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Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure
ACME-302; No. of Pages 13 archives of civil and mechanical engineering xxx (2015) xxx–xxx
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Original Research Article
Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure S.Q. Yang a,*, T. Xu b, L. He c, H.W. Jing a, S. Wen a, Q.L. Yu b a
State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, PR China b Center for Rock Instability & Seismicity Research, Northeastern University, Shenyang 110819, PR China c Department of Civil Engineering, Monash University, Melbourne, Victoria 3800, Australia
article info
abstract
Article history:
To understand deeply the fracture mechanism of brittle rock material, the rock specimen
Received 6 April 2014
containing combined flaws (two square holes and one pre-existing fissure) with seven
Accepted 26 March 2015
different fissure angles is carried out the numerical simulation by RFPA2D. Numerically
Available online xxx
simulated results show a good agreement with the experimental results. The crack coalescence behavior of specimen containing combined flaws under uniaxial compression is
Keywords:
summarized, which is closely dependent to fissure angle. The stable propagation of original
Brittle rock
cracks does not lead to a larger AE event, but the coalescence of new cracks causes a larger AE
Combined flaws
event. The peak strength of specimen containing combined flaws increases with the
Numerical simulation
confining pressure. According to the linear Mohr–Coulomb criterion, the cohesion and
Crack coalescence
internal friction angle of specimen containing combined flaws are obtained, which is found
AE
to take on a distinct nonlinear relation with the fissure angle. The accumulated AE events decreases as the confining pressure increases from 0 to 30 MPa, which results mainly from the restraining of higher confining pressures on the initiation and propagation of tensile cracks at the fissure tips and nearby double squares. # 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
1.
Introduction
Rock is a kind of natural geological material, which usually consists of unequal flaws with different shapes (such as holes, fissures, inclusions) [1–7]. With the development of numerical methods, many simulation softwares were adopted to analyze the crack coalescence process of brittle rock material, such as the code ‘‘FROCK’’ based on displacement discontinuous
method (DDM), 2-D (two-dimensional) particle flow code (PFC), 2-D rock failure process analysis (RFPA2D), boundary element method (BEM) [8–10], cellular automata (CA) [11,12], extended finite element method (X-FEM) [13–15], etc. Mughieda and Omar [16] investigated the stress distribution of rock containing two fissures by using the finite element code with the name of SAP2000. The simulated results showed that tensile stress was mainly responsible for wing crack initiation while the shear stress was responsible for the
* Corresponding author. Tel.: +86 516 83995678. E-mail address: [email protected] (S.Q. Yang). http://dx.doi.org/10.1016/j.acme.2015.03.005 1644-9665/# 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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secondary crack initiation. Vásárhelyi and Bobet [17] adopted FROCK code to model experimental observations on precracked specimens of gypsum. They analyzed mainly the crack initiation stress, the direction and propagation of the new cracks, and type of coalescence in the gypsum containing two open and close fissures. Using PFC2D based on the DEM (Discrete Element Method), Lee and Jeon [18] made a numerical simulation for the crack coalescence characteristics in Hwangdeung granite containing a horizontal fissure and an inclined fissure. Yang et al. [19] carried out a discrete element modeling for the fracture coalescence behavior of red sandstone specimens containing two unparallel fissures under uniaxial compression, which shows a good agreement with the experiment results. On basis of the maximum tensile stress failure criterion, Chen et al. [10] developed a new BEM procedure to predict the crack initiation direction and the crack propagation path in anisotropic rock discs under mixed mode loading, which found a good agreement between crack initiation angles and propagation paths predicted with the BEM and experimental observations reported by previous researchers on isotropic materials. Using a modified G-criterion and DDM, Shen [8] carried out a numerical simulation for the coalescence observed in the tests, which showed that the numerically predicted path and critical load of coalescence were in good agreement with the experimental results. Feng et al. [12] and Pan et al. [11] developed a numerical elasto-plastic cellular automation (EPCA) 2D and 3D code, which can be used to simulate the initiation, propagation and coalescence of cracks in the failure processes of rock material. RFPA2D was developed by Northeastern University, China [20,21], which can reproduce many conventional phenomena of rock mechanics in the laboratory [22–24]. Wong et al. [25,26] analyzed the crack growth in brittle rocks containing a single, triple and multiple fissures under uniaxial compression by using RFPA2D, which found that the fissure length, fissure location and stress interaction between the nearby fissures are important factors affecting the crack initiation, propagation and coalescence behaviors. Tang and Kou [24] conducted numerical simulation on rock-like specimens containing three fissures to investigate the failure mechanism and crack coalescence patterns by RFPA2D. The numerical results replicated most of the phenomena observed in actual experiments, such as initiation and growth of wing and secondary cracks, crack coalescence, and the macro-failure of the specimen. The results obtained in the simulations are in good agreement with experiments [23]. Wang et al. [27] simulated a loading-type failure process and acoustic activities around an underground excavation (model tunnel) in brittle rock by RFPA2D, which was in very in very good agreement with the experimental results. However, in real rock engineering practice, some flaws (such as circular hole, square hole or elliptical hole) all existed, which was very possible to coalesce with the pre-existing fissures under complex stress states. Once the coalescence occurs between the holes and the fissures, rock mass will be able to occur the unstable failure. In the previous studies, the fracture coalescence behaviors of some rock material containing the holes or the fissures have been made some numerical investigations. But less simulations are carried out for real rock
specimen containing combined flaws (i.e. the combination of two square holes and one pre-existing fissure), and the fracture coalescence mechanism of rock material containing combined flaws has not almost been understood. Therefore, the main aim of this research is to analyze the strength and deformation behavior of brittle rock specimen containing combined flaws, and to investigate its fracture coalescence process. Moreover, the influence of confining pressure on strength and deformation failure behavior of rock specimen containing combined flaws is also investigated.
2.
Numerical model and micro-parameters
Before the numerical simulation is discussed, the numerical model and micro-parameters for brittle rock material will be illustrated in this section. RFPA2D is chosen to simulate the fracture coalescence of brittle rock specimens containing preexisting combined flaws. The essential features of RFPA2D are described as follows [20–24]. The RFPA2D code can be used to perform the stress analysis for each element by FEM. Using tensorial notation, the mechanical equilibrium equation for the solid is expressed as s i j; j ¼ Fi ;
i; j ¼ 1; 2; 3;
(1)
where sij is the stress tensor (Pa) in the solid and Fi the component of the body force (N/m3). The constitutive equation defines the relation between the total bulk stress components (Pa), sij and strain components, eij. The stress–strain law is given by s i j ¼ Di jkl ei j
i; j ¼ 1; 2; 3;
(2)
where Dijkl is the elasticity tensor (Pa), and is related to the Young's modulus E and Poisson's ratio v for isotropic elastic media, which is regarded with damage initiation and development. eij = (Ui,j + Uj,i)/2 and Ui represents the displacement vector of the solid. In RFPA2D, the model is discretized into a large number of small elements. Taking into account the heterogeneity of real rock material, element local mechanical microscopic parameters are assumed to follow a Weibull distribution [28], i.e. Eq. (3): PðuÞ ¼
m m u m1 u exp u 0 u0 u0
(3)
where u is a random variable, representing strength, elastic modulus, etc. u0 is the mean value of element parameters for the whole specimen and m is the shape parameter of distribution function which is referred to as homogeneity index. Fig. 1 shows the influence of the shape parameter m on the distribution function, which indicates that a smaller m means a more heterogeneous rock material and vice versa. However for the same m, the Weibull's randomness indicated that the numerical simulated results of different rock specimens are very approximate, but not identical [29]. An elastic constitutive model with linear behavior is employed for all elements, which have been assigned different strength and elastic modulus according to the heterogeneity of rock material. The elastic constitutive relation for an element under uniaxial compressive stress and tensile stress is
Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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P (θ )
an element in order to capture the principal modes of damage and fracture of rock, respectively, and the tensile stress criterion is considered preferentially. When its stress state satisfies one criterion, the element begins to accumulate damage in tension or shear. For tension, the maximum tensile stress criterion is chosen as the strength criterion for the elements:
m m m
m m
m
s t f tp
θ Fig. 1 – The influence of the shape parameter m on the distribution function.
illustrated in Fig. 2 [30]. In Fig. 2, ftr and ftp are respectively the residual and peak tensile strength of the elements, which is given as ftr = lftp = lE0et0. l is the residual strength coefficient. fcr and fcp are respectively the residual and peak compressive strength of the elements. It is assumed ftr/ftp = fcr/fcp = l. et0 and etu are the tensile threshold strain of damaged element and the ultimate tensile strain of the failed element. The ultimate tensile strain is defined as etu = j et0, where j is the ultimate tensile strain coefficient. ec0 and ecu are the compressive threshold strain of damaged element and the ultimate compressive strain of the failed element. The ultimate compressive strain is defined as ecu = d ec0, where d is the ultimate compressive strain coefficient. When the stress of an element exceeds the strength criterion, the element is assumed to be irreversibly damaged. The stiffness and strength of damaged elements will be reduced to the residual strength level. In elastic damage mechanics, the elastic modulus of the element may degrade gradually as damage progresses, and the elastic modulus of damaged element is defined as follows [31]: E ¼ ð1 DÞE0
(4)
where D represents the damage variable. E and E0 are elastic modulus of the damaged and undamaged element, respectively. Here, the element and its damage are assumed to be isotropic elastic. Therefore the parameters E, E0 and D are all scalar. In this code, the maximum tensile stress criterion and the Mohr–Coulomb criterion are used to check the stress state of
σ Compression
f cp
E0 ε tu
ε t0
f cr − f tr
ε c0
ε cu ε
− f tp
Tension Fig. 2 – Elastic damage constitutive model for element under the compression and tension [30].
