#### Abstract

Joint surfaces are widely distributed in natural rock mass, and their shear mechanical properties play an important role in determining the safety and stability of rock mass. Previous studies rarely discussed the contribution degree of different joint protrusions to resisting shear stress. In this study, seven irregular dentate joint profiles were proposed to represent the geometric morphology of natural joints. Joint samples were subjected to direct shear tests under constant normal stress using the particle flow code (PFC). First, the reliability of the research scheme was verified by routine test results. Secondly, based on the microcrack tracking module and the force chain analysis, the local failure modes of the joint sample after and during the shearing process are discussed in detail. The relationship between the shear stress and the number of microcracks was studied. Finally, based on the measuring circle function, the variation law of the mean stress at different joint protrusions during the shearing process was tracked. The maximum stress of each protrusion before the shear stress peak was introduced to quantitatively describe the contribution of local protrusions to shear stress. There was an important link between the size of the protrusion and the stress it can withstand. During the shearing process, the local shearing mechanism of the joint surface is controlled by the distribution of joint protrusions. The research results of this paper can provide a good idea for the follow-up joint surface research.

#### 1. Introduction

In natural rock mass, joint surfaces with different morphologies, scales, and directions widely exist [1–3]. Multifractured rock masses have many typical characteristics, such as discontinuity, anisotropy, and heterogeneity. This is distinctly different from a complete rock block or metal material [4–6].Two important factors for evaluating the overall stability of rock mass are the deformation law and failure characteristics. Through a series of engineering practices, it is verified that the joints in the rock mass play a very important role in the deformation and damage of the surrounding rock of the tunnel and the rock slope [7, 8]. The failure of rock mass usually starts from the joint surface and eventually results in a series of large-scale joint rock mass failures. The joint surface is generally shear failure. Therefore, in practical engineering, an accurate understanding of the shear mechanics mechanism of the joint surface is an important reference for evaluating the safety and stability of rock mass [9–11].

The joint surfaces traditionally used for research are obtained from natural rock masses. In most of the existing schemes, there is only one shearing process for each joint under the same conditions except for repeated shearing tests [12, 13]. After the shearing process, the joint surface will be damaged more or less. Therefore, it is not convenient to systematically analyze the influence of different factors on the shear characteristics of joints, such as normal stress and roughness.

To this end, many researchers use rock-like materials to study the shear mechanical properties of joints, such as mortar and gypsum [14–16]. Through the mold, different geometries, including regular and irregular, can be set on the joint surface. Joint samples with the same roughness can be replicated infinitely. Therefore, it is continuously revealed that the shear mechanical properties of joints are affected by factors such as roughness, shear speed, and filling degree [17–19]. However, it is worth noting that the researchers did not understand the failure mechanism of the joint morphology during the shearing process based on physical experimental studies. Therefore, it is difficult to reveal the local shearing mechanism of different protrusions on the joint surface. In addition, the different protrusions set in the existing literature have the same inclination angle on the dentate joint, which cannot well reflect the complex and rough natural joint.

Recently, in the field of geotechnical engineering, the application of numerical simulation technology has become more and more extensive [20, 21]. Among them, the discrete element method (DEM) derived from the particle and bulk theoretical systems has developed rapidly [22, 23]. In the DEM-based numerical test, the complex mechanical response of the jointed rock mass can be realized through the particle contact logic, and the failure law of the sample during the shearing process can be well presented [24, 25]. To date, scholars had conducted many useful DEM-based studies on the shear mechanical properties of joint samples [24–28].

An important factor affecting the shear characteristics of rock joints is the joint roughness coefficient (JRC) proposed by Barton et al. [29–31]. In order to achieve accurate estimation of joint roughness and even shear strength, many scholars had established functions between JRC and roughness parameters such as the roughness profile index () and the root mean square of the first deviation of profiles (Z_{2}) [32, 33].In the definition of most of the roughness parameters, all morphologies on the joint surface were involved, and the mean method is used. Obviously, this procedure ignored the effect of the protrusion size on overall roughness. The distribution law of the external force borne by the joint protrusion was rarely mentioned in the existing literature, and there was no quantitative result for reference [34–36]. Moreover, the size of joint protrusion in rough joints is difficult to be clearly simulated, which leads to poor correlation of corresponding analysis results [37].

