# Effect of Rheological Mesoparameters on Shear Mechanical Behavior of Joints

Authors: Kefeng Zhou; Sheng Liu; Yanhui Cheng
Journal: Geofluids
 (2022)

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## Abstract

Under natural conditions, joint failure usually occurs after long-term loading rather than short-term loading. In order to prevent rock mass failure caused by creep behavior, bolt is widely used as a mature and effective reinforcement method. Therefore, it is necessary to consider the rheological effect of bolted joint. In this paper, the particle flow code (PFC) was used to study the effect of mesoparameters on the rheological shear mechanical behavior of joints without bolts and with bolts. The effects of mesoparameters such as Maxwell elastic coefficientEm, Maxwell viscosity coefficient Cm, Kelvin elastic coefficient Em, Kelvin viscosity coefficient Cm, and friction coefficient Fs were analyzed. The results show that Em and Fs mainly affect the instantaneous shear displacement, but have little effect on the rheological shear displacement. However, Cm, Ek, and Ck have little effect on the instantaneous shear displacement, and the sensitivity is mainly reflected in the rheological shear displacement. Em and Fs mainly affect the instantaneous shear displacement, but have little effect on the rheological shear displacement. However, Cm, Ek, and Ck have little effect on the instantaneous shear displacement, and the sensitivity is mainly reflected in the rheological shear displacement. Compared with the parameter sensitivity characteristics without bolt, the instantaneous shear displacement, shear stiffness, and steady rheological displacement under anchor condition are less affected by parameters, indicating that anchor can reduce the influence of joint parameters.


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## Body

## 1. Introduction

The mechanical properties of joints are important factors affecting the stability of rock mass engineering [1–3]. Under natural conditions, joint failure usually occurs after long-term loading rather than short-term loading. Under the condition of constant external stress, the displacement and stress field of joint are constantly adjusted and reorganized with time, resulting in the creep phenomenon that the strain increases continuously with time, which leads to the failure of jointed rock mass [4–6]. In order to prevent rock mass failure caused by creep behavior, bolt is widely used as a mature and effective reinforcement method [7, 8]. The mechanical properties of bolted joints can be studied by experiment, theoretical analysis, and numerical simulation [9–13]. At present, numerical calculation methods have been widely used in the field of rock mechanics, forming a variety of numerical calculation methods, including finite element method, discrete element method, motion element method, and meshless method [14–16]. Each method has its own advantages and disadvantages, which can solve geotechnical engineering-related problems. In the process of joint shear, due to the fluctuation of rock rupture, the cracks will occur in the rock mass and lead to stress and strain change [17–19]. However, the numerical calculation method based on continuum cannot easily simulate crack development, joint shear slip of continuous deformation of the line. Therefore, it is necessary to consider the material discontinuity numerical calculation method, such as particle flow code (PFC) [20, 21]. In order to study the mesomechanical characteristics of bolted joints, some scholars adopted PFC to simulate the macroscopic mechanical response of bolted joints through the interaction between mesoparticles. Shang et al. [22] established a numerical calculation model of discontinuous joints, studied the characteristics of joint mesoscopic parameters, and simulated the failure mode and strength characteristics of joints under equal normal stress and equal normal stiffness. The previous work mainly studied the shear mechanical properties of joints under conventional conditions, and the rheological effects of joint shear were seldom considered. Therefore, this paper adopts discrete element numerical calculation method to conduct rheological shear simulation tests on both anchored and unbolted joints and adopts control variable method in parameter sensitivity analysis. The effects of rheological mesoscopic parameters such as Maxwell body elastic coefficient Em, Maxwell body viscosity coefficient Cm, Kelvin body elastic coefficient Em, Kelvin body viscosity coefficient Cm, and friction coefficient Fs on shear mechanical behavior of joints were investigated.


