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Unloading-induced failure of brittle rock and implications for excavation-induced strain burst
Tunnelling and Underground Space Technology 84 (2019) 495–506
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Unloading-induced failure of brittle rock and implications for excavationinduced strain burst
T
Kang Duana, Yinlin Jib, Wei Wub, , Chung Yee Kwokc ⁎
a
School of Civil Engineering, Shandong University, Jinan, China School of Civil and Environmental Engineering, Nanyang Technological University, Singapore c Department of Civil Engineering, The University of Hong Kong, Hong Kong b
ARTICLE INFO
ABSTRACT
Keywords: Unloading-induced failure Brittle rock Discrete element model Strain burst
Understanding the unloading-induced failure characteristics of brittle rock is essential for predicting excavationinduced strain burst. We conduct both the laboratory experiments and discrete element simulations to investigate the deformation, failure pattern, and strain energy evolution of the Bukit Timah granite. The laboratory tests with radial unloading show that the radial strain mainly contributes to the change of volumetric strain in the unloading process. The numerical tests with the flat-jointed contact model reproduce the Hoke-Brown failure envelope of the rock, and simulate the unloading-induced rock failure under different combinations of initial confining pressure and unloading rate. The evolutions of strain energy release and failure pattern show that higher unloading rate likely induces violent failure, in terms of greater lateral expansion and more ejected fragments. The micromechanical analyses on the particle velocity and tensile contact force reveal that the nonuniform rock deformation concentrates at lateral surfaces and results in strain burst with fragment ejection. The Hoek-Brown failure criterion well predicts the confining pressure at failure when the rock is laterally unloaded at lower unloading rate, but likely overestimates the stress level at higher unloading rate.
1. Introduction Creating urban underground space and extracting deep natural resources are the next frontier for social development and environmental sustainability. However, these anthropogenic disturbances deep underground may perturb the initial equilibrium of rock masses and lead to the occurrence of unpredictable geohazards (Wu et al., 2017a, 2017b). At greater depth, rocks are subjected to higher in-situ stress. Field observations indicate that rock failure under high in-situ stress conditions can be either conditionally stable, which is accompanied by the progressive formation of layered structure (e.g., spalling failure), or abruptly unstable, which occurs along with the violent release of strain energy (e.g., rock burst) (Zhao and Cai, 2014). Rock failure is also dependent on various excavation work progresses. For instance, drill and blast can induce instantaneous unloading at tunnel sidewalls and result in catastrophic rock failure. Hence, both in-situ stress and unloading rate significantly affect the unloading-induced rock failure. Rock burst is a typical geohazard induced by deep rock excavation and defined as sudden rupture and expulsion of rock mass at an excavation face. In recent years, the destructive geohazard has been frequently encountered in underground space development. For example,
⁎
Xu et al. (2016) reported that a considerable length of tunnel section experienced strain burst during the deep tunnel excavation of the China Jinping II hydropower station project. Feng et al. (2018) symmetrically studied the failure mechanism of rock spalling in the same project and found that the thickness of split slabs increased in thickness from the sidewall inwards. In the construction of the Gotthard Base Tunnel in Switzerland, the fault slip burst was attributed to the reduction of both normal and shear stresses induced by the excavation activities (Wu et al., 2017a). Among these rock bursts, strain burst is the most common type (Kaiser and Cai, 2012), and may immediately occur after underground excavation (Su et al., 2017a) or unexpectedly evolve for several days (Feng et al., 2012, 2015). The delay of strain burst is likely unpredictable. Although several field studies have been conducted to reveal the occurrence of strain burst using the microseismic monitoring technique (Hirata et al., 2007; Xu et al., 2016; Su et al., 2017a), the mechanism that drives this event still eludes explanation. Experimental attempts have been made to reproduce strain burst under a controlled manner. Based on a true triaxial rock test system with acoustic emission (AE) monitoring, He et al. (2010) conducted a set of unloading tests on limestone to simulate strain burst. This study showed that the frequency and amplitude of the strain burst were
Corresponding author. E-mail address: [email protected] (W. Wu).
https://doi.org/10.1016/j.tust.2018.11.012 Received 17 May 2018; Received in revised form 20 October 2018; Accepted 17 November 2018 0886-7798/ © 2018 Elsevier Ltd. All rights reserved.
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study, and provides the mechanical parameters for the calibration of micro-parameters in the numerical model. Once the numerical model is validated, in-depth analysis is carried out to quantitatively capture the strength envelope and systematically evaluate the effects of initial confining pressure and unloading rate on the failure characteristics of the rock. Additionally, the micromechanical analyses examine the spatial distribution and magnitude of particle velocity as well as the orientation distribution of tensile contact force to reveal the mechanism of excavation-induced strain burst. 2. Experimental study The Bukit Timah granite used in this study is sourced from the central part of Singapore Island. The medium-grained and light gray granite has a P-wave velocity of 5900 m/s. The bulk density, water content, and porosity of the granite are 2660 kg/m3, 0.07%, and 0.26%, respectively. The mineral composition of the granite includes feldspar (60–65%), quartz (30%), as well as biotite and hornblende (5–10%) (Zhao, 1996). We prepared a group of intact samples with 50 mm diameter and 100 mm height as well as no visible cracks on the surfaces. We carried out a suite of triaxial tests using the MTS rock mechanics test system. The axial and radial loading capacities are 2600 kN and 140 MPa, respectively, and both have a precision of ± 0.3%. The extensometers measure the axial and radial deformation of the sample with an uncertainty of ± 0.2%. In the meantime, we recorded the acoustic signals using the AE system provided by the Physical Acoustic Corporation. Three ceramic piezoelectric sensors were attached onto the axial loading column to record the signals at a sampling rate of 1 MHz. Both pre-amplitude gain and amplitude threshold were set to 40 dB. We conducted the triaxial tests with two stress paths, one with axial loading and the other with radial unloading (Fig. 1). We first raised the confining pressure to a predetermined level (e.g., 10, 20 or 30 MPa). For the triaxial test with axial loading, we fixed the confining pressure and increased the axial stress at a radial displacement rate of 0.001 mm/s until the sample failed. For the triaxial test with radial unloading, we selected an intact sample adjacent to the one used in the previous test with axial loading. Under the same confining pressure, we fixed the axial load at 70%–80% of the peak strength collected from the previous test, and subsequently reduced the confining pressure at an unloading rate of 0.1 MPa/s until the sample failed. We set 70%–80% of the peak strength as the axial load at the beginning of the unloading process to make sure the sample failed before the confining pressure approached zero. The axial stress, confining pressure, sample deformation, and AE signals were simultaneously recorded in the triaxial tests.
Fig. 1. Triaxial tests with two stress paths in Stage III: axial loading (blue dashed line) and radial unloading (magenta solid line).
highly related to in-situ stress state. Gu et al. (2002) observed three distinct stages (i.e., crack cleavage extension, rock slab rupturing and fragment ejecting) during the rock burst in the construction of the Qinling Tunnel. Most recently, numerous triaxial tests with unloading paths have been conducted to evaluate the influence of various factors, such as sample size (Zhao and Cai, 2014; Li et al., 2018), unloading rate (Zhao et al., 2014), loading and unloading paths (Li et al., 2017), in-situ stress (Su et al., 2017a), and radial stress gradient (Su et al., 2017b), on the evolution of strain burst. In these tests, AE monitoring system and high-speed camera were used to record AE activities and fracture development, respectively. By analyzing the recorded images, the kinetic energy of ejected fragments can be estimated (Su et al., 2017a). However, the end effect originating from loading platens inevitably exists in the experimental system and may affect the unloading-induced failure process (Zhao and Cai, 2014). Numerical approaches can provide alternative avenues to address the mechanism of strain burst from different perspectives. A mathematical-physical model was proposed based on the finite element method to characterize the unloading-induced rock failure (Tao et al., 2012). An enhanced cohesion weakening and friction strengthening model was implemented based on the finite difference method to simulate the progressive failure of hard rock pillars (Rafiei Renani and Martin, 2018). Cai (2008a, 2008b) and Cai and Kaiser (2005) showed a set of numerical simulations to investigate the rock failure adjacent to an excavation face. Most of these studies used the continuum method with pre-determined failure mechanism. As the failure of brittle rock is a progressive process, the discrete element method (DEM) (Cundall and Strack, 1979) provides a proper avenue to explicitly represent the initiation and propagation of fractures at both micro- and macro-scales (Duan et al., 2015, 2018; Potyondy and Cundall, 2004; Zhao, 2016, 2017). So far, the mechanical responses of brittle rocks subjected to the uniaxial compression (Potyondy and Cundall, 2004), true triaxial compression (Duan et al., 2017), direct shear condition (Cho et al., 2008), direct and indirect tensile conditions (Shang et al., 2017) have been extensively investigated based on the DEM analysis. However, modeling the unloading-induced failure characteristics of brittle rock has been rarely reported. We conduct both the experimental and numerical studies on the Bukit Timah granite to investigate the unloading-induced failure of brittle rock. The laboratory experiments investigate the sample deformation and strain energy release during the unloading process. The experimental study serves as the physical background of the numerical
3. Numerical model 3.1. Flat-joint contact model It has been widely recognized that the classic DEM model with parallel bonds suffers from several intrinsic limitations in modeling the mechanical responses of brittle rock (Ding and Zhang, 2014; Duan et al., 2018; Potyondy and Cundall, 2004; Wu and Xu, 2016). First, the tensile strength is significantly overestimated if one calibrates the micro-parameters to match the uniaxial compressive strength of brittle rock. Second, if the DEM model is calibrated to match the compressive strength under a confining pressure (for instance, 20 MPa as illustrated in Fig. 2a), the slope of the strength envelope produced by the DEM model is lower than that obtained from the laboratory experiments. These intrinsic limitations may be attributed to the idealized geometry of spherical grains adopted in the model. If the strength envelope of brittle rock (the magenta line in Fig. 2a) cannot be quantitatively captured, the reduction of confining pressure may not trigger rock 496
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Fig. 2. (a) Schematic diagram showing the limitations of the DEM model with parallel bonds in the triaxial test with radial unloading; (b) Flat-jointed contact model for brittle rock simulation; (c) Adjacent particles with a flat and notional interface in two-dimension (after Itasca, 2015).
