## 2 Materials and Methods
### Mechanical Model and Fracture Analysis of the Roof After Coal Seam Mining
#### 2.2.2 Establishment and Solution of the Mechanical Model  
Based on the mining geological conditions of the 22,031 working face in XinDeng (Zhengzhou, China) Coal Mine, to simplify the calculation process, the load acting on the main roof is idealized as a uniformly distributed load, and the effect of plastic deformation failure in the underlying strata on elastic foundation parameters is not considered. Selecting the section at a certain distance behind the working face as the research object, a mechanical model is developed to characterize the interaction between the roof and underlying coal-rock strata following coal seam extraction, as shown in Figure 2. In this notation, \(q\) represents the load exerted by the overlying strata on the main roof; \(k\) signifies the Winkler elastic foundation coefficient; \(h\) denotes the main roof thickness; \(L\) indicates the exposed length of the main roof along the working face direction; and \(D\) stands for the flexural rigidity of the main roof.  
1. Differential Equation of Main Roof Deflection For the main roof above the goat, the differential equation of roof deflection is \[D_{j}\frac{d^{4}\omega_{1}}{dx^{4}}=q\] (1)  
Figure 2: Roof stress analysis diagram.  
In the equation, \(\omega(x)\) denotes the deflection of the main roof, while \(q(x)\) represents the load imposed by the overlying strata.  
For the main roof sections flanking the gob area, the differential equation governing roof deflection is  
\[D_{j}\frac{d^{4}\omega_{2}}{dx^{4}}+K\frac{d^{2}w_{2}}{dx^{2}}=q \tag{2}\]  
In the equation, \(\omega_{2}(x)\) is the vertical deflection within the support area; the extra term \(Kd^{2}\omega^{2}/dx^{2}\) represents the restoring force of the Winkler spring.  
Model Boundary Conditions  
In the far-field region away from the gob area, the main roof is subject to a fixed constraint between the overlying and underlying strata, where both the deflection and rotation angle of the main roof remain zero. At the boundary of the gob area, the main roof exhibits continuity in deflection, rotation angle, bending moment, and shear force.  
Model Solution Process  
The model depicted in Figure 2 is axisymmetric with respect to the o-z axis, so only the left half of the main roof's deflection equation requires a solution. The general solution forms of Equations (1) and (2) are derived as follows:  
\[\omega_{1}(x)=\frac{q}{24D_{j}}+C_{1}x^{3}+C_{2}x^{2}+C_{3}x+C_{4} \tag{3}\]  
\[\begin{array}{c}\omega_{2}(x)=e^{ax}(C_{5}sin(\alpha x)+C_{6}cos(\alpha x)) \\ +e^{-ax}(C_{7}sin(\alpha x)+C_{8}cos(\alpha x))\end{array} \tag{4}\]  
In the equation, \(C_{1}\)-\(C_{8}\) are unknown coefficients; \(\alpha=\sqrt[]{\frac{K}{D_{j}}}\).  
Based on the model's boundary conditions, the following equations can be derived:  
\[\lim_{x\rightarrow-\infty}\omega_{1}(x)=0,\lim_{x\rightarrow-\infty}\omega_ {1}^{\prime}(x)=0 \tag{5}\]  
\[\begin{cases}\omega_{1}(0)=\omega_{2}(0)\\ \omega_{1}^{\prime}(0)=\omega_{2}^{\prime}(0)\\ \omega_{1}^{\prime\prime}(0)=\omega_{2}^{\prime\prime}(0)\\ \omega_{1}^{\prime\prime}(0)=\omega_{2}^{\prime\prime}(0)\end{cases} \tag{6}\]  
Given that the rotation angle and shear force at the central axis of the model vary, the following equations can be derived:  
\[\begin{cases}\omega_{2}^{\prime}(L)=0\\ \omega_{1}^{\prime\prime}(L)=0\end{cases} \tag{7}\]  
By substituting the above conditions into the general solutions (1)-(2), the constants \(C_{1}\) to \(C_{1}\) can be determined in sequence:  
\[C_{2}=\frac{q\big{(}k^{2}L^{2}+2 kL+4D_{j}\big{)}}{24D_{j}(kL+2D_{j})} \tag{8}\]  
\[C_{4}=-\frac{qk^{2}L^{3}\big{(}kL+2D_{j}\big{)}}{384D_{j}(kL+2D_{j})} \tag{9}\]  
\[C_{5},\ C_{6},\ C_{7},\ C_{8}\ldots\]  
In the equation, \(D=\frac{E_{j}h_{j}^{3}}{\left[12\left(1-p_{j}^{2}\right)\right]}\).