Dynamic shear modulus and damping ratio of thawed saturated clay under long-term cyclic loading

Cold Regions Science and Technology 145 (2018) 93–105

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

Dynamic shear modulus and damping ratio of thawed saturated clay under long-term cyclic loading

MARK

Bo Lina, Feng Zhanga,⁎, Decheng Fenga, Kangwei Tanga, Xin Fengb a b

School of Transportation Science and Engineering, Harbin Institute of Technology, 150090 Harbin, China Liaoning Communications Research Institute Co., Ltd, 110015 Shenyang, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Freeze-thaw cycle Saturated clay Dynamic shear modulus Damping ratio Dynamic tri-axial test Empirical model Seasonally frozen region

Dynamic properties of subgrade soil under the long-term traffic loading are crucial for designing the subgrade structures and evaluating the long-term performance of traffic infrastructures in seasonally frozen regions. In this study, the dynamic shear modulus and damping ratio were employed to evaluate the long-term dynamic behaviors of thawed saturated clay by conducting a series of cyclic tri-axial tests. Effects of freeze-thaw cycles, dynamic stress amplitude, confining pressure and multi stage cyclic loading on the evolution rules of dynamic shear modulus and damping ratio versus shear strain were analyzed. The results indicate that both dynamic shear modulus and damping ratio decrease with the increasing shear strain during the long-term cyclic loading. Repeated freeze-thaw cycles have tremendous effects on decreasing the dynamic shear modulus and increasing the damping ratio. Increasing dynamic stress amplitude has a decreasing effect on dynamic shear modulus, and an increasing effect on damping ratio. Increasing confining pressure has no obvious effect on the evolution of damping ratio, but has an increasing effect on increasing the dynamic shear modulus before the shear strain reaches a certain level. Multi stage cyclic loading can be an alternative method to determine the evolution of dynamic shear modulus during the long-term cyclic loading, but it is not applicable for the damping ratio. Finally, the Martin-Davidenkov model and a hyperbolic model which can be used for predicting the evolution of dynamic shear modulus and damping ratio, respectively, were proposed and validated. The results show that both the two models can give satisfactory prediction. The results achieved in this study contribute to a better understanding of the long-term dynamic behavior for cohesive subgrade soils in seasonally frozen region.

1. Introduction Long-term dynamic behaviors of subgrade soil in seasonally frozen regions are more complicated than that of subgrade soil in non-frozen regions, because the mechanical properties of the subgrade soil could be changed by the effects of repeated freeze-thaw cycles (Cui et al., 2014; Qi et al., 2006; Wang et al., 2007, 2015a). Moisture content has been proved to be one of the main influencing factors that determine the effects of freeze-thaw cycles on the properties of soils (Feng et al., 2017; Wang et al., 2015b; Zhao et al., 2009; Zheng et al., 2015). The moisture within the soil structures would migrate and accumulate due to the effects of freeze-thaw cycles (Harlan, 1973; Kane and Stein, 1983; Othman and Benson, 1993; Perfect and Williams, 1980; Sheng et al., 2014; Zhang and Michalowski, 2015). These mean that the subgrades, commonly built at the optimum moisture contents, typically stay in an unsaturated condition, have a high risk of turning into saturated state during the spring thawing period due to the water and moisture accumulation occurred in winter freezing period. Especially in the clay-rich ⁎

Corresponding author. E-mail address: [email protected] (F. Zhang).

http://dx.doi.org/10.1016/j.coldregions.2017.10.003 Received 18 April 2017; Received in revised form 7 August 2017; Accepted 7 October 2017 Available online 08 October 2017 0165-232X/ © 2017 Elsevier B.V. All rights reserved.

seasonally frozen area, many constructed or being constructed subgrades are suffering from special disasters involving freeze-thaw effects, such as frost heave, thawing settlement, frost boiling and uneven settlement, because the clays are more sensitive to the impact of repeated freeze-thaw cycles compared with the coarse grained soils (Hansson and Lundin, 2006; Özgan et al., 2015). These special disasters not only affect the stability of subgrades, but also destroy the pavement structures and threaten the traffic safety. Thus, it is urgent to investigate the evolution of dynamic properties for thawed saturated clay under longterm cyclic loading. Dynamic shear modulus and damping ratio are the two main parameters that contribute to variation in the stiffness and energy dissipation during the cyclic loading of soils, which are being used for defining the dynamic properties of soil induced by earthquake, wave and traffic cyclic loading. In the past decades, many advances have been made in investigating the evolution of dynamic shear modulus and damping ratio for soils (Brennan et al., 2005; Ishibashi and Zhang, 1993; Rollins et al., 1998; Seed et al., 1986; Wang et al., 2012), these

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100

Percent finer/%

investigation were mostly conducted on unfrozen soils. It appears that few contributions could be found in literature that the soils in cold regions were focused on. Compared with thawed soils in cold region engineering, some studies have been done to investigate the dynamic shear modulus and damping ratio of frozen soils. Stevens (1975) studied the stiffness and damping properties of frozen soils subjected to vibratory loads and to define the significant factors affecting these parameters by using of one dimensional wave test. Czajkowski and Vinson (1980) found that the dynamic Young's modulus of frozen silt decreases with increasing axial strain amplitude and temperature, increases with increasing frequency, and is unaffected by confining pressure. Al-Hunaidi et al. (1996) used the resonant-column test to determine the dynamic shear modulus and damping ratio of naturally frozen soil. Ling et al. (2013, 2015) found that, for frozen clay the stiffness increases and damping ratio decreases with the increasing number of repeated loading cycles; For the frozen sand, the dynamic shear modulus increases with increasing initial water content, temperature, loading frequency and confining pressure. The damping ratio increases with increasing initial water content, while decreases with increasing temperature and loading frequency. In general, these findings mentioned above can be helpful in understanding the evolution of dynamic shear modulus and damping ratio for soils subjected to freezing conditions. But for the soils which experienced thawing conditions, only few contributions could be found in recent years. Zhang and Hulsey (2014) found that the freeze-thaw cycles resulted in an increase in both the dynamic shear modulus and the damping ratio of the Marble Creek silt. Wang et al. (2014) studied the changes in dynamic modulus and damping ratio of compacted subgrade soil after been subjected to different freeze-thaw cycles, the results showed that the dynamic modulus decreases with the increase of freezethaw cycles, but the relationship between damping ratio and freezethaw cycles was not clear. Tang et al. (2014) proved that the freezethaw action can actually decrease the dynamic elastic modulus of mucky clay, and the dynamic elastic modulus decreases remarkably with the increasing of cyclic stress amplitude, while the accumulated plastic strain behaves adversely. Wang et al. (2015a, 2015b) observed that the dynamic modulus of silty sand sharply decreases while the damping ratio increases with incremental increase in freeze-thaw cycles, the changes level off after six freeze-thaw cycles. It could be noticed that almost all the evolutions of dynamic shear modulus and damping ratio in these studies were just employed to evaluate the effects of freeze-thaw cycles. The experimental data usually obtained by means of a series of multi stage cyclic loading, that is, dynamic stress was imposed by steps and each step corresponding to a stress amplitude and lasted for several cycles. However, little research has been directed towards understanding the evolution of dynamic properties for thawed saturated clay. In this study, the evolution of dynamic shear modulus and damping ratio for thawed saturated clay under long-term cyclic loading were investigated firstly. Secondly, the relationships between the dynamic shear modulus G and the cyclic shear strain γ, which are typically expressed by the curves of G/Gmax versus γ, together with the relationships between the damping ratio λ and shear strain γ were employed to analyze the dynamic characteristics of thawed saturated clay under long-term repeated cyclic loading. Thirdly, the influencing factors such as dynamic stress amplitude and confining pressure were discussed. Effects of multi stage cyclic loading on the dynamic shear modulus and damping ratio were also compared with that of the long-term loading. Finally, empirical models for dynamic shear modulus and damping ratio were proposed and validated, respectively.

