Experimental investigation of dynamic shear modulus and damping ratio of Qinghai-Tibet frozen silt under multi-stage cyclic loading

Cold Regions Science and Technology 170 (2020) 102938

Contents lists available at ScienceDirect

Cold Regions Science and Technology journal homepage: www.elsevier.com/locate/coldregions

Experimental investigation of dynamic shear modulus and damping ratio of Qinghai-Tibet frozen silt under multi-stage cyclic loading

T

Futang Zhao, Lijun Chang , Wuyu Zhang ⁎

School of Civil Engineering, Qinghai University, Xining 810016, China

ARTICLE INFO

ABSTRACT

Keywords: Frozen silt soil Cryogenic dynamic triaxial test Dynamic shear modulus Damping ratio Hyperbolic model

The transportation infrastructure built in cold regions is mostly above frozen soil, whose stability is of concern, and which has been studied as a subgrade for highways and railways. Based on cryogenic cyclic dynamic triaxial tests, the subgrade frozen silt from the Qinghai-Tibet Plateau is taken as a research object. The dynamical characteristics of frozen silt are analyzed under different freezing temperatures, initial moisture contents, compaction degrees, and confining pressures. The results show that a hyperbolic constitutive model can well express the dynamic stress-strain response relationship of frozen silt under external dynamic loading. The dynamic shear modulus increased with decreasing freezing temperature, increasing moisture content, compaction degree, and confining pressure; the damping ratio had exactly the opposite relationship; and frozen silt energy consumption was gradually weakened. However, when the moisture content exceeded a certain threshold (11.4%), the damping ratio tended to increase instead. The unfrozen moisture content of frozen silt under different water contents was measured, which proved that the dynamic shear modulus increased due to the increase of ice crystals. In addition, variance analysis was introduced to analyze the significant influences of mechanical parameters. Four experimental conditions were compared for the influence volatility of the maximum dynamic shear modulus, final shear strain amplitude, and maximum damping ratio. Obviously, the dynamic properties of frozen silt were influenced by the freezing temperature and moisture content, followed by the compaction degree and confining pressure. Finally, the empirical formulas of the dynamic shear modulus and damping ratio were established by verifying the reliability and analyzing the experimental results.

1. Introduction China has the third largest frozen soil area in the world, accounting for 68.6% of the country's territory, including 2.15 million km2 of permafrost area and 5.14 million km2 of seasonally frozen soil area (Zhou et al., 2000). The high-altitude frozen soil on the Qinghai-Tibet Plateau is the most widely distributed frozen soil area in the world (Zhao et al., 2003; Li et al., 2012a, 2012b; Wu et al., 2018). To keep pace with people expanding their activity space and increasing resource demands, engineering construction projects, such as railways, highways, oil pipelines, airports, and mineral mines, have expanded gradually in the cold region. Mechanical vibration, vehicle vibration of high-speed trains or automobiles, and seismic action will exert dynamic loading on road subgrades and structure foundations in cold regions, resulting in varying degrees of damage, such as road-surface cracking and boiling, roadbed uplift and subsidence, slope instability, and pipeline breakage. Therefore, it is important to study the dynamic characteristics of frozen soil for engineering construction and anti-vibration

⁎

design in cold regions. As is well known, frozen soil is a four-phase system composed of soil particles, ice, air, and unfrozen water. The existence and dynamic equilibrium for unfrozen water and ice make the mechanical properties of frozen soil extremely sensitive to temperature, so frozen soil differs greatly from conventional soil in its mechanical properties (Liu et al., 2018). Frozen soil dynamics research started late, and has mainly addressed the effects of different soil materials, temperatures, moisture contents, confining pressures, vibration frequencies, freeze-thaw cycles, and admixtures on the frozen soil dynamic parameters, dynamic strength, deformation, and dynamic creep. In particular, experimental studies on the dynamic shear modulus and damping ratio began around the early 1970s (Ling et al., 2015). Xu et al. (1998), Li et al. (1979), and Ling et al. (2009) studied the dynamic elastic modulus, shear modulus, and damping ratio of frozen soil under different temperatures and frequencies. Shen and Zhang (1998), Wu et al. (2003), Wang et al. (2005), Zhao et al. (2006), and Zhang et al. (2019) studied the dynamic strength characteristics of frozen soil and carried out dynamic and static

Corresponding author. E-mail address: [email protected] (L. Chang).

https://doi.org/10.1016/j.coldregions.2019.102938 Received 15 April 2019; Received in revised form 17 September 2019; Accepted 31 October 2019 Available online 05 November 2019 0165-232X/ © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

Cold Regions Science and Technology 170 (2020) 102938

F. Zhao, et al.

strength comparison tests. Ling et al. (2010) and Ling et al. (2013) conducted dynamic triaxial tests on frozen silty clay and discussed the vibration characteristics of the subgrade rail traffic load, dynamic nonlinear constitutive relationship, and dynamic shear modulus. Zhu et al. (2009) and Zhu et al. (2010) analyzed the deformation characteristics of silty clay along the Qinghai-Tibet Railway under the longterm loading of reciprocating cycles, and conducted exploratory research on the vibration subsidence model. Jiao et al. (2010) and Hazirbaba et al. (2011) obtained the variation of the dynamic characteristics of Qinghai-Tibet artificial frozen silt with the frequency and dynamic stress amplitude at higher freezing temperature after freezethaw cycles. Naeini and Gholampoor (2014) and Muge et al. (2017) observed the dynamic properties of reinforced fiber soil under freezethaw cycles and noted that the dynamic shear modulus increased distinctly with increasing of fiber content. Regarding the damping ratio, Nakano and Amold (1973) studied the variation of Ottawa sand with moisture content under different damping. Vinson's (1978) results showed that the damping ratio of frozen soil was more dispersed (irregular) with the change of moisture content, and the damping ratio increased slightly with an increase in the confining pressure. Czajkowski and Vinson (1980) showed that frequency had no effect on the damping ratio of frozen soil. Luo et al. (2015) observed that the frozen clay damping ratio firstly decreased and then slowly increased under the action of a graded load, and that the frozen loess damping ratio first decreased gradually and then remained unchanged. In summary, there has been considerable research on frozen-soil dynamics. However, some studies consider the dynamic characteristics of frozen soil based on a single environmental factor. Some research methods apply only to a specific project, and there is no universality. In addition, the consensus on variations of some frozen-soil dynamic parameters has not been unified, and different development rules have appeared along with variations of experimental conditions, due to the distinct physical, thermal, mechanical properties of the frozen soil and the specificity of the dynamic load. Therefore, the research of frozen soil dynamics requires the accumulation of large amounts of experimental data and empirical methods to better reflect the dynamic characteristics. In particular, the dynamic shear modulus and the damping ratio are important reference values of engineering design that must be improved and studied in-depth. In this study, a series of cryogenic dynamic triaxial tests are conducted under graded cyclic loading of subgrade frozen soil while considering the deficiencies of existing research. We further consider the important factor of subgrade soil compaction degree, which has been the subject of little research. The variation rules of the backbone curve, dynamic shear modulus, and damping ratio for different freezing temperatures, moisture contents, compaction degrees, and confining pressure conditions are discussed. Variance analysis is applied to comprehensively consider the degree of influence of each single test factor on the dynamic characteristic parameters. On this basis, empirical models of the dynamic shear modulus and damping ratio following shear strain under multi-factor influences are proposed and verified. Ultimately, this research is expected to draw some meaningful conclusions for coldregion geotechnical engineering design, construction, and protection, and to provide a meaningful contribution to these fields.

