Shear Behavior and Asperity Damage of 3D Rough Joints under CNS Boundary Conditions Based on CZM Simulation

Abstract

Although boundary conditions can significantly impact the shear behaviors and asperity damage evolution of jointed rocks, numerical studies on the damage of 3D rough rock joints under the constant normal stiffness (CNS) boundary condition have rarely been reported. In this work, the three-dimensional model of the irregular joint surface is established by using point cloud reconstruction technology. Based on the cohesive zone model (CZM), we simulate the shear behavior of three-dimensional rough rock joints under the CNS boundary condition, which is realized by using embedded spring elements implemented with a Python subroutine. We conducted laboratory direct shear tests under CNS boundary conditions. The agreement with the laboratory experimental results verifies the fidelity of the numerical method. Our results show that boundary conditions can significantly affect the shear behavior of rock joints, especially in the post-peak stage. Under the same initial normal stress, the peak shear stress and the number of microcracks in the asperities increase significantly with the increase of normal stiffness. The proportion of shear cracks positively correlates with the normal stiffness, indicating that the normal stiffness affects the joint failure mode. The damaged area and the volume of asperities increase with the increase of normal stiffness. Moreover, the distribution of shear-induced asperity loss becomes more nonuniform, and the loss of joint roughness increases rapidly and nonlinearly.

1. Introduction

Understanding the shear characteristic and behavior of rock joints is essential for analyzing and designing rock geotechnical engineering [1]. Roughness plays an important role in the jointed rocks’ wear [2]. In addition to roughness [3], which affects the shear strength of rock joints, shear behavior is also affected by boundary conditions. Boundary conditions, under which natural rock joints are sheared, are generally divided into the constant normal load (CNL) and constant normal stiffness (CNS) boundary conditions [4,5]. CNL boundary conditions are suitable for analyzing the mechanical properties of unreinforced near-surface rock masses, such as natural rock slopes [6]. For heavily buried constructions or bolt-reinforced rock slopes, it is necessary to consider the constraints of the surrounding rock or the reinforcing system, which makes the CNS boundary condition more suitable [7,8]. Therefore, to better understand the rough joints, it is essential to investigate the shear behavior of joints under CNS boundary conditions.

To date, laboratory test has been the primary method of investigating the shear properties of joints. Compared with the CNS boundary condition, most direct shear tests were carried out under the CNL boundary condition due to the ease of experimentation [9,10,11]. Initially, Obert et al. [12] designed shear loading under the CNS boundary condition by adding springs to the shear box. Subsequently, Jiang et al. [13] used varying normal stress generated by hydraulic servo control instead of springs to apply the CNS boundary condition, which dramatically improved the experiment’s efficiency. In this setup, the effects of roughness and normal stiffness on the shear behavior were studied. According to the existing CNS shear tests, scholars have conducted many laboratory experiments, and significant advances have been achieved. However, defining the damage behavior of asperities is difficult because the joint surfaces were undetectable during shearing. Hence, some researchers have explored the numerical method to study the shear behavior of rock joints. For example, Park et al. [14] used the PFC to research direct shear numerical simulations on rough joints under the CNL boundary condition. They studied the influence of joint topography on shear behavior. So far, the direct shear simulations of rock joints have primarily focused on the CNL boundary condition, and studies under the CNS boundary condition have been rarely reported [15,16]. Using FLAC3D, Nguyen [17] simulated the shear response of 3D rough joints under the CNS boundary condition and found that the contact area and stress state significantly impacted the shear behavior of joints. Although the finite difference method (FDM) can simulate the 3D rock joint shearing, it is difficult to accurately reflect the damage behavior of asperities on the joint surfaces. Therefore, a few scholars have applied the discrete element method (DEM) to explore the evolution of joint surface damage. For instance, Saadat et al. [18] conducted direct shear experiments on rough joints under the CNS boundary condition using PFC2D to study the influence of boundary conditions on the shear behavior of rock joints. Gutiérrez [19] applied PFC2D to perform numerical simulations on saw tooth joints under CNL and CNS to analyze the failure behavior of the rock-concrete interface. However, these studies were only conducted in 2D, and the 3D topology of the joints was not considered. In addition, as the amount of particles increases in the model, the computational efficiency of PFC would be significantly reduced in 3D. Therefore, developing a more computationally efficient and robust 3D DEM to explore the shear behavior of rough joints is important.

