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Experimental and theoretical studies on the creep behavior of warm ice-rich frozen sand
Cold Regions Science and Technology 63 (2010) 61–67
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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
Experimental and theoretical studies on the creep behavior of warm ice-rich frozen sand Yugui Yang, Yuanming Lai ⁎, Xiaoxiao Chang State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou, 730000, China
a r t i c l e
i n f o
Article history: Received 1 March 2010 Accepted 22 April 2010 Keyword: Warm ice-rich frozen sand Creep Constitutive model Stability of embankment
a b s t r a c t It is recognized experimentally that the creep property of warm ice-rich frozen soils will cause a significant deformation, which affects the stability of infrastructure constructed in permafrost regions. In this paper, the creep behavior of warm ice-rich frozen sand is investigated through a series of experimental data under different stress levels at temperatures of −1.0 °C, −1.5 °C and −2.0 °C, respectively. The results showed that the creep characteristic of warm ice-rich frozen sand is greatly affected by the stress levels. The creep curves mainly present the decaying and the stable creep stages under low stress levels. However, under high stress levels, once the strain increasing to critical value, the creep strain velocity gradually increases and the specimen quickly happen to destroy. To reproduce the creep process of warm ice-rich frozen sand, a statistical damage constitutive model, in which the Weibull function is employed to describe the random distribution of inner flaw in creep process, is proposed based on the experimental results. The validity of the model is verified by comparing its modeling results with the results of creep tests both under low and high stress levels. It is found that the results predicted by this model agree well with the corresponding experimental data. © 2010 Elsevier B.V. All rights reserved.
1. Introduction According to Cheng (2003), the Qinghai–Tibetan Railway is about 134 km laid on a region of warm ice-rich permafrost. The warm icerich frozen soil, whose temperature usually ranges from −1.5 °C to the soil's freezing temperature and ice content is greater than 25%, is made up of soil particle, ice inclusions, liquid water and air (Tsytovich, 1985; Ma et al., 2008). Fig. 1 shows the electron microscope image of ice-rich frozen soil. With global climate warming, the engineering constructions in permafrost regions will encounter a lot of problems in the embankment stability of warm ice-rich frozen soil. The mechanical behavior of warm ice-rich frozen soil is crucial to the stability of construction of embankment engineering, such as highways, railways and other engineering activities in permafrost regions (Ma et al., 2005). To meet the needs of the application of engineering activities in permafrost such as the prediction of long-term deformation of embankment for Qinghai–Tibet railroad, a series of investigations has been carried out. In view of warm and ice-rich frozen soil is sensitive to temperature and water content, Ma et al. (2007) investigated uniaxial creep tests of frozen clay with different water contents (40%, 80%, 120%) at warm temperatures (− 0.3 °C, −0.5 °C, −1.0 °C). They found that the strain rate of warm ice-rich frozen clay always decreases with the increasing in time regardless of loading; the long-term strength of frozen clay decreases with the increasing
⁎ Corresponding author. E-mail address: [email protected] (Y. Lai). 0165-232X/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2010.04.011
water content at first, and then increases with the further increase of water content. To study the deformation mechanism of warm ice-rich frozen clay under constant load and variable temperature, which greatly influences the stability of embankment in permafrost, Zheng et al. (2009) carried out a series of investigations on the creep deformation of warm ice-rich frozen clay. They also verified that the creep deformation of warm ice-rich frozen clay are greatly affected by temperature, and found that the compressibility at − 1.5 °C or −1.0 °C is smaller than that at −0.5 °C or −0.3 °C. Consider that the permafrost with ice-rich content under the embankment poses an adverse effect on the operation of road, Qin et al. (2009) carried out experimental study on the compressible behavior of warm ice-rich frozen soil. They found that the warm ice-rich frozen clay is essentially sensitive to both load and temperature; the settlement incurred by the additional load from the built embankment is of considerable magnitude; with the increase in temperature, the settlement will be greatly magnified. To reproduce the creep deformation of warm icerich frozen clay, most often cited in the literature (Ma et al., 2007; Qin et al., 2009) and implemented in the finite element method (Majda and Skrodzewicz, 2009), the following phenomenological constitutive mode was employed to describe the creep behavior of frozen clay: B C
D
ε˙ = Aσ t eT
ð1Þ
where ε̇, σ and T is the strain rate, axial stress, and temperature of specimen, respectively; A, B, C, D are four experimental determined coefficients.
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Y. Yang et al. / Cold Regions Science and Technology 63 (2010) 61–67
ice-rich frozen soil, the detail process of preparation for specimens was presented in the following content. Firstly, according to the water content of specimen, we weigh the dry sand and the water weight for specimens. Then, a well calculated amount of water was added into the dry sand and mixed thoroughly. After preparation of the mixtures of soil and water, the specimen was prepared in a split mold and subsequently put into the refrigerator with a temperature of about −35 °C to freeze for 24 h, in which the specimen was frozen quickly to prevent the ice lens formation because of the water transmission. After that, the mould was dismantled and the specimen was subsequently coated by a plastic film and covered by an epoxy resin cap to avoid moisture evaporation. After preparation of the specimen, the specimen was placed into the pressure cell of MTS-810 low temperature testing machine for 24 h at given temperature. Temperature was controlled using a refrigerated circulating bath with a precision of 0.1 °C. Then, the axial pressure was applied on the specimen until reaching the given pressure and kept constant. 2.2. Test results and discussions Fig. 1. The electron microscope image of ice-rich frozen soil.
As can be seen from the references mentioned above, the creep curve shapes of warm ice-rich frozen clay only present both the decaying and the stable creep stages in spite of loading conditions in those papers. In fact, the creep curves of warm ice-rich frozen sand present the decaying and the stable creep stages under low stress level, however, under high stress level, once the strain increasing to a critical value, the creep strain velocity gradually increases and the specimen quickly happen to destroy. Based on the studies mentioned above, it is can be seen that the present researches on the creep characteristic of warm ice-rich frozen sand are still at the primary stage and only some phenomenological constitutive models are presented to describe the deformation behavior up to the present. However, the theoretical research, especially considering the effect of the random distribution of inner flaw in creep process, has been rarely reported for warm ice-rich frozen sand. So, the theoretical research on the creep characteristic of warm ice-rich frozen sand is becoming an important subject in the current development of mechanical theory of frozen soil. In this paper, the creep behavior of warm ice-rich frozen sand is investigated through a series of experimental data under different stress levels at temperatures of − 1.0 °C, − 1.5 °C and −2.0 °C, respectively. Based on experimental creep results, a statistical creep damage constitutive model is presented. The validity of the model is verified by comparing its modeling results with the results of tests.