failure when the confining pressure is reduced to zero. Moreover, obtaining proper tensile cutoff is important for the strength envelope to simulate the unloading-induced tensile failure of brittle rock (Zhu et al., 2014). To date, considerable efforts have been taken to tackle these intrinsic limitations by modifying particle texture (Cho et al., 2007), changing bond contact (Scholtès and Donzé, 2012), and introducing new constitutive contact model (Ding and Zhang, 2014). Among these approaches, the flat-jointed contact model (FJCM) (Potyondy, 2012) can represent the polygonal shape of rock crystals, and mimic rock damage at both micro- and macro-scales. FJCM has been successfully applied to solve the intrinsic problems of a classic DEM model (Wu and Xu, 2016). A flat-jointed material and FJCM are shown in Fig. 2b and c, respectively. A particle-particle contact simulates the behavior of a finite-length interface with locally flat and notional surfaces. By considering the contact between two particles as a planar interface, the particles in the flat-jointed material are able to maintain their polygonal grain structures. FJCM is thus able to provide grain interlocking and rotational resistance even after the bonds break. In a 2D model, the interface is a flat line, which is discretized into several elements (Fig. 2c). On the notional surface, the force and moment (F (e) and M(e) ) act at the centroid of the element. At the center of the interface, the equivalent force and moment (Fc and Mc ) can be calculated statically based on all the element forces and moments:
F (e ) , M c =
Fc = e
{(r (e) × F (e) ) + M(e) }
The element normal and shear stresses, which act at the element center, can be calculated as: (e )
M (e ) =
Fn(e) n c + F (se)
(2)
M(be)
(3)
Mt(e) n c
+
(e )
=
F (se) A(e)
(4)
where is the area of the element. More details regarding the constitutive relationship of FJCM can be found in Itasca Consulting Group (2015) and Potyondy (2012). 3.2. Calibration of micro-parameters Following the procedure recommended by Wu and Xu (2016), we calibrated the micro-parameters of FJCM to match the strength envelope of the Bukit Timah granite. The mechanical properties of the granite obtained from the triaxial tests under the confining pressures ranging from 0 MPa to 30 MPa are summarized in Table 1. The uniaxial compressive and direct tensile strengths are 186.8 MPa and 10.1 MPa, respectively (Peng et al., 2017). The size of the DEM sample is the same as that used in the experimental study. The sample consists of 20,694 particles, following the uniform distribution with the minimum particle size (Rmin) of 0.2 mm and particle size ratio (Rmax/Rmin) of 1.66. The corresponding micro-parameters are listed in Table 2. The excellent agreement can be found between the strength envelopes obtained numerically and experimentally (Fig. 3). For an intact rock sample, the Hoek-Brown failure criterion is expressed as (Hoek and Brown, 1997):
where r (e) is the position vector. During the calculation, the state of the element bond may be modified by the element force and moment according to the forcedisplacement law. The element force is resolved into the normal force (Fn(e) ) and shear force (F (se) ), while the element moment is resolved into the twisting moment (Mt(e) ) and bending moment (M(be) ):
F (e ) =
Fn(e) , A(e)
A(e)
(1)
e
=
1
=
3
+
c (mi
3
+ 1)0.5
(5)
c
where c is the uniaxial compressive strength, 1 and 3 are the major and minor principal stresses, respectively, mi is a material constant. The results of curve fitting are also provided in Fig. 3, in which both the experimental and numerical results are consistent with those predicted by the failure criterion. 3.3. Procedure of numerical simulation
where Fn(e) > 0 is in tension, and n c is the unit normal vector.
We conducted a series of DEM simulations to mimic the triaxial tests
Table 1 Summary of triaxial tests on the Bukit Timah granite. Triaxial test with axial loading
Triaxial test with radial unloading
Confining pressure (MPa)
Peak strength (MPa)
Elastic modulus (GPa)
Poisson’s ratio
Initial confining pressure (MPa)
Initial axial stress (MPa)
Confining pressure at failure (MPa)
10 20 30
330.5 442.0 527.1
72.35 73.23 73.42
0.164 0.153 0.158
10 20 30
267.1 306.8 427.0
10.2 15.2
* The sample does not fail when the confining pressure is reduced to 0 MPa. 497
*
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Hazzard et al., 2000). Hazzard et al. (2000) found that the strain energy released by bond breakage may have a significant influence on rock behavior, because the waves emanating from cracks are capable of inducing more cracks if nearby bonds are closed to failure. The influence of the degree of damping on the unloading-induced rock failure is worth exploring in the future study.
Table 2 Micro-parameters calibrated for the flat-jointed contact model to represent the Bukit Timah granite. Micro-parameter
Value
Minimum particle radius, Rmin (mm) Ratio between maximum to minimum particle size, Rmax/Rmin Installation gap ratio Bonded element fraction Circumferential elements Effective modulus of both particle and bond (GPa) Ratio of normal to shear stiffness of both particle and bond Mean and standard deviation of bond tensile strength (MPa) Mean and standard deviation bond cohesion (MPa) Local friction angle (° )
0.2 1.66 0.4 1.0 4 115.5 1.9 20±0 355±35.5 36
Residual friction angle (° )
4. Results and discussion 4.1. Experimental results We conduct the triaxial tests with radial unloading under three initial confining pressures, i.e., 10 MPa, 20 MPa, and 30 MPa. The experimental results are summarized in Table 1. The samples under initial confining pressures of 20 MPa and 30 MPa fail when the confining pressures decrease to 10.2 MPa and 15.2 MPa, respectively. The sample under an initial confining pressure of 10 MPa does not fail immediately when the confining pressure reaches 0 MPa. Fig. 4 shows the typical results obtained from the triaxial test with radial unloading at a rate of 0.1 MPa/s under 20 MPa initial confining pressure. The positive volumetric strain, positive axial strain, and negative radial strain indicate volumetric compaction, axial compression, and radial expansion, respectively. We set both the initial cut-offs of axial and radial strain as zero. The volumetric strain ( v ) is calculated based on the axial strain ( 1) and radial strain ( 3 ) as:
38.7
with radial unloading illustrated in Fig. 1. At Stage I, we raised both the confining pressure and axial stress simultaneously to predetermined values. At Stage II, we kept the confining pressure ( c0 ) constant, and increased the axial stress monotonically to 70%–80% of the peak strength obtained from the triaxial test with axial loading. At Stage III, we maintained the axial stress and reduced the confining pressure at a constant unloading rate ( c ). The confining pressure after nt cycles can be determined by: ct
=
c0
+
c
× nt
(6)
v
Note that the DEM simulations were performed through the timestepping algorithm with time cycle as low as 1 × 10−7 s. In the experimental study, a triaxial test with 0.1 MPa/s unloading rate lasted about 1000 s. Hence, a numerical test with the same unloading rate could take about 1 × 1010 cycles. However, as running 1 × 105 cycles needed 20 h, we were unable to strictly follow the unloading rate adopted in the experimental study. To save the computation cost, we used the unloading rates of 1 × 103 MPa/s, 1 × 104 MPa/s and 1 × 105 MPa/s to represent 1.97 × 10−5 MPa/cycle, −4 −3 1.97 × 10 MPa/cycle and 1.97 × 10 MPa/cycle, respectively, to uncover the mechanism of unloading-induced rock failure. All the numerical simulations reported in this study were carried out with a damping coefficient of 0.7 to explore the influence of unloading rate on the unloading-induced failure characteristics of brittle rock. Previous studies have revealed that the degree of damping may have a significant influence on rock behavior (Potyondy and Cundall, 2004;
=
1
+2×
3
(7)
The initial loading processes of the confining pressure and axial stress at Stage I are similar to those shown in Fig. 1. At Stage II, the axial strain increases linearly at a rate of 4×10−6/s, while the radial strain decreases at a much lower rate of 6×10−7/s. The ratio of radial strain reduction to axial strain increment is 0.15, which is approximately
Fig. 3. Comparison between the strengths obtained from the experimental and numerical studies, together with the strength envelope predicted by the HoekBrown failure criterion.
