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A one-dimensional creep model for frozen soils taking temperature as an independent variable
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ScienceDirect Soils and Foundations xxx (2018) xxx–xxx www.elsevier.com/locate/sandf
A one-dimensional creep model for frozen soils taking temperature as an independent variable Xiaoliang Yao a,⇑, Jilin Qi b, Jianming Zhang a, Fan Yu b a
State Key Laboratory of Frozen Soil Engineering, Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, China b School of Civil and Transportation Engineering, Beijing University of Architecture and Civil Engineering, China Received 16 March 2017; received in revised form 8 February 2018; accepted 15 February 2018
Abstract In this paper, a one-dimensional creep model that takes temperature as an independent variable was proposed for frozen soils. A series of K0 compression tests was conducted under different constant surcharge loads with stepped increases in temperature. An analysis of the test results indicated that the characteristics of creep strain, developing due to the stepped increases in temperature, matched well with the parallel lines postulate of the isotache model for unfrozen soils. The independent variable (stress) in the original model was replaced with a reciprocal of the temperature’s absolute value, and a novel model for directly describing the effect of the increase in temperature on the creep behavior of frozen soils was established. It was verified by the test results in this study and in previous research work that the tendency of the stepped development of creep strain, due to the stepped increases in temperature, can be reasonably captured by the model. Based on further analysis of the test results, a simplified parameter-obtaining method was recommended and its applicability was verified. Ó 2018 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society.
Keywords: Frozen soils; Creep model; Temperature; Independent variable
1. Introduction Over the past few decades, a rise in the large-scale ground temperature, characterized by permafrost degradation, has been observed in cold regions (Li and Cheng, 1999; Yu et al., 2013). Due to this rise in ground temperature, warmer permafrost layers have often formed and led to considerable creep settlement in the layers (Qi et al., 2007; Ma et al., 2008). In certain areas of cold regions, where the deformations of infrastructure foundations need to be strictly controlled, such as along high-grade highways and high-speed railways, the creep of the permafrost layers must be taken into consideration. Therefore, in the design
Peer review under responsibility of The Japanese Geotechnical Society. ⇑ Corresponding author. E-mail address: [email protected] (X. Yao).
of permafrost foundations and for the prevention of engineering damage, the making of precise predictions of the creep of frozen soils has become an urgent task. To get a good understanding of the creep behavior of frozen soils, tremendous experimental work has been carried out (Fish, 1980; Ladanyi, 1983; Zhu and Carbee, 1983; Vyalov, 1986). Based on their test results, some empirical models have been proposed to predict creep at different stages (i.e., primary, secondary and tertiary stages) (Goughnour and Andersland, 1968; Ladanyi, 1972; Ting and Martin, 1979; Zhu and Carbee, 1983), where the effects of the stress level, the ice content, the temperature and the long-term strength are taken into account. These empirical models were established under certain testing conditions which reduce their engineering applicability. This situation certainly does not help the application of modern simulation strategies in cold regions. Therefore,
https://doi.org/10.1016/j.sandf.2018.03.001 0038-0806/Ó 2018 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society.
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theoretical models have been proposed for more general stress states. Viscoelastoplastic models were deduced to describe the changes in microscopic damage during creep based on thermodynamics and damage mechanics (Miao et al., 1995; He et al., 1999). By combining the parabolic yield criteria, plastic flow rules and the long-term strength of frozen soils and mechanical elements, elemental rheology models were proposed to describe the creep behavior during different creep stages (Yang et al., 2010; Li et al., 2011; Wang et al., 2014; Liao et al., 2016). Zhou et al. (2016) extended the hyperplasticity theory with parabolashaped strength loci to describe the rate-dependent behavior of frozen loess. Based on an extended hypoplastic model (Xu et al., 2016a), Xu et al. (2016b) decomposed the strain of frozen soils into ‘solid’ and ‘fluid’ components, and developed a rate-dependent model for frozen soils with the second time derivative of strain. In these previous modeling works, the focus was mainly placed on describing the effects of the stress states on the creep behavior of frozen soils under certain temperatures, i.e., stress is taken as an independent variable, while some parameters were related to temperature so as to describe the influence of temperature. When these models were applied to engineering problems, a series of mechanical tests was carried out to obtain the parameters at different temperatures, and the relationships between the model parameters and the temperature were established through data fitting (Wang et al., 2014). This is an indirect and approximated method for reflecting the effect of temperature and may reduce the calculation accuracy to some extent. In some practical engineering problems, the underground stress at a certain layer is usually constant, while the ground temperature increases continuously due to global warming and the thermal disturbance of infrastructures in permafrost regions, such as the Qinghai-Tibet Highway and Railway (Qi et al., 2007; Qin et al., 2009). It is tedious work to calculate the creep settlement under such conditions, for the model parameters under different temperatures must firstly be independently determined. To directly investigate the effect of the increasing temperature on the creep behavior of frozen soils, Qi and Zhang (2008) conducted a series of K0 temperature step-increase tests under different constant surcharge loads. It was indicated that with the stepped increases in temperature, a similar tendency is seen in the change in creep strain as it is applied with surcharge loads. In other words, the rise in temperature plays a role equivalent to that when the increase in stress acts on the stress-strain curves. Considering the phenomenological effects of temperature on the development of creep strain, it can also be taken as an independent variable. For unfrozen soils, a large number of models have been proposed based on the results of stress-strain tests, such as the Modified Cam-Clay and isotache creep models (Roscoe and Burland, 1968; Den Haan, 1996; Yin and Wang, 2012). Therefore, it would surely be worthwhile to try establishing a model taking temperature as an independent variable to directly describe the influence of the
increase in temperature on the creep behavior of frozen soils. In this paper, a series of temperature step-increase K0 compression tests is carried out on a Chinese standard sand under different constant surcharge loads. The test results are analyzed to obtain the characteristics of the developing creep strain. Based on an analysis of the test results, a creep model is proposed, taking temperature as an independent variable, and the applicability is verified with the test results of both this study and the previous research work. 2. Testing procedure A Chinese standard sand was taken as the study object. The grain size distribution curve for this sand is shown in Fig. 1. Sand samples were prepared with the multiple sieving pluviation (MSP) method (Miura and Toki, 1982; Baker and Konrad, 1985). As is shown in Fig. 2, with the MSP method, the dry unit weight of the samples was controlled by the falling height of the sand particles passing through the sieves. In this study, eleven sieves were used and the height of each was 3 cm; thus, the total falling height was 33 cm. The samples were formed to have the dimensions of 125.0 mm in height and 61.8 mm in diameter within a steel tube (Fig. 2). Then, the samples were fixed, together with the steel tube, onto two sides with porous stones and a steel frame, and saturated with distilled water by vacuum. The saturated samples were then placed into a container full of distilled water and put in a refrigerator for quick freezing, so that the water in the samples would be frozen in its original position and the ice would be distributed evenly in the sample (Ma et al., 2015). Four samples were prepared according to the above method. The dry unit weight, the water content and the degree of saturation of each sample are listed in Table 1. All the K0 temperature step-increase tests were conducted on a multifunction environmental testing apparatus (Fig. 3) with the controlling accuracy of the displacement and stress of 0.001 mm and 1 kPa, respectively. Details on the apparatus can be found in literature (Yao et al.,
Fig. 1. Grain size distribution curve for the Chinese standard sand.
Please cite this article in press as: Yao, X. et al., A one-dimensional creep model for frozen soils taking temperature as an independent variable, Soils Found. (2018), https://doi.org/10.1016/j.sandf.2018.03.001
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Fig. 2. Sketch of the sample-preparation device with the multiple sieving pluviation method.
Table 1 Physical parameters of the four prepared samples. Dry unit weight (kN/m3)
Water content (%)
Saturation degree (%)
17.7 17.5 17.7 17.6
16.3 16.5 16.8 16.3
86.8 85.0 89.5 85.4
2013). This apparatus was originally developed for triaxial tests on frozen soils. To keep the samples in the K0 state (i.e., lateral strain is zero), the soil samples together with the steel tube were put onto the pedestal of the apparatus.
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In this way, it was not necessary to control the lateral strain by applying cell pressure. To guarantee the probable drainage of unfrozen water during the K0 compression, two steel disks with small holes (with a diameter of 1 mm and spacing of 10 mm) were put onto the bottom and top of the samples. The vent valve on the top of the pressure cell was switched on so as to release the cell pressure generated during compression (Fig. 3). At that time, a drainage boundary similar to that commonly used in frozen soil K0 compression tests (Qi and Zhang, 2008; Qi et al., 2010; Zhang et al., 2016), i.e., porous stones or steel discs with small holes, was put on one or two sides of the samples for the probable drainage of unfrozen water. With such a drainage boundary, the unfrozen pore water that had drained out would be frozen and would block the drainage channel through the porous stones. According to the research work of Zhang et al. (2016), the unfrozen pore water dissipation process was notably observed when the test temperature was higher than freezing. This indicates that the commonly used drainage boundary is effective under negative temperatures. Considering the fact that the drained out unfrozen pore water will be frozen and will block the drainage channel through the porous stones, the drainage process for the pore water at the drainage boundary might be different from that of unfrozen soils. When unfrozen water is drained out of the soil, it is frozen and firstly becomes ice crystals in the pores of the porous stones or steel discs. Then the ice crystals may migrate to lower stress regions due to the effect of the stress gradient (See the discussion on creep mechanism in Section 3.1). This might be the mechanism that the experimental setup allows, namely, ‘consolidation’ or ‘drainage’. As for the extremely small amount of unfrozen water, its ejection onto the drainage boundary cannot be visually observed or measured accurately with the soil mechanical testing techniques available at present, at least as far as the authors know. This is why the researchers (Zhang et al.,
Fig. 3. Sketch of the apparatus for the K0 compression tests. Please cite this article in press as: Yao, X. et al., A one-dimensional creep model for frozen soils taking temperature as an independent variable, Soils Found. (2018), https://doi.org/10.1016/j.sandf.2018.03.001
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2016) measured the unfrozen pore water pressure by embedding small pore water transducers inside the soil samples. Therefore, the drained out unfrozen water was not measured in this study. As for the test temperature, it can be seen in Fig. 3 that three cooling baths were used to control the sample temperature. Two were directly connected with cool oil that was circulating in the plates on the top and the bottom of the sample. The third was connected to a cooling ring around the sample with cool oil that was circulating so as to cool the oil surrounding the steel tube. During the K0 compression tests, the temperatures on the top and the bottom of the soil samples were monitored. K0 temperature step-increase tests were conducted on four samples under different constant surcharge loads of 0.2, 0.5, 1.0 and 2.0 MPa, respectively. For each sample under a certain surcharge load, five levels of temperatures were applied by stepped increases from about 10 °C, as is shown in Fig. 4. The duration of each temperature step was 1 day. As can be seen in Fig. 4, there were minor vibrations due to changes in the environmental temperature during each step. Therefore, the average value at each step was taken as the representative temperature. The representative temperatures under different surcharge loads are listed in Table 2. 3. Analysis of test results, model setup and verification 3.1. Strain classification and mechanism analysis Fig. 5 presents the strain vs. duration curves under different surcharge loads (0.2, 0.5, 1.0 and 2.0 MPa). As can be seen in the figure, the strain under different surcharge loads includes two parts, i.e., instantaneous strain (esl) and time-dependent strain. The time-dependent strain might be due to the primary or secondary (creep) consolidation process. For unfrozen soils, it is generally considered that primary consolidation occurs during the stage at which the pore water pressure decreases considerably, while secondary consolidation occurs during the stage at which most of the pore water pressure is dissipated and
Fig. 4. Duration curve for the stepped increases in temperature (0.2 MPa).
