# Influence of Flaw Inclination Angle on Unloading Responses of Brittle Rock in Deep Underground

Authors: Zhenghong Chen; Xibing Li; Lei Weng; Shaofeng Wang; Longjun Dong
Journal: Geofluids
 (2019)

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## Abstract

Excavation unloading is a primary stress condition for engineering rock mass in deep underground. Based on the unloading stress condition during the excavation operation, this paper employed a distinct element method (DEM) to simulate the unloading responses of intact and preflawed rock specimens. The simulation results revealed that the unloading failure strength, unloading damage thresholds, and cracking characteristics were largely dependent on the inclination angleα of the preflaws. With the increase in the flaw inclination angle, the unloading failure strength of a preflawed specimen exhibited a sigmoidal curve increasing trend, and it decreased by 5.5%-20% compared to the unloading failure strength of an intact specimen. Based on the crack accumulation in specimens, three damage thresholds were identified under unloading condition and two damage thresholds σci and σdi were found to be increased with the increase in the flaw inclination angle. Furthermore, when the flaw inclination angle was smaller, cracks were initiated around the preflaws, and there were obvious axial splitting cracks in the failure modes of preflawed specimens, while axial splitting cracks were few in the preflawed specimen with a larger flaw inclination angle and none in the intact specimen. These unloading responses indicate that inducing preflaws can reliably reduce the unloading failure strength and promote the cracking process of hard rock during an excavation unloading process. Moreover, inducing preflaws with a smaller inclination angle (e.g., vertical to the unloading direction) will be more helpful for the unloading failure and rock cracking during an excavation unloading process.


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## Body

## 1. Introduction

With the increasing demand of minerals and energy resources, mining operations and other resource excavations turned to deep underground. In deep underground operations or excavations, the construction environment is complex, usually involving high in situ stress, high geotemperature, and high water pressure [1–3]. The excavation activities may induce an important effect on the deformation and failure of rock mass in such complex environment. Particularly, when the drill-and-blast method is used in deep underground excavations, the high in situ stress in the horizontal direction is unloaded instantly, accompanied by a stress concentration in the vertical direction. In this process, the transient released strain energy dissipates completely in the rock mass [4]. Therefore, the underground mining may lead to sudden and severe rock mass failure, posing a significant threat to the safety and stability of underground structures [5–7]. In this regard, the mechanical excavation method with controllable energy release is considered to be superior to the drill-and-blast method in the operations of underground excavation such as mining and tunneling [8]. Nowadays, mining equipment manufacturers are developing new machine concepts with the aim of using a mechanical excavation method in underground mining and tunneling operations [9]. However, the applications of mechanical cutting and excavation tools (e.g., road headers and tunnel boring machines (TBMs)) are limited due to the high strength and high integrity of the hard rock. Particularly in underground deep mining, it has been clearly demonstrated that high rock strength and integrity will significantly influence the excavation performance of road headers [10] and TBMs [11, 12]. Therefore, by reducing the rock strength and integrity, it would be possible that the small and flexible continuous excavation machine can perform well in hard rock underground excavation.One of the simplest and most straightforward ways of reducing rock strength and integrity on purpose is to induce preflaws in the hard rock. Over the last few years, extensive studies have been conducted to investigate the mechanical behavior of rock materials containing preflaws. For instance, Yang and Jing [13], Li et al. [14], and Zhang et al. [15] explored the cracking behavior of rock containing preexisting flaws, and the influences of different flaw geometries (flaw inclination angle, flaw length, and flaw size) on the rock failure were discussed. Bi et al. [16] studied the effect of confining pressure on failure behaviors of preflawed rock specimens, and the results indicated that the growth of tensile cracks and wing cracks was restrained with the increase in confining pressure. Lee and Jeon [17] discussed the fracture coalescence of three different preflawed materials (i.e., PMMA, molded gypsum, and granite) and summarized the common and different features of the cracking process depending on the materials. Zhuang et al. [18] and Park and Bobet [19] comparatively investigated the different mechanisms of crack coalescence between open and closed flaws, and they found that the peak stress and initiation and coalescence stresses for closed flaws were higher than those for open flaws. Huang et al. [20] conducted experiments on sandstones with two preexisting flaws; the results show that the arrangement of the flaw pair and the confining pressure have remarkable impacts on the deformation, strength, and crack coalescence pattern of rock. These investigations revealed that the flaws in rock played a significant role in the cracking and failure behavior of rock material. However, the fracturing and failure behavior of preflawed rock in previous studies were obtained from uniaxial or biaxial compression conditions, in which the fracturing and failure characteristics of preflawed rock under excavation unloading condition have not been involved.Under unloading condition, both experimental and numerical investigations have been performed to study the mechanical responses of rock materials. For example, Zhao et al. [21] investigated the influence of the unloading rate on strain burst under unloading conditions, and they found that granite was more prone to occur strain burst with a higher unloading rate. Huang and Li [22] conducted unloading tests on marbles, and the results indicated that higher initial confining pressure and unloading rate will result in severer “flying fragment” phenomenon. Du et al. [23] investigated the unloading behavior of granite, red sandstone, and cement mortar, showing that slabbing occurred in the granite and sandstone specimens, whereas typical shear failures were observed in the cement mortar samples. Zhao and He [24] investigated the effects of granite size on its unloading behavior; they found that the rock strength exhibits an increasing trend with the decrease in a height-to-thickness ratio. Manouchehrian and Cai [25] performed a series of numerical studies to explore the effects of loading system stiffness, aspect ratio of a specimen, and intermediate principal stress on the unloading failure behavior of granite model specimens. The associated simulation results showed that low-stiffness loading system, tall specimen, and low intermediate principle stress could induce violent unloading failure. In addition, Li et al. [26] presented a theoretical solution to obtain the stress state of highly stressed rock under quasi-static and intensive unloading conditions and found that the initial stress was a critical factor that governs the rock failure. So far, most of the previous studies concerning the unloading responses were performed on intact rock specimens, while the differences in the unloading responses between intact rock and preflawed rock are unclear.In recent years, the numerical method plays an important role in researches related to rock materials [27–29]. For example, the distinct element method (DEM) provides an efficient access to characterize the mechanical behavior of brittle rocks. Compared with laboratory tests, using DEM to investigate the unloading failure behavior of preflawed rocks has the following superiorities: (1) apart from the variation of the flaw inclination angle, the preflawed numerical specimen is exactly identical, eliminating the disturbances of the geometrical errors of induced preflaws or the material heterogeneities of rock specimen in laboratory experiments; (2) all the numerical data are accessible at any arbitrarily instantaneous moment during the unloading process; and (3) the cracking process of rocks involving the initiation, propagation, and coalescence can be explicitly exhibited. Moreover, the numerical simulation is highly efficient, repeatable, and low-cost.In the present study, the unloading failure responses of intact and preflawed rocks in the excavation unloading process are systematically investigated by DEM. First, the fundamentals of DEM are briefly introduced. Then, the unloading stress condition during excavation operations is analyzed, followed by a detailed simulation description of the unloading process on rock specimen, involving the calibration of an intact specimen, the induction of preflaws, and the reproduction of an unloading stress path. Finally, the unloading responses including unloading failure strength, damage thresholds, and cracking process were investigated with the consideration of the effect of the preflaws.


