Numerical study of crack propagation in an indented rock specimen

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Research Paper

Numerical study of crack propagation in an indented rock specimen ⁎

Jie Liua,b, , Jun Wanga, Wen Wanc a b c

Department of Building Engineering, Hunan Institute of Engineering, Xiangtan, China Hunan Provincial Key Laboratory of Safe Mining Techniques of Coal Mines, Hunan University of Science and Technology, Xiangtan, Hunan, China School of Resource, Environment and Safety Engineering, Hunan University of Science and Technology, Xiangtan, Hunan, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Stress evolution Indentation Crack propagation Chip Discrete element method

Indentations were simulated to investigate the stress evolution characteristics of the indentation process using the discrete element method (DEM). The maximum principle stress and the shear stress were recorded by applying measurement circle logic. The results indicate that an increase in indentation force contributes to the concentration of shear and tensile stresses at the crack tips. The indentation force decreases because of the crack propagation, which is accompanied by stress dissipation at the crack tips. In addition, tensile and shear-tensile cracks, propagating in different modes, have been observed. The results show that the shear-tensile cracks are responsible for chip formation.

1. Introduction Rock indentation is likely to cause rock breakage that affects indentation efficiency and tool wear. Thus, the rock breakage mechanism of indentation has been attracting interest for decades. Extensive investigations have contributed to the understanding of rock breakage by indentation. For example, Paul and Sikarshie proposed that rock fails because of concentrated shear force. They further stated that the slopes determined by the fluctuations in the indentation force and depth are constant [1]. In addition, Miller and Sikarshie proposed that tensile crack propagation is responsible for rock breakage due to indentation [2]. Considering the shear and tensile failure in the indentation process, the frequently used cavity model for a blunt indenter or a disc cutter demonstrates that a crushed core first forms because of the nonuniform contact force by the tool [3,4]. Then, a plastic zone forms around the core because of the compression induced by the crushed core. With further indentation, internal cracks initiate from the rim of this plastic zone [5]. The propagation and coalescence of these internal cracks determine the extent of chip formation between adjacent indentations. The indentation energy (characterized by the indentation force) and chipping volume (influenced by the propagation of cracks) codetermine the indentation efficiency. With respect to the characteristics of indentation forces, the laboratory tests of Chen and Labuz indicated that indentation force fluctuates with an increase in the indentation depth when the indentation depth is greater than a critical value [6]; recently, Yin et al. and Li et al. obtained similar results [7,8]. To obtain the crack propagation characteristics of the indentation process, acoustic emission (AE) events that reflect the extent of crack propagation were ⁎

recorded by Yin et al. during biaxial indentation tests [7]. The increased AE counts at indentation depths greater than a critical value indicate that crack propagation also increases. In addition, Liu et al. and Entacher et al. proposed that the fluctuations in the cutting force are accompanied by a considerable increase in AE counts [9,10]. They concurred that the fluctuations of the indentation force are likely to be correlated with the crack propagation induced by the indentations. However, because of the ineffective observation of cracks in laboratory tests, the relation between force fluctuations and crack propagation is still an open topic. Thus, numerical simulations, particularly those based on the discrete element method (DEM), have been widely used to investigate crack propagation during the indentation process. For example, Huang et al. proposed that tensile cracks initiate from the rim of the compressive zone and stated that the initiation point is influenced by the confinement [11]. According to the variation in the initiation points, they also classified cracks that were induced by indentation into vertical and lateral cracks. In addition, Ma et al. stated that indentation tests are capable of simulating linear cutting tests because indentation tests are conducted under a 2D plane strain condition [12]. They also observed fluctuations in the indentation force and AE events. Then, they measured the length and deflection angle of the crack to investigate the indentation efficiency. Additionally, a few other numerical studies investigated the initiation and connection of the internal cracks caused by indentations [13,14]. However, few studies investigated the dynamic evolution of the stress at crack tips after crack initiation, possibly correlative to the fluctuation in the indentation force. The aforementioned studies agreed that tensile stress concentrations

Corresponding author at: Department of Building Engineering, Hunan Institute of Engineering, Xiangtan, China. E-mail address: [email protected] (J. Liu).

http://dx.doi.org/10.1016/j.compgeo.2017.10.014 Received 9 March 2017; Received in revised form 18 October 2017; Accepted 25 October 2017 0266-352X/ © 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: Liu, J., Computers and Geotechnics (2017), http://dx.doi.org/10.1016/j.compgeo.2017.10.014

