# An FDM-Based Dynamic Zoning Method for Disturbed Rock Masses above a Longwall Mining Panel
## 2 The Dynamic Zoning Method
### Establishing the Simplified Complete Stress-Strain Curve  
In general, there are several common characteristics for rock mass complete stress-strain curves [23; 24; 25] (Figure 3). (1) Before reaching the peak strength, the rock mass is in the elastic deformation stage, and the deformation at this stage gradually develops with increasing stress. (2) After reaching the peak strength, the rock mass is in the softening stage. With increasing deformation, the strength of the rock mass gradually decreases. (3) After reaching the residual strength, the rock mass is in the residual strength stage. With the continuous development of plastic deformation, the strength of the rock mass remains at a residual value. (4) With increasing confining pressure, the peak strength and residual strength of the rock mass gradually increase, and the peak strain and residual strain increase correspondingly. Figure 2: Outline of the dynamic zoning method. On the basis of the above characteristics, this paper simplifies the complete stress-strain curve of the overlying disturbed rock mass into four stages (Figure 3(b)). Stage I (Section OA) is the elastic deformation stage, i.e., \(\epsilon^{p}=0\); Stage II (Section AB) is the post-peak softening stage, i.e., \(0<\epsilon^{p}<\epsilon^{p}_{residual}\); Stage III (Section BC) is the residual strength stage, i.e., \(\epsilon^{p}_{residual}\leq\epsilon^{p}<\epsilon^{p}_{loosened}\); and Stage IV (Section CD) is the caved material deformation stage, i.e., \(\epsilon^{p}\geq\epsilon^{p}_{loosened}\). More details are shown in Figure 4.  
(1) Elastic deformation stage (Stage I): Before the rock mass yields, its strain develops linearly with increasing stress; see Section OA of the simplified complete stress-strain curve (Figure 3(a)). (2) Post-peak softening stage (Stage II): After the rock mass yields, internal cracks gradually develop, expand, and coalesce. With increasing rock mass strain, the strength of the rock mass degenerates; see Section AB of the simplified stress-strain curve (Figure 3(a)). The corresponding strength of Point A is the peak strength \(\sigma_{peak}\), and the corresponding strain is the maximum elastic strain \(\epsilon^{e}_{peak}\); the corresponding strength of Point B is the residual strength \(\sigma_{residual}\), and the corresponding strain is \(\epsilon^{t}_{residual}\). See Equation (1):  
\[\epsilon^{t}_{residual}=\epsilon^{e}_{peak}+\epsilon^{p}_{residual} \tag{1}\]  
Figure 4: The simplified complete stress-strain curve of the rock mass. Figure 3: Complete stress-strain curves of rock: (**a**) sandstone; (**b**) mudstone. For any point M located on Section AB (Figure 4a), its strain should satisfy Equation (2):  
\[\varepsilon^{t}=\varepsilon^{e}_{peak}+\varepsilon^{p} \tag{2}\]  
In addition, when subjected to different confining pressures, both the complete stress-strain curves and their simplifications are different. With increasing confining pressure, the peak strength \(\sigma_{peak}\) and residual strength \(\sigma_{residual}\) of the rock mass gradually increase, and the corresponding peak strain \(\varepsilon^{e}_{peak}\) and residual strain \(\varepsilon^{t}_{residual}\) also increase gradually (Figure 4a). \(\varepsilon^{p}_{residual}\) represents the threshold plastic strain of the rock mass from the post-peak softening stage to the residual strength stage, which depends on the property and stress state of the rock mass, and it is difficult to obtain the value of \(\varepsilon^{p}_{residual}\) directly. As \(\varepsilon^{e}_{peak}\) and \(\varepsilon^{t}_{residual}\) change with increasing confining pressure (Figure 4a), the critical value of the plastic principal strain \(\varepsilon^{p}_{residual}\) also changes with the confining pressure.