Dynamic stress–strain behavior of frozen soil: Experiments and modeling

Cold Regions Science and Technology 106–107 (2014) 153–160

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Dynamic stress–strain behavior of frozen soil: Experiments and modeling Qijun Xie a, Zhiwu Zhu a,b,c,⁎, Guozheng Kang a a b c

School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academic Sciences, Lanzhou 730000, China State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China

a r t i c l e

i n f o

Article history: Received 14 March 2014 Accepted 10 July 2014 Available online 23 July 2014 Keywords: Frozen soil Dynamic stress–strain behavior Constitutive model SHPB Impact

a b s t r a c t The dynamic stress–strain behavior of artificial frozen soil (with a moisture content of 30%) was tested using a split Hopkinson pressure bar (SHPB) under various impact compressive loading conditions. The tests were performed at strain rates 400–1000 s−1 and different temperatures (i.e., −3, −8, −18, and −28 °C). The experimental results show that the dynamic stress–strain responses of the artificial frozen soil exhibit a positive strain rate sensitivity and negative temperature dependence. An energy-based dynamic constitutive model was constructed to simulate the dynamic stress–strain behavior of the frozen soil. It is shown that the proposed model can describe the positive strain-rate sensitivity and negative temperature dependence of the artificial frozen soil reasonably, and predict the dynamic stress–strain curves of the frozen soil well. © 2014 Elsevier B.V. All rights reserved.

1. Introduction The total area of the Earth’s permafrost (3.576 × 107 km2) makes up approximately 24% of its land area (French, 2007). Frozen soil consists of mineral particles, ice inclusions, liquid water, and gaseous inclusions (T S ytovich, 1975). Ice inclusions, whose stress–strain response is sensitive to the ambient temperature, make the mechanical behavior of frozen soil complex and different from that of ordinary soil (Qi and Ma, 2010) and dependent upon both the strain rate and the temperature. Many studies have observed the static or quasistatic mechanical behavior of frozen soil (Ma, 2009). However, the dynamic mechanical properties of frozen soil have not yet been studied in detail, especially for high-velocity impact loading conditions. Recently, many engineering activities, such as building highways, railways and tunnels, are occurring in the cold regions of the earth. The frozen soil is subjected not only to the static or quasi-static loading, but also to the dynamic loading, such as blasting and excavation. However, the traditional dynamic models, such as Johnson-Cook model (Johnson and Cook, 1983), Z-WT model (Wang and Shi, 2000) and Steinberg model (Steinberg et al., 1980), were constructed for describing the dynamic stress–strain responses of metal materials, and cannot be directly used to describe those of frozen soil. So, it is necessary to observe the dynamic mechanical behaviors of the frozen soil experimentally and theoretically. The split Hopkinson pressure bar (SHPB) is a device used in the dynamic testing of materials. Because little research has been conducted on the dynamic mechanical behavior of frozen soil under impact

loading, only a few researchers have used the SHPB to study the dynamic stress–strain responses of frozen soil. The most representative studies in the field are the following: the Sandia National Laboratory, USA used SHPB to test Alaska natural frozen soil and proposed a constitutive model (Lee et al., 2002); Ma (Ma, 2010), Chen (Chen et al., 2005) and Zhang (Zhang et al., 2013) used SHPB to study the dynamic mechanical properties of frozen soil and then explored the corresponding constitutive models. However, the model proposed by Sandia National Laboratory has too many parameters, whose values cannot be determined easily. A comparison of the constitutive models proposed by Ma(Ma, 2010), Chen(Chen et al., 2005), and Zhang(Zhang et al., 2013) shows that they do not describe the dynamic stress–strain relations of frozen soil very well. Therefore, a simple and accurate dynamic constitutive model of frozen soil is needed; this model must have as few parameters as possible to facilitate its engineering applications. In this work, the dynamic stress–strain responses of artificial frozen soil are observed by the SHPB tests at various strain rates from 400 s−1 to 1000 s−1 and different temperatures (i.e., −3, −8, − 18 and −28 °C). The effects of strain rate and ambient temperature on the stress–strain responses of the frozen soil are investigated. Then, an energy-based dynamic constitutive model is proposed to describe the dynamic compressive behavior of the frozen soil. 2. Experimental observations 2.1. Dynamic experimental procedure

⁎ Corresponding author at: School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China. E-mail address: [email protected] (Z. Zhu).

http://dx.doi.org/10.1016/j.coldregions.2014.07.004 0165-232X/© 2014 Elsevier B.V. All rights reserved.