When the testing was completed, the specimen's shape is shown in Fig. 2 and the typical creep curves of warm ice-rich frozen sand at a temperature of −1.0 °C are shown in Fig. 3, respectively. From Fig. 3, it is can be seen that three stages can be distinguished in the creep process of warm ice-rich frozen sand: the decaying creep stage, the initial creep strain rate of warm ice-rich frozen sand is very high, but with the increasing in time, the creep strain velocity decreases rapidly; the stable creep stage, in this stage, the creep strain rate almost unalters with the increase of time; the accelerating creep stage, once the strain increasing to a critical value, the creep strain velocity gradually increases and the specimen quickly happen to destroy under high stress levels. It is also found that the creep characteristic of warm ice-rich frozen sand is greatly affected by the load and temperature. The creep strain velocity under high stress level is larger than that under low stress level at same temperature and time. With the decreasing in temperature, the stable strain velocity decreases at the same stress levels, namely the stability of warm ice-rich frozen increases with the decreasing in temperature. As can be seen from the experimental result, the creep curve shapes of warm ice-rich frozen sand in the present work are quite different from those in Ma et al. (2007) and Qin et al. (2009) because the
2. Test conditions and results 2.1. Description of soil samples and testing technique In this paper, the soil used in test was sand taken from Qinghai– Tibet Railway constructions' site and physical parameters were listed in Table 1. The specimens were prepared as cylinders with 6.18 cm in diameter and 12.5 cm in height. The water content and the average dry density of the specimens tested were 30.0% and 1.43 g/cm3, respectively. Considering that it is difficult to prepare specimens for
Table 1 Basic parameters of warm ice-rich frozen sand. N 1.0 mm
1.0– 0.5 mm
0.5– 0.25 mm
0.25– 0.1 mm
0.1– 0.075 mm
b0.075 mm
0.869
2.241
4.692
54.41
31.081
6.241
Fig. 2. The specimen's shape after testing of warm ice-rich frozen sand.
Y. Yang et al. / Cold Regions Science and Technology 63 (2010) 61–67
63
microstructure damage development of frozen soil during the creep process using computerized tomography technique. The stress-strain states of CT scanning at decaying creep stage, stable creep stage and accelerating creep stage are shown in Fig. 4 (a)–(c). Recently, the stochastic theoretical investigations on creep damage have been developed. Considering that the defects are distributed randomly, which make some mechanical behaviors exhibit, Lai et al. (2008) presented a stochastic damage constitutive model to describe the great uncertainty of mechanical behavior of warm ice-rich frozen clays under uniaxial compressive tests. Li et al. (2009) presented an improved statistical damage constitutive model for warm frozen clay based on Mohr–Coulomb criterion. The predictive results showed that the stochastic method can establish its validity to describe the mechanical property of warm ice-rich frozen soil. The purpose of this study is to establish a creep stochastic damage constitutive model within the continuum damage mechanics to describe the creep process for warm ice-rich frozen soil. Some results are presented in order to verify the validity of the model. Fig. 3. The typical creep curves of warm ice-rich frozen sand at a temperature of −1.0 °C.
creep curves of warm ice-rich frozen clay only present both the decaying and the stable creep stages in spite of loading conditions in those papers. In fact, the creep curves of warm ice-rich frozen sand present the decaying and the stable creep stages under low stress level, namely the strain increases slowly with the increasing of time and the creep strain velocity gradually decreases. However, under high stress level, once the strain increasing to a critical value, the creep strain velocity gradually increases and the specimen quickly happen to destroy. The reason is that the deformation of ice-rich frozen clay is affected by cohesion in soil particles, internal friction and cementing force between ice and soil particles. The specimen of ice-rich frozen clay could endure large deformation under external load while the specimen is not destroyed. However, it is because of the lack of cohesion in sand particles, the cracks will appear in inside or on the surface of the specimen of ice-rich frozen sand once the deformation reaches a certain degree, which causes the specimen of warm ice-rich frozen sand is quickly destroyed. 3. Statistical damage creep constitutive model for warm ice-rich frozen sand In order to illustrate the approach of damage method proposed in this paper to be absolutely necessary and significant, Wu et al.'s researching work is introduced now. Wu et al. (1995) monitored the
3.1. The creep theory The Nishihara's model (Wang et al., 2006), which consists of the Hooke element, the Kelvin and Bingham model in series, has been widely used in rock materials. The model could well describe the viscoelastic-plastic deformation process for most of geomaterials. However, the experimental results have shown that Nishihara's model would never be able to match the data well for warm ice-rich frozen sand tested in this paper. To ensure a very good description of observed behavior of the warm ice-rich frozen sand within the range of the first two stages of creeping, the modified Nishihara's model is adopted by adding a Kelvin body into Nishihara's model in series (Fig. 5). From Fig. 5, it can be inferred that, if 0 b σ b σs4, the stress–strain relationship can be given by: σ1 = E1 ε1 σ2 = E2 ε2 + η2 ε˙ 2 σ3 = E3 ε3 + η3 ε˙ 3 σ = σ1 = σ2 = σ3 ε = ε1 + ε2 + ε3
g
ð2Þ
Based on Eq. (2), we have the creep strain vs. time dependence as follows: ε=
σ σ E σ E + 1− exp − 2 t + 1− exp − 3 t η2 η3 E1 E2 E3
Fig. 4. The CT diagrams of frozen sand at different creep stages (after Wu et al., 1995).