In this paper, seven tooth-shaped irregular rock joints were proposed to reflect the geometry of natural joints. Numerical joint samples were built in DEM using the particle flow code (PFC) using the modified smooth joint model (MSJM). The direct shear tests under constant normal stress of joint samples were successfully carried out on the basis of the servomechanism. The routine experimental results of these seven dentate joints were first discussed. Then, the failure modes of the joint samples after and during the shearing process were discussed in detail. The relationship between the numbers of microcracks and the shear stress was also proposed. Finally, the variation law of the mean stress at different joint protrusions during the shearing process was tracked according to the measuring circle function. The maximum stress index was used to quantify the contribution of local joint protrusions to shear stress. After discussion, the local shearing mechanism of different joints had been well presented, which provided important support for subsequent research.

#### 2. Numerical Direct Shear Test of Joint Samples

##### 2.1. Irregular Dentate Joint Profiles

The geometric morphology of dentate joints with a single inclination angle is very different from that of natural joints. In order to reflect the roughness characteristics of the natural rock joint, seven irregular dentate joint profiles with different combinations are designed, as shown in Figure 1.With the increase of joint numbers from 1 to 7, the maximum inclination angle of joint protrusion increases from 5° to 35°.The number of joint protrusions in joints 1 to 4 is 1 to 4, respectively, and the number of joint protrusions in joints 4 to 7 is 4. Significantly, the inclination angle of the protrusions in the joint profile decreases by 5°. This combination method can better simulate the protrusions of different sizes in natural joints. Compared with the 10 standard roughness joint profiles proposed by Barton et al. [29–31], the research on protrusion size is easy to carry out based on these seven joint models.

##### 2.2. Numerical Test Model

In this paper, the modified smooth joint model (MSJM) proposed by Bahaaddini et al. [26] was adopted to establish the numerical model of the joint sample. Compared with the bond removal method (BRM) [22, 23] and the smooth joint model (SJM) [24–26] used previously, the MSJM does not cause abnormal interlocking between particles [26].

First, two vertically stacked boxes were constructed from 4 walls without friction according to the right-hand rule. The adjacent wall geometry was consistent with the dentate joints, and the positions are coincident. The total height of the two boxes was 60 mm and the total length was 100 mm. Second, particles with uniform distribution were randomly generated within the two boxes. The particle size was between 0.15 mm and 0.24 mm. At a porosity of 0.15, approximately 41883 particles were produced in both boxes. When setting the initial stiffness, previously overlapping particles were scattered and redistributed. The particles reach an approximate equilibrium state without friction. To reduce the interlocking phenomenon in the numerical model, we removed particles with contacts less than 1.

Then, the BPM was packed into the particles in the upper and lower boxes. At this point, the intact rock was simulated in the joint sample. After removing the two tooth walls, a suitably low normal force was applied to the upper surface of the model. After that, there will be new contacts between the intact upper and lower rocks. Finally, a discrete fracture network (DFN) was used to determine the position of the dentate joint profile, which was performed in the SJM [38]. When new contacts are generated at the joint surface during the shear process, these new contacts will be automatically set as the SJM model. Through the above process, seven joint samples with different dentate joint surfaces were established. A typical illustration of the seventh joint sample is shown in Figure 2, which also shows how the external load is applied.

As shown in Figure 2, the upper shear box included 3# wall, 4# wall, and 5# wall, and the lower shear box includes 1# wall, 2# wall, and 6# wall. The lower shear box was constrained, and a constant horizontal speed of 4 mm/s is applied to the upper shear box during the shearing process. The calculation process in the PFC software is controlled by a time-stepping algorithm. The actual time value between two consecutive time steps is very small, around .

That is, the moving speed of the 5# wall is per time step. Such a rate of movement ensured that the shearing process of the joint samples was carried out in a quasistatic equilibrium state. The shearing direction is left to right by default. The horizontal force of the 2# and 6# walls can calculate the shear stress. The shear displacement is the 6# wall with the horizontal displacement.

##### 2.3. Calibration Process of Microscopic Parameters

In the PFC software, the macroscopic mechanical behavior of the simulated material is obtained through the interaction of components such as particles and bonds. Microscopic parameters of particles and bonds must be calibrated through trial and error of typical mechanical testing. On this basis, the conclusion that the numerical test simulation effect is reasonable and reliable can be drawn.

