# Study on Dynamic Evolution Law of Blasting Cracks in Elliptical Bipolar Linear Shaped Charge Blasting

Authors: Bo Wu; Shixiang Xu; Guowang Meng; Junhua Cai; Han Wei; Hualong Li; Jinglong Zhang
Journal: Shock and Vibration
 (2021)

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## Abstract

Based on LS-DYNA numerical simulation analysis and comparison with laboratory tests, the blasting crack development dynamic evolution mechanism of elliptical bipolar linear shaped charge is analyzed. The development law of rock crack and optimal radial decoupling coefficient under different blast hole diameters were studied. The results revealed that the blasting with elliptical bipolar linear shaped charge had a remarkable effect on the directional crack formation, and the maximum effective stress of rock close to the position of shaped charge in the direction of concentrating energy is about 2.3 times of that in the direction of nonconcentrated energy. Moreover, the directional crack could be formed by blasting with elliptical bipolar linear shaped charge with different hole diameters, whilst the length of the main crack was related to the radial decoupling coefficient. Particularly, the main crack reached the longest when the radial decoupling coefficient was 3.36.


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## Body

## 1. Introduction

China has become the country with the largest scale, the largest number, and the highest difficulty of tunnel construction in the world [1]. According to statistics, by the end of 2020, China’s railway operating mileage had reached 145000 km, of which 16798 railway tunnels had been put into operation, with a total length of about 19630 km [2]. In addition to the tunnel boring machine, the drilling and blasting method is still widely utilized in rock tunnel construction. Improper control of traditional drilling and blasting methods would easily lead to engineering and social problems such as vibration hazard, environmental pollution, instability of surrounding rock, and serious overbreak and underbreak, which seriously restrict the construction process of tunnel engineering [3].In order to solve the above problems, directional fracture controlled blasting technology is often used at present. Many scholars have conducted in-depth research around the rock failure mechanism and directional fracture effect of directional fracture controlled blasting. For instance, Cho et al. [4] combined model test and numerical simulation method to study the influence of empty hole on crack propagation under different blasting and comprehensively analyzed the relationship between fracture energy and crack propagation. Yang et al. [5–8] adopted a testing system of digital laser dynamic caustics to study the influence of different cutting angle, depth, initial stress field, and other factors on the crack propagation of slotted cartridge blasting and then analyzed the mechanism of crack extension and penetration. Yue et al. [9–12] used a new testing system of digital laser dynamic caustics to carry out the experimental research on the development of blasting crack under slotted cartridge blasting, whilst obtained the crack extension velocity, acceleration, dynamic stress intensity factor at the front end of the crack, and the law of dynamic energy release rate. Wang [13] explored the formation of detonation and initial crack of slotted cartridge blasting and the relationship between decoupling coefficient and blasting damage based on numerical simulation. Luo et al. [14, 15] made a preliminary study on the formation of guided crack, crack initiation, propagation, and penetration of shaped charge. In 2006, the Sinohydro Engineering Bureau 8 Co., Ltd. developed an elliptical bipolar linear shaped charge (EBLSC), which performed well in practical engineering applications [16]. Li et al. [17, 18] further conducted theoretical analysis, numerical simulation, and experimental research on the blasting of EBLSC, and the research showed that this charge structure has a good application effect and prospect in presplit blasting. Subsequently, the elliptical bipolar linear shaped charge blasting technology has been widely promoted and applied in engineering. Wu et al. [19–21] carried out a preliminary study on the blasting mechanism, influencing factors, and crack development of elliptical bipolar linear shaped charge blasting.However, for the directional controlled blasting technology, most of the research studies are focused on the slotted cartridge blasting, while the research on the evolution law of shaped charge blasting crack is less. The rock breaking mechanism of elliptical bipolar shaped charge blasting excavation is not clear. In this paper, the rock failure mechanism, the temporal and spatial law of crack development, and decoupling coefficient of elliptical bipolar linear shaped charge blasting were studied. Moreover, the optimal decoupling coefficient was obtained via analyzing the crack development law of different blast hole diameters, which provide an important reference for practical engineering application.


