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Effect of temperature and strain rate on mechanical characteristics and constitutive model of frozen Helin loess
Cold Regions Science and Technology 136 (2017) 44–51
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Effect of temperature and strain rate on mechanical characteristics and constitutive model of frozen Helin loess Xiangtian Xu a, Yubing Wang b,⁎, Zhenhua Yin a, Hongwei Zhang c a b c
Institute of Transportation, Inner Mongolia University, Hohhot 010070, China Fugro Consultants, Inc., 8613 Cross Park Drive, Austin, TX 78754, United States Inner Mongolia Communications Constructions Engineering Quality Supervision Bureau, Hohhot 010020, China
a r t i c l e
i n f o
Article history: Received 23 June 2016 Received in revised form 14 January 2017 Accepted 31 January 2017 Available online 02 February 2017 Keywords: Frozen Helin loess Uniaxial compressive Stress-strain Strength Elastic modulus
a b s t r a c t To investigate the effect of temperature and strain rate on the mechanical properties of frozen loess, a series of uniaxial compressive tests were conducted on saturated frozen Helin loess under five different strain rates (1 × 10−2/s, 1 × 10−3/s, 1 × 10−4/s, 5 × 10−5/s, and 1 × 10−5/s) and at four different temperatures (−2 °C, −4 °C, −5 °C, and −7 °C). From the stress-strain curves under different testing condition, the stain-softening behavior is observed and the yield point can be obviously seen. The yield stress and strength increase linearly with the decrease of temperature and increase exponentially with the increase of strain rate. The initial tangent modulus and the secant modulus corresponding to half the strength are taken to describe the stiffness of frozen loess. And the effects of temperature and strain rate on the tangent and secant modulus are analyzed. Furthermore, an elasto-plastic constitutive model considering the effects of temperature and strain rate has been proposed to describe the strength and deformation behavior of frozen Helin loess. © 2017 Elsevier B.V. All rights reserved.
1. Introduction The mechanical properties of frozen soil have gained importance on engineering design process, construction and maintenance phase after construction in cold regions (Qi and Ma, 2010; Li et al., 2016). Therefore, various experiments have been performed to explore the mechanical properties of frozen soil (Yang et al., 2010; Xu et al., 2011; Lai et al., 2014; Yang et al., 2016a, 2016b; Zhou et al., 2016). Presently, to study the mechanical characteristics of frozen soil under different testing conditions involving temperature, strain rate, and confining pressure (Bragg and Andersland, 1981; Lai et al., 2013), the majority of laboratory tests are uniaxial compressive test and triaxial compressive tests (Baker et al., 1982; Andersen et al., 1995; Arenson and Springman, 2005). Since triaxial compression test is capable to simulate the effect of earth pressure on the mechanical performance of frozen soil, so it is applicable to investigate the mechanical properties of frozen soil with deep embedment (Sayles, 1973; Jones and Parameswaran, 1983; Zhang et al., 2007; Lai et al., 2010; Zhao et al., 2011). In contrast, for the frozen soil with ice-rich and ice-saturated frozen soil, confining pressure has little impact on its strength and deformation characteristics. Thus, it is time-saving and cost-effective to explore the mechanical properties of the ice-rich and ice-saturated frozen soil by uniaxial compressive test. The uniaxial compressive test is generally capable to characterize the ⁎ Corresponding author. E-mail address: [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.coldregions.2017.01.010 0165-232X/© 2017 Elsevier B.V. All rights reserved.
mechanical properties of frozen soil except the examination of the effect by confining pressure. Especially, for the layer frozen soil at shallow embedment depth, the stiffness and strength characteristics determined by uniaxial compressive tests can be easily applied to engineering practice. Furthermore, the unixial compressive test is usually used to investigate other geomaterials under lower temperature condition (Martin and Jun, 2010; Yang et al., 2015; Yang et al., 2016a, 2016b). The characterizations in the strength and deformation of frozen soil were firstly started by performing uniaxial compressive tests at the early stage. For example, in 1930s, Tsytovich pioneered to study mechanical properties of frozen sand under different temperatures and strain rates by carrying out series of uniaxial compressive tests. He found that, with the decrease of temperature and the increase in strain rate, the strength of frozen sand increased nonlinearly, but the increasing rate of strength gradually mitigated (Tsytovich, 1972). Tsytovich, Vialov, Ladanyi and Haynes et al. (Tsytovich, 1975; Vialov, 1959; Ladanyi, 1981; Haynes et al., 1975; Haynes and Karalius, 1977) conducted a series of unconfined compressive tests on frozen soil under different temperatures, strain rates and unit weight and arrived at similar conclusion by Tsytovich (1972), Wu et al. (1994) also attained that the strength of frozen sand was related to temperature and strain rate, and he accomplished an empirical relationship that strength is a function of temperature based on series of unconfined compressive tests on frozen sand. Chen et al. (1988) found that there exist changes in both physical and chemical properties of frozen soil with the decrease of temperature, which resulted in an increase in soil strength. Zhu and
X. Xu et al. / Cold Regions Science and Technology 136 (2017) 44–51
the variation of temperature and strain rate, which provides reference for the construction projects in cold regions.
Nomenclature σy σy0 σf σf0 σij ε_a _ εa0 εij εeij εpij T T0 K G E υ Y k Deijkl Dpijkl
45
yield stress reference yield stress strength reference strength stress tensor strain rate reference strain rate strain tensor elastic strain tensor plastic strain tensor temperature reference temperature elastic volumetric modulus elastic shear modulus elastic modulus poisson's ratio yield function hardening parameter elastic flexibility tensor plastic flexibility tensor
2. Test program 2.1. Test material The soil material was collected from Helin County, Hohhot City, Inner Mongolia Province, China. The grain size distribution determined by a laser particle size analyzer was shown in Table 1. From the distribution, the soil is classified as silty sand. The specific gravity is 2.526 by specific gravity test.
2.2. Sample preparation
Carbee (1984) completed a series of uniaxial compressive tests on frozen sand under different temperatures and different strain rates, and proposed a prediction model for strength which is a function of different temperature. Yang et al. (2006) discovered three characteristics of saturated saline frozen silty clay from a series of uniaxial compressive tests: (1) the negative temperature had a great effect on the strength of frozen soil; (2) the uniaxial compressive strength increased linearly with the decrease of temperature; (3) at a certain strain rate, the uniaxial strength was twice higher than the yield stress. Li et al. (2004) conduct a series of uniaxial compressive tests on saturated frozen clay to investigate the effects of temperature, strain rate and dry density on its strength. Zhao et al. (2013) applied the uniaxial compressive test to study the deformation and strength of frozen clay under thermal gradient condition. All of the conclusions from these existing research results point out the temperature and strain rate are the primary external factors affecting the mechanical properties of frozen soil. However, the mechanical response of frozen soil under various thermal-mechanical conditions is not only determined by those external factors, the internal factors, such as water content, dry unit weight, salt content, and degree of saturation, are also critical to the mechanical properties of frozen soil. These internal factors result in the measured mechanical characteristics of the same type of frozen soil are different from each other even under the same experimental conditions. These disagreements made it difficult when applying those experimental results to solving practical engineering problems and referencing those researchers' work for future study. The purpose of this paper is to report mechanical properties of frozen loess collected for a road project in Helin County, Hohhot City, Inner Mongolia Autonomous region. A series of uniaxial compressive tests were conducted on the frozen loess under different strain rates (1 × 10−2/s, 1 × 10–3/s, 1 × 10−4/s, 5 × 10−5/s, and 1 × 10−5/s) at different temperatures (−2 °C, −4 °C, −5 °C, and −7 °C). Based on the tests results, attention has focused in particular on the stress-strain curves, yield strength, elastic modulus and constitutive model with
In this paper, the preparation of frozen loess sample was followed by the sample preparation method by Chang et al. (1995). Firstly, the impurities such as small pebbles in the natural loess collected from field were screened out by the sieve with 2 mm mesh size. Then, the loess was reconstituted by mixing thoroughly with distilled water (appropriate 10% by weight of soil), and sealed for 12 h to ensure that water content was uniform in the mixed soil. Next, the mixture was compacted into a cylinder with the diameter of 6.18 cm and the height of 12.5 cm and filled into a three-piece split mold. We placed one porous stone at the top and one at the bottom of specimen. Afterwards, the specimens were evacuated and saturated with water for 24 h and then placed into refrigerator for 48 h. The three-piece split mold was removed from the specimen and epoxy resin platens were place on top and bottom of specimen while a rubber membrane being wrapped around. The specimens are divided into two groups for different testing scopes: the specimens in the first group were used to test in different temperature (− 2 °C, − 4 °C, − 5 °C, and − 7 °C) under the same strain rate (1.67 × 10−4/s); the specimens in the second group were used to test at different strain rate (1 × 10− 2/s, 1 × 10− 3/s, 1 × 10− 4/s, 5 × 10−5/s, and 1 × 10−5/s) but at a constant temperature at − 4 °C. All of the specimens were kept at the target temperature at least for 12 h so that temperature inside each specimen is homogenous. The water content of saturated specimen is 24.58%, and dry density is 1.97 g/cm3. 2.3. Test apparatus and test procedure A frozen soil creep apparatus used in this study is capable of applying axial force ranging from 0kN up to 100kN, and controlling temperature from −25 °C to room temperature with resolution of 0.1 °C. The apparatus can be either strain-controlled or stress-controlled. In this study, the strain-controlled mode was used. The testing procedure was summarized as following. First, we tested the specimens in the first group. We placed the specimen into the loading chamber where the temperature was set up to the same one as in the specimen preparation stage. Then we waited for a couple minutes for the temperature reached equilibrium state, and we slowly lowered down the loading piston until the platen of piton was perfectly in contact with the top of specimen. Last, we started loading the sample at strain rate of 1.67 × 10−4/s, and stopped loading at 20% strain limit. We followed same procedure to test the second group, except that the loading rate was set up to 1 × 10− 2/s, 1 × 10− 3/s, 1 × 10− 4/s,
Table 1 Particle fraction of Helin loess. b0.005 mm
0.005–0.05 mm
0.05–0.075 mm
0.075–0.10 mm
0.10–0.25 mm
N0.25 mm
1.91%
12.59%
20.76%
20.71%
40.56%
3.43%
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X. Xu et al. / Cold Regions Science and Technology 136 (2017) 44–51
5 × 10−5/s, and 1 × 10−5/s, respectively and temperature was setting to the −4 °C for all specimens in the second group. 3. Test results and analysis
Table 2 Yield strain and failure strain of frozen Helin loess under different temperatures. Temperature (°C)
−2
−4
−5
−7
Yield strain (%) Failure strain (%)
1.50 15.96
0.77 13.79
0.90 13.79
0.92 14.27
3.1. The effect of temperature on mechanical characteristics of frozen Helin loess 3.1.1. Deformation characteristics Fig. 1 shows that the stress-strain curves of frozen Helin loess obtained from the uniaxial compressive tests at different temperatures are presented. It can be seen that the frozen Helin loess present strainsoftening and brittle failure under different temperature. The stressstrain curves have three stages: a quasi-elastic stage, a plastic deformation stage and a softening stage. An obvious yield stress point and peak stress can be seen from each stress-strain curve. Before the yield stress point, the stress-strain curve is approximately as a straight line, which is prescribed as the quasi-elastic stage. The slope of this straight line in this stage increases with the decrease of temperature which means the stiffness of frozen soil becomes higher while the temperature is lower. This phenomenon can be explained that the lower temperature makes the unfrozen water less and cementation effect by ice stronger. After the yield stress point, the stress-strain curve enters the plastic deformation stage. At this stage, the soil has less capacity of resistance to deformation. The stress keeps increasing with the increase of strain, however, the rate of the stress increase becomes smaller than the rate in the first stage (the quasi-elastic stage). This is because of the increasing stress in this stage leads to a crush of ice which makes the cementation effect by ice weaker. The loss of cementation results in a decrease in the ability to resist deformation and produces the plastic deformation. With a further increase in strain, the stress reaches the peak value, and the stress-strain curve enters the softening stage. From Fig. 1, it shows that the frozen Helin loess at − 2 °C does not exhibit obvious strain-softening. In comparison with other specimens under lower temperatures, the strain-softening phenomenon is more obvious and the peak stress is more evidently with the decrease of temperature. These two observations indicate that the brittleness of frozen Helin loess increases with the decrease of temperature. As shown in Fig. 1, a distinct yield point can be picked from each stress-strain curve under different temperatures. In this paper, we define the strain at yield point as the yield strain, and the strain at peak value as failure strain. The values of yield strain and failure strain of frozen Helin loess at different temperatures are presented in Table 2. As shown in Table 2, with the decrease in temperature, the yield strain and failure strain decrease first and then increase subsequently. This
Fig. 1. Stress-strain curves of frozen loess under different temperatures.
