Evolution of Rock Cracks Under Unloading Condition ...

Evolution of Rock Cracks Under Unloading Condition

R. Q. Huang & D. Huang

Rock Mechanics and Rock Engineering ISSN 0723-2632 Volume 47 Number 2 Rock Mech Rock Eng (2014) 47:453-466 DOI 10.1007/s00603-013-0429-0

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Author's personal copy Rock Mech Rock Eng (2014) 47:453–466 DOI 10.1007/s00603-013-0429-0

ORIGINAL PAPER

Evolution of Rock Cracks Under Unloading Condition R. Q. Huang • D. Huang

Received: 23 June 2011 / Accepted: 23 April 2013 / Published online: 9 May 2013 Ó Springer-Verlag Wien 2013

Abstract Underground excavation normally causes instability of the mother rock due to the release and redistribution of stress within the affected zone. For gaining deep insight into the characteristics and mechanism of rock crack evolution during underground excavation, laboratory tests are carried out on 36 man-made rock specimens with single or double cracks under two different unloading conditions. The results show that the strength of rock and the evolution of cracks are clearly influenced by both the inclination angle of individual cracks with reference to the unloading direction and the combination geometry of cracks. The peak strength of rock with a single crack becomes smaller with the inclination angle. Crack propagation progresses intermittently, as evidenced by a sudden increase in deformation and repeated fluctuation of measured stress. The rock with a single crack is found to fail in three modes, i.e., shear, tension–shear, and splitting, while the rock bridge between two cracks is normally failed in shear, tension–shear, and tension. The failure mode in which a crack rock or rock bridge behaves is found to be determined by the inclination angle of the original crack, initial stress state, and unloading condition. Another observation is that the secondary cracks are relatively easily created under high initial stress and quick unloading.

R. Q. Huang (&) D. Huang State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu 610059, China e-mail: [email protected] D. Huang College of Civil Engineering, Chongqing University, Chongqing 400045, China

Keywords Unloading Rock with crack Rock bridge Crack evolution Failure mode

1 Introduction Engineering practice indicates that the instability of rock mass is generally associated with human engineering activities. This is usually due to the stress release and redistribution near to the excavation surface, leading the rock mass to deform, especially causing the expansion and/ or extension of existing discontinuities to connect each other, and, finally, produce macroscopic fractures. The failure of Malpasset Dam in France and the collapse of the Vajont landslide of Italy, for instance, are closely related to human engineering activities which weaken the rock mass by gradually altering the feature of discontinuities, in addition to disturbing the original stress field (Post and Bonazzi 1987; Kilburn and Petley 2003; Huang 2012; Gong et al. 2012). Features of existing discontinuity (such as length, strike, and dip), mechanical properties of intact rock, existence of rock bridge between cracks, and the spatial relationship between the excavation face and discontinuity determine the expansion and extension process of existing discontinuity during external loading, and, in turn, impact the deformation and strength of rock mass (Brady and Brown 2007; Gambarotta and Monetto 2002). In general, underground excavation causes stress unloading at the direction towards the excavation face, and the stress state of rock mass near to the excavation surface gradually changes from triaxial compression to biaxial/ uniaxial compression. This would introduce, to some extent, differential rebounding deformation to a certain depth from the excavation face (Zhang et al. 1994). The heterogeneity and inelasticity of the rock mass, especially

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Author's personal copy 454

in the vicinity of discontinuity, further increase the variation of rebounding deformation, resulting in tensile stress concentration and crack propagation (Li and Meng 2001). The initiation and propagation of cracks in the affected zone due to underground excavation are, therefore, necessary to be studied so as to satisfactorily assess the rock mass behavior (Cai and Kaiser 2005). Researches on the mode and mechanism of deformation/failure of rock with cracks under loading condition have been extensively reported (Gambarotta 1993; Lei et al. 2000; Adams and Sines 1978). Deformation characteristics and mechanical properties of rocks under unloading condition have also been widely studied (Huang et al. 2001; He et al. 2010; Xie and He 2004). However, investigation into the failure mode and the expansion and evolution mechanism of rock with cracks under unloading condition are rarely found. This paper takes the mother rock of the power house of the Three Gorges Project as the prototype and generalizes it into six conceptual physical models, which are tested under two different unloading conditions. The aim is to gain deep insight into the strength and deformation characteristics of such rocks, as well as the evolution/propagation pattern of rock cracks.

