J. Cent. South Univ. (2014) 21: 3302−3308 DOI: 10.1007/s11771-014-2303-y

Numerical simulation on effects of embedded crack on rock fragmentation by a tunnel boring machine cutter

LIU Jie(刘杰)1, CAO Ping(曹平)1, JIANG Zhe(蒋喆)1, ZHAO Yan-lin(赵延林)2, CAO Ri-hong(曹日红)1

1. School of Resources and Safety Engineering, Central South University, Changsha 410083, China;

2. School of Energy and Safety Engineering, Hunan University of Science and Technology, Xiangtan 411201, China

© Central South University Press and Springer-Verlag Berlin Heidelberg 2014

Abstract: Based on the simplification of cutting process, a series of numerical simulations were conducted using a 2-D discrete element method to explore the effects of embedded cracks with different dip angles on the rock fragmentation process, cutting characteristics and breaking efficiency. The results show that the simulated results are in a good agreement with previous theoretical study. The main crack propagates to the top tip of embedded crack, except when the dip angle is 90°. Side cracks which are more fully developed in the rocks containing embedded cracks tend to propagate towards the free surface. According to the history of vertical cutting force, it is shown that the peak force is decreased by embedded cracks. The study on cutting efficiency was conducted by combining the quantity of crack and cutting energy. And the results show that breaking efficiency can be treated as a decreasing or a increasing function when the dip angle is less or larger than 30°, respectively. Breaking efficiency is higher than that in intact rock when the dip angle is larger than 45°. Key words: numerical simulation; embedded crack; rock fragmentation; breaking efficiency

1 Introduction

Nowadays, tunnel boring machines (TBMs) are widely utilized in mining and civil engineering [1]. In the past years, many researches were conducted on the rock fragmentation by TBM cutters. According to their different focuses on rocks, they can be briefly divided into two categories. The researches which are focused on external factors of rocks are included in the first category. In this category of researches, the studies on driving system of cutters [2], efficient cutting [3], effect of confining stress on cutting process [4−5] and loading method [6] are conducted. The other category of studies is focused on internal factors of rocks. In 1976, the influence of the orientation of joints was studied by NTUN via observations [7]. In 2013, study on the effects of quality of the rock on penetration rate was conducted [8]. Recently, numerical study was applied by GONG et al [9] and MA and JI [10] to investigate the effects of the orientation of joints on rock breaking systematically. In practical conditions, many defects such as cracks may not extend to the free surface of rocks, but the joints included in the studies mentioned above extend to the free surfaces of the rock. Few studies were conducted on the effects of embedded cracks on rock breaking by

TBM cutters. Furthermore, experiments conducted by LI et al [11] showed that the induced cracks propagate along the direction that the load acts. But, the loads mentioned in their studies behave as surface force, and the force generated by TBM cutters acts at a very small area which can be treated as concentrated force. In the field of fracture mechanics, few studies are conducted on the effects of concentrated loads on the propagation of cracks.

In this work, the simulation of the crack propagation of the rock specimen containing one embedded crack with weak fillings under the concentrated load exerted by a TBM cutter is conducted. A 2-D discrete element method code, particle flow code (PFC), is applied in this simulation. 2 Cavity model of single blunt cutter

Nowadays, blunt cutters are wildly equipped in

TBMs. The cavity expansion model [12] of single blunt cutter (see Fig. 1) shows that a core is formed by crushed rock beneath the cutter. The sphere space that is adjacent to the core called plastic zone is composed of damaged but relatively intact rock, and the other space is called elastic zone in which flaws exist. Based on the assumption that rock acts as an elastic-perfectly plastic

Foundation item: Project(2013CB035401) supported by the National Basic Research Program of China; Project(51174228) supported by the National

Natural Science Foundation of China; Project(71380100003) supported by Hunan Provincial Innovation Foundation for Postgraduate, China

Received date: 2013−04−07; Accepted date: 2013−06−25 Corresponding author: CAO Ping, Professor; Tel: +86−13973128263; E-mail: pcao_csu@sina.com

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Fig. 1 Cavity expansion model for a blunt cutter

material that obeys the Mohr-Coulomb yield criterion (F=0) and the non-associated Mohr-Coulomb potential (G=0), the radius of plastic zone * formed before the brittle fracture is given by [12]

1/)1(*

/1*

ppd)1(KKK

(1) where is a number characterizing the tool properties and the problem dimension:

tan

π

2 (2)

The variables introduced in above equation are

defined as follows:

)1( p

KG

q

dp

p

KK

K

p

dpdp

)1(2

)1)(1)(21()1)(1(

K

KKKK

sin1

sin1p

K

sin1

sin1d

K

where is the friction angle, is the dilatancy angle and q is the uniaxial compressive strength.