## 2. Analysis of the Rock Failure Mechanism of Shaped Charge Blasting

The detonation products of conventional blasting scattered irregularly around the blast hole, and the cracks also expanded irregularly [22]. The shaped charge blasting uses a layer liner to change the structure of the explosive to make the detonation products accumulate in a specific direction and improve the destructive effect in a specific direction [23].An energy cavity is set at the symmetrical position on both sides of the shaped charge. The detonation products generated by blasting will accumulate along the axis of the energy cavity to form a high-density, high-speed, and high-pressure air flow, which is called shaped charge jet [24]. The shaped charge jet penetrates the rock and produces the initial guide crack, which provides a directional effect for the subsequent explosion stress wave and blasting gas to further expand the crack. According to the rock fracture mechanics, a dynamic fracture mechanics model of shaped charge blasting is established, as shown in Figure 1.Figure 1
Mechanical model of cracking due to shaped charge blasting.During crack propagation, the stress intensity factor at the crack tip is as follows [25]:(1)K1=PFπrb+a+σθπa.Thereinto,P is the explosive gas pressure in the fracture; F is the correction factor of stress intensity factor; rb is the blast hole radius; a is the fracture length; and σθ is the tangential stress.According to the theory of fracture mechanics,K1 > KIC, the crack initiates and propagates, where KIC represents the fracture toughness of the rock. Therefore, to ensure that the crack continues to grow, the pressure of detonation gas should meet the following conditions:(2)P>KIC−σθπaFπrb+a.The guiding crack penetrated by the shaped energy jet is much larger than the other small cracks in the crushing zone. After blasting, a large amount of high-pressure explosive gas will be introduced, and the pressure of the explosive gas in the concentrating energy direction will increase; that is, theP will increase; according to the law of conservation of energy, the effect of explosive gas in the nonconcentrated energy direction will be weakened, and the P will decrease. Therefore, while the structure of the shaped charge leads to an increase in the evolution ability of cracks in the direction of concentrating energy, it also reduces the ability of evolution of cracks in the direction of nonconcentrated energy.


## 3. Validation of Numerical Solution

### 3.1. Explosion Test of PMMA

Polymethyl methacrylate (PMMA) is usually used as an ideal test material to study the crack propagation process of PMMA under laboratory conditions. Its main advantage is transparency, which makes the crack morphology easy to be observed directly by naked eyes. The fracture mechanical behavior of this material is similar to that of brittle rock [26, 27]. The test results of Rossmanith et al. [28] also suggested that the blasting cracks could be divided into the following three areas: crushing zone, radial microcrack zone, and radial crack zone. The crack morphology of PMMA is very similar to that of rock under dynamic loading. Hence, it can be considered that the blasting test results of PMMA are consistent with those of rock materials, which is suitable for exploring the mechanism of crack initiation and propagation near the blast hole and in the far-field area.Che [29] carried out the blasting test of a shaped charge with PMMA, wherein the outer diameter of the shaped charge was 7 mm, the inner diameter was 5 mm, the energy gathering tube was made of PVC material, and the thickness of the shell of the energy gathering tube was 1 mm. The structure design of the shaped charge is shown in Figure 2. The geometric size of the specimen was 300 mm × 200 mm × 100 mm. The blast hole was located in the center of the specimen, and the diameter of the blast hole was 12 mm. The specimen of PMMA is shown in Figure 3. The crack development was recorded by the digital laser dynamic caustics test system.Figure 2
Structure design of shaped charge [29].Figure 3
The specimen of PMMA [29].The test results are shown in Figure4. Next, numerical simulation analysis would be carried out for the blasting test to verify the effectiveness of the finite element solution.Figure 4
Test results of blasting crack of shaped charge [29]. (a) t = 0 μs, (b) t = 20 μs, (c) t = 80 μs, (d) t = 120 μs, (e) t = 400 μs, (f) t = 600 μs.
(a)(b)(c)(d)(e)(f)


