Effect of Adjacent Hole on the Blast-Induced Stress Concentration in Rock Blasting

To investigate the effect of an adjacent hole on the blast-induced stress concentration in rock blasting, a rock blasting model with an adjacent hole is explored through theoretical analysis and numerical simulation.*e commercial software LS-DYNA is utilized to simulate adjacent hole effect in rock blasting, in which the Johnson–Holmquist concrete material model is used to simulate rock and the high-explosive-burn-explosive and the equation of state of JWL are used to simulate explosive. Influences of the key parameters of adjacent hole effect in rock blasting, pitch of holes, adjacent hole diameter, and uncoupledmedium in a blasting hole are extensively explored. According to the simulation results, when the explosion stress wave spreads to the adjacent hole wall, the tangential stress on the adjacent hole wall induced by the explosion stress wave is always greater than the radial stress. Adjacent hole diameter has a major effect on stress concentration, but with the adjacent hole diameter increasing, the stress concentration phenomenon weakens and the free surface effect of the adjacent hole plays a more important role.


Introduction
With the continuous development of the national economy, more and more mining and tunnel projects are being constructed in China.Under some ideal conditions, mechanized excavation has the advantages of speediness and low impact on the environment [1,2].But under unstable ground and complex geological conditions, mechanized excavation, compared with drilling and blasting, is inefficient and poor adaptability.Especially in hard rock, the wear of cutting tools of the excavating equipment is severe and the removal and reinstallation of cutting tools takes a long time, so mechanized excavation in hard rock is uneconomic and time-consuming.Although drilling and blasting is a traditional excavation method, it still plays an important role in tunnel excavation on account of its good applicability and economy [3][4][5].At present, the mainstream blasting method is controlled blasting, which includes directional blasting, smooth blasting, splitting blasting, and millisecond blasting [6][7][8].In the process of engineering blasting, explosive detonation is an instantaneous chemical reaction with tremendous energy release, which triggers effective vibration to break the rock.
Setting empty holes near the blasting hole is a valid way for directional blasting.When explosive detonates in the blasting hole, the adjacent empty hole close to the blasting hole not only can provide free face for rock deformation but also can change stress distribution in rocks around the empty hole.After explosive detonates, blasting-induced shock waves act on the blasting hole wall in the first place, activating randomly primary fissures and fresh fissures in the rock near the blasting hole, and then outspread across external rock [9,10].e pressure on the blast hole wall generally exceeds the dynamic compressive strength of the rock, making the rock mass to deform plastically and forming the crushed zone [11][12][13].Because the production of crushed zone consumes a tremendous amount of explosive energy, compressive shock waves turn into compressive stress waves after passing this zone.When the stress waves propagate to the adjacent empty hole wall, compressive stress waves turn into tension stress wave due to reflection on empty hole wall [14][15][16].If reflected tension stress waves exceed the dynamic tension strength of the rock, fissure will arise [17,18].Because of the induction effect of the empty hole, blasting-induced rock fracture will propagate towards the empty hole, which is called as empty hole effect.Due to the complexity and instantaneity of explosive load, the crack propagation law in rock with empty hole is still unclear.erefore, it is of great significance to study the mechanism of the empty hole in rock blasting and its interaction with blasting-induced cracks.
Experimental studies of rock blasting are very expensive and time-consuming [16].It has been shown that theoretical research and numerical simulation can serve as an efficient approach to reveal the crack propagation law in rock blasting [19][20][21][22].In this research, the following topics are discussed by the methods of theoretical research and numerical simulation: (a) the influence of the pitch of holes on the crack propagation law, (b) the influence of the adjacent empty hole diameter on the crack propagation law, and (c) the effect of uncoupled medium on blasting load acting on blasting hole wall.
ese research results have reference value in selecting blasting parameters.

