- Email: [email protected]
Cellular Automata Simulation Of The Interaction Mechanism Of Two Cracks In Rock Under Uniaxial Compression
Paper 2B 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
CELLULAR AUTOMATA SIMULATION OF THE INTERACTION MECHANISM OF TWO CRACKS IN ROCK UNDER UNIAXIAL COMPRESSION Ming-Tian Li, Xia-Ting Feng, Hui Zhou Institute of Rock and Soil Mechanics, the Chinese Academy of Sciences, Wuhan, 430071, China [email protected]
Abstract: Crack coalescence, which is responsible for some engineering disasters such as rock burst, and so on, is studied by one new numerical method-lattice cellular automata. A cellular automaton is an efficient method that can simulate the process of self-organization of the complex system by constructing some local simple rules. It has advantages of locality, parallelization etc. Lattice model can transfer a complex triaxial problem into a simple uniaxial problem. In the meantime, it can consider the heterogeneity of the materials. Lattice cellular automata are presented which have the advantages of both cellular automata and lattice model. In this paper the experimental and numerical results on rock failure, fundamentals and applications of cellular automata are introduced briefly firstly. In the second part lattice cellular automata is presented and its basic thoughts are introduced. Then interaction mechanisms of two cracks with different geometries are studied. The numerical simulation results are in good accordance with the experimental results. Keywords: Cellular automata, lattice model, lattice cellular automata, rock, failure, crack coalescence
1. INTRODUCTION There are cracks with all scales in the natural rock mass. The failure of the rock mass is interrelated with the formation, extension and coalescence of the cracks. The coalescence of cracks is responsible for some engineering disasters such as rock burst, and so on. Mechanism of rock failure is the base to solve, control these engineering disasters. However, because of the complexity of the coalescence of cracks though extensive research has been done on the mechanism of rock failure, Bobet and Einstein (1998), Wong et al. (2001) pointed out that the fundamental mechanisms of crack coalescence, especially coalescence for non-overlapping geometries under uniaxial compression were not well understood. The studies of coalescence mechanisms of cracks need the collective advances of experimental methods, theory analysis and numerical simulation. This paper concentrates on the interaction mechanisms of two cracks with different rock bridge angle under uniaxial compression. Extensive experiments have been done on crack coalescence with different materials. Horri and Nemat-Nasser (1985) studied the influence of the length of pre-existing crack on interaction of cracks under uniaxial and biaxial compression on CR39. Reyes and Einstein (1991) studied the interaction of two crack in gypsum. He concluded that the
pattern of crack coalescence of two cracks was controlled by the initial geometry of two cracks. Shen (1995) studied crack coalescence under shear stress and in compression on gypsum. Bobet and Einstein (1998) studied the crack coalescence of two pre-existing cracks in gypsum under uniaxial and biaxial compression. The results indicated that the pattern of crack coalescence depends on the geometry of the cracks. Wong et al. (1998,2001) studied the crack coalescence on rock-like material, which is a mixture of barite, sand, plaster and water with a mass ratio of 2:4:1:1.5. The influence of the frictional coefficient on the crack surface on the crack coalescence was considered in this study. These experimental results provide us a direct approach to understand interaction mechanisms of cracks. As examples, the typical experimental results are shown in Figure 1. Experiments and numerical simulation is a whole that cannot be divided. On one hand, numerical simulation must be based on the experiments, its mechanical parameters must come from the experiments and the results of the numerical simulation must be verified by the experiments. On the other hand, numerical simulation can help us understand the fundamental mechanisms of experimental phenomena. With the development of computer techniques and theory analysis more and more numerical methods are used to study the crack coalescence. On finite
1
Paper 2B 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
element methods discrete crack model (Hillerborg, 1976) and smeared crack model (Ba ant, 1983) were presented to simulate the fracture process of rock-like materials. Boundary element method (Shen, 1995) is successfully used to simulate the extension, coalescence of cracks. Tang et al. (2001) develop numerical simulation software called as RFPA2D based on meso-failure mechanics to simulate the fracture process and crack coalescence of the rock. Lattice model is introduced to simulate the fracture process of rock-like materials such as concrete. Lattice model does not discretize the continuum equation, but discretizes the continuum into pole system. Murat et al. (1992) proved that lattice model corresponded to a discretization of the continuum equation. It can transfer a complex triaxial problem into a simple uniaxial problem.
