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Mechanical and damage evolution properties of sandstone under triaxial compression
International Journal of Mining Science and Technology xxx (2016) xxx–xxx
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International Journal of Mining Science and Technology journal homepage: www.elsevier.com/locate/ijmst
Mechanical and damage evolution properties of sandstone under triaxial compression Zong Yijiang a,b,⇑, Han Lijun a, Wei Jianjun b, Wen Shengyong a a b
State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China Jiangsu Collaborative Innovation Center for Building Energy Saving and Construction Technology, Jiangsu Jianzhu Institute, Xuzhou 221116, China
a r t i c l e
i n f o
Article history: Received 3 November 2015 Received in revised form 12 January 2016 Accepted 5 March 2016 Available online xxxx Keywords: Rock mechanics Mechanical properties Dilatation Damage evolution Failure characteristics
a b s t r a c t To study the mechanical and damage evolution properties of sandstone under triaxial compression, we analyzed the stress strain curve characteristics, deformation and strength properties, and failure process and characteristics of sandstone samples under different stress states. The experimental results reveal that peak strength, residual strength, elasticity modulus and deformation modulus increase linearly with confining pressure, and failure models transform from fragile failure under low confining pressure to ductility failure under high confining pressure. Macroscopic failure forms of samples under uniaxial compression were split failure parallel to the axis of samples, while macroscopic failure forms under uniaxial compression were shear failure, the shear failure angle of which decreased linearly with confining pressure. There were significant volume dilatation properties in the loading process of sandstone under different confining pressures, and we analyzed the damage evolution properties of samples based on acoustic emission damage and volumetric dilatation damage, and established damage constitutive model, realizing the real-time quantitative evaluation of samples damage state in loading process. Ó 2016 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
1. Introduction Due to the constant movement and development of the crust, there are inevitably a large number of initial defects such as micropores, joints and cracks in the rock mass. The deformation and damage of the rock mass, along with the propagation, development and connection of original cracks, is strongly affected and restricted by internal initial defects, especially structural planes with joints scale [1,2]. Therefore, research on fracture damage evolution progress and the characteristics of rock under the condition of triaxial compression is of great significance to mining engineering, tunnel engineering and geotechnical engineering. Damage is a phenomenon whereby micro defects in a material under monotonic loading or reloading leads to a progressive decrease in the cohesion and damage of volume units. Many scholars at home and abroad have carried out systematic research into the damage evolution characteristics and constitutive models of rock, and have achieved some remarkable results in this field. Lu et al. [3] studied the complete stress–strain curve characteristics of marble under triaxial compression, and established the bilinear
⇑ Corresponding author. Tel.: +86 13685188146. E-mail address: [email protected] (Y. Zong).
elastic-linear strain-softening residual ideal plastic damage constitutive model. Ren [4] studied the damage evolution laws of coal and the rock mass using a computerized tomography triaxial loading system, realizing the quantitative evaluation of the damage state. Jin et al. [5] studied damage evolution of rock under uniaxial compression and built the coal-rock damage evolution model which considered residual strength based on electromagnetic radiation characteristics. Zhang et al. [6] studied the deformation and failure mechanism of strong weathered sandstone by triaxial compression testing, and analyzed the damage evolution processes and established damage evolution equations based on the density method. Li et al. [7] studied the microscopic damage characteristics of siltstone under triaxial compression by scanning electron microscopy and digital image technology, and analyzed the statistical distribution characteristics of azimuth angle, length and width of cracks. Zhou et al. [8] studied the strength, deformation and fracture damage characteristics of sandstone by uniaxial cyclic loading and unloading tests, and defined a damage variable based on linear damage mechanics theory and acoustic emission (AE). Cao et al. [9] established a new damage model and a statistical damage constitutive model of rock under specific confining pressure according to the force of the damaged and undamaged parts in rock on the base of statistical damage theory. Li et al. [10] studied the damage evolution characteristics in the whole uniaxial compression
http://dx.doi.org/10.1016/j.ijmst.2016.05.011 2095-2686/Ó 2016 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
Please cite this article in press as: Zong Y et al. Mechanical and damage evolution properties of sandstone under triaxial compression. Int J Min Sci Technol (2016), http://dx.doi.org/10.1016/j.ijmst.2016.05.011
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Y. Zong et al. / International Journal of Mining Science and Technology xxx (2016) xxx–xxx
process of sandstone by electrical resistivity and AE, and proposed a status qualitative criterion for rock damage. The indicators such as AE, CT values, density, electrical resistivity and crack length were adopted to evaluate the degree of damage and to study the damage evolution laws of rock in related research. However, research on the damage evolution process of rock on the basis of the dilatation properties is less well developed. Therefore, in this paper, the deformation and failure process, mechanical properties and evolution laws of sandstone under different stress states by uniaxial and triaxial compression tests are described, damage evolution based on AE and dilatation properties is analyzed, and a damage constitutive model, realizing real-time and quantitative evaluation of the damage state in the whole compression process, is established. 2. Experimental 2.1. Equipment and sample preparation Uniaxial and triaxial compression tests of sandstone were carried out on a TAW-2000 rock mechanics electro-hydraulic servo testing system, having a maximum axial force of 2000 kN, maximum confining pressure of 60 MPa, axial deformation measurement range from 0 to 10 mm, and radial deformation measurement range from 0 to 5 mm, and which can be used for uniaxial compression tests, uniaxial compression creep tests, triaxial compression tests and triaxial compression creep tests. Red sandstone, taken from Linyi of Shandong province, has a fine, blocky structure, good homogeneity, with a main mineral composition of quartz 17%, feldspar 42%, andesitic debris 25%, cements 15% and zircons 1%. The diameter distribution of the sandstone is: 0.10–0.25 mm 60%, 0.25–0.50 mm 35% and 0.50–1.00 mm 5%. The cementing type of the sandstone is pore cementation with grain point-line contact. Saturated red sandstone was processed into standard samples with diameters of 50 mm and height of 100 mm according to the standard testing method of Methods for Determining the Physical and Mechanical Properties of Coal and Rock (GB/T23561.1-2009) and ISRM. Sample specifications are shown in Table 1. 2.2. Schemes and process The confining pressure of triaxial compression tests were 5, 10, 15, 20 and 30 MPa respectively. The experimental process was as follows. The confining pressure was loaded to the set value at a rate of 0.05 MPa/s after the installation of samples. With a constant confining pressure, axial pressure was increased by the displacement control method at a loading rate of 0.002 mm/s until failure of the sample occurred. The axial pressure loading rate of the T-6
sample after peak strength was reached was 0.001 mm/s. The loading control methods of samples are shown in Table 1. 3. Results and discussion Fig. 1 shows the complete stress–strain curves of sandstone samples under uniaxial and triaxial compression; experimental results are shown in Table 2. 3.1. Characteristics of complete stress–strain curves As shown in Fig. 1, complete stress–strain curves of sandstone samples under uniaxial compression can be divided into five stages, including fissure compression, elastic deformation, steady crack propagation, unsteady fracture propagation and a strainsoftening stage. The residual deformation stage is not included. Fig. 1 shows good elastic property in the pre-peak stage of the stress–strain curves, and the peak point and strain-softening stage can only hold for a remarkably short time, presenting the characteristics of brittleness failure. There is no obvious fissure compression stage in the complete stress–strain curves of the sandstone samples under triaxial compression. Deviatoric stress increases linearly with increasing axial strain, radial strain and volumetric strain in the stress–strain curves up to the yield point, and the slope of the curves increases with increasing confining pressure. After the yield point, with increasing axial stress, new cracks gradually initiate in the samples with increasing axial stress, with further propagation of the original cracks and energy dissipation; this leads to the up-convex morphology of the stress–strain curves, a decrease in the stress–strain curve slope and an increase in plastic deformation. When the deviatoric stress reaches the failure strength, new macroscopic cracks initiate in the samples leading to failure of the samples, which shows the characteristics of a sharp increase in axial strain, radial strain and volumetric strain, and a decrease in sample strength. With increasing confining pressure, the slopes of the stress–strain curves, peak strength and residual strength have a great improvement, the samples at the strain-softening stages have smoother stress–strain curves, showing certain behavior of plastic deformation. The residual strength of the samples under a confining pressure of 30 MPa and a loading rate of 0.001 mm/s is higher, and the slopes of the stress–strain curves are becoming smaller at the strain-softening stage, showing clear behavior of plastic deforma-
σ 1 − σ 3 (MPa) 180
150
Table 1 Sample specification and loading control methods.