80 60 40 20 0 0 10

10-1

10-2

10-3

Grain size/mm Fig. 1. Distribution curves of particle size.

used to fill the subgrade in northeast of China. The grain size distribution is showed in Fig. 1. According to guidance of the Test Methods of Soils for Highway Engineering (JTG E40-2007) issued by the Ministry of Transport, China. The optimal water content and maximum dry density were obtained by the heavy compaction apparatus, together with the other physical properties of the soil, namely specific gravity, liquid limit, plastic limit are shown in Table 1. The soil is named as low liquid limit clay and classified as CL according to the classification method of the Test Methods of Soils for Highway Engineering (JTG E402007).

2.2. Preparation of specimen 2.2.1. Specimens preparation All the procedures on preparing the thawed saturated specimen were performed in a laboratory environment. The air dried clay powder was sieved firstly from a sieve with pores of 2 mm and was mixed with distilled water to achieve the optimal water content of 17.4%. Then the soil was stored in a closed container for 12 h to make sure the uniform distribution of the water. Specimens were obtained by compacting the wet clay powder in a steel cylinder though equal five layers. All the specimens have the sizes of 100 mm in diameter, 200 mm in height, shown in Fig. 2(a), and the compaction degrees of the specimens are controlled at 94%. During the compaction procedure in the steel cylinder, the layer height was controlled to ensure the compactness degrees among the five layers are uniform. Saturating process on the specimens was conducted by the suggestion of Standard for Soil Test Method (GB/T 50123-1999(2008)). The specimens were fixed with the customized saturators first, shown in Fig. 2(b), and then were placed in a seal barrel for vacuuming, the deaired water was supplied for the barrel to ensure the specimens were saturated in the water, shown in Fig. 2(c). The vacuuming and saturating procedures were lasted for 2 h and 12 h respectively. Saturation degrees of the specimens were also checked after the saturation procedures to make sure all the saturation degrees of the specimens reach more than 95%. The eligible specimens were sealed with plastic wrap and adhesive tape to prevent water evaporation, shown in Fig. 2(d). After 7 freeze-thaw cycles, 12 specimens were selected randomly to check the changes in volume and mass. Results showed that changes in volumes and masses of 12 specimens range from 0.32% to 2.92%, and from 0.11% to 0.44%, respectively. Table 1 Basic physical properties of soil specimens.

2. Experimental procedures

Specific gravity

Liquid limit/%

Plastic limit/%

Plastic index

Maximum dry density/g/cm3

Optimal water content/%

2.75

36.98

25.19

11.79

1.74

17.4

2.1. Properties of soil The soil used in this study was obtained from Harbin, it is widely 94

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Fig. 2. Specimens preparation and the UTM100 test system. (a) Specimens prepared for saturation; (b) specimens were fixed in customized saturators; (c) the saturating process; (d) specimens prepared for freeze-thaw cycles; (e) the freezing process in a low temperature freezer; (f) the thawing process in a water bath; (g) the tri-axial pressure chamber.

performed. The loading form for multi state loading is showed in Fig. 3(b). Before the cyclic loading, a tiny axial static load and confining pressure were applied against the specimen for 10 min for the purpose of obtaining a stable situation. All specimens were tested under an unconsolidated undrained condition. Dynamic shear modulus and damping ratio of soil could be influenced by 13 kinds of parameters (Hardin and Drnevich, 1972). In this study, axial stress amplitude, confining pressure, cyclic loading number and multi-stage loading were selected as the main parameters to evaluate the dynamic behaviors for thawed saturated clay. The test program is showed in Table 2. Meanwhile, a group of dynamic tests were carried out to determine the effects of freeze-thaw cycles on the dynamic behaviors of saturated clay subjected to 0, 1, 3, 5, 7 freeze-thaw cycles, respectively.

These changes could be negligible compared with the magnitude of the initial volumes and masses of the specimens. So the specimens were considered to be saturated after freeze-thaw cycles. 2.2.2. Freeze-thaw cycles Due to the physical and mechanical properties of fine-grained soils tend towards stability after been subjected to 6–8 cycles of freeze-thaw (Chang et al., 2014; Wang et al., 2007; Yu et al., 2010), the number of the freeze-thaw cycles in this study were confirmed as 7. Freezing temperature, consideration of the extreme low temperature in cold seasons of Harbin, was confirmed as − 25 °C (Kong et al., 2014; Yao et al., 2009). Thawing temperature was confirmed as 25 °C. A pre-experiment was carried out to determine the freeze-thaw conditions. The results showed that the temperature inner one of 12 specimens can reach at − 25 °C from 25 °C in 14 h during the freezing process in a low temperature freezer. The thawing process in a water bath at 25 °C lasted for 10 h. The same freezing and thawing processes were performed in this study, that is, the freezing process lasts for 14 h at a temperature of − 25 °C, the thawing process lasts for 10 h at a temperature of 25 °C. Fig. 2(e) (f) show the freezing process and thawing process, respectively.

3. Determination of dynamic shear modulus and damping ratio The typical relationship between the cyclic axial stress σ and cyclic axial strain ε is showed in Fig. 4. The loading curve and the unloading curve cannot combine a closed cycle because the residual strain developed. The ellipse that enclosed by the unloading curve and reloading curve usually is defined as the hysteresis loop (Finno and ZapataMedina, 2013; Ling et al., 2013). The loop enclosed by abcd showed in Fig. 4, is the first hysteresis loop, slope of the line which joins the point a and point c is the secant elastic modulus. The secant elastic modulus is defined as the elastic modulus Ee, 1 in this study. Similarly, the secant elastic modulus Ee, n represents the elastic modulus during the Nth cycle, and Ee, 0 is the initial elastic modulus. The secant elastic modulus Ee, n can be calculated as follows,

2.3. Apparatus and test procedures The Universal Testing Machine (i.e. UTM-100) produced by IPC Global Company was employed to conduct the dynamic tri-axial tests in this study, as shown in Fig. 2. The maximum axial load is 100 kN and maximum axial displacement is 100 mm; The maximum frequency of axial dynamic load is 70 Hz; The maximum working pressure of the pressure chamber is 400 kPa when filled with air. In order to measure the pore water pressures of saturated specimen during the test process, the pressure acquisition system which is composed of a high accuracy sensor and a data collector was adopted, shown as Fig. 2. Full scale of the pore water pressure sensor is 150 kPa and the accuracy is 0.15 kPa. A series of long-term cyclic loading tests were conducted by adopting a haversine wave loading. The vibration frequency of the load was set at 1 Hz, because most the vibration frequencies tested in situ are less than 1 Hz. The loading form is showed in Fig. 3(a) For the purpose of investigating the deformation behaviors of thawed saturated clay in multi stage loading, a group of four stages cyclic loading tests were

