# Numerical Simulation on Selection of Optimal Delay Time for Precise Delay Blasting

Authors: Jun Ma; Xianglong Li; Jianguo Wang; Qiang Li; Ting Zuo; Xiao Wu; Meng Hou
Journal: Shock and Vibration
 (2021)

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## Abstract

In the traditional presplitting blasting, the presplit holes are generally uniformly initiated, which causes local damage to the retained rock mass while forming the damping ditch. In order to determine the optimal delay time of the precise delay initiation hole by hole, the finite element software ANSYS/LS-DYNA is used to build a blasting model of the concrete, which includes concrete, explosive, and air to simulate the crack forming process of the presplitting hole under various initiation modes. Four kinds of initiation modes for blasting, namely, simultaneous initiation with 0 ms, 9 ms, 12 ms, and 15 ms of delay between adjacent holes, are set up to determine the exact delay time of the best presplitting effect. The simulation results show that when the prehole detonates simultaneously, the inner hole crack penetrates the fastest, but the peak stress around the hole is up to 147.9 MPa. When interhole delayed initiation is used, although the time of interhole crack penetration is prolonged, the stress coupling is generated around the precrack and the maximum stress is obviously reduced. The maximum stress generated under the three delayed initiation conditions is only 76.8 MPa. Considering the requirement of damage control of surrounding rock mass and the rapid formation of precrack, 9 ms delay time is determined as the precise delay time of this test.


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## Body

## 1. Introduction

When presplitting blasting is used in the engineering practice, it is important to form satisfying precracks of the presplit holes and minimize the damage on the surrounding rock by blasting. Traditionally, the presplitting blasting mainly adopts the simultaneous initiation method, and the explosion stress waves superimpose onto each other during the joint formation process, which causes great damage to the surrounding rock. It is generally believed that the precrack is formed due to the combined action of the explosion shock wave and the explosion gas, and the strain wave also makes a contribution to the crack development process [1]. Liu et al. [2, 3] studied the crack propagation from two aspects, namely, the charge amount and charge mode, and found that the mechanical characteristics of the primary cracks and the secondary cracks are similar when the same charge amount is adopted. Through analysis by LS-DYNA, Qu and Liu [4] found the joint effect is the best when the joints are perpendicular to the connection line of the blast holes. Besides, some researchers studied the mechanism and propagation regularity of the precrack cracking through other methods and influencing factors [5–7], and the results show that when the presplitting blasting is implemented, the simultaneous blasting exhibits the best cracking effect, but the construction environment is very complex which requires controlled presplitting. According to the maximum single-shot charge of the split blasting, segmented initiation of the presplit holes is required, which reduces the blasting vibration and has no influence on the effect of the presplitting blasting at the same time. Yu et al. [8, 9] used numerical simulation software to simulate the blasting situation of double hole under various ground stress conditions and joint angle and obtained the influence of these two factors on the expansion of blasting lines. Wang and Chen [10] used numerical methods to study the influence of charge structures on surrounding rock damage caused by blasting; the numerical simulation results show that the charging structure of water and slotted pipe can achieve a good blasting effect and reduce the damage of surrounding rock. Guo et al. [11] used finite element software ANASYS and LS-DYNA to analyze the effect of coal mass on cracking under shaped energy blasting, and the expansion mechanism was studied through this numerical model. In the work of Gao and Lei [12], finite element software was used to simulate the cutting and excavation process and analyze the stress field and domino effects (such as von Mises) in the rock during the millisecond delay blasting process, along with the propagation regularity of the particle vibration velocity. In addition, numerical software can also be applied to simulate the tunnel excavation in the blasting tests and the crack propagation under blasting load [13–15].However, there are too many influencing factors in minefield tests and model tests that could not be accurately analyzed singly through the tests. In comparison, this problem can be solved by numerical simulation analysis methods which can set various model parameters and experimental conditions. The research needs for a single factor can be achieved by combining the results of the field tests and model tests, thus obtaining comprehensive and accurate analysis results for specific research purposes. In recent years, the application of digital electronic detonators makes the research on precise time-delay controlled blasting technology a hot spot. In the past, presplitting blasting was usually blasted at the same time due to its characteristics, but the damage on surrounding rock is more serious when it is blasted at the same time, so the time-delay controlled blasting technology is proposed. The application of the controlled blasting technology of presplit blasting is a breakthrough technology. This paper intends to use the finite element software ANSYS/LS-DYNA to study the cracking mechanism of hole-by-hole presplitting blasting and the effect of delay time on the cracking mechanism and then to determine the optimal delay time.


## 2. Numerical Simulation Process

### 2.1. Parameters Setting of the Numerical Model

The concrete model test by Wu et al. [16] is based on the similarity theorem, and in this test, the effect of hole spacing and the noncoupling coefficient on the precrack width and half hole rate was analyzed. It was found that when the noncoupling coefficient is 2.16 and the hole spacing is 8 times of the aperture, the effect of crack formation and protection of surrounding rock is the best. According to this model test, the parameters such as noncoupling coefficient, blast hole diameter, hole distance, explosive type, and required charge amount were determined through numerical simulations. The actual pouring model picture is shown in Figure 1, where the hole distance is 104 mm and the hole diameter is 13 mm. The influence of related parameters on the blasting is studied. The size of the numerical model is 700 mm × 600 mm × 5 mm, and the hole diameter is 13 mm, while the diameter of the cartridge is 6 mm. Besides, the hole distance is set as 104 mm and the explosive parameter in the calculation model is set as hexogen.Figure 1
The model cast by Wu Xiao.