## 2. Brief Description of DEM

The DEM-based software particle flow code (PFC) is employed to investigate the unloading behavior of a rock specimen. In the DEM model, the rock specimen is simulated as an assembly of densely packed and bonded rigid spherical particles, in which two adjacent particles are cemented together by a bond (see Figure1). The mechanical properties of the model are controlled by the microparameters of particles and bonds [30]:
(1)Ec,knks,μ,particlemicroparameters,E¯c,k¯nk¯s,λ¯,σ¯c,τ¯c,bondmicroparameters,where Ec and E¯c are Young’s moduli of the particles and bonds, respectively, kn/ks and k¯n/k¯s are the ratios of normal-to-shear stiffness of the particles and bonds, respectively, λ¯ is the ratio multiplier of a parallel bond, μ is the particle friction coefficient, and σ¯c and τ¯c are the tensile and shear strengths of the bonds, respectively.Figure 1
Sketch map of the bond particle model.The mechanical behavior of the particle-bond system consisted of the particle-particle contact behavior and the parallel-bond behavior. The normal forceFn in the particle-particle contact behavior is calculated by
(2)Fn=knUn,where Un is the overlap between particles and kn is the normal stiffness of the particles.The shear forceFs is computed in an incremental fashion. When the contact is formed, Fs is initialized to zero. Each subsequent shear-displacement increment ΔUS will result in an increment of shear force:
(3)ΔFs=−ksΔUs.The normal forceF¯n, shear force F¯s, and the bending moment M¯ in the parallel-bond behavior are also computed in an incremental fashion:
(4)ΔF¯n=k¯nAΔUn,ΔF¯s=−k¯sAΔUs,ΔM¯=−k¯nIΔθs,where ΔUn, ΔUs, and Δθs are the subsequent displacement and rotation increments and A and I are the area of bond across section and the moment of inertia, respectively.The following breakage criterion of the model is employed in this study:(5)σmaxσ¯c≥1,τmaxτ¯c≥1,where σmax and τmax denote the maximum tensile stress and maximum shear stress, respectively, and they are calculated from
(6)σmax=−F¯nA+M¯IR¯,τmax=F¯sA,where R¯ is the bond radius. These calculation equations have received wide acceptance to simulate the fracture mechanism of rocks [31, 32].


## 3. Unloading Description

### 3.1. Unloading Stress Condition

Uniaxial, biaxial, and triaxial compression tests are often performed to investigate rock mechanical property and behavior, whereas the stress conditions of these compression tests are quasi-static loading, and none of them can reflect the in situ stress variation process of rock mass during excavation [33]. In order to better understand the unloading process of in situ stress during the excavation operations, the stress condition of excavation unloading is analyzed.During the excavation process, the stress distribution of rock masses ahead of the working face is shown in Figure2, where σz is the vertical stress, σx is the horizontal stress, p0 is the in situ geostress, x is the distance between the working face and rock mass, z is the height from the tunnel center, and R represents a rock mass unit ahead of the working face. As illustrated in Figure 2, the rock masses ahead of the working face can be divided into three zones: undisturbed zone, elastic zone, and failure zone. Rock mass in the undisturbed zone is under an in situ stress condition. Rock mass in the elastic zone is under an elastic deformation condition, while the rock mass in the failure zone is damaged. With the advancing of the working face, the stress distribution ahead of the working face also moves forward (as shown in Figures 2(a)–2(c)). Therefore, for the rock mass unit R in an underground engineering which is ready for excavation, it will undergo three stages of stress states during the advance process of the working face (Figure 3). When the excavation is just starting and the working face is far away from the rock mass unit 0≤t<t0, the rock mass unit is under a hydrostatic in situ stress state; with the advancing of the working face, the horizontal in situ stress on it is unloaded or released, accompanied by an increase in the vertical in situ stress t0≤t<tp; when the working face advances to the vicinity area of the rock mass unit, the rock failure will happen tp≤t<tn. During the advance process of the working face, the variation process of horizontal in situ stress can be given by
(7)σx=p0,0≤t<t0,p0ft,t0≤t≤tn,where ft is a decreasing function and represents the unloading path of horizontal in situ stress, and different excavation methods will induce different unloading paths.Figure 2
Stress distribution of rock mass ahead of the working face.
(a)
Working face is far away from the rock mass unit (R)
(b)
Working face advances to the nearby region of the rock mass unit (R)
(c)
Working face locates in the boundary line of the rock mass unit (R)Figure 3
Stress variation of the rock mass unit (R) with the advancing of the working face, and t is the time of the stress variation process.The variation process of the vertical in situ stress is a stress concentration process, and it will reach a peak stress before the rock failure. The peak stress can be written as(8)σzmax=σz/t=tp=kp0,where k is the stress concentration factor, and the value of k depends on the unloading process of horizontal in situ stress.Such stress variation processes are the practical stress condition that rock mass will undergo during the excavation in high geostress, which are defined as the unloading process. In this study, the mechanical response of a preflawed rock specimen under such unloading process is investigated, in order to understand the unloading failure mechanism of rock mass containing preflaws.


