Crack Propagation and Fragmentation Characteristics of Single-Flawed Sandstone Samples under Different Loading Conditions

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Introduction
Te macroscopic cracks are staggered and dispersed in engineering rock mass. Te fractured rock mass may be unstable and fail under static and dynamic loads such as geostress feld, earthquakes, blasts, and mechanical vibration. Te failure of rock is often related to existing cracks. Under diferent loading conditions, the failure modes of the rock mass are usually diferent. Terefore, it is of great practical signifcance for the safety of rock engineering to study the infuences of precrack on rock fracturing behavior and energy dissipation.
In the past few decades, the mechanical properties and crack propagation of rocks with precracks have been studied by many scholars. It is very difcult to study the fracturing behavior of rocks using theoretical analysis methods because rock is usually heterogeneous [1,2]. Terefore, experiments are thought to be the main method for studying the fracturing behavior of rocks containing precracks under static or dynamic loading. Under static loading, the discussion on the fracture of fractured rocks started earlier and was studied more thoroughly. For instance, Yang et al. [3] studied the efect of precrack shape on the compressive strength and fracture behavior of rocks under quasi-static compression condition. Bobet et al. [4] made rock-like samples with double precracks, analyzed the law of fracture dislocation, initiation, and propagation of new cracks, and discussed the relationship between the distribution of cracks and the fnal failure mode under static compression loading. Huang et al. [5] and Haeri et al. [6], respectively, summarized 5 crack growth modes and 9 crack types of fawed rock under static loading conditions through experiments. Zhou et al. [7] discussed 5 types of cracks in rock samples, and 10 types of crack coalescence forms are summarized based on the analysis of crack evolution process. Shen-Qi and Hong-Wen [8] analyzed the infuence of single precrack on the compressive strength and deformation behavior of rock samples based on the experimental results of static stress-strain curves, and nine crack types were summarized. Zhao et al. [9] took rock-like specimens containing two precracks as the research objects and conducted a number of static compression tests. Four diferent types of cracks were observed in the experiments.
Strength characteristics and the fracture behavior of intact rock under impact load have been well understood in previous studies. It is long recognized that the rock strength is rate-dependent under dynamic loading [10,11]. Te fracture behavior of rock with precracks is diferent under dynamic load and static load [12]. Te dynamic failure mode of intact rock is rate-dependent [13,14]. Although the research on dynamic mechanical properties and dynamic crack propagation of rock with precracks started later comparatively, many results have been obtained. For instance, Li et al. [15] took slab marble as the research object and studied the efect of precrack angle on the dynamic mechanical properties using a modifed split-Hopkinson pressure bar (SHPB) device at strain rates of 37.8-83.9 s −1 . Six crack growth types were summarized and compared with the crack growth modes under static loading by other scholars. Zou et al. [16] obtained seven crack types of rectangular plate samples with one preexisting crack and summarized the fracture modes under various strain rates of 120-170 s −1 . Li et al. [12] took the square slab marble with one or two preexisting cracks as the research object, analyzed the rock mass fracture mode under dynamic impact loading, and summarized 9 failure types of the samples with two preexisting cracks. Dong et al. [17] recorded and studied the crack propagation process of rock-like plates with one precrack using SHPB and ultra-high-speed camera system. Recently, Han et al. [18] did a lot of experiments for investigating the efects of precrack angle and initial static load on the crack propagation process of rock specimens under dynamic loading. You et al. [19] studied the infuence of precrack on the dynamic mechanical properties and failure modes of sandstone under diferent confning pressures and diferent strain rates. Te numerical simulation study shows that the failure mode of rectangular specimens with single faw under the impact loading condition is related to the loading rate [20].
Under the condition of medium strain rate, the infuences of preexisting faw parameters (i.e., size and inclination angle) on dynamic mechanical properties and crack propagation have been discussed, and the infuences of strain rate on crack progressive growth behavior have also been studied. However, relevant experimental results within the range of medium strain rate are not sufcient. Specially, the gas pressure for driving the striker of the SHPB device is often kept constant during SHPB tests in the past study, resulting in neglect of strain rate efect of failure modes of fawed rock samples. In view of this, studying the related problems under dynamic loading in the range of medium strain rate is essential for rocks with precracks. It is known that mechanical properties of samples are closely related to their size, and relevant studies also show that the crack propagation of rock is closely related to its size too [12]. Square or cylindrical specimens are used in SHPB experiments for discussing the problem of crack propagation frequently. Some scholars generally apply dynamic impact load in the direction of sample length in order to compare with the static experimental results when the rectangular plate specimen is used in SHPB experiment. However, according to the principle of SHPB experiment, it is important to have sufcient number of stress waves propagating back and forth in the sample. If rock samples with a length-to-width ratio greater than 1 : 1 are subjected to impact dynamic load along the length direction, it is not easy for long rock samples to meet the stress equilibrium condition in the SHPB test. Te International Society for Rock Mechanics (ISRM) recommends height-diameter ratios between 0.5 : 1 and 1 : 1 for rock samples in SHPB tests, so the impact dynamic load is applied along the width direction of the sample in this study; at the same time, the inertial efect can be better weakened.
In this study, the static and dynamic load tests were carried out using the same fnite size rock samples with one precrack. Te corresponding static and dynamic loads were applied along the length or width of the samples. Te angle between static load direction and precrack is the same as that between dynamic load direction and precrack by setting the inclination angle of the faw with a reciprocal relationship. Te inclination angle of the precrack includes 0°and 90°, 30°a nd 60°, and 45°in this study. Te static mechanical properties and fracture mode under the infuence of precrack were studied using the hydraulic servo loading system with a camera. Te dynamic mechanical properties and crack evolution under the infuence of precrack and strain rate were investigated using the SHPB equipment with a high-speed camera system. Te results of mechanical properties, fracture behavior, and energy dissipation under these diferent loading conditions are compared and analyzed.