Rocks are neither completely continuous nor completely discontinuous, but in between, so a hybrid continuous-discontinuous model can represent rock masses. The cohesive zone model (CZM) is a method that meshes a rock into crystal particles of different shapes and uses cohesive elements to simulate the interaction between crystal particles to characterize various mechanical behaviors, including material breakage and failure. Many scholars have used this method to study rock mechanics and damage mechanics. Wu et al. [20] used viscoelastic CZM to recreate the rock mechanical failure, such as uniaxial compression and Brazilian splitting test. Then they studied the dynamic damage process of mesoscopic concrete models [21]. Recently, Zhang et al. [22] researched the macro-meso shear failure response of crystalline rock joints with standard Barton’s roughness by embedding zero-thickness interface elements. However, such application on the shear failure of 3D rough joints under the CNS boundary condition has not been reported, if any.

In our paper, the three-dimensional model of the irregular joint surface was established by using point cloud reconstruction technology. A numerical simulation study was conducted to investigate the shear and asperity damage of 3D rough joints by using the CZM method. A novel method was developed to realize the CNS boundary condition by Python subroutine for embedded spring elements. The effect of normal stiffness on the shear behavior of 3D rough joints was investigated, and the evolution of asperity damage characteristics was analyzed.

2. Research Methodology

2.1. Constitutive Model

The bilinear cohesive law describes the degradation of the bearing capacity of a material as a separation function. The initial strengthening of the cohesive element is linear elastic behavior, and the elastic constitutive matrix can account for the relationship of nominal stress and strain across the interface. Cohesive elements in modeling are usually structural elements with a thickness of 0. However, the thickness in the constitutive response can be set as 1, which has the advantage that the nominal strain equals the separation amount. The initial linear elastic behavior of the interface is expressed as:

$$t=\left\{\begin{array}{c}{t}_n\\ t_s\\ t_t\end{array}\right\}=\left[\begin{array}{ccc}{E}_{nn}& E_{ns}& E_{nt}\\ E_{ns}& E_{ss}& E_{st}\\ E_{nt}& E_{st}& E_{tt}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_n\\ {\epsilon }_s\\ {\epsilon }_t\end{array}\right\}=\left[\begin{array}{ccc}{k}_{nn}& k_{ns}& k_{nt}\\ k_{ns}& k_{ss}& k_{st}\\ k_{nt}& k_{st}& k_{tt}\end{array}\right]\left\{\begin{array}{c}{\delta }_n\\ {\delta }_s\\ {\delta }_t\end{array}\right\}$$

(1)

where t stands for the traction load vector, which can be divided into one normal direction component ($t_n$) and two shear direction components ($t_s \mathrm{and} t_t$). The nominal strain in the corresponding direction is ε_n, ε_s, and ε_t. The separation spacing in the corresponding direction is δ_n, δ_s, and δ_t.

When the external load reaches the predetermined traction force, the cohesion element will enter the softening stage. Figure 1 shows the constitutive model of traction separation criterion [23].

Where t_no, t_so(t_to), and t_mo are the initial damage traction load in the normal and shear directions, respectively. δ_no, δ_so, δ_to, and δ_mo are the initial damage displacement in the normal and shear directions, respectively. δ_nf, δ_sf, and δ_tf are the final shear displacement in the normal and shear directions, respectively.

Under different working conditions, the crack can be divided into three different failure modes, namely pure tensile failure (mode I), shear failure under tension (mode Ⅱ), and shear failure under compression (mode III). The fracture energy $G_I^c$ of pure tensile failure is related to fracture toughness of rock $K_{Ic}$.

$$G_I^c=\frac{K_{IC}^2}{E}\left(1-v^2\right).$$

(2)

When the cohesive element enters the softening phase. ${\delta }_n^0$, ${\delta }_s^0, \mathrm{and} {\delta }_t^0$ are:

$$\begin{gathered} {\delta }_n^0=t_{no}/k_n \\ {\delta }_s^0=t_{so}/k_n \\ {\delta }_t^0=t_{to}/k_n \end{gathered}$$

(3)

In ABAQUS, stiffness degradation represents the damage evolution of cohesive elements. The minimum damage degree is 0, and the maximum damage degree is 1, which is expressed by scalar D. Effective displacement δ_m is the main variable of damage evolution, and the expression is as follows:

$${\delta }_m=\sqrt{\langle {\delta }_n\rangle {}^2+{\delta }_s^2+{\delta }_t^2},$$

(4)

When ${\delta }_{no}$ represents tension, $\langle {\delta }_n\rangle$ is ${\delta }_n$, and when ${\delta }_{no}$ represents pressure, $\langle {\delta }_n\rangle$ is zero.