(5)
where st is the tensile strength of elements. Correspondingly, the damage variable D under tension can be described as follows [30]: 8 e > et0 > < 0; le (6) D ¼ 1 t0 ; etu e < eto > e : 1; e etu Under multi-axial stress states the element is still damaged in tensile mode when the equivalent major tensile strain e attains the above-threshold strain et0 . The equivalent principal strain e is defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (7) e ¼ he1 i2 þ he2 i2 þ he3 i2 where e1, e2 and e3 are three principal strains, and h i is a function defined as follows: x x 0; (8) hxi ¼ 0 x < 0: The constitutive law for an element subjected to multiaxial stresses can be easily obtained by replacing the strain e in Eq. (6) with equivalent strain e. The damage variable is expressed as: 8 e > et0 ; > <0 le (9) D ¼ 1 t0 etu < e et0 ; > e : 1 e etu : In order to describe the damage of an element when it is under compressive and shear stress, the Mohr–Coulomb criterion, expressed as follows, is chosen to define the second damage threshold: s1
1 þ sin ’ s 3 f c0 1 sin ’
(10)
where s1 and s3 are the major and minor principal stress, respectively, fc0 is the uniaxial compressive strength and w is the internal friction angle of the mesoscopic element. In the same way as for uniaxial tension, when the element is under uniaxial compression but damaged according to the Mohr– Coulomb criterion, the expression for damage variable D can be described as: ( 0 e < ec0 ; lec0 D¼ (11) e ec0 : 1 e When the element is under a multi-axial stress state and its strength satisfies the Mohr–Coulomb criterion, the maximum principal strain (i.e. maximum compressive principal strain) may be evaluated at the peak value of the maximum principal stress (i.e. maximum compressive principal stress) ec0:
Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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ec0 ¼
1 E0
f c0 þ
1 þ sin ’ s 3 nðs 2 þ s 3 Þ 1 sin ’
(12)
where s2 is the intermediate principal stress. In RFPA2D, the loads were applied by incrementally raising the vertical loads (under displacement control) until the model collapsed. At each step, elements may undergo damage due to excessive tension or yielding from the Mohr–Coulomb criterion. The stress and deformation distribution throughout the specimen is then adjusted instantaneously after each element damages to reach the equilibrium state. It is noted that the damaged elements are not removed from the domain, but the material properties (Strength, Young' modulus) will be very low, something like the air elements. The color of damaged elements is changed to be black automatically. At points which undergo increased load due to a redistribution of the stresses, further damages may be caused. The process is repeated until no further failures in the elements occur [22]. The numerical simulation for rock specimens are carried out on 2-D plane strain model. In the simulation, an external displacement is applied to the top of the rock specimens at a constant rate of 0.0032 mm/step (the strain rate is 2.0 105 step1) in the axial direction. It should be noted that during the experiment, the strain rate is about 8.125 106 s–1. Therefore, the experimental and numerical strain rates all belong to quasistatic loading rate, which has almost no any effect on the mechanical behavior of rock material. We computed the stress and strain in each mesolevel element. The lower boundary of numerical specimens is kept fixed, as is generally done during laboratory tests. In the present study, all the numerical specimens, with geometry of W H = 80 mm 160 mm in size, were discretized into 160 320 = 51,200 elements. The geometry of brittle specimens containing pre-existing combined flaws is described in Fig. 3. The geometry of pre-existing flaws is defined by five geometrical parameters – single fissure angle a
Table 1 – The micro-parameters of RFPA2D simulation for brittle rock specimens in this research. Micro-parameters
Value
Homogeneity index, m Average value of elastic modulus, E0, GPa Average value of compressive strength, fcp, MPa Internal friction angle, F8 Ratio of compressive to tensile strength, C/T Poisson's ratio, m Ultimate tensile strain coefficient, j Ultimate compressive strain coefficient, d Residual strength coefficient, l Loading rate in the present study, mm/step
10 35 370 40 8 0.25 1.5 200 0.1 0.0032
(the angle of single fissure with the direction of the horizontal direction), single fissure length 2a, ligament length 2b between two squares , ligament angle b between two squares , square side length 2c. In this research, 2a, 2b and 2c are fixed to 15, 40 and 14 mm, respectively. The b is fixed to 608, and a is increased from 08 to 908 at an interval of 158. As it is well-known to all of us, it is very difficult to determine micro-parameters by experiment [19]. However, in order to validate the micro-parameters used in the numerical modeling, it is essential to establish a correlation between the macro-behavior (i.e. the axial stress–axial strain curve, the peak strength and elastic modulus, the ultimate failure mode and the crack coalescence process) and micro-parameters. During the calibration process, the macroscopic behavior of intact rock obtained by experiment [32] was used in this research to calibrate the micro-parameters. The macroscopic results obtained by numerical simulation after each trial was used to check the micro-parameters. This process was repeated until the numerical results achieved a good agreement with the experimental results. Table 1 listed the micro-parameters of RFPA2D simulation for brittle rock specimens.
Fig. 3 – Geometry of pre-existing rock specimen in this research. Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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In addition, it is noted that in the RFPA2D simulation, the mesh dependency is unavoidable, which has been detailed in previous publications [33]. However, if the rock specimen is simulated with the relatively small element size, the mesh dependency is ignorable to some degree. Under this small size of elements, the mesh dependency may affect the quantity of the numerical results. However, it does qualitatively reflect the basic behavior of rock that is shown in the laboratory experiments. We just use an optimal element size here to address the crack issues here [33].
3. Simulated results under uniaxial compression Numerical simulated results on the mechanical behavior of brittle rock specimens are herein discussed and compared with the experimental results. Fig. 4 shows the comparison between experimental and numerical results of intact rock specimen under uniaxial compression. During the experiment, in order to decrease the effect of the end friction effects on testing results, two rigid steel blocks (33 mm 83 mm 15 mm) were placed between the loading frame and rock specimen and two antifriction gaskets between two rigid steel blocks and the end surface of specimen [32]. In the previous numerical study [23], Tang et al. have adopted RFPA2D to carry out the numerical simulation on the rock material with five different loading platens. Their simulated results have demonstrated that the loading platens have a minor effect on the peak strength and axial stress–strain curves of brittle rock material. Therefore, in the present numerical simulation, we used a zero friction coefficient between the walls and boundary particles [23], which are in a good agreement with the experiment. From Fig. 4, it can be seen that the numerically simulated curve under uniaxial compression agrees well with the experimental curve, except for the initial phase of the curve. The experimental stress–strain curve of intact specimen at low stress levels shows the downward concave and the initial
210
Numerical result Experimental result
Axial stress σ1 / MPa
180 150
Intact specimen
120 90 60 30 0 0.0
1.5
3.0
4.5
Axial strain ε1 / 10
6.0
7.5
-3
Fig. 4 – Stress–strain curves from the experimental test [32] and the numerical prediction (uniaxial displacement control) of intact rock specimen.
5
non-linear deformation, which mainly results from the closure of some primary cracks, pores and voids in tested rock specimen. It should be noted that in the experimental stress–strain curve, there was a distinct stress drop, which resulted from the gradual coalescence of tensile cracks in the process of deformation [32]. However, the axial stress drop was not observed in the curve obtained by numerical simulation due to obvious brittle characteristics. It is noted that, in RFPA2D code, although the mechanical response of a single mesoscopic element is linear, the macroscopic behavior of a numerical specimen (containing a lot of mesoscopic elements) can be non-linear. The nonlinear material behavior arises from its heterogeneous material properties. The more heterogeneous the material properties are, the stronger is the non-linearity in the stress– strain response. The numerical simulation method can reproduce better the complete process of deformation and failure of brittle rock specimen, especially the localization of deformation and failure [30]. Fig. 5 further illustrates the comparison between experimental and numerical stress–strain curves of brittle rock specimen containing combined flaws under uniaxial compression. In accordance with Fig. 5, it can be seen that the numerical simulated stress–strain curves of specimen containing combined flaws are in a good agreement with the experimental stress–strain curves. Moreover, it should be noticed that the stress drop in the numerical simulated stress– strain curves of the flawed specimen also agrees very well with the experimental results, which indicates that numerical simulated coalescence processes are also similar to the experimental crack coalescence processes. Besides, if not taking into account the initial nonlinear deformation, the simulated elastic modulus of the specimen is approximately equal to the experimental elastic modulus. Fig. 6 shows the uniaxial compressive strength in brittle rock specimens containing combined flaws by numerical simulation and experiment. It is very clear that the peak stress of rock specimen simulated by RFPA2D has the similar trend with the increase of fissure angle a. But for a = 308 and 608, the simulated uniaxial compressive strength is higher about 21% than that obtained by experiment, which can result from the numerical simulation are two-dimensional analysis and can not reflect fully three-dimensional physical phenomenon of heterogeneous rock material [18]. However, for a = 908, the simulated uniaxial compressive strength is approximately equal to that obtained by experiment. From Fig. 6, it can also be seen that the uniaxial compressive strength of rock specimen containing combined flaws takes on a nonlinear relation, i.e. first decreases for a = 0–308 and then increases for a = 30–908 with increasing fissure angle a. Fig. 7 presents the effect of fissure angle on ultimate failure mode of specimen containing combined flaws obtained by numerical simulation under uniaxial compression. For comparison, the experimental failure mode is also given in Fig. 7. In accordance with the compared result, it is clear that the ultimate failure mode in the numerical specimen is very approximate to that in the experimental specimen, e.g. for a = 608, which indicates that the RFPA2D software can be used to simulate the crack coalescence process of brittle rock material [24,25].
Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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Axial stress σ1 / MPa
60 Axial stress σ1 / MPa
(b)
Numerical result Experimental result
(a)
45
30 Specimen containing combined flaws (α= 30°)
15
0
Numerical result Experimental result
60 A
40 B
20
Specimen containing combined flaws (α= 60°)
0 0.0
1.0
2.0
3.0
Axial strain ε1 / 10
4.0
0.0
1.0
2.0
3.0
Axial strain ε1 / 10
-3
Numerical result Experimental result
(c) 100 Axial stress σ1 / MPa
80
4.0
-3
(d)
80
3 a
2
60 b
2 40 Specimen containing combined flaws (α= 90°)
20
Stress drop A
Stress drop B
0 0.0
1.0
2.0
3.0
Axial strain ε1 / 10
4.0
5.0
-3
Crack coalescence in the specimen containing combined flaws (α= 60°) due to larger stress drops after peak strength (by experiment)
Fig. 5 – Comparison between experimental and numerical stress–strain curves of rock specimen containing combined flaws under uniaxial compression. In this figure, the crack coalescence of specimen containing combined flaws (a = 608, b = 608) obtained by experiment is also shown due to larger stress drops after peak strength.
Uniaxial compressive strength σ c / MPa
120 105
Experimental data Numerical result
90 Specimen containing combined flaws (β= 60°)
75 60 45 30 0
15
30
45
60
75
90
Fissure angle α / °
Fig. 6 – Effect of fissure angle on uniaxial compressive strength of rock specimen containing combined flaws (b = 608) obtained by experiment and numerical simulation.
Table 2 summarizes the crack coalescence process of rock specimen obtained by RFPA2D simulation under uniaxial compression. In accordance with Table 2, we can analyze detailed the effect of fissure angle on the crack coalescence process of rock specimen containing combined flaws. When a = 0–158, the crack coalescence process can be illustrated as follows. Tensile crack in the specimen is initiated from a certain distance from the fissure tips, which is a good agreement with the experiment findings for the crack coalescence of rock specimen containing a single fissure (a = 158) [7]. Meantime, tensile cracks are also initiated from the top and bottom edge of two squares and bottom edge of two squares. All the cracks all propagate along the direction of axial stress. Afterwards, with the increase of axial deformation, the specimen undergoes an obvious coalescence in the ligament area between fissure and double squares and bottom edge of two squares. Finally, the specimen takes place the unstable failure and the cracks originate from the right side of and the left side of square . square However, the cracks in the specimens containing combined flaws for a = 30–908 are initiated from the top and bottom edge of two squares and bottom edge of two squares, which is
Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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Fig. 7 – Effect of fissure angle on ultimate failure mode of brittle rock specimen containing combined flaws (b = 608) obtained by experiment and numerical simulation under uniaxial compression.
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Table 2 – Crack coalescence process of brittle rock specimen containing combined flaws obtained by RFPA2D simulation under uniaxial compression. a
b
08
608
158
608
308
608
458
608
608
608
758
608
908
608
Crack coalescence process of specimen containing combined flaws
Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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similar to those for a = 0–158, but the cracks nearby the fissure for a = 30–908 are initiated at the upper and under tip of fissure. It should be noted that the crack coalescence process of specimens containing combined flaws for different fissure angles (a = 30–758) are very similar. Here, we take the specimen containing combined flaws (a = 608) for the examples to analyze in detail the crack coalescence process. Fig. 8 further illustrates numerically simulated results of AE events and accumulated AE events of rock specimen containing combined flaws (a = 608). In Fig. 8, all of the symbols correspond to those denoted in Table 2. From Fig. 8 and Table 2, it can be seen when the axial stress is loaded to point a (s1 = 31.46 MPa = 48.3%sc), the specimen (a = 608) begins to initiate tensile crack 1 at the upper and under tip of fissure, the top and bottom edge of two squares. The initiation of tensile crack 1 linked to 118 AE events. Afterwards, as the deformation progressed, the length of crack from the top and bottom edge of two squares begins to propagate toward the end surface of the specimen along the direction of major principal stress. But the tensile crack 1a, 1c–1d, and 1f did not propagate a lot, which was due to the stress concentration field between fissure and double squares. Furthermore, the propagating speed of tensile crack 1b and 1e was higher than that of tensile crack 1a, 1c–1d, and 1f. When the specimen was loaded to peak point b (s1 = 65.13 MPa = 100%sc), crack 2a coalesced and the upper tip between the left bottom corner of square of fissure, which led to an obvious stress drop from 65.13 to 57.36 MPa. The coalescence of crack 2a was linked to 234 AE events. The continuous increase of axial deformation led to that the axial stress of the specimen began to increase slowly with a lower deformation modulus than that before point b, which resulted from some minor damages of the supporting structure in the specimen. When the specimen was loaded to point c (s1 = 61.78 MPa = 94.9%sc) after the peak strength, crack 2b coalesced between the under tip of fissure and the right top corner of square , which led to an obvious stress drop from 61.78 to 50.39 MPa. The coalescence of crack 2b was
linked to 136 AE events. After that, with the increase of axial deformation, the axial stress of the specimen began to increase slowly with a lower deformation modulus than that before point c, which resulted from more damages of the supporting structure in the specimen. When the specimen was loaded to point d (s1 = 55.31 MPa = 84.9%sc) after the peak strength, a larger stress drop from 55.31 to 32.64 MPa was observed, which resulted from the propagation of crack 3 and more AE events were also observed. It should be noted that and crack 3a was initiated from the left top corner of square propagated toward the left edge of the specimen in the downward direction. At the same time, crack 3b was also in the initiated from the right bottom corner of square upward direction. Afterwards, with the increase of axial deformation, the specimen was difficult to support higher axial stress. When the axial stress was decreased to point e, a minor stress drop was also observed and the corresponding AE events were 170, which resulted from the propagation of crack 3b. When the axial load was decreased to point f (s1 = 30.60 MPa = 47%sc) after the peak strength, the specimen occurred an obvious stress drop from 30.60 to 16.33 MPa, which resulted from the formation of far-field crack nearby crack 3b. After the last stress drop, the specimen entered rapidly the stage of residual strength. From Fig. 8, it is very clear that the increase of AE events and accumulated AE events in the specimen are all closely related to distinct stress drops. It should be noted that the stable propagation of original cracks does not lead to a larger AE event, the coalescence of new cracks results in a larger AE event. Moreover, the maximum AE event in the specimen does not induce at the peak strength, but at a certain position after the peak stress. In accordance with Figs. 5 and 7 and Table 2, it can be seen that in the experimental specimen (a = 608), cracks 2a and 2b are simultaneously initiated in the deformation process, which leads to a distinct stress drop A (see Fig. 5b and d). Whereas in the numerical specimen, crack 2a is first initiated
60 Axial stress σ1 / MPa
b
AE events Axial stress Acc umulated AE events
c
3000
d
50
2500
40
2000
a
Specimen containing combined flaws (α= 60°)
30
e f
1500 1000
20
500
10
AE events (n), Accumulated AE events (n)
3500
70
1b 3b
2a 1d 1a
1f 1c 2b 3a
1e
0
0 0
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
Axial strain ε1 / 10-3 Fig. 8 – Numerically simulated results of AE events and accumulated AE events of rock specimen containing combined flaws (a = 608) under uniaxial compression (RFPA2D). All of the symbols correspond to those denoted in Table 2. Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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and then crack 2b is initiated, which results in two axial stress drops as shown in Fig. 5. In the experiment, crack 3 originates from the right side of square , which also leads to a distinct stress drop B (see Fig. 5b and d). But in the numerical specimen, the cracks are initiated from the left top corner of square and then from the right bottom corner of square , which also results in two larger stress drops as shown in Fig. 8. When a = 908, even though the crack coalescence process obtained by experiment and numerical simulation are a little different (see Fig. 7), the axial stress–strain curves shown in Fig. 5c are very similar. In the experimental specimen, the coalescence occurs in the ligament area between the fissure and two squares, but which are not observed in the numerical specimen. Besides, tensile crack initiated from the right in the experimental specimen is bottom corner of square also observed in the numerical specimen.