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During the indentation process, the energy such as boundary work done by the walls and strain work stored in the bonds can be recorded. Thus, the crack propagation energy can be obtained by subtracting the stored energy from the input energy. This method of calculating the crack propagation energy was reported by Moon and Oh and Moon et al. [26,27]. In the indentation process, measurement circles with specific radii can be installed at designated locations. Within a measurement circle, the average stress tensor is as follows [28]:

are responsible for chip formation. However, shear, tensile and sheartensile cracks have been reported in rock and rock-like materials under different driving stress characteristics [15–19]. In addition, different confinements, joint distributions, and water contents are likely to result in a variation in the stress conditions and thus influence the crack pattern [20–24]. Nevertheless, few studies have investigated crack patterns produced during the indentation process. Thus, in the present study, according to the simplification of stress conditions for rock breakage, a widely used DEM code, PFC 2D [25], was applied to investigate the dynamic stress evolution during the indentation process and to study the crack pattern of the internal cracks under a plane strain condition [12]. Considering the characteristics of indentation force, crack propagation energy, and recorded number of cracks, the stress concentrations and dissipations at the crack tips were discussed. In addition, to analyze the stress evolution characteristics in the chipping process, indentation simulations were conducted at higher confinements.

⎛ ⎞ 1−n ⎟ ⎜ σij = ⎜ ∑ VP ⎟ ⎜ ⎟ ⎝ NP ⎠

∑∑ NP

|x i(C )−x i(P ) |ni(C,P ) F j(C )

NC

(1)

where NP and NC are the numbers of the particles and contacts within the measurement circle, V P is the volume of the particle; x i(P ) and x i(C ) are the locations of the particles and their contacts, ni(C ,P ) is the unit normal vector determined by the particle center, and F j(C ) is the contact force. Thus, the horizontal stress, σh , vertical stresses, σv , and shear stress, τ , can be obtained within the measurement circles. The recorded shear stress can characterize the shear crack initiation and propagation. In addition, in the PFC 2D model, the tensile and compressive stresses are positive and negative, respectively. Therefore, the maximum principle stress can characterize the tensile crack initiation and propagation. The maximum principle stress can be expressed as follows [29]:

2. Numerical model and preparation 2.1. Brief introduction to PFC 2D and measurement circle logic The Particle Flow Code in Two Dimensions (PFC 2D), which is an efficient and rigorous software, has been successfully used to simulate crack initiation, propagation, and coalescence in rock and rock-like specimens [15,25]. The PFC 2D model consists of particles, bonds, and walls. The geometry and mechanical properties of these elements can be predetermined. Bonds, divided into contact and parallel bonds, mainly determine the failure characteristics of the model. In natural rock, shear, tensile, and bending stresses are likely to act on the bonds between particles. Thus, parallel bonds that are capable of resisting moment can more accurately simulate rock and rock-like material. Therefore, to simulate rock specimens, circular particles with defined elastic moduli, friction coefficients, and radii are commonly connected by parallel bonds (Fig. 1) [15]. In the loading process, microcracks form because of bond breakages that occur when the stresses reach the shear or tensile strength of the material. Simultaneously, the numbers of the tensile and shear cracks can be dynamically recorded and output. In addition, indentation adds energy into the calculation model; the energy is then consumed by crack propagation or stored in the model.

σmax =

σh + σv + 2

⎛ ⎝

σh−σv 2 ⎞ + τ2 2 ⎠

(2)

where σmax is the maximum principle stress, σh and σv are the horizontal and vertical stresses, and τ is the shear stress. Thus, according to the measurement circle coordinates, the recorded stresses and the calculated maximum principle stress (Eq. (2)), the stress contours can be drawn by postprocessing. 2.2. Numerical model Before the indentation simulations, the uniaxial compressive strength and fracture toughness of the specimen were measured (Fig. 2). Fracture toughness, KIC , is a critical index representing the resistance of a material to fracture propagation [26,27]:

KIC =

EGIC

(3)

where E is the plane strain Young’s modulus and GIC is the strain energy release rate, which is obtained by deriving the crack energy, UC , in terms of crack area, Ac :

GIC = dUc / dAc

(4)

where Uc is the crack energy and can be expressed as follows:

Uc = Ut −Us

(5)

where Ut is the input energy and Us is the strain energy. In PFC, the input energy and the energy stored in the form of strain energy can be recorded and output every few steps. In PFC 2D, the fractured area is as follows:

Ac = Nc DL

(6)

where Nc is the number of fractured particles, D is the average diameter of the assemblage, and L is the unit depth. Accordingly, the number of fractured particles can be recorded every few steps. By compressing a Brazilian disc with a thoroughgoing crack (Fig. 2(b)), the strain energy release rate can be obtained based on the recorded fractured area and crack energy. The micro- and macroparameters are listed in Table 1. To simulate the indentation process, a parallel bonded model (Fig. 3) with a width and height of 70 and 200 mm, respectively, was

Fig. 1. Parallel bonded particles.

