Fractal Analysis of Fragmentation Distribution of Rockbursts Induced by Low-Frequency Seismic Disturbances

Low-frequency seismic disturbances frequently induce violent rockburst hazards, seriously threatening the safety of deep excavation and mining engineering. To investigate the characteristics and mechanisms of rockbursts induced by seismic disturbances, in this study a series of true triaxial experiments, including the moderate seismically induced, the weak seismically induced, and the self-initiated rockburst experiments under diﬀerent conditions were conducted. The fractal geometry theory was applied to study rockbursts and the fractal dimensions of fragmentation distribution of diﬀerent types of rockbursts were calculated. The results show that the fragmentation distributions of both the seismically induced and self-initiated rockbursts exhibit fractal behaviors. For the moderate seismically induced rockbursts, as the static stresses (i.e., the maximum and minimum static stresses) and disturbance amplitude increase, the fractal dimension increases, whereas, as the disturbance frequency increases, the fractal dimension decreases ﬁrst and then increases. Under similar static loading conditions, the moderate seismically induced rockbursts have the largest fractal dimension, followed by the self-initiated rockbursts, and the weak seismically induced rockbursts have the smallest fractal dimension. There is a linear relationship between the average fractal dimension and kinetic energy of these rockbursts, implying that the fractal dimension can serve as an indicator for estimating rockburst intensity. Furthermore, from a fractal point of view, the energy input, dissipation, and release of these rockbursts are all linear processes.


Introduction
In recent decades, with the worldwide expansion of mining and civil engineering to the deep ground, many large deep-buried tunnels/openings have been built [1]. Excavation of deep hard rocks causes the near-boundary rock mass to be in a high-stress state. Meanwhile, there are many dynamic disturbances during the excavation. Under the combined effects of high static stress and dynamic disturbances, violent rockburst disasters are frequently encountered [2][3][4][5][6][7][8][9][10][11]. e rockbursts caused by dynamic disturbances often bring about equipment damage, casualties, and significant economic losses.
Due to the high incidence and dangerousness, the rockburst hazards caused by dynamic disturbances have aroused widespread concern of researchers. ere are many related terms for rockbursts caused by dynamic disturbances, for example, "remotely triggered rockbursts [6,8]," "dynamically induced/triggered rockbursts [2,4,5,9,10]," "impact rockbursts [3]," and "seismically induced/triggered rockbursts [12]." e common feature of these rockbursts is that the rock mass is subjected to coupled static and dynamic loading [7]. e dynamic disturbances causing rockbursts can be high strain rate shock waves from blasting [7] or lowfrequency seismic waves/disturbances (or analogous stress waves [2-6, 8, 9]). ere are many sources of the seismic disturbances in deep engineering, such as natural seismic events, stress waves attenuated from remote blasting, seismic events caused by fault-slip, rockbursts in nearby excavation, and ground vibration caused by huge vehicles [6,[8][9][10]. e low-frequency seismic disturbances have a long propagation distance and they carry a large amount of energy. erefore, rockbursts caused by low-frequency seismic disturbances are the most common. In fact, a rockburst is defined by [12] as "damage to an excavation that occurs in a sudden and violent manner and is associated with a seismic event." In this study, we focus on the seismically induced rockbursts. Various methods, such as analytical [3,4,7], experimental [2][3][4][5][6][8][9][10], and numerical methods [4,5] have been used to investigate the seismically induced rockbursts. However, due to the high complexity of the seismically induced rockbursts, the driving mechanisms are still not clear and effective control of them is a challenge.
A seismically induced rockburst is a complex dynamic instability phenomenon, which involves the processes of cracking, damage accumulation, and energy dissipation and release of rocks. Because rocks are geomaterials with good self-similar characteristics, their fracturing and failure processes exhibit statistic self-similar fractal behavior. e fractal theory is widely applied to study rock failures [13,14] and rockbursts [15][16][17][18][19][20][21][22][23][24][25][26]. For example, fractal analyses on characteristics of rockburst cracks [16,20], acoustic emissions (AE) [21,[24][25][26], and sounds [22] have been performed by researchers. Since the fragments generated from rockbursts contain a variety of information about the rockburst process and can reflect the mechanism of rockbursts, fractal studies have also been conducted on the fragment geometric characteristics and the fragmentation distribution of rockbursts [15,[17][18][19]23]. In general, previous fractal studies focus on rockbursts without dynamic disturbances (i.e., the self-initiated rockbursts [12]). Fractal analyses on the fragmentation distribution of the seismically induced rockbursts have been rarely reported [19,23]. Moreover, previous studies do not link the fractal dimension and related physical quantities (e.g., the kinetic energy) of different types of rockbursts, which is urgent for the practical application of fractal theory in predicting rockbursts. erefore, further study is needed to investigate the fractal characteristics of fragmentation distribution of the seismically induced rockbursts.
In this study, the fractal geometry theory was applied to investigate the characteristics and mechanisms of the seismically induced rockbursts. e experiments of seismically induced and self-initiated rockbursts were carried out by a novel servo control dynamic true triaxial rockburst testing system. e fragments generated in rockburst experiments were collected, sieved, and classified. e fractal dimensions of fragmentation distribution of different types of rockbursts were calculated according to the mass-equivalent size relationship. e effects of static and dynamic loading conditions on fractal dimensions of the seismically induced rockbursts were studied. By comparing the fractal dimensions, characteristics and mechanisms of different types of rockbursts were investigated. Moreover, the relationship between the fractal dimension and kinetic energy of different types of rockbursts was established.

