#### Abstract

The surrounding rock is divided into the elastic zone and the plastic zone according to the motion and deformation characteristics of the medium in rock blasting. The vibration caused by the cylindrical charge blasting is outlined on the basis of previous research concerning the blasting elastic-plastic theory. Meanwhile, a superposition model is used to calculate the vibration of cylindrical charge blasting on the premise of considering the influence of detonation velocity, the number of drug columns, and the time delay between holes. The analysis results show that the blasting causes the uneven spatial distribution of vibration, containing both strong and weak directions. The velocity of the vibration along weak directions decreases slowly with the increase of distance. But generally, the velocity along weak directions is faster. The vibration distribution of a cylindrical charge is closely related to the detonator location. Furthermore, more vibration distributes towards the forward direction of the detonation wave.

#### 1. Introduction

The influence of blasting seismic effect on the safety of adjacent buildings has become a key concern in blasting engineering projects. Safety problems caused by blasting vibration often induce disputes and lawsuits in either densely populated coastal areas or sparsely populated mountainous areas. Therefore, understanding and controlling the blasting vibration effect has become an important academic research content [1–9].

The gas with both high temperature and pressure is instantly generated at the time of the blasting. Moreover, the strong compression shock waves act directly on the rock materials, which not only causes serious damage to the rock layer near the explosion cavity but also forms the blasting cavity near the explosion area. With the propagating of the plastic stress waves in the rock medium, the overpressure value decreases rapidly till below the failure strength of the rock. This forms elastic seismic waves in the rock mass. As a matter of fact, the characteristics of detonation sources could affect the size and action time of gas detonation pressure. Different types of explosives can form different blasting cavities. In addition, the geometry of the blasting cavity is also subjected to the charge shape. Since the properties of the detonation cavity determine the characteristics of seismic waves, it can be considered that the explosion source has a significant effect on the characteristics of seismic waves. In blasting engineering, the formation and propagation of the blasting vibration effect will be greatly affected by the topographic and geological conditions of the blasting site, the type of explosives, the charge quantity and charge structure, the buried depth of charge, and the initiation mode.

Seismic radiation generated by cylindrical explosives plays a significant role in exploration, mining geophysics, and mining fields. The research on cavity models began in the early twentieth century. Jeffreys [10] was the first to give out the analytical solution of the full-space cavity problem by the method of mathematical symbols. To describe the relationship between the explosion source and the seismic wave and study the process of how the explosive source forms seismic waves, Sharpe [11] established a cavity-sourced model. Figure 1 illustrates the idealized model of the problem of the generation of elastic waves by explosion pressures. The theory of elastic wave in elasticity is applied to address the problems of explosion source and seismic wave, while the effect of explosion source is expressed by the equivalent cavity with pressure on the wall. Different wall pressures and radii lead to different effects of blast sources. In an equivalent cavity, the stress waves in the medium undergo large strain nonlinear attenuation. And the radiation outside this cavity is considered to be elastic. Blake [12] calculated the analytical solution of the spherical wave propagation in viscoelastic media based on the theory of elastic wave propagation, which was generated by the spherical cavity in elastic media. Jiang et al. [13] simulated the elastic wave and the viscoelastic wave emitted from the spherical detonation source in the dynamic finite element model (DFEM). The calculation results of the above actually confirm Blake’s analytical theory. Duvall [14] calculated the shape of displacement, velocity, acceleration, and strain wave pulses in an isolated elastic medium, among which the pressure pulses are applied in a spherical cavity. According to the calculation results, it can be seen that the shapes of the strain wave pulses in cavities at different distances are similar to those of the strain wave pulses recorded in a typical experiment. This indicates that the stress wave in the medium acts as the pressure pulses, cavity radius, and dielectric properties. According to the propagation theory of spherical waves in elastic media, Xiao and Sun [15] calculated and deduced the amplitude and frequency of spherical waves. Then, Lin and Bai obtained the pressure solution in the blasting chamber by the explosive process in rock and soil. Blair [16] expanded such load to the pressure load caused by real seismic sources. Subsequently, Blair [17] probed the influences of the detonation velocity and completely scale-independent parameters on the characteristics of seismic waves caused by the superposition of point spherical charges. Then, the model was analyzed by combining it with other available experimental results. Applying the Laplace transform, Eason [18] obtained the displacement solution of spherical wave in isotropic media and the cavity pressure expression of the cylindrical wave in transversely isotropic media by solving the elastic wave equation. Favreau [19] considered the process of expanding the gas with high temperature and pressure after the spherical charge explosion. The wave equation of elastic wave was obtained by a combined solution established based on the thermodynamics theory, the rock failure theory, and the spherical wave motion equation. It gave out the expression formula of dominant frequency and amplitude and analyzed the main influencing factors. The spherical cavity source model was used to establish the relationship between the explosive parameters, the elastic medium parameters, and the cavity size and seismic wavefield, which, despite simplifying the physical and chemical process of blasting, could not establish the direct relationship between the characteristics of the explosion source and the characteristics of amplitude and frequency in the seismic wavefield. Yu et al. [20] developed the cavity source model, introduced the prediction model describing the initial conditions of the cavity and the viscoelastic model describing the absorption attenuation of the medium, and established the theoretical model of the seismic wavefield excited by the explosive source.