###### 2.3.1. Microscopic Parameter Calibration of the BPM

It is proposed that the microscopic parameters of BPM applied to intact rock be calibrated by uniaxial compression test. A numerical sample with a length of 50 mm and a height of 100 mm was generated for obtaining mechanical parameters. The particle distribution characteristics and contact parameter settings are in full compliance with Section 2.2.

Compression tests were carried out without confining pressure. The tests showed that the uniaxial compressive strength of the intact rock was 59.09 MPa and the deformation modulus was 8.83 GPa. The mechanical indexes and failure modes of the numerical samples were consistent with the physical test results of real rocks, as shown in Figure 3. The calibrated BPM microscopic parameters are shown in Table 1.

###### 2.3.2. Microscopic Parameter Calibration of the SJM

The microscopic parameters of the SJM applied to the rock joints were calibrated by direct shear tests. The construction steps of the joint sample are the same as those in Section 2.2, and the position of the dentate joint surface is changed to the flat joint surface. In a direct shear test, a servomechanism is used to apply a constant normal stress to the top surface of the joint sample. The shear stress-shear displacement curves under the normal stress of 2 MPa, 4 MPa, 6 MPa, and 8 MPa are shown in Figure 4(a). It was confirmed that the shear stress curve showed the characteristics of slip failure, which was consistent with the indoor physical test.

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As shown in Figure 4(b), the shear strength results from the numerical test and the physical test were very close. The friction angle of the joint surface was around 40.22°. The calibrated microscopic parameters of SJM are shown in Table 2.

##### 2.4. Servomechanism

In this study, we wish to discuss the mechanical properties of joint samples under constant normal stress. In DEM, constant load cannot be achieved by direct application, only by servofunction to maintain a constant load within a certain range. This function automatically changes the normal moving speed of the upper wall between successive time steps.

As shown in Figure 2, the normal velocity of the 4# wall could be set as follows:

Among them, represented the servoparameter, represented the normal stress actually applied to the 4# wall, represented the expected normal stress, and *Δσ* represented the difference between and . The maximum value of *Δσ* was defined like this:

In Equation (2), represented the mean stiffness of the particles in contact with 4# wall, represented the particle numbers, and represented the area of the wall. To minimize the difference between and , a release factor was set as shown in Equation (6).

When substituting Equations (1) and (2) into Equation (3), Equation (4) could be obtained.

Thus, the servocoefficient could be defined as the following equation:

When running a calculation step, the servoparameter of the next calculation step would be calculated by the servofunction in advance. The normal velocity of 4# wall will be updated by Equation (1). In the whole shear process, the servofunction will work all the time.

In order to verify the reliability of the above-mentioned servomechanism, the constant load direct shear test was carried out on the plane joint, and the normal stress was 1 MPa, 2 MPa, and 3 MPa. The normal stress-shear displacement curves are shown in Figure 5, which confirms that is very close to and is very stable during the application of normal stress. In the follow-up study, the actual normal stress applied to the joint sample was 2 MPa, 4 MPa, 6 MPa, and 8 MPa, respectively.

#### 3. Routine Test Results

##### 3.1. Shear Stress-Shear Displacement Curves

Based on the numerical direct shear test, the shear stress-shear displacement curves are shown in Figure 6. Among them, Figure 6(a) is the effect of roughness under 4 MPa normal stress, and Figure 6(b) is the effect of normal stress based on joint 7. The shear stress curve can be divided into the prepeak stage and the postpeak stage according to the peak value of the shear stress.

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In the prepeak stage, the shear stress started to increase rapidly and was less affected by the joint roughness. As the shear displacement increases, the growth rate of the shear stress gradually decreased until the shear stress reaches a peak value. Due to the increase in the roughness of the joint and the normal stress, the growth rate of the shear stress decreased slowly. In the stage after the peak, the shear stress gradually decreased and enters the residual stage.

With the increase of joint roughness and normal stress, the reduction of the shear stress was larger, especially in joint 7 and normal stress of 8 MPa. According to the law of shear stress, the failure type of the joint changed from sliding failure to shear failure with the increase of roughness and normal stress.

##### 3.2. Shear Strength

Table 3 summarized the shear strength values of different joint samples under different normal stresses. It can be seen that with the increase of joint roughness and the effect of normal stress, the shear strength of the joint sample showed an increasing trend.

