## 3 Stress Analysis of the Wellbore Surrounding Rocks in Broken Coal Seams
### Induced Stress Analysis  
According to the research, we find that the internal filling material is not completely filled, and it appears to be intermittent along the orientation of the cleat direction. Therefore, it could be assumed that the filler is composed of multiple rod structures, which is like a spring, as shown in Figure 2. There is no connection between these rods. The strong effect of the rod structure is only connected with the upper side \(\Gamma_{1}\) and the lower side \(\Gamma_{2}\) on the cleat. When the normal displacement occurs on both sides of the cleat, the filler can be considered as a rod structure. When the cleat surface undergoes a tangent displacement, the filler can be considered as a beam structure. Therefore, under the external loading, the relationship between the filler and the normal load P and the tangent load Q generated on the cleat surface, and the relative displacement on both sides of the cleat surface, can be written as [24]:  
\[P=E_{F}\cdot\frac{\left|u_{y^{\prime}}^{\Gamma_{1}}-u_{y^{\prime}}^{\Gamma_{2}} \right|}{D_{F}}, \tag{5}\]  
\[Q=\frac{E_{F}d_{b}}{4}\cdot\frac{\left|u_{x^{\prime}}^{\Gamma_{1}}-u_{x^{\prime }}^{\Gamma_{2}}\right|}{D_{F}^{3}}, \tag{6}\]  
where, \(E_{F}\) is the elastic modulus of the cleat filler; \(x^{\prime}\), \(y^{\prime}\) are directions along the cleat and the vertical cleat, respectively; \(u_{x^{\prime}}\) and \(u_{y^{\prime}}\) are displacements in the \(x^{\prime}\) and \(y^{\prime}\) directions, respectively; \(d_{b}\) is the width of the filler, which can be set as the width of the unit; due to the length of the cleat, along the cleat direction, the normal load P and the tangent load Q will change with different positions, so the cleat load and the cleat distribution load are the function of positions, seen in Figures 3 and 4. Figure 4: A tangential load model with asymmetric distribution for cleats. Figure 3: A normal load model with asymmetric distribution for cleats. Figure 2: Mechanical model of the cleat filler. According to the strength factor under a single normal load and a tangent load in the stress field intensity factor manual, and using the superposition principle, the strength factor under an asymmetric distribution load is obtained by integrating [25], seen in Equations (6)and (7):  
\[K_{I}=\sqrt{\frac{1}{\pi a}}\underset{-a}{\overset{a}{\int}}\sqrt{\frac{a+b}{a- b}}p(b)db, \tag{7}\]  
\[K_{II}=\sqrt{\frac{1}{\pi a}}\underset{-a}{\overset{a}{\int}}\sqrt{\frac{a+b}{a- b}}Q(b)db, \tag{8}\]  
Then the stress field expression near the crack tip can be written [25] in Equation (9):  
\[\left\{ \begin{array}{l}\sigma^{\prime\prime}{}_{x^{\prime}}=\frac{K_{I}}{ \sqrt{2\pi\rho}}\cos\frac{\varphi}{2}\Big{(}1-\sin\frac{\varphi}{2}\sin\frac{3 \varphi}{2}\Big{)}-\frac{K_{II}}{\sqrt{2\pi\rho}}\sin\frac{\varphi}{2}\Big{(}2+ \cos\frac{\varphi}{2}\cos\frac{3\varphi}{2}\Big{)}\\ \sigma^{\prime\prime}{}_{y^{\prime}}=\frac{K_{I}}{\sqrt{2\pi\rho}}\cos\frac{ \varphi}{2}\Big{(}1+\sin\frac{\varphi}{2}\sin\frac{3\varphi}{2}\Big{)}+\frac{ K_{II}}{\sqrt{2\pi\rho}}\sin\frac{\varphi}{2}\cos\frac{3\varphi}{2}\\ \tau^{\prime\prime}{}_{x^{\prime}y^{\prime}}=\frac{K_{I}}{\sqrt{2\pi\rho}} \sin\frac{\varphi}{2}\cos\frac{\varphi}{2}\cos\frac{3\varphi}{2}+\frac{K_{II}} {\sqrt{2\pi\rho}}\cos\frac{\varphi}{2}\Big{(}1-\sin\frac{\varphi}{2}\sin\frac {3\varphi}{2}\Big{)}\end{array} \right., \tag{9}\]  
where, \(\sigma^{\prime\prime}{}_{x}\), \(\sigma^{\prime\prime}{}_{y}\) and \(\tau^{\prime\prime}{}_{xy}\) are the normal stress and the shearing stress in the additional stress field, respectively. The stress state is shown in Figure 5, where \(\rho\), \(\varphi\) are the radius and angle of the point to the cleat tip, respectively. The induced displacement field distribution near the cutting tip can be written as [23]:  
\[\left\{