Calculation of Blasting Damage Zone Radius of Different Charge Structures in Burnt Rock

Abstract

The radius of the failure area after a blasting fracture process of burnt rock is affected by joint fissures, does not conform to the existing theoretical calculation formula and the distribution law of the failure area also changes. The fracture area is large, and the fracture extension and expansion area are small. Therefore, in order to describe the damage of blasting to a fractured rock mass more objectively and accurately, on the basis of summarizing the previous research results, a damage variable was introduced to characterize the initial crushing degree of the fractured rock mass, and the corresponding rock failure criterion was used to derive the calculation formula of a blasting crushing circle and fracture circle radius of burnt rock with different charge structures. The results show that the blasting failure zone of fractured rock mass with different charge structures was not only related to the radius of the blast hole and the explosive and rock properties, but also had a strong relationship with the initial damage degree of the rock mass. Taking an open-pit coal mine in Xinjiang as an example, the radius of the fracture zone with different charge structures was obtained by using the obtained calculation formula, and it was applied to the determination of row spacing and hole spacing.

1. Introduction

Quantitative blasting technology, careful blasting construction and fine blasting management can effectively control the process of explosive energy release and rock crushing, as well as meet the requirements of open-pit mines for controllable, safe, and economical blasting processes and effects. At the same time, the application of blasting technology in open-pit mines efficiently reduces the labor intensity of workers and accelerates the advancement of engineering tasks. It is an important measure to realize the rapid development of open-pit mines and even all economic mining constructions.

The blasting of a rock mass in an open-pit mine is a complex dynamic process, which involves the high-speed detonation of explosives, dynamic expansion of the explosion cavity, nonlinear deformation of rock under the explosion load and so on [1]. Taking the explosive explosion in an infinite rock mass as an example, the dynamic process of rock-blasting failure can be divided into three stages: the explosion shock wave action stage, explosion stress wave action stage and explosion gas action stage. The first stage is the stage of explosion shock wave action. The rapid reaction of explosives propagates detonation waves in explosive materials, and the detonation velocity of most high-energy explosives is 1500–9000 m/s [2]. The explosion pressure caused by an explosive’s explosion is usually between 1 GPa and 14 GPa [3], which is far beyond the compressive strength of the surrounding rock, resulting in the large nonlinear deformation and failure of surrounding rock similar to a fluid, and then forming a crushing zone with a range of about 1.5 to 4 times the radius of the explosive [4]. The second stage is the stage of the explosion stress wave. When the explosion shock wave attenuates into a stress wave, the radially propagating compressive stress wave propagates outward, and the associated tensile stress is generated in the tangential direction. Because the dynamic tensile strength of rock is much lower than the dynamic compressive strength, the tangential tensile stress can form radial cracks. With the further propagation of the explosion stress wave, when the intensity of the stress wave produced by the blasting is attenuated to the point where the rock cannot form permanent deformation, a far-field elastic vibration zone is formed [5]. The vibration wave propagating in this zone is measured to have a constant wave velocity, which is considered to be elastic [6]. The third stage is the stage of detonation gas action. The gas product produced by the explosion of the explosive expands dynamically and penetrates into the initial crack formed by the stress wave, which makes the crack expand further. Some studies suggest that the detonation gas forms a quasi-static stress field similar to the stress wave action [7]. The explosive blasting process is described in the three stages shown in Figure 1 below.

According to the different characteristics of rock failure, Kutter and Fairhurst [8] first proposed in 1971 that after the explosion of explosives, a rock-blasting failure zoning model with a crushing zone, usually divided into a crushing zone, fracture zone and vibration zone, will be formed around the blast hole, as shown in Figure 2. This model has been recognized by the majority of scholars and has been continuously developed and improved [9,10,11]. The typical partition model of rock-blasting failure is shown in Figure 1 [12]. The quantitative calculation of the crushing zone and fracture zone is of great significance and function for the study of the blasting fracture mechanism, explosive performance and blasting technology. Many scholars carried out a lot of fruitful research work: Zong Langari et al. [13] proposed a new formula for calculating the radius of the fracture zone by considering the effect of the shock wave and the existence of a crushing zone. Huang et al. [14] studied the radius of the crushing ring and fracture ring under the conditions of coupled and uncoupled charges in blasting on the basis of considering the actual three-dimensional complex stress state in rock and obtained the corresponding radius calculation formula. On the basis of cavity expansion theory, Zhang et al. [15] used the Mohr–Coulomb strength criterion as the constitutive model of rock near the blasting area, modified the velocity field relationship in cavity expansion theory and established the mechanical control equation of cylindrical charge blasting in rock. Combined with the quasi-static solution of the blasting elastic zone, the theoretical calculation formulas of the radius of the blasting cavity, crushing zone and radial crack zone were given. Jayasinghe et al. [16] derived the stress time history expression on the rock-blasting failure interface based on the rock-blasting failure partition model [17]. Armaghani et al. [18], in order to study the failure mechanism of the rock around the blast hole and accurately predict the range of the crushing zone, proposed an improved model for calculating the range of the crushing zone of borehole blasting. Lu et al. [19] proposed a modified model for the size of the crushing zone around the hole in borehole blasting by considering the effect of ring compressive stress and cavity expansion. However, in the above research, the research was mostly based on conventional and homogeneous sandstone and conglomerate. In addition, the main way to perform the analysis and calculation of blasting engineering is to design and preview the blasting scheme by means of computer numerical calculation and image presentation [20,21,22,23,24]. At present, SHOTPlus (Suite 2023), JKSimBlast (2023), ANASYS (2022)/LS-DYNA (R14.0, FLAC (Fast Lagrangian Analysis of Continua, 1.3.3), PFC (Particle Flow Code, 6.0), CDEM (Continuum Discontinuum Element Method) and so on are the most classic and widely used blasting software at home and abroad. At present, PFC (6.0), ANASYS (2022)/LS-DYNA (R14.0) and FLAC (1.3.3) are the most widely used numerical simulation software in the field of blasting research. These three are based on three numerical simulation methods: the discrete element method, finite element method and finite difference method. Due to the different methods, there are advantages and disadvantages in the study of mine blasting [25,26,27]. However, in actual open-pit coal mine blasting engineering, there are often a large number of joint fissures, faults, bedding, schistosity and other weak structural planes in the rock mass. Under the influence of these weak structural planes, the blasting theory studied under ideal conditions (homogeneous rock) has deviations in practical blasting applications and cannot play a precise guiding role in engineering practice.