indicates that as the temperature decreases, the unfrozen water content decreases, the brittleness of frozen Helin soil increases, and yield strain together with failure strain decrease. However, the yield strain and failure strain will not keep decreasing with further decrease of temperature. As the temperature reaches at the critical point (− 4 °C is the critical point in this paper), the further decrease of temperature reduces the unfrozen water content only in a small amount, but the soil particles are more compact due to thermal contraction, which makes frozen soil capable to resist higher plastic deformation and failure, therefore the yield strain and failure strain increase instead of keeps decreasing with the decrease of temperature. Meanwhile,we can find the differences between yield strain or failure strain at different temperature are small. The yield strain at different temperature ranges between 0.77% and 1.50%. The maximum yield strain occurs at − 2 °C, and the minimum yield strain is at − 4 °C. The yield strain firstly decreases with the decrease of temperature, however, it increases with the continuous decrease of temperature after reaching its minimum at −4 °C. The failure strain has similar trend, that is, the maximum failure strain occurs at −2 °C (15.96%), and starts decreasing with the decrease of temperature. At − 4 °C and − 5 °C, the failure strain also reaches its minimum value, and back to increase again with decrease of temperature. 3.1.2. Yield limit and strength characteristics Fig. 2 illustrates the yield limit and strength of the saturated frozen Helin loess at different temperatures. As shown in Fig. 2, the yield limit and strength decrease linearly with the increase of temperature. Because the decrease in temperature results in the amount of unfrozen water decreases and more ice forms in the voids between soil particle, which intensifies the cementation effect by ice in the frozen soil. Thus, frozen soil attains higher strength to resist failure and is more capable to resist plastic deformation. The relationships between yield limit of frozen loess and temperature as well as relationship between strength and temperature can be expressed by the linear functions as follows: σ y =σ y0 ¼ ay ðT=T 0 Þ þ by
ð1Þ
σ f =σ f 0 ¼ a f ðT=T 0 Þ þ b f
ð2Þ
Fig. 2. Strength and yield limit of frozen loess under different temperatures.
X. Xu et al. / Cold Regions Science and Technology 136 (2017) 44–51
According to the test results, the Eqs. (1) and (2) can be applied in the temperature ranges from − 7 °C to − 2 °C. Where σy and σf is yield stress and strength, respectively, T is temperature, T0 = 1 °C, σy0 = σf0 = 1 MPa are reference temperature, reference yield stress and reference strength, respectively. ay, by, af, and bf are material parameters. In this paper ay = −0.1256, by = 0.5934, af = −0.4784, and bf = 1.8631 by fitting the test data. 3.1.3. Effect of temperature on the elastic modulus of frozen loess In this paper, two elastic moduli are analyzed, the initial tangent modulus and the secant modulus corresponding to half the strength, which are generally used to characterize the elastic modulus of frozen soil. The secant modulus corresponding to half the strength is determined according to the People's of Republic of China coal industry standard (MT/593. 5-2011, 2011). As illustrated in the previous section, a distinct yield point can be picked from each stress-stain curve, which means the frozen loess has obvious elastic stage. Therefore, to determine the initial tangent modulus, the data before the yield point are used for linear regression. Fig. 3 shows the initial tangent modulus and the secant modulus corresponding to half the strength of saturated frozen Helin loess at different temperatures. From Fig. 3, at different temperature, the initial tangent modulus is always greater than the secant modulus. The same conclusion is also obtained for other types of frozen soil, which indicates that the stiffness of frozen soil in the initial deformation stage is greater than that in the late deformation stage. Both the initial tangent modulus and the secant modulus corresponding to half the strength increase with the decrease of temperature, and their variations with temperature can be divided into two stages. The first stage starts from − 2 °C to −4 °C, in which the initial tangent modulus and the secant modulus corresponding to half the strength increase rapidly with the decrease of temperature. In details, the increasing rate of the initial tangent modulus and the secant modulus corresponding to half the strength is 52.26 MPa/°C and 13.98 MPa/°C, respectively. This rapid increase can be explained by that there is a relatively large amount of unfrozen water at − 2 °C, thus the soil sample which mostly constituted with silts is still saturated; however, the unfrozen water content decreases while more ice forms when the temperature decreases to − 4 °C, which increases the cementation effect by ice and soil resistance to deformation. As a result, the stiffness of frozen soil is greatly improved. On the other hand, it is found that the increasing rate of the initial tangent modulus is greater than that of the secant modulus corresponding to half the strength, which indicates that the effect of unfrozen water on the initial tangent modulus is greater than that on the secant modulus corresponding to half the strength. The temperature range of the second stage is from −4 °C to −7 °C. In this stage, the initial tangent modulus and the secant modulus corresponding to half the strength continues
Fig. 3. Elastic moduli of frozen loess under different temperatures.
47
increasing with the decrease of temperature, but with a lower increasing rates (2.84 MPa/°C and 5.31 MPa/°C, respectively) compared with the rates in the first stage. The unfrozen water content becomes less when temperature is lower than −4 °C. A further decrease in temperature has slight impact on the ice content in the frozen soil, which results in a hardly improvement of the stiffness. It can be predicted the soil stiffness is not expected to improve by the further decrease of temperature. It should be noted that the increasing rates of both moduli are roughly same in the second stage, which indicates that both two moduli can accurately describe the deformation characteristics of frozen Helin loess under low unfrozen water content. 3.2. The effect of strain rate on mechanical characteristics of the frozen Helin loess 3.2.1. Deformation characteristics Fig. 4 illustrates that the stress-strain curves of frozen Helin loess under different strain rates at −4 °C. The stress-strain curves of frozen Helin loess present strain-softening under each different strain rate, and they are also can be divided into three stages: a quasi-elastic stage, a plastic deformation stage and a softening stage. At the initial loading stage, it can be seen that the stress-strain curve has an obvious yield point, before which the stress-strain curve is a straight line, and after which is more nonlinear. The straight line before the yield point can be taken as the quasi-elastic stage. The slope of the straight line increases with the increase of strain rate. The stress-strain curve enters the plastic deformation stage after the yield point. The stress increases nonlinearly with the increase of strain, and the stress reaches a peak value with the continuous increase of strain. At the peak of stress-strain curve, the specimen behaves brittle failure and then enters the softening stage. As shown in Fig. 4, the strain rate has a significant impact on the deformation characteristics of frozen Helin loess. Specifically, under lower strain rate (1 × 10−5/s to 1 × 10−4/s), the peak stress is not apparent and the degree of strain-softening is moderate, which can be concluded that the sample exhibits more plasticity. In contrast, under the strain rate 1 × 10−3/s to 1 × 10−2/s, the strain-softening is obvious and peak stress is explicit. The frozen Helin soil exhibits obvious plastic deformation and lower degree of strain softening at lower strain rate, and presents obvious brittleness and higher degree of strain softening at higher strain rate. This can be explained by that loading period of tests with lower-strain rate is usually longer which allows the deformation can be developed uniformly along the specimen. It is less likely that this uniform deformation causes the sample reaches its peak stress due to local failure and exhibits obvious brittleness failure and strain softening within a small strain range. However, at higher strain rate, the
Fig. 4. Stress-strain curves of frozen loess under different strain rates.
