Shear Failure of Bolted Joints considering Mesoscopic Deformation Characteristics of Rock

In rock engineering of the cold region, there are a lot of rock joints.)e shear characteristics of joints play a decisive role in the stability of rock engineering in the cold area. In this paper, based on the numerical simulation method of particle flow, reasonable microscopic parameters are selected for the numerical simulation of the direct shear test of bolted joints. )e results show that the shear stiffness and contact modulus are linearly and positively correlated. )e greater the contact modulus, the greater the residual stress, the better the synergetic effect between rock and bolt, and the more developed the microcrack. )e smaller the contact stiffness ratio, the greater the residual stress. )e shear stiffness decreases with the increase in the contact stiffness ratio, and the larger the contact stiffness ratio, the slower the shear stiffness decreases, while the shear strength does not change with the contact stiffness ratio. )e contact stiffness ratio has a weak effect on the number of cracks in the model. )e shear stiffness increases with the increase in the parallel bond modulus, and the shear strength decreases with the increase in the parallel bond modulus. )e binding stiffness is independent of the shear stiffness, and the peak shear stress decreases with the increase in the binding stiffness ratio. )e greater the bond stiffness ratio, the greater the number of cracks.


Introduction
ere are various types of geological defects and discontinuities, such as faults, joints, slices, and unconformities. To a large extent, the movement and deformation of rock mass are mainly subject to these discontinuities, which have a controlling influence on the stability of rock mass engineering [1][2][3][4][5]. For a long time, in order to facilitate the research, rock mass is often idealized as a uniform continuous medium [6][7][8][9][10][11] in the research of exploring the mechanical properties of rock mass, and the theory describing the deformation and failure process of rock mass is the macroscopic elastic-plastic theory. Such a theoretical model has a certain significance for the research on the mechanical properties of rock mass [12][13][14][15][16][17]. e laboratory test is the main method to study jointed rock mass [16,[18][19][20][21]. Based on mass shear tests, Barton [22] proposed a JRC-JCS model for estimating the shear strength of rock joints. Lee et al. [23] studied and discussed in depth the characteristics of shear deformation, failure mechanism, and attenuation law of strength of specimens subjected to cyclic shearing. Jiang et al. [24] studied the influence of joint roughness on shear mechanical characteristics. Ghazvinian et al. [25] studied the variation rules of peak shear strength and shear strength of different sawtooth joints. Jahanian and Sadaghiani [26] studied joint shear curves and peak shear strength at different undulation angles. With the development of computer technology and the appearance of a large computer, the numerical simulation method is more and more abundantly used to study the mechanical properties of the rock [27][28][29][30][31]. Numerical methods for studying the mechanical properties of the rock mass, such as particle flow code (PFC), have been widely used in the design and construction of rock mass engineering [32][33][34][35]. Park and Song [36] carried out a series of direct shear tests of joints based on PFC3D and studied the effects of geometric characteristics and microscopic properties of joints on joint shear performance. e relationship between shear mechanical characteristics of jointed rock mass and normal stress and joint surface topography has been studied from a variety of perspectives [37][38][39][40]. However, it is not enough just to fit the mechanical properties of the rock mass from the macroscopic phenomenon without studying the intrinsic nature of the mechanical properties produced by rock mass. At present, there are few studies on the variation rules of shear mechanical characteristics of bolted jointed rock mass under different rock microscopic parameters [41,42]. Due to the difficulties in obtaining and researching methods and means of bolted joint samples with different microscopic parameters, this paper adopts the PFC numerical calculation method to study in detail the effects of different microscopic parameters on the macroscopic shear mechanical behavior of jointed rock mass and the development and evolution characteristics of microcracks. e research conclusion has a certain reference value for the laboratory test.

Numerical Model.
e numerical model is composed of homogeneous particles which are linked by parallel bonding [43][44][45][46]. In the parallel bond contact model, the bond is equivalent to a number of springs acting side by side, which can transmit both force and torque and which is more in line with the mechanical properties of rock materials, and it is the most widely used model to simulate the mechanical properties of the rock. e final model is established in this paper. e direct shear boundary conditions and particle generation are shown in Figure 1.
Different from other methods of continuum analysis, the simulation of the macroscopic mechanical behavior of the particle discrete element model must be constructed by assigning values to a series of mesoscopic parameters related to material properties [47][48][49][50][51]. e selection of model mesoscopic parameters is related to the accuracy of the final simulation results. For the microscopic parameters of the particle flow model of rock materials, it is usually necessary to select the macroscopic mechanical parameters (shear mechanical parameters and elastic modulus) and the stressstrain relationship in the laboratory physical tests [52]. In this paper, the macroscopic shear mechanical behavior of jointed rock mass under different joint microparameters is studied. Relevant physical and mechanical parameters obtained through laboratory tests are shown in Table 1. e microscopic parameters of rock were obtained through repeated debugging, as shown in Table 2. e peak shear strength of the rock and joints obtained by numerical calculation and the laboratory test was further compared by about 150 times, and the corresponding shear strength parameters, adhesive force and friction angle, were obtained through the Mohr-Coulomb criterion fitting [53][54][55], as shown in Table 3, indicating that the values of the two are relatively consistent.