The experimental material is the artificial frozen sand. The moisture content of each specimen is 30% and the dry density of

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Q. Xie et al. / Cold Regions Science and Technology 106–107 (2014) 153–160

the soil is 1.6 g/cm3. The dimensions of the specimen bars are all ϕ 30 mm × 18 mm. We prepared the frozen soil specimens in accordance with the Chinese standard MT/T593-2011 for the physical and mechanical performance testing of artificial frozen soil (Part I: Artificial frozen soil experiment sample and specimen preparation method). The process is described below. Firstly, the sand was crushed down by a wooden hammer. Then, the large soil particles would be sifted out by a sieve with a mesh size of 2 mm. The remaining soil was put in an oven at 110 °C for 12 h and dehydrated. The soil was then placed in a sealed vessel and cooled to room temperature. The particle size of the soil was listed in Table 1. After that, the water was mixed with the dehydrated soil to satisfy the required water content of 30%, and then the mixture is stored in a closed container for 24 h. The specimens were made from the mixture by molds and smeared Vaseline on its surface in order to keep the water content of the specimens unchanged. Finally, the well-made specimens were placed in a freezer at the prescribed temperatures (i.e., −3, −8, −18 and −28 °C, respectively) for 24 h (See Fig. 1). The specimen was quickly removed for testing. The SHPB used for this testing was an air pressure-driven device. The frozen soil was tested three times at each temperature and air pressure. The three experimental curves obtained are in good agreement. The three stress–strain curves obtained were averaged to obtain the stress–strain curve of the frozen soil as a function of the temperature and strain rate. The strain rates selected were approximately 400, 600, 800, and 1000 s− 1. The reason for selecting such strain rates is based on the actual needs of this study. Moreover, in order to study the dynamic mechanical behavior of frozen soil, the strain rate used in SHPB tests must meet the test strain rate range and accuracy requirements of SHPB equipment.

2.2. Experimental equipment Dynamic testing of the frozen soil specimens was performed using an SHPB setup (Fig. 2). An SHPB is a device used to test the dynamic mechanical behavior of materials under impact loading. The SHPB setup used in this study consisted of a striker, an incident bar, a transmission bar, and an acquisition system. The material parameters of the striker and bars and the length of the striker are important experimental parameters. The striker is made of the 35CrMnSi steel whose Young’s modulus E is 210GPa and density ρ is 7900 kg/m3 , and the length of the striker is 200 mm, the incident and transmission bars are made of the 7075-T6 aluminum whose E is 71 GPa and density ρ is 2810 kg/m3, and the diameters of them are 30 mm. The incident and transmission bars are instrumented with strain gauges to capture the elastic stress waves generated by the striker bar. Frozen soil specimens are placed between the incident and transmission bars. The striker is launched by compressed air. An elastic compression stress wave (incident wave) is generated by the impact of the striker bar striking the incident bar. The incident wave is transmitted through the specimen as a transmitted wave and is partially reflected at the interface between the specimen and the incident bar. Both the incident and reflected waves are recorded by the strain gauge on the incident bar, and the transmitted wave is recorded by the strain gauge on the transmission bar. By using the theory of one-dimensional elastic wave propagation and the continuity of the displacement and stress equilibrium at the interface, the following equations can be derived to describe the stress, strain, and strain rate in the specimen. Table 1 Particle size distribution. Size/mm

b0.1

0.1-0.25

0.25-0.5

0.5-2

N2

Proportion/%

21

19

25

34

1

Fig. 1. A frozen soil specimen before testing.