2

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(a)

(b) 25

Axial stress (MPa)

20

15

10

5

0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Fig. 2. The measurement of the uniaxial compressive strength and fracture toughness.

increased with the increase in indentation depth. Then, a peak, P1, was recorded at an indentation depth of 0.4 mm. The crack propagation energy simultaneously increased, indicating that the number of bonds that failed increased with the increase in indentation depth. This inference is validated in Fig. 5, where the dominance of the tensile microcracks over the shear microcracks can be observed. Fig. 4 indicates that the indentation force considerably decreased with a further increase in indentation depth; however, the crack propagation energy remained high as the indentation depth increased to approximately 0.6 mm. This high crack propagation energy was likely due to the large amount of bond breakages per unit indentation depth (Fig. 5). When the indentation depth reached 1.8 mm, a trough in the indentation force, B1, was recorded. The crack propagation energy and the recorded number of microcracks remained relatively low near this trough. When indentation proceeded, the indentation force increased significantly until an indentation depth of 3.4 mm. The crack propagation energy and the number of generated microcracks remained relatively low as the indentation depth increased from 1.8 mm to 3.4 mm. With a further increase in the indentation depth, two peaks and two troughs were observed in the indentation force; each peak was followed by increases and each trough was followed by decreases in both the crack propagation energy and number of generated microcracks. It can be concluded from the above descriptions that decreases in the indentation force are followed by more bond breakages, whereas increases in the indentation force are accompanied by the generation of fewer microcracks. However, the relation between these fluctuations in indentation force and the variations in microcrack initiation remains unclear. Thus, further investigations are to be conducted on crack propagation and stress variation.

generated. This model consisted of 56683 particles and 143089 parallel bonds. Before the indentation test, 3800 measure circles with a radius of 2 mm were installed in the model to monitor the stresses. During the indentation process, the average horizontal, vertical, and shear stresses were recorded at every 200 steps. By writing FISH functions, the stresses in the measurement circle at various locations for specific indentation depths can be dynamically recorded and output. Then, the contours of the shear and maximum principle stresses can be drawn by postprocessing. The upper and lower walls were servo-regulated to apply a confining stress of 1 MPa (Fig. 3). Two indenters, each with a tip width of 13 mm, were displacement-regulated with a constant horizontal velocity of 0.5 mm/s. The cutter shape and the penetration velocity were validated by previous studies [7,9]. According to the reasonable spacing/penetration ratio proposed by Cho et al. [30], the spacing and final penetration were 70 and 5 mm, respectively. During the indentation process, the counts of microcracks resulting from the tensile and shear bond breakages were recorded. In addition, the input and stored energy were recorded. 3. Numerical results and discussion 3.1. Characteristics of the indentation force and crack propagation energy The average penetration force of the two indenters is presented in Fig. 4, where each column represents the crack propagation energy consumed when the indentation depth is 0.06 mm. This energy was obtained from the recorded boundary energy and strain energy stored in particles and parallel bonds. First, the indentation force rapidly Table 1 Micro and macro synthetic parameters. Micro-parameters

Values

Macro parameters

Value

Parameters of the indenter

value

Minimum radius (mm) Rmax/Rmin Particle contact modulus (GPa) The ratio of the normal stiffness to shear stiffness of particles Friction coefficient Parallel bond modulus (GPa) The ratio of the normal stiffness to shear stiffness of parallel bonds Parallel bond normal strength (MPa) Parallel bond shear strength (MPa)

0.24 1.66 9.5 2.5

Uniaxial compression stress, UCS (MPa)

22.3

Normal stiffness (GPa)

5e10

Young’s modulus, E (GPa)

2.1

The ratio of the normal stiffness to shear stiffness of indenters

1

0.5 1. 95 2.5

Poisson ratio

0.24

Friction coefficient

0.1

15.7 15.7

GIC (Nm/m2) KIC (MPa*m1/2)

50.2 0.32

3

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Servo-controlled force

Fig. 3. Indentation model and measurement circles.

70 mm

200 mm

Measure circles

70 mm

Particles

Parallel bonds

Servo-controlled force

whose magnitudes are reflected by their absolute values, are denoted by the blue shear stress contour. At an indentation depth of 0.2 mm, four shear stress concentration zones, where the maximum shear force was 14.4 MPa, formed ahead of the indenters. The plastic zones, which contain microcracks and are denoted by the purple outlines in Fig. 7(a), likely resulted from the concentrated shear stresses. It is noteworthy that tensile microcracks were more dominant than shear microcracks in the approximately triangular plastic zones. This phenomenon can be attributed to the fact that tensile failure occurs under compression [19], and these triangular plastic zones were remarkably consistent with the results of laboratory tests by Liu et al. and Entacher et al. [9,10].