From the data in Table 3, Figure 7 shows the variation curve of shear strength with the maximum inclination angle (Figure 7(a)) and normal stress ((Figure 7(b)). It can be seen that with the increase of the inclination angle and the normal stress, the shear strength showed an approximate linear upward trend, which was basically consistent with the results of the previous [14–19]. This showed that the numerical shear test done in this paper can well reflect the influence of joint roughness and normal stress on shear strength. In addition, the seven dentate joint profiles set in Section 2.1 are able to distinguish the rough morphology of the joint surface well.

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##### 3.3. Normal Displacement

Figure 8 showed normal displacement-shear displacement curves of joint samples. Among them, Figure 8(a) showed the influence of roughness at normal stress of 4 MPa, Figure 8(b) showed the influence of normal stresses based on joint 3. At the beginning of the shearing process, the phenomenon of shearing and shrinkage occurred. In most schemes, with the increase of shear displacement, the normal displacement of the joint sample increased continuously, and the phenomenon of shear and dilatation was obvious.

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As the joint roughness increased, the mean growth rate of the normal displacement was faster. Under 4 MPa normal stress, the final normal displacement of joints 1 to 7 after the shear process was -0.08 mm, 0.08 mm, 0.31 mm, 0.44 mm, 0.66 mm, 0.82 mm, and 0.87 mm, respectively. The increase of joint roughness made the final normal displacement increase accordingly. This showed that the climbing effect of rough joints was more obvious. With the increase of the normal stress, the mean growth rate of the normal displacement was slower.

For joint 3, the final normal displacement was 0.54, 0.31 mm, 0.24 mm, and 0.18 mm, respectively, with the normal stresses increased from 2 MPa to 8 MPa. The increase of normal stress made the final normal displacement decrease accordingly. This showed that the increase of the normal stress leads to the intensification of compression in the joint surface, which further inhibited the growth of normal displacement.

#### 4. Failure Mode

##### 4.1. Failure Scale of Joint Samples after Shearing Process

In the numerical joint sample established by DEM, when the local stress between particles is greater than the bond strength listed in Table 1, the bond is damaged and microcracks appear. During the shearing process, the microcracks generated in the joint sample can be tracked by the microcrack monitoring module [38].

Figure 9 showed the failure diagram of different joint samples after the shearing process, and the normal stress was 2 MPa. The blue part in the failure diagram was the upper sample, the green part was the lower sample, the red part represents the microcrack, and the white part of the joint surface was the gap.

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For different joint samples, the microcracks are mainly concentrated in the joint parts near the joint surface. For joints 1 to 5, the microcracks are the most at the larger protrusions. For the joints 6 and 7, the microcracks are mainly concentrated at the protrusions 3 and 4. Generally, when there are different protrusions on the joint surface, the failure generally occurs at the larger protrusions.

Figure 10 showed the variation curves of microcrack numbers with the maximum inclination. Under the action of 4 normal stresses, the numbers of microcracks in joint samples increased 938, 1164, 1627, and 1804 with the maximum inclination increased from 5° to 35°. With the increase of the joint roughness, the failure scale of the joint sample showed a nonlinear increase as a whole.

Figure 11 showed the failure diagram of the 7th joint sample after the shearing process under 4 normal stresses to describe the effect of normal stress. With the normal stress increasing from 2 MPa to 8 MPa, the microcracks in the joint sample increased from 964 to 1909. Correspondingly, the numbers of large damaged protrusions increased from 2 to 4.

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Figure 12 showed the variation curves of the numbers of microcracks under the action of normal stress. Under the normal stress of 2 MPa, 4 MPa, 6 MPa, and 8 MPa, the numbers of microcracks in the seven joint samples increased by 79, 350, 641, 894, 837, 789, and 945, respectively. The failure scale of the joint sample also increases with the increase of the normal stress. This was because the increase of normal stress intensifies the compression between joint surfaces.

##### 4.2. Failure and Stress Concentration of Joint Samples during Shear Process

Figure 13 was a description of the local failure mechanism of the joint sample during the shearing process. It is a comparison diagram of the microcrack expansion and the contact force distribution of the joint sample under different shear displacements. The seventh joint sample under 6 MPa normal stress is selected for description, and the shear displacements are 0 mm, 0.5 mm, 0.75 mm, 1 mm, 2 mm, and 3 mm. In the contact force distribution diagram, black represented the contact force chain. The force chain is darker, indicating that the contact force is concentrated here.