Fig. 4. Experimental results of the triaxial test at a radial unloading rate of 0.1 MPa/s under an initial confining pressure of 20 MPa. 498
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Fig. 5. Numerical results of the triaxial tests at a radial unloading rate of 1 × 104 MPa/s under initial confining pressure of (a) 10 MPa, (b) 20 MPa, and (c) 30 MPa.
equal to the Poisson’s ratio determined by the triaxial test with axial loading (Table 1), indicating that the sample behaves elastically before the unloading process. During this stage, the positive volumetric strain increases linearly, implying that the sample is being continuously compacted. The remarkable AE signals are detected when the axial stress reaches 292.4 MPa, which reflects that the sample is approaching the failure stage. At Stage III, the confining pressure reduces at an unloading rate of 0.1 MPa/s, and both the magnitudes of axial and
radial strains remarkably raise. The volumetric strain turns from increasing to decreasing until the sample fails. This observation indicates that the marked increase in the magnitude of radial strain contributes dominantly to the change of volumetric strain at this stage, leading to a transition of sample deformation from compaction to dilation. In the meantime, the cumulative AE count jumps when the confining pressure starts to reduce and subsequently increases until the sample collapses at 10.2 MPa confining pressure (defined as the confining pressure at 499
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Fig. 6. Evolution of stored strain energy and strain energy increment rate (change of stored strain energy every 200 cycles) obtained from the numerical tests at a radial unloading rate of 1 × 104 MPa/s under initial confining pressure of (a) 10 MPa, (b) 20 MPa, and (c) 30 MPa. Table 3 Effects of unloading rate and initial confining pressure on the magnitude of stored strain energy and strain energy increment rate. Initial confining pressure (MPa)
Unloading rate (MPa/ s)
Magnitude of stored strain energy (J)
Increment of stored strain energy (10−3 J/step)
Point A
Point B
Point C
A-B
B-C
4
Influence of initial confining pressure
10 20 30
1 × 10 1 × 104 1 × 104
1408.0 1824.2 3389.6
1483.9 1987.1 3534.8
1936.1 2523.6 4293.8
1.52 1.79 2.10
12.39 15.88 13.71
Influence of unloading rate
20 20 20
1 × 103 1 × 104 1 × 105
1824.2 1824.2 1824.2
1964.3 1987.1 1808.1
2394.4 2523.6 2528.3
0.20 1.79 −1.45
4.87 15.88 22.16
Fig. 7. Failure patterns of the DEM samples obtained from the numerical tests at a radial unloading rate of 1 × 104 MPa/s under initial confining pressure of (a) 10 MPa, (b) 20 MPa, and (c) 30 MPa.
500
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Fig. 8. Numerical results of the triaxial tests under an initial confining pressure of 20 MPa at radial unloading rates of (a) 1 × 103 MPa/s, (b) 1 × 104 MPa/s, and (c) 1 × 105 MPa/s. Note the scale of horizontal axial in (a) is different from the other two cases.
failure). The sample fails accompanied by the sudden drops of axial stress, radial strain, and volumetric strain, as well as the dramatic increases of cumulative AE counts and axial strain.
induced failure of brittle rock, we conduct the numerical tests at an unloading rate of 1 × 104 MPa/s under varying initial confining pressures (e.g., 10 MPa, 20 MPa, and 30 MPa). As illustrated in Fig. 5, the sample deformation and micro-crack evolution of the three cases are similar at Stage II. The ratio of radial strain reduction to axial strain increment is slightly larger than the Poisson’s ratio given in Table 1. This discrepancy can be attributed to the failure of flat-jointed material additionally contributing to the lateral deformation. At the beginning of
4.2. Numerical results 4.2.1. Influence of initial confining pressure To evaluate the effect of initial confining pressure on the unloading501
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Fig. 9. Evolution of stored strain energy and strain energy increment rate (change of stored strain energy every 200 cycles) obtained from the numerical tests under an initial confining pressure of 20 MPa at radial unloading rate of (a) 1 × 103 MPa/s, (b) 1 × 104 MPa/s, and (c) 1 × 105 MPa/s.
Stage III, the axial strain stays almost constant, implying that the sample experiences limited deformation along the axial direction. Meanwhile, the magnitude of radial strain remarkably raises, and mainly contributes to the volume change. In other words, the unloading process induces the formation of tensile micro-cracks parallel with the direction of major principal stress. When the confining pressure approaches zero, the number of micro-cracks increases dramatically, accompanied by the rapid increases of both axial and radial strains. The volume change transits from compaction to dilation. Finally, the sample collapses with the loss of loading bearing capacity. As the linear contact between particles are adopted in the DEM model, the strain energy stored in the sample (EC) can be calculated as (Itasca, 2008):
EC =
1 2
(|Fin |2 /k n + |Fis |2 /k s ) NC
orientation is approaching the direction of the minor principal stress. 4.2.2. Influence of unloading rate The unloading-induced failure of brittle rock can be also affected by varying unloading rates associated with various excavation work progresses. We conduct the numerical tests under 20 MPa initial confining pressure at different unloading rates of 1 × 103 MPa/s, 1 × 104 MPa/s, and 1 × 105 MPa/s. At Stage II, the numerical results show that the sample deformation and micro-crack evolution are the same in the three cases as they follow the same loading path (Fig. 8). The ratio of radial strain reduction to axial strain increment is also slightly higher than the Poisson’s ratio as discussed in Section 4.2.1. At Stage III, the axial strain keeps constant, and the decrease of radial strain mainly contributes to the reduction of volumetric strain. The number of microcracks then dramatically increases with the sharp reduction of volumetric strain. At the end, the sample fails accompanied by the increase of axial strain. The axial stress suddenly jumps because the servomechanism fails to maintain the axial stress during the catastrophic failure of the sample. The evolutions of stored strain energy and strain energy increment rate are plotted in Fig. 9. At Stage II, the same amount of strain energy is stored in the sample (Point A). The stored strain energy stays constant at the beginning of Stage III (from Point A to Point B), but the duration of the case at higher unloading rate remarkably reduces. When the strain energy increment rate starts to fluctuate, the stored strain energy increases until the peak value is reached (Point C). The three cases with different unloading rates show the similar peak value but different increments of stored strain energy (Table 3). After the peak point, the catastrophic failure is characterized by a rapid release of strain energy. Fig. 10a–c shows that the rate-dependent failure patterns can be observed after the numerical tests with radial unloading. At a low unloading rate (e.g., 1 × 103 MPa/s), the sample fails into several relatively large blocks with the major fractures sub-parallel with respect to
(8)
where NC is the number of contacts; |Fin | and |Fis | are the magnitudes of normal and shear components of a contact force; kn and ks are the normal and shear stiffness of a particle contact, respectively. At the end of Stage II (Point A), the stored strain energy is larger in the sample under higher initial confining pressure (Fig. 6). At Stage III, the stored strain energy is first stable (from Point A to Point B), as the number of micro-cracks shows no obvious increase (Fig. 5). Subsequently, the stored strain energy jumps to the peak value (Point C). According to Table 3, the peak value is greater in the case under higher initial confining pressure. However, our results do not show the stress dependence of strain energy increment rate during the unloading process, which is likely due to the random failure of bond contacts. Both ejected fragments from the lateral surfaces and major fractures in the failed sample can be observed in Fig. 7. In the three cases, the distribution of induced factures are similar, and the most major fractures are sub-vertical. When the sample is subjected to higher initial confining pressure, the number of major fractures increases, and the 502
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Fig. 10. Failure patterns of the DEM samples obtained from the numerical tests under an initial confining pressure of 20 MPa at radial unloading rate of: (a) 1 × 103 MPa/s, (b) 1 × 104 MPa/s, and (c) 1 × 105 MPa/s, (d) failure pattern of the Bukit Timah granite sample from our experimental test under an initial confining pressure of 20 MPa. (e) Ejection failure from a lateral surface of granite sample recorded by a high speed camera (Su et al., 2017b).
the direction of major principal stress (Fig. 10a). This failure mode obtained from the numerical test is similar to that observed from our experimental test (Fig. 10d). At higher unloading rates (e.g., 1 × 104 and 1 × 105 MPa/s), small rock flakes are ejected from the lateral surfaces (Fig. 10b and c). The fragment ejection from the lateral surfaces is similar to the phenomenon recorded by a high-speed camera (Su et al., 2017b), as shown in Fig. 10e. The fragment ejection is further discussed in Section 4.3. Apart from small rock flakes, several inclined fractures develop in the sample and split it into relatively small blocks.