ð3Þ
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Y. Yang et al. / Cold Regions Science and Technology 63 (2010) 61–67
The stress–strain relationship can be obtained as follows: ε=
σ σ E σ E σ−σs4 + 1− exp − 2 t + 1− exp − 3 t + t η2 η3 η4 E1 E2 E3 ð5Þ
In fact, the creep behavior is assumed to be isotropic for many conditions. To be convenient in further use, the uniaxial model should be extended to the complex stress states and the following equations are given by: σm = 3Kεm G=
E 2ð1 + υÞ
ð6Þ
Sij = 2Geij K=
E 3ð1−2υÞ
ð7Þ
Based on Eqs. (6) and (7), substituting the tensor Sij for the uniaxial stress σ, and the furthermore formulae can be obtained as follows: eij =
Sij Sij Sij G G + 1− exp − 2 t + 1− exp − 3 t ; ð0bFbF0 Þ 2G1 2G2 H2 2G3 H3
ð8Þ Fig. 5. The creep model for warm ice-rich frozen sand.
Sij Sij Sij G G + 1− exp − 2 t + 1− exp − 3 t eij = 2G1 2G2 H2 2G3 H3 1 ∂Q ϕðF Þ〉 t; ðF ≥ F0 Þ + 2H4 〈 ∂fσ g
ð9Þ
where the function ϕ(F) is given by (Sun, 1999): ϕðF Þ = F−F0
ð10Þ
F−F0 −1 ϕðF Þ = exp M F0
ð11Þ
ϕðF Þ =
F−F0 N F0
ð12Þ
where M and N are parameters relating to yield function, M = 1, N = 1 for frictional geomaterials. 3.2. Statistical damage theory
Fig. 6. The damage schematic diagram.
If σ ≥ σs4, the stress–strain relationship can be given by: σ1 = E1 ε1 σ2 = E2 ε2 + η2 ε˙ 2 σ3 = E3 ε3 + η3 ε˙ 3 σ4 = η4 ε˙ 4 + σs4 σ = σ1 = σ2 = σ3 = σ4 ε = ε1 + ε2 + ε3 + ε4
g
ð4Þ
The continuum damage mechanics has been developed to describe the effect of microvoids or microcracks growth on the mechanical behavior of materials (Murakami, 1983). Kachanov (1958) proposed the concept of the effective stress theory, which has been used to formulate constitutive equations for damaged materials (Lemaitre and Chaboche, 1990). The effective stress theory postulates that the material damage is caused mainly by the decrease in the load-carrying effective area because of the growth of microvoids or microcracks as shown in Fig. 6 (Murakami, 1988). Here A0 is the initial cross-sectional area of undamaged specimen; AD is the area of microvoids or microcracks of damaged specimen; the damage parameter D is defined as the ratio of voids to the total crosssectional area of the specimen; σ is nominal stress applied to the
Table 2 Creep parameters of warm ice-rich frozen sand. Temperature
σ (MPa)
σs4 (MPa)
E1 (MPa)
E2 (MPa)
E3 (MPa)
η2 (MPa h)
η3 (MPa h)
− 1.0 °C
0.50 0.57 0.51 0.73 0.81 1.01
0.568
90.9 90.9 124.4 124.4 168.8 168.8
8.588 3.289 9.091 25.166 11.839 11.712
5.595 9.748 5.636 6.127 36.344 39.559
1.996 6.389 86.912 2.18 205.558 20.715
18.235 1.141 18.097 11.026 15.639 3.119
− 1.5 °C − 2.0 °C
0.600 0.987
η4 (MPa h)
m
n
0.559
0.279
3917.9
10.307
1.352
15700.8
3.969
1.181
31096.9
Y. Yang et al. / Cold Regions Science and Technology 63 (2010) 61–67
damaged specimen, σ = specimen. A˜ = A0 −AD = A0 ð1−DÞ
F A0
; σ˜ is the effective stress for the damaged
ð13Þ
σ˜ =
F σ = 1−D A˜
65
ð14Þ
Considering that the large randomness of the distribution and growth of the microvoids or microcracks for geomaterials, the
Fig. 7. Comparisons of creep strain curves between test results and predicted results.
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Y. Yang et al. / Cold Regions Science and Technology 63 (2010) 61–67
statistical methods give a powerful tool for the randomness. The Weibull distribution (Cao et al., 2003; Lai et al., 2008; Li et al., 2009) has been widely applied to model the damage behavior of geomaterials. In statistical damage mechanics, the damage variable D is defined as:
Then, the creep damage constitutive equations under complex stress state can be obtained as follows:
eij = N ðt Þ D= N
ð
Þ
ð
Þ
G2 G3 − t − t Sij Sij Sij + 1−e H2 + 1−e H3 ; 2G1 2G2 2G3
ð0 bFbF0 Þ
ð15Þ
ð22Þ
where N(t) is the number of damaged elements; N denotes all the elements. If the stress less than the value of σs4, we assume that there is no damage in specimen. The damage probability density function observes the Weibull distribution when stress is greater than or equal to the value of σs4 as follows:
G2 G3 − t − t Sij Sij Sij H H 2 3 + 1−e + 1−e eij = 2G1 ð1−DÞ 2G2 ð1−DÞ 2G3 ð1−DÞ ∂Q 1 + ϕ F˜ 〉 ðF≥F0 Þ t ð23Þ 〈 2H4 ∂fσ g
8 > <0 f ðt Þ = m ðt−tR Þm m−1 > exp − : ðt−tR Þ n n
9 0 b σ b σs4 > = σ ≥ σs4
> ;
ð16Þ
The number of damaged elements N(t) can be obtained from Eq. (16): t m m−1 Nðt Þ= ∫tR N ðt−tR Þ
n
8 <0 ðt−tR Þm exp − dt = ðt−tR Þm : N 1− exp − n n
ð17Þ where tR is the damage parameter. tR = 0 for warm ice-rich frozen sand in this paper. Substituting Eq. (17) into Eq. (15), the following expression can be obtained:
Dðt Þ =
8 > <0
9 0 b σ b σs4 > =
ðt−tR Þm > : 1− exp − n
σ ≥ σs4
> ;
ð18Þ
Based on the analysis, the uniaxial creep damage constitutive model can be derived from Eqs. (3), (5), (14) and (18): E E σ σ σ − 2t − 3t + 1−e η2 + 1−e η3 ð0 bσbσs4 Þ E1 E2 E3
ε=
E E σ σ σ − 2t − 3t + 1−e η2 + 1−e η3 E1 ð1−DÞ E2 ð1−DÞ E3 ð1−DÞ +
σ−ð1−DÞσs4 t η4 ð1−DÞ
ðσ≥σs4 Þ
ð19Þ
Þ
3.4. Model verification and analysis The proposed statistical creep damage model is now used to simulate some typical laboratory tests to check its validity, and the whole creep processes both for low and high stress levels of warm icerich frozen sand are calculated using the values of parameters given in Table 2, and some representative results are shown in Fig. 7. Fig. 7 shows the stress–strain curves obtained by both computation and experiment, and a good agreement between the numerical simulation and experimental data both for low and high stress levels can be seen. The influences of stress level and temperature on the creep behavior of warm ice-rich frozen sand are correctly described by the proposed model. This model also possesses a method that is relatively simple, which makes it feasible to be used for the purposes of engineering design and optimization.
