# Research on the Strength Characteristics and Crack Propagation Law of Noncoplanar Nonthrough Jointed Rock Mass by PFC2D

Authors: Qingzhi Chen; Yuanming Liu; Shilong Mei; Kai Cao; Bin Du; Wei Wang
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

In this study, five groups of numerical models with different conditions were established by using PFC2D (particle flow code) to simulate the direct shear tests of noncoplanar nonthrough jointed rock mass. It is proved that normal stress and shear rate, as well as the connectivity rate, relief angle, and inclination angle of joints, have significant influence on the strength characteristics, number of cracks, and the stress of the rock mass according to measurement taken at five different measurement circles in the rock mass. Moreover, it is determined that in the process of shearing, no matter which group of tests are conducted, the number of cracks in the rock mass caused by tension is far more than that caused by the shear action. In other words, the failure of rock mass with different planes and discontinuous joints is mainly caused by the tension in the process of the direct shear test.


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## Body

## 1. Introduction

A rock mass is a complex geological body with obvious nonlinearity, discontinuity, heterogeneity, and anisotropy. Within a rock mass, there can be joints, cracks, bedding, schistosity, faults, folds, and other structural planes [1–4]. All these different types of structures are collectively referred to as joints, and a rock mass containing various joints is termed as the jointed rock mass. The existence of joints makes the mechanical properties of a jointed rock mass to differ greatly from those of an intact rock [5–9]. The jointed rock mass is the most common complex engineering medium in construction engineering, water conservancy and hydropower engineering, underground space engineering, mineral resource exploration and development engineering, transportation engineering, bridge and tunnel engineering, oil port engineering, and other projects. Its strength and extended through characteristics play a vital role in construction in civil engineering. The mechanical properties of the jointed rock mass are important factors that must be considered in the analysis, evaluation, and design of rock engineering. The existence of joints makes the stability and safety of the built underground engineering projects to have great hidden dangers [10–12]. In addition, with the development of large engineering projects in recent years, the number, scale, difficulty, and complexity of rock mass engineering during the construction period have increased significantly, and the problems in rock mechanics in the construction process have become more complex. These mechanical problems of jointed rock mass will directly increase the technical difficulty of construction and affect the quality and cost of the project [13–15]. If the problems cannot be handled well, it will even cause casualties and serious losses due to the destruction of the complex rock mass. In order to facilitate the study, and based on whether the internal joint plane of the rock mass is through and the degree of joint opening or closing, a jointed rock mass is divided into two types: a through jointed rock mass and a nonthrough jointed rock mass [16–20]. On the basis of the spatial arrangement of rock masses and joints, or whether the joint surface and shear plane are on the same plane, the nonthrough jointed rock mass can be further divided into coplanar nonthrough jointed rock mass (CNJRM) and noncoplanar nonthrough jointed rock mass (NNJRM) [21]. The schematics of CNJRM and NNJRM are shown in Figures 1 and 2, respectively. In this study, numerical simulation experiments are carried out to study the law of crack propagation and the strength characteristics of rock mass in the process of the direct shear test.Figure 1
The schematic diagram of CNJRM.Figure 2
The schematic diagram of NNJRM.The physical model test or numerical simulation test is widely used in the study of nonthrough jointed rock mass. This is because it can be quite difficult and costly to conduct the test in the field or by using natural rock specimens. In addition, the test repeatability will be poor, and the difference between individual and overall measurements would be greater, and so would the error of the test result. However, since the physical model test is greatly affected by the machinery used in the test and the operation mode of the operator in the experimental process, the accuracy of the experimental results has certain discreteness. Moreover, the cost of the physical model test is higher than that of the numerical simulation test. Therefore, if the parameters of the simulated material can be determined, many experts and scholars tend to use the numerical simulation test to study the mechanical properties of the incoherent jointed rock mass [22–29].There are a variety of simulation programs for performing numerical simulation tests, among which the PFC2D (particle flow code) for particle discrete element analysis developed by ITASCA has been widely used [30–34]. This program studies the mechanical properties of the medium from the perspective of microstructure, which is very suitable for the in-depth study of high-level topics such as the expansion, fracture, failure, failure impact, and microseismic response of meso- or macrojoints in solid materials. At present, many experts and scholars have shown that the simulation test results of the PFC program are stable and reliable. For example, Zhou et al. [35] simulated the direct shear test of nontransfixion jointed rock mass based on the particle discrete element theory and the PFC2D program, and also analyzed the mechanical properties and breaking mechanism of joints in experiments from both the macro- and microaspects. The results of the simulation test were compared with the results of the indoor model test, and it was found that the new particle flow calculation method is very suitable for the simulation test of nonthrough jointed rock mass. This method has high reliability and thus can provide good reference for direct shear test of jointed rock mass in the laboratory and also for parameter selection when using PFC to simulate the joint model. Zhou et al. [36, 37] also took the natural slope as the research background and used the granular model and smooth joint model to simulate a rock block and joint, respectively, based on the particle discrete element theory and by using the PFC2D numerical simulation program. By repeatedly comparing and debugging the test results of the uniaxial compression test, direct shear test, and numerical test, the micromechanical parameters of the granular model and smooth joint model were determined. In addition, the mesoanalysis model of the intermittent jointed rock slope was established successfully. From the mesoscopic point of view, they successfully studied the mechanical properties of the rock mass model with two discontinuous joints and the bedding rock slope model with discontinuous joints in the failure process. Yang et al. [38] used the PFC3D program to study the effect of joint direction on the strength, deformation, and failure mode of joint blocks. Wang et al. [39] used the PFC3D program to establish the experimental model of coal gangue with different particle sizes, and simulated the triaxial compression test based on the model. By comparing the stress-strain curve, volume strain curve, and microcrack development curve under different confining pressures, he studied the strength characteristics and deformation law of coal gangue with different particle sizes. Hu et al. [40] used the PFC program to establish a rock slope model containing nonpenetrating jointed rock mass, and calculated the relevant mechanical parameters of the rock mass through simulated direct shear test, biaxial test, Brazil split test, and other conventional mechanical tests, so as to study various failure modes of the rock slope and its internal mesoscopic mechanism during the failure of the rock slope. In addition, many experts, including Huan et al. [41], Jiang et al. [42], Regassa et al. [43], Liu et al. [44], Cao et al. [45], Ghazvinian et al. [46], and Tao et al. [47], have used the PFC program or DEM (discrete element method) program to study various properties of rock masses.Therefore, the PFC2D program was also used in this study to simulate the direct shear test, so as to investigate the influence of the undulating angle, inclination angle, joint connectivity rate, normal stress, and shear rate on the transfixion mechanism, strength, and deformation characteristics of NNJRM. By studying the stress-strain development state of the specimen as a whole based on the data collected from five measurement points inside the specimen (as shown in Figure3) during the test, and the change in the number of cracks in the specimen during the test process, the mechanical properties of the nonpenetrating jointed rock mass under different working conditions were investigated. It needs to be explained that the joint relief angle of the noncoplanar nonthrough jointed rock mass, represented by i, refers to the unevenness of a single structural plane in the joint relative to the whole joint, as shown in Figure 4. The joint inclination angle, represented by ϕ, refers to the inclined angle between the joint tendency and the stress direction of the rock mass [21], as shown in Figure 5. The joint connectivity of the noncoplanar nonthrough jointed rock mass, represented by j, is defined as the ratio of the sum of the projection lengths of each joint segment in the direction of the shear plane to the length of the survey line. Its calculation formula is j=2S/L. The numerical model established in this study adopts L as a fixed value, L = 200 mm. The joint connectivity of the rock mass with different plane joints is shown in Figure 6.Figure 3
The positions of the five measurement circles.Figure 4
The schematic diagram of the joint relief angle.Figure 5
The loading mode and the joint inclination angle of NNJRM.Figure 6
The joint connectivity of NNJRM.


