1 Introduction

Shear failure of rock masses subjected to shear loading is a common occurrence during the construction of geotechnical engineering projects. Such failures can result in instability and ultimately lead to the failure of underground structures, such as tunnels and mines (Borana et al. 2016; Fan et al. 2018; Sun et al. 2020; Wu et al. 2022; Xie et al. 2020). Given that underground engineering projects are often exposed to external factors, such as disturbance and varying boundary conditions, it is imperative to comprehend the role of rock materials in the fracturing process to ensure the safety of such projects (Cho et al. 2008; Huang et al. 2011). Therefore, studying the generation and expansion of cracks in rock specimens, as well as analyzing their shear response under dynamic loading and boundary conditions, is crucial to understanding shear failure.

Previous studies have pointed out the significant impact of normal stress on the shear behaviors of rock material (Cai et al. 2020; Ma et al. 2020; Xu and Dai 2017). The effect of normal stress on the shear behaviors of rocks has been considered as a fundamental factor affecting the shear behaviors of rocks (Dang et al. 2017; Hoek et al. 2002; Shen et al. 2018), and numerous experimental investigations have been carried out to explore its influence on intact rocks (Colmenares and Zoback 2002; Feng et al. 2021; Zou and Li 2021), open jointed rock specimens (Barton 2013; Li et al. 2020; Lin et al. 2019), and infilled ones (Nasir and Fall 2008; Zhao et al. 2020; Zhou et al. 2022a). For instance, Feng et al. (2021) conducted direct shear tests on hard rocks under both normal and lateral stresses and reported that the normal and lateral stresses restrained the lateral deformation and increased the peak shear strength. Tian et al. (2015) conducted experimental and numerical tests to investigate the impact of normal pressure on the shear strength of cemented concrete rock joints. Their findings revealed that if normal stress is less than 6 MPa, the shear strength depended on the bond strength of the interfaces. Similarly, Zhao et al. (2020) conducted direct shear tests on infilled rock joints and analyzed the influence of normal stress on its deformability. These studies focused on the shear strength and deformability of rock materials and demonstrated the close relationship between the shear response of rock specimens and normal stress.

Another threat to the safety of underground engineering is the external disturbance of different strain/loading rates. Researchers have widely investigated the impact of those conditions on the mechanical properties of rocks. Several studies have looked into this issue (Cao et al. 2023; Fukui et al. 2004; Huang et al. 2011; Kleepmek et al. 2016; Li et al. 2005; Xia and Yao 2015; Xie et al. 2021; Zhang and Zhao 2013; Zhou et al. 2020, 2022a, b). For instance, Kato et al. (2003) conducted shear tests on rock specimens at various strain rates and their findings revealed that the shear strength increased logarithmically as the strain rate increased. Similarly, under a strain rate of 10–6–10–2 s−1, Fukui et al. (2004) found that increasing strain rate enhanced the cohesion of Andesite rather than the internal friction angle. While previous studies have primarily focused on the mechanical properties of rock specimens, limited attention has been given to investigating the fracturing behavior of rocks under various loading conditions. The unclear understanding of the fracturing process makes it difficult to predict unforeseen disasters, posing a threat to the safety and lives of workers. It is therefore necessary for future research to explore the influence of external factors on shear behaviors, as this would contribute to a better understanding of the mechanical behavior of rocks. Although experimental tests are valuable, they can be time-consuming and limited by their ability to capture only a narrow range of monitoring instruments. In contrast, the numerical methodology can help address some of these limitations by enabling virtual testing of different conditions in a more controlled manner. Most importantly, it can provide more detailed insights into the underlying mechanisms and failure patterns of rock specimens, thereby offering a deeper understanding of the complex nature of rock mechanics.

With the advancement of computer technology in recent decades, numerically intensive methods, such as finite element method (FEM) (Pradhan and Siddique 2020) and discrete element method (DEM) (Cundall and Strack 1979), have provided researchers with new opportunities to explore the mechanical properties, fracture characteristics, and even failure behaviors of rock that were previously difficult to attain using traditional experimental methods (Qiu et al. 2023). One of the superiorities of numerical modeling is its ability to directly obtain microscopic features, such as stress distribution and real-time displacement, which can be used to reveal the failure behaviors of rock materials. Consequently, researchers can analyze the damage process of rock material and its failure mechanism. As one of the most common numerical methods, DEM has been widely employed to simulate the mechanical behavior of rock specimens subjected to shear loading (Chiu et al. 2016; Gutiérrez-Ch et al. 2018). For instance, Shang et al. (2018) investigated the effect of boundary conditions on the shear behavior of rock discontinuities using DEM modeling and found that boundary conditions had a significant impact on the deformability of incipient rock discontinuities. Jiang et al. (2015) proposed a novel DEM bond contact model, which was further adopted to conduct direct shear tests on specimens containing joints considering chemical erosion. Additionally, Cao et al. (2018) and Harthong et al. (2012) established the numerical models and discussed the shear response of rock material under the influence of joint size and distribution of non-persistent joints together with experimental tests. The results of these studies illustrated that the bonded-particle model (BPM) is a feasible method for studying the shear behaviors of rock materials.

