Experimental and numerical study of granite blocks containing two side flaws and a tunnel-shaped opening

Theoretical and Applied Fracture Mechanics 104 (2019) 102394

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Experimental and numerical study of granite blocks containing two side flaws and a tunnel-shaped opening ⁎

T

⁎

Hao Chena, Xiang Fana,b,c, , Hongpeng Laia, , Yongli Xiea,b, Zhongming Hec a

School of Highway, Chang’an University, Xi’an 710064, PR China Key Laboratory for Bridge and Tunnel of Shaanxi Province, Chang’an University, Xi’an 710064, PR China c Engineering Research Center of Catastrophic Prophylaxis and Treatment of Road & Traffic Safety of Ministry of Education, Changsha University of Science & Technology, Changsha 410004, PR China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Side flaw Tunnel-shaped opening Uniaxial compression Cracking Principal stress distribution

In this study, uniaxial compression tests were conducted on granite specimens containing two side flaws. Based on the experimental results, the numerical models containing two side flaws and a tunnel-shaped opening were created to investigate the peak stress, cracking process and stress distribution under uniaxial compression. It is found that the flaw inclination angle strongly affects the strength, cracking process and stress distribution, while the slight change of the opening shape has little influence. The cracks initiate from the flaw tips and the top and bottom of openings. Unlike the case of a circular opening, in addition to arch waists, large compressive stress occurs at the arch foots of the tunnel-shaped opening.

1. Introduction With the rapid development of transportation infrastructure, more and more excavation projects, such as tunnels, subways and mines, have been carried out within rock masses in recent years. However, rock discontinuities widely exist in rock masses, including joints, bedding planes, fissures and weak surfaces, which have a significant influence on the strength and stability of a natural rock mass. Tunnel excavation in these jointed rock masses often encounters various serious problems, such as excessive deformation of surrounding rock and cracking and spalling of supporting concrete, and even tunnel collapse may eventually occur [1–5]. Therefore, it is necessary to have a deep understanding of the stress distribution and cracking development regulated by excavation to ensure the safety of tunnel construction and operation. The in-situ deformation and stress monitoring of tunnels is an effective means to investigate the stress distribution of surrounding rock. It can also predict the potential tunnel distresses accurately [6–9]. However, in-situ monitoring is affected by many uncertain factors, and it also consumes too much manpower and material resources. Thus, laboratory tests and numerical simulation methods have become popular approaches for investigating tunnel excavation in jointed and fractured rock. For laboratory tests, due to limitations of test

equipment’s size and similar material types, many studies have regularly generated the joints and circular openings in natural rock or rock-like specimens for uniaxial, biaxial or triaxial compression [10–13]. For specimens containing a single flaw, previous studies have found that cracks often first occur at the flaw tips [14,15]. For two or three flaws, the arrangement of flaws, such as the flaw dip angle and flaw length, has a great impact on the crack development, strength and failure mode of the flawed specimens [16–19]. For multiple flaws, the crack propagation and failure mode are more complicated [20–23]. For example, from a uniaxial test of multi-fractured rock-like specimens, Cao et al. [23] found that there were three types of fissure coalescence modes, S-mode, T-mode and M-mode, and four types of failure modes, mixed failure, shear failure, path failure and intact failure. Alternately, the circular opening as the most common hole shape has been widely studied via uniaxial compression tests [24–27]. For instance, Zhao et al. [27] studied the failure characteristics of circular-opening specimens via laboratory test and found that the strength of rock specimens was closely related to the diameters of the circular openings. In addition, some scholars have performed in-depth studies of the failure characteristics of specimens containing multiple holes [28,29] or both flaws and holes [30]. Among them, Huang et al. [29] conducted uniaxial compression on granite specimens containing three non-coplanar holes and summarized four typical crack coalescence patterns. Because the

Abbreviations: PFC, particle flow code; RFPA, rock failure process analysis; TCC, three-centered circular; FCC, five-centered circular; FLAC, fast lagrangian analysis of continua; MP, monitoring point; TSZ, tensile stress zone; SSZ, shear stress zone ⁎ Corresponding authors at: Chang’an University, The Middle Section of Second Ring in South, Xi'an, Shaanxi Province, PR China (X. Fan). E-mail addresses: [email protected] (X. Fan), [email protected] (H. Lai). https://doi.org/10.1016/j.tafmec.2019.102394 Received 25 May 2019; Received in revised form 18 October 2019; Accepted 21 October 2019 Available online 23 October 2019 0167-8442/ © 2019 Published by Elsevier Ltd.

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(a) Intact specimen

(b) A-0

(c) A-25

(d) A-45

Fig. 1. Schematic diagrams of laboratory test specimens.

