## 2 Theoretical Analysis of BBT
### Crack Propagation of BBT  
After the explosive is blasted, it first produces a detonation shock wave. When the strength of the wave reaches the compressive strength of the roof rock mass, the blashhole wall will be crushed. The shock wave gradually weakens in the process of passing through the rock medium and the damage gap and gradually evolves into a stress wave. Since the tensile strength of the rock mass is much smaller than its compressive strength, the expansion of cracks is mainly formed by stress wave tension. Taking a symmetrical crack on the wall of energy-gathering blasting hole as an example, the fracture mechanics model is shown in Figure 4[27; 28]. During the crack propagation process, when the energy flow contacts the initial crack tip, the stress intensity factor of the crack tip (\(K_{I}\)) is [29; 30]:  
\[K_{I}=P_{s}F\sqrt{\pi(r_{b}+a)}+\sigma_{s}\sqrt{\pi a} \tag{1}\]  
where: \(a\) is the length of the initial energy-gathering crack, \(r_{b}\) is the radius of the blashhole, \(P_{s}\) is the pressure when the explosive particles are filled with the blast hole, and \(\sigma_{s}\) is the tangential stress generated by the penetration of energy flow. \(F\) is the stress intensity factor correction factor, which is a function of the blashhole radius and the crack length, as Equation (2). It increases with the increase in the crack propagation length. Define \(r\) as the equation \(r=(r_{b}+a)/r_{b}\). When \(r<1.5\), the change is relatively large, and when \(r>1.5\), the change is relatively small and approaches 1. \[F=f[(r_{b}+a)/r_{b}] \tag{2}\]  
Figure 4: Mechanical model of BBT. (**a**) BBT is applied in the working face; (**b**) A-A section; (**c**) crack propagation model [27]. According to the theory of fracture mechanics, when the stress intensity factor (\(K_{I}\)) at the directional crack tip is greater than the fracture toughness (\(K_{IC}\)) of the rock, the crack will start; otherwise, the crack will stop. The conditions for crack initiation and crack propagation are shown in Equation (3):  
\[P_{s}>\frac{K_{IC}-\sigma_{s}\sqrt{\pi a}}{F\sqrt{\pi(r_{b}+a)}} \tag{3}\]  
When the initial crack is formed in the directional direction, the subsequent gas wedge action of the explosive gas further forces the crack to expand further, and the crack expansion causes the pressure of the explosive gas to drop. In order to ensure the continuous growth of the crack, the instantaneous pressure (\(P_{e}\)) of the explosive gas needs to satisfy Equation (4):  
\[K_{Ie}=P_{e}F_{e}\sqrt{\pi(r_{b}+a_{e})}+\sigma_{e}\sqrt{\pi a_{e}} \tag{4}\]  
where: \(K_{Ie}\) is the stress intensity factor at the crack tip when the fracture stops, and \(P_{e}\) is the explosion gas pressure when the fracture stops. \(F_{e}\) is a certain value close to 1. \(\sigma_{e}\) is the tangential stress when the fracture stops. \(a_{e}\) is the end length of the crack when the directional crack stops growing. When the explosive gas pressure drops to satisfy Equation (5), the directional cracks stop expanding. \[P_{e}\leq\frac{K_{Ie}-\sigma_{e}\sqrt{\pi a_{e}}}{F_{e}\sqrt{\pi(r_{b}+a_{e})}} \tag{5}\]  
Under the action of the stress wave, the peak tensile stress of the rock element in the hoop direction (\(\sigma_{m}\)) can be expressed as Equation (6):  
\[\sigma_{m}=bP/\overline{r}^{\alpha} \tag{6}\]  
where \(P\) is the peak of stress under the action of stress wave and \(\alpha\) is the stress attenuation index. \(b\) is the ratio of the tangential stress to the radial stress coefficient, in which \(b\) is related to the Poisson's ratio of the rock and the propagation distance of the stress wave. The \(b\) value near the explosion zone is relatively large, but the \(b\) value decreases rapidly as the distance increases and tends to depend only on the fixed value of Poisson's ratio, as shown in Equation (7):  
\[b=v/(1-v) \tag{7}\]  
Since the initial stage of blasting has caused some damage to the rock mass and taking into account the defects of the rock mass itself, defined \(\lambda_{0}\) as the damage factor.