Rock Mech Rock Eng (2011) 44:629–634 DOI 10.1007/s00603-011-0151-8
TECHNICAL NOTE
Dynamic Properties of Intact Rock Samples Subjected to Cyclic Loading under Confining Pressure Conditions EnLong Liu • Siming He • Xinhua Xue Jin Xu
•
Received: 20 October 2010 / Accepted: 11 April 2011 / Published online: 27 April 2011 Ó Springer-Verlag 2011
Keywords Dynamic properties Cyclic loading Confining pressure Dynamic stiffness
1 Introduction Recently, attention has focused on the dynamic properties of rocks with the goal of understanding their dynamic mechanical features under different loading histories and conditions (e.g., Stavrogin and Tarasov 2001; Bagde and Petrosˇ 2009). Researchers have carried out studies on different types of rocks to examine the loading effects and loading strain rates on their strength and deformation characteristics (e.g., Zhao 2000; Cho et al. 2003; Mahmutog˘lu 2006; Wang et al. 2009; Liang et al. 2010). More extensive work on cyclic loading has explored whether rocks are subject to weakening from fatigue (e.g., Burdine 1963; Prost 1988; Singh 1989; Tien et al. 1990; Li et al. 2001; Fuenkajorn and Phueakphum 2010). The experimental studies cited above primarily focus on the influence of loading rates or uniaxial cyclic loading on the dynamic mechanical properties of rock samples (e.g., Bagde and Petrosˇ 2005, 2009; Fuenkajorn and Phueakphum 2010; Liang et al. 2010), but few studies have addressed rock samples subjected to cyclic loading under E. Liu S. He (&) Key Laboratory of Mountain Hazards and Surface Process, Institute of Mountain Hazards and Environment, CAS, Chengdu 610041, People’s Republic of China e-mail:
[email protected] E. Liu X. Xue J. Xu School of Hydraulic and Hydropower Engineering, Sichuan University, Chengdu 610065, People’s Republic of China
confining pressure conditions. Rock masses encountered in applied engineering are usually in a stressed state and are confined by pressure. Therefore, it is necessary to study the effects of confining pressures on the dynamic mechanical characteristics of rock samples upon cyclic loading, which is the type of study performed here. In this work, sandstone samples were subjected to axially cyclic loading to experimentally determine the effects of confining pressures on their dynamic residual deformation and dynamic mechanical properties. Five levels of confining pressure (10, 20, 30, 40 and 50 MPa) were applied for axial cyclic loading. Finally, the influence of confining pressures on the dynamic features of sandstone samples with cyclic loading was comprehensively analyzed.
2 Equipment and Test Scheme Dry sandstone samples were cut to a diameter to length ratio of 1:2, with an average diameter of 48.9 mm and an average rock mass density of 2.33 g/cm3. The samples were prepared and tested according to ISRM testing procedures and all relevant guidelines. An MTS-815 Rock and Concrete Test System was used for testing. The MTS controller consists of hardware components and software applications that provide a closed-loop control of the servo-hydraulic test equipment. This test equipment consisted of the following three parts: a compression loading frame, an axial dynamic loading system and a data acquisition system. The equipment was capable of conducting triaxial static and dynamic compression testing of rock specimens. The axial dynamic loading system was driven hydraulically with a 40-lpm flow rate and 21 MPa of output pressure. The data
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acquisition system consisted of signal and acquisition units that interfaced with a computer. Multiple and single data acquisition processes were able to collect data on all channels at a sampling rate up to 6 kHz with a 16-bit resolution. The tests were conducted with an axial displacementcontrolled loading system. The static and dynamic triaxial tests were performed at the following confining pressures: 10, 20, 30, 40 and 50 MPa. For the dynamic test, the axial dynamic load was specified as the sinusoidal cyclic compressive load. The loading frequency was set to 1.0 Hz, and the load path that was employed is illustrated in Fig. 1. During the axial dynamic loading process, cyclic loading was applied at a constant controlled stress rate of 60 kN/ min. The samples that were tested are summarized in Table 1.