## 2. Numerical Model

The numerical calculation model of joint was established using PFC, and the size of the model was100mm×100mm. Based on the established intact rock mass geometry model, the corresponding joint geometry is drawn in CAD. The geometry is imported into PFC by the geometry import command, and it is set to DFN fracture to establish the rock mass geometry model with joints. In the model, different particle parameters are set to simulate the bolt, as shown in Figure 1. In this paper, a flat-joint model is used to simulate the upper and lower rock blocks. Due to shear box in the rheological shear test parts in addition to a constant load, loading wall outside wall is consistent with the only wall loading rate of unconstrained free wall.Figure 1
Numerical model of bolted joint subjected to rheological direct shear.In the study of shear mechanical behavior, the experimental phenomenon at the moment of peak shear displacement is an important research object, and it is necessary to keep the simulated peak shear displacement consistent with the real test as much as possible [23–25]. Therefore, in this paper, the results of numerical simulation when normal stress is 1 MPa and laboratory test are the same as the standard, and the trial-and-error method is adopted to calibrate the mesoscopic parameters of rock. Firstly, a complete sample numerical model of flat joints is established to simulate the shear stress-shear displacement curve of flat joints. Finally, the mesoscopic parameters matching the actual situation are calibrated. The microscopic parameters of the numerical model are shown in Table 1. According to the bolt parameters in reference [26], a parallel bond model is used to describe the bolt. The friction coefficient is 0.5, which is smaller than that of rock. The tensile strength of bolt particles is 4.0 GPa, the bond force is 0.9 GPa, and the internal friction angle is 20°. Set a 5 cm long bolt with a diameter of 2 mm, as shown in Table 2 for specific parameters.Table 1
Mesoscopic parameters of the numerical model.
CategoryMicroparameter (unit)ValueParticleDensity (kg/m3)2500Radius (mm)0.48-0.9f5.0Krat (kn/ks)0.5Porosity0.16Emod (109)0.425Flat-joint contact modelfj_ten (106)4.10fj_coh (106)20.8fj_fa (°)58.47fj_n4Burger contact modelbur_knm (109)0.06bur_ksm (109)0.06bur_cnm (109)12.4bur_csm (109)12.4bur_knk (109)1.6bur_ksk (109)1.6bur_cnk (109)13.1bur_csk (109)13.1bur_fs0.6Table 2
Mesoscopic parameters of bolt.
CategoryMesoscopic parametersValueParticleDensity (kg/m3)7850Radius (mm)0.18-0.3f0.5PorosityP0.16Normal stiffness kn8e9Shear stiffness ks2.286e9Flat-joint contact modelTensile strength pb_ten (GPa)4.25Cohesion pb_coh (MPa)900Friction angle pb_fa (°)0Normal stiffness pb_kn5.9e14Shear stiffness pb_ks5.9e14


## 3. Calculation Results and Analysis

### 3.1. Relationship between Maxwell Elastic Coefficient and Macroscopic Shear Stiffness

In the rheological contact model, Maxwell volume elastic coefficientEm controls the instantaneous displacement of intergranular contact. It is of important reference value to analyze the influence of Em, the microscopic parameter of intergranular contact, on the macroscopic shear mechanical properties of joints. The control variable method was adopted, set to 1E7, 5E7, 1E8, and 5E8 (unit: Pa), respectively, to analyze the sensitivity of elastic coefficient Em of Maxwell body. Shear displacement-time curves of bolt-free joints under different Em conditions are shown in Figure 2. As can be seen from the figure, the instantaneous shear displacement decreases with the increase of Em, and Em has a significant effect on the instantaneous shear displacement. Different Em corresponds to the same rheological shear displacement, and Em has no effect on the rheological shear displacement. By analyzing the elastic moduli corresponding to different Em, the macroscopic shear stiffness Ks corresponding to different Em is obtained, and the relationship between Em and shear stiffness is shown in Figure 3. The shear stiffness is obtained from the curve of relation between shear stress and shear displacement; the curve slope represents the shear stiffness. It can be seen that there is a nonlinear relationship between Em and macroscopic shear stiffness, and the shear stiffness changes greatly and converges to a value, indicating that Em has a significant influence on macroscopic shear stiffness. The nonlinear expression is used to fit it, and the results are shown in Equation (1); R2 is 0.98, indicating that the formula can well predict the macroscopic shear stiffness under different Em conditions.
(1)Ks=3.11−exp−0.04Em−0.719.Figure 2
Shear displacement-time relation of unbolted joints under differentEm.Figure 3
Relationship betweenEm and macroscopic shear stiffness of joints without bolt.In the case of bolted joint, shear displacement-time curves under differentEm conditions were recorded, as shown in Figure 4. It can be seen that, similar to the case without bolt, the instantaneous shear displacement decreases with the increase of Em, and Em has a significant effect on the instantaneous shear displacement. Different Em corresponds to the same rheological shear displacement, and Em has no effect on the rheological shear displacement.Figure 4
Shear displacement-time relation under differentEm with bolts.By analyzing the elastic moduli corresponding to differentEm, the macroscopic shear stiffness corresponding to different Em is obtained, and the relationship between Em and shear stiffness is shown in Figure 5. It can be seen that there is a nonlinear relationship between Em and macroscopic shear stiffness, and the shear stiffness changes greatly and converges to a value, indicating that Em has a significant influence on macroscopic shear stiffness. Nonlinear expression was used to fit them, and the results are shown in Equation (2). R2 is 0.96, indicating that this formula can well predict the macroscopic shear stiffness under different Em. Compared with Formula (1) without bolt, the shear stiffness with bolt is less affected by Em, and the convergence value is also smaller, indicating that the bolt can reduce the influence brought by the change of joint parameters.
(2)Ks=2.61−exp−0.03Em−0.207.Figure 5
Shear stiffness of differentEm with bolt.