100

Passing percentage/%

80

60

40

20

0 -4 10

10

-3

10

-2

10

-1

0

10

Soil particle diameter /mm Fig. 1. Particle size distribution curve of soil sample.

The particle size distribution curve is shown in Fig. 1. After preparing the set moisture content, compaction degrees (C) of 87%, 90%, 93%, and 97% were used, the sample density ρdmax × C was controlled, and the cylindrical sample had a diameter of 39.1 mm and a height of 80 mm. 2.2. Test equipment The test equipment included a Global Digital Systems DYNTTS servo motor control dynamic triaxial test system, shown in Fig. 2, including devices for triaxial loading, temperature control, pressure volume control, data acquisition, and software. The dynamic load could be applied with a 10 Hz maximum frequency. When the axial displacement range was up to 100 mm, the load cell range was 1 kN to 60 kN. A liquid refrigeration compression cycle machine was used to change the inside triaxial cell temperature, which ranged from 10 °C to −30 °C. 2.3. Method and loading condition setting The control situations for each parameter of the frozen silt dynamic triaxial test are shown in Table 2. Four dynamic parameters were determined under different freezing temperatures, moisture contents, compaction degrees, and confining pressures. The samples were artificially frozen. First, the finished samples were frozen at a temperature of −30 °C for 48 h and placed in a constant-temperature triaxial cell for 12 h. The samples were then consolidated for 2 h in a triaxial cell. Finally, a cyclic dynamic load was applied for testing. The dynamic triaxial loading condition simulated the traffic cycle load, and it was loaded as a sine wave:

(t ) =

3

+

di max

di min

2

[1 + sin(2 ft )]

(1)

where σ3 is the confining pressure, f is the loading frequency, t is the loading time, σdi is the amplitude of the i-th dynamic stress, σdimax is the maximum dynamic stress, and σdimin is the minimum dynamic stress. The loading indication is shown in Fig. 3 and Table 1. The dynamic load was incremented stepwise from a small value, and each stage of the load was vibrated 12 times in a stepwise cyclic loading until the deformation termination standard was loaded. Generally, the failure standard for a sample at normal temperature is 5% of the axial dynamic strain, according to the Standard for Soil Test Method (GB/T 501231999). However, the test sample was cryogenic frozen soil, and when the dynamic strain reached 5%, the axial dynamic load could be added gradually. Therefore, the sample failure standard took 10% of the axial dynamic strain as the loading termination condition (Zhu et al., 2009, 2010; Ling et al., 2015).

2. Experimental program 2.1. Basic physical property test and sample preparation Test silt samples were taken from a highway in the Qinghai-Tibet area. The basic physical parameters were measured according to Test Methods of Soils for Highway Engineering (JTG E40-2007), issued by the Ministry of Transport in China. The liquid and plastic limits were 21.03% and 4.5%, respectively, and the maximum dry density and optimum moisture content were 1.82 g/cm3 and 11.4%, respectively. 2

Cold Regions Science and Technology 170 (2020) 102938

F. Zhao, et al.

Fig. 2. DYNTTS GDS dynamic triaxial test system.

3. Constitutive relation and fitting parameter

Table 1 Dynamic triaxial test load grading standard(σ3 = 0.2 MPa).

Through experimental data, the dynamic stress and strain relationships of the hysteresis loop under each load were obtained. Their amplitudes were calculated with Eq. (2) and Eq. (3), which were expressed by Eq. (4) from Hardin and Drnevich (1972a, 1972b). At this time, the dynamic load hysteresis loop curve could be described by Fig. 4(a). The average slope of the curve is the dynamic shear modulus Gd, and its relational expression is Eq. (5). d

=

d (1

d

=

d /2

d

=

Gd =

(4) (5)

d/ d

where μ is Poisson's ratio, μ = 0.3 is appropriate for silt (Xu et al.,1998), γd is the dynamic shear strain, and a and b are the experimental fitting parameters, a > 0, b > 0. Formula (6) is obtained From Eqs. (4) and (5), we obtain

1 =a+b Gd

dult

=

d

|

d

+

0

= 1/ a

(7)

= 1/ b

(8)

5

6

7

8

9

…

σdimax/MPa σdimin/MPa σdi/MPa

0.26 0.22 0.04

0.35 0.22 0.13

0.44 0.22 0.22

0.53 0.22 0.31

0.62 0.22 0.40

0.71 0.22 0.49

0.80 0.22 0.58

0.89 0.22 0.67

0.98 0.22 0.76

… … …

=

dult / Gd max

= a/ b

(9)

Gd max 1 + d/ dr

(10)

Finally, the basic functional relationship between the dynamic shear modulus and dynamic shear strain amplitude can be stated in Eq. (10), which is used to depict the dynamic constitutive relationship of frozen silt. According to Hardin and Drnevich (1972a, 1972b), the damping ratio λ can be described by Fig. 4(b), and

Furthermore, Eqs. (7) and (8) are obtained from the special conditions using γd → 0, γd → +∞ in Eq. (6), as follows:

Gd max = Gd | d

4

Gd =

(6)

d

3

where Gdmax is the maximum dynamic shear modulus, τdult is the final shear stress amplitude, and γdr is the reference shear strain amplitude. The experimental data are described by the relationship between 1/Gd and γd, and the fitting parameters of a and b are obtained. Then, Eq. (10) is acquired from Eqs. (6)–(9), we obtain

d d

2

dr

(3)

a+b

1

Combining Eqs. (7) and (8), we obtain

(2)

+ µ)

Loading series

d

=

S 4 S

(11)

where S is the area enclosed by the hysteresis loop after a stress cycle

Fig. 3. Schematic diagram of axial multistage cyclic loading. 3

Cold Regions Science and Technology 170 (2020) 102938

F. Zhao, et al.

τd 1

Gdmax

1

τd

Gd

1

Gd

Backbone curve

Backbone curve

γd

γd

Hysteresis loop

Hysteresis loop

(a)

(b)

Fig. 4. Conceptual graph for dynamic shear modulus and damping ratio measurement.

