# Investigation of Rock Joint and Fracture Influence on Delayed Blasting Performance

^{1}

^{2}

^{3}

^{4}

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## Abstract

**:**

## 1. Introduction

## 2. Current Status of Blasting on Fractured Rock Mass

## 3. Change in Stress Field in Fractured Rock Due to Blasting

#### 3.1. Attenuation of Stress Wave in Fractured Rock Mass

_{1}= 2 + b(D)

_{2}= 2 − b(D)

_{1}is the attenuation index of shock wave pressure; α

_{2}is the attenuation index of stress wave; b(D) is the lateral stress coefficient of damaged rock mass; C

_{d}is the crack density caused by damage; and μ

_{α}is the dynamic Poisson’s ratio, where μ

_{α}= 0.8 μ

_{c}and μ

_{c}is the static Poisson’s ratio of rock.

_{2}, the change in stress [14] can be represented by the shock pressure attenuation formula. Based on the relationship between stress and wave velocity [15,16,17], the change in the vibration velocity of the incident and transmitted particles at the joint interface can be expressed as

_{0}C

^{2}e

^{−α2t}/8

_{s}and t

_{p}are propagation times of the P-wave and S-wave, respectively; C is the detonation velocity, which is between 2000 and 4000 m/s; μ is the Poisson’s ratio, which is taken as 0.3; G is the shear modulus of the rock, which is taken as 30 GPa; P(t) is the impact pressure; ρ

_{0}is the density of the explosive, which is assumed as 1.63 g/cm

^{3}; α is attenuation index, which equals 4000; and t is the control load input termination time (0.05 s).

_{IP}and v

_{Ts}are particle vibration velocities of the incident P wave and transmitted S wave, respectively; and α is the incident angle, reflection angle, and transmission angle of the P wave. β is the reflection angle of the reflected S-wave and the transmission angle of transmitted S-wave. σ

_{d}is stress at the end of the joint under in situ stress, and τ

_{d}is the shear stress at the end of the joint under in situ stress. In this study, σ

_{1}and τ

_{1}are approximately equivalent to the in situ stress values at the midpoint of the joint.

#### 3.2. Constitutive Model of Fractured Rock Mass Blasting

_{1}represents the location of the axis of symmetry, and β 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

_{a}is the dynamic elastic modulus of rock, which is taken as 10 GPa. σ

_{x}, σ

_{y}

_{,}and τ

_{xy}are the stress components at the joint ends. u and v are the displacement components at the joint ends; r and θ are the polar diameter and polar angle from the joint ends, respectively. This paper studies the plane strain problem, where k = 3 to 4 μ.

## 4. Analysis of Stress and Displacement Distribution around Joints

_{x}is positive in the direction of 0°–330° and subjected to tensile stress, whereas it is negative in the direction between 330° and 360° under compressive stress. σ

_{x}is not symmetrical along the joint interface, and σ

_{x}stress gradient increases rapidly in the direction of 300°–330°, forming a stress concentration area. Figure 3b shows that at the joint end, σ

_{y}is positive in the direction of 0°–180° under tensile stress. On the other hand, it is negative in the direction of 180°–360° and subjected to compressive stress. Although stress is symmetrically distributed along the joint interface, it can be clearly found from the figure that the σ

_{y}stress field is not distributed symmetrically. Comparing Figure 3a,b, however, there are normal stress components in σ

_{x}, and σ

_{y}and σ

_{x}had a larger stress range. It can be seen from Figure 3c that the lateral displacement of the joint end is affected by the stress components in both directions. Although an intuitive understanding of the stress field distribution at the joint end can be obtained from the stress contour diagram, the asymmetry and continuous changes in the whole process still cannot be analyzed from the contour plot. Thus, the numerical simulation method is used to analyze the stress change in the wave acting on the mass point on the left side of the joint. Based on the following three figures, the stress field of the joint particle is asymmetric.

## 5. Analysis of Blasting Simulation

_{0}is the initial yield strength; C and P are constants related to material properties and C = 35, P = 3; $\stackrel{.}{\epsilon}$ is the strain rate; β is an adjustable parameter, β = 1; E

_{P}is the plastic hardening modulus, 23.7 MPa; ${\stackrel{.}{\epsilon}}_{\mathrm{ij}}^{\mathrm{p}}$ is the plastic strain rate; and ${\epsilon}_{\mathrm{eff}}^{\mathrm{p}}$ is the equivalent plastic strain.