(a)
(c)
(b)
Based on the failure mechanics, lattice cellular automata (LCA) are presented to simulate the fracture process, crack coalescence of rock. This model has the advantages of both cellular automata and lattice model. The structure of the lattice cellular automata is shown in Figure 1. The lattice grid point shows the cell, which connects with its neighbours with beams that have three freedom degrees; the red region around the cell shows the influencing region of this cell. The states set of cell can be expressed as below,
φi = {{u , v ,ϕ}, { f x , f y , m}}i (1) The local rules, which are used to update the states of the cell, may be attained according to the equilibrium equations of each cell, the deformation equations and constitute equations. In order to simulate the fracture process of rock we must introduce some fracture laws to judge whether the beams starts to fail or not. Here two types of failure, tension failure and shearing failure, will be considered. To judge tension failure, the maximum tensional strain criteion is adopted and it can be expressed as below,
ε ≤ εt 0
Shear crack Wing crack Wing crack
(d)
2. LATTICE CELLULAR AUTOMATA
(e)
Figure 1. Typical experimental results, (a),(b),(c) from Bobet and Einstein (1998), (d) and (e) from Wong et al.(1998) Cellular automata were presented by Von Neumann in 1950’s to simulate self-organization among biological cells. Cellular automaton is made up of cell, cell lattice, neighbour and rules. The state of one cell is decided only by the states at the previous step of itself and its neighbours. Cellular automata have the advantages of time, space discretization, locality and parallelization and so on. These years, with the rapid development of computer techniques, cellular automata have been widely used in fluid mechanics (Wolframe, 1986), earthquake (Chen, Bak, 1991), solid mechanics (Tatting, Gurdal, 2000) etc. Based on the scalar cellular automata—physical cellular automata Zhou et al. (2002) simulate the rock fracture.
(2)
Where ε is the strain of the beams, εt 0 is the strain corresponding with the uniaxial tensional strength. Here the ultimate tensional strain εu is introduced to judge the fully tensional fracture. To judge shearing failure, Mohr-Coulomb criterion is adopted and it can be expressed as below,
F=
1 + sin θ σ1 − σ 3 ≥ f c 1 − sin θ
(3)
Where θ is frictional angle of the meso-element; f c is uniaxial compression strength of mesoelement; σ1 and σ3 are the major principal stress and minor principal stress respectively.
cell
Figure 2. Sketch map of cell structure
2
Paper 2B 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
So the basic thoughts of lattice cellular automata to simulate rock failure and crack coalescence can be expressed as below, 1. to discretize the rock into equivalent lattice model; 2. to introduce the heterogeneity of the rock by generating the mechanical parameters such as strength, elastic modulus etc. for each pole; 3. to update the states of all the beams according to local rules; 4. to judge whether the beams fail or not according to fracture law; 5. to perform failure analysis to the failed beams with failure mechanics so that we can analyse the strain-softening, simulate the discontinuity caused by fracture based on the thought of lattice cellular automata.
3. NUMERICAL SIMULATION AND DISCUSSION This paper mainly studies the influences of the geometry of two cracks on the interaction mechanisms. Several samples with two cracks of different geometries (shown in Figure 3) will be studied.
The fracture process and stress-strain curve of the sample with two overlapping cracks as shown in Figure 3(a) are shown in Figure 4. In this paper the unit of stress in all stress-strain curves is MPa. And its corresponding shearing failures are shown in Figure 5. The rock bridge angle is about 135 degree.
(a) Step 40
(b) Step 48
(c) Step 59 20 16 12 8 4 0
c d a 0
(d) Step 64
(e) Step 80
e
b 0.001
0.002
Stress-strain curve
Figure 4. Fracture process of Figure 3(a)
40mm
α β h
40mm
g
(a) Step 40
(b) Step 48
(c) Step 59
(a) g=8mm,h=4mm, α=45°,β=135°
Structure of sample a
(d) Step 64
(e) Step 80
Figure 5. Shearing failures of Figure 3(a) (b) g=8mm,h=4mm,
(c) g=8mm,h=4mm,
α=45°,β=90°
α=45°,β=63°
(d) g=8mm,h=4mm,
(e) g=8mm,h=2mm,
α=45°,β=53°
α=45°,β=45°
Figure 3. Rock samples with two cracks of different geometries
According to Figures 4 and 5, at the beginning of loading wing tensile cracks initiate at crack tips. Secondary shearing cracks propagate parallel with the pre-existing cracks. With the development of loading the wing tensile cracks initiating at the inner tip of two cracks join in the bridge area at step 70 as shown in Figure 4(d). Shearing failures of cells in the rock bridge area do not occur as shown in Figure 5. Therefore, the mode of the coalescence of the overlapping cracks should belong to tension mode. This numerical result is consistent with the experimental result shown in Figure 1(e) from Wong et al. (1998).