Ee, n =

qd,max − qd,min ε d,max − ε d,min

(1)

where C (qd, max, εd, max) and D (qd, min, εd, min) are the coordinate values of the top cusp and bottom cusp of an enclosed hysteresis loop, respectively, showed in Fig. 4. When the Nth cyclic unloading finishes, the corresponding residual axial strain εd, min is defined as the accumulated axial strain, which is affected by the load level and initial state of specimens. The area enclosed by hysteresis loop reflects energy consumption during the loading and unloading processes, which is used to calculate 95

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(b)

(a)

1s …… max

Fig. 3. Scheme of the cyclic loading: (a) for DHC-01 ~ 22, (b) for the DHC-23 ~ 25.

max,4

1s …… max,3

1s ……

d

max,2

1s ……

max,1

1s 1s 1s ……

min

min

3

3

Number of cyclic loading /N

Number of cyclic loading /N the damping ratio λ. The damping ratio λ is calculated as follows,

Table 2 Summary of the test program. Specimen number

Dynamic stress amplitude/kPa

DHC-01 ~ 05

50, 57, 70, 89, 114 50, 57, 70, 89, 114 50, 70, 89, 114 50, 70, 89, 114 70 50 → 70 → 89 → 114

DHC-06 ~ 10 DHC-11 ~ 14 DHC-15 ~ 18 DHC-19 ~ 22 DHC-23 ~ 25

Number of freeze-thaw cycles

Number of cyclic loading/ N

60

7

10,000

90

7

10,000

120 150 60 60, 90, 120

7 7 0, 1, 3, 5 7

10,000 10,000 10,000 Each stage for 2500 cycles

Confining pressure/kPa

λ=

Aloop π⋅Atriangle

where Aloop = area of the enclosed hysteresis loop, which reflects the energy dissipated during one cyclic loading; Atriangle = area of the triangle CDE, as shown in Fig. 4, which represents the peak energy during one cyclic loading. Considering that most studies on dynamic behavior adopt the dynamic shear modulus G and shear strain γ to evaluate the dynamic properties of materials. The dynamic shear modulus G and shear strain γ can be obtained based on elastic theory:

Ge, n =

1

Ee,0

1

Ee,1

1

Cyclic axial stress

C(

a

b

γ = (1 + υ)(εd,max − εd,min )

Ee,n

……

D(

E(

,q ) d,min d,min

d,max

4.1. Characteristics of pore water pressures

,qd,min)

Due to the low permeability of clay, the evolution of the pore water pressure under dynamic loading is complicated, as it usually not fluctuate with the cyclic loading. Evolution of pore water pressure for cohesive soils can be taken into consideration as separate components of the instantaneous and the residual excess pore pressures (Cui and Zhang, 2015). The pore water pressures versus number of cyclic loading for saturated clay subjected to 7 freeze-thaw cycles are showed in

Cyclic axial strain

Pore water pressure/kPa

Pore water pressure /kPa

20

1

10

Fig. 5. Evolution of pore water pressures versus the number of cyclic loading.

(b) 160

σd=50kPa σd=57kPa σd=70kPa σd=89kPa σd=114kPa

40

0

(4)

where υ = the Poisson's ratio, it could be 0.5 during the situation of undrained tests for saturated clay (Ishihara, 1996). It should be noted that Ge, n is the maximum shear modulus during the Nth cyclic loading, and γ is the shear strain corresponding to the maximum shear modulus in this study.

,qd,max)

d,max

Fig. 4. Schematic illustration for the elastic modulus and hysteresis loop.

60

(3)

4. Experimental results and analysis

0

80

Ee, n 2(1 + υ)

d

c

(a) 100

(2)

100

1000

10000

Number of cyclic loading(σ3=90kPa)

σd=50kPa σd=70kPa σd=89kPa σd=114kPa

120 80 40 0

1

10

100

1000

10000

Number of cyclic loading(σ3=150kPa) 96

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0 1

Layer number

Meanwhile, the damping ratio of thawed saturated clay is affected by the freeze-thaw cycles obviously. Damping ratio increases with the increase of freeze-thaw cycles, decreases with the increasing shear strain. When confining pressure is 60 kPa and dynamic stress amplitude is 70 kPa, values of the damping ratio for thawed saturated clay under long-term cyclic loading range from 0.05–0.08, 0.06–0.10, 0.065–0.12, 0.065–0.125 and 0.08–0.13, respectively. Taking average values as the evaluation standard, the damping ratio of thawed saturated clay increases appropriately 35.0%, 45.3%, 50.8% and 54.3% after been subjected to 1, 3, 5, 7 freeze-thaw cycles compared with the 0 freezethaw cycle. Effects of freeze-thaw cycles on the evolution rules of dynamic shear modulus and damping ratio can be explained by the phase change of the water or moisture during freeze-thaw cycles (Chamberlain and Gow, 1979; Li et al., 2013). During the freezing process, water or moisture started to form ice crystal and the ice crystal pushed the connected clay particles apart, which could increase the void spaces of specimens, and weaken the structures of the saturated specimens. Meanwhile, the physical changes induced by the freezing effects in the specimens could not recover completely during the thawing process. Thus, with the increase of freeze-thaw cycles, the dynamic shear modulus which reflects the strength of soils will decrease, the damping ratio which reflects the extent of energy propagation will increase.

Degree of compaction 90% 94% 98%

2 3 4 20

25

30

35

40

Water content /% Fig. 6. Water contents inner different layers of specimens at the end of tests.

Fig. 5. It could be seen that the pore water pressures under different confining pressures and dynamic stress amplitudes increase with the increase of loading cycles before appropriately 1000th cycles. After that, the increase tends to be level off, and the values of the pore water pressures almost close to the corresponding confining pressures. This phenomenon might mean the specimens have been liquefied after appropriately 1000th cycles. In fact, the specimens still not lose the strengths after been loaded for more than 1000th cycles. The main reason might be that the pore water pressures measured by the sensor only represent the bottom of the specimens. The pore water pressures inner the specimens can not distribute uniformly during the cyclic loading because the specimen have the height of 200 mm. It is possible that the whole specimens still have the dynamic strengths. For the purpose of proving that, water distributions inner the specimens were measured at the end of test. The specimens obtained from the finished tests were divided into four equal layers. From the top to the bottom of the specimens, the layers were named 1, 2, 3 and 4, respectively. As shown in Fig. 6, the water contents inner the middle of the specimens (layer 2) under 3 kinds of compaction degrees are lower than that of top and bottom layers. This phenomenon is meaningful in explaining why the pore water pressure measured on the bottom of the specimens cannot represent the whole pore water state of the specimens. Differences in water content inner different layers of specimen might be deprived from the processes of saturation, freeze-thaw cycles or long term cyclic loading, but what is certain is that the effective stress theory is not suitable for analyzing the mechanical behavior for the thawed saturated clay in this study. The following discussions related to the cyclic stress all refer to the total stress. At the same time, pore water pressures under confining pressure of 90 kPa are obvious than that of 150 kPa might be due to the uncertain distribution in water contents inner the specimens.