120
Sample No.
Diameter (mm)
Height (mm)
Test type
Loading rate (mm/s)
U-1 U-2 U-3
49.92 49.72 49.58
100.38 100.62 100.12
Uniaxial compression
0.002
T-1 T-2 T-3 T-4 T-5 T-6
49.62 49.72 49.72 49.62 49.60 49.72
100.28 99.98 99.94 100.40 101.48 100.80
Triaxial compression
0.002
Pre-peak 0.002 Post-peak 0.001
60
30
σ 3 =0 MPa
-80 -60 ε 3 (10-3)
σ 3 =30 MPa σ 3 =20 MPa σ 3 =15 MPa σ 3 =10 MPa σ 3 =5 MPa
90
σ 3=30 MPa σ 3 =20 MPa σ 3 =15 MPa σ 3=10 MPa σ 3=5 MPa
-40
-20
σ 3 =0 MPa
0
20
40 ε 1 (10-3)
Fig. 1. Complete stress–strain curves of sandstone samples in uniaxial and triaxial compression tests.
Please cite this article in press as: Zong Y et al. Mechanical and damage evolution properties of sandstone under triaxial compression. Int J Min Sci Technol (2016), http://dx.doi.org/10.1016/j.ijmst.2016.05.011
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Y. Zong et al. / International Journal of Mining Science and Technology xxx (2016) xxx–xxx Table 2 Uniaxial and triaxial compression test results of sandstone samples. Sample No.
Confining pressure (MPa)
Peak strength (MPa)
U-1 U-2 U-3 T-1 T-2 T-3 T-4 T-5 T-6
0 0 0 5 10 15 20 30 30
70.14 69.19 68.34 126.54 137.69 155.94 171.53 176.77 177.24
Residual strength (MPa)
Elastic modulus (GPa)
Deformation modulus (GPa)
Peak strain (103) Axis strain
Radial strain
Volumetric strain
58.14 59.73 70.79 76.15 81.55 100.20
11.80 11.84 11.87 14.14 14.67 16.06 16.28 17.06 17.12
7.11 8.22 8.83 14.76 15.00 16.99 17.15 18.56 18.85
8.26 7.85 7.16 10.56 11.31 12.18 13.63 14.11 14.72
11.81 14.14 11.98 19.86 12.63 16.77 15.41 20.16 20.13
15.36 20.45 16.79 29.17 13.91 21.36 17.19 26.21 25.54
tion. It is 3.2 times the duration of the samples under a loading rate of 0.002 mm/s. Under a confining pressure of 30 MPa, accurate control of loading and unloading tests at a certain stress state can be achieved.
3.2. Failure characteristics Figs. 2 and 3 show the failure modes of samples under uniaxial and triaxial compression respectively. Here we can see in Figs. 2 and 3 that the macroscopic fracture models of sandstone samples under triaxial compression are mainly split failure parallel to the axial direction of the samples, although some shear failure surfaces and lateral failure surfaces also exist, which shows that there are several forces inside the samples leading to two main models of axial split failure and shear failure. The macroscopic fracture models of sandstone samples in triaxial compression tests under different confining pressures are mainly shear failure. There is only one main shear surface throughout the whole sample after failure, thereby dividing the sample into two parts. From the two failure surfaces, we can see many scratches and the appearance of rock powder which is formed by the secondary shear failure of uneven parts on the surface in the shear slip process.
(a) 5 MPa
(d) 20 MPa
(b) 10 MPa
(e) 30 MPa
(c) 15 MPa
(f) Fracture morphology
Fig. 3. Failure modes of samples under triaxial compression.
3.3. Strength properties
(a) U-1
(b) U-2
(c) U-3
Table 2 shows that the peak strengths of the three samples under uniaxial compression are 70.14, 69.19 and 68.34 MPa, the degree of dispersion being only 2.60%, showing the good homogeneity of the samples. The selected rock thus easily achieves the requirement for tests. The peak strength and residual strength of samples under triaxial compression increase linearly with increasing confining pressure. Multiple regression analysis shows that the influence coefficient of confining pressure on peak strength is 3.064, and 1.302 on residual strength, illustrating the higher sensitivity of peak strength to confining pressure, which shows good agreement with those of related experiments [11]. 3.4. Deformation properties
(d) Fracture morphology
Fig. 2. Failure modes of samples under uniaxial compression.
(1) Elastic modulus and deformation modulus The elastic modulus E (the slope of the straight-line portion of deviatoric stress-axial strain curves), and deformation modulus (the slope of the connecting line between the point of 50% deviatoric stress and origin on the deviatoric stress-axial strain curves), increase linearly with increasing confining pressure. The influence
Please cite this article in press as: Zong Y et al. Mechanical and damage evolution properties of sandstone under triaxial compression. Int J Min Sci Technol (2016), http://dx.doi.org/10.1016/j.ijmst.2016.05.011
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Y. Zong et al. / International Journal of Mining Science and Technology xxx (2016) xxx–xxx 20
ε (10 -3)
10
0
-10
ε 1 =0.202σ 3 +9.400 R =0.987
5
10 ε 3 =0.0225σ
15 σ 3 (MPa)
20
25
3 2 3 0.902σ 3 +11.043σ3 −55.33
R=1
-20 -30
ε v =0.046σ 33 1.827σ 23 +22.453σ 3 −101.470 R=1
Fig. 4. Relationship between peak strain and confining pressure.