### 3.2. Numerical Specimen

In order to numerically reproduce the unloading process of rock material, it is critical to establish a suitable numerical model which can reproduce the mechanical behavior of rock. The rock material for investigation is called “Leiyang white” marble, which was collected from Leiyang City in China. The marble has a connected porosity of approximately 0.56% and an average density of 2810 kg/m3. The mechanical behaviors of the marble were obtained through a laboratory uniaxial compression test and triaxial compression test at 40 MPa confining pressure. In laboratory tests, all the marble specimens were cored from an identical block to minimize the scatter effect, and specimens were prepared with dimensions of 50 mm in diameter and 100 mm in height following the ISRM-suggested method [34] (Figure 4(a)). The tests were conducted using the MTS 815 rock mechanics system in the Advanced Research Center at Central South University. The specimens were loaded till failure occurs under a displacement control module, and the loading rate is 0.15 mm/min.Figure 4
Intact marble specimens.
(a)
(b)The numerical model was established with the same size as the marble specimen (Figure4(b)). Through adjusting the values of microparameters (as in equation (1)) in the numerical model, a series of uniaxial compression tests and triaxial compression tests at 40 MPa confining pressure were numerically simulated. In the uniaxial compression numerical test, the specimen was placed between the top and bottom load walls. The loading rate of load walls was set to 0.05 m/s. In the numerical triaxial compression test, the specimen was surrounded by four load walls. The confining pressure applied to the two lateral walls was set to a constant value of 40 MPa, and the loading rate for top and bottom walls was 0.05 m/s. After a series of trial and error, a suitable set of microparameters (Table 1) was obtained to establish the numerical marble specimen which can reproduce the mechanical behavior of the real marble specimen.Table 1
Microparameters of the numerical specimen.
Particle microparameters
Bond microparameters
Minimum radiusRmin (mm)
0.25
Parallel bond modulesE¯c (GPa)
58
Particle size ratioRmax/Rmin
1.5
Tensile strengthσ¯c (MPa)
94±5
Contact modulesEc (GPa)
58
Shear strengthτ¯c (MPa)
188±5
Frictional coefficientμ
0.65
Stiffness ratiok¯n/k¯s
2.5
Stiffness ratiokn/ks
2.5
Ratio multiplierλ¯
1.0Table2 shows the comparisons of the macromechanical behavior of marble specimens from laboratorial and numerical tests. In Table 2, UCS is the uniaxial compression strength and TCS-40 refers to triaxial compression strength at confining pressures of 40 MPa. It is seen that the mechanical properties of numerical marble specimens are much close to those of real marble specimens. Therefore, the established numerical marble specimen commendably reproduced the mechanical properties of the real marble specimen. Figure 5 shows the comparison of laboratorial and numerical stress-strain curves of the marble specimen. From Figure 5, it can be seen that the numerical curves under compression tests agree well with the laboratorial curves, including the crack initiation and growth stage before the peak strength and the unstable failure stage after the peak strength. However, the numerical curve does not mirror the initial nonlinearity deformation (crack closure stage) in the uniaxial compression test. This is because the real rock specimen generally contains natural microfractures while the numerical rock specimen consisted of a compacted assembly of rigid particles. Moreover, the numerical curve does not show the postpeak strain softening in the triaxial compression test at 40 MPa confining pressure. Nevertheless, the discrepancies in the nonlinearity deformation and the postpeak strain softening between the numerical and laboratorial curves do not cause large differences of failure modes. As shown in Figure 6, the failure mode of the numerical marble specimen in the uniaxial compression test showed a typical splitting failure, and the failure mode in the triaxial compression test is in a shear fashion. These failure modes are highly consistent with those obtained in laboratory. The comparisons presented in Table 2 and Figures 5 and 6 justified that the calibrated numerical marble specimen has the capability of simulating the mechanical behavior of the marble.Table 2
Laboratorial and numerical macromechanical behavior of the marble specimen.
Experimental results
Numerical results
Errors (%)
Density (g/cm3)
2.81
2.81
—
Young’s modulus (GPa)
76.4
75.6
1.05
Poisson’s ratio
0.21
0.21
—
UCS (MPa)
135.3
136.1
0.59
TCS-40 (MPa)
273.5
264.2
3.40Figure 5
Laboratorial and numerical stress-strain curves of the marble specimen under compression.Figure 6
Laboratorial and numerical failure modes of marble specimens under compression (the red short lines in numerical specimens represent microcracks).
(a)
Laboratorial failure mode
(b)
Numerical failure modeAfter establishing the suitable intact numerical marble specimen, preflaws were created in the numerical specimen to investigate the influence of the flaw inclination angle on unloading responses. Technologies under consideration for inducing preflaws in real underground rock include thermal perturbation, high-pressure water jet, and high-power microwaves. These technologies use different damage mechanisms to generate flaw networks in the rock. For simplification, in this study, just two straight flaws were added at the center of the numerical specimen. The geometry of the two flaws in the numerical marble specimens is shown in Figure7. The flaw length L1 is 10 mm, and the bridge length between the flaws L2 is 14 mm. The bridge angle β is 30°, and the flaw inclination angle α varies from 0° to 90° in 30° interval. The flaws were created by deleting particles in a specific range. Since the particles in the model cannot be divided, the surfaces of the induced flaws are not smooth as can be observed in Figure 7.Figure 7
Numerical marble specimens containing two preflaws.


### 3.3. Unloading Procedure

During the drill-and-blast excavation, the release of in situ stress is a transient process. While during the mechanical excavation, the unloading of in situ stress is a quasi-static process [35, 36]. In the present study, to analyze the quasi-static unloading process during the mechanical excavation, the typical stress condition of the quasi-static unloading process as illustrated in Section 3.1 was simulated by varying the boundary stresses on the rock specimens. To this end, a set of self-defined functions was encoded in the PFC program. The procedures for the unloading simulation were presented in detail as follows:Step 10≤t<t0.
Increase the lateral pressureσ3 to the in situ stress level, and simultaneously load the axial pressure σ1 to the in situ stress level; then, keep them unchanged for a while.Step 2t0≤t<tp.
Unload the lateral pressureσ3, while the axial pressure σ1 increases at a specified rate until the peak stress σ1max.Step 3tp≤t≤tn.
After the peak stress of the specimen, the lateral pressureσ3 is unloaded continuously to zero to finish the unloading, while the axial pressure σ1 will decrease spontaneously due to the failure of the specimen.To eliminate the unloading wave effect and to keep a quasi-static unloading process, a high-damp parameter was adopted in the modelling. The axial pressureσ1 was applied in a displacement-controlled manner at a rate of 0.05 m/s. Through recoding the servo-controlled program, the lateral pressure was unloaded in a linear form following
(9)σ3=p0,0≤t<t0,p01−t−t0tn−t0,t0≤t≤tn,where p0 is the in situ stress level, initially set as 40 MPa; t is the unloading time; t0 is the initiation moment of unloading, set as 1 ms; tn is the end moment of the unloading process, set as 3 ms. Therefore, the unloading period in the simulation was 2 ms. During the numerical simulation process in DEM, a critical and reasonable time step was internally assigned in the program before each program cycle. The time step for the current simulation is merely about 2.2×10−8s/step; thus, the unloading rate of the lateral pressure can be translated to 4.4×10−4MPa/step, while the loading rate of the axial pressure can be translated to 1.1×10−6mm/step. Such unloading and loading rates are low enough to simulate the quasi-static unloading process similar to the tunnel excavation with a mechanical method.


## 3.1. Unloading Stress Condition

Uniaxial, biaxial, and triaxial compression tests are often performed to investigate rock mechanical property and behavior, whereas the stress conditions of these compression tests are quasi-static loading, and none of them can reflect the in situ stress variation process of rock mass during excavation [33]. In order to better understand the unloading process of in situ stress during the excavation operations, the stress condition of excavation unloading is analyzed.During the excavation process, the stress distribution of rock masses ahead of the working face is shown in Figure2, where σz is the vertical stress, σx is the horizontal stress, p0 is the in situ geostress, x is the distance between the working face and rock mass, z is the height from the tunnel center, and R represents a rock mass unit ahead of the working face. As illustrated in Figure 2, the rock masses ahead of the working face can be divided into three zones: undisturbed zone, elastic zone, and failure zone. Rock mass in the undisturbed zone is under an in situ stress condition. Rock mass in the elastic zone is under an elastic deformation condition, while the rock mass in the failure zone is damaged. With the advancing of the working face, the stress distribution ahead of the working face also moves forward (as shown in Figures 2(a)–2(c)). Therefore, for the rock mass unit R in an underground engineering which is ready for excavation, it will undergo three stages of stress states during the advance process of the working face (Figure 3). When the excavation is just starting and the working face is far away from the rock mass unit 0≤t<t0, the rock mass unit is under a hydrostatic in situ stress state; with the advancing of the working face, the horizontal in situ stress on it is unloaded or released, accompanied by an increase in the vertical in situ stress t0≤t<tp; when the working face advances to the vicinity area of the rock mass unit, the rock failure will happen tp≤t<tn. During the advance process of the working face, the variation process of horizontal in situ stress can be given by
(7)σx=p0,0≤t<t0,p0ft,t0≤t≤tn,where ft is a decreasing function and represents the unloading path of horizontal in situ stress, and different excavation methods will induce different unloading paths.Figure 2
Stress distribution of rock mass ahead of the working face.
(a)
Working face is far away from the rock mass unit (R)
(b)
Working face advances to the nearby region of the rock mass unit (R)
(c)
Working face locates in the boundary line of the rock mass unit (R)Figure 3
Stress variation of the rock mass unit (R) with the advancing of the working face, and t is the time of the stress variation process.The variation process of the vertical in situ stress is a stress concentration process, and it will reach a peak stress before the rock failure. The peak stress can be written as(8)σzmax=σz/t=tp=kp0,where k is the stress concentration factor, and the value of k depends on the unloading process of horizontal in situ stress.Such stress variation processes are the practical stress condition that rock mass will undergo during the excavation in high geostress, which are defined as the unloading process. In this study, the mechanical response of a preflawed rock specimen under such unloading process is investigated, in order to understand the unloading failure mechanism of rock mass containing preflaws.


