# Experimental and Numerical Study of Strength and Failure Behavior of Precracked Marble under True Triaxial Compression

Authors: Yong Han; Yuemao Zhao; Jinglong Li
Journal: Shock and Vibration
 (2021)

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## Abstract

Cracks play an important role in evaluating the strength and failure behavior of engineering rock mass. In order to increase the understanding of strength and failure mechanism of precracked rock, crack propagation and coalescence from preexisting cracks under true triaxial compression are investigated using true triaxial compression tests and Cellular Automata Software for engineering Rockmass fracturing process (CASRock). Three types of specimens were studied experimentally and numerically. Experimental and numerical results show that both the preferential angle and areal intensity of preexisting cracks can affect the compressive strength and failure behavior of the specimens. The peak strength firstly decreases and then increases with increase of the preferential angle. Also, the peak strength nonlinearly decreases with the increase of cracks’ areal intensity. The numerical results show that the crack initiation and coalescence are observed and characterized from the inner and outer tips of preexisting cracks in specimens containing single crack and multiple parallel cracks. The main shear failure in the specimen containing multiple unparallel preexisting cracks initiate and propagate from one of the macroscopic preexisting cracks, and other preexisting cracks do not initiate, propagate, and coalesce until reaching the peak strength.


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## Body

## 1. Introduction

Rock mass is a complex combination of multiphase materials, which usually contains many joints, cracks, and other defects. These defects greatly reduce the integrity and continuity of rock mass, resulting in changes in mechanical properties and failure mechanism [1–3]. In the construction of tunnel and mine, excavation will cause stress concentration and rotation in the surrounding rock. When the force is loaded on rock mass, preexisting cracks and discontinuities would open and propagate, and new cracks will start to grow at or near the tips of preexisting cracks, even coalescing with preexisting cracks, which would lead to rock mass instability and engineering disasters, such as large deformation and collapse.With the progress of theory and technology, many scholars have carried out the research on the mechanical properties and failure mechanism of cracked rock using laboratory tests and numerical simulation and have gained some research achievements.In laboratory test research, many scholars initially used rock-like materials (glass, gypsum, cement mortar, and so on) and prefabricated one or more cracks on the specimens to conduct uniaxial compression and biaxial compression tests in order to study the influence of cracks on the failure mode and mechanical properties of materials [4–6]. Bobet and Einstein [4] studied the fracture coalescence behavior by loading prefractured specimens of gypsum under uniaxial and biaxial compression, and law slippage, wing crack initiation, secondary crack initiation, and crack coalescence were observed. Wong et al. [5] investigated experimentally the peak strength and the mechanisms of crack coalescence of rock-like materials containing three parallel frictional cracks.Although the influence of cracks on the mechanical properties and failure mechanism of materials has been well illustrated by the results rock-like materials, the composition and structure are different from those of real rock materials. Therefore, many scholars use precracked rocks (marble, granite, sand stone, and so on) with good homogeneity to study the mechanical properties of cracked rocks. Lee and Jeon [7] used three materials to understand the characteristics of cracking and crack coalescence, and the results can be used to analyze the stability of rock structures. Li et al. [8] explored the propagation and coalescence of preexisting cracks in marble specimens under uniaxial compressive, and two types of newborn cracks were observed: wing cracks and secondary cracks. Yang et al. [9] analyzed the effect of the coplanar crack angle on mechanical parameters of sandstone containing two coplanar oval cracks and constructed the relation between ligament angle and mechanical parameters. Zhou et al. [10] used the DIC technique to measure fracture characteristics on specimen surface and investigated the coalescence behavior around rectangular cavities in rock under uniaxial compressive. Huang et al. [2] found that the arrangement of the flaws pair would have greater impact on the rock strength, deformation, and crack coalescence pattern than the confining pressure through the conventional triaxial compression tests on sandstone with two preexisting closed cracks.These laboratory results have enhanced our understanding of the strength and failure modes within precracked rocks under uniaxial, biaxial, and triaxial compression. However, it is difficult to observe the detailed process of crack propagation and coalescence due to the opacity of rock, sealing in the test process, and the rapid development of failure. Therefore, many numerical methods have been developed to simulate crack initiation and propagation under loading. These numerical methods include the discrete element method [11], boundary element method [12, 13], and displacement discontinuity method [5]. In recent decades, a numerical simulation code, the cellular automaton (CA) method [14–18], with local consideration and parallel characteristics has been proposed to simulate the failure processes of rock materials under the uniaxial and polyaxial compression condition. Cellular Automata Software for engineering Rockmass fracturing processes (CASRock) was developed, which can commendably simulate the failure process of heterogeneous rocks and the fracturing process of engineering rock mass [19].Work in the past mainly focused on propagation and coalescence of single or parallel cracks under uniaxial and biaxial compression, but few researches studied the coalescence of unparallel cracks under true triaxial compression. In fact, cracks in rock are often unparallel or intersecting. In this study, experimental simulation and numerical simulation were adopted to investigate the effects of cracks on compressive strength and failure mode of marble specimens under true triaxial compression. Firstly, true triaxial compression tests were conducted with precracked marble specimens to investigate the effects of cracks on strength and failure behavior. Then, CASRock was used to simulate the specimen containing cracks, aimed at revealing the evolution process of crack initiation, propagation, and coalescence under true triaxial compression.