### 3.2. Numerical Model

Using LS-DYNA nonlinear dynamic analysis software to carry out numerical simulation analysis, the geometric size of the model is the same as the above model test. A nonreflective boundary condition is added to simulate an infinite plane to eliminate the interference of reflected waves at the boundary. In the numerical model, the material model of the explosive is characterized by MAT_HIGH_EXPLOSIVE_BURN, and the relationship between the pressure and volume of the explosive after detonation is described by the JWL state equation.(3)p=A1−ωR1V−R1V+B1−ωR2V+ωEV,where p is the pressure, V is the volume, A, B, ω, R1, and R2 are the basic parameters of the equation of state, and E is the initial internal energy per unit volume. Explosive and state equation parameters are shown in Table 1.Table 1
Explosive and parameters of equation of state.
ρ (g·cm−3)νD (cm·μs−1)A (GPa)B (GPa)R1R2ωE (GPa)1.30.4214.40.1824.20.90.154.192The PVC energy gathering tube material adopts the model MAT_PLASTIC_KINEMATIC, and the mechanical parameter of the PVC is shown in Table2.Table 2
Mechanical parameter of PVC.
MaterialDensity (g·cm−3)Elastic modulus (GPa)Poisson’s ratio (μ)PVC1.33.00.38The MAT_NULL model and the EOS_LINEAR_POLYNOMIAL state equation are used to simulate air, the HJC constitutive model is used for rock, and the failure mode is added to analyze crack development. The main rock parameter is shown in Table3.Table 3
Material parameters of the HJC model.
ParametersPropertyValueρ (g·cm−3)Density2.18G (GPa)Shear modulus14.86ANormalized cohesive strength0.79BNormalized pressure hardening1.60CStrain rate coefficient0.007SfmaxNormalized maximum strength7εfminAmount of plastic strain before fracture0.01NPressure hardening exponent0.61T (MPa)Maximum tensile hydrostatic pressure4D1Damage constant 10.04D2Damage constant 21K1 (GPa)Pressure constant 185K2 (GPa)Pressure constant 2−171K3 (GPa)Pressure constant 3200


### 3.3. Analysis of Numerical Simulation Results

Figure5 shows the development of blasting cracks at different times. The blasting cavity is formed after 20 μs of detonation, and the length of the cracks in the concentrating energy direction is significantly greater than the length of the cracks in other directions. At 80 μs, microcracks appear in other directions, and the crack in the concentrating energy direction keeps growing. At 120 μs, the crack in the direction of energy accumulation and other directions keeps growing. At 600 μs, the crack tends to stop and the crack in the concentrating energy direction is always larger than that of the other directions.Figure 5
Numerical simulation results of blasting crack development of shaped charge. (a)t = 20 μs, (b) t = 80 μs, (c) t = 120 μs, (d) t = 600 μs.
(a)(b)(c)(d)Compared with Figure4, the results in Figure 5 indicate that the numerical simulation completely reproduces the crushing zone around the hole, the crack initiation, and development process in the concentrating energy direction and other directions of the PMMA under the shaped charge blasting. The final distribution of blasting cracks is in good agreement with the experimental results, thus proving the correctness of the established model and its numerical solution. Next, the numerical model will be used to study the evolution law of blasting cracks in elliptical bipolar linear shaped charge blasting.


## 3.1. Explosion Test of PMMA

Polymethyl methacrylate (PMMA) is usually used as an ideal test material to study the crack propagation process of PMMA under laboratory conditions. Its main advantage is transparency, which makes the crack morphology easy to be observed directly by naked eyes. The fracture mechanical behavior of this material is similar to that of brittle rock [26, 27]. The test results of Rossmanith et al. [28] also suggested that the blasting cracks could be divided into the following three areas: crushing zone, radial microcrack zone, and radial crack zone. The crack morphology of PMMA is very similar to that of rock under dynamic loading. Hence, it can be considered that the blasting test results of PMMA are consistent with those of rock materials, which is suitable for exploring the mechanism of crack initiation and propagation near the blast hole and in the far-field area.Che [29] carried out the blasting test of a shaped charge with PMMA, wherein the outer diameter of the shaped charge was 7 mm, the inner diameter was 5 mm, the energy gathering tube was made of PVC material, and the thickness of the shell of the energy gathering tube was 1 mm. The structure design of the shaped charge is shown in Figure 2. The geometric size of the specimen was 300 mm × 200 mm × 100 mm. The blast hole was located in the center of the specimen, and the diameter of the blast hole was 12 mm. The specimen of PMMA is shown in Figure 3. The crack development was recorded by the digital laser dynamic caustics test system.Figure 2
Structure design of shaped charge [29].Figure 3
The specimen of PMMA [29].The test results are shown in Figure4. Next, numerical simulation analysis would be carried out for the blasting test to verify the effectiveness of the finite element solution.Figure 4
Test results of blasting crack of shaped charge [29]. (a) t = 0 μs, (b) t = 20 μs, (c) t = 80 μs, (d) t = 120 μs, (e) t = 400 μs, (f) t = 600 μs.
(a)(b)(c)(d)(e)(f)