## 3.2. Numerical Specimen

In order to numerically reproduce the unloading process of rock material, it is critical to establish a suitable numerical model which can reproduce the mechanical behavior of rock. The rock material for investigation is called “Leiyang white” marble, which was collected from Leiyang City in China. The marble has a connected porosity of approximately 0.56% and an average density of 2810 kg/m3. The mechanical behaviors of the marble were obtained through a laboratory uniaxial compression test and triaxial compression test at 40 MPa confining pressure. In laboratory tests, all the marble specimens were cored from an identical block to minimize the scatter effect, and specimens were prepared with dimensions of 50 mm in diameter and 100 mm in height following the ISRM-suggested method [34] (Figure 4(a)). The tests were conducted using the MTS 815 rock mechanics system in the Advanced Research Center at Central South University. The specimens were loaded till failure occurs under a displacement control module, and the loading rate is 0.15 mm/min.Figure 4
Intact marble specimens.
(a)
(b)The numerical model was established with the same size as the marble specimen (Figure4(b)). Through adjusting the values of microparameters (as in equation (1)) in the numerical model, a series of uniaxial compression tests and triaxial compression tests at 40 MPa confining pressure were numerically simulated. In the uniaxial compression numerical test, the specimen was placed between the top and bottom load walls. The loading rate of load walls was set to 0.05 m/s. In the numerical triaxial compression test, the specimen was surrounded by four load walls. The confining pressure applied to the two lateral walls was set to a constant value of 40 MPa, and the loading rate for top and bottom walls was 0.05 m/s. After a series of trial and error, a suitable set of microparameters (Table 1) was obtained to establish the numerical marble specimen which can reproduce the mechanical behavior of the real marble specimen.Table 1
Microparameters of the numerical specimen.
Particle microparameters
Bond microparameters
Minimum radiusRmin (mm)
0.25
Parallel bond modulesE¯c (GPa)
58
Particle size ratioRmax/Rmin
1.5
Tensile strengthσ¯c (MPa)
94±5
Contact modulesEc (GPa)
58
Shear strengthτ¯c (MPa)
188±5
Frictional coefficientμ
0.65
Stiffness ratiok¯n/k¯s
2.5
Stiffness ratiokn/ks
2.5
Ratio multiplierλ¯
1.0Table2 shows the comparisons of the macromechanical behavior of marble specimens from laboratorial and numerical tests. In Table 2, UCS is the uniaxial compression strength and TCS-40 refers to triaxial compression strength at confining pressures of 40 MPa. It is seen that the mechanical properties of numerical marble specimens are much close to those of real marble specimens. Therefore, the established numerical marble specimen commendably reproduced the mechanical properties of the real marble specimen. Figure 5 shows the comparison of laboratorial and numerical stress-strain curves of the marble specimen. From Figure 5, it can be seen that the numerical curves under compression tests agree well with the laboratorial curves, including the crack initiation and growth stage before the peak strength and the unstable failure stage after the peak strength. However, the numerical curve does not mirror the initial nonlinearity deformation (crack closure stage) in the uniaxial compression test. This is because the real rock specimen generally contains natural microfractures while the numerical rock specimen consisted of a compacted assembly of rigid particles. Moreover, the numerical curve does not show the postpeak strain softening in the triaxial compression test at 40 MPa confining pressure. Nevertheless, the discrepancies in the nonlinearity deformation and the postpeak strain softening between the numerical and laboratorial curves do not cause large differences of failure modes. As shown in Figure 6, the failure mode of the numerical marble specimen in the uniaxial compression test showed a typical splitting failure, and the failure mode in the triaxial compression test is in a shear fashion. These failure modes are highly consistent with those obtained in laboratory. The comparisons presented in Table 2 and Figures 5 and 6 justified that the calibrated numerical marble specimen has the capability of simulating the mechanical behavior of the marble.Table 2
Laboratorial and numerical macromechanical behavior of the marble specimen.
Experimental results
Numerical results
Errors (%)
Density (g/cm3)
2.81
2.81
—
Young’s modulus (GPa)
76.4
75.6
1.05
Poisson’s ratio
0.21
0.21
—
UCS (MPa)
135.3
136.1
0.59
TCS-40 (MPa)
273.5
264.2
3.40Figure 5
Laboratorial and numerical stress-strain curves of the marble specimen under compression.Figure 6
Laboratorial and numerical failure modes of marble specimens under compression (the red short lines in numerical specimens represent microcracks).
(a)
Laboratorial failure mode
(b)
Numerical failure modeAfter establishing the suitable intact numerical marble specimen, preflaws were created in the numerical specimen to investigate the influence of the flaw inclination angle on unloading responses. Technologies under consideration for inducing preflaws in real underground rock include thermal perturbation, high-pressure water jet, and high-power microwaves. These technologies use different damage mechanisms to generate flaw networks in the rock. For simplification, in this study, just two straight flaws were added at the center of the numerical specimen. The geometry of the two flaws in the numerical marble specimens is shown in Figure7. The flaw length L1 is 10 mm, and the bridge length between the flaws L2 is 14 mm. The bridge angle β is 30°, and the flaw inclination angle α varies from 0° to 90° in 30° interval. The flaws were created by deleting particles in a specific range. Since the particles in the model cannot be divided, the surfaces of the induced flaws are not smooth as can be observed in Figure 7.Figure 7
Numerical marble specimens containing two preflaws.


## 3.3. Unloading Procedure

During the drill-and-blast excavation, the release of in situ stress is a transient process. While during the mechanical excavation, the unloading of in situ stress is a quasi-static process [35, 36]. In the present study, to analyze the quasi-static unloading process during the mechanical excavation, the typical stress condition of the quasi-static unloading process as illustrated in Section 3.1 was simulated by varying the boundary stresses on the rock specimens. To this end, a set of self-defined functions was encoded in the PFC program. The procedures for the unloading simulation were presented in detail as follows:Step 10≤t<t0.
Increase the lateral pressureσ3 to the in situ stress level, and simultaneously load the axial pressure σ1 to the in situ stress level; then, keep them unchanged for a while.Step 2t0≤t<tp.
Unload the lateral pressureσ3, while the axial pressure σ1 increases at a specified rate until the peak stress σ1max.Step 3tp≤t≤tn.
After the peak stress of the specimen, the lateral pressureσ3 is unloaded continuously to zero to finish the unloading, while the axial pressure σ1 will decrease spontaneously due to the failure of the specimen.To eliminate the unloading wave effect and to keep a quasi-static unloading process, a high-damp parameter was adopted in the modelling. The axial pressureσ1 was applied in a displacement-controlled manner at a rate of 0.05 m/s. Through recoding the servo-controlled program, the lateral pressure was unloaded in a linear form following
(9)σ3=p0,0≤t<t0,p01−t−t0tn−t0,t0≤t≤tn,where p0 is the in situ stress level, initially set as 40 MPa; t is the unloading time; t0 is the initiation moment of unloading, set as 1 ms; tn is the end moment of the unloading process, set as 3 ms. Therefore, the unloading period in the simulation was 2 ms. During the numerical simulation process in DEM, a critical and reasonable time step was internally assigned in the program before each program cycle. The time step for the current simulation is merely about 2.2×10−8s/step; thus, the unloading rate of the lateral pressure can be translated to 4.4×10−4MPa/step, while the loading rate of the axial pressure can be translated to 1.1×10−6mm/step. Such unloading and loading rates are low enough to simulate the quasi-static unloading process similar to the tunnel excavation with a mechanical method.