2 Experimental Program 2.1 Fractured Rock Model The power house in the right bank of the Three Gorges Project was built within hard granite, where there exist sets of well-developed tectonic fractures approximately parallel to the axis of the power house (axis azimuth 43.5°) (Fig. 1). These tectonic fractures likely form local weak zones by the connection of existing joints, and, therefore, jeopardize the stability of the mother rock mass. Figure 1 presents a typical cross-section and shows the main faults and discontinuities in the affected zone. Two faults, F10 and F84, both intersect the power house axis at low angles and dip towards the excavation face. Moreover, there are intermittent discontinuities, which nearly parallel to the axis of the power house. As shown in Fig. 2, Set 1 joints strike towards the NNW, intersecting the axis of the power house at large angles, and present to be tightly packed. These cracks, therefore, impose little effect on the stability of the power house. Set 2 strikes at NEE (the trend azimuth ranges from 30° to 90°) and dips at a wide range of angles (0°–60°). These joints present tension characteristics, rough and fluctuant joint surface, with no filling or partially filled with a small amount of weathered debris. Set 2 joints may impose strong effects on the stability of the power house, as they all strike in almost the same direction.

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R. Q. Huang, D. Huang

Fig. 1 Typical engineering geological profile of the surrounding rock of the power house

Two sets of physical models, i.e., single-crack and double-crack, are set up based on the joint survey, in order to investigate the crack propagation and failure of the rock bridge. As shown in Figs. 3 and 4, the physical model is a 100 mm 9 100 mm 9 100 mm cube. The single-crack model has a crack 50 mm in length and the crack inclination angles are set at 30°, 60°, and 90°, respectively. The double-crack model has two cracks, each of length 20 mm, with a spacing of 20 mm, horizontally and vertically, making the rock bridge inclination angle 45°. The two cracks in the double-crack model are inclined at 30°–30°, 80°–80°, and 80°–30°, respectively (Table 1). 2.2 Sample Preparation Experimental simulation in this study guarantees the following four similarity criteria: (1) geometry; (2) mechanical property of the materials; (3) stress state; and (4) excavation process. The similarity constant is defined as the ratio of the mechanical parameter of the prototype to that of the physical model, and meets the relationship shown in Eq. 1: 9 Cr ¼ Cc Cl > > > = Cl ¼ Ce ¼ Cf ¼ 1 ð1Þ Cr ¼ CE ¼ Cc ¼ CRt ¼ CRc > > > ; Cd ¼ Cl where Cr is the stress similarity constant; Cc the bulk density similarity constant; Cl the geometric similarity constant; Cl the Poisson’s ratio similarity constant; Ce the strain similarity constant; Cf the friction strength similarity

Author's personal copy Evolution of Rock Cracks Under Unloading Condition

455

Fig. 2 Occurrence of cracks in the rock mass of the power house: a rose diagram of the dip direction and b histogram of the dip angle

N 35

W

E

Number of cracks

30 25 20 15 10 5 0 0

10

20

30

40

50

60

70

80

90

Dip angle(°)

S

(a)

(b)

Table 1 The tested specimens 3

(a)

(b)

3

Model

9

6

Specimen number* (Program A)

Specimen number* (Program B)

Sa-Bl, Sa-B2, Sa-B3

Single-crack

(c)

6

Dip angle (°)

8

a

30

Sa-Al, Sa-A2, Sa-A3

b

60

Sb-Al, Sb-A2, Sb-A3

Sb-Bl, Sb-B2, Sb-B3

c

90

Sc-Al, Sc-A2, Sc-A3

Sc-Bl, Sc-B2, Sc-B3

Double-crack 3

8

d

30, 30

Dd-Al, Dd-A2, Dd-A3

Dd-Bl, Dd-B2, Dd-B3

(f)

e f

80, 80 80, 30

De-Al, De-A2, De-A3 Df-Al, Df-A2, Df-A3

De-Bl, De-B2, De-B3 Df-Bl, Df-B2, Df-B3

6

(d)

(e)