Rock Sample Preparation.
Rockbursts usually occur in deep hard brittle rocks. Granite and granodiorite are frequently encountered during the excavation of deep tunnels in China. erefore, a typical hard brittle coarsegrained granodiorite is used for rockburst experiments. e size of rectangular prism rock samples is 100 mm (xdirection) × 100 mm (y-direction) × 200 mm (z-direction). e integrity and homogeneity of the rock samples are relatively good. e tested basic physical and mechanical properties of the samples are as follows [4,9]: the density is 2665 kg/m 3 , the P wave velocity is 5.2 km/s, Young's modulus is 30.55 GPa, Poisson's ratio is 0.26, and the uniaxial compressive strength (UCS) is 180 MPa.

True Triaxial Experiments of the Seismically Induced
Rockbursts.
e true triaxial rockburst testing system developed at Guangxi University, China [9,27,28], is used to conduct rockburst experiments ( Figure 1). e rockburst testing system consists of a true triaxial testing machine, a high-speed camera system, an acoustic emission (AE) system, a control system, and a data acquisition system. e testing machine is an integral frame type press machine, which can independently load and unload in three mutually perpendicular directions, and conventional rock mechanics tests, such as uniaxial/triaxial compression, tensile, and shear tests can be performed with this testing machine. A specific function of keeping a rock sample in the state of "one face free and the other five faces loaded" is designed for the testing machine to simulate rockbursts [9,27,28]. A series of self-initiated rockburst experiments have been performed by the researchers [22,27,28].
In order to simulate the seismically induced rockbursts, the testing machine is also equipped with a dynamic disturbance system in three directions (Figure 1(b)). A variety of disturbance waveforms are available [4,5,9]. e in situ monitoring and previous studies [5,[8][9][10]29] show that the frequency of the seismic disturbances inducing rockbursts is low (e.g., several to tens of Hz), and the induced dynamic stress increment (i.e., the amplitude) is in the range of 0-30 MPa. erefore, cyclic loads with frequencies of 0.2-15 Hz and amplitudes of 2-40 MPa are used in the experiments to simulate the seismic disturbances. Among them, the cyclic load with frequencies of 0.2-3 Hz and amplitudes of 10-40 MPa is used to simulate a relatively intense seismic disturbance (termed as moderate seismic disturbance), and the cyclic load with a frequency of 15 Hz and amplitude of 2 MPa is used to simulate a weak seismic disturbance.
e seismically induced rockbursts are failures of highly stressed rocks induced by seismic disturbances, that is, rock failures under coupled static and dynamic loading. e test processes of the seismically induced rockbursts are summarized as follows [4,5,9]: (1) Simulation of initial in situ stress state: Maintain one face of the rock sample free and apply a preset true triaxial stress state to the rock sample; (2) Simulation of stress concentration after excavation: Maintain the lateral (the x-and y-direction) stresses unchanged and gradually increase the vertical (z-2 Advances in Civil Engineering direction) stress to a predetermined high stress at a rate of 0.5 MPa/s; (3) Simulation of seismic disturbance loading: A cyclic load is applied in the maximum static stress direction (z-direction) until a rockburst occurs or the disturbance duration reaches 15 minutes.
After rockburst experiments, the fragments generated are collected, classified, and measured, so as to study the characteristics of rockburst fragments.