## 3.2. Numerical Model

Using LS-DYNA nonlinear dynamic analysis software to carry out numerical simulation analysis, the geometric size of the model is the same as the above model test. A nonreflective boundary condition is added to simulate an infinite plane to eliminate the interference of reflected waves at the boundary. In the numerical model, the material model of the explosive is characterized by MAT_HIGH_EXPLOSIVE_BURN, and the relationship between the pressure and volume of the explosive after detonation is described by the JWL state equation.(3)p=A1−ωR1V−R1V+B1−ωR2V+ωEV,where p is the pressure, V is the volume, A, B, ω, R1, and R2 are the basic parameters of the equation of state, and E is the initial internal energy per unit volume. Explosive and state equation parameters are shown in Table 1.Table 1
Explosive and parameters of equation of state.
ρ (g·cm−3)νD (cm·μs−1)A (GPa)B (GPa)R1R2ωE (GPa)1.30.4214.40.1824.20.90.154.192The PVC energy gathering tube material adopts the model MAT_PLASTIC_KINEMATIC, and the mechanical parameter of the PVC is shown in Table2.Table 2
Mechanical parameter of PVC.
MaterialDensity (g·cm−3)Elastic modulus (GPa)Poisson’s ratio (μ)PVC1.33.00.38The MAT_NULL model and the EOS_LINEAR_POLYNOMIAL state equation are used to simulate air, the HJC constitutive model is used for rock, and the failure mode is added to analyze crack development. The main rock parameter is shown in Table3.Table 3
Material parameters of the HJC model.
ParametersPropertyValueρ (g·cm−3)Density2.18G (GPa)Shear modulus14.86ANormalized cohesive strength0.79BNormalized pressure hardening1.60CStrain rate coefficient0.007SfmaxNormalized maximum strength7εfminAmount of plastic strain before fracture0.01NPressure hardening exponent0.61T (MPa)Maximum tensile hydrostatic pressure4D1Damage constant 10.04D2Damage constant 21K1 (GPa)Pressure constant 185K2 (GPa)Pressure constant 2−171K3 (GPa)Pressure constant 3200


## 3.3. Analysis of Numerical Simulation Results

Figure5 shows the development of blasting cracks at different times. The blasting cavity is formed after 20 μs of detonation, and the length of the cracks in the concentrating energy direction is significantly greater than the length of the cracks in other directions. At 80 μs, microcracks appear in other directions, and the crack in the concentrating energy direction keeps growing. At 120 μs, the crack in the direction of energy accumulation and other directions keeps growing. At 600 μs, the crack tends to stop and the crack in the concentrating energy direction is always larger than that of the other directions.Figure 5
Numerical simulation results of blasting crack development of shaped charge. (a)t = 20 μs, (b) t = 80 μs, (c) t = 120 μs, (d) t = 600 μs.
(a)(b)(c)(d)Compared with Figure4, the results in Figure 5 indicate that the numerical simulation completely reproduces the crushing zone around the hole, the crack initiation, and development process in the concentrating energy direction and other directions of the PMMA under the shaped charge blasting. The final distribution of blasting cracks is in good agreement with the experimental results, thus proving the correctness of the established model and its numerical solution. Next, the numerical model will be used to study the evolution law of blasting cracks in elliptical bipolar linear shaped charge blasting.


## 4. Analysis on Crack Evolution of Elliptical Bipolar Linear Shaped Charge Blasting

### 4.1. Geometric Model of Numerical Calculation

In order to analyze the dynamic evolution law of blasting cracks with different blast hole diameters, the quasi two-dimensional calculation models with various blast hole diameters of 42 mm, 50 mm, 60 mm, 70 mm, 80 mm, 90 mm, and 100 mm were established, respectively. The shaped charge is an elliptical bipolar linear structure, the energy gathering tube is made of PVC material, the thickness of the shell of the energy gathering tube is 2 mm, the thickness of the layer liner is 1.4 mm, and the angle of the shaped charge groove is 70°. The calculation model is shown in Figure6, and no reflective boundary condition is set around the model.Figure 6
Schematic of the numerical calculation model.Select a measuring point every 10 mm along the blast hole radial in the concentrating energy direction and the nonconcentrated energy direction. The concentrating energy direction is numbered as #G1 ∼ #G5 from near to far, and the nonconcentrated energy direction is numbered as #N1 ∼ #N5 from near to far. The layout of each measuring point is shown in Figure7.Figure 7
The layout of measuring points.


