#### Abstract

There are a lot of fissures, holes, and other defects in the formation of natural rocks. Under the influence of the external loads, these defects may cause engineering problems. Therefore, it is of great significance to analyze the characteristics of damage evolution of the defective rocks. In the study, the double-hole defective rocks with different angles of the center connection line are considered and the numerical models are established firstly. Then the mechanical behavior and acoustic emission (AE) characteristics are analyzed systematically. Finally the laws of damage evolution of the defective rock materials are investigated based on the AE characteristics. The research results show that the stress-strain behavior of the defective rocks can be divided into elastic stage, plastic stage and failure stages. The characteristics of acoustic emission evolution and laws of damage evolution are closely related to the stress-strain relationship. The elastic modulus of the double-hole defective rocks is similar with different angles of the center connection line, but the peak strength is different. The shape of the peak strength of these defective rocks is a W type owing to the different failure modes. The influences of different angles of the center connection line on the characteristics of AE evolution include the maximum events number, the strain value of the initial AE events and the maximum AE events, and the strain range of the serious AE events. Different angles of the center connection line have different influences on the laws of damage evolution of the double-hole defective rocks.

#### 1. Introduction

As the main medium of mine, tunnel, hydropower, and other engineering, the stability of rock is very important to the safety of the project [1–3]. Under the influence of geological process, natural rock materials contain a large number of holes, joints, fractures and other defects, which lead to the heterogeneity and anisotropy of the rock mechanical behaviors [4, 5]. Therefore, analysis on the characteristics of the damage evolution of the defective rocks is of great significance for the prevention and control of rock mass engineering disasters.

Many researches have conducted several investigations on the characteristics of damage evolution of defective rocks. Justo et al. [6] used the rock mass classification systems to estimate the modulus and strength of jointed rock. Lee and Jeon [7] conducted a series of uniaxial compression tests on Poly Methyl MethAcrylate (PMMA), Diastone, and Hwangdeung granite to study the coalescence behavior of two nonparallel flaws, which consist of a horizontal flaw and an underlying inclined flaw. Wang and Tian [8] investigated the influence of the combined fracture holes on the mechanical and characteristics of crack evolution in coal by means of microparticle flow code. Sammis and Ashby [9] used the plate specimens containing a single hole of the same size or an array of holes with various diameters to study the interaction of growing cracks with the surfaces of the specimen and the interaction with secondary cracks. Steen et al. [10] carried out both experimental and numerical tests to analyze the damage pattern for a loaded disc with eccentric holes. Li et al. [11] discussed the dynamic compressive strength and characteristics of crack propagation of specimens with prefabricated holes. Han et al. [12] carried out the uniaxial compression tests for the sandstone samples of two circular holes with different diameters and different geometries of holes. The interaction mechanism between two holes and the influence of holes on rock mechanical behavior were obtained. Jiang et al. [13] implemented a joint fractal dimension into the numerical modeling of a jointed rock mass to assess the behaviors of deformation of rock mass under opening. The generated joint network has various fractal dimensions and models different joint traces, space, and gaps.

Existing researches play a positive role in analyzing the damage mechanism of defective rocks. However, the cognition of the damage mechanism of the defective rock materials is not enough to well solve the engineering problems. Therefore, more conditional forms of defective rock damage characteristics need to be studied. In the study, the double-hole defective rocks with different angles, which are between the center connection line of the two holes and the horizontal plane, are considered. The numerical defective rock models are established in Section 2 firstly, and then the mechanical behavior and AE characteristics are analyzed in Section 3. Finally, the laws of damage evolution of defective rocks are investigated based on the AE characteristics in Section 4.