Testing Design.
e testing design is listed in Table 1. Different test groups are designed to study the effects of various static stresses and disturbance characteristics on rockbursts. For a weak seismically induced rockburst, it is essentially a dynamic failure of critically stressed rock triggered by a rather small disturbance [5]. e purpose of static loading is to reach a critical stress state of the rock. It is difficult to compare the critical stress states of rock under different static stress states. erefore, only two groups of tests with different minimum static stresses are considered. e effect of disturbance on the weak seismically induced rockbursts is to activate and aggravate the rock damage, and the induced stress increment and the input energy to the rock by the disturbance are very small. erefore, the frequency and amplitude of the weak seismic disturbance are set as constant values. Furthermore, self-initiated rockburst experiments under similar static conditions are conducted for contrast.  Figure 1: True triaxial experiments of the seismically induced rockbursts (modified after [5,9]): (a) the true triaxial rockburst testing system, and (b) the loading configuration and the stress state of the rock sample, where σ x , σ y , and σ z are the intermediate, minimum, and maximum static stresses acting on the rock sample; τ xy , τ yz , and τ zx are the shear stresses; and σ d is the seismic disturbance simulated by low-frequency cyclic disturbance.

Application of Fractal Theory in Studying Rockbursts
Rocks are geomaterials with micropores, cracks, and macroscale geological structures (e.g., joints, faults, and weak sides). erefore, at different scales, the mechanical behaviors of rocks exhibit uncertainty and irregularity. e classical mathematical and mechanical models are powerless in dealing with the irregularity of rock behaviors. Fortunately, the development of the fractal geometry theory provides an available method for studying the complex mechanical behaviors of rock materials. Fractal geometry theory is proposed by Mandelbrot in the 1980s [30], which is different from the traditional Euclidean geometry system. e fractal geometry theory provides an effective quantitative description for complex, irregular, and seemingly disordered phenomena and systems that are widespread in natural and social activities. Irregular curves or shapes with self-similar and irregular figures with self-inverse characteristics are the main research objects of fractal geometry. Although the mechanical behaviors of rock materials are complex, there are still statistical self-similar characteristics. erefore, fractal geometry is widely used in rock mechanics, such as in the study of rock fracturing, crack propagation, roughness of rock joints, fragmentation distribution, and rockbursts. In fact, a discipline of fractal rock mechanics has formed now [31].
Rockbursts are violent failure forms of hard brittle rocks, which exhibit fractal behaviors in many aspects (Figure 2), such as microfracturing (Figure 2(a)) [32], fragmentation distribution of rock samples in the laboratory (Figure 2(b)) [17], and fragmentation distribution on the project site ( Figure 2(c)) [33]. e fracturing process of rockbursts can be illustrated using the fractal geometry of Sierpinski gasket ( Figure 2(d)).
In previous studies, researchers have conducted fractal analyses on rockbursts from the perspectives of microfracturing and crack propagation [16,18,20], fragments [15,[17][18][19]23], and AE [21,22,24]. Among these studies, since rockburst fragments are easy to collect and analyze, it is an effective way to study rockburst fractal characteristics from the aspect of fragments. e fractal characteristics of rockburst fragments include the fragment geometric features (e.g. the length-width and width-thickness ratios) and fragmentation distribution. Research results have shown that the fractal dimensions of geometric features of rockburst fragments calculated by different methods are very different and difficult to compare [17]. e fragmentation distribution contains a variety of important information about rockbursts, such as fracturing, damage accumulation, energy dissipation, and intensity (i.e., the kinetic energy), and the calculation methods of the fractal dimension of fragmentation distribution are unified [5,15]. It is an effective way to investigate the fractal characteristics of rockbursts from the perspective of fragmentation distribution. e fractal dimension of fragmentation distribution of rockbursts is affected by the types of rockbursts (e.g., the seismically induced or the self-initiated rockbursts), and the mechanical conditions of rockbursts (e.g., the coupled static and disturbance conditions of the moderate seismically induced rockbursts). Furthermore, in order to make the fractal dimension have a physical meaning and reveal the rockburst mechanisms, it is necessary to establish the relationship between the calculated fractal dimension and important rockburst physical quantities (e.g., the kinetic energy).
In the following sections, we will calculate the fractal dimensions of fragmentation distribution of the self-initiated and the seismically induced rockbursts, study the  Figure 3 shows the fragmentation distribution of a typical seismically induced rockburst (CD-Z-3). It can be seen that the fragments are different in shape and size. In this study, we propose a new method for calculating the equivalent size to quantitatively describe the geometric and size characteristics of rockburst fragments. In general, for small fragments (e.g., maximum size less than 10 mm), it is difficult to directly measure their sizes in three directions, and they can be sieved to estimate their equivalent sizes. For fragments larger than 10 mm, it is easy to measure their sizes in three directions. Considering that the fragments are irregularly shaped, we define the largest dimensions of the fragments in three directions from large to small as length, width, and thickness.

Equivalent Size of Rockburst Fragments.
In order to calculate the equivalent size, the fragments are sieved using sieves with diameters of 0.075, 0.150, 0.300, 0.600, 1.180, 2.360, 4.750, and 9.500 mm. Fragments with a maximum size less than 9.50 mm sequentially fall on each level of the sieve. For these fragments, the average size of diameters of the current and upper level sieves is taken as the equivalent size of fragments falling on a certain sieve.
For fragments with a maximum size larger than 9.50 mm, in previous studies [15,17,19,23], the equivalent sizes of these fragments are usually calculated according to the volume equivalent principle; that is, the side length of an equal volume cube is taken as the equivalent size. However, this approach is too simplified and does not reflect the true geometric characteristics of rockburst fragments.
Observation of large fragments (Figure 3(b)) shows that these fragments are mostly in the form of plates, sheets, or flakes. ReLiterature [17] proposed that the representative size of self-initiated rockburst fragments is length : width : thickness � 7.5 : 5.0 : 1.0. e authors statistically analyzed the measurement data of a large number of seismically induced rockburst fragments and found that the length and width are close (length : width � 1-1.5), and the thickness (1-10 mm) is much less than the length and width. erefore, the large fragments are plate shaped with close length and width, and the thickness has little effect on the geometric feature of these fragments. en, when calculating the equivalent size of fragments larger than 9.50 mm, the thickness is ignored, and the equivalent size is chosen as the side length of a square with an area equal to the product of length and width. Based on the above analysis, the equivalent size of rockburst fragments can be calculated as where R eq is the equivalent size and R is the maximum size of a fragment in three directions; r 1 and r 2 are the diameters of the current and upper level sieves, respectively; and l and w are length and width, respectively.