To research the stability of construction in permafrost, in this paper, the creep behavior of warm ice-rich frozen sand is investigated through a series of experimental data under different stress levels at temperatures of − 1.0 °C, −1.5 °C and − 2 °C, respectively. The results showed that the creep characteristic of warm ice-rich frozen sand is greatly affected by the load and temperature. The creep curves mainly present the decaying and the stable creep stages under low stress levels. However, under high stress levels, once the strain increasing to critical value, the creep strain velocity gradually increases and the specimen quickly happen to destroy. To reproduce the creep deformation of warm ice-rich frozen sand, a statistical damage constitutive model, in which the Weibull function is employed to describe the random distribution of inner flaw in creep process, is proposed based on the experimental results. All parameters of the model are identified by experimental results. It is found that the results predicted by this model agree well with the corresponding experimental data. Acknowledgements
E E σ σ σ − 2t − 3t + 1−e η2 + 1−e η3 E1 E2 E3 h i ðt−tR Þm σs4 σ− exp − n ðt−tR Þm t exp + η4 n
f
ð
ð20Þ
Substituting Eq. (18) into Eq. (20), the statistical damage creep constitutive model for warm ice-rich frozen sand under uniaxial compressive condition can be obtained as follows:
ε=
Þ
4. Summaries and conclusions
3.3. Damage constitutive model for warm ice-rich frozen sand
ε=
ð
g
ð21Þ
We would like to thank very much the anonymous reviewers whose constructive comments are helpful for this paper revision. This research was supported by the National Hi-Tech Research and Development Plan (2008AA11Z103), the Western Project Program of the Chinese Academy of Sciences (KZCX2-XB2-10), the Project for Incubation of Specialists in Glaciology and Geocryology of National Natural Science Foundation of China (J0930003/ J0109), National Natural Science Foundation of China (40730736, 40971045) and the Program for Innovative Research Group of Natural Science Foundation of China (No. 40821001), and the foundation of State Key Laboratory of Frozen Soil Engineering (SKLFSE-ZY-03).
Y. Yang et al. / Cold Regions Science and Technology 63 (2010) 61–67
References Cao, W.G., Zhao, M.H., Tang, X.J., 2003. Study on simulation of statistical damage in the full process of rock failure. Chinese Journal of Geotechnical Engineering 25 (2), 184–187. Cheng, G.D., 2003. Construction of Qinghai–Tibetan railway with cooled roadbed. China Railway Science 24 (3), 1–4. Kachanov, L.M., 1958. Time of the rupture process under creep conditions. Izvestia Akademii Nauk SSSR, Otdelenie tekhnicheskich nauk no. 8, 26–31. Lai, Y.M., Li, Shuangyang, Qi, J.L., et al., 2008. Strength distributions of warm frozen clay and its stochastic damage constitutive model. Cold Regions Science and Technology 53, 200–215. Lemaitre, J., Chaboche, J.L., 1990. Mechanics of Solid Materials. Cambridge University Press, Cambridge. Li, S.Y., Lai, Y.M., Zhang, S.J., et al., 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., Cheng, G.D., Wu, Q.B., Wang, D.Y., 2005. Application on idea of dynamic design in Qinghai–Tibet Railway construction. Cold Regions Science and Technology 41 (3), 165–173. Ma, X.J., Zhang, J.M., Chang, X.X., et al., 2007. Experimental study on creep of warm and ice-rich frozen soil. Chinese Journal of Geotechnical Engineering 29 (6), 848–852. Ma, W., Feng, G.L., Wu, Q.B., et al., 2008. Analyses of temperature fields under the embankment with crushed-rock structures along the Qinghai–Tibet Railway. Cold Regions Science and Technology 53 (3), 259–270.
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Majda, P., Skrodzewicz, J., 2009. A modified creep model of epoxy adhesive at ambient temperature. International Journal of Adhesion and Adhesives 29, 396–404. Murakami, S., 1983. Notion of continuum damage mechanics and its application to anisotropic creep damage theory. Journal of Engineering Materials and Technology 105, 99–105. Murakami, S., 1988. Mechanical modeling of material damage. Journal of Applied Mechanics 55 (1), 24–27. Qin, Y.H., Zhang, J.M., Zheng, B., et al., 2009. Experimental study for the compressible behavior of warm and ice-rich frozen soil under the embankment of Qinghai–Tibet Railroad. Cold Regions Science and Technology 57, 148–153. Sun, J., 1999. Rethology of Geotechnical Material and its Engineering Application. China Architecture and Building Press. in Chinese. Tsytovich, N.A., 1985. The Mechanics of Frozen Ground. Science Press, Beijing. (translated by Zhang Changqing, Zhu Yuanling). Wang, R.H., Li, D.W., Wang, X.X., 2006. Improved Nishihara model and realization in ADINA FEM. Rock and Mechanics 27 (11), 1954–1958. Wu, Z.W., Ma, W., Pu, Y.B., et al., 1995. CT analysis on structure of frozen soil in creep process. CT Theory and Applications 4 (3), 31–34. Zheng, B., Zhang, J.M., Ma, X.J., et al., 2009. Study on Compression deformation of warm and ice-enriched frozen soil. Chinese Journal of Rock Mechanics and Engineering 28 (Supp. 1), 3063–3068.
