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Damage evolution and crack propagation in rocks with dual elliptic flaws in compression
Accepted Manuscript
Damage evolution and crack propagation in rocks with dual elliptic flaws in compression Jun Xu , Zhaoxia Li PII: DOI: Reference:
S0894-9166(17)30236-7 10.1016/j.camss.2017.11.001 CAMSS 64
To appear in:
Acta Mechanica Solida Sinica
Received date: Revised date: Accepted date:
27 July 2017 12 November 2017 17 November 2017
Please cite this article as: Jun Xu , Zhaoxia Li , Damage evolution and crack propagation in rocks with dual elliptic flaws in compression, Acta Mechanica Solida Sinica (2017), doi: 10.1016/j.camss.2017.11.001
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Damage evolution and crack propagation in rocks with dual elliptic flaws in compression Jun Xua, b, Zhaoxia Lia, b a
Department of Engineering Mechanics, Southeast University, Nanjing 210096, China
b
Jun Xu
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Jiangsu Key Laboratory of Engineering Mechanics, Southeast University, Nanjing 210096, China
Department of Engineering Mechanics, Southeast University No.2 Sipailou, Nanjing 210096, China
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E-mail: [email protected]
Zhaoxia Li (Corresponding author)
Department of Engineering Mechanics, Southeast University No.2 Sipailou, Nanjing 210096, China
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E-mail: [email protected]
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Abstract
To give an insight into the understanding of damage evolution and crack propagation in rocks, a series of uniaxial
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and biaxial compression numerical tests are carried out. The investigations show that damage evolution occurs firstly in the weak rock, the area around the flaw and the area between the flaw and the neighboring rock layer.
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Cracks mostly generate as tensile cracks under uniaxial compression and shear cracks under biaxial compression. Crack patterns are classified and divided. The relationship between the accumulated lateral displacement and the
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short radius (b) is fitted, and the equation of crack path is also established.
Key words
Damage evolution; Crack propagation; Elliptic flaw
1 Introduction As a common complicated geological mass, rocks are encountered in most engineering practices. In rocks, inhomogeneities such as flaws, joints and mineral grains or particles are ubiquitous, which play a key role in the
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failure process (damage evolution, crack initiation, crack propagation and coalescence) and could directly influence the physical and mechanical properties of rocks. Ashby and Sammis [1] reported that the cracks nucleate and propagate from inhomogeneities, and the development of micro-crack damage in rocks under compression always initiates from the largest flaws, independent of the confining stress. The investigation by Weinberger [2] suggested that the joint initiation points are governed by the isolated, relatively large cavities in rocks. Larsen and Gudmundsson [3] reported that the coalescence of existing flaws and the crack propagation paths depend largely on
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the mechanical layering of the rocks. Gross and Eyal [4] pointed out that the through-going fractures in rocks are formed by the linkage and coalescence of pre-existing, bed-confined joints. Besides, a lot of previous investigations [5-12] have also reported that fracture initiation and coalescence in rocks or rock-type materials are highly dependent on the geometry of the flaw, and three types of cracks can be observed in the failure process when the
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rock specimens are under uniaxial compression. Wing cracks usually initiate as tensile cracks at points of tensile stress concentration and propagate along a curvilinear direction that becomes roughly parallel to the far field compression. Secondary cracks initiate as shear cracks at the flaw tips and in a plane roughly co-planar with the flaw. Coalescence cracks initiate as tensile cracks or shear cracks or the mixed tensile and shear cracks, as shown in
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Fig. 1a. Furthermore, the spacing between the pre-existing flaws is also an important parameter which can influence the type of coalescence crack, such as the coalescence cracks in Fig. 1b. But, in the field observations
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[13], many of the veins penetrate shale layers, mostly as (inclined) shear fractures, and similar observations are also found in numerical simulations [14-17], which means the behaviors of cracks in rocks could be influenced by the
( b)
Pre - existing flaw parameters : inclinatio n ( ) : 45 2a
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(a )
2a
s
wing crack
secondary crack
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coalescence crack
70 Unit : mm
wing crack
length : 2a 20 2mm
140
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rock layers.
width : 1.6mm s : spacing 0mm, 10mm, 20mm
30
T : tension; S : shear
S
S
T
T
S
1#
T
2#
3#
Fig. 1 Three types of cracks in the overlapping geometry under uniaxial compression, and the coalescence crack
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paths in gypsum specimens under uniaxial compression. The coalescence crack paths are obtained by changing the spacing between the two pre-existing flaws, and one of the tension cracks (wing cracks) in specimen 3# gradually closes and stops to propagate with the increase of load. According to most of the previous investigations, flaw geometry (e.g. the flaw length, width, inclination angle and spacing, etc.) was usually considered and investigated, but less work has been done on the flaw shapes and the
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influence of flaw shapes on the damage evolution and crack propagation path in layered rocks. Damage evolution always initiates from the largest defects [1], which means flaw shape could affect damage evolution when rocks are subjected to tension or compression. Some studies [18-20] suggested that crack shape might be an important parameter in the fracture process of brittle solids, which has a significant influence on the mechanical properties of rocks, such as fracture toughness, material strength, crack pattern, etc. Therefore, in order to give an insight into the
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understanding of damage evolution and crack propagation in rocks, different shape flaws as well as rock layers are considered in our investigation, and the concept of “flaw-control” established by Griffith is also used. Moreover, a damage point of view is adopted to describe the evolution of damage and the initiation and coalescence of crack
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propagation.
2 Elliptic flaw in rocks and the tangential stress at the flaw surface
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As mentioned in the introduction, crack shape is an important parameter in brittle solid fracture, which has a significant influence on the mechanical properties of rocks. In rock fracture mechanics, defects are usually treated
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as varied elliptic flaws in plate [21]. The stresses around the elliptic flaw surface was first solved by Inglis using Airy stress functions, and the complex potentials for an elliptic flaw were established by Stevenson and
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Muskhelishvili. Now, we may consider an infinite rock mass containing an elliptic flaw subjected to one or two principle stresses, i.e., 𝑝1∞ acting at an angle 𝛽 to the x-axis and 𝑝2∞ acting at an angle (𝛽 +π/2), the boundary
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of which corresponds to 𝜉 = 𝜉0 , as shown in Fig. 2.
p1 y
p2
0 x
p1
p2
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Fig. 2 Elliptic flaw subjected to a far-field stress Therefore, the semi-axes of the elliptic flaw are 𝑎 = 𝑐 ∙ cosh𝜉0 , 𝑏 = 𝑐 ∙ sinh𝜉0
(1)
where the semi-axes are related through 𝑎2 − 𝑏2 = 𝑐 2 . The parametric equation of the elliptic flaw can be written as 𝑥 = 𝑎cos𝜂, 𝑦 = 𝑏sin𝜂
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(2)
and the polar equation of the elliptic flaw is 𝑥 = 𝑟cos𝛾, 𝑦 = 𝑟sin𝛾
(3)
According to [21], the tangential stresses of the elliptic flaw surface under uniaxial compression and biaxial compression can be written as 𝑝1∞
(𝜏𝜂𝜂 )𝜉=𝜉0 =
=
2𝑎𝑏+(𝑎2 −𝑏2 )cos2𝛽−(𝑎+𝑏)2 cos2(𝛽−𝜂) 𝑎2 +𝑏2 −(𝑎2 −𝑏2 )cos2𝜂
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(𝜏𝜂𝜂 )𝜉=𝜉0
2𝑎𝑏(𝑝1∞ +𝑝2∞ )+(𝑝1∞ −𝑝2∞ )[(𝑎2 −𝑏2 )cos2𝛽−(𝑎+𝑏)2 cos2(𝛽−𝜂)] (𝑎2 +𝑏2 )−(𝑎2 −𝑏2 )cos2𝜂
(4)
(5)
When 𝜉 = 𝜉0 , the elliptical coordinate 𝜂 is related to the polar coordinate 𝛾 by 𝑦
𝑏
tan𝛾 = 𝑥 = 𝑎 tan𝜂
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(6)
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Moreover, to evaluate the shape of the elliptic flaw, the aspect ratio of the elliptic flaw can be written as 𝑏 𝑎
= tanh𝜉0
(7)
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In Fig. 3, the variation of tangential stress and the variation of aspect ratio are presented.