Classification, Characteristics, and Mass Distribution of
Rockburst Fragments. Different fragments are produced by macrofracture cutting of rock at different stages during the development of rockbursts. erefore, the classification of fragments is helpful for understanding the fracturing mechanism of rockbursts. In addition, classification also helps to qualitatively estimate the fragmentation distribution and calculate the fractal dimensions of rockbursts. e fragments are classified into coarse, medium, fine, and micro Advances in Civil Engineering grains, with the equivalent size falling in the ranges of R eq ≥ 40 mm, 4.75mm ≤ R eq < 40mm, 0.075 mm ≤ R eq <4.75 mm, and R eq < 0.075 mm, respectively. According to Figure 3, the coarse grains are mostly plate, flake, or block shaped, which are mainly composed of rock slabs formed by splitting near the free face, blocks formed by breakage of rock slabs, and flakes formed by shearing in the burst pit [4,9,34]. e medium grains are mostly flake shaped with relatively small volume, which are mainly formed by shearing in the burst pit. e fine grains are mostly rice grain and fine sand shaped, which are mainly damaged from comminution of large fragments. e micro grains are powder shaped. e cumulative mass distributions of different types of rockbursts are presented in Figure 4. It can be that the percentage of the micro grains is the smallest, that of the fine and medium grains is slightly larger, and that of the coarse grains is the largest. For different types of rockbursts, the mass percent of coarse grains is about 60%-70%, and that of the moderate seismically induced rockbursts is the largest, that of the self-initiated rockbursts follows, and that of the weak seismically induced rockbursts is the lowest. For the moderate seismically induced rockbursts (Figures 4(b)-4(e)), the mass distribution is affected by the coupled static and disturbance conditions. at is, with the increase of static loads (Figures 4(b) and 4(c)) and disturbance amplitude (Figure 4(d)), the mass percent of fine and medium grains increases, whereas with the increase of disturbance frequency (Figure 4(e)), the mass percent of fine and medium grains decreases first and then increases. e increase of mass percent of fine and medium grains indicates that the distribution of rockburst fragments is more uniform and the degree of fragmentation is higher.

Fractal Dimension Calculation of Fragmentation
Distribution.
e fractal dimension of fragmentation distribution can be calculated according to the equivalent sizefrequency and mass-frequency relationships [13,31]: where, in equation (2), N is the number of fragments with an equivalent size equal to or larger than R eq , N 0 is the number of fragments with the maximum equivalent size of R max , D is the fractal dimension, and C 1 is a constant; in equation (3), N ′ is the number of fragments with a mass equal to or larger than M, N 0 ′ is the number of fragments with the maximum mass of M max , b is the mass-frequency distribution index, and C 2 is a constant. Since the mass and equivalent size of a fragment satisfy M ∝ R 3 eq , according to equations (2) and (3), we have Due to the fact that there are a large number of fragments and many of them are powder shaped, it is difficult to accurately determine the number of rockburst fragments. It is usually difficult to directly calculate the fractal dimension according to equation (2) or (3). e mass of fragments is easy to measure. It will be an effective way to calculate the fractal dimension if we can derive the mass-equivalent size relationship from equations (2) and (3). e derivation process is described as follows.
Let M(R eq ) represent the cumulative mass of fragments with equivalent size less than or equal to R eq , M represents the mass of fragments with equivalent size larger than R eq , and M T represents the total mass of all fragments. In general, the percent of the cumulative mass of fragments approximately follows the Weibull distribution:  where α is the shape parameters and β is a parameter related to the average equivalent size. If (R eq /β) ≪ 1, then equation (5) can be rewritten as   Taking derivatives with respect to R eq on both sides of equation (6), we have dM R eq ∝ R eq α− 1 dR eq .
Because M � M T − M(R eq ) and M T is a constant, Since M is the mass of fragments with equivalent size larger than R eq , then where m i (R eq,i ) and N i are the mass and number of ith group of fragments with equivalent size larger than R eq . erefore, a general formula can be concluded from equation (9): On the other hand, according to equation (2), we have Substituting equation (11) into equation (10), we have Comparison of equations (8) and (12) gives en, the fractal dimension of fragmentation distribution of rockbursts can be calculated according to the massequivalent size relationship (equation (13)). To illustrate the meaning of parameter α, take natural logarithms of both sides of equation (6); we have It can be seen from equation (14) that if ln R eq is taken as the abscissa and ln[M(R eq )/M T ] is taken as the ordinate, then the ln[M(R eq )/M T ] ∼ ln R eq curve is a straight line, and the slope of this straight line is the parameter α.
Once the fractal dimension is calculated, it can be used to quantitatively evaluate the characteristics of rockbursts [15][16][17][18][19][20][21][22][23][24]. In general, the larger the fractal dimension, the larger the number and the smaller the size of rockburst fragments, the greater the fragmentation degree of rockbursts, and the stronger the brittleness (i.e., the rockburst intensity) of rockbursts.