does not change notably. For frozen soils, according to the available research work by Zhang et al. (2016) on silty clay with a high ice content, the dissipation process for pore water pressure can be observed notably in the range in temperature from 0 to 0.4 °C. When the temperature is lower than 0.4 °C, the pore water pressure remains mostly invariant and is less than 5% of the surcharge load. This indicates that, when the temperature approaches or is higher than freezing (for this kind of fine-grained soil, the freezing temperature is about 0.4 °C (Xu et al., 2001; Zhou et al., 2015)), the permeability is not very small and the dissipation process for unfrozen water is notable. Correspondingly, primary consolidation occurs first. When the temperature is lower than the freezing point, the unfrozen water and the permeability decrease considerably and the unfrozen water dissipation process is not notable or extremely long. Previous studies have also confirmed this point under low temperatures (Bishop and Henkel, 1962; Williams and Burt, 1974). As for the drained out unfrozen water, there is no direct evidence at the present time showing that it can be visibly observed. That is why when the temperature is lower than the freezing point, the previous research works usually classified the deformation after applying a surcharge load as either instantaneous or viscous (or creep) (Fish, 1980; Zhu and Carbee, 1983). For the tested sand in this study, the freezing point is about 0.2 °C (Xu et al., 2001; Zhou et al., 2015), and the test temperatures of this study are lower than 0.40 °C (Table 2). Therefore, the time-dependent strain in the test results of this study can be classified as creep strain. In Fig. 5, the creep strain of the samples under the K0 condition is actually due to the volumetric compression, which comes from two aspects. On the one hand, a part of the volumetric compression might come from the drainage of unfrozen water. At current, although there is no direct evidence showing that unfrozen water drained out, it has been proven that unfrozen water migration or drainage does exist in frozen soils (Williams and Burt, 1974; Bishop and Henkel, 1962). It is an extremely long process for the small permeability and unfrozen water content of frozen soils. On the other hand, a part of the volumetric compression comes from the migration of pore ice crystals. According to experimental investigations (Vyalov, 1965; Vyalov et al., 1970), the migration of pore ice crystals from regions of concentrated stress to regions of lower stress can be observed under a constant surcharge load. For the frozen sand samples in this study, it can be seen in Table 1 that they are not fully saturated. Obviously, some void pores exist in the samples. Under the effects of the surcharge load, both the pore ice crystals and the unfrozen pore water can migrate to the void pores. In addition, for the porous drainage boundaries permitted on both sides of the samples, it is possible for the pore ice crystals and the unfrozen water to migrate to the porous drainage boundaries with lower stress. During the process of the migration of the pore ice crystals and the unfrozen water to the void pores and drainage boundaries, the soil particles are rearranged
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Table 2 Representative temperatures under different surcharge loads. Surcharge load (MPa)
T1 (°C)
T2 (°C)
T3 (°C)
T4 (°C)
T5 (°C)
0.2 0.5 1.0 2.0
9.95 10.01 10.02 9.97
5.32 5.39 5.33 5.27
2.15 2.48 2.12 2.11
0.96 1.10 1.14 1.09
0.48 0.55 0.55 0.61
Fig. 5. Strain vs. duration curves under different surcharge loads.
and volumetric compression occurs correspondingly. The above analysis indicates that the mechanism for the creep compression of frozen soils is different from that of unfrozen soils, i.e, both the pore unfrozen water drainage and the ice crystal migration contribute to the creep compression of frozen soils, and that this mechanism is not just due to the drainage of the unfrozen pore water. Therefore, the pore unfrozen water drainage (although this process is extremely slow when the test temperature is lower than the freezing point) and the ice crystal migration are the creep compression mechanisms of the test results. 3.2. Analysis of test results It can be seen that under different surcharge loads (Fig. 5), the development of strain shows a stepped pattern due to the stepped increases in temperature (Table 2), which is similar to the effect of the step loads. The strain vs. time curves at different temperatures are shown in Fig. 6 for each curve with the same surcharge load, when the starting time of each temperature step is reset to zero. On these curves with the same surcharge load, four group points with different strain rates are identified (Fig. 6(a)– (d)). After the strain and corresponding temperature of the points under different strain rates are plotted in the elog(|T|1) space (where |T|1 is the reciprocal of the temperature absolute value (RTAV)), the RTAV vs. strain curves under different strain rates are given, as shown in Fig. 7. For ease of elaboration, the RTAV vs. strain curves at a strain rate of 1.25 108 1/s under different surcharge loads are taken as examples. It can be seen that the RTAV vs. strain curves can generally be represented by two
straight lines (with slopes of a and b, respectively) in the e-log(|T|1) space. The intersection of the two sections is defined as |TP|1. With a decrease in strain rate, the positions of the first sections do not change notably, while the second sections move down (Fig. 7) and the |TP|1 decreases (Table 3). At the same time, the second sections with different strain rates form a series of parallel lines (Fig. 7). In other words, with an increase in temperature or |T|1, the strain in the first section is rate-independent and that in the second section is rate-dependent. Under a constant temperature condition (here, taking the strain vs. time curve at 0.5 MPa and T5 as an example (Fig. 6 (b)), the difference between two strain points is the creep strain which is equal to the vertical space between two parallel lines in Fig. 7(b). The space between two parallel lines can be expressed as follows: ei e0 ¼ ci ln
e_ 0 e_ i
ð1Þ
where ci (i = 1, 2 and 3) is the parameter representing the effect of strain rate on creep strain, and e0 ð_e0 Þ and ei ð_ei Þ (i = 1, 2 and 3) are the strains and strain rates of the points on the strain vs. time curve at 0.5 MPa and T5 (Fig. 6(b)). In the e-log(|T|1) space (Fig. 7(b)), these points are on different parallel lines at the same temperature T5. Along with these strains and strain rates of the points (Fig. 7(b)), ci is calculated by Eq. (1) and the corresponding values at different surcharge loads are listed in Table 4. It can be seen that under different surcharge loads, the ci values calculated with different parallel lines are basically the same, which means that the vertical space between two parallel lines is constant when the strain rate decreases by one order of magnitude (Eq. (1)) and that the average value (c) of ci can be used to represent it. Through the above analysis on the effects of stepped increases in temperature on the strain development, an idealized isotache model is summarized in Fig. 8, i.e., an initial compression line (with slope a) starting from an initial temperature (|Ti|1) is used to represent the first rateindependent section, while a series of creep isotaches (with slope b) is used to represent the second rate-dependent section, and the space between two parallel lines is c when the strain rate decreases by one order of magnitude. With the decrease in strain rate, the parallel lines moves down and the intersection of the two sections (|TP|1) decreases (i.e., a unique |TP|1 can be determined by a creep isotache at a certain strain rate e_ 0 ). This is basically identical to the parallel lines postulate of the isotache model developed
Please cite this article in press as: Yao, X. et al., A one-dimensional creep model for frozen soils taking temperature as an independent variable, Soils Found. (2018), https://doi.org/10.1016/j.sandf.2018.03.001
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(a) 0.2 MPa
(c) 1.0MPa
(b) 0.5 MPa
(d) 2.0MPa
Fig. 6. Strain vs. time curves at different temperatures and surcharge loads.
for unfrozen soil models (Den Haan, 1996; Vermeer and Neher, 1999), where the only difference is that the stress is taken as an independent variable. After the independent variable is replaced with temperature (|T|1), it would be worthwhile to try and develop a novel model capable of directly describing the effect of temperature on the creep of frozen soils. 3.3. Model setup and verification In the original isotache model, the total strain is stressdependent (Den Haan, 1996; Vermeer and Neher, 1999). After replacing the stress with the reciprocal of the absolute value of the temperature (|T|1), the total strain due to the increase in temperature can be expressed as ! ! 1 1 jT p j jT j e_ 0 e ¼ a ln þ b ln þ c ln ð2Þ 1 1 e_ jT i j jT P j 1
where jT P j is the intersection of the initial compression line and the creep isotache with strain rate e_0 (Fig. 8). Here, e_0 and |TP|1 are the reference strain rate and temperature, respectively.