The nominal variable D can be derived from the following equation:

$$D=\frac{{\delta }_{mf}\left({\delta }_{mm}-{\delta }_{mo}\right)}{{\delta }_{mm}\left({\delta }_{mf}-{\delta }_{mo}\right)}$$

(5)

where ${\delta }_{mm}$ is the maximum displacement, ${\delta }_{mo}$ is the displacement of the initial crack; ${\delta }_{mf}$ is the displacement of the separation.

Analogously, the normal stress in the softening phase is expressed as:

$$\begin{gathered} t_s=\left\{\begin{array}{c}\left(1-D\right)t_{o,}\\ t_{o,}\end{array}\right. \begin{array}{c}{t}_0\ge 0\\ t_0<0\end{array} \\ {t}_s=\left(1-D\right)t_{so} \\ {t}_t=\left(1-D\right)t_{to} . \end{gathered}$$

(6)

The damage evolution criterion uses the secondary nominal stress criterion, which can be expressed as:

$${\left\{\frac{\langle t_n\rangle }{t_{no}}\right\}}^2+{\left\{\frac{t_s}{t_{so}}\right\}}^2+{\left\{\frac{t_t}{t_{to}}\right\}}^2=1$$

(7)

When $T_n$ is bigger than zero, it means that it is pulled, $\langle t_n\rangle$ is equal $t_n$, otherwise $\langle t_n\rangle$ means that the pressure is equal to zero.

2.2. Cohesive Element Embedding Technology

Su et al. [24] point out that meshing using ABAQUS generates an input file. Because the simulated crack patterns depend on the initial mesh, triangular elements are preferred in the domain of interest so that curved crack paths can be modeled accurately. To simulate the dynamic crushing behavior of the rock under compressional shear, the model is globally embedded with zero-thickness cohesive elements. The particle interaction in the microscopic and fragmentation in macroscopic, such as fracture development and extension, can be modeled by substituting “crystal bonds” between particles with cohesive elements. The process of embedding the cohesive elements is shown in Figure 2. The embedding process is divided into three steps: 1. From the “inp” file, split the adjacent solid elements in the existing finite element model to achieve the node information; 2. The zero-thickness cohesive element is defined by the spatial position of the node; 3. The calculation model is created by writing the node and element information gathered in the previous two phases into an “inp” file and importing it into the ABAQUS user interface [25].

3. Shear Experiment

3.1. Joint Surface Roughness Characterization

Granite rock specimens were collected and cut into standard test blocks of 200 × 100 × 100 (mm). A universal testing machine was used to split the specimens to obtain the rough joint surfaces. The 3D surface structure laser scanner (Revscan 3D, Creaform, Levis, QC, Canada) was used to capture the joint surface profile information and create a point cloud dataset. Revscan 3D has a scanning accuracy of 0.05 mm, which can collect 50,000 location points per second. Figure 3 shows rock joint surface preparation and scanning.

JRC index was employed to quantify the rock joint surface’s roughness. The normal method for determining a JRC was to visually compare the measured profile to Barton and Choubey’s JRC profiles in 1977 [26]. We calculated the Z₂ values using Equation (8), proposed by Tse and Cruden, and used the empirical relationship in Equation (9) to estimate the JRC values [27].

$$Z_2=\sqrt{\frac{1}{\left(n-1\right){\left(\Delta x\right)}^2}\sum {\left(Z_{i+1}-Z_i\right)}^2}$$

(8)

JRC = 32.2 + 32.47lg Z₂

(9)

where Z_i and Z_i+1 are the height of the i-th and i+1-th sur-face points, respectively; Δ is the sampling interval of points in x direction, and n is the no. of points contained in a slice.

To obtain the average, enter the JRC_i value for the i-th joint profile from Equation (9) into Equation (10):

$$\mathrm{JRC}=\frac{1}{m}{\sum }_{i=1}^m{\mathrm{JRC}}_i.$$

(10)

where m is the number of spatial points in one profile; JRC_i is the roughness coefficient of the i-th profile.

The computed JRC in this study is 5.58 based on the average value obtained using 0.5 mm intervals to measure rock joint roughness.

3.2. Experiment Configurations

To avoid the instability of test results caused by natural rock mass, we chose rock-like material for the jointed rock mass specimens, which were formed by mixing the plaster, water, and retardant at a mass ratio of 1:0.2:0.005 [13]. Artificial specimens were created using inverted molds and left to cure at room temperature for roughly ten days. The size of jointed rock is 200 mm × 100 mm × 100 mm (long × wide × high). Standard Brazilian disc specimens and cylindrical specimens were made at the same time to test the material’s mechanical properties, as shown in

Table 1. Mechanical parameters of test specimen.

