# 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 Propagation Characteristics  
According to rock and soil mechanics [30] and [PERSON]'s criterion [45], a certain amount of microcracks or weak structural planes exist in rock and soil mass, assuming that there is only one fissure in a unit rock and soil mass. Under the action of force, the unit rock and soil mass are subjected to unidirectional tensile stress, compressive stress, or confining pressure, and stress concentration will occur at the end of the fissure, with the tensile stress at the tip of the fissure being many several times the principal stress. The relationship between the tensile stress and the principal stress at the fissure tip is  
\[\left\{ \begin{array}{c}\frac{(\sigma_{1}-\sigma_{3})^{2}}{\sigma_{1}+\sigma _{3}}=8\sigma_{\text{t}},\sigma_{1}+3\sigma_{3}\geq 0\\ \sigma_{3}-\sigma_{\text{t}}=0,\sigma_{1}+3\sigma_{3}\leq 0\end{array} \right. \tag{10}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) are, respectively, the maximum principal stress and the minimum principal stress in the fissure area of the weak plane, MPa, and \(\sigma_{\text{t}}\) is the tensile stress at the fissure tip, MPa. The type I and type II mining fissures formed by the breaking of the key strata are all caused by the failure and expansion of the fissure tip. According to the stress relationship in the direction of mining fissure propagation, the fissure propagation characteristics of rock and soil can be obtained, as shown in Figure 9.  
According to the composite fissure propagation criterion of linear elastic fracture mechanics [46], the expressions of \(\beta_{1}\) and \(\beta_{2}\) can be obtained as  
\[\left\{ \begin{array}{c}\beta_{1}\\ \beta_{2}\end{array} \right.=\arctan\left(\frac{\left(\sigma_{1}+\sigma_{3} \right)\mp\sqrt{\left(\sigma_{1}+\sigma_{3}\right)^{2}+4(\tan\phi\sigma_{3}+ \Omega\sigma_{\text{t}})(\tan\phi\sigma_{1}+\Omega\sigma_{\text{t}})}}{\tan \phi(\sigma_{3}-\sigma_{1})+2\Omega}\right) \tag{11}\]  
Figure 9: Calculation model of mining fissure propagation characteristics. where \(\Omega\) is the mining fissure damage coefficient. It can be seen from Figure 9 that if mining fissures expand, they must develop along the action lines of horizontal principal stress and vertical principal stress. At this time, the mining fissure propagation angle \(\phi_{0}\) has the following relationship:  
\[\tan\phi_{0}=\tan(\beta_{2}-\beta_{1}) \tag{12}\]  
According to the relation of an inverse trigonometric function, parallel vertical Formulas (11) and (12) can give \(\phi_{0}\) as  
\[\phi_{0}=\arctan\left(\frac{\sqrt{(\sigma_{1}+\sigma_{3})^{2}+4(\tan\phi \sigma_{3}+\Omega\sigma_{t})(\tan\phi\sigma_{1}+\Omega\sigma_{t})}}{\tan\phi( \sigma_{3}-\sigma_{1})+2\Omega}\right) \tag{13}\]  
According to the relationship between the fissure propagation angle and the arc length, the opening degree \(d\) of the mining fissure can be calculated as follows:  
\[d=2\sigma_{0}\sin\frac{\phi_{0}}{2} \tag{14}\]  
Under the assumption of unit rock and soil mass, the fissure azimuth \(\beta_{0}\) satisfies the following formula:  
\[\tan 2\beta_{0}=\tan(\beta_{2}+\beta_{1}) \tag{15}\]  
Then, the expression of fissure azimuth \(\beta_{0}\) is  
\[\beta_{0}=\frac{1}{2}\text{arctan}\bigg{(}\frac{\tan\beta_{2}+\tan\beta_{1}}{1 -\tan\beta_{1}\tan\beta_{2}}\bigg{)} \tag{16}\]  
If the directions of horizontal stress, vertical stress, maximum principal stress, and minimum principal stress are the same, then the fissure azimuth is the rock-soil fracture angle.