3.2. Crack propagation patterns and stress distribution in the indentation process Parallel bonds break when the stress exceeds their normal or shear strength. To investigate the stress distributions during the indentation process, the maximum principle stress and the shear stress causing tensile and shear breakages were measured and recorded at every 200 steps. Then, the recorded data were output and processed. Fig. 6 depicts the stress evolution during the indentation process; tensile stress concentrations are represented by the positive contour of the maximum principle stress, whereas the shear stress concentrations, 40

Fig. 4. Average indentation force and energy for crack propagation.

Energy for bond breakage Indentation force

P1

P3

30 P2

300000

Energy (J)

25 B2

20

B3

200000 M1

15

B1

10

100000

5 0

0

1

2

3

4

0 5

Pentration depth (mm) 4

Average force of adjacent indenters (N)

400000

35

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Number of total micro cracks Number of tensile micro cracks Number of shear micro cracks

70

Number of micro cracks

60

initiated from the upper tensile stress concentration zone, where the maximum value of the maximum principle stress was 3.4 MPa. When the indentation depth increased to 0.4 mm, the shear and tensile stress concentration zones significantly expanded, as illustrated in Fig. 6(b). The triangular plastic zones simultaneously expanded, and a few cracks were formed in the proximity of the plastic zone. These cracks, denoted by blue lines, consisted of both tensile and shear microcracks; according to the crack classification by Zhang and Wong [31], these cracks were shear-tensile cracks. However, the crack depicted in green was a tensile crack that initiated from the rim of the plastic zone and did not contain shear microcracks. At an indentation depth of 1.8 mm, the shear and tensile stress concentration zones were significantly smaller, as illustrated in Fig. 6(c). These smaller zones likely resulted from the prominent crack propagation, as illustrated in Fig. 7(c). This significant increase in the number of cracks was consistent with the increased crack propagation energy shown in Fig. 4 and the increased number of microcracks shown in Fig. 5 as the indentation depth increased from 0.4 mm to 1.8 mm. When the indentation depth increased to 3.4 mm, the shear and tensile stress concentration zones expanded, as illustrated in Fig. 6(d). However, the increased concentrations of shear and tensile stress exerted a negligible influence on the prominent crack propagation. The cracks, denoted by blue lines in Fig. 7(d), developed near the plastic zones. When the indentation depth increased to 3.6 mm, cracks developed near the plastic zone of the upper indenter and propagated far away from the plastic zone (Fig. 7(e)). With a further increase in the indentation depth, increases and decreases in the tensile and shear

P1

50

40

30

P3

20

10

0

0

1

2

B3

P2 B2

B1 3

4

Indentation depth (mm) Fig. 5. Bond breakages per unit indentation depth.

Simultaneously, four tensile stress concentration zones formed under compression. Two tensile stress concentration zones were coincident with the shear stress concentration zones, whereas the other two tensile stress concentration zones, which formed approximately on the symmetrical axis of the shear zones, did not coincide with the shear zones. Fig. 7(a) indicates that certain tensile micro-cracks in the blue circle

(a)

(c)

(b)

Fig. 6. Stress distributions in the indentation process: (a), (b), (c), (d), (e), (f) and (g) are the stress distributions at Points M1, P1, B1, P2, B2, P3 and B3 on the indentation force curve.

(d)

The maximum

(e)

(f)

principle stress

(g)

5

shear stress

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Fig. 7. Crack propagation in the indentation process: (a), (b), (c), (d), (e), (f) and (g) are the crack distributions at Points M1, P1, B1, P2, B2, P3 and B3 on the indentation force curve; the red and black micro cracks result from shear and tensile failure, respectively.

tip of the crack. However, the maximum shear stress was observed in the plastic zones ahead of the indenters. Thus, it is inferred that the concentrated stresses at the crack tips and in the crushed zone may correlate with the fluctuations in the indentation force. These descriptions of the stress concentrations indicated that the increase in indentation force first resulted in shear or tensile stress concentrations in the crushed zone or at the crack tips. Subsequently, the concentrated stresses resulted in the initiation and propagation of the cracks when

concentration zones were observed, respectively, accompanied by crack propagation both in the vicinity of and far from the plastic zones. Along with the indentation force in Fig.4, Fig. 8 indicates that similar decreases and increases in the maximum values of the shear and tensile stresses were observed. Compared to the stress contours in Fig. 6, the values of the maximum principle stress were approximately equal to the corresponding values in Fig. 8. Thus, during the indentation process, the maximum tensile stress in the model occurred at the 6