## 2. Numerical Simulation Test Scheme for NNJRM

### 2.1. Establishment of Model

In the numerical simulation part of this study, PFC2D was used to investigate and analyze the NNJRM. The establishment process of the test model is as follows: (1) six walls were defined to form a rectangular range of 200 mm × 200 mm, (2) relevant mesomechanical parameters were calibrated, and the particles were randomly generated and the suspended ones were eliminated (a total of 9356 effective spherical particles were generated inside the wall), (3) joints with connectivity, undulation, and inclination were introduced into the model to form the NNJRM, (4) direct shear test was conducted through FISH language simulation, and (5) five measuring circles were introduced to measure the level of stress (the five measuring circles are shown in Figure3). The entire modeling process is illustrated in Figure 7.Figure 7
The model establishment process. (a) The range of 6-wall forming models. (b) Spherical particles formed in the wall. (c) The sample model after sphere stabilization. (d) The model after importing the different joints. (e) The model after the direct shear test.
(a)(b)(c)(d)(e)


### 2.2. Selection of Microscopic Mechanical Parameters

It should be noted that the calibration of mesomechanical parameters is the core of the model establishment process. It is only when these parameters are set accurately can the test results obtained by the numerical simulation test be true and reliable and can the numerical model be applied to the study of the mechanical properties of NNJRM. The specific method of parameter calibration is as follows: a group of parameters, which can make the numerical simulation test results to match with the experimental results of the physical model test, was obtained through multiple preassignment of micromechanical parameters such as the adhesion stiffness, bond strength, particle density, particle radius, elastic modulus of particles, and Poisson's ratio between particles and walls. Since a set of parameters that meet the requirements were already determined in previous studies [27, 28], the calibration process of the parameters was not repeated here. The set of parameters used in this model is shown in Table 1.Table 1
Some relevant parameters of numerical simulation.
Partial parameters of spherical particlesPartial parameters of the wallPartial parameters of the bondPorosity distributionRadius (mm)Attribute density (kg·m−3)Damping coefficientNormal stiffness (N·m−1)Tangential stiffness (N·m−1)Friction factorNormal stiffness (N·m−1)Tangential stiffness (N·m−1)Radius coefficientNormal stiffness (N·m−1)Tangential stiffness (N·m−1)Friction angle (°)0.020.03∼0.1827500.73 × 1081 × 1080.31 × 10101 × 10811 × 10101 × 101055


### 2.3. Description of the Test Conditions

The numerical simulation tests of a rock mass with different discontinuity joints under different working conditions are divided into five groups, as shown in Table2.Table 2
Grouping of numerical simulation tests.
GroupConnectivity rateNormal stress (MPa)Horizontal displacement rate (mm/s)The joint relief angleThe joint inclination angle10.50.50.0615°15°1.01.52.03.020.11.00.0615°15°0.20.30.40.530.51.00.0215°15°0.040.060.080.1040.51.00.060°15°15°30°45°60°50.51.00.0615°0°15°30°45°60°


#### 2.3.1. Group 1: Simulation Test of NNJRM under Different Normal Stresses

When the shear rate is 0.06 mm/s, the numerical simulation tests with normal stresses of 0.5mpa, 1.0 MPa, 1.5mpa, 2.0mpa, and 3.0mpa were carried out for the NNJRM with a joint relief angle of 15°, inclination angle of 15°, and connectivity rate of 0.5.


#### 2.3.2. Group 2: Simulation Test of NNJRM with Different Connectivity Rates

When the shear rate is 0.06 mm/s, the numerical simulation tests with a normal stress of 1.0 MPa were carried out for the NNJRM with a joint relief angle of 15°, inclination angle of 15°, and connectivity rates of 0.1, 0.2, 0.3, 0.4, and 0.5, successively.


#### 2.3.3. Group 3: Simulation Test of NNJRM at Different Shear Rate States

When the normal stress is 1.0 MPa, the numerical simulation tests with horizontal shear rates of 0.02 mm/s, 0.04 mm/s, 0.06 mm/s, 0.08 mm/s, and 0.10 mm/s were carried out for the NNJRM with a joint relief angle of 15°, inclination angle of 15°, and connectivity rate of 0.5.


#### 2.3.4. Group 4: Simulation Test of NNJRM with Different Joint Relief Angles

When the shear rate is 0.06 mm/s and the normal stress is 1.0 MPa, the numerical simulation tests were carried out for the NNJRM with joint inclination angle of 15°, connectivity rate of 0.5, and joint relief angles of 0°, 15°, 30°, 45°, and 60°, successively.


#### 2.3.5. Group 5: Simulation Test of NNJRM with Different Joint Inclination Angles

When the shear rate is 0.06 mm/s and the normal stress is 1.0 MPa, the numerical simulation tests were carried out for the NNJRM with a joint relief angle of 15°, connectivity rate of 0.5, and joint inclination angles of 0°, 15°, 30°, 45°, and 60°, successively.


## 2.1. Establishment of Model

In the numerical simulation part of this study, PFC2D was used to investigate and analyze the NNJRM. The establishment process of the test model is as follows: (1) six walls were defined to form a rectangular range of 200 mm × 200 mm, (2) relevant mesomechanical parameters were calibrated, and the particles were randomly generated and the suspended ones were eliminated (a total of 9356 effective spherical particles were generated inside the wall), (3) joints with connectivity, undulation, and inclination were introduced into the model to form the NNJRM, (4) direct shear test was conducted through FISH language simulation, and (5) five measuring circles were introduced to measure the level of stress (the five measuring circles are shown in Figure3). The entire modeling process is illustrated in Figure 7.Figure 7
The model establishment process. (a) The range of 6-wall forming models. (b) Spherical particles formed in the wall. (c) The sample model after sphere stabilization. (d) The model after importing the different joints. (e) The model after the direct shear test.
(a)(b)(c)(d)(e)


## 2.2. Selection of Microscopic Mechanical Parameters

It should be noted that the calibration of mesomechanical parameters is the core of the model establishment process. It is only when these parameters are set accurately can the test results obtained by the numerical simulation test be true and reliable and can the numerical model be applied to the study of the mechanical properties of NNJRM. The specific method of parameter calibration is as follows: a group of parameters, which can make the numerical simulation test results to match with the experimental results of the physical model test, was obtained through multiple preassignment of micromechanical parameters such as the adhesion stiffness, bond strength, particle density, particle radius, elastic modulus of particles, and Poisson's ratio between particles and walls. Since a set of parameters that meet the requirements were already determined in previous studies [27, 28], the calibration process of the parameters was not repeated here. The set of parameters used in this model is shown in Table 1.Table 1
Some relevant parameters of numerical simulation.
Partial parameters of spherical particlesPartial parameters of the wallPartial parameters of the bondPorosity distributionRadius (mm)Attribute density (kg·m−3)Damping coefficientNormal stiffness (N·m−1)Tangential stiffness (N·m−1)Friction factorNormal stiffness (N·m−1)Tangential stiffness (N·m−1)Radius coefficientNormal stiffness (N·m−1)Tangential stiffness (N·m−1)Friction angle (°)0.020.03∼0.1827500.73 × 1081 × 1080.31 × 10101 × 10811 × 10101 × 101055


## 2.3. Description of the Test Conditions

The numerical simulation tests of a rock mass with different discontinuity joints under different working conditions are divided into five groups, as shown in Table2.Table 2
Grouping of numerical simulation tests.
GroupConnectivity rateNormal stress (MPa)Horizontal displacement rate (mm/s)The joint relief angleThe joint inclination angle10.50.50.0615°15°1.01.52.03.020.11.00.0615°15°0.20.30.40.530.51.00.0215°15°0.040.060.080.1040.51.00.060°15°15°30°45°60°50.51.00.0615°0°15°30°45°60°


### 2.3.1. Group 1: Simulation Test of NNJRM under Different Normal Stresses

When the shear rate is 0.06 mm/s, the numerical simulation tests with normal stresses of 0.5mpa, 1.0 MPa, 1.5mpa, 2.0mpa, and 3.0mpa were carried out for the NNJRM with a joint relief angle of 15°, inclination angle of 15°, and connectivity rate of 0.5.


### 2.3.2. Group 2: Simulation Test of NNJRM with Different Connectivity Rates

When the shear rate is 0.06 mm/s, the numerical simulation tests with a normal stress of 1.0 MPa were carried out for the NNJRM with a joint relief angle of 15°, inclination angle of 15°, and connectivity rates of 0.1, 0.2, 0.3, 0.4, and 0.5, successively.