techniques, more and more numerical approaches, including the finite element method (FEM) [9,33–35], discrete element method (DEM) [36–40], boundary element method (BEM) [41], numerical manifold method (NMM) [42] and discontinuous deformation analysis (DDA) [43,44], are being employed to explore the failure mechanism of fractured rock. As known, numerical simulation is a good way to further explain the mechanical behavior and fracture characteristics of rock specimens in laboratory tests. Many researchers have attempted to reveal the cracking mechanism and fracture behavior of real rock or rocklike materials containing flaws or holes under uniaxial compression by numerical simulation. For instance, by using the particle flow code (PFC) approach, Zhang et al. [45] studied the cracking process in rocklike material containing a single flaw and found that the flaw inclination angle has a significant impact on the crack initiation and propagation patterns. Mondal et al. [46] simulated fracture propagation and damage evolution in sandstone specimens containing a preexisting 3-D flaw with varying flaw angle, flaw length and flaw depth under uniaxial compression. Jin et al. [14] investigated the influence of a single flaw

shape of the excavation section is usually not a standard circular one in actual engineering projects, some scholars have performed physical tests on irregularly shaped openings, such as an inverted U-shape or an elliptical or arc shape. Zhong et al. [31] conducted biaxial compression on a rhyolite rock block containing various natural cracks and an inverted U-shaped hole. It was found that tensile fracture first occurred at the floor of the hole and then the roof. The compressive stress was concentrated at the sidewall and appeared to be “dumbbell-like”. Liu et al. [32] analyzed the granite specimens containing an inverted Ushaped opening via an acoustic emission technique and moment tensor analysis. The results showed that shear cracks mainly appeared at the sidewall of the hole, while tensile cracks mainly appeared at the roof. However, due to limitations of the rock properties and observation equipment, it is not possible to observe the entire failure process of rock specimens in laboratory tests, including the initiation, propagation and coalescence of microcracks. As a supplement, numerical simulations may be used to capture entire failure processes, local stress distributions and other details. Recently, with the rapid progress in computer 2

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surrounding the flaws and opening using a numerical approach.

Table 1 Arrangement of specimens. Series

Specimen type

The flaw inclination angle (°)

Sample ID

– A

Intact specimen (Fig. 1a) Two side flaws only (Fig. 1b–d)

B

Containing a three-centeredcircular opening (TCC opening)

C

Containing a five-centered-circular opening (FCC opening)

– 0 25 45 0 25 45 0 25 45

– A-0 A-25 A-45 B-0 B-25 B-45 C-0 C-25 C-45

2. Laboratory test and numerical simulation 2.1. Sample preparation In this laboratory uniaxial test, granite blocks were chosen to prepare specimens containing two side flaws. The size of the specimens is 150 mm × 100 mm × 30 mm (height × length × width); see Fig. 1a. The two side flaws were produced through a high-pressure water cutting technique. As presented in Fig. 1b, c and d, each single flaw is 30 mm long and 2 mm wide. The left flaw is 50 mm from the top of the specimen, while the right flaw is 50 mm from the bottom. The line connecting the tips of the two side flaws passes through the center of the specimens. A-0, A-25, and A-45 are used to name the specimens with inclination angles of the side flaw of 0°, 25° and 45° to the X-axis, respectively. In addition, the numerical models containing a differently shaped opening are named Series B and C. In Series B, a three-centered circular opening was created in the flawed specimens, while a fivecentered circular opening was created in Series C. The detailed numbers of all specimens are listed in Table 1.

on the failure modes and energy mechanism of a rock-like material with a combination of the laboratory tests and numerical simulation. The results demonstrated that, with the increase of flaw angle, the input energy, strain energy and dissipation energy increased. Based on experimental results, Fan et al. [47] conducted uniaxial compression simulations on rock specimens with flaws and circular openings via PFC3D. The results showed that the circular opening arrangement and the flaws’ inclination had a marked effect on the crack development and failure modes of the specimens. Wang et al. [48] adopted rock failure process analysis (RFPA2D) to simulate specimens with pre-existing holes under static and dynamic load. It was found that the area around the hole was fractured by tensile cracks when the lateral pressure was zero, and the tensile cracks gradually became shear cracks with an increase in the lateral pressure. In addition, many scholars have carried out uniaxial or biaxial compression simulations on rock specimens containing noncircular openings. For example, RFPA [49,50] code was used to simulate the progressive fracturing processes of a tunnel with three common cross-sections (circular, elliptical and inverted U-shaped) under biaxial compression. The simulation results showed that the fracturing patterns of the openings were affected not only by the opening shape but also by the lateral loading coefficients. On the basis of the abovementioned reports, it can be found that the studies regarding rock specimens containing circular holes and flaws are mature and prolific, while studies containing irregularly shaped openings are still limited, such as three-centered circular (TCC) or fivecentered circular (FCC) tunnel shapes. In this study, two side flaws were pre-prepared within the granite rock block, and then a high-stress zone was formed between the two side flaws using the uniaxial compression test. Then, a TCC or FCC opening was created in the high-stress zone to further explore the cracking mechanism and stress distributions

2.2. Laboratory test Fig. 2 shows the testing system adopted in the laboratory tests. The loading device was an electrohydraulic servo control testing machine that can control the loading velocity with high precision. To reduce the end effect during the test, the contact areas between the specimen ends and loading platforms were daubed with butter. The specimens were loaded at a loading rate of 0.1 mm/s during the load process. The failure processes of specimens were recorded by a high-definition video camera, and the axial stress and displacement were recorded by a computer program. The test results of intact specimens showed that the average uniaxial compression strength was 121.08 MPa, the elastic modulus was 68.87 GPa, and the Poisson’s ratio was 0.2215. 2.3. Numerical simulation In the study, the fast lagrangian analysis of continua (FLAC3D) was used for numerical simulations. Based on the actual dimensions of the intact specimens and Series A specimens in the laboratory test, numerical models were established for calculations. First, an intact cuboid specimen was generated. The element size follows a uniform distribution. The intact numerical model consists of 112,500 elements and 123,848 nodes. Then, a flaw with a length of 30 mm was cut on both

