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Time-dependent strength degradation of granite
ARTICLE IN PRESS International Journal of Rock Mechanics & Mining Sciences 46 (2009) 1103–1114
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Time-dependent strength degradation of granite Q.X. Lin a, Y.M. Liu b, L.G. Tham a,, C.A. Tang c, P.K.K. Lee a, J. Wang b a b c
Department of Civil Engineering, The University of Hong Kong, Hong Kong, China Beijing Research Institute of Uranium Geology, Beijing, China Center for Rock Instability and Seismicity Research, Northeastern University, Shenyang, China
a r t i c l e in fo
abstract
Article history: Received 12 September 2007 Received in revised form 11 March 2009 Accepted 14 July 2009 Available online 8 August 2009
Understanding the time-dependent strength degradation and associated creep behavior is essential to the safety evaluation of a radioactive waste disposal system in rock. In this study, a series of constant loading tests under different confining pressures and temperatures with acoustic emission monitoring have been performed to study evolution of dilatancy (damage) and strength degradation. Source location analysis and moment tensor analysis were carried out to reproduce the progressive damage process during constant load helping to understand the failure mechanism from a microscopic point of view. In this paper, we report on the findings of the experimental results. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Constant loading test Strength degradation Acoustic emission Damage Dilatancy Granite
1. Introduction Nuclear power is cheap, clean and without the shortcomings of finite hydrocarbon and hydroelectric resources. However, generation of nuclear power produces high-level radioactive waste (HLW), which must be isolated from the biosphere until it has decayed to a level that poses no significant risk to human beings. Deep geological disposal is considered to be the best option to deal with the HLW. However, long-lived radionuclide must be isolated over thousands of years after emplacement and heat generated from the waste in the repository will result in an increase in rock temperature. Preferred hydraulic pathways could be created if the rock around excavations is damaged under sustained loading and high temperature and therefore it is important to minimize potential pathways for the transport of radionuclides in the rock surrounding a repository. These damages appear in the form of cracking in the rock. Therefore a better understanding of long-term strength of rock and associated creep behavior is essential for the design and construction of excavations for nuclear waste repositories. Since the long-term deformation of soft rocks is more apparent than hard rocks, such as granite, studies were conducted on more deformable rocks such as sedimentary rock [1], sandstone [2] and salt [3,4]. On the other hand, the effect of long-term behavior of brittle rock is often neglected in the traditional rock mechanics
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design. Only few papers can be found in the literature. When a brittle rock is held at constant loading, its long-term strain is typically composed of three phases: primary, secondary and tertiary phases. During primary and secondary phases, the nucleation and growth of microcracks occur uniformly throughout the rock and the cracks have little tendency to link but instead interact to eliminate regions of high local stress [5,6]. During the tertiary phase, the crack density reaches a critical level and crack grows further in a weakened region and ultimately develops into a macroscopic fault [7]. Scholz [8] studied the failure mechanism of brittle rock under constant load. However, in his theory, there is no tensile stress within the specimen therefore it is difficult to envisage the occurrence of a fracture. Cruden [9] developed a new theory based on Charles’ theory [10] of subcritical growth of preexisting cracks by stress-corrosion. However, Cruden neglected the dislocation motion and assumed that the material should contain pre-existing cracks. Kranz [5] studied the development of cracks in Barre granite under constant loading by scanning electron microscope from micro-structure point of view. As the production of sections often introduces new cracks, making the identification of natural and experimentally stress-induced cracks difficult. Furthermore, cracks viewed in the un-stressed state may give false impression about their dimensions when stressed. Recently, granite was selected as foundation of high dams and potential parent rock for nuclear waste repositories, and the interest of studying the long-term behavior of granite was raised [11,12]. Lau and Chandler [13] developed three laboratory test techniques to study and quantify damage development in loaded granite specimen and to provide data for use in the calibration and
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solution of numerical models. Lin et al. [14] studied the failure process of granite under constant loading and quantified the damage evolution by inelastic volumetric strain, degradation of elastic properties and cumulative acoustic emission under uniaxial condition. However, there is only little information on longterm behavior of brittle rock in the intermediate temperature range and under high confining pressure. One of the early works was conducted by Kranz et al. [15] who indicated that the effects of increasing temperatures from 24 to 200 1C were to weaken the rock considerably and to reduce the time to failure by about two orders of magnitude. Due to the brittle nature, failure of granite was accompanied by micro-cracking or micro-fracturing which could be detected by acoustic emission system. Since acoustic emission monitoring is essentially passive, it provides an ideal non-destructive method for studying crack growth at its actual stress state. The failure process could be monitored by an acoustic emission system. Acoustic emission studies in the literature include: (1) simple counting of the number of AE events prior to specimen failure showing a correlation between AE rate and inelastic strain rate [16], (2) location of hypocenters of AE source events [7] and (3) analysis of full waveform data as recorded at receiver sites. One aspect of this research was to determine fault plane solutions of AE source events from first motion data [17,18]. This paper aims to study the evolution of dilatancy (damage) and strength degradation of granite by a series of constant loading tests. Damage micro-mechanism during constant loading tests and effects of confining pressures and temperatures on the associated long-term behavior of granite will be discussed.
2. Laboratory investigation 2.1. Rock specimens For this study, rock specimens were prepared from granite cores obtained from Beishan, China, and they are referred to as BS granite. BS granite is classified as porphyritic monzonitie granite. The grain density is 2.71 g/cm3 and the natural density is 2.64 g/ cm3. The porosity and water content of specimens are low. The average compressive wave velocity is about 5423 m/s (range: 5197–5549 m/s) (Table 1). The specimens are about 50 mm in diameter and 125 mm in height. The tolerance of straightness and Table 1 Properties of the BS specimens. Specimen no.
