## Abstract

To investigate the dynamic response and attenuation law of rock slope sites subjected to blasting, three lithological numerical models, including slate (hard rock), tuff (relatively soft rock), and shale (soft rock), are established by using MatDEM. By analyzing the wave field, velocity, and acceleration response of the models and their Fourier spectrum, combined with stress and energy analysis, their dynamic response characteristics are investigated. The results show that blasting waves propagate from near field to far field in a circular arc, and the attenuation effect of waves in soft rock is less than that in hard rock. The influence of lithology on the dynamic response of the ground surface and bedrock is different. Blasting waves mainly affect the dynamic response in the near-field area of the blasting source. In addition, the dynamic amplification effect of slopes is as follows: hard rock > relatively soft rock > soft rock. The slope surface has an elevation attenuation effect. A dynamic amplification effect appears in the slope interior within the relative elevation (0.75, 1.0). The Fourier spectrum has an obvious predominant frequency, and that of the slope crest and interior is less than that of the slope surface. Moreover, the total energy generated by the rocky sites gradually changes into kinetic energy, gravitational potential energy, elastic potential energy, and heat. Energy-based analysis shows that the attenuation effect of blasting waves in hard rock is larger than that in soft rock overall. This work can provide a reference for revealing the blasting vibration effect of rock sites.

## 1. Introduction

Because of the advantages of fast construction, low cost, and high efficiency, the blasting method has become the main construction method of slope and tunnel engineering [1]. Nevertheless, due to the influence of the propagation medium, the waveforms and propagation characteristics of blasting seismic waves become very complicated [2]. Blasting seismic waves will lead to slope instability and other geological disasters; in particular, in coal mining areas, under the influence of human blasting over the years, geological disasters, such as mountain cracking and creep, will occur on slopes, seriously threatening the safety of people’s lives and property [3, 4]. Moreover, seismic exploration blasting technology is an important method in geophysical exploration [5]. The seismic effect of explosive blasting has become a key problem in land oil and gas exploration and foundation construction. The propagation law and damage effect of seismic waves in different geological bodies are the main basis of engineering blasting design [6, 7]. Therefore, explosion-induced seismic waves have been one of the most active subjects in the field of civil engineering blasting.

Blasting seismic waves are a complex physical phenomenon [8, 9]. It is affected by many factors, such as the location of the source of detonation, the amount of explosive, the mode of explosion, the state of charge, different media in the transmission path, and local site conditions [10-12]. In the process of propagation, the intensity of blasting seismic waves will gradually weaken with increasing propagation distance, especially the propagation characteristics of far-site seismic waves, which are the key factors affecting the quality of seismic exploration [13, 14]. For the region along the Sichuan‒Tibet Railway, the terrain geological and climatic conditions along the railway construction are extremely complicated. Geological structures and landforms have become important factors affecting the propagation characteristics of blasting seismic waves. In particular, in the rock medium site, the basic properties of rock composition, structure, and so on are different, which essentially affect the propagation characteristics of blasting seismic waves [15]. It is necessary to study the influence of lithology on the near- and far-field waveforms and propagation characteristics of blasting waves.

Numerical simulation has become one of the commonly used methods in the field of explosion impact [16, 17]. At present, the finite element method (FEM), finite difference method (FDM), and discrete element method (DEM) are commonly used in the numerical simulation of explosion impact. Many scholars have studied the propagation characteristics and attenuation law of blasting seismic waves by using FEM [18, 19]. FEM has great limitations for discontinuous media, infinite domains, large deformations, and stress concentrations. Aiming at the large deformation of discontinuous media, many scholars have begun to use the FDM to study the dynamic response characteristics of rock-soil masses [20]. However, the FDM has difficulty simulating the failure process of rock-soil masses because of its arbitrary division and boundary conditions. Some scholars began to use the DEM to study the dynamic response law of rock slopes [21-23]. The research results show that the DEM is mainly suitable for solving discontinuous media and large deformation problems and can better simulate the dynamic response and failure process of complex rock-soil masses. Therefore, many achievements have been made in the numerical simulation of blasting seismic waves in previous studies. However, due to the limitations of the particle number, computing speed, and computing efficiency of the previous discrete-element software, it is urgent to propose a more efficient and convenient discrete-element numerical simulation method.