### 4.2. Analysis of Rock Crack Development with Blast Hole Diameter 42 mm

#### 4.2.1. Analysis of Initial Crack Formation

After the shaped charge is detonated in the blast hole, the detonation wave acts on the shaped charge cover with huge pressure at 5μs to form a high-temperature, high-pressure, high-energy shaped charge jet. The jet first acts on the blast hole wall and forms a guide crack on the rock in this direction, as shown in Figure 8. In other directions of the blast hole, the shell of the shaped charge has instantaneous buffering and inhibiting effect on the detonation products and the air medium between the shell of the shaped charge and the blast hole wall has a buffering effect, which greatly reduces the direct effect and damage degree of the shock wave on the blast hole wall, thus inhibiting the development of cracks.Figure 8
Initial crack formation. (a)t = 5 μs, (b) t = 8 μs.
(a)(b)


#### 4.2.2. Stress Time History Analysis

The stress time history curves of each measuring point are shown in Figures9 and 10.Figure 9
Effective stress time history curve in the concentrating energy direction.Figure 10
Effective stress time history curve in the nonconcentrated energy direction.The above curves suggested that the peak attenuation rate of rock equivalent stress is very fast along the center of the blast hole outward in both the concentrating energy and nonconcentrated energy directions. Especially, the maximum equivalent stress in the direction of shaped charge is 39.25 MPa, the maximum equivalent stress in the direction of nonshaped charge is 17.33 MPa, and the maximum effective stress in the direction of concentrating energy is about 2.3 times of that in the direction of nonconcentrated energy. The results indicated that the ability of rock penetration in the direction of concentrating energy is much greater than that in the direction of nonconcentrated energy.


#### 4.2.3. Crack Propagation Analysis

After the initial guide crack is formed, the explosive detonation product fills the whole blasting cavity, and the quasi-static load is applied to the rock on the blast hole wall. Under the quasi-static load and stress concentration, the tip of the guide crack forms a long crack and propagates.Figure11 shows the crack development of 42 mm blast hole in elliptical bipolar linear shaped charge blasting. At 80 μs, microcracks begin to appear outside the direction of concentrating energy. At 120 μs, both main cracks and microcracks keep growing. At 300 μs, the cracks outside the direction of concentrating energy tend to stop growing. At 600 μs, all cracks tend to stop growing. The cracks in the direction of concentrating energy are always much larger than those in other directions.Figure 11
Crack development with blast hole diameter 42 mm. (a)t = 80 μs, (b) t = 120 μs. (c) t = 300 μs, (d) t = 600 μs.
(a)(b)(c)(d)


### 4.3. Analysis of Rock Cracks Development with Different Blast Hole Diameter

Based on the established numerical model, the crack development of rock with various diameters of 50 mm, 60 mm, 70 mm, 80 mm, 90 mm, and 100 mm is analyzed. The results of crack development are shown in Figure12.Figure 12
Crack development with different blast hole diameters. (a)D = 50 mm, (b) D = 60 mm, (c) D = 70 mm, (d) D = 80 mm, (e) D = 90 mm, (f) D = 100 mm.
(a)(b)(c)(d)(e)(f)The results in Figure12 indicate that under the blasting of an elliptical bipolar linear shaped charge with different blast hole diameters, two main cracks are formed in the left and right concentrating energy direction, and random secondary cracks are also formed in other directions. With the increase in blast hole diameter, the length of the main crack first increases and then decreases, indicating that there is an optimal blast hole diameter.


### 4.4. Analysis of Radial Decoupling Coefficient

The radial decoupling coefficient changes with the change of blast hole diameter. The main crack length of different blast hole diameter blasting is shown in Table4.Table 4
Main crack length of different hole diameter blasting.
Blast hole diameter42 mm (cm)50 mm (cm)60 mm (cm)70 mm (cm)80 mm (cm)90 mm (cm)100 mm (cm)Left main crack length49.2754.4360.1765.2167.1465.5462.16Right main crack length49.2757.4062.1665.3267.4465.6463.16The curve fitting between the length of the main crack and the diameter of the blast hole is shown in Figure13. The correlation coefficients between the blast hole diameter and the length of the left and right main cracks have reached more than 0.98. The best blast hole diameter is 82 mm.Figure 13
Relation curve between main crack length and blast hole diameter.Moreover, the equivalent charge diameter of the elliptical bipolar linear shaped charge blasting is 24.4 mm based on the above established numerical simulation model, and thus the best radial decoupling coefficient calculated is 3.36.


