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Effects of cyclic dynamic loading on the mechanical properties of intact rock samples under confining pressure conditions
Engineering Geology 125 (2012) 81–91
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Effects of cyclic dynamic loading on the mechanical properties of intact rock samples under confining pressure conditions Enlong Liu a, Siming He b,⁎ a b
State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resources and Hydropower, Sichuan University, Chengdu 610065, PR China Key Laboratory of Mountain Hazards and Surface Process, Institute of Mountain Hazards and Environment, CAS, Chengdu 610041, PR China
a r t i c l e
i n f o
Article history: Received 17 July 2011 Received in revised form 11 October 2011 Accepted 19 November 2011 Available online 1 December 2011 Keywords: Fatigue Damage evolution Cyclic dynamic loading Confining pressure Cyclic dynamic stiffness
a b s t r a c t A series of laboratory tests were performed to assess the effects of confining pressure on the mechanical properties and fatigue damage evolution of sandstone samples subjected to cyclic loading. Six levels of confining pressure (2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa) were applied during axial cyclic loading at a 1.0 Hz frequency using a MTS-815 Rock and Concrete Test System. Results from the cyclic dynamic loading tests indicated that the level of confining pressure had a significant influence on the cyclic dynamic deformation and fatigue damage evolution of the sandstone samples tested. With increasing confining pressure, the axial strain at failure increased, as did the residual volumetric strain at the initiation of dilatancy. The residual axial strains of sandstone samples obtained at a confining stress state can be described as three deformational stages, namely, the initial phase, uniform velocity phase and accelerated phase. Both the residual strain method and the axial secant modulus method proposed here could be used to describe the initial fatigue damage and degradation process of sandstone samples subjected to fatigue loading under a confining stress state; however, the latter also considers the influence of stress level on fatigue damage evolution when fatigue loads are applied. At a constant confining pressure, the shear fracture plane can form under static and cyclic dynamic loading conditions, and the higher the confining pressure, the wider the shear fracture planes become under cyclic dynamic loadings. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Intractable rock engineering problems encountered in practice are related to the behavior of rocks in dynamic loading and fatigue (cyclic) loading, which result from rock-burst, earthquake or excavation engineering. For this reason, great attention has recently been focused on studying the dynamic mechanical features of rocks under different loading histories and loading conditions (Stavrogin and Tarasov, 2001). When subjected to dynamic or cyclic loads, different materials respond in different ways. Some of these materials become stronger and more ductile, while others become weaker and more brittle (Bagde and Petroš, 2005). For rock samples, a summary of research work related to the fatigue behavior for various intact and jointed rock samples by different researchers in a laboratory setting can be found in the article by Bagde and Petroš (2005). Many researchers have carried out studies on different rock types, examining the effect of the loading rate and the loading strain rate on rock strength and deformation characteristics. For example, Zhao (2000) conducted a series of dynamic tests, including uniaxial and triaxial compression, uniaxial tension and unconfined shear tests on the Bukit Timah granite of Singapore with a varying loading rate. Zhao found that rock
⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (S. He). 0013-7952/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2011.11.007
material strength under dynamic loads could be approximately described by the Mohr–Coulomb criterion when the confining pressure was low and that the variation of cohesion with loading rate mainly led to a change in strength. Under a compressive stress state, the rock material strength under dynamic loading was better described by the Hoek–Brown criterion. Cho et al. (2003) performed dynamic tension tests on Inada granite and Tage tuff to investigate the strain-rate dependency of the rock dynamic tensile strength based on Hopkinson's effect combined with the spalling phenomena. Their work indicated that the dynamic tensile strength of the two rock types increased rapidly with strain rate. Mahmutoğlu (2006) performed compressive tests at different stress and strain rates based on a small-scale physical model of rock mass in a laboratory environment. It was found experimentally that the strength would decrease under the action of parameters such as joint frequency, time and saturation; however, these parameters affected the strength and rock behavior to different degrees. Wang et al. (2009) performed tests to measure the dynamic tensile strength of a brittle rock by diametrically impacting the Flattened Brazilian Disc specimens using a pulse shaping split Hopkinson pressure bar. Using the one-dimensional stress wave theory, they analyzed the stress waves traveling through the incident bar, the transmission bar and the FBD specimens, which were compared with results recorded using a strain gauge technique. Liang et al. (2010) conducted a series of experiments on the mechanical response of salt rock to study the effect of strain
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
Deviatoric stress q
(a) a typical stress path
(b) the loading sequence
deviatoric stress q =( σ1-σ3) mean stress p =( σ1-2σ3) /3
Deviatoric stress q
82
Variable Δq
dozens to thousands cycles
Mean stress p
Time
Fig. 1. Load path for (a) a typical stress path and (b) the loading sequence.
rate effects. They found that the loading strain rate affected the strength of salt rock slightly and the elastic modulus increased slightly with an increase in strain rate. Furthermore, extensive work on cyclic loading was performed to determine whether rocks are subjected to fatigue weakening. Burdine (1963) tested cylindrical rock samples using a laboratory designed and constructed dynamic-stress apparatus to investigate the cumulative damage of rock samples subjected to cyclic stresses under various loading conditions. Prost (1988) performed tests, including uniaxial tension and compression, triaxial compression, and triaxial cyclic compression–tension, on monolithologic and sandstone-granite composite samples to study the effect of a pre-existing joint on the initiation and propagation of cracks. Prost compared the results of these tests with observations made about outcrops of basement-sandstone contacts. Singh (1989) performed cyclic load tests on Graywacke rock samples to investigate their fatigue and strain hardening behaviors. He found that the fatigue life of a rock increased with decreasing stress amplitude and increasing strain hardening percentage up to a limit with an increasing number of load cycles. Tien et al. (1990) performed tests on a saturated sandstone under quasistatic, repeated and cyclic loadings to study the strain, pore water pressure and fatigue characteristics. The relationship between the accumulation of axial strain and the fatigue life of the sandstone was established based on their experimental results. Li et al. (2001) carried out a model test with intermittently jointed model samples subjected to dynamic cyclic loads with different frequencies. They investigated the dynamic fatigue damage properties of jointed rock mass materials and presented a fatigue damage model for the semicracked media. Later, they also studied the fatigue properties of cracked, saturated and frozen sandstone samples under cyclic loading (Li et al., 2003). By introducing the new RMT test apparatus and conducting cyclic dynamic tests of sandstone samples, Ge (2008) studied the fatigue
failure mechanism of rock samples and concluded that the whole stress–strain process curve of a rock sample controlled the strain of fatigue failure of the rock sample. Bagde and Petroš (2009) performed uniaxial compression tests on rock samples subjected to dynamic cyclic loading with different amplitudes and frequencies. They found that the dynamic strength was affected by the features of the dynamic loading and its loading rate. The fatigue strength of the rock was found to be influenced mainly by quartz content, texture and structure of the rock in dynamic cyclic loading. With increasing loading frequency and amplitude, the average Young's modulus and the dynamic axial stiffness of the rock was reduced, but the dynamic energy sustained by the rock tended to increase. Fuenkajorn and Phueakphum (2010) conducted a series of laboratory tests on salt rock samples under static and cyclic loads. They found experimentally that with an increasing number of loading cycles, the salt compressive strength decreased. During the first few cycles, the salt elastic modulus decreased slightly and then remained constant until failure. Xiao et al. (2010) proposed a damage variable to describe the actual evolution process of granite fatigue damage by analyzing the test results of the uniaxial cyclic dynamic tests. The experimental and theoretical studies cited above primarily focus on the influence of loading rates or uniaxial cyclic loading on the dynamic mechanical properties of rock samples (Bagde and Petroš, 2005, 2009; Fuenkajorn and Phueakphum, 2010; Liang et al., 2010; Xiao et al., 2010), but few studies have addressed rock samples subjected to cyclic loading under 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 cyclic dynamic mechanical properties and fatigue damage evolution of rock samples under cyclic dynamic loading, which is the type of study performed here.
Table 1 Summary of the samples tested and the axial and volumetric strain upon initiation of dilatancy. Test no.
Confining pressure (MPa)
Loading type
Loading condition
The axial strain (%)
The volumetric strain (%)
S2 F2
2.0 2.0
Static Cyclic, dynamic
0.449 0.587
0.305 0.229
S10 F10
10.0 10.0
Static Cyclic, dynamic
0.517 0.707(0.209)
0.323 0.208(0.094)
S20 F20
20.0 20.0
Static Cyclic, dynamic
0.640 0.832
0.359 0.284
S30 F30
30.0 30.0
Static Cyclic, dynamic
0.751 0.935(0.237)
0.387 0.413(0.162)
S40 F40
40.0 40.0
Static Cyclic, dynamic
0.880 1.137(0.533)
0.447 0.516(0.229)
S50 F50
50.0 50.0
Static Cyclic, dynamic
Triaxial, compression Triaxial, 1 Hz, stress path (a), axial dynamic Loading:20–165 kN, failed after 495 cycles Triaxial, compression Triaxial, 1 Hz, stress path (a), axial dynamic Loading: 20–230 kN, failed after 161 cycles Triaxial, compression Triaxial, 1 Hz, stress path (a), axial dynamic Loading:20–280 kN, failed after 257 cycles Triaxial, compression Triaxial, 1 Hz, stress path (a), axial dynamic loading:20–320 kN, failed after 629 cycles Triaxial, compression Triaxial, 1 Hz, stress path (a), axial dynamic Loading: 20–350kN, failed after 241 cycles Triaxial, compression Triaxial, 1 Hz, stress path (a), axial dynamic Loading:20–380 kN, failed after 347 cycles
0.987 1.230(1.180)
0.507 0.597(0.309)
Note: the values in parentheses are residual strain.