0
( b ) 1 .0 Aspect ratio, b/a
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/ p1
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(a ) 5 4 3 2 1 0 1 2
a 2b
45
/3
2 / 3
0 .8 0 .6 0 .4 0 .2 0 .0
0
1
2
0
3
4
5
Fig. 3 (a) is the variation of the tangential stress under uniaxial compression, and (b) is the variation of the aspect ratio, b/a; when the value of constant 𝜉0 increases to 3, the aspect ratio 𝑏/𝑎 → 1, the elliptic flaw becomes a circle.
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According to [22], the maximum tangential stress on the surface of the elliptic flaw under uniaxial compression can be written as 𝜉0 𝜏𝜂𝜂m = 𝑝1∞ sin𝛽(1 − sin𝛽) when 𝜉0 → 0, equation (7)
𝑏 𝑎
(8)
= tanh𝜉0 ≈ 𝜉0 , the inclination angle 𝛽 = 45° , we may obtain the relationship
between the aspect ratio (b/a) and the tangential stress at the critical failure as 0.207𝑝1∞ (𝑏⁄𝑎)
(9)
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𝜏𝜂𝜂c =
3 Numerical simulations on the damage evolution and crack propagation in rocks with dual elliptic flaws
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3.1 Modeling of the material properties
The microstructure of rocks and the inhomogeneities in rocks give the material a strong heterogeneity. Lan et al. [23] pointed out that the micro-heterogeneity of material plays a significant role in determining both the micromechanical behavior and the macroscopic response, such as damage evolution and crack propagation. In
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RFPA-2D [24], Weibull’s theory was used to solve the problem of heterogeneity of rocks or rock-like materials. Weibull distribution function is given as 𝑚
𝑢 𝑚−1
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𝜑(𝑢) = 𝑢 (𝑢 ) 0
0
𝑢
𝑚
exp *− (𝑢 )+ , 𝑚 > 1 0
(10)
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where u is the element parameter, such as elastic modulus, strength and density, u0 is the even value of the parameter of all the elements, and m is the heterogeneity index (shape parameter) which reflects the material
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heterogeneity.
In Fig. 4, the distribution of meso-elements is presented, which comes to a narrow range around u0 with the
(u )
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increase of heterogeneity index. Fig. 5 shows the distribution of meso-element strength.
m 10.0 m 5 .0 m 3.0 m 1.5
u0
u
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Fig. 4 Weibull distribution of meso-elements with different heterogeneity indices (m)
10000
8000
4000
2000
0
0
50
100
150
200
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Strength/MPa
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Number
6000
Fig. 5 Distribution of meso-element strength
3.2 Numerical model of specimen and computational parameters
In Fig. 6, the numerical model of the specimen with dual pre-existing flaws is presented. The geometry of the
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numerical model is 50 mm×100 mm in scale, and the mesh for the model is 200×400=80,000 elements. Other
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computational parameters adopted are presented in Table 1. Moreover, Mohr-Coulomb criterion is chosen as the failure criterion for the element with the advantages in rock failure. It should be noted that the numerical model is
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different from the layered rock, and in the uniaxial compression the axial load is controlled at 0.002 mm/step, and in the biaxial compression, the axial load is still controlled at 0.002 mm/step, but the lateral loads are controlled at
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0.001 mm/step, 0.002 mm/step and 0.003 mm/step, respectively.
Ⅰ
165.6
15 20
Ⅱ
110.4
25
55.19
25
Ⅲ
2b
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2a
220.8
Elliptic flaw b 45
a 2
0 Strength (MPa )
7 2a 14 Unit : mm
Fig. 6 Numerical model of the specimen and the pre-existing flaw; 2a is the length of the elliptic flaw, and 2b is the width of the elliptic flaw, it should be noted that the flaw inclination angle is 45°.
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Table 1 Control parameters in numerical simulation
Heterogeneity index Meso elastic modulus / GPa Meso compressive strength / MPa Poission’s ratio Internal friction angle/° Compression/Tension ratio
3.3 Calculated results and analysis on damage evolution
3 50 100 0.25 30 10
RockⅡ 5 6 150 0.30 30 10
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RocksⅠand Ⅲ
Control parameters
According to [25], damage evolution in materials usually occurs before the occurrence of macro-crack or at the
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peak load of the specimen. In this section, in order to analyze the damage evolution, we investigate the axial stress and the number of failure elements under uniaxial compression, and the situation of b=2 mm is specifically selected for analysis.
In Fig. 7, the failure elements are accumulated and presented, as well as the curve of axial stress and strain
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under uniaxial compression. Fig. 8 shows the change of shear stress and the accumulation of failure elements under uniaxial compression. From Figs. 7 and 8, it can be seen that at the beginning of the numerical test, there is no
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failure element, and the evolution of damage begins (point A in Fig. 7) when the axial stress increases to 16.6% of the peak stress (PS, point B in Fig. 7), and then the damage grows rapidly with the increase of strain. In Fig. 8a, as
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the load is increased, two “X”-shape bands emerge from the tips of the pre-existing flaws and approach the boundary of the specimen, and the areas of the bands are simply divided into four parts, in which the elements are
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more likely to fail because of the concentration of tensile and shear stresses, and the smaller the area is, the easier the damage grows. This phenomenon is similar to the observation [26]. Moreover, in Fig. 8b, the failure elements
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(damage evolution, mainly tensile failure) occur in rocksⅠand Ⅲ, and they initiate mainly around the pre-existing flaws and near the surface of rockⅡ.
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14
400
12
E
350
F
300 250
b=2mm
8
200 6 150 4
100
2 A 0 0.00
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Axial stress /MPa
10
Failure element number
Stress Failure element B C
50
0.02
0.04
0.06 0.08 0.10 Axial strain (%)
0.12
0.14
0 0.16
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Fig. 7 Curve of axial stress – axial strain under uniaxial compression and the accumulation of failure elements
(a )
S3 0
S4
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( b)
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16.6%PS
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27.45
41.43
89.7%PS
55.91
PS
Shear stress (MPa)
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4.12
43.3%PS
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16.6%PS S2 S1
43.3%PS
: tensile failure
89.7%PS
PS
: compressive failure
Fig. 8 (a) is the change of shear stress, and (b) is the accumulation of failure elements under uniaxial compression (b=2 mm), PS is the peak stress (point B in Fig. 7), 𝐴1 , 𝐴2 , 𝐴3 and 𝐴4 are the four areas of the “X”-shape bands, and the relationship among 𝐴1 , 𝐴2 , 𝐴3 and 𝐴4 is 𝐴1 < 𝐴2 < 𝐴3 < 𝐴4.