Specimen Preparation.
Te experiments took red sandstone as the research object in this paper. Red sandstone used in these experiments is made of quartz, feldspar, calcite, and illite. Te red sandstone samples used in these tests were taken from the same rock block of an underground mine in Hunan Province of China, which is a common type of mine surrounding rock in this area.
Under axial load, crack mode on the surface of cylindrical specimen is diferent from that inside the sample [20,21], so the rock samples were processed into thin plate shape rather than a cylinder in this study. Following the method recommended by ISRM [22] and considering the diameter of the bar of the SHPB device, the ratio of length to width of the samples is set about 0.65 with a nominal size of 46 × 30 × 15 mm 3 but with diferent faws. Te two ends of the sample were smoothed to a nonuniformity less than ±0.05 mm, and the nonperpendicularity of the adjacent two sides was within ±0.25°.
Before the faw of sample was prefabricated, P-wave velocityof the samples was measured using wave velocity measuring instrument, dry density was measured by a highprecision electronic balance, electric blast drying oven, and caliper, and saturated water absorption rate was measured by high-precision electronic balance and vacuum water saturation instrument. Te dry density of the samples is 2.37 g/ cm 3 -2.39 g/cm 3 (the average is 2.38 g/cm 3 ), the saturated water absorption rate is 3.11%-3.25% (the average is 3.18%), and dry p-wave velocity is 2286 m/s−2324 m/s (the average is 2299 m/s).
An 8 mm penetrating fssure is prefabricated in the center of the sample using the central drilling method. Te angles between the precrack and length direction of the sample are, respectively, 0°, 30°, 45°, 60°, and 90°. Tese angles are complementary to each other. Te pictures of the parts of samples are shown in Figure 1.
For reducing the end efect in tests, the samples were loaded along the length direction in the static load test, while the samples were loaded along the width direction in the dynamic impact test. Tis ensures that the geometry of the specimens and prefabricated faws is identical in static and dynamic tests. In the static load tests, the samples were divided into 5 groups according to the inclination angles of prefabricated faws, and 3 samples constituted a group. A total of 15 specimens were tested under static uniaxial compression. In the SHPB test, the samples are divided into 20 groups according to the peak of incident wave and the inclination angles of prefabricated faws. A total of 60 specimens were tested under impact loading.