### 3.2. Influence of Maxwell Volume Viscosity Coefficient on the Relationship between Joint Shear Displacement and Time

In Burger’s contact model, Maxwell viscosity coefficientCm controls the creep displacement of interparticle contact, and the displacement is closely related to time. Analyzing the microscopic parameter Cm of interparticle contact can effectively reflect the macroscopic mechanical characteristics of Burger’s model parameters in the process of joint shear. The control variable method was adopted, set to 1E9, 5E9, 1E10, and 5E10, respectively, to analyze the sensitivity of Maxwell body viscosity coefficient Cm. The shear displacement-time curves of bolt-free joints under different Cm are shown in Figure 6. It can be seen that Cm has little influence on instantaneous shear displacement, but significant influence on rheological shear displacement. The larger Cm is, the smaller the rheological shear displacement is, and there is a negative correlation between the two. The rheological shear simulation results of unbolted joints obtained by changing Cm show that the adjustment of Cm has no significant effect on the joint numerical model displacement, crack. The results show that Cm affects the shear mechanical properties under rheological conditions, but the effect is small when no damage occurs. Cm is a factor affecting particle contact, but different from the displacement affected by Em. The displacement affected by Cm is controlled by time. Therefore, when the time scale is not very large and Cm is not very small, the displacement affected by Cm is not particularly obvious.Figure 6
Shear displacement-time relation of unbolted joints under differentCm conditions.In the case of bolted joint, shear displacement-time curves under differentCm conditions were recorded, as shown in Figure 7. It can be seen that Cm has little influence on instantaneous shear displacement, but significant influence on rheological shear displacement. The larger Cm is, the smaller the rheological shear displacement is, and there is a negative correlation between the two. Compared with Figure 6 without bolt, the discretization of shear displacements and time curves of different Cm in Figure 7 with bolt is smaller, indicating that anchor can reduce the influence of joint parameter Cm on joint aging deformation. The rheological shear simulation results of bolted joints obtained by changing Cm show that adjusting Cm has no significant effect on the joint numerical model displacement and crack. The results show that Cm affects the shear mechanical properties under rheological conditions, but the effect is small when no damage occurs. Compared with the condition without bolt, the influence of Cm on displacement is less obvious in the condition with bolt, and the influence of Cm on displacement can be weakened by bolt.Figure 7
Shear displacement-time curves of bolted joints at differentCm.


