# Investigation of Rock Joint and Fracture Influence on Delayed Blasting Performance
## 3 Change in Stress Field in Fractured Rock Due to Blasting
### Constitutive Model of Fractured Rock Mass Blasting  
By combining the classical joint nonlinear normal deformation constitutive model (BB) and the cylindrical cavity Heelan solution [20; 21], a delayed blasting model in the hole of fractured rock mass is constructed [22; 23]. The stress change in the red particles on both sides of the joint interface is analyzed, where \(\gamma\) is the polar angle, \(a\) is the length of the joint, \(l\) is the length of the explosive column, \(c\) is the horizontal distance between the blast hole and the joint, \(b\) is the distance between the lower explosive column and the middle of blasting hole, and \(d\) is the radius of the explosives, as shown in Figure 1 below. In addition, \(x_{1}\) represents the location of the axis of symmetry, and \(\beta\) is any number of adjustment functions. To simplify the change in the vibration velocity of the incident particle at the joint interface, as shown in Figure 2, the change form is simplified as  
\[v_{IP}=\frac{1}{\sqrt{\beta\pi}}e^{\frac{-(x-xt)^{2}}{\beta}} \tag{10}\]  
Therefore, time \(t=0\) to \(4000\) ms of the incident wave function is defined. To ensure the accuracy of numerical calculation, a total of \(2000\) times is selected, in which the peak time \(b=2000\) ms and the corresponding peak value \(\mathcal{G}(t-2)\text{max}=0.2\) m/s. After the initial directional crack is formed, the crack continues to expand under the explosive gas. Based on the theory of fracture mechanics, the stress intensity factors of types I and II inside the crack under quasi-static pressure are  
\[\left\{ \begin{matrix}K_{I}\\ K_{II}\end{matrix} \right.=\frac{1}{\sqrt{\pi a}}\int_{-a}^{a}\sqrt{\frac{a+x}{a- x}}\left\{ \begin{matrix}\sigma_{(i)}\\ \tau_{(i)}\end{matrix} \right.\ dx \tag{11}\]  
Figure 1: Delay blasting hole model within fractured rock mass. By substituting \(\sigma\) and \(\tau\) into Equation (11), the stress intensity factor can be expressed in terms of distance and time after detonation. By bringing the results into Equations (12) and (13), the crack tip stress and displacement can be calculated as  
\[ \begin{cases}\sigma_{x}=\frac{K_{\text{L}}}{\sqrt{2\pi r}}\cos\frac{\theta}{2} \Big{(}1-\sin\frac{\theta}{2}\cdot\sin\frac{3\theta}{2}\Big{)}-\frac{K_{\text{ L}}}{\sqrt{2\pi r}}\sin\frac{\theta}{2}\Big{(}2+\cos\frac{\theta}{2}\cdot\cos \frac{3\theta}{2}\Big{)}\\ \sigma_{y}=\frac{K_{\text{L}}}{\sqrt{2\pi r}}\cos\frac{\theta}{2}\Big{(}1+\sin \frac{\theta}{2}\cdot\sin\frac{3\theta}{2}\Big{)}+\frac{K_{\text{L}}}{\sqrt{2 \pi r}}\sin\frac{\theta}{2}\cdot\cos\frac{\theta}{2}\cdot\cos\frac{3\theta}{2} \\ \tau_{xy}=\frac{K_{\text{L}}}{\sqrt{2\pi r}}\cos\frac{\theta}{2}\cdot\sin\frac{ \theta}{2}\cdot\cos\frac{3\theta}{2}+\frac{K_{\text{L}}}{\sqrt{2\pi r}}\cos \frac{\theta}{2}\Big{(}1-\sin\frac{\theta}{2}\cdot\sin\frac{3\theta}{2}\Big{)} \end{cases}  \tag{12}\]  
\[ \begin{cases}u=\frac{(1+\mu)K_{\text{t}}}{2E_{a}}\sqrt{\frac{\pi}{2\pi}}\Big{[} (2k-1)\cos\frac{\theta}{2}-\cos\frac{3\theta}{2}\Big{]}+\frac{2(1+\mu)K_{ \text{t}}}{4E_{a}}\sqrt{\frac{\pi}{2\pi}}\Big{[}(2k+3)\sin\frac{\theta}{2}+ \sin\frac{3\theta}{2}\Big{]}\\ \
u=\frac{(1+\mu)K_{\text{t}}}{2E_{a}}\sqrt{\frac{\pi}{2\pi}}\Big{[}(2k+1) \sin\frac{\theta}{2}-\sin\frac{3\theta}{2}\Big{]}-\frac{2(1+\mu)K_{\text{t}}} {4E_{a}}\sqrt{\frac{\pi}{2\pi}}\Big{[}(2k-3)\cos\frac{\theta}{2}-\cos\frac{3 \theta}{2}\Big{]}\end{cases}  \tag{13}\]  
where \(E_{a}\) is the dynamic elastic modulus of rock, which is taken as 10 GPa. \(\sigma_{x}\), \(\sigma_{y}\), and \(\tau_{xy}\) are the stress components at the joint ends. \(u\) and \(v\) are the displacement components at the joint ends; \(r\) and \(\theta\) are the polar diameter and polar angle from the joint ends, respectively. This paper studies the plane strain problem, where \(k=3\) to 4 \(\mu\). Since the integral does not have an explicit solution for the stress and displacement of the particle around the crack, MATLAB numerical integration is used to obtain the numerical solution of the vibration velocity.