Under a constant temperature, Eq. (1) is the incremental creep strain, which can be written as e e e_ 0 0 ð3Þ ¼ exp c e_ and exp
e e 0
c
de ¼ e_ 0 dt
where e is the strain at strain rate e_ . The integration of Eq. (4) is Z e Z t e e 0 exp e_ 0 dt de ¼ c e0 t0
ð4Þ
ð5Þ
According to Eq. (5), the relationship between creep strain and time is deduced as follows: e e e_ 0 0 ð6Þ exp ¼ ðt t0 Þ þ 1 c c where a new parameter s0 is defined as c s0 ¼ e_ 0
ð7Þ
and Eq. (6) is written as
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(a) 0.2 MPa
(c) 1.0 MPa
(b) 0.5 MPa
(d) 2.0 MPa
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Fig. 7. Relationship between RTAV (|T|1) and strain at different strain rates.
increases in temperature), t is the real time and (t t0 + s0) is defined as intrinsic time s, i.e.,
Table 3 TP at different strain rates. Surcharge load (MPa)
Strain rate (1/s)
Tp (°C)
0.2
4.00 109 1.25 108
2.90 2.17
0.5
4.00 109 1.25 108
3.77 2.70
1.0
5.00 109 1.25 108
6.06 5.26
2.0
1.25 108 4.00 108
5.88 4.87
exp
e e t t þ s 0 0 0 ¼ c s0
s ¼ t tr
ð9Þ
where tr = t0 s0 is the difference between real time t and intrinsic time s. A detailed discussion on the intrinsic time can be found in Den Haan (1996). With Eqs. (8) and (9), the incremental creep strain from e0 can be written as e e0 ¼ c ln
s s0
ð10Þ
Substituting Eq. (10) into Eq. (2), the total strain is written as ! ! jT p j1 jT j1 s e ¼ a ln ð11Þ þ b ln þ c ln 1 1 s 0 jT i j jT P j
ð8Þ
in which, s0 is the reference intrinsic time, t0 is the starting time of the creep strain (or the ending time of the stepped Table 4 Calculated parameter ci with different parallel lines. Surcharge load (MPa)
c1
c2
c3
c
0.2 0.5 1.0 2.0
2.85 104 2.93 104 3.81 104 6.03 104
2.85 104 2.93 104 2.46 104 6.85 104
2.85 104 2.99 104 3.20 104 5.93 104
2.85 104 2.95 104 3.16 104 6.27 104
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Fig. 8. Idealized isotache model.
Under a constant temperature, the time derivative of Eq. (11) is the creep strain rate, i.e., c ð12Þ e_c ¼ s According to Eq. (12), it is found that for a creep isotache with a certain strain rate, there is a corresponding unique intrinsic time and the two variables are inversely related. With retrospect on the physical meaning of the idealized isotache model in Fig. 8, the total strain in Eq. (11) can be decomposed into two parts, namely, the rateindependent initial compression strain (ei) and the ratedependent strain (er). Therefore, the rate-dependent strain is er ¼ e ei
ð13Þ
where the initial compression strain (ei) is defined as follows: ! ! 1 1 jT p j jT j ei ¼ a ln þ a ln ð14Þ jT i j1 jT P j1 When Eqs. (11), (12) and (14) are substituted into Eq. (13), the creep strain rate can be further expressed as !ðbaÞ=c c er =c jT j1 ð15Þ e_ c ¼ e 1 s0 jT P j Together with Eqs. (14) and (15), the total strain rate is expressed as !ðbaÞ=c 1 1 jT_ j c er =c jT j ð16Þ e_ ¼ e_ i þ e_ c ¼ a 1 þ e 1 s0 jT j jT P j Besides the incremental strain due to the increase in temperature and the creep process (Eq. (16)), the instantaneous strain (esl) due to the applied surcharge load should also be taken into consideration (Fig. 5). Therefore, the total strain can be calculated as follows: 0 !ðbaÞ=c 1 Z t 1 1 _ @a jT j þ c eer =c jT j Adt ð17Þ e ¼ esl þ 1 1 s0 jT j jT P j 0
Eqs. (1)–(17) are set up as a creep model by taking the temperature as an independent variable. In the following, the applicability of this model will be verified based on the test results in this study and in previous research work. The model parameters for the Chinese standard sand are listed in Table 5, where parameters such as a, b, TP and s0 are determined with the RTAV vs. strain curve at a strain rate of 1.25 108 1/s (Fig. 7 and Eq. (12)). Parameter c is taken as the average value of ‘ci’ obtained under different strain rates (Table 4). Table 6 presents the model parameters obtained from the results of tests on the Qinghai-Tibet silty clay with a water content of 40% and a degree of saturation of about 80% (Qi and Zhang, 2008). The parameter-obtaining method is consistent with that in Section 3.2. Figs. 9 and 10 are the test and calculated straindevelopment curves due to the stepped increases in temperature. It can be seen that the calculated results agree well with the Chinese standard sand test results, while for the silty clay, the calculated results do not agree very well with the test results, especially in the last two temperature steps. This might be due to the primary consolidation occurring first during the last two temperature steps. Just as analyzed in Section 3.1, the consolidation stage of frozen soils is mainly influenced by the freezing temperature, i.e., primary consolidation occurs first when the temperature approaches or is higher than the freezing point, while secondary consolidation (creep) mainly occurs below the freezing point. As is shown in Fig. 10, when the temperature is lower than the freezing point (approximately at 0.4 °C for the silty clay) (Table 7), the calculated results agree well with the test results during the first two steps. When the temperature approaches or is higher than the freezing point (Table 7), there is a large discrepancy between the calculated and the test results during the last two steps, which is due to the primary consolidation occurring first in the soil. Therefore, the creep model proposed in this paper is applicable to the creep compression process when temperatures are below the freezing point. As for the applicable duration of this creep model, the model is verified to be applicable within a creep time up to 1.6 days with the available test results (see the second step in Fig. 10). As for longer applicable durations of this model, more tests should be conducted. With retrospect on the test results of different soils in Figs. 9 and 10, the effects of the imperfect K0 condition and saturation on the law of the creep compression can be further analyzed. As described in Section 2, the test results in Fig. 9 were obtained by restraining the samples in the K0 state within a steel tube; the test results in Fig. 10 were obtained by restraining the samples in the K0 state within a plexiglass tube (Qi and Zhang, 2008). The degrees of saturation of the two group test results are also different (the degree of saturation of the silty clay is about 80%, while that of standard sand is in the range of 85–90% (Table 1)). Generally, when the degree of saturation is larger than 85%, the soil samples can be considered
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Table 5 Creep model parameters for Chinese standard sand. Surcharge load (MPa)
a 4.20 10 2.20 104 2.50 104 6.00 104
0.2 0.5 1.0 2.0
Tp (°C)
esl
3
4
2.85 10 2.95 104 3.16 104 6.27 104
4
2.20 10 2.50 104 2.50 104 5.00 104
2.17 2.70 5.26 5.88
1.00 104 1.00 104 3.00 104 1.00 103
b 4
s0 (s)
c
1.25 10 1.35 103 1.35 103 2.45 103
Table 6 Creep model parameters for Qinghai-Tibet silty clay. Surcharge load (MPa)
a
b
c
s0 (s)
Tp (°C)
esl
0.1 0.2
1.20 102 2.30 102
3.50 102 4.00 102
1.40 103 3.20 103
1.87 104 4.27 104
0.66 0.90
1.00 104 3.00 103
(a) 0.2 MPa
(c) 1.0 MPa
(b) 0.5 MPa
(d) 2.0 MPa
Fig. 9. Calculated and test curves for strain vs. time at different surcharge loads (Chinese standard sand).
as saturated in unfrozen soil mechanics. With these tests results under different test conditions (with different K0restrained measures and saturation degrees), the results with the proposed creep model agree well with the test results when the test temperature is lower than the freezing point, as shown in Figs. 9 and 10. This indicates that the law described by the proposed model is an intrinsic creep development characteristic of frozen soils, and is not due to the effects of imperfect test conditions, such as the imperfect K0-restrained measures and saturation.
4. Simplified parameter-obtaining method In Section 3.2, the physical meaning of the model parameters is clarified through an analysis of the law of creep development due to the increase in temperature. However, with the parameter-obtaining method shown in Section 3.2, it is a tedious work to apply this model in practice, for the RTAV vs. strain curves at different strain rates must firstly be determined (Figs. 6 and 7). In this section, the characteristic of the development of creep strain will
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ence between the two curves gradually disappears (Fig. 11(d)). That is to say, when t tr, the real time (t) is approximately equal to the intrinsic time (s), and Eq. (18) can be further expressed as follows:
(a) 0.1 MPa
e e0 ¼ c ln
(b) 0.2 MPa
Fig. 10. Calculated and test curves for strain vs. time at different surcharge loads (Qinghai-Tibet silty clay).
Table 7 Representative temperatures for Qinghai-Tibet silty clay under different surcharge loads. Surcharge load (MPa)
T1 (°C)
T2 (°C)
T3 (°C)
T4 (°C)
0.1 0.2
1.45 1.54
1.02 0.98
0.53 0.50
0.42 0.32
be further analyzed so as to provide a more straightforward parameter-obtaining method. As shown in Fig. 11, all the strain vs. time logarithm curves at high temperatures tend to be a straight line, the slope of which is generally equal to the value of c in Table 5. Therefore, c can be determined by the average values of straight line slopes at high temperatures on the strain vs. time logarithm curves (from T3 to T5). In addition, according to Eqs. (8) and (10), the creep strain under certain temperatures is expressed as e e0 ¼ c ln
s t tr ¼ c ln s0 s0
ð18Þ
For the effect of the time difference (tr = t0 s0), there is a large difference between the strain vs. ln(t) and strain vs. ln(s) curves at the initial stage (Fig. 11(d)). With an increase in time, the strain vs. ln(t) curve tends to be a straight line represented by strain vs. ln(s), and the differ-
s t c ln s0 s0
ð19Þ
It can be concluded from the above analysis that, for these creep strain curves under high temperatures (from T3 to T5), when the time step of each temperature is long enough, the real time (t) at the end of each step is approximately equal to the intrinsic time (s). In another words, for the strain at the end of each temperature step under high temperatures, the duration of each temperature step (i.e., 1 day) can be taken as their reference intrinsic time (s0). As shown in Figs. 6 and 7, parameter a is determined with the strains that do not change notably with the develop of strain rate or time under low temperatures (T1 and T2), while parameters b and Tp are determined with the strains under high temperatures (T3 to T5) at a certain reference strain rate or intrinsic time (Eq. (12)). This means that the reference intrinsic time (s0) of the ending strains at each temperature step (T1 to T5) can be taken as approximately 1 day. According to the analysis in Section 3.2, the RTAV vs. strain curves under different constant strain rates or intrinsic times are a series of parallel lines, and the model parameters are determined with one of them. Thus, parameters a, b and Tp can also be determined with the ending strains at each temperature step. To show the validity of this inference, the results of the tests on the Chinese standard sand at 2.0 MPa are taken as an example (Fig. 12). The RTAV vs. strain curve obtained with the ending strains (Fig. 11(d)) is compared with the one at a strain rate of 1.25 108 1/s (Fig. 7(d)) in Fig. 12. It can be seen that the two curves are generally parallel to each other. This indicates that the RTAV vs. strain curve obtained with the ending strains can be used to represent the one at the intrinsic time of 1 day or the strain rate of c/1 day (according to Eq. (12)). It is can be seen in Fig. 12 that the a and b obtained from the two curves are basically the same, while the Tp obtained from the ending strain curve is lower than that at 1.25 108 1/s. This is due to the fact that the Tp is uniquely determined by the strain rate or the intrinsic time (Eq. (12)). Obviously, with the ending strains at each temperature step, model parameters a, b and Tp can be determined simply instead of tediously obtaining the RTAV vs. strain curves under different constant strain rates. From the above analysis, the simplified parameterobtaining method can be summarized as follows: (1) a, b and |TP|1 can be determined with the RTAV vs. strain curve obtained with the ending strains at each temperature step. (2) c can be determined by the average value of the constant creep slopes under high temperatures on the strain vs. time logarithm curves. (3) The reference intrinsic time (s0) is the duration of each temperature step.