3 Test Results and Discussion 3.1 Triaxial Compressive Tests Triaxial compression tests were conducted to obtain the empirical basis for the sandstone sample and to determine the test parameters for subsequent cyclic loading tests. The test procedure followed the relevant ASTM standard Fig. 1 Load path: a typical stress path and b loading sequence
practices (ASTM D 7012). The uniaxial compressive strength of the sandstone was 71.7 MPa. Figure 2 presents the stress–strain curves of tests with confining pressures of 10, 20, 30, 40 and 50 MPa. The sandstone samples exhibited brittle behavior, which changed to ductile behavior at confining pressures ranging from 10 to 50 MPa (Jaeger et al. 2007). All the samples first contracted and then dilated. The lower the confining pressure, the more dilatant the sample was. Table 2 lists the magnitudes of the principal stresses from the triaxial compression tests under static loading conditions at the point of failure. 3.2 Cyclic Dynamic Tests In the dynamic tests, deviatoric stress, axial strain and lateral strain were obtained for the loading duration when confining pressures of 10, 20, 30, 40 or 50 MPa were applied. For example, Fig. 3a–c present the experimental effects of a 30 MPa-confining pressure when the axial dynamic load changes from the peak value to the valley value. From the test results, we deduced the following conclusions: (1) during initial axial cyclic loading, the samples were almost elastic, and with an increase in the number of cycles, the samples became elastic–plastic and developed irreversible deformations, including axial, volumetric and lateral strains, and their magnitudes became σ1-σ3
Deviatoric stress q
Deviatoric stress q
deviatoric stress q =
Mean stress p
(a)
Variable q
dozens to thousands cycles
Time
(b)
Table 1 Summary of the samples tested Test no.
Confining pressure (Mpa)
Loading type
Loading condition
S10
10.0
Static
Triaxial, compression
D10
10.0
Cyclic, dynamic
Triaxial, 1 Hz, stress path (a), axial dynamic loading 20–230 kN, failed after 161 cycles
S20
20.0
Static
Triaxial, compression
D20
20.0
Cyclic, dynamic
Triaxial, 1 Hz, stress path (a), axial dynamic loading 20–280 kN, failed after 257 cycles
S30 D30
30.0 30.0
Static Cyclic, dynamic
Triaxial, compression Triaxial, 1 Hz, stress path (a), axial dynamic loading 20–320 kN, failed after 629 cycles
S40
40.0
Static
Triaxial, compression
D40
40.0
Cyclic, dynamic
Triaxial, 1 Hz, stress path (a), axial dynamic loading 20–350kN, failed after 241 cycles
S50
50.0
Static
Triaxial, compression
D50
50.0
Cyclic, dynamic
Triaxial, 1 Hz, stress path (a), axial dynamic loading 20–380 kN, failed after 347 cycles
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Fig. 2 Results of triaxial tests with static loading: a stress–strain curves and b volumetric strain-axial strain curves
Table 2 Summary of the stress state at failure
Fig. 3 Results of triaxial tests upon axial dynamic loading (D30): a dynamic deviatoric stress-axial strain curves (value of peak/valley), b dynamic volumetric strain-axial strain curves and c dynamic lateral strain-axial strain curves
Test no.
The maximal principal stress (MPa)
The minimal principal stress (MPa)
S10
132.8
10.0
S20
173.4
20.0
Table 3 Summary of the axial strain and volumetric strain when dilatancy happening
S30
208.9
30.0
Test no.
S40
228.7
40.0
S50
259.6
50.0
greater; and (2) with increased confining pressure, the axial strain at failure increased. Table 3 presents the volumetric strain when dilatancy occurred and the corresponding axial strain under static and dynamic triaxial loading conditions. When the samples became dilatant, the corresponding axial strain was greater for dynamic loading. When dilatancy took place, the volumetric strains of the samples at 10 and 20 MPa confining pressures were lower for dynamic loading conditions, but they were higher for those at confining pressures of 30, 40 and 50 MPa for dynamic loading conditions. The reason for this phenomenon is that when the confining pressure is low, the brittleness of the sample impedes contraction when dynamic loading is applied.
The axial strain (%)
The volumetric strain (%)
S10
0.517
0.323
D10
0.707 (0.209)
0.208 (0.094)
S20
0.640
0.359
D20
0.832
0.284
S30 D30
0.751 0.935 (0.237)
0.387 0.413 (0.162)
S40
0.880
0.447
D40
1.137 (0.533)
0.516 (0.229)
S50
0.987
0.507
D50
1.230 (1.180)
0.579 (0.309)
The values in parentheses are residual strain
When the confining pressure is high and when dynamic loading is applied, the ductility of the sample fosters contraction.