Testing Installation.
Static load tests were carried out according to the suggestions proposed by the professional standards compilation group of people's republic of China (SL264-2001). Te quasi-static compression test is performed on a hydraulic servo loading system. A camera with a resolution of 4096 × 3000 and a refresh rate of 30 Hz was used to record crack propagation path. Te change of strain rates has little efect on the strength of rock under the condition of quasistatic strain rate [16]. In the compression test, the loading rate is fxed at 0.5 MPa/s until the specimens were damaged. In order to reduce the transverse frictional resistance between the specimen and the equipment, some lubricant was spread on the surfaces of the specimen. Te device of static loading experiment and the relationship between the specimen placement direction and loading direction are shown in Figure 2, where P s denotes static load and β denotes the angle between the precrack and loading direction.
Te SHPB was used for dynamic load tests, which is made of 40 Cr alloy steel. Te SHPB device consists of an air gun, an incident bar (3 m in length), a transmitted bar (2 m), an absorbing bar (1 m), and a set of data acquisition and analysis system. Te diameter, P-wave velocity, density, and elastic modulus of the bars of SHPB equipment are 5 cm, 5410 m/s, 7810 kg/m 3 , and 210 GPa, respectively. Te coneshaped striker was used for producing a half-sine wave aimed to eliminate wave dispersion [23]. Crack propagation path was recorded by an ultra-high-speed camera during impact loading. Considering the resolution and image brightness, the acquisition frequency of the camera was set to 75,000 fps. Te SHPB device is shown in Figure 3, where P d denotes dynamic load. Te specimen number represents loading mode (P s or P d ), faw inclination angle (β), and sample serial number. "S" indicates static loading and "D" indicates dynamic loading in the sample number. For example, D-30-3 denotes the third single-fawed sample with β � 30°under dynamic load.
Te peak of incident wave (strain rate of specimens) depends on the speed of the cone-shaped bullet. Te striker velocities in SHPB tests were set as 6 m/s, 8 m/s, and 10 m/s by adjusting the pressure of the air gun. Under diferent striker velocities, the peaks of incident waves were 40.7 MPa, 61.1 MPa, and 76.5 MPa, respectively. Te strain rates of the samples were 37.8 s −1 ∼ 45.8 s −1 , 55.3 s −1 65.8 s −1 , and 75.3 s −1 ∼ 83.9 s −1 , respectively. Due to the diverse dip angle of precracks, the propagation of stress waves in the samples is diferent. Te generated transmitted and refected waves are diferent because of the precrack, so the strain rate of the samples with diverse precracks changes under the same incident wave condition.

Static Mechanical Properties.
Te static stress-strain curves (σ-ε curves) of the samples under static compression are shown in Figure 4. Te specimens generally show the characteristics of brittle fracture in the tests, and there was loud sound when the specimens broke. Te static σ-ε curve can be roughly divided into 4 parts. In the frst stage (compaction stage), the σ-ε curve presents a microcurved shape, and the micropores in the specimen are gradually closed. However, it can be seen from the photographic images that there is no obvious compression collapse phenomenon in the precrack of the sample. In the second stage (elastic deformation), the σ-ε curve is approximately straight, and linear elastic deformation occurs along the load direction. In the third stage (crack propagation), the new crack starts and develops around the preexisting faw, leading to nonlinear deformation of the rock. In the last  Table 1.
With the increase of β angle, the UCS decreases frst and then increases. Te strength of 45°sample is minimum, and the strength of 0°sample is maximum. Te values of strength of all the samples for static uniaxial compressive test are between 33.15 MPa and 44.73 MPa. Te static elastic modulus ranges from 3.22 GPa (45°sample) to 4.65 GPa (0°s ample), which has a similar tendency with the UCS.

Stress Balance.
Te dynamic stress balance is the premise for efective results of SHPB experiment, especially for samples containing penetrating fssure [24]. In this study, a cone-shaped bullet was used for producing a half-sine wave aimed to help with stress equilibrium at the two ends of the specimen and help with providing constant strain rate loading [15].
Te incident wave, refected wave, and transmitted wave were picked up by the strain gauges on the bars of SHPB equipment. Te typical waves obtained by SHPB test in this study are shown in Figure 5 (signal-i, signal-r, signal-t, and signal-i + r represent incident wave, refected wave, transmitted wave, and superposition of the incident wave and refected wave, respectively). Te superposition of the   incident wave and refected wave basically overlaps with the transmitted wave, which indicates that the sample meets the stress balance condition. All the SHPB test results of this study were verifed by stress equilibrium before the test data analysis. "3-wave analysis" method [25] containing incident wave ε I (t), refected wave ε R (t), and transmitted wave ε T (t) can be used to calculate the strain (σ(t)), stress (ε(t)), and strain rate (_ ε (t)). Te formulas of the "3-wave analysis" method are expressed as where A s and A e denote the cross-sectional area of the sample and the steel bars of SHPB equipment, respectively, L s denotes the length of the specimen, and E e and C e denote Young's modulus and longitudinal wave velocity of the steel bars, respectively.

