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Laboratory investigation on strength and deformation characteristics of ice-saturated frozen sandy soil
Cold Regions Science and Technology 69 (2011) 98–104
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Cold Regions Science and Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o l d r e g i o n s
Laboratory investigation on strength and deformation characteristics of ice-saturated frozen sandy soil Xiangtian Xu a, Yuanming Lai a,⁎, Yuanhong Dong a, b, Jilin Qi a a b
State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Gansu, Lanzhou, 730000, China CCCC First Highway Consultants CO., LTD, Shaanxi, Xi'an, 710075, China
a r t i c l e
i n f o
Article history: Received 9 June 2011 Accepted 16 July 2011 Keywords: Frozen sandy soil Strength criterion Stress–strain behavior Pressure melting Confining pressure
a b s t r a c t To investigate the properties of ice-saturated frozen sandy soil, a series of triaxial compression tests on frozen sandy soil with a volumetric ice content of about 50% were carried out at a temperature of − 2.0 °C. The effect of confining pressure on strength and deformation features is analyzed according to the experimental results. The results show that the strength changes with increasing confining pressure in three distinct phases. According to the effective stress principle, the mechanism of strength is explained. A strength criterion is proposed to describe the strength characteristic. The equivalent stress versus axial strain curve shows strainsoftening under each confining pressure, and the extent of strain-softening decreases with the increase in confining pressure, until it behaves the so-called perfect elasto-plastic feature when the confining pressure is large enough. The improved Duncan–Chang hyperbolic model is taken to simulate the stress–strain behaviors. The simulation shows that the model can well describe the strain-softening. The dependency of the volumetric deformation on the confining pressure is also discussed. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In the engineering construction of cold regions and artificial ground freezing engineering, the mechanical property of frozen soil is an important scientific subject, especially strength and deformation. Frozen soil is composed of solid mineral particles, ice, liquid water and air (Tsytovich, 1985). The ice, which is sensitive to temperature, makes the mechanical properties of frozen soil become complex and different from unfrozen soil (Qi and Ma, 2010). The experimental studies on the mechanical properties of frozen soil started in the beginning of the last century (Zhu, 1988). Large amount of previous studies have shown that strength of frozen soils is related to temperature, strain rate, confining pressure as well as ice content. The influence of temperature on strength of frozen soils has been studied overwhelmingly. Sayles and Haines (1974) studied the influence of temperature on the strength of frozen silt and clay under constant strain rate, and fitted the relationship between strength and temperature through a power function. Haynes and Karalius (1977) investigated the influence of temperature on the strength of frozen clay under the condition that loading rate varied from 0.0423 cm/s to 4.23 cm/s. Furthermore, Bragg and Andersland (1981), Bourbonnais and Ladanyi (1985) and Wu et al. (1994) investigated the effect of temperature on the strength of frozen soil, respectively. Due to the
⁎ Corresponding author. Tel.: + 86 931 4967288. E-mail address: [email protected] (Y. Lai). 0165-232X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2011.07.005
existence of ice, frozen soils show strong rheological features, thus strain rate plays an important role in mechanical properties of frozen soils. Haynes et al. (1975) studied the influence of strain rate on the strength of frozen Fairbanks silty clay under the temperature of −9.4 °C, and concluded that unconfined compressive strength was highly sensitive to strain rate but not the initial tangent modulus. Baker (1979) carried out uniaxial compression experiment on frozen saturated Ottawa fine sand with the strain rate ranging from 10 −7/s to 10 −2/s, and concluded that the uniaxial strength of frozen Ottawa fine sand was an exponential function of strain rate. Parameswaran and Jones (1981) suggested that the relationship between strength of frozen soil and strain rate could be described linearly. The study on the frozen Wedron sand by Bragg and Andersland (1981) showed that the strength of frozen sand is independent of strain rate when strain rate is less than 10 −5/s. Confining pressure is another important influencing factor on strength of frozen soil. Chamberlain et al. (1972) carried out triaxial compression test on frozen saturated Ottawa sandy soil and New Hampshire silt under −10 °C with confining pressure varying from 0 MPa to 275.6 MPa, and investigated strength characteristic of frozen soil under different confining pressures. Fish (1991) and Ma et al. (1994) presented a parabolic strength criterion for frozen soil, considering the nonlinear influence of confining pressure on the strength of frozen soil. Ma et al. (1999) analyzed micro-mechanism of the weakening of frozen soil under high confining pressure, and the quantitative relation between confining pressure and unfrozen water content after pressure melting of Lanzhou fine sand was obtained. Additionally, some authors studied the influence of ice content on mechanical properties of
X. Xu et al. / Cold Regions Science and Technology 69 (2011) 98–104
frozen soil (Arenson et al., 2004; Sayles and Carbee, 1981; Vyalov, 1962). As for deformation of frozen soil, the current studies mainly focus on the creep and elasto-plastic deformation behavior under uniaxial/triaxial stress. Andersland and Akili (1967) described the creep behavior of frozen soil through Rate Process Theory (RPT). Fish (1984) proposed a creep model for frozen soil based on thermodynamics. Zhu and Carbee (1987) studied the creep of frozen soil under uniaxial stress, and an empirical equation for the creep of frozen soil was deduced. Zhu and Zhang (1982) investigated compressive deformation of frozen clay from the Qinghai–Tibetan Plateau. Zhu et al. (1992) proposed a series of models describing the constitutive relationship of frozen soil under uniaxial compression. Ma et al. (1998) examined deformation characteristic of frozen sand under triaxial test, and described the coupling response between shear stress and mean stress. Ma et al. (2000) studied the deformation characteristics of frozen clay through normalization method. Zhang et al. (2007) analyzed the volumetric deformation characteristics of frozen silty clay under triaxial compression. Li et al. (2009) proposed a statistical damage constitutive model based on Mohr–Coulomb criterion. Lai et al. (2009) established an elasto-plastic damage constitutive relationship for frozen sandy soil. Lai et al. (2010) presented an elasto-plastic constitutive relationship for frozen clay based on generalized plastic mechanics. With the development of studies on mechanical properties of frozen soils, mechanical test approaches of frozen soils have also been developed, some of which are standardized (Baker et al., 1981; Sayles et al., 1987). However, the soil types and test conditions of various researchers are not the same. So it is inappropriate to extrapolate from one to others (Arenson et al., 2007). Some engineering constructions, e.g. the Qinghai–Tibetan Railway and the Qinghai–Tibetan Expressway, have been or will be located in ice-saturated and ice-rich frozen soil regions (Cao, 2003; Zhang et al., 2004). It should be noticed that the ice-saturated frozen soil is not equivalent to the ice-rich frozen soil. Arenson et al. (2007) introduced a classification to frozen soils based on the volumetric ice content, and proposed a definition for icerich frozen soils. They also have carried out a number of studies on icerich permafrost (Arenson et al., 2003; Arenson and Springman, 2005a, b). Because the ice-saturated frozen soil will largely influence the stability of engineering constructions above it, it is therefore necessary to investigate its engineering properties, for instance strength and deformation characteristics of ice-saturated frozen soil. In this study, triaxial compression tests were conducted on a frozen Qinghai–Tibetan sandy soil. Stress–strain curves and strength changes with confining pressure were analyzed. On this basis, a strength criterion was proposed in p–q plane. An improved Duncan–Chang hyperbolic model (Lai et al., 2007) was employed to describe the stress–strain relationship. 2. Triaxial compression tests on frozen Qinghai–Tibetan sandy soil 2.1. Test equipment In this study, the test equipment is a cryogenic triaxial apparatus improved from MTS-810 material test machine which can measure the volumetric strain of specimen, as shown in Fig. 1. The axial strain of the specimen can be calculated according to the displacement of axial loading piston (‘7’ in Fig. 1) which was measured by displacement transducer (‘10’ in Fig. 1). Fig. 1 shows that after the displacements of pistons 5 and 7 were measured, the volumetric strain of the specimen can be calculated by the following formula: εv =
ΔV S ðL −L0 Þ + S5 ðlt −l0 Þ = 7 t V0 V0
where, V0 is the initial volume of specimen; S5 and S7 are the crosssectional areas of pistons 5 and 7, respectively, S5 = 38.32 cm 2,
99
Fig. 1. Schematic diagram of triaxial apparatus.