3
Paper 2B 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
The fracture process and stress-strain curve of sample as shown in Figure 3(b) is shown in Figure 6. Its corresponding shearing failures are shown in Figure 7. The rock bridge angle β of two cracks is 90 degree.
(a) Step 23
(b) Step 28
(c) Step 34
16 12 8 4 0
c d b a 0
(d) Step 43
(e) Step 69
0.001 0.002
Stress-strain curve
(d) Step 43
(b) Step 28
(c) Step 34
(e) Step 69
(c) Step 60
(b) Step 50
(a) Step 33
20 15 10 5 0
e
Figure 6. Fracture process of Figure 3(b)
(a) Step 23
The fracture process and stress-strain curve of sample as shown in Figure 3(c) is shown in Figure 8. The corresponding shearing failures are shown in Figure 9. The two cracks are non-overlapping cracks. The rock bridge angle is about 63 degree.
a
b
0
(d) Step 77
(e) Step 84
c d e
0.001
0.002
Stress-strain curve
Figure 8. Fracture process of Figure 3(c)
(a) Step 33
(d) Step 77
(b) Step 50
(c) Step 60
(e) Step 84
Figure 7. Shearing failures of Figure 3(b)
Figure 9. Shearing failures of Figure 3(c)
From Figures 6 and 7, wing tensile cracks also emanate firstly at the tips of the pre-existing cracks and they have a bigger angle with pre-existing cracks. Secondary shearing cracks appear in uncertain position but most of them are parallel with the pre-existing crack or have a smaller angle with two pre-existing cracks. At step 43 the wing tensile cracks join in bridge area (shown in Figure 6(d)). From Figure 7 we can obviously find that the shearing failures do not occur at the bridge area during the fracure. So we can conclude the coalescence mode of non-overlapping cracks with β=90° belongs to tension mode. The experimental result as shown in Figure 1(c) from Bobet and Einstein (1998) also supports this result of numerical simulation.
When the two cracks are arranged as shown in Figure 3(c). At the beginning of loading the wing tensile cracks also appear at the tips of the two preexisting cracks firstly. With the development of loading the wing cracks initiating from inner tips of two pre-existing cracks cannot join. At step 84 a new wing tensile crack and new shearing failures join together (as shown in Figure 8(e)). From Figure 9 we can find obviously in the joining area several shearing failures occur at step 84 (shown in Figure 9(e)). So we can conclude that the mode of the coalescence of two cracks with this geometry belongs to mixed mode of tension and shearing. The experimental result from Bobet and Einstein (1998) as shown in Figure 1(b) also supports the numerical simulation result.
4
Paper 2B 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
The fracture process and stress-strain curve of sample as shown in Figure 3(d) is shown in Figure 10, the corresponding shearing failures are shown in Figure 11. The two cracks as shown in Figure 3(d) are non-overlapping cracks and its rock bridge angle is about 53 degree.
(b) Step 25
(a) Step 20
(c) Step 50
20
ab
10
c
d
two cracks with such geometry belongs to mixed mode of tension and shearing. The fracture process and stress-strain curve of sample shown in Figure 3(e) is shown in Figure 12. The corresponding shearing failures are shown in Figure 13. The two cracks are cracks that are in a line. And its rock bridge angle is 45 degree.
(a) Step 31
40 30 20 10 0
e
0 0
(d) Step 66
(e) Step 75
0.001
0.002
Stress-strain curve
Figure 10. Fracture process of Figure 3(d)
(a) Step 20
(d) Step 66
(b) Step 25
(c) Step 50
(e) Step 75
(c) Step 70
(b) Step 50
c d a 0
(d) Step 99
(e) Step 120
e
b 0.001 0.002 0.003
Stress-strain curve
Figure 12. Fracture process of Figure 3(e)
(a) Step 31
(d) Step 99
(b) Step 50
(c) Step 70
(e) Step 120
Figure 11. Shearing failures of Figure 3(d)
Figure 13. Shearing failures of Figure 3(e)
Contrasting Figure 10 with Figure 11, we can find that at the crack tips the wing tensile cracks emanate prior to the secondary shearing cracks. The wing tensile cracks at the inner tips of two preexisting cracks extend with the development of loading. However the two wing tensile crack do not join in bridge area before step 75 (shown in figure 10(e)). At step 75 near the tip of pre-existing crack shearing cracks appear (shown in Figure 11(e)) and the shearing cracks join to the wing tensile crack. The numerical result is in accordance with the experimental result as shown in Figure 1(d) from Wong et al. (1998). The mode of coalescence of
When two pre-existing cracks in a sample are in line, wing tensile cracks firstly appear at the tips of two pre-existing cracks under uniaxial compression. Before step 120 the wing tensile cracks extend but they cannot join. The secondary shearing cracks do not emanate in the bridge area. But at step 120, shearing cracks suddenly appear from the inner tips of two pre-existing cracks and join together (shown in Figure 13(e)). The shearing cracks are in plane with the two pre-existing cracks. So we can conclude that the pre-existing cracks, which are in line, converge suddenly by secondary shearing cracks. The mode of
5
Paper 2B 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
coalescence of these two cracks belongs to shearing mode. The experimental result as shown in Figure 1(c) supports this result.
