1 Introduction

As one of the first building materials used by humans and one of the most widely used stone materials in the built environment, granite has a lengthy history, exceptional performance, and a variety of applications [1, 2]. Tensile strength is typically the determining factor for brittle damage because it is substantially lower in brittle construction materials like granite than its compressive strength. Brittle damage must be avoided since it might immediately result in injuries, equipment loss, and construction delays [3,4,5]. However, it is necessary to promote crack extension to increase production for projects such as oil or shale gas extraction. Thus, brittle damage of rocks is a double-edged sword that can only be controlled by a thorough understanding. It is vital to figure out the accurate tensile strength of rock materials and to recognize the micromechanism of brittle rupture to understand rock mechanics and the design and support of rock construction engineering.

Natural rock masses contain different types of discontinuous structural surfaces, such as fractures, joints, laminae, faults, and shear zones. These discontinuous structural surfaces are highly susceptible to crack initiation, extension, penetration, and slip of faults under external loads, which significantly affect the rock mass stability [6,7,8,9]. Therefore, the research on the mechanism and characteristics of crack extension has essential engineering significance.

Scholars have conducted theoretical, experimental, and numerical researches to comprehend the fracture mechanism of the tensile behavior of brittle construction materials. In terms of theoretical studies, Kong et al. [10] proposed an intrinsic model that can take into account the fine-scale damage and plasticity mechanisms of quasi-brittle materials by coupling microplanes and micromechanics to create a microplane arrangement capable of visually characterizing the fine-scale progression of damage. Yu et al. [11] modified the smooth particle hydrodynamics (SPH) method's standard kernel function by proposing a meshless numerical methodology to model the brittle fracture properties of rock masses. Moreover, they created a penalty force algorithm for boundary particles in the SPH program to actualize the displacement boundary. Liu et al. [12] proposed a micromechanics-based wing fracture extension model, studied the dynamic fracturing properties of saturated rock under pre-tension, and developed a theoretical model for determining the tensile strength of saturated rock under dynamic and static load coupling. Karimi et al. [13] compared and tested the theoretical frameworks to evaluate their applicability and accuracy in predicting the trend of test data obtained from experiments. Xia et al. [14] introduced a novel two-dimensional discontinuous deformation analysis DDA approach for the joint tensile fracture model (UTFM). The improved DDA (I-DDA) can forecast the Brazilian disk's crack development and extension routes and simulate rock fracture in different brittle materials by incorporating the UTFM into the original DDA framework.

Regarding experimental studies, Meng et al. [15] performed dynamic tensile strength deformation, energy evolution, and crack development during impact testing on Brazilian coal samples using a Hopkinson split pressure bar (SHPB) test device. Wu et al. [16] investigated transversely isotropic millipedes with five laminar face angles using split Hopkinson pressure bars and high-speed digital image correlation (DIC) methods. To learn more about the dynamic damage mechanism of transversely isotropic rocks, they conducted full-field strain Brazilian disk tests at various strain rates. For static impact load splitting tensile tests to elucidate the fracture mechanism of sandstone materials under stable and dynamic loads, Zhou et al. [17] employed a series of Brazilian disk samples containing green sandstone, red sandstone, and black sandstone. They investigated and examined how damage behaved under two different loading conditions. Liu et al. [18] studied the dynamic tensile qualities of rocks under freeze–thaw cycles, the microscopic damage features of rocks by nuclear magnetic resonance techniques, and the dynamic mechanical characteristics of SHPB systems utilizing dynamic Brazilian splitting tests. Kou et al. [19] used digital image correlation (DIC) combining micro-X-ray to study the mechanism of the influence of various temperatures and thin layer angles on the tensile strength of Gneiss in Brazilian fracture tests. To examine the small-scale tensile characteristics of rocks, Zhao et al. [20] conducted systematic direct tensile testing on granite, marble, and 3D printed materials utilizing microscopic mechanical test equipment. These materials were evaluated using dog bone samples that were mainly created, and it was discovered that the tensile strength was lower than that found using the Brazilian disk experiment, which is more accurate, indicating the benefits of tiny-scale direct tensile testing. Li et al. [21] used SHPB equipment to perform dynamic Brazilian disk tests on granite at various strain rates with three strain measuring devices set on the sample surface to explore the fracture pattern, and the experimental findings showed that the tensile strength of the rock was dependent on the strain rate. Zhang et al. [22] used the Brazilian disk splitting tensile test method to study the dynamic splitting tensile properties and damage characteristics of basalt fiber reinforced concrete (BFRC) under different loading rates and basalt fiber content.