The cracks induced in the elastic zone obey the Griffith theory which is characterized by brittle fracture. The crack will continue to propagate when the resistance exerted by rock is conquered by the elastic energy released by the crack propagation. Based on energy analysis, the relation between the fracture strength and crack size can be written as [13]

)1(π

22f

a

E (3)

where f is the fracture stress; E is the elastic modulus; is the poison ratio; is the surface energy; a is the crack size.

Combined with the radius of plastic zone shown above with Griffith theory, the length of crack after brittle fracture [14] can be written by

*p22/1

2/11

*

)]1()[(

tan)(2

Kml

lmd

(4)

2/1Ic )(

q

Kl (5)

where *d is the critical penetration depth; m1 and m2 are coefficients dependent on loading and geometry; IcK is the coefficient dependent on crack roughness. 3 PFC modeling

PFC is a discontinuous code proposed by CUNDALL and STRACK. Compared with other finite element methods, in the simulation process of PFC, both the micro mechanic properties and macro constitutive properties can be studied more directly. So, in recent years, PFC is extensively used to conduct researches on micro and macro properties of rocks and soils. The process of PFC simulation of rock breaking by TBM is composed of the generation of particles, the determination of boundary conditions and the loading of the cutters. 3.1 Simplification of cutting process

In the cutting process by a TBM cutter, the forces generated by the cutter composed of rolling force, side force and normal force are shown in Fig. 2. According to previous studies [2], it is concluded that grind breaking is caused by rolling force, while the propagation of cracks is dominated by normal force. So, in order to investigate the propagation of cracks, it is feasible to conduct study on the rock fragmentation process by a TBM cutter in a plane condition. Due to the difference of stiffness between cutters and rocks, the cutter can be treated as rigid walls. The edge of the cutter is 10 mm in length and the edge angle is 20°. As shown in Fig. 3, the top surface of the model is free while the other three surfaces are confined by rigid walls. It is shown in the schematic diagram of indentation model (see Fig. 3) that a rock specimen containing 30124 particles with 200 mm in length and 100 mm in height is generated. And a band of particles with 4 cm in length and at least 3-particle width are created to represent the embedded crack containing weak fillings.

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Fig. 2 Forces generated by TBM cutters

Fig. 3 Cutter indentation model

3.2 Calibration of rock specimen

In general, in order to get a good agreement with practical rock properties, mechanical parameters of particles should be modified in PFC simulation. After the calibration process by conducting biaxial compress test and Brazilian test in PFC, the properties of particles are listed in Table 1, and the corresponding properties of practical rock specimen are listed in Table 2. The corresponding properties of weak fillings are also listed Table 1 Properties of particles and weak fillings

Parameter Particle Weak filling

Friction coefficient 0.3 0.1

kn/(N·m−1) 9.5×108 1×108

kn/ ks 2.5 1

σn/MPa 14 0.02

σs/MPa 16 0.02

Table 2 Corresponding properties of intact rock specimen

Property Value

Bulk density/(kg·m−3) 2450

Uniaxial compressive strength/MPa 20.92

Poison ratio 0.23

Elastic modulus/GPa 1.26

in Table 1. In this work, the effects of dip angle on the propagation by PFC are focused on. 4 Results and analysis 4.1 Rock fragmentation at different dip angles of

embedded crack Figure 4 shows the fragmentation by the cutter

when the dip angle is 45° at the penetration depth of 9 mm. In this model, the initiation angle is defined as the angle between vertical direction and the direction determined by the crack initiation position and the former middle point of rock surface beneath the cutter, while the deflection angle is defined as the angle between vertical direction and the direction determined by the initiation position and the end of the crack. The main study is conducted on two cracks called main crack and side crack (see Fig. 4) in this work. As shown in Figs. 5−10, in order to observe the initiation and propagation of cracks directly at different penetration depths under the effects of different dip angles of embedded cracks, the zone beneath the cutter is zoomed in.

Fig. 4 Rock fragmentation at 45 with penetration depth

of 9 mm

Fig. 5 Rock fragmentation at selected penetration depth in intact rock (Red lines denote tensile failures; Black ones denote

compressive failure): (a) 1.5 mm; (b) 2 mm; (c) 3 mm; (d) 6 mm; (e) 9 mm; (f) 11 mm