4. CONCLUSIONS Based on the lattice cellular automata the interaction mechanisms of two cracks with different geometries are studied. The numerical simulation results are in good accordance with the experimental results. The conclusion can be drawn as follows, 1. Coalescence mode of two overlapping cracks belongs to tension mode. 2. Coalescence mode of two non-overlapping cracks with β=900 belongs to tension mode. 3. Coalescence mode of two non-overlapping cracks that are not in a line and β<900 belongs to mixed mode of tension and shearing. 4. Coalescence mode of two non-overlapping cracks that are in a line belongs to shearing mode. Tabel 1. Coalescence mode of two cracks with different geometries. Coalescence Geometry of two cracks mode g=8mm,h=4mm, Tension α=45°,β=135° g=8mm,h=4mm,α=45°,β=90° Tension g=8mm,h=4mm,α=45°,β=63° tension+shear g=8mm,h=4mm,α=45°,β=53° tension+shear g=8mm,h=2mm,α=45°,β=45° Shear
5. ACKNOWLEGEMENT This paper is financial supported by the Special Funds for Major State Basic Research Project under Grand no. 2002CB412708 and National Natural Science Foundation of China under Grand no. 50179034.
Chen, K. & Bak, P. 1991 Self-organized criticality in a crack-propagation model of earthquakes. Phy. Rev. A 43(2): pp. 625 – 630. Hilerborg, A. Modeer, M. & Peterson, P.E. 1976. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research. 6: pp 773 – 782. Horii, H. & Nemat-Nasser, S. 1985. Compressioninduced microcrack growth in brittle solids: axial splitting and shear failure. J. Geophys. Res. 90(B4): pp. 3105 – 3125. Murat, M. Anholt, M. & Wagner, H.D. 1992. Fracture Behavior of Short-fiber Reinforced Materials. J. Mater. Res. 7(11): pp 3120 – 3131. Reyes, O. & Einstein, H.H. 1991. Fracture mechanics of fractured rock—A fracture coalescence model. 7th Int. Congress on Rock Mech. pp. 333 – 340. Shen, B. 1995. The mechanism of fracture coalescence in compression experimental study and numerical simulation. Eng. Fract. Mech. 51(1): pp 73 – 85. Tang, C.A. Lin, P. Wong, R.H.C. & Chau, K.T. 2001. Analysis of crack coalescence in rock-like material containing three flaws—Part II: Numerical approach. Int. J. Rock Mech. Min. Sci. 38(7): pp 925 – 936. Tatting T. & Gurdal Z. 2000. Cellular automata for design of two-dimensional continuum structure, 8 th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. Wolfram, S. 1986. Theory and applications of Cellular automata. World Publishing CO. PTE. LTD. Wong, R.H.C. & Chau, K.T. 1998. Crack coalescence in a rock-like material containing two cracks. Int. J. Rock Mech. Min. Sci. 35(2): pp. 147 – 164.
Ba ant, Z.P. & Oh, B.H. 1983 Crack band model for concrete. Material and Structures.16: pp. 155 – 177.
Wong, R.H.C. Chau, K.T. Tang, C.A. & Lin, P. 2001. Analysis of crack coalescence in rock-like material containing three flaws—Part I: experimental approach. Int. J. Rock Mech. Min. Sci. 38(9): pp. 909 – 924.
Bobet, A. & Einstein, H.H. 1998. Fracture coalescence in rock-type materials under uniaxial and biaxial compression. Int. J. Rock Mech. Min. Sci. 35(7): pp. 863 – 888.
