1 Introduction

Numerous engineering observations indicate that when an unstable failure occurs in a rock mass, the rock typically shears and slides along the surfaces of joints, and the rock material itself remains undamaged throughout this process (Shen et al. 2020). The shear mechanical properties of rock joints play a crucial role in assessing the stability of rock masses. In deep underground engineering, these properties tend to be highly complex due to various factors such as intricate geological conditions and structural characteristics (Han et al. 2018). Therefore, exploring the shear behavior of rock joints under diverse surface morphology and boundary conditions is tremendously significant.

In deep underground rock engineering, when shear slip failure occurs, the normal load exerted on the rock mass will keep escalating owing to the inability of the rock mass to dilate freely due to the confinement imposed by the surrounding rock mass or anchor rods (Li et al. 2008). Consequently, the constant normal load (CNL) boundary conditions become inadequate to fulfill field requirements, and the constant normal stiffness (CNS) boundary conditions are more appropriate and realistic under such circumstances.

Since the 1870s, researchers have been carrying out theoretical investigations on rock joints under CNS boundary conditions. In 1979, Heuze (1979) presented that normal stress on rock joints rises when the shear dilation of rock joints transpires as a result of the limitation imposed by the external normal stiffness. Additionally, A formula has been proposed to predict the peak shear stress of rock joints under CNS boundary conditions. In 1985, Leichnitz's (1985) believed that the correlation between shear stress and normal displacement (δv) is not influenced by the stress path, and is exclusively determined by shear displacement (δh) and normal stress (σn). Therefore, he derived a system of partial differential equations for anticipating the shear response of materials under CNS boundary conditions. In 1990, Saeb and Amadei (1990) generalized the original graphical method proposed by Goodman (1989) and applied it to predict the shear performance of joints under CNS boundary conditions. Li et al. (2016; 2017; 2018) introduced a novel analytical model that estimates the shear behavior of rock joints under CNS boundary conditions. This model quantitatively dissects the JRC (joint roughness coefficient) of rock joints into waviness and unevenness components. Taking into account the variations in normal stress, the evolution of asperities, and the magnitude of shear displacement, the model provides a comprehensive illustration of the continuous degradation process of asperities on the surface of rock joints. Liu et al. (2022) employed numerical simulations to investigate the response of double-rough joints under CNS boundary conditions concerning shear behavior.

It was not until the end of the last century that CNS shear laboratory tests began to be developed. Indraratna et al. (1999; 2015) pioneered the development of a CNS testing machine that utilized springs to replicate normal stiffness, and formulated an equation based on the principles of energy conservation and Fourier series analysis that was applicable for predicting shear stress in regular saw-tooth joints. Jiang et al. (2004) utilized a newly developed servo-controlled shearing apparatus to reveal, through experiments, the variation of shear strength, volume expansion mechanism, and energy change law of rock joints with different roughness during the shearing process. Shrivastava et al. (2013, 2018) investigated the influence of filling material on the shear behavior of rock joints and proposed a model for predicting the shear strength of filled jointed rock based on experimental tests of serrated filled rock joints with different waviness angles under two different boundary conditions. Liu et al. (2021) conducted cyclic shear tests on rough rock joints under CNS boundary conditions and characterized the acoustic emission response, shear expansion, shear stress and surface wear of the joints. Jiang et al. (2021) examined how CNS boundary conditions affect the shear mechanical properties of anchored joints and the characteristics of shear failure on the joint surface, displacement at the anchor-failure point, and anchor deformation failure. Han et al. (2020) conducted research on the shear mechanical properties of rock joints under cyclic loading and CNS boundary conditions. He proposed empirical equations that could predict the shear stress, normal displacement, and normal stress under cyclic shear loading. In summary, due to the limitation of test conditions, the CNS shear laboratory tests started late, and previous studies have demonstrated the shear mechanical properties of rock joints, which include the shear strength, shear expansion characteristics and surface damage of the joints under CNS boundary conditions. In the CNS shearing field, there is currently no relevant research that combines numerical simulation methods and laboratory tests for post-validation.

First, this paper proposes an innovative cyclic loading approach for the smooth joint model to address the issue of model failure during shear testing of rock joints. Then, the study established a rock joint model under CNS boundary conditions and investigated the impact of four variables, namely JRC, initial normal stress (σn0), normal stiffness (kn), and shear rate (v), on the shear characteristics of the rock joint model separately. Empirical formulas for predicting shear stress (τ), normal displacement (δv), and normal stress (σn) of jointed rock mass are proposed based on the numerical simulation results, the multivariate regression algorithm and the basic principles of the CNS shear test. Finally, shear laboratory tests of rock joints under CNS boundary conditions are carried out, and the results are used to verify the empirical formula proposed by the numerical simulation test.