## 2. Experimental Study of Mechanical Properties of Precracked Marble under True Triaxial Compression

### 2.1. A Method Analyzing the Natural Cracks’ Geometric Characteristics

In order to better explore the effect of natural cracks on rock mechanical properties, the true triaxial compression tests were conducted by marble containing natural cracks in this study. It is important to measure the cracks’ geometric characteristics of natural rock, which is the foundation for the study of rock mechanical properties. The statistical dimension of cracks that were observed was 10–100 mm in length and 0.01–1 mm in width [20]. Zhong et al. [21] established an analysis method of natural crack geometric characteristics by two-dimensional plane sketch in the lateral face of rock specimens and considered the comprehensive influence of crack length and dip angle on mechanical properties of rock under conventional compression tests. Gao et al. [22] investigated the crack effect on rock peak strength and found the peak strength shows a U-shaped variation characteristic trend with increasing of dip angle. Wang et al. [23] studied the effect of crack geometric characteristics on the failure mode and strength of composite rock mass, and the uniaxial compression simulation test of composite rock sample with single fracture was carried out. The results showed that the strength of the composite rock sample had a linear negative correlation with the fracture length and the fracture dip. Liu et al. [24] investigated the mechanical behaviors of specimens containing combined flaws with various inclination angles, and the crack initiation, propagation process, and its corresponding mechanisms were studied.Based on the previous studies [21–28], the intensity and the preferential angle of cracks have important effects on rock mechanical properties. Therefore, these two parameters are calculated by the following method in this paper, as shown in Figure 1.Figure 1
Schematic view of statistical method for evaluating the natural cracks’ geometric characteristics of rock specimens.The length and dip angle in each face of marble specimens are counted by the super-depth-of-field three-dimensional optical microscope. The areal intensity (DL) is defined as the crack length divided by the specimen area, as shown in the following equation:(1)DL=∑i=1NLiS,where Li is the length of crack, N is the total number of cracks in lateral face of specimen, and S is the area of lateral face of the specimen. Based on the typical failure modes under true triaxial tests [29], it is found that the final fracture plane is parallel to the direction of the intermediate principal stress (σ2) due to the constraint of the intermediate principal stress on the crack opening in the σ2 direction [29, 30]. Considering that the final fracture plane after test is parallel to the direction of the intermediate principal stress, in this study, a couple of lateral faces of marble specimen where the total areal intensity is larger than the other two lateral faces are placed on the intermediate principal stress loading plane in order to better explore the effect of natural cracks on the rock mechanical properties.Then, the total length of all cracks in the intermediate principal stress loading plane is projected in a certain direction, as shown in Figure1, and the total length of the all cracks projected in this direction is calculated:(2)Lθ=∑i=1Nlicosθk−αi,where αi is the dip angle of the crack I, θk is the preselected angle between projected line and the σ1 loading plane, and the range of θk is −90°∼+90°. The increment of θk in this paper is 5, and the maximum of k is 36.According to determination method of the preferential angle of cracks [21, 22], the angle with the maximum projection length is the preferential angle of cracks:(3)θp=θk|MaxLθθk,αi,li.The areal intensity (DL) and the preferential angle (θp) are used to analyze the effect of crack geometric characteristics on the mechanical properties of marble.


### 2.2. Specimens and Testing Apparatus

The marble used for tests came from Jinchuan nickel mine, located in Jinchuan, Gansu Province, China. Rectangular prismatic specimens of marble, 50 × 50 × 100 mm in size, were prepared for true triaxial compression tests. The experimental investigation of specimens was accomplished considering the three cases: (a) intact specimens, (b) specimens containing a single crack, and (c) specimens containing multiple cracks, as shown in Figure2. The density of specimens is about 2.75 g/cm3. The tests were conducted on the novel Mogi type of the true triaxial testing system (Lavender 508) of Northeast University developed by Feng et al. [31]. The tests were carried out at a stress level of σ2 = 50 MPa and σ3 = 30 MPa.Figure 2
View of the specimens: (a) intact specimen, (b) specimen containing a single crack, and (c) specimen containing multiple cracks.
(a)(b)(c)


### 2.3. Experimental Results

The experimental results show that the peak strength of marble containing single natural crack (DL is kept around 0.00953–0.01038 m−1) firstly decreases and then increases with increase of the preferential angle, as shown in Figure 3. The peak compressive strength is 361 MPa at θp = 3°, which is close to the compressive strength of intact marble. The peak compressive strength is 285 MPa at θp = 75°. Compared with the compressive strength of intact marble at the same stress level, it is reduced by 86 MPa.Figure 3
Experimental results of the preferential angle (θp) effect on the peak stress of marble containing single crack.Figure4 shows that peak strength nonlinearly decreases with the increase of cracks’ areal intensity, when θp is kept at 65°. When the areal intensity is 0.0457 m−1, the tested specimen has a compressive strength of about 258 MPa. It is reduced by 113 MPa, compared with the compressive strength of intact marble at the same stress level. Therefore, it can be seen from the experimental results that the cracks’ geometric characteristics have a great influence on the mechanical properties of marble.Figure 4
Experimental results of the areal intensity (DL) effect on the peak stress of marble containing multiple cracks.