(red region), indicating the energy dissipated within the cycle, and SΔ is the area of a triangle (black region), indicating the maximum strain energy stored in the period. The damping ratio (Hardin et al., 1972a, b; Seed et al., 1984) can be determined more simply as d

=

max

Gd Gd max

1

=

max

d / dr

1+

(12)

d / dr

where λmax is the maximum damping ratio. Eq. (12) is the dynamic response analysis relationship proposed for seismic loads. While based on the dynamic triaxial test fitting of frozen soil under a traffic load (Zhu et al., 2011), the relationship between the damping ratio and dynamic shear strain amplitude is fitted more consistently by d

=

max

Gd

1

n

=

Gd max

max

d / dr

1+

n

(13)

d / dr

Fig. 5. Backbone curve of frozen silt under different temperatures.

where n is the experimental fitting parameter. The results of the related fitting parameters in the experiment are a, b, Gdmax, λmax and n, as shown in Table 2.

amplitude increased from 0.66 MPa to 6.07 MPa when the temperature was lowered from −1 °C to −15 °C. At a warm temperature of −1 °C, the backbone curve flattened, indicating that the frozen silt was obviously deformed and the dynamic strength was significantly reduced with increasing temperature. Fig. 6(a) shows the relationship between the dynamic shear modulus and the dynamic shear strain under temperatures of −1 °C, −5 °C, −10 °C, and − 15 °C. The dynamic shear modulus increased with decreasing temperatures for a given dynamic shear strain value. The maximum dynamic shear modulus increased from 276.24 MPa to

4. Experimental results and analysis 4.1. Effects of temperature Fig. 5 shows the frozen silt backbone curves under temperatures of −1 °C, −5 °C, −10 °C, and −15 °C. From this, it can be seen that the dynamic shear stress increased greatly with decreasing temperatures under an identical dynamic shear strain value. The final shear stress

Table 2 Frozen silt test conditions and the fitting values of backbone curve parameters ((1) Test nomber, (2) Temperature-T, (3) Initial moisture content-W, (4) compaction degree-C, (5) Frequency-ƒ, (6) confining pressure-σ3, (7) Fitting parameters). (1)

QT1 QT2 QT3 QT4 QT5 QT6 QT7 QT8 QT9 QT10 QT11 QT12 QT13

R2

(2) T

(3) W

(4) C

(5) ƒ

(6) σ3

(7)

°C

%

%

Hz

MPa

a × 10−2

b

Gdmax/MPa

λmax

n

(a,b)

(λmax,n)

−1 −5 −10 −15 −10

11.4

93

2

0.2

93

2

0.2

−10

7 14 17 11.4

2

0.2

−10

11.4

87 90 97 93

2

0.1 0.3 0.4

0.362 0.340 0.158 0.115 0.194 0.114 0.085 0.213 0.178 0.123 0.195 0.144 0.123

1.5067 0.3800 0.1833 0.1649 0.2969 0.1200 0.0981 0.1606 0.1611 0.1983 0.1505 0.1283 0.1221

276.24 294.12 632.91 869.57 515.46 877.19 1174.36 469.48 561.80 813.01 512.82 694.44 813.01

0.4350 0.2778 0.1772 0.1690 0.2625 0.0818 0.0874 0.2383 0.2018 0.1700 0.2144 0.1733 0.1584

0.7966 0.4333 0.3980 0.4100 0.5242 0.3469 0.2926 0.3663 0.3731 0.5517 0.3738 0.4037 0.4396

0.9897 0.9912 0.9913 0.9948 0.9962 0.9938 0.9948 0.9976 0.9958 0.9924 0.99311 0.99081 0.99119

0.9658 0.9703 0.9511 0.9508 0.9432 0.9805 0.9713 0.9759 0.9679 0.9676 0.9252 0.9625 0.9489

4

Cold Regions Science and Technology 170 (2020) 102938

F. Zhao, et al.

Fig. 6. Dynamic shear modulus, damping ratio and dynamic shear strain under different temperatures.

869.57 MPa when the temperature was lowered from −1 °C to −15 °C. Fig. 6(b) shows the relationship between the damping ratio and dynamic shear strain under various temperatures. The damping ratio decreased with decreasing temperatures at a given shear strain value. The maximum damping ratio decreased from 0.435 to 0.169 when the temperature was lowered from −1 °C to −15 °C. However, when the temperature was below −5 °C, the relationship between the different damping ratios and the dynamic shear strain was fairly close, and when the temperature was above −5 °C, the trends of the different curves varied greatly. It was observed that the temperature influence was sensitive to the mechanical properties of the frozen soil. The effect of the temperature on the frozen soil dynamics characteristics might have been related to the free volume water between the soil particles. When the moisture content was constant and the temperature was lower, more free water froze to form ice crystals, the adhesion between the soil particles and the ice crystals was greater, and the bite was tighter. As a result, the soil stiffness increased and deformation was relatively less, so the maximum dynamic shear modulus increased and the damping ratio decreased.