ε˙ ðt Þ ¼

C0 ½ε ðt Þ−ε r ðt Þ−ε t ðt Þ LS i

εðt Þ ¼

C0 LS

σ ðt Þ ¼

A0 E ½ε ðt Þ−ε r ðt Þ−ε t ðt Þ 2AS 0 i

Z

t 0

½εi ðt Þ−ε r ðt Þ−ε t ðt Þdt

ð1Þ

ð2Þ

ð3Þ

where σ(t) is the stress in the sample; εi(t), εr(t), and εt(t) are the incident strain, reflected strain, and transmitted strain, respectively; E0 is the Young’s modulus of the bar; C0 is the elastic wave speed in the bar; AS and LS are the initial cross-sectional area and length of the specimen, respectively; and A0 is the cross-sectional area of the bar. In this study, the frozen soil specimen diameter was the same as the bar diameter, i.e., As = A0.When the stress in the specimen is uniform, the strain components are related as follows: ε i ðt Þ þ εr ðt Þ ¼ ε t ðt Þ

ð4Þ

Eqs. (1)–(3) can be simplified as follows: ε˙ ðt Þ ¼ −2

C0 ε ðt Þ LS r

εðt Þ ¼ −2

C0 LS

Z

t 0

εr ðt Þdt

σ ðt Þ ¼ E0 εt ðt Þ

ð5Þ

ð6Þ

ð7Þ

Therefore, once the incident, reflected, and transmitted signals are measured, the stress–strain data for the material under investigation can be obtained. 2.3. Experimental results 2.3.1. Strain rate effect The strain–stress curves of frozen soil obtained at different strain rates are shown in Fig. 3. It is seen that in the process of impacting, as the specimen deforms within a small strain, the stress–strain curves which obtained from dynamic tests are nearly linear; however, as the strain becomes larger and the specimen deforms within a larger strain, the strain-hardening behavior of the frozen soil specimen becomes notable. Different from the literature (Zhang et al., 2013), there is not an

Q. Xie et al. / Cold Regions Science and Technology 106–107 (2014) 153–160

155

Fig. 2. Sketch of SHPB and its testing system.

obvious plateau when the stress reaches to its peak value. In other words, there is not an obvious plastic flow phenomenon. This difference should be caused by the difference in the soil and its moisture content. Based on the experimental results, the dynamic stress–strain curve is divided into three sections and is shown in Fig. 4: the first part is a linear section which increases sharply at initial stage; the second part decreases slowly; the third part decreases sharply. The critical dynamic compressive strength is set to be the initial fracture strength of frozen soil, and it is approximately equal to the peak strength of linear section of stress–strain curves. As shown in Fig. 3, beyond the peak stress point, the strain softening occurs and the frozen soil specimens are destroyed. With the increase of strain rate, the peak stress of frozen soil increases. Frozen soil specimens are shown after testing in Fig. 5. As shown in Fig. 6, with the increase of strain rate, the dynamic peak stress of frozen soil increases. So the relationship between peak stress and strain rate can be assumed to be exponential.

2.3.2. Temperature effect The effect of temperature on the strength of frozen soil could not be neglected. Under the static, quasi-static and vibration loading conditions, the mechanical properties of frozen soil are significantly influenced by temperature (Li et al., 2009)-(Zhu et al., 2010). Fig. 7 shows that, under the impact loading conditions, the strength of frozen soil increases as the temperature decreases, when the applied high strain rates are approximately identical. The results in Fig. 7 exhibit a reasonable repeatability in the dynamic compressive tests of frozen soil. Frozen soil consists of mineral particles, ice inclusions, liquid water and gaseous inclusions. The strength of the ice is much higher than that of the soil. As the test temperature decreases, the content of ice in the frozen soil increases and the soil particles are linked together more closely, so the strength of frozen soil increases, as shown in Fig. 7. The peak stresses of frozen soil at different temperatures are shown in Fig. 8. The results show the peak stress of frozen soil approximately linearly increases with the increase of negative temperature.

(a) -3°C

(b) -8°C

10

20

402S-1 590S-1 825S-1 1082S-1

Stress,σ,Mpa

Stress,σ,Mpa

8 6 4 2 0

15

10

5

0 0

2

4

6

8

10

12

0

Strain,ε,%

2

4

6

(c) -18°C 462S-1 668S-1 849S-1 1049S-1

10

5

2

4

6

Strain,ε,%

12

8

10

361S-1 518S-1 596S-1 851S-1

20

Stress,σ,Mpa

Stress,σ,Mpa

15

0

10

(d) -28°C

20

0

8

Strain,ε,%

12

10

0

0

4

Strain,ε,%

Fig. 3. Strain–stress curves of frozen soil at different stain rates.