In addition to the spherical detonation source, cylindrical detonation source is also a major form of detonation. In the case of the ball charge, the source radiations generated by blasting are the same in all directions, and the source radiation mode of cylindrical charge is very different in the radial and axial directions. Starfield and Pugliese [21] decomposed cylindrical charge into several equivalent spherical charge bags, discussed the blasting vibration effect of spherical charge bag by using superposition principle, and solved the blasting vibration effect of the cylindrical charge bag equivalently. Heelan [22] discovered for the first time that both P and S waves are generated simultaneously from a short cylindrical explosive, having specific modes and directions of propagation at different speeds. Figure 2 shows the calculation of Heelan solution about the cylindrical container. Heelan simplified the impact force generated by the explosion of short cylindrical charge into three groups of instantaneous loads acting on the short cylindrical cavity according to the direction of action, namely, the lateral pressure of the hole wall, the pressure at both ends of the cylinder, and the shear force. Ignoring the influence of the pressure and shear force at both ends, only the blasting action on the side of the cavity is considered. Blair developed the scale-independent parameters for the seismic-wave transfer functions and established the frequency criteria necessary for valid use of the traditional solution given by Heelan. Though the finite-length explosive source model proposed by Blair can be used to calculate the pointing vibration velocity, it can hardly be used to determine the pressure on the inner surface of the finite-length cavity. By carrying out the explosion test of cylindrical explosives, Rossmanith and Kouzniak [23] solved the problem of inner surface pressure of long column cavity and gave the expression of inner surface pressure, which conforms to the practical conditions. Then, Blair used Rossmanith’s test results to acquire the numerical solution of the elastic wave of the long cylindrical cavity source. Abozena [24] and Blair [25] found some limitations of the Heelan solution, pointing out that the Heelan solution is more consistent with the actual blasting vibration only in the far field and at a certain source frequency. Blair and Minchinton [26] and Minchinton developed a nonlinear superposition model based on the linear superposition model, which is used to predict the vibration waveform of the cylindrical charge blasting vibration. Blair [27] studied the propagation law of P-Mach wave and S-Mach wave in cylindrical charge blasting and their characteristics under different charge lengths. These waves indicated a high near-field directivity of vibration within a uniform viscoelastic medium. Based on previous studies on blasting elastoplastic theory, Chen et al. [28] constructed the vibration frame of the cylindrical charge blasting according to the motion and deformation characteristics of the rock blasting medium and calculated the vibration characteristics of the long cylindrical charge blasting by superposition model. The results show that when the explosive length reaches a certain degree, the peak particle velocity stops increasing with the enlargement of explosive length. Gao et al. [29] analyzed the action mechanism of detonation position of cylindrical explosive from the perspective of energy distribution and phase delay effect of cylindrical explosive source and extended the vibration field of cylindrical charge explosion based on the vibration field of Heelan blasting model. A comparison was conducted on the blasting vibration fields of the cylindrical charge at different initiation positions. Gao et al. [30] analyzed the mathematical and mechanical laws of detonator position effects from the perspectives of the explosion energy distribution and the explosion stress field of the cylindrical charge. Based on the spherical cavity source model, Xu and Qi [31] used the superposition method to calculate the seismic wavefield excited by the horizontally distributed charge and revealed the amplitude and frequency characteristics of seismic wave excited by horizontally distributed explosive. Compared with the numerical simulation, the error of this method was controlled within 7%. The results show that the theoretical model can accurately describe the amplitude-frequency characteristics of the seismic wavefield excited by the horizontally distributed charge.