coefficients of confining pressure on elastic modulus and deformation modulus are 0.10 and 0.16 respectively, which shows that the internal structure of the samples is relatively dense, and the closure of micropores and microfractures leads to the slight increase in elastic modulus and deformation modulus. (2) Peak strain and residual strain The peak strain of samples under triaxial compression is closely related to the confining pressure. Samples under uniaxial compression and triaxial compression with low confining pressure will undergo brittle failure after small deformation. With increasing confining pressure, the peak strain of samples increases, and the failure models of samples transform from brittle failure to ductility failure. The axial peak strain and residual strain of samples increase linearly with increasing confining pressure (Fig. 4). However, the radial and volumetric peak strain and residual stain present a non-linear relationship with confining pressure due to the double effects of axial stress and confining pressure. (3) Dilatation properties Dilatation refers to the phenomenon that the volume of samples under uniaxial stress or unequal triaxial stress turns from a compression state to a dilatation state due to the initiation and propagation of cracks in samples under increasing stress [12]. The failure processes in many geotechnical engineering applications are accompanied by the dilatation phenomenon of the rock mass. Therefore, research into the dilatation properties of rock is of great significance to rock mass stability. The nonlinear deformation of samples under uniaxial and triaxial compression mainly includes elastic deformation, deformation caused by the closure of initial cracks, deformation caused by the initiation and propagation of cracks, shear slip and broken- expansion deformation after failure. The nonlinear deformation of samples in the initial loading stage is caused mainly by the closure of initial cracks, and by the initiation and propagation of cracks, and shear slip under high stress. Fig. 5 shows the relationship between radial strain, volumetric strain and axial strain of sandstone samples under different confining pressures. As shown as in Fig. 5, the axial strain, radial strain and volumetric strain of samples increases linearly with increasing axial stress before the yield point, and samples are under compression. The samples gradually turn from a state of compression to a state of dilatation due to the faster increase in the rate of radial strain. The volumetric strain will reach a maximum when the deviatoric stress is about 5.90–34.42% of the peak strength. With the initiation and propagation of cracks in the samples, the volumetric strain will gradually decrease, and the samples turn from compres-
sion to dilatation states, showing obvious dilatation properties. After the yield point, the radial and volumetric strains of sandstone samples increase sharply with increasing axial stress. Volumetric deformation can be divided into four stages, including: a volumetric compression stage, a pre-peak tiny volumetric dilatation stage, a post-peak rapid-increase stage and a steady-increase stage. There is remarkable shear slip and broken-expansion deformation in samples post-peak, showing significant dilatation properties. The dilatation deformation of samples pre-peak under uniaxial compression is 8.79–9.97% of the total, and 21.70–45.66% of the total under triaxial compression. However, the dilatation deformation post-peak is up to 54.34–91.21% of the total. 4. Damage evolution properties 4.1. Damage evolution properties based on AE There is some local deformation and micro fracturing in rock under external load or internal force, and the instantaneous strain energy rapidly releases in the form of elastic wave, resulting in the AE phenomenon [13,14]. The AE technique is a new unconventional dynamic detection method, which can continuously monitor damage and deformation in rock. Rock failure leads to the initiation and propagation of cracks and the formation of the macro structure surface. AE is the external evidence for the evolution characteristics of cracks, reflecting the damage which occurs in the rock when subjected to an external force. Therefore, the damage evolution laws of rock during the loading process can be studied using AE [15–18]. The damage evolution property of rock under uniaxial compression was studied using a PXWAE waveform acoustic emission detector. The AE sensor was a PXR15, with a bandwidth of 100–300 kHz and a sensitivity of 65 dB. The pre-amplifier was a PXPAI low noise amplifier, with a gain of 40 dB and a noise voltage of 2.7 uVRMS. The AE acquisition card had four AE channels and a PCI interface, with a sampling precision of 12 bits. The AE of the rock during the failure process obeys the statistical distribution of internal defects.N m and N d are respectively regarded as the accumulated acoustic emission ringing count when the whole cross-section fails completely and during failure of the rock. The damage variable D can be expressed as:
D¼
m Nd F ¼ 1 exp F0 Nm
ð1Þ
where F is a random variable of the Weibull distribution, m and F 0 are Weibull distribution parameters. Eq. (1) shows significant consistency between the damage variable and the accumulated acoustic emission ringing count. Therefore we can use the accumulated acoustic emission ringing count to express the damage variable of samples under uniaxial compression. The statistical damage constitutive equation can be obtained by introducing the Drucker–Prager failure criterion as the failure criterion of microelements.
r ¼ Eeemðec Þ 1
e m
ð2Þ
where r, e and E are axial stress, axial strain and elasticity modulus respectively. The damage evolution equation of samples under uniaxial compression can be expressed as: e m
D ¼ 1 emðec Þ 1
ð3Þ
Fig. 6 shows the damage evolution curve drawn according to Eq. (3) and a comparison of the theoretical and test curves.
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Y. Zong et al. / International Journal of Mining Science and Technology xxx (2016) xxx–xxx
120
120
90 40 60
30 20
Volumetric dilatation starting point
10
50
(σ 1 − σ 3 ) − ε 1 ε 3 −ε 1 ε v −ε 1
40 30
90
20 60
Volumetric dilatation starting point
30
10
30 0
0
0
2
4
6
-3
σ 1 (MPa)
50
150
ε (10 )
60
150
σ 1 − σ3 (MPa)
σ1 ε1 ε3 ε1 εv ε1
ε (10-3)
70
8
10
12
14
0
5
10
ε 1 (10 -3 )
15
20
25
ε 1 (10 -3)
(a) U-3 (σ3=0 MPa)
(b) T-2 (σ3 =10 MPa)
Fig. 5. Relationship between radial strain, volumetric strain and axial strain of sandstone samples.
70
is defined as the ratio of volumetric dilatation strain at some point in the strain–stress curves to the volumetric dilatation strain of the initial residual strength stage. The damage variable D can be expressed as:
1.0
Test curve
60
0.8 0.6
40
D
σ 1 (MPa)
50
30 Theoretical curve
20
0
AE demage curve 2
4
6
e1 6 e1d e1 > e1d
D¼0 D ¼ eevvreev d
vd
0.2 10
(
0.4
8
10
0 12
14
ε1 (10 -3 )
Fig. 6. Comparison of theoretical curve and test curve.
It can be seen in Fig. 6 that there is hardly any damage in samples at the fissure compression and elastic deformation stages, with the damage variable D ¼ 0. When the axial stress reaches 72.4% of the peak strength, the internal cracks of samples begin to initiate and propagate, resulting in obvious AE phenomenon, and damage of the samples. So, 0:724rc can be regarded as the damage threshold stress of samples under uniaxial compression. When the axial stress reaches the peak stress, the internal cracks propagate and connect continuously to form macro fractures resulting in accelerating damage of samples, and the damage variable rapidly increases to 1. The theoretical curves based on the AE damage constitutive model have coherence with the test curves, showing that the AE damage constitutive model can reflect the damage evolution properties of samples under uniaxial compression. Because of the continuous function of the damage constitutive model and fixed elastic modulus, the theoretical curves fail to fully reflect fissure compression and post-peak brittle failure of samples under uniaxial compression, having some deviations between theoretical and test curves.
ð4Þ
where D, ev , evd , ev r , e1 and e1d are damage variable, volumetric strain, maximum compressive volumetric strain, volumetric strain of initial residual strength stage, axial strain and axial strain corresponding to the maximum compressive volumetric strains respectively. The complete failure of samples under uniaxial and triaxial compression has not occurred at the residual strength stage, and the samples still have great residual strength. So Eq. (4) can be corrected as:
(
e1 6 e1d e1 > e1d
D¼0 D¼
Kðev ev d Þ ev r ev d
ð5Þ
where K is a correction factor.
K ¼ 1 rr =rp
ð6Þ
where rr and rp are residual strength and peak strength. The damage variable of sandstone samples under different confining pressures can be obtained by substituting Eq. (6) into (5). Fig. 7 shows a comparison of AE damage with volumetric dilatation damage under uniaxial compression. As shown in Fig. 7, volumetric dilatation damage has preferably coherence with AE damage. Therefore it is possible to determine the damage variable of samples in the whole loading process.