### 2.2. Establishment of the Numerical Model

The numerical model takes the model test as the prototype, which consists of three components: air, explosives, and concrete. Referring to the literature [17], keywords are used to define concrete and explosives, and the JWL equation of state is used to describe the relevant parameters of the explosives, as shown in the following equation:(1)P=A1−ωR1Ve−R1V+B1−ωR2Ve−R2V+ωE0V,where V is the relative specific volume of the detonation product of dimensionless; E0 is the initial internal energy of the explosive per unit volume (Pa); ρ is the explosive density (g/cm3); D is the detonation velocity (m/s); P is the explosion pressure (kPa); and A, B, R1, R2, and ω are all parameters related to explosive materials. The simulated explosive is set to hexogen, and the core diameter is about 6 mm. The corresponding model parameters are shown in Table 1.Table 1
Main parameters of the material model of hexogen explosive.
Parameterρ (g/cm3)D (m/s)P (GPa)ABR1R2ωEo (GJ·m−3)Value1.5700023.1611.310.654.41.20.328.13During the explosion process, the loading strain rate of the explosive medium, i.e., concrete, increases as the burst center distance of the media decreases, the maximum value of which can reach 104 s−1. The material keyword ∗MAT_JOHNSON_HOLMGUIST_CONCRETE is used to define the concrete medium. When the strain rate increases, the dynamic compressive strength of the concrete also increases, with the concrete material deforming at the same time.Equation (2) is used to express the strength of the HJC material model [18, 19]:(2)σ∗=A1−D+BP∗N1+Clnε∗.In the formula,σ∗=σ/fc is the ratio of the actual equivalent stress to the static yield strength, where σ is the actual equivalent stress and fc is the quasistatic uniaxial compressive strength; P∗=P/fc is the dimensionless pressure, where p is the actual hydrostatic pressure of the unit; ε∗=ε/ε0 is the dimensionless strain rate, where ε˙ is the true strain rate and ε˙0 is the reference strain rate; A is the normalized viscous strength; B is the normalized pressure hardening coefficient; C is the strain rate influence coefficient; N is the normalized pressure hardening index; D is the damage factor; and Smax is the maximum value that the normalized equivalent stress can reach.The calculation of damage factorD (0 ≤ D ≤ 1) is shown in equations (3) and (4):(3)D=∑Δεp+ΔUpεpf+μpf,(4)fp=εpf+μpf=D1p∗+T∗D2,where Δεp is the increment of the equivalent plastic strain, Δμp is the increment of the equivalent volumetric strain, f(p) is the plastic strain when the material breaks under normal pressure p, p∗ is the normalized pressure, T∗ is the maximum tensile static pressure which the material can withstand, and D1 and D2 are damage constants.The HJC model uses a segmented equation of state to describe the relationship between the static water pressure and volume strain of concrete and uses different polynomial equations to describe the relationship between the static water pressure and volume strain in the elastic zone, the crushing zone, and the compaction zone in the concrete compression process.In the elastic zone0<μ≤μcrush,(5)P=Kμ,where P is the hydrostatic pressure of the element, K is the elastic modulus of the concrete element, and µ is the volume strain of the element.In the crushing area (μcrush<μ≤μlock),(6)P=Pcrush+Klockμ−μcrush,where Pcrush is the hydrostatic pressure in the unit corresponding; Klock=Plock−Pcrush/μlock−μcrush, where μcrush is the volume strain at the crush point, Plock is the hydrostatic pressure in the unit corresponding to μlock, and μlock is the volume strain at compaction point.In the compaction zone (μ≥μlock),(7)P=K1μ¯+K2μ¯2+K3μ¯3.In the formula,μ¯=μ−μlock/1+μlock is the corrected volume strain and K1, K2, and K3 are the material constants of concrete.Table2 shows the main parameters of the HJC concrete constitutive model. Among them, G, FC, PC, and T are the mechanical test results. The remaining parameters are derived from the test with large strain, high strain rate, and high pressure of reference [20].Table 2
Parameters of the HJC constitutive model of concrete.
Item nameValueItem nameValueDensityρo/(g·cm−3)2.300Reference strain rate EPSO1.000Shear modulusG/Pa9.681Minimum plastic strain EFMIN0.010Strength parameter 1A0.300Strength parameter 5 SFMAX7.000Strength parameter 2B1.730Volume pressure during crushingPC/1011 Pa8.000E − 05Strength parameter 3C0.005Volume strain during crushing UC6.200E − 04Strength parameter 4N0.790Ultimate volume stress PL/1011 Pa3.470E − 02Static yield strengthFc/1011 Pa2.400E − 04Limit volume strain UL8.696E − 02Tensile strengthT/1011 Pa3.037E-05Damage constant 1D10.04Damage constant 2D21.0Pressure parameter 2K2−2.430Pressure parameter 1K11.160Pressure parameter 3K35.06The model is divided into a mapped grid, and the number of grid cells is 46104. A nonreflective boundary constraint is applied on the top and bottom of the model with a total of 4 parts of concrete, air, explosive 1, and explosive 2. The maximum compressive strength of the C30 concrete around the blast hole, i.e., 29.96 MPa, is taken as the failure value of the element under dynamic load, and the crack propagation is described by element failure. The model mesh is shown in Figure2. Four sets of delay times of 0, 9, 12, and 15 ms are carried out through comparison experiments by the numerical model. The unit used for numerical model calculation is cm-g-μs.Figure 2
The partition of model mesh.