In recent years, the simulation of acoustic emission (AE) has been proposed as an effective method to monitor the fracture process and behavior of rock materials. Hazzard (1998) proposed a methodology based on the moment tensor theory to extract the seismic source information from the bond breaks between particle in the bonded-particle model (BPM), which can record the spatial and temporal information, intensity of AE events, and distinguish crack types (Hazzard and Young 2000; Tang et al. 2000; Zhang et al. 2019). The adoption of AE simulation in BPM has been shown to be useful in deducing the initiation, propagation, and coalescence of cracks under external loading, thus revealing rock fracture behavior. Shimizu et al. (2010, 2009) and Raziperchikolaee et al. (2021) have successfully utilized AE simulation in BPM to investigate the failure pattern and mode of different rock specimens under various loading conditions. The results indicate that the application of AE simulation in BPM can significantly enhance our understanding of the mechanical behavior and fracturing processes of rock materials. In conclusion, the development of numerical methods has greatly expanded our ability to explore the mechanical properties, fracture characteristics, and failure behavior of rock materials. However, while AE simulation has been used to investigate failure behavior in uniaxial compression, there has been limited research on its use to study the shear behavior of rock material under complex loading conditions. The introduction of the above acoustic emission simulation method into the study on shear numerical tests of rock material can monitor the rock shear damage process from the perspective of micro-crack generation, allowing the identification of the crack expansion characteristics and damage patterns before the final damage of the rock mass and deepen the understanding of the shear damage mode of rock materials.

This study aims to investigate the mechanical behaviors and failure process of rock specimens under direct shear loading conditions by utilizing the innovative BPM. Specifically, it explores the combined impact of loading rate and normal stress and provides insights into the spatiotemporal development of AE and fracture evolution. To ensure consistency between experimental and numerical results, a calibration methodology for direct shear test is proposed based on experimental results, which introduces shear performance parameters in the calibration process. The study established the specimen model and direct shear model parameters based on the experimental test results. Then, through direct shear tests on rock specimens using AE simulation, the study analyzed the shear responses and failure modes of the specimens. From a microscopic perspective, the influence of normal stress and loading rate on the mechanical properties and spatiotemporal development of AE are analyzed, providing valuable insights into the shear failure process of rock specimens under various loading conditions. The findings of this study provide important contributions to our understanding of shear behavior and the cracking process under different rates and normal stress conditions. It also demonstrates the potential of the BPM as an advanced numerical tool in studying the complex behaviors of rock material.

2 Numerical Methodology

2.1 Bonded-Particle Modeling (BPM)

BPM is a numerical method used to simulate the mechanical behavior of rock specimens, which assumes that the material consists of spherical particles with fracture potential, allowing for the description of interactions between particles in contact (Cundall and Strack 1979). Potyondy and Cundall (2004) were the first to apply BPM to mimic the mechanical tests of rock specimens, which significantly expanded the applications of the DEM method in geotechnical engineering (Holt et al. 2005; Yang and Huang 2014). One of the most important features is that the manifestation of its mechanical properties depends on the particle and contact parameters at the microscopic scale rather than defining the macroscopic constitutive relationship of the material. Thus, it provides an effective means of reflecting the mechanical properties (e.g., strength and deformability) of rock specimens under different loading conditions. Moreover, the relative motion between particles can be recorded and utilized to reveal the crack evolution process of rock material, forming the basis for AE simulations.

In utilizing the BPM, the selection of contact model and microscopic parameters becomes crucial as they dictate the macroscopic mechanical behavior. The choice of contact model and microscopic parameters becomes the key when using BPM. The parallel bond model (PBM), a widely adopted model in geotechnical material simulation, was chosen here as it can reflect the mechanical behaviors of rock. The PBM consists of two contact types: the bonded and unbonded modes (as shown in Fig. 1), which can transmit forces and moments between particles, as well as forces acting at the contact point, respectively. Contact bonds can only transmit forces acting at the contact point. Once stress applied to the bonded contact exceeds its stress limit, the bond contact will break. It allows the PBM to accurately depict the cracking characteristics of rocks before and after shear damage.

Fig. 1
figure 1

Schematic diagram and rheological components of the linear parallel bond model with inactive dashpots

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2.2 Acoustic Emission (AE) Modeling

To investigate the fracturing and failure patterns of rock specimens under direct shear loading conditions, AE simulation was employed in this study. AE is a phenomenon where material micro-rupture occurs, accompanied by the release of strain energy in the form of elastic waves (Cai et al. 2004). The moment tensor theory, which is used in seismology to obtain source information by recording waves released from the source, can be complicated and time-consuming to calculate if based on dynamic waves (Feignier and Young 1992; Knopoff and Randall 1970). However, the force and motion states of the particles in BPM can be accessed, and Hazzard (1998) proposed a calculation method based on the moment tensor theory, which is related to the change in the contact force of surrounding particles. Therefore, the basic principle used in the AE simulation in this study equates the corresponding displacement produced by all contact forces on the particle surface to the same effect produced by the physical force.

It is assumed that the area of micro-rupture is a circular area with a radius equal to the largest source particle diameter and a center of microfracture. The moment tensor component (Mij) of this micro-rupture can be obtained by summing the change in contact force on the source particle, multiplied by the distance between the position of the microcrack and the contact:

$${M}_{ij}={\sum }_{s}(\Delta {F}_{i}{R}_{j}),$$
(1)

where \(\Delta {F}_{i}\) is the component of contact force change and \({R}_{j}\) is the component of distance between the contact point position and the microcrack center.