Researchers favor numerical simulations as an effective method for rock rupture analysis. For example, Sun et al. [23] employed a combination of active and passive ultrasonic methods as well as a three-dimensional flat-joint model (FJM3D) in laboratory and computational experiments with the Brazilian test to assess the influence of these elements on the location and focal mechanism of the earthquake. Ma et al. [24] used laboratory testing and numerical simulations to evaluate the fracture behavior of fractured chevron-notched Brazilian disk (CCNBD) sandstone specimens under static and cyclic loads. By using experimental and computational research, Asadizadeh et al. [25] investigated the effects of lay spacing and pitch angle on the indirect tensile strength, fracture formation, damage mechanisms, and interaction force chains of laminated disks under radial loading. Bahaaddin et al. [26] studied the impact of load shape on stress distribution and fracture mechanisms in Brazilian tests by numerical modeling, simulating Brazilian disks with a range of contact loading angles between 1° and 30° in the FLAC. Wang et al. [27] used a DEM model to investigate the dynamic tensile performance of tilted rock-concrete bi-materials made up of various rock types, complete and vacuous structures. The bending moment and the failure mechanism resulting in an acoustic emission were inferred from the micro-mechanical details of cracks after analysis of the numerical findings of nominal tensile strength, damage mode, and force–strain relationships of typical samples. Yang et al. [28] explored the damage mechanism of laminated rock-like materials with embedded hard–soft layers in numerical Brazilian experiments, combining a newly developed rock-like material ontological model to explore the mechanical behavior of layers and layer interfaces. Fu et al. [29] completed numerical simulations of granite hollow disk samples with five different pore sizes in the two-dimensional particle flow code (PFC2D), and they found that the optimum tensile loading condition was for a specimen with a diameter of 15 cm.

In summary, the current research results on samples containing prefabricated fractures using Brazilian splitting tests are abundant, but how to improve the efficiency and accuracy of rock rupture process simulation and realize the detailed simulation of rupture path is a pressing issue in research that needs to be resolved. Based on an effective technique for creating a 3D-Clump numerical model for pre-cracked granite using the discrete element method (DEM), a numerical model was established that can accurately reflect the internal structure and mechanical characteristics of granite multi-angle centrally single-cracked Brazilian disks (CSCBD) in this paper. The damage strength, crack distribution and extension, and acoustic emission characteristics of multi-angle CSCBD in laboratory tests were analyzed. The failure pattern of samples and the evolution characteristics of interparticle parallel bond force in the numerical simulation were investigated to reveal the fracture characteristics of the granite multi-angle CSCBD. The other part of the article is arranged as follows. Section 2 describes the selected CSCBD sample and experimental device. The test findings are detailed in Sect. 3, and mechanics characteristic analysis in the three-dimensional simulations of the Brazilian test is illustrated in Sect. 4. Ultimately, Sect. 5 summarizes the main conclusions.

2 Methodology

2.1 Preparation of samples and testing system

Brazilian disks are important samples in rock mechanics tests. Rock samples containing central cracks are often used to study the process and properties of crack initiation, propagation, and connection in rocks. According to thorough research, samples of typical granite were taken at a vertical depth of -1080 m, with uniform grain shape and good overall integrity. The collected rock samples were refined in the laboratory and prepared into complete disk samples of ϕ50mm × 25 mm. The pre-cracked samples were obtained by hydrodynamically cutting the penetration fissures with different dips at the center of the sample based on the complete samples. The samples' preparation accuracy complies with the requirements of the suggested test guidelines provided by the International Rock Mechanics Society. The single fissures in the pre-cracked disk samples have an angle of 0°, 20°, 40°, 60°, or 90° concerning the horizontal direction. As illustrated in Fig. 1, there are five different pre-cracked granite disk samples: S-0, S-20, S-40, S-60, and S-90. Each sample's radius R, thickness t, fissure length a are all 25 mm, and fissure width w 1.0 mm.