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0e-006m/step 1e-006m/step 2e-006m/step 3e-006m/step
40
I
b=2mm
47.2
J
30 K 20
H(B) 10
11.7
0 0.00
0.02
0.04
0.06 0.08 0.10 Axial strain (%)
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Axial stress /MPa
50
0.12
0.14
H(B)
J
K
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I
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(a) Curves of axial stress – axial strain under biaxial compression
(b) Accumulation of failure elements at different points under biaxial compression
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Fig. 9 Curves of axial stress – strain, and accumulation of failure elements at different points under biaxial compression. It should be noted that the axial load is 2e-006 m/step, but the lateral loads in I, J and K are 1e-006
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m/step, 2e-006 m/step and 3e-006 m/step, respectively. Fig. 9 gives the curves of axial stress – axial strain and the accumulation of failure elements at different points
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under biaxial compression. From Fig. 9a, it can be found that the curve slope increases with the increase of lateral load, which means the damage evolution rate could be promoted by increasing the lateral load. In Fig. 9b, we can
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clearly see that the number of compressive failure elements in figures I, J and K is more than that in figure H (with no lateral confinement). Moreover, the peak stress of the specimen is improved with the increase of lateral load, and the smaller is the lateral load, the higher is the peak stress. In Fig. 9a, the highest peak stress (47.2 MPa) is about four times as much as the stress under the condition with no lateral confinement (11.7 MPa). In order to further investigate the damage evolution, we quantify the damage according to the number of failure elements, and the damage (D) is simply defined as 𝐷 = 𝑛/𝑁 where n is the count of element failure, and N is the total count of element.
(11)
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Fig. 10 gives the damage evolution in the specimen with different flaw shapes. It can be seen that the damage evolves slowly when the axial strain is less than 0.025 (%), and then grows rapidly with the increase of axial strain. However, the largest damage is less than 0.05, indicating that macro-cracks in brittle solids occur quickly after the damage evolution, which is in good agreement with the physical truth and investigations [25]. Moreover, it can also be found that when the axial strain is less than 0.058, the damage increases with the increase of short radius (b), but the increment is small. In Fig. 11, the schematic diagram of mesoscopic damage evolution under uniaxial
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compression is presented. From Fig. 11, we can simply and clearly see that in the mesoscopic crack formation, the elements of weak strength begin to failure firstly under compression, which means the beginning of damage evolution; and then the elements of ordinary strength start to fail with the increase of load, which means the expansion of damage evolution. If the elements of high strength fail, meso-cracks occur, propagate and coalesce,
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followed by the occurrence of macro-cracks. 0.05
b=0mm b=2mm b=4mm b=7mm
0.04
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Damage
0.03 0.02
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0.01
0.00 0.00
0.02
PT
0.01
0.058
0.03 0.04 0.05 Strain (%)
0.06
0.07
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CE
Fig. 10 Curves of damage evolution in specimen with different flaw shapes
damage expansion
damage occurence
Ⅰ
Ⅱ initiation
Ⅲ
Ⅳ
meso - crack propagation
high strength element ordinary strength element weak strength element failure element Fig. 11 Schematic diagram of (mesoscopic) damage evolution under uniaxial compression
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3.4 Calculated results and analysis on crack propagation Propagation of initial defects and initiation of mesoscopic cracks are the damage evolution process [27], and the concentration of damage evolution usually leads to the occurrence of macro-cracks. As mentioned and introduced above, flaw shape has a significant influence on the mechanical properties of rocks. Therefore, in this section, the crack propagation in specimen with different shapes of elliptic flaw under compression is investigated and analyzed.
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In order to facilitate the calculation, the case of b=2 mm is selected. In Fig. 12, the process of crack propagation in specimen with dual elliptic flaws (b=2 mm) is presented.
180
coalescence crack
external tip
b=2mm
C'
90
internal tip
G'
60
②
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①
30 0 0.00
0.02
F'
E'
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B' 120
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Lateral displacement /μm
150
0.04
E' F'
B' C'
0.06 0.08 0.10 Axial strain (%)
0.12
0.14
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Fig. 12 Process of crack propagation in specimen with pre-existing flaws and the curve of lateral displacement –
Fig. 7.
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axial strain (b=2 mm) under uniaxial compression, where Figures B ′, C ′, E ′ and F ′ correspond to the curve in
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According to Fig. 12, at point B ′, the wing and coalescence cracks initiate and generate at or near the elliptic flaw tips; after the peak stress (from point B ′ to point C ′), the wing cracks propagate to the top and bottom of the specimen, and the coalescence cracks propagate to the neighboring rock (rock Ⅱ) in a nonlinear path. At point C ′, the coalescence cracks propagate to the surface of rockⅡ, and then the propagation bifurcates: one along the foregoing path (path ①), the other along the surface of rock Ⅱ (path ②), as shown in Fig. 12F ′ and G′ . Besides, the curve of lateral displacement – axial strain shows that the lateral displacement increases slowly before point B ′; from point B ′ to point C ′, the growth rate increases; and after point C ′, the lateral displacement has a fast growth rate; but from point E ′ to point F ′ , the growth rate decreases and then increases again, which indicates that the
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speed of crack propagation is not immutable in rocks, and the crack propagation could be influenced by the physical and mechanical properties of rock. We may also find that the coalescence cracks occur at the places where damage mainly concentrates, as shown in Fig. 8. Moreover, some branch cracks occur in the progress of crack propagation, which could also increase the lateral displacement. Table 2 shows the crack patters of crack propagation in rocksⅠand Ⅲ in specimen as the strain is 0.09 (%), and the crack patterns are summarized and classified simultaneously. When the short radius b is equal to or smaller
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than 3 mm, two nonlinear coalescence cracks occur between the pre-existing flaw and rock Ⅱ, while three nonlinear coalescence cracks generate when the short radius b is equal to or larger than 4 mm. Obviously, the initiation and propagation of crack can be influenced by the shape of flaw, which means the tangential stress of the elliptic flaw surface is also consistent with the shape of flaw. When the short radius b is 3.5 mm (2𝑏 = 𝑎), the
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tangential stress of the elliptic flaw surface under uniaxial compression is shown in equation (4), and the variation of tangential stress with polar coordinate 𝛾 is presented in Fig. 3. From the crack patterns, it is also found that some branch cracks emerge from the coalescence cracks, which could either promote or stop the propagation of coalescence cracks. Moreover, we may also find that the cracks (wing and coalescence cracks) initiate and generate
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mostly near the flaw tips, which is consistent with the theory of Griffith 2D. Besides, it should be noted that when the short radius b approaches or equals to 0, the pre-existing flaw degrades into a straight flaw but is still an open
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flaw, and there is no force acting on the flaw surfaces during loading. Table 2 Two types of crack patterns during crack propagation in layersⅠand Ⅲ with different flaw shapes
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Type
Crack pattern,strain : 0.09 (%)
b 0mm
b 1mm
b 2mm
b 3mm
b 4mm
b 5mm
b 6mm
b 7 mm
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CE
①
②
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4 Discussion on the lateral displacement and crack path 4.1 The lateral displacement due to damage evolution The lateral displacement usually comes from the initiation of micro-cracks in materials at the beginning of the loading and the nucleation and propagation of micro-cracks before the occurrence of macro-cracks, so the growth of lateral displacement could reflect the rate of damage evolution in the materials to certain extent. Therefore, we
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could further analyze the damage evolution through the lateral displacement. Fig. 13 shows the curve of lateral displacement – axial strain of specimen with different flaw shapes under uniaxial compression.
b=0mm b=1mm b=2mm b=3mm b=4mm b=5mm b=6mm b=7mm
Lateral displacement /μm
7 6 5 4 3
1 0.01
0.02
ED
0 0.00
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2
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8
0.03 0.04 0.05 Axial strain (%)
0.06
0.07
120
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Accumulated lateral displacement (d) /μm
Fig. 13 Curve of lateral displacement – axial strain, where the largest axial strain is 0.07 (%)
Simulation result Fitting result
110
AC
CE
100 90 80
d2=9.814·(b/1000)+46.860
70
R2=0.998
d1=4.351·(b/1000)+49.118
60 R2=0.956 50 40
b=3.5mm 0
1 2 3 4 5 6 Short radius (b) of the elliptic flaw/mm
7
Fig. 14 Curve of accumulated lateral displacement – short radius, where it should be noted that the accumulated lateral displacement is the accumulation of the lateral displacements in Fig. 13.