There are many factors affecting the propagation characteristics of blasting seismic waves [24, 25]. Blasting vibration involves not only the interaction between the explosive and rock mass but also the propagation of waves in the stratum [26]. The influence of the explosion source on the propagation law of blasting seismic waves is complicated. Many scholars have studied the influence of the excitation factors of explosive sources on the propagation characteristics of seismic waves, including the charge, length-diameter ratio, and coupling medium [2, 27]. It can be seen that explosive source excitation factors have a great influence on the near-field waveform and propagation characteristics of waves [28]. In addition, blasting seismic waves in practice are a very complicated problem that is related to the properties of strata rock, which itself is a complex body [29, 30]. Some scholars have shown that the characteristics of rock masses have become one of the main factors affecting the propagation characteristics of blasting seismic waves, including rock type, rock weathering, groundwater level, topography, and geomorphology [31]. Some scholars have studied the effect of landforms on blasting vibration wave propagation [32]. The research results mainly focus on the blasting vibration and attenuation law under flat terrain, the attenuation effect of concave landforms, the amplification effect of convex landforms, and so on. Previous studies have paid more attention to the influence of explosive source excitation factors such as charge amount, type, aspect ratio, coupling medium, and delay time. However, the study of lithology, geological structure, and landforms is insufficient. Current studies have not fully revealed the near- and far-field characteristics and propagation rules of blasting waves in rock medium sites, and there is a lack of efficient discrete-element numerical simulation methods and algorithms to realize the numerical simulation of blasting seismic waves from near field to far field. The study of lithology on explosive source excitation wavelet shape and its propagation characteristics is insufficient, so it is urgent to systematically explore the influence mechanism of lithology and propagation characteristics of blasting seismic waves.

According to the hardness of rocks, they can be divided into four types, including hard rock, relatively hard rock, relatively soft rock, and soft rock. This work takes medium sites with different lithologies as research objects, including slate (hard rock), tuff (soft rock), and shale (soft rock), and focuses on the propagation characteristics of blasting seismic waves with different lithologies under explosive blasting. The high-performance matrix discrete-element software MatDEM was used to carry out numerical simulation tests, and three discrete element models of rock slope sites were established. By applying a blasting source to study the waveform characteristics of blasting seismic waves in different lithological medium sites, the propagation characteristics and attenuation law of near- and far-field waves generated by blasting are explored. The dynamic response of seismic waves in different lithological medium sites was investigated. The influence of different lithology conditions on the shape characteristics of blasting seismic waves, the propagation characteristics of near- and far-field waves, and the dynamic response law of the site are revealed. This study can deepen the understanding of seismic wave propagation characteristics and disaster-causing mechanisms of explosive sources in rock media under complex conditions and has important scientific significance and application value.

## 2. Methodology

### 2.1. Basic Principles of the DEM

The DEM constructed a rock-soil mass model by stacking and cementing a series of particles with specific mechanical properties [33]. Particles interact with each other through different contact models. The linear elastic model used in this model is shown in Figure 1 [34]. The model consists of a series of stacked particles following Newton’s second law of motion. The elements are connected to each other by breakable springs, and the force can only occur at the contact points between adjacent elements, as shown in Figure 1(b) and (c). On this basis, numerical simulation is carried out by a time-step iteration algorithm. In the most basic linear elastic model, it is assumed that particles rely on springs to contact each other and produce forces. The normal force (*F _{n}*) and the normal deformation (

*X*) between particles can be simulated by the normal spring between particles [34]:

_{n}where *K _{n}* is the normal stiffness,

*X*is the normal relative displacement (Figure 1(b)), and

_{n}*X*is the fracture displacement. Initially, the particles are connected to their neighbors and subjected to a spring force of tension or pressure.

_{b}When *X _{n}* between the two particles exceeds the fracture displacement (

*X*), the spring breaks, and the tension between the particles disappears. Only the pressure effect can exist.

_{b}Tangential springs were used to simulate shear forces (*F _{s}*) and shear deformation (

*X*) between particles [35]:

_{s}where *K _{s}* is the tangential stiffness and

*X*is the tangential displacement.

_{s}Similarly, there is a failure criterion in the cutting direction of the spring, which is based on the Mohr-Coulomb criteria [34, 35]:

where *F*_{Smax} is the maximum shear force, *F*_{S0} is the shear resistance between particles, and μ_{p} is the friction coefficient between particles.

In the Mohr-Coulomb criterion, the maximum shear resistance between cells is related to the initial shear resistance (*F _{s}*

_{0}).

*F*

_{s}_{0}is the maximum shear force that can be borne between elements without applying normal pressure, which is similar to the cohesion of the rock-soil mass. The greater the normal pressure is, the greater the shear resistance. When the tangential force exceeds the maximum shear force, the tangential connection breaks, and only the sliding friction force

*−μ*·

_{p}*F*exists between particles.

_{n}In the numerical simulation, a normal spring and tangential spring are introduced to equilibrate the cementation between grains in the real world, such as ferric and calcareous cementation, when sand gravel and other sediments are deposited during diagenesis. Therefore, when the normal spring breaks in the numerical simulation, it corresponds to cement fracture in the real world. At this point, the tangential spring should also be disconnected and vice versa [33, 36].

The stacking model of a unit defined by linear elastic contact has the elastic characteristics of a whole. If the elastoplastic and creep properties of materials are to be simulated, different contact models need to be defined. For example, for elastoplastic materials, the element needs to be defined as approximately elastoplastic as well. Macro and micro research is a very important branch of the DEM, that is, how to establish an appropriate cell contact model and determine the corresponding parameters according to the macroscopic mechanical properties of materials [37, 38].