4. Simulated results under different confining pressure To investigate the influence of confining pressure on strength and deformation failure behavior of brittle rock specimen containing combined flaws, three different confining pressures
of 10, 20 and 30 MPa are applied to the specimen. Fig. 9 presents numerically simulated axial stress–strain curves of rock specimen containing combined flaws under different confining pressures (RFPA2D). From Fig. 9, it can be seen that the strength and deformation behavior of specimen containing combined flaws depend on not only the confining pressure, but also fissure angle a. Fig. 10a illustrates the effect of fissure angle a on triaxial peak strength sS of rock specimen containing combined flaws (RFPA2D simulation). From Fig. 10a, it can be seen that the effect relation of fissure angle a on triaxial peak strength sS under different confining pressures are very similar to that under uniaxial compression. In a word, the peak strength of the specimen containing combined flaws at the same confining pressure first decreases and then increases with the increasing fissure angle a. However for the specimen containing combined flaws with the same fissure angle, as the increase of confining pressure from 0 to 30 MPa, the peak strength and elastic modulus of specimen all increase gradually. For example, the peak strengths of specimen containing combined flaws with a = 308 are 55.11, 66.8, 82.1 and 97.3 MPa at the confining pressures of 0, 10, 20 and 30 MPa, respectively. The linear Mohr–Coulomb criterion can be used to describe the relation between the triaxial peak strength sS of specimen containing
100
100
(a) α= 0°
(b) α= 30° 80
60
σ3 =30 MPa
40
σ3 =20 MPa
20
σ3 =10 MPa
Axial stress σ1 / MPa
Axial stress σ1 / MPa
80
60
σ3 =30 MPa
40
σ3 =20 MPa σ3 =10 MPa
20
σ3 =0 MPa
σ3 =0 MPa
0
0 0.0
1.5
3.0
4.5
Axial strain ε1 / 10
6.0
0.0
3.0
4.5
Axial strain ε1 / 10
6.0
-3
140
120
(c) α= 60°
(d) α= 90°
120 Axial stress σ1 / MPa
100 Axial stress σ1 / MPa
1.5
-3
80 60
σ3 =30 MPa
40
σ3 =20 MPa σ3 =10 MPa
20
100
σ3 =30 MPa
80 σ3 =20 MPa
60 σ3 =10 MPa
40 20
σ3 =0 MPa
σ3 =0 MPa
0
0 0.0
1.5
3.0 Axial strain ε1 / 10
4.5 -3
6.0
0.0
1.5
3.0
4.5
6.0
Axial strain ε1 / 10-3
Fig. 9 – Numerically simulated axial stress–strain curves of brittle rock specimen containing combined flaws under different confining pressures. Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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0MPa 20MPa
130
(b) 50
10MPa 30MPa
Peak strength parameters
Triaxial peak strength σS / MPa
(a) 145
115 100 85 70 55 40
Cohesion Internal friction angle
40 30 20 10 0
0
15
30
45
60
75
90
15
0
30
45
60
75
90
Fissure angle α / °
Fissure angle α / °
Fig. 10 – Effect of fissure angle a on triaxial peak strength and strength parameters of brittle rock specimen containing combined flaws.
combined flaws and the confining pressure s3. As it is wellknown to us, linear Mohr–Coulomb criterion can be expressed with the following equation, i.e. Eq. (13). s S ¼ s 0 þ qs 3 ¼
2C cos f þ s 3 ð1 þ sin fÞ 1 sin f
In accordance with Eq. (13), the cohesion and internal friction angle of specimen containing combined flaws in accordance with linear Mohr–Coulomb criterion (by RFPA2D simulation) can be obtained, which are listed in Table 3. From Table 3, it can be seen that there are good linear regression coefficients (R = 0.980–0.998) for the relation between the triaxial peak strength of specimen containing combined flaws and the confining pressure. Fig. 10b further plots the influence of fissure angle on the cohesion and internal friction angle of specimen containing combined flaws. From Fig. 10b and Table 3, it can be seen that the values of C range from 21.00 to
(13)
where s0 is usually regarded as the uniaxial compressive strength (UCS) of rock material; q is an influence coefficient of confining pressure on sS of rock. C and F, respectively, represent the cohesion and internal friction angle of rock material.
Table 3 – Strength parameters of rock specimen containing combined flaws in accordance with linear Mohr–Coulomb criterion. Strength parameters
a = 08
a = 158
a = 308
a = 458
a = 608
a = 758
a = 908
s0 (MPa) q C (MPa) F (8)
60.62 1.33 26.28 8.1
59.76 1.43 24.99 10.2
50.05 1.42 21.00 10.0
60.02 1.40 25.36 9.6
66.03 1.76 24.89 16.0
84.75 1.58 33.71 13.0
101.33 1.18 46.64 4.7
2500
2000 Accumulated AE events (n)
σ3 =0 MPa
Specimen containing combined flaws (α= 30°)
σ3 =10 MPa
1500
1000 σ3 =30 MPa σ3 =20 MPa
500
0 0.0
1.5
3.0 Axial strain ε1 / 10
4.5
6.0
-3
Fig. 11 – Effect of confining pressure on accumulated AE events of rock specimen containing combined flaws. Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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Fig. 12 – Effect of confining pressure on ultimate failure mode of rock specimen containing combined flaws (a = 308).
46.64 MPa, while the values of F are between 4.78 and 16.08. Moreover, the values of C and F of the specimen are closely dependent to fissure angle. The value of C first decreases from 26.28 to 21.00 MPa as the fissure angle increases from 08 to 308, while increases from 21.00 to 46.64 MPa as the fissure angle increases from 308 to 908. While, the value of F first increases from 8.18 to 10.28 as the fissure angle increases from 08 to 158, and keeps basically a constant 108 in the range of a = 15–458. However, the value of F increases from 9.68 to 168 as the fissure angle increases from 458 to 608, and decreases from 168 to 4.78 as the fissure angle increases from 608 to 908. Fig. 11 shows the effect of confining pressure on accumulated AE events of rock specimen containing combined flaws (a = 308) obtained by RFPA2D. The accumulated AE events of specimen containing combined flaws for other fissure angles are approximate to that for a = 308. From Fig. 11, it is clear that the accumulated AE events decreases as the confining pressure increases from 0 to 30 MPa, which results mainly from the restraining of higher confining pressures on the initiation and propagation of tensile cracks at the fissure tips and nearby double squares (Fig. 12). In order to result in the damage failure of the elements in the numerical specimen, higher axial stresses are also required for higher confining pressures.
5.
Conclusions
(1) Numerically simulated results (including the stress–strain curve, the peak strength and the crack evolution mode) for rock specimen containing combined flaws (two square holes and one pre-existing fissure) under uniaxial compression show a good agreement with the experimental results. The crack evolution behavior of specimen containing combined flaws obtained by RFPA2D simulation under uniaxial compression are summarized, which is closely dependent to fissure angle. When a = 0–158, tensile crack in the specimen is initiated from a certain distance from the fissure tips. Meantime, tensile cracks are also initiated . But in from the top and bottom edge of two squares
the specimens containing combined flaws for a = 30–908, the cracks are initiated from the top and bottom edge of , which is similar to those for a = 0–158, two squares but the cracks nearby the fissure for a = 30–908 are initiated at the upper and under tip of fissure. After crack initiation, with the increase of axial deformation, the specimen undergoes an obvious coalescence in the ligament area . Finally, the between fissure and double squares specimen takes place the unstable failure. (2) The peak strength of specimen containing combined flaws increases with the confining pressure. According to the linear Mohr–Coulomb criterion, the cohesion and internal friction angle of specimen containing combined flaws are obtained, which is found to take on a distinct nonlinear relation with fissure angle. The value of C of specimen containing combined flaws first decreases from 26.28 to 21.00 MPa as the fissure angle increases from 08 to 308, while increases from 21.00 to 46.64 MPa as the fissure angle increases from 308 to 908. While, the value of F first increases from 8.18 to 10.28 as the fissure angle increases from 08 to 158, and keeps basically a constant 108 in the range of a = 15–458. However, the value of F increases from 9.68 to 168 as the fissure angle increases from 458 to 608, and decreases from 16 to 4.78 as the fissure angle increases from 608 to 908. (3) The stable propagation of original cracks does not lead to a larger AE event, but the coalescence of new cracks causes a larger AE event. The accumulated AE events decreases as the confining pressure increases from 0 to 30 MPa, which results mainly from the restraining of higher confining pressures on the initiation and propagation of tensile cracks at the fissure tips and nearby double squares. At the same time, it should be noted that in the present study, we only simulate the fracture coalescence behavior at a laboratory scale, but due to obvious scale effect of rock material, the simulated results at a laboratory scale is very difficult to be predicted directly the strength of rock mass at a engineering scale, therefore in the future, the scale effect of rock containing combined flaws will be further strengthened.
Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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Acknowledgments This research was supported by the National Basic Research 973 Program of China (Grant No. 2014CB046905), the Fundamental Research Funds for the Central Universities (China University of Mining and Technology) (Grant No. 2014YC10), for which the authors are very grateful. The authors would like to express their sincere gratitude to the editor and two anonymous reviewers for their valuable comments, which have greatly improved this paper.