With the deepening of the research, it has become accepted that the influence of the actual joint cracks in the rock mass on the propagation of stress waves must be considered. Given that the research object of this study was burnt rock, the influence of joints and fissures in the blasting process was obvious. Compared with the blasting process of a homogeneous rock mass, the expansion of radial cracks after burnt rock blasting is affected by joints and fissures, which cannot be better extended, resulting in a smaller range of fracture zone and no damage to the rock mass in the vibration zone [28]. When the stress wave propagates to the joint fissure, reflection and refraction occur, which increases the reflection of the stress wave, and, at the same time, aggravates the attenuation of the stress wave and the energy of the detonation gas, which leads to a large degree of fragmentation of the burnt rock in the area near the explosion source. When the stress wave cannot cause the radial movement of the rock, the rock will not be damaged again. In summary, the radius of the failure zone after the blasting crushing process of burnt rock is affected by joints and fissures and does not conform to the existing theoretical calculation formula, and the distribution law of the failure zone also changes. The crushing zone is larger, and the crack extension and expansion zone are smaller. Therefore, in view of this kind of blasting situation of burnt rock, it is necessary to combine the damage theory to deduce the radius of the damage area of burnt rock. The core goal was to deeply explore and accurately quantify the specific influence of different charge structures on the blasting effect of burnt rock. The purpose of this study was to establish an accurate mathematical model and calculation formula through scientific methods and means to predict and evaluate the boundary and range of key damage areas, such as the crushing area and fracture area generated by blasting with different charge structures. By comparing and analyzing the radius of the blasting failure area with different charge structures, this research aimed to reveal the internal relationship between the charge structure and blasting effect, and then guide the optimization of the charge structure in practice so as to improve the blasting efficiency and reduce the damage to the surrounding environment while ensuring the safety of the blasting operation. In addition, this research also aimed to provide a solid theoretical basis and technical support for mining, water conservancy construction and other related fields that need frequent blasting operations, as well as promote technological progress and sustainable development in these fields.

2. Materials and Methods

The overall technical route of this study is shown in Figure 3.

2.1. Explosion Stresses of Different Charge Structures

The charge structure of explosives can be divided into a coupled continuous charge, coupled interval charge, uncoupled continuous charge and uncoupled interval charge according to different charge structure combinations, as shown in Figure 4.

In the mining regulations, when there are two or more free faces in the blasting operation, the minimum resistance line and the length of the filling shall not be less than 0.3 m.

There is no gap between the column and the blast hole, and the charging structure is simple, time-saving and labor-saving. So far, this kind of charging structure is still used in most open-pit mine blasting designs. However, because the explosive is concentrated at the bottom of the blasthole, the force on the upper part of the blasthole is uneven after the explosion of the explosive, which produces more bulks, and thus, reduces the shoveling efficiency and increases the cost of secondary crushing.

The radial uncoupled charge means that a certain interval is kept between the grain and the hole wall when the explosive is filled, and the grain diameter is smaller than the hole diameter. After the explosion of the uncoupled charge, during the propagation of the detonation wave through the air medium to the hole wall, part of the energy of the detonation gas product is compressed and stored by the air medium, thereby reducing the initial pressure peak during the explosion of the explosive. Then, the compressed stored energy is released to work on the rock around the borehole, and finally, the purpose of improving the blasting effect is achieved. An uncoupled charge is often used in pre-splitting blasting and smooth blasting.

The axial uncoupled charge (interval charge) uses materials such as air columns or gun mud to fill the explosives in sections. This method can adjust the center of gravity of the explosives and reasonably distribute the energy of the explosives. It is more suitable to use an interval charge for rock benches with hard or fractured rocks in the middle and upper parts of the blasting face.