### 2.3.3. Group 3: Simulation Test of NNJRM at Different Shear Rate States

When the normal stress is 1.0 MPa, the numerical simulation tests with horizontal shear rates of 0.02 mm/s, 0.04 mm/s, 0.06 mm/s, 0.08 mm/s, and 0.10 mm/s were carried out for the NNJRM with a joint relief angle of 15°, inclination angle of 15°, and connectivity rate of 0.5.


### 2.3.4. Group 4: Simulation Test of NNJRM with Different Joint Relief Angles

When the shear rate is 0.06 mm/s and the normal stress is 1.0 MPa, the numerical simulation tests were carried out for the NNJRM with joint inclination angle of 15°, connectivity rate of 0.5, and joint relief angles of 0°, 15°, 30°, 45°, and 60°, successively.


### 2.3.5. Group 5: Simulation Test of NNJRM with Different Joint Inclination Angles

When the shear rate is 0.06 mm/s and the normal stress is 1.0 MPa, the numerical simulation tests were carried out for the NNJRM with a joint relief angle of 15°, connectivity rate of 0.5, and joint inclination angles of 0°, 15°, 30°, 45°, and 60°, successively.


## 2.3.1. Group 1: Simulation Test of NNJRM under Different Normal Stresses

When the shear rate is 0.06 mm/s, the numerical simulation tests with normal stresses of 0.5mpa, 1.0 MPa, 1.5mpa, 2.0mpa, and 3.0mpa were carried out for the NNJRM with a joint relief angle of 15°, inclination angle of 15°, and connectivity rate of 0.5.


## 2.3.2. Group 2: Simulation Test of NNJRM with Different Connectivity Rates

When the shear rate is 0.06 mm/s, the numerical simulation tests with a normal stress of 1.0 MPa were carried out for the NNJRM with a joint relief angle of 15°, inclination angle of 15°, and connectivity rates of 0.1, 0.2, 0.3, 0.4, and 0.5, successively.


## 2.3.3. Group 3: Simulation Test of NNJRM at Different Shear Rate States

When the normal stress is 1.0 MPa, the numerical simulation tests with horizontal shear rates of 0.02 mm/s, 0.04 mm/s, 0.06 mm/s, 0.08 mm/s, and 0.10 mm/s were carried out for the NNJRM with a joint relief angle of 15°, inclination angle of 15°, and connectivity rate of 0.5.


## 2.3.4. Group 4: Simulation Test of NNJRM with Different Joint Relief Angles

When the shear rate is 0.06 mm/s and the normal stress is 1.0 MPa, the numerical simulation tests were carried out for the NNJRM with joint inclination angle of 15°, connectivity rate of 0.5, and joint relief angles of 0°, 15°, 30°, 45°, and 60°, successively.


## 2.3.5. Group 5: Simulation Test of NNJRM with Different Joint Inclination Angles

When the shear rate is 0.06 mm/s and the normal stress is 1.0 MPa, the numerical simulation tests were carried out for the NNJRM with a joint relief angle of 15°, connectivity rate of 0.5, and joint inclination angles of 0°, 15°, 30°, 45°, and 60°, successively.


## 3. Results and Discussion

In this study, five groups of numerical models of rock mass with different conditions were simulated. The peak stress of the specimens, the total number of cracks in the rock mass (expressed by DFN), the number of cracks produced by tension (expressed by DFN-t), the number of cracks produced by shear (expressed by DFN-s), and the stress-strain relationship of five measuring points in the specimens were all studied to analyze the law of crack propagation and the mechanical properties of the rock mass.


### 3.1. Results of the First Experiment

The stress-strain curves and peak shear stress of NNJRM under different normal stresses of the first group of specimens are shown in Figure8 and Table 3.Figure 8
Stress-strain curves obtained from tests under different normal stresses.Table 3
Peak shear stress of NNJRM under different normal stresses.
Normal stress (MPa)0.51.01.52.03.0Peak stress (MPa)3.763.914.194.504.98According to the data in Figure8 and Table 3, it can be observed that when the normal stress increases from 0.5 MPa to 3.0 MPa, the peak stress of NNJRM increases from 3.76 MPa to 4.98 MPa, and the residual stress increases from 0.47 MPa to 3.15 MPa. It can be seen that with the increase in the normal stress in the test, both the shear strength and peak stress of NNJRM increase.The number of DFN, DFN-t, and DFN-s produced in the first group of specimens under different normal stresses are shown in Figure9 and Table 4.Figure 9
The number of cracks in the first group of specimens. (a) Total number of cracks. (b) The number of cracks caused by tension. (c) The number of cracks caused by shear action.
(a)(b)(c)Table 4
The number of cracks generated under different normal stresses.
Normal stress (MPa)0.51.01.52.03.0DFN (strip)293457355522515DFN-t (strip)198343271401409DFN-s (strip)9511484121106It can be seen from Figure9 and the data in Table 4 that the number of cracks generated in the rock mass increased continuously in the course of the experiment until the end of the test. When the normal stress is gradually increased from 0.5 MPa to 3.0 MPa, the total number of cracks in the rock mass increases from 293 to 515. With the gradual increase in the normal stress from 0.5 MPa to 3.0 MPa, the number of DFN-t in the rock mass increases from 198 to 409, showing an obvious trend of increase. The number of DFN-s also changes with the change in normal stress, increasing from 95 to 106, but the increase trend is not obvious. In addition, by comparing the number of DFN-t and DFN-s in Table 4, it is obvious that no matter how the normal stress changes, the number of cracks in each rock mass caused by tensile action is more than those caused by shear action. Therefore, during the direct shear test, the damage caused by the tensile action of the rock mass is more obvious. Thus, it can be said that the failure of rock mass in the test is mainly caused by tension.In addition, the experiment under different normal stresses, the stress-strain curves, and the peak strength of each measuring circle are shown in Figure10 and Table 5, respectively. After sorting out the data of the experiment under different normal stresses, the obtained curves of peak stress of each measured circle are shown in Figure 11.Figure 10
The stress-strain curves of the five measured circles in the first group of specimens when the normal stress is (a) 0.5 MPa, (b) 1.0 MPa, (c) 1.5 MPa, (d) 2.0 MPa, and (e) 3.0 MPa.
(a)(b)(c)(d)(e)Table 5
The peak stress of each measurement circle in the first group of specimens (MPa).
Number of measuring circlesNormal stress (MPa)0.51.01.52.03.014.755.082.455.644.9334.294.685.205.315.8656.276.3210.2510.27.4373.323.684.044.373.4394.905.426.026.188.44Figure 11
Curves of the peak stress of each measured circle in the first group of specimens.It can be seen from the data in Figures10 and 11 and Table 5 that with the change in the normal stress, the stress curve of each measured point in the rock mass changes significantly. On the one hand, for the same measurement point under different normal stresses: the peak stress of the rock mass will change with the increase in normal stress. The change in the peak stresses of measurement circles 1 and 5 is the most obvious. Here, the measurement circle 1 is taken as an example for analysis. When the normal stress is gradually increased from 0.5 MPa to 3.0 MPa, the measured peak stresses are 6.27 MPa, 6.32 MPa, 10.25 MPa, 10.2 MPa, and 7.43 MPa, respectively. The difference between the maximum peak stress and the minimum peak stress is 3.98 MPa. For the three other measurement circles, i.e. 3, 7, and 9, although their peak stress also changes with the increase in normal stress, the range of change is not obvious and is relatively stable. On the other hand, for the experiment with different measurement points under the same normal stress, by observing the stress curves of the five measurement points, it can be observed that no matter how much normal stress is applied during the direct shear test, the peak stress of the measurement point 5 is almost always the largest among the five measurement points. In addition, for this experiment, there is a big difference in the peak stress of measurement point 5 and that of the other measurement circles. When the normal stress is 3.0 Mpa, the peak stress at the measurement point 5 is 7.43 MPa, and although this is not the maximum value, the difference between this peak stress and the maximum peak stress of 8.44 MPa is still small. It can be seen that during the shear test, the stress of the rock mass is more concentrated at the middle part of the rock mass, that is, around the measuring circle 5, which bears the main shear stress.