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As can be seen from Fig. 13, when the axial strain is smaller than 0.04, the lateral displacement has an almost linear growth, which indicates that the evolution of damage is stable, but the rates of damage evolution of the specimens with different flaw shapes are different, which increases with the increase of the short radius b. When the axial strain is larger than 0.04, differences of the lateral displacement growth generate: 1) the growth increases as the short radius 𝑏 equals to 4~7 mm; 2) the growth slowes down and then increases again when the short
damage evolution.
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radius 𝑏 = 0~3 mm. The differences imply that the aspect ratio (𝑏⁄𝑎) of the elliptic flaw can influence the
When the semi-major axis (a) equals to 7 mm, the relationship between the accumulated lateral displacement (d) and the semi-minor axis or short radius (b) can be written as 𝑏
𝑑1 = 4.351 ∙ 1000 + 49.118 (0 ≤ 𝑏 < 3.5)
(12)
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{ 𝑏 𝑑2 = 9.814 ∙ 1000 + 46.860 (3.5 ≤ 𝑏 ≤ 7)
According to equation (12) and Fig. 14, the growth of lateral displacement accumulation is linear with the increase of short radius and has a cut-off point when 𝑏 = 3.5; and the slope of 𝑑2 is twice as much as the slope of 𝑑1 , which means the damage evolution when 3.5 ≤ 𝑏 ≤ 7 is more quickly and more likely to lead to the initiation of
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macro-cracks.
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4.2 The crack path and path equation in rocks
In rock specimen or rock-type materials, many investigations [28-31, 5-12] have pointed out that cracks always
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initiate at or near the pre-existing flaw tips and coalesce eventually by tension cracks or shear cracks or both of tension and shear cracks; and the patterns of tension cracks and shear cracks, as well as the types of coalescence,
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depend on the geometry and inclination angle of the flaws and the stress conditions to some extent. However, when the specimen consists of two (or more) types of rocks, the propagation of crack could be influenced by the types of
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rocks, such as the layered rock [9, 12-14] and the one investigated in this paper. In Fig. 15, the crack paths in layered rocks and the specimen with one and two types of rocks are shown. As presented in Fig. 15(1), (2) and (4), the cracks penetrate weak rocks, mostly as inclined shear cracks, which
is in good agreement with field observations. From Fig. 15(3) - (5), it can be clearly seen that the crack paths in specimen have been changed by the flaws, and the crack paths ① and ② are similar in the morphological configuration and propagate in a curve way from the tips of the flaw. Moreover, according to Section 3.4, the shape of flaw could also have an effect on the crack propagation path. When the short radius b is larger than 3.5 mm, one
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more crack initiates from the internal tip of the flaw and propagates to rockⅡ.
(1)
(4)
(3)
external tip flaws
(5)
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(2)
②
①
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internal tip
Fig. 15 Crack paths in layered rocks and the specimens, where Figures (1) and (2) show the field observation, and
2a
y
M( xM , y M ) ①
AC
CE
PT
ED
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Figures (3) -(5) are the numerical results of RFPA-2D.
③
x ② Tangent
Fig. 16 The inverse proportion function model of crack path
In order to describe the path of crack propagation, the inverse proportion function is considered. Here we only
consider the situation of 2𝑏 < 𝑎; therefore, the equation of crack path ① can be written as 𝑘
𝑦 = 𝑥−𝑥 + 𝑦0 0
(13)
where the parameters 𝑘, 𝑥0 and 𝑦0 are unknown. Moreover, the flaw tip, point M is 𝑥 = −𝑎cos𝛽 { M 𝑦M = −𝑎sin𝛽
(14)
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According to [32], we may have the crack path ① equation as 𝑦=−
𝑎2 cos2 𝛽 tan(𝛽+𝜃) 𝑥
− [𝑎cos𝛽 tan(𝛽 + 𝜃) + 𝑎sin𝛽]
(15)
where the parameter 𝜃 is the fracture angle which can be obtained using the maximum tensile-stress criterion. The equation of crack path ② can be obtained via the translation of equation (15). Equations of crack paths ① and ③ are symmetric about origin. It should be noted that the above equations are appropriate in the condition of
very complicated, which will be investigated and discussed in the future.
5 Conclusions
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uniaxial compression. In fact, the crack propagation and the equations of crack paths in rock layers are multiple and
This paper numerically investigates damage evolution and crack propagation in rocks with different shapes of
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flaws under compression, and several important findings are as follows.
(1) Damage occurs firstly in weak rocks, the area around the pre-existing flaw and the area between the flaw and the neighboring rock layer under compression. As the load increases, the “X”-shape bands can be observed at the tips of the pre-existing flaws according to the numerical tests, and the bands of “X” could be divided into four parts
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with different sizes, where the small the part size is, the easier the damage evolves. Moreover, the damage is quantified according to the number of failure elements, which grows slowly when the strain is less than 0.025 and
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grows rapidly with the increase of strain.
(2) Under uniaxial compression, cracks mostly occur as tensile cracks which generate at or near the flaw tips and
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propagate to the maximum compressive stress in a nonlinear path; and under biaxial compression, many shear cracks initiate at the tips of the flaws but the extended directions of the cracks are random. Under uniaxial
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compression, when the coalescence cracks propagate to rock Ⅱ, they may propagate along the foregoing path or along the interface of rock Ⅱ. Moreover, the propagation of crack is not immutable in different rocks, which could
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be influenced by the physical and mechanical properties of the rocks. The crack patterns are summarized and classified into two types according to the flaw shape or aspect ratio, b/a. (3) The relationship between the lateral displacement and damage evolution (before the occurrence of macro cracks) and crack path equations (the situation of 2𝑏 < 𝑎) are investigated and discussed. The evolution of damage could be influenced by the aspect ratio. When the axial strain is smaller than 0.04, the lateral displacement has an almost linear growth, and when the axial strain is larger than 0.04, the growth of lateral displacement changes. Besides, the relationship between the accumulated lateral displacement (d) and the short radius (b) is fitted according to the numerical results, and the equation of crack path is also established, as shown in equations (10) and (13),
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respectively.
Acknowledgements The work described in this paper was substantially supported by the National Program on Major Research Project (No. 2016YFC0701301-02) and Jiangsu Higher Education Institutions for the Priority Academic Development Program (CE02-1-34), and the calculation program was provided by Prof. Xiaochun Xiao at Liaoning
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Technical University, all of which are gratefully acknowledged.