Fig. 2. Schematic diagram of the testing system. 3

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(a) B-0

(b) B-25

(c) B-45

(d) C-0

(e) C-25

(f) C-45

Fig. 3. The established numerical models. Table 2 The parameters of numerical model. Density (kg/m3)

Elastic modulus (GPa)

Poisson’s ratio

Internal friction angle (°)

Cohesion (MPa)

Tensile strength (MPa)

2710

68.87

0.2215

56

26

2.018

Fig. 3, the TCC or FCC opening was cut in the center of the Series A numerical model by deleting elements. The sizes of the TCC and FCC were drawn in accordance with the 1: 400 in the standard section of highway tunnel at 100 km/h [51]. In the TCC model, R1 = 14.25 mm, R2 = 20.5 mm and R3 = 37.5 mm. The net width of the TCC opening was 28.5 mm, and the net height was 21.8 mm (Fig. 3a). In the FCC model, R1 = 18.675 mm, R2 = 14.25 mm, R3 = 20.5 mm, R4 = 3.75 mm and R5 = 45 mm. The net width of the FCC opening was 34.7 mm, and the net height was 23.8 mm (Fig. 3d). In the FLAC3D program, the Strain-Hardening/Softening model has been widely adopted for the simulation of uniaxial compression tests [52,53]. The Strain-Hardening/Softening model is based on the MohrCoulomb model, which is not related to the shear flow law but related to the tensile flow law. The difference between these two models is that the cohesion, friction angle, dilation angle and tensile strength may change after plastic yield begins in the Strain-Hardening/Softening model. This model calculates the total plastic shear strain and tensile strain by increasing/decreasing the hardening parameters at each time

Table 3 The softening parameters of numerical model. Plastic strain

Softened internal friction angle (°)

Softened cohesion (MPa)

0 0.010 0.011 0.012 0.015 0.020

56.0 52.3 44.1 29.6 11.7 11.7

26.0 23.0 17.8 10.4 0.6 0.6

sides of the intact specimen. The left side flaw was 50 mm from the top of the specimen, and the right one was 50 mm from the bottom. By rotating the two sides flaw in the clockwise direction at angles of 25° and 45°, three inclination cases of Series A can be obtained. Moreover, to study further on the basis of the Series A specimens, numerical models containing three-centered circular openings (Series B) and fivecentered circular openings (Series C) were established. As shown in 4

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90

140 Numerical curve

120

Experimental curve

80

Peak stress (MPa)

Axial stress (MPa)

70 100 80 60 40

60 50 40 30 Experimental results for Series A Numerical results for Series A Numerical results for Series B Numerical results for Series C

20 20

10

0 0.00

0.05

0.10

0.15

0.20

0.25

0

0.30

0

Axial strain (%)

10

20

30

40

50

Inclination angle of flaw (degrees)

Fig. 4. Numerical and experimental curves of the intact specimen.

Fig. 6. Curves of peak stresses of specimens under different cases.

step. The stress-strain curve is linear before reaching the yield point. At this stage, only the elastic strain is generated, namely, e = ee. After the material yields, the total strain is composed of elastic strain and plastic strain, namely, e = ee + ep. Variables such as cohesion, friction angle, dilation angle and tensile strength are the functional parameters of the plastic strain ep in the total strain [54]. The plastic shear strain is calculated by the shear hardening parameter eps, and the incremental form of eps is defined as follows:

The tensile hardening parameter ept is used to calculate the accumulated tensile plastic strain, and its increment is defined as follows:

Δe ps =

{ 12 (Δe

ps 1

− Δemps )2 +

1 1 (Δemps )2 + (Δe3ps − Δemps )2 2 2

}

Δe pt = Δe3pt Δe3pt

is the tensile plastic strain increment in the direction of the principal stress. The plastic strain increments Δe1ps , Δe3ps and Δe3pt can be obtained based on the flow rule. For the yield function and potential function, the plastic flow rule is completely consistent with the stress-corrected Mohr-Coulomb model. During the uniaxial simulation, only the displacements of the top and bottom boundaries of the model were limited, which was equivalent to the loading head. The displacements of the other four vertical boundaries were unconstrained. In the laboratory test, the loading rate was set to 0.1 mm/s. However, the loading rate set in the experiments is different from the rate set in the numerical simulation. In FLAC3D,

1 2

(1)

where

Δemps =

1 (Δe1ps + Δe3ps ) 3

Δejps ,

(3)

(2)

j = 1, 3 is the principal increment of plastic shear strain.

(a) Monitoring points around the left flaw (b) Monitoring points around the right flaw

(c) Monitoring points around the TCC opening opening

(d) Monitoring points around the FCC

Fig. 5. The arrangement of monitoring points. 5

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Fig. 7. Experimental and numerical failure process of A-0. (The red area represents the tensile stress zone while the green area represents the shear stress zone.) (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

specimens, as shown in Tables 2 And 3. Fig. 4 is the comparison of the stress-strain curves of intact specimens between the experimental test and numerical simulation. As shown in Fig. 4, the peak stress of the intact specimen was 119.78 MPa in numerical simulation, while the experimental test result was 121.08 MPa. The difference between the two results was only 1.30 MPa. The trends of the two stress-strain curves are basically similar, with only differences at the beginning stage. This is because there is a compaction process of rock material at the beginning stage in the laboratory tests but not in numerical simulations.