## 2.1. A Method Analyzing the Natural Cracks’ Geometric Characteristics

In order to better explore the effect of natural cracks on rock mechanical properties, the true triaxial compression tests were conducted by marble containing natural cracks in this study. It is important to measure the cracks’ geometric characteristics of natural rock, which is the foundation for the study of rock mechanical properties. The statistical dimension of cracks that were observed was 10–100 mm in length and 0.01–1 mm in width [20]. Zhong et al. [21] established an analysis method of natural crack geometric characteristics by two-dimensional plane sketch in the lateral face of rock specimens and considered the comprehensive influence of crack length and dip angle on mechanical properties of rock under conventional compression tests. Gao et al. [22] investigated the crack effect on rock peak strength and found the peak strength shows a U-shaped variation characteristic trend with increasing of dip angle. Wang et al. [23] studied the effect of crack geometric characteristics on the failure mode and strength of composite rock mass, and the uniaxial compression simulation test of composite rock sample with single fracture was carried out. The results showed that the strength of the composite rock sample had a linear negative correlation with the fracture length and the fracture dip. Liu et al. [24] investigated the mechanical behaviors of specimens containing combined flaws with various inclination angles, and the crack initiation, propagation process, and its corresponding mechanisms were studied.Based on the previous studies [21–28], the intensity and the preferential angle of cracks have important effects on rock mechanical properties. Therefore, these two parameters are calculated by the following method in this paper, as shown in Figure 1.Figure 1
Schematic view of statistical method for evaluating the natural cracks’ geometric characteristics of rock specimens.The length and dip angle in each face of marble specimens are counted by the super-depth-of-field three-dimensional optical microscope. The areal intensity (DL) is defined as the crack length divided by the specimen area, as shown in the following equation:(1)DL=∑i=1NLiS,where Li is the length of crack, N is the total number of cracks in lateral face of specimen, and S is the area of lateral face of the specimen. Based on the typical failure modes under true triaxial tests [29], it is found that the final fracture plane is parallel to the direction of the intermediate principal stress (σ2) due to the constraint of the intermediate principal stress on the crack opening in the σ2 direction [29, 30]. Considering that the final fracture plane after test is parallel to the direction of the intermediate principal stress, in this study, a couple of lateral faces of marble specimen where the total areal intensity is larger than the other two lateral faces are placed on the intermediate principal stress loading plane in order to better explore the effect of natural cracks on the rock mechanical properties.Then, the total length of all cracks in the intermediate principal stress loading plane is projected in a certain direction, as shown in Figure1, and the total length of the all cracks projected in this direction is calculated:(2)Lθ=∑i=1Nlicosθk−αi,where αi is the dip angle of the crack I, θk is the preselected angle between projected line and the σ1 loading plane, and the range of θk is −90°∼+90°. The increment of θk in this paper is 5, and the maximum of k is 36.According to determination method of the preferential angle of cracks [21, 22], the angle with the maximum projection length is the preferential angle of cracks:(3)θp=θk|MaxLθθk,αi,li.The areal intensity (DL) and the preferential angle (θp) are used to analyze the effect of crack geometric characteristics on the mechanical properties of marble.


## 2.2. Specimens and Testing Apparatus

The marble used for tests came from Jinchuan nickel mine, located in Jinchuan, Gansu Province, China. Rectangular prismatic specimens of marble, 50 × 50 × 100 mm in size, were prepared for true triaxial compression tests. The experimental investigation of specimens was accomplished considering the three cases: (a) intact specimens, (b) specimens containing a single crack, and (c) specimens containing multiple cracks, as shown in Figure2. The density of specimens is about 2.75 g/cm3. The tests were conducted on the novel Mogi type of the true triaxial testing system (Lavender 508) of Northeast University developed by Feng et al. [31]. The tests were carried out at a stress level of σ2 = 50 MPa and σ3 = 30 MPa.Figure 2
View of the specimens: (a) intact specimen, (b) specimen containing a single crack, and (c) specimen containing multiple cracks.
(a)(b)(c)


## 2.3. Experimental Results

The experimental results show that the peak strength of marble containing single natural crack (DL is kept around 0.00953–0.01038 m−1) firstly decreases and then increases with increase of the preferential angle, as shown in Figure 3. The peak compressive strength is 361 MPa at θp = 3°, which is close to the compressive strength of intact marble. The peak compressive strength is 285 MPa at θp = 75°. Compared with the compressive strength of intact marble at the same stress level, it is reduced by 86 MPa.Figure 3
Experimental results of the preferential angle (θp) effect on the peak stress of marble containing single crack.Figure4 shows that peak strength nonlinearly decreases with the increase of cracks’ areal intensity, when θp is kept at 65°. When the areal intensity is 0.0457 m−1, the tested specimen has a compressive strength of about 258 MPa. It is reduced by 113 MPa, compared with the compressive strength of intact marble at the same stress level. Therefore, it can be seen from the experimental results that the cracks’ geometric characteristics have a great influence on the mechanical properties of marble.Figure 4
Experimental results of the areal intensity (DL) effect on the peak stress of marble containing multiple cracks.