It can be seen from the above literature review that the theoretical solution of long column charge through equal superposition can be obtained by either the elastic cavity expansion model or the Heelan theoretical model. Moreover, the error is proved to be relatively small by the comparative analysis of test and numerical solution. In this article, by introducing the equivalent stress of the cylindrical propellant explosion, our model is proposed based on the development of the Yu cavity source model, which takes into account the initial conditions, explosive cavity parameters, dielectric parameters, and the blasting parameters, solves the calculation model of the long cylindrical blasting vibration superposition for any of the blasting vibration velocity of particles in space-time curve, and solves the peak vibration velocity distribution.

#### 2. Materials and Methods

##### 2.1. Single Element of Explosive

The explosion process from blasting to forming elastic waves in the distance is finally accompanied by a series of chemical and physical changes. The energy keeps attenuating during the transmission process due to various dissipation mechanisms. This indicates the evolution from powerful blast waves to elastic waves finally. This process is composed of four stages: the hydrodynamic stage, the geomaterial crushing stage, the dynamic expansion stage, and the elastic wave propagation stage. Meanwhile, the medium would have irreversible deformation under the strong explosion effects during the attenuation process so that the geomaterial near the explosive source shows certain fluid properties under the impacts of huge energy. At the moment of explosion, a wavefront could be pushed out from inside the explosion cave, which is in the same shape as the explosive. With the development of the blast wave, its peak stress attenuates rapidly during the outward propagation process until below the ultimate geomaterial failure strength. At this moment, the geomaterial turns from the fluid stress state to the elastic-plastic stress state, and the attenuation of the stress wave continues until its peak value falls below a certain value. And the geomaterial transforms from the plastic state to the elastic state. In this case, the explosive source forms the explosion cavity area and elastic area and an elastic area in sequence when it blasts in geomaterial along the energy transformation direction. Figure 3 shows the distribution of different damage zones under spherical charge conditions.

Yu [32] and Xu [33] believed that the explosion process of spherical charge is completed instantaneously. After the explosion, the space where the explosive is located is filled with high-temperature and high-pressure gas. High-temperature and high-pressure gas acts on the surrounding rock mass, and the blasting cavity process is completed instantaneously. Assuming that this process is very short, a quasistatic model can be used to predict spherical explosive cavities.

The radius of the fine crushing zone iswhere

The radius of the radial fissures zone iswhere is the radius of the spherical charge, is the pressure of an explosive at the moment of its explosion, is the explosive expansion index, is the cohesion of the soil medium, *c* is the internal friction angle of soil medium, is the compressive strength of the medium, is the tensile strength of the medium, and is Lame’s coefficient.

When the disturbance generated by the explosion propagates to the elastic deformations zone, the explosion wave will propagate as an elastic wave. The propagation of the elastic wave by a spherical cavity could be described by the cavity theory. A mathematical solution was presented by Favreau to predict the properties of the strain waves generated when an explosive detonates inside a spherical cavity in an infinite medium.

In the above model, a spherical explosive of radius was assumed to be surrounded by an infinite, isotropic, homogeneous medium with density , Young's modulus , Poisson’s ratio , and an explosion pressure on the boundary of the elastic zone, but the value of the radius of the elastic zone is not known. The whole process of explosion waves can be described by combining the elastic wave generation and propagation with the elastic zone boundary.

Among all the parameters, the particle vibration velocity , radial strain , and circumferential strain are expressed aswhere

Equation (4) is an ordinary differential equation applied in the field of dynamics to describe a viscously damped spring undergoing forced oscillation. Yu modified (4) as follows:where

The expression for the vibration amplitude can be obtained as follows:

Bring the simplified symbolic formulas A and B into equation (9) to output the amplitude

##### 2.2. Extended Columns of Explosives

For the short column of explosion, the charge length is very small and the blast velocity of the explosives is generally within the 2,000–7,000 m/s range. Under the assumption of a short column, the explosives are considered to be detonated simultaneously and the blast load is applied to the hole wall at the same time. Moreover, as a matter of fact, the Heelan solution is established based on this assumption.