The stiffness described above is the stiffness (*K _{n}*) of connections between elements, and each element has its own stiffness (

*K*). When two elements touch, there are actually two springs in a series. For two elements with normal stiffnesses

_{n}*K*

_{n}_{1}and

*K*

_{n}_{2}, the equivalent normal stiffness (

*K*) of their connection is [33, 36]:

_{n}For two elements with tangential stiffnesses of *K _{s}*

_{1}and

*K*

_{s}_{2}, the equivalent tangential stiffness (

*K*) of their connection is [33, 36]:

_{s}Similarly, each element has its own breaking displacement and friction coefficient, and the mechanical properties of the connection depend on the tensile or shear resistance of the smaller element. Hence, if the stiffness of the two elements is the same, the stiffness of the series connection (*K _{n}*,

*K*) is half the stiffness of the element (

_{s}*K*,

_{n}*K*), and the breaking displacement of the series connection (

_{s}*X*) is twice the breaking displacement of the element (

_{b}*K*). In the numerical calculation, the stiffness and fracture displacement of the element are used, and the mechanical properties of the connection are obtained through calculation.

_{b}On the basis of the force of each particle, the displacement of the particle is calculated by the time-step iteration algorithm. Set the time step *dT* to calculate the force, acceleration, velocity, and displacement of particles. After the calculation of the current time step is completed, another time step is advanced to realize the iteration of the DEM. The specific steps are as follows: based on the traditional Newtonian mechanics method, on the basis of the known resultant force on each particle, divide the resultant force by the particle mass to obtain the acceleration of the particle at this moment. In time step *dT*, add the current velocity plus the increment in velocity. That is, the initial velocity of the next time step can be obtained, and the corresponding element displacement can be calculated by the average velocity within the time step. Then, the new iterative calculation is entered through repeated iterations to achieve the DEM dynamic simulation. For example, if a small displacement is applied to the upper surface of a cubic discrete element model, the first layer of particles on the upper surface will move down slightly and squeeze the adjacent lower layer of particles, causing them to move down. Through continuous iteration of time steps, the propagation of stress waves can be achieved while the force is transferred to the bottom. Hence, there are concepts of time and motion in discrete-element numerical simulation, which also exist in the real world.

### 2.2 Modeling Process of Discrete Element Models

The detailed process of discrete element modeling is as follows (Figure 2). (a) Import data and cut models: generate random units and conduct gravity deposition and compaction on them to simulate the diagenetic process of rocks in nature. First, the discrete element accumulation body is used to construct the slope surface, and the stratigraphic model is cut out according to the elevation data. The model is 600 m high and 1500 m wide. (b) Set the material and balance model: a homogeneous material is set, whose mechanical properties and density are recorded in the notepad document under the Mats folder. The micromechanical parameters of the material are calculated using the transformation formula of macro- and micromechanical properties of the discrete element model. Generate material objects by directly specifying the properties of the material. Finally, the discrete element model is obtained by balancing the models. (c) Set the detonation point and blasting energy: define the location and radius of the detonation point on the slope. To obtain the blasting element, the blasting energy is generated by increasing its radius. (d) Iteration calculation and simulation results: set the number of cycles and conduct a standard balance for each cycle, with a total simulation time of 0.35 seconds in the real world. Monitoring points were set up to record acceleration/displacement time-history data at different positions of the slope to simulate the dynamic process. Stratum and slope materials refer to the macromechanical properties of natural rock materials (Table 1), and the corresponding micromechanical parameters are obtained through a conversion formula (Table 2). The conversion formula of the macro- and micromechanical properties of the discrete element model is as follows [33, 36]:

For the linear elastic model, the normal stiffness (*K _{n}*), tangential stiffness (

*K*), fracture displacement (

_{s}*X*), initial shear force (

_{b}*F*

_{s}_{0}), and friction coefficient (

*μ*) can be represented by five macroscopic mechanical properties of materials. That is, Young’s modulus (

_{p}*E*), Poisson’s ratio (

*v*), compressive strength (

*C*), tensile strength (

_{u}*T*), and coefficient of internal friction (

_{u}*μ*) are calculated by the above formula. In the above formula,

_{i}*d*is the diameter of the element.

To study the influence of the rock slope site on the propagation characteristics and attenuation law of the blasting source, slope site models with a slope angle of 60° and a slope height of 200 m were established, including Model 1 (slate), Model 2 (tuff), and Model 3 (shale), as shown in Figure 2(b). The total number of units in the models is 204,667, and 67 measuring points are set in the models (Figure 2(c)). First, the purpose of setting measuring points on the horizontal and vertical axes of the explosion source is to study the propagation characteristics and attenuation laws of the blasting wave as a function of the distance from the explosion source. Second, to study the influence of the slope area on the propagation characteristics and attenuation law of blasting waves, measuring points are set around the slope. The blasting source position is the 27 (MP-27) measuring point 100 m away from the slope toe. The seismic wave was excited by blasting with 2172.67 kg of explosives.