_{vm}of the rock in the near explosion source area exceeds the set dynamic compressive strength σ

_{cd}, the point fails. In the far zone of the explosion source, when the calculated tensile stress σ

_{t}of the point in the rock mass is greater than the set dynamic tensile strength σ

_{td}, the point fails, and in the near zone of the explosion source, the calculated cumulative plastic strain εp of the point in the rock mass exceeds the set failure strain ε

_{pf}, the point fails. In short, the possibility of rock mass failure increases, which is convenient to more realistically simulate the dynamic force of rock mass after the detonation wave and symbiotic gas.

_{cd}and the static compressive strength σ

_{c}is as follows.

_{2}μ

^{2}and C

_{6}μ

^{3}are 0; C

_{0}= C

_{1}= C

_{2}= C

_{3}= C

_{6}= 0, C

_{4}= C

_{5}= γ − 1, where γ is the ratio of two pressures and volume specific heat.

_{1}, R

_{2,}and w are empirical parameters; and the explosives are selected according to the site conditions. The parameters are detailed in Table 3.

## 6. Conclusions

- (1)
- By constructing a nonlinear joint blasting model and introducing the detonation wave propagation velocity simplification into the vibration velocity of the incident particle at the joint interface, the incident P-wave incident joint is obtained. The peak value is at 3.0 s with a peak vibration velocity of 0.33 m/s; the S-wave reflected from the joint interface is first reflected backward and then forward. The peak vibration velocity of the particle is 0.027 m/s.
- (2)
- By combining with the relevant theories of stress and displacement field at the crack end of type I and II cracks, it is obtained that the joint presents asymmetric characteristics around the stress field. The end σ
_{x}is positive in the direction of 0°–330° subject to tensile stress, whereas σ_{y}is positive in the direction of 0°–180° under tensile stress; the longitudinal stress σ_{y}of the joint is low around the compressive stress distribution area. At this point, the rock material does not fail, and the stress concentration appears in the lower-right position. The lateral displacement of the joint ends is significantly affected by the stress components in both directions. - (3)
- Based on the analysis results of ANSYS, it is found that the intensity of the shock wave after detonation is greater than the strength of the rock. Then, the sub-layer shock wave supplements the energy of the shock wave that is not enough to break the rock and induces further cracking. Based on the analysis results on the attenuation of detonation wave energy, the stress exhibits a decreasing trend in the process. By constructing a variation diagram of the peak effective stress, it is found that the peak value first increases to 10–12 MPa and then shows a downward trend.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 4.**Blasting model of micro-delayed holes. A—Stemming 7 m, B—Explosive 6 m, C—Interval 5 m, D—Explosive 8 m, E—Joint.

**Figure 5.**Contours of effective stress. (

**a**) t = 1197 μs, (

**b**) t = 1498 μs, (

**c**) t = 2298 μs, (

**d**) t = 2999 μs, (

**e**) t = 3598 μs, (

**f**) t = 4798 μs.

**Figure 6.**Change in peak effective stress at selected points. (

**a**) Finite element simulation results, (

**b**) theoretical calculation results.

Density (g/cm^{3}) | Young’s Modulus (MPa) | Poisson’s Ratio | Compressive Strength (MPa) | Tangent Modulus (MPa) |
---|---|---|---|---|

1.85 | 1.2 | 0.38 | 0.8 | 0.1 |

Density (g/cm^{3}) | Young’s Modulus (×10^{4} MPa) | Poisson’s Ratio | Tensile Strength (MPa) | Compressive Strength (MPa) |
---|---|---|---|---|

2.43 | 5 | 0.26 | 5 | 130 |

Density (g/cm^{3}) | Blasting Speed (cm/us) | Blasting Pressure (GPa) | A | B | R_{1} | w | R_{2} |
---|---|---|---|---|---|---|---|

1.20 | 0.40 | 50 | 2.14 × 10^{11} | 1.82 × 10^{8} | 4.15 | 0.3 | 0.95 |

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**MDPI and ACS Style**

Zhang, P.; Bai, R.; Sun, X.; Wang, T.
Investigation of Rock Joint and Fracture Influence on Delayed Blasting Performance. *Appl. Sci.* **2023**, *13*, 10275.
https://doi.org/10.3390/app131810275

**AMA Style**

Zhang P, Bai R, Sun X, Wang T.
Investigation of Rock Joint and Fracture Influence on Delayed Blasting Performance. *Applied Sciences*. 2023; 13(18):10275.
https://doi.org/10.3390/app131810275

**Chicago/Turabian Style**

Zhang, Pengfei, Runcai Bai, Xue Sun, and Tianheng Wang.
2023. "Investigation of Rock Joint and Fracture Influence on Delayed Blasting Performance" *Applied Sciences* 13, no. 18: 10275.
https://doi.org/10.3390/app131810275