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
(a) Test No. S2, S20 and S40
(a) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) Deviatoric stress (MPa)
deviatoric stress (MPa) 200 S40 S40
150 S20
S20
100 axial strain S2
50
S2
83
volumetric strain
100 80
Confining pressure:2 MPa; Failured at 495 cycles; Frequency:1Hz; Dynamic load: 20~165kN.
60 40 Test No. F2
20 0 0.0
Axial strain (%)
0.2
0.4
0.6
0.8
1.0
lateral strain
-2.0
-1.0
0 0.0
1.0
2.0
3.0
0.5
Volumetric strain (%)
strains (%)
(b) Dynamic volumetric strain-axial strain curves(value of thepeak/valley)
(b) Test No. S10, S30 and S50 deviatoric stress (MPa) 250
0.0 0.0 -0.5 -1.0 -1.5 -2.0 -2.5
S50
Axial strain (%)
0.2
0.8
1.0
Test No. F2 Confining pressure:2 MPa; Failured at 495 cycles; Frequency:1Hz; Dynamic load: 20~165kN.
200
S50
(c) Dynamic lateral strain-axial strain curves(value of the peak/valley) 150
Axial strain (%)
S30
100 S10
S10
50
axial strain volumetric strain lateral strain
-2.0
-1.0
0 0.0
1.0
2.0
3.0
strains (%) Fig. 2. Stress–strain curves from triaxial compressive testing with static loading of (a) tests No. S2, S20 and S40 and (b) tests No. S10, S30 and S50.
In this work, sandstone samples were subjected to axial cyclic loading to experimentally determine the effects of confining pressure on their dynamic residual deformation and fatigue mechanical properties. Six levels of confining pressure (2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa) were applied for axial cyclic loading at a frequency of 1.0 Hz. Finally, the fatigue damage evolution of sandstone samples under cyclic dynamic loading was investigated in detail. 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/cm 3. The samples were prepared and tested according to International Society for Rock Mechanics (ISRM) testing procedures and all relevant guidelines. An MTS-815 Rock and Concrete Test System was used for testing. The MTS controller consisted of hardware components and software applications that provided a closed-loop control of the servo-hydraulic Table 2 Summary of the stress state at failure. Test no.
The maximal principal stress (MPa)
The minimal principal stress (MPa)
S2 S10 S20 S30 S40 S50
86.6 132.8 173.4 208.9 228.7 259.6
2.0 10.0 20.0 30.0 40.0 50.0
Lateral strain (%)
S30
0.0 0.0
0.2
0.8
1.0
-0.5 Test No. F2
-1.0 -1.5 -2.0
Confining pressure:2 MPa; Failured at 495 cycles; Frequency:1Hz; Dynamic load: 20~165kN.
Fig. 3. Results of triaxial tests under axial cyclic dynamic loading (Test No. F2) showing (a) dynamic deviatoric stress–axial strain curves (value of the peak/valley), (b) dynamic volumetric strain–axial strain curves (value of the peak/valley) and (c) dynamic lateral strain–axial strain curves (value of the peak/valley).
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 and concrete specimens. The loading frame had a 4600 kN compression load capacity and a 250 kN tensile load capacity with a 100 mm stroke. Additionally, it had the design features of a double-acting, single-ended actuator and a 140 MPa confining pressure capacity. The axial dynamic loading system was driven hydraulically with a 40-Lpm flow rate and 21 MPa of output pressure. The data 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 equipment was facilitated with an automated dynamic control mode switching between any connected transducer. Any transducer could be selected for control (typically load, strain, or displacement) including load-limited displacement during specimen loading. The tests were conducted with an axial displacement-controlled loading system. The static and dynamic triaxial tests were performed at the following confining pressures: 2, 10, 20, 30, 40 and 50 MPa. For the dynamic test, the axial dynamic load was specified as a sinusoidal cyclic compressive load. The loading frequency was set to 1.0 Hz, and the load path that was employed is schematically illustrated in Fig. 1 and the specific experimental one will be given later. During the axial
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
3. Experimental results 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 uniaxial compressive strength of the sandstone was 71.7 MPa. Fig. 2a–b presents the stress–strain curves of tests with the confining pressures of 2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa. The sandstone samples exhibited brittle behavior for all confining pressures. The lower the confining pressure, the more brittle behavior the samples exhibited. 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.
(a) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) Confining pressure:10 MPa; Failured at 161 cycles; Frequency:1Hz; Dynamic load: 20~230kN.
120
60
Test No. F20
Axial strain (%)
0.5
1.0
1.5
2.0
(b) Dynamic volumetric strain-axial strain curves(value of the peak/valley) 0.5 0.0 0.0
Axial strain (%)
0.5
1.0
1.5
2.0
-0.5 -1.0 pressure:20 MPa; -1.5 Confining Failured at 275 cycles;
-2.0
Frequency:1Hz; Dynamic load: 20~280kN. Test No. F20
(c) Dynamic lateral strain-axial strain curves(value of thepeak/valley) Axial strain %)
0.0 0.0
50
1.0
1.5
2.0
Test No. F10 Axial strain (%)
0.5
1.0
1.5
(b) Dynamic volumetric strain-axial strain curves(value of the peak/valley) 1.0 Axial strain (%)
Volumetric strain (%)
Failured at 275 cycles; Frequency:1Hz; Dynamic load: 20~280kN.
100
0 0.0
0.0 0.0
0.5
1.0
-1.5 Confining pressure:20 MPa; Failured at 275 cycles; Frequency:1Hz; Dynamic load: 20~280kN.
Test No. F20
Fig. 5. Results of triaxial tests under cyclic axial dynamic loading (Test No. F20) showing (a) dynamic deviatoric stress–axial strain curves (value of the peak/valley), (b) dynamic volumetric strain–axial strain curves (value of the peak/valley) and (c) dynamic lateral strain–axial strain curves (value of the peak/valley).
-2.0 -3.0 Confining pressure:10 MPa; Failured at 161 cycles; Frequency:1Hz; Dynamic load: 20~230kN.
3.2. Cyclic dynamic tests
Test No. F10
(c) Dynamic lateral strain-axial strain curves(value of the peak/valley) Axial strain (%)
0.0 0.0
1.0
1.5
-1.0
-2.0
-1.0
-2.0
-1.0
-4.0
-0.5
1.5
-5.0
Lateral strain (%)
180 Confining pressure:20 MPa;
0 0.0
Lateral strain (%)
Deviatoric stress (MPa)
150
(a) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) Deviatoric stress (MPa)
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.
Volumetric strain (%)
84
Confining pressure:10 MPa; Failured at 161 cycles; Frequency:1Hz; Dynamic load: 20~230kN. Test No. F10
-3.0 Fig. 4. Results of triaxial tests under axial cyclic dynamic loading (Test No. F10) showing (a) dynamic deviatoric stress–axial strain curves (value of the peak/valley), (b) dynamic volumetric strain–axial strain curves (value of the peak/valley) and (c) dynamic lateral strain–axial strain curves (value of the peak/valley).
In the fatigue tests following the load path shown in Fig. 1, deviatoric stress, axial strain and lateral strain were recorded for the loading duration with applied confining pressures of 2.0, 10.0, 20.0, 30.0, 40.0 or 50.0 MPa. Fig. 3 through Fig. 8 present the experimental results of samples F2, F10, F20, F30, F40 and F50, respectively, when the axial dynamic load changes from the peak value (the maximal value in a cycle) to the valley value (the minimal value in a cycle). From the test results, the following conclusions were made: (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 the deformation magnitudes became greater; (2) with increased confining pressure, the axial strain at failure increased. Table 1 presents the volumetric strain when dilatancy initiated and the corresponding axial strain under static and dynamic triaxial loading conditions. When dilatancy took place under dynamic loading conditions, the volumetric strains of the samples at 2.0, 10.0 and 20.0 MPa confining pressures were lower, but
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
they were higher for samples at confining pressures of 30.0, 40.0 and 50.0 MPa. 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. When the confining pressure is high and dynamic loading is applied, the ductility of the sample fosters contraction.
3.3. Residual strain of fatigue tests Residual strain is defined as the strain (including axial, volumetric and lateral strains) at which the axial load reaches the minimal value in a cycle after N cycles; that is to say, when the axial dynamic load reduces to the minimal value in a cycle, some strain will be reversible and the remaining irreversible portion of the strain is the residual strain. Fig. 9a–f presents the relationship curves for the residual axial strain and the number of cycles (N) for samples F2, F10, F20, F30, F40 and F50, respectively. As N increased, the residual axial strain gradually increased during its initial cycles and then rapidly increased until failure. Fig. 10a–f presents the relationship curves for the residual volumetric strain and the number of cycles (N) for samples F2, F10, F20, FA30, F40 and F50, respectively. The residual volumetric strain contracted during the initial loading cycles and then dilated until failure.
85
The lower the confining pressure, the greater the dilatancy of the sample became. With an increase in the confining pressure, the residual volumetric strain increased when dilatancy initiated (see Table 1). With an increase in confining pressure, the residual axial and volumetric strains increased as well. According to the development of the residual axial strain and its increasing velocity, the residual axial strains of sandstone samples obtained at a confining stress state shown in Fig. 9a–f can be described as three deformational stages (see Figure 11a); these stages are the initial phase in which the axial residual deformation first increases and approaches gradually to a constant velocity, the uniform velocity phase in which the axial residual deformation increases at a constant velocity and the accelerated phase in which the axial residual deformation increases rapidly with an increasing velocity until failure (Ge, 2008). The following is evident: the residual deformation increases rapidly during the initial phase; after some cycles, the rate of residual deformation development tends to be stable; near failure, the residual deformation increases rapidly again to eventual failure. Fig. 11b presents the relationship of εa,re/εa,re, max and N/Nmax, which demonstrates that for confining pressures from 2.0 MPa to 50.0 MPa, the residual axial strains exhibit a three phase behavior like that obtained during uniaxial fatigue loading (Fuenkajorn and Phueakphum, 2010).
