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

College of Civil Engineering and Architecture, Guangxi University, Nanning, Guangxi 530004, China School of Civil and Architectural Engineering, East China University of Technology, Nanchang, Jiangxi 330013, China School of Architectural Engineering, Guangzhou City Construction College, Guangzhou, Guangdong 510925, China &e Key Laboratory of Disaster Prevention and Structural Safety of Ministry of Education, Guangxi University, Nanning, Guangxi 530004, China Sanming Puyan Expressway Co. Ltd., Sanming, Fujian 365000, China


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][6][7][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][10][11][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][20][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. e 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.

Analysis of the Rock Failure Mechanism of
Shaped Charge Blasting e detonation products of conventional blasting scattered irregularly around the blast hole, and the cracks also expanded irregularly [22]. e 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. e detonation products generated by blasting will accumulate along the axis of the energy cavity to form a high-density, high-speed, and highpressure air flow, which is called shaped charge jet [24]. e 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.
During crack propagation, the stress intensity factor at the crack tip is as follows [25]: (1) ereinto, P is the explosive gas pressure in the fracture; F is the correction factor of stress intensity factor; r b is the blast hole radius; a is the fracture length; and σ θ is the tangential stress.
According to the theory of fracture mechanics, K 1 > K IC , the crack initiates and propagates, where K IC represents the fracture toughness of the rock. erefore, to ensure that the crack continues to grow, the pressure of detonation gas should meet the following conditions: (2) e 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, the P 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. erefore, 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.

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. e fracture mechanical behavior of this material is similar to that of brittle rock [26,27]. e 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. e 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. e structure design of the shaped charge is shown in Figure 2. e geometric size of the specimen was 300 mm × 200 mm × 100 mm. e blast hole was located in the center of the specimen, and the diameter of the blast hole was 12 mm. e specimen of PMMA is shown in Figure 3. e crack development was recorded by the digital laser dynamic caustics test system. e test results are shown in Figure 4. Next, numerical simulation analysis would be carried out for the blasting test to verify the effectiveness of the finite element solution.

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.
where p is the pressure, V is the volume, A, B, ω, R 1 , and R 2 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. e PVC energy gathering tube material adopts the model MAT_PLASTIC_KINEMATIC, and the mechanical parameter of the PVC is shown in Table 2.
e MAT_NULL model and the EOS_LINEAR_PO-LYNOMIAL 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.
e main rock parameter is shown in Table 3. Figure 5 shows the development of blasting cracks at different times. e 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.

Analysis of Numerical Simulation Results.
Compared with Figure 4, 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. e 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.  Shock and Vibration

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. e 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°. e calculation model is shown in Figure 6, and no reflective boundary condition is set around the model. Select a measuring point every 10 mm along the blast hole radial in the concentrating energy direction and the nonconcentrated energy direction. e 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. e layout of each measuring point is shown in Figure 7.

Analysis of Rock Crack Development with Blast Hole
Diameter 42 mm

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 5 μs to form a high-temperature, high-pressure, high-energy shaped charge jet. e 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.     Shock and Vibration e 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. e 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.

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. Figure 11 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      Figure 12. e results in Figure 12 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.

Analysis of Radial Decoupling Coefficient.
e radial decoupling coefficient changes with the change of blast hole diameter. e main crack length of different blast hole diameter blasting is shown in Table 4. e curve fitting between the length of the main crack and the diameter of the blast hole is shown in Figure 13. e correlation coefficients between the blast hole diameter and the length of the left and right main cracks have reached more than 0.98. e best blast hole diameter is 82 mm.
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.

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. e 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) ere 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. e 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.

Data Availability
e data supporting the results of this study can be obtained upon request to the corresponding author.

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of this paper.  Shock and Vibration 9