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(a) 0.2 MPa
(c) 1.0 MPa
(b) 0.5 MPa
(d) 2.0 MPa
11
Fig. 11. Strain vs. time logarithm curves at different temperatures and surcharge loads.
5. Discussions
Fig. 12. Relationship between strain and |T|1 at different strain rates (2.0 MPa).
With this simplified method, the parameters of the four test cases for standard sand are listed in Table 8. The corresponding calculated results for strain with stepped increases in temperature are compared with the test results in Fig. 13. It can be seen that the calculated results match the test results well, which indicates that the simplified method is valid. For performing practical calculations with the proposed model, this method is recommended.
In the previous research work, Qi and Zhang (2008) conducted the temperature step-increase tests from 1.5 to 0.3 °C. After an analysis of the change tendency of the compression coefficient against temperature, Qi and Zhang (2008) concluded that when the temperature is higher than 1.0 °C, the compressibility increases considerably; and thus, 1.0 °C is taken as the boundary temperature for defining ‘‘warm” frozen soils. This is approximately consistent with the results given in Table 6, where |TP|1 falls in the range from 0.90 to 0.66 °C. According to the definition in Section 3.2, |TP|1 is the inflection point of the two compression stages; it can also be taken as an index for identifying the compression stage of frozen soils. However, taking a constant temperature value for defining ‘warm’ or ‘cold’ frozen soils is only partially valid due to the narrow range in surcharge loads applied in the previous study by Qi and Zhang (2008) (from 0.1 to 0.3 MPa). It can be seen from the test results on the Chinese standard sand that, when the surcharge load increases from 0.2 MPa to 2.0 MPa, T P decreases from 3.13 °C to 6.25 °C (Table 8); for Qinghai-Tibet silty clay, when the surcharge load increases from 0.1 MPa to 0.2 MPa, the TP decreases from 0.66 °C to 0.90 °C
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Table 8 Model parameters of the standard sand obtained with the simplified method. Surcharge load (MPa)
a
b 4
3.80 10 2.20 104 3.00 104 7.00 104
0.2 0.5 1.0 2.0
s0 (s)
c 3
1.25 10 1.35 103 1.40 103 2.50 103
4
3.70 10 4.30 104 3.30 104 7.00 104
(a) 0.2 MPa
(c) 1.0 MPa
(b) 0.5 MPa
(d) 2.0 MPa
TP (°C)
8.64 10
4
3.13 3.70 5.88 6.25
Fig. 13. Calculated (with simplified parameter-obtaining method) and test curves for strain vs. time at different surcharge loads.
(Table 6). It can be concluded that temperature TP, for distinguishing frozen soils at different compression stages (‘‘cold” or ‘‘warm” frozen soils), is not a constant, which is inversely related to the surcharge load. In other words, with the increase in surcharge load, the boundary temperature TP decreases and the range in temperature of frozen soils under high compressibility is enlarged. As seen above, in the discussion on Tp, all the parameters in this model need to be determined independently for different stresses (Table 8). They all have the common physical meaning under different stresses, such as a, b, c and Tp. These parameters are obtained by analyzing the law of creep development, due to the increases in temperature, not just through simple data fitting. According to Section 3.2, it can be found that the creep development at different surcharge loads shows the same law despite the
effect of instantaneous strain. The total strain is equal to the sum of the creep and instantaneous strains (Eq. (17)), in which the instantaneous strain is related to the applied constant stress and initial temperature and can be taken as a parameter in the model. From this viewpoint, the model proposed in this paper can be taken as a complete model at least in terms of the same position of the models taking stress as an independent variable (Yang et al., 2010; Li et al., 2011; Wang et al., 2014; Liao et al., 2016), and has the capacity to predict the creep behavior due to increases in temperature. Ideally, temperature and stress should both be included as independent variables in a model. However, for the complex mechanical behavior of frozen soils, it is an extremely hard task to reasonably describe the influences of stress and temperature with a model. This is why most of the
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proposed models for frozen soils at the current time take the stress as an independent variable (Yang et al., 2010; Li et al., 2011; Wang et al., 2014; Liao et al., 2016), while the parameters need to be defined independently at different temperatures. Obviously, it is inconvenient to use these models to describe the creep development when the case of constant stress and an increasing temperature is involved. In other words, one model cannot accommodate all conditions. Whether or not the stress or temperature is taken as an independent variable in a model, some parameters must be defined independently for different stresses or temperatures. Based on this consideration, a creep model, which takes temperature as an independent variable, is proposed for the case of constant stress and an increasing temperature, while the simultaneous effects of temperature and stress are not within the scope of this study. 6. Conclusions In this paper, a series of temperature step-increase K0 compression tests were conducted under different constant surcharge loads. Based on an analysis of strain development, a novel creep model was proposed by taking temperature as an independent variable, and the applicability of the model was verified with test results. The following conclusions can be drawn: (1) Based on an analysis of the characteristics of creep strain development, due to stepped increases in temperature, it was indicated that the relationship between the reciprocal of the temperature’s absolute value (RTAV) and the strain at different strain rates fits well with the parallel lines postulate of the isotache model for unfrozen soils. (2) After taking the RTAV as an independent variable in the isotache model, it was shown that the tendency of the stepped pattern development of strain due to the stepped increases in temperature can be reasonably described, which indicates the applicability of the novel model. The applicable range in temperature for the proposed model is below freezing. (3) A simplified parameter-obtaining method was proposed, whereby a, b and |TP|1 can be determined by the RTAV vs. ending strain curve, c can be determined by the average constant slopes under high temperature steps, and the duration of each temperature step in the K0 compression tests can be taken as the intrinsic reference time s0. After verification with the test results, it was indicated that the simplified method is applicable and recommended. (4) The physical meaning of TP is the temperature boundary for distinguishing different compression stages, which is inversely related to the surcharge load. With the increase in surcharge load, the range in temperature of frozen soils under high compressibility is enlarged.
13
Acknowledgements This work was supported in part by the National Natural Science Foundation of China (Nos. 41671061, 41572268, 41501084, 41471063 and 41702333), the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions (granted to Dr. Jilin Qi) (CIT&TCD20150101), the Key Project of Natural Science Foundation of China (No. 41630636) and the 100-Talent Program of the Chinese Academy of Sciences (granted to Dr. Mingyi Zhang). References Baker, T.H., Konrad, J.M., 1985. Effect of sample preparation on the strength of artificially frozen sand, vol. 2. In: Proceedings of the 4th International Symposium on Ground Freezing, Sapporo, Japan, pp. 171–176. Bishop, A.W., Henkel, D.J., 1962. The Measurement of Soil Properties in the Triaxial Test. Edward Arnold, London. Den Haan, E.J., 1996. A compression model for non-brittle soft clays and peat. Ge´otechnique 46 (1), 1–16. https://doi.org/10.1016/S0148-9062 (97)87398-6. Fish, A.M., 1980. Kinetic nature of the long-term strength of frozen soils. In: Proceedings of the 2nd International Symposium on Ground Freezing, Trondheim, Norway, pp. 95–108. Goughnour, R.R., Andersland, O.B., 1968. Mechanical properties of a sand-ice system. ASCE J. Soil Mech. Found. Div. Pro. 94 (4), 923–950. He, P., Cheng, G.D., Zhu, Y.L., 1999. Constitutive theories on viscoelastoplasticity and damage of frozen soil. Sci. China (Ser. D) 42, 38–43. https://doi.org/10.1007/BF02878851. Ladanyi, B., 1972. An engineering theory of creep of frozen soils. Can. Geotech. J. 9 (1), 63–80. https://doi.org/10.1139/t72-005. Ladanyi, B., 1983. Shallow foundations on frozen soil: creep settlement. J. Geotech. Eng. 109 (11), 1434–1448. https://doi.org/10.1061/(ASCE) 0733-9410(1983) 109:11(1434). Li, D.W., Fan, J.H., Wang, R.H., 2011. Research on visco-elastic-plastic creep model of artificial frozen soil under high confining pressures. Cold Reg. Sci. Technol. 65 (2), 219–225. https://doi.org/10.1016/ j.coldregions.2010.08.006. Li, X., Cheng, G.D., 1999. A GIS-aided response model of high altitude permafrost to global change. Sci. China (Ser. D) 42 (1), 72–79. https:// doi.org/10.1007/BF02878500. Liao, M.K., Lai, Y.M., Yang, J.J., Li, S.Y., 2016. Experimental study and statistical theory of creep behavior of warm frozen silt. KSCE J. Civ. Eng. 20 (6), 2333–2344. https://doi.org/10.1007/s12205-015-0374-4. Ma, L., Qi, J.L., Yu, F., Yao, X.L., 2015. Experimental study on variability in mechanical properties of a frozen sand as determined in triaxial compression tests. Acta Geotech. 11 (1). https://doi.org/ 10.1007/s11440-015-0391-y. Ma, W., Qi, J.L., Wu, Q.B., 2008. Analysis of the deformation of embankments on the Qinghai-Tibet railway. J. Geotech. Geoenviron. Eng. 134 (11), 1645–1654. https://doi.org/10.1061/(ASCE)1090-0241 (2008) 134:11(1645). Miao, T.D., Wei, X.X., Zhang, C.Q., 1995. Creep of frozen soil by damage mechanics. Sci. China (Ser. B) 38 (8), 996–1002. Miura, S., Toki, S., 1982. Sample preparation method and its effect on static and cyclic deformation - strength properties of sand. Soils Found. 22 (1), 61–77. https://doi.org/10.1016/0148-9062(83)91658-3. Qi, J.L., Hu, W., Ma, W., 2010. Experimental study of a pseudopreconsolidation pressure in frozen soils. Cold Reg. Sci. Technol. 60 (3), 230–233. https://doi.org/10.1016/j.coldregions.2009.10.008. Qi, J.L., Sheng, Y., Zhang, J.M., Wen, Z., 2007. Settlement of embankments in permafrost regions in the Qinghai-Tibetan plateau.