Deviatoric stress (MPa)
(a) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) 210
Confining pressure:20 MPa; Failured at 629 cycles; Test No. F30 Frequency:1Hz; Dynamic load: 20~320kN.
140
70
Deviatoric stress (MPa)
(a) Dynamic deviatoric stress-axial strain curves(value of the peak/valley)
Axial strain (%)
0 0.0
1.0
2.0
3.0
4.0
4.0
Test No. F30
Axial strain (%)
Lateral strain (%)
2.0
3.0
-3.0
100 50 0 0.0
Test No. F30
Confining pressure:30 MPa; Failured at 629 cycles; Frequency:1Hz; Dynamic load: 20~320kN.
-4.0 Fig. 6. Results of triaxial tests under axial cyclic dynamic loading (Test No. F30) showing (a) dynamic deviatoric stress–axial strain curves (value of the peak/valley), (b) dynamic volumetric strain–axial strain curves (value of the peak/valley) and (c) dynamic lateral strain–axial strain curves (value of the peak/valley).
Axial strain (%)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.3 0.0 0.0
Axial strain (%)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
-0.3 -0.6
Confining pressure:40 MPa; Failured at 241 cycles; Frequency:1Hz; Dynamic load: 20~340kN.
Test No. F40
(c) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) 0.0 0.0
4.0
-1.0 -2.0
150
-0.9
(c) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) 0.0 0.0
Volumetric strain (%)
3.0
Confining pressure:40 MPa; Failured at 241 cycles; Frequency:1Hz; Dynamic load: 20~340kN. Test No. F40
0.6
Lateral strain (%)
Volumetric strain (%)
Axial strain (%)
2.0
200
(b) Dynamic deviatoric stress-axial strain curves(value of the peak/valley)
(b) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) 0.5 0.0 0.0 -0.5 -1.0 -1.5 -2.0 Confining pressure:30 MPa; -2.5 Failured at 629 cycles; -3.0 Frequency:1Hz; Dynamic load: 20~320kN. -3.5
250
Axial strain (%)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
-0.5 Test No. F40
-1.0 -1.5 -2.0 -2.5
Confining pressure:40 MPa; Failured at 241 cycles; Frequency:1Hz; Dynamic load: 20~340kN.
Fig. 7. Results of triaxial tests under axial cyclic dynamic loading (Test No. F40) showing (a) dynamic deviatoric stress–axial strain curves (value of the peak/valley), (b) dynamic volumetric strain–axial strain curves (value of the peak/valley) and (c) dynamic lateral strain–axial strain curves (value of the peak/valley).
86
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
4.1. Residual axial strain method
4. Fatigue damage evolution When establishing a fatigue theory for rock materials, one crucial issue is the definition of a damage variable reflecting the fatigue loading. Damage is the creation and growth of microvoids or microcracks that are discontinuities in a medium considered to be continuous at a larger scale. Damage in rock materials is mainly the process of the initiation and growth of microcracks or microvoids. At that scale, the phenomenon is discontinuous, but at a larger scale, damage can be considered to be continuous. Damage can degrade many properties of rock materials, including the elastic modulus, hardness, ultrasonic wave velocity and residual strength (Lemaitre, 1996). No matter the parameter used when describing the damage process, damage evolution must be consistent with the initiation and stable or unstable propagation of microcracks. Therefore, Xiao et al. (2010) proposed that a reasonable damage variable must meet the following basic requirements: 1) it should have a distinct physical meaning; 2) it can be measured easily and conveniently applied in engineering; 3) its evolution law coincides well with the actual degradation process of the material; and 4) it can take the initial damage into account. The following two fatigue damage variables are employed to investigate the fatigue damage evolution of sandstone samples subjected to axial fatigue loads under a confining stress state.
The damage variable can be defined using residual strain as N
f
D ¼ ε a;re =εa;re ;
ð1Þ
N f where εa, re and εa, re are the residual axial strain after N cycles and the ultimate residual axial strain at fatigue failure, respectively (Xiao et al., 2010). We can see that the damage evolution curves show an obvious three-stage developing process in Fig. 12. The initial fatigue damage, which includes the original damage before loading and the resulted damage during static loading, ranges from 0.0717 at a 2.0 MPa confining pressure to 0.6659 at a 50.0 MPa confining pressure. The initial fatigue damage is affected by the stress level and magnitude of residual strain at which fatigue load is applied, so the initial fatigue damage obtained by Eq. (1) may not consider these correctly in some conditions. The residual strain method is proven to be more appropriate because it has a distinct physical meaning and can describe the initial fatigue damage and the degradation process of sandstone samples subjected to fatigue loading under a confining stress state.
4.2. Axial secant modulus method The damage variable can also be defined with residual strain as
(a) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) Deviatoric stress (MPa)
250
Confining pressure:50 MPa; Failured at 347 cycles; Frequency:1Hz; Dynamic load: 20~380kN.
150 Test No. F50
100 50
Axial strain (%)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
(b) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) Volumetric strain (%)
0.8 0.6
Confining pressure:50 MPa; Failured at 347 cycles; Frequency:1Hz; Dynamic load: 20~380kN.
0.4 0.2 Axial strain (%)
0.0 0.0 -0.2
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Test No. F50
-0.4
(c) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) Lateral strain (%)
Axial strain (%)
0.0 0.0 -0.5
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
-2.5
ð2Þ
ð3Þ
where Δstress is the stress difference presented in GPa, and Δstrain is the strain difference obtained from the corresponding peak-valley data under cyclic dynamic conditions; Asd,max is the maximal stiffness, which can be calculated using the stress and strain in the elastic range upon static or axial cyclic loading; and Asd,min is the minimal stiffness which is equal to Asd when failure happens upon fatigue loading. The damage evolution curves calculated using Eq. (3) are shown in Fig. 13, which presents the three-stage developing process of fatigue damage at different confining pressures. The samples have different initial fatigue damage ranging from 0.423 to 0.818, which depends on the stress level when fatigue loads are applied. The axial secant modulus method proposed here has distinct physical meaning and can also describe the initial fatigue damage and the degradation process of sandstone samples subjected to fatigue loading under a confining stress state. 4.3. The effect of stress level on fatigue damage evolution Rs is defined as the stress ratio and calculated as follows:
4.5 q max;dyn 2τstat
ð4Þ
Test No. F50
-1.5 -2.0
Asd ¼ Δstress =Δstrain ;
Rs ¼
-1.0
Asd;max −Asd Asd;max −Asd;min
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, which can be calculated using the following formula (Bagde and Petrosš, 2005):
200
0 0.0
D¼
Confining pressure:50 MPa; Failured at 347 cycles; Frequency:1Hz; Dynamic load: 20~380kN.
Fig. 8. Results of triaxial tests under axial cyclic dynamic loading (Test No. F50) showing (a) dynamic deviatoric stress–axial strain curves (value of the peak/valley), (b) dynamic volumetric strain–axial strain curves (value of the peak/valley) and (c) dynamic lateral strain–axial strain curves (value of the peak/valley).
In Eq. (4), qmax, dyn is the maximal deviatoric stress under axial dynamic loading, and τstat = (σ1 − σ3)/2, in which σ1 and σ3 are the maximal principal stress and the minimal principal stress in Table 2, respectively, is the strength computed using the Tresca Criterion for static triaxial loading at the same confining pressure as the axial dynamic loading. Fig. 14a–b presents Rs plotted versus the initial fatigue damage and the number of cycles at failure, respectively, under 2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa confining pressures. When Rs is
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
(a)
(b) 0.45
Residual axial strain (%)
Residual axial strain (%)
0.35
Test No. F2
0.30
0.25 Confining pressure:2 MPa; Failured at 495 cycles; Frequency:1Hz; Dynamic load: 20~165kN.
0.20
0
100
200
300
400
N
Confining pressure:10 MPa; Failured at 161 cycles; Frequency:1Hz; Dynamic load: 20~230kN.
0.35
0.25 Test No. F10
0.15
500
(c)
N 0
50
100
150
200
(d)
0.90
1.0
Confining pressure:20 MPa; Failured at 257 cycles; Frequency:1Hz; Dynamic load: 20~280kN.
Residual axial strain (%)
Residual axial strain (%)
87
0.70
0.50
0.30 Test No. F20
Confining pressure:30 MPa; Failured at 629 cycles; Frequency:1Hz; Dynamic load: 20~320kN.
0.8 0.6 0.4
Test No. F30
0.2
N 0.10
0
50
100
150
200
250
(e)
N 0
100
200
300
400
500
600
700
(f) Confining pressure:40 MPa; Failured at 241 cycles; Frequency:1Hz; Dynamic load: 20~350kN.
1.5
3.0
Residual axial strain (%)
2.0
Residual axial strain (%)
0.0
300
Test No. F40
1.0
0.5
N 0.0
0
50
100
150
200
250
Confining pressure:50 MPa; Failured at 347 cycles; Frequency:1Hz; Dynamic load: 20~380kN.