## 4. Simulation Results and Discussions

In an underground mechanical excavation process, the rock mass undergoes the unloading process caused by stress redistribution before subjecting to the cutting force of machines. If the rock failure happens during the unloading process, the cutting using an excavation machine will be easier, and it will be helpful to improve the excavation efficiency. This means that the unloading of minimum principal stress and the concentration of maximum stress throughout the unloading process greatly affect the subsequent cutting process of excavation machines. Therefore, it is important to investigate the failure behavior of rock mass during the unloading process. In this section, the failure behavior of intact and preflawed specimens during the unloading process will be discussed in detail.


### 4.1. Unloading Failure Strength of Preflawed Specimens

The stress variations against time for the specimens during unloading simulations are shown in Figure8. It can be found that the specimens experience the typical three stress stages which have been specified in Section 3.1. At first, the specimen was under a hydrostatic pressure state for 1 ms. Then, the lateral stress σ3 was unloaded and the axial stress σ1 increased to a peak value. Finally, the specimen underwent the unloading failure stage, in which the lateral stress σ3 was unloaded to zero and the axial stress σ1 dropped due to specimen failure. The peak stress σ1 was defined as unloading failure strength σus. Note that the unloading failure strength is different from the failure strength in the conventional compression test, which is the actual failure strength for rock mass during the excavation process.Figure 8
Stress–time curves of specimens under unloading condition.
(a)
(b)
(c)
(d)
(e)It is seen in Figure8 that the unloading failure strength varies significantly for different specimens depending on the geometry of the specimen. For the intact specimen, the unloading failure strength is greater than that for the preflawed specimen. Furthermore, an increase in the unloading failure strength can be observed for the preflawed specimen with the increase in the flaw inclination angle. In order to better understand the influence of preflaws on the strength properties of rock specimens under unloading condition, quantitative investigations on the unloading failure strength were carried out. Figure 9 presented the correlation between the unloading failure strength and flaw inclination angle. In addition, the strength degradation parameter (SDP) was introduced to compare the unloading failure strength characteristics of the preflawed specimen and intact specimen. The SDP is defined as
(10)SDP=Δσusσusi×100%=σusi−σusfσusi×100%,where σusi is the unloading failure strength of the intact specimen and σusf is the unloading failure strength of the preflawed specimen under the same unloading condition.Figure 9
Unloading failure strengthσus of preflawed specimens.The SDP values can show the reduction degrees of the unloading failure strength of preflawed marble specimens relative to that of the intact marble specimen. The SDP values of the preflawed specimens with various flaw inclination angles indicate that the unloading failure strengths of the preflawed specimens decrease by 5.5%–20% compared with those of the intact specimen. Such phenomena indicate that the existence of preflaws in the rock mass can reliably decrease the unloading failure strength. Therefore, inducing preflaws in intact hard rock is helpful to cause rock failure during the unloading process and to improve the cutting efficiency.As shown in Figure9, the unloading failure strength of the preflawed specimen exhibits an “S”-shape rising trend with the increase in the flaw inclination angle. This means that the preflaws with a smaller inclination angle can reduce the unloading failure strength to a greater extent. Previous studies also showed that the strength of the preflawed specimen was closely dependent on the inclination angle of flaws [37, 38]. To further explain the influences of preflaws on the unloading failure strength, the stress state on the flaws was theoretically analyzed. The stress state sketch of preflaws under unloading condition is shown in Figure 10. The normal stress σn and the shear stress τn on the flaw surface can be expressed as
(11)σn=σ1cos2α+σ3sin2α,τn=σ1−σ32sin2α,where α is the inclination angle of flaw and σ1 and σ3 are the axial pressure and lateral pressure, respectively. It should be noted that the flaws are open and the normal stress σn on the flaw can induce tensile stress σb.Figure 10
Stress state sketch of preflaws.The normal stressesσn and shear stresses τn on the flaws with the four different inclination angles can be calculated by substituting α=0°, 30°, 60°, and 90° into equation (11). It is known that the cracking of the rock material usually presents two forms: tensile cracking or shear cracking. If the extreme value of the tensile or shear stress appears and satisfies the tensile or shear strength, then the tensile cracking or shear cracking will happen. For the brittle rock specimen, the tensile strength is much smaller than the shear strength, and the tensile stress plays a dominate role in the cracking of the rock specimen. Comparing the calculated results in equation (11) for flaws with inclination angles of 0°, 30°, 60°, and 90°, it can be found that for the flaw with α=0°, its shear stress is zero τn0°=0 and its normal stress is much larger than that of flaws with other inclination angles (σn0°=σ1, σn0°>σn30°>σn60°>σn90°). Thus, the induced tensile stress σb0° is the largest and tensile cracks are most likely to initiate and propagate around the flaws, hence resulting in the lowest unloading failure strength. For flaws with α=30° and α=60°, they suffer nearly the same shear stresses τn30°≈τn60°; however, the flaw with α=30° suffers a larger normal stress than the flaw with α=60°. Therefore, the induced tensile stress on the flaw with α=30° is larger σb30°>σb60°, which means that it is more likely to initiate cracks around the flaw with α=30°, and it may result in a lower failure strength for the preflawed specimen with α=30°. For flaws with α=90°, their shear stress is zero τn0°=0 and their normal stress is the smallest σn90°=σ3. In this regard, the normal stress is decreasing and it is not great enough to initiate cracks around the flaw. Thus, the failure of the specimen with a flaw inclination angle of 90° is not significantly affected by the flaw, and this is the reason why its unloading failure strength is close to the strength of the intact specimen. These findings indicate that if we induce preflaws with a smaller inclination angle in the specimen (i.e., vertical to the unloading direction), cracks are more likely to initiate and propagate around the flaws and the failure of the specimen is more easily to happen. Therefore, during the unloading process, preflaws with a smaller inclination angle can result in the failure of the specimen that happens more easily.Figure11 shows the comparison between the unloading failure strength and compression failure strength of specimens. It can be seen that the unloading failure strength of the intact specimen is merely 60% of the triaxial compressive strength of the intact specimen, which is, however, approximately 1.2 times the UCS of the intact specimen. Meanwhile, the average value of the unloading failure strength of the preflawed specimen is approximately 50% of the triaxial compression strength of the intact specimen and about 90% of the UCS. The differences between the unloading failure strength and compression strength indicate that the compression strength obtained from laboratory quasi-static compression experiments cannot fully represent the peak failure strength for rock mass under excavation unloading condition. Therefore, it can be considered to use the unloading failure strength obtained from a quasi-static unloading test to evaluate the failure strength properties of underground engineering rock.Figure 11
Comparisons of the unloading failure strengthσus and the compressive strength (UCS, TCS-40).