e Stress Concentration Effect of Adjacent Empty Hole.
When an explosive is detonated, the impact load generated by explosive explosion creates a shock stress field in the rock.
In the rock around the blasting hole, the impact load acts on it and creates pressure, which can be expressed as where P 0 is the pressure on the blasting hole wall generated by the shock wave, ρ 0 is the explosive density, D v is the explosive velocity, and c is the exponent of adiabatic expansion of the detonation gas.
Based on the principle of wave's transmission and reflection, the pressure created by the shock wave transmitting into the rock can be expressed as where c p is the acoustic wave speed in the rock and ρ is the rock density.
After shock waves transmit across the interface between explosion and rock, it will spread along the outer rock.With the increase of distance, shock waves translate into stress waves in no time.e stress waves' peaks decrease as follows: where σ r is the radial stress at a point in the rock, σ θ is the tangential stress at a point in the rock, P is the pressure created by the shock wave transmitting in the rock, r 1 is the blasting hole diameter, r is the distance between a point in the rock and the center of the blasting hole, and α is the attenuation coefficient in the rock, which can be expressed as where μ d is the dynamic Poisson ratio of the rock.It has been revealed by correlational researches that the dynamic Poisson ratio is in correlation with the rock strain rate [23][24][25].With the increase of the rock strain rate, the dynamic Poisson ratio of rock will decrease.In the loading rate range of engineering blasting, the relation between the dynamic Poisson ratio and the static Poisson ratio can be expressed as where μ is the static Poisson ratio of rock.e researches about the physical process of rock blasting show that the attenuation of the explosive stress wave is not only related to the physical mechanical properties of the rock but also depends on the explosive detonation parameters [26][27][28].Effects of explosive detonation parameters on stress not only reflect on the pressure on blasting hole wall but also on the stress wave attenuation coefficient.So far, the certain way to determine the attenuation coefficient of stress wave propagation has not been understood completely.
When the stress wave reaches empty hole wall, the stress near the empty hole wall is greater than that of the rock without hole due to the stress concentration phenomenon.On the basis of elastic mechanics theory [29], the peak stress near the empty hole wall can be expressed as where σ rr is the radial stress in adjacent rock around the empty hole, σ θθ is the tangential stress in adjacent rock around the empty hole, T rθ is the shearing stress in adjacent rock around the empty hole, r 2 is the empty hole radius, r is the distance between a point in the rock and the center of the empty hole, and θ is the angle between the ligature of the calculated point and blasting hole and the ligature of the blasting hole and the empty hole, as shown in Figure 1. e stress state on the empty hole wall is that σ rr � 0, T rθ � 0, and By taking the derivative of σ θθ , we can get dσ θθ /dθ � −4 sin 2θ(σ θ + σ r ).When dσ θθ /dθ � 0, that is, θ � ± π, we can get the peaks of tensile stresses, which can be expressed as 2 Advances in Civil Engineering e empty hole can make tensile stress along the ligature between the blasting hole and the empty hole to be the maximal tensile stress in all tensile stresses at the same distance from the blasting hole [30].

2.2.
e Free Surface Effect of Adjacent Empty Hole.When the stress wave propagates to the empty hole wall, the stress wave, which is incident wave, will reflect and transmit in the boundary of rocks and air, causing reflected wave in rock and transmitted wave in air.e reflected waves impact on rock and make rock fracture.
e following analysis focuses the reflection regularity of stress wave when it propagates vertically to the empty hole wall.
As shown in Figure 2, the wave impedance of rock is ρ r0 C r0 , in which ρ r0 is the rock density and C r0 is the speed of wave propagation in the rock.e wave impedance of air is ρ a0 C a0 , in which ρ a0 is the air density and C a0 is the speed of wave propagation in air.According to the conservation of momentum, the particle velocity increment on boundary of rock and air can be expressed as where u I is the velocity increment of rock particle caused by incident waves, u R is the velocity increment of rock particle caused by reflected waves, u T is the velocity increment of air particle in the empty hole, σ I is the intensity of the incident wave, σ R is the intensity of the reflected wave, and σ T is the intensity of the transmitted wave.e stress and particle velocity on the boundary surface meets the continuous condition, and this can be expressed as Considering Equations (10)(11)(12), we can get where F and T are called the reflection coefficient and transmission coefficient, respectively, whose values are determined by the wave impedance of the rock and air, which can be expressed as Because air density is much smaller than rock density, that is ρa0/ρk0 � 0, so F � −1 and T � 0. When compression wave spreads to a free surface and reflects, the compression wave changes into a tensile wave, and then, tensile wave acts on rock.Because the tensile strength of the rock medium is much lower than the shear strength and compressive strength of the rock, the reflected tensile wave coming from the empty hole wall has an important effect on breaking rock.With the adjacent empty hole diameter increasing, more stretching wave is reflected into the rock medium, as shown in Figure 3, and it is helpful for rock breaking.