### 3.3. Influence of Nonlinear Kelvin Elastic Coefficient on Shear Mechanical Behavior of Joints

In the rheological contact model, the Kelvin elastic coefficient (Ek) affects the aging displacement of interparticle contact. The analysis of the micro parameter (Ek) of interparticle contact can effectively study the reflection of Burger’s model parameters in macroscopic shear mechanical properties of joints. The control variable method was adopted, set to 1E8, 5E8, 1E9, and 5E9 (unit: Pa), respectively, to analyze the sensitivity characteristics of the elastic coefficient of Kelvin body Ek. The shear displacement-time curves of boltless joints under different Ek conditions are shown in Figure 8. It can be seen that Ek has little effect on the instantaneous shear displacement, but has an obvious effect on the rheological shear displacement. The larger Ek is, the smaller the rheological shear displacement is, and there is a negative correlation between them.Figure 8
Shear displacement-time curves under differentEk conditions without bolt.The rheological shear simulation results of unbolted joints obtained by changingEk show that adjusting Ek has no obvious effect on the numerical model displacement, crack of the joints. The results show that Ek affects the shear mechanical properties under rheological conditions, but the effect is small when no damage occurs. Ek affects particle displacement, but the displacement is controlled by time. Therefore, when the time range is not very large and Ek is not very small, the displacement affected by Ek is not particularly obvious. The relationship between shear displacement and time at different Ek was recorded, as shown in Figure 9. It can be seen from Figure 9 that Ek has little influence on instantaneous shear displacement, but has obvious influence on rheological shear displacement. The larger Ek is, the smaller the rheological shear displacement is, and there is a negative correlation between them. The rheological shear simulation results of bolted joints obtained by changing Ek show that adjusting Ek has no obvious effect on the numerical model displacement and crack of the joints. The results show that Ek affects the shear mechanical properties under rheological conditions, but the effect is small when no damage occurs. Compared with the condition without bolt, the rock mass displacement with bolt is less affected by Ek.Figure 9
Shear displacement-time relation under differentEk conditions with bolt.


### 3.4. Relationship between Joint Shear Displacement and Time under Different Kelvin Body Viscosity Coefficients

The Kelvin viscosity coefficientCk affects the aging displacement of the contact between particles. The analysis of the microscopic parameters Ck of the contact between particles can effectively study the reflection of the parameters of Burger’s model on the macroscopic shear mechanical properties of joints. Control variable method was adopted, set as 1E9, 5E9, 1E10, and 5E10, respectively, to analyze the sensitivity characteristics of Kelvin body viscosity coefficient Ck. Shear displacement-time curves under different Ck conditions are shown in Figure 10. It can be seen that Ck has little influence on instantaneous shear displacement, but significant influence on rheological shear displacement. The larger Ck is, the smaller the rheological shear displacement is, and there is a negative correlation between the two.Figure 10
Shear displacement-time relation under differentCk.The rheological shear simulation results of unbolted joints obtained by changingCk show that adjusting Ck has no significant effect on the joint numerical model displacement, crack. The results show that Ck affects the shear mechanical properties under rheological conditions, but the effect is small when no damage occurs. Ck is a factor affecting particle contact, but different from the displacement affected by Em, the displacement affected by Ck is controlled by time. Therefore, when the time scale is not very large and Ck is not very small, the displacement affected by Ck is not particularly obvious. Shear displacement-time curves of bolted joints under different Ck conditions are shown in Figure 11. It can be seen that Ck has little influence on instantaneous shear displacement, but significant influence on rheological shear displacement. The larger Ck is, the smaller the rheological shear displacement is, and there is a negative correlation. The rheological shear simulation results of bolted joints obtained by changing Ck show that adjusting Ck has no significant effect on the joint numerical model displacement and crack. The results show that Ck affects the shear mechanical properties under rheological conditions, but the effect is small when no damage occurs. Compared with the condition without bolt, the rock mass displacement with bolt is less affected by Ck.Figure 11
Shear displacement-time curves with bolts under differentCk conditions.