S7 = 30 cm 2 in this study; lt and l0 are the current and initial positions of piston 5, respectively; Lt and L0 are the current and initial positions of piston 7, respectively. The equipment provides three controlling modes: load control, displacement control and time control. The test process is automatically controlled by program and test data are collected automatically. The maximum axial load is 100 kN. The ranges of axial displacement and confining pressure are −85 cm–85 cm and 0 MPa–20 MPa, respectively. The loading liquid used in confining pressure loading system is an aircraft hydraulic oil. The refrigeration medium is ethyl alcohol. The controlled temperature range is from − 30 °C to 25 °C. 2.2. Test material and test procedures The test material was fine sandy soil collected from the Beiluhe test site along the Qinghai–Tibetan Railway, the grain size distribution is shown in Table 1. To decrease the difference between specimens, the specimens were made on a specially-made specimen machine. To ensure the uniformity of specimen, this sand was mixed with 15% moisture content (corresponding to volumetric ice content 21.8%) by weight at first, and then kept for 6 h with no evaporation. Afterward, the soil was put in a cylindrical mold to make cylindrical soil specimens with target dry densities. The specimens have a height of 12.5 cm and a diameter of 6.18 cm. The specimens were then saturated using a vacuum. The saturated specimens were put in refrigeration together with the mold. They were quickly frozen from top to bottom under − 30 °C to avoid frost heave. After the specimens were completely frozen, which usually took 48 h, the molds were removed and the specimens were mounted with epoxy resin platen on two ends and covered with rubber sleeve. After this, the specimens were kept in an incubator for over 12 h under the target testing temperature so that the specimen had a uniform temperature. The average volumetric ice content of the frozen specimens at the time of testing was about 50% and their dry density was about 1.33 g/cm 3. In order to investigate the influence of pressure melting on frozen soil strength, a series of undrained triaxial compression tests were carried out. Before the triaxial compression, the temperature in the Table 1 Particle fractions of the soil (%). <0.075 mm 0.075–0.1 mm 6.241 31.081
0.1–0.25 mm 54.410
0.25–0.5 mm 4.692
0.5–1 mm 2.241
>1 mm 0.869
100
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pressure chamber was set at −2.0 °C. After the temperature became stable, the specimen was put in a pressure chamber and a certain confining pressure was applied for 5 min, then axial loading started to be applied under the same confining pressure. The confining pressure ranges from 0.3 MPa to 18 MPa in the tests. The loading rate was 1.25 mm/min. The loading was stopped until the axial strain reached about 30%.
2.3. Experimental results In the triaxial compression test, the axial force, axial displacement, confining pressure and the piston displacement of the confiningpressure loading system were automatically collected by data logger. The piston displacement of the confining-pressure loading system was used to compute the volumetric deformation of the specimen (Zhang et al., 2007). The axial stress, axial strain and volumetric strain under different confining pressures can be calculated based on the testing data. The equivalent stress–axial strain curves and volumetric strain–axial strain curves under different confining pressures are shown in Figs. 2 and 3, respectively. Equivalent stress q is computed by:
1 q = pffiffiffi 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðσ1 −σ2 Þ2 + ðσ1 −σ3 Þ2 + ðσ2 −σ3 Þ2 :
ð1Þ
Volumetric strain εv = ε1 + ε2 + ε3, σ1, σ2 and σ3 are principal stresses, ε1, ε2 and ε3 are principal strains. Under the triaxial conditions applied in this paper, q = σ1 − σ3, εv = ε1 + 2ε3.
3. Strength characteristic of frozen sandy soil 3.1. Effects of confining pressures on strength From Fig. 2, it can be seen that the stress–strain curve of icesaturated frozen sandy soil behaves softening under each confining pressure. Therefore, the peak stress of each stress–strain curve is taken as strength. Experimental results show that the strength of frozen soil does not always increase with increase of confining pressure, which is different from unfrozen sandy soil. The relation between strength and confining pressure of frozen sandy soil is shown in Fig. 4, which indicates that the strength-confining pressure curve of the frozen sandy soil can be divided into three phases. In the first phase, the confining pressure range is from 0.3 MPa to 2 MPa. In this phase, the strength of the frozen sandy soil rises with the increase of confining pressure, and the maximum strength appears at σ3 = 2 MPa. In this phase, due to the relatively low confining pressure, pressure melting and crushing may not happen. The increase of confining pressure increases the strength of the frozen sandy soil in two aspects. One is that it increases the normal pressure on the shear surface, which increases the sliding frictional resistance; the other one is that it makes the solid particles (including soil particles and ice particles) tighter, and therefore increases the interlocking frictional force between solid particles. In the second phase, the confining pressure ranges from 2 MPa to 12 MPa. In this phase, the strength of the frozen sandy soil decreases with increase of confining pressure. This is because after confining pressure excesses 2 MPa, the increase of confining pressure makes the ice in the specimen crushed and pressure melting occurs. The crushed ice decreases the ice bonding force between solid grains. The pressure
Fig. 2. Relationships between axial strain and equivalent stress.
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Fig. 3. Relationships between axial strain and volumetric strain.
melting generates pore water, which decreases the internal frictional angle. At the same time, when pressure melting develops, excess hydrostatic pressure appears and the effective normal pressure on the shear surface is reduced. These are the intrinsic reasons that the strength of frozen soil decreases with increase of confining pressure in this phase. In the third phase, the confining pressure range is from 12 MPa to 18 MPa. After the confining pressure excesses 12 MPa, the strength of frozen sandy soil does not obviously decrease with further increase of confining pressure. There is a slight fluctuation near the strength of σ3 = 2 MPa. This illustrates that the strength of frozen sandy soil is hardly influenced by confining pressure when confining pressure excesses 12 MPa. Based on this strength feature of frozen soil, the minimum confining pressure under which the strength does not change with confining pressure is defined as the critical confining pressure σcr. In this test, the critical confining pressure is σcr = 12 MPa. The reason for the strength characteristic in the third phase is that when confining pressure reaches critical confining pressure σcr, a great quantity of pore water appears because of the pressure melting of the ice. Noticing that the test was undrained, when the confining pressure has an increment Δσ3, the excess hydrostatic pressure induced can be expressed as (Chen et al., 1994): Δu = BΔσ3 :
ð2Þ
For the saturated soil in the test, there is B ≈ 1. According to effective stress principle (Chen et al., 1994), when confining pressure excesses σcr, the effective confining pressure from the solid particles in frozen soil is: 0
σ3 = σ3 − u:
ð3Þ
Thus: 0
Δσ3 = Δσ3 −Δu:
ð4Þ
Based on Eqs. (2) and (4), when confining pressure excesses σcr, it can be obtained that: 0
Δσ3 ≈0:
ð5Þ
It is illustrated in Eq. (5) that when confining pressure reaches critical value σcr, the further increase of total confining pressure does not increase effective confining pressure, and the effective confining pressure in frozen soil keeps at σ’cr (the effective confining pressure corresponding to σcr). Therefore, after confining pressure excesses σcr, the increase of total confining pressure does not change the strength of frozen sandy soil any more, and neither crushing nor pressure melting happens. Therefore, the frozen soil strength keeps constant. 3.2. Strength criterion For unfrozen soil, several strength criteria considering effects of confining pressure have been presented, such as Mohr–Coulomb criterion, Drucker–Prager criterion and Matsuoka–Nakai criterion. However, these strength criteria can only describe the characteristic that strength increases with increase of confining pressure, but not for the strength changes with pressure melting in frozen soils. In order to illustrate reasonably the strength characteristic of frozen soil under various confining pressure, a nonlinear strength criterion in the p–q plane is proposed in this study, as follows: 0 −b
q−qu B = a@e pa
Fig. 4. Relationship between confining pressure and strength.