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Fig. 6 Rock fragmentation at selected penetration depth at 90 (Red lines denote tensile failures; Black ones denote

compressive failure): (a) 1.5 mm; (b) 3 mm; (c) 6 mm; (d) 9 mm; (e) 11 mm

Fig. 7 Rock fragmentation at selected penetration depth at 60 (Red lines denote tensile failures; Black ones denote compressive

failure): (a) 1.5 mm; (b) 3 mm; (c) 6 mm; (d) 9 mm; (e) 11mm

Fig. 8 Rock fragmentation at selected penetration depth at α=45° (Red lines denote tensile failures; Black ones denote compressive

failure): (a) 1.5 mm; (b) 3.5 mm; (c) 6 mm; (d) 11 mm

Fig. 9 Rock fragmentation at selected penetration depth at α=30° (Red lines denote tensile failures; Black ones denote compressive

failure): (a) 1.5 mm; (b) 4 mm; (c) 9 mm; (d) 11 mm

As shown in Figs. 5(a), 6(a), 7(a), 8(a), 9(a) and 10(a), crushed zones with tensile and compressive failures are formed at penetration depth of 1.5 mm. Immediately beneath the cutter is the zone of compressive failure. The further the zone is, the more the

tensile failures exist. As the penetration increases, the so- called main crack formed by tensile failures exists and propagates (see Figs. 5(b), 5(c), 6(b), 9(b) and 10(b)). Due to the effects of embedded cracks, it is interesting to note that the main crack propagates to the top tip of the

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Fig. 10 Rock fragmentation at selected penetration depth at α=0° (Red lines denote tensile failures; Black ones denote compressive

failure): (a) 1.5 mm; (b) 5 mm; (c) 9 mm; (d) 11 mm embedded crack except when the dip angle is 90° (see Figs. 7(b), 8(b), 9(b) and 10(b)). But it is shown in Figs. 6(d) and 6(e) that another crack which does not exist in other models containing different embedded cracks initiates at the edge of plastic zone and propagates to the top tip of embedded crack. As shown in Figs. 5(d), 5(e), 5(f), 6(c), 6(d), 6(e), 7(c), 7(d) and 7(e), side cracks exist and propagate rapidly along the tensile failure element while the main crack propagates slowly. Side cracks tend to propagate upwards to the free surface, and the side cracks are more fully developed in rocks containing embedded cracks (see Figs. 5(f), 6(e), 7(e), 8(d), 9(d) and 10(d)). At the early stage of crack propagation, the plastic zones beneath the cutter expand as the penetration increases, but when the depth reaches to 9 mm, the radius of the crushed zones becomes stable (see Figs. 5(e), 5(f), 6(d), 6(e), 7(d), 7(e), 8(c), 8(d), 9(c), 9(d), 10(c) and 10(d)).

The initiation angle , the deflection angle and the length are the main factors of main crack. The corresponding data in intact rock and rocks containing different cracks with different dip angles at penetration depth of 11 mm are given in Table 3. As shown in Table 3, the initiation angle is affected by the embedded crack, but little influence is exerted on the initiation angle by the dip angle among the rocks containing different cracks with different dip angles. Table 3 Main factors of main crack at penetration depth of 11 mm

Rock θ/(°) ο/(°) Length/mm

Intact rock 14.8 21.2 31.5

α=90° 23.3 29.6 36.9

α=60° 22.7 29.3 23.2

α=45° 21.9 32.5 21.4

α=30° 22.3 33.2 24.9

α=0° 22.1 27.7 38.8

It is worth noting that the main crack continues to propagate after connecting to the top tip of embedded crack (see Figs. 7(c), 7(d) and 7(e)) when the dip angle is 60°, but compared with the length of main crack in intact rock, because of the ‘prevention’ exerted by the embedded crack, it is shown that the length of the main crack is reduced. As shown in Figs. 8(b), 8(c), 8(d), 9(b) 9(c), 9(d), 10(c) and 10(d), the main crack which propagates to the top tip of embedded crack ceases to propagate. Based on the assumption that the main crack propagates approximately linearly and combined with the cavity model of single blunt cutter [12] introduced in Section 2, in this model, after the propagation to the top tip of embedded crack of the main crack, the length of main crack η when the dip angle α is less than 45° can be written as

22* )

)cos(

sin()

)cos(

cos[(

2

1

* )])cos(

sin1)(

)cos(

cos)(sin(2

(6)

where is the vertical length between the center of the embedded crack and the rock surface, * is the radius of plastic zone and is the length of the embedded crack.

It can be concluded from above analysis that the rock fragmentation is composed of compressive failure and tensile failure. A good agreement on the distribution of the crushed core, plastic zone and elastic zone between the simulation and theory study [12, 15] is made. The main crack propagates to the top tip of embedded crack except when the dip angle is 90°. The initiation and the propagation of main crack are affected by the embedded crack. Side cracks are more fully developed under the effects of embedded cracks. And the side cracks tend to propagate towards the free surface of the rock.