## 2.1. Parameters Setting of the Numerical Model

The concrete model test by Wu et al. [16] is based on the similarity theorem, and in this test, the effect of hole spacing and the noncoupling coefficient on the precrack width and half hole rate was analyzed. It was found that when the noncoupling coefficient is 2.16 and the hole spacing is 8 times of the aperture, the effect of crack formation and protection of surrounding rock is the best. According to this model test, the parameters such as noncoupling coefficient, blast hole diameter, hole distance, explosive type, and required charge amount were determined through numerical simulations. The actual pouring model picture is shown in Figure 1, where the hole distance is 104 mm and the hole diameter is 13 mm. The influence of related parameters on the blasting is studied. The size of the numerical model is 700 mm × 600 mm × 5 mm, and the hole diameter is 13 mm, while the diameter of the cartridge is 6 mm. Besides, the hole distance is set as 104 mm and the explosive parameter in the calculation model is set as hexogen.Figure 1
The model cast by Wu Xiao.


## 2.2. Establishment of the Numerical Model

The numerical model takes the model test as the prototype, which consists of three components: air, explosives, and concrete. Referring to the literature [17], keywords are used to define concrete and explosives, and the JWL equation of state is used to describe the relevant parameters of the explosives, as shown in the following equation:(1)P=A1−ωR1Ve−R1V+B1−ωR2Ve−R2V+ωE0V,where V is the relative specific volume of the detonation product of dimensionless; E0 is the initial internal energy of the explosive per unit volume (Pa); ρ is the explosive density (g/cm3); D is the detonation velocity (m/s); P is the explosion pressure (kPa); and A, B, R1, R2, and ω are all parameters related to explosive materials. The simulated explosive is set to hexogen, and the core diameter is about 6 mm. The corresponding model parameters are shown in Table 1.Table 1
Main parameters of the material model of hexogen explosive.
Parameterρ (g/cm3)D (m/s)P (GPa)ABR1R2ωEo (GJ·m−3)Value1.5700023.1611.310.654.41.20.328.13During the explosion process, the loading strain rate of the explosive medium, i.e., concrete, increases as the burst center distance of the media decreases, the maximum value of which can reach 104 s−1. The material keyword ∗MAT_JOHNSON_HOLMGUIST_CONCRETE is used to define the concrete medium. When the strain rate increases, the dynamic compressive strength of the concrete also increases, with the concrete material deforming at the same time.Equation (2) is used to express the strength of the HJC material model [18, 19]:(2)σ∗=A1−D+BP∗N1+Clnε∗.In the formula,σ∗=σ/fc is the ratio of the actual equivalent stress to the static yield strength, where σ is the actual equivalent stress and fc is the quasistatic uniaxial compressive strength; P∗=P/fc is the dimensionless pressure, where p is the actual hydrostatic pressure of the unit; ε∗=ε/ε0 is the dimensionless strain rate, where ε˙ is the true strain rate and ε˙0 is the reference strain rate; A is the normalized viscous strength; B is the normalized pressure hardening coefficient; C is the strain rate influence coefficient; N is the normalized pressure hardening index; D is the damage factor; and Smax is the maximum value that the normalized equivalent stress can reach.The calculation of damage factorD (0 ≤ D ≤ 1) is shown in equations (3) and (4):(3)D=∑Δεp+ΔUpεpf+μpf,(4)fp=εpf+μpf=D1p∗+T∗D2,where Δεp is the increment of the equivalent plastic strain, Δμp is the increment of the equivalent volumetric strain, f(p) is the plastic strain when the material breaks under normal pressure p, p∗ is the normalized pressure, T∗ is the maximum tensile static pressure which the material can withstand, and D1 and D2 are damage constants.The HJC model uses a segmented equation of state to describe the relationship between the static water pressure and volume strain of concrete and uses different polynomial equations to describe the relationship between the static water pressure and volume strain in the elastic zone, the crushing zone, and the compaction zone in the concrete compression process.In the elastic zone0<μ≤μcrush,(5)P=Kμ,where P is the hydrostatic pressure of the element, K is the elastic modulus of the concrete element, and µ is the volume strain of the element.In the crushing area (μcrush<μ≤μlock),(6)P=Pcrush+Klockμ−μcrush,where Pcrush is the hydrostatic pressure in the unit corresponding; Klock=Plock−Pcrush/μlock−μcrush, where μcrush is the volume strain at the crush point, Plock is the hydrostatic pressure in the unit corresponding to μlock, and μlock is the volume strain at compaction point.In the compaction zone (μ≥μlock),(7)P=K1μ¯+K2μ¯2+K3μ¯3.In the formula,μ¯=μ−μlock/1+μlock is the corrected volume strain and K1, K2, and K3 are the material constants of concrete.Table2 shows the main parameters of the HJC concrete constitutive model. Among them, G, FC, PC, and T are the mechanical test results. The remaining parameters are derived from the test with large strain, high strain rate, and high pressure of reference [20].Table 2
Parameters of the HJC constitutive model of concrete.
Item nameValueItem nameValueDensityρo/(g·cm−3)2.300Reference strain rate EPSO1.000Shear modulusG/Pa9.681Minimum plastic strain EFMIN0.010Strength parameter 1A0.300Strength parameter 5 SFMAX7.000Strength parameter 2B1.730Volume pressure during crushingPC/1011 Pa8.000E − 05Strength parameter 3C0.005Volume strain during crushing UC6.200E − 04Strength parameter 4N0.790Ultimate volume stress PL/1011 Pa3.470E − 02Static yield strengthFc/1011 Pa2.400E − 04Limit volume strain UL8.696E − 02Tensile strengthT/1011 Pa3.037E-05Damage constant 1D10.04Damage constant 2D21.0Pressure parameter 2K2−2.430Pressure parameter 1K11.160Pressure parameter 3K35.06The model is divided into a mapped grid, and the number of grid cells is 46104. A nonreflective boundary constraint is applied on the top and bottom of the model with a total of 4 parts of concrete, air, explosive 1, and explosive 2. The maximum compressive strength of the C30 concrete around the blast hole, i.e., 29.96 MPa, is taken as the failure value of the element under dynamic load, and the crack propagation is described by element failure. The model mesh is shown in Figure2. Four sets of delay times of 0, 9, 12, and 15 ms are carried out through comparison experiments by the numerical model. The unit used for numerical model calculation is cm-g-μs.Figure 2
The partition of model mesh.