### 4.2. Unloading Damage Thresholds of Preflawed Specimens

In fact, fracturing of the rock specimen is a complex process and simply measuring the peak stress does not capture this fracturing process [39]. In order to capture some unique features in the fracturing process, i.e., crack initiation, onset of fracture localization, and collapse peak, some damage thresholds are pointed out. The fracturing process of rock specimens can be recorded by many different ways, such as acoustic emission measurements and high-speed photography. In laboratory compression experiments, previous scholars determine the damage initiation and interaction thresholds for brittle rock based on the acoustic emissions [40, 41]. In the present study, the crack accumulation amount was used to reflect the fracturing process, and the damage thresholds of specimens under unloading condition were identified by the variation of the crack accumulation amount. Figure 12 gives the relations of the deviatoric stress, the crack accumulation, and the associated axial strain response of specimens under unloading conditions. It can be seen that there are two critical points on the crack accumulation curves, i.e., point A and point B. Point A denotes the first intersection of two tangent lines in the crack accumulation curve, which can mark the onset of a slight increase in cracks. Point B denotes the second intersection of two tangent lines in the crack accumulation curve, which can mark the onset of a sharp increase in cracks. Cracks increase in a stable manner between points A and B, while cracks begin to increase in an unstable manner after point B. The corresponding stress at point A is denoted as σci, which represents the threshold stress for stable crack initiation. The stress at point B is denoted as σcd, which is a threshold stress for unstable crack development. Moreover, the other critical point was the point where the first observable macrocrack appears in the specimen [42]. In this study, the stress where the first observable macrocrack appears is defined as the other threshold, i.e., the damage initiation stress σdi. The three damage thresholds (σci, σcd, and σdi) for different specimens are presented in Figure 12. It can be found that σci>σdi>σcd>σpk and σpk is the peak deviatoric stress.Figure 12
Relations among the deviatoric stress, crack accumulation, and strain of specimens under unloading condition.
(a)
(b)
(c)
(d)
(e)For preflawed specimens, the damage threshold ratios are plotted against the flaw inclination angle (Figure13). It can be seen in Figure 13 that the ratios of σci, σdi, and σcd to the peak deviatoric stress σpk are approximately 0.75, 0.85, and 0.97, respectively. According to previous studies, for the intact rock specimen under uniaxial compression, the crack initiation stress typically ranges from 0.3-0.5 UCS, and the unstable crack development stress is approximately 0.7-0.9 UCS [43]. It can be found that the damage threshold ratios of numerical preflawed rock specimens in the unloading tests are higher than those of real intact rock specimens in the uniaxial compression tests. This is because compared with the uniaxial compression test, the lateral stress in the unloading test constrains the initiation of cracks. In addition, the numerical rock specimen is composed of rigid particles, while the real rock specimen usually contains natural microfractures. As shown in Figure 13, the ratios of the stable crack initiation stress σci and damage initiation stress σdi rise with the increase in the flaw inclination angle, while the ratios of σcd change little with the increase in the flaw inclination angle. It indicates that the crack initiation stress is sensitive to the flaw inclination angle, and the preflawed specimen with a smaller flaw inclination angle is more prone to induce cracks under unloading condition. On the contrary, the stress of unstable crack development has no clear change trend with the variation of the flaw inclination angle.Figure 13
Damage threshold ratios of preflawed specimens under unloading condition.


### 4.3. Cracking Characteristics of Preflawed Specimens under Unloading Condition

The cracking characteristics of preflawed specimens at any moment under unloading condition, including crack distribution, crack extension, and crack amount, can be recorded in numerical simulations. To analyze the cracking characteristics in detail, Figure14 shows the crack distributions in the specimens at some specific points during the unloading process. These specific points are denoted by numbers which correspond to the points in stress-strain curves in Figure 12. For example, point ① refers to the damage initiation stress in the stress-strain curve. Point ② corresponds to a stress before the peak deviatoric stress, and the crack count at point ② is approximately 50% of the crack count at the peak deviatoric stress (point ③). Point ④ is a stress after the peak deviatoric stress, and the crack count at point ④ is approximately 2.5 times the crack approximately at the peak deviatoric stress. Point ⑤ is the stress at the end of unloading.Figure 14
Cracking process of specimens under unloading condition (the red short lines in numerical specimens represent microcracks).
(a)
Intact specimen
(b)
Specimen withα=0°
(c)
Specimen withα=30°
(d)
Specimen withα=60°
(e)
Specimen withα=90°As shown in Figure14, the cracking distribution and crack extension for different specimens under unloading condition are obviously different. To make it clearer, the cracking process was divided into two stages, i.e., stage I—prior to the peak deviatoric stress (including the peak deviatoric stress, ①, ②, and ③)—and stage II—after the peak deviatoric stress (④ and ⑤).In stage I, the specimens undergo the elastic deformation and the crack initiation and propagation before reaching the peak deviatoric stress. For the intact specimen, the cracks distribute randomly in the specimen, while for the preflawed specimen with an inclined angle of 0°, cracks intensively distribute around the flaw tips and flaw surfaces. This phenomenon verifies the results of stress state analyses for preflaws based on Figure10 and equation (11). At an inclined angle of 0°, the normal stress on the flaw is large enough, making the cracks to easily initiate and propagate around the flaw. For the preflawed specimen with an inclined angle of 30°, cracks mainly initiate and extend along the flaws tips. For the preflawed specimen at an inclined angle of 60°, cracks mainly distribute at the area of flaw inner tips and make a direct coalescence between the two preflaws. For the preflawed specimen with an inclined angle of 90°, a small number of cracks accumulate around the flaws, while many cracks distribute randomly in the specimens. This phenomenon verifies that the crack initiation of this specimen is less affected by the flaws.In stage II, for the intact specimen, the accumulation of cracks shows as discrete shear planes, which is totally different from the failure modes of the intact specimen under compression condition. For the preflawed specimen with an inclined angle of 0°, cracks continually extend from the flaw tips and flaw surfaces. The cracks at the inner tips form a “V”-shaped indirect coalescence between the two preflaws. For the preflawed specimen with an inclined angle of 30°, the cracks at the inner tip of the left flaw extend downwards while those at the inner tip of the right flaw extend upwards; therefore, there is no coalescence between the two preflaws. For the preflawed specimen with an inclined angle of 60°, the cracks which extend along the flaw tips make the specimen fail at a shear failure plane. For the preflawed specimen with an inclined angle of 90°, the two preflaws do not exhibit coalescence, and very little cracks accumulate around the preflaws. By comparing the ultimate failure modes of specimens at point ⑤, it can be found that for preflawed specimens with an inclined angle of 0° and 30°, there are a lot of axial splitting cracks (marked by dashed lines in Figure14), and for preflawed specimens with an inclined angle of 60° and 90°, the axial splitting cracks are little. Meanwhile, there are nonsplitting cracks in the intact specimen. Moreover, at the end of unloading (point ⑤), the ultimate crack amounts of the intact specimen and preflawed specimens with inclined angles of 0°, 30°, 60°, and 90° are 2747, 2402, 2395, 2356, and 2826, respectively. It indicates that the ultimate crack amount in specimens does not show a strong correlation with the flaws. This is possibly because two preflaws are not enough to make a significant difference in the crack amount for the preflawed specimen. If more flaws (or flaw networks) are created at a smaller inclination angle (such as 0° or 30°), then more cracks would initiate and propagate around the flaws, which may significantly influence the crack amount of the preflawed specimen under the unloading condition.