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4.2 Effects of embedded cracks on cutting characteristics and breaking efficiency

4.2.1 Effects of embedded cracks on vertical cutting force

It is introduced above that the vertical cutting force is a critical factor affecting the propagation of cracks. Relation curves between penetration depth and vertical cutting force at different dip angles of embedded crack is shown in Fig. 11. At the early stage of cutting process, the vertical force of cutter increases linearly as the penetration depth accumulates, while in the later period of cutting process, the vertical force fluctuates in a small range. Based on the study of TAN et al [15] and the characteristics of jump crushing, it can be obtained from Fig. 11 that the jump and drop of vertical force of cutter are resulted from the cycle composed of the crushing and burst crack in the rock. The force increases linearly as the cutter penetrates, but when the penetration depth reaches to a certain degree, the burst cracks initiate at the frontier of the cutter, and the force decreases immediately when the burst cracks initiate. The fluctuation in small range is caused by the cuttings generated at the frontier of the cutter. The contact area increases as the burst cracks accumulate. When the contact area equals the total area of the cutter section, the cracks burst in larger scale. So, the jump crushing is composed of the cycles containing several crack bursts in small scale and a burst in large scale.

Fig. 11 Relation between penetration depth and vertical cutting

force at different dip angles

It is worth noting that the maximum value of

vertical cutting force exists at penetration depth of 6 mm in intact rock. And in the later period of cutting process, the vertical force of the cutter in intact rock is higher than that in rocks containing embedded cracks. So, it can be concluded that the peak vertical force in the cutting force is decreased by embedded cracks.

4.2.2 Effects of embedded cracks on crack numbers and cutting energy

Damage of rock is a deterioration process caused by the micro defects. In the deterioration process, more defects such as micro cracks are formed. The number of the crack is a critical index of the degree of rock damage. As shown in Fig. 12, little difference is shown in the number of compressive cracks at different dip angles. According to the cavity model for a blunt cutter, compressive failures are formed within the plastic zone. Combined with above analysis, the little difference can show that the little effects are exerted by the dip angle on the distribution of plastic zone from another perspective. But, the number of tensile cracks varies greatly. When the dip angle is less than 30°, the number of tensile cracks formed at penetration depth of 11 mm decreases as the angle increases. While the number increases as the angle increases when the angle is larger than 30° at the penetration depth of 11 mm.

Fig. 12 Number of cracks formed at penetration depth of

11 mm

In the study of TAN et al [15], the ratio of the energy to the chipping volume is introduced to determine the optimum space between two cutters. Compared with their study, because single cutter is analyzed in this model, no chipping volume is formed, so the ratio of cutting energy to crack number is proposed to conduct the study on the breaking efficiency. The cutting energy is recorded in the simulation process, and the corresponding cutting energy at penetration depth of 11 mm is shown in Fig. 13.

It can be obtained from Fig. 13 that the cutting energy consumed in intact rock at penetration depth of 11 m is the maximum, but the average energy consumed per crack is not the maximum in the intact rock. In the rocks containing embedded cracks, the tendency of cutting energy is opposite to that of total crack number in Fig. 12. Based on above results, it can be proposed by the ratio of cutting energy to crack number that the breaking efficiency decreases as the dip angle increases when the angle is less than 30°. And in the dip span

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Fig. 13 Cutting energy and average energy consumed by a

crack at penetration depth of 11 mm

between 30° and 90°, the breaking efficiency is a increasing function. When the dip angle reaches to 45°, the breaking efficiency is close to that in intact rock. This means that the breaking efficiency is higher than that in intact rock when the dip angle is larger than 45°.

It is interesting to note that the results obtained above make a part agreement with the conclusions obtained by BARTON [16] that the breaking efficiency increases as the dip angle of joints increases. 5 Conclusions

1) The simulated distribution of crushed zone, plastic zone and the initiation and propagation of cracks in intact rock is in a good agreement with the cavity model for blunt cutter. The feasibility of numerical simulation is verified. It is known from above analysis that the rock fragmentation is composed of compressive failure and tensile failure.

2) The main crack propagates to the top tip of embedded crack, except when the dip angle is 90°. When the dip angle α is less than 45°, the main crack ceases to propagate. And the length of main crack is reduced by embedded crack when α is 60°. Side cracks tend to propagate towards the free surface, and the side cracks are more fully developed in rocks containing embedded cracks.

3) The jumps and drops of vertical cutting force are resulted from the crack bursts in small and large scales. Peak vertical cutting force is decreased by embedded cracks.

4) Little difference exists in the number of compressive cracks at different dip angles at penetration depth of 11 mm. But, the number of tensile cracks varies greatly. When the dip angle is less than 30°, the number of tensile cracks formed at penetration depth of 11 mm decreases as the dip angle increases. While the number is

an increasing function when the angle is larger than 30°. 5) Breaking efficiency can be treated as a

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(Edited by YANG Bing)