Vp (m/s)
E (GPa)
u
s3 (MPa)
scd (MPa)
Stress ratio b
0-RT-1 0-RT-2 0-RT-3 0-RT-4 10-RT-1 10-RT-2 10-RT-3 10-RT-4 30-RT-1 30-RT-2 30-RT-3 30-RT-4
5314 5313 5453 5279 5545 5549 5548 5523 5477 5197 5441 5442
44.50 42.57 42.61 42.99 46.00 45.61 46.54 46.22 45.49 41.77 45.44 48.49
0.28 0.25 0.28 0.25 0.30 0.31 0.32 0.31 0.31 0.35 0.26 0.30
0 0 0 0 10 10 10 10 30 30 30 30
104.65 109.57 106.71 101.27 164.18 189.38 194.41 196.99 251.04 104.43* 257.78 268.72
1.253 1.190 1.168 1.113 1.350 1.357 1.307 1.277 1.578 – 1.385 1.319
0-50-1 0-50-2 0-50-3 0-50-4
– – – –
43.40 40.82 40.07 38.52
0.28 0.31 0.32 0.30
0 0 0 0
97.86 108.40 104.16 94.98
1.278 1.092 1.161 1.237
0-90-1 0-90-2 0-90-3
– – –
41.12 44.22 39.52
0.25 0.32 0.30
0 0 0
97.84 94.49 105.00
1.239 1.234 1.212
flatness of the specimens meet the specifications of ASTM designation: D4543-85. The straightness tolerance of the sides is o0.50 mm, the flatness tolerance of the ends is o0.025 mm, and the perpendicularity tolerance is o0.251. 2.2. Testing facilities Constant loading tests were conducted using a MTS 815 rock mechanics test system which has a maximum capacity of 4600 kN. The MTS 815 is a computer controlled, servo-hydraulic compression machine, load frame, hydraulic power supply, triaxial cell, confining pressure subsystem, digital controller, test processor and a PC workstation. It is also equipped with a pair of end platen containing ultrasonic sensors for the determination of wave velocities when a specimen is loaded. Two linear variable differential transducers are used to measure the axial deformation, while the circumferential deformation is measured by an extensometer. In addition, the MISTRAS 2001 is used to record the AE events. The system is a fully digital, multi-channel, computerized, acoustic emission system that performs AE waveform and signal measurement and stores, displays and analyzes the resulting data. It consists of an IBM compatible PC, an AEDSP-32/16 card and the MISTRAS software, sensors, preamplifiers and cables. 2.3. Testing procedures To study the evolution of dilatancy (damage) and associated long-term behavior under different confining pressures and temperatures, a series of constant loading tests were conducted at the Rock Engineering Research Centre of The University of Hong Kong. The test programme consisted of 36 cases conducted at confining pressures of 0, 10 and 30 MPa and at temperatures 23 (room temperature, RT), 50 and 90 1C (Table 1). The in situ stress measured by hydrofracturing method [19] indicated that the vertical stress changed from 4.36 to 13.31 MPa, and the horizontal stress ranged from 7.72 to 25.66 MPa at a depth of 161.5 to 493 m. Therefore, the confining pressure of 10 and 30 MPa were chosen to simulate the in situ stress. A temperature of 50 1C was used to simulate the in-situ rock temperature, and 90 1C was the potential temperature of the surrounding under the heating of the nuclear waste in future. The test consisted of two stages: initial loading and constant loading. In the initial loading stage, the specimen was axially loaded at a rate that produced a constant lateral deformation rate of 0.006 mm/min. The crack damage stress scd was determined during the initial loading stage. scd is defined as the stress at the point of volumetric strain reversal, which marks the onset of unstable crack growth and may be related to the long-term strength of rock [20]. When the predetermined stress level bscd (b ¼ applied axial stress/scd) was achieved, the load was maintained at the predetermined stress level by the closed-loop operation of the servo-controlled machine until the specimen failed. b is, thereafter, referred to as stress ratio. If the specimen did not fail after a long period of time, up to several days, the constant loading test was terminated. In the study, the stress ratio varies from 1.05 to 1.20. Six piezoelectric transducers were attached to the specimen to detect AE during the sudden growth of microcracks or slip along existing crack surfaces. Each transducer was attached to the specimen through a copper adapter, of which one surface was machined to the same curvature as the specimen surface, and cemented to the side with cyanoacrylate (CN adhesive). Sonotech Soundsafe ultrasonic couplant was used as the couplant between the polished contact surfaces of both transducers and adapters in
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Fig. 2. A typical creep test and its three creep phases (test on BS granite 10-RT-3).
Fig. 1. Sensors setup and acoustic emission monitoring system.
order to achieve good coupling. The transducers were pressed tightly to the adapters with rubber bands. Signals are amplified and recorded by the MITRAS 2001 system as shown in Fig. 1. Required data and AE signals were recorded automatically by computer for further analysis.
3. Test results During initial loading stage, the cracks inside the specimen became unstable and the specimen started to dilate after the stress level was increased above the crack damage stress (scd). During the constant loading stage, the specimen keeps dilated further until it is fractured to a state that it cannot sustain the designed constant loading after a certain period of time. A typical creep curve during the constant load stage is shown in Fig. 2. The process can be divided into three phases: primary (or transient), secondary (or steady state), and tertiary (or accelerating) creep phases. In the transient creep phase, the strain rate monotonically decays and then remains constant during the steady state creep phase. The strain rate increases suddenly and rapidly until the specimen fractures in the accelerating creep phase. In the subsequent sections, the effect of confining pressure and temperature on the long-term strength of granite as well as the development and propagation of micro-fractures will be discussed in detail. 3.1. Effect of confining pressures during constant load at room temperature Volumetric strain, which reflects the change in ‘pore volume’ caused by cracking, was adopted to study the evolution of dilatancy and damage under different confining pressures. Fig. 3 shows the evolution of volumetric strain (negative volumetric strain, in the present study, represents dilatancy) under different confining pressures. Specimen tends to become more stable under higher confining pressure. Unconfined specimen with stress ratio higher than 1.15 failed in a few hours and all phases can be observed. Under the confining pressure of 10 MPa, the specimen with stress ratio of 1.28 failed after about 18 days. On the other hand, the specimen under confining pressure of 30 MPa and having stress ratio of 1.32 was still quite stable. It had no sign of
failure even after 14 days: only the primary and secondary phases were observed. Though specimens 10-RT-3 and 30-RT-4 have a similar stress ratio of 1.31, the former one failed in about 4 days while the latter one was still quite stable for 14 days. Fig. 4 summarizes the volumetric strain rate of a series of tests under uniaxial loading. The strain rate increases as the stress ratio increases. The initial strain rate of specimen 0-RT-3 is high and then decreases gradually until it reaches a minimum strain rate at the end of primary phase. The minimum strain rate remains almost constant during secondary phase. In the tertiary phase, the strain rate increases suddenly and the specimen eventually ruptured. The stress ratio of specimen 0-RT-4 is lower than that for specimen 0-RT-3. The strain rate of specimen 0-RT-4 decreases gradually to a minimum value (steady state strain rate) and then remains constant. No acceleration of strain rate was observed during the testing period. The stress ratio of specimen 0-RT-2 is higher than that for specimen 0-RT-3 and the strain rate decreases rapidly to the minimum value and then accelerates to failure. Fig. 5 shows the variation of the steady state volumetric strain rate with axial stress. In the figure, the results for Westerly granite reported by Lockner [21] under unconfined conditions are also plotted. The Westerly granite has finer grains and higher ultimate strength [21]. It is noted that the Westerly granite has a lower steady state volumetric strain rate.
3.2. Effect of temperatures during long-term under uniaxial condition The volumetric strains as a function of time at different temperatures are shown in Fig. 6. At the same stress ratio, increasing temperature tends to increase the strain rate and shortens the time to failure. Specimens 0-90-1, 0-90-2 and 0-50-4 have the same stress ratio of 1.23. Specimens 0-90-1 and 0-90-2 were tested under 90 1C and almost failed within a short period (2.8 h), while specimen 0-50-4 tested under 50 1C failed after 84.7 h.
3.3. Critical strain at failure The creep deformation is composed of the elastic deformation and inelastic deformation. If it is assumed that the elastic modulus and Poisson’s ratio are the same throughout the test, one can show that the elastic strains are 1 E
ð1aÞ
1 E
ð1bÞ
ee1 ¼ ½s1 uðs2 þ s3 Þ ee2 ¼ ½s2 uðs1 þ s3 Þ
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Fig. 3. Creep volumetric strain under different confining pressures: (a) s3 ¼ 0 MPa, (b) s3 ¼ 10 MPa and (c) s3 ¼ 30 MPa.
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Substituting Eqs. (1a)–(1c) into Eq. (8), we have
eiev ¼ ev
1 2u ðs1 þ 2s3 Þ E
ð9Þ
Based on Eqs. (7) and (9), we can define the strain path in the eieq 2eiev space (inelastic shear strain–inelastic volumetric strain space). As shown in Fig. 7a, the path is nearly linear. This phenomenon is also observed in Hong Kong granite (HK granite) as well (Fig. 7b) [22]. If dilatancy index is defined as DI ¼ j
Fig. 4. Volumetric strain rate varies with time under different loading levels.
Fig. 5. Comparison of steady volumetric strain rate under unconfined and room temperature condition.