different intensity signals of soil samples. The relationship between the unfrozen moisture content and temperature can be obtained by converting the nuclear magnetic resonance intensity signal into mass moisture content at different temperatures. The experimental results of unfrozen moisture content are shown in Table 3, from which it is known that both the freezing temperature and unfrozen moisture content increase with increasing initial moisture content. When the temperature dropped to −10, the unfrozen moisture content decreased from 7%, 11.4%, 14% and 17% to 1.29%, 1.33%, 1.39% and 1.43%, respectively. It is verified that the moisture content in frozen silt increases, the amount of unfrozen moisture increases slightly, and the amount of ice crystals is increases. Hence the frozen silt stiffness increases. 4.2.2. Impact analysis of experimental results Fig. 8 shows the frozen silt backbone curves under initial moisture contents of 7%, 11.4%, 14%, and 17%. The backbone curves are steeper with increasing water content at a freezing temperature of −10 °C. The final shear stress amplitude increased from 3.37 MPa to 10.19 MPa when the moisture content increased from 7% to 17%. Fig. 9(a) shows the relationship between the dynamic shear modulus and dynamic shear strain under initial moisture contents of 7%, 11.4%, 14%, and 17%. The dynamic shear modulus increased with the initial moisture content at an identical shear strain value. The maximum dynamic shear modulus depended on the initial moisture content, which rose from 7% to 17%, during which the maximum dynamic shear modulus rose from 515.46 MPa to 1174.36 MPa. Fig. 9(b) shows the relationship between the damping ratio and dynamic shear strain at different initial moisture contents. The damping ratio decreased with increasing initial moisture contents under an identical shear strain value. The maximum damping ratio decreased from 0.263 to 0.087 when the initial moisture content increased from 7% to 17%. However, the damping ratio and the dynamic shear strain showed a slight tendency to change when the initial water content exceeded 11.4%, indicating that there was an initial moisture content threshold that controlled the change of the damping ratio, and the

4.2. Effects of moisture content 4.2.1. Unfrozen moisture content experiment Unfrozen moisture content in frozen soil is an important basis for the study of frozen soil mechanics and thermodynamics. This experiment measured the unfrozen moisture content of frozen silt under different initial moisture contents according to the nuclear magnetic resonance instrument, shown in Fig. 7. The nuclear magnetic resonance method excites hydrogen nuclei in sample by radio frequency pulse, causing the hydrogen nuclei to resonate, and absorb energy to obtain

Table 3 Unfrozen moisture content test results of frozen silt (tf -soil sample freezing temperature, wn -unfrozen moisture content).

Fig. 7. Nuclear magnetic resonance instrument. 5

Test no.

W (%)

tf (°C)

T = tf , wn (%)

T = −2 °C, wn (%)

T = −5 °C, wn (%)

T = −10 °C, wn (%)

QT5 QT3 QT6 QT7

7 11.4 14 17

−2.79 −1.37 −1.05 −0.89

3.54 7.54 10.16 13.12

4.87 5.45 5.88 6.48

2.20 2.36 2.40 2.48

1.29 1.33 1.39 1.43

Cold Regions Science and Technology 170 (2020) 102938

F. Zhao, et al.

6

τd /MPa

4

Compaction = 87%, QT 8 Compaction = 90%, QT 9 Compaction = 93%, QT 3 Compaction = 97%, QT 10 Fitting curve of 87%,Eq.(4) Fitting curve of 90%,Eq.(4) Fitting curve of 93%,Eq.(4) Fitting curve of 97%,Eq.(4)

4

3

τd /MPa

5

5 W= 7%, QT 5 W= 11.4%, QT 3 W= 14%, QT 6 W= 17%, QT 7 Fitting curve of 7%,Eq.(4) Fitting curve of 11.4%,Eq.(4) Fitting curve of 14%,Eq.(4) Fitting curve of 17%,Eq.(4)

3

2

2

1

1 0 0.000

0.001

0.002

0.003

γd

0.004

0.005

0.006

0 0.000

0.007

0.001

0.002

0.003

γd

0.004

0.005

0.006

0.007

Fig. 8. Backbone curve of frozen silt under different initial moisture contents.

Fig. 10. Backbone curves of frozen silt under different compaction degrees.

threshold value was approximately 11.4%, which was the optimal moisture content. The test results show that the change of the moisture content also had a significant impact on the mechanical properties of frozen soil. The frozen soil consisted of soil particles, ice crystals, unfrozen water, and air. Under the same temperature and freezing history, the lower moisture content induced the ice content and cohesive bond to decrease, which weakened the cementation ability of the soil particles and ice crystals. Thus the frozen soil had low rigidity, low strength, large deformation, and increased energy consumption. In contrast, a higher moisture content had the exact opposite result.

Under the same conditions of temperature, moisture content, and confining pressure, the variation of the compaction degree had little influence on the mechanical properties of frozen soil. The pore space between soil particles decreased as the compaction degree increased, and the arrangement of the soil skeleton became more compact and close, resulting in a reduction of the dislocation space of the soil particles and the ice crystals, and a slightly increased stiffness of the frozen soil. This increase was limited, as was its contribution to the strength of frozen soil was limited. As the compaction degree increased, the movement space and the energy consumption of the soil particles and ice crystals under a dynamic load decreased, so the damping ratio tended to decrease.

4.3. Effects of compaction degree

4.4. Effects of confining pressure

Backbone curves at compaction degrees of 87%, 90%, 93%, and 97% are shown in Fig. 10. The final shear stress amplitude decreased from 6.23 MPa to 5.04 MPa when the compaction degree increased from 87% to 97%. Moreover, the 90% and 93% backbone curves show overlapping trends, indicating that the compaction degrees had little impact on the dynamic strength and deformation of the frozen silt. The relationships between the dynamic shear modulus and shear strain at compaction degrees of 87%, 90%, 93%, and 97% are shown in Fig. 11(a). At a given identical shear strain value, the dynamic shear modulus increased with the compaction degree. The maximum dynamic shear modulus improved from 469.48 MPa to 813.01 MPa, and the compaction degree increased from 87% to 97%. The relationship curves between the damping ratio and dynamic shear strain under different compaction degrees are shown in Fig. 11(b). The maximum damping ratio decreased from 0.238 to 0.17 when the compaction degree increased from 87% to 97%. 1400 1200

The backbone curves with confining pressures of 0.1 MPa, 0.2 MPa, 0.3 MPa, and 0.4 MPa are shown in Fig. 12. The final shear stress amplitude increased from 6.64 MPa to 8.19 MPa when the confining pressure increased from 0.1 MPa to 0.4 MPa under identical experimental conditions. It can be observed that the confining pressure had a slight influence on the dynamic strength and deformation of the frozen silt, and the variation trends of the backbone curves were similar. The correlations between the dynamic shear modulus and shear strain at confining pressures of 0.1 MPa, 0.2 MPa, 0.3 MPa, and 0.4 MPa are shown in Fig. 13(a). The dynamic shear modulus rose as the confining pressure increased, and the shear strain value remained constant. When the confining pressure increased from 0.1 MPa to 0.4 MPa, the maximum dynamic shear modulus value increased from 512.82 MPa to 813.01 MPa. The relationship between the damping ratio and dynamic shear strain under different confining pressures is shown in Fig. 13(b). 0.5