negative magnitude implies the particle velocity in the opposite direction. For the three unloading rates, 1 × 103 MPa/s, 1 × 104 MPa/s, and 1 × 105 MPa/s, the confining pressures at failure are 6.18 MPa, 1.63 MPa, and 0 MPa, respectively. At an unloading rate of 1 × 103 MPa/s, the horizontal component of particle velocity approaches zero with limited fluctuation. At higher unloading rates, particle movements with remarkable velocities can be observed, which are similar to the fragment ejection observed in the experimental study. The maximum particle velocity at the unloading rate of 1 × 104 MPa/s is about 6.4 m/s, while the estimated maximum ejection velocity is obtained between 3.85 and 5.11 m/s in an experimental study (He et al., 2010). Additionally, the non-uniform distribution of horizontal component of particle velocity indicates that the particles with high velocities concentrate close to the lateral surfaces, which explains the appearance of detached fragments shown in Fig. 10. Fig. 12 shows the orientation and magnitude distributions of tensile contact force, when the confining pressure reduces from 20 MPa to 10 MPa at varying unloading rates. With the same change of confining pressure magnitude, both the number of particle contacts in tension and magnitude of tensile contact force increase with higher unloading rate. The orientations of high tensile contact force are along the horizontal and sub-horizontal directions (maximum 20° away from the horizontal axis). The generation of high tensile contact force is likely attributed to
4.3. Micromechanical analysis In deep rock excavation, the unloading-induced rock failure is probably more related to our excavation work progresses rather than uncontrollable in-situ stress. To further interpret the effect of unloading rate on rock failure, we look into the grain-scale mechanism in this section and take particle velocities at Point B (see Fig. 9) in the three cases as examples. The spatial distributions of particle velocities at point B from the numerical tests at different unloading rates are provided in the top row of Fig. 11. The magnitudes of horizontal components of particle velocities are shown in the bottom row. The positive magnitude indicates the horizontal direction of particle velocity towards right, while the
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Fig. 11. Spatial distributions of particle velocities (the top row) and corresponding horizontal components (the bottom row) at point B (marked in Fig. 9) from the numerical tests under 20 MPa initial confining pressure at radial unloading rate of: (a) 1 × 103 MPa/s; (b) 1 × 104 MPa/s, and (c) 1 × 105 MPa/s. The arrow orientation is along the particle velocity and the length is proportional to the magnitude of the velocity. Note that the horizontal component of particle velocity (vertical axis) in (a) is one order of magnitude lower than that in (b) and (c).
1 × 105 MPa/s), the confining pressure at failure falls below the value predicted by the failure criterion. This phenomenon is consistent with the trend reported by Zhao et al. (2014), in which the true-triaxial unloading tests on the Beishan granite show that the peak strength increases with higher unloading rate. The discrepancy is time-dependent (He et al., 2010) and associated with the evolutions of rock deformation and strain energy release during the unloading process. Our study shows that unloading rate, which corresponds to different excavation work progresses, plays an important role in the excavationinduced failure of brittle rock. Although the Hoek-Brown failure criterion has been widely adopted in the prediction of rock failure induced by underground excavation (Carranza-Torres and Fairhurst, 1999; Hajiabdolmajid et al., 2002; Martin et al., 1999), our simulation results indicate that the failure criterion likely overestimates the confining pressure at failure at relatively high unloading rates. In other words, the stress reduction rate could be faster than the reduction rate predicted by the failure criterion. Under this circumstance, intensive monitoring and timely support are recommended as the failure with violent release of strain energy occurs in a short period after the excavation activities.
the fact that the immediate reduction of confining pressure induces relatively large radial deformation close to the sample surfaces, and the deformation is non-uniformly distributed from the surface to the interior. The non-uniform deformation causes the concentration of tensile contact force close to the surfaces, and ultimately results in the ejection of fragments along the horizontal or sub-horizontal direction. 4.4. Applicability of failure criterion The peak strength ( cpeak 0 ) of a rock sample at an initial confining pressure ( c0 ) can be predicted by the Hoek-Brown failure criterion (Eq. (5)). If the triaxial test is performed on a rock sample subjected to the axial stress equal to a certain proportion of the peak strength ( × cpeak 0 , is defined as the ratio of initial axial stress to peak strength), the confining pressure at failure ( c1) can be calculated as:
×
peak c0
=
c1
+ 186 × (38.59 ×
c1
186
+ 1)0.5
(9)
The confining pressures at failure obtained from our experimental and numerical studies are compared in Fig. 13. The solid lines represent the confining pressure at failure when the initial confining pressure varies from 0 MPa to 30 MPa and α = 60%, 70%, 80% and 90%. The experimental and numerical results at the unloading rate of 1 × 103 MPa/s approximately fall into the area between two lines with α = 70% and 80%. At higher unloading rates (e.g., 1 × 104 MPa/s and
5. Conclusions We conduct the experimental and numerical studies on the Bukit Timah granite to investigate the unloading-induced failure of brittle rock. The experimental results show that the radial expansion
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Fig. 12. Orientation distributions of tensile contact force when the confining pressure is reduced from 20 MPa to 10 MPa under a radial unloading rate of: (a) 1 × 103 MPa/s, (b) 1 × 104 MPa/s, and (c) 1 × 105 MPa/s. The length of each bin is proportional to the number of contacts in tension in the corresponding orientation, and the color represents the magnitude of averaged tensile contact force.
dominates the volumetric dilation during the unloading process. We adopt the flat-jointed contact model to reproduce quantitatively the strength envelope of the rock, which matches well with the HoekBrown failure criterion. The numerical results show that the unloading rate associated with excavation work progresses has a significant effect on the unloading-induced rock failure. The numerical results agree well with the experimental results, in terms of confining pressure at failure and failure pattern. The micromechanical analyses on the particle
velocity and tensile contact force reveal that the sudden reduction of confining pressure causes the nonuniform deformation in the sample and leads to the concentration of tensile contact force around the sample surfaces. This observation explains the formation of fragment ejection and provides insights into the mechanism of excavation-induced strain burst. Based on our studies, the Hoek-Brown failure criterion likely overestimates the stress level at higher unloading rate.
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Su, G., Zhai, S., Jiang, J., Zhang, G., Yan, L., 2017b. Influence of radial stress gradient on strainbursts: an experimental study. Rock Mech. Rock Eng. 50 (10), 2659–2676. Tao, M., Li, X., Wu, C., 2012. Characteristics of the unloading process of rocks under high initial stress. Comput. Geomech. 45, 83–92. Wu, S., Xu, X., 2016. A study of three intrinsic problems of the classic discrete element method using flat-joint model. Rock Mech. Rock Eng. 49 (5), 1813–1830. Wu, W., Zhao, Z., Duan, K., 2017a. Unloading-induced instability of a simulated granular fault and implications for excavation-induced seismicity. Tunn. Undergr. Space Technol. 63, 154–161. Wu, W., Reece, J.S., Gensterblum, Y., Zoback, M.D., 2017b. Permeability evolution of slowly slipping faults in shale reservoirs. Geophys. Res. Lett. 44, 11368–11375. Xu, N.W., Li, T.B., Dai, F., Zhang, R., Tang, C.A., Tang, L.X., 2016. Microseismic monitoring of strainburst activities in deep tunnels at the Jinping II hydropower station. China. Rock Mech. Rock Eng. 49 (3), 981–1000. Zhao, J., 1996. Construction and utilization of rock caverns in Singapore Part A: the Bukit Timah granite bedrock resource. Tunn. Undergr. Space Technol. 11 (1), 65–72. Zhao, G.F., 2017. Developing a four-dimensional lattice spring model for mechanical responses of solid. Comp. Meth. Appl. Mech. Eng. 315, 881–895. Zhao, X.G., Cai, M., 2014. Influence of specimen height-to-width ratio on the strainburst characteristics of Tianhu granite under true-triaxial unloading conditions. Can. Geotech. J. 52 (7), 890–902. Zhao, X.G., Wang, J., Cai, M., Cheng, C., Ma, L.K., Su, R., Zhao, F., Li, D.J., 2014. Influence of unloading rate on the strainburst characteristics of Beishan granite under true-triaxial unloading conditions. Rock Mech. Rock Eng. 47 (2), 467–483. Zhao, Z.H., 2016. Thermal influence on mechanical properties of granite: a microcracking perspective. Rock Mech. Rock Eng. 49 (3), 747–762. 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Fig. 13. Comparison between the experimental results, numerical results, and theoretical results predicted by the Hoek-Brown failure criterion (black lines).