## 3. Numerical Study of Mechanical Properties of Precracked Marble under True Triaxial Compression

Based on the experimental study of intact and precracked marble specimens, the effect of cracks’ geometric characteristics on the strength of marble has been obtained. However, it is difficult to observe the detailed process of rock fracture and crack propagation. Therefore, a numerical simulation using Cellular Automata Software for engineering Rockmass fracturing processes (CASRock) was carried out for the precracked marble specimens.


### 3.1. Descriptions of Cellular Automata Software for Engineering Rockmass Fracturing Processes (CASRock)

Cellular Automata Software for engineering Rockmass fracturing processes (CASRock) was developed to solve the problem of the fracturing process for engineering rock mass, which is a combination of cellular automaton, rock mechanics, engineering geology, elastoplastic mechanics, and fracture mechanics [16]. The process is a “down-top” way of handling the fracturing behavior of rock mass. A cellular automaton is mainly composed of cell, cell space, cell state, neighborhood, and updating rule [16–18]. The rock mass to be simulated is put into a system composed of cells. Figure 5 shows the relationship between cell and its neighbors in the development of updating rule [17]. The purpose of this paper is to study the failure process of cracked marble under true triaxial compression, so all the components should be defined in a three-dimensional domain.Figure 5
Numerical model and three-dimensional domain to be solved.In order to model the precracked specimens, a rock discontinuous cellular automaton [18] was developed. Its updating rule is more complex because there are internal boundaries, as shown in Figure 6 [18].Figure 6
Representation of the discontinuous cellular automaton model and its different states.The cell is the basic component of the cellular automaton [15], and it can be defined as(4)Di=Ni∪Ei1,…,Eimi;Ni1,…,Nini,Ni∈Eij|Nik∈Eij,j=1,…mi,where Di is the cell, which is composed of cell nodes Ni, corresponding to cell elements Eij, and neighboring cell nodes Nik.In the cellular automaton model, the equilibrium equation [15, 17, 18] of each cell can be defined as follows:(5)KijΔuj=ΔFi,where Kij is the local stiffness on the cell node, Δuj is the incremental degree of freedom, and ΔFi is the nodal force. The dimension of Kij, Δuj, and ΔFi is dependent on the cell type.The increment nodal forceΔFik on the neighboring cells [17] is calculated by the following equation:(6)ΔFik=BimjΔum,where Bij is the cell element stiffness.In CASRock, each cell follows updating rule described in equations (5) and (6), and the system can get static equilibrium state when the self-organization phenomenon of Δui ⟶ 0 or ΔFi ⟶ 0 is reached [15, 17, 18].In this paper, the 3D elastic-plastic-ductile-brittle mode [32] is selected to describe the response of marble materials. This model considers the asymmetric effect of intermediate principal stress on the strength of hard rock, and it can be more consistent with true triaxial compression. The failure function can be expressed as [32](7)F=σ1−σ3sinφb2−σ1+σ3+2c0cotφ02+2σt+2c0cotφ02,sinφb=sinφ01−b+sb2+t1−1−b+sb2sinφ0,where b is intermediate principal stress coefficient, which can be expressed as(8)b=σ2−σ3σ1−σ3,where c0, φ0, and σt are the cohesion, friction angle, and uniaxial tension strength, respectively, s is the strength-difference coefficient between generalized triaxial extension and generalized triaxial compression, and t is the material constant controlling the maximum strength-increasing ratio affected by σ2.