W= 7%, QT 5 W= 14%, QT 6

W= 11.4%, QT 3 W= 17%, QT 7

0.4

1000 0.3

λ

Gd /MPa

800 600 400 200 0 -5 10

W= 7%, QT 5 W= 11.4%, QT 3 W= 14%, QT 6 W= 17%, QT 7 Fitting curve of 7%,Eq.(13) Fitting curve of 11.4%,Eq.(13) Fitting curve of 14%,Eq.(13) Fitting curve of 17%,Eq.(13)

0.2 Fitting curve of 7%,Eq.(10) Fitting curve of 11.4%,Eq.(10) Fitting curve of 14%,Eq.(10) Fitting curve of 17%,Eq.(10)

10

-4

0.1

-3

γd

10

-2

10

(a)

0.0 -5 10

-4

10

-3

10 γd

-2

10

(b)

Fig. 9. Dynamic shear modulus, damping ratio and dynamic shear strain under different initial moisture content. 6

10

-1

Cold Regions Science and Technology 170 (2020) 102938

F. Zhao, et al.

1000

0.5 Compaction= 87%,QT8 Compaction= 93%,QT3

Compaction= 90%,QT9 Compaction= 97%,QT10

0.4

600

0.3

λ

Gd /MPa

800

Compaction = 87%, QT 8 Compaction = 90%, QT 9 Compaction = 93%, QT 3 Compaction = 97%, QT 10 Fitting curve of 87%,Eq.(13) Fitting curve of 90%,Eq.(13) Fitting curve of 93%,Eq.(13) Fitting curve of 97%,Eq.(13)

0.2

400 Fitting curve of 87%,Eq.(10) Fitting curve of 90%,Eq.(10) Fitting curve of 93%,Eq.(10) Fitting curve of 97%,Eq.(10)

200

0 -5 10

10

0.1

-4

10

γd

-3

10

0.0 -5 10

-2

-4

-3

10

-2

10 γd

(a)

10

-1

10

(b)

Fig. 11. Dynamic shear modulus, damping ratio and dynamic shear strain under different compaction degrees.

5

τd /MPa

4

3

σ3= 0.1MPa, QT 11

σ3= 0.2MPa, QT 3

σ3= 0.3MPa, QT 12

σ3= 0.4MPa, QT 13

decreasing energy consumption when the dynamic load was applied. 4.5. Impact analysis of test factors

Fitting curve of 0.1MPa,Eq.(4) Fitting curve of 0.2MPa,Eq.(4) Fitting curve of 0.3MPa,Eq.(4) Fitting curve of 0.4MPa,Eq.(4)

The four experimental conditions have different effects on results, and the magnitude of effects can be analyzed by variance(Li et al., 2012; Lui et al., 2016). The calculation formulas for the variance analysis are as follows:

2

X =

1

0 0.000

1 n

Dk = 0.001

0.002

0.003

γd

0.004

0.005

0.006

n

Xi

(14)

i=1

S=

Fig. 12. Backbone curves of frozen silt under different confining pressures.

S2 =

(15)

X )2

(Xk

0.007

n

1 n

1

X )2

(Xk

(16)

k=1

where Xi is the experimental value; n = 4; X is the data average; Dk is the calculated variance, k = 1, 2, 3, 4; and S is the sample standard deviation. S can be used to judge the stability and fluctuation of test data. Thus S can reflect the significance of different influencing factors in the experiment. A larger S indicates that test data fluctuate more and are more unstable, the greater influence of instability caused by a test condition, the test condition is more conspicuous. A smaller S indicates that test data fluctuate less and are more stable, the lighter influence of instability caused by a test condition, the test condition is more weaker. The calculated values of X and S are shown in Table 4. Figs. 14 and 15 show variance histograms of the different influencing factors of Gdmax and γdr. Obviously, the influences of temperature and moisture-content factors on the frozen silt, Gdmax differ significantly

Under the same shear strain value, the damping ratio decreased with the increase of the confining pressure. The maximum damping ratio was reduced from 0.214 to 0.158 as the confining pressure increased from 0.1 MPa to 0.4 MPa. The variation of the confining pressure also had little influence on the mechanical properties of the frozen soil. The slip and orientation arrangement between the soil particles and ice crystals increased less with the confining pressure, there were fewer open pores, some microcracks were closed, and the cementing force tended to be uniform. Therefore, the strength of the frozen soil was not obviously changed. Moreover, the soil's structural performance was strengthened due to the increase of the confining pressure, and the damping ratio reduced with

0.5 σ3= 0.1MPa, QT 11

800 0.4

0.3

λ

Gd /MPa

600

400

200

0 -5 10

σ3= 0.1MPa, QT 11

σ3= 0.2MPa, QT 3

σ3= 0.3MPa, QT 12

σ3= 0.4MPa, QT 13

0.2

Fitting curve of 0.1MPa,Eq.(10) Fitting curve of 0.2MPa,Eq.(10) Fitting curve of 0.3MPa,Eq.(10) Fitting curve of 0.4MPa,Eq.(10) -4

10

σ3= 0.3MPa, QT 12 σ3= 0.4MPa, QT 13 Fitting curve of 0.1MPa,Eq.(13) Fitting curve of 0.2MPa,Eq.(13) Fitting curve of 0.3MPa,Eq.(13) Fitting curve of 0.4MPa,Eq.(13)

0.1

-3

γd

σ3= 0.2MPa, QT 3

10

-2

10

(a)

0.0 -5 10

-4

10

-3

10 γd

-2

10

(b)

Fig. 13. Dynamic shear modulus, damping ratio and dynamic shear strain under different confining pressures. 7

10

-1

Cold Regions Science and Technology 170 (2020) 102938

F. Zhao, et al.

Table 4 Data variance analysis results under different conditions (A -temperature condition, B -initial moisture content condition, C -compaction degree condition, D -confining pressure condition). Factor

A B C D

Gdmax/MPa

γdr

λmax

n

X

S

X

S

X

S

X

S

518.209 799.982 619.300 663.296

285.991 291.537 145.441 125.100

0.0067 0.0083 0.0098 0.0107

0.0030 0.0013 0.0030 0.0018

0.265 0.152 0.197 0.181

0.124 0.086 0.031 0.024

0.509 0.390 0.422 0.404

0.192 0.099 0.087 0.027

Fig. 16. Histogram of maximum damping ratio calculation variance.