Mechanical Parameters	Unit	Value
Density (ρ)	kg/m3	2066
Ultimate compressive strength (σc)	MPa	47.4
Ultimate tensile strength (σt)	MPa	2.5
Elastic Modulus (E)	MPa	28,700
Poisson’s ratio (υ)	-	0.23
Cohesion (c)	MPa	5.3
Friction angle (φ)	°	33.3

.

TJW-1000 servo-controlled shearing equipment was used to perform direct shear experiments. Figure 4 shows the experimental system, including the MIS test, control, and data processing systems. The frame of the shear test equipment is cast in one piece. The straight shear box is placed on the base to facilitate pushing out the main frame. The loading system is loaded by a straight-line hydraulic cylinder and the controller is equipped with a high-precision digital controller imported from Germany and corresponding measuring board. Force and displacement measurements are fed back to the controller to form a closed-loop control system for precise control. The ultimate normal load and ultimate shear load of the shearing equipment are both 200 kN. The equipment feeds back the normal dilation information in real-time, calculates the normal stress according to the set constant normal stiffness, and performs servo control through the normal jack. Assuming that, at time t, the normal stress of the specimen is σ_n(t), and at time t + Δt the dilatation deformation of the joint surface is Δδ_v, the normal stress at this time is σ_n(t + Δt) and can be calculated as [28]:

σ_n(t + Δt) = σ_n(t) + k_nΔδ_v,

(11)

where Δσ_n is the normal stress increment; k_n is the normal stiffness; Δδ_n is the normal displacement; and t is the time.

This study mainly investigates the mechanical properties of 3D rough joint surfaces under the CNS boundary condition. Wang et al. [9] pointed out that 1, 3, 5 GPa/m represent most of the working conditions under constant normal stiffness. Four different normal stiffnesses (k_n = 0, 1, 3, 5 Gpa/m) are set in the test. Where k_n = 0 is the CNL boundary condition, the initial normal load σ_n0 is 1 Mpa, shear rate v is 0.5 mm/min, and the max shear displacement is l_max = 10 mm.

4. Modeling Method

4.1. Parameter Calibration

To investigate whether the numerical model can accurately predict tensile and compressive-shear damage of rocks, two different laboratory tests were performed: the Brazilian disc test and the uniaxial compression test. Figure 5 depicts the rock fracture in the Brazilian disc test, and the rock fracture in the cylindrical uniaxial compression test is shown in Figure 6. It can be seen that the numerical simulation and the experiment are highly consistent. It is verified that the proposed method can simulate the mixed fracture mode in the rock, and the physical and mechanical properties of the rock-like are obtained, as shown in

Table 2. Mechanical parameters of numerical rock model.

Cohesive Element Parameters	Value	Macro Parameters	Experiment	Simulation
Initial tensile stiffness	5 GPa/m	Tensile strength	3.5 MPa	3.4 MPa
Initial shear stiffness	2 GPa/m	Compressive strength	40.5 MPa	41.2 MPa
Normal traction	4.5 MPa
Shear traction	18 MPa
Mode-I (tensile) fracture energy	0.1 N/mm
Mode-Ⅱ (shear) fracture energy	0.3 N/mm

.

4.2. CNS Boundary Condition Establishment

“Geomagic” was used for inverse point cloud modeling, and the model was imported into ABAQUS for meshing. The mesh is divided by tetrahedral elements (C3D4), and cohesive elements are globally embedded in solid elements. Figure 7 describes the 3D numerical joint model and CNS boundary conditions. “General contact” is for the interface of the hanging wall and footwall with a coefficient of friction of 0.2. The parameters in Table 2 are used in the numerical model. In the CNS direct shear system, the upper shear box and the top wall are fixed with horizontal displacement, the lower shear box is deformed, and the shear rate is 0.5 mm/min. The linear spring element in ABAQUS is used to realize constant normal stiffness. A subroutine for the batch embedding of spring elements is written in Python to better reflect the loading condition in practice. The number of springs in the numerical model is 693, and the length of the spring is 10 mm. The total number of elements in the numerical model is 365,224: 228,356 cohesive elements (COH3D6) and 136,868 tetrahedral elements (C3D4). The mesh of the joint surface is controlled by the nodes of the point cloud file. In order to simulate the random distribution of particles in rock, the free meshing technique in ABAQUS is adopted for the internal grid. By modifying the spring stiffness value, the normal stiffness can be set. In this simulation, the stiffness (k_n) of the spring is set to 1, 3, and 5 GPa/m. The normal load (1 MPa) is applied to the spring with preload. Figure 8 shows the flow of the simulation of rock joint shearing under CNS conditions.