Results and Discussion
In this section, fractal dimensions of fragmentation distribution of different types of rockbursts are presented. e effects of different static and disturbance factors on the fractal dimensions of the moderate seismically induced rockbursts are investigated. en, the fractal dimensions of different types of rockbursts are compared, and a quantitative relationship between the fractal dimension and kinetic energy of these types of rockbursts is established. Finally, based on the fractal analysis, the characteristics and mechanisms of different types of rockbursts are discussed.

Fractal Dimensions under Different Maximum Static
Stress σ z . Figure 5 shows the calculation results of fractal dimensions of the moderate seismically induced rockbursts under different maximum static stress σ z . It can be seen that the fractal dimension increases with the increase of σ z . is means that, under the preset high static stress, increasing the maximum static stress σ z can effectively improve the utilization of the energy input by the cyclic disturbance, so that the rock consumes more energy during the failure process, thus leading to a higher fragmentation degree.

Fractal Dimensions under Different Minimum Static
Stress σ y . Figure 6 shows the calculation results of fractal dimensions of the moderate seismically induced rockbursts under different minimum static stress σ y . It can be seen that, as σ y increases, the fractal dimension increases. is shows that, under the conditions of low minimum stress σ y (1-5 MPa) and rockburst occurrence, increasing σ y can strengthen the restraint of the rock and increase its strength, so that the rock can make full use of the energy input by the cyclic disturbance, thus leading to a higher fragmentation degree.

Fractal Dimensions under Different Disturbance Am-
plitude Δσ. Figure 7 shows the calculation results of fractal dimensions of the moderate seismically induced rockbursts under different disturbance amplitude Δσ. It can be seen that the fractal dimension increases as the disturbance amplitude Δσ increases. erefore, an increase of disturbance amplitude Δσ can aggravate the fragmentation degree of rockbursts.

Fractal Dimensions under Different Disturbance
Frequency f. Figure 8 shows the calculation results of fractal dimensions of the moderate seismically induced rockbursts under different disturbance frequency f. It can be seen that, in the range of 0.2-3 Hz, as the disturbance frequency increases, the fractal dimension decreases first and then increases; that is, the fragmentation degree decreases first and then increases. Figure 9 shows the calculation results of fractal dimensions of the weak seismically induced rockbursts. It can be seen that the fractal 8 Advances in Civil Engineering dimension increases with increasing minimum static stress, which is similar to that of the moderate seismically induced rockbursts ( Figure 6). It is noted that the maximum fractal dimension of the weak seismically induced rockbursts is 1.802, which is much lower than that of the moderate seismically induced rockbursts (mainly in the range of 2.3-2.5). e reason is that these two seismically induced rockbursts are controlled by different failure mechanisms, which will be explained in detail in Section 5.3.

