J. Cent. South Univ. Technol. (2007)02-0210-06
DOI: 10.1007/s11771-007-0042-z 
Strength and elastic properties of sandstone under different testing conditions
CHEN Yun-ping(陈运平)1,2, WANG Si-jing(王思敬)1, WANG En-zhi(王恩志)1
(1. State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China;
2. Computational Geosciences Research Centre, Central South University, Changsha 410083, China)
Abstract: A laboratory experimental program performed on Wuhan sandstones was presented under monotonic loading, partial cyclic loading during loading path and sine wave cyclic loading with different strain rates to compare uniaxial compression strength and elastic properties (elastic modulus and Poisson ratio) under different conditions and influence of pore fluid on them. When the loading strain rates are 10-5, 10-4 and 10-3/s, uniaxial compression strengths of dry sandstones are 82.3, 126.6 and 141.6 MPa, respectively, and that of water saturated sandstones are 70.5, 108.3 and 124.1 MPa, respectively. The above results show that the uniaxial compression strength increases with the increase of strain rate, however, variation of softening coefficient is insignificant. Under monotonic loading condition, tangent modulus increases with an increment of stress (strain) to a maximum value at a certain stress level, beyond which it starts to decline. Under the partial cyclic loading during loading path condition, unloading or reloading modulus is larger than loading modulus, and unloading and reloading moduli are almost constants with respect to stress level, especially unloading modulus. Under the sine wave cyclic loading condition, tangent modulus and Poisson ratio display asymmetric ‘X’ shape with various strain, and the average unloading modulus is larger than the average loading modulus.
Key words: strain rate; strength; deformation modulus; Poisson ratio; softening coefficient; rock
1 Introduction
A granular material such as sandstone subjected to an external stress displays complex stress—strain characteristics of nonlinearity, hysteresis and stress- induced anisotropy[1-4], which are mainly caused by nonlinear deformation and frictional sliding of grain contacts.
Design of pillars, underground caverns, support, drilling, blast, freight highway and so on, requires understanding and research of rock mechanical properties under various loading conditions[5]. Strain can be changed in a few seconds, such as blast and earthquake, or in a few decades, such as formation of coals. The rock mechanical characteristics would be of distinct difference under different strain rates. In recent rock engineering designs, rock mechanical behaviors under different strain rates have been paid much attention[6-7]. Rock physical and mechanical properties under cyclic loading can be different from that under monotonic loading, such as stress-strain loop, weakening or strengthening (strain hardening) of compressive strength[2,8-9]. Hence it is helpful to investigate rock cyclic loading in engineering design for stabilizing structure and eliminating accidental breakage.
Fatigue strength of rock subjected to cyclic loading can be applied to rockburst prediction. Elastic constants (elastic modulus and Poisson ratio) are basic mechanical properties that is necessary to be considered in rock engineering project analysis and design, and they are widespread applied to simulation technique to predict stress and strain behavior of rock subjected to various loading conditions. The objective of the experiment is to study the elastic modulus and Poisson ratio under different types of loading-unloading with different strain rates.
2 Experimental
2.1 Specimen processing and experimental set-up
Testing materials were quartz sandstones from Wuhan, China, whose composition was 86% of quartz, 8% of feldspar, and 6% of others. The physical properties of rock samples included air-dry density of 2.59×103 kg/m3, porosity of 5.34%, and permeability of 7.45×10-15 m2. The cylindrical rock samples for uniaxial compression tests, 50 mm in diameter and 100 mm in length with ends ground flat and parallel to within 2.5×10-2 mm, were cored from intact blocks. All the bedding of sandstone should be parallel to the end of specimen to avoid the influence of bedding orientation on rock’s properties[7,10]. Samples were placed in an oven at 120 ℃ for 48 h to make them dry, then were taken to the vacuum chamber to vacuumize them. Some samples were chosen to infuse into distilled water for 24 h, when the water saturated samples were removed from the chamber, they were immediately weighed, and jacketed with epoxy resin to avoid fluid dissipating.
An MTS 810, servo-controlled rock mechanics test system was used for uniaxial stress cycling tests with 250 kN compression loading capacity and 150 mm stroke, and loading sensor precision excelled 0.5%. The system had strain-controlled loading frame, which was equipped with a load cell. Rock specimen was mounted under the loading frame. Vertical displacement was measured by a ring assembly with linear variable differential transformer (LVDT) connecting sample endcaps via fused quartz rods inside the pressure vessel, and lateral strain was measured by the strain gauge mounted to the middle section of specimen. The load cell, LVDTs and strain gauge were connected to the computer system, through which the program control of scheduled loading spectrum will be realized and data will be acquired automatically to the pointed file. During specimen loading, loading time, distance between two ends of specimen, loading cell, axial strain and lateral displacement were continuously recorded.