#### 2. Numerical Modeling of Double-Hole Defective Rocks

##### 2.1. Particle Flow Code (PFC)

In 1979, Cundall and Strack [14] established particle flow code (PFC) theory using the discrete element method. The basic compositions of the PFC model are particles and bonds, and the geometrical and mechanical properties of the particles and bonds determined the macroscopic mechanical properties of the models. The particle is a disk in the two-dimensional model and a ball in the three-dimensional model. There are two kinds of modes of the bonds, named contact bond (CB), and parallel bond (PB) [8, 15, 16]. The CB model (Figure 1(a)) has little resistance to the moment induced by particle rotation or shearing while PB model (Figure 1(b)) can resist to such particle movements since PB model is acting like a beam that resists the bending moment occurring within the bonded area [8, 17]. In this study, the models of the double-hole defective rocks are built by using the parallel bond.

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##### 2.2. Acoustic Emission Simulation by PFC

When the rock numerical models are established, the normal and tangential bonding strength can directly reflect the macroscopic strength of the rock. Under the influence of the loading, the bond fractures create microcracks in the rock specimen when the stress intensity transmitted between the particles exceeds the bonding strength between the particles [18]. As the microcrack propagating in the rock specimen, the damage energy is rapidly released as acoustic waves, which is the AE phenomenon [19, 20]. Therefore, the AE events can be simulated by counting the particle bond breaking number during the numerical tests. Due to the limitation of computing capacity, the particle size and particle number of PFC2D cannot reach the level of mechanical response directly to real macroscopic rock, but the mechanical law reflected is very helpful for understanding the AE phenomenon of rocks [21].

##### 2.3. Parameters of Numerical Rock Models

In PFC theory, the macromechanical property of the rock model is determined by the micromechanical property of the particles and bonds. However, these parameters cannot be directly derived directly from in-situ tests and indoor experiments. Therefore, the microscopic parameters selection and verification are needed before the numerical simulation [8]. Usually, the microparameters of PFC rock models are calibrated by simulating the uniaxial compression experiments [17, 22]. During the process of calibration, the microparameters of the particles and bonds are adjusted many times through “trial and error” method until these parameters can better reflect the mechanical properties of the real rocks [22–24]. The uniaxial compression calibration procedure of PFC (version 5.0) is shown in Figure 2.

Based on the uniaxial compression laboratory tests of sandstone [12], three PFC numerical models (I–III) are established, as shown in Figure 3. The model I is corresponding with the real intact rock specimen, and the other two models of II and III are corresponding with the test scheme A45–1 and B20-1, respectively. The size of the numerical intact rock model is 80 mm (length) × 160 mm (height), and the total number of particles is 14,456. The angle *α*, which is between the center connection line of the two holes and the horizontal plane, is 45°. The distance (*L*) between the centers of two holes is 25 mm, and the diameter of the center hole (*D*) is 20 mm. The diameter of the secondary hole (*d*) in the model I and the model III is 6 mm and 20 mm, respectively. The load of the numerical models is applied by moving the upper wall and the loading rate is 0.0017 mm/s. All conditions of the numerical tests are same as the laboratory tests. Through “trial and error” method with repeated check comparisons, the physical mechanical parameters listed in Table 1 can well reflect the macroscopic mechanical property of the real sandstone. The stress-strain curves (Figure 4) and the failure modes (Figure 5) of PFC models are in good agreement with the laboratory results of real sandstone.

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##### 2.4. Numerical Double-Hole Defective Rock Models

In order to study the characteristics of damage evolution of the double-hole defective rocks with different center connection line angle, seven PFC numerical rock models with the same *L*, *D*, and *d* are built with different *α*, as shown in Figure 6. These models are firstly established as intact rock based on the parameters listed in Table 1. Then some balls of the intact rock models are deleted to form the double-hole defective rocks. There are seven kinds of the double-hole defective rock with different *α*, which are −90°, −60°, −30°, 0°, 30°, 60°, and 90°. It is assumed that the positive value of *α* is along the counter clockwise direction. *L*, *D*, and *d* of these double-hole defective models are fixed in 25 mm, 20 mm, and 10 mm. Loads are controlled by applying a compress displacement at the top wall of the specimen, with the rate of 0.0017 mm/s. The numerical tests will be ended until residual stress of the numerical specimens reached 10% of the peak strength controlling by the FISH language.