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Before shearing (Figure 13(a)), the sample had no microcracks, and the contact force was evenly distributed. When the shear displacement reached 0.5 mm (Figure 13(b)), there were a few microcracks near the joint surface and the contact force was concentrated at the protrusions. When the shear displacement reached 0.75 mm (Figure 13(c)), the microcracks at protrusions 3 and 4 had increased and gradually extended inward. The concentrated effect of the contact force of the joint sample is more obvious.

When the shear displacement reached 1 mm (Figure 13(d)), there were more microcracks on the 3th and 4th protrusions, especially the 4th protrusion. In addition, the increase of the shear displacement caused the dislocation of the upper and lower joint surfaces. The concentration range of the contact force is reduced, and the fourth protrusion was the main one.

When the shear displacement reached 2 mm (Figure 13(e)), the place where the contact force was mainly concentrated had been transferred from the 4th protrusion to the 2nd protrusion. Accordingly, some new microcracks appeared in the 2nd protrusion, and the damage degree of the 3rd and 4th protrusions was also intensified. When the shear displacement reached 3 mm (Figure 13(f)), the main contact force was transferred from the second protrusion to the first protrusion. The new microcracks in the joint sample mainly occurred at the 1st and 2nd protrusions.

Generally, the expansion of microcracks is closely related to the concentration of contact force, and they all occur first at the largest protrusion. According to our analysis, the largest contribution in terms of shear resistance is the largest protrusion (4th protrusion) of the joint surface. Not only is the failure time early, but the failure scale is also large. In contrast, the other relatively small protrusions (the 1st, 2nd, and 3rd protrusions) contributed correspondingly less in resisting shear forces. They failed even later, with only a tiny fraction of the damage. During the shearing process, the joint surface behaved as a progressive failure mechanism. The size of the protrusions was critical to the effect of shear stress.

##### 4.3. Relationship between Microcrack Numbers and Shear Stress

During the shearing process, only sufficient shearing force can microcrack the joint protrusion. Therefore, there must be an obvious relationship between the development of shear stress and the expansion of microcracks. Figure 14 showed the variation curves of shear stress and microcrack numbers with shear displacement under 8 MPa normal stress. Among them, Figures 14(a)–14(g) show the 1st to 7th joint samples, respectively.

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At the beginning of shearing, the shear stress of the joint sample increased the fastest, but no microcracks occurred. As the shear displacement increased, some microcracks began to appear in the sample. Then, the microcrack number variation curve entered the high-speed expansion stage, and the microcrack numbers increased at the fastest speed in the sample at this time. At the same time, before the shear stress reaches its peak value, the shear stress growth rate decreased significantly. Subsequently, the shear stress curve continued to decrease, and the number of microcracks continued to extend downward at a high speed. Finally, the shear stress curve entered the residual stage, and the variation curve of the microcrack numbers also transferred to the low-speed expansion stage.

For rough joints such as the 4th, 5th, 6th, and 7th joints, there were obvious inflection points on the variation curve of microcrack numbers and shear stress. For smooth joints such as the 1st, 2nd, and 3rd joints, the inflection point was not obvious. When the shear stress rose rapidly, the microcrack development of the sample was slower. When the growth rate of the shear stress decreased rapidly, the number of microcracks grew quickly. In general, the shear stress will decrease after the sample fails, because the growth of the shear stress will be inhibited by the expansion of the microcrack in the sample. The hysteresis effect of the damaged sample on the change of shear stress is obvious.

#### 5. Distribution Law of Mean Stress in Joint Protrusions

In Section 4.2, we found the progressive failure mechanism of the joint surface in the shear process. However, there were no quantitative data to evaluate the contribution of different joint protrusions to resist external shear stress. Using the measuring circle function in PFC, we had monitored the variation law of the mean stress on joint protrusion during the shear process.

##### 5.1. Layout Scheme of Measuring Circles

During shear test, the horizontal and normal move of the upper rock block will occur. But the measuring circle cannot easily move after being arranged. This makes it difficult for the measuring circle to monitor the data in a certain area. Therefore, the layout of the measuring circle in the joint sample was based on the fixed lower rock block, as shown in Figure 15 (taking the 4th joint as an example). During the shear process, only the left surface of the joint protrusion makes contact, and the right surface will be void. Therefore, the arrangement of the measuring circle only needs to cover the left contact surface. A measuring circle was set on each protrusion. The center of the measuring circle was located on the joint profile, and the diameter was the same as the length of the hypotenuse of protrusion.