### 3.2. Results of the Second Experiment

The stress-strain curves obtained by the direct shear test of the rock mass under different shear rates are shown in Figure12. The peak stresses of the rock mass shown in Figure 12 are statistically analyzed, and the results are shown in Table 6.Figure 12
The stress-strain curves obtained from experiments under different shear rates.Table 6
The peak shear stress of NNJRM under different shear rates.
Shear rate (mm/s)0.020.040.060.080.10Peak stress (MPa)3.854.004.034.154.30According to the data in Figure12 and Table 6, it can be found that the shear strength of the rock mass changes according to the change in the shear rate in the direct shear test. When the shear rate is 0.02 mm/s and 0.06 mm/s, the peak shear stress of the rock mass is 3.85 Mpa and 4.03 Mpa, respectively. Furthermore, when the shear rate is gradually increased to 0.10 mm/s, the peak shear stress of the rock mass increases to 4.3 MPa. Therefore, the peak stress of the rock mass increases with the increase in the shear rate, and the shear strength of the rock mass is also enhanced.In addition, the number of DFN, DFN-t, and DFN-s produced in the second group of specimens for the experiment under different shear rates are shown in Figure13 and Table 7.Figure 13
The number of cracks in the second group of specimens. (a) Total number of cracks. (b) The number of cracks caused by tension. (c) The number of cracks caused by shear action.
(a)(b)(c)Table 7
The number of cracks generated under different shear rates.
Shear rate (mm/s)0.020.040.060.080.10DFN (strip)344343449462371DFN-t (strip)241249335354289DFN-s (strip)1039411410882According to the data in Figure13 and Table 7, the shear rate has significant influence on the number of cracks in the rock mass during the direct shear test. First, the total number of cracks (DFN) is analyzed. It can be seen that when the shear rate is 0.02 mm/s and 0.04 mm/s, the total number of cracks in the rock mass is 344 and 449, respectively. However, when the shear rate continues to increase, the total number of cracks tends to decrease. For instance, when the shear rate is increased to 0.10 mm/s, the total number of cracks in the rock mass decreases to 371. Therefore, it can be said that with the increase in the shear rate, the total number of cracks in the rock mass increases first and then decreases. Second, the quantity of DFN-t is analyzed, and it is found that the variation law of DFN-t is similar to that of DFN; the total number of cracks in the rock mass first increases and then decreases with the increase in the shear rate. For DFN-s, although the number of cracks also increases first and then decreases, the change is not obvious. Finally, by comparing the number of DFN-t and DFN-s, it is found that the number of DFN-t is much higher than that of DFN-s, the change of shear rate notwithstanding. Therefore, it can be concluded that, for the experiment under different shear rates, the number of cracks in the rock mass caused by tension is far higher than that caused by shear action. This means that the ultimate failure of a rock mass in the direct shear test is mainly caused by tension, rather than shear.The stress-strain curves of each measurement circle under different shear rates are shown in Figure14. The peak stress of each measurement circle is determined and presented in Table 8. The change curve of the peak stress of each measurement point under the condition of different shear rates is shown in Figure 15.Figure 14
Stress-strain curves of each measurement circle in the second group of specimens when the shear rate is (a) 0.02 mm/s, (b) 0.04 mm/s, (c) 0.06 mm/s, (d) 0.08 mm/s, and (e) 0.10 mm/s.
(a)(b)(c)(d)(e)Table 8
The peak stress of each measurement circle in the second group of specimens.
Number of measuring circlesShear rate (mm/s)0.020.040.060.080.1012.104.503.94.434.4834.454.334.644.324.5659.998.716.385.274.9171.341.512.212.742.8095.265.495.415.815.97Figure 15
Curves of peak stress of each measured circle in the second group of specimens.It can be seen from the data in Figures14 and 15 and Table 8 that the shear rate has significant influence on rock mass stress.(1)
On the one hand, for the stress at the same measuring point under different shear rates: there is little change in peak stress of the four measurement circles 1 (except at 0.02 mm/s, the peak stress of the measurement circle 1 is discrete), 3, 7, and 9, when the shear rate is adjusted. However, for the measurement circle 5, there is obvious change in the peak stress when the shear rate is adjusted. When the shear rate is increased from 0.02 mm/s to 0.10 mm/s, the peak stress of measurement circle 5 decreases from 9.99 MPa to 4.91 MPa. This means that the peak stress of the measurement circle 5 gradually decreases with the increase in the shear rate. Therefore, with the increase in the shear rate, both the shear capacity and shear stress at the center point of the rock mass decrease. In addition, as can be seen from the data in Figure12 and Table 6, the peak stress and the shear strength of the whole rock mass increase with the increase in the shear rate during the test. In summary, as the shear rate increases, the shear stress borne by the center of the rock mass decreases, and the force borne by the interior of the rock mass gradually balances.(2)
On the other hand, for the stress of different measurement circles under the same shear rate condition: the peak stresses of the measurement circle 5 at the shear rates of 0.02 mm/s, 0.04 mm/s, and 0.06 mm/s are 9.99 MPa, 8.71 MPa, and 6.38 MPa, respectively, and these are the maximum values of the peak stress of the five measurement circles. In addition, when the shear rates are 0.08 mm/s and 0.10 mm/s, the peak stresses at the measurement circle 5 are 5.27 MPa and 4.91 MPa, respectively. Although these values differ slightly with the maximum value, they are still relatively large compared with those of other measurement circles. Therefore, in general, under different shear rates, the central position of a rock mass specimen is subjected to greater shear action and the stress variation is more obvious.


### 3.3. Results of the Third Experiment

The stress-strain curve obtained after the direct shear test of the rock mass with different joint connectivity rates is shown in Figure16. The peak stress of the rock mass in Figure 16 is sorted out, and the results are presented in Table 9.Figure 16
The stress-strain curve of the rock mass with different joint connectivity rates.Table 9
The peak shear stress of NNJRM with different joint connectivity rates.
Connectivity rate0.10.20.30.40.5Peak shear stress (MPa)5.905.805.775.564.03According to the data in Figure16 and Table 9, it can be found that the test results of NNJRM with different joint connectivity rates have obvious consistency. When the connectivity of joints within the rock mass is 0.1, the peak shear stress of the rock mass is 5.90 MPa, and the residual stress is 1.34 MPa. When the joint connectivity rate increases gradually, the peak stress and residual stress of the rock mass tend to decrease. When the joint connectivity rate increases to 0.5, the peak shear stress and residual stress decrease to 4.03 MPa and 0.75 MPa, respectively. Therefore, the smaller the joint connectivity, the greater the peak stress and residual stress of the rock mass. Furthermore, the stronger the shear capacity of a rock mass, the stronger the ability to resist external forces.In addition, the number of DFN, DFN-t, and DFN-s produced in the third group of specimens with different joint connectivity rates are shown in Figure17 and Table 10.Figure 17
The number of cracks in the third group of specimens. (a) Total number of cracks. (b) The number of cracks caused by tension. (c) The number of cracks caused by shear action.
(a)(b)(c)Table 10
The number of cracks generated at different connectivity rates.
Connectivity rate0.10.20.30.40.5DFN (strip)829681720654457DFN-t (strip)640526563502343DFN-s (strip)189155157152114By summarizing and sorting out the data in Figure17 and Table 10, it can be found that no matter the value of the joint connectivity rate, the number of cracks generated in the rock mass gradually increases with the progress of the test process. However, different joint connectivity rates generate different increment rates of cracks. When the connectivity of joints was 0.1, the number of DFN, DFN-s, and DFN-t increased to 829, 189, and 640, respectively. When the joint connectivity rate gradually increased to 0.5, the final total number of cracks in the experimental rock mass decreased to 457, among which 114 are DFN-s and 343 are DFN-t.According to these data, the number of these two kinds of cracks, DFN-s, and DFN-t, which are caused by different forms of forces, decreases with the increase in joint connectivity, leading to a decrease in the total amount of cracks in the rock mass. Therefore, it can be concluded that the smaller the connectivity of joints within the rock mass, the better the integrity of the rock mass and the stronger the shear resistance capability of the rock mass. This is consistent with practical experiences in engineering practice.The stress-strain curves of each measuring circle in the specimens with different joint connectivity rates are shown in Figure18. The peak stress of each measuring circle is sorted out as shown in Table 11, and the change curve of peak stress of each measurement point under the condition of different joint connectivity rates is obtained, as shown in Figure 19.Figure 18
Stress-strain curves of each measurement circle in the third group of specimens when the connectivity rate is (a) 0.1, (b) 0.2, (c) 0.3, (d) 0.4, and (e) 0.5.
(a)(b)(c)(d)(e)Table 11
The peak stress of each measurement circle in the third group of specimens.
Number of measuring circlesConnectivity rate0.10.20.30.40.516.286.054.314.64.00310.09.178.306.514.5452.232.533.657.096.3475.515.726.264.813.8195.916.386.796.865.60Figure 19
Curves of peak stress of each measurement circle in the third group of specimens.Based on the data shown in Figures18 and 19 and Table 11, we find that, unlike direct shear tests conducted under the above controlled conditions, the maximum peak shear stress is not always at the measurement circle 5 located at the center of the specimen when the test is carried out at different joint connectivity rates. When the connectivity of the joints in the rock mass ranges from 0.1 to 0.3, the peak shear stress of the measurement circle 3 in the rock mass is the largest among the five measuring points. The peak stresses of measurement circles 1, 7, and 9 near both sides of the rock mass are large, while that of measuring circle 5 in the middle of the specimen is the smallest. Therefore, when the connection rate of joints is 0.1 to 0.3, the rock mass close to both sides of the rock mass bears more stress. However, when the connectivity of joints increases to 0.4 or 0.5, the measurement circle 5 experiences the highest peak stress among the five measuring points, and the peak stress of the five measurement circles is more uniform.