2.2. Explosion Stress of Coupled Charge

When the charge structure adopts a columnar coupling charge, the explosive energy directly acts on the initial peak pressure P of the hole wall. According to the acoustic approximation principle, it can be expressed as follows [29]:

$$P={\displaystyle \frac{2\rho {{\displaystyle C}}_P}{\rho C_p+{{\displaystyle \rho }}_0{D}_p}}{P}_0$$

(1)

$$P_0={\displaystyle \frac{1}{1+\gamma }}{{\displaystyle \rho }}_0{D}_p^2$$

(2)

After integrating Formulas (1) and (2):

$$P={\displaystyle \frac{2\rho {{\displaystyle C}}_p{{\displaystyle \rho }}_0{D}_p^2}{\left(\rho C_p+{{\displaystyle \rho }}_0{D}_p\right)\left(1+\gamma \right)}}$$

(3)

In the formulas, P₀—detonation pressure of the explosive, MPa; D_p—detonation velocity, m/s; C_p—wave velocity of the rock, m/s; ρ/ρ₀—rock/explosive density, kg/m³; and γ—expansion adiabatic index, where the value is 3.

2.3. Explosion Stress of Uncoupled Charge

When the charge structure adopts a cylindrical uncoupled charge, the air between the explosive and the hole wall is compressed into an air shock wave after the detonation of the explosive, which, in turn, exerts a force on the rock.

The initial pressure P_m during the expansion is

$$P_m={\displaystyle \frac{1}{2}}{P}_H={\displaystyle \frac{1}{8}}{\rho }_0{D}_p^2$$

(4)

In this formula, P_H—blasting wave pressure, Pa; ρ₀—explosive density, kg/m³; and D_p—explosive detonation velocity, m/s.

Therefore, when the charge structure is radially uncoupled, the initial pressure value generated after the explosion of the explosive is

$$P=P_m{\left({\displaystyle \frac{V_c}{V_b}}\right)}^3={\displaystyle \frac{1}{8}}{\rho }_0{D}_p^2{\left({\displaystyle \frac{V_c}{V_b}}\right)}^3$$

(5)

In this formula, V_c—explosive volume, m³, and V_b—borehole volume, m³.

For the uncoupled charge structure, the actual pressure on the hole wall is about 8~11 times the initial pressure due to the combined action of the air compression wave formed by the explosion and the explosion shock wave [30]. n can be used to indicate the increase in pressure; then, the actual impact pressure on the hole wall is

$$P=n{P}_m{\left({\displaystyle \frac{V_c}{V_b}}\right)}^3={\displaystyle \frac{1}{8}}n{\rho }_0{D}_p^2{\left({\displaystyle \frac{V_c}{V_b}}\right)}^3$$

(6)

When $V_c={\displaystyle \frac{1}{4}}\pi d_c^2$ and $V_b={\displaystyle \frac{1}{4}}\pi d_b^2$ in the above formula, the formula can be used to calculate the pressure on the hole wall when the uncoupled charge is loaded. In addition, the charge density needs to be converted to account for uncoupling (air gap): $\rho ={\rho }_0{\displaystyle \frac{l_e}{l_b}}$. In summary, when the uncoupled charge structure is used, the pressure on the hole wall is

$$P={\displaystyle \frac{1}{8}}n{\rho }_0{D}_p^2{\left({\displaystyle \frac{d_c}{d_b}}\right)}^6{\left({\displaystyle \frac{l_e}{l_b}}\right)}^3$$

(7)

In this formula, d_c—charge diameter, d_b—blasthole diameter, l_e—blasting hole length and l_b—charge length (all units are m).

3. The Introduction of the Damage Factor in Burnt Rock

3.1. Damage Factor in Damage Mechanics

Under the action of an external load, the internal microstructure of rock is further deteriorated, which is called rock damage. Damage variables or damage factors are usually used to represent the degree of deterioration of rock. There are two kinds of damage variables commonly used in rock mechanics:

(1)

Defining the damage variable according to the crack area

The meso-defects inside the rock reduce the effective bearing area of the rock. If the bearing area of the intact rock is A and the effective bearing area of the damaged rock is reduced to A₁, the damage factor defined by the fracture area can be expressed as

$$D={\displaystyle \frac{A-A_1}{A}}$$

(8)

In Equation (8), parameter D represents the degree of rock damage. It should be noted here that when D = 0, the rock is complete; when D = 1, the rock is completely destroyed.

(2)

Defining the damage variable according to the decrease in the elastic modulus

The elastic modulus of intact rock is defined as E₀, and the elastic modulus of damaged rock is defined as E₁; then, the damage factor D is

$$D=1-{\displaystyle \frac{E_1}{E_0}}$$

(9)

After defining the damage factor D, the stress σ/σ’ relationship in the material before and after damage can be obtained:

$${\sigma }^{\prime }={\displaystyle \frac{\sigma }{1-D}}$$

(10)

The equivalent effect becomes

$$\epsilon ={\displaystyle \frac{\sigma }{E_1}}={\displaystyle \frac{\sigma }{\left(1-D\right)E_0}}={\displaystyle \frac{{\sigma }^{\prime }}{E_0}}$$

(11)

The damage factor in rock can be obtained by direct or indirect measurement methods. Direct measurement refers to the direct observation of the number, shape, size and distribution of various microdefects in the rock by means of an optical microscope, X-ray imaging and an infrared camera. An indirect measurement method involves determining the damage factor of rock according to the principle that the microstructure of rock determines the macroscopic behavior by measuring the resistance, wave velocity, stiffness, strength and plastic deformation of rock.