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X. Xu et al. / Cold Regions Science and Technology 136 (2017) 44–51
deformation of sample is not uniform and usually localized in part of the sample. This localized deformation results in the local stress in the part of sample where carries the most of load rapidly reaches its peak stress and then the brittleness failure occurs, which indicates the sample is at the softening stage. This non-uniformity of deformation limits the development of plastic deformation inside the sample and leads to the increase of yield limit. The yield strain and failure strain of frozen Helin loess under different strain rate are presented in Table 3. It is found that the strain rate has less effect on the yield strain and failure strain than the influence by temperature. The yield rate ranges from 0.61% to 0.96%, and the failure strain ranges from 7.05% to 11.09%, respectively. 3.2.2. Yield limit and yield strength Fig. 5 illustrates log-log relationships between yield limit/strength and strain rate. As shown in Fig. 5, the yield limit and yield strength increases linearly with the increase of strain rate in the log-log coordinates. Zhang et al. (2003) found similar tendency for Lanzhou frozen silty soil. The relationships between yield limit/strength of saturated frozen Helin loess and the strain rate in the log-log ssas follows: lg σ y =σ y0 ¼ Ay þ By lgðε_ a =ε_ a0 Þ
ð3Þ
lg σ f =σ f 0 ¼ A f þ B f lgðε_ a =ε_ a0 Þ
ð4Þ
where σy and σf are the yield stress and strength, respectively, ε_ a is the strain rate,ε_ a0 ¼ 1s−1 is the reference strain rate. Ay, By, Af, and Bf are material parameters. In this study, Ay = 1.4154, By = 0.3897, Af = 1.2928, and Bf = 0.1871 by fitting the test data. The substance of the linear relationship between logarithmic of yield stress/strength and logarithmic of strain rate is the exponential relationship between the yield limit/ strength and the strain rate. 3.2.3. The effect of strain rate on elastic modulus of frozen Helin soil The initial tangent modulus and the secant modulus corresponding to half the strength are determined by the method in Section 3.1.3. Fig. 6 shows the relationship between the initial tangent modulus and the strain rate as well as the relationship between the secant modulus corresponding to half the strength and the strain rate. The initial tangent modulus increases linearly with the increase of strain rate in the log-log coordinates. The secant modulus corresponding to half the strength also increases with the increase in the strain rate. When the strain rate is 10−2/s, the values of the initial and the secant tangent modulus corresponding to half the strength are approximately equal. This is because the yield stress is approximately half of the failure stress at a higher strain rate, thus the data segment used to determine the initial tangent modulus coincides with the part to determine the secant modulus corresponding to half the strength. As a result, the value of the secant modulus corresponding to half the strength and the initial tangent modulus are approximately equal to each other.
Fig. 5. Relationships between yield limit/strength and strain rate under log-log coordinates.
general elastic-plastic theory (Chen, 2007), the incremental constitutive model is used to characterize the elasto-plastic behavior. Then, the strain (εij) can be divided as the sum of elastic strain (εeij) and plastic strain (εpij) in the incremental form (Xu et al., 2014): dεij ¼ dεeij þ dεpij
ð5Þ
In order to involve the effects of temperature and strains, the variables of temperature and strain rate have been introduced to constitutive model: dε ij ¼ Deijkl ðT; ε_ a Þdσ kl þ Dpijkl ðT; ε_ a Þdσ kl
ð6Þ
where σij is the stress tensor, Deijkl ðT; ε_ a Þ and Dpijkl ðT; ε_ a Þ are the elastic flexibility tensor and plastic flexibility tensor coupling with temperature and strain rate, respectively. By comparing Eqs. (5) and (6), we have dεeij ¼ Deijkl ðT; ε_ a Þdσ kl
ð7Þ
dε pij ¼ Dpijkl ðT; ε_ a Þdσ kl
ð8Þ
Eqs. (7) and (8) indicate that determining the specific forms of Deijkl ðT; ε_ a Þ and Dpijkl ðT; ε_ a Þ is the key step for constructing the
4. Elasto-plastic constitutive model considering temperature and rate strain for frozen Helin loess 4.1. Constitutive model The previous studies show that the frozen soil behaves obviously elasto-plastic deformation (Lai et al., 2014; Yang et al., 2016a, 2016b). Therefore, we adopt an elasto-plastic constitutive model to describe the stress-strain relationship of frozen Helin soil. According to the Table 3 Yield strain and failure strain of frozen Helin loess under different strain rates. Strain rate (s−1)
1 × 10−2
1 × 10−3
1 × 10−4
5 × 10−5
1 × 10−5
Yield strain (%) Failure strain (%)
0.96 7.05
0.88 10.54
0.73 10.44
0.63 11.09
0.61 10.76
Fig. 6. Relationships between elastic moduli and strain rate under log-log coordinates.
X. Xu et al. / Cold Regions Science and Technology 136 (2017) 44–51
constitutive model coupling with temperature and strain rate. Due to the frozen soil is usually undergoing axisymmetric load, we can adopt the constitutive model for frozen Helin loess under two dimensions (i.e. at p-q plane). In this case, the K-G model (Li, 2004) can be used to describe the elastic behavior of frozen Helin loess as following forms:
dε ev ¼ dp=K ðT; ε_ a Þ dεeγ ¼ dq=3GðT; ε_ a Þ
ð9Þ
And the matrix form is:
dεev dεeγ
¼
1=K ðT; ε_ a Þ 0 0 1=3GðT; ε_ a Þ
dp dq
Y q; εpγ ¼ q−k T; ε_ a ; εpγ ¼ 0
ð10Þ
ð11Þ
dλ in Eq. (11) is a non-negative multiplier which can be derived from yield criterion, flow rule and hardening law. The first formulation of Eq. (11) shows the plastic volumetric strain is zero since hydrostatic pressure has no effect on the plastic behavior of frozen Helin loess. Applying the consistency conditions on Eq. (10), we obtain dY ¼ dq−
∂k p dε ¼ 0 ∂εpγ γ
∂k ∂εpγ
ð13Þ
dεpv ¼ 0
p : dεγ
¼ dq=
∂k ∂εpγ
ð14Þ
2 ¼4
0 0
3 0 ∂k 5 dp 1= p dq ∂εγ
ð14’Þ
Combining Eq. (9’) with Eq. (14’), we obtain the incremental form constitutive model with the effects of temperature and strain rate for frozen Helin loess in matrix form as following:
dε v dεγ
p dεev dεv þ e dε dεpγ 2 γ 3 0 1=K ðT; ε_ a Þ ∂k 5 dp ¼4 0 1=3GðT; ε_ a Þ þ 1= p dq ∂εγ
¼
ð15Þ
This model includes three parameters coupling with temperature and strain rate. Those three parameters can be determined by uniaxial compression test of frozen Helin loess under different temperatures and strain rates from Sections 2 and 3. 4.2. Determination of model parameters and model verification The elastic parameters K and G can be expressed by elastic modulus and Poisson's ratio by the following transformation in elastic theory (Chen, 2007): K¼
E 3ð1−2ν Þ
ð16Þ
G¼
E 2ð1 þ ν Þ
ð17Þ
In this paper, we use the initial tangent modulus as the elastic modulus of frozen Helin loess. Based on the results of triaxial compression tests of the same frozen Helin loess (Xu et al., 2016), the elastic volumetric strain is ignored because it is far less than the elastic equivalent shear strain. Thus, the Poisson's ratio ν = 0.5. Based on the uniaxial compression test data of frozen Helin loess under different temperatures and strain rates, the elastic modulus of frozen Helin loess under different temperatures and strain rates can be determined by regressing testing temperature on elastic modulus. For instance, based on the tests results, the relationship between elastic modulus and temperature when the strain rate at 1.67 × 10−4/s is EðT Þ ¼
−52:26T−55:49 −2:84T þ 141:48
−4 ≤T ≤−2 −7 ≤T ≤−4
ð18Þ
Substituting Poisson's ratio ν = 0.5 and Eq. (18) into Eqs. (16) and (17), we can obtain that the elastic bulk modulus is infinity and the elastic shear modulus G depending on temperature: GðT Þ ¼
Combining Eq. (11) with Eq. (13), the plastic strain increment can be calculated as following: 8 <
dε pv dεpγ
ð12Þ
Substituting the second formulation in Eq. (11) into Eq. (12), we have dλ ¼ dq=
ð9’Þ
Here kðT; ε_ a ; εpγ Þ is the hardening parameter with the effects of temperature and strain rate on the plastic deformation of frozen Helin loess. Based on the associated flow rule (Chen, 2007), we have 8 ∂Y > > ¼0 < dε pv ¼ dλ ∂p ∂Y > > : dεpγ ¼ dλ ¼ dλ ∂q
And the matrix form is:
where εev =εe1 +2εe3 and εeγ =2/3(εe1 −εe3) are elastic bulk strain and elastic equivalent shear strain, respectively, p=(σ1 +2σ3)/3 and q=σ1 −σ3 are the mean stress and equivalent stress, respectively, KðT; ε_ a Þ and GðT; ε_ a Þ are elastic bulk modulus and elastic shear modulus affected by temperature and strain rate. To calculate the plastic deformation of frozen Helin loess, the yield criterion, flow rule and hardening law of frozen Helin loess need to be determined. Xu et al. (2016) performed triaxial test on the same type of frozen Helin loess, and they found the influence of confining pressure on the yield limit and strength of the frozen Helin loess can be ignored. Thus, the Mise's yield criterion is applicable to characterize the yield and plastic behavior of frozen Helin loess. Furthermore, we can assume the frozen Helin loess obeys the associated flow rule and the isotropic hardening. Therefore, the Mise's yield criterion can be written as following:
49
ð−52:26T−55:49Þ=3 ð−2:84T þ 141:48Þ=3
−4≤T ≤−2 −7 ≤T ≤−4
ð19Þ
When the strain rate at 1.67 × 10−4/s, the hardening parameter k can be determined by fitting k on the plastic shear strain from the uniaxial compressive test data under different temperatures as following: k T; ε pγ ¼ −0:117T þ 0:728 þ
εpγ
2 0:033e0:143T εpγ −0:06T 2 −0:338T
þ 1:978
ð20Þ
The Eqs. (19) and (20) provide the elastic and plastic parameters of frozen Helin loess under different temperatures at strain rate of 1.67 × 10−4/s.
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X. Xu et al. / Cold Regions Science and Technology 136 (2017) 44–51
Fig. 7. Comparison of stress-strain curves between experimental results (symbols) and predicted results (lines) under different temperatures.
Fig. 8. Comparison of stress-strain curves between experimental results (symbols) and predicted results (lines) under different strain rates.