### 3.4. Results of the Fourth Experiment

The curves of stress-strain and peak shear stress of NNJRM with different joint relief angles of the fourth group specimens are shown in Figure20 and Table 12.Figure 20
The stress-strain curves of NNJRM with different relief angles.Table 12
The peak shear stress of NNJRM with different relief angles.
Relief angle0°15°30°45°60°Peak shear stress (MPa)3.964.124.283.834.09According to the data in Figure20 and Table 12, it can be found that the relief angle of joints has an influence on the stress-strain curve of NNJRM. If the joint relief angle changes, the peak shear stress and residual stress of the rock mass show obvious changes. However, there is no obvious regularity to this change.The number of DFN, DFN-t, and DFN-s produced in the fourth group of specimens with different relief angles are shown in Figure21 and Table 13.Figure 21
The number of cracks in the fourth group of specimens. (a) Total number of cracks. (b) The number of cracks caused by tension. (c) The number of cracks caused by shear action.
(a)(b)(c)Table 13
The number of cracks generated with different relief angles.
Relief angle0°15°30°45°60°DFN (strip)414447342510634DFN-t (strip)292335263386504DFN-s (strip)12211279124130By comparing the number of DFN, DFN-s, and DFN-t in the rock mass with five kinds of joint relief angles, it can be found that the change of joint relief angle has obvious influence on the direct shear test of a jointed rock mass. The difference between the maximum value (634) and the minimum value (342) of the total number of cracks produced by the direct shear test on the rock mass with different joint relief angles is 292. Similarly, the difference between the maximum value (504) and the minimum value (263) of the number of DFN-t is 241. Furthermore, under the same joint relief angle, the number of DFN-s generated in the specimen is significantly less than that of DFN-t. Therefore, the following conclusions can be drawn: in the direct shear tests of a rock mass with different joint relief angles, the main reason for its failure is the action of tension rather than the shear action.The stress-strain curves of each measurement circle in the fourth group of specimens with different joint relief angles are shown in Figure22. The peak stress of each measurement circle of the specimens under each joint relief angle is sorted out as shown in Table 14, and the change curve of the peak stress of each measured point with different joint relief angles is obtained, as shown in Figure 23.Figure 22
The stress-strain curves of each measurement circle in the fourth group of specimens when the relief angle is (a) 0°, (b) 15°, (c) 30°, (d) 45°, and (e) 60°.
(a)(b)(c)(d)(e)Table 14
The peak stress of each measurement circle in the fourth group of specimens.
Number of measuring circlesRelief angle0°15°30°45°60°12.074.094.013.070.2932.554.464.983.64.7158.496.355.875.247.5672.224.372.042.253.8494.495.655.724.464.32Figure 23
Curves of peak stress of each measurement circle in the fourth group of specimens.According to the data in Figures22 and 23 and Table 14, when conducting direct shear test on the rock mass with joints with different undulating angles, the measurement circle 5 of the rock mass experiences the largest peak stress among the five measurement circles. It is proved that the central part of the rock mass with discontinuous joints bears more forces during the direct shear test. In addition, according to the curve in Figure 23, when the joint relief angle is 0°or 60°, the peak stress of each measurement circle in the rock mass varies greatly and the stress distribution is not uniform.


### 3.5. Results of the Fifth Experiment

The curve of stress-strain and peak shear stress of NNJRM with different inclination angles of the fifth group of specimens are shown in Figure24 and Table 15.Figure 24
The stress-strain curves of NNJRM with different joint inclination angles.Table 15
The peak shear stress of NNJRM with different joint inclination angles.
Joint inclination angle0°15°30°45°60°Peak shear stress (MPa)3.954.015.926.526.00We can see from the data in Figure24 and Table 15 that when the joint inclination angle is 0° and 15°, the peak stress of the rock mass is 3.95 MPa and 4.01 MPa, respectively. If the angle is increased to 60°, the peak shear stress increases to 6.00 MPa. It can be seen that the peak stress of the rock mass increases with the increase in the joint inclination angle. Therefore, when the inclined angle of the joints is increased, the shear strength of the rock mass with different plane joints is enhanced.The number of cracks generated by the fifth group of specimens during the experiment is shown in Figure25 and Table 16.Figure 25
The number of cracks in the fifth group of specimens. (a) Total number of cracks. (b) The number of cracks caused by tension. (c) The number of cracks caused by shear action.
(a)(b)(c)Table 16
The number of cracks generated with different joint inclination angles.
Joint inclination angles0°15°30°45°60°DFN (strip)7654537811307935DFN-t (strip)5573405741040726DFN-s (strip)208114207267209From the chart and table above, it can be found that the dip angle of the joints has a significant effect on the crack propagation of the incoherent jointed rock mass. When the inclination angle of joints is changed, the number of cracks generated in the rock mass shows a very obvious change. However, there is no specific rule for the change in the number of cracks. Although the variation law of DFN, DFN-t, and DFN-s is not obvious, the number of DFN-t is always more than that of DFN-s. It is proved that the final failure of the NNJRM with different inclination angles in the direct shear test is also caused by tension.The stress-strain curves of each measurement circle in the specimens with different joint inclination angles are shown in Figure26. The peak stress of each measuring circle is sorted out as shown in Table 17, and the change curve of peak stress of each measured point under the condition of different joint inclination angles is obtained, as shown in Figure 27.Figure 26
The stress-strain curves of each measurement circle in the fifth group of specimens when the joint inclination angle is (a) 0°, (b) 15°, (c) 30°, (d) 45°, and (e) 60°.
(a)(b)(c)(d)(e)Table 17
The peak stress of each circle in the fifth group of specimens.
Number of measuring circlesJoint inclination angle0°15°30°45°60°10.583.976.493.935.7137.264.896.379.039.33511.064.624.343.242.5379.663.725.586.275.5390.845.526.708.197.71Figure 27
Curves of peak stress of each measurement circle in the fifth group of specimens.Based on the data obtained through the above simulation experiment, it can be seen that the joint inclination angle has significant influence on the strength characteristics of the rock mass. Although it is not possible to establish a specific law that reflects the influence of the joint inclination angle on the peak stress at each position in the rock mass, it is certain that the peak stress of the five measurement circles in the rock mass varies with the change in the joint inclination angle. Moreover, when the joint inclination angle is 15°, the peak stresses of the five measurement circles in the rock mass are not significantly different, and the stress distribution in the model specimen is more uniform.