Influence of Rock Mesoscopic Parameters on Macroscopic Shear Mechanical Behavior
In order to study the influence of rock mesoscopic parameters on the macroscopic shear mechanical behavior of bolted jointed rock mass, the control variable method was adopted to conduct the direct shear test numerical simulation and result analysis for the bolted jointed rock mass specimens with different rock mesoscopic parameters. Without changing other parameters, only the microscopic parameters of the research object were changed to observe the change of the shear mechanical response.

Contact Modulus.
e contact modulus E * is the parameter that controls the elastic modulus of the particle contact. e larger the E * is, the greater the elastic modulus of the particle contact is. e contact modulus E * can be combined with the contact stiffness ratio k * . By obtaining the contact area conversion between the two particles, the normal stiffness kn and tangential stiffness ks of the two particles can be obtained, respectively. Only the contact modulus E * was changed, which was set as 0.1, 0.3, 1, and 3 (GPa), respectively. e shear deformation curve of the bolted jointed rock mass was obtained, as shown in Figure 2. By analyzing the elastic modulus corresponding to different contact modulus, the shear stiffness and shear strength corresponding to different contact modulus are obtained, as shown in Table 4 and Figure 3. Although in general, the contact modulus and the shear strength increase, the contact modulus has a major impact on the stiffness of the model, so only the relationship between the contact modulus and the shear stiffness is emphatically analyzed.
Since the contact modulus E * mainly controls the elastic modulus between particle contacts, the larger the E * is, the harder the lithology is. It can be seen from Figure 2 that the slope of the linear segment of the shear stress-shear displacement curve increases with the increase in the contact modulus. From Figure 3, it can be seen that the relationship between shear stiffness and contact modulus is also basically linear. e larger the contact modulus, the greater the rock stiffness and the smaller the shear displacement corresponding to the shear stress peak without changing the other parameters. In addition, the larger the contact modulus E * , the greater the postpeak curve fluctuation, the greater the postpeak residual stress, and the better the synergetic effect between rock bolt and rock. e smaller the contact       modulus, the smaller the residual stress. is is because the bolt has a regulating effect on the comprehensive shear stiffness of the structural plane. When the shear modulus of the rock is small, the bolt can increase the comprehensive shear stiffness of the structural plane. When the shear modulus of the rock is large, the bolt can reduce the comprehensive shear stiffness of the structural plane.

Contact Stiffness Ratio.
e contact stiffness ratio k * is the ratio of the normal contact stiffness kn and the tangential contact stiffness ks in the parallel bonding linear group, namely, kn/ks, which is a dimensionless quantity. e contact stiffness ratio k * needs to be assigned only when the contact method command is used, and kn and ks can be directly assigned when the contact property command is directly used, so it is not necessary to specify the contact stiffness ratio k * . e PFC program uses the contact stiffness ratio k * to automatically calculate the normal stiffness kn and the tangential stiffness ks of two contact particles in the software through formula (1) and formula (2): where A is the contact surface area and L is the distance between the centers of the two particles. e contact stiffness ratio k * means that the contact modulus E * is used to distribute the normal stiffness kn and the tangential stiffness ks. e larger the contact modulus E * , the larger the normal stiffness and the smaller the ks. e smaller the k * is, the smaller the normal stiffness kn is and the larger the tangential stiffness ks is. rough the control variable method, only k * was changed, and the contact stiffness ratio k * was set as 0.5, 1.0, 1.5, and 2.2. e direct shear test was conducted on the structural plane model with bolt and without bolt, and the influence of the contact stiffness ratio k * on the mechanical behavior of the structural plane was observed. e straight shear-simulated shear deformation curve of rock mass with bolted joints is shown in Figure 4. e corresponding shear stiffness and shear strength of each contact stiffness ratio were obtained as shown in Table 5. e relationship between contact stiffness ratio and shear stiffness and shear strength was plotted as shown in Figure 5.
According to the shear deformation curve of bolted joints in Figure 4, the larger the contact stiffness ratio, the smaller the slope of the linear segment before the peak of the curve, that is, the smaller the shear stiffness. e smaller the contact stiffness ratio, the higher the slope of the linear section of the stress deformation curve. is is because the larger the contact stiffness ratio is, the smaller the tangential stiffness ks of the particles will be, and the decreasing tangential stiffness of the particles will reflect the decreasing shear stiffness of the structural plane on the macrolevel. e smaller the contact stiffness ratio, the greater the residual stress and the more obvious the strain hardening characteristics.
It can be seen from Table 5 and Figure 5 that the shear stiffness decreases with the increase in the contact    stiffness ratio, and the model shear stiffness is negatively correlated with the contact stiffness ratio. e larger the contact stiffness ratio, the slower the shear stiffness decreases, which shows the nonlinear change. e shear strength does not change with the contact stiffness ratio.