2 PFC Numerical Model Establishment

The model size is set to 200 mm × 100 mm (in length and height, respectively), and the joints were located in the middle of the height direction of the specimen. Taking a joint (JRC = 10–12, the standard JRC contour line proposed by Barton and Choubey (1977)) as an example, the steps for establishing the deep rock joint model are as follows:

The first step is to generate a wall. The upper and lower frictionless walls are generated separately using the “wall” command in PFC. To generate a complex joint profile in PFC, the geometry command is used to import the joint contour. However, because only one side of the wall is active and valid, the creation of a complex joint profile requires the generation of two walls.

The second step is to generate particles. The uniformly distributed collection of particles is generated that meet a certain pore density in the upper and lower walls, respectively. The minimum and maximum radius of the particles applied in this paper are 0.5 mm and 0.75 mm, respectively. Under these conditions, the computer calculation requirements are met, while the calculation results are more accurate and stable. Sufficient time for the resulting particle aggregate is required to reach a state of static equilibrium in this step.

The third step is the parallel bonding model assignment. In this step, suspended particles with fewer than three contacts in the particle assembly generated in the second step are removed to improve computational efficiency. Subsequently, the mechanical behavior of the intact rock was simulated using the parallel bond model.

The fourth step is the smooth joint model assignment. The assignment method for the smooth joint model is the key aspect of this modeling approach. The assignment of the smooth joint model is carried out by distinguishing the joint surface between the upper and lower particle groups and continuously assigning the smooth joint model to the joint surface during the normal stress and shear process. The orientation of the smooth joint at each contact is maintained consistent with the orientation of the overall joint. This method can ensure that a smooth joint model is always present where the joint surfaces touch during the joint shear process. Otherwise, the accuracy and reliability of the joint model may be compromised, leading to potential errors in the shear strength obtained from the simulation as the shear displacement increases. The figure in this paper shows the numerical model and boundary conditions that have been established, as illustrated in Fig. 1. The specific model parameters are shown in Table 1.

Fig. 1
figure 1

Numerical model of rock joints under CNS boundary conditions

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Table 1 Micro-mechanical parameters of PFC numerical model
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Calibration of parameters is the key to numerical simulation tests. Firstly, the rock-like materials were made of mixtures of plaster, water and retardant with a weight ratio of 1:0.2:0.005, with a uniaxial compressive strength and modulus of elasticity of 47.4 MPa and 69.95 GPa, respectively, the material is representative of harder rocks commonly found in deep rock mass engineering. As shown in Fig. 2a, the uniaxial compressive strength, peak strain, and elastic modulus of the standard sample in the numerical simulation test are mainly consistent with laboratory test results. Secondly, as shown in Fig. 2c, the shear stress–shear displacement curves as well as the final damage modes of samples in the numerical simulation tests are mainly consistent with laboratory test results when σn0 is 1, 3 and 5 MPa, respectively, at JRC = 10 to 12 (the standard JRC contour line proposed by Barton and Choubey (1977)). Thus, we assert that the parameters utilized in the numerical simulation test are justifiable.

Fig. 2
figure 2

Uniaxial compression test: a stress–strain curve b splitting failure in test and PFC CNS shear test: c shear stress–shear displacement curve d final failure modes in test and PFC

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3 Numerical Calculation Scheme for Rock Single Joint

The numerical simulation test includes the following four research contents:

  1. 1.

    The influence of JRC on the shear effect of rock joints under CNS boundary conditions.

  2. 2.

    The influence of σn0 on the shear effect of rock joints under CNS boundary conditions.

  3. 3.

    The influence of kn on the shear effect of rock joints under CNS boundary conditions.

  4. 4.

    The influence of v on the shear effect of rock joints under CNS boundary conditions.

Among them, the maximum shear displacement in this test is set to 10mm and the joint roughness coefficient of rock joints adopts the standard JRC contour line proposed by Barton and Choubey (1977). The specific implementation scheme of test is shown in Table 2:

Table 2 Numerical calculation scheme for rock single joint
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Condition 1 Research on the shear effect of JRC on rock joints: σn0, kn, and v are set to 1 MPa, 3 GPa/m, and 0.1 m/s, respectively. In addition, JRC is set to five levels: 2–4, 6–8, 10–12, 14–16 and 18–20.