## 4.1. Geometric Model of Numerical Calculation

In order to analyze the dynamic evolution law of blasting cracks with different blast hole diameters, the quasi two-dimensional calculation models with various blast hole diameters of 42 mm, 50 mm, 60 mm, 70 mm, 80 mm, 90 mm, and 100 mm were established, respectively. The shaped charge is an elliptical bipolar linear structure, the energy gathering tube is made of PVC material, the thickness of the shell of the energy gathering tube is 2 mm, the thickness of the layer liner is 1.4 mm, and the angle of the shaped charge groove is 70°. The calculation model is shown in Figure6, and no reflective boundary condition is set around the model.Figure 6
Schematic of the numerical calculation model.Select a measuring point every 10 mm along the blast hole radial in the concentrating energy direction and the nonconcentrated energy direction. The concentrating energy direction is numbered as #G1 ∼ #G5 from near to far, and the nonconcentrated energy direction is numbered as #N1 ∼ #N5 from near to far. The layout of each measuring point is shown in Figure7.Figure 7
The layout of measuring points.


## 4.2. Analysis of Rock Crack Development with Blast Hole Diameter 42 mm

### 4.2.1. Analysis of Initial Crack Formation

After the shaped charge is detonated in the blast hole, the detonation wave acts on the shaped charge cover with huge pressure at 5μs to form a high-temperature, high-pressure, high-energy shaped charge jet. The jet first acts on the blast hole wall and forms a guide crack on the rock in this direction, as shown in Figure 8. In other directions of the blast hole, the shell of the shaped charge has instantaneous buffering and inhibiting effect on the detonation products and the air medium between the shell of the shaped charge and the blast hole wall has a buffering effect, which greatly reduces the direct effect and damage degree of the shock wave on the blast hole wall, thus inhibiting the development of cracks.Figure 8
Initial crack formation. (a)t = 5 μs, (b) t = 8 μs.
(a)(b)


### 4.2.2. Stress Time History Analysis

The stress time history curves of each measuring point are shown in Figures9 and 10.Figure 9
Effective stress time history curve in the concentrating energy direction.Figure 10
Effective stress time history curve in the nonconcentrated energy direction.The above curves suggested that the peak attenuation rate of rock equivalent stress is very fast along the center of the blast hole outward in both the concentrating energy and nonconcentrated energy directions. Especially, the maximum equivalent stress in the direction of shaped charge is 39.25 MPa, the maximum equivalent stress in the direction of nonshaped charge is 17.33 MPa, and the maximum effective stress in the direction of concentrating energy is about 2.3 times of that in the direction of nonconcentrated energy. The results indicated that the ability of rock penetration in the direction of concentrating energy is much greater than that in the direction of nonconcentrated energy.


### 4.2.3. Crack Propagation Analysis

After the initial guide crack is formed, the explosive detonation product fills the whole blasting cavity, and the quasi-static load is applied to the rock on the blast hole wall. Under the quasi-static load and stress concentration, the tip of the guide crack forms a long crack and propagates.Figure11 shows the crack development of 42 mm blast hole in elliptical bipolar linear shaped charge blasting. At 80 μs, microcracks begin to appear outside the direction of concentrating energy. At 120 μs, both main cracks and microcracks keep growing. At 300 μs, the cracks outside the direction of concentrating energy tend to stop growing. At 600 μs, all cracks tend to stop growing. The cracks in the direction of concentrating energy are always much larger than those in other directions.Figure 11
Crack development with blast hole diameter 42 mm. (a)t = 80 μs, (b) t = 120 μs. (c) t = 300 μs, (d) t = 600 μs.
(a)(b)(c)(d)


## 4.2.1. Analysis of Initial Crack Formation

After the shaped charge is detonated in the blast hole, the detonation wave acts on the shaped charge cover with huge pressure at 5μs to form a high-temperature, high-pressure, high-energy shaped charge jet. The jet first acts on the blast hole wall and forms a guide crack on the rock in this direction, as shown in Figure 8. In other directions of the blast hole, the shell of the shaped charge has instantaneous buffering and inhibiting effect on the detonation products and the air medium between the shell of the shaped charge and the blast hole wall has a buffering effect, which greatly reduces the direct effect and damage degree of the shock wave on the blast hole wall, thus inhibiting the development of cracks.Figure 8
Initial crack formation. (a)t = 5 μs, (b) t = 8 μs.
(a)(b)