4.3. Simulation Result Verification

Figure 9 shows the contrast of the direct shear test results with the numerical simulation results under 1 MPa normal stress and 1 GPa/m normal stiffness. The test and numerical simulation results are surface wear and damage, and the position and shape of the wear area are roughly the same. Comparing the shear stress-shear displacement curve and the normal displacement-shear displacement curve, it is evident that they have similar curve shapes during the shearing. The peak shear stress of the specimen in the simulation and test is 1.44 MPa and 1.31 MPa, respectively. The corresponding shear displacements are 0.64 mm and 1.21 mm, respectively. Therefore, the numerical simulation method is reliable for simulating jointed rock mass shear.

5. Result

5.1. Macroscopic Shear Mechanical Behavior under CNS Conditions

Figure 10 shows the relationship between the shear strength and the displacement of the joint surface under different normal stiffnesses. From a macroscopic perspective, normal stiffness can significantly affect shear strength and failure mechanisms. For both CNL and CNS conditions, shear stress shows a rapid linear increase until reaching the peak. After the peak, the shear stress exhibits an apparent stress-softening behavior under low normal stiffness conditions (e.g., 0, 1GPa/m). As the normal stiffness increases, the softening behavior gradually becomes insignificant, and the stress-hardening behavior becomes more obvious. These results are similar to those of Jiang et al. [13].

With an increase in normal stiffness, the peak shear stress shows a rising trend. When the normal stiffness k_n = 1 GPa/m, 3 GPa/m, and 5 GPa/m, the peak shear stress is 1.75 MPa, 2.69 MPa, and 3.06 MPa, respectively. When the normal stiffness is increased, the measured ultimate shear strength rises. Among them, when the normal stiffness is 3 GPa/m and 5 GPa/m, and the residual stage is not obvious, characterized by shear stress increases with an increase in shear displacement. In addition, Li et al. [29] noted that the shear stress decreased after a peak at lower normal stiffness. When the normal stiffness is increased to a moderate level, the post-peak shear stress decreases slightly and then increases steadily. If the normal stiffness is high, the shear stress increases significantly with the shear displacement (Figure 10b). The same mechanical laws are also depicted in the numerical simulation result, which further confirms the reliability of our research method.

Figure 11 shows the relationship between dilatancy and shear displacement under different normal stiffness conditions. As the normal stiffness increases, the normal shear displacement decreases gradually. When σ_n = 1 MPa, normal stiffness k_n = 0 GPa/m (CNL), 1 GPa/m, 3 GPa/m, 5 GPa/m, and the maximum normal shear displacements are 1.50 mm, 1.17 mm, 0.73 mm, and 0.44 mm, respectively. Having a high normal stiffness can significantly decrease the normal deformation between the joined rocks and raise the degree of occlusion with them, enhancing the shear strength of the jointed rocks.

Figure 12 shows the relationship between normal stress and shear displacement under different normal stiffness conditions. The normal stress remained constant in the CNL situation. However, for the CNS conditions, the normal stress variation increases linearly with the normal displacement. When the normal stiffness is 1 GPa/m, 3 GPa/m, and 5 GPa/m, the maximum normal stress is 2.39 MPa, 3.37 MPa, and 4.27 MPa, respectively. It can be inferred that the normal stress is constantly changing in CNS direct shearing, and increasing the normal stiffness can increase the normal stress in shearing, thereby increasing the final shear strength.

5.2. Mesoscopic Crack Evolution under CNS Conditions

Figure 13 describes the shear stress and crack evolution characteristics under a 1 MPa constant normal load. Figure 14 and Figure 15 are the shear stress and crack evolution characteristics under the 1 MPa constant normal load and 1 GPa/m, 3 GPa/m, and 5 GPa/m constant normal stiffness. The figure also shows five key points: the test initial point (A), the shear peak strength point (B), the initial point in the residual stage (C), the development point in the residual stage (D), and the final point (E), respectively.

Figure 13d, Figure 14d, Figure 15d, and Figure 16d are the crack evolutions monitored at key points. From point (A) to (B), sporadic tensile and shear cracks appear on joint asperity under compressive shear, and most of the cohesive elements are in a low-stress state in CNS and CNL direct shear. At this stage, the specimens under different stiffness conditions are all in the linear elastic deformation stage, and the damage degree of the jointed rock is relatively low. Hence, their number of cracks is roughly the same. From point (B) to (C), the number of cracks on the joint surface gradually increases, and the larger asperities are the first to suffer degradation. Both tensile and shear cracks are alternately distributed on the joint surface. At this stage, concentrated shear cracks appear where the asperities are larger. From point (C) to point (E), the pattern of CNS boundary exhibited distinctly different from that of the CNL. In CNL direct shear, the crack grows steadily and spreads based on the existing damage. However, the increase of cracks on the rough joint surface in the CNS direct shear is more rapid. With the increasing normal stiffness, the proportional distribution of shear cracks on the CNS direct shear is significantly higher than that of the CNL direct shear. At this stage, most of the areas on the joint surface are shear cracks, and the distribution of shear cracks is more obvious under the CNS condition.