4.3. Effect of dynamic stress amplitude 4.3.1. Dynamic shear modulus Fig. 8 shows the evolution of the dynamic shear modulus under different dynamic stress amplitudes and confining pressures. It can be observed that the dynamic shear modulus at four kinds of confining pressure decreases with the increase of shear strain. The curve characteristics for dynamic shear modulus at various dynamic stress amplitudes are similar, which could be divided into two stages by the turning point. The dynamic shear modulus decreases gradually with the increasing shear strain during the initial stage until the shear strain reaches a certain level, and then decreases linearly until the end of cyclic loading. Values of dynamic shear modulus at the turning point could be used to evaluate the effect of dynamic stress amplitude. As shown in Fig. 8, when the confining pressures are 60 kPa, 90 kPa and 120 kPa, the dynamic shear modulus at the turning point generally decreases with the increase of dynamic stress amplitude. When the confining pressure turns into 150 kPa, dynamic shear modulus at the dynamic stress amplitude of 50 kPa is larger than that of 70 kPa, but both are lower than the dynamic stress amplitudes are 89 kPa and 114 kPa. It seems that the decreasing trend of dynamic shear modulus versus dynamic stress amplitude disappears. Follow the contribution of (Hardin and Drnevich, 1972), the rate at which the dynamic modulus decreases with increasing shear amplitude depends primarily on the values of maximum dynamic modulus and on the shear strength of the soil. So it is supposed that the mechanical states among the four specimens in this study might be somehow vary. Even so, the increasing dynamic stress amplitude has a decreasing effect on the dynamic shear modulus is confirmable.

4.2. Effect of freeze-thaw cycles Fig. 7 shows the variations of the dynamic shear modulus and damping ratio for saturated clay after been subjected to 0, 1, 3, 5, 7 freeze-thaw cycles (FT). In general, the dynamic shear modulus decreases with the increase of freeze-thaw cycles. The relationship between dynamic shear modulus with shear strain can be divided into two stages. At the initial stage, the dynamic shear modulus decreases gradually with the increasing shear strain. At the second stage, dynamic shear modulus decreases linearly with the increasing shear strain. Taking the average values during the initial stage as the evaluation standard, the dynamic shear modulus of thawed saturated clays decreases appropriately 3.1%, 18.4%, 31.1% and 37.7%, respectively, after been subjected to 1, 3, 5, 7 freeze-thaw cycles compared with the 0 freeze-thaw cycle.

4.3.2. Damping ratio It could be observed from the turning point in Fig. 9 that evolution of the damping ratio generally increases with the increasing dynamic stress amplitude at four kinds of confining pressure. For the thawed saturated clay in this study, effect of dynamic stress amplitude on the evolution of damping ratio agrees with the results of saturated cohesive soils investigated by Hardin and Drnevich (1972) and soft clay by Idriss et al. (1978). The long-term evolution for damping ratio during the cyclic loading should be paid more attention in this study. From Fig. 9, it could also be observed that damping ratio decreases with the increasing shear strain during the long-term cyclic loading. This result is consist with the 97

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14

(b) 0.14

FT=0 FT=1 FT=3 FT=5 FT=7

12 10 8 6

0.3

0.10 0.08 0.06

σ3=60kPa, σd=70kPa

0.2

σ3=60kPa, σd=70kPa

0.4

0.5

0.6 0.7

0.04 0.2

0.3

13 12 11

(b)

σd=50kPa σd=57kPa σd=70kPa σd=89kPa σd=114kPa

10 9 8 7 6 0.2

0.4

Dynamic shear modulus/MPa

Dynamic shear modulus/MPa

Turning point

11 10 9 8 7 6 0.2

10 9 σd=50kPa σd=70kPa σd=89kPa σd=114kPa

0.2

0.4

0.6

0.8

Shear strain/% (σ3=120kPa)

Dynamic shear modulus/MPa

Dynamic shear modulus/MPa

(d)

11

6

Fig. 8. Relations between dynamic shear modulus versus shear strain.

σd=50kPa σd=57kPa σd=70kPa σd=89kPa σd=114kPa

0.4

0.6

0.8

1

Shear strain/% (σ3=90kPa)

Turning point

7

4.4.1. Dynamic shear modulus Fig. 11 shows the effect of confining pressure on the evolution of dynamic shear modulus versus shear strain. It can be observed that the relationship between dynamic shear modulus with shear strain could be studied by means of the turning point mentioned above. At the initial stage, the dynamic shear modulus increases with the increasing confining pressure. This evolution characteristic is consistent with the result of Delfosse-Ribay et al. (2004). At the second stage, the dynamic shear modulus decrease linearly with the increasing shear strain, the slopes of the linear decreasing trend among different dynamic stress amplitudes almost coincide with each other. It can be deduced that the effect of confining pressure on the dynamic shear modulus will disappear when the shear strain reach a certain level.

Turning point

0.6 0.8 1

12

8

0.6 0.7

4.4. Effect of confining pressure

12

Shear strain/% (σ3=60kPa) (c)

0.5

decreases with the increasing number of cyclic loading is acceptable.

research of Hardin and Drnevich (1972), who found that the damping ratio for cohesive soil decreases appropriately with the logarithm of the number of cycles of loading when the loading cycles less than 50,000. Evolution characteristics of the hysteresis loops during the long-term cyclic loading can be the evidence for explaining the result that damping ratio decrease with the increasing shear strain, as shown in Fig. 10(a). The hysteresis loops at different number of cyclic loading are moved together for a better understanding on the evolution rule, as shown in Fig. 10(b). It could be observed that the area of the hysteresis loops of different loading cycles almost are uniform when the loading cycles exceed 100. Meanwhile, the hysteresis loops are falling towards the axis of cyclic axial strain when the cyclic strain increases. This evolution characteristics for hysteresis loops means that the damping ratio indeed decrease with the increasing of cyclic loadings based on the calculated formula showed in the Eq. (2). From the perspective of specimens at the end of tests, it was found that almost all the specimen heights were compressed into lower ones. The compressed specimens are benefit for propagating the vibration energy, so the damping ratio

14

0.4

Shear strain/%

Shear strain/%

(a)

Fig. 7. Evolutions of dynamic shear modulus and damping ratio after different freeze-thaw cycles.

FT=0 FT=1 FT=3 FT=5 FT=7

0.12

Damping ratio

(a)

Dynamic shear modulus/Mpa

B. Lin et al.

16 Turning point

14 12 10 8 6

σd=50kPa σd=70kPa σd=89kPa σd=114kPa

0.2

1

0.4

0.6 0.8 1

Shear strain/% (σ3=150kPa) 98

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(a)

0.12 0.10

σd=50kPa σd=57kPa σd=70kPa σd=89kPa σd=114kPa

0.08 0.06

0.16

Damping ratio

Damping ratio

(b)

Turning point

0.14

0.14 0.12 σd=50kPa σd=57kPa σd=70kPa σd=89kPa σd=114kPa

0.10 0.08 0.06

0.2

0.4

0.6 0.8 1

0.2

Shear strain/% (σ3=60kPa)

0.4

0.6 0.8 1

Shear strain/% (σ3=90kPa)

0.16

(c)

Fig. 9. Relations between damping ratio versus shear strain.

Turning point

(d) 0.14

Turning point

Turning point σd=50kPa σd=70kPa σd=89kPa σd=114kPa

Damping ratio

Damping ratio

0.14 0.12 0.10

σd=50kPa σd=70kPa σd=89kPa σd=114kPa

0.08 0.06

0.2

0.4

0.6

0.8

0.12 0.10 0.08 0.06

1

0.2

Shear strain/% (σ3=120kPa) (b) 80

Cyclic axial stress/kPa

Cyclic axial stress/kPa

60 40 20 σ3 = 60kPa, σd = 70kPa

1

2

3

60 40

Turning point(σ3=120kPa)

8 7 6 0.2

Turning point(σ3=90kPa) Turning point(σ3=60kPa)

σ3=60kPa σ3=90kPa σ3=120kPa σ3=150kPa

0.4

0.6

Shear strain/% (σd=70kPa)

0.8

Dynamic shear modulus/MPa

Dynamic shear modulus/MPa

Turning point(σ3=150kPa)

9

0.1

0.2

0.3

0.4

0.5

0.6

Cyclic axial strain/%

(b)

10

Fig. 10. Typical characteristics of the hysteresis loops under different cyclic loadings.