Dynamic Stress-Strain Curves.
Taking the specimens with strain rate around 60 s −1 as examples, the dynamic σ-ε curves of specimens with diferent precrack angles are shown in Figure 6. Taking samples with precrack angle β � 60°as examples, the σ-ε curves of samples with diferent strain rates are shown in Figure 7. Te dynamic σ-ε curves have no compaction stages compared to static stress-strain curves. Clearly, the preexisting faw and strain rate have important impact on the σ-ε curves of the rock samples. With the increase of strain rate, the peak stress and the dynamic modulus of elasticity increase signifcantly when the faw angle is constant. Meanwhile, the peak stress and the dynamic modulus of elasticity change, depending on the increase of faw angle when the peak of incident wave is constant. However, the precrack angle has no obvious impact on the peak strain of σ-ε curves.

Dynamic Strength and Modulus of Samples.
Te mechanical parameters of the tested samples with diferent β values and strain rates are listed in Table 2. Te parameters _ ε, σ d , and E d are denoted as strain rate, dynamic compressive strength, and dynamic modulus of elasticity of samples in Table 2, respectively.
Te dynamic strength and the modulus of elasticity of the samples with diferent strain rates are plotted against the β values in Figure 8. With the increase of precrack angle, strength (σ d ) decreases frst and then increases, which is similar to the static strength. Te dynamic strength of samples with β � 45°is the minimum, which is similar to the static experiment results. Te dynamic strength of samples with β � 60°or β � 0°is the maximum with the change of peak of incident wave. Under static loading, the static strength of 0°sample is the highest, which is 134.6% of that of 45°sample. When the peak of incident wave is about 40.7 MPa, the strength of the rock specimens with β � 60°is the highest, which is 123.5% of that of the 45°sample. When the peak of incident wave is 61.1 MPa, the dynamic strength of 0°sample is the highest, which is 123.7% of that of 45°s ample. When the peak of incident wave is 76.5 MPa, the strength of the rock specimens with β � 0°is the highest, which is 122.6% of that of the 45°sample. Te ratio of minimum and maximum of strength of rock samples with diferent β values is very similar, which is less than that of static strength slightly. With the increase of peak of incident wave, the dynamic modulus of elasticity increases, which is consistent with strain rate impact of dynamic compression strength.
Under impact load, the growth factor of dynamic strength (DIF) is the ratio of dynamic strength and static strength, which can be used as an indicator of strength growth [26]. Te DIF values of the samples are listed in Table 3. Equation (2) is suitable to refect the strain rate impact of rock strength in the range of medium strain rate [26].
where a and b are ftting parameters. Te parameter b refects the strain rate correlation of samples. Te ftting curves of DIF values against the strain rate of samples with diferent β values are plotted in Figure 9. Te relevant parameters of equation (2) are listed in Table 4.

Shock and Vibration
It is evident that the sensitivity of dynamic strength to strain rate of samples with diferent faws is diferent. In the range of 40 s −1 -80 s −1 strain rate, the b value of sample with faw of β � 30°is the largest relatively. It indicates that the dynamic strength of sample with β � 30°increases the most with the increase of strain rate. In contrast, b value of rock specimen with β � 60°is relatively minimum, indicating that for this rock specimen with β � 60°, the impact of strain rate on strength is the weakest. Many studies in the past have been conducted on the crack propagation path of rocks by static or dynamic load tests. Many results of quasi-static tests showed that 3 molds of new crack could be observed during static loading: wing crack, antiwing crack, and secondary crack [8,27]. Tensile crack includes the wing crack and antiwing crack; meanwhile, shear crack includes oblique secondary crack and coplanar secondary crack ( Figure 10). More secondary cracks (shear cracks) have been observed during many dynamic load tests [16,28]. Tose kinds of shear cracks are the secondary cracks shown in Figure 10. Te tensile cracks are initiated frstly during static load tests [8]. On the contrary, the tensile cracks initiated later than shear cracks    during dynamic load tests [15,28]. Terefore, in this paper, crack types are divided into shear cracks and tensile cracks according to crack propagation mechanism, rather than naming "secondary crack." Tensile cracks are usually clean and straight in character and generally exhibit plumose textures. In contrast, shear cracks are usually rough, which are often flled with pulverized rock debris.