## 3.1. Results of the First Experiment

The stress-strain curves and peak shear stress of NNJRM under different normal stresses of the first group of specimens are shown in Figure8 and Table 3.Figure 8
Stress-strain curves obtained from tests under different normal stresses.Table 3
Peak shear stress of NNJRM under different normal stresses.
Normal stress (MPa)0.51.01.52.03.0Peak stress (MPa)3.763.914.194.504.98According to the data in Figure8 and Table 3, it can be observed that when the normal stress increases from 0.5 MPa to 3.0 MPa, the peak stress of NNJRM increases from 3.76 MPa to 4.98 MPa, and the residual stress increases from 0.47 MPa to 3.15 MPa. It can be seen that with the increase in the normal stress in the test, both the shear strength and peak stress of NNJRM increase.The number of DFN, DFN-t, and DFN-s produced in the first group of specimens under different normal stresses are shown in Figure9 and Table 4.Figure 9
The number of cracks in the first group of specimens. (a) Total number of cracks. (b) The number of cracks caused by tension. (c) The number of cracks caused by shear action.
(a)(b)(c)Table 4
The number of cracks generated under different normal stresses.
Normal stress (MPa)0.51.01.52.03.0DFN (strip)293457355522515DFN-t (strip)198343271401409DFN-s (strip)9511484121106It can be seen from Figure9 and the data in Table 4 that the number of cracks generated in the rock mass increased continuously in the course of the experiment until the end of the test. When the normal stress is gradually increased from 0.5 MPa to 3.0 MPa, the total number of cracks in the rock mass increases from 293 to 515. With the gradual increase in the normal stress from 0.5 MPa to 3.0 MPa, the number of DFN-t in the rock mass increases from 198 to 409, showing an obvious trend of increase. The number of DFN-s also changes with the change in normal stress, increasing from 95 to 106, but the increase trend is not obvious. In addition, by comparing the number of DFN-t and DFN-s in Table 4, it is obvious that no matter how the normal stress changes, the number of cracks in each rock mass caused by tensile action is more than those caused by shear action. Therefore, during the direct shear test, the damage caused by the tensile action of the rock mass is more obvious. Thus, it can be said that the failure of rock mass in the test is mainly caused by tension.In addition, the experiment under different normal stresses, the stress-strain curves, and the peak strength of each measuring circle are shown in Figure10 and Table 5, respectively. After sorting out the data of the experiment under different normal stresses, the obtained curves of peak stress of each measured circle are shown in Figure 11.Figure 10
The stress-strain curves of the five measured circles in the first group of specimens when the normal stress is (a) 0.5 MPa, (b) 1.0 MPa, (c) 1.5 MPa, (d) 2.0 MPa, and (e) 3.0 MPa.
(a)(b)(c)(d)(e)Table 5
The peak stress of each measurement circle in the first group of specimens (MPa).
Number of measuring circlesNormal stress (MPa)0.51.01.52.03.014.755.082.455.644.9334.294.685.205.315.8656.276.3210.2510.27.4373.323.684.044.373.4394.905.426.026.188.44Figure 11
Curves of the peak stress of each measured circle in the first group of specimens.It can be seen from the data in Figures10 and 11 and Table 5 that with the change in the normal stress, the stress curve of each measured point in the rock mass changes significantly. On the one hand, for the same measurement point under different normal stresses: the peak stress of the rock mass will change with the increase in normal stress. The change in the peak stresses of measurement circles 1 and 5 is the most obvious. Here, the measurement circle 1 is taken as an example for analysis. When the normal stress is gradually increased from 0.5 MPa to 3.0 MPa, the measured peak stresses are 6.27 MPa, 6.32 MPa, 10.25 MPa, 10.2 MPa, and 7.43 MPa, respectively. The difference between the maximum peak stress and the minimum peak stress is 3.98 MPa. For the three other measurement circles, i.e. 3, 7, and 9, although their peak stress also changes with the increase in normal stress, the range of change is not obvious and is relatively stable. On the other hand, for the experiment with different measurement points under the same normal stress, by observing the stress curves of the five measurement points, it can be observed that no matter how much normal stress is applied during the direct shear test, the peak stress of the measurement point 5 is almost always the largest among the five measurement points. In addition, for this experiment, there is a big difference in the peak stress of measurement point 5 and that of the other measurement circles. When the normal stress is 3.0 Mpa, the peak stress at the measurement point 5 is 7.43 MPa, and although this is not the maximum value, the difference between this peak stress and the maximum peak stress of 8.44 MPa is still small. It can be seen that during the shear test, the stress of the rock mass is more concentrated at the middle part of the rock mass, that is, around the measuring circle 5, which bears the main shear stress.


## 3.2. Results of the Second Experiment

The stress-strain curves obtained by the direct shear test of the rock mass under different shear rates are shown in Figure12. The peak stresses of the rock mass shown in Figure 12 are statistically analyzed, and the results are shown in Table 6.Figure 12
The stress-strain curves obtained from experiments under different shear rates.Table 6
The peak shear stress of NNJRM under different shear rates.
Shear rate (mm/s)0.020.040.060.080.10Peak stress (MPa)3.854.004.034.154.30According to the data in Figure12 and Table 6, it can be found that the shear strength of the rock mass changes according to the change in the shear rate in the direct shear test. When the shear rate is 0.02 mm/s and 0.06 mm/s, the peak shear stress of the rock mass is 3.85 Mpa and 4.03 Mpa, respectively. Furthermore, when the shear rate is gradually increased to 0.10 mm/s, the peak shear stress of the rock mass increases to 4.3 MPa. Therefore, the peak stress of the rock mass increases with the increase in the shear rate, and the shear strength of the rock mass is also enhanced.In addition, the number of DFN, DFN-t, and DFN-s produced in the second group of specimens for the experiment under different shear rates are shown in Figure13 and Table 7.Figure 13
The number of cracks in the second group of specimens. (a) Total number of cracks. (b) The number of cracks caused by tension. (c) The number of cracks caused by shear action.
(a)(b)(c)Table 7
The number of cracks generated under different shear rates.
Shear rate (mm/s)0.020.040.060.080.10DFN (strip)344343449462371DFN-t (strip)241249335354289DFN-s (strip)1039411410882According to the data in Figure13 and Table 7, the shear rate has significant influence on the number of cracks in the rock mass during the direct shear test. First, the total number of cracks (DFN) is analyzed. It can be seen that when the shear rate is 0.02 mm/s and 0.04 mm/s, the total number of cracks in the rock mass is 344 and 449, respectively. However, when the shear rate continues to increase, the total number of cracks tends to decrease. For instance, when the shear rate is increased to 0.10 mm/s, the total number of cracks in the rock mass decreases to 371. Therefore, it can be said that with the increase in the shear rate, the total number of cracks in the rock mass increases first and then decreases. Second, the quantity of DFN-t is analyzed, and it is found that the variation law of DFN-t is similar to that of DFN; the total number of cracks in the rock mass first increases and then decreases with the increase in the shear rate. For DFN-s, although the number of cracks also increases first and then decreases, the change is not obvious. Finally, by comparing the number of DFN-t and DFN-s, it is found that the number of DFN-t is much higher than that of DFN-s, the change of shear rate notwithstanding. Therefore, it can be concluded that, for the experiment under different shear rates, the number of cracks in the rock mass caused by tension is far higher than that caused by shear action. This means that the ultimate failure of a rock mass in the direct shear test is mainly caused by tension, rather than shear.The stress-strain curves of each measurement circle under different shear rates are shown in Figure14. The peak stress of each measurement circle is determined and presented in Table 8. The change curve of the peak stress of each measurement point under the condition of different shear rates is shown in Figure 15.Figure 14
Stress-strain curves of each measurement circle in the second group of specimens when the shear rate is (a) 0.02 mm/s, (b) 0.04 mm/s, (c) 0.06 mm/s, (d) 0.08 mm/s, and (e) 0.10 mm/s.
(a)(b)(c)(d)(e)Table 8
The peak stress of each measurement circle in the second group of specimens.
Number of measuring circlesShear rate (mm/s)0.020.040.060.080.1012.104.503.94.434.4834.454.334.644.324.5659.998.716.385.274.9171.341.512.212.742.8095.265.495.415.815.97Figure 15
Curves of peak stress of each measured circle in the second group of specimens.It can be seen from the data in Figures14 and 15 and Table 8 that the shear rate has significant influence on rock mass stress.(1)
On the one hand, for the stress at the same measuring point under different shear rates: there is little change in peak stress of the four measurement circles 1 (except at 0.02 mm/s, the peak stress of the measurement circle 1 is discrete), 3, 7, and 9, when the shear rate is adjusted. However, for the measurement circle 5, there is obvious change in the peak stress when the shear rate is adjusted. When the shear rate is increased from 0.02 mm/s to 0.10 mm/s, the peak stress of measurement circle 5 decreases from 9.99 MPa to 4.91 MPa. This means that the peak stress of the measurement circle 5 gradually decreases with the increase in the shear rate. Therefore, with the increase in the shear rate, both the shear capacity and shear stress at the center point of the rock mass decrease. In addition, as can be seen from the data in Figure12 and Table 6, the peak stress and the shear strength of the whole rock mass increase with the increase in the shear rate during the test. In summary, as the shear rate increases, the shear stress borne by the center of the rock mass decreases, and the force borne by the interior of the rock mass gradually balances.(2)
On the other hand, for the stress of different measurement circles under the same shear rate condition: the peak stresses of the measurement circle 5 at the shear rates of 0.02 mm/s, 0.04 mm/s, and 0.06 mm/s are 9.99 MPa, 8.71 MPa, and 6.38 MPa, respectively, and these are the maximum values of the peak stress of the five measurement circles. In addition, when the shear rates are 0.08 mm/s and 0.10 mm/s, the peak stresses at the measurement circle 5 are 5.27 MPa and 4.91 MPa, respectively. Although these values differ slightly with the maximum value, they are still relatively large compared with those of other measurement circles. Therefore, in general, under different shear rates, the central position of a rock mass specimen is subjected to greater shear action and the stress variation is more obvious.