Fig. 1
figure 1

Schematic of the CSCBD sample

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The multi-angle CSCBD splitting test was carried out on the TAW-2000 microcomputer-controlled electro-hydraulic servo rigid pressure tester produced by Chaoyang Instruments Co. A sensor, a preamplifier, a filter, a signal processing system, and a display system normally make up an AE detection device. Figure 2 presents the AE detector and the Brazilian disk splitting test equipment.

Fig. 2
figure 2

Diagram of the Brazilian disk splitting tensile test system and AE detection system

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2.2 Modeling procedure in PFC3D

Aiming to construct a refined numerical calculation model that efficiently matches the pre-cracked disk samples in laboratory tests, the 3D-Clump model construction approach developed in the literature [30] was utilized in this research. The materials used in building engineering come in a variety of forms. The 3D-Clump pre-cracked granite numerical model construction method can be used to reflect the various mechanical characteristics of the materials when studying the cracking behavior of other types of materials, such as natural rocks like sandstone, shale, and mudstone, or cement, concrete, and bituminous materials.

The three-dimensional CSCBD samples were developed using the built-in FISH language in PFC program. The geometric parameters of the fissure match those samples in the laboratory. The modeling process can be organized into four parts in Fig. 3 (using the type S-40 sample as an example):

  1. (1)

    Step 1 For generating the fracture, a horizontal 25 mm × 50 mm × 1 mm cubical wall (Box wall) is first constructed in the sample's center and configured to move 40° counterclockwise along the y-axis. By creating cubical geometry (Geometry) of the same shape using the same technique, a cylindrical wall (Cylinder wall) of about ϕ50 mm × 25 mm is then put up, as shown in Fig. 3a. The Geometry is more practical for eliminating the particles inside the contour to make fissures, while the Box wall is more capable of constraining the sphere to be generated at the desired location.

  2. (2)

    Step 2 As shown in Fig. 3b, the bigger spheres (Ballmax) with the radius ratio Rmin: Rmax = 1: x (x > 1) form in the Box wall. As shown in Fig. 3c, a fissure is created by deleting the spheres in the Geometry. The bigger spheres (Ballmax) are then removed once the 3D coordinate locations and radii of these spheres are saved in the database using the FISH language.

  3. (3)

    Step 3 The smaller spheres (Ballmin) with rmin: rmax = 1: y (rmin < Rmin, y > 1) are regenerated in the Box wall, as in Fig. 3d. The spheres in the Geometry are removed to form a fissure, as in Fig. 3e. The smaller spheres (Ballmin) within the Cylinder wall are indicated, and the Ballmax's coordinates and radius information are obtained from the database. Each bigger sphere has a range of smaller spheres (Ballmin) that are separated into groups (Group), and the spheres that are not designated for grouping are separated into groups by the FISH language for all the ungrouped spheres connected with that sphere.

  4. (4)

    Step 4 The Clump structure is then organized once each collection of smaller spheres has been turned into a Clump template using the FISH language, and the smaller spheres (Ballmin) are eventually removed. The type S-40 sample model is shown in Fig. 3f.

Fig. 3
figure 3

CSCBD sample modeling procedure

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3 Experimental results and analysis

3.1 Failure strength analysis

Before the test, it is crucial to understand the degree of influence of the non-homogeneity of the samples on the test results first. For this purpose, three intact granite disk samples, S–N-1, S–N-2, and S–N-3, were selected and subjected to Brazilian disk splitting tests. Table 1 presents the results from the Brazilian disk splitting experiment on granite intact samples.

Table 1 Test mechanical parameters of complete granite samples
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The non-homogeneity of the samples was evaluated by calculating the dispersion coefficient of each parameter and the similarity of the failure pattern of the samples, where the dispersion coefficient was calculated by subtracting the minimum value from the maximum value of the corresponding parameter and dividing it by its average value. The average peak strength of the three complete samples \(\overline{\sigma }_{t}\)  = 10.18 MPa, the moderate peak strain \(\overline{\varepsilon }_{c}\)  = 0.45%, and the average elastic modulus \(\overline{E}\)  = 2.37 GPa. The dispersion coefficients of the peak strength, peak strain, and elastic modulus were 32.42%, 6.67%, and 28.27%, respectively. In summary, the granite used in the test has good homogeneity, and the non-homogeneity has a minor influence on the strength and deformation properties of the intact granite.