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#### 3. Numerical Test Results Analysis

##### 3.1. Mechanical Properties

Figures 7 and 8 show the stress-strain and peak strength-*α* relationship curves of the double-hole defective rocks. It should be explained that the stress-strain curves refer to the average stress-strain behaviors of the whole specimen, rather than that of one part area of the specimen. The stress-strain of the defective rocks can be divided into three stages, which are elastic stage, plastic stage, and failure stage. Owing to that the particles of PFC rock models are rigid and there is no initial damage in the models, the stress-strain curves of the numerical rock models have not the closure stage at the beginning of the elastic stage such as the real rocks. PFC models eliminated the discreteness in the real rock and are more conducive to the study of the influence of the double-hole defects on mechanical properties of the rocks.

It can be seen from these two figures that as *α* varyies from −90° to 0° and form 90° to 0°, the peak strength of rocks is decreasing firstly and then increasing. As the *α* increasing from −90° to 90°, the peak strength of the defective rocks is 64.05 MPa, 52.40 MPa, 44.63 MPa, 52.79 MPa, 45.58 MPa, 53.43 MPa, and 65.21 MPa, respectively. The shape of the peak strength-*α* curve is a W type or an approximately symmetric √ type, as shown in Figure 8. It is because that the secondary hole of double-hole defective rock is symmetric to the horizontal plane which is along the direction when *α* is 0°. The symmetry of the defects leads to the symmetry of peak strength. The peak strength of the defective double-hole rocks when *α* is positive is higher than that when *α* is negative, even the same absolute value. The differences of the peak strength are decreasing as the absolute value of *α* from 90° to 0°, which are 1.16 MPa, 1.03 MPa, and 0.95 MPa, respectively. The reason is the loads applied by moving the top wall and the secondary hole near the loading wall, which has a boundary effect partly.

With different *α*, the elastic modulus of the double-hole defective rocks is similarly and about 15.59 GPa. However, the stress-strain characteristics are quite discrete after the peak strength. Along with decreasing of the absolute value of *α*, the post peak stress-strain curves change from smoothness with one peak to fluctuate with several peaks. For example, when *α* is −90°, the stress-strain curve has one peak and suddenly drops down to the minimum value after the peak. But the stress-strain curve of the double-hole defective rock with *α* of −30° has three peaks and displays as a saw-tooth shape.

By comparing the failure modes (Figure 9), several common phenomena can be concluded, especially the defective double-hole rocks with the symmetrical secondary holes to the horizontal axis. When the absolute value of *α* is 90°, the failure modes of the double-hole defective rock are symmetrical to the horizontal axis or the longitudinal axis. There is a huge penetrating fracture through the center hole. The fractures of the rocks are obviously an X type without overlying the secondary hole. Owing to that the secondary hole is not affected by the main fracture, the peak strength of these two defective rock specimens is higher than the others.

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When the absolute values of *α* are 60° and 30°, the failure modes of the double-hole defective rock are approximately symmetrical to the horizontal axis and the main fractures are through the double holes. The failure shape is a Y type when the absolute value of *α* is 60° and is a U type when the absolute value of *α* is 30°. When *α* is 0°, the shape of the failure mode shows a V type. The secondary hole plays an important effect on the peak strength of the defective rock mass.

##### 3.2. Acoustic Emission Characteristics

Figure 10 shows the curves of stress-strain-AE events of the double-hole defective rocks with different *α*. It can be seen from the figures that the evolutionary characteristics of AE events are closely related to the stress-strain relationship. When the stress-strain behavior is in the elastic stage, the number of AE events is very little. When the stress-strain behavior is in the plastic stage, the number of AE events increases and shows an incidental characteristic. When the stress-strain behavior is in the failure stage, the number of AE events reaches the peak and falls quickly, which means that the double-hole defective rock has damaged greatly in this period.