Condition 2 Research on the shear effect of σn0 on rock joints: JRC, kn, and v are set to 10–12, 3 GPa/m, and 0.1 m/s, respectively. σn0 includes 5 levels, which are 1, 2, 3, 4 and 5 MPa, respectively.

Condition 3 Research on the shear effect of kn on rock joints: JRC, σn0, and v are set to 10–12, 1 MPa, and 0.1 m/s, respectively. kn includes 5 levels, which are 1, 3, 5, 7 and 9 GPa/m, respectively.

Condition 4 Research on the shear effect of v on rock joints: JRC, σn0, and kn are set to 10–12, 1 MPa, and 3 GPa/m, respectively. v includes 7 levels, which are 0.1, 0.3, 0.5, 0.7, 1.0, 2.0 and 4.0 m/s, respectively.

It is a remarkable fact that m/s is the default speed unit in PFC, which is not equivalent to the loading speed in real laboratory tests. However, the loading speed of 1.0 m/s can fully simulate the quasi-static loading condition, it takes about 375 calculation steps to produce 1mm shear displacement when the loading speed is 1.0 m/s, which is a relatively slow shear rate.

4 Analysis of the Numerical Simulation Results

4.1 Influence of JRC on Shear Behavior of Rock Joints

For shear stress, the relationship between shear stress and shear displacement of rock joint samples with different JRC under CNS boundary conditions is shown in Fig. 3a. The shear stress of all samples did not appear obvious peaks and rapid decline with the increase of δh, which showed a more stable rising trend and multiple peaks. In addition, the shear stress of the samples showed an overall increasing trend with the increase of JRC. For example, when δh is 2 mm, τ corresponding to JRC of 2–4, 6–8, 10–12, 14–16, and 18–20 is 0.53, 1.02, 1.61, 1.44, and 1.52 MPa, respectively.

Fig. 3
figure 3

Relationship between a shear stress b normal stress c normal displacement and shear displacement with different JRC under CNS boundary conditions

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For normal stress, the relationship between normal stress and shear displacement of rock joint samples with different JRC under CNS boundary conditions is shown in Fig. 3b. The normal stress of the samples can be divided into two stages with the increase of δh: slowly increasing stage (stage 1) and slowly decreasing stage (stage 2). In addition, the increasing rate gradually decreases in stage 1. The occurrence of stage 1 is due to the inhibition of shear expansion of rock joints during the first half of the shear test. The reason for the occurrence of stage 2 is that rock joints are damaged during the shearing process, and some particles are sheared off and separated from the sample body. With the increase of JRC, the normal stress of samples showed a trend of increasing first and then decreasing. That means the normal stress increased gradually when JRC increased from 2–4 to 10–12. Meanwhile, the normal stress decreased gradually when JRC increases from 10–12 to 18–20. It was analyzed that the excessive JRC results in irreversible failure of the sample during shearing.

For normal displacement, the relationship between normal displacement and shear displacement of rock joint samples with different JRC under CNS boundary conditions is shown in Fig. 3c. Since numerical simulation tests in this section are carried out under CNS boundary conditions, there is a linear relationship between δh and σn. In the range of JRC from 2 to 20, samples exhibit shear expansion during the whole shearing process.

For failure mode, the final failure modes of rock joint samples with different JRC under CNS boundary conditions are shown in Fig. 4a–e. The green, blue and red lines in the figure indicate the standard JRC curve, tensile cracks and shear crack, respectively. Obviously, the cracks generated near the joints are mainly tensile cracks, and there are almost no shear cracks. Therefore, the number of tensile cracks is counted, as shown in Fig. 4f. With the increase of shear displacement or JRC, the number of shear cracks in all samples shows a gradual increase. Taking shear displacement is equal to 2 mm as an example, the number of tensile cracks corresponding to JRC of 2–4, 6–8, 10–12, 14–16, and 18–20 is 77, 147, 204, 149, and 415, respectively. The increase in the number of tensile cracks indicates more severe damage to the joints, which in turn generates greater shear stresses as well as normal stresses. What’s more, the damage of the joints is mainly concentrated in the protruding part of the joints.