At present, there are two simple methods used in many test methods:

(1)

Elastic modulus measurement method

By indirectly measuring the elastic modulus of rock before and after damage, the damage factor of rock can be obtained according to the calculation of Formula (9) above. This method is simple and easy to operate.

(2)

Wave velocity measurement method

The relationship between the elastic wave velocity c, elastic modulus E and rock density ρ can be expressed by the following formula:

$$E=\rho c^2$$

(12)

The relationship between the rock elastic wave velocity and damage factor can be obtained by combining Formula (9) and Formula (12):

$$D=1-{\displaystyle \frac{{\rho }_1{c}_1^2}{{\rho }_2{c}_2^2}}$$

(13)

In this formula, ρ₁/ρ₂—density of rock before and after the damage, kg/m³, and c₁/c₂—wave velocity before and after the rock damage, m/s.

3.2. Introduction of Damage Factor

Under the action of high temperature and long-term geological conditions, there are a large number of joint fissures, beddings, joints and other structural planes in burnt rock, which leads to the deterioration of rock properties, thus increasing the complexity of the propagation of explosive stress waves in this fractured rock mass. In order to reflect the degree of deterioration of rock, the physical and mechanical properties of burnt rock are expressed by introducing the damage factor D into the mechanical properties of conventional rock:

$$S_{ce}=\left(1-D\right)S_c$$

(14)

$$S_{te}=\left(1-D\right)S_t$$

(15)

$${\alpha }_c={\displaystyle \frac{\alpha }{1-D}}$$

(16)

$${\mu }_c={\mu }_d\left(1-{\displaystyle \frac{16}{9}}{C}_d\right)$$

(17)

In these formulas, S_c/S_ce—uniaxial static compressive strength of intact/fractured rock mass, MPa; C_d—crack density caused by damage; S_t/S_te—uniaxial static tensile strength of the intact/fractured rock mass, MPa; μ_d/μ_c—dynamic Poisson’s ratio of the intact/fractured rock; and α/α_c—stress wave attenuation coefficient of the intact/fractured rock mass.

4. Attenuation Law of Stress Wave in Burnt Rock

The strong shock wave generated in the rock after the explosion of the explosive propagates over a distance at a supersonic speed, where the propagation distance is generally 3~7 R. With the propagation of the shock wave, the stress amplitude and wave velocity decrease gradually, and gradually decay into a compressive stress wave. The propagation distance of the compressive stress wave is generally 120~150 R. When the strength of the compressive stress wave attenuates to the point where it can no longer damage the rock, it will be transformed into seismic waves. The formation and attenuation processes are shown in Figure 5 below. The attenuation curve of the stress wave is shown in Figure 6 below.

The shock wave in the rock will rapidly attenuate into a stress wave. The attenuation law of radial stress caused by any point in the rock mass within the range of the shock wave can be expressed by the following formula:

$${{\displaystyle \sigma }}_{\mathrm{r}}=\mathrm{P}{\overline{\mathrm{r}}}^{-\alpha }$$

(18)

$$\alpha =2\pm {\displaystyle \frac{{\mu }_d}{1-{\mu }_d}}$$

(19)

In these formulas, $\overline{\mathrm{r}}$—the ratio of the distance from the gun hole and the radius of the gun hole; ${{\displaystyle \sigma }}_r$—radial stress, MPa; $\alpha$—stress wave attenuation coefficient; ±—shock wave area/stress wave zone; and ${\mu }_d$—Poisson’s ratio, ${\mu }_d=0.8\mu$.

By introducing the damage variable D, the attenuation law of radial stress and tangential stress caused by any point in the burnt rock can be obtained:

$${\sigma }_{re}=\mathrm{P}{\overline{r}}^{-{\displaystyle \frac{\alpha }{1-D}}}$$

(20)

$${\sigma }_{\theta e}=-b_e{\sigma }_{re}$$

(21)

$${\sigma }_{ze}={\mu }_c\left({\sigma }_{re}+{\sigma }_{\theta e}\right)={\mu }_c\left(1-b_e\right){\sigma }_{re}$$

(22)

$$\alpha =\left\{\begin{array}{c}{\alpha }_1\begin{array}{c}:\hspace{1em}2+b_e,\hspace{1em}Shock wave zone\end{array}\\ {\alpha }_2\begin{array}{c}:\hspace{1em}2-b_e,\hspace{1em}Stress wave zone\end{array}\end{array}\right.$$

(23)

$$b_e={\displaystyle \frac{{\mu }_c}{1-{\mu }_c}}={\displaystyle \frac{{\mu }_d\left(1-\frac{16}{9}{C}_d\right)}{1-{\mu }_d\left(1-\frac{16}{9}{C}_d\right)}}$$

(24)

In these formulas, ${\sigma }_{re}/{\sigma }_{\theta e}$—radial stress and tangential stress in the burnt rock, MPa, and $b_e$—lateral stress coefficient.