### 3.5. Influence of Joint Friction Coefficient on Joint Mechanical Behavior

In the rheological contact model, the friction coefficient affects the contact force between particles, and the analysis of the friction coefficient can effectively study the reflection of the rheological model parameters on the macroscopic shear mechanical properties of joints. The control variable method was adopted, set to 0.1, 0.3, 0.5, and 0.7, respectively, to analyze the sensitivity characteristics of friction coefficientFs. Shear displacement-time curves under different Fs conditions are shown in Figure 12. As can be seen from the figure, the instantaneous shear displacement decreases with the increase of Fs, and Fs has a significant impact on the instantaneous shear displacement. The rheological shear displacement corresponding to different Fs is roughly the same, and Fs has no effect on the rheological shear displacement. By analyzing the elastic moduli corresponding to different Fs, the macroscopic shear stiffness corresponding to different Fs is obtained, and the relationship between Fs and shear stiffness is shown in Figure 13. It can be seen that there is a nonlinear relationship between Fs and the macroscopic shear stiffness, and the shear stiffness changes greatly, indicating that Fs has a significant impact on the macroscopic shear stiffness. The nonlinear expression is used to fit it, and the results are shown in Equation (3); R2 is 0.99, indicating that the formula can well predict the macroscopic shear stiffness under different Fs conditions.
(3)Ks=2.771−exp−7.3Fs−0.59.Figure 12
Shear displacement-time curves under differentFs conditions.Figure 13
Relationship betweenFs and shear stiffness.Respectively, change the interparticle friction coefficient ofFs, by calculation, the joint can be obtained corresponding crack number, displacement, as shown in Figure 14. It shows that, in the process of shearing, joint displacement and number of crack are significantly affected by Fs. The greater the shear displacement is smaller, the less crack. Fs has a negative relationship with shear displacement and crack number. In the rheological model, the shear strength is controlled by the normal stress and friction coefficient Fs. Therefore, the larger Fs is, the greater the static friction force to overcome under the condition of the same displacement caused by particle contact is, and it is less likely to produce sliding and cracks.Figure 14
Number of sample cracks under different friction coefficientsFs.
(a)
Fs=0.1(b)
Fs=0.7In the case of bolted joint, shear displacement-time curves under differentFs conditions are shown in Figure 15. As can be seen from the figure, the instantaneous shear displacement decreases with the increase of Fs, and Fs has a significant impact on the instantaneous shear displacement. The rheological shear displacement corresponding to different Fs is roughly the same, and Fs has no effect on the rheological shear displacement. By analyzing the elastic moduli corresponding to different Fs, the macroscopic shear stiffness corresponding to different Fs is obtained, and the relationship between Fs and shear stiffness is shown in Figure 16. It can be seen that there is a nonlinear relationship between Fs and the macroscopic shear stiffness, and the shear stiffness changes greatly, indicating that Fs has a significant impact on the macroscopic shear stiffness. The nonlinear expression is used to fit it, and the results are shown in Equation (4); R2 is 0.99, indicating that the formula can well predict the macroscopic shear stiffness under different Fs conditions. Compared with Formula (3) without bolt, the shear stiffness with bolt is less affected by Fs, and the anchor can reduce the influence of joint parameter Fs on mechanical properties.Figure 15
Shear displacement-time curves of differentFs with bolt.Figure 16
Relationship betweenFs and shear stiffness.For bolted joints,Fs is changed separately, and it is found that the larger Fs is, the smaller the shear displacement is, the fewer the cracks are, and the less obvious the tension chain concentration on the joints is. Fs has a negative relationship with shear displacement and crack number. Shear strength is controlled by normal stress and friction coefficient Fs and is proportional to Fs. Therefore, the larger Fs is, the more static friction force to be overcome under the condition of the same displacement caused by particle contact and the less prone to sliding and cracking of rock mass. However, the rock mass under the condition of anchor is less affected by Fs than that without bolt, and the anchor can reduce the influence brought by the change of rock mass mesoscopic parameters, which is conducive to enhancing the stability of rock mass.
(4)Ks=1.061−exp−3.86Fs+1.09.