Fig. 3 Layout of the cracks in the tested models

1

Excavated surface

3

Fig. 4 Three-dimensional sketch of the tested model

constant; CE the elasticity modulus similarity constant; Cc the cohesion similarity constant; CRt the anti-tensile strength similarity constant; and CRc is the compressive strength similarity constant. Taking into account the size of the experimental equipment and the scale of the measured discontinuities, a geometric similarity ratio of 20 is used. Through trial and error, the mass ratio of materials making up the physical

* Sa-A1 stands for the first specimen of the single-crack model with a crack inclined at 30°, which is tested under loading scheme Program A

model is set as: barite:quartz sand:cement:plaster:water = 50:20:5:2:6, in order to achieve the mechanical properties shown in Table 2. The quartz sands used are well graded, with particle size ranging from 0.1 to 0.84 mm. The plaster and cement of 425 grade ordinary silicate are used to make the model achieve higher elasticity modulus, high compressive strength, and similar brittleness of rock. Boric acid at a concentration of 1 % is used as the retarder during the sample preparation. As shown in Table 1, a total of 36 specimens are prepared, 18 of which are used in the single-crack model and the other 18 are used in the double-crack model. Each specimen is prepared by compaction in three layers of equal thickness. Special effort is paid to placing the muscovite film in cracks during the preparation of specimens. An additional set of specimens without cracks is prepared in the same manner and tested according to the Chinese standard for rock tests (SL264-2001, Ministry of Water Resources of the People’s Republic of China 2001) for obtaining the physical and mechanical properties of the specimens with cracks (Table 2). According to Table 2, the

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Table 2 Physical and mechanical parameters of the materials Material

Density (kg/m3)

Elasticity modulus (GPa)

Compressive strength (MPa)

Tensile strength (MPa)

Cohesion (MPa)

Friction angle (°)

Poisson’s ratio

Model

2,300

3.45

4.73

0.37

0.51

46.54

0.19

Prototype

2,700

79.74

116

8.41

12.40

60.53

0.20

similarity constants can be obtained as: CE = 23.11, CRc = 24.52, CRt = 22.73, Cc = 24.31, Cf = 1.74, Cl = 1.05, Cc = 1.17, and Cl = 20. These results meet the similarity requirements in Eq. 1. 2.3 Loading and Unloading Scheme The specimen is restrained in the sample chamber except for the two faces where the major principal stress r1 and the minor principal stress r3 are applied. In other words, only strain in the r1 and r3 directions is allowed during the tests (Fig. 4). Such a test scheme is used to approximately simulate the process of underground excavation. Two loading and unloading programs are employed and the detailed procedures are given as follows. 2.3.1 Program A: Loading r1 and Unloading r3 Simultaneously Step 1 Step 2

Step 3

Step 4

Increase both r1 and r3 simultaneously to the initial stress level of 0.5 MPa; Further increase r1 to 1.0 MPa and keep this stress condition for 5 min to retrieve the initial stress state; Increase r1 by 0.1 MPa while decreasing r3 by 0.5 MPa. The loading and unloading both happen quickly and simultaneously. The stress state is kept for 1 min before the next round of loading and unloading actions. Such a loading and unloading scheme is to simulate the secondary stress field induced by underground excavation; If the specimen failed before r3 is decreased to 0, r3 is then immediately unloaded to 0 to finish the test. Otherwise, r1 is increased continuously to make it fail.

2.3.2 Program B: Unloading r3 While r1 Remains Unchanged The procedure for Program B again consists of four steps. During Step 1, both r1 and r3 are increased simultaneously to the initial stress level of 1.05 MPa. r1 is further increased to 2.0 MPa in Step 2. In Step 3, r1 remains unchanged while unloading r3 by 0.15 MPa steps to 0. The

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unloading happens quickly and there is a pause of 1 min before each unloading. Step 4 is exactly the same as that for Program A. With such unloading schemes, both programs provide the same differential stress (r1 - r3) at each stage of unloading, though the unloaded magnitude (change of r3) of Program A (0.05 MPa) is less than that of Program B (0.15 MPa). Program B is used to simulate the relatively high initial stress and quick unloading during excavation compared with Program A.

3 Analysis of the Experiment Results 3.1 Strength Characteristics Figure 5 shows the peak stress ratio [(r1 rc)/rc, where rc is the uniaxial compressive strength] of all the tested specimens. Observations from Fig. 5 can be summarized as follows: 1.