## 3. Simulation Results and Analysis

Before running the LS-DYNA 3D program, five elements are designated as observation points to facilitate analyzing and processing the numerical results [21–40]. As shown in Figure 3, points H69786, H68778, H68787, H62503, and H61486 are selected at the lower part of the centerline of two blast holes, which are signed as A, B, C, D, and E, respectively. The left blast hole is defined as the No. 1 blast hole, and the right one is No. 2. The unit of the distance between elements is mm. After the calculation is completed, the blasting process is described intuitively and detailedly through stress cloud charts and stress curves in time.Figure 3
Location of the observation points in the present model.


### 3.1. Stress Cloud Charts

The stress cloud of simultaneous initiation of prehole and interhole delay time of 12 ms was taken as an example to describe and analyze. In the stress cloud charts, the stress value of different areas is distinguished by colour.(1)
Simultaneous initiation stress cloud analysis: Figure4 shows the stress cloud of simultaneous initiation of the presplitting blast hole. It can be seen from Figure 4 that the detonation wave propagates outward in a circular shape in the homogeneous material. At the time of 9 μs after the numerical simulation initiation, the crushing zone and fracturing zone are formed around the hole wall. At the time of 14 μs, the stress waves generated by the two blast holes are gradually superimposed together, forming a stress concentration zone between the two blast holes. The explosion stress waves of the two blast holes have a promoting effect on each other; thus, the cracks are produced between the two blast holes. The hole is connected to the centerline; the colour of the stress superimposed area changes to red at 17 μs, which means that the stress in this area rises and reaches the highest of 147.6 MPa. After that, the cracks penetrate at 19 μs, and there is still a high-stress zone around the holes. At 39 μs, the stress wave diffuses to the empty surface of the concrete medium, the reflected tensile stress wave begins to act, the stress concentration phenomenon begins to dissipate, and a “calabash” unloading wave is gradually formed.(2)
Analyze the stress cloud chart of the case with a 12 ms delay time. The stress cloud chart of the batch with 12 ms delayed time is shown in Figure5. It can be seen from Figure 5 that, after 0.1 ms of the detonation of No. 1 blast hole, the stress wave breaks the blast hole wall and surrounding rock and the detonation stress wave propagates to the entire homogeneous material. A high-stress area appears at the left side of the No. 1 blast hole. When the stress wave reaches the material boundary, a reflected tensile wave is generated, inducing the material boundary stress slightly greater than that of the surrounding. At 9 ms after the explosion, the detonation stress wave of the No. 1 blast hole gradually dissipates on the right side of the homogeneous material and the right side of the No. 2 blast hole remains crushed. The stress around the No. 1 blast hole is about 43.6 MPa. At 12 ms of blasting, the stress generated by the explosion of the No. 1 blast hole gradually dissipates in the entire homogeneous material. At this time, the No. 2 blast hole is detonated, causing the wall of No. 2 blast hole to break. At 12.1 ms, the stress wave generated by the explosion of the No. 2 hole has diffused on the surface of the material. At 12.3 ms, the stress around the No. 2 blast hole was concentrated, the maximum stress is about 56.6 MPa, and the crack between the two holes is completely connected, forming precrack.Figure 4
Stress cloud charts of explosive detonation propagation with time.Figure 5
Stress cloud chart of explosive detonation propagation in the case with 12 ms delayed time.Table3 lists the crack penetration time and the maximum stress around the blast holes under the four delay times. It can be seen from Table 3 that although the fastest crack penetration occurs in the simultaneous blasting, the stress value generated around the blast hole with the delayed detonation is significantly less than that of the simultaneous blasting. Under the four delay times, the smallest stress around the blast hole appears at the case of 15 ms delay with a value of 49.1 MPa and the fastest crack penetration is found at the case of 9 ms delay with a maximum stress value of only 50.7 MPa. In this situation, the damage to the surrounding rock is greatly reduced and the crack can be penetrated as soon as possible.Table 3
Comparison of parameters under four kinds of delay time.
Interhole delay time (ms)Crack penetration time (μs)Maximum stress (MPa)019147.6910.3 × 10350.71213 × 10356.61517 × 10349.1