References
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345-349 (in Chinese)
Damage evolution and crack propagation in rocks with dual elliptic flaws in compression Jun Xu , Zhaoxia Li PII: DOI: Reference:
S0894-9166(17)30236-7 10.1016/j.camss.2017.11.001 CAMSS 64
To appear in:
Acta Mechanica Solida Sinica
Received date: Revised date: Accepted date:
27 July 2017 12 November 2017 17 November 2017
Please cite this article as: Jun Xu , Zhaoxia Li , Damage evolution and crack propagation in rocks with dual elliptic flaws in compression, Acta Mechanica Solida Sinica (2017), doi: 10.1016/j.camss.2017.11.001
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Damage evolution and crack propagation in rocks with dual elliptic flaws in compression Jun Xua, b, Zhaoxia Lia, b a
Department of Engineering Mechanics, Southeast University, Nanjing 210096, China
b
Jun Xu
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Jiangsu Key Laboratory of Engineering Mechanics, Southeast University, Nanjing 210096, China
Department of Engineering Mechanics, Southeast University No.2 Sipailou, Nanjing 210096, China
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E-mail: [email protected]
Zhaoxia Li (Corresponding author)
Department of Engineering Mechanics, Southeast University No.2 Sipailou, Nanjing 210096, China
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E-mail: [email protected]
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Abstract
To give an insight into the understanding of damage evolution and crack propagation in rocks, a series of uniaxial
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and biaxial compression numerical tests are carried out. The investigations show that damage evolution occurs firstly in the weak rock, the area around the flaw and the area between the flaw and the neighboring rock layer.
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Cracks mostly generate as tensile cracks under uniaxial compression and shear cracks under biaxial compression. Crack patterns are classified and divided. The relationship between the accumulated lateral displacement and the
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short radius (b) is fitted, and the equation of crack path is also established.
Key words
Damage evolution; Crack propagation; Elliptic flaw
1 Introduction As a common complicated geological mass, rocks are encountered in most engineering practices. In rocks, inhomogeneities such as flaws, joints and mineral grains or particles are ubiquitous, which play a key role in the
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failure process (damage evolution, crack initiation, crack propagation and coalescence) and could directly influence the physical and mechanical properties of rocks. Ashby and Sammis [1] reported that the cracks nucleate and propagate from inhomogeneities, and the development of micro-crack damage in rocks under compression always initiates from the largest flaws, independent of the confining stress. The investigation by Weinberger [2] suggested that the joint initiation points are governed by the isolated, relatively large cavities in rocks. Larsen and Gudmundsson [3] reported that the coalescence of existing flaws and the crack propagation paths depend largely on
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the mechanical layering of the rocks. Gross and Eyal [4] pointed out that the through-going fractures in rocks are formed by the linkage and coalescence of pre-existing, bed-confined joints. Besides, a lot of previous investigations [5-12] have also reported that fracture initiation and coalescence in rocks or rock-type materials are highly dependent on the geometry of the flaw, and three types of cracks can be observed in the failure process when the
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rock specimens are under uniaxial compression. Wing cracks usually initiate as tensile cracks at points of tensile stress concentration and propagate along a curvilinear direction that becomes roughly parallel to the far field compression. Secondary cracks initiate as shear cracks at the flaw tips and in a plane roughly co-planar with the flaw. Coalescence cracks initiate as tensile cracks or shear cracks or the mixed tensile and shear cracks, as shown in
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Fig. 1a. Furthermore, the spacing between the pre-existing flaws is also an important parameter which can influence the type of coalescence crack, such as the coalescence cracks in Fig. 1b. But, in the field observations
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[13], many of the veins penetrate shale layers, mostly as (inclined) shear fractures, and similar observations are also found in numerical simulations [14-17], which means the behaviors of cracks in rocks could be influenced by the
( b)
Pre - existing flaw parameters : inclinatio n ( ) : 45 2a
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(a )
2a
s
wing crack
secondary crack
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coalescence crack
70 Unit : mm
wing crack
length : 2a 20 2mm
140
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rock layers.
width : 1.6mm s : spacing 0mm, 10mm, 20mm
30
T : tension; S : shear
S
S
T
T
S
1#
T
2#
3#
Fig. 1 Three types of cracks in the overlapping geometry under uniaxial compression, and the coalescence crack
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paths in gypsum specimens under uniaxial compression. The coalescence crack paths are obtained by changing the spacing between the two pre-existing flaws, and one of the tension cracks (wing cracks) in specimen 3# gradually closes and stops to propagate with the increase of load. According to most of the previous investigations, flaw geometry (e.g. the flaw length, width, inclination angle and spacing, etc.) was usually considered and investigated, but less work has been done on the flaw shapes and the
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influence of flaw shapes on the damage evolution and crack propagation path in layered rocks. Damage evolution always initiates from the largest defects [1], which means flaw shape could affect damage evolution when rocks are subjected to tension or compression. Some studies [18-20] suggested that crack shape might be an important parameter in the fracture process of brittle solids, which has a significant influence on the mechanical properties of rocks, such as fracture toughness, material strength, crack pattern, etc. Therefore, in order to give an insight into the
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understanding of damage evolution and crack propagation in rocks, different shape flaws as well as rock layers are considered in our investigation, and the concept of “flaw-control” established by Griffith is also used. Moreover, a damage point of view is adopted to describe the evolution of damage and the initiation and coalescence of crack
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propagation.
2 Elliptic flaw in rocks and the tangential stress at the flaw surface
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As mentioned in the introduction, crack shape is an important parameter in brittle solid fracture, which has a significant influence on the mechanical properties of rocks. In rock fracture mechanics, defects are usually treated
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as varied elliptic flaws in plate [21]. The stresses around the elliptic flaw surface was first solved by Inglis using Airy stress functions, and the complex potentials for an elliptic flaw were established by Stevenson and
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Muskhelishvili. Now, we may consider an infinite rock mass containing an elliptic flaw subjected to one or two principle stresses, i.e., 𝑝1∞ acting at an angle 𝛽 to the x-axis and 𝑝2∞ acting at an angle (𝛽 +π/2), the boundary
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of which corresponds to 𝜉 = 𝜉0 , as shown in Fig. 2.
p1 y
p2
0 x
p1
p2
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Fig. 2 Elliptic flaw subjected to a far-field stress Therefore, the semi-axes of the elliptic flaw are 𝑎 = 𝑐 ∙ cosh𝜉0 , 𝑏 = 𝑐 ∙ sinh𝜉0
(1)
where the semi-axes are related through 𝑎2 − 𝑏2 = 𝑐 2 . The parametric equation of the elliptic flaw can be written as 𝑥 = 𝑎cos𝜂, 𝑦 = 𝑏sin𝜂
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(2)
and the polar equation of the elliptic flaw is 𝑥 = 𝑟cos𝛾, 𝑦 = 𝑟sin𝛾
(3)
According to [21], the tangential stresses of the elliptic flaw surface under uniaxial compression and biaxial compression can be written as 𝑝1∞
(𝜏𝜂𝜂 )𝜉=𝜉0 =
=
2𝑎𝑏+(𝑎2 −𝑏2 )cos2𝛽−(𝑎+𝑏)2 cos2(𝛽−𝜂) 𝑎2 +𝑏2 −(𝑎2 −𝑏2 )cos2𝜂
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(𝜏𝜂𝜂 )𝜉=𝜉0
2𝑎𝑏(𝑝1∞ +𝑝2∞ )+(𝑝1∞ −𝑝2∞ )[(𝑎2 −𝑏2 )cos2𝛽−(𝑎+𝑏)2 cos2(𝛽−𝜂)] (𝑎2 +𝑏2 )−(𝑎2 −𝑏2 )cos2𝜂
(4)
(5)
When 𝜉 = 𝜉0 , the elliptical coordinate 𝜂 is related to the polar coordinate 𝛾 by 𝑦
𝑏
tan𝛾 = 𝑥 = 𝑎 tan𝜂
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(6)
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Moreover, to evaluate the shape of the elliptic flaw, the aspect ratio of the elliptic flaw can be written as 𝑏 𝑎
= tanh𝜉0
(7)
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In Fig. 3, the variation of tangential stress and the variation of aspect ratio are presented.