2.

3.

For single-crack models, the peak stress ratio decreases with increasing inclination angle of the crack (Fig. 5a). In other words, the greater the angle between the crack and the excavation surface, the higher the peak strength. The average strength ratios of models a, b, and c tested under Programs A and B are 0.62, 0.59, 0.53 and 0.55, 0.48, 0.43, respectively. For double-crack models, the strength of model e (80°–80° combination) is the highest, that of model d (30°–30°) is intermediate, and the strength of model f (80°–30°) is the lowest (Fig. 5b). In other words, the rock bridge between steep and gentle cracks is the most susceptible to excavation unloading. The average strength ratios of models d, e, and f in Programs A and B are 0.64, 0.70, 0.55 and 0.54, 0.58, 0.46, respectively. High initial stress and great unloading magnitude favor low strength of cracked rocks. Low initial stress and small unloading magnitude show the opposite. This indicates that the excavation speed and initial stress field have clear impacts on the unloading strength of crack rock, since the faster the excavation and the higher the initial stress, the lower the strength of the adjoining rocks.

Author's personal copy Evolution of Rock Cracks Under Unloading Condition 0.7

457

(a)

3

(MPa)

0.50 0.5

0.41

0.32

0.23

0.14

0.05

(a)

0.6 3

0.4

c

90

1

0.3

(mm)

60

0.5

Program A

30

h

Program B

0.2 0.1

0.4 c - 30

0

b - 60

d - 90

0.0

(b)

v

0.8

(mm)

Dip angle of cracks (°)

0.7

90

0.1 30

0.2 0.3

60 0.50

0.80 (

1

c

3

0.4 0.6

0.5

Program A

3

Program B

1

1.10 (MPa) 3)

1.40

1.70

2.00

(MPa) 0.90

1.05

0.75

0.60

0.30

0.45

0.15

0.00

0.4

0.4 f - 80° ,30°

Dip angle of cracks (°) Fig. 5 Peak strength of the tested specimens: a single-crack model and b double-crack model

(b)

0.3

(mm)

e - 80° ,80°

90 30 60

0.2

h

d - 30° ,30°

0.1

3.2 Deformation Characteristics

0.0

Figures 6 and 7 show the curves of displacement (both horizontal and vertical) versus stress of the tested specimens for both single-crack and double-crack models. The numbers next to each curve denotes the dip angle of the crack. At the beginning of unloading, horizontal (r3 direction) and vertical (r1 direction) displacements are both small. With further unloading, a sudden stress drop appears, which indicates initial damage. The stress drop repeats until the peak stress occurs. This indicates that damage and crack extension of the rock with crack happen intermittently and suddenly. Another observation is that the stress drop repeats more frequently in Program B than in Program A, as shown in Figs. 6 and 7, indicating that secondary cracks are relatively easily created under the condition of high initial stress and quick unloading (Fig. 8). Table 3 shows the average displacements in the horizontal and vertical directions and their ratios at peak stress for the tested models. For the single-crack models, when the crack angle is 90°, the horizontal displacement is

0.1 0.2

v

(mm)

90

0.3 30

60

0.4 0.95 (

1

0.10

1.25

1.40

1.55

1.70

1.85

2.00

3) (MPa)

Fig. 6 Curves of horizontal (dh) and vertical displacement (dv) vs. stress of the single-crack models: a Program A and b Program B

obviously larger than the vertical displacement (Fig. 6; Table 3). This is in agreement with the apparent tension crack perpendicular to the unloading direction (Fig. 8e, f). The sharp change of the vertical displacement with stress decrease is more pronounced than that of the horizontal displacement (Fig. 6) and the vertical displacement exhibits a clear increase when the stress is approaching its

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3

(MPa) 0.40

0.50

0.30

0.20

0.00

0.10

0.5

(a) 80 -30 0.3

h

(mm)

0.4

30 -30

0.2

80 -80 0.1

3.3 Fracture Morphology 80 -80

0.1

v

(mm)

0.0

30 -30

0.2 80 -30

0.3 0.50 (

0.80 1

1.05

1.10

1.40

1.70

2.00

2.30

2.60

0.60

0.45

0.30

0.15 0.00

3) (MPa)

3

(MPa)