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Xu, G.F., Wu, W., Qi, J.L., 2016b. Modeling the viscous behavior of frozen soil with hypoplastic. Int. J. Numer. Anal. Methods 40 (15), 2061–2075. https://doi.org/10.1002/nag.2516. Xu, X.Z., Wang, J.C., Zhang, L.X., 2001. Frozen Soil Physics. Science Press, Beijing. Yang, Y.G., Lai, Y.M., Chang, X.X., 2010. Experimental and theoretical studies on the creep behavior of warm ice-rich frozen sand. Cold Reg. Sci. Tech. 63 (1), 61–67. https://doi.org/10.1016/j.coldregions. 2010.04.011. Yao, X.L., Qi, J.L., Yu, F., Ma, L., 2013. A versatile triaxial apparatus for frozen soils. Cold Reg. Sci. Technol. 92, 48–54. https://doi.org/ 10.1115/1.4025546. Yin, Z.Y., Wang, J.H., 2012. A one-dimensional strain-rate based model for soft structured clays. Sci. China (Ser. E) 55 (1), 90–100. https://doi. org/10.1007/s11431-011-4513-. Yu, F., Qi, J.L., Yao, X.L., Liu, Y.Z., 2013. In-situ monitoring of settlement at different layers under embankments in permafrost regions on the Qinghai-Tibet Plateau. Eng. Geol. 160, 44–53. https://doi.org/ 10.1016/j.enggeo.2013.04.002. Zhang, H., Zhang, J.M., Zhang, Z.L., Chai, M.T., 2016. Investigation of the pore water pressure of saturated warm frozen soils under a constant load. J. Offshore Mech. Arct. 138 (6). https://doi.org/10.1115/ 1.4033936, 062001-1–6. Zhou, J.Z., Tan, L., Wei, C.F., Wei, H.Z., 2015. Experimental research on freezing temperature and super-cooling temperature of soil. Rock Soil Mech. 36 (3), 777–785. https://doi.org/10.16285/j.rsm.2015.03.023. Zhou, Z.W., Ma, W., Zhang, S.J., Du, H.M., Mu, Y.H., Li, G.Y., 2016. Multiaxial creep of frozen loess. Mech. Mater. 95, 172–191. https:// doi.org/10.1016/j.mechmat.2015.11.020. Zhu, Y.L., Carbee, D.L., 1983. Creep behavior of frozen silt under constant uniaxial stress. In: Proceedings of the 4th International Conference on Permafrost, Fairbanks, Alaska, pp. 1507–1512.
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Available online at www.sciencedirect.com
ScienceDirect Soils and Foundations xxx (2018) xxx–xxx www.elsevier.com/locate/sandf
A one-dimensional creep model for frozen soils taking temperature as an independent variable Xiaoliang Yao a,⇑, Jilin Qi b, Jianming Zhang a, Fan Yu b a
State Key Laboratory of Frozen Soil Engineering, Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, China b School of Civil and Transportation Engineering, Beijing University of Architecture and Civil Engineering, China Received 16 March 2017; received in revised form 8 February 2018; accepted 15 February 2018
Abstract In this paper, a one-dimensional creep model that takes temperature as an independent variable was proposed for frozen soils. A series of K0 compression tests was conducted under different constant surcharge loads with stepped increases in temperature. An analysis of the test results indicated that the characteristics of creep strain, developing due to the stepped increases in temperature, matched well with the parallel lines postulate of the isotache model for unfrozen soils. The independent variable (stress) in the original model was replaced with a reciprocal of the temperature’s absolute value, and a novel model for directly describing the effect of the increase in temperature on the creep behavior of frozen soils was established. It was verified by the test results in this study and in previous research work that the tendency of the stepped development of creep strain, due to the stepped increases in temperature, can be reasonably captured by the model. Based on further analysis of the test results, a simplified parameter-obtaining method was recommended and its applicability was verified. Ó 2018 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society.
Keywords: Frozen soils; Creep model; Temperature; Independent variable
1. Introduction Over the past few decades, a rise in the large-scale ground temperature, characterized by permafrost degradation, has been observed in cold regions (Li and Cheng, 1999; Yu et al., 2013). Due to this rise in ground temperature, warmer permafrost layers have often formed and led to considerable creep settlement in the layers (Qi et al., 2007; Ma et al., 2008). In certain areas of cold regions, where the deformations of infrastructure foundations need to be strictly controlled, such as along high-grade highways and high-speed railways, the creep of the permafrost layers must be taken into consideration. Therefore, in the design
Peer review under responsibility of The Japanese Geotechnical Society. ⇑ Corresponding author. E-mail address: [email protected] (X. Yao).
of permafrost foundations and for the prevention of engineering damage, the making of precise predictions of the creep of frozen soils has become an urgent task. To get a good understanding of the creep behavior of frozen soils, tremendous experimental work has been carried out (Fish, 1980; Ladanyi, 1983; Zhu and Carbee, 1983; Vyalov, 1986). Based on their test results, some empirical models have been proposed to predict creep at different stages (i.e., primary, secondary and tertiary stages) (Goughnour and Andersland, 1968; Ladanyi, 1972; Ting and Martin, 1979; Zhu and Carbee, 1983), where the effects of the stress level, the ice content, the temperature and the long-term strength are taken into account. These empirical models were established under certain testing conditions which reduce their engineering applicability. This situation certainly does not help the application of modern simulation strategies in cold regions. Therefore,
https://doi.org/10.1016/j.sandf.2018.03.001 0038-0806/Ó 2018 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society.
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theoretical models have been proposed for more general stress states. Viscoelastoplastic models were deduced to describe the changes in microscopic damage during creep based on thermodynamics and damage mechanics (Miao et al., 1995; He et al., 1999). By combining the parabolic yield criteria, plastic flow rules and the long-term strength of frozen soils and mechanical elements, elemental rheology models were proposed to describe the creep behavior during different creep stages (Yang et al., 2010; Li et al., 2011; Wang et al., 2014; Liao et al., 2016). Zhou et al. (2016) extended the hyperplasticity theory with parabolashaped strength loci to describe the rate-dependent behavior of frozen loess. Based on an extended hypoplastic model (Xu et al., 2016a), Xu et al. (2016b) decomposed the strain of frozen soils into ‘solid’ and ‘fluid’ components, and developed a rate-dependent model for frozen soils with the second time derivative of strain. In these previous modeling works, the focus was mainly placed on describing the effects of the stress states on the creep behavior of frozen soils under certain temperatures, i.e., stress is taken as an independent variable, while some parameters were related to temperature so as to describe the influence of temperature. When these models were applied to engineering problems, a series of mechanical tests was carried out to obtain the parameters at different temperatures, and the relationships between the model parameters and the temperature were established through data fitting (Wang et al., 2014). This is an indirect and approximated method for reflecting the effect of temperature and may reduce the calculation accuracy to some extent. In some practical engineering problems, the underground stress at a certain layer is usually constant, while the ground temperature increases continuously due to global warming and the thermal disturbance of infrastructures in permafrost regions, such as the Qinghai-Tibet Highway and Railway (Qi et al., 2007; Qin et al., 2009). It is tedious work to calculate the creep settlement under such conditions, for the model parameters under different temperatures must firstly be independently determined. To directly investigate the effect of the increasing temperature on the creep behavior of frozen soils, Qi and Zhang (2008) conducted a series of K0 temperature step-increase tests under different constant surcharge loads. It was indicated that with the stepped increases in temperature, a similar tendency is seen in the change in creep strain as it is applied with surcharge loads. In other words, the rise in temperature plays a role equivalent to that when the increase in stress acts on the stress-strain curves. Considering the phenomenological effects of temperature on the development of creep strain, it can also be taken as an independent variable. For unfrozen soils, a large number of models have been proposed based on the results of stress-strain tests, such as the Modified Cam-Clay and isotache creep models (Roscoe and Burland, 1968; Den Haan, 1996; Yin and Wang, 2012). Therefore, it would surely be worthwhile to try establishing a model taking temperature as an independent variable to directly describe the influence of the
increase in temperature on the creep behavior of frozen soils. In this paper, a series of temperature step-increase K0 compression tests is carried out on a Chinese standard sand under different constant surcharge loads. The test results are analyzed to obtain the characteristics of the developing creep strain. Based on an analysis of the test results, a creep model is proposed, taking temperature as an independent variable, and the applicability is verified with the test results of both this study and the previous research work. 2. Testing procedure A Chinese standard sand was taken as the study object. The grain size distribution curve for this sand is shown in Fig. 1. Sand samples were prepared with the multiple sieving pluviation (MSP) method (Miura and Toki, 1982; Baker and Konrad, 1985). As is shown in Fig. 2, with the MSP method, the dry unit weight of the samples was controlled by the falling height of the sand particles passing through the sieves. In this study, eleven sieves were used and the height of each was 3 cm; thus, the total falling height was 33 cm. The samples were formed to have the dimensions of 125.0 mm in height and 61.8 mm in diameter within a steel tube (Fig. 2). Then, the samples were fixed, together with the steel tube, onto two sides with porous stones and a steel frame, and saturated with distilled water by vacuum. The saturated samples were then placed into a container full of distilled water and put in a refrigerator for quick freezing, so that the water in the samples would be frozen in its original position and the ice would be distributed evenly in the sample (Ma et al., 2015). Four samples were prepared according to the above method. The dry unit weight, the water content and the degree of saturation of each sample are listed in Table 1. All the K0 temperature step-increase tests were conducted on a multifunction environmental testing apparatus (Fig. 3) with the controlling accuracy of the displacement and stress of 0.001 mm and 1 kPa, respectively. Details on the apparatus can be found in literature (Yao et al.,
Fig. 1. Grain size distribution curve for the Chinese standard sand.
