## 5 Drilling Pressure Relief Mechanism
### Borehole Mechanics Model Establishment and Pressure Relief Analysis  
The three-dimensional rectangular coordinate system (\(x,y,z\)) and the two-dimensional polar coordinate system (\(\rho\), \(\theta\)) are established as the auxiliary coordinate system of the  
Figure 12: (**a**) is the failure diagram of the sample with 1 cm drilling spacing, (**b**) is the failure diagram of the sample with 2 cm drilling spacing, (**c**) is the failure diagram of the sample with 3 cm drilling spacing, (**d**) is the failure diagram of the sample with 4 cm drilling spacing, and (**e**) is the failure diagram of the sample with 5 cm drilling spacing. Figure 13: Failure diagram of specimens with different hole spacing. (**a**) 1 cm; (**b**) 2 cm; (**c**) 3 cm; (**d**) 4 cm; and (**e**) 5 cm. drilling mechanics model, as shown in Figure 14. In order to simplify the calculation, it is assumed that the internal force of the rock mass is isotropic and the rock mass is an elastic-plastic material. Before the drilling construction, the rock mass of the roadway side is in an elastic state, the stress (\(p\)) of the rock mass edge is the confining pressure of the borehole, and the stress (\(p\)) is valued according to the distribution law of the abutment pressure (\(\sigma_{p}\)) before the drilling in the z-axis direction. According to the theory of classical mechanics, under the action of equal confining pressure, the rock mass around the hole will form a plastic zone and an elastic zone, and the radial stress expressions of the plastic zone, the elastic zone, and the junction of the two zones are shown as follows [26]:  
\[\sigma_{p}^{p}=\frac{\sigma_{c}}{\xi-1}\Big{[}(\rho/R)^{\xi-1}-1\Big{]} \tag{1}\]  
\[\sigma_{\rho}^{c}=\sigma_{1}\Big{[}1-(r/\rho)^{2}\Big{]}+\sigma_{r}(r/\rho)^{2} \tag{2}\]  
\[\sigma_{r}=(r^{\xi-1}/R-1)\sigma_{c}/(\xi-1) \tag{3}\]  
In the formula: the radial stress of the rock mass (\(\sigma_{\rho}^{p}\)) in the plastic zone (the unit is MPa); the radial stress of rock mass (\(\sigma_{\rho}^{c}\)) in the elastic zone (the unit is MPa); \(\rho\) is the distance from the point to the center of the borehole in m; \(\xi\) is the plastic coefficient of the rock mass; \(\sigma_{r}\) is the radial stress at the junction of the two zones, and the unit is MPa; \(\sigma_{c}\) is the uniaxial compressive strength of the rock mass (unit is MPa); \(\sigma_{1}\) is the maximum principal stress in MPa; \(r\) is the radius of the plastic zone (unit is m); \(R\) is the radius of the borehole in m.  
From the plastic zone radius equation in classical mechanics, it can be seen that [27]:  
\[r=R[(p+c\cot\theta)(1-\sin\theta)/(c\cot\theta)]^{\frac{1-\sin\theta}{2\sin \theta}} \tag{4}\]  
In the formula: \(c\) is the cohesion of the rock mass, the internal friction angle. Taking \(\sigma_{1}=p=\sigma_{p}\), the radial stress of the rock mass in the elastic zone in the z-axis direction of the borehole is the stress of the rock mass of the two sides of the roadway after unloading above the borehole (\(\sigma_{p}^{\prime}\)) [12]:  
\[\sigma_{p}^{\prime}=\sigma_{p}\Big{[}1-(r/\rho)^{2}\Big{]}+\sigma_{r}(r/\rho )^{2} \tag{5}\]  
Figure 14: Schematic diagram of borehole mechanics model of roadway surrounding rock. The difference in unloading stress after drilling can be obtained (\(\Delta\sigma\)) from Equation (5):  
\[\Delta\sigma=\sigma_{p}-\sigma_{p}^{\prime}=(\sigma_{1}-\sigma_{r})(r/\rho)^{2} \tag{6}\]  
Assuming that the plastic zone of the rock mass is no longer elastic, there is no change in kinetic energy and non-mechanical energy during drilling, and the work performed by external forces is completely converted into strain energy; the strain energy formula of the unit body can be obtained [28]:  
Figure 13:0  
In the formula: the strain of rock mass; the Poisson's ratio of rock mass; \(E\) is the elastic modulus of the rock mass (unit is MPa); \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) are the principal stresses in three directions, respectively, and the unit is MPa.