Acoustic emission damage Volumetric dilatation damage
4.2. Damage evolution properties based on dilatation
70
1.0
60 0.8 50 0.6
40
D
σ 1 (MPa)
The samples show obvious dilatation properties under uniaxial and triaxial compression. Dilatation is the result of the initiation and propagation of micro fractures in rock, and reflects the damage in rock under external load. Therefore, the damage evolution laws of rock during the whole loading process, based on the dilatation properties, can be studied. The nonlinear deformation of samples in the initial loading stage is caused mainly by the closure of initial cracks, and by the initiation and propagation of cracks, and shear slip under high stress. So we can consider the maximum compressive volumetric strains evd as the volumetric dilatation starting point, and assume that there is no damage before that point. The damage variable D
30
0.4
20 0.2
10 0
2
4
6
8
10
12
14
0
ε 1 (10 -3 )
Fig. 7. AE damage and volumetric dilatation damage of sandstone samples under uniaxial compression.
Please cite this article in press as: Zong Y et al. Mechanical and damage evolution properties of sandstone under triaxial compression. Int J Min Sci Technol (2016), http://dx.doi.org/10.1016/j.ijmst.2016.05.011
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Y. Zong et al. / International Journal of Mining Science and Technology xxx (2016) xxx–xxx
The three-dimensional damage constitutive model of rock material can be obtained based on the strain equivalence hypothesis.
r1 ¼ Ee1 þ 2lr3 e1 6 e1d r1 ¼ Ee1 ð1 DÞ þ 2lr3 e1 > e1d
ð7Þ
Fig. 8 shows a comparison of theoretical curves based on the damage constitutive model in Eq. (7) with the test curves. Prepeak theoretical curves based on the dilatation damage constitutive model are in good agreement with the test curves. The theoretical curves at the peak point and post-peak reflect the strainsoftening properties of samples to some extent, but there are some increasing deviations with confining pressure between theoretical curves and test curves. Therefore we can introduce the porosity n into the constitutive equation to correct the damage constitutive model. Rock mass under the action of load can be abstracted into an object composed of pores, a damaged part and an undamaged part. Here we assume that the undamaged part will bear the whole load, without considering the bearing load of pores and the damaged part [19,20].
8 > < A0 ¼ nA A ¼ A0 þ A1 þ A2 > : 0 ri A1 ¼ ri A ¼ ri ðA0 þ A1 þ A2 Þ
ð8Þ
where ri and r0i are the stress in the rock mass and the effective stress of the undamaged part respectively, A, A0 , A1 and A2 action area of ri , action area of pore, action area of r0i and action area of the damaged part, i ¼ 1; 2; 3. If the damage variable is defined as:
D ¼ A2 =ðA1 þ A2 Þ
σ 1−σ3 (MPa)
ð10Þ
It is assumed that the undamaged part of the rock mass obeys the generalized Hook’s law. The constitutive equation of the undamaged part can be expressed as:
r0i ¼ E0 e0i þ l0 ðr0j þ r0k Þ
ð11Þ
where E0 , l0 and e0i are the elastic modulus of the undamaged part, Poisson’s ratio and micro strain respectively, i, j, k ¼ 1; 2; 3. Considering the deformation coordination of the rock mass during the loading process, we can get:
150 120
T-6 (σ 3=30 MPa) 90
T-2 (σ 3 =10 MPa) 30
0
U-3 (σ 3 =0 MPa) 5
10
15
20
Fig. 9. Comparison of correctional volumetric dilatation damage theoretical curves and test curves.
ei ¼ e0i l ¼ l0
Test curve Theoretical curve
(
ri ¼ P=A ¼ Eei þ lðrj þ rk Þ ¼ Eei r0i ¼ P=A3 ¼ E0 e0i þ l0 ðr0j þ r0k Þ ¼ E0 e0i
r ¼ P=½ð1 nÞA ¼ E0 ei
σ 1 (MPa)
T-2 (σ 3 =10 MPa)
50
E ¼ ð1 nÞE0
r ¼ Eei =ð1 nÞ þ lðr0j þ r0k Þ
10
15
20
25
ε 1 (10 ) -3
Fig. 8. Comparison of volumetric dilatation damage theoretical curves and test curves.
ð16Þ
With the simultaneous Eqs. (10) and (15), the dilatation damage constitutive equation of the rock material can be obtained as follows:
ð17Þ
The porosity of rock under triaxial compression can be expressed as:
E0 ð1 n0 Þð1 DÞ þ 2r3 ð1 l 2l2 Þ E0 ð1 DÞ½1 ð1 2lÞe1
ð18Þ
where n0 is the initial porosity of the rock. By assuming an initial porosity of zero (n0 ¼ 0), Eq. (18) can be simplified as follows:
n¼1 5
ð15Þ
Substituting Eqs. (12) and (15) into (11): 0 i
U-3 (σ 3 =0 MPa)
0
ð14Þ
With the simultaneous Eqs. (13) and (14), we get the relationship between E and E0 .
T -6 (σ 3 = 30 MPa)
100
ð13Þ
Substituting Eqs. (8) and (12) into (13): 0 i
n¼1
150
ð12Þ
where l and ei are macro Poisson’s ratio and macro strain respectively, i ¼ 1; 2; 3. If the rock mass is abstracted into the object composed of pore and rock material, the action area of rock material is A3 , the deformation of the rock mass and rock material under load P are ei and e0i respectively. The corresponding elastic moduli are E and E0 , and Poisson’s ratios are l and l0 . The constitutive equation of rock mass and rock material can be expressed as follows:
ri ¼ E0 ei ð1 nÞð1 DÞ þ lðrj þ rk Þ 200
25
ε 1 (10-3)
By substituting Eq. (9) into (8), the damage constitutive equation can be corrected as follows:
ri ¼ ð1 nÞð1 DÞr
Test curve Theoretical curve
60
ð9Þ
0 i
180
E0 ð1 DÞ þ 2r3 ð1 l 2l2 Þ E0 ð1 DÞ½1 ð1 2lÞe1
ð19Þ
The three-dimensional constitutive equation of rock material can be obtained by substituting n and D into Eq. (17).
r1 ¼ E0 e1 ð1 nÞ þ 2lr3 e1 6 e1d 0 r1 ¼ E e1 ð1 nÞð1 DÞ þ 2lr3 e1 > e1d
ð20Þ
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Fig. 9 shows the comparison of theoretical curves, based on the correctional damage constitutive model, with the test curves. The theoretical curves based on the dilatation damage constitutive model are in good agreement with the test curves, reflecting the deformation properties and damage evolution process of sandstone samples under different confining pressures. 5. Conclusions (1) The peak strength, residual strength, elastic modulus and deformation modulus increase linearly with increasing confining pressure, and failure models transform from fragile failure to ductility failure. (2) Before the yield point, axial strain, radial strain and volumetric strain of samples increases linearly with increasing axial stress and the samples gradually turn from compression states to dilatation states due to the faster rate of increase of radial strain when the deviatoric stress is about 10.47– 62.64% of the peak strength. After the yield point, there is noticeable shear slip and broken-expansion deformation in samples post-peak, showing significant dilatation properties. The dilatation deformation pre-peak is 21.70–45.66% of the total, and 54.34–78.30% of the total post-peak. (3) The macroscopic fracture models of sandstone samples under triaxial compression are mainly split failure parallel to the axial direction of the samples, but some shear failure surfaces and lateral failure surfaces also exist. The macroscopic fracture models of sandstone samples in triaxial compression tests under different confining pressures are mainly shear failure. (4) Based on acoustic emission damage and volumetric dilatation, the damage evolution properties of samples were analyzed, and damage constitutive models were established, realizing the real-time quantitative evaluation of samples damage state in loading process.