control of the loading rate by displacement has been widely adopted. The unit of the loading rate is m/step in simulation, namely, displacement of the model boundary in each iteration step. According to the number and size of elements, conversion and calibration were conducted, and the loading process between the loading rate controlled by a displacement of 3.95 × 10−8 m/step was determined, which was considered to ensure that the specimen was always in a quasi-static equilibrium state during the entire loading process. The numerical material parameters were determined according to the triaxial compression test of the same type of granite cylindrical 6

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Fig. 8. Experimental and numerical failure process of A-25.

clockwise direction. Eight monitoring points were arranged surrounding the opening at the crown (MP-1), arch shoulders (MP-2, MP8), arch waists (MP-3, MP-7), arch foots (MP-4, MP-6) and invert (MP5). Starting from the crown, the monitoring points were numbered in the clockwise direction (Fig. 5c and d). It is worth noting that these monitoring locations are basically the same as the monitoring locations of stress in the actual in-situ tunnel surrounding rock.

2.4. Arrangement of monitoring points Previous studies [31,47,49] indicated that the cracks initiation often occurs at the tips of the flaws or some specific locations surrounding an opening during the uniaxial or biaxial loading, and the internal stresses are usually concentrated in these locations. To accurately obtain the internal stresses in these locations, the monitoring points (MP) were set up in these parts during numerical uniaxial loading, as shown in Fig. 5. Nine monitoring points were arranged around each flaw in the 7

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Fig. 9. Experimental and numerical failure process of A-45.

3. Uniaxial compressive strength

less than that of the intact specimen, approximately 36.18%. Similarly, the numerically obtained peak stresses of B-45 and C-45 are approximately 52.85 MPa and 48.94 MPa, respectively. Compared to the peak stress of the intact block, it can be found that the tunnel-shaped opening further dramatically reduces the strength of the tested specimens, decreasing by up to 56.35% and 59.58% for B-45 and C-45, respectively. This illustrates that the flaws or openings significantly degrade the bearing capacity of the specimens. In addition, it can be seen that the peak stress of Series B is slightly higher than that of Series C at the same

As known, the arrangement of flaws and the cross-section shape of an opening have significant influences on the uniaxial compressive strength of the specimens. Fig. 6 shows the peak stresses of the tested and numerical specimens. It can be shown that, compared to the intact specimen, the peak stress significantly decreases when the specimens contain the side flaws or/and a tunnel-shaped opening. For example, the experimental peak stress of A-45 is 77.27 MPa, which is 43.81 MPa 8

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Fig. 10. Numerical failure process of specimen B-0.

deformation of the specimen. It causes excessive local stress around the flaw, resulting in a decrease in the strength of the specimen. The second type is shown by Series B and Series C, and as the flaw inclination angle increases, the peak stress first decreases and then increases. That is, the peak stress is a minimum at the inclination angle of 25° while maximum at 45°. The peak stress of B-45 is 4.97 MPa higher than that of B-0 and 10.82 MPa higher than that of B-25; the peak stress of C-45 is 4.78 MPa higher than that of C-0 and 10.76 MPa higher than that of C-25. The reason for this phenomenon may be that the two side flaws tips are closer to the boundary of the tunnel-shaped opening than the other two cases when the inclination angle is 25°, which leads to higher local stresses at the side flaw tips and opening. This causes both the flaw tips and the opening to be more susceptible to cracking, resulting in strength reduction of the specimen. However, the maximum value of peak stress always occurs at the dip angle of 45°, no matter what series of specimen.

dip angle. The reason is that the area of TCC in Series B is slightly smaller than the area of FCC in Series C. A reduction in the cross-sectional area leads to a reduction of the specimen’s bearing ability. This also reasonably explains why the peak stress of Series A is much higher than that of Series B or Series C. According to the results of laboratory and numerical tests, the influence of the flaw inclination angle on the specimen peak stress is divided into two types. The first type can be observed in Series A, and the peak stress increases with the flaw inclination angle. The peak stress of A-45 is 8.08 MPa higher than that of A-25 and 16.10 MPa higher than that of A-0 in laboratory tests, while it is 4.75 MPa higher than that of A-25 and 14.68 MPa higher than that of A-0 in numerical simulations. This is because the local stress surrounding the flaw is closely related to the flaw inclination angle in uniaxial compression. Excessive local stress can easily cause cracking around the flaw, which leads to the reduction of specimen strength. Previous studies [23,47] have revealed that cracking of specimens with small flaw dip angles is usually easier than for those with large dip angles. This is because the loading direction is vertical in uniaxial compression. When the flaw dip angle is 0°, the loading direction is perpendicular to the flaw. At the elastic deformation stage, compared to a flaw with a large dip angle, a flaw with a small dip angle is more likely to exhibit closure, which leads to larger

4. Cracking and failure 4.1. Crack development and failure process of Series A As is well known, the flaws existing inside a rock block 9

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Fig. 11. Numerical failure process of specimen B-25.