8

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Q. Xie et al. / Cold Regions Science and Technology 106–107 (2014) 153–160

linear increases

decreases slowly

10

30

decreases sharply

25

Peek Stress,σ,Mpa

Stress,σ,Mpa

891S-1

5

-3 -8 -18 -28

20

15

10

5 0

0

3

6

9

400

Strain,ε,%

The experimental results show that the dynamic stress–strain responses of the artificial frozen soil are rate-sensitive and temperaturedependent, and remarkable strain softening occurs beyond the peak stress. Such features cannot be described reasonably by the traditional dynamic constitutive models such as Johnson-Cook model (Johnson and Cook, 1983) and Z-W-T model (Wang and Shi, 2000). Therefore, a new constitutive model is necessary to describe the dynamic compressive behavior of frozen soil. It is well known that the frozen soils are different from each other if the temperature and moisture content of the soils are different. The

(a) Strain rate about 400 s-1

800

1000

1200

1400

Strain Rate,ε,S-1

Fig. 4. Three sections of strain–stress curves.

3. Dynamic constitutive model

600

Fig. 6. Relationships of peek stress vs. strain rate at different temperatures.

temperature and moisture content can only affect the material parameters. Due to the complexity of the stress–strain curves of frozen soil, directly establishing a dynamic constitutive model with a clear physical meaning from the experimental stress–strain curves is extremely difficult. So, the energy method can be adopted to establish indirectly the dynamic constitutive model of frozen soil. The energy U absorbed by a unit volume of frozen soil is defined as the area under the stress–strain curve, and can be obtained by a numerical integration (Mukai et al., 1999), i.e., Z U¼

εm 0

σdε

ð8Þ

where, εm is the final strain, σ is the stress. It is shown in Fig. 9 that during the impact loading, the energy absorbed by the frozen soil specimen increases with the increases of compressive strain and strain rate. From Fig. 9, it is seen that the obtained energy absorption curves under the different loading conditions have a nearly consistent evolution rule. So we can obtain the empiric formula of frozen soil by finding the function of energy absorption with respect to the applied strain. From the evolution curves of the energy absorbed by the frozen soil with respect to applied strain shown in Fig. 9, the GuassAmp peak function is suitable for describing such evolution. The curve of GuassAmp peak function is shown in Fig. 10 and its formulation is expressed as: x−xc 2 −1 y ¼ y0 þ Ae 2ð w Þ

ð9Þ

(b) Strain rate about 800 s-1 Where, y0 is the offset; A is the amplitude; xc is the abscissa of the peak point; 2w is the width of the curve at half amplitude, as shown in Fig. 7. Since the absorbed energy increases monotonically and reaches its maximum at the end of each dynamic compressive test, only the first half of the GuassAmp peak function is used to describe the evolution of absorbed energy. Thus, based on the GuassAmp peak function, the absorbed energy U can be formulated by  −12

U ¼ U 0 þ A0 e

Fig. 5. Frozen specimens after testing.

ε−ε c w0

2 ð10Þ

For example, the experimental data of absorbed energy obtained in the dynamic compressive tests at various strain rates and −18 °C can be fitted very well by the GuassAmp peak function, as shown in Fig. 11. The parameters obtained from the fitting are shown in Table 2.

Q. Xie et al. / Cold Regions Science and Technology 106–107 (2014) 153–160

(a) Strain rate about 400 s-1

5

-3oC 627S-1 -3oC 609S-1 -8oC 590S-1 -8oC 554S-1 -18oC 633S-1 -28oC 546S-1 -28oC 596S-1

20

Stress,σ,Mpa

Stress,σ,Mpa

10

(b) Strain rate about 600 s-1

25

-3oC 381 -8oC 414 -18oC 462 -28oC 387

15

157

15 10 5

0 0

1

2

3

0

4

0

2

4

Strain,ε,%

(c) Strain rate about 800 s-1 -3 C 861S -3oC 891S-1 -8oC 825S-1 -8oC 870S-1 -18oC 849S-1 -28oC 851S-1

Stress,σ,Mpa

20 15

8

10

(d) Strain rate about 1000 s-1

30

-1

Stress,σ,Mpa

o

25

6

Strain,ε,%

-3oC 1155S-1 -3oC 1087S-1 -8oC 1082S-1 -8oC 933S-1 -18oC 1049S-1 -28oC 1065S-1

20

10

5 0

0 0

2

4

6

8

10

0

4

8

Strain,ε,%

12

Strain,ε,%

Fig. 7. Strain–stress curves of frozen soil at different temperatures.