Fracture Behaviors
Te crack types according to geometry of crack propagation path under dynamic impact loading are summarized by Li et al. [15] and Zou et al. [16], and those under static load are summarized by Wong et al. [29] and Yang et al. [3]. According to the crack types summarized by those relevant scholars and phenomenon observed in the present experiments, nine crack types with diferent initiation mechanisms (shear or tensile) and geometries were proposed in this paper ( Figure 11). "T" and "S" in Figure 11 represent tensile and shear crack, respectively. Te "separation layer crack" (type IX) is separated from the type of far-feld crack (type VIII proposed by Li et al. [15] and Li et al. [12], which is caused by tensile stress formed by refection of stress wave). Type VIII crack is characterized by propagating along the loading direction; in contrast, type IX crack propagates perpendicularly to the loading direction. According to the angle between direction of crack initiation and precrack, types I, II, and III are separated from the type of tensile crack proposed by Li et al. [15]. Types I, II, III, VIII, and IX are tensile cracks, types V, VI, and VII are shear cracks, and type IV is a mixed tensile-shear crack. Compared with Figure 10, type IV and three kinds of far-feld cracks, VII, VIII, and IX, are proposed in this paper, which are more suitable for describing dynamic experimental results. Characteristics of each type of crack are described as follows.
Type I tensile cracks generate near the precrack tips or at the middle of the faw, propagate perpendicularly to the faw, and then transform to grow along the direction of external load. Type II tensile cracks generate along the precrack direction and then transform to grow along the direction of external load. Type III tensile cracks generate near precrack tips and grow along the direction of load. Type IV mixed tensile-shear cracks frst propagate near the precrack tips as shear cracks and then become tensile cracks and grow along the load direction. Compared with type II tensile cracks, type IV cracks usually are covered with pulverized rock debris near the tips of the precrack. Type V oblique shear cracks usually generate near the precrack tips, at an angle with the loading direction. Te cracks are often characterized by shear bands flled with rubble powder. Type VI quasicoplanar shear cracks usually generate near the precrack tips and grow along the precrack direction. Type VII far-feld shear cracks and type VIII and type IX far-feld tensile cracks      Figure 12). Most of the ultimate broken patterns of the samples with precrack contain a variety of crack forms. Te crack propagation path of samples with diferent precracks is as follows.
Firstly, a new crack initiated and propagated vertically upward along the prefabricated crack in the sample with β � 0°, and then type V oblique shear cracks propagated toward the boundaries of the sample. Te crack propagating vertically upward is sliding mode crack, which can be classifed as type VI quasi-coplanar shear crack because the propagation path of this new crack is consistent with the theoretical solution of the initiation angle of the shear crack [28]. Te two kinds of cracks in the sample with β � 0°initiated continuously, so there are no obvious two stress peaks of the stress-strain curves. Te UCS of samples controlled by shear crack is higher than that controlled by tensile crack, so the strength of specimen with β � 0°is higher than that of other kinds of specimens.
After tensile cracks initiated and propagated, some shear crack branches initiated in the samples with faw angle of 30°, 60°, and 90°. Tensile cracks are easier to initiate around the prefabricated crack than shear cracks. For the sample with β � 30°, a type III tensile crack initiated from tip of precrack frstly, and then multiple shear cracks initiated from upper boundary of the specimen, which are connected with the type III crack. For the sample with β � 60°, after type III and type I tensile cracks initiated from both tips of the prefabricated crack, one shear crack occurred at the upper part of the sample. For the sample with β � 90°, two type III tensile cracks initiated from the upper tip of prefabricated crack; after that, a type I tensile crack initiated and propagated downward toward the boundary of the sample from the middle of the prefabricated crack. Tensile cracks are the main kinds of crack in the samples with faw angle of 30°, 60°, and 90°, and shear cracks are the minor kinds of crack. Te σ-ε curves of samples with β � 30°, 45°, 60°, or 90°have two stress peaks because of phase propagation of cracks.
Tere is no obvious shear crack that appeared in the samples with β � 45°. Type III and I tensile crack initiated from both tips of the precrack, respectively. Te number of new cracks in the sample with β � 45°is less than that of other types of specimens, so the total energy required of sample with β � 45°is the least. Tis is the important reason why the strength of the sample with β � 45°is lower than other types of samples.
In accordance with Figure 12, under static loading, type I tensile crack at the lower part of the specimens with β � 45°, 60°, or 90°is closer to the middle of the precrack with the increase of the angle β. Tis phenomenon is consistent with the research results in [29]. In this paper, there are no obvious type IX far-feld cracks and type IV mixed tensileshear cracks and little type VII-VIII far-feld shear cracks in the samples with diferent precracks under static loading.   Figure 10: Types of new cracks during static loading. 8 Shock and Vibration