### 3.2. Numerical Simulation Schemes

Figure7 shows the setup of the numerical models. In the numerical model, the size of the specimens is designed to match the specimens in the experimental tests. As for the single crack and parallel cracks, the length is 50 mm, and the distance between two cracks is 5 mm, respectively. The parameters of the specimens for numerical simulations are shown in Table 1. In this paper, two cases are stimulated. In Case 1, in order to study the effect of different preferential angles and areal intensity on the strength and failure modes of the specimens, three types of crack geometry characteristics are considered. Type A is the case of intact marble without cracks, and Type B and Type C are specimens containing single crack and multiple parallel cracks with different preferential angles. In Case 2, in order to analyze the process of unparallel crack coalescence, Type D containing multiple unparallel cracks is established, which is set up based on the geometric characteristics of natural cracks in test specimens. A detailed description of these specimens with different crack geometric characteristics is presented in Table 2. The numerical simulation was carried out at a stress level of σ2 = 50 MPa and σ3 = 30 MPa.Figure 7
Numerical model of intact and precracked specimens: (a) intact specimen, (b) specimen containing single crack, (c) specimen containing multiple parallel cracks, and (d) specimen containing multiple unparallel cracks.Table 1
Parameters used in CASRock.
ParametersMarbleCrackParametersMarbleCrackYoung’s modulus (GPa)8060Poisson’s ratio0.350.35Initial cohesion (MPa)155Residual cohesion (MPa)33Initial friction angle (°)50.840Residual frictional angle (°)4545Homogeneous index (m)88Seed parameter (s)1010Table 2
Crack geometries of precracked specimens.
TypeThe preferential angles (°)The areal intensity (m−1)TypeThe preferential angles (°)The areal intensity (m−1)AIntact marble (no crack)B900.010B00.010C450.020300.010450.030450.010450.040600.010450.050750.010D650.045


### 3.3. Simulation Results

#### 3.3.1. Effects of the Preferential Angle on the Strength and Failure Modes

Figure8 shows the complete stress-strain curves of specimens containing single crack with different preferential angles, when DL is kept at 0.01 m−1. The variation trend of strength with preferential angle obtained from simulation is similar to the experimental results. The peak strength of specimens containing single crack firstly decreases and then increases with the increase of the preferential angle (Figure 9).Figure 8
The stress-strain curves of specimens containing single crack with different preferential angles.Figure 9
Effect of preferential angle on the strength of specimens containing single crack.Figure10 compares the crack evolutions of intact and cracked specimens obtained from numerical results. In the intact specimen, the microdamage initiates at the surface of the marble specimen, and the failure mode is shear failure. When specimens contain single crack, failure modes under true triaxial compression include two modes. In mode 1, the failure mode of specimen is main shear failure, when the preferential angle is close to 0° or 90°, which was same as the intact marble, but there are other macrocracks instead of one within intact marble, as shown in Figure 10(a). The microdamage initiated mainly from the tips of the preexisting crack and the surface of the specimens. The shear failure is caused by the crack initiation and coalescence at the surface of the specimens. The preexisting crack initiation results in tensile failure. In mode 2, the failure mode of the specimen partly coincides with the plane of the crack or occurs totally along the preexisting crack. Also, the failure angle is same as the preferential angle (60°). Figure 11 shows the final failure mode comparison between the experimental and numerical simulation results. It can be found that the simulation results are consistent with the experimental results. Therefore, this method can be used to study the failure evolution process of precracked specimens under true triaxial compression.Figure 10
Numerically simulated crack evolution in specimens containing single crack with different preferential angles. (a) Intact specimens. (b)θp = 90°. (c) θp = 60°.
(a)(b)(c)Figure 11
Comparison of experimental results and numerical simulation results.
(a)(b)(c)


#### 3.3.2. Effects of the Areal Intensity on the Strength and Failure Modes

Figure12 shows the complete stress-strain curves of specimens containing multiple parallel cracks, when the preferential angle is kept at 45°. With the increase of the areal intensity, the peak strength of specimens decreases nonlinearly, as shown in Figure 13.Figure 12
The stress-strain curves of specimens containing multiple parallel cracks.Figure 13
Effect of the areal intensity on the strength of specimens containing multiple parallel cracks.Figure14 shows the numerically simulated microdamage evolutions from the macroscopic preexisting cracks with different areal intensities. When the specimens contain multiple parallel cracks, microdamage initiates from the tips of the one or two preexisting cracks and propagates gradually along the preexisting cracks. Then, some shear cracks grow and coalesce. Then, the final macrofailure mode is shear failure, which is caused by preexisting crack coalescence.Figure 14
Numerically simulated crack evolution in specimens containing multiple parallel cracks.
(a)(b)


#### 3.3.3. Analysis of Cracks’ Coalescence in Specimens Containing Unparallel Cracks

In the underground engineering construction, the fracture distribution state is extremely complex, including parallel and cross distribution. In order to better reveal the failure evolution process of cracked rock mass, this paper establishes a model of marble specimen containing intersected cracks to explore the evolution process of crack initiation, propagation, and coalescence under true triaxial compression.Figure15 shows the numerical and experimental failure modes of specimens containing intersected cracks. It can be seen that the numerical failure mode has a good agreement with the experimental results in terms of coalescence characteristics. As indicated previously, it is difficult to observe the microdamage evolutions in laboratory testing.Figure 15
(a) The numerical and experimental failure modes of specimens and (b) crack evolution in specimens containing multiple parallel cracks.
(a)(b)Figure16 shows the crack evolutions during the loading process. During the elastic deformation stage, there is no obvious damage region. At the first peak value, the main shear cracks initiate and propagate from one of the macroscopic preexisting cracks. Due to the penetration of crack, the peak strength drops slowly with increase of ε1. It can be explained that this failure is caused by the penetration of preexisting cracks, and other researchers also obtained similar results by experimental and numerical study [2, 7]. Subsequently, the failure of specimens develops along the preexisting crack and spreads to the intact rock part before the stress level reaches the second peak stress. Entering the postpeak stage, the other preexisting cracks also initiate, propagate, and coalesce to form a macrofailure until reaching the residual strength.Figure 16
Crack initiation and coalescence process during the process of true triaxial compression.