If ε = 0, σ = 0 should be satisfied. It implies that

By Eq. (8), it yields σ¼

∂U ∂ε

ð11Þ

Where, σ is the stress, ε is the strain. If U0 is a constant, in Eq. (11), ε = 0, so that σ ≠ 0, which is not consistent with the reality. It means that U0 must be a function of the strain ε. Therefore, we can set U0 = A1 + B(1 − ε)n and substitute it into Eq. (11) to get:

ðn−1Þ

σ ¼ −nBð1−ε Þ

 þ A0

  c 2 εc −ε −12 ε−ε w0 e w0 2

ð12Þ

30

A0 ¼

Strain rate about 400S Strain rate about 600S-1 Strain rate about 800S-1 Strain rate about 1000S-1

!  2 1

e2

εc w0

ðn−1Þ

σ ¼ −nBð1−ε Þ

ð13Þ

þ

nBðεc −ε Þ 2εε2wc −ε2 e 0 εc

2

ð14Þ

Fitting the stress–strain curves of frozen soil, the parameters in Eq. (14) were obtained and shown in Table 3. The final strain and strain rate are calculated, respectively, by

ε˙ ¼ −2

20

2

Then, Eq. (11) can be rewritten as

C0 LS

ε f ¼ −2

-1

Peek Stress,σ,Mpa

nBwo εc

Z

T 0

εr ðt Þdt

C0 ε ðt Þ LS r

ð15Þ

ð16Þ

where T is loading time, εr(t) is the reflected wave. Z

T

10

ε f =ε˙ ¼

0

10

20

Negative Temperature,T, Fig. 8. The relationship of peak stress vs. temperature.

0

εr ðt Þdt

ε r ðt Þ

ð17Þ

Making T = nΔt, when Δt is very small, it yields

30

Z

T 0

ε r ðt Þdt ¼ ε r ðt 1 ÞΔt þ εr ðt 2 ÞΔt… þ εr ðt n ÞΔt

ð18Þ

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Q. Xie et al. / Cold Regions Science and Technology 106–107 (2014) 153–160

(b) -8°C 381S-1 609S-1 672S-1 861S-1 891S-1 1155S-1

0.6

0.4

0.2

0.0 0

4

8

Energy Absorption Per Unit Volume,U,J/m3

Energy Absorption Per Unit Volume,U,J/m3

(a) -3°C

12

1.2

0.8

402S-1 590S-1 825S-1 1082S-1

0.4

0.0 0

4

1.2

462S-1 668S-1 849S-1 1049S-1

0.8

0.4

0

4

12

(d) -28°C Energy Absorption Per Unit Volume,U,J/m3

Energy Absorption Per Unit Volume,U,J/m3

(c) -18°C

0.0

8

Strain,ε,%

Strain,ε,100%

8

12

2.4 2.0

361S-1 518S-1 596S-1 851S-1 1065S-1

1.6 1.2 0.8 0.4 0.0

0

4

8

12

Strain,ε,%

Strain,ε,%

Fig. 9. The curves of energy absorbed by frozen soil vs. strain at different temperatures.

C0 ε˙ þε˙ … þε˙n ε r ðt Þ ¼ 1 2 L n S  C ε ðt Þ þ εr ðt 2 Þ… þ εr ðt n Þ ¼ −2 0 r 1 LS n

ε˙ ¼¼ −2

ð19Þ

By Eqs. (15) to (19), it gives ε f ¼ Tε˙

ð20Þ

Based on the one dimensional theory of elastic wave (Wang, 2007), T depends on the length of striker and the elastic wave speed in the striker. The aggregation phenomenon (Qi and Ma, 2010)(Ma, 2005) can be explained by Eq. (20). ˙ In Table 3, the variation of εc is approximately linear with the ε(Δε c= Δ ε˙ is constant approximately). It gives ð21Þ

w0 ¼ ε c −ε p

ð22Þ

where εp is the strain where the stress reaches to its peak, εc is the strain where the stress is unloaded to zero again.