Then, the shear stress appeared at the flaw tips and propagated along the loading direction. When the axial stress reached the peak stress, as presented in Fig. 7g, shear stress concentrated in the zone between the two side flaws. Compared to the final tested failure shown in Fig. 7e and the numerical stress status shown in Fig. 7g, the spalling area was in good agreement with the shear stress zone. In the case of A-25, as shown in Fig. 8, similar to the failure process of A-0, a crack also first initiated at an acute angle at the tip of the left side flaw (Fig. 8b). A shear stress zone and tensile stress zone concentrated at the flaw tip and the two sides of the flaw approaching the tip (Fig. 8f). With a further increase in axial stress, the cracks gradually extended to the area between the two side flaws (Fig. 8c, d). Then, surface rock spalling was observed in the high shear stress zone, which caused rock fragment ejection (Fig. 8e, g). As illustrated in Fig. 9, even though the inclination angle increased to 45°, the failure process of A45 (Fig. 9b–e) was analogous to that of A-25. A slight difference was that the angles of the cracks initiating from the side flaw are unequal. The numerical process repeats the physical failure process. Under uniaxial loading, the flawed block first experiences crack initiation, and then surface rock ejection and pit generation occur shortly after crack initiation. The angle of the cracking trajectories to the side flaw varies with the inclination angle of the side flaw. The occurrence of surface

predominantly control the mechanical behavior in the compressive condition. The location of a side flaw is different from an interior flaw. Therefore, to some extent, different cracking may occur for a side flaw under compressive loading. During laboratory testing, cracking evolution was recorded by a camera. In the study, the inclination angles of side flaws were set to 0°, 25° and 45°, respectively. The experimental and numerical cracking processes are presented in Fig. 7, Fig. 8 and Fig. 9, respectively. As seen in Fig. 7a, no obvious crack initiation was observed in the early stage of loading. As the axial stress increased to a level of 90.69%, approximately 55.48 MPa, cracking and rock crumb flaking first occurred at the right tip of the left side flaw (Fig. 7b) and then the left tip of the right side flaw (Fig. 7c). At the early stage of the cracking process, surface cracks were distinctly observed at the tips of the two side flaws (Fig. 7c, d). Namely, internal stress concentrated at the flaw tips. Because the view face is free, the surface rock rapidly spalled and ejected at the peak stress time, a pit was generated, and then the 0° specimen completely failed (Fig. 7e). Due to the high strength of granite specimens, the process from surface crack occurrence to surface rock ejection is very short, and it is hard to observe the cracking development during the tests. Numerical simulation is able to offset this limitation. Compared to the stress distribution shown in Fig. 7f, the tensile stress was mainly distributed around the side flaws. 10

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Fig. 12. Numerical failure process of specimen B-45.

axial stress increased, tensile stress occurred at the bottom of the TCC opening (Fig. 10c), namely, tensile cracks first initiated at the bottom. Then, shear stress was distributed close to the TSZ (Fig. 10d) and gradually extended along the loading direction to the SSZ connected to that surrounding the inner flaw tips (Fig. 10e). Finally, the shear stress distribution around the void became a “butterfly”-shaped zone (Fig. 10f). Highly concentrated shear stress at the waist of an opening can make the inner wall spall. The failure process of B-25 was similar to that of B-0. The TSZ and SSZ were also concentrated at the tips of the two side flaws (Fig. 11a, b). Then, the tensile cracks occurred at the bottom of the TCC opening (Fig. 11c). Finally, due to the coalescence of the SSZ between the side flaws tips and TCC opening, the specimen was crushed and lost bearing capacity (Fig. 11e, f). The failure process of B-45 was analogous to that of B-0 and B-25, as shown in Fig. 12. However, when the flaws angle reached 45°, fewer cracks on the upper side of the left flaw were generated than on the lower side, while the opposite phenomenon occurred for the right flaw (Fig. 12a–d). This indicates that it is harder for cracks to be generated surrounding the flaws with an increase in the inclination angle. The increase of the flaw angle causes a decrease in the area of the flaw along the loading direction. Compared to a flaw with a small dip angle, it is

rock ejection is due to a high shear stress concentration resulting from the two side flaws. Accompanying surface rock ejection, a surface pit is often generated at the high shear stress zone.

4.2. Crack development and failure process of Series B As described above, under compressive loading, cracks first initiated at the tips of the two side flaws. Then the two side flaws resulted in a phenomenon whereby shear stress was distinctly concentrated in the zone between the two inner tips of the side flaws. Obviously, a void created in the highly stress concentrated zone can significantly change the internal stress distribution under the condition of compressive loading, and further affect the cracking process. In the study, to understand the stress disturbance by an irregular opening, on the basis of numerical side flaw models, a three-centered circle opening or a fivecentered circle opening was created at the center of the model. The numerical failure processes of Series B and Series C are shown in Figs. 10–12 and Figs. 13–15, respectively. As shown in Fig. 10, in the case of B-0, similar to the failure process of Series A, a crack first initiated at the tips of the two side flaws. The TSZ first concentrated at the flaw tip (Fig. 10a) and then the SSZ concentrated (Fig. 10b). Apart from the cracking at the flaw tips, as the 11

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Fig. 13. Numerical failure process of specimen C-0.

harder for a flaw with a large dip angle to exhibit closure at the elastic deformation stage, and the deformation of the specimen is smaller. The local stress around flaws with large dip angles is smaller than for those with small ones, which causes it to be more difficult to generate cracks around the flaws with large dip angles. In addition, when the inclination angle was 0°, the SSZ at the left side flaw exhibited coalescence with the SSZ at the top of the TCC first, and the SSZ at the right side flaw exhibited coalescence with the SSZ at the bottom of the TCC (Fig. 10e). The coalescence of SSZ was opposite when the inclination angle was 45°. The left side flaw SSZ exhibited coalescence with the SSZ at the bottom of the opening first, and right side flaw SSZ exhibited coalescence with the SSZ at the top of the opening (Fig. 12e). The increase of the flaw angle made the coalescence of the SSZ between the flaws and TCC opening difficult and changed the coalescence location between the two SSZs.