Building on this assumption, the intensity of AE activities at different loading stages can be derived from the source information of the micro-rupture inversion. However, assuming that each micro-rupture the actual situation due to the similar particle sizes, which results in the rupture strength and release energy being almost identical for all events. To address this limitation, this study assumes that microcracks occurring within a specified spatial and temporal space are the same macroscopic rupture event. Additionally, to simplify the algorithm and save the storage resource, the principal scalar moment (M0) of each AE activity is picked and stored in the form of moment tensor, as follows (Silver and Jordan 1982):

$${M}_{0}=\frac{{(\sum_{j=1}^{3}{m}_{j}^{2}+0.01)}^{1/2}}{2},$$
(2)

where mj represents the jth eigenvalue of the moment tensor.

Finally, the magnitude (M) of each AE activity can be calculated through the following equation (Hanks and Kanamori 1979):

$$M=\frac{2}{3}\mathrm{lg}{M}_{0}-6.$$
(3)

The AE event type is determined by the simulated AE code, which records the moment tensor and calculates its eigenvalue during the numerical simulation. AE events are classified into three types based on the ratio of isotropic and deviatoric components of the moment tensor. If the ratio exceeds 30, it is considered a tensile crack. A tensile crack is characterized by the relative displacement of particles perpendicular to the crack, that is, rapid opening movement. If the ratio is between -30 and 30, it is a shear crack. The shear crack is formed as two particles move in opposite directions. If the ratio is less than -30, it is an implosive crack (Feignier and Young 1992; Knopoff and Randall 1970). Hazzard and Young (2004) suggested that the implosive crack results from particles moving into the formed crack voids under external load.

2.3 Direct Shear Model

The direct shear model using BPM was established to investigate the combined influence of loading rates and normal stresses on shear responses of rock material. As shown in Fig. 2, the model consists of upper and lower shear boxes with a width of 100 mm. To set up the model, particles with various radii were filled into the shear box to achieve the targeted porosity, and microscopic contact properties were applied between particles after attaining static equilibrium. To apply the shear load, the upper shear box was moved horizontally from right to left at a fixed loading rate, while the lower box remained fixed. During the loading process, a constant normal stress was applied using a servo-control routine code mainly following these steps: (1) calculate the difference between the target and actual stress of the servo wall; (2) count the average stiffness of particles in contact with the wall; (3) calculate the maximum force increment for the servo wall; (4) calculate the servo-control parameters; and (5) calculate the loading velocity of servo wall, see Fig. 3.

Fig. 2
figure 2

The schematic diagram of loading device. a Equipment for direct shear test; b the simulation model of direct shear test in PFC2D

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Fig. 3
figure 3

Program diagram of the servo-process

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3 The Validation-Based Calibration Methodology

The calibration of microscopic parameters is crucial in establishing the direct shear model due to the dependence of BPM’s mechanical properties on the particle and contact parameters. To obtain particle and mechanical properties for calibration, various laboratory tests were conducted, including uniaxial compression tests, direct shear tests, and thin-section analysis. The microscopic parameters were then calibrated using the trial-and-error method based on the physical characteristics of the targeted rock material, including model dimension, particle size, density, and porosity. Physical tests were performed to determine the particle dimensions of the BPM. Following the experimental tests, particle radii were set between 0.350 and 0.581 mm, with the numerical model having a density of 2066 kg/m3, porosity of 0.16, and damping of 0.2.

Then, the calibration of contact parameters was performed using direct shear, uniaxial compression, and Brazilian disk tests as benchmarks for the micro-parameters’ calibration methodology. The simulation model for these tests was established to obtain simulation results for comparison with the experimental results. The fundamental concept of calibration is to continuously adjust the microscopic parameters of the simulation model by comparing experimental results with numerical results. Previous researchers have provided some experience in uniaxial compression model, where the calibration sequence typically begins with the contact modulus, followed by the Poisson’s ratio and uniaxial compressive strength (UCS) (Asadi et al. 2022; Sarfarazi et al. 2021). Bahaaddini et al. (2013) calibrated a joint model with the UCS, Young’s modulus, and Poisson’s ratio to study its shear behavior, which provides a basis and evidence of the rationality of numerical simulation of shear tests. However, their calibration process is limited to the uniaxial compression test, neglecting shear failure behavior and shear mechanical properties. A novel calibration methodology for the direct shear test is proposed in this study (Table 1). The specific procedure is described below, and the flowchart is plotted in Fig. 4.

  1. 1.

    First, the effective modulus of bonded contact (\(\overline{E }\)*) is determined. This is the first factor to be identified as it has a significant impact on the mechanical properties of the numerical model. Calibrating it first can quickly determine the general direction of calibration, reducing later adjustments. Uniaxial compression simulations are performed using the established uniaxial compression model to obtain the desired value of \(\overline{E }\)*. The effective modulus of bonded contact is continuously adjusted during calibration to achieve agreement between Young’s modulus obtained from numerical simulation and experimental tests while maintaining other parameters at a high level.

  2. 2.