Energy Absorption Per Unit

  ε˙ þ C1 εc ¼ ε f þ C 1 ¼ A ε˙0

where A is a parameter dependent on the equipment, and A = 0.95 × 10−6 (obtained by anyone of experimental ultimate strain dividing the strain rate ) in this paper. ε˙0 is the dimensionless reference strain rate, and ε˙0 ¼ 0:01. C1 is a material parameter. The critical compressive strain is defined as the strain where the stress reaches to the peak (Li and Xu, 2009). The peak stress is used to describe the deformation properties of frozen soil. By Table 2 and comparison of experimental results, we can approximately assume that the w0 equals the εc minus the strain where the stress reaches its peak, i.e.,

Volume,U,J/m3

ε˙ is the average strain rate.

462S-1 Energy absorption 668S-1 Energy absorption 849S-1 Energy absorption 1049S-1 Energy absorption GausAMP fit curves

1.2

0.8

0.4

0.0 0

4

8

Strain,ε,% Fig. 10. Curve of GuassAmp peak function.

Fig. 11. GuassAmp peak function fitting(−18 °C).

Q. Xie et al. / Cold Regions Science and Technology 106–107 (2014) 153–160 Table 2 The parameters obtained from the fitting.

σ f ¼ C5

Strain rate

U0

A0

εc

w0

1049 849 668 462

−0.286 −0.203 −0.170 −0.045

1.57 0.995 0.687 0.343

0.099 0.082 0.064 0.044

0.052 0.045 0.038 0.021

Based on Table 2, it is assumed ε p =εc ¼ C 2

ð23Þ

where C2 is a material parameter. By Eq. (14), it is seen that the n × B must be a stress, so it yields n × B = σf. Then, the height of the curves is determined by σf. When ε = εp , σ p ¼ σ f ð1−C 2 Þe

C 2 ð2−C 2 Þ 2ð1−C 2 Þ2

    n−1 ! ε˙ þ C1 − 1−C 2 A ε˙ 0

ð24Þ

    n−1 1−C 2 A ε˙ε˙ þ C 1 is very small and can be

The value of neglected, then

0

C 2 ð2−C 2 Þ 2

σ p =σ f ¼ ð1−C 2 Þe 2ð1−C2 Þ

ð25Þ

n determines the initial slope of curves and is defined as n ¼ C3

 C 4 ε˙ ε˙ 0

ð26Þ

σP decreases with the increase of strain rate. From the experiment results, the value of σp can be obtained by σ P ¼ C5

 C 6 ε˙ ε˙ 0

ð27Þ

So the dynamic constitutive model can be obtained as follows: 2

ðn−1Þ

σ ¼ σ f −ð1−ε Þ

þ

2εε c −ε ðεc −ε Þ 2ð1−C 2 2 2 Þ εc e εc

! ð28Þ

Table 3 The parameters obtained by fitting. temperature

strain rate

B

n

εc

w0

peak stress

−3 °C −3 °C −3 °C −3 °C −3 °C −3 °C −8 °C −8 °C −8 °C −8 °C −18 °C −18 °C −18 °C −18 °C −28 °C −28 °C −28 °C −28 °C −28 °C

381 609 627 861 891 1155 402 590 825 1082 462 668 869 1049 361 518 596 851 1065

0.006 0.040 0.009 0.025 0.035 0.069 0.340 0.056 0.030 0.030 0.220 0.017 0.060 0.270 0.020 0.200 0.077 0.054 0.034

356 166 578 233 176.5 134 21.7 90.6 345 597 63.2 498.24 181.65 74.4 432.3 83.9 144.9 287.6 497.2

0.040 0.060 0.070 0.090 0.096 0.117 0.040 0.056 0.080 0.106 0.048 0.067 0.090 0.110 0.030 0.048 0.056 0.084 0.104

0.020 0.040 0.040 0.050 0.056 0.098 0.020 0.027 0.050 0.101 0.038 0.043 0.060 0.090 0.016 0.030 0.029 0.05 0.058

4.340 7.930 7.870 9.430 9.630 9.600 11.89 13.40 14.77 17.00 9.920 11.43 14.18 19.03 17.09 19.63 22.08 23.61 27.66

 C 6 C 2 ð2−C 2 Þ ε˙ 2 = ð1−C 2 Þe 2ð1−C2 Þ ε˙ 0

ð29Þ

 C 4 ε˙ ε˙ 0

ð30Þ

  ε˙ þ C1 ε˙ 0

ð31Þ

n ¼ C3

εc ¼ A

159

!