0
( b ) 1 .0 Aspect ratio, b/a
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/ p1
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(a ) 5 4 3 2 1 0 1 2
a 2b
45
/3
2 / 3
0 .8 0 .6 0 .4 0 .2 0 .0
0
1
2
0
3
4
5
Fig. 3 (a) is the variation of the tangential stress under uniaxial compression, and (b) is the variation of the aspect ratio, b/a; when the value of constant 𝜉0 increases to 3, the aspect ratio 𝑏/𝑎 → 1, the elliptic flaw becomes a circle.
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According to [22], the maximum tangential stress on the surface of the elliptic flaw under uniaxial compression can be written as 𝜉0 𝜏𝜂𝜂m = 𝑝1∞ sin𝛽(1 − sin𝛽) when 𝜉0 → 0, equation (7)
𝑏 𝑎
(8)
= tanh𝜉0 ≈ 𝜉0 , the inclination angle 𝛽 = 45° , we may obtain the relationship
between the aspect ratio (b/a) and the tangential stress at the critical failure as 0.207𝑝1∞ (𝑏⁄𝑎)
(9)
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𝜏𝜂𝜂c =
3 Numerical simulations on the damage evolution and crack propagation in rocks with dual elliptic flaws
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3.1 Modeling of the material properties
The microstructure of rocks and the inhomogeneities in rocks give the material a strong heterogeneity. Lan et al. [23] pointed out that the micro-heterogeneity of material plays a significant role in determining both the micromechanical behavior and the macroscopic response, such as damage evolution and crack propagation. In
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RFPA-2D [24], Weibull’s theory was used to solve the problem of heterogeneity of rocks or rock-like materials. Weibull distribution function is given as 𝑚
𝑢 𝑚−1
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𝜑(𝑢) = 𝑢 (𝑢 ) 0
0
𝑢
𝑚
exp *− (𝑢 )+ , 𝑚 > 1 0
(10)
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where u is the element parameter, such as elastic modulus, strength and density, u0 is the even value of the parameter of all the elements, and m is the heterogeneity index (shape parameter) which reflects the material
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heterogeneity.
In Fig. 4, the distribution of meso-elements is presented, which comes to a narrow range around u0 with the
(u )
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increase of heterogeneity index. Fig. 5 shows the distribution of meso-element strength.
m 10.0 m 5 .0 m 3.0 m 1.5
u0
u
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Fig. 4 Weibull distribution of meso-elements with different heterogeneity indices (m)
10000
8000
4000
2000
0
0
50
100
150
200
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Strength/MPa
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Number
6000
Fig. 5 Distribution of meso-element strength
3.2 Numerical model of specimen and computational parameters
In Fig. 6, the numerical model of the specimen with dual pre-existing flaws is presented. The geometry of the
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numerical model is 50 mm×100 mm in scale, and the mesh for the model is 200×400=80,000 elements. Other
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computational parameters adopted are presented in Table 1. Moreover, Mohr-Coulomb criterion is chosen as the failure criterion for the element with the advantages in rock failure. It should be noted that the numerical model is
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different from the layered rock, and in the uniaxial compression the axial load is controlled at 0.002 mm/step, and in the biaxial compression, the axial load is still controlled at 0.002 mm/step, but the lateral loads are controlled at
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0.001 mm/step, 0.002 mm/step and 0.003 mm/step, respectively.
Ⅰ
165.6
15 20
Ⅱ
110.4
25
55.19
25
Ⅲ
2b
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2a
220.8
Elliptic flaw b 45
a 2
0 Strength (MPa )
7 2a 14 Unit : mm
Fig. 6 Numerical model of the specimen and the pre-existing flaw; 2a is the length of the elliptic flaw, and 2b is the width of the elliptic flaw, it should be noted that the flaw inclination angle is 45°.
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Table 1 Control parameters in numerical simulation
Heterogeneity index Meso elastic modulus / GPa Meso compressive strength / MPa Poission’s ratio Internal friction angle/° Compression/Tension ratio
3.3 Calculated results and analysis on damage evolution
3 50 100 0.25 30 10
RockⅡ 5 6 150 0.30 30 10
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RocksⅠand Ⅲ
Control parameters
According to [25], damage evolution in materials usually occurs before the occurrence of macro-crack or at the
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peak load of the specimen. In this section, in order to analyze the damage evolution, we investigate the axial stress and the number of failure elements under uniaxial compression, and the situation of b=2 mm is specifically selected for analysis.
In Fig. 7, the failure elements are accumulated and presented, as well as the curve of axial stress and strain
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under uniaxial compression. Fig. 8 shows the change of shear stress and the accumulation of failure elements under uniaxial compression. From Figs. 7 and 8, it can be seen that at the beginning of the numerical test, there is no
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failure element, and the evolution of damage begins (point A in Fig. 7) when the axial stress increases to 16.6% of the peak stress (PS, point B in Fig. 7), and then the damage grows rapidly with the increase of strain. In Fig. 8a, as
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the load is increased, two “X”-shape bands emerge from the tips of the pre-existing flaws and approach the boundary of the specimen, and the areas of the bands are simply divided into four parts, in which the elements are
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more likely to fail because of the concentration of tensile and shear stresses, and the smaller the area is, the easier the damage grows. This phenomenon is similar to the observation [26]. Moreover, in Fig. 8b, the failure elements
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(damage evolution, mainly tensile failure) occur in rocksⅠand Ⅲ, and they initiate mainly around the pre-existing flaws and near the surface of rockⅡ.
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14
400
12
E
350
F
300 250
b=2mm
8
200 6 150 4
100
2 A 0 0.00
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Axial stress /MPa
10
Failure element number
Stress Failure element B C
50
0.02
0.04
0.06 0.08 0.10 Axial strain (%)
0.12
0.14
0 0.16
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Fig. 7 Curve of axial stress – axial strain under uniaxial compression and the accumulation of failure elements
(a )
S3 0
S4
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( b)
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16.6%PS
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27.45
41.43
89.7%PS
55.91
PS
Shear stress (MPa)
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4.12
43.3%PS
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16.6%PS S2 S1
43.3%PS
: tensile failure
89.7%PS
PS
: compressive failure
Fig. 8 (a) is the change of shear stress, and (b) is the accumulation of failure elements under uniaxial compression (b=2 mm), PS is the peak stress (point B in Fig. 7), 𝐴1 , 𝐴2 , 𝐴3 and 𝐴4 are the four areas of the “X”-shape bands, and the relationship among 𝐴1 , 𝐴2 , 𝐴3 and 𝐴4 is 𝐴1 < 𝐴2 < 𝐴3 < 𝐴4.