## 3.3. Results of the Third Experiment

The stress-strain curve obtained after the direct shear test of the rock mass with different joint connectivity rates is shown in Figure16. The peak stress of the rock mass in Figure 16 is sorted out, and the results are presented in Table 9.Figure 16
The stress-strain curve of the rock mass with different joint connectivity rates.Table 9
The peak shear stress of NNJRM with different joint connectivity rates.
Connectivity rate0.10.20.30.40.5Peak shear stress (MPa)5.905.805.775.564.03According to the data in Figure16 and Table 9, it can be found that the test results of NNJRM with different joint connectivity rates have obvious consistency. When the connectivity of joints within the rock mass is 0.1, the peak shear stress of the rock mass is 5.90 MPa, and the residual stress is 1.34 MPa. When the joint connectivity rate increases gradually, the peak stress and residual stress of the rock mass tend to decrease. When the joint connectivity rate increases to 0.5, the peak shear stress and residual stress decrease to 4.03 MPa and 0.75 MPa, respectively. Therefore, the smaller the joint connectivity, the greater the peak stress and residual stress of the rock mass. Furthermore, the stronger the shear capacity of a rock mass, the stronger the ability to resist external forces.In addition, the number of DFN, DFN-t, and DFN-s produced in the third group of specimens with different joint connectivity rates are shown in Figure17 and Table 10.Figure 17
The number of cracks in the third group of specimens. (a) Total number of cracks. (b) The number of cracks caused by tension. (c) The number of cracks caused by shear action.
(a)(b)(c)Table 10
The number of cracks generated at different connectivity rates.
Connectivity rate0.10.20.30.40.5DFN (strip)829681720654457DFN-t (strip)640526563502343DFN-s (strip)189155157152114By summarizing and sorting out the data in Figure17 and Table 10, it can be found that no matter the value of the joint connectivity rate, the number of cracks generated in the rock mass gradually increases with the progress of the test process. However, different joint connectivity rates generate different increment rates of cracks. When the connectivity of joints was 0.1, the number of DFN, DFN-s, and DFN-t increased to 829, 189, and 640, respectively. When the joint connectivity rate gradually increased to 0.5, the final total number of cracks in the experimental rock mass decreased to 457, among which 114 are DFN-s and 343 are DFN-t.According to these data, the number of these two kinds of cracks, DFN-s, and DFN-t, which are caused by different forms of forces, decreases with the increase in joint connectivity, leading to a decrease in the total amount of cracks in the rock mass. Therefore, it can be concluded that the smaller the connectivity of joints within the rock mass, the better the integrity of the rock mass and the stronger the shear resistance capability of the rock mass. This is consistent with practical experiences in engineering practice.The stress-strain curves of each measuring circle in the specimens with different joint connectivity rates are shown in Figure18. The peak stress of each measuring circle is sorted out as shown in Table 11, and the change curve of peak stress of each measurement point under the condition of different joint connectivity rates is obtained, as shown in Figure 19.Figure 18
Stress-strain curves of each measurement circle in the third group of specimens when the connectivity rate is (a) 0.1, (b) 0.2, (c) 0.3, (d) 0.4, and (e) 0.5.
(a)(b)(c)(d)(e)Table 11
The peak stress of each measurement circle in the third group of specimens.
Number of measuring circlesConnectivity rate0.10.20.30.40.516.286.054.314.64.00310.09.178.306.514.5452.232.533.657.096.3475.515.726.264.813.8195.916.386.796.865.60Figure 19
Curves of peak stress of each measurement circle in the third group of specimens.Based on the data shown in Figures18 and 19 and Table 11, we find that, unlike direct shear tests conducted under the above controlled conditions, the maximum peak shear stress is not always at the measurement circle 5 located at the center of the specimen when the test is carried out at different joint connectivity rates. When the connectivity of the joints in the rock mass ranges from 0.1 to 0.3, the peak shear stress of the measurement circle 3 in the rock mass is the largest among the five measuring points. The peak stresses of measurement circles 1, 7, and 9 near both sides of the rock mass are large, while that of measuring circle 5 in the middle of the specimen is the smallest. Therefore, when the connection rate of joints is 0.1 to 0.3, the rock mass close to both sides of the rock mass bears more stress. However, when the connectivity of joints increases to 0.4 or 0.5, the measurement circle 5 experiences the highest peak stress among the five measuring points, and the peak stress of the five measurement circles is more uniform.


## 3.4. Results of the Fourth Experiment

The curves of stress-strain and peak shear stress of NNJRM with different joint relief angles of the fourth group specimens are shown in Figure20 and Table 12.Figure 20
The stress-strain curves of NNJRM with different relief angles.Table 12
The peak shear stress of NNJRM with different relief angles.
Relief angle0°15°30°45°60°Peak shear stress (MPa)3.964.124.283.834.09According to the data in Figure20 and Table 12, it can be found that the relief angle of joints has an influence on the stress-strain curve of NNJRM. If the joint relief angle changes, the peak shear stress and residual stress of the rock mass show obvious changes. However, there is no obvious regularity to this change.The number of DFN, DFN-t, and DFN-s produced in the fourth group of specimens with different relief angles are shown in Figure21 and Table 13.Figure 21
The number of cracks in the fourth group of specimens. (a) Total number of cracks. (b) The number of cracks caused by tension. (c) The number of cracks caused by shear action.
(a)(b)(c)Table 13
The number of cracks generated with different relief angles.
Relief angle0°15°30°45°60°DFN (strip)414447342510634DFN-t (strip)292335263386504DFN-s (strip)12211279124130By comparing the number of DFN, DFN-s, and DFN-t in the rock mass with five kinds of joint relief angles, it can be found that the change of joint relief angle has obvious influence on the direct shear test of a jointed rock mass. The difference between the maximum value (634) and the minimum value (342) of the total number of cracks produced by the direct shear test on the rock mass with different joint relief angles is 292. Similarly, the difference between the maximum value (504) and the minimum value (263) of the number of DFN-t is 241. Furthermore, under the same joint relief angle, the number of DFN-s generated in the specimen is significantly less than that of DFN-t. Therefore, the following conclusions can be drawn: in the direct shear tests of a rock mass with different joint relief angles, the main reason for its failure is the action of tension rather than the shear action.The stress-strain curves of each measurement circle in the fourth group of specimens with different joint relief angles are shown in Figure22. The peak stress of each measurement circle of the specimens under each joint relief angle is sorted out as shown in Table 14, and the change curve of the peak stress of each measured point with different joint relief angles is obtained, as shown in Figure 23.Figure 22
The stress-strain curves of each measurement circle in the fourth group of specimens when the relief angle is (a) 0°, (b) 15°, (c) 30°, (d) 45°, and (e) 60°.
(a)(b)(c)(d)(e)Table 14
The peak stress of each measurement circle in the fourth group of specimens.
Number of measuring circlesRelief angle0°15°30°45°60°12.074.094.013.070.2932.554.464.983.64.7158.496.355.875.247.5672.224.372.042.253.8494.495.655.724.464.32Figure 23
Curves of peak stress of each measurement circle in the fourth group of specimens.According to the data in Figures22 and 23 and Table 14, when conducting direct shear test on the rock mass with joints with different undulating angles, the measurement circle 5 of the rock mass experiences the largest peak stress among the five measurement circles. It is proved that the central part of the rock mass with discontinuous joints bears more forces during the direct shear test. In addition, according to the curve in Figure 23, when the joint relief angle is 0°or 60°, the peak stress of each measurement circle in the rock mass varies greatly and the stress distribution is not uniform.