1 E
ee3 ¼ ½s3 uðs1 þ s2 Þ
ð1cÞ
For triaxial tests in the octahedral plane, the strain path could be expressed by the octahedral shear strain goct and octahedral normal strain eoct. The octahedral shear strain goct is pffiffiffi 2 2 ðe1 e3 Þ goct ¼ ð2Þ 3
deie v j deie q
the slope of the strain path is equal to 1/DI, where deie v is increment of inelastic volumetric strain, deie q is increment of inelastic shear strain. The slope of the line defines a flow rule of rock material under constant loading. It should be noted that the inelastic volumetric strain deie v is always negative (dilatant) and inelastic shear strain deie q is always positive during creep in this study, so absolute value is used to represent the dilatancy index DI for easy interpretation. Fig. 8 presents the dilatancy index for a series of uniaxial longterm loading tests on BS granite as well as HK granite [22]. The trend indicates that the dilatancy index does not vary very much. Note that the HK granite specimens were tested at a higher stress ratio. On plotting the normalized dilatancy index (DI/b) against the confining pressure, one can note that the normalized dilatancy index decreases with the confining pressure (Fig. 9) confirming that the confined state is more stable. The critical strains (inelastic volumetric strain and inelastic shear strain) at the onset of fracturing for BS granite and HK granite [22] are calculated and plotted in Fig. 10. As the results fall on a narrow zone, one may deduce that slipping along the slant cracks and propagation of axial cracks occur simultaneously at the onset of fracturing. 3.4. Strength degradation Adopting the Hoek–Brown criterion, one can express rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s s1 ¼ s3 þ sui mb 3 þ 1
sui
ð10Þ
where e1 and e3 are the axial and radial strains. The shear strain eq in this study is defined as octahedral shear strain goct.
where s1 and s3 are the major and minor principal stresses; sui is the uniaxial compressive strength and mb is the material
eq ¼ goct
constant. The Hoek–Brown envelop for peak strength is plotted in Fig. 11. In addition, the Hoek–Brown envelops for the crack damage stress scd at room temperature, 50 and 90 1C are also plotted. For crack damage stress envelopes, scd at unconfined condition replaces sui of Eq. (10). The coefficients of the envelopes are tabulated in Table 2. If the crack damage stress scd is related to the long-term strength of rock, the long-term strength of rock will decrease when the temperature increases from room temperature to 50 or 90 1C. The results also indicate that the stress difference between the peak strength and long-term strength at the same confining pressure will increase when the confining pressure increases. One may assume that the crack damage stress scd, at which the volumetric strain starts to reverse, could be an indicator for the long-term strength of the rock (sN), that is the strength at infinite time. On the other hand, the peak strength of rock, sp, should represent the instant strength of the rock. Under constant load above the crack damage stress, the rock fails, depending on the load level, after being loaded for time t. Mathematically, one
ð3Þ
The octahedral normal strain eoct is
eoct ¼ 13 ðe1 þ e2 þ e3 Þ ¼ 13ðe1 þ 2e3 Þ
ð4Þ
The volumetric strain ev is defined as 3 times of the octahedral normal strain eoct, that is
ev ¼ 3 eoct ¼ ðe1 þ 2e3 Þ The inelastic shear strain eie q is, therefore, pffiffiffi 2 2 e ðe1 ee3 Þ eieq ¼ eq eeq ¼ eq 3 Substituting Eqs. (1a)–(1c) into Eq. (6), we have pffiffiffi 2 21þ u eieq ¼ eq ðs1 s3 Þ 3 E
ð5Þ
ð6Þ
ð7Þ
The inelastic volumetric strain eie v is
eiev ¼ ev eev ¼ ev ðee1 þ 2ee3 Þ
ð8Þ
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Fig. 6. Volumetric strain under different temperatures: (a) RT, (b) 50 1C and (c) 90 1C.
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Fig. 7. Strain path during creep in inelastic shear strain vs volumetric strain space: (a) BS granite at room temperature 23 1C and (b) HK granite at room temperature 23 1C.
can assume
st ¼ s1 þ ðsp s1 Þeqt
ð11Þ
st ¼ sp if t ¼ 0, which means that the specimen fails instantly at the peak strength. When t-N, st ¼ sN ¼ scd. q is a material constant. Rearranging Eq. (11), we obtain st s1 ¼ ðsp s1 Þeqt
ð12Þ
Dividing both sides of Eq. (12) by sN, we have
b 1 ¼ ðC 1Þeqt
ð13Þ
where b is the stress ratio st/scd, C is a constant with the ratio of peak strength sp to crack damage stress scd. Fig. 12 shows the time to failure of specimen at different stress ratios under different confining pressures at room temperatures, respectively. For a given confining pressure, a shorter time is required to fail the specimen if the stress ratio is higher. For a given stress ratio, the time to failure increases when the confining pressure increases. Therefore, the stress state under uniaxial condition is the most dangerous. This is why creep failure always occurs on the surface of large carven, tunnels and slopes. In the figure, the results of HK granite [22] are also plotted. As the HK
Fig. 8. Effect of stress ratio on dilatancy index under uniaxial condition.
granite has a higher strength, the axial stress, for the same stress ratio, that the specimens were load were much higher than those of the unconfined case but comparable to those of the confined case for the BS granite. It is not surprised to find that the time to failure for HK granite is longer.
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Fig. 9. Effect of confining pressure on the normalized dilatancy index.
Fig. 11. Hoek–Brown envelope for peak strength and crack damage stress.
Table 2 Coefficients of the Hoek–Brown envelope.
Peak scd at RT scd at 50 1C scd at 90 1C
Fig. 10. Critical strains at the onset of fracturing: (a) BS granite and (b) HK granite.
3.5. Development and propagation of micro-fractures Fig. 13 shows the relation between volume change (or dilatancy) and acoustic emission counts. During primary phase, acoustic emission activity decays with time and the inelastic volumetric strain rate decreases accordingly. In the secondary phase, the acoustic emission activity levels off and the volumetric strain rate remains constant. During tertiary phase, the acoustic emission activity accelerates rapidly with time and the volumetric strain rate accelerates gradually until the specimen fails.
sui (MPa)
mb
137.9 107.0 97.3 95.9
50 19.3 12.0 9.7
As the waveforms of acoustic emission from six sensors attached to the specimen are recorded simultaneously, the arrival time and the amplitude of the first motion can then be determined. Hence, the location of acoustic emission source can be computed by the arrival time differences (Appendix A). The crack type and crack direction are determined from the amplitude of the first motion (Appendix B). Therefore, the detailed information about the damage process in both space and time can be obtained to predict where and when the damages occur. Fig. 14 showed the orthographic projections of the specimen tested under the confining pressure of 30 MPa. The right projection shows the front view of the specimen, while the left projection is viewed from left to right. The bottom plot is a view from the top of the same specimen. Each symbol represents one acoustic emission event. Different crack types were represented by different symbols. Symbol represents shear crack; symbol & represents tensile crack; symbol D represents mix mode. It is
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interesting to find that most tensile cracks occur on the circumferential surface of the specimen. On the other hand, shear cracks occur mainly inside the specimen while a mix mode crack nucleation zone between tensile crack zone and shear crack zone. To further study the progressive damage process, the complete process was divided into 10 stages from (a) to (j) (Fig. 15). The distribution percentage of crack type during each period is plotted in Fig. 16. The chart shows clearly that tensile crack dominates in
Fig. 12. Time to failure under room temperature conditions.