## 4.1. Unloading Failure Strength of Preflawed Specimens

The stress variations against time for the specimens during unloading simulations are shown in Figure8. It can be found that the specimens experience the typical three stress stages which have been specified in Section 3.1. At first, the specimen was under a hydrostatic pressure state for 1 ms. Then, the lateral stress σ3 was unloaded and the axial stress σ1 increased to a peak value. Finally, the specimen underwent the unloading failure stage, in which the lateral stress σ3 was unloaded to zero and the axial stress σ1 dropped due to specimen failure. The peak stress σ1 was defined as unloading failure strength σus. Note that the unloading failure strength is different from the failure strength in the conventional compression test, which is the actual failure strength for rock mass during the excavation process.Figure 8
Stress–time curves of specimens under unloading condition.
(a)
(b)
(c)
(d)
(e)It is seen in Figure8 that the unloading failure strength varies significantly for different specimens depending on the geometry of the specimen. For the intact specimen, the unloading failure strength is greater than that for the preflawed specimen. Furthermore, an increase in the unloading failure strength can be observed for the preflawed specimen with the increase in the flaw inclination angle. In order to better understand the influence of preflaws on the strength properties of rock specimens under unloading condition, quantitative investigations on the unloading failure strength were carried out. Figure 9 presented the correlation between the unloading failure strength and flaw inclination angle. In addition, the strength degradation parameter (SDP) was introduced to compare the unloading failure strength characteristics of the preflawed specimen and intact specimen. The SDP is defined as
(10)SDP=Δσusσusi×100%=σusi−σusfσusi×100%,where σusi is the unloading failure strength of the intact specimen and σusf is the unloading failure strength of the preflawed specimen under the same unloading condition.Figure 9
Unloading failure strengthσus of preflawed specimens.The SDP values can show the reduction degrees of the unloading failure strength of preflawed marble specimens relative to that of the intact marble specimen. The SDP values of the preflawed specimens with various flaw inclination angles indicate that the unloading failure strengths of the preflawed specimens decrease by 5.5%–20% compared with those of the intact specimen. Such phenomena indicate that the existence of preflaws in the rock mass can reliably decrease the unloading failure strength. Therefore, inducing preflaws in intact hard rock is helpful to cause rock failure during the unloading process and to improve the cutting efficiency.As shown in Figure9, the unloading failure strength of the preflawed specimen exhibits an “S”-shape rising trend with the increase in the flaw inclination angle. This means that the preflaws with a smaller inclination angle can reduce the unloading failure strength to a greater extent. Previous studies also showed that the strength of the preflawed specimen was closely dependent on the inclination angle of flaws [37, 38]. To further explain the influences of preflaws on the unloading failure strength, the stress state on the flaws was theoretically analyzed. The stress state sketch of preflaws under unloading condition is shown in Figure 10. The normal stress σn and the shear stress τn on the flaw surface can be expressed as
(11)σn=σ1cos2α+σ3sin2α,τn=σ1−σ32sin2α,where α is the inclination angle of flaw and σ1 and σ3 are the axial pressure and lateral pressure, respectively. It should be noted that the flaws are open and the normal stress σn on the flaw can induce tensile stress σb.Figure 10
Stress state sketch of preflaws.The normal stressesσn and shear stresses τn on the flaws with the four different inclination angles can be calculated by substituting α=0°, 30°, 60°, and 90° into equation (11). It is known that the cracking of the rock material usually presents two forms: tensile cracking or shear cracking. If the extreme value of the tensile or shear stress appears and satisfies the tensile or shear strength, then the tensile cracking or shear cracking will happen. For the brittle rock specimen, the tensile strength is much smaller than the shear strength, and the tensile stress plays a dominate role in the cracking of the rock specimen. Comparing the calculated results in equation (11) for flaws with inclination angles of 0°, 30°, 60°, and 90°, it can be found that for the flaw with α=0°, its shear stress is zero τn0°=0 and its normal stress is much larger than that of flaws with other inclination angles (σn0°=σ1, σn0°>σn30°>σn60°>σn90°). Thus, the induced tensile stress σb0° is the largest and tensile cracks are most likely to initiate and propagate around the flaws, hence resulting in the lowest unloading failure strength. For flaws with α=30° and α=60°, they suffer nearly the same shear stresses τn30°≈τn60°; however, the flaw with α=30° suffers a larger normal stress than the flaw with α=60°. Therefore, the induced tensile stress on the flaw with α=30° is larger σb30°>σb60°, which means that it is more likely to initiate cracks around the flaw with α=30°, and it may result in a lower failure strength for the preflawed specimen with α=30°. For flaws with α=90°, their shear stress is zero τn0°=0 and their normal stress is the smallest σn90°=σ3. In this regard, the normal stress is decreasing and it is not great enough to initiate cracks around the flaw. Thus, the failure of the specimen with a flaw inclination angle of 90° is not significantly affected by the flaw, and this is the reason why its unloading failure strength is close to the strength of the intact specimen. These findings indicate that if we induce preflaws with a smaller inclination angle in the specimen (i.e., vertical to the unloading direction), cracks are more likely to initiate and propagate around the flaws and the failure of the specimen is more easily to happen. Therefore, during the unloading process, preflaws with a smaller inclination angle can result in the failure of the specimen that happens more easily.Figure11 shows the comparison between the unloading failure strength and compression failure strength of specimens. It can be seen that the unloading failure strength of the intact specimen is merely 60% of the triaxial compressive strength of the intact specimen, which is, however, approximately 1.2 times the UCS of the intact specimen. Meanwhile, the average value of the unloading failure strength of the preflawed specimen is approximately 50% of the triaxial compression strength of the intact specimen and about 90% of the UCS. The differences between the unloading failure strength and compression strength indicate that the compression strength obtained from laboratory quasi-static compression experiments cannot fully represent the peak failure strength for rock mass under excavation unloading condition. Therefore, it can be considered to use the unloading failure strength obtained from a quasi-static unloading test to evaluate the failure strength properties of underground engineering rock.Figure 11
Comparisons of the unloading failure strengthσus and the compressive strength (UCS, TCS-40).


## 4.2. Unloading Damage Thresholds of Preflawed Specimens

In fact, fracturing of the rock specimen is a complex process and simply measuring the peak stress does not capture this fracturing process [39]. In order to capture some unique features in the fracturing process, i.e., crack initiation, onset of fracture localization, and collapse peak, some damage thresholds are pointed out. The fracturing process of rock specimens can be recorded by many different ways, such as acoustic emission measurements and high-speed photography. In laboratory compression experiments, previous scholars determine the damage initiation and interaction thresholds for brittle rock based on the acoustic emissions [40, 41]. In the present study, the crack accumulation amount was used to reflect the fracturing process, and the damage thresholds of specimens under unloading condition were identified by the variation of the crack accumulation amount. Figure 12 gives the relations of the deviatoric stress, the crack accumulation, and the associated axial strain response of specimens under unloading conditions. It can be seen that there are two critical points on the crack accumulation curves, i.e., point A and point B. Point A denotes the first intersection of two tangent lines in the crack accumulation curve, which can mark the onset of a slight increase in cracks. Point B denotes the second intersection of two tangent lines in the crack accumulation curve, which can mark the onset of a sharp increase in cracks. Cracks increase in a stable manner between points A and B, while cracks begin to increase in an unstable manner after point B. The corresponding stress at point A is denoted as σci, which represents the threshold stress for stable crack initiation. The stress at point B is denoted as σcd, which is a threshold stress for unstable crack development. Moreover, the other critical point was the point where the first observable macrocrack appears in the specimen [42]. In this study, the stress where the first observable macrocrack appears is defined as the other threshold, i.e., the damage initiation stress σdi. The three damage thresholds (σci, σcd, and σdi) for different specimens are presented in Figure 12. It can be found that σci>σdi>σcd>σpk and σpk is the peak deviatoric stress.Figure 12
Relations among the deviatoric stress, crack accumulation, and strain of specimens under unloading condition.
(a)
(b)
(c)
(d)
(e)For preflawed specimens, the damage threshold ratios are plotted against the flaw inclination angle (Figure13). It can be seen in Figure 13 that the ratios of σci, σdi, and σcd to the peak deviatoric stress σpk are approximately 0.75, 0.85, and 0.97, respectively. According to previous studies, for the intact rock specimen under uniaxial compression, the crack initiation stress typically ranges from 0.3-0.5 UCS, and the unstable crack development stress is approximately 0.7-0.9 UCS [43]. It can be found that the damage threshold ratios of numerical preflawed rock specimens in the unloading tests are higher than those of real intact rock specimens in the uniaxial compression tests. This is because compared with the uniaxial compression test, the lateral stress in the unloading test constrains the initiation of cracks. In addition, the numerical rock specimen is composed of rigid particles, while the real rock specimen usually contains natural microfractures. As shown in Figure 13, the ratios of the stable crack initiation stress σci and damage initiation stress σdi rise with the increase in the flaw inclination angle, while the ratios of σcd change little with the increase in the flaw inclination angle. It indicates that the crack initiation stress is sensitive to the flaw inclination angle, and the preflawed specimen with a smaller flaw inclination angle is more prone to induce cracks under unloading condition. On the contrary, the stress of unstable crack development has no clear change trend with the variation of the flaw inclination angle.Figure 13
Damage threshold ratios of preflawed specimens under unloading condition.