    The particle contact modulus (E*) is then determined, after \(\overline{E }\)* is selected. E* can be calibrated with the experimental shear modulus. During calibration, the value of \(\overline{E }\)* obtained in the previous step remains stable, and the other parameters are sustained at a high level in the direct shear model. E* is continuously adjusted until the shear modulus of the simulation model matches the desire value.

  3. 3.

    Next, the normal-to-shear stiffness ratio (\(\overline{k }\)* and k*) for parallel bonds and contacts, respectively, are calibrated. To simplify the calibration process, it is assumed that \(\overline{k }\)* and k* are the same value (Yang et al. 2014). In this step, the Poisson’s ratio obtained by uniaxial compression simulation is compared with the experimental result to calibrate \(\overline{k }\)* and k*. Multiple stiffness ratio cases were set up, and the uniaxial compression tests are separately performed to obtain the appropriate stiffness ratio.

  4. 4.

    The ratio of parallel bond cohesion (kn) to parallel bond tensile strength (ks) is then determined. Since both kn and ks are closely related to the strength of the material, it is difficult to calibrate each of them individually. However, the ratio of kn/ks can control the overall damage pattern in direct shear simulation. Therefore, the ratio is determined first by comparing the final failure between experimental and numerical results.

  5. 5.

    After roughly determining the deformation parameters of the model, the parallel bond cohesion (kn) is calibrated using shear strength based on the ratio obtained in the previous step. It is assumed that kn is a specific value, and other parameters are fixed while adjusting kn and ks according to the specific ratio until the shear strength approaches the desired value.

  6. 6.

    The friction coefficient (μ) is then determined using the shear residual strength obtained from the experimental tests. The shear residual strength is closely related to friction properties.

  7. 7.

    Finally, adjustments based on the overall results are made. Since the parameters interact with each other, it is still necessary to adjust the identified parameters considering the overall situation. The overall macro-mechanical properties of the simulation model can be obtained by performing a numerical simulation of compression, tension, and shear tests. Then the parameters are slightly adjusted based on the overall test results to obtain simulation results close to the laboratory test results.

Table 1 Micro-parameters of simulated numerical model
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Fig. 4
figure 4

The flowchart of calibration for direct shear model

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The calibration results obtained from the procedure are listed in Table 2, and the numerical and experimental results are compared in Fig. 5. As shown in Fig. 5a, the shear strength increases with the loading rate, which is consistent with experimental results. The stress–strain curves at three loading rates under normal stress of 10 MPa are shown in Fig. 5b–d, respectively. The comparison indicates that the shear strength difference between numerical and experimental results is 0.37%, 2.43%, and 0.37%, respectively, with loading rates ranging from 0.001 to 0.1 mm/s. Notably, the numerical result under normal stress of 10 MPa at a loading rate of 0.01 mm/s exhibits a failure pattern that agrees well with experimental results, as shown in Fig. 6. However, the compaction stage observed in the experimental test cannot be observed in the numerical model due to the method of computational model construction, which encloses the space between particles with facets that are generally in contact with other particles before shear load loading (Lin et al. 2019; Sarfarazi et al. 2021). Nevertheless, the later loading stages corresponded well with the experimental results well indicating that the BPM simulation can replicate the rock specimen subjected to direct shear testing and reveal its shearing process.

Table 2 Comparison between experimental and numerical results
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Fig. 5
figure 5

Comparison between experimental and numerical results under direct shear loading. a Shear strength vs. normalized loading rate; b–d Stress–strain curves at normalized rates of 0.001, 0.01, and 0.1 mm/s under a normal stress of 10 MPa

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Fig. 6
figure 6

Comparison between numerical and experimental failure patterns of specimen subjected to shear loading at 0.01 mm/s under a normal stress of 10 MPa

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4 Numerical Results

After calibration, direct shear simulations were performed. In this section, the influence of normal stress and loading rate on shear behaviors of sandstone was systematically investigated.

4.1 Stress–Strain Responses

The stress–strain curves under different loading rate and normal stress conditions are presented in Fig. 7. The shear stress–strain curve is typically divided into three stages: linear-elastic, nonlinear, and post-peak stages, respectively. In the linear-elastic stage, the stress of specimen increases linearly with strain, and this stage may cause micro-cracks initiation. After the initiation of micro-cracks, the shear stress–strain curve enters the nonlinear stage, where micro-cracks continue to propagate and coalesce until the specimen breaks into upper and lower blocks after reaching the peak shear stress. The curve then enters the post-peak stage.

Fig. 7
figure 7

Numerical shear test results of different loading rates and normal stresses. a 0.001 m/s, b 0.01 m/s, and c 0.1 m/s

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The impact of the loading rate on shear stress–strain curves is primarily evident in the post-peak stage rather than in linear-elastic and nonlinear stages. The duration of shear stress decreasing from the peak to the residual strength was extended when the specimen experienced final failure. In Fig. 7a, it can be observed that when the specimen was sheared without normal stress at a loading rate of 0.001 mm/s, the shear stress–strain curve dropped abruptly to the residual stress level after reaching its peak at a strain of 0.22%. However, as the loading rate increased to 0.1 mm/s (Fig. 7c), the shear stress began to drop at a strain of approximately 0.22% and continued until around 0.3%. The drop in the post-peak curve altered from violent to moderate with an increase in the loading rate, indicating that the specimen exhibited brittleness at lower loading rates but ductility at higher loading rates (Liu et al. 2020; Zhang et al. 2017).