Then, by substituting Eqs. (19)–(20) into Eq. (15), the influence of temperature on the stress-strain relationship of frozen Helin loess under a defined strain rate (1.67 × 10−4/s) can be predicted by the constitutive model. The comparison between uniaxial compressive test stress-strain curves and the predictive results from constitutive model under different temperatures is shown in Fig. 7. Fig. 7 indicates that the proposed constitutive model can precisely describe the characteristics of stress-strain curves of frozen Helin loess under different temperatures at strain rate of 1.67 × 10−4/s. If temperature is at −4 °C, the constitutive model parameters, G and k, depending on strain rate can also be determined by fitting the uniaxial compressive test data under different strain rates. According to the test data from Fig. 6, the relationship between elastic modulus of frozen Helin loess and strain rate can be expressed as following:
constitutive model can describe the stress-strain curves of frozen Helin loess under different strain rates at temperature of −4 °C.
ln ðEðε_ a ÞÞ ¼ 0:3109 ln ðε_ a Þ þ 1:2732
ð21Þ
Then, substituting ν = 0.5 into Eq. (17) and then combining with (21), we have ln ðGðε_ a ÞÞ ¼ 0:3109 ln ðε_ a Þ þ 1:2732− ln ð3Þ
ð22Þ
When the temperature is −4 °C, the hardening parameter k can be determined by fitting k on the plastic shear strain from the uniaxial compressive test data under different strain rates as following: k ε_ a ; εpγ ¼ P 1 þ
ε pγ p 2
P2 εγ
ð23Þ
þ P3
where, the P1, P2 and P3 are fitting parameters which depend on the strain rates. They can be determined by fitting the experimental data as follow: P 1 ðε_ a Þ ¼ 0:672ð ln ðε_ a ÞÞ þ 5:431 ln ðε_ a Þ þ 11:58 2
ð24Þ
3 2 P 2 ðε_ a Þ ¼ −0:011ð ln ðε_ a ÞÞ −0:095 ln ðε_ a Þ −0:263 ln ðε_ a Þ−0:210 ð25Þ 3
2
P 3 ðε_ a Þ ¼ 0:231ð ln ðε_ a ÞÞ þ 2:228 ln ðε_ a Þ þ 6:256 ln ðε_ a Þ þ 5:865 ð26Þ The effects of strain rates on the stress-strain relationship of frozen Helin loess at − 4 °C can be modeled by combining Eqs. (15) and (22)–(26). The comparison between uniaxial compressive test stressstrain curves and their constitutive modeling results under different strain rates is presented in Fig. 8. The Fig. 8 shows that the proposed
5. Conclusions A series of uniaxial compressive tests were conducted on the saturated frozen Helin loess. The stress-strain curves, yield strength, strength and elastic modulus with the variation of temperature (− 2 °C, − 4 °C, − 5 °C, and − 7 °C) and strain rate (1 × 10− 2/s, 1 × 10−3/s, 1 × 10−4/s, 5 × 10−5/s and 1 × 10− 5/s) are analyzed based on test results. The main findings are as follows: (1) The stress-strain curves of saturated frozen Helin loess present strain-softening under different strain rates and temperatures. Under different temperature, the turning point between quasi-elastic stage and plastic stage appears at strain of 0.77%–1.50%, which marks yield behavior occurring in frozen Helin loess; in contrast, under different strain rate, the yield phenomenon approximately occurs at strain of 1%. With the decrease of temperature and increase of strain rate, the brittleness of frozen Helin loess increases. (2) The yield limit and strength of the saturated frozen Helin loess increase with the decrease of temperature and the increase of strain rate. However, the yield limit and strength are linearly related with temperature, whereas the relationship between the yield limit/strength and strain rate are exponential. (3) The relationships between the initial tangent modulus/the secant modulus corresponding to half the strength and temperature have two stages. In the first stage, the temperature ranges from −4 °C to −2 °C. The initial tangent modulus and the secant modulus corresponding to half the strength increase rapidly with the increase of temperature in this stage, and the values of both moduli are small. In the second stage, the temperature ranges from −7 °C to −4 °C. The initial tangent modulus and the secant modulus corresponding to half the strength are higher than those in the first stage, but with a slower increasing rate. The tendency of both moduli varied with the decrease of temperature is similar. The initial tangent modulus and secant modulus corresponding to half the strength increase exponentially with the increase of the strain rate when strain rate ranges from 10− 5/s to 10−3/s. And both moduli are approximately equal under strain rate of 10−2/s. (4) An elasto-plastic constitutive model was proposed to describe the stress-strain relationship of frozen Helin loess under different temperatures and strain rates. The procedure of determining the model parameters was detailed. All of the model parameters were identified by fitting the experimental data. The comparison of stress-strain curves between experimental results and modeled results showed that the
X. Xu et al. / Cold Regions Science and Technology 136 (2017) 44–51
proposed model can correctly predict the deformation behavior of frozen Helin loess under different temperatures and strain rates. Acknowledgements This research was supported by National Natural Science Foundation of China (41301072, 41230630, 51579157, 51574219), Natural Science Foundation of Inner Mongolia (No. 2016MS0530), the Post-doctor Study Program of Inner Mongolia University (21200-5152604), the Knowledge Innovation Program of the Chinese Academy of Sciences (KZCX2-EW-QN301), and the Program for Frost-heave diseases mitigation technology for the expressways in seasonally frozen regions in Inner Mongolia Autonomous Region (Entrusted by MCC Communication Construction group Co., Ltd., 22200-5166011). References Andersen, G.R., Swan, C.W., Ladd, C.C., Germaine, J.T., 1995. Small-strain behavior of frozen sand in triaxial compression. Can. Geotech. J. 32, 428–451. Arenson, L., Springman, S., 2005. Triaxial constant stress and constant strain rate tests on ice-rich permafrost samples. Can. Geotech. J. 42 (2), 412–430. Baker, T.H.W., Jones, S.J., Parameswaran, V.R., 1982. Confined and unconfined compression tests on frozen sands. Proc. 4th Canadian Permafrost Conference, Roger J.E. Brown Memorial Vol., pp. 387–393 Bragg, R.A., Andersland, O.B., 1981. Strain rate, temperature, and sample size effects on compression and tensile properties of frozen sand. Eng. Geol. 18, 35–46. Chang, X.X., Ma, W., Sheng, Y., et al., 1995. Influence of experimental conditions on the mechanical parameters of frozen soil. J. Glaciol. Geocryol. 17 (suppl), 151–155. Chen, M.X., 2007. Elasticity and Plasticity. Science Press, Beijing. Chen, X.B., Qiu, G.Q., Wang, Y.Q., et al., 1988. The character of physical chemistry and mechanics for soil with abundant salinity under varying temperature. Sci. China A 4, 429–438. Haynes, F.D., Karalius, J.A., 1977. Effect of Temperature on the Strength of Frozen Silt. CRREL Report, Hanover (77–3). Haynes, F.D., Karalius, J.A., Kalafut, J., 1975. Strain rate effect on the strength of frozen silt. Hanover: CRREL Report RR350. Jones, S.J., Parameswaran, V.R., 1983. Deformation behavior of frozen sand-ice materials under triaxial compression. Proc. 4th International Permafrost Conference, pp. 560–565. Ladanyi, B., 1981. Mechanical behavior of frozen soils. Proc. of Int. Symp. on the Mechanical Behavior of Structured Media. A. A. Balkema, Ottawa, pp. 205–245. Lai, Y., Xu, X., Dong, Y., Li, S., 2013. Present situation and prospect of mechanical research on frozen soils in China. Cold Reg. Sci. Technol. 87, 6–18. Lai, Y.M., Xu, X.T., Yu, W.B., Qi, J.L., 2014. An experimental investigation of the mechanical behavior and a hyperplastic constitutive model of frozen loess. Int. J. Eng. Sci. 84, 29–53. Lai, Y.M., Yang, Y.G., Chang, X.X., et al., 2010. Strength criterion and elastoplastic constitutive model of frozen silt in generalized plastic mechanics. Int. J. Plast. 26 (10), 1461–1481. Li, G.X., 2004. Advanced Soil Mechanics. Tsinghua University Press, Beijing.