1 p p −c pa −e pa C A
ð6Þ
where p is average stress, p=(σ1 +2σ3)/3 under axisymmetric condition; Pa is normal atmospheric pressure, taken as Pa =0.101325 MPa; qu is the frozen soil strength when confining pressure is infinite, and it is close
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X. Xu et al. / Cold Regions Science and Technology 69 (2011) 98–104 Table 2 Material parameters of frozen sandy soil under different confining pressures. Confining pressure (MPa)
l (MPa−1)
m (MPa−1)
n (MPa−1)
0.3 0.5 0.8 1 2 3 4 5 6 7 8 10 12 14 16 18
0.01013 0.00707 0.00706 0.0087 0.00519 0.0051 0.00709 0.00832 0.00953 0.00971 0.0125 0.01222 0.00662 0.01289 0.01453 0.01389
0.84641 0.87641 0.85138 0.78041 0.7829 0.81434 0.83264 0.91814 0.83404 0.87789 0.87789 0.95654 1.16031 0.97201 1.12803 1.10385
0.79177 0.61048 0.3852 0.60464 0.45091 0.27808 0.2837 0.58743 0.9898 0.64313 0.78528 0.76322 1.06695 1.11668 1.05776 0.99914
Fig. 5. Comparisons of strength between test value and fitted value.
to the strength corresponding to critical confining pressure; a, b, and c are three dimensionless material parameters, which can be obtained by experimental data. On the p–q plane, the strength criterion (Eq. (6)) can be used to fit the experimental data, the fitted result is shown in Fig. 5. From this figure, the material parameters can be obtained as follows: a = 132.6425, b= 0.03743, c =0.04047, qu = 0.759 MPa. Fig. 5 also shows that the strength criterion can reasonably describe the strength characteristics of frozen soil, i.e., the strength increases with the increase of confining pressure, then decreases with a further increase, and does not change with confining pressure after the confining pressure reaches the critical value. Furthermore, the isotropic tensile strength −f3t can be predicted from the cross point of strength curve and p axis, −f3t =−1.19 MPa, which is consistent with the experimental results of Lai et al. (2007). This further illustrates the rationality of the strength criterion (Eq. (6)).
It can be seen from Fig. 2 that when the temperature was −2 °C, the frozen sandy soil with a volumetric ice content of 50% has a strainsoftening behavior under the confining pressure of 0.3–18.0 MPa. This is different from the ice-poor frozen sandy soil which behaves strainsoftening under relatively low confining pressure and strain-hardening under higher confining pressure (Lai et al., 2009). The experimental results show that the strain-softening extent of ice-saturated frozen sandy soil decreases with the increase of confining pressure. When confining pressure is less than 5 MPa, the strain-softening is obvious, which means that the stress–strain curve declines significantly after the peak stress. When confining pressure is larger than 5 MPa, the strainsoftening is slight, showing a perfect elasto-plastic deformation feature. When confining pressure is over 12 MPa, the stress–strain curves under different confining pressures are almost the same. This illustrates that the deformation characteristics of the frozen sandy soil is slightly influenced by confining pressures, which is consistent with the conclusion in the previous section that the effective confining pressure is not obviously increased with the increase of total confining pressure when confining pressure reaches its critical value. According to the strain-softening characteristics, an improved Duncan–Chang hyperbolic model is adopted to model the stress–strain behavior of frozen sandy soil (Lai et al., 2007): εa : lεa2 + mεa + n
dq −lεa2 + n = 2 : 2 dεa lεa + mεa + n
ð7Þ
In which, l, m and n are material parameters related to confining pressure with a dimension of MPa −1, respectively and their values under different confining pressures are shown in Table 2.
ð8Þ
According to the definition of initial elastic modulus, we can get: dq E0 = 100 dεa
εa = 0
4. Deformation characteristic
q=
The stress–strain curves, based on the improved Duncan–Chang hyperbolic model and experimental data, are shown in Fig. 6. Fig. 6 illustrates that the improved Duncan–Chang hyperbolic model can reasonably simulate the strain-softening behavior of frozen sandy soil under different confining pressures. This can theoretically be explained. The following equation can be obtained by differentiating Eq. (7):
=
100 n
ð9Þ
where, the factor 100 before the derivate of q is introduced because the axial strain is in percent. As the initial elastic modulus is positive, n is positive. Furthermore, according to the numerator in the right of pffiffiffiffiffiffiffiffi ffi Eq. (8), if l is positive, q is an increasing function when ε < n = l; a pffiffiffiffiffiffiffiffiffi while q is a decreasing function when εa > n = l. Therefore, Eq. (7) can be used to describe strain-softening behavior. If l is negative, q′ is above zero, therefore Eq. (7) can also describe strain-hardening behavior. The improved Duncan–Chang hyperbolic model can accurately describe the stress–strain behavior of frozen sandy soil, so an accurate initial elastic modulus can be obtained from Eq. (9). The relation between initial elastic modulus and confining pressure is plotted in Fig. 7. The change of initial elastic modulus with confining pressure is similar to that of strength with confining pressure. At the beginning, the initial elastic modulus increases with the increase of confining pressure. After the initial elastic modulus reaches its peak value, it decreases with a further increase of confining pressure. After confining pressure reaches a certain value, the initial elastic modulus is not influenced by confining pressure but keeps constant. This is because under relatively low confining pressure, the initial elastic modulus increases with the compaction of frozen sandy soil. With a further increase of confining pressure, the ice in the frozen soil begins to be crushed and pressure melting happens, both of which result in decrease in the initial elastic modulus. When the total confining pressure is large enough, the increase in confining pressure is undertaken by the excess hydrostatic pressure generated from pore water, while the effective confining pressure does not increase at all. Therefore, the initial elastic modulus is not changed by confining pressure any more.
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103
Fig. 6. Comparisons between test values and simulated values for stress–strain relationship.
Besides strain-softening, the volumetric deformation is also different from either unfrozen sandy soil or ice-poor frozen sandy soil. It can be seen in Fig. 3 that, at the beginning, the volumetric strain increases with the increase of axial strain (positive volumetric strain means that the specimen is compressed, while negative means expanded), showing that the specimen is compressed. When axial strain reaches a certain value, the volumetric strain reaches its maximum value which corresponds to different axial strains under different confining pressures. Afterward, the volumetric strain begins to decrease with a further increase of axial strain, indicating that the specimen is expanded. The specimen behaves in a contraction first and dilatancy later under each confining pressure; however, the
contraction is different under different confining pressures. The contraction is relatively large when confining pressure is less than 3 MPa (Fig. 3a). When confining pressure is larger than 3 MPa, the specimen experienced a slight contraction at the very initial stage, and followed by overwhelming dilatancy. This is because that when the confining pressure is higher, the contraction of specimen is relatively large during the confining pressure loading process, therefore, it is difficult to compress the specimen in the axial loading process and the dilation happens after failure. 5. Conclusions The strength and deformation characteristics of ice-saturated frozen sandy soil under different confining pressures are studied based on the triaxial compression tests. The following conclusions can be drawn from the present study:
Fig. 7. Initial elastic modulus under varying confining pressures.