As Fig. 7 shows, the mechanical responses of the numerical specimens were significantly influenced by normal stress under all loading rate conditions. With an increase in normal stress, the duration of the linear-elastic stage notably increased under all loading rates. On average, the strain of the specimen without normal stress reached around 0.2% at the end of linear-elastic stage, while it lasted approximately 0.28% for specimens under a normal stress of 20 MPa. Similarly, the nonlinear stage lasted longer under higher normal stress conditions. This phenomenon can be attributed to the increase in constraint on the normal deformation caused by the increase in normal stress. In addition, the shear stress–strain curves fluctuated up and down near the peak point under the normal stress of 15 and 20 MPa, indicating that the failure of the specimens became more complicated under higher normal stresses and demonstrated a typical progressive fracturing process. Furthermore, we observed a positive correlation between the residual strength and normal stress. For instance, taking the specimens at a loading rate of 0.01 mm/s as an example, the residual strength rose from approximately 6–28 MPa as the normal stress increased from 0 to 20 MPa. This may be attributed to the enhancement of the friction between the upper and lower blocks by increasing normal stress.

4.2 Mechanical Properties

4.2.1 Shear Modulus

Based on the stress–strain curves presented in Sect. 4.1, the shear modulus and shear strength can be obtained (Fig. 8). As shown in Fig. 8a, the shear modulus decreases with increasing loading rate. Specifically, the average shear modulus for all specimens decreased by 6.62% as the loading rate increased from 0.001 to 0.1 mm/s. Furthermore, the rate effect on shear modulus was found to be sensitive to the normal stress level. The decrease in shear modulus was less pronounced at higher normal stress conditions. Under a normal stress of 5 MPa, the shear modulus of specimens decreased by 6.72% as the loading rate increased from 0.001 to 0.1 mm/s, while it only decreased by 3.85% for rock specimens subjected to a normal stress of 20 MPa. This suggests that the normal stress reduced the rate effect on the elastic deformation properties of rocks. In contrast, the shear modulus increased with increasing normal stress, see Fig. 8b. The averaged shear modulus under all loading rate conditions increased from 6.11 to 9.69 GPa, an increase of 58.56%, when the normal stress changed from 0 to 5 MPa. Then, the increment in shear modulus slowed down at higher normal stresses and gradually vanished after the normal stress exceeded 15 MPa. The change in shear modulus was less than 5% when the normal stress was beyond 10 MPa. It indicates that normal stress has a significant impact on the elastic deformability of rocks, particularly for those at lower normal stresses.

Fig. 8
figure 8

Numerical shear modulus vs. normalized loading rate curves under different normal stresses

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4.2.2 Shear Strength

Figure 9 presents the shear strength of rocks subjected to different normal stresses and loading rates. As shown in Fig. 9, the shear strength exhibits an increasing trend with increasing loading rate. Specifically, when the loading rate increased from 0.001 to 0.1 mm/s, the average shear strength of all specimens increased by 12.98%, regardless of the normal stress. However, it is noted that the rate-dependent increase in shear strength is limited by the normal stress. Under the loading rate range of 0.001–0.1 mm/s, the shear strength of rocks under normal stress of 0 MPa increased by 19.84%, while that under normal stress of 20 MPa was 9.30%. By fitting the data to Mohr–Coulomb criterion, the curves for specimens under loading rates of 0.001, 0.01, and 0.1 mm/s are plotted in Fig. 9b. The results demonstrate that when loading rate increased from 0.001 to 0.1 mm/s, the cohesive strength increased from 14.42 to 17.27 MPa, while the coefficient of friction remained relatively constant.

Fig. 9
figure 9

Numerical shear strength vs. normalized shear rate under different normal stresses

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In addition, while shear strength is positively correlated with normal stress, its relationship differs from that induced by the rate effect. Specifically, the increase in shear strength gradually slows down with increasing normal stress. As shown in Fig. 9b, the average shear strength of rocks under different loading rates increased from 13.32 MPa to 27.91 MPa as normal stress rose from 0 to 10 MPa, representing a 109.53% increase. However, when normal stress was further increased to 20 MPa, the average shear strength was 34.56 MPa, representing only a 23.83% increase compared to that under a normal stress of 10 MPa. The enhancement of shear strength by normal stress has been widely observed and discussed in both experimental and numerical tests on intact rock specimens and joints (Feng et al. 2021; Karami and Stead 2008; Liu et al. 2021; Park and Song 2009). It is recognized that the increase in shear strength of rocks under direct shear loading conditions may be attributed to the increased friction and constraint in rock specimens with increasing normal stress (Liu et al. 2021).