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stresses. In the complete propagation process, shear stresses also fluctuated with the tensile stresses (Fig. 10). However, the shear stresses remained significantly low and did not induce shear microcracking; at this stage, the shear stresses were likely to negligibly contribute to crack initiation at the monitor points. From the bond breakages depicted in Fig. 5 and failure patterns depicted in Fig. 7 for the specific indentation depths, it can be concluded that the tensile stress concentrations drove the initiation of tensile cracks. After crack initiation, the tensile stress decreased considerably. In contrast, the tensile stress at the farthest monitor point along the crack trajectory increased significantly. In the laboratory tests conducted by Zhao et al. to investigate the crack propagation characteristics of prefissured specimens, strain jumps occurred when the wing (tensile) cracks initiated [32]. This phenomenon is likely to have resulted from the stress concentration and dissipation at crack tips observed in the present study. It is noteworthy that the critical tensile stress for crack initiation decreased as crack length increased. Assuming that tensile stress acts uniformly on crack surfaces, the stress intensity factor for a mode I crack can be expressed as follows [33]:

KI = σ πa

(7)

where σ is the stress acting on the mode I crack and a is the length of the mode I crack. When the stress intensity factor, KI , is equal to the fracture toughness, KIC , the mode I crack initiates. Thus, with an increase in the length of the mode I crack, the stress driving the initiation of the mode I crack decreases. This conclusion is consistent with the decrease in the tensile stress concentrations after the crack propagates from point A to point C (Fig. 11). Similarly, Liu et al. reported equivalent results for the propagation of tensile cracks initiating from a simulated circular hole [28]. Similar fluctuations in the tensile and shear stresses were observed at other monitor points. Fig. 11 depicts the recorded maximum tensile and shear stresses at the monitor points. Evidently, the maximum shear stresses at the monitor points D, E, F, G, H, and I were significantly higher than those in the mode I crack simulation. Thus, these cracks are mode I-II cracks. Contrary to the decreasing tensile stress concentrations as the tensile cracks propagated, the tensile stress concentrations increased as the mode I-II cracks propagated, whereas the shear stress concentrations decreased. The Griffith energy criterion for a mode I-II crack can be expressed as follows [33]:

Fig. 8. The maximum tensile and shear stress in the indentation process.

the stresses were sufficiently large. Then, the indentation force significantly decreased because the prominent crack propagation weakened the firm contact between the indenter and specimen. Simultaneously, the stress concentrations in the crushed zone or at the crack tip dissipated. When the stress concentration at the crack tip dissipated to a value that was insufficient to cause crack propagation, the crack propagation ceased. Then, with a further increase in the indentation depth, firmer contacts formed between the indenters and specimen; thus, continued indentation resulted in an increase in the indentation force and stress concentrations in the crushed zone and at the crack tip. Then, another cycle, characterized by indentation force fluctuation, concentration and dissipation of the stresses, and propagation and cessation of the cracks, occurred with an increase in indentation depth. Thus, it can be concluded that the increase in indentation depth first caused stress concentrations in the crushed zone and at the crack tip. Then, the intense crack initiation and propagation caused by the stress concentrations resulted in the decrease in the indentation force. Finally, as indentation proceeded, the stress dissipation in the propagation process resulted in the cessation of the propagation and increase in the indentation force.

GC =

KI2 K2 + II E E

KII = τ πa

(8) (9)

where GC is the critical strain release rate and τ is the shear stress acting on the crack. Thus, during propagation of a mode I-II crack, an increase in the crack length and the tensile stress concentration results in an increase in KI and a decrease in KII (Fig. 11). Then, according to the fracture criterion, the shear stress, τ , is predicted to decrease. It can also be inferred from Fig. 11 that tensile stress may dominate when the crack propagates. To investigate the propagation velocities of the various cracks, the indentation depths for crack initiation were measured at the monitor to characterize the crack propagation time (Fig. 12). Then, the lengths between adjacent monitor points were measured. The normalized velocity ratios, V, between two monitor points are illustrated in Fig. 12. For the tensile crack, VAB and VCB were approximately equal. However, for the tensile-shear cracks, apart from VEF, the propagation velocity significantly increased. Zhao et al. proposed that wing and secondary cracks initiate from the tip of a prefabricated crack in laboratory tests [32]; they further proposed that wing cracks, initiated by tensile stress, propagate slowly, whereas secondary cracks, generally propelled by shear or tensile-shear stress, propagate abruptly. Thus, the previous laboratory studies on crack propagation in prefissured specimens and the present numerical study concerning crack propagation due to