ANSYS/LS-DYNA Algorithm and Material
Constitutive Model ere are a variety of algorithms to simulate the explosion process in ANSYS/LS-DYNA.e explosion of dynamite in rock is accompanied with the intrusion of explosive gas, which acts as a gas wedge for crack expansion.e key of blasting simulation is to realize the interaction between rock mass and dynamite.
For Lagrange algorithm, materials cannot flow in the grid, and the finite element mesh in this algorithm can become deformed seriously so that this algorithm cannot solve blasting problem.
e Euler algorithm allows the material flowing in the fixed grid.We can make Euler finite element mesh couple with Lagrange finite element mesh to deal with the fluid-structure interaction under different conditions.However, in the Euler algorithm, in order to accurately capture the coupling deformation of gas and rock material in the explosion, the fine partitioning of the grid is required, resulting in the increase of the computational cost and the consumption of a large number of machines.e ALE algorithm combines the advantages of Lagrange algorithm and Euler algorithm.In the calculating order of the ALE method, Lagrange calculation is performed firstly and Advances in Civil Engineering then the grid deforms with the deformation of the material.In contrast, the ALE algorithm is more suitable for solving the problem of large deformation caused by blasting [31].
erefore, the ALE algorithm and its fluid-structure coupling method are selected to simulate rock blasting in the subsequent chapters.

Rock Constitutive Model.
In order to study the deformation and failure mechanism of rock material under dynamic load, many scholars have established lots of rock constitutive models [32,33].In blasting simulation, Johnson-Holmquist concrete material model is representative, which can describe the high strain and large deformation of rock and concrete materials under dynamic load more accurately [34][35][36].e equivalent yield strength of the JHC model is a function of the pressure strain rate and damage factor.Meanwhile, pressure is a function of volume strain, and damage accumulation is a function of plastic volume strain, equivalent plastic strain, and pressure.e intensity of the JHC model is described by the normalized equivalent stress: where σ * � σ/f c ′ is the ratio of the actual equivalent stress σ to the static compressive strength f c ′ , P * � P/f c ′ is the dimensionless pressure, where P is the unit hydrostatic pressure, _ ε * � _ ε/_ ε 0 is the dimensionless strain rate, where _ ε is the true strain rate and _ ε 0 is the reference strain rate, A is the standardized cohesion strength, B is the normalized pressure hardening coefficient, N is the pressure hardening index, and C is the strain rate coefficient.e damage factor D(0 ≤ D ≤ 1) can be obtained by equivalent plastic strain and plastic volume strain as where Δε P is the equivalent plasticity, Δμ P is the equivalent volume strain increment, ε

Explosive Material Model.
e key to study the blasting is to determine the blasting load.e blasting damage on rock can be quantitatively analyzed by determining the pressure change course curve in the blasting hole.In actual measurement, the high temperature and high pressure makes it difficult to measure the pressure change of the blast hole.
e explosive constitutive model provided by numerical simulation technology can effectively solve this obstacle.For the complex characteristics of explosive detonation, the dynamite constitutive model provided in LS-DYNA can accurately describe the physical and chemical properties of explosives.In this paper, the MAT-highexplosive-burn-explosive embedded in the LS-DYNA program is selected as the explosive model.e relationship between pressure and volume change of detonation product is described by using the JWL state equation.e state equation of JWL is described as follows: where P is the pressure of detonation product, V is the relative volume, E 0 is the initial internal energy density, and A, B, R 1 , R 2 , and ω are constants.e related parameters of the explosive material model are shown in Tables 2 and 3.