### 3.2. Analysis of Effective Stress Curve

The destruction effect of explosives is analyzed by intercepting the von Mises equivalent stress of the model element. von Mises is a yield criterion that describes the stress state by constructing stress contours, and the value of the yield criterion is usually called the equivalent stress. The effective stress curve of each delayed initiation is shown in Figure6. Combining Figures 4 and 6(a), it can be seen that when the blast holes are all detonated, the stress wave on the homogeneous material starts to spread out from the blast hole. Therefore, the peak stresses of element A and element E appear first, with the values of 138.8 MPa and 147.6 MPa, respectively. The stress waves of elements B and D intersect and superpose the stress wave of element C, resulting in the effective stress of element C being higher than elements B and D. After the stress wave is transmitted away, the effective stress of each element is greatly reduced.Figure 6
Effective stress curves for each case of delay time. (a) The effective stress curve under simultaneous initiation. The effective stress curve of delayed initiation at (b) 9 ms, (c) 12 ms, and (d) 15 ms.
(a)(b)(c)(d)It can be seen from Figure6(b) that, in the case with the delayed detonation of 9 ms, the stress wave is in a spherical shape when transmitting to No. 2 blast hole after the detonation of No. 1 blast hole. The peak stress first occurs in element A with a value of 42.6 MPa, followed by elements B, C, D, and E, with values of 39.7, 35.3, 20.9, and 13.0 MPa, respectively. The second blast hole detonates at 9 ms after the detonation of the first blast hole, and the produced stress wave transmits to the first blast hole, causing the effective stress of elements to increase successively. The new peak stress appears at element E first with a value of 50.7 MPa, followed by elements D, C, B, and A, with the stress values of 37.5, 34.0, 32.1, and 30.3 MPa, respectively. After that, the stress wave gradually dissipates and the effective stress curve decreases to zero. This is because at the initiation of the No. 2 blast hole, the explosion stress wave of the No. 1 blast hole has not dissipated yet and the stress superposition is formed between the two holes, resulting in a secondary loading phenomenon. Here, the stress superposition is quantitatively described by the coupling degree of stress waves which is defined as the percentage of the overlap value to the first peak stress value when waves propagate. In this study, the coupling degree of the stress wave is calculated to be 51.69% in the case of a 9 ms delay.Combining Figure5 with Figure 6(c), in the case with 12 ms delayed detonation, the stress wave spreads into surrounding homogeneous material after the detonation of No. 1 blast hole and reaches the peak stress within 0.1 ms. At this time, the maximum stress appears in element A with a value of 54.3 MPa and the peak stresses of elements B, C, D, and E are 39.5, 36.1, 21.0, and 18.6 MPa, respectively. Before the time reaching 12 ms, the stress value received by each element is relatively stable, which is due to the reason that the stress wave generated by the explosion of the No. 1 blast hole is gradually spreading to the boundary without other interferences. At 12 ms, the No. 2 blast hole detonates and the maximum stresses at elements E, D, C, B, and A are 57.3, 45.6, 42.3, 34.9, and 31.8 MPa, respectively, due to the superposition of stresses. The stress quickly drops to about 10 MPa within 0.4 ms, and the coupling degree of stress wave is 67.54%.It can be seen from Figure6(d) that, under the condition of delayed detonation of 15 ms, the stress of five elements in the homogeneous material surrounding to the No. 1 blast hole increases rapidly after the detonation of No. 1 blast hole. At 0.2 ms after the explosion, the elements A, B, C, and E reach stress values of 26.2, 28.3, 27.0, and 22.3 MPa, respectively, and the peak stress appears at element D with a value of 31.8 MPa. From 2 ms, the stress fluctuations received by these five elements tend to be balanced with no large changes. It is because the stress wave generated by the detonation of the No. 1 blast hole gradually dissipates from the right side. At 15 ms, the stress received by the detonation elements of the No. 2 blast hole suddenly increases, but the stress superposition is not obvious because the stress due to the blast of the No. 1 blast hole has dissipated. The maximum stress appears at element E at 15 ms with a value of 49.1 MPa, and its coupling degree of stress wave is 54.03%. As the homogeneous material begins to shatter and break, the stress received by each element dissipates and decreases to zero quickly.