Acknowledgments Financial support for this work, provided by the National Natural Science Foundation of China (Nos. 51323004 and 51574223), the Postdoctoral Science Foundation of China (No. 2015M571842) and the Open Research Fund of Research Center of Jiangsu Collaborative Innovation Center for Building Energy Saving and Construction Technology (No. SJXTY1502), are gratefully acknowledged.
7
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Contents lists available at ScienceDirect
International Journal of Mining Science and Technology journal homepage: www.elsevier.com/locate/ijmst
Mechanical and damage evolution properties of sandstone under triaxial compression Zong Yijiang a,b,⇑, Han Lijun a, Wei Jianjun b, Wen Shengyong a a b
State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China Jiangsu Collaborative Innovation Center for Building Energy Saving and Construction Technology, Jiangsu Jianzhu Institute, Xuzhou 221116, China
a r t i c l e
i n f o
Article history: Received 3 November 2015 Received in revised form 12 January 2016 Accepted 5 March 2016 Available online xxxx Keywords: Rock mechanics Mechanical properties Dilatation Damage evolution Failure characteristics
a b s t r a c t To study the mechanical and damage evolution properties of sandstone under triaxial compression, we analyzed the stress strain curve characteristics, deformation and strength properties, and failure process and characteristics of sandstone samples under different stress states. The experimental results reveal that peak strength, residual strength, elasticity modulus and deformation modulus increase linearly with confining pressure, and failure models transform from fragile failure under low confining pressure to ductility failure under high confining pressure. Macroscopic failure forms of samples under uniaxial compression were split failure parallel to the axis of samples, while macroscopic failure forms under uniaxial compression were shear failure, the shear failure angle of which decreased linearly with confining pressure. There were significant volume dilatation properties in the loading process of sandstone under different confining pressures, and we analyzed the damage evolution properties of samples based on acoustic emission damage and volumetric dilatation damage, and established damage constitutive model, realizing the real-time quantitative evaluation of samples damage state in loading process. Ó 2016 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
1. Introduction Due to the constant movement and development of the crust, there are inevitably a large number of initial defects such as micropores, joints and cracks in the rock mass. The deformation and damage of the rock mass, along with the propagation, development and connection of original cracks, is strongly affected and restricted by internal initial defects, especially structural planes with joints scale [1,2]. Therefore, research on fracture damage evolution progress and the characteristics of rock under the condition of triaxial compression is of great significance to mining engineering, tunnel engineering and geotechnical engineering. Damage is a phenomenon whereby micro defects in a material under monotonic loading or reloading leads to a progressive decrease in the cohesion and damage of volume units. Many scholars at home and abroad have carried out systematic research into the damage evolution characteristics and constitutive models of rock, and have achieved some remarkable results in this field. Lu et al. [3] studied the complete stress–strain curve characteristics of marble under triaxial compression, and established the bilinear
⇑ Corresponding author. Tel.: +86 13685188146. E-mail address: [email protected] (Y. Zong).
elastic-linear strain-softening residual ideal plastic damage constitutive model. Ren [4] studied the damage evolution laws of coal and the rock mass using a computerized tomography triaxial loading system, realizing the quantitative evaluation of the damage state. Jin et al. [5] studied damage evolution of rock under uniaxial compression and built the coal-rock damage evolution model which considered residual strength based on electromagnetic radiation characteristics. Zhang et al. [6] studied the deformation and failure mechanism of strong weathered sandstone by triaxial compression testing, and analyzed the damage evolution processes and established damage evolution equations based on the density method. Li et al. [7] studied the microscopic damage characteristics of siltstone under triaxial compression by scanning electron microscopy and digital image technology, and analyzed the statistical distribution characteristics of azimuth angle, length and width of cracks. Zhou et al. [8] studied the strength, deformation and fracture damage characteristics of sandstone by uniaxial cyclic loading and unloading tests, and defined a damage variable based on linear damage mechanics theory and acoustic emission (AE). Cao et al. [9] established a new damage model and a statistical damage constitutive model of rock under specific confining pressure according to the force of the damaged and undamaged parts in rock on the base of statistical damage theory. Li et al. [10] studied the damage evolution characteristics in the whole uniaxial compression
http://dx.doi.org/10.1016/j.ijmst.2016.05.011 2095-2686/Ó 2016 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
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process of sandstone by electrical resistivity and AE, and proposed a status qualitative criterion for rock damage. The indicators such as AE, CT values, density, electrical resistivity and crack length were adopted to evaluate the degree of damage and to study the damage evolution laws of rock in related research. However, research on the damage evolution process of rock on the basis of the dilatation properties is less well developed. Therefore, in this paper, the deformation and failure process, mechanical properties and evolution laws of sandstone under different stress states by uniaxial and triaxial compression tests are described, damage evolution based on AE and dilatation properties is analyzed, and a damage constitutive model, realizing real-time and quantitative evaluation of the damage state in the whole compression process, is established. 2. Experimental 2.1. Equipment and sample preparation Uniaxial and triaxial compression tests of sandstone were carried out on a TAW-2000 rock mechanics electro-hydraulic servo testing system, having a maximum axial force of 2000 kN, maximum confining pressure of 60 MPa, axial deformation measurement range from 0 to 10 mm, and radial deformation measurement range from 0 to 5 mm, and which can be used for uniaxial compression tests, uniaxial compression creep tests, triaxial compression tests and triaxial compression creep tests. Red sandstone, taken from Linyi of Shandong province, has a fine, blocky structure, good homogeneity, with a main mineral composition of quartz 17%, feldspar 42%, andesitic debris 25%, cements 15% and zircons 1%. The diameter distribution of the sandstone is: 0.10–0.25 mm 60%, 0.25–0.50 mm 35% and 0.50–1.00 mm 5%. The cementing type of the sandstone is pore cementation with grain point-line contact. Saturated red sandstone was processed into standard samples with diameters of 50 mm and height of 100 mm according to the standard testing method of Methods for Determining the Physical and Mechanical Properties of Coal and Rock (GB/T23561.1-2009) and ISRM. Sample specifications are shown in Table 1. 2.2. Schemes and process The confining pressure of triaxial compression tests were 5, 10, 15, 20 and 30 MPa respectively. The experimental process was as follows. The confining pressure was loaded to the set value at a rate of 0.05 MPa/s after the installation of samples. With a constant confining pressure, axial pressure was increased by the displacement control method at a loading rate of 0.002 mm/s until failure of the sample occurred. The axial pressure loading rate of the T-6
sample after peak strength was reached was 0.001 mm/s. The loading control methods of samples are shown in Table 1. 3. Results and discussion Fig. 1 shows the complete stress–strain curves of sandstone samples under uniaxial and triaxial compression; experimental results are shown in Table 2. 3.1. Characteristics of complete stress–strain curves As shown in Fig. 1, complete stress–strain curves of sandstone samples under uniaxial compression can be divided into five stages, including fissure compression, elastic deformation, steady crack propagation, unsteady fracture propagation and a strainsoftening stage. The residual deformation stage is not included. Fig. 1 shows good elastic property in the pre-peak stage of the stress–strain curves, and the peak point and strain-softening stage can only hold for a remarkably short time, presenting the characteristics of brittleness failure. There is no obvious fissure compression stage in the complete stress–strain curves of the sandstone samples under triaxial compression. Deviatoric stress increases linearly with increasing axial strain, radial strain and volumetric strain in the stress–strain curves up to the yield point, and the slope of the curves increases with increasing confining pressure. After the yield point, with increasing axial stress, new cracks gradually initiate in the samples with increasing axial stress, with further propagation of the original cracks and energy dissipation; this leads to the up-convex morphology of the stress–strain curves, a decrease in the stress–strain curve slope and an increase in plastic deformation. When the deviatoric stress reaches the failure strength, new macroscopic cracks initiate in the samples leading to failure of the samples, which shows the characteristics of a sharp increase in axial strain, radial strain and volumetric strain, and a decrease in sample strength. With increasing confining pressure, the slopes of the stress–strain curves, peak strength and residual strength have a great improvement, the samples at the strain-softening stages have smoother stress–strain curves, showing certain behavior of plastic deformation. The residual strength of the samples under a confining pressure of 30 MPa and a loading rate of 0.001 mm/s is higher, and the slopes of the stress–strain curves are becoming smaller at the strain-softening stage, showing clear behavior of plastic deforma-
σ 1 − σ 3 (MPa) 180
150
Table 1 Sample specification and loading control methods.