σ3 = 60kPa, σd = 70kPa

0 0.0

4

12 11

1

20

Cyclic axial strain/% (a)

0.8

N(number of cyclic loading)=1 N=100 N=500 N=1500 N=3000 N=6000 N=9000

N=1 N=100 N=500 N=1500 N=3000 N=6000 N=9000

0

0.6

Shear strain/% (σ3=150kPa)

(a) 80

0

0.4

14

Fig. 11. Effect of confining pressure on the evolutions of dynamic shear modulus.

Turning point(σ3=150kPa)

12 Turning point(σ3=120kPa)

10 8 6

Turning point(σ3=90kPa) Turning point(σ3=60kPa)

σ3=60kPa σ3=90kPa σ3=120kPa σ3=150kPa

0.2

0.4

0.6

0.8

Shear strain/% (σd=89kPa)

99

1

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(b)

Turning point σ3=60kPa σ3=90kPa σ3=120kPa σ3=150kPa

0.12

Damping ratio

Damping ratio

(a) 0.14

0.10 0.08

σ3=60kPa σ3=90kPa σ3=120kPa σ3=150kPa

0.12 0.10 0.08

0.06 0.2

0.4

0.6

0.8

0.06 0.2

1

0.4

Shear strain/% (σd=70kPa)

The usual way to determine the dynamic shear modulus and damping ratio in laboratory is by means of applying multi-stages loading on only one specimen. But for the cohesive soil, the accumulated strains induced by the former stage loading would play a great role in effecting the dynamic response during the following stage loading. This kind of test will only be meaningful if the loading history induced by the former-stage loading cause no, or negligible effects on the dynamic properties of soil during the subsequent loading. Otherwise, applying various cyclic loading on different specimens will be the most reasonable way to obtain the dynamic shear modulus and damping ratio for cohesive soil. For the purpose of investigating the effect of multi stage cyclic loading on the thawed saturated clay in this

σd=50kPa σd=70kPa

σd=89kPa

10

σd=114kPa

8 6 0.2

0.4

0.6

0.8

1

4.6. Comparison between the multi stages loading and long-term loading As mentioned above, it might be not advisable to determine the dynamic shear modulus and damping ratio by conducting multi stage loading on cohesive soil. Because the accumulated strain induced by former stage of cyclic loading would cause a negative effect on dynamic

(b) 0.12

σ3=60kPa σ3=90kPa σ3=120kPa

1

Damping ratio

12

0.8

study, the four stages cyclic loading tests were conducted, the results are showed in Fig. 13. It can be observed from Fig. 13(a) that the dynamic shear modulus totally decrease linearly with the increasing shear strain at four kinds of dynamic stress amplitudes. The declining rates of the dynamic shear modulus during the four stages cyclic loading almost coincide with each other among the three kinds of confining pressures. This evolution rule is consistent with the result discussed above. Meanwhile, when the cyclic loading turns into the next loading stage, the dynamic shear modulus quickly reach to a higher value and then linearly decrease with the increasing shear strain. The evolution of damping ratio versus shear strain under the four stages cyclic loading are showed in Fig. 13(b). Effect of confining pressures on the damping ratio during the multi stage loading are also negligible, same as the discussion in Section 4.4.2. The damping ratio decreases with the increasing shear strain in the first stage. For the subsequent three stages, the damping ratio increases first, and then decreases with the increasing shear strain. This phenomenon is different from the evolution of dynamic shear modulus when the cyclic loading changed to the subsequent stage. The reason might be that the dynamic shear modulus is a parameter that reflects the stiffness of the thawed saturated clay, the stiffness can be quickly response to the change of dynamic loading. Damping ratio reflects the extent of energy dissipation during the cyclic loading, which would have a short period of increasing when the dynamic stress amplitude change to a higher one because accumulative plastic strain occurred.

4.5. Multi stage cyclic loading

(a) 14

0.6

Shear strain/% (σd=89kPa)

4.4.2. Damping ratio Fig. 12 shows the effect of confining pressure on the evolution of damping ratio versus shear strain. It can be observed that the confining pressures have negligible effect on the damping ratio. The reason might be that the clay particles sliding and rotating are limited under a lower confining pressure (less than 150 kPa), which can reduce the dependence of damping ratio on confining pressure. This result is consistent with that of Delfosse-Ribay et al. (2004), who found that variation in confining stress has no significant effect on damping either for sand or grouted sand. When the confining pressure increases from 60 kPa to 150 kPa, the ranges of the damping ratio under different confining pressures almost coincide with each other. In this study, value of the damping ratio for thawed saturated clay under long-term cyclic loading ranges from 0.06 to 0.125 when the dynamic stress amplitude is 50 kPa, from 0.065 to 0.135 when the dynamic stress amplitude is 70 kPa, from 0.069 to 0.14 when the dynamic stress amplitude is 89 kPa, and from 0.065 to 0.16 when the dynamic stress amplitude is 114 kPa.

Dynamic shear modulus/MPa

Fig. 12. Effects of confining pressure on the evolutions of damping ratio.

Turning point

0.14

Fig. 13. Evolution of dynamic shear modulus and damping ratio during the four stages loading.

σ3=60kPa σ3=90kPa σ3=120kPa

0.10

σd=50kPa

σd=70kPa

σd=89kPa

σd=114kPa

0.08

0.06

0.2

0.4

0.6

Shear strain/%

Shear strain/% 100

0.8

1

Cold Regions Science and Technology 145 (2018) 93–105

14

(b) 0.16

Multi stage loading, σd=50kPa,70kPa,89kPa,114kPa

13

Long-term loading, σd=50kPa Long-term loading, σd=70kPa Long-term loading, σd=89kPa Long-term loading, σd=114kPa

12 11 10 9 8

Long-term loading, σd=50kPa Long-term loading, σd=70kPa Long-term loading, σd=89kPa Long-term loading, σd=114kPa

0.12 0.10 0.08

7 6

0.2

0.4

0.6

0.8

0.06

1

0.2

(b)

1.0 Tested data σd=50kPa σd=57kPa σd=70kPa σd=89kPa σd=114kPa

0.9 0.8 0.7

Proposed model σd=50kPa σd=57kPa σd=70kPa σd=89kPa σd=114kPa

0.6 0.5

0.4

0.6

0.8

1

Shear strain/% (σ3=120kPa) Normalized shear modulus G/Gmax

Normalized shear modulus G/Gmax

Shear strain/% (σ3=120kPa) (a)

Fig. 14. Comparison of dynamic shear modulus and damping ratio between the multi stages loading and long-term loading.

Multi stage loading,σd=50kPa,70kPa,89kPa,114kPa

0.14

Damping ratio

(a)

Dynamic shear modulus/MPa

B. Lin et al.

0.2 0.4 0.6 0.8 Shear strain/% (σ3=60kPa)

Fig. 15. Comparison between tested data and fitting curves for normalized dynamic shear modulus.