Air and Water Constitutive Model.
In LS-DYNA, air and water are treated as fluid materials.Fluid materials generally need to be described in two aspects: constitutive equation and state equation.e former describes the relation between stress and strain and the latter describes the relationship between volume deformation and pressure.In the calculation of fluid-structure interaction, the NULL material is adopted as the constitutive equation and the Gruneisen equation is adopted as the state equation: 4 Advances in Civil Engineering where ρ 0 is the material density, c 0 is the Gruneisen parameter, E 0 is the internal energy, C is the intercept of a curve, S 1 , S 2 , and S 3 are coefficients of the slope of the curve, and a is the first-order volume correction of c 0 and μ.

The Effect of Pitch of Holes on Stress Concentration
4.1.Calculation Model.In rock blasting, the empty hole effect only plays a role within a certain range around the blasting hole, and the empty hole effect will not be obvious beyond the range.is paper establishes a numerical model to analyze the effect of pitch of holes on empty hole effect.As shown in Figure 4, the model size is 150 cm × 150 cm, the blasting hole whose diameter is 5 cm is placed in the center of the model with a coupling charge, and the diameter of empty hole is 5 cm.e distance between blasting hole and empty hole is a.According to the difference of the hole spacing a, 4 sets of comparison simulation tests were set up.
In Table 4, r 1 is the blasting hole radius and r 2 is the empty hole radius.

Result and Analysis.
Taking model 1 as an example, the explosive load acts on hole wall after the detonation of the explosive in the blasting hole, and the explosion stress wave propagates from the center of the blasting hole to the surrounding rock, as shown in Figure 5(a).In the initial stage of the explosion stress wave propagating, the empty hole has no obvious guiding effect on the explosive stress wave propagating, and the explosion stress wave is centered at the center of the blasting hole and propagates outward in a circular shape.When the explosive stress wave spread around the empty hole, as shown in Figure 5(b), because of the existence of empty hole, explosion stress wave propagating law changes and stress concentration phenomenon appears on empty hole wall near the blasting hole.rough the calculation results of different models with different pitches of holes, we can find that the damage degree of rock mass around the blast hole and empty hole is more serious than other areas.With the further propagation of the explosive stress wave, the effective stress on empty hole wall increases gradually, as shown in Figure 6.We can observe that there exists a small low-damage zone between the blasting and the empty hole.
is phenomenon can be explained by the superposition of waves.When a cylindrical explosive is detonated in the blasting hole, the explosion stress wave propagates outwards in the form of compression wave.e adjacent empty hole wall can be considered as a free surface due to no constraints.When the explosion stress wave propagates to the empty hole wall, the explosion stress wave will reflect and change from compression wave to tension wave.e reflected stress wave will propagate to the blasting hole direction in the form of tension wave.e superposition of compression wave coming from the blasting hole and that of the tension wave coming from the empty hole will affect the damage distribution of rock mass.In the area where the compression wave strength is close to Advances in Civil Engineering that of the tension wave, the compression wave can cancel out the tension wave, so that the damage degree in this area is very small.In the stress analysis of finite element software ANSYS, positive stress is tensile stress and the negative stress is compressive stress.rough the comparison of radial stress and tangential stress of the empty hole wall, which can be found that tangential stress and radial stress are all compressive stress, as shown in Figure 7, and the tangential stress is greater than the radial stress, and the tangential stress peak is usually 3-4 times than the radial stress peak, as shown in Figure 8.
e stress wave produced by the explosion of the explosive in the blasting hole gradually propagates from the center of blasting hole to outside.e effect of stress wave on the empty hole wall is not obvious before it arrives at the empty hole.When the stress wave propagates to the empty hole, the radial stress and tangential stress of the empty hole wall increase sharply.In the initial stage of increasing stress, the growth law of radial stress and tangential stress is the same.After that, the radial stress decreases slightly, then stays the same and reaches the second peak after a period of time.At the same time, the tangential stress has been increasing, and at the time when the radial stress reached the second peak, it also reaches the peak.In the process of stress changing, the tangential stress is always greater than the radial stress.
Under the action of explosion stress wave, empty hole wall has displacement to the empty hole center due to the fact that empty hole wall has no restraint inside; i.e., the pore diameter is reduced.e displacements of the empty hole wall are not uniform, the displacement of point 1 is the biggest, the displacement of point 3 is the least, and the displacement of point 2 is equal approximately to that of point 4, as shown in Figures 9 and 10.