Fig. 14. Histogram of maximum dynamic shear modulus calculation variance.

Fig. 17. Histogram of parameter n calculation variance.

exerts the weakest control over it. 5. Empirical formula and verification 5.1. Empirical presentation of the dynamic shear modulus Previous studies (Hardin and Drnevich, 1972a, 1972b; Seed et al., 1984; Kallioglou et al., 2008; Subramaniam and Banerjee, 2013; Ling et al., 2015; Lin et al., 2018; Yu et al., 2018) have considered many influencing factors in the establishment of an empirical formula for the maximum dynamic shear modulus Gdmax, mainly including soil properties, fabric, temperature, moisture content, confining pressure, stress, and strain history. Based on the experimental data, the main factors of temperature, moisture content, compaction degree, and confining pressure were considered in the dynamic triaxial test of subgrade frozen silt, and the maximum dynamic shear modulus Gdmax can be expressed as

Fig. 15. Histogram of reference shear strain amplitude calculation variance.

from those of compaction-degree and confining-pressure factors. This indicates that the effect of frozen silt stiffness on temperature reduction and increase of moisture content is greater than on the increase in compaction degree and confining pressure. The influence of γdr under the date fluctuation of temperature and compaction-degree factors is greater than moisture content and confining pressure factors. This indicates that frozen silt deformation is controlled by the decrease of temperature and the increase of compaction degree, while the increase of moisture content and confining pressure has a relatively weak influence on deformation. Figs. 16 and 17 are variance histograms of the different influencing factors of λmax and n. As shown in these figures, the change of temperature factors has the most fluctuation effect on frozen silt λmax, followed by moisture-content factors, while compaction-degree and confining-pressure factors are least significant. Also, n is most fluctuation under the influence of temperature factors, followed by moisturecontent and compaction-degree factors, while confining pressure factors are least significant. This shows that frozen silt energy dissipation is mainly controlled by the freezing temperature, and confining pressure

(17)

Gd max = G0 × gT (T ) × gW (W ) × gC (C ) × g ( 3)

where G0 is the simulation parameter, and gT(T), gW(W), gC(C), and gσ(σ3) are functions of the temperature, initial moisture content, compaction degree, and confining pressure, respectively. A regression analysis of the experimental results using Eq. (17) gives the result

Gd max = 276.09 × |T|0.65 × W 1.08 × C5.02 × Pa ×

3

Pa

0.32

(18)

where Pa is the reference stress (usually equal to the atmospheric pressure, 101.3 kPa), the units of T, W, and C are degrees centigrade (°C), percentage (%), and percentage (%), respectively, and σ3 and Gdmax are expressed in kPa. The data are shown in Table 2. 8

Cold Regions Science and Technology 170 (2020) 102938

F. Zhao, et al.

1400

1000 Maximum dynamic shear modulus Gdmax

800

1000 600

G /MPa

Predicted results/MPa

1200

800

Experimental results

400 QT4 QT6 QT9 QT12

600 200

400 200 200

400

600

800

1000

1200

0 -5 10

1400

Fitting curve of QT4,Eq.(17) Fitting curve of QT6,Eq.(17) Fitting curve of QT9,Eq.(17) Fitting curve of QT12,Eq.(17) -4

10

-3

-2

10

γd

Experimental results/MPa

10

Fig. 18. Experimental and predicted values of maximum dynamic shear modulus.

Fig. 20. Experimental and predicted values of dynamic shear modulus versus dynamic shear strain.

Fig. 18 shows a comparison between the experimental and predicted values of the maximum dynamic shear modulus calculated by Eq. (18). It can be seen from these results that the predicted and experimental values have a well linear relationship, indicating that the maximum dynamic shear modulus presentation can be estimated and calculated by Eq. (18). Applying the maximum dynamic shear modulus to normalize the dynamic shear modulus, Fig. 19 shows the normalized results of the dynamic shear modulus versus dynamic strain curves. All the experimental results are plotted in the figure, showing that different curves have similar variation tendencies. The regression analysis of the data indicates that

5.2. Empirical presentation of damping ratio

G Gd max

=

Considering four experimental conditions, the maximum damping ratio of frozen silt can be expressed as max

d/

max

where G/Gdmax is the normalized dynamic shear modulus ratio, γd is the — dynamic shear strain, and dr is the normalized parameter equal to 0.0092. Substituting Eq. (18) into Eq. (19), the dynamic shear modulus prediction of the frozen silt can be obtained as follows:

G=

276.09 × |T|0.65 × W 1.08 × C5.02 × Pa × 1 + d/ dr

3

0.32

(20)

Pa

Fig. 20 shows the relationship between the prediction results and experimental results QT4, QT6, QT9, and QT12. The relationship for the predicted dynamic shear modulus versus the dynamic shear strain is in good agreement with the experimental results. It is thus concluded that Eq. (20) can be used to estimate the dynamic shear modulus of frozen silt under various conditions of temperature, initial moisture content, compaction degree, and confining pressure.

= 0.243 × |T|

= max

1+

0.0 -5 10

×C

3.32

×

3

(22)

0.21

(23)

d / dr

Maximum damping ratio

max

0.4

Predicted results

G /Gdmax

0.2

1.26

0.5

1.0

0.4

×W

where λ/λmax is the normalized damping ratio and n is the normalized parameter, equal to 0.26. Analogously, substituting Eq. (22) into Eq. (23) can produce the damping ratio prediction of frozen silt,

Fitting curve,Eq.(16)

0.6

0.35

n

d / dr

1.2

0.8

(21)

× hT (T ) × hW (W ) × hC (C ) × h ( 3)

where the units of T, W, C, and σ3 are degrees centigrade (°C), percentage (%), percentage (%), and megapascals (MPa), respectively. Fig. 21 shows the connections between the predicted and the experimental results of the maximum damping ratio, which have a well linear relationship, as illustrated by the fact that the maximum damping ratio presentation can be estimated by Eq. (22). The maximum damping ratio regression analysis was performed by the transformation of Eq. (12), as proposed by Hardin and Drnevich et al. (1972a, b). The maximum damping ratio regression analysis was performed in Fig. 22 to obtain the damping ratio normalization formula

(19)

dr

0

where λ0 is the simulated parameter, equal to 0.243, and hT(T), hW(W), hC(C), and hσ(σ3) are functions of the temperature, initial moisture content, compaction degree, and confining pressure, respectively. A regression analysis of the experimental results using Eq. (21) gives the result

1 1+

=

Experimental results QT1 QT3 QT5 QT7 QT9 QT11 QT13

10

QT2 QT4 QT6 QT8 QT10 QT12

-4

0.3

0.2

0.1 -3

10 γd

10

-2

10

-1

0.1

0.2

0.3

0.4

0.5

Experimental results

Fig. 19. Normalized dynamic shear modulus and dynamic shear strain.