3.3. Stress conditions for various cracks The aforementioned analysis indicates that the crack propagating along the indentation axis likely resulted from the tensile stress concentration, whereas the other cracks likely resulted from coincident stress concentrations. To validate this inference, nine monitor points were installed along the crack propagation trajectory, as illustrated in Fig. 9. The initiation of tensile and shear stresses was monitored and plotted in Figs. 10 and 11. Fig. 10 illustrates that the monitor points A, B, and C were initially in compression. When the indentation proceeded, the maximum principle stress significantly increased at point A. Then, the maximum principle stress decreased to approximately 0 MPa. Simultaneously, the increase in the maximum principle stresses at points B and C were significant. As the indentation depth increased to approximately 0.5 mm, the maximum principle stresses at points B and C remained approximately stable. When the indentation depth increased to 3.2 mm, the maximum principle stress at point B increased significantly to approximately 4 MPa and then sharply decreased. With a further increase in the indentation depth to approximately 4.3 mm, an equally sharp increase and decrease were observed in the maximum principle 7

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Fig. 9. Monitor points of the stress condition in the indentation process.

The maximum tensile stress The maximum shear stress

2

2

0

5 5 4

3

4

2 3 1 2

-2

0

0

Monitor points

Indentation depth (mm)

Fig. 11. Concentrated stresses at crack tips for crack initiation.

Fig. 10. Monitored stress conditions of the typical tensile crack.

indenters are consistent.

3.4. Stress evolution in the chipping process In the previous section, two types of cracks, namely, tensile cracks and shear-tensile cracks, were observed. However, no chips formed by lateral crack coalescence during the indentation process. To further analyze the stress characteristics of the crack coalescence process that results in effective chipping between indentations, the confinement is increased to 2.5 MPa and 5 MPa because increased confinement promotes lateral crack coalescence [7,9,12,34]. Similar peaks and troughs in the indentation forces were observed under the confining stresses of 2.5 MPa and 5 MPa (Fig. 13). For a similar indentation depth, the indentation force under a higher confining stress was marginally higher. This increase in indentation force for a higher confining stress was validated by previous laboratory studies [9,34]. The corresponding stress conditions at the peaks and troughs are plotted in Figs. 14 and 15. Similar concentrations of shear and tensile stresses were observed at the peaks. These concentrations were

Fig. 12. Crack propagation velocity.

8

The maximum shear stress (MPa)

The maximum tensile stress (MPa)

4

6

4

6

6

Shear stress (MPa)

The maximum principle stress (MPa)

8

The maximum principle stress at Monitor Point A The maximum principle stress at Monitor Point B The maximum principle stress at Monitor Point C The shear stress at Monitor Point A The shear stress at Monitor Point B The shear stress at Monitor Point C

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coincident tensile and shear stresses. Thus, based on the definition by Zhang and Wong [31], these lateral cracks are shear-tensile cracks. The laboratory indentation tests on sandstone specimens in biaxial states may properly be verified by this conclusion. Fig. 16(a) illustrates the schematic of the indentation tests on a triaxial testing platform [9]. After indentation, the specimens were incised along the plane determined by Points A, B, and C in Fig. 16(a) to investigate the internal crack propagation caused by the indentations. Evidently, the typical internal crack propagation in Fig. 16(b) illustrates that the adjacent plastic zones were connected by a lateral crack. This crack first initiated from the right plastic zone. The white powder in the closed crack in the vicinity of the plastic zone, denoted by the black dashed line, indicated that shear abrasion was likely to occur in the incipient propagation of this crack. As the crack propagated, the crack opened and less abrasion occurred along the crack, indicating that tensile stress is likely to dominate the later stages of propagation. In addition, the abrasive areas enclosed by the red lines were observed in the proximity of the indentations, whereas on the uneven surface in the middle of the crack, less abrasion was observed (Fig. 16(c)). This phenomenon was consistent with the monitored tensile and shear stresses for the mode I-II crack propagation in Fig. 11.

Fig. 13. Average indentation force when the confining stresses are 2.5 and 5 MPa.

responsible for the subsequent crack initiation and propagation. When the indentation force decreased, the number of cracks increased significantly; then, the concentrations of tensile and shear forces dissipated. The stress dissipation ceased when the indentation force reached a trough. When the indentation force further increased, the stress concentrations in the shear zones and at the crack tip increased until another peak was reached. Similar decreases and increases in the concentrated stresses were subsequently observed when the indentation forces decreased and increased with further indentation. It is interesting to note that propagation of the tensile cracks was restrained because of the lesser tensile stress concentration when the confining stress was 2.5 MPa. In addition, when the confining stress was increased to 5 MPa, no concentrated tensile stress formed on the axis of the shear concentration zones. Therefore, no tensile crack formed. This phenomenon was consistent with laboratory tests by Yin et al. [7] and Liu et al. [34]. Furthermore, the #1 and #2 cracks in Fig. 14(f) and the crack connecting adjacent plastic zones in Fig. 15 initiated from the zone of