1111
the early period, but the percentage of shear crack increases rapidly at the late stage of the test. The plots of the locations and types of cracks at each stage are given in Fig. 17. During stages (a)–(c), tensile cracks occur firstly on the circumference of the specimen. The percentages of tensile crack and shear crack are 58.9% and 24.5%, respectively. During stages (d)–(f), cracks start to propagate along the direction of a potential shear fault. A cluster is formed obviously during stage (f). Tensile crack is also the major type of crack in this stage. The percentages of tensile crack and shear crack are about 54.1% and 24.7%. During stages (g)–(i), the activities of the cluster increased continuously. The zone of the cluster continues to grow and enlarge as the cracks propagated. The percentage of tensile crack decreases while percentage of shear crack increases. Shear cracks dominate the crack types finally. After that, the occurrence of acoustic emission events with higher energy was extensively concentrated within the active cluster just before faulting. Clustering of hypocenters around the shear plane of the specimen indicates the location of final fracture zone. The estimation matches well with that observed in the failed specimen. To study the effect of confining pressure and temperature on the acoustic emission activities, the acoustic emission rate at the secondary phase was selected for comparison. Fig. 18 shows that acoustic emission rate increases rapidly when the stress level increases for a given confining pressure. Higher confining pressure
Fig. 13. Comparison of dilatancy versus cumulative count.
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Fig. 16. Distribution of crack type during different creep stages.
Fig. 14. Orthographic projections of specimen tested under high confining pressure.
The effect of temperatures from room temperature to 90 1C is not obvious. Results of the effect of temperatures on the strain rate, acoustic emission rate, dilatancy index and time to failure fall within a narrow band. Strength degradation with time is due to the degradation of internal material properties such as degradation of elastic properties (Young’s modulus and Poisson’s ratio), which is attributed to micro-cracking or micro-fracturing within granite. Source location analysis and moment tensor analysis reproduce the progressive damage process during long term and help to understand the failure mechanism from a microscopic point of view.
Acknowledgments This research is sponsored by the Research Grants Council of the HKSAR (HKU7029/02E), which is gratefully acknowledged. The authors also wish to thank Mr. M.F. Lam, Mr. H. Lee and Mr. N.C. Poon for their assistance in the specimen preparation works.
Appendix A
Fig. 15. Creep curve of specimen 30-RT-2 and different stages.
seems to make the specimen more stable and produces less acoustic emission. However, the effect of temperatures from room temperature to 90 1C on the acoustic emission rate is small and the results fall within a narrow band.
4. Conclusions In this study, a series of constant loading tests under different confining pressures and temperatures with acoustic emission monitoring has been performed to study evolution of dilatancy (damage), strength degradation and damage micro-mechanism of granite with time. At the same stress ratio, increasing confining pressure tends to make the specimen more stable, reduce the strain rate and acoustic emission rate. The dilatancy index decreases when the stress ratio increases, but the effect of confining pressure on the dilatancy index is insignificant. The strength degradation curve indicates that the strength of rock will reduce with time. For the same stress ratio, increasing confining pressure will increase the time to failure.
The travel-time-difference method is employed for source (or hypocenter) location analysis. Computerized methods for determining source locations utilize a least-squares iteration technique that searches for the best average solution for a set of equations that contain the transducer coordinates, the corresponding acoustic emission arrival times and velocities. The method involves the solution of the following set of i distance equations: d2i ¼ ðXi X0 Þ2 þ ðYi Y0 Þ2 þ ðZi Z0 Þ2 ¼ Vðy; FÞ2 ðti T0 Þ2
ðA:1Þ
where di is the distance between the source and the ith transducer; Xi, Yi, and Zi ði ¼ 1; 2; 3; . . . ; nÞ are the coordinates of the ith transducer; X0, Y0, and Z0 are the hypocenter coordinates; V(y,F) is the p-wave velocity in the direction between the source and receiver; ti is the arrival time at the ith transducer; T0 is the time of initiation of the event.
Appendix B AE waveform uð~ x ; tÞ at observation point x [~ x ¼ ðx1 ; x2 ; x3 Þ is the position vector of x] due to crack vector bð~ y ; tÞ [~ y ¼ ðy1 ; y2 ; y3 Þ is the position vector of a point y on crack surface F] is represented by [21,22] Z ui ð~ x ; tÞ ¼ Gip;q ð~ x; ~ y ; tÞmpq SðtÞ dF ðB:1Þ F
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Fig. 17. Crack propagation process during creep: (a) 0–5 ks, (b) 5–10 ks, (c) 10–15 ks, (d) 15–20 ks, (e) 20–25 ks, (f) 25–30 ks, (g) 30–35 ks, (h) 35–40 ks, (i) 40–42 ks and (j) 42 ks.
References
Fig. 18. Variation of steady state acoustic emission rate with stress ratio.
mpq ¼ Cpqkl bk nl x; ~ y ; tÞ SðtÞ ¼ Gip;q ð~
ðB:2Þ Z
þ1
Gip;q ð~ x; ~ y ; tÞSðt tÞ dt
ðB:3Þ
1
Gip,q is the spatial derivative of Green’s function, S(t) represents the source-time function, and * represents the convolution integral. n is the direction normal to the crack and b is the direction of the crack displacement. Cpqkl is the elastic constants. To perform the de-convolution analysis and clarify crack kinetmatics, a computer code named SiGMA (simplified Green’s function for moment tensor analysis) is used to determine the moment tensor components. A unified decomposition of eigenvalues of the moment tensor is then adopted for the classification of crack type. Based on proportions of a double-couple (DC) part, a compensated linear vector dipole (CLVD) part, and an isotropic part in the eigenvalues, an AE source is classified into a shear crack, or a tensile crack, or mix mode.
[1] Maranini E, Brignoli M. Creep behavior of a weak rock: experimental characterization. Int J Rock Mech Min Sci 1999;36(1):127–38. [2] Cristescu N. Rock rheology. In: Fan J, Murakami S, editors. Proceedings of ICCLEM on advances in constitutive laws for engineering materials, vol. 2. International Academic Publisher; 1989. p. 906–19. [3] Hunsche U, Mingerzahn G, Schulze O. The influence of textural parameters and mineralogical composition on the creep behavior of rock salt. In: Ghoreychi M, Berest P, Hardy Jr H, Langer M, editors. Proceedings of the third conference on the mechanical behavior of salt. Trans Tech Publications; 1996. p. 143–51. [4] Hunsche U, Schulze O. Effect of humidity and confining pressure on creep of rock salt. In: Ghoreychi M, Berest P, Hardy Jr H, Langer M, editors. Proceedings of the third conference on the mechanical behavior of salt, Trans Tech Publications; 1996. p. 237–48. [5] Kranz RL. Crack growth and development during creep of Barre granite. Int J Rock Mech Min Sci Geomech Abstr 1979;16:23–35. [6] Ashby MF, Sammis CG. The damage mechanics of brittle solids in compression. Pure Appl Geophys 1990;133:489–521. [7] Lockner DA, Byerlee JD, Kuksenko V, Ponomarev A, Sidorin A. Observations of quasistatic fault growth from acoustic emissions. In: Evans B, Wong TF, editors. Fault mechanics and transport properties of rocks. London: Academic Press; 1992. p. 3–1. [8] Scholz CH. Mechanism of creep in brittle rock. J Geophys Res 1968;73(10): 3295–3302. [9] Cruden DM. A theory of brittle creep in rock under uniaxial compression. J Geophys Res 1970;75(17):3431–42. [10] Charles RJ. Strength of silicate glasses and some crystalline oxides. In: Proceedings of the international conference on fracture. Cambridge, Massachusetts: Massachusetts Institute of Technology Press; 1959. p. 225–50. [11] Li Y, Xia C. Time-dependent tests on intact rocks in uniaxial compression. Int J Rock Mech Min Sci 2000;37(3):465–75. [12] Chandler N, Lau JSO. In situ granite strength examined using non-traditional laboratory techniques. In: Proceedings of the fifth North American rock mechanics symposium and the 17th tunnelling association of Canada conference, Toronto, Ontario, Canada, 710 July 2002. p. 489–96. [13] Lau JSO, Chandler NA. Innovative laboratory testing. Int J Rock Mech Min Sci 2004;41:1427–45. [14] Lin QX, Tham LG, Yeung MR, Lee PKK. Failure of granite under constant loading. Int J Rock Mech Min Sci 2004;41(3):49–54. [15] Kranz RL, Harris WJ, Carter NL. Static fatigue of granite at 200 1C. Geophys Res Lett 1982;9(1):1–4.