Calculation Model.
In rock blasting, when the empty hole diameter increases, the more tensile wave is reflected into the rock medium, and the more favorable for fracture extension in rock.It is helpful for enhancing free surface effect of empty hole to enlarge the empty hole diameter.In this paper, a numerical calculation model is established to analyze the effect of empty hole diameter on empty hole effect.As shown in Figure 11, the model size is 150 cm × 150 cm. e blasting hole with a diameter of 5 cm and coupling charge is placed in the center of the model.e empty hole was placed at the point which is 37.5 cm from the blasting hole center, and the radius r 2 was 1.25 cm, 2.5 cm, 3.75 cm, and 5 cm.According to the difference of the empty hole radius r 2 , a total of 4 sets of comparison simulating tests were set up, whose parameters are shown in Table 5.

Result and Analysis.
e plastic area around the empty hole increases as the empty hole radius increases, as shown 6 Advances in Civil Engineering in Figure 12. e high stress area around the empty hole has a tendency to move towards blasting hole. is indicates that with the empty hole diameter increasing, the stress concentration phenomenon weakens, and the free surface effect of empty hole plays a major role.
In order to analyze the stress distribution of the empty hole wall, point 1 and point 2 are selected, which are the nearest and the farthest points to the blasting hole, as shown in Figure 11.e growth of effective stress at point 2 lags behind that of point 1, as shown in Figure 13.When empty hole diameter is smaller than the blasting hole, although the effective stress on point 1 begins to increase earlier than point 2, the effective stress growth law sand effective peak values of point 1 and point 2 are the same.With the increase of the empty hole diameter, the time interval of effective stress between point 2 and point 1 increases.When empty hole diameter is bigger than the blasting hole, the effective stress of point 1 has obvious crest and effective stress change of point 2 is flat.

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When the adjacent empty hole diameter is larger than the blasting hole diameter, the moment at which the effective stress begins to increase at the point 2 lags behind that of point 1, and with the increase of the adjacent empty hole diameter, the retardation time increases, as shown in Figure 14.

Comparison of Different Uncoupled
Medium Charges

Calculation Model.
ere are two kinds of blasting hole charging mode: coupling charge and uncoupling charge.Due to the different coupling media, there are many kinds of uncoupling mode, and the modes that are commonly used are air-uncoupling charge and water-uncoupling charge.In the case of uncoupled charge blasting, the detonation wave and the detonation gas act on the uncoupling medium first.Advances in Civil Engineering e load is then applied to the blast hole wall through the transmission of the uncoupled medium.
e different uncoupling media have different performances and the way to transmit explosive energy is different, so the transmission pressure on the rock near blasting hole wall is very different.In the water-medium uncoupled charge, the detonation wave compresses water medium causing the shock wave in the water, which transmits to rock through water.Because of the cushion effect of water, the pressure on the blasting hole wall is reduced compared with coupling charge.In the airmedium uncoupled charge, because the compressibility of air is very strong, the detonation product expands to fill the blasting hole and then acts on the blasting hole wall, so the pressure on the blasting hole wall is also reduced compared with coupling charge.
e different charging forms have great influence on pressure of blasting hole wall.In addition, the dynamic stress field in the surrounding rock is affected, which can affect the extent of blasting damage and destruction of rock.
Numerical simulation is used to reconstruct the stress wave during explosion, which is one kind of main methods to study blasting.In this paper, LS-DYNA numerical simulation platform is proposed to establish a single hole model of air-medium uncoupled charging blasting and watermedium uncoupled charging blasting, as shown in Figure 15, and the decouple coefficients two models are all 1.5.