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##### 5.2. Mean Stress Variation Curve of Different Joint Protrusions

Through the measurement of the circular function, the porosity, stress, strain rate, particle size distribution, coordination numbers, etc. of a specific area of the sample can be monitored. This section selected the stress results for subsequent analysis. Stress, as a common variable in the continuum model, can be approximated by the averaging method in discrete media [38]. The specific calculation formula is

where represents the volume of the measuring circle, which is the area in the two-dimensional model; represents the numbers of active contacts; represents the vector of contact force; represents the branch vector connecting the centroids of two particles; and represents the vector product [38]. The mean stress monitored by the measuring circle function is mainly divided into horizontal stress (*stress*), tangential stress (stress or stress ), and vertical stress (stress ). The 1st and 3rd indexes were selected for representative description. The mean stress property monitored by the measuring circle is compression in all scenarios.

Figure 16 showed the variation curves of horizontal and vertical stresses for different joint protrusions. Results are listed for the 7th joint samples with a normal stress of 2 MPa. The left picture shows a horizontal force, and the right picture shows a vertical force.

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It could be seen that the horizontal stress of each joint protrusion increased from 0 MPa at the beginning of the shear process. In contrast, the vertical stress value increased by approximately 2 MPa, which was relatively close to the normal stress applied on the sample.

For joint 1, the horizontal stress and vertical stress at the protrusion increase continuously with the increase of shear displacement. This is because the pressure surface between the joint surfaces becomes smaller, resulting in the concentration of stress. For joint 2, the horizontal stress and vertical stress at protrusion 2 both showed an increasing trend, while the horizontal stress and vertical stress at protrusion 1 increased first and then decreased.

Conversely, during shearing, the stress on protrusion 2 was always greater than that on protrusion 1. For joint 3, during the shearing process, the horizontal stress and vertical stress on the three protrusions both increased first and then decreased, but the turning points were different. The force on protrusion 3 was the largest, followed by protrusion 2 and protrusion 1, in the same order as the inclination angle.

For joints 4 and 5, the magnitude of the stress and the variation range of the joint protrusion were still closely related to its own morphology. The stress borne by protrusion 4 was the largest, followed by protrusion 3, protrusion 2, and protrusion 1.

For joints 6 and 7, the larger protrusions (3 and 4) experienced more stress during shear than the smaller protrusions (1 and 2). It is worth noting that the horizontal and vertical stress curves corresponding to protrusions 3 and 4 were crossed. When intersecting back and forth, the stress of protrusion 4 was continuously decreasing, while the stress of protrusion 3 was continuously increasing. This showed that the stress-bearing capacity of protrusion 4 decreased after failure. Then, the stress concentration position was transferred to protrusion 3, which greatly increased the stress at protrusion 3. This phenomenon is basically consistent with the results of the force chain analysis in Section 4.2. Through the above analysis, it can be verified that the force distribution of each joint protrusion had a strong correlation with its surface morphology.

The joint surfaces at the small protrusions gradually separated with increasing shear displacement. The vertical stress on the small protrusions gradually decreased around 0 MPa, but the horizontal stress still existed. Due to the reduction of the contact surface, the vertical stress on the large protrusion was significantly larger than the external normal stress on the sample. Comparing the stress values in different directions, the stress borne by joint protrusions was gradually dominated by horizontal stress and supplemented by vertical stress.

##### 5.3. Distribution Law of Maximum Stress in Different Joint Protrusions

As an important index of joint surface, shear strength is widely used in rock mass engineering. Therefore, it is more meaningful to discuss the force magnitude of different protrusions before the shear stress peak. Then, the maximum stress of each protrusion in the prepeak stage of shear stress was obtained. Figure 17 showed the maximum stress distribution of joint protrusions under different normal stresses. The results of 7 joint samples were listed. The left graph shows the maximum horizontal stress and the right graph shows the maximum vertical stress.

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For each joint surface, the maximum horizontal stress of different protrusions was very similar to the distribution law of maximum vertical stress. With the increase of the normal stress, both the maximum horizontal stress and the maximum vertical stress at the joint protrusions increased. In addition, it was verified again that the horizontal stress on the joint protrusion was significantly larger than the vertical stress. As the joint roughness increased, the difference between them became larger.

The size of the protrusion had a nonnegligible influence on the maximum value of the horizontal stress and vertical stress that it can bear. As the inclination angle of protrusion increased, the maximum values of the horizontal and vertical stresses at the prepeak stage of the shearing process will also rise. This law was not affected by the amount of joint protrusion. It was verified that the contribution of different protrusions against the external shear stress on the joint surface was significantly different. This provided a good idea to study for the subsequent joint surface roughness evaluation.

