# 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
\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}=\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{)}\end{array} \right. \tag{32}\]  
According to the small-scale yield theory in fracture mechanics, when \(r_{\rm I}\to 0\), \(\sigma_{x}\rightarrow\infty\), \(\sigma_{z}\rightarrow\infty\). It is obviously inconsistent with the reality under mining conditions. When the above situation occurs in the rock formation, it shows that the rock formation has undergone plastic deformation, that is, plastic failure. According to the [PERSON] criterion, the force on the unit body satisfies the following formula:  
\[\left(\sigma_{1}-\sigma_{2}\right)^{2}+\left(\sigma_{2}-\sigma_{3}\right)^{2}+ \left(\sigma_{3}-\sigma_{1}\right)^{2}=2{\sigma_{\rm s}}^{2} \tag{33}\]  
According to the stress relation of Formula (22), we can obtain  
\[{\sigma_{\rm s}}^{2}=\left(\sigma_{x}+\Delta\sigma_{x}\right)^{2}+{\sigma_{\rm y }}^{2}+{\sigma_{z}}^{2} \tag{34}\]  
Substituting the stress at the tip of the type I fissure, under the plane stress, \(r_{\rm I}\) can be obtained as  
\[r_{\rm I}=\frac{1}{2\pi}\left(\frac{K_{\rm I}}{\sigma_{\rm s}}\right)^{2}\left[ \cos^{2}\frac{\beta_{c}}{2}\Big{(}1+3\sin^{2}\frac{\beta_{c}}{2}\Big{)}\right] \tag{35}\]  
It can be seen from Formula (35) that the value of \(r_{\rm I}\) is related to the stress intensity response at the tip of the type I fissure, the stress of the unit body, and the fracture angle of the rock formation. The unit body stress and rock formation failure angle can be calculated by Formulas (19) and (22), respectively. Therefore, only the stress intensity at the tip of the fissure should be an unknown parameter. Plastic failure occurs at the tip of the fissure under the mining action of the rock formation, and the \(K_{\rm I}\) plastic zone can be corrected to obtain  
\[K_{\rm I}=\frac{\sigma\sqrt{\pi a}}{\sqrt{1-\frac{1}{2}\Big{(}\frac{\sigma}{ \sigma_{\rm s}}\Big{)}^{2}}} \tag{36}\]where \(\sigma\) is the stress without fissures, MPa, and \(a\) is the length of \(1/2\) of the major axis of the fissure, m.  
It can be seen from Formula (36) that when \(\sigma=\sigma_{\rm s}\) is the largest, \(K_{\rm I}\) is the largest, that is, the mining fissure is the longest, and the probability of sudden disasters in the mining process of the working face is the largest. Substituting \(\sigma=\sigma_{\rm s}\) into Formula (36) can obtain  
\[K_{\rm I}=\sigma_{\rm s}\sqrt{2\pi a} \tag{37}\]  
Due to different geological conditions and different mining thicknesses, there are differences in the rotation of key strata. According to the relationship between mining thickness and hinged rock beam rotation and subsidence, we can obtain  
\[\frac{a}{b}=\frac{h\sin\beta_{\rm c}}{h\cos\beta_{\rm c}/2}=\lambda \tag{38}\]  
where \(b\) is the length of \(1/2\) of the minor axis of the fissure, m.  
By substituting Formulas (37) and (38) into Formula (35), the evolution height of upward type I mining fissures under polar coordinate conditions can be obtained. By transforming it into rectangular coordinates, the evolution height \(H_{\rm st}\) of upward mining fissures can be obtained as  
\[H_{\rm st}=\lambda a\left[\cos^{2}\frac{\beta_{\rm c}}{2}\left(1+3\sin^{2} \frac{\beta_{\rm c}}{2}\right)\right]\cdot\cos\beta_{\rm c} \tag{39}\]  
(3) Upward II type mining fissure evolution height  
The analysis of the migration characteristics of the fractured overlying strata shows that when the key strata produce a stepped rock beam structure, the overlying strata will undergo shear failure and form upward type II mining fissures.