Fig. 21. Experimental and predicted values of maximum damping ratio. 9

Cold Regions Science and Technology 170 (2020) 102938

F. Zhao, et al.

during the stage of large dynamic shear strain (10−3). The influence of the four test conditions on the fluctuation of Gdmax mainly considered the freezing temperature and moisture content, and secondly considered the compaction degree and confining pressure. 3) Damping ratio-shear strain curves increased and then gradually flattened. Furthermore, the growth rate was evident in the dynamic shear strain range from 10−3 to 10−2. The damping ratio increased with decreasing moisture content, compaction degree, and confining pressure, and increased with increasing freezing temperature. The influence of the four test conditions on the fluctuation of λmax considered obvious distinction. The values of λmax are SA = 0.124, SB = 0.086, SC = 0.031, SD = 0.024, which are in descending order of freezing temperature condition > moisture content condition > compaction degree condition > confining pressure condition. The values of n fluctuation have same tendency as the λmax order effect. 4) Considering the four factors in the test, two empirical expressions for the dynamic shear modulus and the damping ratio comply with shear strain are established. After verification of the experimental data, the dynamic shear modulus and damping ratio of the frozen silt are predicted by these two empirical models when subjected to multistage cyclic loading.

1.4 Experimental results

1.2 QT1 QT3 QT5 QT7 QT9 QT11 QT13

1.0 0.8 0.6

QT2 QT4 QT6 QT8 QT10 QT12

0.4 0.2 Fitting curve,Eq.(20)

0.0 -5 10

10

-4

10

-3

-2

10

10

-1

Fig. 22. Normalized damping ratio and dynamic shear strain.

0.5 Experimental results

0.4

Fitting curve of QT4,Eq.(21) Fitting curve of QT6,Eq.(21) Fitting curve of QT9,Eq.(21) Fitting curve of QT12,Eq.(21)

QT4 QT6 QT9 QT12

Declaration of Competing Interest

λ

0.3

None.

0.2

Acknowledgments 0.1

0.0 -5 10

10

-4

10 γd

-3

10

-2

This research is supported by the Science and Technology Department Project of Qinghai Province (Grant No. 2017-ZJ-791) and the Qinghai Province Innovation Service Platform Construction Special (Grant No. 2018-ZJ-T01).

-1

10

Fig. 23. Experimental and predicted values of damping ratio versus dynamic shear strain.

= 0.243

Czajkowski, R.L., Vinson, T.S., 1980. Dynamic properties of frozen silt under cyclic loading. J. Geotech. Eng. Div. 106 (9), 963–980. Hardin, B.O., Drnevich, V.P., 1972a. Shear modulus and damping in soils: designing equations and curves. J. Soil Mech. Found. Div. 98 (sm7), 667–692. Hardin, B.O., Drnevich, V.P., 1972b. Shear modulus and damping in soils: measurements and parameter effect. J. Soil Mech. Found. Div. 98 (6), 603–624. Hazirbaba, K., Zhang, Y., Hulsey, J.L., 2011. Evaluation of temperature and freeze-thaw effects on pore pressure generation of fine-grain soils. Soil Dyn. Earthq. Eng. 31, 372–384. Jiao, G.D., Zhao, S.P., Ma, W., 2010. Experimental study of the dynamic characteristics of warm-frozen silt after freeze-thaw cycles under cyclic loading. Chin. Civ. Eng. J. 43 (12), 107–113 (in Chinese). Kallioglou, P., Tika, T.H., Pitilakis, K., 2008. Shear modulus and damping ratio of cohensive soils. J. Earthq. Eng. 12, 879–913. Li, J.C., Baladi, G.Y., Andersland, O.B., 1979. Cyclic triaxial tests on frozen Sand. Eng. Geol. 13 (1–4), 233–246. Li, S., Gao, L., Chai, S., 2012a. Significance and interaction of factors on mechanical properties of frozen soil. Rock Soil Mech. 33 (4), 1173–1177 (in Chinese). Li, S.Y., Lai, Y.M., Zhang, S.J., Yang, Y.G., Yu, W.B., 2012b. Dynamic response of QinghaiTibet railway embankment subjected to train loading in different seasons. Soil Dyn. Earthq. Eng. 32, 1–14. Lin, B., Zhang, F., Feng, D.C., Tang, K.W., Xin Feng, X., 2018. Dynamic shear modulus and damping ratio of thawed saturated clay under long-term cyclic loading. Cold Reg. Sci. Technol. 145 (1), 93–105. Ling, X.Z., Zhu, Z.Y., Zhang, F., et al., 2009. Dynamic elastic modulus for frozen soil from the embankment on Beiluhe Basin along the Qinghai-Tibet Railway. Cold Reg. Sci. Technol. 57 (1), 7–12. Ling, X.Z., Chen, S.J., Zhu, Z.Y., Zhang, F., Wang, L.N., Zou, Z.Y., 2010. Field monitoring on the train-induced vibration response of track structure in the Beiluhe permafrost region along Qinghai-Tibet railway in China. Cold Reg. Sci. Technol. 60 (1), 75–83. Ling, X.Z., Wang, Z.Y., Zhnag, F., et al., 2013. Experimental investigation on dynamic shear modulus of frozen clay from subgrade of Beijing-Harbin Railway. Chin. J. Geotech. Eng. 35 (S2), 38–43 (in Chinese). Ling, X.Z., Zhang, F., Li, Q.L., An, L.S., Wang, J.H., 2015. Dynamic shear modulus and damping ratio of frozen compacted sand subjected to freeze thaw cycle under multistage cyclic loading. Soil Dyn. Earthq. Eng. 76, 111–121. Liu, J.K., Chang, D., Yu, Q.M., 2016. Influence of freeze-thaw cycles on mechanical properties of a silty sand. Eng. Geol. 210 (8), 23–32. Liu, G.Y., Xie, C.W., Yang, S.H., 2018. Spatial and temporal variation characteristics on the onset dates of freezing and thawing of active layer and its influence factors in