Figure 4 shows the results of the Brazilian disk splitting experiment for the five CSCBD samples. When testing intact and CSCBD samples side by side, it can be found that the centrally single fissure can seriously weaken the damage strength of the samples. CSCBD samples have lower average peak strengths, peak strains, and elastic moduli than intact samples. According to Fig. 5, the sample's peak strength shows a slight fluctuation as increasing single fissure angle, with a "W" type decreasing trend. The highest peak strength is 4.58 MPa when the single crack angle γ = 0° and the lowest peak strength is 2.48 MPa when the single crack angle γ = 60°. The elastic modulus first decreases, then increases, and then falls again, and the maximum elastic modulus is 1.29 GPa when the single crack angle γ = 0°. As the single crack angle rises, the peak strain steadily diminishes.

Fig. 4
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CSCBD granite samples test result

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Fig. 5
figure 5figure 5

Relationship diagram of stress, AE hits, AE event accumulation, and strain

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To sum up, the single crack surface significantly affects the damage strength of the disk sample. On the one hand, it causes the strength to weaken. And on the other, it causes the strength to become anisotropic. In other words, there is a significant disparity in the samples' tensile strength at various single crack angles.

3.2 Crack extension and distribution

The pictures of the CSCBD samples after damage and the crack sketches are shown in Table 2, where the tensile crack is shown by the red line and the shear crack is represented by the green line.

Table 2 Fracture characteristics of CSCBD samples
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The starting crack points and expansion paths of each type of sample are as follows:

  1. (1)

    When γ = 0°, the sample's upper and lower starting crack points are located at the center of the prefabricated fissure. The main cracks of the samples all extend from the starting crack point to the end of the sample, and the cracks extend along the vertical direction.

  2. (2)

    When γ = 20°, the sample's upper and lower starting crack points are located about 5 mm inward from the prefabricated fissure endpoint, and the cracks expand parallel to the stress loading direction at an angle of about 27°. Apparent shear cracks appear near the lower end of the sample.

  3. (3)

    When γ = 40°, the lower cracking point of the sample is located at the endpoint of the precast fissure, and the cracking angle is 82°. The top starting crack point is about 2 mm away from the endpoint of the precast fissure, and the crack expands approximately parallel to the stress loading direction.

  4. (4)

    When γ = 60°, the upper cracking point of the sample deviates from the endpoint of the prefabricated fissure by about 4 mm, and the lower cracking point of the sample is located at the endpoint of the precast fissure with a cracking angle of 64°. The crack expands parallel to the stress loading direction at an angle of about 30°.

  5. (5)

    When γ = 90°, the crack begins from the endpoint of the precast fissure, and the crack initiation angle is 0°. The crack extends to the end of the sample along the stress loading direction, which indicates that the prefabricated fissure at this time affects the sample's main starting crack point.

As shown in Table 3, the starting crack point of the sample deviates from the endpoint of the precast fissure when the angle of the precast fissure is 0°, 20°, 40°, and 60°, indicating that the endpoint position of the precast fissure is not the maximum tendency point of the primary crack initiation at this time.

Table 3 Micro-parameters in the PFC3D model
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3.3 Acoustic emission test results

Many microcracks generate during the loading of samples, and the release of acoustic waves and energy accompanies this process. The acoustic emission signals during sample loading are monitored and analyzed, and the analysis results contribute to understanding the microscopic rock damage process. The acoustic emission counts of the damage process of each sample were monitored and recorded, and the relationship between stress, acoustic emission counts, cumulative acoustic emission counts, and strain was plotted based on the observed results, as shown in Fig. 5.

The acoustic emission counting curves of multi-angle CSCBD samples during the complete loading process can be classified into two modes. One is the peak distribution type (Type I), which means that the acoustic emission ringing counts show only a limited number of peak intervals during the complete loading process. The other is the uniform distribution type (Type II), which means that the acoustic emission counts are uniformly distributed or nearly uniformly distributed throughout the loading process.

According to Fig. 5, the acoustic emission ringing counts are low for all the disk samples during the first and middle loading stages. As the loading progresses, the ringing counts start to increase gradually. Before the peak stress is reached, the ringing counts increase abruptly at a low level for a short period. When the radial stress reaches the peak, the ringing counts reach the maximum value.