4. Conclusions In this work, we conducted a numerical investigation on the stress evolution characteristics of the crack initiation, propagation, and coalescence caused by indentations. In numerical simulations, the shear and tensile stresses were recorded by applying measurement circle logic in PFC 2D. We found that the crack propagation correlated with the fluctuations in the indentation force and the stress concentrations at the crack tips. The average indentation force first increased during the early indentation stage. Simultaneously, shear stress and tensile stress concentrations, which were responsible for the crack propagation, increased. Then, a peak in the indentation force at significantly concentrated stresses at the crack tips was observed. With further indentation, the concentration zones contracted because of the propagation of tensile and shear-tensile cracks, and the indentation force

Fig. 14. Stress distributions in the indentation process: (a), (b), (c), (d) and (e) are the stress distributions at Points P1′, B1′, P2′, B2′, and P3′ on the indentation force curve when the confining stress is 2.5 MPa, (f) is the crack distribution at Point P3′.

9

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Fig. 15. Stress distributions in the indentation process: (a), (b), (c), and (d) are the stress distributions at Points P1″, B1″, P2″ and B2″ on the indentation force curve when the confining stress is 5 MPa, (e) is the crack distribution at Point B2″, (f) is the final crack propagation condition.

(a)

(b)

Plastic zones

(c)

Fig. 16. Laboratory indentation specifications and results (a): indentation schematic; (b): internal crack propagation; (c): typical chips.

observation of the internal cracks and the chips, formed by indentations in biaxial states, verified this conclusion. For the tensile cracks, which usually propagated along the indentation direction, the increase in the crack length resulted in a decreasing tensile stress concentration as the crack propagated. For the tensile-shear cracks, the tensile stress increased as the crack propagated, whereas the shear stress decreased.

decreased. Then, with a further increase in the indentation depth, similar fluctuations in the indentation force and stress concentration magnitudes were observed, accompanied with crack propagation and cessation. Numerical analysis indicates that two kinds of cracks, namely, tensile and shear-tensile cracks, were created by the indentations. The 10

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[14] Gong QM, Zhao J, Jiao YY. Numerical modeling of the effects of joint orientation on rock fragmentation by TBM cutters. Tunn Undergr Space Technol 2005;20:183–91. [15] Cao RH, Cao P, Lin H, Pu CZ, Ou K. Mechanical behavior of brittle rock-like specimens with pre-existing fissures under uniaxial loading: experimental studies and particle mechanics approach. Rock Mech Rock Eng 2016;49(3):763–83. [16] Zhou XP, Bi J, Qian QH. Numerical simulation of crack growth and coalescence in rock-like materials containing multiple pre-existing flaws. Rock Mech Rock Eng 2014;48(3):1097–114. [17] Zhang XP, Wong LNY. Crack initiation, propagation and coalescence in rock-like material containing two flaws: a numerical study based on bonded-particle model approach. Rock Mech Rock Eng 2013;46(5):1001–21. [18] Cao P, Liu TY, Pu CZ, Lin H. Crack propagation and coalescence of brittle rock-like specimens with pre-existing cracks in compression. Eng Geol 2015;187:113–21. [19] Wong LNY, Einstein HH. Systematic evaluation of cracking behavior in specimens containing single flaws under uniaxial compression. Int J Rock Mech Min Sci 2009;46(2):239–49. [20] Manouchehrian A, Marji MF. Numerical analysis of confinement effect on crack propagation mechanism from a flaw in a pre-cracked rock under compression. Acta Mech Sin 2012;28(5):1389–97. [21] Wang M, Cao P. Experimental study of Crack growth in rock-like materials containing multiple parallel pre-existing flaws under biaxial compression. Geotech Geol Eng 2017;35(3):1023–34. http://dx.doi.org/10.1007/s10706-017-0158-3. [22] Amann F, Ündül Ö, Kaiser PK. Crack initiation and crack propagation in heterogeneous sulfate-rich clay rocks. Rock Mech Rock Eng 2013;47(5):1849–65. [23] Zhai SF, Zhou XP, Bi J. The effects of joints on rock fragmentation by TBM cutters using General Particle Dynamics. Tunn Undergr Space Technol 2016;57:162–72. [24] Abu Bakar MZ, Gertsch LS, Rostami J. Evaluation of fragments from disc cutting of dry and saturated sandstone. Rock Mech Rock Eng 2014;47(5):1891–903. [25] Jin J, Cao P, Chen Y, Pu CZ, Mao DW, Fan X. Influence of single flaw on the failure process and energy mechanics of rock-like material. Comput Geotech 2017;86:150–62. [26] Moon T, Oh J. A study of optimal rock-cutting conditions for hard rock TBM using the discrete element method. Rock Mech Rock Eng 2012;45:837–49. [27] Moon T, Nakagawa M, Berger J. Measurement of fracture toughness using the distinct element method. Int J Rock Mech Min Sci 2007;44:449–56. [28] Liu T, Lin BQ, Yang W, Zou QL, Kong J, Yan FZ. Cracking process and stress field evolution in specimen containing combined flaw under uniaxial compression. Rock Mech Rock Eng 2016;49:3095–113. [29] Xie YS, Cao P, Jin J, Wang M. Mixed mode fracture analysis of semi-circular bend (SCB) specimen: a numerical study based on extended finite element method. Comput Geotech 2017;82:157–72. [30] Cho JW, Jeon SJ, Ho YC, Soo H. Evaluation of cutting efficiency during TBM disc cutter excavation within a Korean granitic rock using linear-cutting-machine testing and photogrammetric measurement. Tunn Undergr Space Technol 2013;35:37–54. [31] Zhang XP, Wong LNY. Cracking Processes in Rock-Like Material Containing a Single Flaw Under Uniaxial Compression: A Numerical Study Based on Parallel BondedParticle Model Approach. Rock Mech Rock Eng 2012;45:711–37. [32] Zhao YL, Zhang LY, Wang WJ, Pu CZ, Wan W, Tang JZ. Cracking and stress–strain behavior of rock-like material containing two flaws under uniaxial compression. Rock Mech Rock Eng 2016;49(7):2665–87. [33] Li SY, He TM, Yin XC. Introduction of rock fracture mechanics. Press of Chinese University of Science and Technology 2010; Hefei: 157–169 (in Chinese). [34] Liu J, Cao P, Li KH. A study on isotropic rock breaking with TBM cutters under different confining stresses. Geotech Geol Eng 2015;33(6):1–16.