## 4.3. Cracking Characteristics of Preflawed Specimens under Unloading Condition

The cracking characteristics of preflawed specimens at any moment under unloading condition, including crack distribution, crack extension, and crack amount, can be recorded in numerical simulations. To analyze the cracking characteristics in detail, Figure14 shows the crack distributions in the specimens at some specific points during the unloading process. These specific points are denoted by numbers which correspond to the points in stress-strain curves in Figure 12. For example, point ① refers to the damage initiation stress in the stress-strain curve. Point ② corresponds to a stress before the peak deviatoric stress, and the crack count at point ② is approximately 50% of the crack count at the peak deviatoric stress (point ③). Point ④ is a stress after the peak deviatoric stress, and the crack count at point ④ is approximately 2.5 times the crack approximately at the peak deviatoric stress. Point ⑤ is the stress at the end of unloading.Figure 14
Cracking process of specimens under unloading condition (the red short lines in numerical specimens represent microcracks).
(a)
Intact specimen
(b)
Specimen withα=0°
(c)
Specimen withα=30°
(d)
Specimen withα=60°
(e)
Specimen withα=90°As shown in Figure14, the cracking distribution and crack extension for different specimens under unloading condition are obviously different. To make it clearer, the cracking process was divided into two stages, i.e., stage I—prior to the peak deviatoric stress (including the peak deviatoric stress, ①, ②, and ③)—and stage II—after the peak deviatoric stress (④ and ⑤).In stage I, the specimens undergo the elastic deformation and the crack initiation and propagation before reaching the peak deviatoric stress. For the intact specimen, the cracks distribute randomly in the specimen, while for the preflawed specimen with an inclined angle of 0°, cracks intensively distribute around the flaw tips and flaw surfaces. This phenomenon verifies the results of stress state analyses for preflaws based on Figure10 and equation (11). At an inclined angle of 0°, the normal stress on the flaw is large enough, making the cracks to easily initiate and propagate around the flaw. For the preflawed specimen with an inclined angle of 30°, cracks mainly initiate and extend along the flaws tips. For the preflawed specimen at an inclined angle of 60°, cracks mainly distribute at the area of flaw inner tips and make a direct coalescence between the two preflaws. For the preflawed specimen with an inclined angle of 90°, a small number of cracks accumulate around the flaws, while many cracks distribute randomly in the specimens. This phenomenon verifies that the crack initiation of this specimen is less affected by the flaws.In stage II, for the intact specimen, the accumulation of cracks shows as discrete shear planes, which is totally different from the failure modes of the intact specimen under compression condition. For the preflawed specimen with an inclined angle of 0°, cracks continually extend from the flaw tips and flaw surfaces. The cracks at the inner tips form a “V”-shaped indirect coalescence between the two preflaws. For the preflawed specimen with an inclined angle of 30°, the cracks at the inner tip of the left flaw extend downwards while those at the inner tip of the right flaw extend upwards; therefore, there is no coalescence between the two preflaws. For the preflawed specimen with an inclined angle of 60°, the cracks which extend along the flaw tips make the specimen fail at a shear failure plane. For the preflawed specimen with an inclined angle of 90°, the two preflaws do not exhibit coalescence, and very little cracks accumulate around the preflaws. By comparing the ultimate failure modes of specimens at point ⑤, it can be found that for preflawed specimens with an inclined angle of 0° and 30°, there are a lot of axial splitting cracks (marked by dashed lines in Figure14), and for preflawed specimens with an inclined angle of 60° and 90°, the axial splitting cracks are little. Meanwhile, there are nonsplitting cracks in the intact specimen. Moreover, at the end of unloading (point ⑤), the ultimate crack amounts of the intact specimen and preflawed specimens with inclined angles of 0°, 30°, 60°, and 90° are 2747, 2402, 2395, 2356, and 2826, respectively. It indicates that the ultimate crack amount in specimens does not show a strong correlation with the flaws. This is possibly because two preflaws are not enough to make a significant difference in the crack amount for the preflawed specimen. If more flaws (or flaw networks) are created at a smaller inclination angle (such as 0° or 30°), then more cracks would initiate and propagate around the flaws, which may significantly influence the crack amount of the preflawed specimen under the unloading condition.


## 5. Conclusions

Based on the unloading stress condition of hard rock in underground excavation, numerical simulations were performed to analyze the unloading responses of marble specimens containing preflaws under unloading condition. The unloading strength properties, damage thresholds, and cracking characteristics of preflawed specimens with different flaw inclination angles were deeply investigated. The following conclusions can be summarized:(1)
The unloading failure strength of a specimen containing two preflaws decreases by 5.5%-20% compared to that of the intact specimen. With the increase in the flaw inclination angle, the unloading failure strength of a specimen containing two preflaws exhibits an “S”-shaped increasing trend
(2)
Three damage thresholds (σci, σdi, and σcd) are determined based on the crack accumulation of specimens under unloading condition. For preflawed specimens, the damage threshold ratios of σci and σdi rise with the increase in the flaw inclination angle, while the damage threshold ratio of σcd has no clear change trend with the variation of the flaw inclination angle
(3)
Prior to the peak deviatoric stress, the cracks of preflawed specimens are distributed in a centralized manner around the flaw tips and flaw surfaces when the flaw inclination angle is smaller, while the distribution of the cracks for the intact specimen and preflawed specimen withα=90° is decentralized. After the peak deviatoric stress, the coalescence form between two flaws varies with the change of the flaw inclination angle. Moreover, at the end of unloading, there are obvious axial splitting cracks for preflawed specimens with α=0° and 30°, while axial splitting cracks are little in the preflawed specimen with α=60° and 90° and completely none in the intact specimenThese conclusive remarks can provide useful information for underground mechanical excavation. For one thing, the unloading failure strength defined in this study can be considered a parameter to evaluate the actual failure strength properties of rock mass in a mechanical excavation process. For another, the findings show that inducing flaws can reliably reduce the unloading failure strength and promote crack initiation and propagation of hard rock during the unloading process. Thus, inducing preflaws in hard rock will benefit the underground mechanical excavation. In particular, when inducing preflaws at a smaller flaw inclination angle, the unloading failure and crack initiation are more likely to happen, and this means that inducing preflaws with a smaller inclination angle (e.g., vertical to the unloading direction) will be more helpful to improve the excavation efficiency.



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*Source: 4657645-2019-02-13.xml*