The typical mode of the acoustic emission count curves of the CSCBD samples, except for sample S-60, is peak distribution type (Type I), which indicates that the multi-angle CSCBD samples have prominent brittle fracture characteristics. Meanwhile, the analysis of samples S-0 and S-40 shows that the acoustic emission count curves have several obvious peak points. Compared with sample S-60, these two samples' uniform distribution of acoustic emission counts must be apparent. This indicates that although some microcracks are generated before the samples reach the peak damage, the sharp growth phase of microcracks is still located in the peak damage area. The analysis of other CSCBD samples shows that most samples have only one peak in the acoustic emission counting curve. This indicates that the damage to the samples is a temporary process, and the presence of prefabricated fissures has an induced effect on the formation of cracks. The macroscopic rupture surface of the sample will be generated rapidly at the early loading stage, and the sample will show significant brittle fracture characteristics.

In conclusion, there is a correlation between the fracture mode of the multi-angle CSCBD granite samples and their acoustic emission characteristics. This correlation characteristic mainly shows that when the sample undergoes a significant brittle fracture, the acoustic emission count curve is mainly characterized by peak distribution (Type I). The peak distribution of the acoustic emission count curve becomes more visible as the sample's damage mode becomes more single and the tensile damage becomes more pronounced.

4 Three-dimensional simulations of the Brazilian splitting test

4.1 Numerical model and parameters

The above experimental findings indicate that the variation of precast fissure angles has a substantial impact on the samples' crack extension process and macroscopic failure pattern. To further investigate the damage mechanism of granite under the action of multi-angle single fissures, PFC3D numerical software in the discrete element method (DEM) was used to numerically simulate the splitting experiments on Brazilian disk samples with different angle single fissures. The information on radial stress, radial strain, energy evolution, and number of cracks during the loading process was recorded by tracking to analyze further the fracture process and mechanism of multi-angle granite CSCBD.

The stiffness of the parallel bond model (PBM) is governed by both contact and bond stiffness, which is quickly lowered owing to bond breakage, affecting not only the stiffness of the bonded particles but also the macroscopic stiffness of the whole model. The PBM can successfully simulate the mechanical behavior of rock materials [31,32,33,34,35]. The numerical model of CSCBD using PBM is shown in Fig. 6.

Fig. 6
figure 6

Diagram of the numerical model of centrally single-cracked Brazilian disk (CSCBD)

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The micro-parameters were chosen in accordance with the parameter modification approach described in reference [36] when the simulated results approximately matched the experimental ones. The micro-parameters of the PFC3D model are shown in Table 3.

4.2 Relationship between cracks number and stress–strain response

The correlation between crack extension characteristics and stress–strain response in numerical simulation of CSCBD samples with five different angles is divided into four stages in Fig. 7 as follows:

  1. (1)

    Crack-free stage (OA): The stress–strain curve at this stage is approximately linear, and the sample is in the elastic deformation stage when microcracks have not yet formed inside the disk sample.

  2. (2)

    Crack initiation stage (AB): A few cracks start to sprout at or near the endpoint of the precast fissure starting at point A. The initiation stress is the radial stress corresponding to the appearance of the crack. The initiation stresses of type S-0, type S-20, type S-40, type S-60, and type S-90 samples are 1.98 MPa, 2.28 MPa, 2.14 MPa, 1.57 MPa, and 1.62 MPa, respectively.

  3. (3)

    Crack stable expansion stage (BC): As the radial stress increases after point B, the primary cracks in the upper and lower zones continue to expand slowly along the loading direction. The microcrack generation and expansion are slow in this stage, and the new crack number is still quite small.

  4. (4)

    Crack surge stage (CD): The peak strengths of type S-0, type S-20, type S-40, type S-60 and type S-90 samples are 4.68 MPa, 4.09 MPa, 3.98 MPa, 3.64 MPa and 3.39 MPa, respectively. Once the radial stress reaches the peak stress point C, the stress–strain curve decreases rapidly, accompanied by the rapid generation of many microcracks. When the loading reaches point D, the occurrence of cracks stops, and the final tensile cracks are more than shear cracks.

Fig. 7
figure 7figure 7

Relationship between cracks number and stress–strain response

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4.3 Micro-scale failure patterns

Micro-scale failure patterns in numerical simulation as shown in Fig. 8. The starting crack points and expansion paths of each type of sample are as follows:

  1. (1)

    Cracks in type S-0 sample extend downward from the center of the precast fissure and from approximately 6 mm from the center of the precast fissure and extend up to the end of the sample.