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Fig. 4 Strain-N curves: a strain-N curves (D10), b strain-N curves (D30) and c strain-N curves (D50), where N is the number of cycles
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Fig. 5 Effects of confining pressures on dynamic mechanical properties: a residual strain (%) at different confining pressures, b axial stiffness versus residual axial strain (%) and c the number of cycles at failure with Rs under different confining pressures
3.3 Effects of Confining Pressures on Dynamic Strain Residual strain is defined as the strain (including axial and volumetric strains) at which the axial load reaches the valley value during the process of cyclic loading. Figure 4a–c present the relationship curves for the residual strain, including residual axial and residual volumetric strains, and the number of cycles (N) for samples D10, D30 and D50. As N increases, the residual axial strain gradually increases during its initial cycles, and then it rapidly increases until failure. The residual volumetric strain contracted during the initial loading cycles and then dilated until failure. The lower the confining pressure, the greater the dilatancy of the sample was. With an increase in the
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confining pressure, the residual volumetric strain increased when dilatancy occurred (Table 3). With the increase of confining pressure, the residual axial and volumetric strains increased, as well. Figure 5a presents the residual axial strain and residual volumetric strain with different confining pressures. For residual axial strain ea;re , we can use the following equation to describe how ea;re varies with confining pressure rc : ea;re ¼ 0:29e0:045rc R2 ¼ 0:98 : ð1Þ For residual volumetric strain ev;re , we can use the following equation to describe how ev;re varies with confining pressure rc :
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Table 4 Summary of the failure modes of the sandstone samples tested Failure mode
Pictures of samples at failure Test no.S10
Test no.D10
Test no.S30
Test no.D30
Shear failure. Localized failure bands exist. Compared with static loading with the same confining stress, the localized failure bands are wider upon dynamic loading under higher confining pressure
ev;re ¼ 0:021rc 1:304
R2 ¼ 0:984 :
ð2Þ
In Eqs. 1 and 2, the residual strain is a percentage, and rc is presented in MPa.
failure under different confining pressures. When Rs is large, the sample fails after only a small number of cycles under confining pressure conditions. 3.5 Effects of Confining Pressures on Failure Models
3.4 Effects of Confining Pressures on Dynamic Mechanical Properties The average dynamic axial stiffness (Asd) throughout loading is calculated using the following formula (Bagde and Petrosˇ 2005): Asd ¼ Dstress =Dstrain ;
ð3Þ
where Asd is the average dynamic axial stiffness (i.e., the modulus of deformation or Young’s modulus over the elastic interval) presented in GPa, Dstress is the stress difference presented in GPa, and Dstrain is the strain difference obtained from the corresponding peak-valley data under dynamic conditions. The calculated dynamic axial stiffness using Eq. (3) is plotted against the residual axial strain, as shown in Fig. 5b. The dynamic axial stiffness of the rock sample decreases with an increase in the residual axial strain (Fig. 5b). This decrease in stiffness is the result of micro fracturing and a fatiguing phenomenon in the sample. Rocks subjected to dynamic cyclic loading respond with a significantly higher initial strength and stiffness at higher confining pressures than at lower confining pressures. The stiffness decreases more rapidly at lower confining pressures than at higher confining pressures because the latter can increase the strength and stiffness of the samples upon dynamic loading. Rs is defined as the stress ratio and is calculated as follows: qmax;dyn Rs ¼ ; ð4Þ 2sstat where qmax;dyn is the maximal deviatoric stress upon axial dynamic loading, and sstat is the strength upon static triaxial loading at the same confining pressure of the axial dynamic loading. Figure 5c presents Rs, the number of cycles at
Table 4 presents the failure modes of the sandstone samples tested with confining stresses under static and dynamic loading conditions. Compared with static loading under the same confining stress, the localized failure bands were wider upon dynamic loading under higher confining pressures.
4 Conclusions From these tests, we draw the following conclusions: (1) with increased confining pressure, the residual axial and volumetric strain of the rock samples and the residual volumetric strain when dilatancy occurred both became larger; (2) the rock samples subjected to dynamic cyclic loading responded with a significantly higher initial strength and stiffness at higher confining pressures than at lower confining pressures, and the stiffness decreased more rapidly at lower confining pressures than at higher confining pressures; (3) when the stress ratio (Rs) was larger, the sample failed after fewer cycles under confining pressure conditions; (4) when the samples had dilatant behavior, the corresponding axial strain was greater for dynamic loading than for the static triaxial tests; and (5) the failure modes of the samples were shear failure. Compared with static loading under the same confining stress, the localized failure bands were wider upon dynamic loading under higher confining pressures. Acknowledgments The authors thank the anonymous reviewers for their careful review, contributions and critics which led to the improvement of the manuscript and the funding of Key Laboratory of Mountain Hazards and Surface Process, Chinese Academy of Sciences.
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