(1) The strength of the frozen sandy soil changing with increasing confining pressure can be divided into three phases. In phase I, the confining pressure range is from 0.3 MPa to 2 MPa, strength increases with the increase of confining pressure and the maximum value observed at a confining pressure of 2 MPa; afterward, the strength decreases with the increase of confining pressure, which is phase II, confining pressure ranges from 2 MPa to 12 MPa in this phase; however, with the further increase in confining pressure to a critical value 12 MPa, the strength is not influenced by confining pressure any more, which is called phase III. (2) Based on the strength characteristic of frozen sandy soil, a nonlinear strength criterion for frozen sandy soil is proposed, which can reasonably describe the strength characteristic in all phases.
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(3) The stress–strain curve of ice-saturated frozen sandy soil exhibits strain-softening under all of the testing confining pressures and the strain-softening becomes weaker with the increase of confining pressure. The strain-softening is relatively obvious when confining pressure ranges from 0.3 MPa to 5 MPa, but slight in other confining pressures. Similarly, after the confining pressure of 12 MPa, the stress–strain curve was hardly influenced by the confining pressure. (4) The improved Duncan–Chang hyperbolic model is applied to simulate the deformation of frozen sandy soil. The simulation shows that the model can well describe the strain-softening behavior of frozen sandy soil. After this, the reason that the model can describe both strain-softening and strain-hardening is theoretically explained. (5) Confining pressure also has a profound effect on volumetric deformation and initial elastic modulus. Above a confining pressure of 3 MPa, the volumetric deformation behaves mainly in dilation. The initial elastic modulus changing with the increase of confining pressure is similar to the strength. Acknowledgments We would like to sincerely thank the two anonymous reviewers whose constructive comments are helpful for the publication of this paper. This research was supported by the National Natural Science Foundation of China (40730736, 40821001, 40801029), the Knowledge Innovation Program of the Chinese Academy of Sciences (Grant No. KZCX2-EW-QN301), the Project for Incubation of Specialists in Glaciology and Geocryology of National Natural Science Foundation of China (J0930003/J0109) and the foundation of State Key Laboratory of Frozen Soil Engineering (SKLFSE-ZY-03). References Andersland, O.B., Akili, W., 1967. Stress effect on creep rates of a frozen clay soil. Geotechnique 17 (1), 27–39. Arenson, L.U., Springman, S.M., 2005a. Mathematical descriptions for the behaviour of ice-rich frozen soils at temperatures close to 0 °C. Canadian Geotechnical Journal 42, 431–442. Arenson, L.U., Springman, S.M., 2005b. Triaxial constant stress and constant strain rate tests on ice-rich permafrost specimens. Canadian Geotechnical Journal 42, 412–430. Arenson, L.U., Almasi, N., Springman, S.M., 2003. Shearing response of ice-rich rock glacier material. Eight International Conference on Permafrost Zurich, Switzerland, pp. 39–44. Arenson, L.U., Johansen, M.M., Springman, S.M., 2004. Effects of volumetric ice content and strain rate on shear strength under triaxial conditions for frozen soil specimens. Permafrost and Periglacial Processes 15, 261–271. Arenson, L.U., Springman, S.M., Sego, D.C., 2007. The rheology of frozen soils. Applied Rheology 17 (1), 1–14. Baker, T.H.W., 1979. Strain rate effect on the compressive strength of frozen sand. Proceedings of the First Symposium on Ground Freezing Bochum, Germany, pp. 223–231. Baker, T.H.W., Jones, V.R., Parameswaran, V.R., 1981. Confined and unconfined compression tests of frozen sand. The Roger J.E. Brown Memorial Volume Proc. Fourth Canadian Permafrost Conference, pp. 387–393. Bourbonnais, J., Ladanyi, B., 1985. The mechanical behavior of frozen sand down to cryogenic temperatures. Proceedings of the Fourth International Symposium on Ground Freezing Sapporo, Japan, Vol. 1, pp. 235–244. Bragg, R.A., Andersland, O.B., 1981. Strain rate, temperature, and specimen size effects on compression and tensile properties of frozen soil. Engineering Geology 18, 35–46. Cao, Y.P., 2003. Test investigation on the ventilation duct embankment in permafrost regions. Subgrade Engineering 6, 12–15.
Chamberlain, E., Groves, C., Perham, R., 1972. The mechanical behavior of frozen earth materials under high pressure triaxial test conditions. Journal of Geotechnical 22 (3), 469–483. Chen, Z.Y., Zhou, J.X., Wang, H.T., 1994. Soil Mechanics. Tsinghua University Press, Beijing. Fish, A.M., 1984. Thermodynamic model of creep at constant stress and constant strain rate. Cold Regions Science and Technology 9 (2), 143–161. Fish, A.M., 1991. Strength of frozen soil under a combined stress state. Proceedings of the Sixth International Symposium on Ground Freezing, Beijing, China, Vol. 1, pp. 135–145. Haynes, F.D., Karalius, J.A., 1977. Effect of temperature on the strength of frozen silt. USA Cold Regions Research and Engineering Laboratory, CRREL Report 77-3, pp. 1–21. Haynes, F.D., Karalius, J.A., Kalafut, J., 1975. Strain rate effect on the strength of frozen silt. USA Cold Regions Research and Engineering Laboratory, Research Report, p. 350. Lai, Y.M., Cheng, H.B., Gao, Z.H., Zhang, S.J., Chang, X.X., 2007. Stress–strain relationships and nonlinear Mohr strength criterion of frozen sand clay. Chinese Journal of Rock Mechanics and Engineering 26 (8), 1612–1617. Lai, Y.M., Long, J., Chang, X.X., 2009. Yield criterion and elasto-plastic damage constitutive model for frozen sandy soil. International Journal of Plasticity 25, 1177–1205. Lai, Y.M., Yang, Y.G., Chang, X.X., Li, S.Y., 2010. Strength criterion and elastoplastic constitutive model of frozen silt in generalized plastic mechanics. International Journal of Plasticity 26, 1461–1484. Li, S.Y., Lai, Y.M., Zhang, S.J., Liu, D.R., 2009. An improved statistical damage constitutive model for warm frozen clay based on Mohr–Coulomb criterion. Cold Regions Science and Technology 57, 154–159. Ma, W., Wu, Z.W., Zhang, C.Q., 1994. Strength and yield criterion of frozen soil. Progress in Natural Science 4 (3), 319–322. Ma, W., Wu, Z.W., Chang, X.X., Sheng, Y., 1998. Effects of the shear stress strength and the average normal stress on deformation of frozen soil. Progress in Natural Science 8 (1), 77–81. Ma, W., Wu, Z.W., Zhang, L.X., Chang, X.X., 1999. Analyses of process on the strength decrease in frozen soils under high confining pressures. Cold Regions Science and Technology 29, 1–7. Ma, W., Zhu, Y.L., Ma, W.T., Chang, X.X., 2000. Analysis of deformation in frozen clayey soils. Journal of Glaciology and Geocryology 22 (1), 43–47. Parameswaran, V.R., Jones, S.J., 1981. Triaxial testing of frozen sand. Journal of Glaciology 27 (95), 147–155. Qi, J.L., Ma, W., 2010. State-of-art of research on mechanical properties of frozen soils. Rock and Soil Mechanics 31 (1), 133–143. Sayles, F.H., Carbee, D.L., 1981. Strength of frozen silt as a function of ice content and dry unit weight. Engineering Geology 18, 55–66. Sayles, F.H., Haines, D., 1974. Creep of frozen silt and clay. USA Cold Regions Research and Engineering Laboratory, Technical Report 252. Sayles, F.H., Baker, T.H.W., Gallavresi, F., Jessberger, H.L., Kinosita, S., Sadovskiy, A.V., Sego, D.C., Vyalov, S.S., 1987. Classification and laboratory testing of artificially frozen ground. Journal of Cold Regions Engineering. 