Fracture Mode under Dynamic Impact Loading.
Te failure modes of the specimens are afected by strain rates and faw inclination angles (Table 5). Te crack types of diferent rock specimens under dynamic loading are listed in Table 6. Under dynamic loading, with the increase of strain rate, much smaller sized fragments are produced, and the crack network of sample becomes more complex. When the peak of incident wave is 40.7 MPa, new crack was basically initiated around the prefabricated crack. Only one main crack propagated in each side of the prefabricated crack. Te specimens with β � 60°or β � 90°were broken into two halves at failure, and the specimens with β � 0°, 30°, or 45°w ere broken into several major parts with small amount of pulverized rock debris. Only tensile crack propagation occurred in specimens with β � 45°or 90°mainly. For specimens with β � 0°or β � 30°, tensile and shear cracks are all the dominant kinds of crack. By contrast, pure shear fracture occurred only in specimens with β � 60°. When the peak of incident wave increases to 61.1 MPa, crack network of sample becomes more complex, generally showing "Y" shape crack bands. One or two main cracks propagated in each side of the precrack, and several far-feld cracks initiated. Type V shear crack became the main crack type except for specimens with faw angle of 0°. More far-feld shear cracks initiated in the specimens when the peak of incident wave was 61.1 MPa. Far-feld tensile cracks appeared in specimens with faw angle of 90°and penetrated the whole sample. When the peak of incident wave is 76.5 MPa, more far-feld cracks appeared in the rock specimens. Tere are more shear cracks and fewer tensile cracks.
It was previously thought that shear cracks dominate the fracture mode of samples with single precrack [28], and the "X" shape crack band is generally the macroscopic failure mode when the rectangular plate specimen is used in the SHPB experiment [16]. Te experimental results of this paper illustrate that for some samples of certain size, when the peak of incident wave is not very large, tensile cracks may dominate the fracture mode or several shear cracks form a "Y" shape failure pattern. Relevant studies [30,31] have shown that when the strain rate ranges from 10 −5 s −1 to 10 −2 s −1 (low strain rate), with the increase of strain rate, the shear failure mode of single-fawed specimens changes to the tensile failure mode. By comparison, in the range of medium strain rate, with the increase of strain rate, the failure mode   0 Shock and Vibration of single-fawed specimens changes from tensile failure (except for β � 60°) to X-type shear failure, according to Table 5.

Crack Growth Process.
Crack growth processes of specimens caused by dynamic loading are shown in Figure 13, when the peak of incident wave increases to 76.5 MPa. Various shear and tensile cracks constituted complex crack network of the samples. Tensile crack propagation is often restricted by shear crack. Shear cracks generally form an "X" shape failure pattern, and more far-feld cracks appeared. Te incident energy is relatively high, so more random far-feld cracks generate in sample except for crack network around the prefabricated crack. Due to the nonuniformity of primary defects and new cracks in samples, the travel of stress waves inside the specimen was afected, resulting in the fracture degree of the part of the specimen near the incident bar being generally greater. Tensile stress along the axial direction, caused by the travel of stress wave in the sample, leads to the occurrence of separation layer crack (type IX tensile crack), which is almost absent when the peak of incident wave is 40.7 MPa or 61.1 MPa.
Tere are three stages in the crack growth process of samples with single precrack under impact load. First, cracks initially generate around the preexisting faw tips except for a few samples. Second, the frst shear crack continues to grow, while other tensile or shear cracks initiate and develop. Meanwhile, the precrack collapses and closes. Finally, farfeld shear or tensile cracks propagate and connect with each other. In contrast, under quasi-static loading, collapsion of precrack is not obvious.
Te type and initiation position of the frst cracks for samples with diferent faw are diferent under quasi-static or dynamic loading conditions. Under static loading, the frst crack of all the kinds of rock samples in this paper generates around the precrack. Nevertheless, under impact load, the frst crack may initiate away from the prefabricated crack. Shear cracks initially generate around the precrack tips except for the sample with β = 0°which shows that far-feld cracks initiate frstly when the peak of incident wave is 76.5 MPa. By contrast, when the peak of incident wave is 40.7 MPa or 61.1 MPa, cracks generate frstly at the preexisting crack tips in samples with β = 0°. Under dynamic loading, characteristics of the frst cracks are listed as follows: (1) In general, the frst crack propagation is always accompanied by the collapse of the prefabricated crack. In contrast, under static loading, the collapsion of the precrack almost never occurred. (2) When the peak of incident wave is 76.5 MPa or 61.1 MPa, in most cases, the propagation path of the frst crack is not parallel to the dynamic impact direction. Tis phenomenon is contrary to that of the frst crack propagation under quasi-static loading. When the peak of incident wave drops to 40.7 MPa, the propagation path of the frst crack is roughly parallel to the impact direction except for the samples with β = 60°. (3) When the peak of incident wave is 76.5 MPa, for the samples with β = 90°, the frst cracks generate far away from the precrack, which are roughly parallel to the dynamic impact direction. Te initiation of the frst far-feld cracks is also along with the broken of precrack itself.