Figure 13b, Figure 14b, Figure 15b, and Figure 16b show the cumulative number of total cracks, shear cracks, and tensile cracks versus shear displacement. It can be observed that, from point A to point B, the crack growth rate is faster under both CNS and CNL conditions because the asperities on the rock have not been degraded and are prone to large stress concentrations. The crack growth laws under the CNS and CNL conditions are similar before the peak shear strength. Since the total amount of cracks in this stage is small, the CNS boundary condition has a low effect on the normal strength at the initial shearing. When the shear displacement exceeds point B, the number of cracks continues to grow, but the growth rate decreases. The growth rate of cracks under CNS condition is significantly higher than that of CNL, and the number of cracks positively correlates with the normal stiffness. From points C to E, most of the asperities and particles of the joint surface are destroyed and fail, and the final number of cracks in the CNS direct is significantly higher than that of the CNL. When the normal stiffness k_n = 1 GPa/m, 3 GPa/m, 5 GPa/m, the final number of cracks is 35,538, 43,379, 63,978, and 73,365, respectively. Compared with the CNL condition, the increases were 22.06%, 80.00%, and 106.44%, respectively.

Interestingly, the shear stress monitoring points (A, B, C, D, and E above) correspond to the trend of the number of cracks. The peak shear strength point B in Figure 13a, Figure 14a, Figure 15a, and Figure 16a corresponds to the point (B) in Figure 13b, Figure 14b, Figure 15b, and Figure 16b where the crack growth rate is the largest. The residual stage initial point (C) corresponds to point (C) in Figure 13b, Figure 14b, Figure 15b, andFigure 16b, where the crack growth rate is low. However, the correspondence of point C gradually disappears with the increase of normal stiffness. The reason is that the normal stiffness increases the shear strength of the jointed rock in the nonlinear softening stage, and the shear strength curve gradually evolves into a bilinear constitutive. For instance, when the normal stiffness is 5 GPa/m, the shear stress at yield point (B) does not decrease.

Interestingly, the shear stress monitoring points (A, B, C, D, and E above) correspond to the trend of the number of cracks. The peak shear strength point B in Figure 13a, Figure 14a, Figure 15a, and Figure 16a corresponds to the point (B) in Figure 13b, Figure 14b, Figure 15b, and Figure 16b where the crack growth rate is the largest. The residual stage initial point (C) corresponds to point (C) in Figure 13b, Figure 14b, Figure 15b, andFigure 16b, where the crack growth rate is low. However, the correspondence of point C gradually disappears with the increase of normal stiffness. The reason is that the normal stiffness increases the shear strength of the jointed rock in the nonlinear softening stage, and the shear strength curve gradually evolves into a bilinear constitutive. For instance, when the normal stiffness is 5 GPa/m, the shear stress at yield point (B) does not decrease.

Figure 13, Figure 14, Figure 15 and Figure 16c show the frequencies of tensile and shear cracks. Figure 17 shows the frequency evolution law of tensile and shear cracks under different normal stiffnesses. The cracks of CNL and CNS are mainly shear-type cracks. The difference is that the frequency of shear and tensile cracks in CNL direct shears does not change after the peak strength point. However, the frequency of tensile cracks in CNS direct shears decreases continuously, and the frequency of shear cracks increases continuously. With the increase in shear stiffness, the proportion of tensile cracks decreases, and the proportion of shear cracks increases. The growth of shear cracks and the decrease of tensile cracks are linear with the increase in stiffness. It can be concluded that the frequency of shear crack is positively correlated with normal stiffness, and the frequency of tensile crack growth is negatively correlated with normal stiffness. When the normal stiffness k_n = 1 GPa/m, 3 GPa/m, 5 GPa/m, the final shear crack frequencies are 66.67%, 74.47%, and 76.48%, respectively. Compared with CNL, it increased by 7.23%, 15.30%, and 17.04%, respectively. This is because as the normal stiffness increases, the normal stress increases accordingly, making friction and sliding more difficult. The shearing effect of asperities on the joint surface becomes more pronounced, resulting in a continuous increase in the frequency of shear cracks and a corresponding decrease in the frequency of tensile cracks.