Under the action of the explosion load, any point is in a three-dimensional stress state, and its strength is

$${\sigma }_i={\displaystyle \frac{1}{\sqrt{2}}}{\left[{\left({{\displaystyle \sigma }}_r-{{\displaystyle \sigma }}_{\theta }\right)}^2+{\left({{\displaystyle \sigma }}_{\theta }-{{\displaystyle \sigma }}_z\right)}^2+{\left({{\displaystyle \sigma }}_z-{{\displaystyle \sigma }}_r\right)}^2\right]}^{{\displaystyle \frac{1}{2}}}$$

(25)

Arranged:

$${\sigma }_i={\displaystyle \frac{1}{\sqrt{2}}}{\sigma }_{re}\sqrt{{\left(1+b_e\right)}^2-2{\mu }_c{\left(1-b_e\right)}^2\left(1-{\mu }_c\right)+\left(1+b_e^2\right)}$$

(26)

Since $\sqrt{{\left(1+b_e\right)}^2-2{\mu }_c{\left(1-b_e\right)}^2\left(1-{\mu }_c\right)+\left(1+b_e^2\right)}$ is a constant in the formula and in order to facilitate the derivation of the subsequent formula, this is defined as a constant B.

It can be seen from the analysis that the crushing area in the blasting failure area is compressed and the fracture area is stretched. Therefore, according to the Mises criterion, when the stress ${\sigma }_i$ at any point in the failure area satisfies the following conditions, the rock is destroyed.

$${\displaystyle \frac{{\sigma }_i}{1-D}}\ge {\sigma }_0$$

(27)

$${\sigma }_0=\left\{\begin{array}{c}{\sigma }_{cd}\left(Crushing Circle\right)\\ {\sigma }_{td}\left(Fissured Circle\right)\end{array}\right.$$

(28)

In these formulas, ${\sigma }_0$—rock failure strength, MPa; ${\sigma }_{cd}$—dynamic compressive strength of rock, ${\sigma }_{cd}=S_c\sqrt[3]{\dot{\epsilon }}$, MPa; ${\sigma }_{td}$—dynamic tensile strength, ${\sigma }_{td}={\sigma }_t$, MPa; and $S_c$—static compressive strength, MPa.

The difference between the dynamic compressive and tensile strengths of the rock and the static value is not fixed. It is affected by many factors, including the type, composition, structure and test conditions of the rock. The following is a general description of the difference between the dynamic and static compressive and tensile strengths of rock:

1.

The difference between the dynamic compressive strength and static compressive strength

(1)

Generally speaking, the dynamic compressive strength of rock is greater than the static compressive strength. The ratio of dynamic compressive strength to static compressive strength, namely, the dynamic growth factor (DIF), is widely used to measure the effect of the strain rate on rock strength. This ratio may vary depending on the rock type and test conditions.

(2)

In some cases, the dynamic compressive strength of rock can be 1.5 times or higher than that of the static compressive strength. However, this multiple is not absolute; it may vary from case to case.

2.

Difference between dynamic tensile strength and static tensile strength

(1)

The dynamic tensile strength of rock is usually greater than the static tensile strength. Compared with the compressive strength, the tensile strength is more sensitive to voids and cracks, so the difference between the dynamic and static may be more significant.

(2)

The dynamic tensile strength of rock can be 4~8 times that of the static tensile strength. But likewise, this multiple is influenced by many factors, including the type of rock and the test conditions.

5. Radius Calculation of Burnt-Rock-Blasting Damage Area

5.1. Calculation of Crushing Zone Radius of Burnt Rock

The crushing zone is mainly affected by the compression shock wave formed by the detonation product and has little to do with the effect of the explosive gas. Therefore, it is not necessary to consider the effect of the explosive gas when calculating the crushing zone.

5.1.1. Coupled Charge

The impact load generated by the explosive explosion with the coupled charge structure is calculated by Formula (3). By combining Formula (3) and Formulas (9)–(28), the crushing circle radius $R_1^1$ of fractured rock mass with the coupled charge structure of fractured rock mass can be derived:

$$\begin{array}{l}{R}_1^1=r_b\cdot {\left[{\displaystyle \frac{2\rho C_p}{\rho C_p+{\rho }_0{D}_p}}\cdot {\displaystyle \frac{B{\rho }_0{D}_p^2}{8\sqrt{2}{\left(1-D\right)}^2{S}_c\sqrt[3]{\dot{\epsilon }}}}\right]}^{{\displaystyle \frac{2+b_e}{1-D}}}\\ \end{array}$$

(29)

