## 6 Loose-Layer Fissure Formation Mechanism  
According to the force chain development process observed in the numerical simulation, when the overburden on the working surface was damaged, strong chains extended vertically in an arched shape. The arch's vault was situated in the undisturbed strata, with its foot resting on the coal wall flanking both sides of the working face (Figure 14a). Damage to the hard rock layer 1 on the working surface led to the collapse of the force chain arch structure, resulting in the failure of the hard rock layer within the arch (Figure 14b). As mining damage progressed upward to the hard rock layer 2, a new arch structure formed, with the unbroken portion of hard rock layer 1 serving as the arch foot (Figure 14c). As the working face advanced, the arch structure destabilized, undergoing a repetitive process (Figure 14d). Under conditions of high-strength mining, the force chain arch structure was prone to instability, which eventually led to the formation of surface ground fissures [2; 15].  
Figure 13: Evolution law of mining rock chain: (**a**) excavate 80 m; (**b**) excavate 100 m; (**c**) excavate 180 m; (**d**) excavate 200 m; (**e**) excavate 300 m.  
During mining of the working face, the force chain arch extended upward, reaching the hard rock layer, which manifested at a height of \(\Delta m\), as follows:  
\[\Delta m=h_{\text{i}}+m-p[h_{\text{i}}+(1-C)m] \tag{2}\]  
In this context, \(m\) represents the coal seam thickness, \(h_{i}\) denotes the thickness of the rock seam beneath the hard rock layer, \(C\) is the recovery rate, and \(p\) is the residual rock swelling coefficient.  
Upon the fracture of the hard rock layer, the force chain arch pressure arch re-elevates at the fault position of the hard rock layer; the criterion for determining the occurrence of a hard rock layer fracture is provided as follows:  
\[L_{\text{i}}-2h_{\text{i}}\cot\alpha\geq h_{\text{i}}^{\prime}(R_{\text{t}}/ q_{\text{i}})^{1/2} \tag{3}\]  
In this context, \(\alpha\) represents the bedrock fracture angle, \(h_{\text{i}}^{\prime}\) denotes the thickness of the hard rock layer, \(R_{t}\) signifies the tensile strength of the hard rock strata, and \(q_{i}\) indicates the overburden load.  
The force-chain arch equation is established as follows:  
\[y=y_{0}-Cch(\frac{X}{C}) \tag{4}\]  
\(C\) is the coefficient associated with \(h_{0}\) and \(L_{n}\) of the natural logarithm of the force chain arch, while \(y_{0}\) represents a constant.  
Figure 14: Development model of the force chain arch structure at various excavation depths: (**a**) excavate 100 m, (**b**) excavate 200 m, (**c**) excavate 250 m, (**d**) excavate 300 m. (In the figure, \(L_{n}\) is the propulsion distance of the working face, \(b\) is the arch span, and \(a\) is the force chain arch foot from the working face).  
When the working face progresses to a specific point, the force chain arch reaches its maximum development.  
\[y_{0}=h_{0} \tag{5}\]  
\[C=\frac{L_{n}+2a}{2 arcsh(tan\theta)} \tag{6}\]  
\[x=\frac{L_{n}}{2}+a \tag{7}\]  
\[a=m\tan(45^{\circ}-\phi/2) \tag{8}\]  
Transfer (5)-(8) into Formula (4), as follows:  
\[y=h_{0}-\frac{L_{n}+2m\tan(45^{\circ}-\phi/2)}{2 arcsh(tan\theta)}ch[2 arch(tan \theta)] \tag{9}\]  
Where \(tan\theta\) is the highest point of the force chain arch, \(tan\theta=1\).  
\[y=h_{0}-\frac{L_{n}+2m\tan(45^{\circ}-\phi/2)}{2 arcsh1}ch(arcsh1) \tag{10}\]  
Figure 15 illustrates the approximate exponential relationship between the working face propulsion distance and the corresponding force chain arch height, as follows:  
\[h_{0}=ke^{xL_{\alpha}} \tag{11}\]  
From Equations (4)-(11), obtain  
\[y=ke^{xL_{\alpha}}-\frac{L_{n}-2m\tan(45^{\circ}-\phi/2)}{2 arcsh1}ch(arcsh1) \tag{12}\]  
Based on the numerical simulation  
\[y=54.219e^{0.053L_{n}}-\frac{L_{n}+2m\tan(45^{\circ}-\phi/2)}{2 arcsh1}ch(arcsh1) \tag{13}\]