Acknowledgements Wei Wu gratefully acknowledges the support of the Start-Up Grant from Nanyang Technological University, Singapore. Reference Cai, M., 2008a. Influence of intermediate principal stress on rock fracturing and strength near excavation boundaries – insight from numerical modeling. Int. J. Rock Mech. Min. 45 (5), 763–772. Cai, M., 2008b. Influence of stress path on tunnel excavation response – numerical tool selection and modeling strategy. Tunn. Undergr. Space Technol. 23 (6), 618–628. Cai, M., Kaiser, P.K., 2005. Assessment of excavation damaged zone using a micromechanics model. Tunn. Undergr. Space Technol. 20 (4), 301–310. Carranza-Torres, C., Fairhurst, C., 1999. The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek-Brown failure criterion. Int. J. Rock Mech. Min. 36 (6), 777–809. Cho, N., Martin, C.D., Sego, D.C., 2007. A clumped particle model for rock. Int. J. Rock Mech. Min. 44 (7), 997–1010. Cho, N., Martin, C.D., Sego, D.C., 2008. Development of a shear zone in brittle rock subjected to direct shear. Int. J. Rock Mech. Min. 45 (8), 1335–1346. Cundall, P.A., Strack, O.D., 1979. A discrete numerical model for granular assemblies. Geotechnique 29 (1), 47–65. Ding, X., Zhang, L., 2014. A new contact model to improve the simulated ratio of unconfined compressive strength to tensile strength in bonded particle models. Int. J. Rock Mech. Min. 69, 111–119. Duan, K., Kwok, C.Y., Tham, L.G., 2015. Micromechanical analysis of the failure process of brittle rock. Int. J. Numer. Anal. Meth. Geomech. 39 (6), 618–634. Duan, K., Kwok, C.Y., Ma, X., 2017. DEM simulations of sandstone under true triaxial compressive tests. Acta Geotech. 12 (3), 495–510. Duan, K., Kwok, C.Y., Wu, W., Jing, L., 2018. DEM modeling of hydraulic fracturing in permeable rock: influence of viscosity, injection rate and in situ states. Acta Geotech. Feng, G.L., Feng, X.T., Chen, B.R., Xiao, Y.X., Yu, Y., 2015. A microseismic method for dynamic warning of rockburst development processes in tunnels. Rock Mech. Rock Eng. 48 (5), 2061–2076. Feng, X.T., Xu, H., Qiu, S.L., Li, S.J., Yang, C.X., Guo, H.S., Cheng, Y., Gao, Y.H., 2018. In situ observation of rock spalling in the deep tunnels of the China Jinping underground laboratory (2400 m depth). Rock Mech. Rock Eng. 51 (4), 1193–1213. Feng, X.T., Chen, B.R., Li, S., Zhang, C., Xiao, Y., Feng, G., Zhou, H., Qiu, S., Zhao, Z., Yu, Y., 2012. Studies on the evolution process of rockbursts in deep tunnels. J. Rock Mech. Geotech. Eng. 4 (4), 289–295.
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Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Unloading-induced failure of brittle rock and implications for excavationinduced strain burst
T
Kang Duana, Yinlin Jib, Wei Wub, , Chung Yee Kwokc ⁎
a
School of Civil Engineering, Shandong University, Jinan, China School of Civil and Environmental Engineering, Nanyang Technological University, Singapore c Department of Civil Engineering, The University of Hong Kong, Hong Kong b
ARTICLE INFO
ABSTRACT
Keywords: Unloading-induced failure Brittle rock Discrete element model Strain burst
Understanding the unloading-induced failure characteristics of brittle rock is essential for predicting excavationinduced strain burst. We conduct both the laboratory experiments and discrete element simulations to investigate the deformation, failure pattern, and strain energy evolution of the Bukit Timah granite. The laboratory tests with radial unloading show that the radial strain mainly contributes to the change of volumetric strain in the unloading process. The numerical tests with the flat-jointed contact model reproduce the Hoke-Brown failure envelope of the rock, and simulate the unloading-induced rock failure under different combinations of initial confining pressure and unloading rate. The evolutions of strain energy release and failure pattern show that higher unloading rate likely induces violent failure, in terms of greater lateral expansion and more ejected fragments. The micromechanical analyses on the particle velocity and tensile contact force reveal that the nonuniform rock deformation concentrates at lateral surfaces and results in strain burst with fragment ejection. The Hoek-Brown failure criterion well predicts the confining pressure at failure when the rock is laterally unloaded at lower unloading rate, but likely overestimates the stress level at higher unloading rate.
1. Introduction Creating urban underground space and extracting deep natural resources are the next frontier for social development and environmental sustainability. However, these anthropogenic disturbances deep underground may perturb the initial equilibrium of rock masses and lead to the occurrence of unpredictable geohazards (Wu et al., 2017a, 2017b). At greater depth, rocks are subjected to higher in-situ stress. Field observations indicate that rock failure under high in-situ stress conditions can be either conditionally stable, which is accompanied by the progressive formation of layered structure (e.g., spalling failure), or abruptly unstable, which occurs along with the violent release of strain energy (e.g., rock burst) (Zhao and Cai, 2014). Rock failure is also dependent on various excavation work progresses. For instance, drill and blast can induce instantaneous unloading at tunnel sidewalls and result in catastrophic rock failure. Hence, both in-situ stress and unloading rate significantly affect the unloading-induced rock failure. Rock burst is a typical geohazard induced by deep rock excavation and defined as sudden rupture and expulsion of rock mass at an excavation face. In recent years, the destructive geohazard has been frequently encountered in underground space development. For example,
⁎
Xu et al. (2016) reported that a considerable length of tunnel section experienced strain burst during the deep tunnel excavation of the China Jinping II hydropower station project. Feng et al. (2018) symmetrically studied the failure mechanism of rock spalling in the same project and found that the thickness of split slabs increased in thickness from the sidewall inwards. In the construction of the Gotthard Base Tunnel in Switzerland, the fault slip burst was attributed to the reduction of both normal and shear stresses induced by the excavation activities (Wu et al., 2017a). Among these rock bursts, strain burst is the most common type (Kaiser and Cai, 2012), and may immediately occur after underground excavation (Su et al., 2017a) or unexpectedly evolve for several days (Feng et al., 2012, 2015). The delay of strain burst is likely unpredictable. Although several field studies have been conducted to reveal the occurrence of strain burst using the microseismic monitoring technique (Hirata et al., 2007; Xu et al., 2016; Su et al., 2017a), the mechanism that drives this event still eludes explanation. Experimental attempts have been made to reproduce strain burst under a controlled manner. Based on a true triaxial rock test system with acoustic emission (AE) monitoring, He et al. (2010) conducted a set of unloading tests on limestone to simulate strain burst. This study showed that the frequency and amplitude of the strain burst were
Corresponding author. E-mail address: [email protected] (W. Wu).
https://doi.org/10.1016/j.tust.2018.11.012 Received 17 May 2018; Received in revised form 20 October 2018; Accepted 17 November 2018 0886-7798/ © 2018 Elsevier Ltd. All rights reserved.
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study, and provides the mechanical parameters for the calibration of micro-parameters in the numerical model. Once the numerical model is validated, in-depth analysis is carried out to quantitatively capture the strength envelope and systematically evaluate the effects of initial confining pressure and unloading rate on the failure characteristics of the rock. Additionally, the micromechanical analyses examine the spatial distribution and magnitude of particle velocity as well as the orientation distribution of tensile contact force to reveal the mechanism of excavation-induced strain burst. 2. Experimental study The Bukit Timah granite used in this study is sourced from the central part of Singapore Island. The medium-grained and light gray granite has a P-wave velocity of 5900 m/s. The bulk density, water content, and porosity of the granite are 2660 kg/m3, 0.07%, and 0.26%, respectively. The mineral composition of the granite includes feldspar (60–65%), quartz (30%), as well as biotite and hornblende (5–10%) (Zhao, 1996). We prepared a group of intact samples with 50 mm diameter and 100 mm height as well as no visible cracks on the surfaces. We carried out a suite of triaxial tests using the MTS rock mechanics test system. The axial and radial loading capacities are 2600 kN and 140 MPa, respectively, and both have a precision of ± 0.3%. The extensometers measure the axial and radial deformation of the sample with an uncertainty of ± 0.2%. In the meantime, we recorded the acoustic signals using the AE system provided by the Physical Acoustic Corporation. Three ceramic piezoelectric sensors were attached onto the axial loading column to record the signals at a sampling rate of 1 MHz. Both pre-amplitude gain and amplitude threshold were set to 40 dB. We conducted the triaxial tests with two stress paths, one with axial loading and the other with radial unloading (Fig. 1). We first raised the confining pressure to a predetermined level (e.g., 10, 20 or 30 MPa). For the triaxial test with axial loading, we fixed the confining pressure and increased the axial stress at a radial displacement rate of 0.001 mm/s until the sample failed. For the triaxial test with radial unloading, we selected an intact sample adjacent to the one used in the previous test with axial loading. Under the same confining pressure, we fixed the axial load at 70%–80% of the peak strength collected from the previous test, and subsequently reduced the confining pressure at an unloading rate of 0.1 MPa/s until the sample failed. We set 70%–80% of the peak strength as the axial load at the beginning of the unloading process to make sure the sample failed before the confining pressure approached zero. The axial stress, confining pressure, sample deformation, and AE signals were simultaneously recorded in the triaxial tests.