5.3. Damage Evolution of Joint Surfaces under CNS Conditions

In the process of jointed rock shearing, the rough joint surface is continuously polished, resulting from the shear load. The fragments are peeled off from the parent rock, all belonging to the joint surface deterioration. The status cloud image of the numerical simulation results is processed into a binarized image, and the corresponding threshold is set to count the damaged area on the joint surface. Then, the damage area frequencies of joint surfaces at different monitoring points (B, C, D, E) under different normal stiffnesses were extracted.

Figure 18 shows the frequency evolution law of the damaged area under different normal stiffnesses. From point (B) to (C), the maximum damage area of joint surfaces with different normal stiffnesses is similar, indicating that normal stiffness has little effect on joint damage at this stage. From point (D) to point (E), the damaged area of the joint surface gradually increases with the normal stiffness, and the change is roughly linear. The damage of the joint surface of the point (D) monitoring point increased by 16.56% compared with that of the CNL condition at k_n = 3 GPa/m, and the damage of the joint surface of the E monitoring point increased by 25.31% compared with that of the CNL condition at k_n = 5 GPa/m. This indicates that as the normal stiffness increases, the degradation of the asperities is greater. Moreover, normal stiffness greatly influences the damage to the joint surface after the peak shear strength and the residual stage.

The roughness loss is obtained by extracting the surface morphology of the jointed rock mass after shearing, as shown in Figure 19. There is almost no roughness loss before point (B), and the wear mainly occurs in the residual shear stage, which is similar to the research conclusion of Li et al. [29]. The attenuation of roughness under CNL condition is more in line with the linear distribution. The roughness loss is more exponentially linear under CNL conditions. The reason is that the damage volume of the joints in the residual stage is more minor. As the degree of normal stiffness rises, the wear loss gradually increases, and the law of roughness loss changes from exponential decay to linear. Current studies have shown that the ultimate degradation of roughness is related to the normal strength (σ_n) and the compressive strength of the rock (σ_UCS) [30,31]. Under CNL conditions, the asperity of the joint surface becomes frictional slip in the residual stage, reducing the roughness loss rate. However, the jointed rock mass’s normal stress increases as the normal stiffness increases, leading to further shearing of the asperities and increasing the final degradation.

5.4. Distribution Regularity of Sheared-Off Asperity Fragments under CNS Conditions

During joint shearing, wear occurs when the contact is uneven: gouge marks and peeled rock fragments of different sizes are formed on the two opposite fracture surfaces. As the shear displacement increases, the sheared-off asperity fragments are further crushed into many smaller fragments, and these phenomena can significantly change the roughness of the joint surface. The volume and height of sheared fragments are essential factors affecting roughness, and many theoretical and empirical models have been proposed to predict the volume of worn material during shearing [32]. This study quantitatively analyzes the volume and height of sheared-off asperity fragments with different normal stiffness from the perspective of numerical simulation, which provides a new perspective for the theory of roughness degradation. The sheared-off asperity fragment information at different monitoring points is output, as shown in

Table 3. The sheared-off asperity fragment distribution under different normal stiffnesses.

Normal Stiffness	Monitoring Points	Maximum Sheared-Off Asperity Fragment Volume/cm3	Total Sheared-Off Asperity Fragment Volume/cm3	Maximum Sheared-Off Asperity Fragment Height/cm	Difference between Sheared-Off Asperity Height and Highest Asperity Height/cm	Sheared-Off Asperity Fragment
1 GPa/m	B	0.08	0.34	0.08	0.77
	C	0.13	1.03	0.13	0.72
	D	2.12	7.21	0.24	0.61
	E	6.08	12.25	0.40	0.45
3 GPa/m	B	0.079	0.96	0.09	0.76
	C	0.18	1.56	0.17	0.68
	D	3.13	8.402	0.35	0.50
	E	9.00	19.82	0.54	0.31
5 GPa/m	B	0.10	1.52	0.09	0.76
	C	0.22	2.05	0.21	0.64
	D	5.83	13.55	0.42	0.43
	E	12.32	28.86	0.69	0.16

. The cloud map in the table indicates that the number of sheared-off asperity fragments increases gradually with the increase of shear displacement, and it also increases significantly with the increase of normal stiffness. Before the residual stage (B, C), the sheared-off asperity fragments are displayed sporadically, indicating that only fragments are produced on the joint surface during the stage, and the volume of the asperities on the joint surface has not significantly changed. When the shearing enters the residual stage (D, E), the fractured area on the joint surface increases significantly and gradually connects into one piece. It shows that the sheared-off asperity fragments at this stage have gradually become larger blocks and peeled off from the parent rock. The asperities on the joint surfaces have been largely damaged, and the roughness of the rock has degraded sharply.