4.3 Typical AE Activities

Figure 10 illustrates the evolution of AE activities, cumulative fracture numbers, and shear stress of the specimens under different loading rates while maintaining a constant normal stress of 10 MPa. To facilitate a better comparison of AE activities, the x-axial was divided into 50 segments to calculate the AE event rate. Additionally, the cumulative fracture number curve for each simulation specimen was calculated to reflect the fracturing behavior during the direct shear process. In general, fractures and AE events tend to concentrate around the peak shear stress at lower loading rates. For instance, in the case of the specimen under the loading rate of 0.001 mm/s (Fig. 10a), few AE activities were observed at linear-elastic and nonlinear stages, and the AE event rate was low until the final failure of the specimen, where the AE event rate at the peak was nearly eight times higher than the average one before approaching the peak shear stress point. As the loading rate increased to 0.01 mm/s, AE activities became more active in the nonlinear stage, and several AE rate spike points appeared before approaching the peak shear stress (Fig. 10b). Accordingly, more fractures were generated and accumulated in the nonlinear stage compared to those under the loading rate of 0.001 mm/s. Specifically, the total fracture number was approximately 100 before approaching the peak shear stress under the loading rate of 0.001 mm/s, while it reached 200 in the case of 0.01 mm/s. When the loading rate increased to 0.1 mm/s, these evolution laws still held and became more evident (Fig. 10c). AE events were generated distinctly in the later linear-elastic stage, and the AE event rate curve rose steadily to the maximum at the peak shear stress point. Correspondingly, the cumulative fracture number curve exhibited a stepwise rise during the nonlinear stage, rather than a jumping manner under lower loading rate situations.

Fig. 10
figure 10

AE activities, cumulative micro-crack number, and shear stress vs. time step curves of numerical results under normal stress of 10 MPa at different loading rates

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Figure 11 depicts the changes in AE activities, cumulative fracture numbers, and shear stress for specimens under three different normal stresses, while the loading rate was kept constant at 0.01 mm/s. The results indicate that increasing normal stress enhanced the intensity of the AE event rate and substantially increased the cumulative fracture number. In the absence of normal stress (Fig. 11a), very few AE events occurred in the linear-elastic stage (section O–A) and the AE event rate only slightly increased during the late linear-elastic stage (section A–B). As the shear stress approached its peak, the AE event rate rapidly increased and reached its maximum. During the post-peak stage (section C–D), AE activities almost disappeared, as the upper and lower parts of the specimen were completely separated without the constraint of normal stress. However, this phenomenon vanished when normal stress was applied, as shown in Fig. 11b and c. Under a normal stress of 10 MPa, AE activities became more active in the nonlinear and post-peak stages and both the AE event rate and the cumulative fracture number significantly increased before the shear stress reached its peak. Moreover, considerable AE events still occurred during the post-peak stage, resulting from friction and failure of joint apertures on the shear failure surfaces of the specimen under the constraint of normal stress. Under a normal stress of 20 MPa (Fig. 11c), the specimen experienced a significant period of the nonlinear stage before it reached the peak stress point and eventually broken into upper and lower blocks. This period was characterized by a progressive fracturing process, which resulted in the simultaneous generation of numerous AE events and fractures.

Fig. 11
figure 11

The stress-AE rate cumulative micro-crack number time step curves of numerical model at normalized loading velocity of 0.01 mm/s under normal stress of 0, 10, and 20 MPa, respectively

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4.4 Cracking Characteristics

4.4.1 Effect of Loading Rate on Cracking Characteristics

Figure 12 displays the fracturing behavior of specimens under a normal stress of 10 MPa at three loading rates. The distribution of AE events at virous loading time points is collected from Figs. 10 and 11 and is plotted in the figures as small circles of different colors. The color of circles represents the micro-crack types (Feignier and Young 1992). The circle’s size reflects the rupture intensity calculated by Eq. 3. Figure 12a exhibits the cracking process of the specimen at a loading rate of 0.001 mm/s. During the linear-elastic stage (section O–A), several sporadic micro-cracks were generated at both ends of the shear plane, most of which were tensile micro-cracks. As the shear loading increased, the number of micro-cracks significantly increased. In the nonlinear stage (section B–C), both tensile and shear micro-cracks appeared in the middle part of the specimen, with a small fraction of implosive micro-cracks observed at the left end of the shear plane. After the stress–strain curve reaching the peak, a bunch of micro-cracks were generated, forming the failure plane.

Fig. 12
figure 12

Evolution of micro-cracks inside numerical specimens under normal stress of 10 MPa at loading rate of 0.001, 0.01, and 0.1 mm/s, respectively. Note that points A, B, C, and D correspond to the same letters presented in Fig. 10a. The green, red, and blue circles represent implosive, shear, and tensile micro-cracks, respectively

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In the case of higher loading rates, micro-crack development tends to cluster at both ends of the shear plane before aggregating toward the middle part during the linear-elastic phase. For example, at 0.1-mm/s loading rate (Fig. 12c), as shown in Fig. 12c (B), more shear and implosive micro-cracks appeared at the right side of the shear plane when the stress–strain curve entered the nonlinear stage. This may attribute to the fact that shear stress was unable to be uniformly transferred to the inside of the specimen before reaching the elastic limit at high loading rates. Micro-cracks then propagated along the shear plane to the central part of the specimen during the nonlinear stage, where plenty of implosive micro-cracks were further generated around the right side of the specimen, while the shear and tensile ones mainly propagated along the shear plane. When the shear stress reached its peak, micro-cracks coalesced throughout the shear plane and led to the final failure.