1 (1), 22–39. Tsytovich, N.A., 1985. The Mechanics of Frozen Ground. Translated by Zhang C.Q, Zhu Y.L Beijing, Science Press. Vyalov, S.S., 1962. Strength and creep of frozen soils and calculations in ice-soil retaining structures. USA Cold Regions Research and Engineering Laboratory, Translation 76.AD 484093. Wu, Z.W., Ma, W., Zhang, C.Q., Sheng, Z.Y., 1994. Strength characteristics of frozen sandy soil. Journal of Glaciology and Geocryology 16 (1), 15–20. Zhang, X.F., Lai, Y.M., Yu, W.B., Wu, Y.P., 2004. Forecast and analysis of refreezing in FengHuoShan Permafrost tunnel. Chinese Journal of Rock Mechanics and Engineering 23 (24), 4170–4178. Zhang, S.J., Lai, Y.M., Sun, Z.Z., Gao, Z.H., 2007. Volumetric strain and strength behavior of frozen soils under confinement. Cold Regions Science and Technology 47, 263–270. Zhu, Y.L., 1988. Studies on strength and creep behavior of frozen soils in China. Journal of Glaciology and Geocryology 10 (3), 332–337. Zhu, Y.L., Carbee, D.L., 1987. Creep and strength behavior of frozen silt in uniaxial compression. USA Cold Regions Research and Engineering Laboratory, CRREL Report 87-10, pp. 1–60. Zhu, Y.L., Zhang, J.Y., 1982. Elastic and compressive deformation of frozen soils. Journal of Glaciology and Geocryology 4 (3), 29–39. Zhu, Y.L., Zhang, J.Y., Peng, W.W., Sheng, Z.Y., Miao, L.N., 1992. Constitutive relations of frozen soil in uniaxial compression. Journal of Glaciology and Geocryology 14 (3), 210–217.
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Cold Regions Science and Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o l d r e g i o n s
Laboratory investigation on strength and deformation characteristics of ice-saturated frozen sandy soil Xiangtian Xu a, Yuanming Lai a,⁎, Yuanhong Dong a, b, Jilin Qi a a b
State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Gansu, Lanzhou, 730000, China CCCC First Highway Consultants CO., LTD, Shaanxi, Xi'an, 710075, China
a r t i c l e
i n f o
Article history: Received 9 June 2011 Accepted 16 July 2011 Keywords: Frozen sandy soil Strength criterion Stress–strain behavior Pressure melting Confining pressure
a b s t r a c t To investigate the properties of ice-saturated frozen sandy soil, a series of triaxial compression tests on frozen sandy soil with a volumetric ice content of about 50% were carried out at a temperature of − 2.0 °C. The effect of confining pressure on strength and deformation features is analyzed according to the experimental results. The results show that the strength changes with increasing confining pressure in three distinct phases. According to the effective stress principle, the mechanism of strength is explained. A strength criterion is proposed to describe the strength characteristic. The equivalent stress versus axial strain curve shows strainsoftening under each confining pressure, and the extent of strain-softening decreases with the increase in confining pressure, until it behaves the so-called perfect elasto-plastic feature when the confining pressure is large enough. The improved Duncan–Chang hyperbolic model is taken to simulate the stress–strain behaviors. The simulation shows that the model can well describe the strain-softening. The dependency of the volumetric deformation on the confining pressure is also discussed. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In the engineering construction of cold regions and artificial ground freezing engineering, the mechanical property of frozen soil is an important scientific subject, especially strength and deformation. Frozen soil is composed of solid mineral particles, ice, liquid water and air (Tsytovich, 1985). The ice, which is sensitive to temperature, makes the mechanical properties of frozen soil become complex and different from unfrozen soil (Qi and Ma, 2010). The experimental studies on the mechanical properties of frozen soil started in the beginning of the last century (Zhu, 1988). Large amount of previous studies have shown that strength of frozen soils is related to temperature, strain rate, confining pressure as well as ice content. The influence of temperature on strength of frozen soils has been studied overwhelmingly. Sayles and Haines (1974) studied the influence of temperature on the strength of frozen silt and clay under constant strain rate, and fitted the relationship between strength and temperature through a power function. Haynes and Karalius (1977) investigated the influence of temperature on the strength of frozen clay under the condition that loading rate varied from 0.0423 cm/s to 4.23 cm/s. Furthermore, Bragg and Andersland (1981), Bourbonnais and Ladanyi (1985) and Wu et al. (1994) investigated the effect of temperature on the strength of frozen soil, respectively. Due to the
⁎ Corresponding author. Tel.: + 86 931 4967288. E-mail address: [email protected] (Y. Lai). 0165-232X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2011.07.005
existence of ice, frozen soils show strong rheological features, thus strain rate plays an important role in mechanical properties of frozen soils. Haynes et al. (1975) studied the influence of strain rate on the strength of frozen Fairbanks silty clay under the temperature of −9.4 °C, and concluded that unconfined compressive strength was highly sensitive to strain rate but not the initial tangent modulus. Baker (1979) carried out uniaxial compression experiment on frozen saturated Ottawa fine sand with the strain rate ranging from 10 −7/s to 10 −2/s, and concluded that the uniaxial strength of frozen Ottawa fine sand was an exponential function of strain rate. Parameswaran and Jones (1981) suggested that the relationship between strength of frozen soil and strain rate could be described linearly. The study on the frozen Wedron sand by Bragg and Andersland (1981) showed that the strength of frozen sand is independent of strain rate when strain rate is less than 10 −5/s. Confining pressure is another important influencing factor on strength of frozen soil. Chamberlain et al. (1972) carried out triaxial compression test on frozen saturated Ottawa sandy soil and New Hampshire silt under −10 °C with confining pressure varying from 0 MPa to 275.6 MPa, and investigated strength characteristic of frozen soil under different confining pressures. Fish (1991) and Ma et al. (1994) presented a parabolic strength criterion for frozen soil, considering the nonlinear influence of confining pressure on the strength of frozen soil. Ma et al. (1999) analyzed micro-mechanism of the weakening of frozen soil under high confining pressure, and the quantitative relation between confining pressure and unfrozen water content after pressure melting of Lanzhou fine sand was obtained. Additionally, some authors studied the influence of ice content on mechanical properties of
X. Xu et al. / Cold Regions Science and Technology 69 (2011) 98–104