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[16] Lockner DA, Byerlee JD. Acoustic emission and creep in rock at high confining pressure and differential stress. Bull Seismological Soc Am 1977;67:247–58. [17] Ohtsu M. Simplified moment tensor analysis and unified decomposition of acoustic emission source: application to in situ hydrofracturing test. J Geophys Res 1991;96(B4):6211–21. [18] Ohtsu M. Acoustic emission theory for moment tensor analysis. Res Nondestr Eval 1995;6:169–84. [19] Wang J. Preliminary site characterization at Juijing Block, Beishan Area, Northwest China—the potential site for China’s high-level radioactive waste repository. In: IAEA-TC workshop on reviewing achievements of project CPR/
9/026 ‘‘siting and site characterization study for China’s high-level radioactive waste repository’’ and plans for future activities, 27 May–4 June 2002, Beijing, China. [20] Martin CD, Chandler NA. The progressive fracture of Lac du Bonnet granite. Int J Rock Mech Min Sci Geomech Abstr 1994;31(6):643–59. [21] Lockner DA. Room temperature creep in saturated granite. J Geophys Res 1993;98(B1):475–87. [22] Lin QX. Strength degradation and damage micromechanism of granite under long-term loading. PhD thesis, Department of Civil Engineering, The University of Hong Kong; 2006.
Contents lists available at ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Time-dependent strength degradation of granite Q.X. Lin a, Y.M. Liu b, L.G. Tham a,, C.A. Tang c, P.K.K. Lee a, J. Wang b a b c
Department of Civil Engineering, The University of Hong Kong, Hong Kong, China Beijing Research Institute of Uranium Geology, Beijing, China Center for Rock Instability and Seismicity Research, Northeastern University, Shenyang, China
a r t i c l e in fo
abstract
Article history: Received 12 September 2007 Received in revised form 11 March 2009 Accepted 14 July 2009 Available online 8 August 2009
Understanding the time-dependent strength degradation and associated creep behavior is essential to the safety evaluation of a radioactive waste disposal system in rock. In this study, a series of constant loading tests under different confining pressures and temperatures with acoustic emission monitoring have been performed to study evolution of dilatancy (damage) and strength degradation. Source location analysis and moment tensor analysis were carried out to reproduce the progressive damage process during constant load helping to understand the failure mechanism from a microscopic point of view. In this paper, we report on the findings of the experimental results. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Constant loading test Strength degradation Acoustic emission Damage Dilatancy Granite
1. Introduction Nuclear power is cheap, clean and without the shortcomings of finite hydrocarbon and hydroelectric resources. However, generation of nuclear power produces high-level radioactive waste (HLW), which must be isolated from the biosphere until it has decayed to a level that poses no significant risk to human beings. Deep geological disposal is considered to be the best option to deal with the HLW. However, long-lived radionuclide must be isolated over thousands of years after emplacement and heat generated from the waste in the repository will result in an increase in rock temperature. Preferred hydraulic pathways could be created if the rock around excavations is damaged under sustained loading and high temperature and therefore it is important to minimize potential pathways for the transport of radionuclides in the rock surrounding a repository. These damages appear in the form of cracking in the rock. Therefore a better understanding of long-term strength of rock and associated creep behavior is essential for the design and construction of excavations for nuclear waste repositories. Since the long-term deformation of soft rocks is more apparent than hard rocks, such as granite, studies were conducted on more deformable rocks such as sedimentary rock [1], sandstone [2] and salt [3,4]. On the other hand, the effect of long-term behavior of brittle rock is often neglected in the traditional rock mechanics
Corresponding author. Tel.: +852 28591965; fax: +852 25595337.
E-mail address: [email protected] (L.G. Tham). 1365-1609/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2009.07.005
design. Only few papers can be found in the literature. When a brittle rock is held at constant loading, its long-term strain is typically composed of three phases: primary, secondary and tertiary phases. During primary and secondary phases, the nucleation and growth of microcracks occur uniformly throughout the rock and the cracks have little tendency to link but instead interact to eliminate regions of high local stress [5,6]. During the tertiary phase, the crack density reaches a critical level and crack grows further in a weakened region and ultimately develops into a macroscopic fault [7]. Scholz [8] studied the failure mechanism of brittle rock under constant load. However, in his theory, there is no tensile stress within the specimen therefore it is difficult to envisage the occurrence of a fracture. Cruden [9] developed a new theory based on Charles’ theory [10] of subcritical growth of preexisting cracks by stress-corrosion. However, Cruden neglected the dislocation motion and assumed that the material should contain pre-existing cracks. Kranz [5] studied the development of cracks in Barre granite under constant loading by scanning electron microscope from micro-structure point of view. As the production of sections often introduces new cracks, making the identification of natural and experimentally stress-induced cracks difficult. Furthermore, cracks viewed in the un-stressed state may give false impression about their dimensions when stressed. Recently, granite was selected as foundation of high dams and potential parent rock for nuclear waste repositories, and the interest of studying the long-term behavior of granite was raised [11,12]. Lau and Chandler [13] developed three laboratory test techniques to study and quantify damage development in loaded granite specimen and to provide data for use in the calibration and
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solution of numerical models. Lin et al. [14] studied the failure process of granite under constant loading and quantified the damage evolution by inelastic volumetric strain, degradation of elastic properties and cumulative acoustic emission under uniaxial condition. However, there is only little information on longterm behavior of brittle rock in the intermediate temperature range and under high confining pressure. One of the early works was conducted by Kranz et al. [15] who indicated that the effects of increasing temperatures from 24 to 200 1C were to weaken the rock considerably and to reduce the time to failure by about two orders of magnitude. Due to the brittle nature, failure of granite was accompanied by micro-cracking or micro-fracturing which could be detected by acoustic emission system. Since acoustic emission monitoring is essentially passive, it provides an ideal non-destructive method for studying crack growth at its actual stress state. The failure process could be monitored by an acoustic emission system. Acoustic emission studies in the literature include: (1) simple counting of the number of AE events prior to specimen failure showing a correlation between AE rate and inelastic strain rate [16], (2) location of hypocenters of AE source events [7] and (3) analysis of full waveform data as recorded at receiver sites. One aspect of this research was to determine fault plane solutions of AE source events from first motion data [17,18]. This paper aims to study the evolution of dilatancy (damage) and strength degradation of granite by a series of constant loading tests. Damage micro-mechanism during constant loading tests and effects of confining pressures and temperatures on the associated long-term behavior of granite will be discussed.
2. Laboratory investigation 2.1. Rock specimens For this study, rock specimens were prepared from granite cores obtained from Beishan, China, and they are referred to as BS granite. BS granite is classified as porphyritic monzonitie granite. The grain density is 2.71 g/cm3 and the natural density is 2.64 g/ cm3. The porosity and water content of specimens are low. The average compressive wave velocity is about 5423 m/s (range: 5197–5549 m/s) (Table 1). The specimens are about 50 mm in diameter and 125 mm in height. The tolerance of straightness and Table 1 Properties of the BS specimens. Specimen no.