## 3.1. Descriptions of Cellular Automata Software for Engineering Rockmass Fracturing Processes (CASRock)

Cellular Automata Software for engineering Rockmass fracturing processes (CASRock) was developed to solve the problem of the fracturing process for engineering rock mass, which is a combination of cellular automaton, rock mechanics, engineering geology, elastoplastic mechanics, and fracture mechanics [16]. The process is a “down-top” way of handling the fracturing behavior of rock mass. A cellular automaton is mainly composed of cell, cell space, cell state, neighborhood, and updating rule [16–18]. The rock mass to be simulated is put into a system composed of cells. Figure 5 shows the relationship between cell and its neighbors in the development of updating rule [17]. The purpose of this paper is to study the failure process of cracked marble under true triaxial compression, so all the components should be defined in a three-dimensional domain.Figure 5
Numerical model and three-dimensional domain to be solved.In order to model the precracked specimens, a rock discontinuous cellular automaton [18] was developed. Its updating rule is more complex because there are internal boundaries, as shown in Figure 6 [18].Figure 6
Representation of the discontinuous cellular automaton model and its different states.The cell is the basic component of the cellular automaton [15], and it can be defined as(4)Di=Ni∪Ei1,…,Eimi;Ni1,…,Nini,Ni∈Eij|Nik∈Eij,j=1,…mi,where Di is the cell, which is composed of cell nodes Ni, corresponding to cell elements Eij, and neighboring cell nodes Nik.In the cellular automaton model, the equilibrium equation [15, 17, 18] of each cell can be defined as follows:(5)KijΔuj=ΔFi,where Kij is the local stiffness on the cell node, Δuj is the incremental degree of freedom, and ΔFi is the nodal force. The dimension of Kij, Δuj, and ΔFi is dependent on the cell type.The increment nodal forceΔFik on the neighboring cells [17] is calculated by the following equation:(6)ΔFik=BimjΔum,where Bij is the cell element stiffness.In CASRock, each cell follows updating rule described in equations (5) and (6), and the system can get static equilibrium state when the self-organization phenomenon of Δui ⟶ 0 or ΔFi ⟶ 0 is reached [15, 17, 18].In this paper, the 3D elastic-plastic-ductile-brittle mode [32] is selected to describe the response of marble materials. This model considers the asymmetric effect of intermediate principal stress on the strength of hard rock, and it can be more consistent with true triaxial compression. The failure function can be expressed as [32](7)F=σ1−σ3sinφb2−σ1+σ3+2c0cotφ02+2σt+2c0cotφ02,sinφb=sinφ01−b+sb2+t1−1−b+sb2sinφ0,where b is intermediate principal stress coefficient, which can be expressed as(8)b=σ2−σ3σ1−σ3,where c0, φ0, and σt are the cohesion, friction angle, and uniaxial tension strength, respectively, s is the strength-difference coefficient between generalized triaxial extension and generalized triaxial compression, and t is the material constant controlling the maximum strength-increasing ratio affected by σ2.


## 3.2. Numerical Simulation Schemes

Figure7 shows the setup of the numerical models. In the numerical model, the size of the specimens is designed to match the specimens in the experimental tests. As for the single crack and parallel cracks, the length is 50 mm, and the distance between two cracks is 5 mm, respectively. The parameters of the specimens for numerical simulations are shown in Table 1. In this paper, two cases are stimulated. In Case 1, in order to study the effect of different preferential angles and areal intensity on the strength and failure modes of the specimens, three types of crack geometry characteristics are considered. Type A is the case of intact marble without cracks, and Type B and Type C are specimens containing single crack and multiple parallel cracks with different preferential angles. In Case 2, in order to analyze the process of unparallel crack coalescence, Type D containing multiple unparallel cracks is established, which is set up based on the geometric characteristics of natural cracks in test specimens. A detailed description of these specimens with different crack geometric characteristics is presented in Table 2. The numerical simulation was carried out at a stress level of σ2 = 50 MPa and σ3 = 30 MPa.Figure 7
Numerical model of intact and precracked specimens: (a) intact specimen, (b) specimen containing single crack, (c) specimen containing multiple parallel cracks, and (d) specimen containing multiple unparallel cracks.Table 1
Parameters used in CASRock.
ParametersMarbleCrackParametersMarbleCrackYoung’s modulus (GPa)8060Poisson’s ratio0.350.35Initial cohesion (MPa)155Residual cohesion (MPa)33Initial friction angle (°)50.840Residual frictional angle (°)4545Homogeneous index (m)88Seed parameter (s)1010Table 2
Crack geometries of precracked specimens.
TypeThe preferential angles (°)The areal intensity (m−1)TypeThe preferential angles (°)The areal intensity (m−1)AIntact marble (no crack)B900.010B00.010C450.020300.010450.030450.010450.040600.010450.050750.010D650.045