frozen soil (Arenson et al., 2004; Sayles and Carbee, 1981; Vyalov, 1962). As for deformation of frozen soil, the current studies mainly focus on the creep and elasto-plastic deformation behavior under uniaxial/triaxial stress. Andersland and Akili (1967) described the creep behavior of frozen soil through Rate Process Theory (RPT). Fish (1984) proposed a creep model for frozen soil based on thermodynamics. Zhu and Carbee (1987) studied the creep of frozen soil under uniaxial stress, and an empirical equation for the creep of frozen soil was deduced. Zhu and Zhang (1982) investigated compressive deformation of frozen clay from the Qinghai–Tibetan Plateau. Zhu et al. (1992) proposed a series of models describing the constitutive relationship of frozen soil under uniaxial compression. Ma et al. (1998) examined deformation characteristic of frozen sand under triaxial test, and described the coupling response between shear stress and mean stress. Ma et al. (2000) studied the deformation characteristics of frozen clay through normalization method. Zhang et al. (2007) analyzed the volumetric deformation characteristics of frozen silty clay under triaxial compression. Li et al. (2009) proposed a statistical damage constitutive model based on Mohr–Coulomb criterion. Lai et al. (2009) established an elasto-plastic damage constitutive relationship for frozen sandy soil. Lai et al. (2010) presented an elasto-plastic constitutive relationship for frozen clay based on generalized plastic mechanics. With the development of studies on mechanical properties of frozen soils, mechanical test approaches of frozen soils have also been developed, some of which are standardized (Baker et al., 1981; Sayles et al., 1987). However, the soil types and test conditions of various researchers are not the same. So it is inappropriate to extrapolate from one to others (Arenson et al., 2007). Some engineering constructions, e.g. the Qinghai–Tibetan Railway and the Qinghai–Tibetan Expressway, have been or will be located in ice-saturated and ice-rich frozen soil regions (Cao, 2003; Zhang et al., 2004). It should be noticed that the ice-saturated frozen soil is not equivalent to the ice-rich frozen soil. Arenson et al. (2007) introduced a classification to frozen soils based on the volumetric ice content, and proposed a definition for icerich frozen soils. They also have carried out a number of studies on icerich permafrost (Arenson et al., 2003; Arenson and Springman, 2005a, b). Because the ice-saturated frozen soil will largely influence the stability of engineering constructions above it, it is therefore necessary to investigate its engineering properties, for instance strength and deformation characteristics of ice-saturated frozen soil. In this study, triaxial compression tests were conducted on a frozen Qinghai–Tibetan sandy soil. Stress–strain curves and strength changes with confining pressure were analyzed. On this basis, a strength criterion was proposed in p–q plane. An improved Duncan–Chang hyperbolic model (Lai et al., 2007) was employed to describe the stress–strain relationship. 2. Triaxial compression tests on frozen Qinghai–Tibetan sandy soil 2.1. Test equipment In this study, the test equipment is a cryogenic triaxial apparatus improved from MTS-810 material test machine which can measure the volumetric strain of specimen, as shown in Fig. 1. The axial strain of the specimen can be calculated according to the displacement of axial loading piston (‘7’ in Fig. 1) which was measured by displacement transducer (‘10’ in Fig. 1). Fig. 1 shows that after the displacements of pistons 5 and 7 were measured, the volumetric strain of the specimen can be calculated by the following formula: εv =
ΔV S ðL −L0 Þ + S5 ðlt −l0 Þ = 7 t V0 V0
where, V0 is the initial volume of specimen; S5 and S7 are the crosssectional areas of pistons 5 and 7, respectively, S5 = 38.32 cm 2,
99
Fig. 1. Schematic diagram of triaxial apparatus.
S7 = 30 cm 2 in this study; lt and l0 are the current and initial positions of piston 5, respectively; Lt and L0 are the current and initial positions of piston 7, respectively. The equipment provides three controlling modes: load control, displacement control and time control. The test process is automatically controlled by program and test data are collected automatically. The maximum axial load is 100 kN. The ranges of axial displacement and confining pressure are −85 cm–85 cm and 0 MPa–20 MPa, respectively. The loading liquid used in confining pressure loading system is an aircraft hydraulic oil. The refrigeration medium is ethyl alcohol. The controlled temperature range is from − 30 °C to 25 °C. 2.2. Test material and test procedures The test material was fine sandy soil collected from the Beiluhe test site along the Qinghai–Tibetan Railway, the grain size distribution is shown in Table 1. To decrease the difference between specimens, the specimens were made on a specially-made specimen machine. To ensure the uniformity of specimen, this sand was mixed with 15% moisture content (corresponding to volumetric ice content 21.8%) by weight at first, and then kept for 6 h with no evaporation. Afterward, the soil was put in a cylindrical mold to make cylindrical soil specimens with target dry densities. The specimens have a height of 12.5 cm and a diameter of 6.18 cm. The specimens were then saturated using a vacuum. The saturated specimens were put in refrigeration together with the mold. They were quickly frozen from top to bottom under − 30 °C to avoid frost heave. After the specimens were completely frozen, which usually took 48 h, the molds were removed and the specimens were mounted with epoxy resin platen on two ends and covered with rubber sleeve. After this, the specimens were kept in an incubator for over 12 h under the target testing temperature so that the specimen had a uniform temperature. The average volumetric ice content of the frozen specimens at the time of testing was about 50% and their dry density was about 1.33 g/cm 3. In order to investigate the influence of pressure melting on frozen soil strength, a series of undrained triaxial compression tests were carried out. Before the triaxial compression, the temperature in the Table 1 Particle fractions of the soil (%). <0.075 mm 0.075–0.1 mm 6.241 31.081
0.1–0.25 mm 54.410
0.25–0.5 mm 4.692
0.5–1 mm 2.241
>1 mm 0.869
100
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pressure chamber was set at −2.0 °C. After the temperature became stable, the specimen was put in a pressure chamber and a certain confining pressure was applied for 5 min, then axial loading started to be applied under the same confining pressure. The confining pressure ranges from 0.3 MPa to 18 MPa in the tests. The loading rate was 1.25 mm/min. The loading was stopped until the axial strain reached about 30%.
2.3. Experimental results In the triaxial compression test, the axial force, axial displacement, confining pressure and the piston displacement of the confiningpressure loading system were automatically collected by data logger. The piston displacement of the confining-pressure loading system was used to compute the volumetric deformation of the specimen (Zhang et al., 2007). The axial stress, axial strain and volumetric strain under different confining pressures can be calculated based on the testing data. The equivalent stress–axial strain curves and volumetric strain–axial strain curves under different confining pressures are shown in Figs. 2 and 3, respectively. Equivalent stress q is computed by:
1 q = pffiffiffi 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðσ1 −σ2 Þ2 + ðσ1 −σ3 Þ2 + ðσ2 −σ3 Þ2 :
ð1Þ
Volumetric strain εv = ε1 + ε2 + ε3, σ1, σ2 and σ3 are principal stresses, ε1, ε2 and ε3 are principal strains. Under the triaxial conditions applied in this paper, q = σ1 − σ3, εv = ε1 + 2ε3.