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Li, H.P., Zhu, Y.L., Zhang, J.B., et al., 2004. Effects of temperature, strain rate and dry density on compressive strength of saturated frozen clay. Cold Reg. Sci. Technol. 39 (1), 39–45. Li, Q.L., Ling, X.Z., Sheng, D.C., 2016. Elasto-plastic behavior of frozen soil subjected to long-term low-level repeated loading. Part I: Experimental investigation, Cold Regions Science and Technology. 125, pp. 138–151. Martin, C., Jun, B.P., 2010. Laboratory determination of strength properties of frozen rubber-sand mixtures. Cold Reg. Sci. Technol. 60 (2), 169–175. People's of Republic of China coal industry standard. MT/593. 5-2011, 2011. Artificial Frozen Soil Physics Mechanics Performance Test—Part 5: Artificial Frozen Soil Tri-Axial Shear Test Method. China Coal Industry Press, Beijing. Qi, J.L., Ma, W., 2010. State-of-art of research on mechanical properties of frozen soils. Rock Soil Mech. 31 (1), 133–143. Sayles, F.H., 1973. Triaxial and creep tests on frozen Ottawa sand. Proc. 2nd Int. Conf. On Permafrost, Yakutsk, U.S.S.R., North American Contribution, U.S. National Academy Of Science, pp. 384–391. Tsytovich, H.A., 1975. Mechanics of Frozen Soils. McGraw-Hill, New York. Tsytovich, H.A., 1972. Mechanics of Frozen Soils. Translated by Zhang Changqing, et al. Beijing:Science Press, pp. 25–43. Vialov, S.S., 1959. Theological properties and bearing capacity of frozen soils. USA Snow, Ice and Permafrost Research Establishment, Translation. 74 (AD48156). Wu, Z.W., Ma, W., Zhang, C.Q., et al., 1994. Strength characteristics of frozen sandy soil. J. Glaciol. Geocryol. 16 (1), 15–20. Xu, X.T., Fan, C.X., Zhang, T.Y., 2014. A constitutive model with effect of temperature for frozen soil. Adv. Mater. Res. 919–921, 627–631. Xu, X.T., Lai, Y.M., Dong, Y.H., et al., 2011. Laboratory investigation on strength and deformation characteristics of ice-saturated frozen sandy soil. Cold Reg. Sci. Technol. 69 (1), 98–104. Xu, X.T., Wang, J.W., Zhang, M.Y., 2016. Experimental study of the mechanical properties of frozen Helin loess. J. Glaciol. Geocryol. 38 (3), 685–691. Yang, C.S., He, P., Cheng, G.D., et al., 2006. Uniaxial compressive strength of frozen saline silty clay. Eng. Mech. 23 (1), 144–148. Yang, Y.G., Gao, F., Lai, Y.M., Cheng, H.M., 2016a. Experimental and theoretical investigations on the mechanical behavior of frozen silt. Cold Reg. Sci. Technol. 130, 59–65. Yang, Y.G., Lai, Y.M., Li, J.B., 2010. Laboratory investigation on the strength characteristic of frozen sand considering effect of confining pressure. Cold Reg. Sci. Technol. 60, 245–250. Yang, Y.H., Wei, Z.A., Yin, G.Z., et al., 2016b. Uniaxial compression test of frozen tailings. Cold Reg. Sci. Technol. 129, 60–68. Yang, Z.H., Benjamin, S., Ge, X.X., 2015. Mechanical properties of seasonally frozen and permafrost soils at high strain rate. Cold Reg. Sci. Technol. 113, 12–19. Zhang, J.B., Li, H.P., Lin, C.N., et al., 2003. Compressive strength of saturated frozen silt under constant strain rate. Chin. J. Rock Mech. Eng. 22 (suppl.2), 2865–2870. Zhang, S.J., Lai, Y.M., Sun, Z.Z., et al., 2007. Volumetric strain and strength behavior of frozen soils under confinement. Cold Reg. Sci. Technol. 47 (3), 263–270. Zhao, X.D., Zhou, G.Q., Chen, G.Z., et al., 2011. Triaxial compression deformation for artificial frozen clay with thermal gradient. Cold Reg. Sci. Technol. 67 (3), 171–177. Zhao, X.D., Zhou, G.Q., Wang, J.Z., 2013. Deformation and strength behaviors of frozen clay with thermal gradient under uniaxial compression. Tunn. Undergr. Space Technol. 38, 550–558. Zhou, Z.W., Ma, W., Zhang, S.J., et al., 2016. Multiaxial creep of frozen loess. Mech. Mater. 95, 172–191. Zhu, Y.L., Carbee, D.L., 1984. Uniaxial compressive strength of frozen silt under constant defornation rate. Cold Reg. Sci. Technol. 9 (1), 3–15.
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Cold Regions Science and Technology journal homepage: www.elsevier.com/locate/coldregions
Effect of temperature and strain rate on mechanical characteristics and constitutive model of frozen Helin loess Xiangtian Xu a, Yubing Wang b,⁎, Zhenhua Yin a, Hongwei Zhang c a b c
Institute of Transportation, Inner Mongolia University, Hohhot 010070, China Fugro Consultants, Inc., 8613 Cross Park Drive, Austin, TX 78754, United States Inner Mongolia Communications Constructions Engineering Quality Supervision Bureau, Hohhot 010020, China
a r t i c l e
i n f o
Article history: Received 23 June 2016 Received in revised form 14 January 2017 Accepted 31 January 2017 Available online 02 February 2017 Keywords: Frozen Helin loess Uniaxial compressive Stress-strain Strength Elastic modulus
a b s t r a c t To investigate the effect of temperature and strain rate on the mechanical properties of frozen loess, a series of uniaxial compressive tests were conducted on saturated frozen Helin loess under five different strain rates (1 × 10−2/s, 1 × 10−3/s, 1 × 10−4/s, 5 × 10−5/s, and 1 × 10−5/s) and at four different temperatures (−2 °C, −4 °C, −5 °C, and −7 °C). From the stress-strain curves under different testing condition, the stain-softening behavior is observed and the yield point can be obviously seen. The yield stress and strength increase linearly with the decrease of temperature and increase exponentially with the increase of strain rate. The initial tangent modulus and the secant modulus corresponding to half the strength are taken to describe the stiffness of frozen loess. And the effects of temperature and strain rate on the tangent and secant modulus are analyzed. Furthermore, an elasto-plastic constitutive model considering the effects of temperature and strain rate has been proposed to describe the strength and deformation behavior of frozen Helin loess. © 2017 Elsevier B.V. All rights reserved.
1. Introduction The mechanical properties of frozen soil have gained importance on engineering design process, construction and maintenance phase after construction in cold regions (Qi and Ma, 2010; Li et al., 2016). Therefore, various experiments have been performed to explore the mechanical properties of frozen soil (Yang et al., 2010; Xu et al., 2011; Lai et al., 2014; Yang et al., 2016a, 2016b; Zhou et al., 2016). Presently, to study the mechanical characteristics of frozen soil under different testing conditions involving temperature, strain rate, and confining pressure (Bragg and Andersland, 1981; Lai et al., 2013), the majority of laboratory tests are uniaxial compressive test and triaxial compressive tests (Baker et al., 1982; Andersen et al., 1995; Arenson and Springman, 2005). Since triaxial compression test is capable to simulate the effect of earth pressure on the mechanical performance of frozen soil, so it is applicable to investigate the mechanical properties of frozen soil with deep embedment (Sayles, 1973; Jones and Parameswaran, 1983; Zhang et al., 2007; Lai et al., 2010; Zhao et al., 2011). In contrast, for the frozen soil with ice-rich and ice-saturated frozen soil, confining pressure has little impact on its strength and deformation characteristics. Thus, it is time-saving and cost-effective to explore the mechanical properties of the ice-rich and ice-saturated frozen soil by uniaxial compressive test. The uniaxial compressive test is generally capable to characterize the ⁎ Corresponding author. E-mail address: [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.coldregions.2017.01.010 0165-232X/© 2017 Elsevier B.V. All rights reserved.
mechanical properties of frozen soil except the examination of the effect by confining pressure. Especially, for the layer frozen soil at shallow embedment depth, the stiffness and strength characteristics determined by uniaxial compressive tests can be easily applied to engineering practice. Furthermore, the unixial compressive test is usually used to investigate other geomaterials under lower temperature condition (Martin and Jun, 2010; Yang et al., 2015; Yang et al., 2016a, 2016b). The characterizations in the strength and deformation of frozen soil were firstly started by performing uniaxial compressive tests at the early stage. For example, in 1930s, Tsytovich pioneered to study mechanical properties of frozen sand under different temperatures and strain rates by carrying out series of uniaxial compressive tests. He found that, with the decrease of temperature and the increase in strain rate, the strength of frozen sand increased nonlinearly, but the increasing rate of strength gradually mitigated (Tsytovich, 1972). Tsytovich, Vialov, Ladanyi and Haynes et al. (Tsytovich, 1975; Vialov, 1959; Ladanyi, 1981; Haynes et al., 1975; Haynes and Karalius, 1977) conducted a series of unconfined compressive tests on frozen soil under different temperatures, strain rates and unit weight and arrived at similar conclusion by Tsytovich (1972), Wu et al. (1994) also attained that the strength of frozen sand was related to temperature and strain rate, and he accomplished an empirical relationship that strength is a function of temperature based on series of unconfined compressive tests on frozen sand. Chen et al. (1988) found that there exist changes in both physical and chemical properties of frozen soil with the decrease of temperature, which resulted in an increase in soil strength. Zhu and
X. Xu et al. / Cold Regions Science and Technology 136 (2017) 44–51
the variation of temperature and strain rate, which provides reference for the construction projects in cold regions.
Nomenclature σy σy0 σf σf0 σij ε_a _ εa0 εij εeij εpij T T0 K G E υ Y k Deijkl Dpijkl
45
yield stress reference yield stress strength reference strength stress tensor strain rate reference strain rate strain tensor elastic strain tensor plastic strain tensor temperature reference temperature elastic volumetric modulus elastic shear modulus elastic modulus poisson's ratio yield function hardening parameter elastic flexibility tensor plastic flexibility tensor
2. Test program 2.1. Test material The soil material was collected from Helin County, Hohhot City, Inner Mongolia Province, China. The grain size distribution determined by a laser particle size analyzer was shown in Table 1. From the distribution, the soil is classified as silty sand. The specific gravity is 2.526 by specific gravity test.