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Fig. 2. Sketch of the sample-preparation device with the multiple sieving pluviation method.
Table 1 Physical parameters of the four prepared samples. Dry unit weight (kN/m3)
Water content (%)
Saturation degree (%)
17.7 17.5 17.7 17.6
16.3 16.5 16.8 16.3
86.8 85.0 89.5 85.4
2013). This apparatus was originally developed for triaxial tests on frozen soils. To keep the samples in the K0 state (i.e., lateral strain is zero), the soil samples together with the steel tube were put onto the pedestal of the apparatus.
3
In this way, it was not necessary to control the lateral strain by applying cell pressure. To guarantee the probable drainage of unfrozen water during the K0 compression, two steel disks with small holes (with a diameter of 1 mm and spacing of 10 mm) were put onto the bottom and top of the samples. The vent valve on the top of the pressure cell was switched on so as to release the cell pressure generated during compression (Fig. 3). At that time, a drainage boundary similar to that commonly used in frozen soil K0 compression tests (Qi and Zhang, 2008; Qi et al., 2010; Zhang et al., 2016), i.e., porous stones or steel discs with small holes, was put on one or two sides of the samples for the probable drainage of unfrozen water. With such a drainage boundary, the unfrozen pore water that had drained out would be frozen and would block the drainage channel through the porous stones. According to the research work of Zhang et al. (2016), the unfrozen pore water dissipation process was notably observed when the test temperature was higher than freezing. This indicates that the commonly used drainage boundary is effective under negative temperatures. Considering the fact that the drained out unfrozen pore water will be frozen and will block the drainage channel through the porous stones, the drainage process for the pore water at the drainage boundary might be different from that of unfrozen soils. When unfrozen water is drained out of the soil, it is frozen and firstly becomes ice crystals in the pores of the porous stones or steel discs. Then the ice crystals may migrate to lower stress regions due to the effect of the stress gradient (See the discussion on creep mechanism in Section 3.1). This might be the mechanism that the experimental setup allows, namely, ‘consolidation’ or ‘drainage’. As for the extremely small amount of unfrozen water, its ejection onto the drainage boundary cannot be visually observed or measured accurately with the soil mechanical testing techniques available at present, at least as far as the authors know. This is why the researchers (Zhang et al.,
Fig. 3. Sketch of the apparatus for the K0 compression tests. Please cite this article in press as: Yao, X. et al., A one-dimensional creep model for frozen soils taking temperature as an independent variable, Soils Found. (2018), https://doi.org/10.1016/j.sandf.2018.03.001
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2016) measured the unfrozen pore water pressure by embedding small pore water transducers inside the soil samples. Therefore, the drained out unfrozen water was not measured in this study. As for the test temperature, it can be seen in Fig. 3 that three cooling baths were used to control the sample temperature. Two were directly connected with cool oil that was circulating in the plates on the top and the bottom of the sample. The third was connected to a cooling ring around the sample with cool oil that was circulating so as to cool the oil surrounding the steel tube. During the K0 compression tests, the temperatures on the top and the bottom of the soil samples were monitored. K0 temperature step-increase tests were conducted on four samples under different constant surcharge loads of 0.2, 0.5, 1.0 and 2.0 MPa, respectively. For each sample under a certain surcharge load, five levels of temperatures were applied by stepped increases from about 10 °C, as is shown in Fig. 4. The duration of each temperature step was 1 day. As can be seen in Fig. 4, there were minor vibrations due to changes in the environmental temperature during each step. Therefore, the average value at each step was taken as the representative temperature. The representative temperatures under different surcharge loads are listed in Table 2. 3. Analysis of test results, model setup and verification 3.1. Strain classification and mechanism analysis Fig. 5 presents the strain vs. duration curves under different surcharge loads (0.2, 0.5, 1.0 and 2.0 MPa). As can be seen in the figure, the strain under different surcharge loads includes two parts, i.e., instantaneous strain (esl) and time-dependent strain. The time-dependent strain might be due to the primary or secondary (creep) consolidation process. For unfrozen soils, it is generally considered that primary consolidation occurs during the stage at which the pore water pressure decreases considerably, while secondary consolidation occurs during the stage at which most of the pore water pressure is dissipated and
Fig. 4. Duration curve for the stepped increases in temperature (0.2 MPa).
does not change notably. For frozen soils, according to the available research work by Zhang et al. (2016) on silty clay with a high ice content, the dissipation process for pore water pressure can be observed notably in the range in temperature from 0 to 0.4 °C. When the temperature is lower than 0.4 °C, the pore water pressure remains mostly invariant and is less than 5% of the surcharge load. This indicates that, when the temperature approaches or is higher than freezing (for this kind of fine-grained soil, the freezing temperature is about 0.4 °C (Xu et al., 2001; Zhou et al., 2015)), the permeability is not very small and the dissipation process for unfrozen water is notable. Correspondingly, primary consolidation occurs first. When the temperature is lower than the freezing point, the unfrozen water and the permeability decrease considerably and the unfrozen water dissipation process is not notable or extremely long. Previous studies have also confirmed this point under low temperatures (Bishop and Henkel, 1962; Williams and Burt, 1974). As for the drained out unfrozen water, there is no direct evidence at the present time showing that it can be visibly observed. That is why when the temperature is lower than the freezing point, the previous research works usually classified the deformation after applying a surcharge load as either instantaneous or viscous (or creep) (Fish, 1980; Zhu and Carbee, 1983). For the tested sand in this study, the freezing point is about 0.2 °C (Xu et al., 2001; Zhou et al., 2015), and the test temperatures of this study are lower than 0.40 °C (Table 2). Therefore, the time-dependent strain in the test results of this study can be classified as creep strain. In Fig. 5, the creep strain of the samples under the K0 condition is actually due to the volumetric compression, which comes from two aspects. On the one hand, a part of the volumetric compression might come from the drainage of unfrozen water. At current, although there is no direct evidence showing that unfrozen water drained out, it has been proven that unfrozen water migration or drainage does exist in frozen soils (Williams and Burt, 1974; Bishop and Henkel, 1962). It is an extremely long process for the small permeability and unfrozen water content of frozen soils. On the other hand, a part of the volumetric compression comes from the migration of pore ice crystals. According to experimental investigations (Vyalov, 1965; Vyalov et al., 1970), the migration of pore ice crystals from regions of concentrated stress to regions of lower stress can be observed under a constant surcharge load. For the frozen sand samples in this study, it can be seen in Table 1 that they are not fully saturated. Obviously, some void pores exist in the samples. Under the effects of the surcharge load, both the pore ice crystals and the unfrozen pore water can migrate to the void pores. In addition, for the porous drainage boundaries permitted on both sides of the samples, it is possible for the pore ice crystals and the unfrozen water to migrate to the porous drainage boundaries with lower stress. During the process of the migration of the pore ice crystals and the unfrozen water to the void pores and drainage boundaries, the soil particles are rearranged
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5
Table 2 Representative temperatures under different surcharge loads. Surcharge load (MPa)
T1 (°C)
T2 (°C)
T3 (°C)
T4 (°C)
T5 (°C)
0.2 0.5 1.0 2.0
9.95 10.01 10.02 9.97
5.32 5.39 5.33 5.27
2.15 2.48 2.12 2.11
0.96 1.10 1.14 1.09
0.48 0.55 0.55 0.61
Fig. 5. Strain vs. duration curves under different surcharge loads.
and volumetric compression occurs correspondingly. The above analysis indicates that the mechanism for the creep compression of frozen soils is different from that of unfrozen soils, i.e, both the pore unfrozen water drainage and the ice crystal migration contribute to the creep compression of frozen soils, and that this mechanism is not just due to the drainage of the unfrozen pore water. Therefore, the pore unfrozen water drainage (although this process is extremely slow when the test temperature is lower than the freezing point) and the ice crystal migration are the creep compression mechanisms of the test results. 3.2. Analysis of test results It can be seen that under different surcharge loads (Fig. 5), the development of strain shows a stepped pattern due to the stepped increases in temperature (Table 2), which is similar to the effect of the step loads. The strain vs. time curves at different temperatures are shown in Fig. 6 for each curve with the same surcharge load, when the starting time of each temperature step is reset to zero. On these curves with the same surcharge load, four group points with different strain rates are identified (Fig. 6(a)– (d)). After the strain and corresponding temperature of the points under different strain rates are plotted in the elog(|T|1) space (where |T|1 is the reciprocal of the temperature absolute value (RTAV)), the RTAV vs. strain curves under different strain rates are given, as shown in Fig. 7. For ease of elaboration, the RTAV vs. strain curves at a strain rate of 1.25 108 1/s under different surcharge loads are taken as examples. It can be seen that the RTAV vs. strain curves can generally be represented by two
straight lines (with slopes of a and b, respectively) in the e-log(|T|1) space. The intersection of the two sections is defined as |TP|1. With a decrease in strain rate, the positions of the first sections do not change notably, while the second sections move down (Fig. 7) and the |TP|1 decreases (Table 3). At the same time, the second sections with different strain rates form a series of parallel lines (Fig. 7). In other words, with an increase in temperature or |T|1, the strain in the first section is rate-independent and that in the second section is rate-dependent. Under a constant temperature condition (here, taking the strain vs. time curve at 0.5 MPa and T5 as an example (Fig. 6 (b)), the difference between two strain points is the creep strain which is equal to the vertical space between two parallel lines in Fig. 7(b). The space between two parallel lines can be expressed as follows: ei e0 ¼ ci ln
e_ 0 e_ i
ð1Þ
where ci (i = 1, 2 and 3) is the parameter representing the effect of strain rate on creep strain, and e0 ð_e0 Þ and ei ð_ei Þ (i = 1, 2 and 3) are the strains and strain rates of the points on the strain vs. time curve at 0.5 MPa and T5 (Fig. 6(b)). In the e-log(|T|1) space (Fig. 7(b)), these points are on different parallel lines at the same temperature T5. Along with these strains and strain rates of the points (Fig. 7(b)), ci is calculated by Eq. (1) and the corresponding values at different surcharge loads are listed in Table 4. It can be seen that under different surcharge loads, the ci values calculated with different parallel lines are basically the same, which means that the vertical space between two parallel lines is constant when the strain rate decreases by one order of magnitude (Eq. (1)) and that the average value (c) of ci can be used to represent it. Through the above analysis on the effects of stepped increases in temperature on the strain development, an idealized isotache model is summarized in Fig. 8, i.e., an initial compression line (with slope a) starting from an initial temperature (|Ti|1) is used to represent the first rateindependent section, while a series of creep isotaches (with slope b) is used to represent the second rate-dependent section, and the space between two parallel lines is c when the strain rate decreases by one order of magnitude. With the decrease in strain rate, the parallel lines moves down and the intersection of the two sections (|TP|1) decreases (i.e., a unique |TP|1 can be determined by a creep isotache at a certain strain rate e_ 0 ). This is basically identical to the parallel lines postulate of the isotache model developed
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(a) 0.2 MPa
(c) 1.0MPa
(b) 0.5 MPa
(d) 2.0MPa
Fig. 6. Strain vs. time curves at different temperatures and surcharge loads.