1.0 0.9 Tested data σd=50kPa σd=70kPa σd=89kPa σd=114kPa

0.8 0.7

Proposed model σd=50kPa σd=70kPa σd=89kPa σd=114kPa

0.6 0.5

0.2 0.4 0.6 0.8 Shear strain/% (σ3=120kPa)

Fig. 14(b). The damping ratio during the long-term loading always larger than that of four stages loading at the confining pressure of 120 kPa. This implies that loading history during the former stages seriously impact the evolutions of damping ratio during the subsequent loading. The values of damping ratio determined from multi cyclic loading would be totally lower than it supposed to be if taking the shear strain as the reference standard. In this perspective, the multi stages loading is not reasonable for determining the damping ratio for the thawed saturated clay.

Table 3 Values of fitting parameters in the Martin-Davidenkov model. Number

A

B

γ0

R2

Number

A

B

γ0

R2

DHC-02 DHC-03 DHC-04 DHC-05 DHC-06 DHC-07 DHC-08 DHC-09 DHC-10

21.09 15.70 16.91 22.02 53.45 29.27 4.27 63.11 41.78

1.15 0.83 0.89 1.30 1.48 1.21 1.14 1.08 1.08

0.08 0.08 0.13 0.28 0.25 0.10 0.23 0.10 0.14

0.98 0.99 0.98 0.88 0.98 0.97 0.99 0.96 0.98

DHC-11 DHC-12 DHC-13 DHC-14 DHC-15 DHC-16 DHC-17 DHC-18 DHC-19

162.00 192.65 21.52 157.28 48.30 24.88 40.28 244.35 264.43

1.92 1.05 0.93 1.04 1.16 1.19 1.05 0.95 0.84

0.26 0.03 0.10 0.06 0.19 0.09 0.09 0.03 0.02

0.98 0.98 0.99 0.98 0.96 0.98 0.99 0.98 0.99

1

5. Empirical models and validation 5.1. Empirical model for dynamic shear modulus

Table 4 Values of p1 ~ p5 for fitting parameters in Martin-Davidenkov model. Parameter

p1

p2

p3

p4

p5

R2

A B γ0

−0.828 1.87E-2 2.1E-3

− 0.574 3.93E-3 0

0 − 3.83E −5 0

0 8.71E-5 3.73E-5

0.025 − 1.57E-4 − 4.29E-5

0.654 0.632 0.768

The model considered in this study was proposed by (Martin and Seed, 1982), which derived from the Hardin-Drnevich model (Hardin and Drnevich, 1972), the formula of Martin-Davidenkov model is written as follows, A

(γ / γ0 )2B ⎤ G =1−⎡ 2B ⎥ ⎢ Gmax ⎣ 1 + (γ / γ0 ) ⎦

(5)

where A, B, γ0 are the fitting parameters related to the soil properties. Gmax is the maximum shear modulus of the soil during the dynamic loading, and γ is the shear strain. When A = 1, B = 0.5, γ0 refers to the references strain, the Martin-Davidenkov model degenerates into the Hardin-Drnevich model. The relationship between dynamic shear modulus and shear strain, as shown in Fig. 8, are fitted by Martin-Davidenkov model in terms of the normalized dynamic shear modulus G/Gmax versus the shear strain. The typical comparison between the tested data and fitting curves are showed in Fig. 15. Table 3 shows the fitting parameters of all the tested specimens. It could be seen that the fitting curves match well with the tested curves before and after the turning points, which means that the

properties during the subsequent stages. For the thawed saturated clay, the evolution rules of dynamic shear modulus and damping ratio during long-term cyclic loading were selected and compared with that of multi stage cyclic loading. The results are showed in Fig. 14. Evolutions of dynamic shear modulus versus shear strain during the four stages loading are almost coincide with that of the long-term loading. It seems that the dynamic shear modulus during the four stages loading could be replaced by the corresponding long-term loading, which means that for the thawed saturated clay, the dynamic shear modulus might not be effected by the multi stage loading. For the damping ratio, the comparison is obvious, as shown in 101

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Fig. 16. Regression correlations between parameters in Martin-Davidenkov model with test condition.

(a)

(b) 0.14

0.12 0.10 0.08 0.06

Tested data σd=50kPa σd=57kPa σd=70kPa σd=89kPa σd=114kPa

Damping ratio

Damping ratio

0.14

Proposed model σd=50kPa σd=57kPa σd=70kPa σd=89kPa σd=114kPa

Fig. 17. Comparison between tested data and fitting curves for the damping ratio.

Tested data σd=50kPa σd=70kPa σd=89kPa σd=114kPa

0.12

Proposed model σd=50kPa σd=70kPa σd=89kPa σd=114kPa

0.10 0.08 0.06

0.2 0.4 0.6 0.8 Shear strain/% (σ3=150kPa)

0.2 0.4 0.6 0.8 1 Shear strain/% (σ3=60kPa)

dynamic shear modulus of thawed saturated clay could be predicted well by Martin-Davidenkov model. Meanwhile, it was found that evolutions of the fitting parameter A, B, γ0 are affected by the test condition, i.e., dynamic stress amplitude σd and confining pressure σ3. In general, parameter A increases with increasing σd and σ3. Parameter B fluctuate when σd and σ3 change. Parameter γ0 increases with increasing σd and decreases with increasing σ3. Based on these evolutions, a binary quadratic equation is used for describing the relationships between fitting parameters and test condition,

Table 5 Values of fitting parameters in the hyperbolic function model. Number

a

γr

c

R2

Number

a

γr

c

R2

DHC-02 DHC-03 DHC-04 DHC-05 DHC-06 DHC-07 DHC-08 DHC-09 DHC-10

0.04 0.05 0.05 0.05 0.06 0.04 0.05 0.06 0.06

0.20 0.24 0.32 0.38 0.36 0.25 0.28 0.33 0.33

0.23 0.28 0.36 0.41 0.39 0.26 0.25 0.29 0.30

0.88 0.98 0.98 0.98 0.94 0.93 0.96 0.98 0.97

DHC-11 DHC-12 DHC-13 DHC-14 DHC-15 DHC-16 DHC-17 DHC-18 DHC-19

0.06 0.06 0.06 0.06 0.07 0.07 0.06 0.07 0.07

0.49 0.28 0.34 0.38 0.53 0.25 0.34 0.32 0.39

0.35 0.13 0.18 0.19 0.23 0.08 0.11 0.09 0.12

0.98 0.93 0.85 0.99 0.99 0.84 0.94 0.97 0.98

1

Zpara = p1 σ3 + p2 σd + p3 σ32 + p4 σd2 + p5 σ3 σd

(6)

where Zpara represents the fitting parameters in the Martin-Davidenkov model. p1 ~p5 are the fitting parameters in Eq. (6). Values of p1 ~ p5 102

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Fig. 18. Regression correlations between parameters in hyperbolic function model with test condition.

cyclic loading would be built directly in this study based on the evolution evidences discussed above. It seems that little study directed towards proposing the damping ratio model for large shear strain under long-term cyclic loading. In this study, a hyperbolic function model was proposed, written as follows,

Table 6 Values of p1 ~ p5 for fitting parameters in hyperbolic function model. Parameter

p1

p2

p3

p4

p5

R2

γr c

4.74E-3 0

0 8.25E-3

− 3.08E-5 0

0 −1.99E-5

2.56E-5 − 3.44E-5

0.737 0.927

λ= were obtained by fitting the values of A, B, γ0 by means of Eq. (6) in three-dimensional spaces (Table 4). Fig. 16 shows the three-dimensional relations between fitting parameters with test condition. It can be observed that Eq. (6) could be reasonable in describing the relations between fitting parameters in Martin-Davidenkov model and test condition, as values of the r-squared under each regression are relatively high. In the case of this study, value of the fitting parameter A, B, γ0 ranges from 4.27–264.43, 0.83–1.92 and 0.02–0.28, respectively.