## 3. Wave Propagation Characteristics of Seismic Waves Induced by Blasting at Slope Sites

### 3.1. Influence of Lithology on the Characteristics of the Blasting Seismic Wave Field

To investigate the influence of lithology on the characteristics of the blasting seismic wave field, the velocity and displacement fields of the blasting wave are analyzed as examples (Figures 3 and 4). Figure 3 shows that the velocity distributions of the blasting waves are similar in Models 1–3 (slate, tuff, and shale slope sites). The velocity wave field has the following propagation characteristics: First, after the explosive explodes at the blasting source, the velocity wave propagates from near field to far field in a circular arc along the rocky site with the blasting source as the center, such as the velocity field distribution at *t* = 0.038 seconds. Second, with the duration of the explosion, the velocity wave field continues to propagate to the far-field area of the models in a circular arc. The amplitude of the outermost velocity field decreases, such as the distribution characteristics at *t* = 0.076 seconds. Finally, as the explosion time continues, waves continue to propagate to the far field, and their amplitude further attenuates within the propagation process, such as the velocity field distribution at *t* = 0.114 seconds.

In addition, the displacement wave field of the models under explosion conditions shows the following characteristics (Figure 4): First, when *t* = 0.038 seconds in the early stage of the explosion, the displacement of the near-field near the blasting source is large, and the displacement of bedrock under the blasting source is obviously smaller than that of the surface area. Second, with the duration of blasting time *t* = 0.076 seconds, the displacement field is diffused in the far field of the models with circular arc characteristics centered on the blasting source. The displacement amplitude in the area near the ground surface is obviously larger than that of the deep bedrock. This is because the free surface of the earth has little influence on the propagation of waves, while the bedrock has an obvious weakening effect on the energy propagation of waves. The amplitude of the displacement field has obvious attenuation characteristics. As the blasting time lasts, the waves continue to propagate to the far field, and the displacement field exhibits further attenuation characteristics. The displacement of the far-field ground surface area on the left side of the blasting source is obviously greater than that of the slope area. This indicates that the slope has a greater weakening effect on waves than the ground surface.

By comparing the velocity and displacement wave propagation characteristics, the lithology has an effect on the velocity and amplitude of blasting wave propagation in the models. Figure 3 shows that under the same blasting time, different lithologies influence the propagation distance of the velocity wave field. For example, when *t* = 0.114 seconds, the propagation distance in the slope site of Model 1 is relatively close, approximately 80 m from the bottom boundary. The outermost wave field of Model 2 is approximately 40 m away from the bottom boundary. In Model 3, the outermost velocity wave field basically reaches the bottom boundary. Hence, the lithology has an effect on the velocity wave field propagation velocity of waves in the models. The propagation speed of hard rock is the slowest, and that of soft rock is the fastest. This is because the particles in hard rock are denser, and the elastic modulus and density are larger, which has a greater damping effect on the velocity propagation of waves. In addition, in Figure 4, lithology has an impact on the distribution characteristics of the amplitude and propagation distance of the displacement wave field. For example, when *t* = 0.076 seconds, the amplitude of the outermost displacement wave field near the ground surface of Model 1 is the largest. However, compared with Models 2–3, the outermost maximum displacement in Model 1 is the smallest, Model 2 is larger, and Model 3 is obviously larger than that of the hard rock and relatively soft rock sites. This indicates that, compared with hard rock, the attenuation effect of soft rock on the amplitude of the displacement wave field in rocky sites is weaker. When *t* = 0.114 seconds, the displacement wave in Model 3 basically reaches the bottom boundary, and its displacement wave field propagates faster than that of Models 1–2, which is consistent with the analysis results of the velocity wave field in Figure 3.