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Available online at www.sciencedirect.com
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Original Research Article
Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure S.Q. Yang a,*, T. Xu b, L. He c, H.W. Jing a, S. Wen a, Q.L. Yu b a
State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, PR China b Center for Rock Instability & Seismicity Research, Northeastern University, Shenyang 110819, PR China c Department of Civil Engineering, Monash University, Melbourne, Victoria 3800, Australia
article info
abstract
Article history:
To understand deeply the fracture mechanism of brittle rock material, the rock specimen
Received 6 April 2014
containing combined flaws (two square holes and one pre-existing fissure) with seven
Accepted 26 March 2015
different fissure angles is carried out the numerical simulation by RFPA2D. Numerically
Available online xxx
simulated results show a good agreement with the experimental results. The crack coalescence behavior of specimen containing combined flaws under uniaxial compression is
Keywords:
summarized, which is closely dependent to fissure angle. The stable propagation of original
Brittle rock
cracks does not lead to a larger AE event, but the coalescence of new cracks causes a larger AE
Combined flaws
event. The peak strength of specimen containing combined flaws increases with the
Numerical simulation
confining pressure. According to the linear Mohr–Coulomb criterion, the cohesion and
Crack coalescence
internal friction angle of specimen containing combined flaws are obtained, which is found
AE
to take on a distinct nonlinear relation with the fissure angle. The accumulated AE events decreases as the confining pressure increases from 0 to 30 MPa, which results mainly from the restraining of higher confining pressures on the initiation and propagation of tensile cracks at the fissure tips and nearby double squares. # 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
1.
Introduction
Rock is a kind of natural geological material, which usually consists of unequal flaws with different shapes (such as holes, fissures, inclusions) [1–7]. With the development of numerical methods, many simulation softwares were adopted to analyze the crack coalescence process of brittle rock material, such as the code ‘‘FROCK’’ based on displacement discontinuous
method (DDM), 2-D (two-dimensional) particle flow code (PFC), 2-D rock failure process analysis (RFPA2D), boundary element method (BEM) [8–10], cellular automata (CA) [11,12], extended finite element method (X-FEM) [13–15], etc. Mughieda and Omar [16] investigated the stress distribution of rock containing two fissures by using the finite element code with the name of SAP2000. The simulated results showed that tensile stress was mainly responsible for wing crack initiation while the shear stress was responsible for the
* Corresponding author. Tel.: +86 516 83995678. E-mail address: [email protected] (S.Q. Yang). http://dx.doi.org/10.1016/j.acme.2015.03.005 1644-9665/# 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
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secondary crack initiation. Vásárhelyi and Bobet [17] adopted FROCK code to model experimental observations on precracked specimens of gypsum. They analyzed mainly the crack initiation stress, the direction and propagation of the new cracks, and type of coalescence in the gypsum containing two open and close fissures. Using PFC2D based on the DEM (Discrete Element Method), Lee and Jeon [18] made a numerical simulation for the crack coalescence characteristics in Hwangdeung granite containing a horizontal fissure and an inclined fissure. Yang et al. [19] carried out a discrete element modeling for the fracture coalescence behavior of red sandstone specimens containing two unparallel fissures under uniaxial compression, which shows a good agreement with the experiment results. On basis of the maximum tensile stress failure criterion, Chen et al. [10] developed a new BEM procedure to predict the crack initiation direction and the crack propagation path in anisotropic rock discs under mixed mode loading, which found a good agreement between crack initiation angles and propagation paths predicted with the BEM and experimental observations reported by previous researchers on isotropic materials. Using a modified G-criterion and DDM, Shen [8] carried out a numerical simulation for the coalescence observed in the tests, which showed that the numerically predicted path and critical load of coalescence were in good agreement with the experimental results. Feng et al. [12] and Pan et al. [11] developed a numerical elasto-plastic cellular automation (EPCA) 2D and 3D code, which can be used to simulate the initiation, propagation and coalescence of cracks in the failure processes of rock material. RFPA2D was developed by Northeastern University, China [20,21], which can reproduce many conventional phenomena of rock mechanics in the laboratory [22–24]. Wong et al. [25,26] analyzed the crack growth in brittle rocks containing a single, triple and multiple fissures under uniaxial compression by using RFPA2D, which found that the fissure length, fissure location and stress interaction between the nearby fissures are important factors affecting the crack initiation, propagation and coalescence behaviors. Tang and Kou [24] conducted numerical simulation on rock-like specimens containing three fissures to investigate the failure mechanism and crack coalescence patterns by RFPA2D. The numerical results replicated most of the phenomena observed in actual experiments, such as initiation and growth of wing and secondary cracks, crack coalescence, and the macro-failure of the specimen. The results obtained in the simulations are in good agreement with experiments [23]. Wang et al. [27] simulated a loading-type failure process and acoustic activities around an underground excavation (model tunnel) in brittle rock by RFPA2D, which was in very in very good agreement with the experimental results. However, in real rock engineering practice, some flaws (such as circular hole, square hole or elliptical hole) all existed, which was very possible to coalesce with the pre-existing fissures under complex stress states. Once the coalescence occurs between the holes and the fissures, rock mass will be able to occur the unstable failure. In the previous studies, the fracture coalescence behaviors of some rock material containing the holes or the fissures have been made some numerical investigations. But less simulations are carried out for real rock
specimen containing combined flaws (i.e. the combination of two square holes and one pre-existing fissure), and the fracture coalescence mechanism of rock material containing combined flaws has not almost been understood. Therefore, the main aim of this research is to analyze the strength and deformation behavior of brittle rock specimen containing combined flaws, and to investigate its fracture coalescence process. Moreover, the influence of confining pressure on strength and deformation failure behavior of rock specimen containing combined flaws is also investigated.
2.
Numerical model and micro-parameters
Before the numerical simulation is discussed, the numerical model and micro-parameters for brittle rock material will be illustrated in this section. RFPA2D is chosen to simulate the fracture coalescence of brittle rock specimens containing preexisting combined flaws. The essential features of RFPA2D are described as follows [20–24]. The RFPA2D code can be used to perform the stress analysis for each element by FEM. Using tensorial notation, the mechanical equilibrium equation for the solid is expressed as s i j; j ¼ Fi ;
i; j ¼ 1; 2; 3;
(1)
where sij is the stress tensor (Pa) in the solid and Fi the component of the body force (N/m3). The constitutive equation defines the relation between the total bulk stress components (Pa), sij and strain components, eij. The stress–strain law is given by s i j ¼ Di jkl ei j
i; j ¼ 1; 2; 3;
(2)
where Dijkl is the elasticity tensor (Pa), and is related to the Young's modulus E and Poisson's ratio v for isotropic elastic media, which is regarded with damage initiation and development. eij = (Ui,j + Uj,i)/2 and Ui represents the displacement vector of the solid. In RFPA2D, the model is discretized into a large number of small elements. Taking into account the heterogeneity of real rock material, element local mechanical microscopic parameters are assumed to follow a Weibull distribution [28], i.e. Eq. (3): PðuÞ ¼
m m u m1 u exp u 0 u0 u0
(3)
where u is a random variable, representing strength, elastic modulus, etc. u0 is the mean value of element parameters for the whole specimen and m is the shape parameter of distribution function which is referred to as homogeneity index. Fig. 1 shows the influence of the shape parameter m on the distribution function, which indicates that a smaller m means a more heterogeneous rock material and vice versa. However for the same m, the Weibull's randomness indicated that the numerical simulated results of different rock specimens are very approximate, but not identical [29]. An elastic constitutive model with linear behavior is employed for all elements, which have been assigned different strength and elastic modulus according to the heterogeneity of rock material. The elastic constitutive relation for an element under uniaxial compressive stress and tensile stress is
Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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P (θ )
an element in order to capture the principal modes of damage and fracture of rock, respectively, and the tensile stress criterion is considered preferentially. When its stress state satisfies one criterion, the element begins to accumulate damage in tension or shear. For tension, the maximum tensile stress criterion is chosen as the strength criterion for the elements:
m m m
m m
m
s t f tp
θ Fig. 1 – The influence of the shape parameter m on the distribution function.
illustrated in Fig. 2 [30]. In Fig. 2, ftr and ftp are respectively the residual and peak tensile strength of the elements, which is given as ftr = lftp = lE0et0. l is the residual strength coefficient. fcr and fcp are respectively the residual and peak compressive strength of the elements. It is assumed ftr/ftp = fcr/fcp = l. et0 and etu are the tensile threshold strain of damaged element and the ultimate tensile strain of the failed element. The ultimate tensile strain is defined as etu = j et0, where j is the ultimate tensile strain coefficient. ec0 and ecu are the compressive threshold strain of damaged element and the ultimate compressive strain of the failed element. The ultimate compressive strain is defined as ecu = d ec0, where d is the ultimate compressive strain coefficient. When the stress of an element exceeds the strength criterion, the element is assumed to be irreversibly damaged. The stiffness and strength of damaged elements will be reduced to the residual strength level. In elastic damage mechanics, the elastic modulus of the element may degrade gradually as damage progresses, and the elastic modulus of damaged element is defined as follows [31]: E ¼ ð1 DÞE0
(4)
where D represents the damage variable. E and E0 are elastic modulus of the damaged and undamaged element, respectively. Here, the element and its damage are assumed to be isotropic elastic. Therefore the parameters E, E0 and D are all scalar. In this code, the maximum tensile stress criterion and the Mohr–Coulomb criterion are used to check the stress state of
σ Compression
f cp
E0 ε tu
ε t0
f cr − f tr
ε c0
ε cu ε
− f tp
Tension Fig. 2 – Elastic damage constitutive model for element under the compression and tension [30].