Most of the empirical models for damping ratio in previous studies were established by means of the relationship between normalized λ/ λmax and G/Gmax. Considering the fact that the G/Gmax models are also empirical, the predicted model for damping ratio during long-term DHC-07 Test data Model

0.7 0.6

0.9 0.8 0.7 0.6

0.3

0.4

0.5

0.5 0.1

0.6

0.2

0.12 0.10 DHC-05 Test data Model

0.06

0.3

0.4

0.5

0.6

0.8

Shear strain /%

0.7

0.5 0.1

0.6

0.2

0.12 0.10 DHC-09 Test data Model

0.08

1

0.3

0.4

0.5

0.6

Shear strain /%

0.06

0.4

0.8

Shear strain /%

Damping ratio

Damping ratio

Shear strain /%

0.08

0.9

0.6

Damping ratio

0.2

DHC-16 Test data Model

1.0

G/Gmax

0.8

0.5 0.1

DHC-12 Test data Model

1.0

G/Gmax

G/Gmax

0.9

(7)

where γr is the relative shear strain, which reflects the extent of the initial state of specimens and test condition. When γ tends to γr, the damping ratio λ approaches its maximum value λmax. a and c are the fitting parameters which control the shape of the hyperbolic function. The relationship between the damping ratio and shear strain shown in Fig. 9 were fitted by the power law model, and the typical results are shown in Fig. 17. Table 5 shows the values of parameter a, γr and c. It could be seen that the hyperbolic function model can predict the damping ratio well if neglecting several points in the beginning. It can be observed from Table 5 that, value of parameter a generally ranges from 0.04 to 0.07, so the average value, 0.058 could be represented for parameter a. Meanwhile, it is observed that parameter γr increases with increasing dynamic stress amplitude σd, fluctuates with increasing confining pressure σ3. Parameter c decreases with increasing σd and increases with increasing σ3. So Eq. (6) is also used for simulating

5.2. Empirical model for damping ratio

1.0

a (γ − γr )c

0.12 0.10 DHC-13 Test data Model

0.08 0.06

0.4

0.6

0.8

1

0.4

Shear strain /%

Fig. 19. Typical comparison between measured and calculated results.

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0.6

0.8

Shear strain /%

1

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the relations between Parameter γr and c with test condition. The fitting result is showed by Fig. 18, values of the fitting parameters are showed in Table 6. It can be observed that the fitting surface is reasonable in reflecting the evolution of fitting parameter γr and c with test condition. Values of the r-squared under each regression are fairly high. In the case of this study, values of fitting parameter γr and c range from 0.20–0.49 and 0.08–041, respectively.

damping ratio. For the thawed saturated clay in this study, the dynamic shear modulus can be obtained from long-term cyclic loading test or multi cyclic loading test, But it is not reasonable to determine the damping ratio by multi stage cyclic loading test. (5) The Martin-Davidenkov model and the hyperbolic function model were proposed, respectively, to predict the evolutions of dynamic shear modulus and damping ratio of thawed saturated clay during the long-term cyclic loading. Meanwhile, the binary quadratic equation was proved to be reasonable in describing the relations between fitting parameters in the two models and test condition.

5.3. Validation and evolution for empirical models A prediction analysis was attempted by using the empirical models with the correlation coefficients determined from the experimental data of specimens DHC-01 ~18. Dynamic shear modulus in this study can be calculated by Eq. (5) and Eq. (6). Damping ratio can be calculated by Eq. (6) and Eq. (7). Typical comparisons between measured dada and calculated results on dynamic shear modulus and damping ratio are showed in Fig. 19. It can be observed that the calculated dynamic shear modulus may fluctuate around the measured curve more or less, but the trend coincides well with the evolution characteristic of measured curve, that is, dynamic shear modulus gradually decreases at the initial stage, linearly decreases at the second stage. Calculated damping ratio shows a good agreement with the measured dada during the long-term cyclic loading, which suggests that the empirical model provides good predictions. In general, the Martin-Davidenkov model and the proposed hyperbolic function model are proved to be capable of predicting the dynamic shear modulus and damping ratio for thawed saturated clay under longterm cyclic loading, respectively. Experimental evidences and empirical model studies in this study provide useful data for evaluating the dynamic properties of thawed saturated subgrade soil in seasonally frozen regions.

Acknowledgements The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 51408163, 51578200 and 41430634), and Heilongjiang Province Science Foundation of China (Grant No. ZD201218). The authors thankful to the Key Laboratory of Highway Construction and Maintenance Technology in seasonal frozen soil regions (Changchun), China. References Al-Hunaidi, M., Chen, P., Rainer, J., et al., 1996. Shear moduli and damping in frozen and unfrozen clay by resonant column tests. Can. Geotech. J. 33 (3), 510–514. Brennan, A., Thusyanthan, N., Madabhushi, S., 2005. Evaluation of shear modulus and damping in dynamic centrifuge tests. J. Geotech. Geoenviron. 131 (12), 1488–1497. Chamberlain, E.J., Gow, A.J., 1979. Effect of freezing and thawing on the permeability and structure of soils. Eng. Geol. 13 (1), 73–92. Chang, D., Liu, J., Li, X., et al., 2014. Experiment study of effects of freezing-thawing cycles on mechanical properties of Qinghat-Tibet silty sand. Chin. J. Rock Mech. Eng. 33 (7), 1496–1502. Cui, Z., Zhang, Z., 2015. Comparison of dynamic characteristics of the silty clay before and after freezing and thawing under the subway vibration loading. Cold Reg. Sci. Technol. 119, 29–36. Cui, Z., He, P., Yang, W., 2014. Mechanical properties of a silty clay subjected to freezing–thawing. Cold Reg. Sci. Technol. 98, 26–34. Czajkowski, R.L., Vinson, T.S., 1980. Dynamic properties of frozen silt under cyclic loading. J. Geotech. Eng. Div. 106 (9), 963–980. Delfosse-Ribay, E., Djeran-Maigre, I., Cabrillac, R., et al., 2004. Shear modulus and damping ratio of grouted sand. Soil Dyn. Earthq. Eng. 24 (6), 461–471. Feng, D., Lin, B., Zhang, F., et al., 2017. A review of freeze-thaw effects on soil geotechnical properties. Scientia Sinica Technologica 47 (2), 111–127. Finno, R., Zapata-Medina, D., 2013. Effects of construction-induced stresses on dynamic soil parameters of bootlegger cove clays. J. Geotech. Geoenviron. Eng. 140 (4), 4013051. Hansson, K., Lundin, L., 2006. Equifinality and sensitivity in freezing and thawing simulations of laboratory and in situ data. Cold Reg. Sci. Technol. 44 (1), 20–37. Hardin, B.O., Drnevich, V.P., 1972. Shear modulus and damping in soils: measurement and parameter effects. Journal of Soil Mechanics & Foundations Div 98 (sm6), 603–624. Harlan, R., 1973. Analysis of coupled heat-fluid transport in partially frozen soil. Water Resour. Res. 9 (5), 1314–1323. Idriss, I.M., Dobry, R., Sing, R., 1978. Nonlinear behavior of soft clays during cyclic loading. J. Geotech. Geoenviron. 104 (GT12), 1427–1447. Ishibashi, I., Zhang, X., 1993. Unified dynamic shear moduli and damping ratios of sand and clay. Soils Found. 33 (1), 182–191. Ishihara, K., 1996. Soil Behaviour in Earthquake Geotechnics. Oxford University Press, New York. Kane, D.L., Stein, J., 1983. Water movement into seasonally frozen soils. Water Resour. Res. 19 (6), 1547–1557. Kong, Q., Wang, R., Song, G., et al., 2014. Monitoring the soil freeze-thaw process using piezoceramic-based smart aggregate. J. Cold Reg. Eng. 28 (2), 6014001. Li, Q., Ling, X., Wang, L., et al., 2013. Accumulative strain of clays in cold region under long-term low-level repeated cyclic loading: experimental evidence and accumulation model. Cold Reg. Sci. Technol. 94, 45–52. Ling, X., Li, Q., Wang, L., et al., 2013. Stiffness and damping radio evolution of frozen clays under long-term low-level repeated cyclic loading: experimental evidence and evolution model. Cold Reg. Sci. Technol. 86, 45–54. Ling, X., Zhang, F., Li, Q., et al., 2015. Dynamic shear modulus and damping ratio of frozen compacted sand subjected to freeze–thaw cycle under multi-stage cyclic loading. Soil Dyn. Earthq. Eng. 76, 111–121. Martin, P.P., Seed, H.B., 1982. One-dimensional dynamic ground response analyses. J. Geotech. Eng. Div. 108 (7), 935–952. Othman, M.A., Benson, C.H., 1993. Effect of freeze-thaw on the hydraulic conductivity and morphology of compacted clay. Can. Geotech. J. 30 (2), 236–246. Özgan, E., Serin, S., Ertürk, S., et al., 2015. Effects of freezing and thawing cycles on the engineering properties of soils. Soil Mech Found Eng 52 (2), 95–99. Perfect, E., Williams, P., 1980. Thermally induced water migration in frozen soils. Cold Reg. Sci. Technol. 3 (2), 101–109.