2.0
1.0 Test No. F50
N
0.0 0
100
200
300
400
Fig. 9. Residual axial strain–N curves (N: the number of cycles) for (a) Test No. F2, (b) Test No. F10, (c) Test No. F20, (d) Test No. F30, (e) Test No. F40, and (f) Test No. F50.
large, the initial fatigue damage is greater, and the sample fails after only a small number of cycles under confining pressure conditions. From the fatigue damage evolution shown in Figs. 12 and 13, we can draw the conclusion that the axial secant modulus method can consider the influence of Rs on the damage evolution of rock samples under axial cyclic dynamic loading but the residual axial strain method cannot. 4.4. The fatigue failure mechanisms When subjected to static loads at confining pressures of 2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa, the sandstone specimens failed with clearly defined planes of fracture. The angle formed between the fracture plane and the plane of the maximum principal stress was greater than 45°. These planes are referred to as shear fractures and are
characterized by shearing displacement along their surfaces. From the failure patterns of samples S2, S10, S20, S30 and S40 shown in Fig. 15, we can conclude that the lower the confining pressure, the larger the angle formed at failure between the horizontal plane, where the maximum principal stress was applied, and the plane of the shear fracture. When the confining pressure was 2.0 MPa, the sample failure was accompanied by longitudinal splitting, but when the confining pressure was more than 10 MPa, the sample failure was accompanied by lateral expansion. When subjected to dynamic cyclic loads at a frequency of 1.0 Hz and confining pressures of 2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa, shear fractures also formed, as shown in Fig. 15. Compared with static loading under the same confining stress, when the confining pressures were either 2.0 or 10.0 MPa, the shear fracture planes were almost the same for cyclic dynamic loading, but when the confining
88
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
(a) 0.1 0
100
200
400
300
500
-0.2
Confining pressure:2 MPa; Failured at 495 cycles; Frequency: 1Hz; Dynamic load: 20~165kN.
-0.5
Test No. F10
0.2
N
N 0.0
Residual volumetric strain (%)
Residual volumetric strain (%)
(b)
Test No. F2
0
30
60
90
120
150
180
-0.2 -0.4 -0.6
Confining pressure:10 MPa; Failured at 161 cycles; Frequency: 1Hz; Dynamic load: 20~230kN.
-0.8
-0.8 -1.0
(c)
(d) Test No. F20
0.2
N 0.0 0
40
80
120
160
200
240
280
-0.2 -0.4
Confining pressure:20 MPa; Failured at 257 cycles; Frequency: 1Hz; Dynamic load: 20~280kN.
-0.6 -0.8
Residual volumetric strain (%)
Residual volumetric strain (%)
0.2
N 0.0 0
100
200
300
400
500
600
700
-0.2
-0.4
Confining pressure:30 MPa; Failured at 629 cycles; Frequency: 1Hz; Dynamic load: 20~320kN.
-0.6
-0.8
-1.0
(e)
(f) 0.3
0.4
0.2
Test No. F40
0.1
N
0.0 0
50
100
150
200
250
-0.1 -0.2 -0.3
Confining pressure:40 MPa; Failured at 241 cycles; Frequency: 1Hz; Dynamic load: 20~350kN.
-0.4
Residual volumetric strain (%)
Residual volumetric strain (%)
Test No. F30
Test No. F50 0.3 0.2 0.1 0.0 0
Confining pressure:50 MPa; Failured at 347 cycles; Frequency: 1Hz; Dynamic load: 20~380kN.
100
200
N 300
400
-0.1 -0.2
Fig. 10. Residual volumetric strain–N curves (N: the number of cycles) for (a) Test No. F2, (b) Test No. F10, (c) Test No. F20, (d) Test No. F30, (e) Test No. F40, and (f) Test No. F50.
pressures were greater than 10.0 MPa, the shear fractured planes were wider upon cyclic dynamic loading. The reason for this phenomenon is that when the confining pressure is lower, the brittleness of the sandstone specimens contributes to the formation of fracture planes and when the confining pressure is higher, they become less brittle, resulting in the formation of shear fracture bands upon dynamic cyclic loadings. 5. Conclusions From these tests of sandstone samples performed at 2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa confining stress states with axial cyclic
dynamic loading at a 1.0 Hz frequency, the following conclusions may be drawn: (1) With increased confining pressure, the axial strain at failure increased. When dilatancy took place, the volumetric strains of the samples at 2.0, 10.0 and 20.0 MPa confining pressures were lower for cyclic dynamic loading conditions but were higher for samples at confining pressures of 30.0, 40.0 and 50.0 MPa for cyclic dynamic loading conditions (2) 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
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
89
1.0
(a) Residual axial strain
F10
F40
0.9 F20
Test No. F2
F50
Damage, D
0.8
Stage I:initialphase Stage II:uniform velocity phase Stage III:accelerated phase
F30
0.7 0.6 0.5 0.4
N
0
100
200
300
400
500
600
700
Fig. 13. D–N curves (N: the number of cycle) using the axial secant modulus method.
Stage I
Stage II
Stage III
0
(b)
1.0 Test No. F2
0.8
εa,re/εa,re, max
an increase in the confining pressure, the residual volumetric strain increased when dilatancy occurred (4) The residual axial strains of sandstone samples obtained at a confining stress state could be described as three deformational stages: the initial phase, the uniform velocity phase and the accelerated phase (5) The residual strain method was proved more appropriate for its distinct physical meaning and ability to describe the initial fatigue damage and the degradation process of sandstone samples subjected to fatigue loading under a confining stress state. Additionally, the axial secant modulus method proposed here had distinct physical meaning and also described the initial fatigue damage and the degradation process of sandstone samples subjected to fatigue loading under a confining stress
N
0.6 Test No. F10
0.4
Test No. F20
0.2
Test No. F30 Test No. F40 Test No. F50
0.0 0.0
0.2
0.4
N/Nmax
0.6
0.8
1.0
(a)
Fig. 11. Plot showing (a) the three phases of residual axial strain (N: the number of cycles) and (b) the relationship of εa,re/εa,re, max ~ N/Nmax (εa,re is residual axial strain, εa,re, max the maximum of residual axial strain, N the number of cycles and Nmax the maximum of N ).
1.00
including axial, volumetric and lateral strains, whose magnitudes became greater (3) 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 became. With
0.97
0.99
Rs
0.98
0.96
Rs =
qmax,dyn 2τstat
Test No.F2 Test No.F10 Test No.F20 Test No.F30 Test No.F40 Test No.F50
0.95 0.94 0.0
0.2
0.4
0.6
0.8
1.0
Initial fatigue damage, D0 1.0
(b)
Test No. F2
0.8
1.00 0.99 F10
0.98 F30
0.4
Rs
Damage, D
0.6
0.97
Rs =
qmax,dyn 2τstat Test No.F2 Test No.F10 Test No.F20 Test No.F30 Test No.F40 Test No.F50
0.96
F20
0.95 F40
0.2
0.0
F50
0.94 100.0
0
100
200
300
400
500
600
1,000.0
The number of cycle at failure Nf
N
700
Fig. 12. D–N curves (N: the number of cycle) using the residual axial strain method.
Fig. 14. The influence of Rs on (a) initial fatigue damage and (b) the number of cycles at failure.
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E. Liu, S. He / Engineering Geology 125 (2012) 81–91
(a)
(b)
Test No.S2
Test No.F2
(c)
Test No.S10
Test No.F10
(d)
Test No.S20
Test No.F20
Test No.S30
Test No.F30
(e)
Test No.S40
Test No.F40
Fig. 15. Pictures of samples at failure with a confining pressure of (a) 2.0 MPa, (b) 10.0 MPa, (c) 20.0 MPa, (d) 30.0 MPa and (e) 40.0 MPa.
state. Comparing these two methods, the axial secant modulus method considers the influence of level stress when fatigue loads applied, while the residual axial strain method does not (6) The shear fracture planes can form under both static and cyclic dynamic loadings. When the confining pressure is lower, the shear fracture planes formed in the sandstone specimens are almost the same under both static and cyclic dynamic loading conditions. However, when the confining pressure is higher,
the shear fracture planes formed are wider under cyclic dynamic loading conditions than under static loading conditions. Acknowledgments The authors thank the two anonymous reviewers and the editor for their careful review, contributions and critiques, which led to the improvement of the manuscript. The authors also thak the
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
funding of Key Laboratory of Mountain Hazards and Surface Process, Chinese Academy of Science. References Bagde, M.N., Petroš, V., 2005. Fatigue properties of intact sandstone samples subjected to dynamic uniaxial cyclical loading. International Journal of Rock Mechanics and Mining Sciences 42, 237–250. Bagde, M.N., Petroš, V., 2009. Fatigue and dynamic energy behavior of rock subjected to cyclical loading. International Journal of Rock Mechanics and Mining Sciences 46, 200–209. Burdine, N.T., 1963. Rock failure under dynamic loading conditions. Society of Petroleum Engineers Journal 3, 1–8. Cho, S.H., Ogata, Y., Kaneko, K., 2003. Strain-rate dependency of the dynamic tensile strength of rock. International Journal of Rock Mechanics and Mining Sciences 40, 763–777. Fuenkajorn, K., Phueakphum, D., 2010. Effects of cyclic loading on mechanical properties of Maha Sarakham salt. Engineering Geology 112, 43–52. Ge, X., 2008. Deformation control law of rock fatigue failure, real-time X-ray CT scan of geotechnical testing, and new method of stability analysis of slopes and dam foundations. Chinese Journal of Geotechnical Engineering 30 (1), 1–20. Lemaitre, J., 1996. A course on damage mechanics. Springer Publishing. Li, N., Chen, W., Zhang, P., Swoboda, G., 2001. The mechanical properties and a fatiguedamage model for jointed rock masses subjected to dynamic cyclical loading. International Journal of Rock Mechanics and Mining Sciences 38, 1071–1079.