0.26

d/

1+

References

dr d / dr

× |T |

0.35

×W

1.26

×C

3.32

×

3

0.21

(24)

Fig. 23 shows the relationship between the prediction results and experimental results QT4, QT6, QT9, and QT12. The relationship of the predicted damping ratio and dynamic shear strain was in good agreement with the experimental results. It was thus concluded that Eq. (24) could be used to estimate the damping ratio of frozen silt under various conditions of temperature, initial moisture content, compaction degree, and confining pressure. 6. Conclusions In this study, a series of cryogenic cyclic dynamic triaxial tests under a closed system of frozen silt among traffic cyclic loading was carried out. The backbone curves, dynamic shear modulus, damping ratio, dynamic shear stress, and strain of frozen silt under different experimental conditions were analyzed. The following conclusions are obtained within the scope of the experimental conditions. 1) The dynamic shear stress versus dynamic shear strain curves of the Qinghai-Tibet frozen silt conformed to the Hardin hyperbola model. With the increase of initial moisture content, compaction degree, and confining pressure, as well as the decrease of temperature, the development trend of the backbone curve was steeper. The unfrozen moisture content controlled the quantity change of ice crystals inside the frozen silt, affecting the development trend of the backbone curve. 2) The dynamic shear modulus curves tended to be stationary during the stage of small dynamic shear strain (10−4) and declined sharply 10

Cold Regions Science and Technology 170 (2020) 102938

F. Zhao, et al. permafrost regions along the Qinghai-Tibet Highway. J. Glaciol. Geocryol. 40 (6), 1067–1078 (in Chinese). Luo, F., He, Y.T., Zhao, S.P., et al., 2015. Experimental study of damping ratio of frozen soil under stepwise loading. Rock Soil Mech. 36 (11), 3143–3149 (in Chinese). Muge, E.O., Liu, J.K., Niu, F.J., 2017. Dynamic behavior of fiber-reinforced soil under freeze-thaw cycles. Soil Dyn. Earthq. Eng. 101, 269–284. Naeini, S.A., Gholampoor, N., 2014. Cyclic behaviour of dry silty sand reinforced with a geotextile. Geotext. Geomembr. 42 (6), 611–619. Nakano, Y., Amold, R., 1973. Acoustic properties of frozen Ottawa sand. Water Resour. Res. 9 (1), 178–184. Seed, H.B., Wong, R.T., Idriss, I.M., Tokimatsu, K., 1984. Moduli and Damping Factors for Dynamic Analyses of Cohesionless Soils. Earthquake Engineering Research Center, University of California, pp. 1–37 Berkeley Report No. Shen, Z.Y., Zhang, J.Y., 1998. Loading effect of dynamic strength and limit of long-term dynamic strength of frozen silt. J. Glaciol. Geocryol. 20 (1), 42–45 (in Chinese). Subramaniam, P., Banerjee, S., 2013. Shear modulus degradation model for cohesive soils. Soil Dyn. Earthq. Eng. 53, 210–216. Vinson, T.S., 1978. Geotechnical Engineering in Cold Regions. Mc Graw-Hill Book Company, New York, pp. 405–458. Wang, L.X., Ling, X.Z., Xu, X.Y., et al., 2005. Comparison of dynamic and static triaxial test on frozen silty clay of Qinghai-Tibet railway. Chin. J. Geotech. Eng. 27 (2), 202–205 (in Chinese). Wu, Z.J., Chen, T., Zhao, T., et al., 2018. Dynamic response analysis of railway embankments under train loads in permafrost regions of the Qinghai-Tibet Plateau. Soil Dyn. Earthq. Eng. 112 (1–7). Wu, Z.J., Ma, W., Wang, L.M., et al., 2003. Testing study on the effect of temperature and

confining pressure on strength of frozen soil under seismic dynamic loading. J. Glaciol. Geocryol. 25 (6), 648–652 (in Chinese). Xu, X.Y., Zhong, C.L., Chen, Y.M., et al., 1998. The dynamic characteristic of frozen soil and determination of parameters. Chin. J. Geotech. Eng. 20 (5), 77–81 (in Chinese). Yu, X.B., Liu, H.B., Sun, R., Yuan, X.M., 2018. Improved Hardin-Drnevich model for the dynamic modulus and damping ratio of frozen soil. Cold Reg. Sci. Technol. 153 (9), 64–77. Zhang, D., Li, Q.M., Liu, E.L., Liu, X.Y., Zhang, G., Song, B.T., 2019. Dynamic properties of frozen silty soils with different coarse-grained contents subjected to cyclic triaxial loading. Cold Reg. Sci. Technol. 157 (1), 64–85. Zhao, S.P., Zhu, Y.L., He, P., et al., 2003. Testing study on dynamic mechanics parameters of frozen soil. Chin. J. Rock Mech. Eng. 22 (2), 2677–2681 (in Chinese). Zhao, S.P., He, P., Zhu, Y.L., 2006. Comparison between dynamic and static creep characteristics of frozen silt. Chin. J. Geotech. Eng. 28 (12), 2160–2163 (in Chinese). Zhou, Y.W., Guo, D.X., Cheng, G.D., 2000. Frozen Soil of China. Science Press, Beijing. Zhu, Z.Y., Ling, X.Z., Hu, Q.L., et al., 2009. Experimental study of vibration excited subsidence of frozen soil under long-term dynamic loads. Rock Soil Mech. 30 (4), 955–959 (in Chinese). Zhu, Z.Y., Ling, X.Z., Chen, S.J., Zhang, F., Wang, L.N., Wang, Z.Y., Zou, Z.Y., 2010. Experimental investigation on the train-induced subsidence prediction model of Beiluhe permafrost subgrade along the Qinghai-Tibet Railway in China. Cold Reg. Sci. Technol. 62 (1), 67–75. Zhu, Z.Y., Ling, X.Z., Wang, Z.Y., Lu, Q.R., Chen, S.J., Zou, Z.Y., et al., 2011. Experimental investigation of the dynamic behavior of frozen clay from the Beiluhe subgrade along QRT. Cold Reg. Sci. Technol. 69, 91–97.

11