Contents lists available at ScienceDirect
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
Experimental and theoretical studies on the creep behavior of warm ice-rich frozen sand Yugui Yang, Yuanming Lai ⁎, Xiaoxiao Chang State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou, 730000, China
a r t i c l e
i n f o
Article history: Received 1 March 2010 Accepted 22 April 2010 Keyword: Warm ice-rich frozen sand Creep Constitutive model Stability of embankment
a b s t r a c t It is recognized experimentally that the creep property of warm ice-rich frozen soils will cause a significant deformation, which affects the stability of infrastructure constructed in permafrost regions. In this paper, the creep behavior of warm ice-rich frozen sand is investigated through a series of experimental data under different stress levels at temperatures of −1.0 °C, −1.5 °C and −2.0 °C, respectively. The results showed that the creep characteristic of warm ice-rich frozen sand is greatly affected by the stress levels. The creep curves mainly present the decaying and the stable creep stages under low stress levels. However, under high stress levels, once the strain increasing to critical value, the creep strain velocity gradually increases and the specimen quickly happen to destroy. To reproduce the creep process of warm ice-rich frozen sand, a statistical damage constitutive model, in which the Weibull function is employed to describe the random distribution of inner flaw in creep process, is proposed based on the experimental results. The validity of the model is verified by comparing its modeling results with the results of creep tests both under low and high stress levels. It is found that the results predicted by this model agree well with the corresponding experimental data. © 2010 Elsevier B.V. All rights reserved.
1. Introduction According to Cheng (2003), the Qinghai–Tibetan Railway is about 134 km laid on a region of warm ice-rich permafrost. The warm icerich frozen soil, whose temperature usually ranges from −1.5 °C to the soil's freezing temperature and ice content is greater than 25%, is made up of soil particle, ice inclusions, liquid water and air (Tsytovich, 1985; Ma et al., 2008). Fig. 1 shows the electron microscope image of ice-rich frozen soil. With global climate warming, the engineering constructions in permafrost regions will encounter a lot of problems in the embankment stability of warm ice-rich frozen soil. The mechanical behavior of warm ice-rich frozen soil is crucial to the stability of construction of embankment engineering, such as highways, railways and other engineering activities in permafrost regions (Ma et al., 2005). To meet the needs of the application of engineering activities in permafrost such as the prediction of long-term deformation of embankment for Qinghai–Tibet railroad, a series of investigations has been carried out. In view of warm and ice-rich frozen soil is sensitive to temperature and water content, Ma et al. (2007) investigated uniaxial creep tests of frozen clay with different water contents (40%, 80%, 120%) at warm temperatures (− 0.3 °C, −0.5 °C, −1.0 °C). They found that the strain rate of warm ice-rich frozen clay always decreases with the increasing in time regardless of loading; the long-term strength of frozen clay decreases with the increasing
⁎ Corresponding author. E-mail address: [email protected] (Y. Lai). 0165-232X/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2010.04.011
water content at first, and then increases with the further increase of water content. To study the deformation mechanism of warm ice-rich frozen clay under constant load and variable temperature, which greatly influences the stability of embankment in permafrost, Zheng et al. (2009) carried out a series of investigations on the creep deformation of warm ice-rich frozen clay. They also verified that the creep deformation of warm ice-rich frozen clay are greatly affected by temperature, and found that the compressibility at − 1.5 °C or −1.0 °C is smaller than that at −0.5 °C or −0.3 °C. Consider that the permafrost with ice-rich content under the embankment poses an adverse effect on the operation of road, Qin et al. (2009) carried out experimental study on the compressible behavior of warm ice-rich frozen soil. They found that the warm ice-rich frozen clay is essentially sensitive to both load and temperature; the settlement incurred by the additional load from the built embankment is of considerable magnitude; with the increase in temperature, the settlement will be greatly magnified. To reproduce the creep deformation of warm icerich frozen clay, most often cited in the literature (Ma et al., 2007; Qin et al., 2009) and implemented in the finite element method (Majda and Skrodzewicz, 2009), the following phenomenological constitutive mode was employed to describe the creep behavior of frozen clay: B C
D
ε˙ = Aσ t eT
ð1Þ
where ε̇, σ and T is the strain rate, axial stress, and temperature of specimen, respectively; A, B, C, D are four experimental determined coefficients.
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ice-rich frozen soil, the detail process of preparation for specimens was presented in the following content. Firstly, according to the water content of specimen, we weigh the dry sand and the water weight for specimens. Then, a well calculated amount of water was added into the dry sand and mixed thoroughly. After preparation of the mixtures of soil and water, the specimen was prepared in a split mold and subsequently put into the refrigerator with a temperature of about −35 °C to freeze for 24 h, in which the specimen was frozen quickly to prevent the ice lens formation because of the water transmission. After that, the mould was dismantled and the specimen was subsequently coated by a plastic film and covered by an epoxy resin cap to avoid moisture evaporation. After preparation of the specimen, the specimen was placed into the pressure cell of MTS-810 low temperature testing machine for 24 h at given temperature. Temperature was controlled using a refrigerated circulating bath with a precision of 0.1 °C. Then, the axial pressure was applied on the specimen until reaching the given pressure and kept constant. 2.2. Test results and discussions Fig. 1. The electron microscope image of ice-rich frozen soil.
As can be seen from the references mentioned above, the creep curve shapes of warm ice-rich frozen clay only present both the decaying and the stable creep stages in spite of loading conditions in those papers. In fact, the creep curves of warm ice-rich frozen sand present the decaying and the stable creep stages under low stress level, however, under high stress level, once the strain increasing to a critical value, the creep strain velocity gradually increases and the specimen quickly happen to destroy. Based on the studies mentioned above, it is can be seen that the present researches on the creep characteristic of warm ice-rich frozen sand are still at the primary stage and only some phenomenological constitutive models are presented to describe the deformation behavior up to the present. However, the theoretical research, especially considering the effect of the random distribution of inner flaw in creep process, has been rarely reported for warm ice-rich frozen sand. So, the theoretical research on the creep characteristic of warm ice-rich frozen sand is becoming an important subject in the current development of mechanical theory of frozen soil. In this paper, the creep behavior of warm ice-rich frozen sand is investigated through a series of experimental data under different stress levels at temperatures of − 1.0 °C, − 1.5 °C and −2.0 °C, respectively. Based on experimental creep results, a statistical creep damage constitutive model is presented. The validity of the model is verified by comparing its modeling results with the results of tests.