### 3.2. Influence of Lithology on the Waveform Characteristics of Blasting Seismic Waves in the Models

To investigate the influence of different lithologies on the near- and far-field waveform characteristics of explosion seismic waves in the models, with the explosion source as the center, the velocity and acceleration waveforms of the left, right, and lower areas of the explosion source of the models are shown in Figures 5 and 6. Figures 5(a) and (b) and Figures 6(a) and (b) indicate that in the process of the velocity and acceleration waveform propagating from the near field to the far field in Model 1, there is no obvious attenuation phenomenon of the seismic wave in the surface area on the left and right sides of the blasting source. However, the velocity and acceleration waveforms in Models 2–3 have obvious attenuation effects during propagation from the near field to the far field. In addition, Figures 5(a) and (b) show that as the lithology changes from hard rock to soft rock, the near-site wave velocity and acceleration amplitude near the blasting source gradually increase. This shows that lithology has an impact on the energy attenuation of waves in the surface area near the blasting source. The waves in the hard rock field mainly attenuated and dissipated rapidly in the ground surface area near the blasting source, while the near-site waves in the soft rock attenuated slowly. Figure 5(c) shows that in the deep bedrock below the explosion source, the velocity and acceleration waveform of Model 1 attenuated rapidly in the near-field area of the explosion source, while Models 2–3 attenuated slowly. This indicates that the influence of lithology in the deep bedrock area on the waveform and attenuation effect is opposite to that in the ground surface. This is because the damping ratio of soft rock in bedrock is greater than that of hard rock, and the energy dissipation and attenuation effects of waves are more likely to occur in soft rock than in hard rock in the near-field area. This is consistent with the wave field analysis results in Figures 3 and 4.

### 3.3. Influence of Lithology on the Attenuation Law of the Slope Area

To explore the influence of lithology on the dynamic response characteristics of slopes under blasting, the peak ground velocity (PGV) and peak ground acceleration (PGA) were used as analysis indexes. The PGV and PGD slopes in different lithology sites are shown in Figure 7. At the slope surface, PGV and PGA gradually decrease with relative elevation (*h*/*H*). In particular, the PGA and PGV at the slope surface decreased rapidly when *h*/*H* = 0–0.2 but decreased slowly when *h*/*H* = 0.2–1.0. This indicates that the blasting wave has a typical elevation attenuation effect at the slope surface. In particular, the attenuation rate is at a maximum within *h*/*H* = 0–0.2, and the attenuation rate is small within *h*/*H* = 0.2–1.0. In the inner slope, PGV and PGA decreased gradually in *h*/*H* = 0–0.75 but increased rapidly in *h*/*H* = 0.75–1.0. At the slope crest, PGA and PGV showed a rapid increase in a certain range with distance from the slope shoulder and then showed a rapid decrease at the slope crest outside MP-58. PGV and PGA gradually decrease to stable values far from the slope shoulder. Hence, at the slope surface, slope elevation has an obvious elevation attenuation effect on the blasting waves, especially the attenuation effect, which is the largest within *h*/*H* = 0–0.2. Inside the slope, elevation first attenuates and then amplifies the waves, and the critical point is *h*/*H* = 0.75. At the slope crest, with the distance from the slope shoulder, the blasting wave first shows an amplification effect and then an attenuation effect.

### 3.4. Influence of Lithology on the Attenuation Law of Seismic Waves in the Models

To analyze the influence of lithology on the attenuation law of blasting seismic waves in the models, with the blasting source as the center, the PGV and PGA of the horizontal and vertical axes of the blasting source are shown in Figures 8 and 9. Figure 8 shows that in the horizontal axis direction of the blasting source, PGV and PGA gradually decrease with distance from the blasting source. In particular, the PGV and PGA rapidly decrease within 0–33.3 m and gradually decrease and tend to be stable when the distance is >33.3 m. Figure 9 shows that in the vertical axial direction of the blasting source, PGV and PGA in the bedrock decay rapidly within 0–50 m with depth to the blasting source, and the attenuation effect gradually becomes stable when it is >50 m. This indicates that the rapid attenuation effect appears in the small range of the blasting source (near-field area), and the attenuation effect gradually becomes stable in the far-field area with distance from the source. Figures 10 and 11 show the PGV and PGA distributions of waves inside the slope area under blasting. Figures 10 and 11 show that the explosion source has a limited influence on the dynamic response of the rock slope site, and the maximum values of PGA and PGV are mainly concentrated in the near-field area of the blasting source, while the wave has little influence in the far-field area, which is consistent with the analysis results in Figures 8 and 9. In addition, from Figures 7-10, the PGV and PGA in Model 1 are the largest, followed by Model 2, and the smallest in Model 3. The PGVs of Models 1–3 were 22.0, 18.7, and 16.5 m/s, respectively. The PGAs of Models 1–3 were 0.60, 0.48, and 0.39 g, respectively. This shows that lithology has an influence on the dynamic response characteristics of blasting waves in the slope. With the lithology changing from hard to soft rock, in the slope area, their dynamic amplification effect decreases gradually, and the dynamic amplification effect of hard rock is greater than that of soft rock.

## 4. Influence of Lithology on the Dynamic Response Characteristics of Blasting Seismic Waves in the Frequency Domain

### 4.1. Influence of Lithology on the Fourier Spectrum Characteristics of Blasting Seismic Waves in the Rock Sites

The fast Fourier transform (FFT) is a combination of seismic signals into multiple harmonic signals and is widely used in signal analysis and processing [39]. Based on frequency domain signal analysis, the spectrum response of the rock-soil mass and the spectrum characteristics of waves can be obtained. The Fourier transform has the advantages of good frequency positioning and clear identification of different frequency components of the signal. The FFT can quickly identify the main components of the signal and can also quickly filter and become a common means of processing seismic signals [40]. The mathematical expression of the FFT is shown as follows [41]:

where a(*t*) is the acceleration history of the time domain, and *F*(a) is the Fourier transform of the acceleration history a(*t*).