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0e-006m/step 1e-006m/step 2e-006m/step 3e-006m/step
40
I
b=2mm
47.2
J
30 K 20
H(B) 10
11.7
0 0.00
0.02
0.04
0.06 0.08 0.10 Axial strain (%)
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Axial stress /MPa
50
0.12
0.14
H(B)
J
K
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I
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(a) Curves of axial stress – axial strain under biaxial compression
(b) Accumulation of failure elements at different points under biaxial compression
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Fig. 9 Curves of axial stress – strain, and accumulation of failure elements at different points under biaxial compression. It should be noted that the axial load is 2e-006 m/step, but the lateral loads in I, J and K are 1e-006
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m/step, 2e-006 m/step and 3e-006 m/step, respectively. Fig. 9 gives the curves of axial stress – axial strain and the accumulation of failure elements at different points
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under biaxial compression. From Fig. 9a, it can be found that the curve slope increases with the increase of lateral load, which means the damage evolution rate could be promoted by increasing the lateral load. In Fig. 9b, we can
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clearly see that the number of compressive failure elements in figures I, J and K is more than that in figure H (with no lateral confinement). Moreover, the peak stress of the specimen is improved with the increase of lateral load, and the smaller is the lateral load, the higher is the peak stress. In Fig. 9a, the highest peak stress (47.2 MPa) is about four times as much as the stress under the condition with no lateral confinement (11.7 MPa). In order to further investigate the damage evolution, we quantify the damage according to the number of failure elements, and the damage (D) is simply defined as 𝐷 = 𝑛/𝑁 where n is the count of element failure, and N is the total count of element.
(11)
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Fig. 10 gives the damage evolution in the specimen with different flaw shapes. It can be seen that the damage evolves slowly when the axial strain is less than 0.025 (%), and then grows rapidly with the increase of axial strain. However, the largest damage is less than 0.05, indicating that macro-cracks in brittle solids occur quickly after the damage evolution, which is in good agreement with the physical truth and investigations [25]. Moreover, it can also be found that when the axial strain is less than 0.058, the damage increases with the increase of short radius (b), but the increment is small. In Fig. 11, the schematic diagram of mesoscopic damage evolution under uniaxial
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compression is presented. From Fig. 11, we can simply and clearly see that in the mesoscopic crack formation, the elements of weak strength begin to failure firstly under compression, which means the beginning of damage evolution; and then the elements of ordinary strength start to fail with the increase of load, which means the expansion of damage evolution. If the elements of high strength fail, meso-cracks occur, propagate and coalesce,
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followed by the occurrence of macro-cracks. 0.05
b=0mm b=2mm b=4mm b=7mm
0.04
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Damage
0.03 0.02
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0.01
0.00 0.00
0.02
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0.01
0.058
0.03 0.04 0.05 Strain (%)
0.06
0.07
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Fig. 10 Curves of damage evolution in specimen with different flaw shapes
damage expansion
damage occurence
Ⅰ
Ⅱ initiation
Ⅲ
Ⅳ
meso - crack propagation
high strength element ordinary strength element weak strength element failure element Fig. 11 Schematic diagram of (mesoscopic) damage evolution under uniaxial compression
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3.4 Calculated results and analysis on crack propagation Propagation of initial defects and initiation of mesoscopic cracks are the damage evolution process [27], and the concentration of damage evolution usually leads to the occurrence of macro-cracks. As mentioned and introduced above, flaw shape has a significant influence on the mechanical properties of rocks. Therefore, in this section, the crack propagation in specimen with different shapes of elliptic flaw under compression is investigated and analyzed.
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In order to facilitate the calculation, the case of b=2 mm is selected. In Fig. 12, the process of crack propagation in specimen with dual elliptic flaws (b=2 mm) is presented.
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coalescence crack
external tip
b=2mm
C'
90
internal tip
G'
60
②
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①
30 0 0.00
0.02
F'
E'
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B' 120
ED
Lateral displacement /μm
150
0.04
E' F'
B' C'
0.06 0.08 0.10 Axial strain (%)
0.12
0.14
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Fig. 12 Process of crack propagation in specimen with pre-existing flaws and the curve of lateral displacement –
Fig. 7.
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axial strain (b=2 mm) under uniaxial compression, where Figures B ′, C ′, E ′ and F ′ correspond to the curve in
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According to Fig. 12, at point B ′, the wing and coalescence cracks initiate and generate at or near the elliptic flaw tips; after the peak stress (from point B ′ to point C ′), the wing cracks propagate to the top and bottom of the specimen, and the coalescence cracks propagate to the neighboring rock (rock Ⅱ) in a nonlinear path. At point C ′, the coalescence cracks propagate to the surface of rockⅡ, and then the propagation bifurcates: one along the foregoing path (path ①), the other along the surface of rock Ⅱ (path ②), as shown in Fig. 12F ′ and G′ . Besides, the curve of lateral displacement – axial strain shows that the lateral displacement increases slowly before point B ′; from point B ′ to point C ′, the growth rate increases; and after point C ′, the lateral displacement has a fast growth rate; but from point E ′ to point F ′ , the growth rate decreases and then increases again, which indicates that the
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speed of crack propagation is not immutable in rocks, and the crack propagation could be influenced by the physical and mechanical properties of rock. We may also find that the coalescence cracks occur at the places where damage mainly concentrates, as shown in Fig. 8. Moreover, some branch cracks occur in the progress of crack propagation, which could also increase the lateral displacement. Table 2 shows the crack patters of crack propagation in rocksⅠand Ⅲ in specimen as the strain is 0.09 (%), and the crack patterns are summarized and classified simultaneously. When the short radius b is equal to or smaller
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than 3 mm, two nonlinear coalescence cracks occur between the pre-existing flaw and rock Ⅱ, while three nonlinear coalescence cracks generate when the short radius b is equal to or larger than 4 mm. Obviously, the initiation and propagation of crack can be influenced by the shape of flaw, which means the tangential stress of the elliptic flaw surface is also consistent with the shape of flaw. When the short radius b is 3.5 mm (2𝑏 = 𝑎), the
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tangential stress of the elliptic flaw surface under uniaxial compression is shown in equation (4), and the variation of tangential stress with polar coordinate 𝛾 is presented in Fig. 3. From the crack patterns, it is also found that some branch cracks emerge from the coalescence cracks, which could either promote or stop the propagation of coalescence cracks. Moreover, we may also find that the cracks (wing and coalescence cracks) initiate and generate
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mostly near the flaw tips, which is consistent with the theory of Griffith 2D. Besides, it should be noted that when the short radius b approaches or equals to 0, the pre-existing flaw degrades into a straight flaw but is still an open
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flaw, and there is no force acting on the flaw surfaces during loading. Table 2 Two types of crack patterns during crack propagation in layersⅠand Ⅲ with different flaw shapes
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Type
Crack pattern,strain : 0.09 (%)
b 0mm
b 1mm
b 2mm
b 3mm
b 4mm
b 5mm
b 6mm
b 7 mm
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①
②
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4 Discussion on the lateral displacement and crack path 4.1 The lateral displacement due to damage evolution The lateral displacement usually comes from the initiation of micro-cracks in materials at the beginning of the loading and the nucleation and propagation of micro-cracks before the occurrence of macro-cracks, so the growth of lateral displacement could reflect the rate of damage evolution in the materials to certain extent. Therefore, we
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could further analyze the damage evolution through the lateral displacement. Fig. 13 shows the curve of lateral displacement – axial strain of specimen with different flaw shapes under uniaxial compression.
b=0mm b=1mm b=2mm b=3mm b=4mm b=5mm b=6mm b=7mm
Lateral displacement /μm
7 6 5 4 3
1 0.01
0.02
ED
0 0.00
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2
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8
0.03 0.04 0.05 Axial strain (%)
0.06
0.07
120
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Accumulated lateral displacement (d) /μm
Fig. 13 Curve of lateral displacement – axial strain, where the largest axial strain is 0.07 (%)
Simulation result Fitting result
110
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100 90 80
d2=9.814·(b/1000)+46.860
70
R2=0.998
d1=4.351·(b/1000)+49.118
60 R2=0.956 50 40
b=3.5mm 0
1 2 3 4 5 6 Short radius (b) of the elliptic flaw/mm
7
Fig. 14 Curve of accumulated lateral displacement – short radius, where it should be noted that the accumulated lateral displacement is the accumulation of the lateral displacements in Fig. 13.