120
Sample No.
Diameter (mm)
Height (mm)
Test type
Loading rate (mm/s)
U-1 U-2 U-3
49.92 49.72 49.58
100.38 100.62 100.12
Uniaxial compression
0.002
T-1 T-2 T-3 T-4 T-5 T-6
49.62 49.72 49.72 49.62 49.60 49.72
100.28 99.98 99.94 100.40 101.48 100.80
Triaxial compression
0.002
Pre-peak 0.002 Post-peak 0.001
60
30
σ 3 =0 MPa
-80 -60 ε 3 (10-3)
σ 3 =30 MPa σ 3 =20 MPa σ 3 =15 MPa σ 3 =10 MPa σ 3 =5 MPa
90
σ 3=30 MPa σ 3 =20 MPa σ 3 =15 MPa σ 3=10 MPa σ 3=5 MPa
-40
-20
σ 3 =0 MPa
0
20
40 ε 1 (10-3)
Fig. 1. Complete stress–strain curves of sandstone samples in uniaxial and triaxial compression tests.
Please cite this article in press as: Zong Y et al. Mechanical and damage evolution properties of sandstone under triaxial compression. Int J Min Sci Technol (2016), http://dx.doi.org/10.1016/j.ijmst.2016.05.011
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Y. Zong et al. / International Journal of Mining Science and Technology xxx (2016) xxx–xxx Table 2 Uniaxial and triaxial compression test results of sandstone samples. Sample No.
Confining pressure (MPa)
Peak strength (MPa)
U-1 U-2 U-3 T-1 T-2 T-3 T-4 T-5 T-6
0 0 0 5 10 15 20 30 30
70.14 69.19 68.34 126.54 137.69 155.94 171.53 176.77 177.24
Residual strength (MPa)
Elastic modulus (GPa)
Deformation modulus (GPa)
Peak strain (103) Axis strain
Radial strain
Volumetric strain
58.14 59.73 70.79 76.15 81.55 100.20
11.80 11.84 11.87 14.14 14.67 16.06 16.28 17.06 17.12
7.11 8.22 8.83 14.76 15.00 16.99 17.15 18.56 18.85
8.26 7.85 7.16 10.56 11.31 12.18 13.63 14.11 14.72
11.81 14.14 11.98 19.86 12.63 16.77 15.41 20.16 20.13
15.36 20.45 16.79 29.17 13.91 21.36 17.19 26.21 25.54
tion. It is 3.2 times the duration of the samples under a loading rate of 0.002 mm/s. Under a confining pressure of 30 MPa, accurate control of loading and unloading tests at a certain stress state can be achieved.
3.2. Failure characteristics Figs. 2 and 3 show the failure modes of samples under uniaxial and triaxial compression respectively. Here we can see in Figs. 2 and 3 that the macroscopic fracture models of sandstone samples under triaxial compression are mainly split failure parallel to the axial direction of the samples, although some shear failure surfaces and lateral failure surfaces also exist, which shows that there are several forces inside the samples leading to two main models of axial split failure and shear failure. The macroscopic fracture models of sandstone samples in triaxial compression tests under different confining pressures are mainly shear failure. There is only one main shear surface throughout the whole sample after failure, thereby dividing the sample into two parts. From the two failure surfaces, we can see many scratches and the appearance of rock powder which is formed by the secondary shear failure of uneven parts on the surface in the shear slip process.
(a) 5 MPa
(d) 20 MPa
(b) 10 MPa
(e) 30 MPa
(c) 15 MPa
(f) Fracture morphology
Fig. 3. Failure modes of samples under triaxial compression.
3.3. Strength properties
(a) U-1
(b) U-2
(c) U-3
Table 2 shows that the peak strengths of the three samples under uniaxial compression are 70.14, 69.19 and 68.34 MPa, the degree of dispersion being only 2.60%, showing the good homogeneity of the samples. The selected rock thus easily achieves the requirement for tests. The peak strength and residual strength of samples under triaxial compression increase linearly with increasing confining pressure. Multiple regression analysis shows that the influence coefficient of confining pressure on peak strength is 3.064, and 1.302 on residual strength, illustrating the higher sensitivity of peak strength to confining pressure, which shows good agreement with those of related experiments [11]. 3.4. Deformation properties
(d) Fracture morphology
Fig. 2. Failure modes of samples under uniaxial compression.
(1) Elastic modulus and deformation modulus The elastic modulus E (the slope of the straight-line portion of deviatoric stress-axial strain curves), and deformation modulus (the slope of the connecting line between the point of 50% deviatoric stress and origin on the deviatoric stress-axial strain curves), increase linearly with increasing confining pressure. The influence
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Y. Zong et al. / International Journal of Mining Science and Technology xxx (2016) xxx–xxx 20
ε (10 -3)
10
0
-10
ε 1 =0.202σ 3 +9.400 R =0.987
5
10 ε 3 =0.0225σ
15 σ 3 (MPa)
20
25
3 2 3 0.902σ 3 +11.043σ3 −55.33
R=1
-20 -30
ε v =0.046σ 33 1.827σ 23 +22.453σ 3 −101.470 R=1
Fig. 4. Relationship between peak strain and confining pressure.