2.2. Sample preparation
Carbee (1984) completed a series of uniaxial compressive tests on frozen sand under different temperatures and different strain rates, and proposed a prediction model for strength which is a function of different temperature. Yang et al. (2006) discovered three characteristics of saturated saline frozen silty clay from a series of uniaxial compressive tests: (1) the negative temperature had a great effect on the strength of frozen soil; (2) the uniaxial compressive strength increased linearly with the decrease of temperature; (3) at a certain strain rate, the uniaxial strength was twice higher than the yield stress. Li et al. (2004) conduct a series of uniaxial compressive tests on saturated frozen clay to investigate the effects of temperature, strain rate and dry density on its strength. Zhao et al. (2013) applied the uniaxial compressive test to study the deformation and strength of frozen clay under thermal gradient condition. All of the conclusions from these existing research results point out the temperature and strain rate are the primary external factors affecting the mechanical properties of frozen soil. However, the mechanical response of frozen soil under various thermal-mechanical conditions is not only determined by those external factors, the internal factors, such as water content, dry unit weight, salt content, and degree of saturation, are also critical to the mechanical properties of frozen soil. These internal factors result in the measured mechanical characteristics of the same type of frozen soil are different from each other even under the same experimental conditions. These disagreements made it difficult when applying those experimental results to solving practical engineering problems and referencing those researchers' work for future study. The purpose of this paper is to report mechanical properties of frozen loess collected for a road project in Helin County, Hohhot City, Inner Mongolia Autonomous region. A series of uniaxial compressive tests were conducted on the frozen loess under different strain rates (1 × 10−2/s, 1 × 10–3/s, 1 × 10−4/s, 5 × 10−5/s, and 1 × 10−5/s) at different temperatures (−2 °C, −4 °C, −5 °C, and −7 °C). Based on the tests results, attention has focused in particular on the stress-strain curves, yield strength, elastic modulus and constitutive model with
In this paper, the preparation of frozen loess sample was followed by the sample preparation method by Chang et al. (1995). Firstly, the impurities such as small pebbles in the natural loess collected from field were screened out by the sieve with 2 mm mesh size. Then, the loess was reconstituted by mixing thoroughly with distilled water (appropriate 10% by weight of soil), and sealed for 12 h to ensure that water content was uniform in the mixed soil. Next, the mixture was compacted into a cylinder with the diameter of 6.18 cm and the height of 12.5 cm and filled into a three-piece split mold. We placed one porous stone at the top and one at the bottom of specimen. Afterwards, the specimens were evacuated and saturated with water for 24 h and then placed into refrigerator for 48 h. The three-piece split mold was removed from the specimen and epoxy resin platens were place on top and bottom of specimen while a rubber membrane being wrapped around. The specimens are divided into two groups for different testing scopes: the specimens in the first group were used to test in different temperature (− 2 °C, − 4 °C, − 5 °C, and − 7 °C) under the same strain rate (1.67 × 10−4/s); the specimens in the second group were used to test at different strain rate (1 × 10− 2/s, 1 × 10− 3/s, 1 × 10− 4/s, 5 × 10−5/s, and 1 × 10−5/s) but at a constant temperature at − 4 °C. All of the specimens were kept at the target temperature at least for 12 h so that temperature inside each specimen is homogenous. The water content of saturated specimen is 24.58%, and dry density is 1.97 g/cm3. 2.3. Test apparatus and test procedure A frozen soil creep apparatus used in this study is capable of applying axial force ranging from 0kN up to 100kN, and controlling temperature from −25 °C to room temperature with resolution of 0.1 °C. The apparatus can be either strain-controlled or stress-controlled. In this study, the strain-controlled mode was used. The testing procedure was summarized as following. First, we tested the specimens in the first group. We placed the specimen into the loading chamber where the temperature was set up to the same one as in the specimen preparation stage. Then we waited for a couple minutes for the temperature reached equilibrium state, and we slowly lowered down the loading piston until the platen of piton was perfectly in contact with the top of specimen. Last, we started loading the sample at strain rate of 1.67 × 10−4/s, and stopped loading at 20% strain limit. We followed same procedure to test the second group, except that the loading rate was set up to 1 × 10− 2/s, 1 × 10− 3/s, 1 × 10− 4/s,
Table 1 Particle fraction of Helin loess. b0.005 mm
0.005–0.05 mm
0.05–0.075 mm
0.075–0.10 mm
0.10–0.25 mm
N0.25 mm
1.91%
12.59%
20.76%
20.71%
40.56%
3.43%
46
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5 × 10−5/s, and 1 × 10−5/s, respectively and temperature was setting to the −4 °C for all specimens in the second group. 3. Test results and analysis
Table 2 Yield strain and failure strain of frozen Helin loess under different temperatures. Temperature (°C)
−2
−4
−5
−7
Yield strain (%) Failure strain (%)
1.50 15.96
0.77 13.79
0.90 13.79
0.92 14.27
3.1. The effect of temperature on mechanical characteristics of frozen Helin loess 3.1.1. Deformation characteristics Fig. 1 shows that the stress-strain curves of frozen Helin loess obtained from the uniaxial compressive tests at different temperatures are presented. It can be seen that the frozen Helin loess present strainsoftening and brittle failure under different temperature. The stressstrain curves have three stages: a quasi-elastic stage, a plastic deformation stage and a softening stage. An obvious yield stress point and peak stress can be seen from each stress-strain curve. Before the yield stress point, the stress-strain curve is approximately as a straight line, which is prescribed as the quasi-elastic stage. The slope of this straight line in this stage increases with the decrease of temperature which means the stiffness of frozen soil becomes higher while the temperature is lower. This phenomenon can be explained that the lower temperature makes the unfrozen water less and cementation effect by ice stronger. After the yield stress point, the stress-strain curve enters the plastic deformation stage. At this stage, the soil has less capacity of resistance to deformation. The stress keeps increasing with the increase of strain, however, the rate of the stress increase becomes smaller than the rate in the first stage (the quasi-elastic stage). This is because of the increasing stress in this stage leads to a crush of ice which makes the cementation effect by ice weaker. The loss of cementation results in a decrease in the ability to resist deformation and produces the plastic deformation. With a further increase in strain, the stress reaches the peak value, and the stress-strain curve enters the softening stage. From Fig. 1, it shows that the frozen Helin loess at − 2 °C does not exhibit obvious strain-softening. In comparison with other specimens under lower temperatures, the strain-softening phenomenon is more obvious and the peak stress is more evidently with the decrease of temperature. These two observations indicate that the brittleness of frozen Helin loess increases with the decrease of temperature. As shown in Fig. 1, a distinct yield point can be picked from each stress-strain curve under different temperatures. In this paper, we define the strain at yield point as the yield strain, and the strain at peak value as failure strain. The values of yield strain and failure strain of frozen Helin loess at different temperatures are presented in Table 2. As shown in Table 2, with the decrease in temperature, the yield strain and failure strain decrease first and then increase subsequently. This
Fig. 1. Stress-strain curves of frozen loess under different temperatures.
indicates that as the temperature decreases, the unfrozen water content decreases, the brittleness of frozen Helin soil increases, and yield strain together with failure strain decrease. However, the yield strain and failure strain will not keep decreasing with further decrease of temperature. As the temperature reaches at the critical point (− 4 °C is the critical point in this paper), the further decrease of temperature reduces the unfrozen water content only in a small amount, but the soil particles are more compact due to thermal contraction, which makes frozen soil capable to resist higher plastic deformation and failure, therefore the yield strain and failure strain increase instead of keeps decreasing with the decrease of temperature. Meanwhile,we can find the differences between yield strain or failure strain at different temperature are small. The yield strain at different temperature ranges between 0.77% and 1.50%. The maximum yield strain occurs at − 2 °C, and the minimum yield strain is at − 4 °C. The yield strain firstly decreases with the decrease of temperature, however, it increases with the continuous decrease of temperature after reaching its minimum at −4 °C. The failure strain has similar trend, that is, the maximum failure strain occurs at −2 °C (15.96%), and starts decreasing with the decrease of temperature. At − 4 °C and − 5 °C, the failure strain also reaches its minimum value, and back to increase again with decrease of temperature. 3.1.2. Yield limit and strength characteristics Fig. 2 illustrates the yield limit and strength of the saturated frozen Helin loess at different temperatures. As shown in Fig. 2, the yield limit and strength decrease linearly with the increase of temperature. Because the decrease in temperature results in the amount of unfrozen water decreases and more ice forms in the voids between soil particle, which intensifies the cementation effect by ice in the frozen soil. Thus, frozen soil attains higher strength to resist failure and is more capable to resist plastic deformation. The relationships between yield limit of frozen loess and temperature as well as relationship between strength and temperature can be expressed by the linear functions as follows: σ y =σ y0 ¼ ay ðT=T 0 Þ þ by
ð1Þ
σ f =σ f 0 ¼ a f ðT=T 0 Þ þ b f
ð2Þ
Fig. 2. Strength and yield limit of frozen loess under different temperatures.
X. Xu et al. / Cold Regions Science and Technology 136 (2017) 44–51
According to the test results, the Eqs. (1) and (2) can be applied in the temperature ranges from − 7 °C to − 2 °C. Where σy and σf is yield stress and strength, respectively, T is temperature, T0 = 1 °C, σy0 = σf0 = 1 MPa are reference temperature, reference yield stress and reference strength, respectively. ay, by, af, and bf are material parameters. In this paper ay = −0.1256, by = 0.5934, af = −0.4784, and bf = 1.8631 by fitting the test data. 3.1.3. Effect of temperature on the elastic modulus of frozen loess In this paper, two elastic moduli are analyzed, the initial tangent modulus and the secant modulus corresponding to half the strength, which are generally used to characterize the elastic modulus of frozen soil. The secant modulus corresponding to half the strength is determined according to the People's of Republic of China coal industry standard (MT/593. 5-2011, 2011). As illustrated in the previous section, a distinct yield point can be picked from each stress-stain curve, which means the frozen loess has obvious elastic stage. Therefore, to determine the initial tangent modulus, the data before the yield point are used for linear regression. Fig. 3 shows the initial tangent modulus and the secant modulus corresponding to half the strength of saturated frozen Helin loess at different temperatures. From Fig. 3, at different temperature, the initial tangent modulus is always greater than the secant modulus. The same conclusion is also obtained for other types of frozen soil, which indicates that the stiffness of frozen soil in the initial deformation stage is greater than that in the late deformation stage. Both the initial tangent modulus and the secant modulus corresponding to half the strength increase with the decrease of temperature, and their variations with temperature can be divided into two stages. The first stage starts from − 2 °C to −4 °C, in which the initial tangent modulus and the secant modulus corresponding to half the strength increase rapidly with the decrease of temperature. In details, the increasing rate of the initial tangent modulus and the secant modulus corresponding to half the strength is 52.26 MPa/°C and 13.98 MPa/°C, respectively. This rapid increase can be explained by that there is a relatively large amount of unfrozen water at − 2 °C, thus the soil sample which mostly constituted with silts is still saturated; however, the unfrozen water content decreases while more ice forms when the temperature decreases to − 4 °C, which increases the cementation effect by ice and soil resistance to deformation. As a result, the stiffness of frozen soil is greatly improved. On the other hand, it is found that the increasing rate of the initial tangent modulus is greater than that of the secant modulus corresponding to half the strength, which indicates that the effect of unfrozen water on the initial tangent modulus is greater than that on the secant modulus corresponding to half the strength. The temperature range of the second stage is from −4 °C to −7 °C. In this stage, the initial tangent modulus and the secant modulus corresponding to half the strength continues
Fig. 3. Elastic moduli of frozen loess under different temperatures.
