(c\) and friction angle \(\varphi\) reduce to the parameters of the caved materials.  
According to the simplified complete stress-strain curve, the adaptive adjustment of the mechanical parameters of rock masses at different stages is as follows:  
(1) Elastic deformation stage (Stage I): the mechanical parameters of rock masses in this stage remain unchanged.  
(2) Post-peak softening stage (Stage II): the mechanical parameters of rock masses in this stage need to be reduced.  
The principles of damage mechanics are employed to interpret the characteristics of the post-peak softening stage of the disturbed overlying strata. More specifically, the formation, expansion, and coalescence of cracks in rock masses can be considered a damage process of the mechanical properties of rock masses. That is, the post-peak softening stage of the coal-seam roof can be regarded as a progressive damage process along with the development of plastic strains.  
In the post-peak softening stage, with increasing plastic strain \(\varepsilon^{p}\), the cohesion \(c\) decreases from an initial value \(c_{initial}\) to a residual value \(c_{residual}\), and the friction angle \(\varphi\) decreases from an initial value \(\varphi_{initial}\) to a residual value \(\varphi_{residual}\).  
More specifically, \(\varepsilon^{p}_{residual}\) can be determined according to the confining pressure \(\sigma_{3}\) (Figure 5); thus, the linear softening curves of cohesion \(c\) and friction angle \(\varphi\) are determined (Figure 6). According to the \(\varepsilon^{p}\) of any point in the post-peak softening stage, the corresponding cohesion \(c_{new}\) and friction angle \(\varphi_{new}\) can be achieved.  
(3) Residual strength stage (Stage III): the mechanical parameters of rock masses in this stage are adjusted to those parameters in the residual state (i.e., \(c_{residual}\) and \(\varphi_{residual}\)).  
In the residual strength stage, with increasing plastic strain, the cohesion \(c\) and friction angle \(\varphi\) remain at residual values of \(c_{residual}\) and \(\varphi_{residual}\), respectively (Figure 6).  
(4) Caved material deformation stage (Stage IV): the mechanical parameters of rock masses in this stage are adjusted to those parameters of the caved materials (i.e., \(c_{loosened}\) and \(\varphi_{loosened}\)).  
When the roof rock masses fall and break into irregular blocks, the cohesion of the rock mass decreases significantly, while the friction angle changes little. In some studies [17; 18; 19; 20], the cohesion of the caved material is reduced to zero, and the friction angle is not changed or is maintained at the residual value. In this paper, it is assumed that \(c_{loosened}=0\) and \(\varphi_{loosened}=\varphi_{residual}\) (Figure 6).  
Figure 6: Softening model of strength parameters: (**a**) softening model of cohesion; (**b**) softening model of friction angle.