To study the propagation characteristics and attenuation law of blasting waves in the models from the frequency domain, Fourier spectra of acceleration-time histories in different lithology models are analyzed. The Fourier spectra of the blasting waves are shown in Figure 12. Figures 12(a) and (b) show that the Fourier spectrum characteristics along the horizontal axis of the blasting source have the following characteristics. The spectrum amplitudes of Model 1 are uniformly distributed along the horizontal frequency axis with abundant frequency components, and no obvious abrupt change occurs in the spectrum amplitudes. The spectral characteristics of Models 2–3 are obviously different from those of Model 1. The spectrum amplitudes of the measuring points (MP-26, MP-28, and MP-29) in Models 2–3 have an obvious surge phenomenon along the horizontal frequency axis. This indicates that soft rock has an obvious amplification effect on some frequency components of blasting waves in the near-field area of the blasting source. Meanwhile, in Figure 12(a) and (b), the peak Fourier spectrum amplitude (PFSA) of Model 1 is smaller than that of Model 2, and the PFSA of Model 3 is the largest. This indicates that the dynamic amplification effect of the hard rock model is smaller than that of the soft rock model in the horizontal axial direction of the blasting source. In other words, the blasting wave has a fast attenuation effect in the near-field area of the horizontal axis of the source in the hard rock, while the attenuation effect is weak in the soft rock. Figure 12(c) shows that in the vertical direction of the blasting source, the PFSA of the Model 1 hard rock model is obviously larger than that of Models 2–3. The Fourier spectral characteristics of the models are similar overall. This indicates that in the vertical axial bedrock, the lithology has little influence on the spectral characteristics of the frequency axis, and the wave attenuation effect of bedrock in the hard rock is less than that in the soft rock. The above analysis is consistent with the results of the time domain.

In addition, to study the propagation characteristics and dynamic response characteristics of blasting waves in the slope area, Fourier spectra of typical measuring points on the slope surface, interior, and crest are selected for analysis, as shown in Figure 13. Figure 13(a) shows that at the slope surface of the models, no obvious superior frequency segment appears in the Fourier spectrum of MP-30 (slope toe). The distribution of spectral amplitude along the horizontal frequency axis does not have obvious change rules, and the amplification features are relatively obvious along the whole frequency axis. However, other measuring points on the slope surface (MP-53-MP-56) have obvious excellent frequencies, which have an obvious amplification effect on the spectrum amplitudes of MP-53–MP-56 when the frequency component is between 8 and 13 Hz. In Figures 13(b) and (c), the measuring points inside the slope and at the slope crest have obvious excellent frequency segments, the spectral amplitude between 5 and 10 Hz has an obvious amplification effect, and the PFSA also appears between 5 and 10 Hz overall. In Figure 13, the PFSA of the slope shows the following characteristics: Model 1 > Model 2 > Model 3. This indicates that the dynamic response characteristics of hard rock are more intense than those of soft rock in the slope area, and the attenuation effect of blasting waves is greater in the slope area of soft rock.

### 4.2. Influence of Lithology on the Dynamic Response Characteristics of the Slope Area

To further study the dynamic response characteristics of the slope under blasting, the PFSA of the slope is shown in Figure 14. Figure 14(a) shows that the PFSA of the slope surface gradually decreases with increasing *h*/*H*; in particular, the fastest attenuation rate appears within *h*/*H* = 0–0.2, and the attenuation rate is small and gradually tends to be stable within *h*/*H* = 0.2–1.0. Figure 14(b) shows that PFSA decreases within *h*/*H* = 0–0.75 inside the slope, while it increases within *h*/*H* = 0.75–1.0. This indicates that inside the slope, the dynamic response first shows an attenuation effect along the elevation and then shows an amplification effect near the slope crest. *h*/*H* = 0.75 is the critical point of the dynamic response change. In Figure 14(c), the PFSA at the slope crest first shows an amplification effect with distance from the slope shoulder and then rapidly decreases and gradually stabilizes outside MP-58. In addition, the PFSA of Model 1 was larger than that of Model 2 overall in the slope area, and the PFSA of Model 3 was the smallest. The PFSA of the slope surface is larger than that of the slope interior. Hence, the slope under blasting has a typical slope surface amplification effect, and the elevation attenuation effect can be found in the slope surface and interior. There is an obvious dynamic amplification effect in the area near the slope shoulder.