bottom and top of the FCC opening (Fig. 13c, d). With a further increase in axial stress, the SSZ was connected between the two side flaws tips and the FCC opening (Fig. 13e). Like B-0, the final shape of the SSZ surrounding the FCC opening was a 'butterfly' (Fig. 13f). With the increase of the inclination angle of the two side flaws, the failure processes of C-25 and C-45 were slightly different from those of Series B and C-0. As shown in Fig. 14a and Fig. 15a, unlike Series B or C0, the TSZ was generated at the side flaw tips and the bottom and top of the FCC opening at the same time. This is probably because the FCC opening has more cross-sectional area loss and a lower high-span ratio compared to the TCC opening. More cross-sectional area loss leads to more strength reduction of the specimen, which causes the specimen to be destroyed more easily. Previous studies [55] indicated that extreme vertical deformation and failure may occur at the crown and invert of large-span tunnels. The low strength of the specimen and the extreme deformation of the opening make the opening more susceptible to stress concentration at the vault and invert. In addition to this feature, the failure processes of C-25 and C-45 were analogous to those of B-25 and B-45, respectively. Similarly, when the inclination angle of side flaw reached 45°, it was harder to generate the cracks at the side flaw tips (Fig. 15a–c). The coalescence location of SSZ between the side flaw and the FCC opening (Fig. 15e) was different from that of C-0 (Fig. 13e) and

4.3. Crack development and failure process of Series C The failure process of Series C specimens is depicted in Figs. 13–15. Comparing to Figs. 10–12, the failure processes of Series C were similar to those of Series B. In the case of C-0, the cracks first generated at the side flaws tips (Fig. 13a, b). Then, the TSZ was concentrated at the 12

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Fig. 14. Numerical failure process of specimen C-25.

flaws, TCC and FCC openings, the eight and nine monitoring points presented in Fig. 5 were arranged around the opening and side flaw in the numerical models, respectively. The maximum principal stress (σ1) and minimum principal stress (σ3) obtained from the monitoring points are plotted to represent the stress distribution. It should be noted that, in the following sections, the tensile stress is negative while the compressive stress is positive.

C-25 (Fig. 14e). Uniaxial compressions of Series B and Series C were simulated in this study, and the failure processes were recorded and studied. Under uniaxial loading, the cracks first occurred at the tips of the two side flaws. Then, the TSZ was concentrated at the bottom and top of the TCC or FCC opening. Finally, the SSZ became a “butterfly” shape after the coalescence between the side flaws and opening. However, with an increase in the side flaw inclination angle, the initiation of cracks at the side flaw tip became more difficult, and the coalescence location of the SSZ changed. Due to the reduction of the cross-sectional area and lower high-span ratio of the FCC opening, the strength of the specimens decreased and the vertical deformation of the opening increased, which caused tensile stress concentration at the crown and invert of the FCC opening more easily.

5.1. Principal stress distribution around the side flaws Fig. 16 shows the principal stress distributions surrounding the left and right 0° flaws when the crack initiation appears around the flaws. Combined with Fig. 5, it can be shown that the maximum σ1 values are both observed at the flaw tips and that the magnitudes of σ1 at the flaw tips are much higher than the other locations. In addition, the σ3 values at the flaw tips are higher than those obtained from the other monitoring points. This indicates that the compressive stress is predominantly concentrated at the flaw tips, which is where the earliest cracking location is also observed in the physical tests. Except for σ1 and σ3 obtained at the flaw tips, the magnitudes of σ1 and σ3 at other monitoring points are approximately equal. In Series B and C, even though two differently shaped openings were created at the centers of

5. Stress analysis The existence of voids within the granite block can significantly affect the internal stress distribution under uniaxial compression. The stress distribution is strongly related to the shape of a void as well as the side flaw inclination angle, as presented in Figs. 10–15. To accurately obtain the stress distribution surrounding the voids, including the side 13

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Fig. 15. Numerical failure process of specimen C-45.

larger than in the cases of 0° and 25°. Similar to the case of 25°, σ1 at MP-1 in C-45 is largest while that in A-45 is smallest. The principal stress distributions around the flaws when the crack initiation appears around the flaws are depicted in Figs. 16–18. It can be seen that the inclination angle heavily affects the principal stress distributions around the side flaw. Although the maximum σ1 always appears at flaw tips, the σ1 values at MPs 6–9 become larger with an increase in the inclination angle. The opening and its shape show little influence on the principal stress distributions around the side flaws when the inclination angle is 0°, while those factors affect the σ1 at flaw tips when the inclination angle is 25° or 45°. This is because the distance between the side flaw tip and opening becomes shorter with the change of the side flaw angle, which intensifies the stress concentration at the side flaw tip.