Fig. 1. Triaxial tests with two stress paths in Stage III: axial loading (blue dashed line) and radial unloading (magenta solid line).
highly related to in-situ stress state. Gu et al. (2002) observed three distinct stages (i.e., crack cleavage extension, rock slab rupturing and fragment ejecting) during the rock burst in the construction of the Qinling Tunnel. Most recently, numerous triaxial tests with unloading paths have been conducted to evaluate the influence of various factors, such as sample size (Zhao and Cai, 2014; Li et al., 2018), unloading rate (Zhao et al., 2014), loading and unloading paths (Li et al., 2017), in-situ stress (Su et al., 2017a), and radial stress gradient (Su et al., 2017b), on the evolution of strain burst. In these tests, AE monitoring system and high-speed camera were used to record AE activities and fracture development, respectively. By analyzing the recorded images, the kinetic energy of ejected fragments can be estimated (Su et al., 2017a). However, the end effect originating from loading platens inevitably exists in the experimental system and may affect the unloading-induced failure process (Zhao and Cai, 2014). Numerical approaches can provide alternative avenues to address the mechanism of strain burst from different perspectives. A mathematical-physical model was proposed based on the finite element method to characterize the unloading-induced rock failure (Tao et al., 2012). An enhanced cohesion weakening and friction strengthening model was implemented based on the finite difference method to simulate the progressive failure of hard rock pillars (Rafiei Renani and Martin, 2018). Cai (2008a, 2008b) and Cai and Kaiser (2005) showed a set of numerical simulations to investigate the rock failure adjacent to an excavation face. Most of these studies used the continuum method with pre-determined failure mechanism. As the failure of brittle rock is a progressive process, the discrete element method (DEM) (Cundall and Strack, 1979) provides a proper avenue to explicitly represent the initiation and propagation of fractures at both micro- and macro-scales (Duan et al., 2015, 2018; Potyondy and Cundall, 2004; Zhao, 2016, 2017). So far, the mechanical responses of brittle rocks subjected to the uniaxial compression (Potyondy and Cundall, 2004), true triaxial compression (Duan et al., 2017), direct shear condition (Cho et al., 2008), direct and indirect tensile conditions (Shang et al., 2017) have been extensively investigated based on the DEM analysis. However, modeling the unloading-induced failure characteristics of brittle rock has been rarely reported. We conduct both the experimental and numerical studies on the Bukit Timah granite to investigate the unloading-induced failure of brittle rock. The laboratory experiments investigate the sample deformation and strain energy release during the unloading process. The experimental study serves as the physical background of the numerical
3. Numerical model 3.1. Flat-joint contact model It has been widely recognized that the classic DEM model with parallel bonds suffers from several intrinsic limitations in modeling the mechanical responses of brittle rock (Ding and Zhang, 2014; Duan et al., 2018; Potyondy and Cundall, 2004; Wu and Xu, 2016). First, the tensile strength is significantly overestimated if one calibrates the micro-parameters to match the uniaxial compressive strength of brittle rock. Second, if the DEM model is calibrated to match the compressive strength under a confining pressure (for instance, 20 MPa as illustrated in Fig. 2a), the slope of the strength envelope produced by the DEM model is lower than that obtained from the laboratory experiments. These intrinsic limitations may be attributed to the idealized geometry of spherical grains adopted in the model. If the strength envelope of brittle rock (the magenta line in Fig. 2a) cannot be quantitatively captured, the reduction of confining pressure may not trigger rock 496
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Fig. 2. (a) Schematic diagram showing the limitations of the DEM model with parallel bonds in the triaxial test with radial unloading; (b) Flat-jointed contact model for brittle rock simulation; (c) Adjacent particles with a flat and notional interface in two-dimension (after Itasca, 2015).
failure when the confining pressure is reduced to zero. Moreover, obtaining proper tensile cutoff is important for the strength envelope to simulate the unloading-induced tensile failure of brittle rock (Zhu et al., 2014). To date, considerable efforts have been taken to tackle these intrinsic limitations by modifying particle texture (Cho et al., 2007), changing bond contact (Scholtès and Donzé, 2012), and introducing new constitutive contact model (Ding and Zhang, 2014). Among these approaches, the flat-jointed contact model (FJCM) (Potyondy, 2012) can represent the polygonal shape of rock crystals, and mimic rock damage at both micro- and macro-scales. FJCM has been successfully applied to solve the intrinsic problems of a classic DEM model (Wu and Xu, 2016). A flat-jointed material and FJCM are shown in Fig. 2b and c, respectively. A particle-particle contact simulates the behavior of a finite-length interface with locally flat and notional surfaces. By considering the contact between two particles as a planar interface, the particles in the flat-jointed material are able to maintain their polygonal grain structures. FJCM is thus able to provide grain interlocking and rotational resistance even after the bonds break. In a 2D model, the interface is a flat line, which is discretized into several elements (Fig. 2c). On the notional surface, the force and moment (F (e) and M(e) ) act at the centroid of the element. At the center of the interface, the equivalent force and moment (Fc and Mc ) can be calculated statically based on all the element forces and moments:
F (e ) , M c =
Fc = e
{(r (e) × F (e) ) + M(e) }
The element normal and shear stresses, which act at the element center, can be calculated as: (e )
M (e ) =
Fn(e) n c + F (se)
(2)
M(be)
(3)
Mt(e) n c
+
(e )
=
F (se) A(e)
(4)
where is the area of the element. More details regarding the constitutive relationship of FJCM can be found in Itasca Consulting Group (2015) and Potyondy (2012). 3.2. Calibration of micro-parameters Following the procedure recommended by Wu and Xu (2016), we calibrated the micro-parameters of FJCM to match the strength envelope of the Bukit Timah granite. The mechanical properties of the granite obtained from the triaxial tests under the confining pressures ranging from 0 MPa to 30 MPa are summarized in Table 1. The uniaxial compressive and direct tensile strengths are 186.8 MPa and 10.1 MPa, respectively (Peng et al., 2017). The size of the DEM sample is the same as that used in the experimental study. The sample consists of 20,694 particles, following the uniform distribution with the minimum particle size (Rmin) of 0.2 mm and particle size ratio (Rmax/Rmin) of 1.66. The corresponding micro-parameters are listed in Table 2. The excellent agreement can be found between the strength envelopes obtained numerically and experimentally (Fig. 3). For an intact rock sample, the Hoek-Brown failure criterion is expressed as (Hoek and Brown, 1997):
where r (e) is the position vector. During the calculation, the state of the element bond may be modified by the element force and moment according to the forcedisplacement law. The element force is resolved into the normal force (Fn(e) ) and shear force (F (se) ), while the element moment is resolved into the twisting moment (Mt(e) ) and bending moment (M(be) ):
F (e ) =
Fn(e) , A(e)
A(e)
(1)
e
=
1
=
3
+
c (mi
3
+ 1)0.5
(5)
c
where c is the uniaxial compressive strength, 1 and 3 are the major and minor principal stresses, respectively, mi is a material constant. The results of curve fitting are also provided in Fig. 3, in which both the experimental and numerical results are consistent with those predicted by the failure criterion. 3.3. Procedure of numerical simulation
where Fn(e) > 0 is in tension, and n c is the unit normal vector.
We conducted a series of DEM simulations to mimic the triaxial tests
Table 1 Summary of triaxial tests on the Bukit Timah granite. Triaxial test with axial loading
Triaxial test with radial unloading
Confining pressure (MPa)
Peak strength (MPa)
Elastic modulus (GPa)
Poisson’s ratio
Initial confining pressure (MPa)
Initial axial stress (MPa)
Confining pressure at failure (MPa)
10 20 30
330.5 442.0 527.1
72.35 73.23 73.42
0.164 0.153 0.158
10 20 30
267.1 306.8 427.0
10.2 15.2
* The sample does not fail when the confining pressure is reduced to 0 MPa. 497
*
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Hazzard et al., 2000). Hazzard et al. (2000) found that the strain energy released by bond breakage may have a significant influence on rock behavior, because the waves emanating from cracks are capable of inducing more cracks if nearby bonds are closed to failure. The influence of the degree of damping on the unloading-induced rock failure is worth exploring in the future study.