### 3.3. Analysis and Discussion of Simulation Results

Numerical simulations are carried out on the above four situations with various delay times, namely, the aligned detonation and detonations with 9 ms, 12 ms, and 15 ms delay time. The analysis and discussion of the simulation results are as follows:(1)
When both holes are detonated simultaneously, the maximum stress value received by elementE is only 147.6 MPa, while the maximum stress is only 56.6 MPa when the detonation is delayed, with the peak stress of each element reduced by 61.65% compared to the simultaneous detonation. The maximum stress value received by the surrounding rock during the delayed detonation is reduced by 61.65% compared with the maximum stress value received when the holes are detonated simultaneously, which effectively reduces the damage on the surrounding rock. In the case of delay time, the maximum stress is generated after the explosion of the No. 2 blast hole. Because the stress wave generated by the detonation of No. 2 blast hole overlaps with the undissipated stress wave of No. 1 blast hole, the maximum stress is generated after the explosion of No. 2 blast hole, which is later detonated. Therefore, a reasonable delay time can effectively control the damage to the surrounding rock mass.(2)
When the delay time is set, the two peak stresses of the case with 9 ms delay time detonation are 42.6 MPa and 50.7 MPa, respectively, and those for the 12 ms delay time detonation are 54.3 MPa and 57.3 MPa, respectively, while those of the case with 15 ms delay time is 31.8 MPa and 49.1 MPa, respectively. Therefore, the peak stress of the case with 12 ms delay is the largest, and that of the 15 ms delay case is the smallest, indicating that the case with 12 ms delay time has the greatest damage on the surrounding rock and the case with 15 ms delay is the smallest. The crack penetration time is 9.1, 12.1, and 15.1 ms, respectively, for the cases with 9, 12, and 15 ms delay time. This result indicates that the case with 9 ms delay time forms precrack faster.(3)
The two stress peaks received by elementE are compared and analyzed. The coupling degrees of stress wave for the cases with 9 ms, 12 ms, and 15 ms delay time are 51.69%, 67.54%, and 54.03%, respectively, so the case with 12 ms delay time has the most coupling degree with higher energy utilization rate and less energy loss. The case with a 9 ms delay has the fastest precrack formatting speed and causes less damage on the surrounding rock but wastes a certain amount of energy. The case with 12 ms delay time forms the precracks in a higher speed, with the highest energy utilization rate but greater damage on the surrounding rock mass, while the 15 ms delay time case has the slowest speed of formation of precracks and the least damage on the surrounding rock and a certain amount of energy will also be wasted. In summary, according to the requirements for reducing damage on the surrounding rock and rapid formation of precracks, the 9 ms delay is more in line with the purpose of the test than other delay times.(4)
The stress cloud chart and stress curve are used to analyze the delayed initiation and precracking processes, and it is found that when No. 1 blast hole is detonated, a large number of cracks are generated around the blast hole, and the stress is generated throughout the material quickly. The cracks induced by the explosions of the No. 1 and No. 2 blast holes develop along the connecting centerline of the two holes due to the guiding effect. The cracks develop gradually as the stress dissipates, but the crushing zone of the No. 2 blast hole still has large stress and generates small cracks. After the detonations, a large number of cracks are generated around the blast hole and develop along the blast hole connecting the centerline. At the same time, the growth rate of the crack generated by the explosion of the No. 1 blast hole on the blast hole connecting centerline accelerates again until it overlaps the precrack produced by the No. 2 blast hole.


## 3.1. Stress Cloud Charts

The stress cloud of simultaneous initiation of prehole and interhole delay time of 12 ms was taken as an example to describe and analyze. In the stress cloud charts, the stress value of different areas is distinguished by colour.(1)
Simultaneous initiation stress cloud analysis: Figure4 shows the stress cloud of simultaneous initiation of the presplitting blast hole. It can be seen from Figure 4 that the detonation wave propagates outward in a circular shape in the homogeneous material. At the time of 9 μs after the numerical simulation initiation, the crushing zone and fracturing zone are formed around the hole wall. At the time of 14 μs, the stress waves generated by the two blast holes are gradually superimposed together, forming a stress concentration zone between the two blast holes. The explosion stress waves of the two blast holes have a promoting effect on each other; thus, the cracks are produced between the two blast holes. The hole is connected to the centerline; the colour of the stress superimposed area changes to red at 17 μs, which means that the stress in this area rises and reaches the highest of 147.6 MPa. After that, the cracks penetrate at 19 μs, and there is still a high-stress zone around the holes. At 39 μs, the stress wave diffuses to the empty surface of the concrete medium, the reflected tensile stress wave begins to act, the stress concentration phenomenon begins to dissipate, and a “calabash” unloading wave is gradually formed.(2)
Analyze the stress cloud chart of the case with a 12 ms delay time. The stress cloud chart of the batch with 12 ms delayed time is shown in Figure5. It can be seen from Figure 5 that, after 0.1 ms of the detonation of No. 1 blast hole, the stress wave breaks the blast hole wall and surrounding rock and the detonation stress wave propagates to the entire homogeneous material. A high-stress area appears at the left side of the No. 1 blast hole. When the stress wave reaches the material boundary, a reflected tensile wave is generated, inducing the material boundary stress slightly greater than that of the surrounding. At 9 ms after the explosion, the detonation stress wave of the No. 1 blast hole gradually dissipates on the right side of the homogeneous material and the right side of the No. 2 blast hole remains crushed. The stress around the No. 1 blast hole is about 43.6 MPa. At 12 ms of blasting, the stress generated by the explosion of the No. 1 blast hole gradually dissipates in the entire homogeneous material. At this time, the No. 2 blast hole is detonated, causing the wall of No. 2 blast hole to break. At 12.1 ms, the stress wave generated by the explosion of the No. 2 hole has diffused on the surface of the material. At 12.3 ms, the stress around the No. 2 blast hole was concentrated, the maximum stress is about 56.6 MPa, and the crack between the two holes is completely connected, forming precrack.Figure 4
Stress cloud charts of explosive detonation propagation with time.Figure 5
Stress cloud chart of explosive detonation propagation in the case with 12 ms delayed time.Table3 lists the crack penetration time and the maximum stress around the blast holes under the four delay times. It can be seen from Table 3 that although the fastest crack penetration occurs in the simultaneous blasting, the stress value generated around the blast hole with the delayed detonation is significantly less than that of the simultaneous blasting. Under the four delay times, the smallest stress around the blast hole appears at the case of 15 ms delay with a value of 49.1 MPa and the fastest crack penetration is found at the case of 9 ms delay with a maximum stress value of only 50.7 MPa. In this situation, the damage to the surrounding rock is greatly reduced and the crack can be penetrated as soon as possible.Table 3
Comparison of parameters under four kinds of delay time.
Interhole delay time (ms)Crack penetration time (μs)Maximum stress (MPa)019147.6910.3 × 10350.71213 × 10356.61517 × 10349.1