Comparison of Fractal Dimensions of Different Types of
Rockbursts. e fractal dimension comprehensively reflects the quantity, size, uniformity of the distribution of rockburst fragments, and the brittleness of rockbursts. erefore, a comparison of the fractal dimension is helpful for understanding the characteristics and mechanisms of different types of rockbursts. e calculation result of a typical selfinitiated rockburst is shown in Figure 10 (D � 1.733). e calculation results of fractal dimensions of different types of rockbursts studied in this study and those in the literature are presented in Table 2 and Figure 11. It can be seen that the fractal dimensions of different types of rockbursts fall in different ranges. According to the statistical results, the fractal dimension of the self-initiated rockbursts falls in the range of 1.7-2.2, that of the moderate seismically induced rockbursts falls in the range of 2.2-2.6, and that of the weak seismically induced rockbursts falls in the range of 1.0-1.8. erefore, the fractal dimension of the moderate seismically induced rockbursts is larger than that of the self-initiated

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rockbursts, and the fractal dimension of the self-initiated rockbursts is larger than that of the weak seismically induced rockbursts.

Characteristics and Mechanisms of Different Types of
Rockbursts. e characteristics of different types of rockbursts can be deduced from the statistical results of their fractal dimensions: the average fragment size of the moderate seismically induced rockbursts is the smallest, followed by that of the self-initiated rockbursts, and that of the weak seismically induced rockbursts is the largest; the number of fragments of the moderate seismically induced rockbursts is the largest, followed by that of the self-initiated rockbursts, and that of the weak seismically induced rockbursts is the least; the distribution of fragments of the moderate seismically induced rockbursts is the most uniform, followed by that of the self-initiated rockbursts, and that of the weak seismically induced rockbursts is the least uniform; the brittleness (i.e., the intensity) of the moderate seismically induced rockbursts is the strongest, followed by that of the self-initiated rockbursts, and that of the weak seismically induced rockbursts is the weakest. e characteristics of different types of rockbursts can also be reflected in their failure modes ( Figure 12). According to Figures 12(a) and 12(b) and previous studies [4,9,27,28], the failure modes of the moderate seismically induced and the self-initiated rockbursts can be described as follows: most close to the free face, rock slabs are formed by tensile dominated splitting, a V-shaped or step-shaped rockburst pit is formed near the free face by combined tensile and shear failures, and several oblique macro shear bands are formed away from the free face. However, according to Figure 12(c) and literature [4,5], the weak seismically induced rockbursts mainly undergo tensile splitting failure, and the rock sample is cut into several large thick rock slabs along the vertical direction.   Microscopically, tensile cracks usually correspond to intergranular failure and shear cracks usually correspond to transgranular failure [16,18,20,32]. Macroscopically, tensile cracks are usually vertical and straight, and shear cracks are usually oblique and irregular [5,28,32,35]. erefore, from the perspective of fractal geometry, the fractal dimension of the relatively simpler tensile cracks is less than that of shear cracks. en, based on the above analysis, it can be deduced that the weak seismically induced rockbursts are mainly controlled by tensile failure mechanism, the self-initiated and the moderate seismically induced rockbursts are controlled by combined tensile and shear failure mechanisms, and the shear failure characteristic of the moderate seismically induced rockbursts is more remarkable.