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The influence of different *α* on the AE characteristics is described as follows.(1)It affects the maximum AE event number. Apart from the maximum AE event number of double-hole defective rock with *α* equal to 0°, the maximum AE event number of these defective rocks decreases with the decreasing of the absolute value of *α*. When *α* is less than or equal to 0°, the maximum AE event number is 579, 307, and 244. And when *α* is bigger than or equal to 0°, the maximum AE event number is 498, 296, and 237. The evolutionary characteristic of the maximum event number of AE with different *α* is similarly with the change of the peak strength of the double-hole defective rocks. However, when the absolute value of *α* is same, the maximum AE event number of the defective rocks with *α* greater than 0° are generally larger than that of the defective rocks with *α* less than 0°.(2)It influences the strain values of the initial AE event and the maximum AE event. As *α* is from −90° to 90°, the strain values of the initial AE event are 0.196%, 0.097%, 0.107%, 0.178%, 0.104%, 0.091%, and 0.170% and the strain values of the maximum AE event are 0.419%, 0.406%, 0.405%, 0.404%, 0.365%, 0.383%, and 0.411%. Apart from the strain value of double-hole defective rock when *α* is equal to 0°, the strain values of the initial AE event and the maximum AE event of these defective rocks are decreasing with decrease of the absolute value of *α*. The strain values of the initial AE event and the maximum AE event are almost equally when the absolute value of *α* is the same. The strain values of the double-hole defective rocks when *α* is positive are less than that of the defective rocks when *α* is negative. However, the strain values of the maximum AE event of the double-hole defective rocks when *α* is positive are larger than that of the defective rocks when *α* is negative.(3)It influences the strain ranges of the drastic AE events. Apart from the strain range of the drastic AE events of the rock with *α* of 0°, the strain range of the drastic AE events increases with the decreasing of the absolute value of α. For *α* of −90° to −30°, the strain ranges of the drastic AE events are 0.059%, 0.067%, and 0.126%. While for *α* of 90° to 30°, the strain ranges of the drastic AE events are 0.075%, 0.086%, and 0.127%. When the absolute value of *α* is same, the strain ranges of the drastic AE events of the double-hole defective rocks are also approximately equally with each other. The strain range of the serious AE events of the defective rocks when *α* is positive is wider than that of the defective rocks when *α* is negative.

#### 4. Damage Evolution Law of Rocks with Double-Hole Defects

##### 4.1. Damage Model Based on AE Events

There are a lot of parameters that can be used to define the damage variable (*D*), such as cracks [25], elastic coefficient [26], yield stress [27], elongation [28], AE [29, 30], energy [31], etc. In this study, *D* is defined by the parameters of AE events as follows:where is the total AE events number of the rock material from the original point to a specific time, and *N* is the total AE events number of the rock material when it failures completely.

Owing to that it is hard to judge if a specimen has been damaged completely during an experiment, the damage variable needs to be modified [32] and the modified damage variable can be expressed aswhere *k* is the damage critical value and it can be normalized aswhere is the residual strength of the rock material and is the peak strength. In this study, the tests will be end when the residual strength of the rock models is 10% of the peak strength, so the *k* is equal to 0.9.

Based on the AE events characteristics and strain equivalence principle [19, 33], the damage model of the rock model under uniaxial compression condition is established as shown in Equation (4).

Figure 11 shows the stress-strain curve fitted by Equation (4) and the numerical test result. It can be seen that for the rock materials, the damage constitutive model based on AE events characteristics can well reflect the stress-strain evolution characteristics.

##### 4.2. Characteristics of Damage Evolution of Double-Hole Defective Rocks

Figure 12 describes the damage variable-strain curves of double-hole defective rocks with different *α*. The damage variable-strain curve can also be divided into three stages according to the classification of the stress-strain curves. The first stage is corresponding with the elastic stage of the stress-strain curve. In this stage, there is no damage occurring and the stage can be named zero damage stage. The second stage is corresponding with the plastic stage of the stress-strain stage. In this stage, the damage increases gradually and the stage can be named stable evolution stage. The third stage is corresponding with the failure stage of the stress-strain curve. In this stage, the damage increases rapidly and the stage can be named failure damage stage.