## 3.2. Analysis of Effective Stress Curve

The destruction effect of explosives is analyzed by intercepting the von Mises equivalent stress of the model element. von Mises is a yield criterion that describes the stress state by constructing stress contours, and the value of the yield criterion is usually called the equivalent stress. The effective stress curve of each delayed initiation is shown in Figure6. Combining Figures 4 and 6(a), it can be seen that when the blast holes are all detonated, the stress wave on the homogeneous material starts to spread out from the blast hole. Therefore, the peak stresses of element A and element E appear first, with the values of 138.8 MPa and 147.6 MPa, respectively. The stress waves of elements B and D intersect and superpose the stress wave of element C, resulting in the effective stress of element C being higher than elements B and D. After the stress wave is transmitted away, the effective stress of each element is greatly reduced.Figure 6
Effective stress curves for each case of delay time. (a) The effective stress curve under simultaneous initiation. The effective stress curve of delayed initiation at (b) 9 ms, (c) 12 ms, and (d) 15 ms.
(a)(b)(c)(d)It can be seen from Figure6(b) that, in the case with the delayed detonation of 9 ms, the stress wave is in a spherical shape when transmitting to No. 2 blast hole after the detonation of No. 1 blast hole. The peak stress first occurs in element A with a value of 42.6 MPa, followed by elements B, C, D, and E, with values of 39.7, 35.3, 20.9, and 13.0 MPa, respectively. The second blast hole detonates at 9 ms after the detonation of the first blast hole, and the produced stress wave transmits to the first blast hole, causing the effective stress of elements to increase successively. The new peak stress appears at element E first with a value of 50.7 MPa, followed by elements D, C, B, and A, with the stress values of 37.5, 34.0, 32.1, and 30.3 MPa, respectively. After that, the stress wave gradually dissipates and the effective stress curve decreases to zero. This is because at the initiation of the No. 2 blast hole, the explosion stress wave of the No. 1 blast hole has not dissipated yet and the stress superposition is formed between the two holes, resulting in a secondary loading phenomenon. Here, the stress superposition is quantitatively described by the coupling degree of stress waves which is defined as the percentage of the overlap value to the first peak stress value when waves propagate. In this study, the coupling degree of the stress wave is calculated to be 51.69% in the case of a 9 ms delay.Combining Figure5 with Figure 6(c), in the case with 12 ms delayed detonation, the stress wave spreads into surrounding homogeneous material after the detonation of No. 1 blast hole and reaches the peak stress within 0.1 ms. At this time, the maximum stress appears in element A with a value of 54.3 MPa and the peak stresses of elements B, C, D, and E are 39.5, 36.1, 21.0, and 18.6 MPa, respectively. Before the time reaching 12 ms, the stress value received by each element is relatively stable, which is due to the reason that the stress wave generated by the explosion of the No. 1 blast hole is gradually spreading to the boundary without other interferences. At 12 ms, the No. 2 blast hole detonates and the maximum stresses at elements E, D, C, B, and A are 57.3, 45.6, 42.3, 34.9, and 31.8 MPa, respectively, due to the superposition of stresses. The stress quickly drops to about 10 MPa within 0.4 ms, and the coupling degree of stress wave is 67.54%.It can be seen from Figure6(d) that, under the condition of delayed detonation of 15 ms, the stress of five elements in the homogeneous material surrounding to the No. 1 blast hole increases rapidly after the detonation of No. 1 blast hole. At 0.2 ms after the explosion, the elements A, B, C, and E reach stress values of 26.2, 28.3, 27.0, and 22.3 MPa, respectively, and the peak stress appears at element D with a value of 31.8 MPa. From 2 ms, the stress fluctuations received by these five elements tend to be balanced with no large changes. It is because the stress wave generated by the detonation of the No. 1 blast hole gradually dissipates from the right side. At 15 ms, the stress received by the detonation elements of the No. 2 blast hole suddenly increases, but the stress superposition is not obvious because the stress due to the blast of the No. 1 blast hole has dissipated. The maximum stress appears at element E at 15 ms with a value of 49.1 MPa, and its coupling degree of stress wave is 54.03%. As the homogeneous material begins to shatter and break, the stress received by each element dissipates and decreases to zero quickly.