## 3.1. Relationship between Maxwell Elastic Coefficient and Macroscopic Shear Stiffness

In the rheological contact model, Maxwell volume elastic coefficientEm controls the instantaneous displacement of intergranular contact. It is of important reference value to analyze the influence of Em, the microscopic parameter of intergranular contact, on the macroscopic shear mechanical properties of joints. The control variable method was adopted, set to 1E7, 5E7, 1E8, and 5E8 (unit: Pa), respectively, to analyze the sensitivity of elastic coefficient Em of Maxwell body. Shear displacement-time curves of bolt-free joints under different Em conditions are shown in Figure 2. As can be seen from the figure, the instantaneous shear displacement decreases with the increase of Em, and Em has a significant effect on the instantaneous shear displacement. Different Em corresponds to the same rheological shear displacement, and Em has no effect on the rheological shear displacement. By analyzing the elastic moduli corresponding to different Em, the macroscopic shear stiffness Ks corresponding to different Em is obtained, and the relationship between Em and shear stiffness is shown in Figure 3. The shear stiffness is obtained from the curve of relation between shear stress and shear displacement; the curve slope represents the shear stiffness. It can be seen that there is a nonlinear relationship between Em and macroscopic shear stiffness, and the shear stiffness changes greatly and converges to a value, indicating that Em has a significant influence on macroscopic shear stiffness. The nonlinear expression is used to fit it, and the results are shown in Equation (1); R2 is 0.98, indicating that the formula can well predict the macroscopic shear stiffness under different Em conditions.
(1)Ks=3.11−exp−0.04Em−0.719.Figure 2
Shear displacement-time relation of unbolted joints under differentEm.Figure 3
Relationship betweenEm and macroscopic shear stiffness of joints without bolt.In the case of bolted joint, shear displacement-time curves under differentEm conditions were recorded, as shown in Figure 4. It can be seen that, similar to the case without bolt, the instantaneous shear displacement decreases with the increase of Em, and Em has a significant effect on the instantaneous shear displacement. Different Em corresponds to the same rheological shear displacement, and Em has no effect on the rheological shear displacement.Figure 4
Shear displacement-time relation under differentEm with bolts.By analyzing the elastic moduli corresponding to differentEm, the macroscopic shear stiffness corresponding to different Em is obtained, and the relationship between Em and shear stiffness is shown in Figure 5. It can be seen that there is a nonlinear relationship between Em and macroscopic shear stiffness, and the shear stiffness changes greatly and converges to a value, indicating that Em has a significant influence on macroscopic shear stiffness. Nonlinear expression was used to fit them, and the results are shown in Equation (2). R2 is 0.96, indicating that this formula can well predict the macroscopic shear stiffness under different Em. Compared with Formula (1) without bolt, the shear stiffness with bolt is less affected by Em, and the convergence value is also smaller, indicating that the bolt can reduce the influence brought by the change of joint parameters.
(2)Ks=2.61−exp−0.03Em−0.207.Figure 5
Shear stiffness of differentEm with bolt.


## 3.2. Influence of Maxwell Volume Viscosity Coefficient on the Relationship between Joint Shear Displacement and Time

In Burger’s contact model, Maxwell viscosity coefficientCm controls the creep displacement of interparticle contact, and the displacement is closely related to time. Analyzing the microscopic parameter Cm of interparticle contact can effectively reflect the macroscopic mechanical characteristics of Burger’s model parameters in the process of joint shear. The control variable method was adopted, set to 1E9, 5E9, 1E10, and 5E10, respectively, to analyze the sensitivity of Maxwell body viscosity coefficient Cm. The shear displacement-time curves of bolt-free joints under different Cm are shown in Figure 6. It can be seen that Cm has little influence on instantaneous shear displacement, but significant influence on rheological shear displacement. The larger Cm is, the smaller the rheological shear displacement is, and there is a negative correlation between the two. The rheological shear simulation results of unbolted joints obtained by changing Cm show that the adjustment of Cm has no significant effect on the joint numerical model displacement, crack. The results show that Cm affects the shear mechanical properties under rheological conditions, but the effect is small when no damage occurs. Cm is a factor affecting particle contact, but different from the displacement affected by Em. The displacement affected by Cm is controlled by time. Therefore, when the time scale is not very large and Cm is not very small, the displacement affected by Cm is not particularly obvious.Figure 6
Shear displacement-time relation of unbolted joints under differentCm conditions.In the case of bolted joint, shear displacement-time curves under differentCm conditions were recorded, as shown in Figure 7. It can be seen that Cm has little influence on instantaneous shear displacement, but significant influence on rheological shear displacement. The larger Cm is, the smaller the rheological shear displacement is, and there is a negative correlation between the two. Compared with Figure 6 without bolt, the discretization of shear displacements and time curves of different Cm in Figure 7 with bolt is smaller, indicating that anchor can reduce the influence of joint parameter Cm on joint aging deformation. The rheological shear simulation results of bolted joints obtained by changing Cm show that adjusting Cm has no significant effect on the joint numerical model displacement and crack. The results show that Cm affects the shear mechanical properties under rheological conditions, but the effect is small when no damage occurs. Compared with the condition without bolt, the influence of Cm on displacement is less obvious in the condition with bolt, and the influence of Cm on displacement can be weakened by bolt.Figure 7
Shear displacement-time curves of bolted joints at differentCm.