Zhou, H. Wang, Y.J. Tan Y.L. & Feng X.T. 2002. Physical cellular automata to simulate rock fracture—part I: fundamental. Chi. J. of Rock Mech. and Eng. 21(4): pp. 475 – 478
6. REFERENCES
6
CELLULAR AUTOMATA SIMULATION OF THE INTERACTION MECHANISM OF TWO CRACKS IN ROCK UNDER UNIAXIAL COMPRESSION Ming-Tian Li, Xia-Ting Feng, Hui Zhou Institute of Rock and Soil Mechanics, the Chinese Academy of Sciences, Wuhan, 430071, China [email protected]
Abstract: Crack coalescence, which is responsible for some engineering disasters such as rock burst, and so on, is studied by one new numerical method-lattice cellular automata. A cellular automaton is an efficient method that can simulate the process of self-organization of the complex system by constructing some local simple rules. It has advantages of locality, parallelization etc. Lattice model can transfer a complex triaxial problem into a simple uniaxial problem. In the meantime, it can consider the heterogeneity of the materials. Lattice cellular automata are presented which have the advantages of both cellular automata and lattice model. In this paper the experimental and numerical results on rock failure, fundamentals and applications of cellular automata are introduced briefly firstly. In the second part lattice cellular automata is presented and its basic thoughts are introduced. Then interaction mechanisms of two cracks with different geometries are studied. The numerical simulation results are in good accordance with the experimental results. Keywords: Cellular automata, lattice model, lattice cellular automata, rock, failure, crack coalescence
1. INTRODUCTION There are cracks with all scales in the natural rock mass. The failure of the rock mass is interrelated with the formation, extension and coalescence of the cracks. The coalescence of cracks is responsible for some engineering disasters such as rock burst, and so on. Mechanism of rock failure is the base to solve, control these engineering disasters. However, because of the complexity of the coalescence of cracks though extensive research has been done on the mechanism of rock failure, Bobet and Einstein (1998), Wong et al. (2001) pointed out that the fundamental mechanisms of crack coalescence, especially coalescence for non-overlapping geometries under uniaxial compression were not well understood. The studies of coalescence mechanisms of cracks need the collective advances of experimental methods, theory analysis and numerical simulation. This paper concentrates on the interaction mechanisms of two cracks with different rock bridge angle under uniaxial compression. Extensive experiments have been done on crack coalescence with different materials. Horri and Nemat-Nasser (1985) studied the influence of the length of pre-existing crack on interaction of cracks under uniaxial and biaxial compression on CR39. Reyes and Einstein (1991) studied the interaction of two crack in gypsum. He concluded that the
pattern of crack coalescence of two cracks was controlled by the initial geometry of two cracks. Shen (1995) studied crack coalescence under shear stress and in compression on gypsum. Bobet and Einstein (1998) studied the crack coalescence of two pre-existing cracks in gypsum under uniaxial and biaxial compression. The results indicated that the pattern of crack coalescence depends on the geometry of the cracks. Wong et al. (1998,2001) studied the crack coalescence on rock-like material, which is a mixture of barite, sand, plaster and water with a mass ratio of 2:4:1:1.5. The influence of the frictional coefficient on the crack surface on the crack coalescence was considered in this study. These experimental results provide us a direct approach to understand interaction mechanisms of cracks. As examples, the typical experimental results are shown in Figure 1. Experiments and numerical simulation is a whole that cannot be divided. On one hand, numerical simulation must be based on the experiments, its mechanical parameters must come from the experiments and the results of the numerical simulation must be verified by the experiments. On the other hand, numerical simulation can help us understand the fundamental mechanisms of experimental phenomena. With the development of computer techniques and theory analysis more and more numerical methods are used to study the crack coalescence. On finite
1
Paper 2B 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
element methods discrete crack model (Hillerborg, 1976) and smeared crack model (Ba ant, 1983) were presented to simulate the fracture process of rock-like materials. Boundary element method (Shen, 1995) is successfully used to simulate the extension, coalescence of cracks. Tang et al. (2001) develop numerical simulation software called as RFPA2D based on meso-failure mechanics to simulate the fracture process and crack coalescence of the rock. Lattice model is introduced to simulate the fracture process of rock-like materials such as concrete. Lattice model does not discretize the continuum equation, but discretizes the continuum into pole system. Murat et al. (1992) proved that lattice model corresponded to a discretization of the continuum equation. It can transfer a complex triaxial problem into a simple uniaxial problem.