Fragmentation Characteristics Afected by the Strain Rate and Preexisting Flaw
Angle. Te sample fragments were screened for quantifying the infuences of preexisting faw and loading rate on the fragment distribution of singlefawed samples, and the fractal theory is used to quantitatively analyze the degree of fragmentation. Te aperture sizes of these sieves are 40 mm, 30 mm, 20 mm, 10 mm, 5 mm, and 1 mm, respectively. Fractal theory is used widely to characterize the distribution of rock fragment [32], which can be expressed as follows: where D b denotes fractal dimension, x and x m denote granular size and maximum fragment dimension, respectively, M(x) and M T , respectively, denote the cumulative mass of broken particles smaller than x and the total mass of rock debris, and Y denotes lumpiness. Photos of rock fragments after sieving under diferent peak of incident waves are shown in Table 7. Under impact load condition, loading rate obviously infuences the debris characteristics of single-fawed samples. Te experiments found that under a lower peak of incident wave, the single-fawed specimens break into two or several large fragments. However, when the peak of incident wave is relatively high, single-fawed specimens break into smaller fragments. With the increasing peak of incident wave, the rock sample is more fragmented. Te high absorption energy promotes the growth of cracks, resulting in more fractured samples. At a given peak of incident wave, it can be observed that the preexisting faw angle has signifcant infuence on the debris characteristics in these tests.  (°)  0  30  45  60  90  Peak of incident wave 40.7 MPa  II, IV  III, V  III  V  I  Peak of incident wave 61.1 MPa  II, V, VII  V, VII  II, V, VII  V, VII  I, V, VII, VIII  Peak of incident wave 76.5 MPa IV, V, II, VII, VIII, IX III, V, VI, VII, VIII, IX II, III, VII, VIII, IX V, VII,   Te fractal dimensions of rock samples are determined using equation (3). In a coordinate system where lg[M(x)/ M T ] is the x-coordinate and lg(x) is the y-coordinate, the slope of the line according to equation (3) Table 8. Infuences of precrack angle and dynamic strain rate on fractal dimension are shown in Figures 14 and 15, respectively.
As peak of incident wave increases, the fractal dimension of single-fawed specimens increases. When the peak of incident wave is 40.7 MPa, the fractal dimension of single-fawed specimens is between 1.22 and 1.40. When the peak of incident wave increases to 76.5 MPa, the fractal dimension is between 2.14 and 2.56, showing an evident strain rate dependence. Fractal dimension is larger, the distinct self-similarity is more, and fragment distribution is more homogeneous. Under a lower peak of incident wave, the energy supplied from the outside moves to preexisting and new faw tips and produces some crack    tracks. Te surface energy of these new appeared faws consumes the energy transferring to faw tips, so there are only a few new cracks. Under a higher peak of incident wave, the external energy is relatively high, resulting in producing abundant new cracks and fracture surface and inducing the samples to form mass rock debris. In this case, samples are characterized by smaller fragments. If fractal dimension is higher, the fragment distribution will be more uniform.
When the peak of incident wave is 40.7 MPa, the fractal dimensions of specimen with diferent faw angles are similar. With the increase of the peak of incident wave, fractal dimension of specimen is obviously afected by the change of faw angle. Under a given incident wave with a peak of 61.1 MPa or 76.5 MPa, the fractal dimensions of specimen with β � 45°are generally lower than those of the other specimens. It indicates that under impact load, the rock debris distribution of the specimens with β � 45°is most inhomogeneous.