## 3.3. Influence of Nonlinear Kelvin Elastic Coefficient on Shear Mechanical Behavior of Joints

In the rheological contact model, the Kelvin elastic coefficient (Ek) affects the aging displacement of interparticle contact. The analysis of the micro parameter (Ek) of interparticle contact can effectively study the reflection of Burger’s model parameters in macroscopic shear mechanical properties of joints. The control variable method was adopted, set to 1E8, 5E8, 1E9, and 5E9 (unit: Pa), respectively, to analyze the sensitivity characteristics of the elastic coefficient of Kelvin body Ek. The shear displacement-time curves of boltless joints under different Ek conditions are shown in Figure 8. It can be seen that Ek has little effect on the instantaneous shear displacement, but has an obvious effect on the rheological shear displacement. The larger Ek is, the smaller the rheological shear displacement is, and there is a negative correlation between them.Figure 8
Shear displacement-time curves under differentEk conditions without bolt.The rheological shear simulation results of unbolted joints obtained by changingEk show that adjusting Ek has no obvious effect on the numerical model displacement, crack of the joints. The results show that Ek affects the shear mechanical properties under rheological conditions, but the effect is small when no damage occurs. Ek affects particle displacement, but the displacement is controlled by time. Therefore, when the time range is not very large and Ek is not very small, the displacement affected by Ek is not particularly obvious. The relationship between shear displacement and time at different Ek was recorded, as shown in Figure 9. It can be seen from Figure 9 that Ek has little influence on instantaneous shear displacement, but has obvious influence on rheological shear displacement. The larger Ek is, the smaller the rheological shear displacement is, and there is a negative correlation between them. The rheological shear simulation results of bolted joints obtained by changing Ek show that adjusting Ek has no obvious effect on the numerical model displacement and crack of the joints. The results show that Ek affects the shear mechanical properties under rheological conditions, but the effect is small when no damage occurs. Compared with the condition without bolt, the rock mass displacement with bolt is less affected by Ek.Figure 9
Shear displacement-time relation under differentEk conditions with bolt.


## 3.4. Relationship between Joint Shear Displacement and Time under Different Kelvin Body Viscosity Coefficients

The Kelvin viscosity coefficientCk affects the aging displacement of the contact between particles. The analysis of the microscopic parameters Ck of the contact between particles can effectively study the reflection of the parameters of Burger’s model on the macroscopic shear mechanical properties of joints. Control variable method was adopted, set as 1E9, 5E9, 1E10, and 5E10, respectively, to analyze the sensitivity characteristics of Kelvin body viscosity coefficient Ck. Shear displacement-time curves under different Ck conditions are shown in Figure 10. It can be seen that Ck has little influence on instantaneous shear displacement, but significant influence on rheological shear displacement. The larger Ck is, the smaller the rheological shear displacement is, and there is a negative correlation between the two.Figure 10
Shear displacement-time relation under differentCk.The rheological shear simulation results of unbolted joints obtained by changingCk show that adjusting Ck has no significant effect on the joint numerical model displacement, crack. The results show that Ck affects the shear mechanical properties under rheological conditions, but the effect is small when no damage occurs. Ck is a factor affecting particle contact, but different from the displacement affected by Em, the displacement affected by Ck is controlled by time. Therefore, when the time scale is not very large and Ck is not very small, the displacement affected by Ck is not particularly obvious. Shear displacement-time curves of bolted joints under different Ck conditions are shown in Figure 11. It can be seen that Ck has little influence on instantaneous shear displacement, but significant influence on rheological shear displacement. The larger Ck is, the smaller the rheological shear displacement is, and there is a negative correlation. The rheological shear simulation results of bolted joints obtained by changing Ck show that adjusting Ck has no significant effect on the joint numerical model displacement and crack. The results show that Ck affects the shear mechanical properties under rheological conditions, but the effect is small when no damage occurs. Compared with the condition without bolt, the rock mass displacement with bolt is less affected by Ck.Figure 11
Shear displacement-time curves with bolts under differentCk conditions.