0.90

0.75

0.5

(b)

h

(mm)

0.4 80 -80

0.3

30 -30 80 -30

0.2 0.1 0.0

(mm)

0.1

v

the r1 direction and the lateral expansion due to the occurrence of tension cracks. The displacement characteristics in both r1 and r3 directions of the double-crack specimens with 30°–30° cracks are similar to that of the single-crack specimen with a 30° crack (Fig. 7). The 80°–80° double-crack specimens show similar characteristics to the 90° single-crack specimen. The stress drop and accompanied displacements (vertical and horizontal) in the 80°–30° double-crack specimens are higher than the 30°–30° and 80°–80° specimens (Fig. 7).

80 -80

0.2 0.3 30 -30

0.4

80 -30

0.5 0.95

1.10 (

1

1.25 3)

1.40

1.55

1.60

1.85

2.00

(MPa)

With a crack inclination angle of 30°, tension cracks develop at the upper end of the original crack, while compressioninduced shear cracks appear at the other end (Fig. 8a, b). For the 60° single-crack specimen, both tension and shear cracks develop at both ends of the original crack (Fig. 8c, d). The tension crack seems to occur along wing cracks and the specimen fails by connecting shear cracks and the original crack. The 90° single-crack specimen is found to be split by extension of the original crack with local tension or shear cracks (Fig. 8e, f). Based upon the above observations, the single-crack rock exhibits three different failure modes, i.e., shear, tension–shear, and splitting. Under unloading condition, the failure mode in which a crack rock behaves is determined by the inclination angle of the original crack. Another observation is that unloading in Program B introduces more cracks in the right half (near to the unloading surface) of the specimen than in Program A. The extension pattern of individual cracks in doublecrack specimens is similar to that in single-crack specimens. Shear cracks are found in the rock bridge of 30°–30° double-crack specimens (Fig. 8g, h). The rock bridge of 80°–80° specimens are broken by a shear crack in Program A, while a tension crack develops in the rock bridge in Program B (Fig. 8i, j). Shear cracks in the rock bridge are only found in 80°–30° specimens unloaded in Program A, while tension and shear cracks appear in Program B (Fig. 8k, l). The above observations indicate that the rock bridge is normally failed in three modes, i.e., shear, tension–shear, and tension. Under the condition of high initial stress and quick unloading (Program B), the rock bridge between steep cracks tends to be broken in tension, while the rock bridge between steep cracks and gentle cracks tends to fail in tension–shear mode.

Fig. 7 Curves of horizontal (dh) and vertical displacement (dv) vs. stress of the double-crack models: a Program A and b Program B

4 Crack Propagation peak value. This represents shear deformation of the specimen, which is also observed in Fig. 8c, d. When the crack angle is 30°, the displacements in both directions are similar. This may be due to the compressive contraction in

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Evolution processes of single crack propagation and rock bridge failure are put described as follows based on the foregoing analysis.


459

Fig. 8 Photographs and sketches of failed crack rocks. S shear crack, T tension crack, W wing crack

1

T

T

W 30 3

S

Original crack

W T

(a) Sa-A1 1

T W T

30

T

3

W

T

T S

(b) Sa-B2 1

S W

T 60 3

S

(c) Sb-A3 4.1 Single Crack Shear mode With unloading, wing cracks appear (Fig. 9a-I). Sets of shear cracks, normally parallel to the original crack, are then introduced in the vicinity of both ends of the

original crack (Fig. 9a-II). These shear cracks often start from the wing cracks and extend to make the rock fail with small secondary tension cracks around them (Fig. 9a-III). Tension–shear mode Tension cracks are developed by extending the wing cracks at the upper end of the

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Fig. 8 continued 1

T

S

W 60 T S

S

T

3

S T

(d) Sb-B2 1

T T

3

S T

S S

T

(e) Sc-A2 1

S

S

T T

S

3

T

S

T

(f) Sc-B3

original crack. At the same time, shear cracks appear at the other end (near to the unloading surface) (Fig. 9bII). The rock fails with the extension of both tension

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and shear cracks, normally accompanied with the occurrence of tension cracks at the lower part (Fig. 9bIII).