for unfrozen soil models (Den Haan, 1996; Vermeer and Neher, 1999), where the only difference is that the stress is taken as an independent variable. After the independent variable is replaced with temperature (|T|1), it would be worthwhile to try and develop a novel model capable of directly describing the effect of temperature on the creep of frozen soils. 3.3. Model setup and verification In the original isotache model, the total strain is stressdependent (Den Haan, 1996; Vermeer and Neher, 1999). After replacing the stress with the reciprocal of the absolute value of the temperature (|T|1), the total strain due to the increase in temperature can be expressed as ! ! 1 1 jT p j jT j e_ 0 e ¼ a ln þ b ln þ c ln ð2Þ 1 1 e_ jT i j jT P j 1
where jT P j is the intersection of the initial compression line and the creep isotache with strain rate e_0 (Fig. 8). Here, e_0 and |TP|1 are the reference strain rate and temperature, respectively.
Under a constant temperature, Eq. (1) is the incremental creep strain, which can be written as e e e_ 0 0 ð3Þ ¼ exp c e_ and exp
e e 0
c
de ¼ e_ 0 dt
where e is the strain at strain rate e_ . The integration of Eq. (4) is Z e Z t e e 0 exp e_ 0 dt de ¼ c e0 t0
ð4Þ
ð5Þ
According to Eq. (5), the relationship between creep strain and time is deduced as follows: e e e_ 0 0 ð6Þ exp ¼ ðt t0 Þ þ 1 c c where a new parameter s0 is defined as c s0 ¼ e_ 0
ð7Þ
and Eq. (6) is written as
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(a) 0.2 MPa
(c) 1.0 MPa
(b) 0.5 MPa
(d) 2.0 MPa
7
Fig. 7. Relationship between RTAV (|T|1) and strain at different strain rates.
increases in temperature), t is the real time and (t t0 + s0) is defined as intrinsic time s, i.e.,
Table 3 TP at different strain rates. Surcharge load (MPa)
Strain rate (1/s)
Tp (°C)
0.2
4.00 109 1.25 108
2.90 2.17
0.5
4.00 109 1.25 108
3.77 2.70
1.0
5.00 109 1.25 108
6.06 5.26
2.0
1.25 108 4.00 108
5.88 4.87
exp
e e t t þ s 0 0 0 ¼ c s0
s ¼ t tr
ð9Þ
where tr = t0 s0 is the difference between real time t and intrinsic time s. A detailed discussion on the intrinsic time can be found in Den Haan (1996). With Eqs. (8) and (9), the incremental creep strain from e0 can be written as e e0 ¼ c ln
s s0
ð10Þ
Substituting Eq. (10) into Eq. (2), the total strain is written as ! ! jT p j1 jT j1 s e ¼ a ln ð11Þ þ b ln þ c ln 1 1 s 0 jT i j jT P j
ð8Þ
in which, s0 is the reference intrinsic time, t0 is the starting time of the creep strain (or the ending time of the stepped Table 4 Calculated parameter ci with different parallel lines. Surcharge load (MPa)
c1
c2
c3
c
0.2 0.5 1.0 2.0
2.85 104 2.93 104 3.81 104 6.03 104
2.85 104 2.93 104 2.46 104 6.85 104
2.85 104 2.99 104 3.20 104 5.93 104
2.85 104 2.95 104 3.16 104 6.27 104
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Fig. 8. Idealized isotache model.
Under a constant temperature, the time derivative of Eq. (11) is the creep strain rate, i.e., c ð12Þ e_c ¼ s According to Eq. (12), it is found that for a creep isotache with a certain strain rate, there is a corresponding unique intrinsic time and the two variables are inversely related. With retrospect on the physical meaning of the idealized isotache model in Fig. 8, the total strain in Eq. (11) can be decomposed into two parts, namely, the rateindependent initial compression strain (ei) and the ratedependent strain (er). Therefore, the rate-dependent strain is er ¼ e ei
ð13Þ
where the initial compression strain (ei) is defined as follows: ! ! 1 1 jT p j jT j ei ¼ a ln þ a ln ð14Þ jT i j1 jT P j1 When Eqs. (11), (12) and (14) are substituted into Eq. (13), the creep strain rate can be further expressed as !ðbaÞ=c c er =c jT j1 ð15Þ e_ c ¼ e 1 s0 jT P j Together with Eqs. (14) and (15), the total strain rate is expressed as !ðbaÞ=c 1 1 jT_ j c er =c jT j ð16Þ e_ ¼ e_ i þ e_ c ¼ a 1 þ e 1 s0 jT j jT P j Besides the incremental strain due to the increase in temperature and the creep process (Eq. (16)), the instantaneous strain (esl) due to the applied surcharge load should also be taken into consideration (Fig. 5). Therefore, the total strain can be calculated as follows: 0 !ðbaÞ=c 1 Z t 1 1 _ @a jT j þ c eer =c jT j Adt ð17Þ e ¼ esl þ 1 1 s0 jT j jT P j 0
Eqs. (1)–(17) are set up as a creep model by taking the temperature as an independent variable. In the following, the applicability of this model will be verified based on the test results in this study and in previous research work. The model parameters for the Chinese standard sand are listed in Table 5, where parameters such as a, b, TP and s0 are determined with the RTAV vs. strain curve at a strain rate of 1.25 108 1/s (Fig. 7 and Eq. (12)). Parameter c is taken as the average value of ‘ci’ obtained under different strain rates (Table 4). Table 6 presents the model parameters obtained from the results of tests on the Qinghai-Tibet silty clay with a water content of 40% and a degree of saturation of about 80% (Qi and Zhang, 2008). The parameter-obtaining method is consistent with that in Section 3.2. Figs. 9 and 10 are the test and calculated straindevelopment curves due to the stepped increases in temperature. It can be seen that the calculated results agree well with the Chinese standard sand test results, while for the silty clay, the calculated results do not agree very well with the test results, especially in the last two temperature steps. This might be due to the primary consolidation occurring first during the last two temperature steps. Just as analyzed in Section 3.1, the consolidation stage of frozen soils is mainly influenced by the freezing temperature, i.e., primary consolidation occurs first when the temperature approaches or is higher than the freezing point, while secondary consolidation (creep) mainly occurs below the freezing point. As is shown in Fig. 10, when the temperature is lower than the freezing point (approximately at 0.4 °C for the silty clay) (Table 7), the calculated results agree well with the test results during the first two steps. When the temperature approaches or is higher than the freezing point (Table 7), there is a large discrepancy between the calculated and the test results during the last two steps, which is due to the primary consolidation occurring first in the soil. Therefore, the creep model proposed in this paper is applicable to the creep compression process when temperatures are below the freezing point. As for the applicable duration of this creep model, the model is verified to be applicable within a creep time up to 1.6 days with the available test results (see the second step in Fig. 10). As for longer applicable durations of this model, more tests should be conducted. With retrospect on the test results of different soils in Figs. 9 and 10, the effects of the imperfect K0 condition and saturation on the law of the creep compression can be further analyzed. As described in Section 2, the test results in Fig. 9 were obtained by restraining the samples in the K0 state within a steel tube; the test results in Fig. 10 were obtained by restraining the samples in the K0 state within a plexiglass tube (Qi and Zhang, 2008). The degrees of saturation of the two group test results are also different (the degree of saturation of the silty clay is about 80%, while that of standard sand is in the range of 85–90% (Table 1)). Generally, when the degree of saturation is larger than 85%, the soil samples can be considered
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Table 5 Creep model parameters for Chinese standard sand. Surcharge load (MPa)
a 4.20 10 2.20 104 2.50 104 6.00 104
0.2 0.5 1.0 2.0
Tp (°C)
esl
3
4
2.85 10 2.95 104 3.16 104 6.27 104
4
2.20 10 2.50 104 2.50 104 5.00 104
2.17 2.70 5.26 5.88
1.00 104 1.00 104 3.00 104 1.00 103
b 4
s0 (s)
c
1.25 10 1.35 103 1.35 103 2.45 103
Table 6 Creep model parameters for Qinghai-Tibet silty clay. Surcharge load (MPa)
a
b
c
s0 (s)
Tp (°C)
esl
0.1 0.2
1.20 102 2.30 102
3.50 102 4.00 102
1.40 103 3.20 103
1.87 104 4.27 104
0.66 0.90
1.00 104 3.00 103
(a) 0.2 MPa
(c) 1.0 MPa
(b) 0.5 MPa
(d) 2.0 MPa
Fig. 9. Calculated and test curves for strain vs. time at different surcharge loads (Chinese standard sand).
as saturated in unfrozen soil mechanics. With these tests results under different test conditions (with different K0restrained measures and saturation degrees), the results with the proposed creep model agree well with the test results when the test temperature is lower than the freezing point, as shown in Figs. 9 and 10. This indicates that the law described by the proposed model is an intrinsic creep development characteristic of frozen soils, and is not due to the effects of imperfect test conditions, such as the imperfect K0-restrained measures and saturation.