coefficients of confining pressure on elastic modulus and deformation modulus are 0.10 and 0.16 respectively, which shows that the internal structure of the samples is relatively dense, and the closure of micropores and microfractures leads to the slight increase in elastic modulus and deformation modulus. (2) Peak strain and residual strain The peak strain of samples under triaxial compression is closely related to the confining pressure. Samples under uniaxial compression and triaxial compression with low confining pressure will undergo brittle failure after small deformation. With increasing confining pressure, the peak strain of samples increases, and the failure models of samples transform from brittle failure to ductility failure. The axial peak strain and residual strain of samples increase linearly with increasing confining pressure (Fig. 4). However, the radial and volumetric peak strain and residual stain present a non-linear relationship with confining pressure due to the double effects of axial stress and confining pressure. (3) Dilatation properties Dilatation refers to the phenomenon that the volume of samples under uniaxial stress or unequal triaxial stress turns from a compression state to a dilatation state due to the initiation and propagation of cracks in samples under increasing stress [12]. The failure processes in many geotechnical engineering applications are accompanied by the dilatation phenomenon of the rock mass. Therefore, research into the dilatation properties of rock is of great significance to rock mass stability. The nonlinear deformation of samples under uniaxial and triaxial compression mainly includes elastic deformation, deformation caused by the closure of initial cracks, deformation caused by the initiation and propagation of cracks, shear slip and broken- expansion deformation after failure. The nonlinear deformation of samples in the initial loading stage is caused mainly by the closure of initial cracks, and by the initiation and propagation of cracks, and shear slip under high stress. Fig. 5 shows the relationship between radial strain, volumetric strain and axial strain of sandstone samples under different confining pressures. As shown as in Fig. 5, the axial strain, radial strain and volumetric strain of samples increases linearly with increasing axial stress before the yield point, and samples are under compression. The samples gradually turn from a state of compression to a state of dilatation due to the faster increase in the rate of radial strain. The volumetric strain will reach a maximum when the deviatoric stress is about 5.90–34.42% of the peak strength. With the initiation and propagation of cracks in the samples, the volumetric strain will gradually decrease, and the samples turn from compres-
sion to dilatation states, showing obvious dilatation properties. After the yield point, the radial and volumetric strains of sandstone samples increase sharply with increasing axial stress. Volumetric deformation can be divided into four stages, including: a volumetric compression stage, a pre-peak tiny volumetric dilatation stage, a post-peak rapid-increase stage and a steady-increase stage. There is remarkable shear slip and broken-expansion deformation in samples post-peak, showing significant dilatation properties. The dilatation deformation of samples pre-peak under uniaxial compression is 8.79–9.97% of the total, and 21.70–45.66% of the total under triaxial compression. However, the dilatation deformation post-peak is up to 54.34–91.21% of the total. 4. Damage evolution properties 4.1. Damage evolution properties based on AE There is some local deformation and micro fracturing in rock under external load or internal force, and the instantaneous strain energy rapidly releases in the form of elastic wave, resulting in the AE phenomenon [13,14]. The AE technique is a new unconventional dynamic detection method, which can continuously monitor damage and deformation in rock. Rock failure leads to the initiation and propagation of cracks and the formation of the macro structure surface. AE is the external evidence for the evolution characteristics of cracks, reflecting the damage which occurs in the rock when subjected to an external force. Therefore, the damage evolution laws of rock during the loading process can be studied using AE [15–18]. The damage evolution property of rock under uniaxial compression was studied using a PXWAE waveform acoustic emission detector. The AE sensor was a PXR15, with a bandwidth of 100–300 kHz and a sensitivity of 65 dB. The pre-amplifier was a PXPAI low noise amplifier, with a gain of 40 dB and a noise voltage of 2.7 uVRMS. The AE acquisition card had four AE channels and a PCI interface, with a sampling precision of 12 bits. The AE of the rock during the failure process obeys the statistical distribution of internal defects.N m and N d are respectively regarded as the accumulated acoustic emission ringing count when the whole cross-section fails completely and during failure of the rock. The damage variable D can be expressed as:
D¼
m Nd F ¼ 1 exp F0 Nm
ð1Þ
where F is a random variable of the Weibull distribution, m and F 0 are Weibull distribution parameters. Eq. (1) shows significant consistency between the damage variable and the accumulated acoustic emission ringing count. Therefore we can use the accumulated acoustic emission ringing count to express the damage variable of samples under uniaxial compression. The statistical damage constitutive equation can be obtained by introducing the Drucker–Prager failure criterion as the failure criterion of microelements.
r ¼ Eeemðec Þ 1
e m
ð2Þ
where r, e and E are axial stress, axial strain and elasticity modulus respectively. The damage evolution equation of samples under uniaxial compression can be expressed as: e m
D ¼ 1 emðec Þ 1
ð3Þ
Fig. 6 shows the damage evolution curve drawn according to Eq. (3) and a comparison of the theoretical and test curves.
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Y. Zong et al. / International Journal of Mining Science and Technology xxx (2016) xxx–xxx
120
120
90 40 60
30 20
Volumetric dilatation starting point
10
50
(σ 1 − σ 3 ) − ε 1 ε 3 −ε 1 ε v −ε 1
40 30
90
20 60
Volumetric dilatation starting point
30
10
30 0
0
0
2
4
6
-3
σ 1 (MPa)
50
150
ε (10 )
60
150
σ 1 − σ3 (MPa)
σ1 ε1 ε3 ε1 εv ε1
ε (10-3)
70
8
10
12
14
0
5
10
ε 1 (10 -3 )
15
20
25
ε 1 (10 -3)
(a) U-3 (σ3=0 MPa)
(b) T-2 (σ3 =10 MPa)
Fig. 5. Relationship between radial strain, volumetric strain and axial strain of sandstone samples.
70
is defined as the ratio of volumetric dilatation strain at some point in the strain–stress curves to the volumetric dilatation strain of the initial residual strength stage. The damage variable D can be expressed as:
1.0
Test curve
60
0.8 0.6
40
D
σ 1 (MPa)
50
30 Theoretical curve
20
0
AE demage curve 2
4
6
e1 6 e1d e1 > e1d
D¼0 D ¼ eevvreev d
vd
0.2 10
(
0.4
8
10
0 12
14
ε1 (10 -3 )
Fig. 6. Comparison of theoretical curve and test curve.
It can be seen in Fig. 6 that there is hardly any damage in samples at the fissure compression and elastic deformation stages, with the damage variable D ¼ 0. When the axial stress reaches 72.4% of the peak strength, the internal cracks of samples begin to initiate and propagate, resulting in obvious AE phenomenon, and damage of the samples. So, 0:724rc can be regarded as the damage threshold stress of samples under uniaxial compression. When the axial stress reaches the peak stress, the internal cracks propagate and connect continuously to form macro fractures resulting in accelerating damage of samples, and the damage variable rapidly increases to 1. The theoretical curves based on the AE damage constitutive model have coherence with the test curves, showing that the AE damage constitutive model can reflect the damage evolution properties of samples under uniaxial compression. Because of the continuous function of the damage constitutive model and fixed elastic modulus, the theoretical curves fail to fully reflect fissure compression and post-peak brittle failure of samples under uniaxial compression, having some deviations between theoretical and test curves.
ð4Þ
where D, ev , evd , ev r , e1 and e1d are damage variable, volumetric strain, maximum compressive volumetric strain, volumetric strain of initial residual strength stage, axial strain and axial strain corresponding to the maximum compressive volumetric strains respectively. The complete failure of samples under uniaxial and triaxial compression has not occurred at the residual strength stage, and the samples still have great residual strength. So Eq. (4) can be corrected as:
(
e1 6 e1d e1 > e1d
D¼0 D¼
Kðev ev d Þ ev r ev d
ð5Þ
where K is a correction factor.
K ¼ 1 rr =rp
ð6Þ
where rr and rp are residual strength and peak strength. The damage variable of sandstone samples under different confining pressures can be obtained by substituting Eq. (6) into (5). Fig. 7 shows a comparison of AE damage with volumetric dilatation damage under uniaxial compression. As shown in Fig. 7, volumetric dilatation damage has preferably coherence with AE damage. Therefore it is possible to determine the damage variable of samples in the whole loading process.
Acoustic emission damage Volumetric dilatation damage
4.2. Damage evolution properties based on dilatation
70
1.0
60 0.8 50 0.6
40
D
σ 1 (MPa)
The samples show obvious dilatation properties under uniaxial and triaxial compression. Dilatation is the result of the initiation and propagation of micro fractures in rock, and reflects the damage in rock under external load. Therefore, the damage evolution laws of rock during the whole loading process, based on the dilatation properties, can be studied. The nonlinear deformation of samples in the initial loading stage is caused mainly by the closure of initial cracks, and by the initiation and propagation of cracks, and shear slip under high stress. So we can consider the maximum compressive volumetric strains evd as the volumetric dilatation starting point, and assume that there is no damage before that point. The damage variable D
30
0.4
20 0.2
10 0
2
4
6
8
10
12
14
0
ε 1 (10 -3 )
Fig. 7. AE damage and volumetric dilatation damage of sandstone samples under uniaxial compression.