As a special form of dynamic failure, the dangerousness of rockbursts mainly comes from the kinetic energy of the ejected fragments. Kinetic energy is also an important reference index for selecting support measures [12,36]. In addition, the purpose of studying the fractal characteristics of rockbursts is not only to study the rockburst mechanisms but also to apply them in engineering practice. erefore, it is necessary to link the fractal dimension with important physical quantities (e.g., the kinetic energy) of rockbursts. Figure 13 displays the data of kinetic energy and fractal dimension of different types of rockbursts. It can be seen that, due to the variation of test conditions and test errors, there is a certain degree of dispersion of the data. However, from a statistical point of view, under similar static loading conditions, the kinetic energy of the moderate seismically induced rockbursts is larger than that of the self-initiated rockbursts, and the kinetic energy of the self-initiated rockbursts is larger than that of the weak seismically induced rockbursts (Figure 13(a)). Furthermore, Figure 13(c) shows that there is a good linear relationship between the average fractal dimension and the kinetic energy of different types of rockbursts.
e linear relationship shown in Figure 13(c) is of great significance for studying the mechanisms of rockbursts. A rockburst is a complex nonlinear process if it is investigated from the perspectives of stress and deformation [5,9,27,28]. However, the problem can be simplified from an energy perspective. For example, Gong et al. [37] observed the linear energy storage and dissipation laws for different rocks under uniaxial compression. For rockbursts, the entire energy process is more complicated, which includes the energy input, storage, and dissipation in the prepeak stage and the energy input, dissipation, and release (i.e., the kinetic energy) in the postpeak stage. erefore, there are no such linear laws similar to those proposed by Gong et al. [37] for the entire energy process of rockbursts [4,38]. However, from a fractal point of view, the problem will turn around. For example, during the experimental study of energy dissipation of rockbursts, Chen et al. [15] found that there is a good linear relationship between the fractal dimension and the total dissipated energy. erefore, from the fractal point of view, the energy behaviors of rock during the entire rockburst process can be linearly described as the following: where E in is the total energy input to the rock, E er is the residual elastic energy, E p is the total dissipated energy, E k is the kinetic energy, D is the fractal dimension, and k 1 and k 2 are constants. Since E er is a very small quantity that can be treated as a constant [4,5,9], we have Equations (15) and (16) indicate that, from the perspective of fractal, the energy input, dissipation, and release of rockbursts are all linear processes. e laws revealed by equations (15) and (16) can also provide a potential method for quantitative estimation of rockburst intensity in engineering practice. Generally speaking, in engineering practice, it is difficult to directly and quantitatively measure the kinetic energy of a rockburst, but semiquantitatively estimate the rockburst intensity based on the mass and ejection distance of the ejected fragments. Fortunately, the fractal dimension of fragmentation distribution of rockbursts studied in this study is a quantitative description of the mass and ejection distance (related to the fracturing process) of rockburst fragments. erefore, according to the linear relationship between fractal dimension and kinetic of rockbursts (equation (15)), the fractal dimension can be used as an effective indicator for quantitatively estimating the intensity of in situ rockbursts.
It should be noted that the observed linear law between the fractal dimension and kinetic energy is suitable for the three types of rockbursts studied in this paper. ese three types of rockbursts are essentially three different forms of rock failures [4,5,9], and their fractal dimensions and kinetic energies are distributed in distinct ranges. However, for a definite type of rockbursts, the fractal dimensions and kinetic energies are distributed in a rather narrow range (Figure 13), due to the variation of test conditions and test errors, there is yet no sufficient evidence supporting a linear relationship between the fractal dimension and kinetic energy. Further studies are needed to figure out the quantitative relationship between the fractal dimension and kinetic energy of rockbursts.