## 3.5. Results of the Fifth Experiment

The curve of stress-strain and peak shear stress of NNJRM with different inclination angles of the fifth group of specimens are shown in Figure24 and Table 15.Figure 24
The stress-strain curves of NNJRM with different joint inclination angles.Table 15
The peak shear stress of NNJRM with different joint inclination angles.
Joint inclination angle0°15°30°45°60°Peak shear stress (MPa)3.954.015.926.526.00We can see from the data in Figure24 and Table 15 that when the joint inclination angle is 0° and 15°, the peak stress of the rock mass is 3.95 MPa and 4.01 MPa, respectively. If the angle is increased to 60°, the peak shear stress increases to 6.00 MPa. It can be seen that the peak stress of the rock mass increases with the increase in the joint inclination angle. Therefore, when the inclined angle of the joints is increased, the shear strength of the rock mass with different plane joints is enhanced.The number of cracks generated by the fifth group of specimens during the experiment is shown in Figure25 and Table 16.Figure 25
The number of cracks in the fifth group of specimens. (a) Total number of cracks. (b) The number of cracks caused by tension. (c) The number of cracks caused by shear action.
(a)(b)(c)Table 16
The number of cracks generated with different joint inclination angles.
Joint inclination angles0°15°30°45°60°DFN (strip)7654537811307935DFN-t (strip)5573405741040726DFN-s (strip)208114207267209From the chart and table above, it can be found that the dip angle of the joints has a significant effect on the crack propagation of the incoherent jointed rock mass. When the inclination angle of joints is changed, the number of cracks generated in the rock mass shows a very obvious change. However, there is no specific rule for the change in the number of cracks. Although the variation law of DFN, DFN-t, and DFN-s is not obvious, the number of DFN-t is always more than that of DFN-s. It is proved that the final failure of the NNJRM with different inclination angles in the direct shear test is also caused by tension.The stress-strain curves of each measurement circle in the specimens with different joint inclination angles are shown in Figure26. The peak stress of each measuring circle is sorted out as shown in Table 17, and the change curve of peak stress of each measured point under the condition of different joint inclination angles is obtained, as shown in Figure 27.Figure 26
The stress-strain curves of each measurement circle in the fifth group of specimens when the joint inclination angle is (a) 0°, (b) 15°, (c) 30°, (d) 45°, and (e) 60°.
(a)(b)(c)(d)(e)Table 17
The peak stress of each circle in the fifth group of specimens.
Number of measuring circlesJoint inclination angle0°15°30°45°60°10.583.976.493.935.7137.264.896.379.039.33511.064.624.343.242.5379.663.725.586.275.5390.845.526.708.197.71Figure 27
Curves of peak stress of each measurement circle in the fifth group of specimens.Based on the data obtained through the above simulation experiment, it can be seen that the joint inclination angle has significant influence on the strength characteristics of the rock mass. Although it is not possible to establish a specific law that reflects the influence of the joint inclination angle on the peak stress at each position in the rock mass, it is certain that the peak stress of the five measurement circles in the rock mass varies with the change in the joint inclination angle. Moreover, when the joint inclination angle is 15°, the peak stresses of the five measurement circles in the rock mass are not significantly different, and the stress distribution in the model specimen is more uniform.


## 4. Summary

In this section, the results of each of the above five tests are summarized separately, and these results of this study are generally consistent with the research conclusions in relevant articles [30, 31, 45–51]. It indicates that the results are reliable. In addition, the influence of the normal stress, shear rate, connectivity of joints, relief angle, and inclination angle of joints on the strength characteristics, crack development state, and the peak shear stress of the five measurement circles in the rock specimens is discussed below:(1)
After the first group of tests, it has been found that normal stress has significant influence on the characteristics of a rock mass. Specifically, the greater the normal stress, the greater the peak stress and residual stress of noncoplanar nonthrough jointed rock mass, and the stronger the shear capacity of the rock mass. Similarly, the larger the normal stress, the more the cracks produced by shear and tension in the rock mass. The peak shear stresses of measurement circles 1 and 5 in the rock mass are hugely affected by the normal stress, whereas the peak shear stresses of measurement circles 3, 7, and 9 are not significantly affected by the normal stress. In the first group of tests, the rock mass near the center experiences more concentrated stress, especially at the position of the measurement circle 5, which bears the main shear stress.(2)
After the second group of tests, it has been determined that both the peak shear stress and shear strength of the rock mass increase with the increase in the shear rate. In addition, with the increase in the shear rate, the number of total cracks, the cracks produced by shear and tension in the rock mass, increases first and then decreases. Moreover, with the increase in the shear rate, the peak shear stress of measurement circle 5 (the measurement circle with the highest peak stress in the rock mass) decreases, the shear stress borne by the central position of rock mass decreases, and so does the shear strength of the central position of the rock specimen. However, since the peak stress of the whole rock mass increases with the increase in the shear rate, the overall shear strength of the rock mass increases. Therefore, it is obvious that the forces within in the rock mass gradually balance with the increase in the shear rate.(3)
After the third group of tests, it has been noted that with the gradual increase in joint connectivity, the peak stress and residual stress of the rock mass decrease and the rock mass resistance to external forces weakens. In addition, the smaller the connection rate of the joints contained in the rock mass, the fewer the cracks generated in the rock mass, the better the integrity of the rock mass, and the stronger the shear resistance of the rock mass. This is consistent with the on-field experience in engineering practice. When the connectivity of joints contained in the rock mass ranges between 0.1 and 0.3, among the five measurement points, the largest peak shear stress is experienced at measurement circle 3, while the smallest is experienced at measurement circle 5. Therefore, when the joint connectivity is between 0.1 and 0.3, the internal stress of the rock mass is distributed close to both sides of the rock mass. When the connectivity of the joints increases to 0.4 or 0.5, among the five measurement points, the maximum peak stress is experienced at measurement circle 5 in the rock mass, and the peak stresses of the five measured circles are also relatively uniform.(4)
After the fourth group of tests, it has been determined that the peak shear stress, residual stress, the number of total cracks in the rock mass, and the cracks produced by shear and tension in the rock mass show obvious changes with the increase in the joint undulation angle. The peak stress of each measurement circle in the rock mass varies greatly with the change in the joint relief angle. However, a very obvious rule in these changes has not been observed.(5)
After the fifth group of tests, it has been noted that the peak stress of the rock mass increases with the increase in the joint inclination angle. In addition, with the increase in the joint inclination angle, the shear capacity of the noncoplanar nonthrough jointed rock mass increases. Furthermore, the joint inclination angle has significant influence on the generation of the number of cracks in the rock mass. However, the results of this simulation test cannot clearly reveal the precise variation law of the number of cracks. In addition, the peak stresses of the five measurement circles in the rock mass vary significantly with different joint inclination angles. The smallest difference in the peak stresses of the five measuring circles is obtained when the inclination angle is 15°, meaning the stress distribution in the specimen is more uniform.(6)
According to the quantitative relationship between the cracks produced by shear and tension in the rock mass under all the different conditions discussed above, it is proved that the failure of a rock mass in the direct shear test is mainly caused by tension rather than shear action.



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*Source: 6935414-2021-08-03.xml*