To further study the attenuation law of blasting waves, the PFSA in the horizontal and vertical axes of the blasting source is shown in Figures 15 and 16. Figure 15 shows that in the horizontal and vertical axis directions of the blasting source, the PFSA gradually decreases with the distance from the blasting source. The phenomenon of rapid attenuation appears in the near-field area of the horizontal axis (0, 33.3 m) and vertical axis (0, 50 m). With the propagation of the blasting wave to the far-field area, the PFSA gradually decreases and tends to be stable. In addition, the PFSA of Model 1 was larger than that of Model 2, and the PFSA of Model 3 was the smallest. This indicates that lithology has an effect on the attenuation characteristics of blasting waves, and the attenuation effect of hard rock is less than that of soft rock. In addition, the PFSA distribution under blasting is shown in Figure 16. The PFSA under blasting is mainly concentrated in the near-field area near the blasting source but has little influence on the dynamic response in the far-field area. The analysis results in the frequency domain are consistent with those in the time domain.

## 5. Attenuation Law of Blasting Seismic Waves Based on Stress and Energy Analysis

To further explore the attenuation law of seismic waves in the models with different lithologies under blasting, the stress distribution of Models 1–3 and their boundary normal stress are shown in Figures 17 and 18. In Figure 17, the stress distribution characteristics of Models 1–3 are similar, which indicates that lithology has little influence on the stress distribution characteristics of the models. Figure 17 shows that the maximum positive stress (tensile stress) near the blasting source, the slope region, and the far-field area on both sides of the boundary is smaller than the maximum negative stress (compressive stress) of the deep bedrock at the bottom of the models overall. This shows that the blasting wave mainly produces tensile stress in the near-field area of the blasting source and slope area, and the deep bedrock mass mainly produces compressive stress under blasting. Compared with other far-field areas, blasting waves have a greater influence on the bottom boundary of deep bedrock in the models. The lithology has an effect on the stress amplitude of the rocky site. The maximum positive stresses (tensile stresses) of Models 1–3 are 5 × 10^{7}, 4 × 10^{7}, and 2 × 10^{7}, respectively. The maximum negative stress (compressive stress) is −20 × 10^{7}, −15 × 10^{7}, and −10 × 10^{7}. Hence, as the lithology changes from hard to soft rock, the tensile stress and compressive stress of the rocky site gradually decrease under blasting. The dynamic response of the blasting wave in the hard rock model is greater than that in the soft rock model; that is, the attenuation effect of the blasting wave in hard rock is smaller than that in soft rock. In addition, in Figure 18, the normal stress at the deep bedrock boundary at the bottom of the models is significantly greater than that at other boundaries, indicating that the blasting load has a greater impact on the deformation characteristics of the bottom boundary of the models. The bottom boundary stresses of Models 1–3 are approximately 3.88 × 10^{10}, 3.6 × 10^{10}, and 3.25 × 10^{10}, respectively. This indicates that the normal stress at the bottom boundary of the hard rock is greater than that of the soft rock under blasting, and the attenuation effect of blasting waves in the hard rock is minimal. This is consistent with the stress analysis of the model in Figure 13.

In addition, the energy method is used to further explore the dynamic response and attenuation law of the models with different lithologies under blasting. The energy and heat time histories of Models 1–3 are shown in Figures 19 and 20. Figure 19 shows that the total energy generated by different models under blasting is gradually transformed into mechanical energy (gravitational potential energy, kinetic energy, and elastic potential energy) and heat. With continuous blasting time, the mechanical energy (kinetic energy and elastic potential energy) of the rocky site decreases, while the gravitational potential energy and heat increase. In other words, the mechanical energy dissipated in the rocky site under blasting is gradually transformed into gravitational potential energy and heat. The total energy in Models 1–3 is approximately 2.45 × 10^{10}, 1.75 × 10^{10}, and 1.45 × 10^{10} J, respectively. This indicates that lithology has an effect on the total energy produced in the rocky sites under blasting, and the total energy produced in the hard rock is the largest, followed by relatively soft rock, and the soft rock is the least. Meanwhile, the mechanical energy (kinetic energy and elastic potential energy) of Model 1 is greater than that of Models 2–3. This indicates that the dynamic amplification effect of blasting in hard rock is greater than that in soft rock. Figure 20 shows that the heat generated by Model 1 is also greater than that of Models 2–3, approximately 1.08 × 10^{10}, 0.69 × 10^{10}, and 0.33 × 10^{10} J, respectively. This phenomenon shows that the attenuation effect of blasting waves in hard rock is larger than that in soft rock. This is consistent with the above analysis results in the time and frequency domains.