When the testing was completed, the specimen's shape is shown in Fig. 2 and the typical creep curves of warm ice-rich frozen sand at a temperature of −1.0 °C are shown in Fig. 3, respectively. From Fig. 3, it is can be seen that three stages can be distinguished in the creep process of warm ice-rich frozen sand: the decaying creep stage, the initial creep strain rate of warm ice-rich frozen sand is very high, but with the increasing in time, the creep strain velocity decreases rapidly; the stable creep stage, in this stage, the creep strain rate almost unalters with the increase of time; the accelerating creep stage, once the strain increasing to a critical value, the creep strain velocity gradually increases and the specimen quickly happen to destroy under high stress levels. It is also found that the creep characteristic of warm ice-rich frozen sand is greatly affected by the load and temperature. The creep strain velocity under high stress level is larger than that under low stress level at same temperature and time. With the decreasing in temperature, the stable strain velocity decreases at the same stress levels, namely the stability of warm ice-rich frozen increases with the decreasing in temperature. As can be seen from the experimental result, the creep curve shapes of warm ice-rich frozen sand in the present work are quite different from those in Ma et al. (2007) and Qin et al. (2009) because the
2. Test conditions and results 2.1. Description of soil samples and testing technique In this paper, the soil used in test was sand taken from Qinghai– Tibet Railway constructions' site and physical parameters were listed in Table 1. The specimens were prepared as cylinders with 6.18 cm in diameter and 12.5 cm in height. The water content and the average dry density of the specimens tested were 30.0% and 1.43 g/cm3, respectively. Considering that it is difficult to prepare specimens for
Table 1 Basic parameters of warm ice-rich frozen sand. N 1.0 mm
1.0– 0.5 mm
0.5– 0.25 mm
0.25– 0.1 mm
0.1– 0.075 mm
b0.075 mm
0.869
2.241
4.692
54.41
31.081
6.241
Fig. 2. The specimen's shape after testing of warm ice-rich frozen sand.
Y. Yang et al. / Cold Regions Science and Technology 63 (2010) 61–67
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microstructure damage development of frozen soil during the creep process using computerized tomography technique. The stress-strain states of CT scanning at decaying creep stage, stable creep stage and accelerating creep stage are shown in Fig. 4 (a)–(c). Recently, the stochastic theoretical investigations on creep damage have been developed. Considering that the defects are distributed randomly, which make some mechanical behaviors exhibit, Lai et al. (2008) presented a stochastic damage constitutive model to describe the great uncertainty of mechanical behavior of warm ice-rich frozen clays under uniaxial compressive tests. Li et al. (2009) presented an improved statistical damage constitutive model for warm frozen clay based on Mohr–Coulomb criterion. The predictive results showed that the stochastic method can establish its validity to describe the mechanical property of warm ice-rich frozen soil. The purpose of this study is to establish a creep stochastic damage constitutive model within the continuum damage mechanics to describe the creep process for warm ice-rich frozen soil. Some results are presented in order to verify the validity of the model. Fig. 3. The typical creep curves of warm ice-rich frozen sand at a temperature of −1.0 °C.
creep curves of warm ice-rich frozen clay only present both the decaying and the stable creep stages in spite of loading conditions in those papers. In fact, the creep curves of warm ice-rich frozen sand present the decaying and the stable creep stages under low stress level, namely the strain increases slowly with the increasing of time and the creep strain velocity gradually decreases. However, under high stress level, once the strain increasing to a critical value, the creep strain velocity gradually increases and the specimen quickly happen to destroy. The reason is that the deformation of ice-rich frozen clay is affected by cohesion in soil particles, internal friction and cementing force between ice and soil particles. The specimen of ice-rich frozen clay could endure large deformation under external load while the specimen is not destroyed. However, it is because of the lack of cohesion in sand particles, the cracks will appear in inside or on the surface of the specimen of ice-rich frozen sand once the deformation reaches a certain degree, which causes the specimen of warm ice-rich frozen sand is quickly destroyed. 3. Statistical damage creep constitutive model for warm ice-rich frozen sand In order to illustrate the approach of damage method proposed in this paper to be absolutely necessary and significant, Wu et al.'s researching work is introduced now. Wu et al. (1995) monitored the
3.1. The creep theory The Nishihara's model (Wang et al., 2006), which consists of the Hooke element, the Kelvin and Bingham model in series, has been widely used in rock materials. The model could well describe the viscoelastic-plastic deformation process for most of geomaterials. However, the experimental results have shown that Nishihara's model would never be able to match the data well for warm ice-rich frozen sand tested in this paper. To ensure a very good description of observed behavior of the warm ice-rich frozen sand within the range of the first two stages of creeping, the modified Nishihara's model is adopted by adding a Kelvin body into Nishihara's model in series (Fig. 5). From Fig. 5, it can be inferred that, if 0 b σ b σs4, the stress–strain relationship can be given by: σ1 = E1 ε1 σ2 = E2 ε2 + η2 ε˙ 2 σ3 = E3 ε3 + η3 ε˙ 3 σ = σ1 = σ2 = σ3 ε = ε1 + ε2 + ε3
g
ð2Þ
Based on Eq. (2), we have the creep strain vs. time dependence as follows: ε=
σ σ E σ E + 1− exp − 2 t + 1− exp − 3 t η2 η3 E1 E2 E3
Fig. 4. The CT diagrams of frozen sand at different creep stages (after Wu et al., 1995).