461

Fig. 8 continued 1

T

T 30 S

W

W 30

3

Original crack T

(g) Dd-A1 1

T

T W

T

30 W

S W 30

3

W

T T

T

T

(h) Dd-B2 1

T 8 W S W

3

8 T

S

(i) De-A1

123



Fig. 8 continued 1

T

T 8 T

T

T

3

8 T

(j) de-B3 1

T 8 S 3

30 W

S

(k) Df-A2 1

T 80 T S

W

T 3

30 T

(l) Df-B1

123

S


463

Table 3 Average displacement at peak stress in the tests Model

Program A

Program B

Horizontal displacement (mm)

Vertical displacement (mm)

Ratio

Horizontal displacement (mm)

Vertical displacement (mm)

Ratio

a

0.19

0.21

0.90

0.28

0.36

0.78

b

0.28

0.24

1.17

0.21

0.29

0.72

c d

0.33 0.17

0.07 0.15

4.71 1.13

0.29 0.23

0.07 0.27

4.14 0.85

e

0.19

0.09

2.11

0.25

0.06

4.16

f

0.29

0.21

1.38

0.24

0.31

0.77

Fig. 9 Evolution model of crack extension: a shear, b tension–shear, and c splitting

(a)

1

1

1

S T

S 3

60 W

I

(b)

S

II

1

T

I

II

T 3 decreased

4.2 Rock Bridge Failure The failure evolution of the rock bridge is shown in Fig. 10. Cracks on other parts except for the rock bridge are not shown for simplification. Shear mode Shear crack occurs at both ends of the rock bridge and grow towards each other to cut through the rock

1

T

90

Splitting mode Splitting cracks develop almost vertically from both ends of the original crack, with small dendritic cracks (Fig. 9c-II). All these cracks further grow towards the rock boundaries to make the rock specimen fail (Fig. 9c-III).

T S

1

3

W

3=0

T III

1

W

I

T

3 decreased

S

W

(c)

S T

III

1

3

3

3=0

1

T

W

3 decreased

W

T II

T

T

T

III

3=0

T T

bridge (Fig. 10a). It is noted that wing cracks may not be developed around the ends of the original cracks, e.g., specimen Df-A2 (Fig. 8k). Tension–shear mode With the extension of wing cracks, a shear crack starts from one end of the rock bridge (Fig. 10b-II). The rock bridge is cut through when the wing crack meets the shear crack originating from the other end (Fig. 10b-III). In rocks with gentle and steep cracks, the unloading-induced shear crack normally starts from the end of the gentle crack. Tension mode With the extension of wing cracks, tension cracks nearly parallel to r1 are developed in the rock bridge (Fig. 10c-II). The accumulation of tension cracks

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Author's personal copy 464 Fig. 10 Evolution model of rock bridge failure: a shear, b tension–shear, and c tension


(a)

1

3

W

S S

W 3

I

II

(b)

1

1

3

T

W 3

T S

I

3 decreased

8

II

(c)

W 8

5 Mechanism of Crack Propagation As shown in Fig. 11, decreasing r3 causes a clear difference in rebounding deformation between the side parts of crack, as the half near to the unloading surface deforms much more than the remaining part. This, in turn, introduces, around the crack ends, tension stress parallel to the rebounding direction, i.e., in the r3 direction. Provided that the angle between the crack and r3 (unloading direction) is a, the shear stress s and normal stress rn on the crack surface are: 3 s ¼ r1 r 2 sin 2a ð2Þ r1 r3 3 rn ¼ r1 þr 2 þ 2 cos 2a

T

T T

II

1

3 decreased

3

causes the rock bridge to break (Fig. 10c-III). Shear failure may occur later in such a broken rock bridge. The foregoing discussion indicates that rock bridge is most susceptible to tension–shear failure under unloading condition. The observation made here is different from the failure mode of rock bridge under biaxial compression condition. Zhu et al. (1998) concluded that rock bridge tends to be failed in shear mode during biaxial compression. This is due to the unloading, which favors tension cracks to occur in the vicinity of the rock bridge.