Table 2 Micro-parameters calibrated for the flat-jointed contact model to represent the Bukit Timah granite. Micro-parameter
Value
Minimum particle radius, Rmin (mm) Ratio between maximum to minimum particle size, Rmax/Rmin Installation gap ratio Bonded element fraction Circumferential elements Effective modulus of both particle and bond (GPa) Ratio of normal to shear stiffness of both particle and bond Mean and standard deviation of bond tensile strength (MPa) Mean and standard deviation bond cohesion (MPa) Local friction angle (° )
0.2 1.66 0.4 1.0 4 115.5 1.9 20±0 355±35.5 36
Residual friction angle (° )
4. Results and discussion 4.1. Experimental results We conduct the triaxial tests with radial unloading under three initial confining pressures, i.e., 10 MPa, 20 MPa, and 30 MPa. The experimental results are summarized in Table 1. The samples under initial confining pressures of 20 MPa and 30 MPa fail when the confining pressures decrease to 10.2 MPa and 15.2 MPa, respectively. The sample under an initial confining pressure of 10 MPa does not fail immediately when the confining pressure reaches 0 MPa. Fig. 4 shows the typical results obtained from the triaxial test with radial unloading at a rate of 0.1 MPa/s under 20 MPa initial confining pressure. The positive volumetric strain, positive axial strain, and negative radial strain indicate volumetric compaction, axial compression, and radial expansion, respectively. We set both the initial cut-offs of axial and radial strain as zero. The volumetric strain ( v ) is calculated based on the axial strain ( 1) and radial strain ( 3 ) as:
38.7
with radial unloading illustrated in Fig. 1. At Stage I, we raised both the confining pressure and axial stress simultaneously to predetermined values. At Stage II, we kept the confining pressure ( c0 ) constant, and increased the axial stress monotonically to 70%–80% of the peak strength obtained from the triaxial test with axial loading. At Stage III, we maintained the axial stress and reduced the confining pressure at a constant unloading rate ( c ). The confining pressure after nt cycles can be determined by: ct
=
c0
+
c
× nt
(6)
v
Note that the DEM simulations were performed through the timestepping algorithm with time cycle as low as 1 × 10−7 s. In the experimental study, a triaxial test with 0.1 MPa/s unloading rate lasted about 1000 s. Hence, a numerical test with the same unloading rate could take about 1 × 1010 cycles. However, as running 1 × 105 cycles needed 20 h, we were unable to strictly follow the unloading rate adopted in the experimental study. To save the computation cost, we used the unloading rates of 1 × 103 MPa/s, 1 × 104 MPa/s and 1 × 105 MPa/s to represent 1.97 × 10−5 MPa/cycle, −4 −3 1.97 × 10 MPa/cycle and 1.97 × 10 MPa/cycle, respectively, to uncover the mechanism of unloading-induced rock failure. All the numerical simulations reported in this study were carried out with a damping coefficient of 0.7 to explore the influence of unloading rate on the unloading-induced failure characteristics of brittle rock. Previous studies have revealed that the degree of damping may have a significant influence on rock behavior (Potyondy and Cundall, 2004;
=
1
+2×
3
(7)
The initial loading processes of the confining pressure and axial stress at Stage I are similar to those shown in Fig. 1. At Stage II, the axial strain increases linearly at a rate of 4×10−6/s, while the radial strain decreases at a much lower rate of 6×10−7/s. The ratio of radial strain reduction to axial strain increment is 0.15, which is approximately
Fig. 3. Comparison between the strengths obtained from the experimental and numerical studies, together with the strength envelope predicted by the HoekBrown failure criterion.
Fig. 4. Experimental results of the triaxial test at a radial unloading rate of 0.1 MPa/s under an initial confining pressure of 20 MPa. 498
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Fig. 5. Numerical results of the triaxial tests at a radial unloading rate of 1 × 104 MPa/s under initial confining pressure of (a) 10 MPa, (b) 20 MPa, and (c) 30 MPa.
equal to the Poisson’s ratio determined by the triaxial test with axial loading (Table 1), indicating that the sample behaves elastically before the unloading process. During this stage, the positive volumetric strain increases linearly, implying that the sample is being continuously compacted. The remarkable AE signals are detected when the axial stress reaches 292.4 MPa, which reflects that the sample is approaching the failure stage. At Stage III, the confining pressure reduces at an unloading rate of 0.1 MPa/s, and both the magnitudes of axial and
radial strains remarkably raise. The volumetric strain turns from increasing to decreasing until the sample fails. This observation indicates that the marked increase in the magnitude of radial strain contributes dominantly to the change of volumetric strain at this stage, leading to a transition of sample deformation from compaction to dilation. In the meantime, the cumulative AE count jumps when the confining pressure starts to reduce and subsequently increases until the sample collapses at 10.2 MPa confining pressure (defined as the confining pressure at 499
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Fig. 6. Evolution of stored strain energy and strain energy increment rate (change of stored strain energy every 200 cycles) obtained from the numerical tests at a radial unloading rate of 1 × 104 MPa/s under initial confining pressure of (a) 10 MPa, (b) 20 MPa, and (c) 30 MPa. Table 3 Effects of unloading rate and initial confining pressure on the magnitude of stored strain energy and strain energy increment rate. Initial confining pressure (MPa)
Unloading rate (MPa/ s)
Magnitude of stored strain energy (J)
Increment of stored strain energy (10−3 J/step)
Point A
Point B
Point C
A-B
B-C
4
Influence of initial confining pressure
10 20 30
1 × 10 1 × 104 1 × 104
1408.0 1824.2 3389.6
1483.9 1987.1 3534.8
1936.1 2523.6 4293.8
1.52 1.79 2.10
12.39 15.88 13.71
Influence of unloading rate
20 20 20
1 × 103 1 × 104 1 × 105
1824.2 1824.2 1824.2
1964.3 1987.1 1808.1
2394.4 2523.6 2528.3
0.20 1.79 −1.45
4.87 15.88 22.16
Fig. 7. Failure patterns of the DEM samples obtained from the numerical tests at a radial unloading rate of 1 × 104 MPa/s under initial confining pressure of (a) 10 MPa, (b) 20 MPa, and (c) 30 MPa.
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Fig. 8. Numerical results of the triaxial tests under an initial confining pressure of 20 MPa at radial unloading rates of (a) 1 × 103 MPa/s, (b) 1 × 104 MPa/s, and (c) 1 × 105 MPa/s. Note the scale of horizontal axial in (a) is different from the other two cases.
failure). The sample fails accompanied by the sudden drops of axial stress, radial strain, and volumetric strain, as well as the dramatic increases of cumulative AE counts and axial strain.
induced failure of brittle rock, we conduct the numerical tests at an unloading rate of 1 × 104 MPa/s under varying initial confining pressures (e.g., 10 MPa, 20 MPa, and 30 MPa). As illustrated in Fig. 5, the sample deformation and micro-crack evolution of the three cases are similar at Stage II. The ratio of radial strain reduction to axial strain increment is slightly larger than the Poisson’s ratio given in Table 1. This discrepancy can be attributed to the failure of flat-jointed material additionally contributing to the lateral deformation. At the beginning of
4.2. Numerical results 4.2.1. Influence of initial confining pressure To evaluate the effect of initial confining pressure on the unloading501
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Fig. 9. Evolution of stored strain energy and strain energy increment rate (change of stored strain energy every 200 cycles) obtained from the numerical tests under an initial confining pressure of 20 MPa at radial unloading rate of (a) 1 × 103 MPa/s, (b) 1 × 104 MPa/s, and (c) 1 × 105 MPa/s.