Vp (m/s)
E (GPa)
u
s3 (MPa)
scd (MPa)
Stress ratio b
0-RT-1 0-RT-2 0-RT-3 0-RT-4 10-RT-1 10-RT-2 10-RT-3 10-RT-4 30-RT-1 30-RT-2 30-RT-3 30-RT-4
5314 5313 5453 5279 5545 5549 5548 5523 5477 5197 5441 5442
44.50 42.57 42.61 42.99 46.00 45.61 46.54 46.22 45.49 41.77 45.44 48.49
0.28 0.25 0.28 0.25 0.30 0.31 0.32 0.31 0.31 0.35 0.26 0.30
0 0 0 0 10 10 10 10 30 30 30 30
104.65 109.57 106.71 101.27 164.18 189.38 194.41 196.99 251.04 104.43* 257.78 268.72
1.253 1.190 1.168 1.113 1.350 1.357 1.307 1.277 1.578 – 1.385 1.319
0-50-1 0-50-2 0-50-3 0-50-4
– – – –
43.40 40.82 40.07 38.52
0.28 0.31 0.32 0.30
0 0 0 0
97.86 108.40 104.16 94.98
1.278 1.092 1.161 1.237
0-90-1 0-90-2 0-90-3
– – –
41.12 44.22 39.52
0.25 0.32 0.30
0 0 0
97.84 94.49 105.00
1.239 1.234 1.212
flatness of the specimens meet the specifications of ASTM designation: D4543-85. The straightness tolerance of the sides is o0.50 mm, the flatness tolerance of the ends is o0.025 mm, and the perpendicularity tolerance is o0.251. 2.2. Testing facilities Constant loading tests were conducted using a MTS 815 rock mechanics test system which has a maximum capacity of 4600 kN. The MTS 815 is a computer controlled, servo-hydraulic compression machine, load frame, hydraulic power supply, triaxial cell, confining pressure subsystem, digital controller, test processor and a PC workstation. It is also equipped with a pair of end platen containing ultrasonic sensors for the determination of wave velocities when a specimen is loaded. Two linear variable differential transducers are used to measure the axial deformation, while the circumferential deformation is measured by an extensometer. In addition, the MISTRAS 2001 is used to record the AE events. The system is a fully digital, multi-channel, computerized, acoustic emission system that performs AE waveform and signal measurement and stores, displays and analyzes the resulting data. It consists of an IBM compatible PC, an AEDSP-32/16 card and the MISTRAS software, sensors, preamplifiers and cables. 2.3. Testing procedures To study the evolution of dilatancy (damage) and associated long-term behavior under different confining pressures and temperatures, a series of constant loading tests were conducted at the Rock Engineering Research Centre of The University of Hong Kong. The test programme consisted of 36 cases conducted at confining pressures of 0, 10 and 30 MPa and at temperatures 23 (room temperature, RT), 50 and 90 1C (Table 1). The in situ stress measured by hydrofracturing method [19] indicated that the vertical stress changed from 4.36 to 13.31 MPa, and the horizontal stress ranged from 7.72 to 25.66 MPa at a depth of 161.5 to 493 m. Therefore, the confining pressure of 10 and 30 MPa were chosen to simulate the in situ stress. A temperature of 50 1C was used to simulate the in-situ rock temperature, and 90 1C was the potential temperature of the surrounding under the heating of the nuclear waste in future. The test consisted of two stages: initial loading and constant loading. In the initial loading stage, the specimen was axially loaded at a rate that produced a constant lateral deformation rate of 0.006 mm/min. The crack damage stress scd was determined during the initial loading stage. scd is defined as the stress at the point of volumetric strain reversal, which marks the onset of unstable crack growth and may be related to the long-term strength of rock [20]. When the predetermined stress level bscd (b ¼ applied axial stress/scd) was achieved, the load was maintained at the predetermined stress level by the closed-loop operation of the servo-controlled machine until the specimen failed. b is, thereafter, referred to as stress ratio. If the specimen did not fail after a long period of time, up to several days, the constant loading test was terminated. In the study, the stress ratio varies from 1.05 to 1.20. Six piezoelectric transducers were attached to the specimen to detect AE during the sudden growth of microcracks or slip along existing crack surfaces. Each transducer was attached to the specimen through a copper adapter, of which one surface was machined to the same curvature as the specimen surface, and cemented to the side with cyanoacrylate (CN adhesive). Sonotech Soundsafe ultrasonic couplant was used as the couplant between the polished contact surfaces of both transducers and adapters in
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Fig. 2. A typical creep test and its three creep phases (test on BS granite 10-RT-3).
Fig. 1. Sensors setup and acoustic emission monitoring system.
order to achieve good coupling. The transducers were pressed tightly to the adapters with rubber bands. Signals are amplified and recorded by the MITRAS 2001 system as shown in Fig. 1. Required data and AE signals were recorded automatically by computer for further analysis.
3. Test results During initial loading stage, the cracks inside the specimen became unstable and the specimen started to dilate after the stress level was increased above the crack damage stress (scd). During the constant loading stage, the specimen keeps dilated further until it is fractured to a state that it cannot sustain the designed constant loading after a certain period of time. A typical creep curve during the constant load stage is shown in Fig. 2. The process can be divided into three phases: primary (or transient), secondary (or steady state), and tertiary (or accelerating) creep phases. In the transient creep phase, the strain rate monotonically decays and then remains constant during the steady state creep phase. The strain rate increases suddenly and rapidly until the specimen fractures in the accelerating creep phase. In the subsequent sections, the effect of confining pressure and temperature on the long-term strength of granite as well as the development and propagation of micro-fractures will be discussed in detail. 3.1. Effect of confining pressures during constant load at room temperature Volumetric strain, which reflects the change in ‘pore volume’ caused by cracking, was adopted to study the evolution of dilatancy and damage under different confining pressures. Fig. 3 shows the evolution of volumetric strain (negative volumetric strain, in the present study, represents dilatancy) under different confining pressures. Specimen tends to become more stable under higher confining pressure. Unconfined specimen with stress ratio higher than 1.15 failed in a few hours and all phases can be observed. Under the confining pressure of 10 MPa, the specimen with stress ratio of 1.28 failed after about 18 days. On the other hand, the specimen under confining pressure of 30 MPa and having stress ratio of 1.32 was still quite stable. It had no sign of
failure even after 14 days: only the primary and secondary phases were observed. Though specimens 10-RT-3 and 30-RT-4 have a similar stress ratio of 1.31, the former one failed in about 4 days while the latter one was still quite stable for 14 days. Fig. 4 summarizes the volumetric strain rate of a series of tests under uniaxial loading. The strain rate increases as the stress ratio increases. The initial strain rate of specimen 0-RT-3 is high and then decreases gradually until it reaches a minimum strain rate at the end of primary phase. The minimum strain rate remains almost constant during secondary phase. In the tertiary phase, the strain rate increases suddenly and the specimen eventually ruptured. The stress ratio of specimen 0-RT-4 is lower than that for specimen 0-RT-3. The strain rate of specimen 0-RT-4 decreases gradually to a minimum value (steady state strain rate) and then remains constant. No acceleration of strain rate was observed during the testing period. The stress ratio of specimen 0-RT-2 is higher than that for specimen 0-RT-3 and the strain rate decreases rapidly to the minimum value and then accelerates to failure. Fig. 5 shows the variation of the steady state volumetric strain rate with axial stress. In the figure, the results for Westerly granite reported by Lockner [21] under unconfined conditions are also plotted. The Westerly granite has finer grains and higher ultimate strength [21]. It is noted that the Westerly granite has a lower steady state volumetric strain rate.