Fig. 4
figure 4

Effect of JRC on failure models and tensile cracks of rock joints under constant normal stiffness conditions: failure models of a JRC = 2–4 b JRC = 6–8 c JRC = 10–12 d JRC = 14–16 e JRC = 18–20 f effect of JRC on tensile cracks

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JRC of 2–4 (Fig. 5), 10–12 (Fig. 6) and 18–20 (Fig. 7) is taken as examples to analyze the inner stress evolution of rock joints during the shear process. In Figs. 5, 6, 7, subgraph (a)–(f) are the inner stress distribution of the samples at shear displacement of 0.2, 2, 4, 6, 8 and 10 mm, respectively. The red area indicates the compression region and the blue indicates the crack. Subgraph (g) shows the relationship between shear stress and shear displacement, as well as the relationship between the number of tensile cracks and shear displacement.

Fig. 5
figure 5

Evolution process of inner stress in samples for JRC = 2–4: a δh = 0.2 mm b δh = 2 mm c δh = 4 mm d δh = 6 mm e δh = 8 mm f δh = 10 mm g the relationship between shear stress and shear displacement, and the relationship between crack number and shear displacement

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Fig. 6
figure 6

Evolution process of internal stress in samples for JRC = 10–12: a δh = 0.2 mm b δh = 2 mm c δh = 4 mm d δh = 6 mm e δh = 8 mm f δh = 10 mm g the relationship between shear stress and shear displacement, and the relationship between crack number and shear displacement

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Fig. 7
figure 7

Evolution process of internal stress in samples for JRC = 18–20: a δh = 0.2 mm b δh = 2 mm c δh = 4 mm d δh = 6 mm e δh = 8 mm f δh = 10 mm g the relationship between shear stress and shear displacement, and the relationship between crack number and shear displacement

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When JRC is 2–4, the compression area inside samples is more uniformly distributed, and the stress concentration exists only in a few contact points due to the low JRC. The shear stress is also relatively stable in the post-peak stage.

When JRC is 10–12, the stress concentration area inside the specimen is obvious due to the joint surface fluctuating greatly. In the process of shearing, the stress concentration area of the specimen is irreversibly destroyed, and a new contact is formed nearby. In addition, the shear stress of the specimen begins to decrease when the shear displacement is equal to 4 mm. According to the analysis, the reduction of shear stress is caused by the destructive destruction of the joint surface protrusion.

When JRC is 18–20, due to the joint surface is very rough, the joint surface at the beginning of the shear occurred more serious damage, and the number of cracks increased significantly. As a result, the shear stress of the specimen began to decrease when the shear displacement was 0.5 mm, and showed a rapid downward trend throughout the post-peak stage.

4.2 Influence of σ n0 on Shear Behavior of Rock Joints

For shear stress, the relationship between shear stress and shear displacement of rock joint samples with different initial normal stress under CNS boundary conditions is shown in Fig. 8a. The initial normal stress has a more obvious effect on the shear stress of samples in the pre-peak, which affect little in the post-peak stage. At δh is equal to 1.8 mm, the shear stress corresponding to σn0 of 1, 2, 3, 4 and 5 MPa are 1.49, 1.90, 2.29, 2.79 and 3.21 MPa, respectively. At δh is equal to 6.3 mm, the shear stress corresponding to σn0 of 1, 2, 3, 4 and 5 MPa are 0.96, 0.57, 1.48, 0.73 and 0.78 MPa, respectively. The results above all indicate that the increase in σn0 significantly increases the shear stress of samples in the pre-peak stage.

Fig. 8
figure 8

Effect of σn0 on a shear stress, b normal stress and c normal displacement of rock joints under CNS conditions

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For normal stress, the relationship between normal stress and shear displacement of rock joint samples with different initial normal stress under CNS boundary conditions is shown in Fig. 8b. The evolutionary trend of σn with the change of δh can be roughly divided into the following two stages: linear increasing stage (stage 1), σn exhibits a linear increasing trend with the increase of δh when δh is in the range of 0–4 mm; and slow declining stage (stage 2), σn exhibits slow decreasing trend with the increase of δh when δh is in the range of 4–10 mm. The initial normal stress significantly increased σn in stage 1, while the increase of σn0 increases the decreasing rate of σn to a certain extent in stage 2.

For normal displacement, the relationship between normal displacement and shear displacement of rock joint samples with different initial normal stress under CNS boundary conditions is shown in Fig. 8c. The changing trend of the δv with δh is consistent with the changing trend of σn with δh. What should pay attention to is that samples exhibit shear shrinkage in the post-half of the shear test when σn0 is greater than or equal to 4 MPa. This is due to the excessive σn0 causing irreversible damage to rock joints. Some protrusions are sheared off and detached from samples, which results in shear shrinkage.