2.2 Experimental procedure
2.2.1 Monotonic loading
Samples were loaded with strain rates of 10-5, 10-4 and 10-3/s, respectively until they were in failure. Dry and water saturated samples were used to do this test. Repetition was made three times for a type of test to reduce the influence of individual discrepancy, since the measurement result of axial compressive strength was dispersive.
2.2.2 Partial cyclic loading during loading path
The specimen was loaded to some stress level with strain rate of 10-4/s, and went along unloading-reloading cycle with amplitude of about 8 MPa (Approximately 9% of axial compressive strength). This test included three cycles at different stress levels before the specimen failed. Dry and water saturated samples were used to do this test. Strain control loading was adopted in this test.
2.2.3 Sine wave cyclic loading
Sine wave whose dynamic loading was smaller than static loading was adopted in uniaxial compressive cyclic loading. Stress amplitude was close to axial compressive strength, and loading frequency was 5 Hz. Stress control loading was adopted in this test, and sampling rate was 100 point/cycle.
3 Result and analysis
3.1 Monotonic loading
Fig.1 shows the stress—strain curves of dry sandstones under various strain rates subjected to monotonic loading. The axial compressive strength of specimen ML01D1 is 83.3 MPa with strain rate of 10-5/s, axial compressive strength of specimen ML02D1 is 121.7 MPa with strain rate of 10-4/s, and axial compressive strength of specimen ML03D1 is 148.5 MPa with strain rate of 10-3/s. Axial compressive strength increases with the increment of loading strain rate.
Fig.1 Stress—strain curves of dry sandstones under different strain rate conditions
1— Specimen ML01D1, strain rate 10-5/s; 2—Specimen ML02D1, strain rate 10-4/s; 3—Specimen ML03D1, strain rate 10-3/s
Sandstone is heterogeneous and the relationship of stress—strain is not linear, hence modulus of deformation can be measured by tangent modulus, average modulus and secant modulus[11]. Tangent modulus (Et) is the slope of tangent line of one point in stress—strain curve; average modulus (E) is the slope of part of stress—strain curve that is almost straight line; and secant modulus (Es) is the ratio of stress to strain of one point in stress—strain curve.
Tangent modulus (Et), shown in Eqn.(1), is obtained by differentiating the formula that describes the relationship between the stress (σ) and axial strain (εa) with respect to εa.
Et=?σ/?εa (1)
Tangent Poisson ratio (υt), shown as Eqn.(2), is obtained by the same way.
υt=-?ε1/?εa (2)
where εl is the lateral strain.
Fig.2 shows the tangent modulus (Et) and Poisson ratio (υt) with respect to axial strain under various strain rates. It can be seen that Et increases with increasing strain rate. These tangent moduli have the same characteristics. Et increases with increasing axial strain (thus increasing stress) until a certain level, beyond which it starts to decrease. For the specimen of ML01D1, tangent modulus changes from 24.3 GPa at the beginning to 31.4 GPa of the maximum value, accordingly the stress level changes from 0 to 56% of axial compressive strength, and then it commences to decline until 21.1 GPa at the very point of failure. For specimens of ML02D1 and ML03D1, tangent modulus can be marked by 3 stages: ascending, smooth and descending stages. However, tangent modulus of specimen ML01D1 is not so evident. In the ascending stage, there are abundant of voids and microcracks in the rock, and they gradually close as subjected to applied force. During the smooth stage, the rock displays approximate linear elastic and the change of modulus is little. During the descending stage, nonlinear deformation that is called ‘rock dilation’ increases, and the increment rate of strain picks up as the rock reaches yielding strength. Anyway, rock gradually becomes stiff and reaches maximum value, and then gradually becomes soft. Contrary to regular changes of tangent modulus, changes of Poisson ratio with respect to strain are complex, as shown in Fig.2(b).
Fig.2 Tangent modulus and Poisson ratio vs strain(a) Tangent modulus；(b) Poisson ratio
1— Specimen ML01D1, strain rate 10-5/s; 2—Specimen ML02D1, strain rate 10-4/s; 3—Specimen ML03D1, strain rate 10-3/s
Axial compressive strength and elastic constants of dry and water saturated sandstones under various strain rates are listed in Table 1. The intenerated degree of rock can be expressed as softening coefficient Kp, Kp= σw/σd, where σw is the axial compressive strength of water saturated sandstone, and σd is the axial compressive strength of dry sandstone. According to the tests, softening coefficients of Wuhan sandstones are 85.7%, 85.5%, and 87.6% under the strain rates of 10-5, 10-4 and 10-3/s, respectively. It seems that there is little influence of strain rate on softening coefficient. Elastic modulus of water-saturated sandstone is smaller than that of dry sandstone at the same strain rate, but the difference between Poisson ratios of water-saturated and dry sandstones is insignificant. It can be seen that Poisson ratio of water-saturated sandstone is a little bigger than that of dry sandstone.