6. Conclusions This paper presented the findings on the evolution of dynamic shear modulus and damping ratio for thawed saturated clay during the longterm cyclic loading based on a series of cyclic tri-axial tests. The effects of dynamic stress amplitude, confining pressure, multi stage loading on the dynamic shear modulus and damping ratio were analyzed. Finally, the empirical models used for predicting the dynamic shear modulus and damping ratio were introduced and validated, respectively. The following conclusions can be drawn: (1) Repeated freeze-thaw cycles have obvious effect on the long-term dynamic properties of saturated clay. The dynamic shear modulus of thawed saturated clay decreases approximately 3.06%, 18.35%, 31.07%, 37.71%, respectively. The damping ratio increases approximately 34.95%, 45.34%, 50.75%, 54.30%, respectively, after been subjected to 1, 3, 5, 7 freeze-thaw cycles compared with the 0 freeze-thaw cycle. (2) Dynamic shear modulus and damping ratio of thawed saturated clay in general decrease with the increase of shear strain during the long-term cyclic loading. Value of dynamic shear modulus and damping ratio of thawed saturated clay ranges from 6 to 16 Mpa and from 0.04 to 0.16, respectively. (3) With an increase of dynamic stress amplitude, the dynamic shear modulus generally decreases, the damping ratio increases. Effect of confining pressure on the dynamic shear modulus depends on the shear strain level. At a lower shear strain, the dynamic shear modulus increases with the increasing confining pressure. When the shear strain reach certain level, effect of confining pressure on the dynamic shear modulus disappears. Damping ratio of thawed saturated clay is not effected by the confining pressure. (4) Multi stages cyclic loading has no obvious effect on the evolution of dynamic shear modulus compared with that of long-term cyclic loading, but have a remarkable effect on decreasing the evolution of 104

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Wang, T., Liu, Y., Yan, H., et al., 2015a. An experimental study on the mechanical properties of silty soils under repeated freeze–thaw cycles. Cold Reg. Sci. Technol. 112, 51–65. Wang, R., Zhu, D., Liu, X., et al., 2015b. Monitoring the freeze-thaw process of soil with different moisture contents using piezoceramic transducers. Smart Mater. Struct. 24 (5), 57003. Yao, X., Qi, J., Ma, W., 2009. Influence of freeze–thaw on the stored free energy in soils. Cold Reg. Sci. Technol. 56 (2), 115–119. Yu, L., Xu, X., Qiu, M., et al., 2010. Influnce of freeze-thaw on shear strength properties of saturated silty clay. Rock Soil Mech. 31 (8), 2448–2452. Zhang, Y., Hulsey, J., 2014. Temperature and freeze-thaw effects on dynamic properties of fine-grained soils. J. Cold Reg. Eng. 29 (2), 4014012. Zhang, Y., Michalowski, R.L., 2015. Thermal-hydro-mechanical analysis of frost heave and thaw settlement. J. Geotech. Geoenviron. 141 (7), 4015027. Zhao, G., Tao, X., Liu, B., 2009. Experimental study on water migration in undisturbed soil during freezing and thawing process. Chinese Journal of Geotechnical Engineering 31 (12), 1952–1957. Zheng, Y., Ma, W., Mu, Y., et al., 2015. Analysis of Soil Structures and the Mechanisms Under the Action of Freezing and Thawing Cycles, Cold Regions Engineering 2015: Developing and Maintaining Resilient Infrastructure. American Society of Civil Engineers, pp. 25–33.

Qi, J., Vermeer, P.A., Cheng, G., 2006. A review of the influence of freeze-thaw cycles on soil geotechnical properties. Permafr. Periglac. Process. 17 (3), 245–252. Rollins, K.M., Evans, M.D., Diehl, N.B., et al., 1998. Shear modulus and damping relationships for gravels. J. Geotech. Geoenviron. 124 (5), 396–405. Seed, H.B., Wong, R.T., Idriss, I., et al., 1986. Moduli and damping factors for dynamic analyses of cohesionless soils. J. Geotech. Eng. 112 (11), 1016–1032. Sheng, D., Zhang, S., Niu, F., et al., 2014. A potential new frost heave mechanism in highspeed railway embankments. Géotechnique 64 (2), 144. Stevens, H.W., 1975. The Response of Frozen Soils to Vibratory Loads. Army Cold Regions Research and Engineering Laboratory, Hanover, NH. Tang, Y., Li, J., Wan, P., et al., 2014. Resilient and plastic strain behavior of freezing–thawing mucky clay under subway loading in Shanghai. Nat. Hazards 72 (2), 771–787. Wang, D., Ma, W., Niu, Y., et al., 2007. Effects of cyclic freezing and thawing on mechanical properties of Qinghai–Tibet clay. Cold Reg. Sci. Technol. 48 (1), 34–43. Wang, Z., Luo, Y., Guo, H., et al., 2012. Effects of initial deviatoric stress ratios on dynamic shear modulus and damping ratio of undisturbed loess in China. Eng. Geol. 143, 43–50. Wang, J., Liu, H., Wu, C., et al., 2014. Influence of freeze-thaw cycles on dynamic characteristics of subgrade soils with different plasticity indices. Chinese Journal of Geotechnical Engineering 36 (4), 633–639.

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