## 3.3. Analysis and Discussion of Simulation Results

Numerical simulations are carried out on the above four situations with various delay times, namely, the aligned detonation and detonations with 9 ms, 12 ms, and 15 ms delay time. The analysis and discussion of the simulation results are as follows:(1)
When both holes are detonated simultaneously, the maximum stress value received by elementE is only 147.6 MPa, while the maximum stress is only 56.6 MPa when the detonation is delayed, with the peak stress of each element reduced by 61.65% compared to the simultaneous detonation. The maximum stress value received by the surrounding rock during the delayed detonation is reduced by 61.65% compared with the maximum stress value received when the holes are detonated simultaneously, which effectively reduces the damage on the surrounding rock. In the case of delay time, the maximum stress is generated after the explosion of the No. 2 blast hole. Because the stress wave generated by the detonation of No. 2 blast hole overlaps with the undissipated stress wave of No. 1 blast hole, the maximum stress is generated after the explosion of No. 2 blast hole, which is later detonated. Therefore, a reasonable delay time can effectively control the damage to the surrounding rock mass.(2)
When the delay time is set, the two peak stresses of the case with 9 ms delay time detonation are 42.6 MPa and 50.7 MPa, respectively, and those for the 12 ms delay time detonation are 54.3 MPa and 57.3 MPa, respectively, while those of the case with 15 ms delay time is 31.8 MPa and 49.1 MPa, respectively. Therefore, the peak stress of the case with 12 ms delay is the largest, and that of the 15 ms delay case is the smallest, indicating that the case with 12 ms delay time has the greatest damage on the surrounding rock and the case with 15 ms delay is the smallest. The crack penetration time is 9.1, 12.1, and 15.1 ms, respectively, for the cases with 9, 12, and 15 ms delay time. This result indicates that the case with 9 ms delay time forms precrack faster.(3)
The two stress peaks received by elementE are compared and analyzed. The coupling degrees of stress wave for the cases with 9 ms, 12 ms, and 15 ms delay time are 51.69%, 67.54%, and 54.03%, respectively, so the case with 12 ms delay time has the most coupling degree with higher energy utilization rate and less energy loss. The case with a 9 ms delay has the fastest precrack formatting speed and causes less damage on the surrounding rock but wastes a certain amount of energy. The case with 12 ms delay time forms the precracks in a higher speed, with the highest energy utilization rate but greater damage on the surrounding rock mass, while the 15 ms delay time case has the slowest speed of formation of precracks and the least damage on the surrounding rock and a certain amount of energy will also be wasted. In summary, according to the requirements for reducing damage on the surrounding rock and rapid formation of precracks, the 9 ms delay is more in line with the purpose of the test than other delay times.(4)
The stress cloud chart and stress curve are used to analyze the delayed initiation and precracking processes, and it is found that when No. 1 blast hole is detonated, a large number of cracks are generated around the blast hole, and the stress is generated throughout the material quickly. The cracks induced by the explosions of the No. 1 and No. 2 blast holes develop along the connecting centerline of the two holes due to the guiding effect. The cracks develop gradually as the stress dissipates, but the crushing zone of the No. 2 blast hole still has large stress and generates small cracks. After the detonations, a large number of cracks are generated around the blast hole and develop along the blast hole connecting the centerline. At the same time, the growth rate of the crack generated by the explosion of the No. 1 blast hole on the blast hole connecting centerline accelerates again until it overlaps the precrack produced by the No. 2 blast hole.


## 4. Conclusions

Based on the numerical simulation blasting model of concrete, the simulation was carried out on the four situations of the simultaneous initiation of two gun holes and the delayed initiation of 9 ms, 12 ms, and 15 ms between the holes, and the research results were as follows:(1)
Two holes were detonated at the same time, and the cracks between the holes penetrated att =19 μs. The maximum stress on both sides of the hole reached 147.6 MPa in the process of forming the cracks, which was greater than the stress generated by the three kinds of hole-by-hole initiation set in the numerical model test, and the maximum damage to the surrounding rock was caused.(2)
When the interhole delay time is 9 ms, the crack penetrates att =9.1 ms, the coupling degree of stress wave is 51.69%, and the peak stress on both sides of the hole is 50.7 MPa, which is reduced by 65.72% compared with simultaneous initiation, and the damage to the surrounding rock mass is obviously weakened.(3)
When the interhole delay time is 12 ms, the crack penetrates att = 12.1 ms, the coupling degree of stress wave is 67.54%, and the peak stress on both sides of the hole is 56.6 MPa, which is 61.65% lower than that of simultaneous initiation.(4)
When the interhole delay time is 15 ms, the crack penetrates att = 15.1 ms, the coupling degree of stress wave is 54.03%, and the peak stress on both sides of the hole is 49.1 MPa, which is 66.73% lower than that of simultaneous initiation.(5)
Taking into account the damage control of surrounding rock mass, energy utilization rate, and precrack penetration rate of blasting load, it is determined that 9 ms delay time is the best accurate delay time between holes of precrack initiation in this numerical test. This method is of guiding significance to the future research of precrack formation and the choice of delay time of precrack blasting by hole.



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*Source: 4593221-2021-09-13.xml*