47
increasing with the decrease of temperature, but with a lower increasing rates (2.84 MPa/°C and 5.31 MPa/°C, respectively) compared with the rates in the first stage. The unfrozen water content becomes less when temperature is lower than −4 °C. A further decrease in temperature has slight impact on the ice content in the frozen soil, which results in a hardly improvement of the stiffness. It can be predicted the soil stiffness is not expected to improve by the further decrease of temperature. It should be noted that the increasing rates of both moduli are roughly same in the second stage, which indicates that both two moduli can accurately describe the deformation characteristics of frozen Helin loess under low unfrozen water content. 3.2. The effect of strain rate on mechanical characteristics of the frozen Helin loess 3.2.1. Deformation characteristics Fig. 4 illustrates that the stress-strain curves of frozen Helin loess under different strain rates at −4 °C. The stress-strain curves of frozen Helin loess present strain-softening under each different strain rate, and they are also can be divided into three stages: a quasi-elastic stage, a plastic deformation stage and a softening stage. At the initial loading stage, it can be seen that the stress-strain curve has an obvious yield point, before which the stress-strain curve is a straight line, and after which is more nonlinear. The straight line before the yield point can be taken as the quasi-elastic stage. The slope of the straight line increases with the increase of strain rate. The stress-strain curve enters the plastic deformation stage after the yield point. The stress increases nonlinearly with the increase of strain, and the stress reaches a peak value with the continuous increase of strain. At the peak of stress-strain curve, the specimen behaves brittle failure and then enters the softening stage. As shown in Fig. 4, the strain rate has a significant impact on the deformation characteristics of frozen Helin loess. Specifically, under lower strain rate (1 × 10−5/s to 1 × 10−4/s), the peak stress is not apparent and the degree of strain-softening is moderate, which can be concluded that the sample exhibits more plasticity. In contrast, under the strain rate 1 × 10−3/s to 1 × 10−2/s, the strain-softening is obvious and peak stress is explicit. The frozen Helin soil exhibits obvious plastic deformation and lower degree of strain softening at lower strain rate, and presents obvious brittleness and higher degree of strain softening at higher strain rate. This can be explained by that loading period of tests with lower-strain rate is usually longer which allows the deformation can be developed uniformly along the specimen. It is less likely that this uniform deformation causes the sample reaches its peak stress due to local failure and exhibits obvious brittleness failure and strain softening within a small strain range. However, at higher strain rate, the
Fig. 4. Stress-strain curves of frozen loess under different strain rates.
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X. Xu et al. / Cold Regions Science and Technology 136 (2017) 44–51
deformation of sample is not uniform and usually localized in part of the sample. This localized deformation results in the local stress in the part of sample where carries the most of load rapidly reaches its peak stress and then the brittleness failure occurs, which indicates the sample is at the softening stage. This non-uniformity of deformation limits the development of plastic deformation inside the sample and leads to the increase of yield limit. The yield strain and failure strain of frozen Helin loess under different strain rate are presented in Table 3. It is found that the strain rate has less effect on the yield strain and failure strain than the influence by temperature. The yield rate ranges from 0.61% to 0.96%, and the failure strain ranges from 7.05% to 11.09%, respectively. 3.2.2. Yield limit and yield strength Fig. 5 illustrates log-log relationships between yield limit/strength and strain rate. As shown in Fig. 5, the yield limit and yield strength increases linearly with the increase of strain rate in the log-log coordinates. Zhang et al. (2003) found similar tendency for Lanzhou frozen silty soil. The relationships between yield limit/strength of saturated frozen Helin loess and the strain rate in the log-log ssas follows: lg σ y =σ y0 ¼ Ay þ By lgðε_ a =ε_ a0 Þ
ð3Þ
lg σ f =σ f 0 ¼ A f þ B f lgðε_ a =ε_ a0 Þ
ð4Þ
where σy and σf are the yield stress and strength, respectively, ε_ a is the strain rate,ε_ a0 ¼ 1s−1 is the reference strain rate. Ay, By, Af, and Bf are material parameters. In this study, Ay = 1.4154, By = 0.3897, Af = 1.2928, and Bf = 0.1871 by fitting the test data. The substance of the linear relationship between logarithmic of yield stress/strength and logarithmic of strain rate is the exponential relationship between the yield limit/ strength and the strain rate. 3.2.3. The effect of strain rate on elastic modulus of frozen Helin soil The initial tangent modulus and the secant modulus corresponding to half the strength are determined by the method in Section 3.1.3. Fig. 6 shows the relationship between the initial tangent modulus and the strain rate as well as the relationship between the secant modulus corresponding to half the strength and the strain rate. The initial tangent modulus increases linearly with the increase of strain rate in the log-log coordinates. The secant modulus corresponding to half the strength also increases with the increase in the strain rate. When the strain rate is 10−2/s, the values of the initial and the secant tangent modulus corresponding to half the strength are approximately equal. This is because the yield stress is approximately half of the failure stress at a higher strain rate, thus the data segment used to determine the initial tangent modulus coincides with the part to determine the secant modulus corresponding to half the strength. As a result, the value of the secant modulus corresponding to half the strength and the initial tangent modulus are approximately equal to each other.
Fig. 5. Relationships between yield limit/strength and strain rate under log-log coordinates.
general elastic-plastic theory (Chen, 2007), the incremental constitutive model is used to characterize the elasto-plastic behavior. Then, the strain (εij) can be divided as the sum of elastic strain (εeij) and plastic strain (εpij) in the incremental form (Xu et al., 2014): dεij ¼ dεeij þ dεpij
ð5Þ
In order to involve the effects of temperature and strains, the variables of temperature and strain rate have been introduced to constitutive model: dε ij ¼ Deijkl ðT; ε_ a Þdσ kl þ Dpijkl ðT; ε_ a Þdσ kl
ð6Þ
where σij is the stress tensor, Deijkl ðT; ε_ a Þ and Dpijkl ðT; ε_ a Þ are the elastic flexibility tensor and plastic flexibility tensor coupling with temperature and strain rate, respectively. By comparing Eqs. (5) and (6), we have dεeij ¼ Deijkl ðT; ε_ a Þdσ kl
ð7Þ
dε pij ¼ Dpijkl ðT; ε_ a Þdσ kl
ð8Þ
Eqs. (7) and (8) indicate that determining the specific forms of Deijkl ðT; ε_ a Þ and Dpijkl ðT; ε_ a Þ is the key step for constructing the
4. Elasto-plastic constitutive model considering temperature and rate strain for frozen Helin loess 4.1. Constitutive model The previous studies show that the frozen soil behaves obviously elasto-plastic deformation (Lai et al., 2014; Yang et al., 2016a, 2016b). Therefore, we adopt an elasto-plastic constitutive model to describe the stress-strain relationship of frozen Helin soil. According to the Table 3 Yield strain and failure strain of frozen Helin loess under different strain rates. Strain rate (s−1)
1 × 10−2
1 × 10−3
1 × 10−4
5 × 10−5
1 × 10−5
Yield strain (%) Failure strain (%)
0.96 7.05
0.88 10.54
0.73 10.44
0.63 11.09
0.61 10.76
Fig. 6. Relationships between elastic moduli and strain rate under log-log coordinates.
X. Xu et al. / Cold Regions Science and Technology 136 (2017) 44–51
constitutive model coupling with temperature and strain rate. Due to the frozen soil is usually undergoing axisymmetric load, we can adopt the constitutive model for frozen Helin loess under two dimensions (i.e. at p-q plane). In this case, the K-G model (Li, 2004) can be used to describe the elastic behavior of frozen Helin loess as following forms:
dε ev ¼ dp=K ðT; ε_ a Þ dεeγ ¼ dq=3GðT; ε_ a Þ
ð9Þ
And the matrix form is:
dεev dεeγ
¼
1=K ðT; ε_ a Þ 0 0 1=3GðT; ε_ a Þ
dp dq
Y q; εpγ ¼ q−k T; ε_ a ; εpγ ¼ 0
ð10Þ
ð11Þ
dλ in Eq. (11) is a non-negative multiplier which can be derived from yield criterion, flow rule and hardening law. The first formulation of Eq. (11) shows the plastic volumetric strain is zero since hydrostatic pressure has no effect on the plastic behavior of frozen Helin loess. Applying the consistency conditions on Eq. (10), we obtain dY ¼ dq−
∂k p dε ¼ 0 ∂εpγ γ
∂k ∂εpγ
ð13Þ
dεpv ¼ 0
p : dεγ
¼ dq=
∂k ∂εpγ
ð14Þ
2 ¼4
0 0
3 0 ∂k 5 dp 1= p dq ∂εγ
ð14’Þ
Combining Eq. (9’) with Eq. (14’), we obtain the incremental form constitutive model with the effects of temperature and strain rate for frozen Helin loess in matrix form as following:
dε v dεγ
p dεev dεv þ e dε dεpγ 2 γ 3 0 1=K ðT; ε_ a Þ ∂k 5 dp ¼4 0 1=3GðT; ε_ a Þ þ 1= p dq ∂εγ
¼
ð15Þ
This model includes three parameters coupling with temperature and strain rate. Those three parameters can be determined by uniaxial compression test of frozen Helin loess under different temperatures and strain rates from Sections 2 and 3. 4.2. Determination of model parameters and model verification The elastic parameters K and G can be expressed by elastic modulus and Poisson's ratio by the following transformation in elastic theory (Chen, 2007): K¼
E 3ð1−2ν Þ
ð16Þ
G¼
E 2ð1 þ ν Þ
ð17Þ
In this paper, we use the initial tangent modulus as the elastic modulus of frozen Helin loess. Based on the results of triaxial compression tests of the same frozen Helin loess (Xu et al., 2016), the elastic volumetric strain is ignored because it is far less than the elastic equivalent shear strain. Thus, the Poisson's ratio ν = 0.5. Based on the uniaxial compression test data of frozen Helin loess under different temperatures and strain rates, the elastic modulus of frozen Helin loess under different temperatures and strain rates can be determined by regressing testing temperature on elastic modulus. For instance, based on the tests results, the relationship between elastic modulus and temperature when the strain rate at 1.67 × 10−4/s is EðT Þ ¼
−52:26T−55:49 −2:84T þ 141:48
−4 ≤T ≤−2 −7 ≤T ≤−4
ð18Þ
Substituting Poisson's ratio ν = 0.5 and Eq. (18) into Eqs. (16) and (17), we can obtain that the elastic bulk modulus is infinity and the elastic shear modulus G depending on temperature: GðT Þ ¼
Combining Eq. (11) with Eq. (13), the plastic strain increment can be calculated as following: 8 <
dε pv dεpγ
ð12Þ
Substituting the second formulation in Eq. (11) into Eq. (12), we have dλ ¼ dq=
ð9’Þ
Here kðT; ε_ a ; εpγ Þ is the hardening parameter with the effects of temperature and strain rate on the plastic deformation of frozen Helin loess. Based on the associated flow rule (Chen, 2007), we have 8 ∂Y > > ¼0 < dε pv ¼ dλ ∂p ∂Y > > : dεpγ ¼ dλ ¼ dλ ∂q
And the matrix form is:
where εev =εe1 +2εe3 and εeγ =2/3(εe1 −εe3) are elastic bulk strain and elastic equivalent shear strain, respectively, p=(σ1 +2σ3)/3 and q=σ1 −σ3 are the mean stress and equivalent stress, respectively, KðT; ε_ a Þ and GðT; ε_ a Þ are elastic bulk modulus and elastic shear modulus affected by temperature and strain rate. To calculate the plastic deformation of frozen Helin loess, the yield criterion, flow rule and hardening law of frozen Helin loess need to be determined. Xu et al. (2016) performed triaxial test on the same type of frozen Helin loess, and they found the influence of confining pressure on the yield limit and strength of the frozen Helin loess can be ignored. Thus, the Mise's yield criterion is applicable to characterize the yield and plastic behavior of frozen Helin loess. Furthermore, we can assume the frozen Helin loess obeys the associated flow rule and the isotropic hardening. Therefore, the Mise's yield criterion can be written as following:
49
ð−52:26T−55:49Þ=3 ð−2:84T þ 141:48Þ=3
−4≤T ≤−2 −7 ≤T ≤−4
ð19Þ
When the strain rate at 1.67 × 10−4/s, the hardening parameter k can be determined by fitting k on the plastic shear strain from the uniaxial compressive test data under different temperatures as following: k T; ε pγ ¼ −0:117T þ 0:728 þ
εpγ
2 0:033e0:143T εpγ −0:06T 2 −0:338T
þ 1:978
ð20Þ
The Eqs. (19) and (20) provide the elastic and plastic parameters of frozen Helin loess under different temperatures at strain rate of 1.67 × 10−4/s.
