# Study on the Co-Evolution Mechanism of Key Strata and Mining Fissures in Shallow Coal Seam Mining
## 4 Analysis of Mining Fissure Propagation Characteristics and Evolution Parameters
### Analysis of Mining Fissure Evolution Parameters
Substituting simultaneous Formulas (22)-(24) into Formula (25), the deformation \(\varepsilon_{\rm xc}\) of type I mining fissures under mining action can be obtained as  
\[\varepsilon_{\rm xc}=\frac{\big{(}1+\mu^{2}\big{)}\gamma H-E\varepsilon_{\rm c} }{E(1-\mu)} \tag{26}\]  
When the amount of deformation generated under mining satisfies Formula (27), type I mining fissures are formed downward from the surface. \[\varepsilon_{\rm xc}=\frac{\big{(}1+\mu^{2}\big{)}\gamma H-E\varepsilon_{\rm c }}{E(1-\mu)}\geq[\varepsilon] \tag{27}\]  
Similarly, the Mohr-Coulomb criterion applicable to the downward type I mining fissure can be obtained by combining the above formulas:  
\[\frac{\mu}{1-\mu}\sigma_{Z}-\frac{E\varepsilon_{\rm c}}{1-\mu}=\sigma_{Z}\tan ^{2}(45+\phi/2)-2c\tan(45+\phi/2) \tag{28}\]  
Substituting Formulas (22) and (26) into Formula (28), the depth \(H_{\rm c}\) of the descending type I mining fissure can be obtained as  
\[H_{\rm c}=\frac{\frac{E\varepsilon_{\rm c}}{1-\mu}-2c\tan(45+\phi/2)}{\gamma \big{(}\frac{\mu}{1-\mu}-\tan^{2}(45+\phi/2)\big{)}} \tag{29}\]  
Evolution height of upward type I mining fissures  
During the mining process of the working face, under the action of tensile stress, upward and downward type I mining fissures are formed. Compared with the soil layer, the rock layer has greater cohesion. When studying the rock layer fracture, it is not easy to continue to use the method of calculating fissures in down mining. According to the nature of the rock formation, the research methods related to fracture mechanics are used for the research on upward mining fissures. From the formation process of fissures in fracture mechanics, it can be seen that the angle at which fissures are generated in the rock formation at this time is the fracture angle of the rock formation. Therefore, combining the fracture angle \(\beta_{\rm c}\) of the rock formation with the unit body at the tip of the fracture can obtain the mechanical calculation model of the upward type I mining fissure, as shown in Figure 12.  
Figure 12: Calculation model of the force of the upward type I mining fissure element. According to the force characteristics of type I mining fissures in the working face [13], in the advancing direction of the working face, the stress field expression of the type I fissure tip in fracture mechanics can be obtained:  
\[\left\{ \begin{array}{l}\sigma_{x}+\Delta\sigma_{x}=\frac{K_{\rm I}}{\sqrt{2\pi r _{\rm I}}}\cos\frac{\beta_{c}}{2}\Big{(}1-\sin\frac{\beta_{c}}{2}\sin\frac{3 \beta_{c}}{2}\Big{)}\\ \sigma_{z}+\Delta\sigma_{z}=\frac{K_{\rm I}}{\sqrt{2\pi r_{\rm I}}}\cos\frac{ \beta_{c}}{2}\Big{(}1+\sin\frac{\beta_{c}}{2}\sin\frac{3\beta_{c}}{2}\Big{)}\\ \tau_{zz}=\frac{K_{\rm I}}{\sqrt{2\pi r_{\rm I}}}\cos\frac{\beta_{c}}{2}\sin \frac{\beta_{c}}{2}\sin\frac{3\beta_{c}}{2}\end{array} \right. \tag{30}\]  
where \(r_{\rm I}\) is the distance between the type I fissure and the fissure tip in polar coordinates, m, and \(K_{\rm I}\) is the stress intensity factor of the type I fissure tip. When it is known that fissures are formed in the rock formation, the fissures tend to pinch out gradually due to the inhibition of the force of the rock formation. At the same time, the occurrence of type I mining fissures needs to reach the critical value of stress intensity \(K_{\rm IC}\), so the judgment condition for determining whether fissures are generated in the rock formation satisfies the following formula:  
\[K_{\rm I}\leq K_{\rm IC} \tag{31}\]  
From the previous analysis, it can be seen that at the same depth of the unit body, there is only an additional stress \(\Delta\sigma_{x}\) in the strike direction of the working face, and at the same time, type I fissures are formed under the action of tensile stress and the effect of shear stress is small. Therefore, Formula (30) can be abbreviated as for  
\[\left\{