Comparison of the cracking processes of specimens at different loading rates reveals that both the location of micro-crack initiation and distribution of different types of micro-cracks are sensitive to rate effect. At higher loading rates, such as 0.1 mm/s, more micro-cracks tend to appear on the right side of the shear plane during the nonlinear stage. This can be attributed to the fact that stress on the right side of the specimen exceeded the elastic limit of the rock material before the force expanded uniformly inside the specimen due to rapid loading. At low loading rates, micro-cracks distributed at the right end of the shear plane were dominated by shear micro-cracks, whereas under higher loading rates, more shear micro-cracks tended to distribute in the middle part of the specimen, and the distribution pattern shifted toward a mixture mode of tensile and implosive micro-cracks.

To illustrate the impact of loading rate on the failure behavior, Fig. 13 plots the number of shear, tensile, and implosive micro-cracks. Under a normal stress of 10 MPa, tensile micro-crack dominated the failure process, with the proportion of tensile, shear, and implosive micro-cracks is roughly 41.1%, 35.0%, and 23.9%, respectively. The magnitude of these micro-cracks fluctuated by less than 2.5% at different loading rates. However, the curves of three micro-crack types behaved differently at different loading rates. At lower loading rates, such as 0.001 mm/s, the curves of the three types of micro-cracks increase sharply at the time of final failure. In contrast, they began to appear from the nonlinear stage under higher loading rate conditions, which is consistent with the results of AE events as shown in Fig. 10. Additionally, the shear and tensile number curves rose in the nonlinear stage before implosive micro-cracks, suggesting that failure at the initial stage was dominated by a mixed mode of tensile and shear failure, with tensile failure being dominant. When the shear stress approached the peak point, micro-cracks began to generate in large quantities and particles moved into the formed crack voids under the action of external load, resulting in the generation of implosive micro-cracks (Hazzard and Young 2004). At this stage, all three types of micro-cracks expanded rapidly, but tensile failure maintained the dominant position.

Fig. 13
figure 13

Stress–strain curve and evolution of crack number of three types of micro-cracks corresponding to the same loading conditions presented in Fig. 12

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4.4.2 Effect of Normal Stress on Cracking Characteristics

Figure 14 exhibits the cracking characteristics of specimens subjected to different normal stresses of 0, 10, and 20 MPa under the same loading rate of 0.01 mm/s. Overall, the normal stress has a more significant influence on the generation and propagation of micro-cracks than the loading rate. Without normal stress, as shown in Fig. 14a, micro-cracks mainly appeared along the shear plane. The initiation of micro-cracks started from the ends of the shear plane in the linear-elastic stage (section O–A). When the stress curve entered the nonlinear stage, the generation of micro-cracks was accelerated, and tensile micro-cracks expanded inclined to the shear plane. Meanwhile, shear micro-cracks extended from the ends to the middle part of the specimen along the shear plane (section B–C). After the shear stress approached the peak, more micro-cracks were generated simultaneously and finally coalesced and penetrated the center of the BPM specimen. As shown in Fig. 14a (D), tensile micro-cracks are evenly distributed along the shear plane, while most shear and implosive micro-cracks are distributed near the ends of the shear plane.

Fig. 14
figure 14

Evolution of micro-cracks inside numerical specimens under normal stresses of 0, 10, and 20 MPa at the loading rate of 0.01 mm/s, where A, B, C, and D correspond to the same letters presented in Fig. 11a. The green, red, and blue circles represent implosive, shear, and tensile micro-cracks, respectively

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The application of normal stress has a significant impact on the total number of micro-cracks increases as illustrated in Fig. 14c. Under a normal stress of 20 MPa, the initiation of micro-cracks occurred in the linear-elastic stage (section O–A), and the distribution area of micro-cracks is widened. Tensile micro-cracks were generated along two curves forming the boundary of the micro-cracks propagation region. During the nonlinear stage (section B–C), shear and tensile micro-cracks expanded mainly along the edge of the oval area, while implosive micro-cracks were observed in a small area at both ends of the shear plane. In addition, a series of tensile micro-cracks extended perpendicular to the shear plane, which expanded the range of micro-crack distribution and increased the fragmentation degree of the shear surface. During the post-peak stage, asperities failure due to friction between the upper and lower parts on the failure surface led to the generation of many implosive micro-cracks.

In Fig. 15a, the evolution of micro-cracks at a loading rate of 0.01 mm/s in the absence of normal stress is shown. The failure process is mainly controlled by shear micro-cracks, which account for 51.8% of the total crack numbers, followed by tensile and implosive micro-cracks of 27.1% and 21.2%, respectively. During the elastic-linear stage, only a few micro-cracks were generated at both ends of the shear plane. In the later linear-elastic stage (Section A–B), the number of shear micro-cracks increased to 18, while the number of tensile and implosive micro-cracks was 9 and 5, respectively. This trend persisted until the end of the nonlinear stage (Section B–C), where the cumulative number of shear micro-cracks increased to 55, while that of tensile and implosive micro-cracks slightly increased to 13 and 12, respectively. After the peak shear stress, all three types of micro-crack showed rapid growth, and the growth rate was similar. Therefore, without normal stress, the cracking at both ends of the shear plane in the initial phase is mainly controlled by shear mode. The simultaneous rise of all three types of micro-cracks at the onset of final failure indicates that the failure is a mixture mode. At this point, tensile and shear micro-cracks control the convergence of micro-cracks, while implosive micro-cracks are mainly generated at the ends of shear plane. After the final failure of the specimen, no micro-cracks were observed.