(5)
where st is the tensile strength of elements. Correspondingly, the damage variable D under tension can be described as follows [30]: 8 e > et0 > < 0; le (6) D ¼ 1 t0 ; etu e < eto > e : 1; e etu Under multi-axial stress states the element is still damaged in tensile mode when the equivalent major tensile strain e attains the above-threshold strain et0 . The equivalent principal strain e is defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (7) e ¼ he1 i2 þ he2 i2 þ he3 i2 where e1, e2 and e3 are three principal strains, and h i is a function defined as follows: x x 0; (8) hxi ¼ 0 x < 0: The constitutive law for an element subjected to multiaxial stresses can be easily obtained by replacing the strain e in Eq. (6) with equivalent strain e. The damage variable is expressed as: 8 e > et0 ; > <0 le (9) D ¼ 1 t0 etu < e et0 ; > e : 1 e etu : In order to describe the damage of an element when it is under compressive and shear stress, the Mohr–Coulomb criterion, expressed as follows, is chosen to define the second damage threshold: s1
1 þ sin ’ s 3 f c0 1 sin ’
(10)
where s1 and s3 are the major and minor principal stress, respectively, fc0 is the uniaxial compressive strength and w is the internal friction angle of the mesoscopic element. In the same way as for uniaxial tension, when the element is under uniaxial compression but damaged according to the Mohr– Coulomb criterion, the expression for damage variable D can be described as: ( 0 e < ec0 ; lec0 D¼ (11) e ec0 : 1 e When the element is under a multi-axial stress state and its strength satisfies the Mohr–Coulomb criterion, the maximum principal strain (i.e. maximum compressive principal strain) may be evaluated at the peak value of the maximum principal stress (i.e. maximum compressive principal stress) ec0:
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ec0 ¼
1 E0
f c0 þ
1 þ sin ’ s 3 nðs 2 þ s 3 Þ 1 sin ’
(12)
where s2 is the intermediate principal stress. In RFPA2D, the loads were applied by incrementally raising the vertical loads (under displacement control) until the model collapsed. At each step, elements may undergo damage due to excessive tension or yielding from the Mohr–Coulomb criterion. The stress and deformation distribution throughout the specimen is then adjusted instantaneously after each element damages to reach the equilibrium state. It is noted that the damaged elements are not removed from the domain, but the material properties (Strength, Young' modulus) will be very low, something like the air elements. The color of damaged elements is changed to be black automatically. At points which undergo increased load due to a redistribution of the stresses, further damages may be caused. The process is repeated until no further failures in the elements occur [22]. The numerical simulation for rock specimens are carried out on 2-D plane strain model. In the simulation, an external displacement is applied to the top of the rock specimens at a constant rate of 0.0032 mm/step (the strain rate is 2.0 105 step1) in the axial direction. It should be noted that during the experiment, the strain rate is about 8.125 106 s–1. Therefore, the experimental and numerical strain rates all belong to quasistatic loading rate, which has almost no any effect on the mechanical behavior of rock material. We computed the stress and strain in each mesolevel element. The lower boundary of numerical specimens is kept fixed, as is generally done during laboratory tests. In the present study, all the numerical specimens, with geometry of W H = 80 mm 160 mm in size, were discretized into 160 320 = 51,200 elements. The geometry of brittle specimens containing pre-existing combined flaws is described in Fig. 3. The geometry of pre-existing flaws is defined by five geometrical parameters – single fissure angle a
Table 1 – The micro-parameters of RFPA2D simulation for brittle rock specimens in this research. Micro-parameters
Value
Homogeneity index, m Average value of elastic modulus, E0, GPa Average value of compressive strength, fcp, MPa Internal friction angle, F8 Ratio of compressive to tensile strength, C/T Poisson's ratio, m Ultimate tensile strain coefficient, j Ultimate compressive strain coefficient, d Residual strength coefficient, l Loading rate in the present study, mm/step
10 35 370 40 8 0.25 1.5 200 0.1 0.0032
(the angle of single fissure with the direction of the horizontal direction), single fissure length 2a, ligament length 2b between two squares , ligament angle b between two squares , square side length 2c. In this research, 2a, 2b and 2c are fixed to 15, 40 and 14 mm, respectively. The b is fixed to 608, and a is increased from 08 to 908 at an interval of 158. As it is well-known to all of us, it is very difficult to determine micro-parameters by experiment [19]. However, in order to validate the micro-parameters used in the numerical modeling, it is essential to establish a correlation between the macro-behavior (i.e. the axial stress–axial strain curve, the peak strength and elastic modulus, the ultimate failure mode and the crack coalescence process) and micro-parameters. During the calibration process, the macroscopic behavior of intact rock obtained by experiment [32] was used in this research to calibrate the micro-parameters. The macroscopic results obtained by numerical simulation after each trial was used to check the micro-parameters. This process was repeated until the numerical results achieved a good agreement with the experimental results. Table 1 listed the micro-parameters of RFPA2D simulation for brittle rock specimens.
Fig. 3 – Geometry of pre-existing rock specimen in this research. Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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In addition, it is noted that in the RFPA2D simulation, the mesh dependency is unavoidable, which has been detailed in previous publications [33]. However, if the rock specimen is simulated with the relatively small element size, the mesh dependency is ignorable to some degree. Under this small size of elements, the mesh dependency may affect the quantity of the numerical results. However, it does qualitatively reflect the basic behavior of rock that is shown in the laboratory experiments. We just use an optimal element size here to address the crack issues here [33].
3. Simulated results under uniaxial compression Numerical simulated results on the mechanical behavior of brittle rock specimens are herein discussed and compared with the experimental results. Fig. 4 shows the comparison between experimental and numerical results of intact rock specimen under uniaxial compression. During the experiment, in order to decrease the effect of the end friction effects on testing results, two rigid steel blocks (33 mm 83 mm 15 mm) were placed between the loading frame and rock specimen and two antifriction gaskets between two rigid steel blocks and the end surface of specimen [32]. In the previous numerical study [23], Tang et al. have adopted RFPA2D to carry out the numerical simulation on the rock material with five different loading platens. Their simulated results have demonstrated that the loading platens have a minor effect on the peak strength and axial stress–strain curves of brittle rock material. Therefore, in the present numerical simulation, we used a zero friction coefficient between the walls and boundary particles [23], which are in a good agreement with the experiment. From Fig. 4, it can be seen that the numerically simulated curve under uniaxial compression agrees well with the experimental curve, except for the initial phase of the curve. The experimental stress–strain curve of intact specimen at low stress levels shows the downward concave and the initial
210
Numerical result Experimental result
Axial stress σ1 / MPa
180 150
Intact specimen
120 90 60 30 0 0.0
1.5
3.0
4.5
Axial strain ε1 / 10
6.0
7.5
-3
Fig. 4 – Stress–strain curves from the experimental test [32] and the numerical prediction (uniaxial displacement control) of intact rock specimen.
5
non-linear deformation, which mainly results from the closure of some primary cracks, pores and voids in tested rock specimen. It should be noted that in the experimental stress–strain curve, there was a distinct stress drop, which resulted from the gradual coalescence of tensile cracks in the process of deformation [32]. However, the axial stress drop was not observed in the curve obtained by numerical simulation due to obvious brittle characteristics. It is noted that, in RFPA2D code, although the mechanical response of a single mesoscopic element is linear, the macroscopic behavior of a numerical specimen (containing a lot of mesoscopic elements) can be non-linear. The nonlinear material behavior arises from its heterogeneous material properties. The more heterogeneous the material properties are, the stronger is the non-linearity in the stress– strain response. The numerical simulation method can reproduce better the complete process of deformation and failure of brittle rock specimen, especially the localization of deformation and failure [30]. Fig. 5 further illustrates the comparison between experimental and numerical stress–strain curves of brittle rock specimen containing combined flaws under uniaxial compression. In accordance with Fig. 5, it can be seen that the numerical simulated stress–strain curves of specimen containing combined flaws are in a good agreement with the experimental stress–strain curves. Moreover, it should be noticed that the stress drop in the numerical simulated stress– strain curves of the flawed specimen also agrees very well with the experimental results, which indicates that numerical simulated coalescence processes are also similar to the experimental crack coalescence processes. Besides, if not taking into account the initial nonlinear deformation, the simulated elastic modulus of the specimen is approximately equal to the experimental elastic modulus. Fig. 6 shows the uniaxial compressive strength in brittle rock specimens containing combined flaws by numerical simulation and experiment. It is very clear that the peak stress of rock specimen simulated by RFPA2D has the similar trend with the increase of fissure angle a. But for a = 308 and 608, the simulated uniaxial compressive strength is higher about 21% than that obtained by experiment, which can result from the numerical simulation are two-dimensional analysis and can not reflect fully three-dimensional physical phenomenon of heterogeneous rock material [18]. However, for a = 908, the simulated uniaxial compressive strength is approximately equal to that obtained by experiment. From Fig. 6, it can also be seen that the uniaxial compressive strength of rock specimen containing combined flaws takes on a nonlinear relation, i.e. first decreases for a = 0–308 and then increases for a = 30–908 with increasing fissure angle a. Fig. 7 presents the effect of fissure angle on ultimate failure mode of specimen containing combined flaws obtained by numerical simulation under uniaxial compression. For comparison, the experimental failure mode is also given in Fig. 7. In accordance with the compared result, it is clear that the ultimate failure mode in the numerical specimen is very approximate to that in the experimental specimen, e.g. for a = 608, which indicates that the RFPA2D software can be used to simulate the crack coalescence process of brittle rock material [24,25].