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Li, N., Zhang, P., Chen, W., Swoboda, G., 2003. Fatigue properties of a cracked, saturated and frozen sandstone samples under cyclic loading. International Journal of Rock Mechanics and Mining Sciences 40, 145–150. Liang, W.G., Zhao, Y.S., Xu, S.G., Dusseault, M.B., 2010. Effects of strain rate on the mechanical properties of salt rock. International Journal of Rock Mechanics and Mining Sciences. doi:10.1016/j.ijrmms. Mahmutoğlu, Y., 2006. The effects of strain rate and saturation on a micro-cracked marble. Engineering Geology 82, 137–144. Prost, G.L., 1988. Jointing at rock contacts in cyclic loading. International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts 25 (5), 263–272. Singh, S.K., 1989. Fatigue and strain hardening behavior of graywacke from the flagstaff formation, New South Wales. Engineering Geology 26, 171–179. Stavrogin, A.N., Tarasov, B.G., 2001. Experimental physics and rock mechanics. Balkem, Rotterdam. (356 pp.). Tien, Y.M., Lee, D.H., Juang, C.H., 1990. Strain, pore pressure and fatigue characteristics of sandstone under various load conditions. International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts 27 (4), 283–289. Wang, Q.Z., Li, W., Xie, H.P., 2009. Dynamic split tensile test of flattened Brazilian disc of rock with SHPB setup. Mechanics of Materials 41, 252–260. Xiao, J.Q., Ding, D.X., Jiang, F.L., Xu, G., 2010. Fatigue damage variable and evolution of rock subjected to cyclic loading. International Journal of Rock Mechanics and Mining Sciences 47, 461–468. Zhao, J., 2000. Applicability of Mohr–Coulomb and Hoek–Brown strength criteria to the dynamic strength of brittle rock. International Journal of Rock Mechanics and Mining Sciences 37, 1115–1121.
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Engineering Geology journal homepage: www.elsevier.com/locate/enggeo
Effects of cyclic dynamic loading on the mechanical properties of intact rock samples under confining pressure conditions Enlong Liu a, Siming He b,⁎ a b
State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resources and Hydropower, Sichuan University, Chengdu 610065, PR China Key Laboratory of Mountain Hazards and Surface Process, Institute of Mountain Hazards and Environment, CAS, Chengdu 610041, PR China
a r t i c l e
i n f o
Article history: Received 17 July 2011 Received in revised form 11 October 2011 Accepted 19 November 2011 Available online 1 December 2011 Keywords: Fatigue Damage evolution Cyclic dynamic loading Confining pressure Cyclic dynamic stiffness
a b s t r a c t A series of laboratory tests were performed to assess the effects of confining pressure on the mechanical properties and fatigue damage evolution of sandstone samples subjected to cyclic loading. Six levels of confining pressure (2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa) were applied during axial cyclic loading at a 1.0 Hz frequency using a MTS-815 Rock and Concrete Test System. Results from the cyclic dynamic loading tests indicated that the level of confining pressure had a significant influence on the cyclic dynamic deformation and fatigue damage evolution of the sandstone samples tested. With increasing confining pressure, the axial strain at failure increased, as did the residual volumetric strain at the initiation of dilatancy. The residual axial strains of sandstone samples obtained at a confining stress state can be described as three deformational stages, namely, the initial phase, uniform velocity phase and accelerated phase. Both the residual strain method and the axial secant modulus method proposed here could be used to describe the initial fatigue damage and degradation process of sandstone samples subjected to fatigue loading under a confining stress state; however, the latter also considers the influence of stress level on fatigue damage evolution when fatigue loads are applied. At a constant confining pressure, the shear fracture plane can form under static and cyclic dynamic loading conditions, and the higher the confining pressure, the wider the shear fracture planes become under cyclic dynamic loadings. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Intractable rock engineering problems encountered in practice are related to the behavior of rocks in dynamic loading and fatigue (cyclic) loading, which result from rock-burst, earthquake or excavation engineering. For this reason, great attention has recently been focused on studying the dynamic mechanical features of rocks under different loading histories and loading conditions (Stavrogin and Tarasov, 2001). When subjected to dynamic or cyclic loads, different materials respond in different ways. Some of these materials become stronger and more ductile, while others become weaker and more brittle (Bagde and Petroš, 2005). For rock samples, a summary of research work related to the fatigue behavior for various intact and jointed rock samples by different researchers in a laboratory setting can be found in the article by Bagde and Petroš (2005). Many researchers have carried out studies on different rock types, examining the effect of the loading rate and the loading strain rate on rock strength and deformation characteristics. For example, Zhao (2000) conducted a series of dynamic tests, including uniaxial and triaxial compression, uniaxial tension and unconfined shear tests on the Bukit Timah granite of Singapore with a varying loading rate. Zhao found that rock
⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (S. He). 0013-7952/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2011.11.007
material strength under dynamic loads could be approximately described by the Mohr–Coulomb criterion when the confining pressure was low and that the variation of cohesion with loading rate mainly led to a change in strength. Under a compressive stress state, the rock material strength under dynamic loading was better described by the Hoek–Brown criterion. Cho et al. (2003) performed dynamic tension tests on Inada granite and Tage tuff to investigate the strain-rate dependency of the rock dynamic tensile strength based on Hopkinson's effect combined with the spalling phenomena. Their work indicated that the dynamic tensile strength of the two rock types increased rapidly with strain rate. Mahmutoğlu (2006) performed compressive tests at different stress and strain rates based on a small-scale physical model of rock mass in a laboratory environment. It was found experimentally that the strength would decrease under the action of parameters such as joint frequency, time and saturation; however, these parameters affected the strength and rock behavior to different degrees. Wang et al. (2009) performed tests to measure the dynamic tensile strength of a brittle rock by diametrically impacting the Flattened Brazilian Disc specimens using a pulse shaping split Hopkinson pressure bar. Using the one-dimensional stress wave theory, they analyzed the stress waves traveling through the incident bar, the transmission bar and the FBD specimens, which were compared with results recorded using a strain gauge technique. Liang et al. (2010) conducted a series of experiments on the mechanical response of salt rock to study the effect of strain
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
Deviatoric stress q
(a) a typical stress path
(b) the loading sequence
deviatoric stress q =( σ1-σ3) mean stress p =( σ1-2σ3) /3
Deviatoric stress q
82
Variable Δq
dozens to thousands cycles
Mean stress p
Time
Fig. 1. Load path for (a) a typical stress path and (b) the loading sequence.
rate effects. They found that the loading strain rate affected the strength of salt rock slightly and the elastic modulus increased slightly with an increase in strain rate. Furthermore, extensive work on cyclic loading was performed to determine whether rocks are subjected to fatigue weakening. Burdine (1963) tested cylindrical rock samples using a laboratory designed and constructed dynamic-stress apparatus to investigate the cumulative damage of rock samples subjected to cyclic stresses under various loading conditions. Prost (1988) performed tests, including uniaxial tension and compression, triaxial compression, and triaxial cyclic compression–tension, on monolithologic and sandstone-granite composite samples to study the effect of a pre-existing joint on the initiation and propagation of cracks. Prost compared the results of these tests with observations made about outcrops of basement-sandstone contacts. Singh (1989) performed cyclic load tests on Graywacke rock samples to investigate their fatigue and strain hardening behaviors. He found that the fatigue life of a rock increased with decreasing stress amplitude and increasing strain hardening percentage up to a limit with an increasing number of load cycles. Tien et al. (1990) performed tests on a saturated sandstone under quasistatic, repeated and cyclic loadings to study the strain, pore water pressure and fatigue characteristics. The relationship between the accumulation of axial strain and the fatigue life of the sandstone was established based on their experimental results. Li et al. (2001) carried out a model test with intermittently jointed model samples subjected to dynamic cyclic loads with different frequencies. They investigated the dynamic fatigue damage properties of jointed rock mass materials and presented a fatigue damage model for the semicracked media. Later, they also studied the fatigue properties of cracked, saturated and frozen sandstone samples under cyclic loading (Li et al., 2003). By introducing the new RMT test apparatus and conducting cyclic dynamic tests of sandstone samples, Ge (2008) studied the fatigue
failure mechanism of rock samples and concluded that the whole stress–strain process curve of a rock sample controlled the strain of fatigue failure of the rock sample. Bagde and Petroš (2009) performed uniaxial compression tests on rock samples subjected to dynamic cyclic loading with different amplitudes and frequencies. They found that the dynamic strength was affected by the features of the dynamic loading and its loading rate. The fatigue strength of the rock was found to be influenced mainly by quartz content, texture and structure of the rock in dynamic cyclic loading. With increasing loading frequency and amplitude, the average Young's modulus and the dynamic axial stiffness of the rock was reduced, but the dynamic energy sustained by the rock tended to increase. Fuenkajorn and Phueakphum (2010) conducted a series of laboratory tests on salt rock samples under static and cyclic loads. They found experimentally that with an increasing number of loading cycles, the salt compressive strength decreased. During the first few cycles, the salt elastic modulus decreased slightly and then remained constant until failure. Xiao et al. (2010) proposed a damage variable to describe the actual evolution process of granite fatigue damage by analyzing the test results of the uniaxial cyclic dynamic tests. The experimental and theoretical studies cited above primarily focus on the influence of loading rates or uniaxial cyclic loading on the dynamic mechanical properties of rock samples (Bagde and Petroš, 2005, 2009; Fuenkajorn and Phueakphum, 2010; Liang et al., 2010; Xiao et al., 2010), but few studies have addressed rock samples subjected to cyclic loading under 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 cyclic dynamic mechanical properties and fatigue damage evolution of rock samples under cyclic dynamic loading, which is the type of study performed here.
Table 1 Summary of the samples tested and the axial and volumetric strain upon initiation of dilatancy. Test no.