Table 1 Uniaxial compression strength and elastic constants of sandstones under different strain rates
Rate of crack propagation is much bigger than loading rate in the porous saturated rock at the low strain rate, which makes pore fluid in rock diffuse to expanded cracks. The pore pressure becomes bigger and activity of brittle micro-fracture in rock is increased that accelerates expansion of cracks. Besides, existence of pore fluid changes the physical state (e.g. softening of gelatine in rock)[12], and weakens the link among rock grains. That’s why strength and stiffness of water saturated rock is lower than those of dry rock.
The elastic constants measured by the above 3 methods are various (see Table 1). Et50 and E show little discrepancy but Es50 is smaller for about 18% than Et50 and E.
Experimental results show that the test data of rock’s strength and elastic properties are scattered. These facts are attributed to lithological property variation and microcracks distribution in the rock specimens, and perhaps to errors in alignment during sample preparation and mounting in addition.
In these failure tests, we noted that the specimens are broken into large fragments at low strain rates, whereas they are broken into small fragments at high strain rates, which means that the brittleness of sandstones increases with the increase of strain rate. This is consistent with the elastic modulus that increases with the increase of strain rate. This is to be further certified by designing some experiments.
3.2 Partial cyclic loading during loading path
Fig.3 shows the dry sandstone stress—strain relationship of 3 cycles of unloading-reloading subjected to different stress levels at strain rate of 10-4/s. The uniaxial compressive strength is 93.7 MPa. Table 2 lists the results of tangent moduli and Poisson ratios of loading, unloading and reloading at the different stress levels of unloading-reloading. It can be seen that the unloading or reloading tangent modulus is larger than the loading one, and their differences increase with increasing stress level.
Fig.3 Stress—strain curves of sandstone under partial cyclic loading during loading path
Table 2 Tangent modulus and Poisson ratio under partial cyclic loading during loading path
Loading tangent modulus varies with stress level because of the irreversible strain of rock at the loading process. At the third cycle (79.8-87.3 MPa of stress level), when the loading strength exceeds the yielding strength, plastic deformation becomes evident, and rock dilatancy appears, hence the loading tangent modulus becomes smaller. Whereas there is little influence of stress level on the loading or reloading tangent modulus, especially the loading tangent modulus. On the other hand, Poisson ratio is less affected by the loading, unloading and reloading because the irreversible strains exist in both the axial and lateral strains that are used to compute the Poisson ratio. The Poisson ratios obtained from the loading and unloading curves have less variation than that obtained from the reloading curve. The similar result was obtained with the water saturated sandstone.
Experiments with partial cyclic loading during loading path testify to the fact that small dynamic hysteresis loop on the big static stress—strain curve changes its slope with changing stress. This means that wave velocity will increase with increasing stress.
3.3 Sine wave cyclic loading
Sine wave cyclic loading test of dry sandstone was applied with harmonic forces of dynamic loading of 34.9 MPa, static loading of 36.8 MPa, and 5 Hz of frequency (Fig.4(a)). The first several hysteresis loops are sparse and then they become dense rapidly with the increasing loop numbers and finally tend to stability. Fig.4(b) shows the tangent modulus and Poisson ratio derived from the stabilized stress—strain hysteresis loops. They display asymmetric ‘X’ shape with discontinuous points at the stress reversal. Fig.4(c) shows the relationship between the average modulus and cyclic number under loading and unloading. Average loading and unloading moduli decrease gradually when the cyclic number increases, and reach the levels of 39.90 GPa and 40.00 GPa, respectively. At the same sine wave cyclic loading, the average unloading modulus is larger than the average loading modulus.
Fig.4 Dynamic response of sandstone under sine wave cyclic loading
(a) Stress—strain curve; (b) Tangent modulus obtained by stabilized hysteresis loop; (c) Average loading and unloading modulus vs cyclic number
Hysteresis loops of the sandstone are not closed during the first several sine wave cyclic loading processes, which means the existence of the residual deformation between the loading and unloading curves. With increasing of cyclic number, the hysteresis loop moves towards the direction of strain accretion, but the changing rate decreases and finally reaches stability. This is ascribed to the fact that there are large residual strain and plastic deformation occurring as a large number of pores and microcracks existing in the rock before loading continuously close when the rock is subjected to loading. With increasing number of cyclic loading, the average loading or unloading modulus decreases, but the difference between them becomes smaller and smaller (Fig.4(c)). This shows that the permanent deformation of a hysteresis loop is less and less with increasing number of cyclic loading, and finally the hysteresis loop reaches stability.