ð3Þ
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Y. Yang et al. / Cold Regions Science and Technology 63 (2010) 61–67
The stress–strain relationship can be obtained as follows: ε=
σ σ E σ E σ−σs4 + 1− exp − 2 t + 1− exp − 3 t + t η2 η3 η4 E1 E2 E3 ð5Þ
In fact, the creep behavior is assumed to be isotropic for many conditions. To be convenient in further use, the uniaxial model should be extended to the complex stress states and the following equations are given by: σm = 3Kεm G=
E 2ð1 + υÞ
ð6Þ
Sij = 2Geij K=
E 3ð1−2υÞ
ð7Þ
Based on Eqs. (6) and (7), substituting the tensor Sij for the uniaxial stress σ, and the furthermore formulae can be obtained as follows: eij =
Sij Sij Sij G G + 1− exp − 2 t + 1− exp − 3 t ; ð0bFbF0 Þ 2G1 2G2 H2 2G3 H3
ð8Þ Fig. 5. The creep model for warm ice-rich frozen sand.
Sij Sij Sij G G + 1− exp − 2 t + 1− exp − 3 t eij = 2G1 2G2 H2 2G3 H3 1 ∂Q ϕðF Þ〉 t; ðF ≥ F0 Þ + 2H4 〈 ∂fσ g
ð9Þ
where the function ϕ(F) is given by (Sun, 1999): ϕðF Þ = F−F0
ð10Þ
F−F0 −1 ϕðF Þ = exp M F0
ð11Þ
ϕðF Þ =
F−F0 N F0
ð12Þ
where M and N are parameters relating to yield function, M = 1, N = 1 for frictional geomaterials. 3.2. Statistical damage theory
Fig. 6. The damage schematic diagram.
If σ ≥ σs4, the stress–strain relationship can be given by: σ1 = E1 ε1 σ2 = E2 ε2 + η2 ε˙ 2 σ3 = E3 ε3 + η3 ε˙ 3 σ4 = η4 ε˙ 4 + σs4 σ = σ1 = σ2 = σ3 = σ4 ε = ε1 + ε2 + ε3 + ε4
g
ð4Þ
The continuum damage mechanics has been developed to describe the effect of microvoids or microcracks growth on the mechanical behavior of materials (Murakami, 1983). Kachanov (1958) proposed the concept of the effective stress theory, which has been used to formulate constitutive equations for damaged materials (Lemaitre and Chaboche, 1990). The effective stress theory postulates that the material damage is caused mainly by the decrease in the load-carrying effective area because of the growth of microvoids or microcracks as shown in Fig. 6 (Murakami, 1988). Here A0 is the initial cross-sectional area of undamaged specimen; AD is the area of microvoids or microcracks of damaged specimen; the damage parameter D is defined as the ratio of voids to the total crosssectional area of the specimen; σ is nominal stress applied to the
Table 2 Creep parameters of warm ice-rich frozen sand. Temperature
σ (MPa)
σs4 (MPa)
E1 (MPa)
E2 (MPa)
E3 (MPa)
η2 (MPa h)
η3 (MPa h)
− 1.0 °C
0.50 0.57 0.51 0.73 0.81 1.01
0.568
90.9 90.9 124.4 124.4 168.8 168.8
8.588 3.289 9.091 25.166 11.839 11.712
5.595 9.748 5.636 6.127 36.344 39.559
1.996 6.389 86.912 2.18 205.558 20.715
18.235 1.141 18.097 11.026 15.639 3.119
− 1.5 °C − 2.0 °C
0.600 0.987
η4 (MPa h)
m
n
0.559
0.279
3917.9
10.307
1.352
15700.8
3.969
1.181
31096.9
Y. Yang et al. / Cold Regions Science and Technology 63 (2010) 61–67
damaged specimen, σ = specimen. A˜ = A0 −AD = A0 ð1−DÞ
F A0
; σ˜ is the effective stress for the damaged
ð13Þ
σ˜ =
F σ = 1−D A˜
65
ð14Þ
Considering that the large randomness of the distribution and growth of the microvoids or microcracks for geomaterials, the
Fig. 7. Comparisons of creep strain curves between test results and predicted results.
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Y. Yang et al. / Cold Regions Science and Technology 63 (2010) 61–67
statistical methods give a powerful tool for the randomness. The Weibull distribution (Cao et al., 2003; Lai et al., 2008; Li et al., 2009) has been widely applied to model the damage behavior of geomaterials. In statistical damage mechanics, the damage variable D is defined as:
Then, the creep damage constitutive equations under complex stress state can be obtained as follows:
eij = N ðt Þ D= N
ð
Þ
ð
Þ
G2 G3 − t − t Sij Sij Sij + 1−e H2 + 1−e H3 ; 2G1 2G2 2G3
ð0 bFbF0 Þ
ð15Þ
ð22Þ
where N(t) is the number of damaged elements; N denotes all the elements. If the stress less than the value of σs4, we assume that there is no damage in specimen. The damage probability density function observes the Weibull distribution when stress is greater than or equal to the value of σs4 as follows:
G2 G3 − t − t Sij Sij Sij H H 2 3 + 1−e + 1−e eij = 2G1 ð1−DÞ 2G2 ð1−DÞ 2G3 ð1−DÞ ∂Q 1 + ϕ F˜ 〉 ðF≥F0 Þ t ð23Þ 〈 2H4 ∂fσ g
8 > <0 f ðt Þ = m ðt−tR Þm m−1 > exp − : ðt−tR Þ n n
9 0 b σ b σs4 > = σ ≥ σs4
> ;
ð16Þ
The number of damaged elements N(t) can be obtained from Eq. (16): t m m−1 Nðt Þ= ∫tR N ðt−tR Þ
n
8 <0 ðt−tR Þm exp − dt = ðt−tR Þm : N 1− exp − n n
ð17Þ where tR is the damage parameter. tR = 0 for warm ice-rich frozen sand in this paper. Substituting Eq. (17) into Eq. (15), the following expression can be obtained:
Dðt Þ =
8 > <0
9 0 b σ b σs4 > =
ðt−tR Þm > : 1− exp − n
σ ≥ σs4
> ;
ð18Þ
Based on the analysis, the uniaxial creep damage constitutive model can be derived from Eqs. (3), (5), (14) and (18): E E σ σ σ − 2t − 3t + 1−e η2 + 1−e η3 ð0 bσbσs4 Þ E1 E2 E3
ε=
E E σ σ σ − 2t − 3t + 1−e η2 + 1−e η3 E1 ð1−DÞ E2 ð1−DÞ E3 ð1−DÞ +
σ−ð1−DÞσs4 t η4 ð1−DÞ
ðσ≥σs4 Þ
ð19Þ
Þ
3.4. Model verification and analysis The proposed statistical creep damage model is now used to simulate some typical laboratory tests to check its validity, and the whole creep processes both for low and high stress levels of warm icerich frozen sand are calculated using the values of parameters given in Table 2, and some representative results are shown in Fig. 7. Fig. 7 shows the stress–strain curves obtained by both computation and experiment, and a good agreement between the numerical simulation and experimental data both for low and high stress levels can be seen. The influences of stress level and temperature on the creep behavior of warm ice-rich frozen sand are correctly described by the proposed model. This model also possesses a method that is relatively simple, which makes it feasible to be used for the purposes of engineering design and optimization.