Confining pressure (MPa)
Loading type
Loading condition
The axial strain (%)
The volumetric strain (%)
S2 F2
2.0 2.0
Static Cyclic, dynamic
0.449 0.587
0.305 0.229
S10 F10
10.0 10.0
Static Cyclic, dynamic
0.517 0.707(0.209)
0.323 0.208(0.094)
S20 F20
20.0 20.0
Static Cyclic, dynamic
0.640 0.832
0.359 0.284
S30 F30
30.0 30.0
Static Cyclic, dynamic
0.751 0.935(0.237)
0.387 0.413(0.162)
S40 F40
40.0 40.0
Static Cyclic, dynamic
0.880 1.137(0.533)
0.447 0.516(0.229)
S50 F50
50.0 50.0
Static Cyclic, dynamic
Triaxial, compression Triaxial, 1 Hz, stress path (a), axial dynamic Loading:20–165 kN, failed after 495 cycles Triaxial, compression Triaxial, 1 Hz, stress path (a), axial dynamic Loading: 20–230 kN, failed after 161 cycles Triaxial, compression Triaxial, 1 Hz, stress path (a), axial dynamic Loading:20–280 kN, failed after 257 cycles Triaxial, compression Triaxial, 1 Hz, stress path (a), axial dynamic loading:20–320 kN, failed after 629 cycles Triaxial, compression Triaxial, 1 Hz, stress path (a), axial dynamic Loading: 20–350kN, failed after 241 cycles Triaxial, compression Triaxial, 1 Hz, stress path (a), axial dynamic Loading:20–380 kN, failed after 347 cycles
0.987 1.230(1.180)
0.507 0.597(0.309)
Note: the values in parentheses are residual strain.
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
(a) Test No. S2, S20 and S40
(a) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) Deviatoric stress (MPa)
deviatoric stress (MPa) 200 S40 S40
150 S20
S20
100 axial strain S2
50
S2
83
volumetric strain
100 80
Confining pressure:2 MPa; Failured at 495 cycles; Frequency:1Hz; Dynamic load: 20~165kN.
60 40 Test No. F2
20 0 0.0
Axial strain (%)
0.2
0.4
0.6
0.8
1.0
lateral strain
-2.0
-1.0
0 0.0
1.0
2.0
3.0
0.5
Volumetric strain (%)
strains (%)
(b) Dynamic volumetric strain-axial strain curves(value of thepeak/valley)
(b) Test No. S10, S30 and S50 deviatoric stress (MPa) 250
0.0 0.0 -0.5 -1.0 -1.5 -2.0 -2.5
S50
Axial strain (%)
0.2
0.8
1.0
Test No. F2 Confining pressure:2 MPa; Failured at 495 cycles; Frequency:1Hz; Dynamic load: 20~165kN.
200
S50
(c) Dynamic lateral strain-axial strain curves(value of the peak/valley) 150
Axial strain (%)
S30
100 S10
S10
50
axial strain volumetric strain lateral strain
-2.0
-1.0
0 0.0
1.0
2.0
3.0
strains (%) Fig. 2. Stress–strain curves from triaxial compressive testing with static loading of (a) tests No. S2, S20 and S40 and (b) tests No. S10, S30 and S50.
In this work, sandstone samples were subjected to axial cyclic loading to experimentally determine the effects of confining pressure on their dynamic residual deformation and fatigue mechanical properties. Six levels of confining pressure (2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa) were applied for axial cyclic loading at a frequency of 1.0 Hz. Finally, the fatigue damage evolution of sandstone samples under cyclic dynamic loading was investigated in detail. 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/cm 3. The samples were prepared and tested according to International Society for Rock Mechanics (ISRM) testing procedures and all relevant guidelines. An MTS-815 Rock and Concrete Test System was used for testing. The MTS controller consisted of hardware components and software applications that provided a closed-loop control of the servo-hydraulic Table 2 Summary of the stress state at failure. Test no.
The maximal principal stress (MPa)
The minimal principal stress (MPa)
S2 S10 S20 S30 S40 S50
86.6 132.8 173.4 208.9 228.7 259.6
2.0 10.0 20.0 30.0 40.0 50.0
Lateral strain (%)
S30
0.0 0.0
0.2
0.8
1.0
-0.5 Test No. F2
-1.0 -1.5 -2.0
Confining pressure:2 MPa; Failured at 495 cycles; Frequency:1Hz; Dynamic load: 20~165kN.
Fig. 3. Results of triaxial tests under axial cyclic dynamic loading (Test No. F2) showing (a) dynamic deviatoric stress–axial strain curves (value of the peak/valley), (b) dynamic volumetric strain–axial strain curves (value of the peak/valley) and (c) dynamic lateral strain–axial strain curves (value of the peak/valley).
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 and concrete specimens. The loading frame had a 4600 kN compression load capacity and a 250 kN tensile load capacity with a 100 mm stroke. Additionally, it had the design features of a double-acting, single-ended actuator and a 140 MPa confining pressure capacity. The axial dynamic loading system was driven hydraulically with a 40-Lpm flow rate and 21 MPa of output pressure. The data 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 equipment was facilitated with an automated dynamic control mode switching between any connected transducer. Any transducer could be selected for control (typically load, strain, or displacement) including load-limited displacement during specimen loading. The tests were conducted with an axial displacement-controlled loading system. The static and dynamic triaxial tests were performed at the following confining pressures: 2, 10, 20, 30, 40 and 50 MPa. For the dynamic test, the axial dynamic load was specified as a sinusoidal cyclic compressive load. The loading frequency was set to 1.0 Hz, and the load path that was employed is schematically illustrated in Fig. 1 and the specific experimental one will be given later. During the axial
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
3. Experimental results 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 uniaxial compressive strength of the sandstone was 71.7 MPa. Fig. 2a–b presents the stress–strain curves of tests with the confining pressures of 2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa. The sandstone samples exhibited brittle behavior for all confining pressures. The lower the confining pressure, the more brittle behavior the samples exhibited. 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.
(a) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) Confining pressure:10 MPa; Failured at 161 cycles; Frequency:1Hz; Dynamic load: 20~230kN.
120
60
Test No. F20
Axial strain (%)
0.5
1.0
1.5
2.0
(b) Dynamic volumetric strain-axial strain curves(value of the peak/valley) 0.5 0.0 0.0
Axial strain (%)
0.5
1.0
1.5
2.0
-0.5 -1.0 pressure:20 MPa; -1.5 Confining Failured at 275 cycles;
-2.0
Frequency:1Hz; Dynamic load: 20~280kN. Test No. F20
(c) Dynamic lateral strain-axial strain curves(value of thepeak/valley) Axial strain %)
0.0 0.0
50
1.0
1.5
2.0
Test No. F10 Axial strain (%)
0.5
1.0
1.5
(b) Dynamic volumetric strain-axial strain curves(value of the peak/valley) 1.0 Axial strain (%)
Volumetric strain (%)
Failured at 275 cycles; Frequency:1Hz; Dynamic load: 20~280kN.
100
0 0.0
0.0 0.0
0.5
1.0
-1.5 Confining pressure:20 MPa; Failured at 275 cycles; Frequency:1Hz; Dynamic load: 20~280kN.
Test No. F20
Fig. 5. Results of triaxial tests under cyclic axial dynamic loading (Test No. F20) showing (a) dynamic deviatoric stress–axial strain curves (value of the peak/valley), (b) dynamic volumetric strain–axial strain curves (value of the peak/valley) and (c) dynamic lateral strain–axial strain curves (value of the peak/valley).
-2.0 -3.0 Confining pressure:10 MPa; Failured at 161 cycles; Frequency:1Hz; Dynamic load: 20~230kN.
3.2. Cyclic dynamic tests
Test No. F10
(c) Dynamic lateral strain-axial strain curves(value of the peak/valley) Axial strain (%)
0.0 0.0
1.0
1.5
-1.0
-2.0
-1.0
-2.0
-1.0
-4.0
-0.5
1.5
-5.0
Lateral strain (%)
180 Confining pressure:20 MPa;
0 0.0
Lateral strain (%)
Deviatoric stress (MPa)
150
(a) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) Deviatoric stress (MPa)
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.
Volumetric strain (%)
84
Confining pressure:10 MPa; Failured at 161 cycles; Frequency:1Hz; Dynamic load: 20~230kN. Test No. F10
-3.0 Fig. 4. Results of triaxial tests under axial cyclic dynamic loading (Test No. F10) showing (a) dynamic deviatoric stress–axial strain curves (value of the peak/valley), (b) dynamic volumetric strain–axial strain curves (value of the peak/valley) and (c) dynamic lateral strain–axial strain curves (value of the peak/valley).
In the fatigue tests following the load path shown in Fig. 1, deviatoric stress, axial strain and lateral strain were recorded for the loading duration with applied confining pressures of 2.0, 10.0, 20.0, 30.0, 40.0 or 50.0 MPa. Fig. 3 through Fig. 8 present the experimental results of samples F2, F10, F20, F30, F40 and F50, respectively, when the axial dynamic load changes from the peak value (the maximal value in a cycle) to the valley value (the minimal value in a cycle). From the test results, the following conclusions were made: (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 the deformation magnitudes became greater; (2) with increased confining pressure, the axial strain at failure increased. Table 1 presents the volumetric strain when dilatancy initiated and the corresponding axial strain under static and dynamic triaxial loading conditions. When dilatancy took place under dynamic loading conditions, the volumetric strains of the samples at 2.0, 10.0 and 20.0 MPa confining pressures were lower, but
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
they were higher for samples at confining pressures of 30.0, 40.0 and 50.0 MPa. 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. When the confining pressure is high and dynamic loading is applied, the ductility of the sample fosters contraction.