Stage III, the axial strain stays almost constant, implying that the sample experiences limited deformation along the axial direction. Meanwhile, the magnitude of radial strain remarkably raises, and mainly contributes to the volume change. In other words, the unloading process induces the formation of tensile micro-cracks parallel with the direction of major principal stress. When the confining pressure approaches zero, the number of micro-cracks increases dramatically, accompanied by the rapid increases of both axial and radial strains. The volume change transits from compaction to dilation. Finally, the sample collapses with the loss of loading bearing capacity. As the linear contact between particles are adopted in the DEM model, the strain energy stored in the sample (EC) can be calculated as (Itasca, 2008):
EC =
1 2
(|Fin |2 /k n + |Fis |2 /k s ) NC
orientation is approaching the direction of the minor principal stress. 4.2.2. Influence of unloading rate The unloading-induced failure of brittle rock can be also affected by varying unloading rates associated with various excavation work progresses. We conduct the numerical tests under 20 MPa initial confining pressure at different unloading rates of 1 × 103 MPa/s, 1 × 104 MPa/s, and 1 × 105 MPa/s. At Stage II, the numerical results show that the sample deformation and micro-crack evolution are the same in the three cases as they follow the same loading path (Fig. 8). The ratio of radial strain reduction to axial strain increment is also slightly higher than the Poisson’s ratio as discussed in Section 4.2.1. At Stage III, the axial strain keeps constant, and the decrease of radial strain mainly contributes to the reduction of volumetric strain. The number of microcracks then dramatically increases with the sharp reduction of volumetric strain. At the end, the sample fails accompanied by the increase of axial strain. The axial stress suddenly jumps because the servomechanism fails to maintain the axial stress during the catastrophic failure of the sample. The evolutions of stored strain energy and strain energy increment rate are plotted in Fig. 9. At Stage II, the same amount of strain energy is stored in the sample (Point A). The stored strain energy stays constant at the beginning of Stage III (from Point A to Point B), but the duration of the case at higher unloading rate remarkably reduces. When the strain energy increment rate starts to fluctuate, the stored strain energy increases until the peak value is reached (Point C). The three cases with different unloading rates show the similar peak value but different increments of stored strain energy (Table 3). After the peak point, the catastrophic failure is characterized by a rapid release of strain energy. Fig. 10a–c shows that the rate-dependent failure patterns can be observed after the numerical tests with radial unloading. At a low unloading rate (e.g., 1 × 103 MPa/s), the sample fails into several relatively large blocks with the major fractures sub-parallel with respect to
(8)
where NC is the number of contacts; |Fin | and |Fis | are the magnitudes of normal and shear components of a contact force; kn and ks are the normal and shear stiffness of a particle contact, respectively. At the end of Stage II (Point A), the stored strain energy is larger in the sample under higher initial confining pressure (Fig. 6). At Stage III, the stored strain energy is first stable (from Point A to Point B), as the number of micro-cracks shows no obvious increase (Fig. 5). Subsequently, the stored strain energy jumps to the peak value (Point C). According to Table 3, the peak value is greater in the case under higher initial confining pressure. However, our results do not show the stress dependence of strain energy increment rate during the unloading process, which is likely due to the random failure of bond contacts. Both ejected fragments from the lateral surfaces and major fractures in the failed sample can be observed in Fig. 7. In the three cases, the distribution of induced factures are similar, and the most major fractures are sub-vertical. When the sample is subjected to higher initial confining pressure, the number of major fractures increases, and the 502
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Fig. 10. Failure patterns of the DEM samples obtained from the numerical tests under an initial confining pressure of 20 MPa at radial unloading rate of: (a) 1 × 103 MPa/s, (b) 1 × 104 MPa/s, and (c) 1 × 105 MPa/s, (d) failure pattern of the Bukit Timah granite sample from our experimental test under an initial confining pressure of 20 MPa. (e) Ejection failure from a lateral surface of granite sample recorded by a high speed camera (Su et al., 2017b).
the direction of major principal stress (Fig. 10a). This failure mode obtained from the numerical test is similar to that observed from our experimental test (Fig. 10d). At higher unloading rates (e.g., 1 × 104 and 1 × 105 MPa/s), small rock flakes are ejected from the lateral surfaces (Fig. 10b and c). The fragment ejection from the lateral surfaces is similar to the phenomenon recorded by a high-speed camera (Su et al., 2017b), as shown in Fig. 10e. The fragment ejection is further discussed in Section 4.3. Apart from small rock flakes, several inclined fractures develop in the sample and split it into relatively small blocks.
negative magnitude implies the particle velocity in the opposite direction. For the three unloading rates, 1 × 103 MPa/s, 1 × 104 MPa/s, and 1 × 105 MPa/s, the confining pressures at failure are 6.18 MPa, 1.63 MPa, and 0 MPa, respectively. At an unloading rate of 1 × 103 MPa/s, the horizontal component of particle velocity approaches zero with limited fluctuation. At higher unloading rates, particle movements with remarkable velocities can be observed, which are similar to the fragment ejection observed in the experimental study. The maximum particle velocity at the unloading rate of 1 × 104 MPa/s is about 6.4 m/s, while the estimated maximum ejection velocity is obtained between 3.85 and 5.11 m/s in an experimental study (He et al., 2010). Additionally, the non-uniform distribution of horizontal component of particle velocity indicates that the particles with high velocities concentrate close to the lateral surfaces, which explains the appearance of detached fragments shown in Fig. 10. Fig. 12 shows the orientation and magnitude distributions of tensile contact force, when the confining pressure reduces from 20 MPa to 10 MPa at varying unloading rates. With the same change of confining pressure magnitude, both the number of particle contacts in tension and magnitude of tensile contact force increase with higher unloading rate. The orientations of high tensile contact force are along the horizontal and sub-horizontal directions (maximum 20° away from the horizontal axis). The generation of high tensile contact force is likely attributed to
4.3. Micromechanical analysis In deep rock excavation, the unloading-induced rock failure is probably more related to our excavation work progresses rather than uncontrollable in-situ stress. To further interpret the effect of unloading rate on rock failure, we look into the grain-scale mechanism in this section and take particle velocities at Point B (see Fig. 9) in the three cases as examples. The spatial distributions of particle velocities at point B from the numerical tests at different unloading rates are provided in the top row of Fig. 11. The magnitudes of horizontal components of particle velocities are shown in the bottom row. The positive magnitude indicates the horizontal direction of particle velocity towards right, while the
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Fig. 11. Spatial distributions of particle velocities (the top row) and corresponding horizontal components (the bottom row) at point B (marked in Fig. 9) from the numerical tests under 20 MPa initial confining pressure at radial unloading rate of: (a) 1 × 103 MPa/s; (b) 1 × 104 MPa/s, and (c) 1 × 105 MPa/s. The arrow orientation is along the particle velocity and the length is proportional to the magnitude of the velocity. Note that the horizontal component of particle velocity (vertical axis) in (a) is one order of magnitude lower than that in (b) and (c).
1 × 105 MPa/s), the confining pressure at failure falls below the value predicted by the failure criterion. This phenomenon is consistent with the trend reported by Zhao et al. (2014), in which the true-triaxial unloading tests on the Beishan granite show that the peak strength increases with higher unloading rate. The discrepancy is time-dependent (He et al., 2010) and associated with the evolutions of rock deformation and strain energy release during the unloading process. Our study shows that unloading rate, which corresponds to different excavation work progresses, plays an important role in the excavationinduced failure of brittle rock. Although the Hoek-Brown failure criterion has been widely adopted in the prediction of rock failure induced by underground excavation (Carranza-Torres and Fairhurst, 1999; Hajiabdolmajid et al., 2002; Martin et al., 1999), our simulation results indicate that the failure criterion likely overestimates the confining pressure at failure at relatively high unloading rates. In other words, the stress reduction rate could be faster than the reduction rate predicted by the failure criterion. Under this circumstance, intensive monitoring and timely support are recommended as the failure with violent release of strain energy occurs in a short period after the excavation activities.
the fact that the immediate reduction of confining pressure induces relatively large radial deformation close to the sample surfaces, and the deformation is non-uniformly distributed from the surface to the interior. The non-uniform deformation causes the concentration of tensile contact force close to the surfaces, and ultimately results in the ejection of fragments along the horizontal or sub-horizontal direction. 4.4. Applicability of failure criterion The peak strength ( cpeak 0 ) of a rock sample at an initial confining pressure ( c0 ) can be predicted by the Hoek-Brown failure criterion (Eq. (5)). If the triaxial test is performed on a rock sample subjected to the axial stress equal to a certain proportion of the peak strength ( × cpeak 0 , is defined as the ratio of initial axial stress to peak strength), the confining pressure at failure ( c1) can be calculated as:
×
peak c0
=
c1
+ 186 × (38.59 ×
c1
186
+ 1)0.5
(9)
The confining pressures at failure obtained from our experimental and numerical studies are compared in Fig. 13. The solid lines represent the confining pressure at failure when the initial confining pressure varies from 0 MPa to 30 MPa and α = 60%, 70%, 80% and 90%. The experimental and numerical results at the unloading rate of 1 × 103 MPa/s approximately fall into the area between two lines with α = 70% and 80%. At higher unloading rates (e.g., 1 × 104 MPa/s and
5. Conclusions We conduct the experimental and numerical studies on the Bukit Timah granite to investigate the unloading-induced failure of brittle rock. The experimental results show that the radial expansion
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Fig. 12. Orientation distributions of tensile contact force when the confining pressure is reduced from 20 MPa to 10 MPa under a radial unloading rate of: (a) 1 × 103 MPa/s, (b) 1 × 104 MPa/s, and (c) 1 × 105 MPa/s. The length of each bin is proportional to the number of contacts in tension in the corresponding orientation, and the color represents the magnitude of averaged tensile contact force.
dominates the volumetric dilation during the unloading process. We adopt the flat-jointed contact model to reproduce quantitatively the strength envelope of the rock, which matches well with the HoekBrown failure criterion. The numerical results show that the unloading rate associated with excavation work progresses has a significant effect on the unloading-induced rock failure. The numerical results agree well with the experimental results, in terms of confining pressure at failure and failure pattern. The micromechanical analyses on the particle
velocity and tensile contact force reveal that the sudden reduction of confining pressure causes the nonuniform deformation in the sample and leads to the concentration of tensile contact force around the sample surfaces. This observation explains the formation of fragment ejection and provides insights into the mechanism of excavation-induced strain burst. Based on our studies, the Hoek-Brown failure criterion likely overestimates the stress level at higher unloading rate.
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Fig. 13. Comparison between the experimental results, numerical results, and theoretical results predicted by the Hoek-Brown failure criterion (black lines).
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