(a)
(c)
(b)
Based on the failure mechanics, lattice cellular automata (LCA) are presented to simulate the fracture process, crack coalescence of rock. This model has the advantages of both cellular automata and lattice model. The structure of the lattice cellular automata is shown in Figure 1. The lattice grid point shows the cell, which connects with its neighbours with beams that have three freedom degrees; the red region around the cell shows the influencing region of this cell. The states set of cell can be expressed as below,
φi = {{u , v ,ϕ}, { f x , f y , m}}i (1) The local rules, which are used to update the states of the cell, may be attained according to the equilibrium equations of each cell, the deformation equations and constitute equations. In order to simulate the fracture process of rock we must introduce some fracture laws to judge whether the beams starts to fail or not. Here two types of failure, tension failure and shearing failure, will be considered. To judge tension failure, the maximum tensional strain criteion is adopted and it can be expressed as below,
ε ≤ εt 0
Shear crack Wing crack Wing crack
(d)
2. LATTICE CELLULAR AUTOMATA
(e)
Figure 1. Typical experimental results, (a),(b),(c) from Bobet and Einstein (1998), (d) and (e) from Wong et al.(1998) Cellular automata were presented by Von Neumann in 1950’s to simulate self-organization among biological cells. Cellular automaton is made up of cell, cell lattice, neighbour and rules. The state of one cell is decided only by the states at the previous step of itself and its neighbours. Cellular automata have the advantages of time, space discretization, locality and parallelization and so on. These years, with the rapid development of computer techniques, cellular automata have been widely used in fluid mechanics (Wolframe, 1986), earthquake (Chen, Bak, 1991), solid mechanics (Tatting, Gurdal, 2000) etc. Based on the scalar cellular automata—physical cellular automata Zhou et al. (2002) simulate the rock fracture.
(2)
Where ε is the strain of the beams, εt 0 is the strain corresponding with the uniaxial tensional strength. Here the ultimate tensional strain εu is introduced to judge the fully tensional fracture. To judge shearing failure, Mohr-Coulomb criterion is adopted and it can be expressed as below,
F=
1 + sin θ σ1 − σ 3 ≥ f c 1 − sin θ
(3)
Where θ is frictional angle of the meso-element; f c is uniaxial compression strength of mesoelement; σ1 and σ3 are the major principal stress and minor principal stress respectively.
cell
Figure 2. Sketch map of cell structure
2
Paper 2B 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
So the basic thoughts of lattice cellular automata to simulate rock failure and crack coalescence can be expressed as below, 1. to discretize the rock into equivalent lattice model; 2. to introduce the heterogeneity of the rock by generating the mechanical parameters such as strength, elastic modulus etc. for each pole; 3. to update the states of all the beams according to local rules; 4. to judge whether the beams fail or not according to fracture law; 5. to perform failure analysis to the failed beams with failure mechanics so that we can analyse the strain-softening, simulate the discontinuity caused by fracture based on the thought of lattice cellular automata.
3. NUMERICAL SIMULATION AND DISCUSSION This paper mainly studies the influences of the geometry of two cracks on the interaction mechanisms. Several samples with two cracks of different geometries (shown in Figure 3) will be studied.
The fracture process and stress-strain curve of the sample with two overlapping cracks as shown in Figure 3(a) are shown in Figure 4. In this paper the unit of stress in all stress-strain curves is MPa. And its corresponding shearing failures are shown in Figure 5. The rock bridge angle is about 135 degree.
(a) Step 40
(b) Step 48
(c) Step 59 20 16 12 8 4 0
c d a 0
(d) Step 64
(e) Step 80
e
b 0.001
0.002
Stress-strain curve
Figure 4. Fracture process of Figure 3(a)
40mm
α β h
40mm
g
(a) Step 40
(b) Step 48
(c) Step 59
(a) g=8mm,h=4mm, α=45°,β=135°
Structure of sample a
(d) Step 64
(e) Step 80
Figure 5. Shearing failures of Figure 3(a) (b) g=8mm,h=4mm,
(c) g=8mm,h=4mm,
α=45°,β=90°
α=45°,β=63°
(d) g=8mm,h=4mm,
(e) g=8mm,h=2mm,
α=45°,β=53°
α=45°,β=45°
Figure 3. Rock samples with two cracks of different geometries
According to Figures 4 and 5, at the beginning of loading wing tensile cracks initiate at crack tips. Secondary shearing cracks propagate parallel with the pre-existing cracks. With the development of loading the wing tensile cracks initiating at the inner tip of two cracks join in the bridge area at step 70 as shown in Figure 4(d). Shearing failures of cells in the rock bridge area do not occur as shown in Figure 5. Therefore, the mode of the coalescence of the overlapping cracks should belong to tension mode. This numerical result is consistent with the experimental result shown in Figure 1(e) from Wong et al. (1998).