Energy Evolution under Dynamic Impact Loading.
Rock failure caused by crack connection and propagation is essentially an energy dissipation process. Energies can be represented by incident wave ε I (t), refected wave ε R (t), and transmitted wave ε T (t) collected during the SHPB tests. Te relevant energy (incident energy (W i ), transmitted energy (W t ), and refected energy (W r )) can be calculated as follows: Te dissipated energy (W s ) of specimens includes three aspects: energy to fracture, the kinetic energy of splinters, and the dissipated energy depending on acoustic emission and heat. If the energy loss between the specimen and the bar is ignored, the energy W s can be calculated as follows: During the SHPB tests, energy absorption is an important feature, which is closely connected with the surface area and number of new defects in samples. Te density of energy dissipation (e) and the energy dissipating rate (η) of specimens can be calculated as follows: where V denotes the volume of rock sample. Te relevant energy indexes obtained by equations (4)-(9) are listed in Table 9.
As the peak of incident wave increases, the W i value and W s value increase signifcantly. Te energy dissipating rate (average of three samples) under various strength impact loads is plotted in Figure 16. Te energy dissipating rate is closely related to peak of incident wave. Te single-fawed specimen with β � 90°is the kind of specimen having most dissipating rate, which varies from 0.40 to 0.53. Obviously, the energy dissipating rate of single-fawed specimens can be improved in a certain range of strain rate under external dynamic impact loading. Figure 17 presents the efect of peak of incident wave on the e value (average of three samples) of single-fawed specimens. As the peak of incident wave increases, the e value increases obviously. Te e value of the single-fawed sample with β � 45°is the lowest among all kinds of samples when the peak of incident wave is 40.7 MPa and 61.1 MPa. With the increase of incident wave peak, the diference of energy dissipation density of samples with various β values is more obvious. When the peak of incident wave is 76.5 MPa, the single-fawed sample with β � 0°has the lowest energy dissipation density. It indicates that the e value of the single-precrack specimens features a rising trend with the increase of incident wave peak, and the infuence of precrack angle on energy dissipation is signifcant for a given peak of incident wave.
Tere are relationships between the energy dissipation and the failure modes. Te number and development of cracks is related to the density of energy dissipation. Te density of energy dissipation is less when the sample has fewer cracks and is broken into several parts. On the contrary, when the sample has more cracks and is completely crushed, the density of energy dissipation will increase signifcantly.

Discussion
Microcrack development from tips of precrack and coalescence with branching cracks in rock specimens lead to the ultimate destruction of samples. Macroproperties of specimens could be shown by dynamic strength, energy dissipation, and fragment distribution. Te interrelations of these parameters are shown in Figure 18.  Te strength, energy dissipation, and fractal dimension of the rock samples are positively correlated with each other under diferent strength impact loading. For all kinds of rock samples with a certain precrack in this study, with the increase of dynamic strength, the average energy dissipation density of samples and fractal dimension increase. For a given peak of incident wave, as the energy dissipation density increases, the fractal dimension also increases. When the peak of incident wave is low (40.7 MPa), the efect of precrack angle on the relationships between these three parameters of rock specimens is not signifcant. As the peak of incident wave increases, the angle of precrack has more important impact on the interrelation of these three parameters.
High strain rate (the peak of incident wave is high) has enhancement efect on the three parameters of rock specimens with precrack. According to the rock fracture theory [33], the relationship between the fragment size, size of the intergranular cracks, and length of crack propagation can be understood. Under stronger impact loading, with the increase of the peak of incident wave, more energy is absorbed for crack initiation, which also causes more tiny cracks to expand, resulting in generation of more small rock debris. Under this condition, more microstructures of the specimen take part in resistance to break, causing increase of the dynamic macrostrength of the specimen. Te precrack guides the initiation of the main crack in the rock specimen, so it infuences the relationship between these three parameters, together with the strain rate.

Conclusions
Under static loading, with the increase of precrack angle, the static strength and static modulus of elasticity of singlefawed rock specimen decrease frst and then increase. Te ultimate crack network of the samples contains several types of cracks, such as types I, II, III, V, and VI. In general, after new tensile cracks initiated around the prefabricated crack, some shear crack branches initiated in the samples except for the specimen with β � 0°.
Under impact dynamic loading, when the strain rate increases, the dynamic strength and modulus increase signifcantly, whose change trend also depends on the angle of precrack in specimens. With the increase of strain rate, crack network of sample becomes more complex. Shear cracks usually dominate the fracture mode of single-fawed specimens and form an "X" shape failure pattern, but when the peak of incident wave is not very large, tensile cracks may dominate the fracture mode or several shear cracks form a "Y" shape failure pattern.
Under impact dynamic loading, with the increasing peak of incident wave, the fractal dimension of single-fawed specimens increases. When the strain rate increases, the fractal dimension of specimen is obviously afected by the change of faw angle. Te energy dissipation density and energy dissipating rate of single-fawed samples show an increasing trend with the increasing peak of incident wave. Te infuence of inclination angle of precrack on the energy dissipation is signifcant for a given peak of incident wave.
Te strength, energy dissipation, and fractal dimension of the rock samples are positively correlated with each other under diferent strength impact loading. Te larger the peak value of incident wave, the more signifcant the precrack angle on the interrelation of these three parameters.

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.