5.1.2. Non-Coupling Explosive Fill

The pressure on the borehole wall with the uncoupled charge structure is given by combining Formulas (7)–(28) to derive the radius of the crushing ring $R_1^2$ of the fractured rock mass with the uncoupled charge structure of the fractured rock mass:

$$R_1^2=r_b\cdot {\left[{\displaystyle \frac{Bn{\rho }_0{D}_p^2{\left(\frac{d_c}{d_b}\right)}^6{\left(\frac{l_e}{l_b}\right)}^3}{8\sqrt{2}{\left(1-D\right)}^2{S}_c\sqrt[3]{\dot{\epsilon }}}}\right]}^{{\displaystyle \frac{1-D}{2+b_e}}}$$

(30)

5.2. Calculation of Fracture Zone Radius of Burnt Rock

The stress wave intensity in the fracture zone is less than the tensile strength of the rock, and the rock cannot be broken. It is necessary to rely on the generated circumferential tensile stress to further damage the rock. The ensuing detonation gas generates a quasi-static stress field in the rock and further expands and extends the formed radial cracks, and the detonation gas acts for a long time. Therefore, it is necessary to consider the combined effect of the stress wave and detonation gas when calculating the range of the fracture zone.

When the shock wave propagates to a certain position (the radius of the crushing zone) away from the center of the borehole under the action of rapid attenuation, the strength acting on the rock at this time is exactly equal to the compressive strength limit of the rock, and then it will be transformed into a stress wave when it continues to propagate outward. Therefore, the radial pressure ${\sigma }_R$ at the junction of the crushing zone and the fracture zone has the following relationship:

$${\sigma }_R={\displaystyle \frac{{\sigma }_r{|}_{r=R_1}}{1-D}}={\displaystyle \frac{\sqrt{2}\left(1-D\right){\sigma }_{cd}}{B}}$$

(31)

5.2.1. Coupled Charge

(1)

The range of the fracture zone under the action of stress wave

From Formulas (20)–(31):

$$\left(1-D\right){\sigma }_{td}={\displaystyle \frac{1}{\sqrt{2}}}\cdot {\sigma }_R\cdot B\cdot {\left({\displaystyle \frac{R_{2\left(1\right)}^1}{R_1^1}}\right)}^{-\left({\displaystyle \frac{{\alpha }_2}{1-D}}\right)}$$

(32)

The initial radius $R_{2\left(1\right)}^1$ of the crack zone caused by the stress wave with the coupled charge structure is

$$R_{2\left(1\right)}^1=R_1^1\cdot {\left[{\displaystyle \frac{B{\sigma }_R}{\sqrt{2}{\sigma }_{td}}}\right]}^{{\displaystyle \frac{1-D}{{\alpha }_2}}}=R_1^1\cdot {\left({\displaystyle \frac{{\sigma }_{cd}}{{\sigma }_{td}}}\right)}^{{\displaystyle \frac{1-D}{2-b_e}}}$$

(33)

(2)

The range of fracture zone under the action of explosive gas

With the coupled charge structure, the detonation gas penetrates into the cracks generated by the stress wave. At this time, the pressure $P_d^1$ on the borehole wall is

$$P_d^1={\displaystyle \frac{1}{2}}P$$

(34)

Because the rock mechanics parameters of the fractured rock mass are quite different from those of the homogeneous rock, the damage effect of the shock wave again leads to serious changes in the rock mechanics parameters at this time. Therefore, according to the theory of damage mechanics, the damage factor of the fracture zone at this time also needs to consider the damage caused by the shock wave to the rock, so the damage factor at this time should be

$$D^{\prime }=D+D_0$$

(35)

In this formula, $D^{\prime }$—the sum of the damage, $D$—initial damage and $D_0$—the damage of the stress wave done to the rock formation.

The damage $D_0$ of stress wave done to the rock can be obtained according to the following formulas:

$$D_0=1-{\displaystyle \frac{1}{1+Ak{\theta }^m{a}^3}}$$

(36)

$$A={\displaystyle \frac{16\left(1-{\mu }_c^2\right)}{9\left(1-2{\mu }_c\right)}}$$

(37)

$$a={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\sqrt{20}{K}_{IC}}{\rho C_p{\dot{\theta }}_{\max }}}\right)}^{{\displaystyle \frac{2}{3}}}$$

(38)

In these formulas, $k$ and $m$—Weibull coefficients of rock; $\theta$—volume tensile strain; $a$—average radius of microcracks, m; ${\mu }_c$—dynamic Poisson’s ratio of fractured rock; $K_{IC}$—fracture toughness; and ${\dot{\theta }}_{\max }$—maximum volume strain rate.

The radial stress and tangential stress under the action of the quasi-static stress field are

$${\sigma }_{rd}={\displaystyle \frac{{\sigma }_r}{1-D^{\prime }}}={\displaystyle \frac{P_d^1}{1-D^{\prime }}}{\left({\displaystyle \frac{R_{2\left(1\right)}^1}{r}}\right)}^2$$

(39)

$${\sigma }_{\theta d}={\displaystyle \frac{{\sigma }_{\theta }}{1-D^{\prime }}}=-{\displaystyle \frac{P_d^1}{1-D^{\prime }}}{\left({\displaystyle \frac{R_{2\left(1\right)}^1}{r}}\right)}^2$$

(40)