Mineral grains of a porous rock interlock to form a solid frame, and the surfaces of these grains are not perfectly smooth, for there are several asperity contacts between two adjacent grains[13]. In the sine wave cyclic loading tests, once the maximum load is reached and the load begins to decrease, the stick-slip motion between the fluid and the surfaces of the asperities and between the fluid molecules at the grain contact prevents immediate relaxation of strain until the force becomes large enough to overcome the frictional force. This is an explanation for the hysteresis loops in stress vs axial strain and stress vs lateral strain diagrams, the discontinuous asymmetric ‘X’ shape of elastic modulus or Poisson ratio, and the unloading modulus being larger than loading modulus.
4 Discussion and conclusions
When a sedimentary rock such as sandstone is subjected to cyclic stress, cracks grow, develop and coalesce each other, and finally produce faults and fracture along the disadvantageous direction. During the stable expansion of cracks, the stress—strain curve shows nonlinear deformation and characteristics of bulk expansion localization. This is the sign of macrofracture in rock, with which the uniaxial compressive strength has intimate relationship. The simulation study of a saturated porous material under compression displays that when the loading rate is increased, the crack growth rate is decreased[6] . That’s why the increasing loading rate will raise the strength of rock.
Uniaxial compressive strength of the sandstone increases with strain rates. However, variation of their softening coefficients is not so distinct. Under the condition of monotonic loading, partial cyclic loading during loading path and sine wave cyclic loading, the elastic constants of sandstone exhibit different characteristics. Under the monotonic loading condition, tangent modulus and Poisson ratio are not invariable, but the function of stress—strain level does. The tangent modulus increases with increasing axial strain (thus increasing stress) until a certain level, beyond which it starts to decrease. At the beginning of loading, the rock undergoes compression and part of microcracks close, and density of rock becomes larger, so modulus increases. In the stage of modulus decreasing, the rock material has induced damage and its integrality lowers. Under the condition of partial cyclic loading during loading path, the unloading or reloading modulus is larger than the loading one, and their difference is more distinct with the increase of stress level. Moreover, the tangent modulus derived from loading stage varies with stress level, but not the tangent modulus derived from unloading or reloading stage, especially from reloading stage. The residual deformation between the loading and unloading curves takes place owing to microplasticity. After the loading, the microplastic strain is exhausted, therefore the unloading modulus is higher than the loading modulus. When the stress exceeds the preceding stress level then microplastic strain appears again. There is hysteresis loop in stress—strain constitution under the sine wave cyclic loading condition. In every stress—strain hysteresis loop, tangent modulus and Poisson ratio appear to be asymmetric ‘X’ shape, and average unloading modulus is larger than average loading modulus.
The experimental data of strength and elastic properties of rock are dispersed, which is ascribed to the lithological variation and distribution of microcracks in rock samples, and to the possible loading misalignment during sample preparation and mounting.
The stress—strain relationship is not completely linear elastic due to heterogeneous material of rock. The elastic constants (elastic modulus and Poisson ratio) measured by above three methods are different, so the measurement approach of elastic constants of rock should be carefully considered. During the loading process, loading curve contains both recoverable and irreversible components, but the elastic constants derived from unloading-reloading curve of a stress level are stable, especially from the unloading part of the curve. We suggest the elastic constants derived from unloading curve.
If the rocks in the earth’s crust are intact, the stress in the earth’s crust is not beyond the failure strength of rock. Therefore, investigating the brittle fracture of intact rock can give the restraints of stress field of the earth’s crust. Strength, softening coefficient and elastic constants of sandstone are useful parameters in oil exploitation, mining engineering and earthquake forecasting, and the investigation is also significant in basic research of application geologies (engineering geology, hazardous geology and environmental geology).
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Foundation item: Project(Z110510) supported by Opening Research Foundation of the Chinese Academy of Sciences Key Laboratory of Rock and Soil
Mechanics; Project(20060390473) supported by China Postdoctoral Science Foudation; Project(40172084) supported by the National Natural Science Foundation of China
Received date: 2006-06-21; Accepted date: 2006-08-27
Corresponding author: CHEN Yun-ping, PhD; Tel: 86-10-62784444; E-mail: chyp@tsinghua.edu.cn
(Edited by LI Xiang-qun)