Result and Analysis.
According to the explosive stress distribution and the dynamic compressive strength of the rock under the blasting load, we can simply regard the zone where the damage factor is greater than 0.9, the red area in Figure 16, as the crushed zone, the zone where the damage

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factor is in the region of 0.1 to 0.9, which is green and shallow blue area in Figure 12, as a fracture zone, the zone where the damage factor is less than 0.1, which is the dark blue area in Figure 12, as the elastic zone.By comparing the rock mass damage distribution of coupling charge, air mediumuncoupling charging, and water-uncoupling charging, we can know that the crushing area produced by the watermedium uncoupled charging is the same as that produced by the coupling charging.e fracture zone produced by the water-medium uncoupled charging is greater than that of the coupling charging.
e crushing zone and fracture zone of air-medium uncoupled charging are less than that of coupling charge and water medium.e blasting e ciency of the water-medium uncoupled charging is the highest, and the rock-breaking e ect is the best, which is suitable for cut hole blasting.e damage to rock mass caused by air uncoupled charging is minimal.Air uncoupled charging can be used in peripheral eye blasting in smoothing blasting to reduce the damage to the surrounding rock.
After explosive detonation, the stress of the blasting hole wall rises to a maximum stress state in a very short time.In coupling charge, the blasting hole wall pressure reached the peak state of 13.08 GPa when it is 3.99 μs after the explosive explosion.In water-uncoupling charge, the blasting hole wall pressure reached the peak state of 11.72 GPa when it is 4.47 μs after the explosive explosion.In air-uncoupling charge, the blasting hole wall pressure reached the peak state of 5.16 GPa when it is 5.49 μs after the explosive explosion.As shown in Figure 17, the pressure of the blasting hole wall produced by coupling charge is the biggest.e blasting hole wall pressure caused by the water-uncoupling charge is slightly less than that of the coupling charge.e 10 Advances in Civil Engineering blasting hole wall pressure generated by the air-uncoupling charge is 0.4 times the pressure of the blasting hole wall produced by the coupling charge, as shown in Figure 18.By comparing the time of blasting hole wall pressure reaching peak value, it is known that the air medium has a better cushioning e ect on the blast than the water medium.
Comparing the stress in the surrounding rock with three kinds of charging methods, coupling charge, wateruncoupling charge, and air-uncoupling charge, we can nd that the stress in the surrounding rock with the three charging methods all decreases exponentially with the distance.e stress produced by coupling charge and wateruncoupling charge is greater than that of air-uncoupling charge on blasting hole wall and in the surrounding rock.
e stresses with the three kinds of loading methods are consistent beyond 15 cm, as shown in Figure 18.

Conclusion
In the present study, LS-DYNA is used to simulate the e ect of adjacent empty hole.e in uences of key parameters of adjacent empty hole in rock blasting, pitch of holes, empty hole diameter, and uncoupled medium are extensively investigated.From the numerical simulation results, we come to the following conclusions.
e stress on adjacent empty hole wall gradually decreases with the increase of the pitch of holes.In the process of stress on empty hole wall changing, the tangential stress is always greater than the radial stress, and the tangential stress peak is usually 3-4 times than the radial stress peak.

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With the empty hole diameter increasing, the high stress area around the empty hole has a tendency to move towards blasting hole, which indicates that the stress concentration phenomenon weakens and the free surface effect of empty hole plays a major role.
e blasting efficiency of the water-medium uncoupled charging is the highest, and the rock-breaking effect is the best, which is suitable for cut hole blasting.e rock mass damage caused by air-uncoupled charging is minimal.Air medium has a better cushioning effect on the blast wave than the water medium.Air-uncoupled charging can be used in peripheral eye blasting in smooth blasting to reduce the surrounding rock damage.12 Advances in Civil Engineering

Figure 1 :
Figure 1: e stress concentration effect of an adjacent empty hole.

Figure 2 :
Figure 2: e free surface effect of an adjacent empty hole.

Figure 11 :
Figure 11: Comparison test of different adjacent empty hole radii.

Figure 13 :
Figure 13: e e ective stress change of point 1 and point 2 on empty hole wall.

Figure 18 :
Figure 18: e pressure attenuates in rock with distance.

Table 3 :
Parameters of the state equation of dynamite JWL.

Table 4 :
Experimental group parameters of different distances between holes.

Table 5 :
Experimental group parameters of different adjacent empty hole radii.