3.3. Residual strain of fatigue tests Residual strain is defined as the strain (including axial, volumetric and lateral strains) at which the axial load reaches the minimal value in a cycle after N cycles; that is to say, when the axial dynamic load reduces to the minimal value in a cycle, some strain will be reversible and the remaining irreversible portion of the strain is the residual strain. Fig. 9a–f presents the relationship curves for the residual axial strain and the number of cycles (N) for samples F2, F10, F20, F30, F40 and F50, respectively. As N increased, the residual axial strain gradually increased during its initial cycles and then rapidly increased until failure. Fig. 10a–f presents the relationship curves for the residual volumetric strain and the number of cycles (N) for samples F2, F10, F20, FA30, F40 and F50, respectively. The residual volumetric strain contracted during the initial loading cycles and then dilated until failure.
85
The lower the confining pressure, the greater the dilatancy of the sample became. With an increase in the confining pressure, the residual volumetric strain increased when dilatancy initiated (see Table 1). With an increase in confining pressure, the residual axial and volumetric strains increased as well. According to the development of the residual axial strain and its increasing velocity, the residual axial strains of sandstone samples obtained at a confining stress state shown in Fig. 9a–f can be described as three deformational stages (see Figure 11a); these stages are the initial phase in which the axial residual deformation first increases and approaches gradually to a constant velocity, the uniform velocity phase in which the axial residual deformation increases at a constant velocity and the accelerated phase in which the axial residual deformation increases rapidly with an increasing velocity until failure (Ge, 2008). The following is evident: the residual deformation increases rapidly during the initial phase; after some cycles, the rate of residual deformation development tends to be stable; near failure, the residual deformation increases rapidly again to eventual failure. Fig. 11b presents the relationship of εa,re/εa,re, max and N/Nmax, which demonstrates that for confining pressures from 2.0 MPa to 50.0 MPa, the residual axial strains exhibit a three phase behavior like that obtained during uniaxial fatigue loading (Fuenkajorn and Phueakphum, 2010).
Deviatoric stress (MPa)
(a) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) 210
Confining pressure:20 MPa; Failured at 629 cycles; Test No. F30 Frequency:1Hz; Dynamic load: 20~320kN.
140
70
Deviatoric stress (MPa)
(a) Dynamic deviatoric stress-axial strain curves(value of the peak/valley)
Axial strain (%)
0 0.0
1.0
2.0
3.0
4.0
4.0
Test No. F30
Axial strain (%)
Lateral strain (%)
2.0
3.0
-3.0
100 50 0 0.0
Test No. F30
Confining pressure:30 MPa; Failured at 629 cycles; Frequency:1Hz; Dynamic load: 20~320kN.
-4.0 Fig. 6. Results of triaxial tests under axial cyclic dynamic loading (Test No. F30) showing (a) dynamic deviatoric stress–axial strain curves (value of the peak/valley), (b) dynamic volumetric strain–axial strain curves (value of the peak/valley) and (c) dynamic lateral strain–axial strain curves (value of the peak/valley).
Axial strain (%)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.3 0.0 0.0
Axial strain (%)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
-0.3 -0.6
Confining pressure:40 MPa; Failured at 241 cycles; Frequency:1Hz; Dynamic load: 20~340kN.
Test No. F40
(c) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) 0.0 0.0
4.0
-1.0 -2.0
150
-0.9
(c) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) 0.0 0.0
Volumetric strain (%)
3.0
Confining pressure:40 MPa; Failured at 241 cycles; Frequency:1Hz; Dynamic load: 20~340kN. Test No. F40
0.6
Lateral strain (%)
Volumetric strain (%)
Axial strain (%)
2.0
200
(b) Dynamic deviatoric stress-axial strain curves(value of the peak/valley)
(b) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) 0.5 0.0 0.0 -0.5 -1.0 -1.5 -2.0 Confining pressure:30 MPa; -2.5 Failured at 629 cycles; -3.0 Frequency:1Hz; Dynamic load: 20~320kN. -3.5
250
Axial strain (%)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
-0.5 Test No. F40
-1.0 -1.5 -2.0 -2.5
Confining pressure:40 MPa; Failured at 241 cycles; Frequency:1Hz; Dynamic load: 20~340kN.
Fig. 7. Results of triaxial tests under axial cyclic dynamic loading (Test No. F40) showing (a) dynamic deviatoric stress–axial strain curves (value of the peak/valley), (b) dynamic volumetric strain–axial strain curves (value of the peak/valley) and (c) dynamic lateral strain–axial strain curves (value of the peak/valley).
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E. Liu, S. He / Engineering Geology 125 (2012) 81–91
4.1. Residual axial strain method
4. Fatigue damage evolution When establishing a fatigue theory for rock materials, one crucial issue is the definition of a damage variable reflecting the fatigue loading. Damage is the creation and growth of microvoids or microcracks that are discontinuities in a medium considered to be continuous at a larger scale. Damage in rock materials is mainly the process of the initiation and growth of microcracks or microvoids. At that scale, the phenomenon is discontinuous, but at a larger scale, damage can be considered to be continuous. Damage can degrade many properties of rock materials, including the elastic modulus, hardness, ultrasonic wave velocity and residual strength (Lemaitre, 1996). No matter the parameter used when describing the damage process, damage evolution must be consistent with the initiation and stable or unstable propagation of microcracks. Therefore, Xiao et al. (2010) proposed that a reasonable damage variable must meet the following basic requirements: 1) it should have a distinct physical meaning; 2) it can be measured easily and conveniently applied in engineering; 3) its evolution law coincides well with the actual degradation process of the material; and 4) it can take the initial damage into account. The following two fatigue damage variables are employed to investigate the fatigue damage evolution of sandstone samples subjected to axial fatigue loads under a confining stress state.
The damage variable can be defined using residual strain as N
f
D ¼ ε a;re =εa;re ;
ð1Þ
N f where εa, re and εa, re are the residual axial strain after N cycles and the ultimate residual axial strain at fatigue failure, respectively (Xiao et al., 2010). We can see that the damage evolution curves show an obvious three-stage developing process in Fig. 12. The initial fatigue damage, which includes the original damage before loading and the resulted damage during static loading, ranges from 0.0717 at a 2.0 MPa confining pressure to 0.6659 at a 50.0 MPa confining pressure. The initial fatigue damage is affected by the stress level and magnitude of residual strain at which fatigue load is applied, so the initial fatigue damage obtained by Eq. (1) may not consider these correctly in some conditions. The residual strain method is proven to be more appropriate because it has a distinct physical meaning and can describe the initial fatigue damage and the degradation process of sandstone samples subjected to fatigue loading under a confining stress state.
4.2. Axial secant modulus method The damage variable can also be defined with residual strain as
(a) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) Deviatoric stress (MPa)
250
Confining pressure:50 MPa; Failured at 347 cycles; Frequency:1Hz; Dynamic load: 20~380kN.
150 Test No. F50
100 50
Axial strain (%)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
(b) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) Volumetric strain (%)
0.8 0.6
Confining pressure:50 MPa; Failured at 347 cycles; Frequency:1Hz; Dynamic load: 20~380kN.
0.4 0.2 Axial strain (%)
0.0 0.0 -0.2
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Test No. F50
-0.4
(c) Dynamic deviatoric stress-axial strain curves(value of the peak/valley) Lateral strain (%)
Axial strain (%)
0.0 0.0 -0.5
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
-2.5
ð2Þ
ð3Þ
where Δstress is the stress difference presented in GPa, and Δstrain is the strain difference obtained from the corresponding peak-valley data under cyclic dynamic conditions; Asd,max is the maximal stiffness, which can be calculated using the stress and strain in the elastic range upon static or axial cyclic loading; and Asd,min is the minimal stiffness which is equal to Asd when failure happens upon fatigue loading. The damage evolution curves calculated using Eq. (3) are shown in Fig. 13, which presents the three-stage developing process of fatigue damage at different confining pressures. The samples have different initial fatigue damage ranging from 0.423 to 0.818, which depends on the stress level when fatigue loads are applied. The axial secant modulus method proposed here has distinct physical meaning and can also describe the initial fatigue damage and the degradation process of sandstone samples subjected to fatigue loading under a confining stress state. 4.3. The effect of stress level on fatigue damage evolution Rs is defined as the stress ratio and calculated as follows:
4.5 q max;dyn 2τstat
ð4Þ
Test No. F50
-1.5 -2.0
Asd ¼ Δstress =Δstrain ;
Rs ¼
-1.0
Asd;max −Asd Asd;max −Asd;min
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, which can be calculated using the following formula (Bagde and Petrosš, 2005):
200
0 0.0
D¼
Confining pressure:50 MPa; Failured at 347 cycles; Frequency:1Hz; Dynamic load: 20~380kN.
Fig. 8. Results of triaxial tests under axial cyclic dynamic loading (Test No. F50) showing (a) dynamic deviatoric stress–axial strain curves (value of the peak/valley), (b) dynamic volumetric strain–axial strain curves (value of the peak/valley) and (c) dynamic lateral strain–axial strain curves (value of the peak/valley).
In Eq. (4), qmax, dyn is the maximal deviatoric stress under axial dynamic loading, and τstat = (σ1 − σ3)/2, in which σ1 and σ3 are the maximal principal stress and the minimal principal stress in Table 2, respectively, is the strength computed using the Tresca Criterion for static triaxial loading at the same confining pressure as the axial dynamic loading. Fig. 14a–b presents Rs plotted versus the initial fatigue damage and the number of cycles at failure, respectively, under 2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa confining pressures. When Rs is
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
(a)
(b) 0.45
Residual axial strain (%)
Residual axial strain (%)
0.35
Test No. F2
0.30
0.25 Confining pressure:2 MPa; Failured at 495 cycles; Frequency:1Hz; Dynamic load: 20~165kN.
0.20
0
100
200
300
400
N
Confining pressure:10 MPa; Failured at 161 cycles; Frequency:1Hz; Dynamic load: 20~230kN.