## 4.2.2. Stress Time History Analysis

The stress time history curves of each measuring point are shown in Figures9 and 10.Figure 9
Effective stress time history curve in the concentrating energy direction.Figure 10
Effective stress time history curve in the nonconcentrated energy direction.The above curves suggested that the peak attenuation rate of rock equivalent stress is very fast along the center of the blast hole outward in both the concentrating energy and nonconcentrated energy directions. Especially, the maximum equivalent stress in the direction of shaped charge is 39.25 MPa, the maximum equivalent stress in the direction of nonshaped charge is 17.33 MPa, and the maximum effective stress in the direction of concentrating energy is about 2.3 times of that in the direction of nonconcentrated energy. The results indicated that the ability of rock penetration in the direction of concentrating energy is much greater than that in the direction of nonconcentrated energy.


## 4.2.3. Crack Propagation Analysis

After the initial guide crack is formed, the explosive detonation product fills the whole blasting cavity, and the quasi-static load is applied to the rock on the blast hole wall. Under the quasi-static load and stress concentration, the tip of the guide crack forms a long crack and propagates.Figure11 shows the crack development of 42 mm blast hole in elliptical bipolar linear shaped charge blasting. At 80 μs, microcracks begin to appear outside the direction of concentrating energy. At 120 μs, both main cracks and microcracks keep growing. At 300 μs, the cracks outside the direction of concentrating energy tend to stop growing. At 600 μs, all cracks tend to stop growing. The cracks in the direction of concentrating energy are always much larger than those in other directions.Figure 11
Crack development with blast hole diameter 42 mm. (a)t = 80 μs, (b) t = 120 μs. (c) t = 300 μs, (d) t = 600 μs.
(a)(b)(c)(d)


## 4.3. Analysis of Rock Cracks Development with Different Blast Hole Diameter

Based on the established numerical model, the crack development of rock with various diameters of 50 mm, 60 mm, 70 mm, 80 mm, 90 mm, and 100 mm is analyzed. The results of crack development are shown in Figure12.Figure 12
Crack development with different blast hole diameters. (a)D = 50 mm, (b) D = 60 mm, (c) D = 70 mm, (d) D = 80 mm, (e) D = 90 mm, (f) D = 100 mm.
(a)(b)(c)(d)(e)(f)The results in Figure12 indicate that under the blasting of an elliptical bipolar linear shaped charge with different blast hole diameters, two main cracks are formed in the left and right concentrating energy direction, and random secondary cracks are also formed in other directions. With the increase in blast hole diameter, the length of the main crack first increases and then decreases, indicating that there is an optimal blast hole diameter.


## 4.4. Analysis of Radial Decoupling Coefficient

The radial decoupling coefficient changes with the change of blast hole diameter. The main crack length of different blast hole diameter blasting is shown in Table4.Table 4
Main crack length of different hole diameter blasting.
Blast hole diameter42 mm (cm)50 mm (cm)60 mm (cm)70 mm (cm)80 mm (cm)90 mm (cm)100 mm (cm)Left main crack length49.2754.4360.1765.2167.1465.5462.16Right main crack length49.2757.4062.1665.3267.4465.6463.16The curve fitting between the length of the main crack and the diameter of the blast hole is shown in Figure13. The correlation coefficients between the blast hole diameter and the length of the left and right main cracks have reached more than 0.98. The best blast hole diameter is 82 mm.Figure 13
Relation curve between main crack length and blast hole diameter.Moreover, the equivalent charge diameter of the elliptical bipolar linear shaped charge blasting is 24.4 mm based on the above established numerical simulation model, and thus the best radial decoupling coefficient calculated is 3.36.


## 5. Conclusions

(1)
Under the elliptical bipolar linear shaped charge blasting, the damage ability of rock penetrating in the concentrating energy direction is much greater than that in the nonconcentrated energy direction. The maximum effective stress of rock closed to the position of shaped charge in the concentrating energy direction is about 2.3 times of that in the direction of nonconcentrated energy.(2)
Under the elliptical bipolar linear shaped charge blasting, the left and right main cracks can be formed in the concentrating energy direction with different hole diameters, and the random secondary cracks can be formed in other directions.(3)
There is a certain relationship between the length of the main crack and the diameter of the blast hole under elliptical bipolar linear shaped charge blasting. The curve fitting shows that the main crack is the longest when the diameter of the blast hole is 82 mm; that is, the best decoupling coefficient is 3.36.



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