Parallel Bonding Modulus.
Parallel bond modulus is the cementation between two particles in each parallel bonding model; the parallel bond modulus is set to be 0.01 GPa, 0.3 GPa, 3 GPa, and 5 GPa while other mesoscopic parameters are kept the same, then the shear deformation of the bolted joint is shown in Figure 6. By analyzing the elastic modulus corresponding to different parallel bond modulus, the shear stiffness corresponding to different contact modulus is obtained, and the relationship between the parallel contact bond modulus, shear stiffness, and shear strength is obtained, as shown in Figure 7.
According to the shear deformation curve of bolted joints in Figure 6, the smaller the parallel bonding modulus is, the smaller the slope of the linear segment before the curve peak is, and the higher the peak strength is, the sharper the curve peak is. e higher the parallel bonding modulus, the higher the slope of the linear segment before the curve peak and the lower the peak strength.
e strain hardening occurred after the peak, and the parallel bond modulus increased. When the shear displacement reaches to the vicinity of 4 mm, the four curves experience a sudden decrease in stress of different degrees, which is caused by the sudden embrittlement of the rock bite. e greater the parallel bonding modulus, the greater the stress reduction degree, and the shear displacement corresponding to the stress reduction decreases with the increase in the bonding modulus. It shows that the bond strength remains unchanged; the greater the parallel bond modulus, the smaller the displacement generated by the contact fracture, and the brittleness of the contact bond increases. According to Figure 7, the relationship between the parallel bonding modulus and shear stiffness and shear strength and the shear stiffness increases with the increase in the parallel bonding modulus, but the increase in the rate of the shear stiffness decreases with the increase in the parallel bonding modulus. e shear strength decreases with the increase in the parallel bond modulus, and the rate of shear strength decreases with the increase in the parallel bond modulus.

Parallel Bonding Stiffness Ratio.
e meaning of parallel bond stiffness ratio k * and contact stiffness is similar, the parallel bond stiffness ratio k * is the ratio of the stiffness k n tothe caking tangential stiffness k s , namely k n /k s ,. When the contact method is used, it is needed to assign value for the parallel bond stiffness ratio k * , PFC software uses formulas (3) and (4) to calculate the bonding stiffness and the tangential stiffness: where L is the distance between the centers of two particles. By using the method of control variable to control other mesoscopic parameters as the same, only the parallel bond stiffness ratio k * is changed and is, respectively, set to 0.5, 1.0, 1.5, and 2.2. e peak shear stress corresponding to different bond stiffness ratios is obtained, and the relationship between the bond stiffness ratio and peak shear stress is shown in Table 6. Figure 8 shows that the bond stiffness ratio does not have much effect on the deformation  Advances in Civil Engineering and stress curve, namely, binder modulus has effect on bonding and tangential stiffness, but almost has no effect on the overall shear stiffness of the model, the stress deformation curves are similar at the elastic stage basic coincidence. As can be seen from the relationship between the bond stiffness ratio and peak shear stress in Figure 9, the peak shear stress decreases with the increase in the bond stiffness ratio, and the decreasing rate of peak shear stress decreases with the increase in the bond stiffness ratio, which tends to be flat. If the bond modulus remains the same, the bond stiffness ratio decreases, that is, the bond tangential stiffness increases. It shows that increasing the tangential stiffness of the bond can affect the shear strength and increase the shear strength.

Conclusions
(1) Shear stiffness and contact modulus are linearly and positively correlated. e smaller the contact stiffness ratio of bolted joints, the greater the residual stress and the more obvious the strain hardening characteristics. e shear stiffness decreases with the increase in the contact stiffness ratio. e influence of the contact stiffness ratio on the number of cracks in the model is positive, but the correlation is weak. (2) e shear stiffness increases with the increase in the parallel bond modulus, but the increased rate of the shear stiffness decreases with the increase in the parallel bond modulus. e strain hardening characteristic is enhanced with the increase in the parallel bond modulus. (3) e peak shear stress decreases with the increase in the bond stiffness ratio, and the reduction rate decreases with the increase in the bond stiffness ratio. e bond stiffness ratio also has an effect on the number of cracks, but the effect is far less than that of the parallel bond modulus.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.     6 Advances in Civil Engineering