For failure modes, Fig. 9a–f show the final failure modes of rock joint samples with different initial normal stress under CNS boundary conditions. The blue area inside samples is expanding with the increase of σn0. The blue area is concentrated near the joint’s surface and on both sides of the large protrusions. The above phenomenon indicates that the damage degree is enhancing with the increase of σn0. The trend of the number of tensile cracks inside samples with δh is shown in Fig. 9g. The number of tensile cracks increases alone with shear displacement for any σn0. In addition, σn0 has less effect on the number of tensile cracks inside the specimen compared with the influence of JRC. Taking δh is equal to 2 mm as an example, the number of tensile cracks corresponding to σn0 of 1, 2, 3, 4, and 5 MPa are 558, 697, 810, 1045 and 1075, respectively.

Fig. 9
figure 9

Effect of σn0 on failure models and tensile cracks of rock joints under constant normal stiffness conditions: failure models of a before failure b σn0 = 1 MPa c σn0 = 2 MPa d σn0 = 3 MPa e σn0 = 4 MPa f σn0 = 5 MPa g effect of σn0 on tensile cracks

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4.3 Influence of k n on Shear Behavior of Rock Joints

For shear stress, the relationship between shear stress and shear displacement of rock joint samples with different normal stiffness under CNS boundary conditions is shown in Fig. 10a. Similar to the effect of σn0 on τ, the increase in kn increases the pre-peak shear stress of samples. At δh of 2 mm, τ corresponding to kn of 1, 3, 5, 7, and 9 GPa/m are 1.35, 1.48, 1.54, 1.62 and 1.70 MPa, respectively. The normal stiffness has little effect on the post-peak shear stress of samples. When the shear displacement is equal to 4 mm, the shear stress of the specimen has a relatively obvious downward trend.

Fig. 10
figure 10

Effect of kn on a shear stress, b normal stress and c normal displacement of rock joints under CNS conditions

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For normal stress, the relationship between normal stress and shear displacement of rock joint samples with different normal stiffness under CNS boundary conditions is shown in Fig. 10b. The influence of kn on the normal stress of the specimen is particularly obvious. At δh of 0–5 mm, the normal stress of the specimen increases approximately linearly with the increase of the shear displacement. In addition, the greater the normal stiffness, the greater the normal stress of the specimen, and the greater the rate of the normal stress increase. At δh of 5–10 mm, the normal stress of each specimen is significantly different. When kn is 1–3 GP/m, the normal stress of the specimen shows a slow rise or stays stable. When kn is 5–9 GP/m, the normal stress of the specimen shows a rapid decline trend after reaching the peak value. The shear expansion of the joint is greatly restrained due to the high normal stiffness, which leads to the excessive normal stress and shear stress of the specimen.

For normal displacement, the relationship between normal displacement and shear displacement of rock joint samples with different normal stiffness under CNS boundary conditions is shown in Fig. 10c. The trend of normal displacement with shear displacement of the specimen is consistent with the trend of normal stress with shear displacement of the specimen. The samples exhibit shear expansion during the whole shearing process when kn ranges from 1 to 9 GPa/m.

For failure modes, the final failure modes of rock joint with different normal stiffness under CNS boundary conditions is shown in Fig. 11a–f.

Fig. 11
figure 11

Effect of kn on failure models and tensile cracks of rock joints under constant normal stiffness conditions: failure models of a before failure b kn = 1 GPa/m c kn = 3 GPa/m d kn = 5 GPa/m e kn = 7 GPa/m f kn = 9 GPa/m g Effect of kn on tensile cracks

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The blue area inside samples is expanding with the increase of normal stiffness. The blue area is concentrated near the joint's surface and on both sides of the large protrusions. The above phenomenon indicates that the damage degree is enhancing with the increase of kn. The variation trend of the number of tensile cracks inside each sample with δh is shown in Fig. 11g. The number of tensile cracks inside samples showed an increasing trend with the increase of δh for any normal stiffness, but the increasing rate gradually decreased. In addition, the increase in kn significantly increases the number of tensile cracks inside samples, especially when the normal stiffness is in the range of 1–5 GP/m. Taking the δh of 5 mm as an example, when kn increases from 1 to 3 GP/m, the number of tensile cracks increases from 613 to 769, improving up to 25%. When kn increases from 3 to 5 GP/m, the number of tensile cracks increases from 769 to 1067, improving up to 39%. When kn increases from 5 to 7 GP/m, the number of tensile cracks increases from 1067 to 1154, improving up to 8%. When kn increases from 5 to 7 GP/m, the number of tensile cracks increases from 1067 to 1154, improving up to 5%.