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Axial stress σ1 / MPa
60 Axial stress σ1 / MPa
(b)
Numerical result Experimental result
(a)
45
30 Specimen containing combined flaws (α= 30°)
15
0
Numerical result Experimental result
60 A
40 B
20
Specimen containing combined flaws (α= 60°)
0 0.0
1.0
2.0
3.0
Axial strain ε1 / 10
4.0
0.0
1.0
2.0
3.0
Axial strain ε1 / 10
-3
Numerical result Experimental result
(c) 100 Axial stress σ1 / MPa
80
4.0
-3
(d)
80
3 a
2
60 b
2 40 Specimen containing combined flaws (α= 90°)
20
Stress drop A
Stress drop B
0 0.0
1.0
2.0
3.0
Axial strain ε1 / 10
4.0
5.0
-3
Crack coalescence in the specimen containing combined flaws (α= 60°) due to larger stress drops after peak strength (by experiment)
Fig. 5 – Comparison between experimental and numerical stress–strain curves of rock specimen containing combined flaws under uniaxial compression. In this figure, the crack coalescence of specimen containing combined flaws (a = 608, b = 608) obtained by experiment is also shown due to larger stress drops after peak strength.
Uniaxial compressive strength σ c / MPa
120 105
Experimental data Numerical result
90 Specimen containing combined flaws (β= 60°)
75 60 45 30 0
15
30
45
60
75
90
Fissure angle α / °
Fig. 6 – Effect of fissure angle on uniaxial compressive strength of rock specimen containing combined flaws (b = 608) obtained by experiment and numerical simulation.
Table 2 summarizes the crack coalescence process of rock specimen obtained by RFPA2D simulation under uniaxial compression. In accordance with Table 2, we can analyze detailed the effect of fissure angle on the crack coalescence process of rock specimen containing combined flaws. When a = 0–158, the crack coalescence process can be illustrated as follows. Tensile crack in the specimen is initiated from a certain distance from the fissure tips, which is a good agreement with the experiment findings for the crack coalescence of rock specimen containing a single fissure (a = 158) [7]. Meantime, tensile cracks are also initiated from the top and bottom edge of two squares and bottom edge of two squares. All the cracks all propagate along the direction of axial stress. Afterwards, with the increase of axial deformation, the specimen undergoes an obvious coalescence in the ligament area between fissure and double squares and bottom edge of two squares. Finally, the specimen takes place the unstable failure and the cracks originate from the right side of and the left side of square . square However, the cracks in the specimens containing combined flaws for a = 30–908 are initiated from the top and bottom edge of two squares and bottom edge of two squares, which is
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Fig. 7 – Effect of fissure angle on ultimate failure mode of brittle rock specimen containing combined flaws (b = 608) obtained by experiment and numerical simulation under uniaxial compression.
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Table 2 – Crack coalescence process of brittle rock specimen containing combined flaws obtained by RFPA2D simulation under uniaxial compression. a
b
08
608
158
608
308
608
458
608
608
608
758
608
908
608
Crack coalescence process of specimen containing combined flaws
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similar to those for a = 0–158, but the cracks nearby the fissure for a = 30–908 are initiated at the upper and under tip of fissure. It should be noted that the crack coalescence process of specimens containing combined flaws for different fissure angles (a = 30–758) are very similar. Here, we take the specimen containing combined flaws (a = 608) for the examples to analyze in detail the crack coalescence process. Fig. 8 further illustrates numerically simulated results of AE events and accumulated AE events of rock specimen containing combined flaws (a = 608). In Fig. 8, all of the symbols correspond to those denoted in Table 2. From Fig. 8 and Table 2, it can be seen when the axial stress is loaded to point a (s1 = 31.46 MPa = 48.3%sc), the specimen (a = 608) begins to initiate tensile crack 1 at the upper and under tip of fissure, the top and bottom edge of two squares. The initiation of tensile crack 1 linked to 118 AE events. Afterwards, as the deformation progressed, the length of crack from the top and bottom edge of two squares begins to propagate toward the end surface of the specimen along the direction of major principal stress. But the tensile crack 1a, 1c–1d, and 1f did not propagate a lot, which was due to the stress concentration field between fissure and double squares. Furthermore, the propagating speed of tensile crack 1b and 1e was higher than that of tensile crack 1a, 1c–1d, and 1f. When the specimen was loaded to peak point b (s1 = 65.13 MPa = 100%sc), crack 2a coalesced and the upper tip between the left bottom corner of square of fissure, which led to an obvious stress drop from 65.13 to 57.36 MPa. The coalescence of crack 2a was linked to 234 AE events. The continuous increase of axial deformation led to that the axial stress of the specimen began to increase slowly with a lower deformation modulus than that before point b, which resulted from some minor damages of the supporting structure in the specimen. When the specimen was loaded to point c (s1 = 61.78 MPa = 94.9%sc) after the peak strength, crack 2b coalesced between the under tip of fissure and the right top corner of square , which led to an obvious stress drop from 61.78 to 50.39 MPa. The coalescence of crack 2b was
linked to 136 AE events. After that, with the increase of axial deformation, the axial stress of the specimen began to increase slowly with a lower deformation modulus than that before point c, which resulted from more damages of the supporting structure in the specimen. When the specimen was loaded to point d (s1 = 55.31 MPa = 84.9%sc) after the peak strength, a larger stress drop from 55.31 to 32.64 MPa was observed, which resulted from the propagation of crack 3 and more AE events were also observed. It should be noted that and crack 3a was initiated from the left top corner of square propagated toward the left edge of the specimen in the downward direction. At the same time, crack 3b was also in the initiated from the right bottom corner of square upward direction. Afterwards, with the increase of axial deformation, the specimen was difficult to support higher axial stress. When the axial stress was decreased to point e, a minor stress drop was also observed and the corresponding AE events were 170, which resulted from the propagation of crack 3b. When the axial load was decreased to point f (s1 = 30.60 MPa = 47%sc) after the peak strength, the specimen occurred an obvious stress drop from 30.60 to 16.33 MPa, which resulted from the formation of far-field crack nearby crack 3b. After the last stress drop, the specimen entered rapidly the stage of residual strength. From Fig. 8, it is very clear that the increase of AE events and accumulated AE events in the specimen are all closely related to distinct stress drops. It should be noted that the stable propagation of original cracks does not lead to a larger AE event, the coalescence of new cracks results in a larger AE event. Moreover, the maximum AE event in the specimen does not induce at the peak strength, but at a certain position after the peak stress. In accordance with Figs. 5 and 7 and Table 2, it can be seen that in the experimental specimen (a = 608), cracks 2a and 2b are simultaneously initiated in the deformation process, which leads to a distinct stress drop A (see Fig. 5b and d). Whereas in the numerical specimen, crack 2a is first initiated
60 Axial stress σ1 / MPa
b
AE events Axial stress Acc umulated AE events
c
3000
d
50
2500
40
2000
a
Specimen containing combined flaws (α= 60°)
30
e f
1500 1000
20
500
10
AE events (n), Accumulated AE events (n)
3500
70
1b 3b
2a 1d 1a
1f 1c 2b 3a
1e
0
0 0
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
Axial strain ε1 / 10-3 Fig. 8 – Numerically simulated results of AE events and accumulated AE events of rock specimen containing combined flaws (a = 608) under uniaxial compression (RFPA2D). All of the symbols correspond to those denoted in Table 2. Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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and then crack 2b is initiated, which results in two axial stress drops as shown in Fig. 5. In the experiment, crack 3 originates from the right side of square , which also leads to a distinct stress drop B (see Fig. 5b and d). But in the numerical specimen, the cracks are initiated from the left top corner of square and then from the right bottom corner of square , which also results in two larger stress drops as shown in Fig. 8. When a = 908, even though the crack coalescence process obtained by experiment and numerical simulation are a little different (see Fig. 7), the axial stress–strain curves shown in Fig. 5c are very similar. In the experimental specimen, the coalescence occurs in the ligament area between the fissure and two squares, but which are not observed in the numerical specimen. Besides, tensile crack initiated from the right in the experimental specimen is bottom corner of square also observed in the numerical specimen.