4. Simplified parameter-obtaining method In Section 3.2, the physical meaning of the model parameters is clarified through an analysis of the law of creep development due to the increase in temperature. However, with the parameter-obtaining method shown in Section 3.2, it is a tedious work to apply this model in practice, for the RTAV vs. strain curves at different strain rates must firstly be determined (Figs. 6 and 7). In this section, the characteristic of the development of creep strain will
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ence between the two curves gradually disappears (Fig. 11(d)). That is to say, when t tr, the real time (t) is approximately equal to the intrinsic time (s), and Eq. (18) can be further expressed as follows:
(a) 0.1 MPa
e e0 ¼ c ln
(b) 0.2 MPa
Fig. 10. Calculated and test curves for strain vs. time at different surcharge loads (Qinghai-Tibet silty clay).
Table 7 Representative temperatures for Qinghai-Tibet silty clay under different surcharge loads. Surcharge load (MPa)
T1 (°C)
T2 (°C)
T3 (°C)
T4 (°C)
0.1 0.2
1.45 1.54
1.02 0.98
0.53 0.50
0.42 0.32
be further analyzed so as to provide a more straightforward parameter-obtaining method. As shown in Fig. 11, all the strain vs. time logarithm curves at high temperatures tend to be a straight line, the slope of which is generally equal to the value of c in Table 5. Therefore, c can be determined by the average values of straight line slopes at high temperatures on the strain vs. time logarithm curves (from T3 to T5). In addition, according to Eqs. (8) and (10), the creep strain under certain temperatures is expressed as e e0 ¼ c ln
s t tr ¼ c ln s0 s0
ð18Þ
For the effect of the time difference (tr = t0 s0), there is a large difference between the strain vs. ln(t) and strain vs. ln(s) curves at the initial stage (Fig. 11(d)). With an increase in time, the strain vs. ln(t) curve tends to be a straight line represented by strain vs. ln(s), and the differ-
s t c ln s0 s0
ð19Þ
It can be concluded from the above analysis that, for these creep strain curves under high temperatures (from T3 to T5), when the time step of each temperature is long enough, the real time (t) at the end of each step is approximately equal to the intrinsic time (s). In another words, for the strain at the end of each temperature step under high temperatures, the duration of each temperature step (i.e., 1 day) can be taken as their reference intrinsic time (s0). As shown in Figs. 6 and 7, parameter a is determined with the strains that do not change notably with the develop of strain rate or time under low temperatures (T1 and T2), while parameters b and Tp are determined with the strains under high temperatures (T3 to T5) at a certain reference strain rate or intrinsic time (Eq. (12)). This means that the reference intrinsic time (s0) of the ending strains at each temperature step (T1 to T5) can be taken as approximately 1 day. According to the analysis in Section 3.2, the RTAV vs. strain curves under different constant strain rates or intrinsic times are a series of parallel lines, and the model parameters are determined with one of them. Thus, parameters a, b and Tp can also be determined with the ending strains at each temperature step. To show the validity of this inference, the results of the tests on the Chinese standard sand at 2.0 MPa are taken as an example (Fig. 12). The RTAV vs. strain curve obtained with the ending strains (Fig. 11(d)) is compared with the one at a strain rate of 1.25 108 1/s (Fig. 7(d)) in Fig. 12. It can be seen that the two curves are generally parallel to each other. This indicates that the RTAV vs. strain curve obtained with the ending strains can be used to represent the one at the intrinsic time of 1 day or the strain rate of c/1 day (according to Eq. (12)). It is can be seen in Fig. 12 that the a and b obtained from the two curves are basically the same, while the Tp obtained from the ending strain curve is lower than that at 1.25 108 1/s. This is due to the fact that the Tp is uniquely determined by the strain rate or the intrinsic time (Eq. (12)). Obviously, with the ending strains at each temperature step, model parameters a, b and Tp can be determined simply instead of tediously obtaining the RTAV vs. strain curves under different constant strain rates. From the above analysis, the simplified parameterobtaining method can be summarized as follows: (1) a, b and |TP|1 can be determined with the RTAV vs. strain curve obtained with the ending strains at each temperature step. (2) c can be determined by the average value of the constant creep slopes under high temperatures on the strain vs. time logarithm curves. (3) The reference intrinsic time (s0) is the duration of each temperature step.
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(a) 0.2 MPa
(c) 1.0 MPa
(b) 0.5 MPa
(d) 2.0 MPa
11
Fig. 11. Strain vs. time logarithm curves at different temperatures and surcharge loads.
5. Discussions
Fig. 12. Relationship between strain and |T|1 at different strain rates (2.0 MPa).
With this simplified method, the parameters of the four test cases for standard sand are listed in Table 8. The corresponding calculated results for strain with stepped increases in temperature are compared with the test results in Fig. 13. It can be seen that the calculated results match the test results well, which indicates that the simplified method is valid. For performing practical calculations with the proposed model, this method is recommended.
In the previous research work, Qi and Zhang (2008) conducted the temperature step-increase tests from 1.5 to 0.3 °C. After an analysis of the change tendency of the compression coefficient against temperature, Qi and Zhang (2008) concluded that when the temperature is higher than 1.0 °C, the compressibility increases considerably; and thus, 1.0 °C is taken as the boundary temperature for defining ‘‘warm” frozen soils. This is approximately consistent with the results given in Table 6, where |TP|1 falls in the range from 0.90 to 0.66 °C. According to the definition in Section 3.2, |TP|1 is the inflection point of the two compression stages; it can also be taken as an index for identifying the compression stage of frozen soils. However, taking a constant temperature value for defining ‘warm’ or ‘cold’ frozen soils is only partially valid due to the narrow range in surcharge loads applied in the previous study by Qi and Zhang (2008) (from 0.1 to 0.3 MPa). It can be seen from the test results on the Chinese standard sand that, when the surcharge load increases from 0.2 MPa to 2.0 MPa, T P decreases from 3.13 °C to 6.25 °C (Table 8); for Qinghai-Tibet silty clay, when the surcharge load increases from 0.1 MPa to 0.2 MPa, the TP decreases from 0.66 °C to 0.90 °C
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Table 8 Model parameters of the standard sand obtained with the simplified method. Surcharge load (MPa)
a
b 4
3.80 10 2.20 104 3.00 104 7.00 104
0.2 0.5 1.0 2.0
s0 (s)
c 3
1.25 10 1.35 103 1.40 103 2.50 103
4
3.70 10 4.30 104 3.30 104 7.00 104
(a) 0.2 MPa
(c) 1.0 MPa
(b) 0.5 MPa
(d) 2.0 MPa
TP (°C)
8.64 10
4
3.13 3.70 5.88 6.25
Fig. 13. Calculated (with simplified parameter-obtaining method) and test curves for strain vs. time at different surcharge loads.
(Table 6). It can be concluded that temperature TP, for distinguishing frozen soils at different compression stages (‘‘cold” or ‘‘warm” frozen soils), is not a constant, which is inversely related to the surcharge load. In other words, with the increase in surcharge load, the boundary temperature TP decreases and the range in temperature of frozen soils under high compressibility is enlarged. As seen above, in the discussion on Tp, all the parameters in this model need to be determined independently for different stresses (Table 8). They all have the common physical meaning under different stresses, such as a, b, c and Tp. These parameters are obtained by analyzing the law of creep development, due to the increases in temperature, not just through simple data fitting. According to Section 3.2, it can be found that the creep development at different surcharge loads shows the same law despite the
effect of instantaneous strain. The total strain is equal to the sum of the creep and instantaneous strains (Eq. (17)), in which the instantaneous strain is related to the applied constant stress and initial temperature and can be taken as a parameter in the model. From this viewpoint, the model proposed in this paper can be taken as a complete model at least in terms of the same position of the models taking stress as an independent variable (Yang et al., 2010; Li et al., 2011; Wang et al., 2014; Liao et al., 2016), and has the capacity to predict the creep behavior due to increases in temperature. Ideally, temperature and stress should both be included as independent variables in a model. However, for the complex mechanical behavior of frozen soils, it is an extremely hard task to reasonably describe the influences of stress and temperature with a model. This is why most of the
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proposed models for frozen soils at the current time take the stress as an independent variable (Yang et al., 2010; Li et al., 2011; Wang et al., 2014; Liao et al., 2016), while the parameters need to be defined independently at different temperatures. Obviously, it is inconvenient to use these models to describe the creep development when the case of constant stress and an increasing temperature is involved. In other words, one model cannot accommodate all conditions. Whether or not the stress or temperature is taken as an independent variable in a model, some parameters must be defined independently for different stresses or temperatures. Based on this consideration, a creep model, which takes temperature as an independent variable, is proposed for the case of constant stress and an increasing temperature, while the simultaneous effects of temperature and stress are not within the scope of this study. 6. Conclusions In this paper, a series of temperature step-increase K0 compression tests were conducted under different constant surcharge loads. Based on an analysis of strain development, a novel creep model was proposed by taking temperature as an independent variable, and the applicability of the model was verified with test results. The following conclusions can be drawn: (1) Based on an analysis of the characteristics of creep strain development, due to stepped increases in temperature, it was indicated that the relationship between the reciprocal of the temperature’s absolute value (RTAV) and the strain at different strain rates fits well with the parallel lines postulate of the isotache model for unfrozen soils. (2) After taking the RTAV as an independent variable in the isotache model, it was shown that the tendency of the stepped pattern development of strain due to the stepped increases in temperature can be reasonably described, which indicates the applicability of the novel model. The applicable range in temperature for the proposed model is below freezing. (3) A simplified parameter-obtaining method was proposed, whereby a, b and |TP|1 can be determined by the RTAV vs. ending strain curve, c can be determined by the average constant slopes under high temperature steps, and the duration of each temperature step in the K0 compression tests can be taken as the intrinsic reference time s0. After verification with the test results, it was indicated that the simplified method is applicable and recommended. (4) The physical meaning of TP is the temperature boundary for distinguishing different compression stages, which is inversely related to the surcharge load. With the increase in surcharge load, the range in temperature of frozen soils under high compressibility is enlarged.
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