samples, the principal stress distributions are not affected, and the same trend can be found. For the case of 25°, as shown in Fig. 17, the principal stress distributions surrounding the two side flaws are similar to the case of 0° when the crack initiation appears around the flaws. Including the TCC and FCC cases, the σ1 values at the flaw tips are much higher than the other locations, and the σ3 values at the flaw tips are slightly higher than the other locations. However, referring to Fig. 17a, c and e, it can be found that σ1 at MP-1 becomes larger relative to the case of 0°. σ1 at MP-1 is smallest in A-25, while σ1 at MP-1 is largest in C-25. This means that the existence of an opening and the shape of the opening have an influence on the principal stress at the side flaw tips for the case of 25°. Additionally, compared to Fig. 16, due to the increase of the side flaw inclination angle, the σ1 values obtained at the top of the left flaw are higher than those at the bottom, while the opposite trend is found for the right flaw. For the case of 45°, as shown in Fig. 18, the σ3 values surrounding the left and right side flaws are approximately equal. However, comparing Fig. 18 to Fig. 16 and Fig. 17, it can be seen that the stress distributions of σ1 surrounding the 45° flaws are quite different from those surrounding the 0° and 25° flaws. Although the maximum σ1 still appears at the tips of the side flaws, the σ1 values at MPs 6–9 become

5.2. Principal stress distribution around the opening As presented in Fig. 10 through Fig. 15, a void significantly disturbs the internal stress distribution. The principal stress distributions surrounding the assumed and real TCC openings in the numerical models at the first crack initiation of openings are presented in Fig. 19 through Fig. 21. In Fig. 19, σ1 for A-0 is approximately uniformly distributed 14

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Fig. 16. The principal stress distributions around the 0° flaws at the flaw crack-initiation stress (Unit: MPa).

small, and the σ3 values are negative. This indicates that tensile stresses mainly occur at the crown and invert for a road-shaped tunnel. These characteristics are consistent with the stress distribution of the specimens containing the circle openings under uniaxial compression.

surrounding the assumed TCC void, and σ3 is quite small but positive. Surrounding the real TCC opening in B-0, σ1 and σ3 are nonuniformly distributed. The σ1 values derived from monitoring points 3, 4 and 7 are large for specimen B-0. At the crown and invert, the σ1 values are quite 15

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Fig. 17. The principal stress distributions around the 25° flaws at the flaw crack-initiation stress (Unit: MPa).

However, because of the existence of side flaws, the stress distribution around the TCC opening is not symmetrical. For example, the σ1 and σ3 at MP-4 are larger than those at MP-6 due to the fact that MP-4 is located very close to the right side flaw. Additionally, unlike the standard circle, because of the special shape of the TCC opening, higher values of σ1 are observed at MP-4 and MP-6, namely, the arch foots of the tunnelshaped opening. In Fig. 20, for A-25, the distributions of σ1 and σ3 surrounding the assumed TCC opening are analogous to those for A-0. For B-25, similar to B-0, the maximum σ1 values surrounding the TCC opening are

observed at MP-3 and MP-7, namely, the arch waist of the tunnelshaped opening. The σ3 values at the crown and invert are also negative. However, due to the change of the inclination angle, the maximum positive σ3 values appear at MP-3 and MP-7, rather than at MP-4 and MP-7 as in B-0. In addition, the σ1 values at MP-3 and MP-7 are much larger than those in B-0, which means that the inclination angles of the side flaws have a significant influence on the principal stress distributions around the TCC opening. As shown in Fig. 21, the same trends of σ1 and σ3 can be found in A45 and B-45. However, due to the increase of inclination angle in B-45, 16

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Fig. 18. The principal stress distributions around the 45° flaws at the flaw crack-initiation stress (Unit: MPa).

the maximum σ1 values are derived from MP-3 and MP-6. That is, significant compressive stress appears at the arch waists and the arch foots, and the maximum σ3 values are now located at MP-6. For the FCC opening (Figs. 22–24), when the inclination angle is the same, the σ1 and σ3 distributions are similar to that for the TCC opening. In Series A, σ1 and σ3 around the assumed FCC opening are

approximately uniform. Comparing the principal stress distributions around the FCC opening with those around the TCC opening, it can be found that the slight change in opening shape does not prominently affect the principal stress distribution. The inclination angle of the side flaw strongly affects the principal stress around the FCC opening. For example, with the increase of inclination angle, the maximum σ3 values 17

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Fig. 20. The principal stress distributions around the TCC opening for A-25 and B-25 at the opening crack-initiation stress (Unit: MPa). Fig. 19. The principal stress distributions around the TCC opening for A-0 and B-0 at the opening crack-initiation stress (Unit: MPa).

tunnels, are generally shallow buried and have no obvious tectonic pressure. During the structural safety assessment, due to the small lateral pressure, only the deformation and stress of the tunnel structure under the overburden soil load are generally considered. In these projects, the lateral pressure is small and has little effect on the failure of the tunnel structure. In this study, two side flaws were pre-prepared within the real granite block, and a high stress concentrated zone was generated. Then the tunnel-shaped opening was created in the highstress zone, and uniaxial compression tests were conducted on the specimens. The results of this study are useful for revealing cracking development and stress distribution characteristics of shallow buried three-centered circular or five-centered circular tunnel structures in the jointed rock mass under only gravity loading of soil, then selecting reasonable supporting measures or taking special reinforcement of stress-concentrated areas to ensure safety. As is well known, the lateral pressure coefficient has a significant influence on the cracking development and stress distribution characteristics around the openings [48–50]. In an actual tunnel engineering, when the buried depth of the tunnel is deep or significant tectonic stress exists in the tunnel, the lateral pressure cannot be ignored and has a great impact on the failure of the tunnel structure. The application of the results presented in this study to those projects is difficult. In that case, it is necessary to consider biaxial tests or largescale model tests for a further study in the future.