According to the single element of the explosive model and under the same conditions, a way to calculate the column charge characteristics of blast-induced seismic wave approximation was proposed using the spherical superposition, which can be approximated that the column charge package consists of a number of suitable spherical packet superposition. In order to carry out an equivalent substitution, the following assumptions are made: (1) column charge package and spherical packet total volume are equal, (2) with the continuity of the blast process, spherical package spacing is zero; (3) if the package shape is similar, spherical package diameter should be as close as possible to the diameter of the column package or equal and the sum of spherical package diameter should be equal to the length of the column package. Figure 4 shows the equivalent superposition model.

Based on the above three assumptions and Figure 4, it is supposed that the radius of the target column charge is , the length is , the number of equivalent ball charges is , and the ball radius is , then a relationship as below can be figured out between them. The column charge parameters are known and the ball charge parameters are unknown. The following equations have two relations and two unknowns, and the parameters of the ball charge can be obtained by solving the system of quadratic equations:

The effect of blast velocity on mass vibration is considered when analyzing the blast vibration effect of the long columnar packages. For long columnar package blasting, the impact load does not act on the hole wall simultaneously but acts sequentially on the hole wall as the blast proceeds.

Assuming that the explosive starts from the bottom, the detonation speed is , and the charge length of the long cylindrical charge package is , the distances from the center of the equivalent spherical charge package to the point of investigation are , , , , , and the calculation diagram for inspection point is shown in Figure 5.

Since the vibration directions of particles acting on spatial positions of each sphere charge are different, it is necessary to decompose the vibration of multiple spherical charges and finally synthesize the velocity after superposition in - and -directions. At inspection point in space, when the seismic wave generated by the explosion of the first charge reaches inspection point, its vibration velocity is , the velocity component in the -direction is , and the velocity component in the -direction is ; then

For *n* spherical charge packages, the sum of velocity components generated at the investigation point is

The final vibration velocity at the inspection point is

Depending on the type of positive or negative value of the vibration of the resultant velocity direction, and due to the fact that the explosive detonation velocity is greater than the propagation velocity of seismic wave in the rock and greater in some point of the space of vibration period than in explosive detonation process, the final speed direction with various seismic vibration produced by the spherical cartridge in the same direction is with its positive and negative value for - and -direction and speed of positive and negative value.

In bench blasting, the explosives charged in blastholes are cylindrically shaped. It is well recognized that explosive detonation is a complex chemical reaction process along with the chemical energy transforming to the internal and kinetic energy of the detonation products. The modeling of the seismic wavefield in a single row of blast holes using a superposition model. In this study, the first equivalent spherical center of the column charge in the first blast hole is taken as the coordinate origin, the direction perpendicular to the first row of blast holes is taken as the -axis, and the direction parallel to the blast holes is taken as the -axis, to determine the -axis according to the right-hand rule. Figure 6 shows a schematic diagram of the bench blasting and the definition of the model axes.

It is assumed that the coordinate of the observation point is , and the distance from the n-th equivalent sphere charge of the m-th blast hole to the observation point is . At inspection point in space, when the seismic wave generated by the explosion of the first equivalent sphere charge in the first blast hole reaches the inspection point, its vibration velocity is , the velocity component in the -direction is , the velocity component in the -direction is , and the velocity component in the -direction is ; then

For *m* blast holes and *n* spherical charge packages, the sum of velocity components generated at the investigation point is

The final vibration resultant velocity at the inspection point is

It depended on the type of positive or negative value for the vibration of the resultant velocity direction.

#### 3. Results and Discussion

##### 3.1. Calculation of the Seismic Wavefield at a Location in Space

Supposing that three long column explosives with charge radius of and charge length of explode in sandstone, the stemming length would be . Taking buildings on the ground as the detection targets, multiple observation points are set up in two directions, namely, the observation points along the vertical direction of the cylindrical charge and the observation points along the radial direction of the cylindrical charge. Figure 7 shows the layout of the charges and the observation points, among which the characteristic parameters of TNT and sandstone are as shown in Tables 1 and 2. And the blasting parameters are listed in Table 3.

**(a)**

**(b)**

According to Figures 6 and 7, the coordinates of the spherical center of the first equivalent ball where the ignition point is located as (0, 0, 0) are defined. In Figure 7(b), the measuring point in the vertical direction is centered relative to the three charge columns. Moreover, the vibration data at the symmetric position is typical. When = 100, the coordinates of the corresponding observation point are . The particle velocity of the three directions at = 100 m was calculated using the equivalent superposition model of the sphere charge package. Figure 8 plots the typical velocity time-history recorded by the vibration senor, which contains three evident differences that are separated at the direction axis. They are, respectively, the *X*-direction, *Y*-direction, and the *Z*-direction. The positive peak particle velocities along three directions are 80 cm/s, 0.25 cm/s, and 0.3 cm/s respectively. Considering the offset distance between *x*, *y* direction and blasting column plane, the distance difference is 20 times and the peak particle velocity difference is 320 times. Due to the axisymmetric characteristics, the velocity component in the *y* direction is partially offset, and the velocity component in the *X*-direction is superimposed.

