# 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  
(1) Evolution depth of type I mining fissures in the downward direction  
The mining action of the mine promotes the counterclockwise rotation of the key strata and the overlying rock and soil layer. When the tensile deformation \(\varepsilon_{xc}(x,y,\theta_{\text{r}})\) and the ultimate tensile deformation \([\varepsilon]\) caused by the rotation and subsidence satisfy Formula (21), the surface forms a downward type I mine moving fissures. \[\varepsilon_{xc}(x,y,\theta_{\text{r}})\geq[\varepsilon] \tag{21}\]  
In order to obtain an effective formula for calculating the development depth of type I mining fissures, the unit body in the fissure was studied and analyzed, and a calculation model for the development depth of type I mining fissures under mining action was established, as shown in Figures 10 and 11. Figure 10: Calculation model for the evolution depth of a type I-I mining fissure. First, it is assumed that the unit body has the following characteristics. The unit body is small enough to ignore its own load. The elevation of rock and soil layers at the same elevation is consistent, there is no undulating shape, and the influence of terrain can be ignored. At this time, the force acting on the top and bottom surfaces of the unit body is \(\sigma_{Z}\), and the forces acting on the left, right, and front surfaces are \(\sigma_{x}\) and \(\sigma_{y}\). Because it is mainly affected by tension, the shear stress effect is neglected in the study of the downward mining fissure. According to the theory of advanced soil mechanics [30], the calculation formulas can be obtained as  
\[\left\{ \begin{array}{l}\sigma_{Z}=\gamma H_{\text{d}}\\ \sigma_{x}=\sigma_{y}=\frac{\mu}{1-\mu}\sigma_{Z}=\frac{\mu}{1-\mu}\gamma H_{ \text{d}}\end{array} \right. \tag{22}\]  
where \(\sigma_{Z}\) is the vertical principal stress of the unit body at the depth \(H_{\text{d}}\), MPa; \(\sigma_{x}\) and \(\sigma_{y}\) are the horizontal principal stresses of the unit body at the depth \(H_{\text{d}}\), MPa; and \(\mu\) is Poisson's ratio. According to the generalized Hooke's law and the Mohr-Coulomb criterion, the following stress-strain relationship can be obtained [48; 49]:  
\[\left\{ \begin{array}{l}\varepsilon_{x}=\frac{1}{E}\big{[}\sigma_{x}-\mu \big{(}\sigma_{Z}+\sigma_{y}\big{)}\big{]}\\ \varepsilon_{y}=\frac{1}{E}\big{[}\sigma_{y}-\mu\big{(}\sigma_{Z}+\sigma_{x} \big{)}\big{]}\\ \varepsilon_{Z}=\frac{1}{E}\big{[}\sigma_{Z}-\mu\big{(}\sigma_{x}+\sigma_{y} \big{)}\big{]}\\ \sigma_{x}=\sigma_{y}=\sigma_{Z}\tan^{2}(45+\phi/2)-2c\tan(45+\phi/2)\end{array} \right. \tag{23}\]  
where \(E\) is the elastic modulus of the soil layer, MPa. Under the condition of the same depth and direction of advancing tensile deformation, the additional stress \(\Delta\sigma_{Z}=\Delta\sigma_{y}=0\) is applied in the vertical and inclined directions of the working face, and the additional stress \(\Delta\sigma_{x}\) is applied in the direction of the working face as follows:  
\[\Delta\sigma_{x}=\frac{E\varepsilon_{\text{c}}}{1-\mu} \tag{24}\]  
where \(\varepsilon_{\text{c}}\) is the critical tensile deformation of the surface, which is 2-3 mm/m. At this time, the deformation \(\varepsilon_{xc}\) and stress \(\sigma_{xc}\) under the mining disturbance in the strike direction of the working face are  
\[\left\{ \begin{array}{l}\varepsilon_{xc}=\frac{1}{E}\big{[}(\sigma_{x}- \Delta\sigma_{x})-\mu\big{(}\sigma_{Z}+\sigma_{y}\big{)}\big{]}\\ \sigma_{xc}=\sigma_{Z}\tan^{2}(45+\phi/2)-2c\tan(45+\phi/2)-\Delta\sigma_{x} \end{array} \right. \tag{25}\]  
Figure 11: Calculation model of the force of the downward type I mining fissure element.