## 3.5. Influence of Joint Friction Coefficient on Joint Mechanical Behavior

In the rheological contact model, the friction coefficient affects the contact force between particles, and the analysis of the friction coefficient can effectively study the reflection of the rheological model parameters on the macroscopic shear mechanical properties of joints. The control variable method was adopted, set to 0.1, 0.3, 0.5, and 0.7, respectively, to analyze the sensitivity characteristics of friction coefficientFs. Shear displacement-time curves under different Fs conditions are shown in Figure 12. As can be seen from the figure, the instantaneous shear displacement decreases with the increase of Fs, and Fs has a significant impact on the instantaneous shear displacement. The rheological shear displacement corresponding to different Fs is roughly the same, and Fs has no effect on the rheological shear displacement. By analyzing the elastic moduli corresponding to different Fs, the macroscopic shear stiffness corresponding to different Fs is obtained, and the relationship between Fs and shear stiffness is shown in Figure 13. It can be seen that there is a nonlinear relationship between Fs and the macroscopic shear stiffness, and the shear stiffness changes greatly, indicating that Fs has a significant impact on the macroscopic shear stiffness. The nonlinear expression is used to fit it, and the results are shown in Equation (3); R2 is 0.99, indicating that the formula can well predict the macroscopic shear stiffness under different Fs conditions.
(3)Ks=2.771−exp−7.3Fs−0.59.Figure 12
Shear displacement-time curves under differentFs conditions.Figure 13
Relationship betweenFs and shear stiffness.Respectively, change the interparticle friction coefficient ofFs, by calculation, the joint can be obtained corresponding crack number, displacement, as shown in Figure 14. It shows that, in the process of shearing, joint displacement and number of crack are significantly affected by Fs. The greater the shear displacement is smaller, the less crack. Fs has a negative relationship with shear displacement and crack number. In the rheological model, the shear strength is controlled by the normal stress and friction coefficient Fs. Therefore, the larger Fs is, the greater the static friction force to overcome under the condition of the same displacement caused by particle contact is, and it is less likely to produce sliding and cracks.Figure 14
Number of sample cracks under different friction coefficientsFs.
(a)
Fs=0.1(b)
Fs=0.7In the case of bolted joint, shear displacement-time curves under differentFs conditions are shown in Figure 15. As can be seen from the figure, the instantaneous shear displacement decreases with the increase of Fs, and Fs has a significant impact on the instantaneous shear displacement. The rheological shear displacement corresponding to different Fs is roughly the same, and Fs has no effect on the rheological shear displacement. By analyzing the elastic moduli corresponding to different Fs, the macroscopic shear stiffness corresponding to different Fs is obtained, and the relationship between Fs and shear stiffness is shown in Figure 16. It can be seen that there is a nonlinear relationship between Fs and the macroscopic shear stiffness, and the shear stiffness changes greatly, indicating that Fs has a significant impact on the macroscopic shear stiffness. The nonlinear expression is used to fit it, and the results are shown in Equation (4); R2 is 0.99, indicating that the formula can well predict the macroscopic shear stiffness under different Fs conditions. Compared with Formula (3) without bolt, the shear stiffness with bolt is less affected by Fs, and the anchor can reduce the influence of joint parameter Fs on mechanical properties.Figure 15
Shear displacement-time curves of differentFs with bolt.Figure 16
Relationship betweenFs and shear stiffness.For bolted joints,Fs is changed separately, and it is found that the larger Fs is, the smaller the shear displacement is, the fewer the cracks are, and the less obvious the tension chain concentration on the joints is. Fs has a negative relationship with shear displacement and crack number. Shear strength is controlled by normal stress and friction coefficient Fs and is proportional to Fs. Therefore, the larger Fs is, the more static friction force to be overcome under the condition of the same displacement caused by particle contact and the less prone to sliding and cracking of rock mass. However, the rock mass under the condition of anchor is less affected by Fs than that without bolt, and the anchor can reduce the influence brought by the change of rock mass mesoscopic parameters, which is conducive to enhancing the stability of rock mass.
(4)Ks=1.061−exp−3.86Fs+1.09.


## 4. Conclusions

(1)
For unbolted joint,Em and Fs mainly affect the instantaneous shear displacement, but have little effect on the rheological shear displacement. However, Cm, Ek, and Ck have little effect on the instantaneous shear displacement, and the sensitivity is mainly reflected in the rheological shear displacement(2)
For bolted joint,Em and Fs mainly affect the instantaneous shear displacement, but have little effect on the rheological shear displacement. However, Cm, Ek, and Ck have little effect on the instantaneous shear displacement, and the sensitivity is mainly reflected in the rheological shear displacement. It should be noted that the instantaneous shear displacement, shear stiffness, and steady rheological displacement with bolt are less affected by parameters than the parameter sensitivity without bolt, indicating that the anchor can reduce the influence of joint parameters



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*Source: 3512507-2022-03-30.xml*