4. Simulated results under different confining pressure To investigate the influence of confining pressure on strength and deformation failure behavior of brittle rock specimen containing combined flaws, three different confining pressures
of 10, 20 and 30 MPa are applied to the specimen. Fig. 9 presents numerically simulated axial stress–strain curves of rock specimen containing combined flaws under different confining pressures (RFPA2D). From Fig. 9, it can be seen that the strength and deformation behavior of specimen containing combined flaws depend on not only the confining pressure, but also fissure angle a. Fig. 10a illustrates the effect of fissure angle a on triaxial peak strength sS of rock specimen containing combined flaws (RFPA2D simulation). From Fig. 10a, it can be seen that the effect relation of fissure angle a on triaxial peak strength sS under different confining pressures are very similar to that under uniaxial compression. In a word, the peak strength of the specimen containing combined flaws at the same confining pressure first decreases and then increases with the increasing fissure angle a. However for the specimen containing combined flaws with the same fissure angle, as the increase of confining pressure from 0 to 30 MPa, the peak strength and elastic modulus of specimen all increase gradually. For example, the peak strengths of specimen containing combined flaws with a = 308 are 55.11, 66.8, 82.1 and 97.3 MPa at the confining pressures of 0, 10, 20 and 30 MPa, respectively. The linear Mohr–Coulomb criterion can be used to describe the relation between the triaxial peak strength sS of specimen containing
100
100
(a) α= 0°
(b) α= 30° 80
60
σ3 =30 MPa
40
σ3 =20 MPa
20
σ3 =10 MPa
Axial stress σ1 / MPa
Axial stress σ1 / MPa
80
60
σ3 =30 MPa
40
σ3 =20 MPa σ3 =10 MPa
20
σ3 =0 MPa
σ3 =0 MPa
0
0 0.0
1.5
3.0
4.5
Axial strain ε1 / 10
6.0
0.0
3.0
4.5
Axial strain ε1 / 10
6.0
-3
140
120
(c) α= 60°
(d) α= 90°
120 Axial stress σ1 / MPa
100 Axial stress σ1 / MPa
1.5
-3
80 60
σ3 =30 MPa
40
σ3 =20 MPa σ3 =10 MPa
20
100
σ3 =30 MPa
80 σ3 =20 MPa
60 σ3 =10 MPa
40 20
σ3 =0 MPa
σ3 =0 MPa
0
0 0.0
1.5
3.0 Axial strain ε1 / 10
4.5 -3
6.0
0.0
1.5
3.0
4.5
6.0
Axial strain ε1 / 10-3
Fig. 9 – Numerically simulated axial stress–strain curves of brittle rock specimen containing combined flaws under different confining pressures. Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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0MPa 20MPa
130
(b) 50
10MPa 30MPa
Peak strength parameters
Triaxial peak strength σS / MPa
(a) 145
115 100 85 70 55 40
Cohesion Internal friction angle
40 30 20 10 0
0
15
30
45
60
75
90
15
0
30
45
60
75
90
Fissure angle α / °
Fissure angle α / °
Fig. 10 – Effect of fissure angle a on triaxial peak strength and strength parameters of brittle rock specimen containing combined flaws.
combined flaws and the confining pressure s3. As it is wellknown to us, linear Mohr–Coulomb criterion can be expressed with the following equation, i.e. Eq. (13). s S ¼ s 0 þ qs 3 ¼
2C cos f þ s 3 ð1 þ sin fÞ 1 sin f
In accordance with Eq. (13), the cohesion and internal friction angle of specimen containing combined flaws in accordance with linear Mohr–Coulomb criterion (by RFPA2D simulation) can be obtained, which are listed in Table 3. From Table 3, it can be seen that there are good linear regression coefficients (R = 0.980–0.998) for the relation between the triaxial peak strength of specimen containing combined flaws and the confining pressure. Fig. 10b further plots the influence of fissure angle on the cohesion and internal friction angle of specimen containing combined flaws. From Fig. 10b and Table 3, it can be seen that the values of C range from 21.00 to
(13)
where s0 is usually regarded as the uniaxial compressive strength (UCS) of rock material; q is an influence coefficient of confining pressure on sS of rock. C and F, respectively, represent the cohesion and internal friction angle of rock material.
Table 3 – Strength parameters of rock specimen containing combined flaws in accordance with linear Mohr–Coulomb criterion. Strength parameters
a = 08
a = 158
a = 308
a = 458
a = 608
a = 758
a = 908
s0 (MPa) q C (MPa) F (8)
60.62 1.33 26.28 8.1
59.76 1.43 24.99 10.2
50.05 1.42 21.00 10.0
60.02 1.40 25.36 9.6
66.03 1.76 24.89 16.0
84.75 1.58 33.71 13.0
101.33 1.18 46.64 4.7
2500
2000 Accumulated AE events (n)
σ3 =0 MPa
Specimen containing combined flaws (α= 30°)
σ3 =10 MPa
1500
1000 σ3 =30 MPa σ3 =20 MPa
500
0 0.0
1.5
3.0 Axial strain ε1 / 10
4.5
6.0
-3
Fig. 11 – Effect of confining pressure on accumulated AE events of rock specimen containing combined flaws. Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
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Fig. 12 – Effect of confining pressure on ultimate failure mode of rock specimen containing combined flaws (a = 308).
46.64 MPa, while the values of F are between 4.78 and 16.08. Moreover, the values of C and F of the specimen are closely dependent to fissure angle. The value of C first decreases from 26.28 to 21.00 MPa as the fissure angle increases from 08 to 308, while increases from 21.00 to 46.64 MPa as the fissure angle increases from 308 to 908. While, the value of F first increases from 8.18 to 10.28 as the fissure angle increases from 08 to 158, and keeps basically a constant 108 in the range of a = 15–458. However, the value of F increases from 9.68 to 168 as the fissure angle increases from 458 to 608, and decreases from 168 to 4.78 as the fissure angle increases from 608 to 908. Fig. 11 shows the effect of confining pressure on accumulated AE events of rock specimen containing combined flaws (a = 308) obtained by RFPA2D. The accumulated AE events of specimen containing combined flaws for other fissure angles are approximate to that for a = 308. From Fig. 11, it is clear that the accumulated AE events decreases as the confining pressure increases from 0 to 30 MPa, which results mainly from the restraining of higher confining pressures on the initiation and propagation of tensile cracks at the fissure tips and nearby double squares (Fig. 12). In order to result in the damage failure of the elements in the numerical specimen, higher axial stresses are also required for higher confining pressures.
5.
Conclusions
(1) Numerically simulated results (including the stress–strain curve, the peak strength and the crack evolution mode) for rock specimen containing combined flaws (two square holes and one pre-existing fissure) under uniaxial compression show a good agreement with the experimental results. The crack evolution behavior of specimen containing combined flaws obtained by RFPA2D simulation under uniaxial compression are summarized, which is closely dependent to fissure angle. When a = 0–158, tensile crack in the specimen is initiated from a certain distance from the fissure tips. Meantime, tensile cracks are also initiated . But in from the top and bottom edge of two squares
the specimens containing combined flaws for a = 30–908, the cracks are initiated from the top and bottom edge of , which is similar to those for a = 0–158, two squares but the cracks nearby the fissure for a = 30–908 are initiated at the upper and under tip of fissure. After crack initiation, with the increase of axial deformation, the specimen undergoes an obvious coalescence in the ligament area . Finally, the between fissure and double squares specimen takes place the unstable failure. (2) The peak strength of specimen containing combined flaws increases with the confining pressure. According to the linear Mohr–Coulomb criterion, the cohesion and internal friction angle of specimen containing combined flaws are obtained, which is found to take on a distinct nonlinear relation with fissure angle. The value of C of specimen containing combined flaws first decreases from 26.28 to 21.00 MPa as the fissure angle increases from 08 to 308, while increases from 21.00 to 46.64 MPa as the fissure angle increases from 308 to 908. While, the value of F first increases from 8.18 to 10.28 as the fissure angle increases from 08 to 158, and keeps basically a constant 108 in the range of a = 15–458. However, the value of F increases from 9.68 to 168 as the fissure angle increases from 458 to 608, and decreases from 16 to 4.78 as the fissure angle increases from 608 to 908. (3) The stable propagation of original cracks does not lead to a larger AE event, but the coalescence of new cracks causes a larger AE event. The accumulated AE events decreases as the confining pressure increases from 0 to 30 MPa, which results mainly from the restraining of higher confining pressures on the initiation and propagation of tensile cracks at the fissure tips and nearby double squares. At the same time, it should be noted that in the present study, we only simulate the fracture coalescence behavior at a laboratory scale, but due to obvious scale effect of rock material, the simulated results at a laboratory scale is very difficult to be predicted directly the strength of rock mass at a engineering scale, therefore in the future, the scale effect of rock containing combined flaws will be further strengthened.
Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005
ACME-302; No. of Pages 13 archives of civil and mechanical engineering xxx (2015) xxx–xxx
Acknowledgments This research was supported by the National Basic Research 973 Program of China (Grant No. 2014CB046905), the Fundamental Research Funds for the Central Universities (China University of Mining and Technology) (Grant No. 2014YC10), for which the authors are very grateful. The authors would like to express their sincere gratitude to the editor and two anonymous reviewers for their valuable comments, which have greatly improved this paper.
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Please cite this article in press as: S.Q. Yang et al., Numerical study on failure behavior of brittle rock specimen containing pre-existing combined flaws under different confining pressure, Archives of Civil and Mechanical Engineering (2015), http://dx.doi.org/10.1016/j. acme.2015.03.005