4.4 Influence of v on Shear Behavior of Rock Joints

For shear stress, the relationship between shear stress and shear displacement of rock joint samples with different shear rate under CNS boundary conditions is shown in Fig. 12a. When v is in the range of 0.1–1.0, v has a weak effect on the shear stress of the sample. However, when v increases to 2.0 and above, τ of samples is significantly improved. Taking δh of 10 mm as an example, when v increases from 0.1 to 1.0 m/s, the shear stress of samples fluctuates in the range of 0.29 to 0.86 MPa. When v is 2.0, the shear stress increases to 1.91 MPa. When v increases to 4.0, the shear stress increases to 4.92 MPa. It is noteworthy that when v is increased to 4.0, a significant peak shear stress appears in samples and the peak is significantly larger than the residual value.

Fig. 12
figure 12

Effect of v on a shear stress, b normal stress and c normal displacement of rock joints under CNS conditions

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For normal stress, the relationship between normal stress and shear displacement of rock joint samples with different shear rate under CNS boundary conditions is shown in Fig. 12b. The effect of v on σn of samples is similar to its effect on shear stress. When v is 0.1–0.5 m/s, the curves of σn and δh of samples almost coincide. As v continues to increase, the normal stress of the specimen has been improved on the whole, especially in the pre-peak stage. Taking δh of 10 mm as an example, the normal stress of samples corresponding to v of 0.1–0.5, 0.7, 1.0, 2.0 and 4.0 m/s are 2.48, 3.64, 5.06, 5.72 and 7.58 MPa, respectively.

For normal displacement, the relationship between normal displacement and shear displacement of rock joint samples with different shear rate under CNS boundary conditions is shown in Fig. 12c. With the increase of δh, the δv of samples shows a trend of decreasing after increasing when v is in the range of 0.1–1.0 m/s, and the turning point is δh of 5 mm. When v is in the range of 2.0–4.0 m/s, the normal displacement of samples increases linearly along with δh. Furthermore, each sample exhibits shear expansion throughout the whole shearing process when v is in the range of 0.1–4.0 m/s. For any shear displacement, the increase of v will decrease the normal displacement of samples. That is, the increase of v restrains shear expansion to a certain extent.

For failure modes, the final failure modes of rock joint with different shear rate under CNS boundary conditions is shown in Fig. 13. An increase in v leads to an increase in the damage degree of the sample, especially when v is varied in the range of 1.0 to 4.0. Taking δh of 6 mm as an example, the number of tensile cracks corresponding to v of 0.1, 0.2, 0.5, 0.7, 2.0 and 4.0 m/s are 858, 880, 968, 940, 1212 and 1433, respectively. It can be clearly seen from Fig. 13 that the blue area inside samples expands with the increase of v (especially when v is higher than 1.0 m/s). The statistics of tensile cracks inside samples at different v are shown in Fig. 14. The number of tensile cracks increases along with δh, and the increase rate gradually slows down. When v is in the range of 0.1–0.7 m/s, the relationship between the number of tensile cracks and δh is almost coincident. The number of tensile cracks increases significantly when v is greater than or equal to 1.0 m/s, and v greatly influences the damage degree of the specimen. In addition, the number of tensile cracks further illustrates the damage degree inside the specimen.

Fig. 13
figure 13

Effect of v on failure models and tensile cracks of rock joints under constant normal stiffness conditions: failure models of a before failure b v = 0.1 m/s c v = 0.3 m/s d v = 0.5 m/s e v = 0.7 m/s f v = 1.0 m/s g v = 2.0 m/s h v = 4.0 m/s

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Fig. 14
figure 14

Effect of v on tensile cracks of rock joints under constant normal stiffness conditions

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4.5 Establishment of Mathematical Model

Although scholars have proposed many empirical models before, such as the prediction model of shear mechanical parameters of regular zigzag joints under CNS boundary conditions, the prediction model of single-time shear mechanical parameters, and cyclic shear mechanical parameters of rock natural joints under CNS conditions, etc. However, the currently existing prediction models do not take the combined effects of JRC, initial normal stress, normal stiffness, and shear rate in all aspects into consideration. In this paper, numerical simulation tests are used to study the effects of JRC, initial normal stress, normal stiffness and shear rate on the shear mechanical properties of rock joints. Based on the results of the 110 groups' tests, a multivariate regression algorithm was used to establish two empirical formulas for predicting the shear stress and normal stress of rock joint samples, respectively. According to the characteristics of the CNS shear test (there is a mathematical relationship between normal stress and normal displacement), an empirical formula for predicting the normal displacement of rock joint samples is given. The effects of JRC, initial normal stress, normal stiffness, shear rate and shear displacement were taken into account in the three empirical formulas above.