3
Paper 2B 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
The fracture process and stress-strain curve of sample as shown in Figure 3(b) is shown in Figure 6. Its corresponding shearing failures are shown in Figure 7. The rock bridge angle β of two cracks is 90 degree.
(a) Step 23
(b) Step 28
(c) Step 34
16 12 8 4 0
c d b a 0
(d) Step 43
(e) Step 69
0.001 0.002
Stress-strain curve
(d) Step 43
(b) Step 28
(c) Step 34
(e) Step 69
(c) Step 60
(b) Step 50
(a) Step 33
20 15 10 5 0
e
Figure 6. Fracture process of Figure 3(b)
(a) Step 23
The fracture process and stress-strain curve of sample as shown in Figure 3(c) is shown in Figure 8. The corresponding shearing failures are shown in Figure 9. The two cracks are non-overlapping cracks. The rock bridge angle is about 63 degree.
a
b
0
(d) Step 77
(e) Step 84
c d e
0.001
0.002
Stress-strain curve
Figure 8. Fracture process of Figure 3(c)
(a) Step 33
(d) Step 77
(b) Step 50
(c) Step 60
(e) Step 84
Figure 7. Shearing failures of Figure 3(b)
Figure 9. Shearing failures of Figure 3(c)
From Figures 6 and 7, wing tensile cracks also emanate firstly at the tips of the pre-existing cracks and they have a bigger angle with pre-existing cracks. Secondary shearing cracks appear in uncertain position but most of them are parallel with the pre-existing crack or have a smaller angle with two pre-existing cracks. At step 43 the wing tensile cracks join in bridge area (shown in Figure 6(d)). From Figure 7 we can obviously find that the shearing failures do not occur at the bridge area during the fracure. So we can conclude the coalescence mode of non-overlapping cracks with β=90° belongs to tension mode. The experimental result as shown in Figure 1(c) from Bobet and Einstein (1998) also supports this result of numerical simulation.
When the two cracks are arranged as shown in Figure 3(c). At the beginning of loading the wing tensile cracks also appear at the tips of the two preexisting cracks firstly. With the development of loading the wing cracks initiating from inner tips of two pre-existing cracks cannot join. At step 84 a new wing tensile crack and new shearing failures join together (as shown in Figure 8(e)). From Figure 9 we can find obviously in the joining area several shearing failures occur at step 84 (shown in Figure 9(e)). So we can conclude that the mode of the coalescence of two cracks with this geometry belongs to mixed mode of tension and shearing. The experimental result from Bobet and Einstein (1998) as shown in Figure 1(b) also supports the numerical simulation result.
4
Paper 2B 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
The fracture process and stress-strain curve of sample as shown in Figure 3(d) is shown in Figure 10, the corresponding shearing failures are shown in Figure 11. The two cracks as shown in Figure 3(d) are non-overlapping cracks and its rock bridge angle is about 53 degree.
(b) Step 25
(a) Step 20
(c) Step 50
20
ab
10
c
d
two cracks with such geometry belongs to mixed mode of tension and shearing. The fracture process and stress-strain curve of sample shown in Figure 3(e) is shown in Figure 12. The corresponding shearing failures are shown in Figure 13. The two cracks are cracks that are in a line. And its rock bridge angle is 45 degree.
(a) Step 31
40 30 20 10 0
e
0 0
(d) Step 66
(e) Step 75
0.001
0.002
Stress-strain curve
Figure 10. Fracture process of Figure 3(d)
(a) Step 20
(d) Step 66
(b) Step 25
(c) Step 50
(e) Step 75
(c) Step 70
(b) Step 50
c d a 0
(d) Step 99
(e) Step 120
e
b 0.001 0.002 0.003
Stress-strain curve
Figure 12. Fracture process of Figure 3(e)
(a) Step 31
(d) Step 99
(b) Step 50
(c) Step 70
(e) Step 120
Figure 11. Shearing failures of Figure 3(d)
Figure 13. Shearing failures of Figure 3(e)
Contrasting Figure 10 with Figure 11, we can find that at the crack tips the wing tensile cracks emanate prior to the secondary shearing cracks. The wing tensile cracks at the inner tips of two preexisting cracks extend with the development of loading. However the two wing tensile crack do not join in bridge area before step 75 (shown in figure 10(e)). At step 75 near the tip of pre-existing crack shearing cracks appear (shown in Figure 11(e)) and the shearing cracks join to the wing tensile crack. The numerical result is in accordance with the experimental result as shown in Figure 1(d) from Wong et al. (1998). The mode of coalescence of
When two pre-existing cracks in a sample are in line, wing tensile cracks firstly appear at the tips of two pre-existing cracks under uniaxial compression. Before step 120 the wing tensile cracks extend but they cannot join. The secondary shearing cracks do not emanate in the bridge area. But at step 120, shearing cracks suddenly appear from the inner tips of two pre-existing cracks and join together (shown in Figure 13(e)). The shearing cracks are in plane with the two pre-existing cracks. So we can conclude that the pre-existing cracks, which are in line, converge suddenly by secondary shearing cracks. The mode of
5
Paper 2B 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
coalescence of these two cracks belongs to shearing mode. The experimental result as shown in Figure 1(c) supports this result.