Fig. 15
figure 15

Stress–strain curve and evolution of crack number of three types of micro-cracks correspond

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When normal stress presents, the failure behavior and evolution of micro-cracks change compared to when normal stress is absent. The dominant failure mode changed from shear to a mixed mode with tensile micro-cracks being dominant, as demonstrated in Fig. 15b and c. The total number of micro-cracks gradually increased at the end of the nonlinear phase under all normal stresses, but the trend of increasing the number of micro-cracks is gentler at a normal stress of 20 MPa, compared to the abrupt increase observed after the peak point with normal stresses of 0 and 10 MPa. Additionally, with the application of normal stress, tensile micro-cracks generate earlier than the other types, which is consistent with the process presented in Fig. 14.

5 Discussion

In this study, the shearing response of the direct shear numerical model was investigated in terms of mechanical properties and AE simulation results. Results suggested that the shear strength increases with both the increase in loading rate and normal stress, which agrees with the results of previous studies (Atkinson 1984; Kato 2003; Lin et al. 2019; Mehrishal et al. 2016; Xu et al. 2019). There are several explanations for the rate effect mechanism: cohesion enhancement, energy release, and fracture mechanics-crack expansion rate (Fukui et al. 2004; Du et al. 2020; Zhou et al. 2018). Fukui et al. (2004) and Tian et al. (2015) suggested that the increase in shear strength with increasing strain rate is attributed to the increase in cohesion, which is also confirmed by the results of the present simulation. As shown in Fig. 9b, the fitting curves of shear strength according to Mohr–Coulomb criterion suggest that the rate effect is related to the enhancement of cohesion. The cohesion rises with the increase of normal stress, while coefficient of friction remains at the same level. In terms of fracture mechanics, Cho et al. (2003) suggested that the sharp increase in rock strength with increasing strain rate is due to the increase in the number of micro-cracks under the action of high strain rate and the hysteresis of extensional cracks caused by the stress release of adjacent micro-cracks. However, it was not evident in this experiment, we observed that the degree of fracturing increased slightly with the loading rate. There is just a weak increase in the number of cracks with increasing loading rate. It may be attributed to the fact that the loading rate in this numerical simulation does not reach the dynamic loading range, and shear tests under dynamic impact will be carried out on the specimens established in this paper in subsequent studies to further investigate the rate effect of shear mechanical properties of rock materials.

In contrast, the increase in normal stress significantly increased the number of micro-cracks and widened its distribution area. Under higher normal stress, the micro-cracks distributed in a broader area rather than that in a relatively uniform line appeared in the case without normal stress. This observation is similar to the results reported by Allersma (2005) and Cho et al. (2008). It can be considered that the normal stress restrains the deformation of the specimen in the normal direction, which results in more deformation and ruptures occurring around the shear plane.

Notably, due to the assumption of the DEM modeling, it is difficult to simulate the characteristics of rock materials during the compaction stage. Instead, it directly entered the linear-elastic deformation stage, resulting in a large initial modulus. In addition, this paper does not consider the lithological, mineralogical, and textural influence on rock properties. Simulation experiments for specimens containing cracks or rocks with special layering structures (such as shale) can be further investigated based on this work in future to guide the mechanical responses of rocks under more specific conditions. Also, external conditions should be considered in further studies, for example, thermal effect, seawater corrosion, weather effect, and freezing. A more comprehensive understanding of rock mass behavior can be carried out by considering these external conditions, which enables better prediction and management of underground engineering projects.

6 Conclusion

In this study, numerical simulations using BPM were employed to investigate the influence of loading rate and normal stress on the mechanical and fracture behavior of rocks subjected to shear loading. The analysis of the strength and cracking characteristics of rock specimens led to the following conclusions:

  1. 1.

    A novel calibration methodology for direct shear is proposed, taking into account loading rates, and uses experimental results to derive the microscopic parameters of PBM for obtaining the macroscopic properties of rocks. This methodology provides a method for the establishment of a direct shear model.

  2. 2.

    It is revealed that both normal stress and loading rate have a positive influence on shear strength. However, normal stress has a greater impact on the shear strength compared to loading rate. Furthermore, the effect of loading rate on shear strength weakens as the normal stress level increases. This indicates that the strength increase with loading rate is more significant at lower normal stress levels.

  3. 3.

    Loading rate and normal stress have a similar influence trend on the AE events in rock specimens, mainly in the nonlinear stage and post-peak stage. With an increase in loading rate and normal stress, the AE event rate was more actively distributed in the nonlinear stage. At low loading rate and normal stress, intense AE activity was mainly concentrated around the peak stress point. Meanwhile, under higher loading rate and normal stress, the nonlinear stage was prolonged, and more AE events were generated in the nonlinear and post-peak stages.

  4. 4.

    The initiation and distribution of micro-cracks during direct shearing are influenced by both normal stress and loading rate. However, normal stress has a greater impact on the generation and propagation of micro-cracks compared to loading rate. With higher normal stress, the failure mode shifts from a shear-dominated to a mixed mode with tensile micro-cracks in a dominant role.