S

1

8

I

3=0

III

1

W

3=0

S

III

1

W

123

1

3 decreased

3

1

T

3=0

III

As unloading causes the stress difference (r1 – r3) to increase, shear stress s on the crack surface increases (Eq. 2). Figure 12 shows the change of r3 with rn during unloading in Program B (r1 = 2.0 MPa, r3 is decreased step by step). It is observed that rn on the crack surface decreases gradually with decreasing r3, and with the same r3, the larger the crack dip angle, the smaller the rn. Therefore, the unloading process would cause the shear stress s of the crack to increase and the shear strength ss to decrease. Based on the above analysis, the evolution of rock crack under unloading condition is controlled by the tension stress rT, caused by differential rebounding deformation and the shear stress s on the crack surface (Fig. 11). When tensile stress rT and normal stress rn meet the following criterion, new tension cracks would be developed: rT sin a ¼ rn ¼

r1 þ r3 r1 r3 þ cos 2a 2 2

ð3Þ

At the beginning of unloading, differential rebounding deformation is small, causing small tensile stress rT. This leads the rock specimen to a three-dimensional compression state. The unloading leads the shear stress to increase and the shear strength to decrease, increasing the potential of

Author's personal copy Evolution of Rock Cracks Under Unloading Condition Fig. 11 The conceptual model of deformation/failure of singlecrack rock under unloading

465

Sa-B2

1

T

s

line 1

n

W =30

T

T 0.5

T T

3

T

line 2 n

W T

s

S n

0.5

T

h

line 1 Accumulating

Small

Large h

h

line 2 Small

Large

h

h

Distance

original crack expand. The relatively high shear stress may cause the original crack to deform by shear. The shear stress is the key factor to control the crack propagation, while unloading tensile stress acts as the supplement. When the crack angle is low (e.g., a = 30° in model a), the small tensile stress cannot make the original crack expand. However, the wing cracks produced at the beginning of unloading are almost vertical. The wing crack far from the unloading face tends to extend by tension. Further unloading favors a shear crack towards the unloading surface, leading to a tension–shear failure mode.

Fig. 12 Change of rn with r3 (r1 = 2 MPa)

6 Conclusions

shear slipping. The driving force of shear slipping would introduce tension cracks (wing cracks) at both ends of the original crack. With unloading, the tensile stress caused by differential rebounding deformation and the shear stress on the crack surface are increasing. Such a change of stresses leads the extension direction and failure mode of the crack to alter. When the crack is steep (e.g., a = 90° in model c), the tensile stress on both ends of the crack is relatively large, while the shear stress s is small (e.g., a = 90°, s = 0). The original crack will be extended along its own direction. The tension stress plays an important role in such kind of crack propagation. When the crack angle is intermediate (e.g., a = 60° in model b), the tensile stress is not enough to make the

The peak strength of rock with a single crack becomes smaller when the inclination angle between the crack and the unloading direction increases. Rocks with two steep cracks have the highest peak strength, while rock with two gentle cracks have an intermediate peak strength. A pair of steep and gentle cracks lead to a minimum peak strength. Crack propagation shows intermittent behavior. This is evidenced by the sudden increase in deformation with repeated fluctuation of stress. The pattern of deformation and stress development (magnitude of stress drop, sudden increase of deformation, etc.) is dominated by the inclination angles of the original crack and their combination. Generally, the horizontal deformation of rock with steep cracks is large, while that of rock with gentle cracks is small.

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The failure mode in which a crack rock behaves under unloading condition is determined by the inclination angle of the original crack. Rocks with a crack at intermediate inclination angles (about 60°) behave in shear mode. Tension–shear mode occurs within the rock with a crack inclined at gentle angles (around 30°). Splitting mode appears in rock with steep cracks (e.g., 90°). This is also verified by the theoretical calculations. The rock bridge is normally failed in three modes, i.e., shear, tension–shear, and tension. Under the condition of high initial stress and quick unloading, the rock bridge between two steep cracks tends to be broken in tension, while the rock bridge between a steep crack and a gentle crack tends to fail in tension–shear mode. High initial stress and quick unloading introduce many cracks in the vicinity of the unloading surface. Acknowledgments This work is supported by the National Natural Science Foundation of China (No. 41130745 and No. 41172243), the Fundamental Research Funds for the Central Universities (No. CDJZR12205501), and the Open Foundation of State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (No. SKLGP2011K003). The authors are deeply grateful to Dr. Yanrong Li of AGECON Ltd. (Hong Kong) for his valuable comments on and constructive revision of the paper, and stimulating discussions on the research.

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