The “Appendix” (Table 3) includes a list of cases and their corresponding parameters used for fitting the regression functions. The resulting best-fitted expressions are presented below:

$$\tau = 0.1177 JRC + 0.0715 \sigma_{{{\text{n}}0}} + 0.0584 k_{{\text{n}}} + 1.1629 v - 0.1318 \delta_{{\text{h}}}$$
(1)
$$\sigma_{{\text{n}}} = 0.0421 JRC + 0.2931 \sigma_{{{\text{n}}0}} + 0.0930 k_{{\text{n}}} + 1.5147 v - 0.1094 \delta_{{\text{h}}} + 2.0469$$
(2)
Table 3 The values of JRC, σn0, kn, v and δh used in the fitting curves, and experimental results and predicted results of τ, σn and δv
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In CNS shear test, the following relationship exists between σn and δv of samples (Jiang et al. 2004).

$$\sigma_{n} = \sigma_{n0} + k_{n} \times \delta_{v}$$
(3)
$$\delta_{v} = \frac{{\sigma_{n} - \sigma_{n0} }}{{k_{n} }}$$
(4)

Substitute formula 3 into formula 4 to get the expression of δv:

$$\delta_{v} = \frac{1}{{k_{n} }}\left( {0.0421 JRC - 0.7069 \sigma_{n0} + 1.5147 v - 0.1094 \delta_{h} + 2.0469} \right) + 0.0930$$
(5)

where τ, σn, and δv are shear stress, normal stress and normal displacement of rock joint samples, respectively, and δh represents shear displacement. The shear and normal stress prediction models showed that τ and σn were proportional to JRC, initial normal stress, normal stiffness and shear rate. The effects of JRC and v are more significant (higher effect coefficient) for τ, while the effects of σn0 and v are more significant for σn. The prediction model of δv shows that δv of the rock joint sample is closely related to kn.

Formulas 1–3 can accurately predict the changes of τ, σn and δv of rock joints. Among them, the correlation coefficients (R2) of formulas 1 and 2 are 0.814 and 0.719, respectively. The test and fitting results are shown in Fig. 15.

Fig. 15
figure 15

Experimental results and predicted results for shear stress and normal stress of rock joints: a shear stress prediction model b normal stress prediction model

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5 Numerical Calculation Scheme for Rock Single Joint

In order to verify the above three prediction models proposed in Sect. 4.5, the CNS laboratory shear test was carried out in this paper. It is worth noting that there are certain differences and connections between samples used here to verify the reliability of the prediction model and to adjust parameter calibration in the previous numerical simulation.

The similarities lie in the same rock-like materials type and pouring ratios: the dimensions of the specimens are 200 × 100 × 100 mm (length × width × height), and the rock-like materials were made of mixtures of plaster, water and retardant with a weight ratio of 1:0.2:0.005.

The difference is that the joint surface of samples used for parameter correction in the previous numerical simulation adopts the standard JRC (Barton and Choubey (1977) reported), while samples used to verify the reliability of the prediction model adopts the natural joint surface.

The specific test conditions are as follows: One casted joint surface from the field of a hydraulic power plant in Japan was used as the prototype for sample preparation. To measure the joint surface, a laser scanning profilometer system was employed, which demonstrated an accuracy of ± 20 μm in the x- and y-directions and ± 10 μm in the height (z) direction. The JRC value was calculated using formulas (6) and (7).

$$Z_{2} = \left[ {\frac{1}{M}\sum \left( {\frac{{z_{i - 1 - } z_{i} }}{{x_{i - 1 - } x_{i} }}} \right)^{2} } \right]^{1/2}$$
(6)
$$JRC = 32.2 + 32.47\log z_{2}$$
(7)

where xi and zi represent the coordinates of the joint surface profile and M is the number of sampling points along the length of a joint surface. The mean JRC value is 5.62. The applied initial normal stress (σn0) was 2 MPa and the shear rate (v) was 1 mm/min. The maximum shear displacement was 10 mm. A normal stiffness kn = 5 GPa/m was adopted to mimic the stiffness boundary conditions.