Figure 20 shows the distribution evolution of the sheared-off asperity fragments. The maximum sheared-off asperity fragment volume, total sheared-off asperity fragment volume, and maximum sheared-off asperity fragment height increase as the normal stiffness increases. As for the regular change of the surface damage area, the change is more obvious in the residual stage.

6. Discussion

Over time, engineers and researchers have recognized that the normal stiffness employed for the shear surface affects the shear behavior of the interface [33,34]. The normal strength and shear strength of the joint surface under the CNS boundary condition change with time. In this work, we performed numerical simulations on shearing joints under the CNS boundary condition. The results show that the CNS boundary condition significantly impacts the shear characteristics of the joint, which is displayed in the enhancement effect of the normal stiffness in the residual shear stage, including shear stress, normal stress, shear dilation, damage characteristics, and crack evolution of the joint.

From a mesoscopic point of view, under the CNL boundary condition, crack propagation and damage evolution are more stable after the peak shear stress. A similar trend has been observed in Shang et al. [8]. However, under the CNS boundary condition, the changes in strength and damage of the rock in the residual strength stage are more pronounced. After the shear slip, the interlocking of asperities is more obvious, the damaged area and ruptured volume increase sharply (Figure 18), and the proportion of shear cracks increases significantly (Figure 17). All of these indicate that, under the CNS boundary condition, the shear evolution of rock joints is more similar to that of the intact rock. In previous studies, the shear strength and the degree of damage of the joint in the residual stage were underestimated under the CNS condition.

Figure 21 shows the reduction of roughness at different stiffnesses. Based on the roughness volumetric loss ratio proposed by Zhao et al. [35], we define a new joint damage coefficient as

$$M=\frac{V_{loss}}{V_0}$$

(12)

where M is the joint damage coefficient, which indicates the loss volume relative to the initial volume of asperities. V₀ is the total volume above the lowest point of the joint surface, which can be directly queried in the modeling software, 73.82147 cm³. As seen from Figure 21, as the normal stiffness increases, the surface roughness of the joint decreases significantly, and the roughness loss gradually changes from linear to quadratic polynomial. This indicates that stiffness is critical to joint surface damage. The joint will bear cyclic load under seismic action and excavation disturbance.

Under cyclic load, the joints will slip and close continuously, resulting in the continuous deterioration of the asperities, and its shear behavior will become more complex. It is essential to study the cyclic shear characteristics of joints for accurately evaluating the long-term stability of rock mass engineering. In future research, we will pay more attention to the nature of wear and continue to use numerical simulation methods to study the shear characteristics of the cyclic jointed rock mass.

7. Conclusions

To simulate the shear damage of 3D rough joints under the CNS boundary condition, we developed a CZM-based hybrid finite element method with point cloud reconstruction technology. This method can replicate the shear damage of rock joints and capture the crack evolution on 3D rough joint surfaces. The simulation results are consistent with the experiment results, which suggests the method’s accuracy. In addition, we analyzed and summarized the macroscopic shear behavior and mesoscopic crack evolution of the joint surface under different stiffnesses. The roughness reduction was described according to the volumetric loss of asperities during shearing.

(1)

Boundary condition significantly impacts the shear characteristics of rock joints. Compared with the CNL boundary condition, the CNS condition inhibits the shear dilation of the joint and improves the shear strength. Under the CNL condition, the peak shear stress occurs in the yield stage, while under the CNS condition, the peak shear stress may appear in the residual stage as the normal stiffness increases.

(2)

From the perspective of mesoscopic crack evolution, the number of cracks under the CNS condition is higher than that under the CNL condition. In the CNL condition, the number of cracks in the residual shear stage no longer changes, while under the CNS condition, the number of cracks gradually increases with the increase of stiffness. At the same time, the proportion of shear cracks also increases as the stiffness increases.

(3)

The damaged area, roughness reduction, and volume of ruptured rock fragments are positively correlated with normal stiffness. Before the yield point, there is almost no roughness reduction, and wear mainly occurs in the residual shear stage. The normal strength remains constant under the CNL condition, and the asperities on the joint surfaces undergo frictional slip after shear failure. Therefore, the decay rate of roughness exhibits a linearly decreasing trend. However, with the increase of normal stiffness, the roughness loss rate increases from linear to nonlinear.