##### 5.4. Influence of Inclination Angle on the Maximum Horizontal Stress

To further discuss the effect of protrusion size on resistance to external shear stress, Figure 18 shows the relationship between the inclination angle of protrusion and the maximum horizontal stress. The results of 7 joints under the same normal stress are listed. From the fitting results, it could be seen that with the increase of the inclination angle, the maximum horizontal stress in protrusion showed an increasing trend as a whole.

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It is particularly noteworthy that the horizontal stress borne by the protrusion with an inclination angle of 5° on joint 1 was significantly greater than that on other joints (joints 2, 3, and 4). Similarly, the horizontal stress borne by the protrusion with an inclination angle of 10° on joint 2 was significantly greater than that on other joints (joints 3, 4, and 5). This phenomenon also exists for protrusions with an inclination angle of 15° and 20°. With the increase of the normal stress, the above difference was less obvious. It could be seen that the ability of the protrusion to bear horizontal stress was not only related to its size but also related to the size of other protrusions. This was caused by the local shear failure mechanism of the joint surface shown in Section 4.2. In addition, this was also the reason why the fitting function correlation between joint inclination angle and the maximum horizontal stress in Figure 18 is not very high. Therefore, the size of local protrusion should be considered in the evaluation work of joint roughness.

#### 6. Conclusion

In order to reveal the local shearing mechanism of rock joints during the shearing process, numerical direct shear tests were carried out on the joint samples using the PFC of DEM under normal stresses of 2 MPa, 4 MPa, 6 MPa, and 8 MPa. The conventional test results, the overall failure scale, the progressive local failure mechanism, and the mean stress distribution are discussed in detail. Here are the main conclusions:

Seven irregular dentate joint profiles were proposed to reflect the geometry morphology of natural rock joints. The MSJM was introduced to simulate joint surfaces. The microscopic parameters of intact rock and joint surfaces were calibrated by comparing numerical tests and laboratory physical tests. The servomechanism was set to keep the normal stress constant. These preparations could ensure that the research results of this paper are reliable.

With the increase of maximum inclination and normal stress, the failure type of the joint surface changed from sliding failure to shear failure and the shear strength showed a near linear growth phenomenon. The normal displacement increased with the increase of the joint inclination angle and decreased with the increase of normal stress. This law was consistent with physical experiments.

After the shearing process, the failure of the joint sample was concentrated on the joint surface, especially in the larger protrusions. With the increase of the maximum inclination and the normal stress, the failure scale of the joint samples showed a nonlinear increasing trend as a whole. In addition, the number of failure protrusions had also increased. During the shearing process, the sample failure and contact force mainly started and concentrated at the largest protrusion and gradually turned to other protrusions in the later stage of the shear process. Microcrack propagation could significantly inhibit the growth of the shear stress. The change of microcrack numbers lag behind that of the shear stress as a whole. The change of shear stress had a good corresponding relationship with the failure speed of the sample.

Using the measuring circle function, the variation law of mean stress in different joint protrusions during the shear process was tracked. For all joints, the stress borne by the largest protrusion in the shear process, especially the horizontal stress, was significantly greater than that of other protrusions. In addition, it had now been demonstrated that protrusions, which were mainly affected by external stress, might be altered during shearing. The maximum stress was chosen to describe the contribution of the local joint protrusions against the shear stress. The external stress borne by the local protrusions had a good corresponding relationship with its protrusion size, which well verifies the difference in the contribution of joint protrusions to the overall roughness.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

#### Authors’ Contributions

Jiuyang Huan designed the study and wrote the main manuscript text. Jiuyang Huan and Mingming He conducted most of the numerical direct shear tests of the joint samples. Zhiqiang Zhang prepared the figures and tables. Mengdie Hu checked the language. All authors reviewed the manuscript.

#### Acknowledgments

This study was funded by the National Natural Science Foundation of China (Grant No. 11872301), Key Research and Development Projects of Shaanxi Province (No. 2022SF-412), and Education Bureau of Shaanxi Province in China (Grant No. 20JS093). We are thankful for the numerical and physical experimental research conditions provided by the State Key Laboratory of Eco-Hydraulics in Northwest Arid Region in Xi’an University of Technology.