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X. Xu et al. / Cold Regions Science and Technology 136 (2017) 44–51
Fig. 7. Comparison of stress-strain curves between experimental results (symbols) and predicted results (lines) under different temperatures.
Fig. 8. Comparison of stress-strain curves between experimental results (symbols) and predicted results (lines) under different strain rates.
Then, by substituting Eqs. (19)–(20) into Eq. (15), the influence of temperature on the stress-strain relationship of frozen Helin loess under a defined strain rate (1.67 × 10−4/s) can be predicted by the constitutive model. The comparison between uniaxial compressive test stress-strain curves and the predictive results from constitutive model under different temperatures is shown in Fig. 7. Fig. 7 indicates that the proposed constitutive model can precisely describe the characteristics of stress-strain curves of frozen Helin loess under different temperatures at strain rate of 1.67 × 10−4/s. If temperature is at −4 °C, the constitutive model parameters, G and k, depending on strain rate can also be determined by fitting the uniaxial compressive test data under different strain rates. According to the test data from Fig. 6, the relationship between elastic modulus of frozen Helin loess and strain rate can be expressed as following:
constitutive model can describe the stress-strain curves of frozen Helin loess under different strain rates at temperature of −4 °C.
ln ðEðε_ a ÞÞ ¼ 0:3109 ln ðε_ a Þ þ 1:2732
ð21Þ
Then, substituting ν = 0.5 into Eq. (17) and then combining with (21), we have ln ðGðε_ a ÞÞ ¼ 0:3109 ln ðε_ a Þ þ 1:2732− ln ð3Þ
ð22Þ
When the temperature is −4 °C, the hardening parameter k can be determined by fitting k on the plastic shear strain from the uniaxial compressive test data under different strain rates as following: k ε_ a ; εpγ ¼ P 1 þ
ε pγ p 2
P2 εγ
ð23Þ
þ P3
where, the P1, P2 and P3 are fitting parameters which depend on the strain rates. They can be determined by fitting the experimental data as follow: P 1 ðε_ a Þ ¼ 0:672ð ln ðε_ a ÞÞ þ 5:431 ln ðε_ a Þ þ 11:58 2
ð24Þ
3 2 P 2 ðε_ a Þ ¼ −0:011ð ln ðε_ a ÞÞ −0:095 ln ðε_ a Þ −0:263 ln ðε_ a Þ−0:210 ð25Þ 3
2
P 3 ðε_ a Þ ¼ 0:231ð ln ðε_ a ÞÞ þ 2:228 ln ðε_ a Þ þ 6:256 ln ðε_ a Þ þ 5:865 ð26Þ The effects of strain rates on the stress-strain relationship of frozen Helin loess at − 4 °C can be modeled by combining Eqs. (15) and (22)–(26). The comparison between uniaxial compressive test stressstrain curves and their constitutive modeling results under different strain rates is presented in Fig. 8. The Fig. 8 shows that the proposed
5. Conclusions A series of uniaxial compressive tests were conducted on the saturated frozen Helin loess. The stress-strain curves, yield strength, strength and elastic modulus with the variation of temperature (− 2 °C, − 4 °C, − 5 °C, and − 7 °C) and strain rate (1 × 10− 2/s, 1 × 10−3/s, 1 × 10−4/s, 5 × 10−5/s and 1 × 10− 5/s) are analyzed based on test results. The main findings are as follows: (1) The stress-strain curves of saturated frozen Helin loess present strain-softening under different strain rates and temperatures. Under different temperature, the turning point between quasi-elastic stage and plastic stage appears at strain of 0.77%–1.50%, which marks yield behavior occurring in frozen Helin loess; in contrast, under different strain rate, the yield phenomenon approximately occurs at strain of 1%. With the decrease of temperature and increase of strain rate, the brittleness of frozen Helin loess increases. (2) The yield limit and strength of the saturated frozen Helin loess increase with the decrease of temperature and the increase of strain rate. However, the yield limit and strength are linearly related with temperature, whereas the relationship between the yield limit/strength and strain rate are exponential. (3) The relationships between the initial tangent modulus/the secant modulus corresponding to half the strength and temperature have two stages. In the first stage, the temperature ranges from −4 °C to −2 °C. The initial tangent modulus and the secant modulus corresponding to half the strength increase rapidly with the increase of temperature in this stage, and the values of both moduli are small. In the second stage, the temperature ranges from −7 °C to −4 °C. The initial tangent modulus and the secant modulus corresponding to half the strength are higher than those in the first stage, but with a slower increasing rate. The tendency of both moduli varied with the decrease of temperature is similar. The initial tangent modulus and secant modulus corresponding to half the strength increase exponentially with the increase of the strain rate when strain rate ranges from 10− 5/s to 10−3/s. And both moduli are approximately equal under strain rate of 10−2/s. (4) An elasto-plastic constitutive model was proposed to describe the stress-strain relationship of frozen Helin loess under different temperatures and strain rates. The procedure of determining the model parameters was detailed. All of the model parameters were identified by fitting the experimental data. The comparison of stress-strain curves between experimental results and modeled results showed that the
X. Xu et al. / Cold Regions Science and Technology 136 (2017) 44–51
proposed model can correctly predict the deformation behavior of frozen Helin loess under different temperatures and strain rates. Acknowledgements This research was supported by National Natural Science Foundation of China (41301072, 41230630, 51579157, 51574219), Natural Science Foundation of Inner Mongolia (No. 2016MS0530), the Post-doctor Study Program of Inner Mongolia University (21200-5152604), the Knowledge Innovation Program of the Chinese Academy of Sciences (KZCX2-EW-QN301), and the Program for Frost-heave diseases mitigation technology for the expressways in seasonally frozen regions in Inner Mongolia Autonomous Region (Entrusted by MCC Communication Construction group Co., Ltd., 22200-5166011). References Andersen, G.R., Swan, C.W., Ladd, C.C., Germaine, J.T., 1995. Small-strain behavior of frozen sand in triaxial compression. Can. Geotech. J. 32, 428–451. Arenson, L., Springman, S., 2005. Triaxial constant stress and constant strain rate tests on ice-rich permafrost samples. Can. Geotech. J. 42 (2), 412–430. Baker, T.H.W., Jones, S.J., Parameswaran, V.R., 1982. Confined and unconfined compression tests on frozen sands. Proc. 4th Canadian Permafrost Conference, Roger J.E. Brown Memorial Vol., pp. 387–393 Bragg, R.A., Andersland, O.B., 1981. Strain rate, temperature, and sample size effects on compression and tensile properties of frozen sand. Eng. Geol. 18, 35–46. Chang, X.X., Ma, W., Sheng, Y., et al., 1995. Influence of experimental conditions on the mechanical parameters of frozen soil. J. Glaciol. Geocryol. 17 (suppl), 151–155. Chen, M.X., 2007. Elasticity and Plasticity. Science Press, Beijing. Chen, X.B., Qiu, G.Q., Wang, Y.Q., et al., 1988. The character of physical chemistry and mechanics for soil with abundant salinity under varying temperature. Sci. China A 4, 429–438. Haynes, F.D., Karalius, J.A., 1977. Effect of Temperature on the Strength of Frozen Silt. CRREL Report, Hanover (77–3). Haynes, F.D., Karalius, J.A., Kalafut, J., 1975. Strain rate effect on the strength of frozen silt. Hanover: CRREL Report RR350. Jones, S.J., Parameswaran, V.R., 1983. Deformation behavior of frozen sand-ice materials under triaxial compression. Proc. 4th International Permafrost Conference, pp. 560–565. Ladanyi, B., 1981. Mechanical behavior of frozen soils. Proc. of Int. Symp. on the Mechanical Behavior of Structured Media. A. A. Balkema, Ottawa, pp. 205–245. Lai, Y., Xu, X., Dong, Y., Li, S., 2013. Present situation and prospect of mechanical research on frozen soils in China. Cold Reg. Sci. Technol. 87, 6–18. Lai, Y.M., Xu, X.T., Yu, W.B., Qi, J.L., 2014. An experimental investigation of the mechanical behavior and a hyperplastic constitutive model of frozen loess. Int. J. Eng. Sci. 84, 29–53. Lai, Y.M., Yang, Y.G., Chang, X.X., et al., 2010. Strength criterion and elastoplastic constitutive model of frozen silt in generalized plastic mechanics. Int. J. Plast. 26 (10), 1461–1481. Li, G.X., 2004. Advanced Soil Mechanics. Tsinghua University Press, Beijing.
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