To research the stability of construction in permafrost, in this paper, the creep behavior of warm ice-rich frozen sand is investigated through a series of experimental data under different stress levels at temperatures of − 1.0 °C, −1.5 °C and − 2 °C, respectively. The results showed that the creep characteristic of warm ice-rich frozen sand is greatly affected by the load and temperature. The creep curves mainly present the decaying and the stable creep stages under low stress levels. However, under high stress levels, once the strain increasing to critical value, the creep strain velocity gradually increases and the specimen quickly happen to destroy. To reproduce the creep deformation of warm ice-rich frozen sand, a statistical damage constitutive model, in which the Weibull function is employed to describe the random distribution of inner flaw in creep process, is proposed based on the experimental results. All parameters of the model are identified by experimental results. It is found that the results predicted by this model agree well with the corresponding experimental data. Acknowledgements
E E σ σ σ − 2t − 3t + 1−e η2 + 1−e η3 E1 E2 E3 h i ðt−tR Þm σs4 σ− exp − n ðt−tR Þm t exp + η4 n
f
ð
ð20Þ
Substituting Eq. (18) into Eq. (20), the statistical damage creep constitutive model for warm ice-rich frozen sand under uniaxial compressive condition can be obtained as follows:
ε=
Þ
4. Summaries and conclusions
3.3. Damage constitutive model for warm ice-rich frozen sand
ε=
ð
g
ð21Þ
We would like to thank very much the anonymous reviewers whose constructive comments are helpful for this paper revision. This research was supported by the National Hi-Tech Research and Development Plan (2008AA11Z103), the Western Project Program of the Chinese Academy of Sciences (KZCX2-XB2-10), the Project for Incubation of Specialists in Glaciology and Geocryology of National Natural Science Foundation of China (J0930003/ J0109), National Natural Science Foundation of China (40730736, 40971045) and the Program for Innovative Research Group of Natural Science Foundation of China (No. 40821001), and the foundation of State Key Laboratory of Frozen Soil Engineering (SKLFSE-ZY-03).
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References Cao, W.G., Zhao, M.H., Tang, X.J., 2003. Study on simulation of statistical damage in the full process of rock failure. Chinese Journal of Geotechnical Engineering 25 (2), 184–187. Cheng, G.D., 2003. Construction of Qinghai–Tibetan railway with cooled roadbed. China Railway Science 24 (3), 1–4. Kachanov, L.M., 1958. Time of the rupture process under creep conditions. Izvestia Akademii Nauk SSSR, Otdelenie tekhnicheskich nauk no. 8, 26–31. Lai, Y.M., Li, Shuangyang, Qi, J.L., et al., 2008. Strength distributions of warm frozen clay and its stochastic damage constitutive model. Cold Regions Science and Technology 53, 200–215. Lemaitre, J., Chaboche, J.L., 1990. Mechanics of Solid Materials. Cambridge University Press, Cambridge. Li, S.Y., Lai, Y.M., Zhang, S.J., et al., 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., Cheng, G.D., Wu, Q.B., Wang, D.Y., 2005. Application on idea of dynamic design in Qinghai–Tibet Railway construction. Cold Regions Science and Technology 41 (3), 165–173. Ma, X.J., Zhang, J.M., Chang, X.X., et al., 2007. Experimental study on creep of warm and ice-rich frozen soil. Chinese Journal of Geotechnical Engineering 29 (6), 848–852. Ma, W., Feng, G.L., Wu, Q.B., et al., 2008. Analyses of temperature fields under the embankment with crushed-rock structures along the Qinghai–Tibet Railway. Cold Regions Science and Technology 53 (3), 259–270.
67
Majda, P., Skrodzewicz, J., 2009. A modified creep model of epoxy adhesive at ambient temperature. International Journal of Adhesion and Adhesives 29, 396–404. Murakami, S., 1983. Notion of continuum damage mechanics and its application to anisotropic creep damage theory. Journal of Engineering Materials and Technology 105, 99–105. Murakami, S., 1988. Mechanical modeling of material damage. Journal of Applied Mechanics 55 (1), 24–27. Qin, Y.H., Zhang, J.M., Zheng, B., et al., 2009. Experimental study for the compressible behavior of warm and ice-rich frozen soil under the embankment of Qinghai–Tibet Railroad. Cold Regions Science and Technology 57, 148–153. Sun, J., 1999. Rethology of Geotechnical Material and its Engineering Application. China Architecture and Building Press. in Chinese. Tsytovich, N.A., 1985. The Mechanics of Frozen Ground. Science Press, Beijing. (translated by Zhang Changqing, Zhu Yuanling). Wang, R.H., Li, D.W., Wang, X.X., 2006. Improved Nishihara model and realization in ADINA FEM. Rock and Mechanics 27 (11), 1954–1958. Wu, Z.W., Ma, W., Pu, Y.B., et al., 1995. CT analysis on structure of frozen soil in creep process. CT Theory and Applications 4 (3), 31–34. Zheng, B., Zhang, J.M., Ma, X.J., et al., 2009. Study on Compression deformation of warm and ice-enriched frozen soil. Chinese Journal of Rock Mechanics and Engineering 28 (Supp. 1), 3063–3068.