3. Strength characteristic of frozen sandy soil 3.1. Effects of confining pressures on strength From Fig. 2, it can be seen that the stress–strain curve of icesaturated frozen sandy soil behaves softening under each confining pressure. Therefore, the peak stress of each stress–strain curve is taken as strength. Experimental results show that the strength of frozen soil does not always increase with increase of confining pressure, which is different from unfrozen sandy soil. The relation between strength and confining pressure of frozen sandy soil is shown in Fig. 4, which indicates that the strength-confining pressure curve of the frozen sandy soil can be divided into three phases. In the first phase, the confining pressure range is from 0.3 MPa to 2 MPa. In this phase, the strength of the frozen sandy soil rises with the increase of confining pressure, and the maximum strength appears at σ3 = 2 MPa. In this phase, due to the relatively low confining pressure, pressure melting and crushing may not happen. The increase of confining pressure increases the strength of the frozen sandy soil in two aspects. One is that it increases the normal pressure on the shear surface, which increases the sliding frictional resistance; the other one is that it makes the solid particles (including soil particles and ice particles) tighter, and therefore increases the interlocking frictional force between solid particles. In the second phase, the confining pressure ranges from 2 MPa to 12 MPa. In this phase, the strength of the frozen sandy soil decreases with increase of confining pressure. This is because after confining pressure excesses 2 MPa, the increase of confining pressure makes the ice in the specimen crushed and pressure melting occurs. The crushed ice decreases the ice bonding force between solid grains. The pressure
Fig. 2. Relationships between axial strain and equivalent stress.
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101
Fig. 3. Relationships between axial strain and volumetric strain.
melting generates pore water, which decreases the internal frictional angle. At the same time, when pressure melting develops, excess hydrostatic pressure appears and the effective normal pressure on the shear surface is reduced. These are the intrinsic reasons that the strength of frozen soil decreases with increase of confining pressure in this phase. In the third phase, the confining pressure range is from 12 MPa to 18 MPa. After the confining pressure excesses 12 MPa, the strength of frozen sandy soil does not obviously decrease with further increase of confining pressure. There is a slight fluctuation near the strength of σ3 = 2 MPa. This illustrates that the strength of frozen sandy soil is hardly influenced by confining pressure when confining pressure excesses 12 MPa. Based on this strength feature of frozen soil, the minimum confining pressure under which the strength does not change with confining pressure is defined as the critical confining pressure σcr. In this test, the critical confining pressure is σcr = 12 MPa. The reason for the strength characteristic in the third phase is that when confining pressure reaches critical confining pressure σcr, a great quantity of pore water appears because of the pressure melting of the ice. Noticing that the test was undrained, when the confining pressure has an increment Δσ3, the excess hydrostatic pressure induced can be expressed as (Chen et al., 1994): Δu = BΔσ3 :
ð2Þ
For the saturated soil in the test, there is B ≈ 1. According to effective stress principle (Chen et al., 1994), when confining pressure excesses σcr, the effective confining pressure from the solid particles in frozen soil is: 0
σ3 = σ3 − u:
ð3Þ
Thus: 0
Δσ3 = Δσ3 −Δu:
ð4Þ
Based on Eqs. (2) and (4), when confining pressure excesses σcr, it can be obtained that: 0
Δσ3 ≈0:
ð5Þ
It is illustrated in Eq. (5) that when confining pressure reaches critical value σcr, the further increase of total confining pressure does not increase effective confining pressure, and the effective confining pressure in frozen soil keeps at σ’cr (the effective confining pressure corresponding to σcr). Therefore, after confining pressure excesses σcr, the increase of total confining pressure does not change the strength of frozen sandy soil any more, and neither crushing nor pressure melting happens. Therefore, the frozen soil strength keeps constant. 3.2. Strength criterion For unfrozen soil, several strength criteria considering effects of confining pressure have been presented, such as Mohr–Coulomb criterion, Drucker–Prager criterion and Matsuoka–Nakai criterion. However, these strength criteria can only describe the characteristic that strength increases with increase of confining pressure, but not for the strength changes with pressure melting in frozen soils. In order to illustrate reasonably the strength characteristic of frozen soil under various confining pressure, a nonlinear strength criterion in the p–q plane is proposed in this study, as follows: 0 −b
q−qu B = a@e pa
Fig. 4. Relationship between confining pressure and strength.
1 p p −c pa −e pa C A
ð6Þ
where p is average stress, p=(σ1 +2σ3)/3 under axisymmetric condition; Pa is normal atmospheric pressure, taken as Pa =0.101325 MPa; qu is the frozen soil strength when confining pressure is infinite, and it is close
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X. Xu et al. / Cold Regions Science and Technology 69 (2011) 98–104 Table 2 Material parameters of frozen sandy soil under different confining pressures. Confining pressure (MPa)
l (MPa−1)
m (MPa−1)
n (MPa−1)
0.3 0.5 0.8 1 2 3 4 5 6 7 8 10 12 14 16 18
0.01013 0.00707 0.00706 0.0087 0.00519 0.0051 0.00709 0.00832 0.00953 0.00971 0.0125 0.01222 0.00662 0.01289 0.01453 0.01389
0.84641 0.87641 0.85138 0.78041 0.7829 0.81434 0.83264 0.91814 0.83404 0.87789 0.87789 0.95654 1.16031 0.97201 1.12803 1.10385
0.79177 0.61048 0.3852 0.60464 0.45091 0.27808 0.2837 0.58743 0.9898 0.64313 0.78528 0.76322 1.06695 1.11668 1.05776 0.99914
Fig. 5. Comparisons of strength between test value and fitted value.
to the strength corresponding to critical confining pressure; a, b, and c are three dimensionless material parameters, which can be obtained by experimental data. On the p–q plane, the strength criterion (Eq. (6)) can be used to fit the experimental data, the fitted result is shown in Fig. 5. From this figure, the material parameters can be obtained as follows: a = 132.6425, b= 0.03743, c =0.04047, qu = 0.759 MPa. Fig. 5 also shows that the strength criterion can reasonably describe the strength characteristics of frozen soil, i.e., the strength increases with the increase of confining pressure, then decreases with a further increase, and does not change with confining pressure after the confining pressure reaches the critical value. Furthermore, the isotropic tensile strength −f3t can be predicted from the cross point of strength curve and p axis, −f3t =−1.19 MPa, which is consistent with the experimental results of Lai et al. (2007). This further illustrates the rationality of the strength criterion (Eq. (6)).
It can be seen from Fig. 2 that when the temperature was −2 °C, the frozen sandy soil with a volumetric ice content of 50% has a strainsoftening behavior under the confining pressure of 0.3–18.0 MPa. This is different from the ice-poor frozen sandy soil which behaves strainsoftening under relatively low confining pressure and strain-hardening under higher confining pressure (Lai et al., 2009). The experimental results show that the strain-softening extent of ice-saturated frozen sandy soil decreases with the increase of confining pressure. When confining pressure is less than 5 MPa, the strain-softening is obvious, which means that the stress–strain curve declines significantly after the peak stress. When confining pressure is larger than 5 MPa, the strainsoftening is slight, showing a perfect elasto-plastic deformation feature. When confining pressure is over 12 MPa, the stress–strain curves under different confining pressures are almost the same. This illustrates that the deformation characteristics of the frozen sandy soil is slightly influenced by confining pressures, which is consistent with the conclusion in the previous section that the effective confining pressure is not obviously increased with the increase of total confining pressure when confining pressure reaches its critical value. According to the strain-softening characteristics, an improved Duncan–Chang hyperbolic model is adopted to model the stress–strain behavior of frozen sandy soil (Lai et al., 2007): εa : lεa2 + mεa + n
dq −lεa2 + n = 2 : 2 dεa lεa + mεa + n
ð7Þ
In which, l, m and n are material parameters related to confining pressure with a dimension of MPa −1, respectively and their values under different confining pressures are shown in Table 2.