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As can be seen from Fig. 13, when the axial strain is smaller than 0.04, the lateral displacement has an almost linear growth, which indicates that the evolution of damage is stable, but the rates of damage evolution of the specimens with different flaw shapes are different, which increases with the increase of the short radius b. When the axial strain is larger than 0.04, differences of the lateral displacement growth generate: 1) the growth increases as the short radius 𝑏 equals to 4~7 mm; 2) the growth slowes down and then increases again when the short
damage evolution.
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radius 𝑏 = 0~3 mm. The differences imply that the aspect ratio (𝑏⁄𝑎) of the elliptic flaw can influence the
When the semi-major axis (a) equals to 7 mm, the relationship between the accumulated lateral displacement (d) and the semi-minor axis or short radius (b) can be written as 𝑏
𝑑1 = 4.351 ∙ 1000 + 49.118 (0 ≤ 𝑏 < 3.5)
(12)
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{ 𝑏 𝑑2 = 9.814 ∙ 1000 + 46.860 (3.5 ≤ 𝑏 ≤ 7)
According to equation (12) and Fig. 14, the growth of lateral displacement accumulation is linear with the increase of short radius and has a cut-off point when 𝑏 = 3.5; and the slope of 𝑑2 is twice as much as the slope of 𝑑1 , which means the damage evolution when 3.5 ≤ 𝑏 ≤ 7 is more quickly and more likely to lead to the initiation of
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macro-cracks.
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4.2 The crack path and path equation in rocks
In rock specimen or rock-type materials, many investigations [28-31, 5-12] have pointed out that cracks always
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initiate at or near the pre-existing flaw tips and coalesce eventually by tension cracks or shear cracks or both of tension and shear cracks; and the patterns of tension cracks and shear cracks, as well as the types of coalescence,
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depend on the geometry and inclination angle of the flaws and the stress conditions to some extent. However, when the specimen consists of two (or more) types of rocks, the propagation of crack could be influenced by the types of
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rocks, such as the layered rock [9, 12-14] and the one investigated in this paper. In Fig. 15, the crack paths in layered rocks and the specimen with one and two types of rocks are shown. As presented in Fig. 15(1), (2) and (4), the cracks penetrate weak rocks, mostly as inclined shear cracks, which
is in good agreement with field observations. From Fig. 15(3) - (5), it can be clearly seen that the crack paths in specimen have been changed by the flaws, and the crack paths ① and ② are similar in the morphological configuration and propagate in a curve way from the tips of the flaw. Moreover, according to Section 3.4, the shape of flaw could also have an effect on the crack propagation path. When the short radius b is larger than 3.5 mm, one
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more crack initiates from the internal tip of the flaw and propagates to rockⅡ.
(1)
(4)
(3)
external tip flaws
(5)
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(2)
②
①
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internal tip
Fig. 15 Crack paths in layered rocks and the specimens, where Figures (1) and (2) show the field observation, and
2a
y
M( xM , y M ) ①
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Figures (3) -(5) are the numerical results of RFPA-2D.
③
x ② Tangent
Fig. 16 The inverse proportion function model of crack path
In order to describe the path of crack propagation, the inverse proportion function is considered. Here we only
consider the situation of 2𝑏 < 𝑎; therefore, the equation of crack path ① can be written as 𝑘
𝑦 = 𝑥−𝑥 + 𝑦0 0
(13)
where the parameters 𝑘, 𝑥0 and 𝑦0 are unknown. Moreover, the flaw tip, point M is 𝑥 = −𝑎cos𝛽 { M 𝑦M = −𝑎sin𝛽
(14)
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According to [32], we may have the crack path ① equation as 𝑦=−
𝑎2 cos2 𝛽 tan(𝛽+𝜃) 𝑥
− [𝑎cos𝛽 tan(𝛽 + 𝜃) + 𝑎sin𝛽]
(15)
where the parameter 𝜃 is the fracture angle which can be obtained using the maximum tensile-stress criterion. The equation of crack path ② can be obtained via the translation of equation (15). Equations of crack paths ① and ③ are symmetric about origin. It should be noted that the above equations are appropriate in the condition of
very complicated, which will be investigated and discussed in the future.
5 Conclusions
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uniaxial compression. In fact, the crack propagation and the equations of crack paths in rock layers are multiple and
This paper numerically investigates damage evolution and crack propagation in rocks with different shapes of
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flaws under compression, and several important findings are as follows.
(1) Damage occurs firstly in weak rocks, the area around the pre-existing flaw and the area between the flaw and the neighboring rock layer under compression. As the load increases, the “X”-shape bands can be observed at the tips of the pre-existing flaws according to the numerical tests, and the bands of “X” could be divided into four parts
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with different sizes, where the small the part size is, the easier the damage evolves. Moreover, the damage is quantified according to the number of failure elements, which grows slowly when the strain is less than 0.025 and
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grows rapidly with the increase of strain.
(2) Under uniaxial compression, cracks mostly occur as tensile cracks which generate at or near the flaw tips and
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propagate to the maximum compressive stress in a nonlinear path; and under biaxial compression, many shear cracks initiate at the tips of the flaws but the extended directions of the cracks are random. Under uniaxial
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compression, when the coalescence cracks propagate to rock Ⅱ, they may propagate along the foregoing path or along the interface of rock Ⅱ. Moreover, the propagation of crack is not immutable in different rocks, which could
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be influenced by the physical and mechanical properties of the rocks. The crack patterns are summarized and classified into two types according to the flaw shape or aspect ratio, b/a. (3) The relationship between the lateral displacement and damage evolution (before the occurrence of macro cracks) and crack path equations (the situation of 2𝑏 < 𝑎) are investigated and discussed. The evolution of damage could be influenced by the aspect ratio. When the axial strain is smaller than 0.04, the lateral displacement has an almost linear growth, and when the axial strain is larger than 0.04, the growth of lateral displacement changes. Besides, the relationship between the accumulated lateral displacement (d) and the short radius (b) is fitted according to the numerical results, and the equation of crack path is also established, as shown in equations (10) and (13),
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respectively.
Acknowledgements The work described in this paper was substantially supported by the National Program on Major Research Project (No. 2016YFC0701301-02) and Jiangsu Higher Education Institutions for the Priority Academic Development Program (CE02-1-34), and the calculation program was provided by Prof. Xiaochun Xiao at Liaoning
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Technical University, all of which are gratefully acknowledged.
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