When ${\sigma }_{\theta d}$ > $S_{te}$, the crack propagation is strengthened, which can be expressed by the following formula:

$${\displaystyle \frac{{\sigma }_{\theta d}}{1-D^{\prime }}}\ge S_{te}$$

(41)

To sum up, combining (34)–(41) gives

$$S_{te}\left(1-D^{\prime }\right)={\displaystyle \frac{P_d^1}{1-D^{\prime }}}\cdot {\left({\displaystyle \frac{R_{2\left(1\right)}^1}{R_{2\left(2\right)}^1}}\right)}^2$$

(42)

Therefore, the radius $R_{2\left(2\right)}^1$ of the fracture zone of the burnt rock with the coupling charge structure is

$$R_{2\left(2\right)}^1={\displaystyle \frac{1}{\left(1-D^{\prime }\right)}}\cdot \sqrt{{\displaystyle \frac{P_d^1}{S_{te}}}}\cdot R_{2\left(1\right)}^1$$

(43)

5.2.2. Non-Coupling Explosive Fill

(1)

The range of the fracture zone under the action of a stress wave

The range of the crack zone under the action of a stress wave after an explosive explosion with the uncoupled charge structure is similar to that under the action of a stress wave with the coupled charge structure described above. However, due to the different impact pressure of the two charge structures on the blasthole wall, the range of the crushing zone under the action of a shock wave is different. Therefore, according to the Mises criterion, the following equation can be established with the uncoupled charge structure:

$$\left(1-D\right){\sigma }_{td}={\displaystyle \frac{1}{\sqrt{2}}}\cdot {\sigma }_R\cdot B\cdot {\left({\displaystyle \frac{R_{2\left(1\right)}^2}{R_1^2}}\right)}^{-\left({\displaystyle \frac{{\alpha }_2}{1-D}}\right)}$$

(44)

The initial radius $R_{2\left(1\right)}^2$ of the crack zone caused by the stress wave with the uncoupled charge structure is

$$R_{2\left(1\right)}^2=R_1^2\cdot {\left[{\displaystyle \frac{B{\sigma }_R}{\sqrt{2}{\sigma }_{td}}}\right]}^{{\displaystyle \frac{1-D}{{\alpha }_2}}}=R_1^2\cdot {\left({\displaystyle \frac{{\sigma }_{cd}}{{\sigma }_{td}}}\right)}^{{\displaystyle \frac{1-D}{{\alpha }_2}}}$$

(45)

(2)

The range of the fracture zone under the action of an explosive gas

With the uncoupled charge structure, the detonation gas penetrates into the cracks generated by the stress wave to form a quasi-static stress field, and the pressure $P_d^2$ on the borehole wall is given as

$$P_d^2={\left({\displaystyle \frac{P_0}{P_k}}\right)}^{{\displaystyle \frac{\gamma }{k}}}\cdot \left({\displaystyle \frac{V_c}{V_b}}\right)\cdot P_k$$

(46)

In this formula, $P_0$—detonation pressure of the explosive, MPa; $P_k$—the critical pressure used in the static pressure calculation, $P_k$ = 200 MPa; $\gamma$—expansion adiabatic index, the value is usually 3; $k$—explosive gas isentropic index, k = 3; $V_c$—explosive volume, m³; and $V_b$—hole volume (excluding mud volume), m³.

Combining Equations (39), (40) and (46) gives

$$S_{te}\left(1-D^{\prime }\right)={\displaystyle \frac{P_d^2}{1-D^{\prime }}}\cdot {\left({\displaystyle \frac{R_{2\left(1\right)}^2}{R_{2\left(2\right)}^2}}\right)}^2$$

(47)

Therefore, the radius $R_{2\left(2\right)}^2$ of the fracture zone of the burnt rock with the uncoupled charge structure is

$$R_{2\left(2\right)}^2={\displaystyle \frac{1}{\left(1-D^{\prime }\right)}}\cdot \sqrt{{\displaystyle \frac{P_d^2}{S_{te}}}}\cdot R_{2\left(1\right)}^2$$

(48)

The above derivation shows that the blasting failure zone of the burnt rock mass has a strong relationship with the initial damage D₀ of the rock mass.

6. Application Example

The special properties of burnt rock determine the ‘excellent and poor’ blasting effects. Due to the existence of a large number of discontinuities, such as cracks and joints, with different scales and directions in the burnt rock, the rock mass in the fire area has typical discontinuities and anisotropy. The explosion stress wave and the energy released by the explosive are affected by the cracks and cannot be fully utilized, resulting in an unsatisfactory blasting effect or even failure to complete the blasting operation. In this study, the bench blasting of burnt rock in an open-pit mine in Xinjiang was used as an example to verify the research results.

The physical and mechanical indexes of burnt rock are shown in

Table 1. Summary of rock property tests in burning area.

Rocks Type	Compressive Strength /MPa	Tensile Strength /MPa	Cohesion /MPa	Angle of Internal Friction/°	Elastic Modulus /GPa	Poisson Ratio	Density /kg·m−3	Porosity /%
Burnt rocks	58.69	5.69	57.48	18.26	5.10	0.30	2030	6.4

below.