**(a)**

**(b)**

**(c)**

###### 3.1.1. Calculation for Observation Points at Different Locations in Space

Figure 7(b) shows the investigation of the explosive seismic wave distribution in the vertical direction based on the the explosive parameters, rock parameters, and blasting parameters in Tables 1–3. Six points of A, B, C, D, E, and F are selected, which are with coordinates of (300, 5, 15). The peak particle velocity of cylindrical blasting vibration can be calculated through the superposition of a microcharge in an axial direction. The calculating data are shown in Figure 8.

The peak particle velocity indicator, as is usually used to evaluate the structural damage, is also applied to study the blast vibration field. Figure 9 shows the peak particle velocity observed along the vertical direction of the charge plane. Figures 9(a) and 9(b), respectively, denote the bottom initiation, which is the synthesis of the radial and vertical peak particle velocity. As shown in Figure 9(a), the *X*-direction vibration velocity decreases with the increase in vertical distance. At the point of *X* = 150 m in the *X*-direction, there is a turning point; before the turning point, the speed drops rapidly, and then after the turning point, the velocity tends to be stable. The vibration velocities along *Y*-direction and *Z*-direction decrease slowly with the increase of the vertical distance, which can offer the exponential relationship.

**(a)**

**(b)**

Figure 9(b) shows the investigation of the explosive seismic wave distribution in the radial direction based on the explosive parameters, rock parameters, and blasting parameters in Tables 1–3. Five points of A′, B′, C′, D′, and E′ are selected in this article, which are with coordinates of (100, −95, 15), (100, −45, 15), (100, 5, 15), (100, 55, 15), and (100, 105, 15) respectively. The calculating data are as shown in Figure 10. Being affected by the symmetry, the minimum velocities of the peak particle in the *X*- and *Y*-directions appear at the symmetry axis. At the same distance from the axis of symmetry, the values of measuring points along the direction of detonation are obviously larger. For the bottom initiation, the observation points are located at the forward direction of the detonation wave, so the contributions from the subsequent charge elements become stronger and stronger. Without the influence of symmetrical charge, the vibration velocity in the *Z* direction has little change.

**(a)**

**(b)**

#### 4. Conclusions

A superposition model is used to calculate the vibration of cylindrical charge blasting while considering the influence of detonation velocity, number of drug columns, and delay time between holes. The following conclusions are obtained by analyzing the calculation superposition model:(1)The vibration waveform of spherical charge blasting is dependent on the blasting source pressure function acting on the elastic-plastic boundary; the smaller the attenuation coefficient of pressure a, the larger the vibration period and amplitude.(2)The distributions of vibration caused by blasting along the *X*-, *Y*-, and *Z*-directions are not uniform, including one strong direction and two weak directions. The main vibration direction (strong direction) is perpendicular to the cylindrical charges plane. The vibration velocity of the weak direction decreases slowly with the increase of distance. Moreover, the vibration velocity of the strong direction decreases fast with the increase of distance in front of the turning point while showing a velocity descending tendency with the same characteristics after the turning point.(3)The in-hole detonator (firing point) is capable of adjusting the spatial distribution of blast vibration in the surrounding rock mass. The vibration distribution of a cylindrical charge is closely related to the detonator location. More vibrations actually distribute to the forward direction of the detonation wave.

The new calculation formula proposed in this article is primarily based on theory and previous research and has not yet been applied in an actual project. As such, further experimental and engineering research is warranted in the future. However, the model proposed in this article provides a theoretical basis and research approach for conducting the refined blasting and production.

#### Data Availability

Data are available and included within this article; readers can access the data supporting the conclusions of this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest.

#### Acknowledgments

The authors would like to acknowledge the State Key Laboratory of Explosion Science and Technology at Beijing Institute of Technology. This work was also supported by the National Natural Science Foundation of China (no. 51678050) and the National Key R&D Program of China (no. 2017YFC0804702).