4. CONCLUSIONS Based on the lattice cellular automata the interaction mechanisms of two cracks with different geometries are studied. The numerical simulation results are in good accordance with the experimental results. The conclusion can be drawn as follows, 1. Coalescence mode of two overlapping cracks belongs to tension mode. 2. Coalescence mode of two non-overlapping cracks with β=900 belongs to tension mode. 3. Coalescence mode of two non-overlapping cracks that are not in a line and β<900 belongs to mixed mode of tension and shearing. 4. Coalescence mode of two non-overlapping cracks that are in a line belongs to shearing mode. Tabel 1. Coalescence mode of two cracks with different geometries. Coalescence Geometry of two cracks mode g=8mm,h=4mm, Tension α=45°,β=135° g=8mm,h=4mm,α=45°,β=90° Tension g=8mm,h=4mm,α=45°,β=63° tension+shear g=8mm,h=4mm,α=45°,β=53° tension+shear g=8mm,h=2mm,α=45°,β=45° Shear
5. ACKNOWLEGEMENT This paper is financial supported by the Special Funds for Major State Basic Research Project under Grand no. 2002CB412708 and National Natural Science Foundation of China under Grand no. 50179034.
Chen, K. & Bak, P. 1991 Self-organized criticality in a crack-propagation model of earthquakes. Phy. Rev. A 43(2): pp. 625 – 630. Hilerborg, A. Modeer, M. & Peterson, P.E. 1976. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research. 6: pp 773 – 782. Horii, H. & Nemat-Nasser, S. 1985. Compressioninduced microcrack growth in brittle solids: axial splitting and shear failure. J. Geophys. Res. 90(B4): pp. 3105 – 3125. Murat, M. Anholt, M. & Wagner, H.D. 1992. Fracture Behavior of Short-fiber Reinforced Materials. J. Mater. Res. 7(11): pp 3120 – 3131. Reyes, O. & Einstein, H.H. 1991. Fracture mechanics of fractured rock—A fracture coalescence model. 7th Int. Congress on Rock Mech. pp. 333 – 340. Shen, B. 1995. The mechanism of fracture coalescence in compression experimental study and numerical simulation. Eng. Fract. Mech. 51(1): pp 73 – 85. Tang, C.A. Lin, P. Wong, R.H.C. & Chau, K.T. 2001. Analysis of crack coalescence in rock-like material containing three flaws—Part II: Numerical approach. Int. J. Rock Mech. Min. Sci. 38(7): pp 925 – 936. Tatting T. & Gurdal Z. 2000. Cellular automata for design of two-dimensional continuum structure, 8 th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. Wolfram, S. 1986. Theory and applications of Cellular automata. World Publishing CO. PTE. LTD. Wong, R.H.C. & Chau, K.T. 1998. Crack coalescence in a rock-like material containing two cracks. Int. J. Rock Mech. Min. Sci. 35(2): pp. 147 – 164.
Ba ant, Z.P. & Oh, B.H. 1983 Crack band model for concrete. Material and Structures.16: pp. 155 – 177.
Wong, R.H.C. Chau, K.T. Tang, C.A. & Lin, P. 2001. Analysis of crack coalescence in rock-like material containing three flaws—Part I: experimental approach. Int. J. Rock Mech. Min. Sci. 38(9): pp. 909 – 924.
Bobet, A. & Einstein, H.H. 1998. Fracture coalescence in rock-type materials under uniaxial and biaxial compression. Int. J. Rock Mech. Min. Sci. 35(7): pp. 863 – 888.
Zhou, H. Wang, Y.J. Tan Y.L. & Feng X.T. 2002. Physical cellular automata to simulate rock fracture—part I: fundamental. Chi. J. of Rock Mech. and Eng. 21(4): pp. 475 – 478
6. REFERENCES
6