## 3.3. Simulation Results

### 3.3.1. Effects of the Preferential Angle on the Strength and Failure Modes

Figure8 shows the complete stress-strain curves of specimens containing single crack with different preferential angles, when DL is kept at 0.01 m−1. The variation trend of strength with preferential angle obtained from simulation is similar to the experimental results. The peak strength of specimens containing single crack firstly decreases and then increases with the increase of the preferential angle (Figure 9).Figure 8
The stress-strain curves of specimens containing single crack with different preferential angles.Figure 9
Effect of preferential angle on the strength of specimens containing single crack.Figure10 compares the crack evolutions of intact and cracked specimens obtained from numerical results. In the intact specimen, the microdamage initiates at the surface of the marble specimen, and the failure mode is shear failure. When specimens contain single crack, failure modes under true triaxial compression include two modes. In mode 1, the failure mode of specimen is main shear failure, when the preferential angle is close to 0° or 90°, which was same as the intact marble, but there are other macrocracks instead of one within intact marble, as shown in Figure 10(a). The microdamage initiated mainly from the tips of the preexisting crack and the surface of the specimens. The shear failure is caused by the crack initiation and coalescence at the surface of the specimens. The preexisting crack initiation results in tensile failure. In mode 2, the failure mode of the specimen partly coincides with the plane of the crack or occurs totally along the preexisting crack. Also, the failure angle is same as the preferential angle (60°). Figure 11 shows the final failure mode comparison between the experimental and numerical simulation results. It can be found that the simulation results are consistent with the experimental results. Therefore, this method can be used to study the failure evolution process of precracked specimens under true triaxial compression.Figure 10
Numerically simulated crack evolution in specimens containing single crack with different preferential angles. (a) Intact specimens. (b)θp = 90°. (c) θp = 60°.
(a)(b)(c)Figure 11
Comparison of experimental results and numerical simulation results.
(a)(b)(c)


### 3.3.2. Effects of the Areal Intensity on the Strength and Failure Modes

Figure12 shows the complete stress-strain curves of specimens containing multiple parallel cracks, when the preferential angle is kept at 45°. With the increase of the areal intensity, the peak strength of specimens decreases nonlinearly, as shown in Figure 13.Figure 12
The stress-strain curves of specimens containing multiple parallel cracks.Figure 13
Effect of the areal intensity on the strength of specimens containing multiple parallel cracks.Figure14 shows the numerically simulated microdamage evolutions from the macroscopic preexisting cracks with different areal intensities. When the specimens contain multiple parallel cracks, microdamage initiates from the tips of the one or two preexisting cracks and propagates gradually along the preexisting cracks. Then, some shear cracks grow and coalesce. Then, the final macrofailure mode is shear failure, which is caused by preexisting crack coalescence.Figure 14
Numerically simulated crack evolution in specimens containing multiple parallel cracks.
(a)(b)


### 3.3.3. Analysis of Cracks’ Coalescence in Specimens Containing Unparallel Cracks

In the underground engineering construction, the fracture distribution state is extremely complex, including parallel and cross distribution. In order to better reveal the failure evolution process of cracked rock mass, this paper establishes a model of marble specimen containing intersected cracks to explore the evolution process of crack initiation, propagation, and coalescence under true triaxial compression.Figure15 shows the numerical and experimental failure modes of specimens containing intersected cracks. It can be seen that the numerical failure mode has a good agreement with the experimental results in terms of coalescence characteristics. As indicated previously, it is difficult to observe the microdamage evolutions in laboratory testing.Figure 15
(a) The numerical and experimental failure modes of specimens and (b) crack evolution in specimens containing multiple parallel cracks.
(a)(b)Figure16 shows the crack evolutions during the loading process. During the elastic deformation stage, there is no obvious damage region. At the first peak value, the main shear cracks initiate and propagate from one of the macroscopic preexisting cracks. Due to the penetration of crack, the peak strength drops slowly with increase of ε1. It can be explained that this failure is caused by the penetration of preexisting cracks, and other researchers also obtained similar results by experimental and numerical study [2, 7]. Subsequently, the failure of specimens develops along the preexisting crack and spreads to the intact rock part before the stress level reaches the second peak stress. Entering the postpeak stage, the other preexisting cracks also initiate, propagate, and coalesce to form a macrofailure until reaching the residual strength.Figure 16
Crack initiation and coalescence process during the process of true triaxial compression.