are observed at MP-7 (Fig. 22b), MP-3 (Fig. 23b) and MP-6 (Fig. 24b) in Series C. In addition, the σ1 values at MP-3 and MP-7 in B-25 or C-25 are larger than those in B-0/C-0 or B-45/C-45. This is probably because the distance between the side flaw tip and the opening boundary is the shortest in specimens containing 25° flaws. The interaction between the side flaw and opening makes the stress concentration at the arch waist more extreme. It causes the specimens containing the 25° flaws and openings to fail more easily, which explains the reason that the peak stress of B-25 or C-25 is smaller than that of B-0/C-0 or B-45/C-45. The principal stress distributions surrounding the opening when the crack initiation appears around the openings are depicted in Figs. 19–24. It can be seen that a high compressive stress is concentrated at the arch waist and the arch foot of the tunnel-shaped opening, while the tensile stress is concentrated at the crown and invert. The inclination angle of the side flaw strongly affects the principal stress distribution around the opening. The location of the maximum σ3 changes with the change of inclination angle. When the angle is 25°, the σ1 at the arch waist reaches its maximum. The slight change of opening shape has little influence on the principal stress distribution surrounding the opening.

6. Discussion 7. Conclusion Zhu [49] and Zhao [50] found that the fracturing patterns of the openings were affected by lateral pressure coefficient. However, they also indicated that the failure process or the path to collapse of underground openings under biaxial compression is similar to that under uniaxial compression when the lateral pressure coefficient is small. Some tunnel projects, such as shallow mountain tunnels or urban metro

Both a laboratory test and numerical simulations of granite specimens with different flaws and tunnel opening geometries were investigated. The effects of these geometries on the strength, cracking process and principal stress distributions surrounding the flaws and tunnel-shaped opening under uniaxial loading were analyzed in this 18

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Fig. 22. The principal stress distributions around the FCC opening for A-0 and C-0 at the opening crack-initiation stress (Unit: MPa).

Fig. 21. The principal stress distributions around the TCC opening for A-45 and B-45 at the opening crack-initiation stress (Unit: MPa).

study. On the basis of the experimental and numerical results, some meaningful conclusions are drawn as follows: (1) The results of laboratory tests and numerical simulations both indicate that the existences of flaws and openings significantly degrade the load bearing capacity of the specimens. The reduction of specimen strength is influenced by the loss of cross-section area in specimens; the greater the loss of area, the greater the decrease in peak strength. Therefore, the largest strength is observed in Series A, the second best in Series B, and the smallest in Series C at the same inclination angle. The influence of flaw inclination angle on the specimen’s peak stress is divided into two types. In Series A, the peak stress increases with the increase of the flaw inclination angle. In Series B and Series C, the peak stress decreases first and then increases with the increase of the flaw inclination angle. (2) For Series A, cracking first occurs at the tips of the side flaws. As the axial stress increases, shear stress is concentrated in the zone between the two side flaws, and then the surface rock rapidly spalls at the peak stress time. For Series B and Series C, the cracking surrounding the side flaws is basically similar to that of Series A. Around the openings, tensile cracks are first initiated at the bottom and top. Then, shear cracks occur near the tensile zone and extend along the loading direction, and the shear cracks surrounding the flaws or openings continue propagating and finally coalesce together. When the flaw inclination angle reaches 45°, crack initiation from the flaw tips becomes more difficult, and it also changes the crack coalescence location between the flaw tips and openings. Compared with the TCC opening in Series B, due to the greater cross-sectional area loss and lower high-span ratio of the FCC opening, the strength of the specimens decreased and the vertical displacement of the opening increased, which caused easier tensile

Fig. 23. The principal stress distributions around the FCC opening for A-25 and C-25 at the opening crack-initiation stress (Unit: MPa).

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Fig. 24. The principal stress distributions around the FCC opening for A-45 and C-45 at the opening crack-initiation stress (Unit: MPa).

stress concentration at the crown and invert of the FCC opening in Series C. (3) Surrounding the side flaws, the maximum σ1 and σ3 are both obtained at the flaw tips, and the magnitude of σ1 is much higher than the other locations. The inclination angle has a strong influence on the principal stress distribution around the side flaws, while the opening and its shape show little impact on it. Surrounding the opening, due to the particular shape of the TCC or FCC opening, high compressive stress appears at the arch waists and arch foots. High tensile stress occurs at the crown and invert. The slight change in opening shape does not prominently affect the principal stress distribution. The flaw inclination angle significantly affects the stress distribution at the arch waists and arch foots. The σ1 at the arch waists reaches its maximum when the inclination angle is 25°.

Declaration of Competing Interest The authors declared that there is no conflict of interest.

Acknowledgements The research reported in this paper was financially supported by National Natural Science Foundation of China (41807241, 51978064) and Natural Science Basic Research Plan of Shaanxi Province (2018JQ4015) and Fundamental Research Funds for the Central Universities, CHD (300102219110, 300102219208) and Open Fund of Engineering Research Center of Catastrophic Prophylaxis and Treatment of Road & Traffic Safety of Ministry of Education (Changsha University of Science & Technology kfj170406) and Research Fund of Department of Transportation of Zhejiang Province (2016019, 2019043). 20

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