Conclusions
In the present study, the fractal characteristics of fragmentation distribution and mechanisms of the seismically induced rockbursts were investigated. e following conclusions can be drawn: (1) e fragmentation distributions of both the seismically induced and self-initiated rockbursts exhibit fractal behaviors. e fractal dimension of the moderate seismically induced rockbursts is affected by the coupled static and disturbance conditions, that is, as the static stresses (i.e., the maximum and minimum static stresses) and disturbance amplitude increase, the fractal dimension increases, whereas as the disturbance frequency increases, the fractal dimension decreases first and then increases. (2) From a statistical point of view, under similar static loading conditions, the moderate seismically induced rockbursts have the largest fractal dimension, followed by the self-initiated rockbursts, and the weak seismically induced rockbursts have the smallest fractal dimension. (3) It can be deduced from the comparison of the fractal dimensions of the three types of rockbursts that the weak seismically induced rockburst is tensile splitting failure dominated, the self-initiated and the moderate seismically induced rockbursts are combined tensile-shear failures dominated, and the shear failure characteristic of the moderate seismically induced rockburst is more remarkable. (4) ere is a linear relationship between the average fractal dimension and kinetic energy of the three types of rockbursts, which implies that the fractal dimension can serve as an indicator for quantitatively estimating rockburst intensity. Furthermore, from a fractal point of view, the energy input, dissipation, and release (kinetic energy) of these three types of rockbursts are all linear processes.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.