3.2. Effect of temperatures during long-term under uniaxial condition The volumetric strains as a function of time at different temperatures are shown in Fig. 6. At the same stress ratio, increasing temperature tends to increase the strain rate and shortens the time to failure. Specimens 0-90-1, 0-90-2 and 0-50-4 have the same stress ratio of 1.23. Specimens 0-90-1 and 0-90-2 were tested under 90 1C and almost failed within a short period (2.8 h), while specimen 0-50-4 tested under 50 1C failed after 84.7 h.
3.3. Critical strain at failure The creep deformation is composed of the elastic deformation and inelastic deformation. If it is assumed that the elastic modulus and Poisson’s ratio are the same throughout the test, one can show that the elastic strains are 1 E
ð1aÞ
1 E
ð1bÞ
ee1 ¼ ½s1 uðs2 þ s3 Þ ee2 ¼ ½s2 uðs1 þ s3 Þ
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Fig. 3. Creep volumetric strain under different confining pressures: (a) s3 ¼ 0 MPa, (b) s3 ¼ 10 MPa and (c) s3 ¼ 30 MPa.
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Substituting Eqs. (1a)–(1c) into Eq. (8), we have
eiev ¼ ev
1 2u ðs1 þ 2s3 Þ E
ð9Þ
Based on Eqs. (7) and (9), we can define the strain path in the eieq 2eiev space (inelastic shear strain–inelastic volumetric strain space). As shown in Fig. 7a, the path is nearly linear. This phenomenon is also observed in Hong Kong granite (HK granite) as well (Fig. 7b) [22]. If dilatancy index is defined as DI ¼ j
Fig. 4. Volumetric strain rate varies with time under different loading levels.
Fig. 5. Comparison of steady volumetric strain rate under unconfined and room temperature condition.
1 E
ee3 ¼ ½s3 uðs1 þ s2 Þ
ð1cÞ
For triaxial tests in the octahedral plane, the strain path could be expressed by the octahedral shear strain goct and octahedral normal strain eoct. The octahedral shear strain goct is pffiffiffi 2 2 ðe1 e3 Þ goct ¼ ð2Þ 3
deie v j deie q
the slope of the strain path is equal to 1/DI, where deie v is increment of inelastic volumetric strain, deie q is increment of inelastic shear strain. The slope of the line defines a flow rule of rock material under constant loading. It should be noted that the inelastic volumetric strain deie v is always negative (dilatant) and inelastic shear strain deie q is always positive during creep in this study, so absolute value is used to represent the dilatancy index DI for easy interpretation. Fig. 8 presents the dilatancy index for a series of uniaxial longterm loading tests on BS granite as well as HK granite [22]. The trend indicates that the dilatancy index does not vary very much. Note that the HK granite specimens were tested at a higher stress ratio. On plotting the normalized dilatancy index (DI/b) against the confining pressure, one can note that the normalized dilatancy index decreases with the confining pressure (Fig. 9) confirming that the confined state is more stable. The critical strains (inelastic volumetric strain and inelastic shear strain) at the onset of fracturing for BS granite and HK granite [22] are calculated and plotted in Fig. 10. As the results fall on a narrow zone, one may deduce that slipping along the slant cracks and propagation of axial cracks occur simultaneously at the onset of fracturing. 3.4. Strength degradation Adopting the Hoek–Brown criterion, one can express rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s s1 ¼ s3 þ sui mb 3 þ 1
sui
ð10Þ
where e1 and e3 are the axial and radial strains. The shear strain eq in this study is defined as octahedral shear strain goct.
where s1 and s3 are the major and minor principal stresses; sui is the uniaxial compressive strength and mb is the material
eq ¼ goct
constant. The Hoek–Brown envelop for peak strength is plotted in Fig. 11. In addition, the Hoek–Brown envelops for the crack damage stress scd at room temperature, 50 and 90 1C are also plotted. For crack damage stress envelopes, scd at unconfined condition replaces sui of Eq. (10). The coefficients of the envelopes are tabulated in Table 2. If the crack damage stress scd is related to the long-term strength of rock, the long-term strength of rock will decrease when the temperature increases from room temperature to 50 or 90 1C. The results also indicate that the stress difference between the peak strength and long-term strength at the same confining pressure will increase when the confining pressure increases. One may assume that the crack damage stress scd, at which the volumetric strain starts to reverse, could be an indicator for the long-term strength of the rock (sN), that is the strength at infinite time. On the other hand, the peak strength of rock, sp, should represent the instant strength of the rock. Under constant load above the crack damage stress, the rock fails, depending on the load level, after being loaded for time t. Mathematically, one
ð3Þ
The octahedral normal strain eoct is
eoct ¼ 13 ðe1 þ e2 þ e3 Þ ¼ 13ðe1 þ 2e3 Þ
ð4Þ
The volumetric strain ev is defined as 3 times of the octahedral normal strain eoct, that is
ev ¼ 3 eoct ¼ ðe1 þ 2e3 Þ The inelastic shear strain eie q is, therefore, pffiffiffi 2 2 e ðe1 ee3 Þ eieq ¼ eq eeq ¼ eq 3 Substituting Eqs. (1a)–(1c) into Eq. (6), we have pffiffiffi 2 21þ u eieq ¼ eq ðs1 s3 Þ 3 E
ð5Þ
ð6Þ
ð7Þ
The inelastic volumetric strain eie v is
eiev ¼ ev eev ¼ ev ðee1 þ 2ee3 Þ
ð8Þ
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Fig. 6. Volumetric strain under different temperatures: (a) RT, (b) 50 1C and (c) 90 1C.
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Fig. 7. Strain path during creep in inelastic shear strain vs volumetric strain space: (a) BS granite at room temperature 23 1C and (b) HK granite at room temperature 23 1C.
can assume
st ¼ s1 þ ðsp s1 Þeqt
ð11Þ
st ¼ sp if t ¼ 0, which means that the specimen fails instantly at the peak strength. When t-N, st ¼ sN ¼ scd. q is a material constant. Rearranging Eq. (11), we obtain st s1 ¼ ðsp s1 Þeqt
ð12Þ
Dividing both sides of Eq. (12) by sN, we have
b 1 ¼ ðC 1Þeqt
ð13Þ
where b is the stress ratio st/scd, C is a constant with the ratio of peak strength sp to crack damage stress scd. Fig. 12 shows the time to failure of specimen at different stress ratios under different confining pressures at room temperatures, respectively. For a given confining pressure, a shorter time is required to fail the specimen if the stress ratio is higher. For a given stress ratio, the time to failure increases when the confining pressure increases. Therefore, the stress state under uniaxial condition is the most dangerous. This is why creep failure always occurs on the surface of large carven, tunnels and slopes. In the figure, the results of HK granite [22] are also plotted. As the HK
Fig. 8. Effect of stress ratio on dilatancy index under uniaxial condition.
granite has a higher strength, the axial stress, for the same stress ratio, that the specimens were load were much higher than those of the unconfined case but comparable to those of the confined case for the BS granite. It is not surprised to find that the time to failure for HK granite is longer.
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Fig. 9. Effect of confining pressure on the normalized dilatancy index.
Fig. 11. Hoek–Brown envelope for peak strength and crack damage stress.
Table 2 Coefficients of the Hoek–Brown envelope.