The stress evolutions for these cracks can be accurately verified based on linear fracture mechanics theory. In addition, the results indicated that the shear-tensile cracks were responsible for the chipping between indentations. Acknowledgments The authors would like to express their gratitude to the reviewers and editors for their hard work and beneficial comments on this study. Thanks to Professor Hang Lin for the suggestions during revision. The authors would like to acknowledge the financial support from the following groups: the China Postdoctoral Science Foundation (2017M612557), the Open Fund of the Safe Coal Mining Techniques of the Hunan University of Science and Technology (E21731) and the National Natural Science Foundation of China (51774131, 51774132). References [1] Paul B, Sikarshie DL. A preliminary theory of static penetration by a rigid wedge into a brittle material. Tran Soc Min Engrs 1965:232, 372–383. [2] Miller MH, Sikarshie DL. On the penetration of rock by three-dimensional indentors. Int J Rock Mech Min Sci 1968;5:375–98. [3] Labra C, Rojek J, Oñate E. Discrete/finite element modelling of rock cutting with a TBM disc cutter. Rock Mech Rock Eng 2017;50(3):621–38. [4] Rostami J. Study of pressure distribution within the crushed zone in the contact area between rock and disc cutters. Int J Rock Mech Min Sci 2013;57:172–86. [5] Alehoseein H, Detournay E, Huang H. An analytical model for the indentation of rocks by blunt tools. Rock Mech Rock Eng 2000;33(4):267–84. [6] Chen LH, Labuz JF. Indentation of rock by wedge-shaped tools. Int J Rock Mech Min Sci 2006;43(7):1023–33. [7] Yin LJ, Gong QM, Ma HS, Zhao J, Zhao XB. Use of indentation tests to study the influence of confining stress on rock fragmentation by a TBM cutter. Int J Rock Mech Min Sci 2014;72:261–76. [8] Li XF, Li HB, Liu YQ, Zhou QC, Xia X. Numerical simulation of rock fragmentation mechanisms subject to wedge penetration for TBMs. Tunn Undergr Space Technol 2016;53:96–108. [9] Liu J, Cao P, Han DY. Sequential indentation tests to investigate the influence of confining stress on rock breakage by tunnel boring machine cutter in a biaxial state. Rock Mech Rock Eng 2016;49(4):1479–95. [10] Entacher M, Winter G, Galler R. Cutter force measurement on tunnel boring machines – implementation at Koralm tunnel. Tunn Undergr Space Technol 2013;38:487–96. [11] Huang H, Damjanal B, Detournay E. Normal wedge indentation in rocks with lateral confinement. Rock Mech Rock Eng 1998;31(2):81–94. [12] Ma HS, Yin LJ, Ji HG. Numerical study of the effect of confining stress on rock fragmentation by TBM cutter. Int J Rock Mech Min Sci 2011;48:1021–33. [13] Liu HY, Kou SQ, Lindqvist PA, Tang CA. Numerical simulation of the rock fragmentation process induced by indenters. Int J Rock Mech Min Sci 2002;39(4):491–505.

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