## 6. Discussion

Based on the above analysis, lithology influences the dynamic response characteristics and attenuation law of rocky sites under blasting. The dynamic response characteristics and attenuation law of relatively soft and soft rock are similar. The dynamic response characteristics of relatively soft and soft rock are obviously different from those of hard rock. The energy generated by the explosion source in the hard rock is the largest, and its dynamic response is the strongest. Based on the energy analysis, the attenuation effect of seismic waves in hard rock is greater than that in soft rock overall. The analyses of the time and frequency domains show that the blasting wave mainly has a great influence on the near-field area near the blasting source but has little influence on the dynamic response in the far-field area. The attenuation effect of blasting waves in hard rock is greater than that in soft rock in the ground surface and slope area, while the attenuation effect of waves in hard rock is smaller than that in soft rock in the bedrock area. Therefore, the dynamic response characteristics and attenuation law of blasting waves in different lithology sites are very complicated and need to be studied from multiple angles. It is difficult to fully reveal the dynamic response characteristics and attenuation rule of rocky sites only in the time and frequency domains; hence, further research should be carried out from the perspective of energy conversion and transformation. However, it is difficult to precisely identify the propagation characteristics and attenuation rules of blasting waves in the ground surface, slope, and deep bedrock area below the blasting source only from the perspective of energy. Therefore, it is necessary to carry out multiangle analysis from the perspectives of the time domain, frequency domain, and energy to reveal the dynamic response characteristics of rocky sites. In addition, the discrete-element numerical simulation method is used to explore the dynamic response characteristics of different lithological sites. Although useful research conclusions are obtained, the reliability of the conclusions of this work needs to be improved due to the lack of verification of model tests or field tests. This is also the next research plan to further carry out physical model tests of rocky sites under blasting, which has important scientific significance for the systematic study of this scientific problem.

## 7. Conclusions

MatDEM is used to investigate the propagation characteristics and attenuation rules of blasting seismic waves in rock slope sites. Some main conclusions can be drawn as follows:

The blasting wave shows circular arc attenuation propagation characteristics from the near- to far-field area with the source as the center. The dynamic attenuation effect of the slope is greater than that of the ground surface. The lithology influences the propagation velocity and amplitude of waves in the models. The propagation velocity of waves in soft rock is greater than that in hard rock, and the attenuation propagation characteristics of the wave field in soft rock are smaller than those in hard rock.

Lithology has different effects on the dynamic response characteristics of waves in the ground surface and bedrock. At the ground surface, the dynamic amplification effect of hard rock is smaller than that of soft rock, and the attenuation rate in hard rock is larger than that of soft rock. The attenuation effect of waves in the bedrock is smaller in hard rock than in soft rock. The blasting load mainly affects the dynamic response in the near-field area of the blasting source but has little effect on the far-field area. In the near-field area, waves appear to have a rapid attenuation effect in the ground surface (<33.3 m) and bedrock area (<50 m). In the far-field area, the attenuation effect gradually becomes stable with distance from the blasting source.

Lithology has an influence on the dynamic response of slopes. The dynamic amplification effect of slopes is as follows: hard rock > relatively soft rock > soft rock. The dynamic attenuation effect in soft rock slopes is greater than that in hard rock slopes. The elevation has an obvious attenuation effect on waves, and the attenuation rate within

*h*/*H*= 0–0.2 is the largest. In the slope interior, the dynamic attenuation effect is presented first within*h*/*H*= 0–0.75, and the amplification effect is presented within*h*/*H*= 0.75–1.0. At the slope crest, with the distance from the slope shoulder, it is characterized by first amplification and then attenuation. Lithology has little influence on the Fourier spectrum characteristics of the ground surface and bedrock of the blasting source but has a great influence on those of the slope.The stress and energy analysis shows that blasting waves mainly produce tensile stress on the ground surface and slope area and compressive stress in deep bedrock. The blasting load has more influence on the dynamic response of the bottom boundary of the models, and the normal stress of the bottom boundary of the hard rock field is greater than that of the soft rock. Under blasting, the total energy produced by different rock sites gradually changes into kinetic energy, gravitational potential energy, elastic potential energy, and heat. Lithology has an effect on the total energy and mechanical energy generated in the rocky site: hard rock > relatively soft rock > soft rock. The attenuation effect of blasting seismic waves in hard rock fields is larger than that in soft rock fields.

### Highlights

The dynamic response characteristics and attenuation law of rock slope sites are investigated via MatDEM in the time and frequency domains.

The propagation and waveform characteristics of explosion seismic waves in rock slope sites from near field to far field are discussed.

The influence of lithology on the dynamic response characteristics of rock sites under blasting is revealed.

## Data Availability

Data will be made available on request.

## Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Authors’ Contributions

Danqing Song: methodology, investigation, visualization, and writing—original draft. Xuerui Quan: software, numerical modeling, and writing—original draft (numerical modeling). Zhuo Chen: conceptualization and supervision. Dakai Xu, Chun Liu, Xiaoli Liu, and Enzhi Wang: writing—review and editing.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (52109125, 52208359, and 41941019), the Fundamental Research Funds for the Central Universities (2023ZYGXZRx2tjD2231010), the Natural Science Foundation of Jiangsu Province (Grant No. BK20231217), Science and Technology Service Network Initiative (2022T3051), and the Natural Science Foundation of Sichuan Province (24NSFSC4541).