Peak scd at RT scd at 50 1C scd at 90 1C
Fig. 10. Critical strains at the onset of fracturing: (a) BS granite and (b) HK granite.
3.5. Development and propagation of micro-fractures Fig. 13 shows the relation between volume change (or dilatancy) and acoustic emission counts. During primary phase, acoustic emission activity decays with time and the inelastic volumetric strain rate decreases accordingly. In the secondary phase, the acoustic emission activity levels off and the volumetric strain rate remains constant. During tertiary phase, the acoustic emission activity accelerates rapidly with time and the volumetric strain rate accelerates gradually until the specimen fails.
sui (MPa)
mb
137.9 107.0 97.3 95.9
50 19.3 12.0 9.7
As the waveforms of acoustic emission from six sensors attached to the specimen are recorded simultaneously, the arrival time and the amplitude of the first motion can then be determined. Hence, the location of acoustic emission source can be computed by the arrival time differences (Appendix A). The crack type and crack direction are determined from the amplitude of the first motion (Appendix B). Therefore, the detailed information about the damage process in both space and time can be obtained to predict where and when the damages occur. Fig. 14 showed the orthographic projections of the specimen tested under the confining pressure of 30 MPa. The right projection shows the front view of the specimen, while the left projection is viewed from left to right. The bottom plot is a view from the top of the same specimen. Each symbol represents one acoustic emission event. Different crack types were represented by different symbols. Symbol represents shear crack; symbol & represents tensile crack; symbol D represents mix mode. It is
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interesting to find that most tensile cracks occur on the circumferential surface of the specimen. On the other hand, shear cracks occur mainly inside the specimen while a mix mode crack nucleation zone between tensile crack zone and shear crack zone. To further study the progressive damage process, the complete process was divided into 10 stages from (a) to (j) (Fig. 15). The distribution percentage of crack type during each period is plotted in Fig. 16. The chart shows clearly that tensile crack dominates in
Fig. 12. Time to failure under room temperature conditions.
1111
the early period, but the percentage of shear crack increases rapidly at the late stage of the test. The plots of the locations and types of cracks at each stage are given in Fig. 17. During stages (a)–(c), tensile cracks occur firstly on the circumference of the specimen. The percentages of tensile crack and shear crack are 58.9% and 24.5%, respectively. During stages (d)–(f), cracks start to propagate along the direction of a potential shear fault. A cluster is formed obviously during stage (f). Tensile crack is also the major type of crack in this stage. The percentages of tensile crack and shear crack are about 54.1% and 24.7%. During stages (g)–(i), the activities of the cluster increased continuously. The zone of the cluster continues to grow and enlarge as the cracks propagated. The percentage of tensile crack decreases while percentage of shear crack increases. Shear cracks dominate the crack types finally. After that, the occurrence of acoustic emission events with higher energy was extensively concentrated within the active cluster just before faulting. Clustering of hypocenters around the shear plane of the specimen indicates the location of final fracture zone. The estimation matches well with that observed in the failed specimen. To study the effect of confining pressure and temperature on the acoustic emission activities, the acoustic emission rate at the secondary phase was selected for comparison. Fig. 18 shows that acoustic emission rate increases rapidly when the stress level increases for a given confining pressure. Higher confining pressure
Fig. 13. Comparison of dilatancy versus cumulative count.
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Fig. 16. Distribution of crack type during different creep stages.
Fig. 14. Orthographic projections of specimen tested under high confining pressure.
The effect of temperatures from room temperature to 90 1C is not obvious. Results of the effect of temperatures on the strain rate, acoustic emission rate, dilatancy index and time to failure fall within a narrow band. Strength degradation with time is due to the degradation of internal material properties such as degradation of elastic properties (Young’s modulus and Poisson’s ratio), which is attributed to micro-cracking or micro-fracturing within granite. Source location analysis and moment tensor analysis reproduce the progressive damage process during long term and help to understand the failure mechanism from a microscopic point of view.
Acknowledgments This research is sponsored by the Research Grants Council of the HKSAR (HKU7029/02E), which is gratefully acknowledged. The authors also wish to thank Mr. M.F. Lam, Mr. H. Lee and Mr. N.C. Poon for their assistance in the specimen preparation works.
Appendix A
Fig. 15. Creep curve of specimen 30-RT-2 and different stages.
seems to make the specimen more stable and produces less acoustic emission. However, the effect of temperatures from room temperature to 90 1C on the acoustic emission rate is small and the results fall within a narrow band.
4. Conclusions In this study, a series of constant loading tests under different confining pressures and temperatures with acoustic emission monitoring has been performed to study evolution of dilatancy (damage), strength degradation and damage micro-mechanism of granite with time. At the same stress ratio, increasing confining pressure tends to make the specimen more stable, reduce the strain rate and acoustic emission rate. The dilatancy index decreases when the stress ratio increases, but the effect of confining pressure on the dilatancy index is insignificant. The strength degradation curve indicates that the strength of rock will reduce with time. For the same stress ratio, increasing confining pressure will increase the time to failure.
The travel-time-difference method is employed for source (or hypocenter) location analysis. Computerized methods for determining source locations utilize a least-squares iteration technique that searches for the best average solution for a set of equations that contain the transducer coordinates, the corresponding acoustic emission arrival times and velocities. The method involves the solution of the following set of i distance equations: d2i ¼ ðXi X0 Þ2 þ ðYi Y0 Þ2 þ ðZi Z0 Þ2 ¼ Vðy; FÞ2 ðti T0 Þ2
ðA:1Þ
where di is the distance between the source and the ith transducer; Xi, Yi, and Zi ði ¼ 1; 2; 3; . . . ; nÞ are the coordinates of the ith transducer; X0, Y0, and Z0 are the hypocenter coordinates; V(y,F) is the p-wave velocity in the direction between the source and receiver; ti is the arrival time at the ith transducer; T0 is the time of initiation of the event.
Appendix B AE waveform uð~ x ; tÞ at observation point x [~ x ¼ ðx1 ; x2 ; x3 Þ is the position vector of x] due to crack vector bð~ y ; tÞ [~ y ¼ ðy1 ; y2 ; y3 Þ is the position vector of a point y on crack surface F] is represented by [21,22] Z ui ð~ x ; tÞ ¼ Gip;q ð~ x; ~ y ; tÞmpq SðtÞ dF ðB:1Þ F
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1113
Fig. 17. Crack propagation process during creep: (a) 0–5 ks, (b) 5–10 ks, (c) 10–15 ks, (d) 15–20 ks, (e) 20–25 ks, (f) 25–30 ks, (g) 30–35 ks, (h) 35–40 ks, (i) 40–42 ks and (j) 42 ks.
References
Fig. 18. Variation of steady state acoustic emission rate with stress ratio.
mpq ¼ Cpqkl bk nl x; ~ y ; tÞ SðtÞ ¼ Gip;q ð~
ðB:2Þ Z
þ1
Gip;q ð~ x; ~ y ; tÞSðt tÞ dt
ðB:3Þ
1
Gip,q is the spatial derivative of Green’s function, S(t) represents the source-time function, and * represents the convolution integral. n is the direction normal to the crack and b is the direction of the crack displacement. Cpqkl is the elastic constants. To perform the de-convolution analysis and clarify crack kinetmatics, a computer code named SiGMA (simplified Green’s function for moment tensor analysis) is used to determine the moment tensor components. A unified decomposition of eigenvalues of the moment tensor is then adopted for the classification of crack type. Based on proportions of a double-couple (DC) part, a compensated linear vector dipole (CLVD) part, and an isotropic part in the eigenvalues, an AE source is classified into a shear crack, or a tensile crack, or mix mode.
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