  2. (2)

    Cracks in type S-20 sample all generate from about 5 mm from the precast fissure’s endpoint and expand up and down along an angle of about 25° nearly parallel to the stress loading direction.

  3. (3)

    Cracks at the upper end of type S-40 sample all generate and expand upward from a location approximately 2 mm from the endpoint of the precast fissure, and cracks at the lower part all generate and develop downward from the endpoint of the precast fissure. The expansion direction is approximately 23° and nearly parallel to the loading direction.

  4. (4)

    The primary cracks at the upper and lower ends of type S-60 sample generate from the endpoints of the prefabricated fissure and expand up and down, with the expansion direction close to parallel to the loading direction. The secondary cracks also generate from the endpoints of the precast fissure, with an angle of about 75° to the primary cracks.

  5. (5)

    Cracks at both the top and lower ends of type S-90 sample generate from the endpoints of the prefabricated fissure and extend along the loading direction. A larger area of crack aggregation appears at both ends of the precast fissure.

Fig. 8
figure 8

Micro-scale failure patterns in numerical simulation

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According to Fig. 8 and Table 3, the failure mode in numerical simulation is quite similar to the laboratory test. The CSCBD samples in the Brazilian splitting experiment are mainly tensile damage, so there are more red tensile cracks than green shear cracks in the monitored microcracks.

4.4 Evolution characteristics of parallel bond force

According to the simulation results, the evolution of the interparticle parallel bond force during crack extension for the CSCBD Brazilian splitting test is shown in Fig. 9. For the type S-0 sample, the prominent blank areas appear in the central attachment of the sample, revealing that the bonding has broken and cracks have formed at that location. For type S-20 and S-40 samples, the parallel bond force increases near the endpoints of the precast fissure, and the increased area is close to the crack expansion area. For the type S-60 sample, the parallel bonding force increases near the lower endpoint of the precast fissure, and the increased area is close to the crack expansion area. The parallel bond force increases insignificantly near the upper endpoint of the precast fissure. For the type S-90 sample, the parallel bonding force increases near the two endpoints of the precast crack, and the increased area is close to the crack expansion area. There is an apparent blank area near the upper endpoint of the precast fissure.

Fig. 9
figure 9figure 9

Evolution of parallel bond force

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In summary, due to the end effect of the loading plate, a larger area of stress concentration occurs at the top and lower loading ends of each sample. There is also a significant stress concentration at both ends of the precast fissures in each sample, and the contact force on the particles inside the sample will increase at the damage site. Hence, the parallel bond force between the particles increases at the corresponding location simultaneously to resist the contact force. As the damage increases, the contact force inside the sample increases, reaching the parallel bond force bearing limit when the bond breaks. The local damage of the sample gets its limit, forming macroscopic cracks.

5 Conclusion

According to the damage strength, crack distribution and extension, and acoustic emission characteristics of CSCBD in laboratory tests, it is confirmed that the established particle flow numerical calculation model of granite multi-angle CSCBD based on the discrete element method (DEM) can realize the efficient and accurate calculation of the Brazilian splitting test. The following main conclusions can be drawn from this work:

  1. (1)

    The precast fissure surface significantly influences the damage strength of the disk sample, and the strength of the sample has a significant difference with different angles of the precast fissure. When the angle of the precast fissure is 0°, 20°, 40°, and 60°, the cracking point of the sample deviates from the endpoint of the precast crack, which means that the endpoint position of the precast fissure is not the maximum tendency point of the central crack initiation at this time.

  2. (2)

    The acoustic emission properties of the multi-angle CSCBD granite samples have a relative correlation with the fracture mode. This correlation characteristic is primarily shown that when the sample undergoes a significant brittle fracture, the acoustic emission count curve is mainly characterized by peak distribution (Type I). The peak distribution of the acoustic emission count curve becomes more visible as the sample's damage mode becomes more single and the tensile damage becomes more pronounced.

  3. (3)

    The numerical simulation is quite similar to the failure pattern of Brazilian disk samples from the laboratory test, where the samples show mainly tensile damage. Applying the efficient three-dimensional clump (3D-Clump) numerical model construction method for pre-cracked granite can realize the microscopic mechanism study of brittle fracture of multi-angle CSCBD. It is significant to analyze the microscopic mechanism of brittle fracture of various types of rocks and the design and maintenance of building engineering.