The mine used a #2 emulsion explosive, and the specific explosive parameters are shown in

Table 2. Explosive material model parameters.

Name of Explosive	Density/kg·m−3	Detonation Velocity/m·s−1	Detonation Pressure/GPa
Emulsified explosive	1.1 × 103	4500	5.6

below.

According to the derived formulas for calculating the blasting failure area of burnt rock with different charge structures (Formulas (29), (30), (33) and (36)), under the condition that the blasthole (150 mm), explosive and rock parameters are known, and assuming that the initial damage of the rock was D = 0.2 and the damage of the blasting stress wave to rock was D₀ = 0.5, the blasting failure radius of the burnt rock could be obtained. The calculation formula of the radius of the failure zone under the condition of the coupling charge structure was not affected by the variable factors. The calculated radius was 2.736 m, which was less than the failure radius of the homogeneous rock and was in line with the actual blasting situation. Under the condition of an uncoupled charge structure, the formula for the radius of the failure area is affected by the charge diameter and charge interval. Assuming that the radial uncoupling coefficient was 1.5, the calculated radius was 2.472 m. For the burnt rock step with a large number of random joint fissures, the radial uncoupled charge structure is more suitable. However, at present, most open-pit mines use mixed-emulsion explosives, which cannot be made into cartridges in advance, and the radial uncoupled charging process is complex and takes a long time. Therefore, considering the problems of charging efficiency and application, the coupling interval charging structure was selected.

Since the damage radius was known, its diameter could be regarded as the row spacing in the blasting hole network parameters. Then, triangular holes were usually used in the blasting design of the open-pit mines. The relationship between the hole spacing and the row spacing was

$$b=a\sin 60°=0.866a$$

(49)

Therefore, the hole spacing of the burnt rock step with the coupled charge structure was 6.4 m. It can be seen that after determining the damage caused by the advanced propagation of the shock wave to the rock mass, the appropriate coupling/uncoupling coefficient value could be determined according to the actual situation of the mine, and then the corresponding blasting hole spacing/row spacing parameter setting could be obtained.

In this field blasting test, the layout of the blast hole and the schematic diagram of the network laying, the profile of the blasting area, and the schematic diagram of the charge structure are shown in Figure 7, Figure 8 and Figure 9.

The test site was arranged on the burnt rock bench in the +455 fire area of the open-pit coal mine. After determining that all the equipment and personnel had evacuated from the dangerous area, the blasting command was issued by the blasting commander, and the blaster performed the initiation. The moment of explosion is shown in Figure 10 below.

After the blasting test, the data were collected in the blasting area under the premise of ensuring that there were no abnormal conditions and absolute safety in the blasting area after strict and careful inspection by the blaster. The blasting results are shown in Figure 11.

It can be seen from the figure that the blasting fragmentation in the test results was relatively uniform, the blasting pile formed on the surface was more flat, there was no blasting crater with a large depth, and the fragmentation of the flying rock produced by the blasting was smaller. It should be noted that the majority of small rocks after the bench blasting in the fire area were due to the fact that the bench in the fire area belonged to the fragmentation structure, which was not caused by the excessive explosive energy.

The radius r of the blasting crater was measured at the test site, and the size of the crater surface is shown in Figure 12.

The hole spacing calculated earlier was 6.4 m, and the radius of the blasting funnel obtained by the test was 6.15 m, which shows that the theoretical formula was successfully deduced and could be applied to the field blasting design of the burnt rock bench.

7. Conclusions

The radius of the failure area after the blasting fracture process of burnt rock when affected by a joint fissure does not conform to the existing theoretical calculation formula, and the distribution law of the failure area also changes. The fracture area is large, and the fracture extension and expansion area are small. Therefore, in view of the blasting situation of burnt rock, based on the combined action of stress wave and explosive gas, combined with damage theory, the damage area radius of burnt rock was deduced by introducing a damage factor. The main conclusions of this study are as follows:

• There were three stages of blasting failure mode: shock wave, compressive stress wave and seismic wave, and then the rock failure area was divided into a crushing zone, fracture zone and vibration zone. The propagation of radial cracks after the blasting of burnt rock was affected by joint fissures, which could not be better extended, which resulted in a smaller range of crack areas and no damage to the rock mass in the vibration area.
• Combined with the theory of rock damage mechanics, a damage factor characterizing the deterioration of rock was introduced into the rock mechanics parameters, and some mechanical properties of fractured rock mass and the calculation formula of stress wave attenuation were obtained.
• According to the Mises criterion, through the stress wave propagation law of fractured rock mass and the expression of three-dimensional stress state of rock, the formulas for calculating the radii of blasting failure area of burnt rock with coupled charge and uncoupled charge structures were derived.
• In addition, it is recommended that the mine adhere to the principle of safety first regarding blasting design to ensure the crushing effect while saving explosives and reducing the impact on the environment. The blasting parameters were designed reasonably and accurately according to the rock characteristics and working conditions, and the construction management and personnel training were strengthened. At the same time, we should actively introduce advanced technology and equipment, improve blasting efficiency and safety, formulate emergency plans to deal with emergencies and comprehensively ensure the smooth progress of blasting operations.