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The three-dimensional damage constitutive model of rock material can be obtained based on the strain equivalence hypothesis.
r1 ¼ Ee1 þ 2lr3 e1 6 e1d r1 ¼ Ee1 ð1 DÞ þ 2lr3 e1 > e1d
ð7Þ
Fig. 8 shows a comparison of theoretical curves based on the damage constitutive model in Eq. (7) with the test curves. Prepeak theoretical curves based on the dilatation damage constitutive model are in good agreement with the test curves. The theoretical curves at the peak point and post-peak reflect the strainsoftening properties of samples to some extent, but there are some increasing deviations with confining pressure between theoretical curves and test curves. Therefore we can introduce the porosity n into the constitutive equation to correct the damage constitutive model. Rock mass under the action of load can be abstracted into an object composed of pores, a damaged part and an undamaged part. Here we assume that the undamaged part will bear the whole load, without considering the bearing load of pores and the damaged part [19,20].
8 > < A0 ¼ nA A ¼ A0 þ A1 þ A2 > : 0 ri A1 ¼ ri A ¼ ri ðA0 þ A1 þ A2 Þ
ð8Þ
where ri and r0i are the stress in the rock mass and the effective stress of the undamaged part respectively, A, A0 , A1 and A2 action area of ri , action area of pore, action area of r0i and action area of the damaged part, i ¼ 1; 2; 3. If the damage variable is defined as:
D ¼ A2 =ðA1 þ A2 Þ
σ 1−σ3 (MPa)
ð10Þ
It is assumed that the undamaged part of the rock mass obeys the generalized Hook’s law. The constitutive equation of the undamaged part can be expressed as:
r0i ¼ E0 e0i þ l0 ðr0j þ r0k Þ
ð11Þ
where E0 , l0 and e0i are the elastic modulus of the undamaged part, Poisson’s ratio and micro strain respectively, i, j, k ¼ 1; 2; 3. Considering the deformation coordination of the rock mass during the loading process, we can get:
150 120
T-6 (σ 3=30 MPa) 90
T-2 (σ 3 =10 MPa) 30
0
U-3 (σ 3 =0 MPa) 5
10
15
20
Fig. 9. Comparison of correctional volumetric dilatation damage theoretical curves and test curves.
ei ¼ e0i l ¼ l0
Test curve Theoretical curve
(
ri ¼ P=A ¼ Eei þ lðrj þ rk Þ ¼ Eei r0i ¼ P=A3 ¼ E0 e0i þ l0 ðr0j þ r0k Þ ¼ E0 e0i
r ¼ P=½ð1 nÞA ¼ E0 ei
σ 1 (MPa)
T-2 (σ 3 =10 MPa)
50
E ¼ ð1 nÞE0
r ¼ Eei =ð1 nÞ þ lðr0j þ r0k Þ
10
15
20
25
ε 1 (10 ) -3
Fig. 8. Comparison of volumetric dilatation damage theoretical curves and test curves.
ð16Þ
With the simultaneous Eqs. (10) and (15), the dilatation damage constitutive equation of the rock material can be obtained as follows:
ð17Þ
The porosity of rock under triaxial compression can be expressed as:
E0 ð1 n0 Þð1 DÞ þ 2r3 ð1 l 2l2 Þ E0 ð1 DÞ½1 ð1 2lÞe1
ð18Þ
where n0 is the initial porosity of the rock. By assuming an initial porosity of zero (n0 ¼ 0), Eq. (18) can be simplified as follows:
n¼1 5
ð15Þ
Substituting Eqs. (12) and (15) into (11): 0 i
U-3 (σ 3 =0 MPa)
0
ð14Þ
With the simultaneous Eqs. (13) and (14), we get the relationship between E and E0 .
T -6 (σ 3 = 30 MPa)
100
ð13Þ
Substituting Eqs. (8) and (12) into (13): 0 i
n¼1
150
ð12Þ
where l and ei are macro Poisson’s ratio and macro strain respectively, i ¼ 1; 2; 3. If the rock mass is abstracted into the object composed of pore and rock material, the action area of rock material is A3 , the deformation of the rock mass and rock material under load P are ei and e0i respectively. The corresponding elastic moduli are E and E0 , and Poisson’s ratios are l and l0 . The constitutive equation of rock mass and rock material can be expressed as follows:
ri ¼ E0 ei ð1 nÞð1 DÞ þ lðrj þ rk Þ 200
25
ε 1 (10-3)
By substituting Eq. (9) into (8), the damage constitutive equation can be corrected as follows:
ri ¼ ð1 nÞð1 DÞr
Test curve Theoretical curve
60
ð9Þ
0 i
180
E0 ð1 DÞ þ 2r3 ð1 l 2l2 Þ E0 ð1 DÞ½1 ð1 2lÞe1
ð19Þ
The three-dimensional constitutive equation of rock material can be obtained by substituting n and D into Eq. (17).
r1 ¼ E0 e1 ð1 nÞ þ 2lr3 e1 6 e1d 0 r1 ¼ E e1 ð1 nÞð1 DÞ þ 2lr3 e1 > e1d
ð20Þ
Please cite this article in press as: Zong Y et al. Mechanical and damage evolution properties of sandstone under triaxial compression. Int J Min Sci Technol (2016), http://dx.doi.org/10.1016/j.ijmst.2016.05.011
Y. Zong et al. / International Journal of Mining Science and Technology xxx (2016) xxx–xxx
Fig. 9 shows the comparison of theoretical curves, based on the correctional damage constitutive model, with the test curves. The theoretical curves based on the dilatation damage constitutive model are in good agreement with the test curves, reflecting the deformation properties and damage evolution process of sandstone samples under different confining pressures. 5. Conclusions (1) The peak strength, residual strength, elastic modulus and deformation modulus increase linearly with increasing confining pressure, and failure models transform from fragile failure to ductility failure. (2) Before the yield point, axial strain, radial strain and volumetric strain of samples increases linearly with increasing axial stress and the samples gradually turn from compression states to dilatation states due to the faster rate of increase of radial strain when the deviatoric stress is about 10.47– 62.64% of the peak strength. After the yield point, there is noticeable shear slip and broken-expansion deformation in samples post-peak, showing significant dilatation properties. The dilatation deformation pre-peak is 21.70–45.66% of the total, and 54.34–78.30% of the total post-peak. (3) The macroscopic fracture models of sandstone samples under triaxial compression are mainly split failure parallel to the axial direction of the samples, but some shear failure surfaces and lateral failure surfaces also exist. The macroscopic fracture models of sandstone samples in triaxial compression tests under different confining pressures are mainly shear failure. (4) Based on acoustic emission damage and volumetric dilatation, the damage evolution properties of samples were analyzed, and damage constitutive models were established, realizing the real-time quantitative evaluation of samples damage state in loading process.
Acknowledgments Financial support for this work, provided by the National Natural Science Foundation of China (Nos. 51323004 and 51574223), the Postdoctoral Science Foundation of China (No. 2015M571842) and the Open Research Fund of Research Center of Jiangsu Collaborative Innovation Center for Building Energy Saving and Construction Technology (No. SJXTY1502), are gratefully acknowledged.
7
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Please cite this article in press as: Zong Y et al. Mechanical and damage evolution properties of sandstone under triaxial compression. Int J Min Sci Technol (2016), http://dx.doi.org/10.1016/j.ijmst.2016.05.011