In these three stages, *α* has different influence on the characteristics of damage evolution of the double-hole defective rocks. In zero damage stage, *α* affects the strain range of zero damage. Along with decreasing of the absolute value of *α*, the strain range of zero damage is decreasing firstly and then increasing. When *α* is less than or equal to 0°, the strain ranges are 0.196%, 0.098%, 0,108%, and 0.179%. While when *α* is bigger than 0°, the strain ranges are 0.177%, 0.091%, and 0.136%.

In the stable evolution stage, *α* affects the damage rate of the double-hole defective rocks. When the strain value is less than 0.242%, with decreasing of the absolute value of *α*, the damage rate shows a trend of increasing firstly and then decreasing. When the strain value is larger than 0.242%, it shows an increasing trend except when *α* is 0°. Meanwhile, the damage variable-strain curves are fluctuated when *α* is different. The damage fluctuations of double-hole defective rocks when the absolute values of *α* equal to 30° and 60° are higher than that of rocks when the absolute values of *α* equal to 90° and 0°.

In failure damage stage, the increase rates of damage of defective rocks are higher than that in the stable evolution stage. *α* also affects the increasing rate of damage. Along with decreasing of the absolute value of *α*, the damage rate is decreasing gradually. The slope is almost vertical when *α* is 90° or −90°.

#### 5. Conclusions

In the study, the damage evolution characteristics of double-hole defective rock with different angles of the center connection line are studied systematically based on particle flow code. Several conclusions are obtained as the following.

The stress-strain behavior of the defective rock can be divided into the three stages of elastic, plastic and failure. The evolution characteristics of AE events and evolution laws of damage are closely related to the stress-strain relationship. According to the stress-strain curve, the damage variable-strain curve can be grouped zero damage stage, stable evolution stage, and failure damage stage.

With decrease of the absolute value of *α*, the peak strength of the defective rocks is decreasing firstly and then increasing. The shape of the peak strength curve as *α* increasing from −90° to 90° is a W type. The stress-strain characteristics are very discrete after peak strength. Along with the decreasing of the absolute value of *α*, the post peak stress-strain curves changing from smoothness with one peak to fluctuate with several peaks. The failure modes of the double-hole defective rocks with the same absolute value of *α* are symmetrical to the horizontal axis or the longitudinal axis approximately. The main failure modes are oblique, Y type, U type, and V type when the absolute values of *α* are 90°, 60°, 30° and 0° respectively.

The influences of different center connection line angles on the characteristics of AE events include the maximum AE event number, the strain values of the initial AE event and the maximum AE event, and the strain range of the serious AE events. With the decreasing of the absolute value of *α*, the maximum AE event number and the strain values of the initial AE event and the maximum AE event are decreasing, but the strain ranges of the serious AE events are increasing.

Different *α* has different influence on the characteristics of damage evolution of the double-hole defective rocks. In zero damage stage, *α* affects the strain range of zero damage. With decreasing of the absolute value of *α*, the strain ranges of zero damage are decreasing firstly and then increasing. In the stable evolution stage, *α* affects the damage rate of the double-hole defective rocks. When the strain value is less than 0.242%, with decreasing of the absolute value of *α*, the damage rate shows a trend of increasing firstly and then decreasing. When the strain value is larger than 0.242%, it shows an increasing trend except when *α* is 0°. In failure damage stage, the damage rates of defective rocks are higher than the stable evolution stage. With decreasing of the absolute value of *α*, the damage rates are decreasing gradually. The slope is almost vertical when *α* is 90° or −90°.

#### Data Availability

We declare that all the data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This study is supported partially by National Natural Science Foundation of China (Grant nos. 41602300 and 51704179), Natural Science Foundation of Jiangsu Province (BK20150618), Open Research Foundation of the State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology (SKLGDUEK1503), Natural Science Foundation of Shandong Province (ZR2016EEB23), Science and Technology Program of Shandong Province University (J15lh02), the Priority Academic Program Development of Jiangsu Higher Education Institutions (CE01-3-7), and the Key Research and Development Project of Shandong Province.