0.35
0.25 Test No. F10
0.15
500
(c)
N 0
50
100
150
200
(d)
0.90
1.0
Confining pressure:20 MPa; Failured at 257 cycles; Frequency:1Hz; Dynamic load: 20~280kN.
Residual axial strain (%)
Residual axial strain (%)
87
0.70
0.50
0.30 Test No. F20
Confining pressure:30 MPa; Failured at 629 cycles; Frequency:1Hz; Dynamic load: 20~320kN.
0.8 0.6 0.4
Test No. F30
0.2
N 0.10
0
50
100
150
200
250
(e)
N 0
100
200
300
400
500
600
700
(f) Confining pressure:40 MPa; Failured at 241 cycles; Frequency:1Hz; Dynamic load: 20~350kN.
1.5
3.0
Residual axial strain (%)
2.0
Residual axial strain (%)
0.0
300
Test No. F40
1.0
0.5
N 0.0
0
50
100
150
200
250
Confining pressure:50 MPa; Failured at 347 cycles; Frequency:1Hz; Dynamic load: 20~380kN.
2.0
1.0 Test No. F50
N
0.0 0
100
200
300
400
Fig. 9. Residual axial strain–N curves (N: the number of cycles) for (a) Test No. F2, (b) Test No. F10, (c) Test No. F20, (d) Test No. F30, (e) Test No. F40, and (f) Test No. F50.
large, the initial fatigue damage is greater, and the sample fails after only a small number of cycles under confining pressure conditions. From the fatigue damage evolution shown in Figs. 12 and 13, we can draw the conclusion that the axial secant modulus method can consider the influence of Rs on the damage evolution of rock samples under axial cyclic dynamic loading but the residual axial strain method cannot. 4.4. The fatigue failure mechanisms When subjected to static loads at confining pressures of 2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa, the sandstone specimens failed with clearly defined planes of fracture. The angle formed between the fracture plane and the plane of the maximum principal stress was greater than 45°. These planes are referred to as shear fractures and are
characterized by shearing displacement along their surfaces. From the failure patterns of samples S2, S10, S20, S30 and S40 shown in Fig. 15, we can conclude that the lower the confining pressure, the larger the angle formed at failure between the horizontal plane, where the maximum principal stress was applied, and the plane of the shear fracture. When the confining pressure was 2.0 MPa, the sample failure was accompanied by longitudinal splitting, but when the confining pressure was more than 10 MPa, the sample failure was accompanied by lateral expansion. When subjected to dynamic cyclic loads at a frequency of 1.0 Hz and confining pressures of 2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa, shear fractures also formed, as shown in Fig. 15. Compared with static loading under the same confining stress, when the confining pressures were either 2.0 or 10.0 MPa, the shear fracture planes were almost the same for cyclic dynamic loading, but when the confining
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E. Liu, S. He / Engineering Geology 125 (2012) 81–91
(a) 0.1 0
100
200
400
300
500
-0.2
Confining pressure:2 MPa; Failured at 495 cycles; Frequency: 1Hz; Dynamic load: 20~165kN.
-0.5
Test No. F10
0.2
N
N 0.0
Residual volumetric strain (%)
Residual volumetric strain (%)
(b)
Test No. F2
0
30
60
90
120
150
180
-0.2 -0.4 -0.6
Confining pressure:10 MPa; Failured at 161 cycles; Frequency: 1Hz; Dynamic load: 20~230kN.
-0.8
-0.8 -1.0
(c)
(d) Test No. F20
0.2
N 0.0 0
40
80
120
160
200
240
280
-0.2 -0.4
Confining pressure:20 MPa; Failured at 257 cycles; Frequency: 1Hz; Dynamic load: 20~280kN.
-0.6 -0.8
Residual volumetric strain (%)
Residual volumetric strain (%)
0.2
N 0.0 0
100
200
300
400
500
600
700
-0.2
-0.4
Confining pressure:30 MPa; Failured at 629 cycles; Frequency: 1Hz; Dynamic load: 20~320kN.
-0.6
-0.8
-1.0
(e)
(f) 0.3
0.4
0.2
Test No. F40
0.1
N
0.0 0
50
100
150
200
250
-0.1 -0.2 -0.3
Confining pressure:40 MPa; Failured at 241 cycles; Frequency: 1Hz; Dynamic load: 20~350kN.
-0.4
Residual volumetric strain (%)
Residual volumetric strain (%)
Test No. F30
Test No. F50 0.3 0.2 0.1 0.0 0
Confining pressure:50 MPa; Failured at 347 cycles; Frequency: 1Hz; Dynamic load: 20~380kN.
100
200
N 300
400
-0.1 -0.2
Fig. 10. Residual volumetric strain–N curves (N: the number of cycles) for (a) Test No. F2, (b) Test No. F10, (c) Test No. F20, (d) Test No. F30, (e) Test No. F40, and (f) Test No. F50.
pressures were greater than 10.0 MPa, the shear fractured planes were wider upon cyclic dynamic loading. The reason for this phenomenon is that when the confining pressure is lower, the brittleness of the sandstone specimens contributes to the formation of fracture planes and when the confining pressure is higher, they become less brittle, resulting in the formation of shear fracture bands upon dynamic cyclic loadings. 5. Conclusions From these tests of sandstone samples performed at 2.0, 10.0, 20.0, 30.0, 40.0 and 50.0 MPa confining stress states with axial cyclic
dynamic loading at a 1.0 Hz frequency, the following conclusions may be drawn: (1) With increased confining pressure, the axial strain at failure increased. When dilatancy took place, the volumetric strains of the samples at 2.0, 10.0 and 20.0 MPa confining pressures were lower for cyclic dynamic loading conditions but were higher for samples at confining pressures of 30.0, 40.0 and 50.0 MPa for cyclic dynamic loading conditions (2) 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
E. Liu, S. He / Engineering Geology 125 (2012) 81–91
89
1.0
(a) Residual axial strain
F10
F40
0.9 F20
Test No. F2
F50
Damage, D
0.8
Stage I:initialphase Stage II:uniform velocity phase Stage III:accelerated phase
F30
0.7 0.6 0.5 0.4
N
0
100
200
300
400
500
600
700
Fig. 13. D–N curves (N: the number of cycle) using the axial secant modulus method.
Stage I
Stage II
Stage III
0
(b)
1.0 Test No. F2
0.8
εa,re/εa,re, max
an increase in the confining pressure, the residual volumetric strain increased when dilatancy occurred (4) The residual axial strains of sandstone samples obtained at a confining stress state could be described as three deformational stages: the initial phase, the uniform velocity phase and the accelerated phase (5) The residual strain method was proved more appropriate for its distinct physical meaning and ability to describe the initial fatigue damage and the degradation process of sandstone samples subjected to fatigue loading under a confining stress state. Additionally, the axial secant modulus method proposed here had distinct physical meaning and also described the initial fatigue damage and the degradation process of sandstone samples subjected to fatigue loading under a confining stress
N
0.6 Test No. F10
0.4
Test No. F20
0.2
Test No. F30 Test No. F40 Test No. F50
0.0 0.0
0.2
0.4
N/Nmax
0.6
0.8
1.0
(a)
Fig. 11. Plot showing (a) the three phases of residual axial strain (N: the number of cycles) and (b) the relationship of εa,re/εa,re, max ~ N/Nmax (εa,re is residual axial strain, εa,re, max the maximum of residual axial strain, N the number of cycles and Nmax the maximum of N ).
1.00
including axial, volumetric and lateral strains, whose magnitudes became greater (3) 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 became. With
0.97
0.99
Rs
0.98
0.96
Rs =
qmax,dyn 2τstat
Test No.F2 Test No.F10 Test No.F20 Test No.F30 Test No.F40 Test No.F50
0.95 0.94 0.0
0.2
0.4
0.6
0.8
1.0
Initial fatigue damage, D0 1.0
(b)
Test No. F2
0.8
1.00 0.99 F10
0.98 F30
0.4
Rs
Damage, D
0.6
0.97
Rs =
qmax,dyn 2τstat Test No.F2 Test No.F10 Test No.F20 Test No.F30 Test No.F40 Test No.F50
0.96
F20
0.95 F40
0.2
0.0
F50
0.94 100.0
0
100
200
300
400
500
600
1,000.0
The number of cycle at failure Nf
N
700
Fig. 12. D–N curves (N: the number of cycle) using the residual axial strain method.
Fig. 14. The influence of Rs on (a) initial fatigue damage and (b) the number of cycles at failure.
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E. Liu, S. He / Engineering Geology 125 (2012) 81–91
(a)
(b)
Test No.S2
Test No.F2
(c)
Test No.S10
Test No.F10
(d)
Test No.S20
Test No.F20
Test No.S30
Test No.F30
(e)
Test No.S40
Test No.F40
Fig. 15. Pictures of samples at failure with a confining pressure of (a) 2.0 MPa, (b) 10.0 MPa, (c) 20.0 MPa, (d) 30.0 MPa and (e) 40.0 MPa.
state. Comparing these two methods, the axial secant modulus method considers the influence of level stress when fatigue loads applied, while the residual axial strain method does not (6) The shear fracture planes can form under both static and cyclic dynamic loadings. When the confining pressure is lower, the shear fracture planes formed in the sandstone specimens are almost the same under both static and cyclic dynamic loading conditions. However, when the confining pressure is higher,
the shear fracture planes formed are wider under cyclic dynamic loading conditions than under static loading conditions. Acknowledgments The authors thank the two anonymous reviewers and the editor for their careful review, contributions and critiques, which led to the improvement of the manuscript. The authors also thak the
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