Figure 16a shows the variation curves of shear stress, normal stress and normal displacement with shear displacement for samples tested in the laboratory. Shear stress, normal stress and normal displacement of samples were extracted from the curve (in Fig. 16b) when shear displacement was 4, 6, 8, and 10 mm, respectively. The results are compared with the data calculated by the prediction model, as shown in Fig. 16b. What should pay more attention to is that there is a great difference between shear rate applied in laboratory test and PFC simulation software.

Fig. 16
figure 16

Comparison of CNS laboratory shear test and prediction model: a variation law of shear mechanical parameters of samples during CNS laboratory shear test. b Comparison of test results and prediction results under specific shear displacement

Full size image

To ensure that the laboratory test results are comparable to the numerical simulation results as much as possible, it is necessary to assume that the shear behavior of the laboratory shear test and the PFC shear test under quasi-static conditions (1 mm/min in laboratory shear test and 1 m/s in PFC shear test) remains consistent. Hence, when the prediction model was applied for calculations, the real shear rate needs to be replaced by 1m/s. The calculated results of the prediction model are in good overall agreement with the laboratory test results, and the error decreases with the increase of shear displacement. When shear displacement is in the range of 4 to 10 mm, the error of shear stress, normal stress and normal displacement calculated by the prediction model is in the range of 9–40%, 1–27% and 1.7–64%, respectively. In addition, the calculated results of prediction models are generally higher than the results of laboratory tests.

The error mainly comes from two aspects: on the one hand, the standard JRC contour line proposed by Barton and Choubey was used in numerical simulation, while the natural joint surface was used in laboratory tests and JRC was calculated by formulas (6) and (7). In short, there are differences in the calculation principles and calculation methods of the two types of JRC. On the other hand, the shear rate assumptions in laboratory tests and numerical simulations are different. In order to ensure the comparability of results in laboratory tests and numerical simulation tests to the greatest possible extent, it is necessary to assume that the shear behavior of the two tests remains consistent under quasi-static conditions. The above two reasons lead to the overestimation of the shear mechanical parameters of the sample in the calculation results of the prediction model.

6 Conclusion

In this paper, the effects of four variables (JRC, initial normal stress, normal stiffness and shear rate) on the shear properties of rock joint models are investigated based on a self-programmed PFC program. Based on the multivariate regression algorithm and the basic principle of the CNS shear test, three empirical formulas for predicting shear stress, normal stress and normal displacement of rock mass containing a single joint are proposed. Finally, laboratory shear tests of rock joints under CNS boundary conditions were carried out to verify the reliability of the three empirical formulations. The main conclusions are as follows:

  1. 1.

    The increase of JRC significantly improves the shear resistance of rock joints, shear stress, normal stress, normal displacement and the number of tensile cracks is significantly increased. Taking the shear displacement of 2 mm as an example, shear stress corresponding to JRC of 2–4, 10–12, and 18–20 is 0.53, 1.61, and 1.52 MPa, respectively.

  2. 2.

    When shear displacement is less than 4 mm, the increase of initial normal stress enhances the shear mechanical parameters of the joints. Some protrusions of rock joints are sheared or ground and detached from samples as shear displacement continuously increase, which leads to the decrease of shear stress, normal stress and normal displacement, and even shear shrinkage occurs in some samples. This is due to the irreversible damage of rock joints caused by the excessive initial normal stress.

  3. 3.

    The increase in normal stiffness lowers normal displacement of samples while elevating shear stress and normal stress. In addition, with the increase of normal stress, the final failure modes became more and more severe. Since the change of normal stiffness during the test directly affected the change of normal stress, the internal damage of rock joints was more severe when shear expansion was suppressed.

  4. 4.

    When shear rate is in the range of 0.1–1.0 m/s, shear rate has little effect on τ of samples. However, shear stress has been significantly improved when shear rate increases to 2.0 m/s or above. Furthermore, the increase of shear rate will reduce normal displacement of samples, which means the increase of shear rate restrains the shear expansion of samples to a certain extent.

  5. 5.

    Three empirical formulas for predicting shear stress, normal stress and normal displacement of rock joint samples under CNS boundary conditions were proposed and validated in laboratory tests. The calculated results of the prediction model are in overall agreement with the results of laboratory tests, and the error decreases with the increase of shear displacement. When the shear displacement is in the range of 4–10 mm, the error of shear stress, normal stress and normal displacement calculated by the prediction model is in the range of 9–40%, 1–27%, 1.7–64%, respectively. Moreover, the error mainly comes from the “calculation method of JRC” and “assumption of shear rate”.