## 3.3.1. Effects of the Preferential Angle on the Strength and Failure Modes

Figure8 shows the complete stress-strain curves of specimens containing single crack with different preferential angles, when DL is kept at 0.01 m−1. The variation trend of strength with preferential angle obtained from simulation is similar to the experimental results. The peak strength of specimens containing single crack firstly decreases and then increases with the increase of the preferential angle (Figure 9).Figure 8
The stress-strain curves of specimens containing single crack with different preferential angles.Figure 9
Effect of preferential angle on the strength of specimens containing single crack.Figure10 compares the crack evolutions of intact and cracked specimens obtained from numerical results. In the intact specimen, the microdamage initiates at the surface of the marble specimen, and the failure mode is shear failure. When specimens contain single crack, failure modes under true triaxial compression include two modes. In mode 1, the failure mode of specimen is main shear failure, when the preferential angle is close to 0° or 90°, which was same as the intact marble, but there are other macrocracks instead of one within intact marble, as shown in Figure 10(a). The microdamage initiated mainly from the tips of the preexisting crack and the surface of the specimens. The shear failure is caused by the crack initiation and coalescence at the surface of the specimens. The preexisting crack initiation results in tensile failure. In mode 2, the failure mode of the specimen partly coincides with the plane of the crack or occurs totally along the preexisting crack. Also, the failure angle is same as the preferential angle (60°). Figure 11 shows the final failure mode comparison between the experimental and numerical simulation results. It can be found that the simulation results are consistent with the experimental results. Therefore, this method can be used to study the failure evolution process of precracked specimens under true triaxial compression.Figure 10
Numerically simulated crack evolution in specimens containing single crack with different preferential angles. (a) Intact specimens. (b)θp = 90°. (c) θp = 60°.
(a)(b)(c)Figure 11
Comparison of experimental results and numerical simulation results.
(a)(b)(c)


## 3.3.2. Effects of the Areal Intensity on the Strength and Failure Modes

Figure12 shows the complete stress-strain curves of specimens containing multiple parallel cracks, when the preferential angle is kept at 45°. With the increase of the areal intensity, the peak strength of specimens decreases nonlinearly, as shown in Figure 13.Figure 12
The stress-strain curves of specimens containing multiple parallel cracks.Figure 13
Effect of the areal intensity on the strength of specimens containing multiple parallel cracks.Figure14 shows the numerically simulated microdamage evolutions from the macroscopic preexisting cracks with different areal intensities. When the specimens contain multiple parallel cracks, microdamage initiates from the tips of the one or two preexisting cracks and propagates gradually along the preexisting cracks. Then, some shear cracks grow and coalesce. Then, the final macrofailure mode is shear failure, which is caused by preexisting crack coalescence.Figure 14
Numerically simulated crack evolution in specimens containing multiple parallel cracks.
(a)(b)


## 3.3.3. Analysis of Cracks’ Coalescence in Specimens Containing Unparallel Cracks

In the underground engineering construction, the fracture distribution state is extremely complex, including parallel and cross distribution. In order to better reveal the failure evolution process of cracked rock mass, this paper establishes a model of marble specimen containing intersected cracks to explore the evolution process of crack initiation, propagation, and coalescence under true triaxial compression.Figure15 shows the numerical and experimental failure modes of specimens containing intersected cracks. It can be seen that the numerical failure mode has a good agreement with the experimental results in terms of coalescence characteristics. As indicated previously, it is difficult to observe the microdamage evolutions in laboratory testing.Figure 15
(a) The numerical and experimental failure modes of specimens and (b) crack evolution in specimens containing multiple parallel cracks.
(a)(b)Figure16 shows the crack evolutions during the loading process. During the elastic deformation stage, there is no obvious damage region. At the first peak value, the main shear cracks initiate and propagate from one of the macroscopic preexisting cracks. Due to the penetration of crack, the peak strength drops slowly with increase of ε1. It can be explained that this failure is caused by the penetration of preexisting cracks, and other researchers also obtained similar results by experimental and numerical study [2, 7]. Subsequently, the failure of specimens develops along the preexisting crack and spreads to the intact rock part before the stress level reaches the second peak stress. Entering the postpeak stage, the other preexisting cracks also initiate, propagate, and coalesce to form a macrofailure until reaching the residual strength.Figure 16
Crack initiation and coalescence process during the process of true triaxial compression.


## 4. Conclusions

In this study, firstly, a series of true triaxial compression tests and numerical simulations using CASRock were conducted on intact marble and precracked marble specimens. Based on the results, the effects of cracks’ geometric characteristics on strength and failure modes were analyzed, and the following conclusions are drawn:(1)
With the increase of the preferential angle, the peak strength firstly decreases and then increases. Also, the peak strength nonlinearly decreases with the increase of cracks’ areal intensity.(2)
The crack initiation and coalescence are observed and characterized from the inner and outer tips of preexisting cracks in specimens containing single crack and multiple parallel cracks.(3)
The failure mode of the specimen is main shear failure, when the preferential angle is close to 0° or 90°. The failure mode of the specimen partly coincides with the plane of the crack or occurs totally along the preexisting crack, when the preexisting crack dip angle is close to the failure angle of the intact specimen.(4)
For the specimens containing multiple unparallel cracks, the main shear failure initiate and propagate from one of the macroscopic preexisting cracks, and other preexisting cracks do not initiate, propagate, and coalesce until reaching the peak strength.



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*Source: 3869045-2021-12-30.xml*