ð8Þ
According to the definition of initial elastic modulus, we can get: dq E0 = 100 dεa
εa = 0
4. Deformation characteristic
q=
The stress–strain curves, based on the improved Duncan–Chang hyperbolic model and experimental data, are shown in Fig. 6. Fig. 6 illustrates that the improved Duncan–Chang hyperbolic model can reasonably simulate the strain-softening behavior of frozen sandy soil under different confining pressures. This can theoretically be explained. The following equation can be obtained by differentiating Eq. (7):
=
100 n
ð9Þ
where, the factor 100 before the derivate of q is introduced because the axial strain is in percent. As the initial elastic modulus is positive, n is positive. Furthermore, according to the numerator in the right of pffiffiffiffiffiffiffiffi ffi Eq. (8), if l is positive, q is an increasing function when ε < n = l; a pffiffiffiffiffiffiffiffiffi while q is a decreasing function when εa > n = l. Therefore, Eq. (7) can be used to describe strain-softening behavior. If l is negative, q′ is above zero, therefore Eq. (7) can also describe strain-hardening behavior. The improved Duncan–Chang hyperbolic model can accurately describe the stress–strain behavior of frozen sandy soil, so an accurate initial elastic modulus can be obtained from Eq. (9). The relation between initial elastic modulus and confining pressure is plotted in Fig. 7. The change of initial elastic modulus with confining pressure is similar to that of strength with confining pressure. At the beginning, the initial elastic modulus increases with the increase of confining pressure. After the initial elastic modulus reaches its peak value, it decreases with a further increase of confining pressure. After confining pressure reaches a certain value, the initial elastic modulus is not influenced by confining pressure but keeps constant. This is because under relatively low confining pressure, the initial elastic modulus increases with the compaction of frozen sandy soil. With a further increase of confining pressure, the ice in the frozen soil begins to be crushed and pressure melting happens, both of which result in decrease in the initial elastic modulus. When the total confining pressure is large enough, the increase in confining pressure is undertaken by the excess hydrostatic pressure generated from pore water, while the effective confining pressure does not increase at all. Therefore, the initial elastic modulus is not changed by confining pressure any more.
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103
Fig. 6. Comparisons between test values and simulated values for stress–strain relationship.
Besides strain-softening, the volumetric deformation is also different from either unfrozen sandy soil or ice-poor frozen sandy soil. It can be seen in Fig. 3 that, at the beginning, the volumetric strain increases with the increase of axial strain (positive volumetric strain means that the specimen is compressed, while negative means expanded), showing that the specimen is compressed. When axial strain reaches a certain value, the volumetric strain reaches its maximum value which corresponds to different axial strains under different confining pressures. Afterward, the volumetric strain begins to decrease with a further increase of axial strain, indicating that the specimen is expanded. The specimen behaves in a contraction first and dilatancy later under each confining pressure; however, the
contraction is different under different confining pressures. The contraction is relatively large when confining pressure is less than 3 MPa (Fig. 3a). When confining pressure is larger than 3 MPa, the specimen experienced a slight contraction at the very initial stage, and followed by overwhelming dilatancy. This is because that when the confining pressure is higher, the contraction of specimen is relatively large during the confining pressure loading process, therefore, it is difficult to compress the specimen in the axial loading process and the dilation happens after failure. 5. Conclusions The strength and deformation characteristics of ice-saturated frozen sandy soil under different confining pressures are studied based on the triaxial compression tests. The following conclusions can be drawn from the present study:
Fig. 7. Initial elastic modulus under varying confining pressures.
(1) The strength of the frozen sandy soil changing with increasing confining pressure can be divided into three phases. In phase I, the confining pressure range is from 0.3 MPa to 2 MPa, strength increases with the increase of confining pressure and the maximum value observed at a confining pressure of 2 MPa; afterward, the strength decreases with the increase of confining pressure, which is phase II, confining pressure ranges from 2 MPa to 12 MPa in this phase; however, with the further increase in confining pressure to a critical value 12 MPa, the strength is not influenced by confining pressure any more, which is called phase III. (2) Based on the strength characteristic of frozen sandy soil, a nonlinear strength criterion for frozen sandy soil is proposed, which can reasonably describe the strength characteristic in all phases.
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(3) The stress–strain curve of ice-saturated frozen sandy soil exhibits strain-softening under all of the testing confining pressures and the strain-softening becomes weaker with the increase of confining pressure. The strain-softening is relatively obvious when confining pressure ranges from 0.3 MPa to 5 MPa, but slight in other confining pressures. Similarly, after the confining pressure of 12 MPa, the stress–strain curve was hardly influenced by the confining pressure. (4) The improved Duncan–Chang hyperbolic model is applied to simulate the deformation of frozen sandy soil. The simulation shows that the model can well describe the strain-softening behavior of frozen sandy soil. After this, the reason that the model can describe both strain-softening and strain-hardening is theoretically explained. (5) Confining pressure also has a profound effect on volumetric deformation and initial elastic modulus. Above a confining pressure of 3 MPa, the volumetric deformation behaves mainly in dilation. The initial elastic modulus changing with the increase of confining pressure is similar to the strength. Acknowledgments We would like to sincerely thank the two anonymous reviewers whose constructive comments are helpful for the publication of this paper. This research was supported by the National Natural Science Foundation of China (40730736, 40821001, 40801029), the Knowledge Innovation Program of the Chinese Academy of Sciences (Grant No. KZCX2-EW-QN301), the Project for Incubation of Specialists in Glaciology and Geocryology of National Natural Science Foundation of China (J0930003/J0109) and the foundation of State Key Laboratory of Frozen Soil Engineering (SKLFSE-ZY-03). References Andersland, O.B., Akili, W., 1967. Stress effect on creep rates of a frozen clay soil. Geotechnique 17 (1), 27–39. Arenson, L.U., Springman, S.M., 2005a. Mathematical descriptions for the behaviour of ice-rich frozen soils at temperatures close to 0 °C. Canadian Geotechnical Journal 42, 431–442. Arenson, L.U., Springman, S.M., 2005b. Triaxial constant stress and constant strain rate tests on ice-rich permafrost specimens. Canadian Geotechnical Journal 42, 412–430. Arenson, L.U., Almasi, N., Springman, S.M., 2003. Shearing response of ice-rich rock glacier material. Eight International Conference on Permafrost Zurich, Switzerland, pp. 39–44. Arenson, L.U., Johansen, M.M., Springman, S.M., 2004. Effects of volumetric ice content and strain rate on shear strength under triaxial conditions for frozen soil specimens. Permafrost and Periglacial Processes 15, 261–271. Arenson, L.U., Springman, S.M., Sego, D.C., 2007. The rheology of frozen soils. Applied Rheology 17 (1), 1–14. Baker, T.H.W., 1979. Strain rate effect on the compressive strength of frozen sand. Proceedings of the First Symposium on Ground Freezing Bochum, Germany, pp. 223–231. Baker, T.H.W., Jones, V.R., Parameswaran, V.R., 1981. Confined and unconfined compression tests of frozen sand. The Roger J.E. Brown Memorial Volume Proc. Fourth Canadian Permafrost Conference, pp. 387–393. Bourbonnais, J., Ladanyi, B., 1985. The mechanical behavior of frozen sand down to cryogenic temperatures. Proceedings of the Fourth International Symposium on Ground Freezing Sapporo, Japan, Vol. 1, pp. 235–244. Bragg, R.A., Andersland, O.B., 1981. Strain rate, temperature, and specimen size effects on compression and tensile properties of frozen soil. Engineering Geology 18, 35–46. Cao, Y.P., 2003. Test investigation on the ventilation duct embankment in permafrost regions. Subgrade Engineering 6, 12–15.
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