where A is the parameter dependent on the experimental equipment. C1, C2, C3, C4, C5 and C6 are material parameters. 4. Parameters determination The constitutive model proposed in this paper has seven parameters. One of them is related to the experimental device and the other six parameters are dependent on the experimental material. The determination method is stated as follows: From Eqs. (20) and (21), it is seen that the parameter A is equal to a   fixed value A ¼ ε fε˙0 =ε˙ for the tests using the same experimental apparatus and data processing method. So the parameter A can be determined by calibrating the laboratory equipment. C1 is the distance from the final strain εf to the strain εc. By extending the stress–strain curves to the stress axis, the εc can be obtained and the εf can be determined by Eq. (20). So the parameter C1 can be determined. C2 is the ratio of εp to εc, and its value can be affected by the parameter A and the experimental material, which must be obtained by experiment. The parameters C3 and C4 control the elastic modulus of the initial section. Their values are influenced by strain rate and can be obtained by fitting the initial section of stress–strain curves. The parameters C5 and C6 control the relationship of peak stress and strain rate, and characterize the strain rate hardening effect of the material. Their values can be obtained by fitting the curves of peak stress vs. strain rate. 5. Validation of model The proposed model can reflect the variation of elastic modulus of the frozen soil at various strain rates. With the increase of strain rate, the strength of frozen soil increases. The proposed model also reflects a complete stress–strain history of frozen soil under the impact loading conditions, including the initial elasticity, strengthening stage and ultimately softening process. After the parameters used in the proposed model are determined (as shown in Table 4), the dynamic stress–strain curves of frozen soil can be obtained by Eqs. (27) to (30). Comparison of experimental and theoretical results is shown in Fig. 12. It can be seen that the results simulated by the proposed constitutive model are in well agreement with the experimental ones. Therefore, the proposed dynamic constitutive model can reflect well the deformation characteristics of frozen soil under the impact loading. 6. Conclusions The dynamic compressive behavior of frozen soil is experimentally observed by the SHPB at various temperatures of − 3, − 8, − 18 and Table 4 The constitutive model parameters. temperature

A

−3 °C −8 °C −18 °C −28 °C

0.95 0.95 0.95 0.95

× × × ×

10−6 10−6 10−6 10−6

C1

C2

C3

C4

C5

C6

0.01 0.002 0.007 0.003

0.40 0.36 0.37 0.43

1.33 × 106 7.2 × 10−11 7.07 × 1020 3.66 × 10−9

−0.78 2.57 −3.76 2.21

0.012 0.259 0.0008 0.34

0.58 0.359 0.87 0.38

160

Q. Xie et al. / Cold Regions Science and Technology 106–107 (2014) 153–160

(a) -3°C

(b) -8°C 1082S-1 1082S-1prediction 825S-1 825S-1prediction 590S-1 590S-1prediction

-1

Stress,σ,Mpa

6

3

16

Stress,σ,Mpa

609S 609S-1prediction 627S-1 627S-1prediction 861S-1 861S-1prediction 891S-1 891S-1prediction

9

12 8 4

0 0

4

8

0

12

0

4

Strain,ε,%

(c) -18°C 15

10

12

(d) -28°C 462S-1 462S-1prediction 668S-1 668S-1prediction 849S-1 849S-1prediction 1049S-1 1049S-1prediction

5

30

361S-1 361S-1prediction 518S-1 518S-1prediction 851S-1 851S-1prediction 1065S-1 1065S-1prediction

24

Stress,σ,Mpa

Stress,σ,Mpa

20

8

Strain,ε,%

18 12 6

0 0

4

8

12

0

0

Strain,ε,%

4

8

12

Strain,ε,%

Fig. 12. Theoretical and experimental curves of frozen soil at different temperatures.

− 28 °C first. Then, an energy-based dynamic constitutive model is established. The following conclusions can be drawn out: (1) The dynamic stress–strain behavior of artificial frozen soil (with a moisture content of 30%) was tested by the split Hopkinson pressure bar (SHPB) under various impact compressive loading conditions. The dynamic compressive stress–strain curves of frozen soil can be divided approximately into three stages: i.e., the initial approximately linear increases stage (elastic stage), the decreases slowly stage (plastic stage) and the decreases sharply stage (softening stage). (2) The dynamic compressive responses of frozen soil are temperature-dependent and rate-sensitive. The strength of frozen soil increases with the increase of strain rate and the decrease of temperature. (3) In analyzing the curve and assessing its physical meaning, we obtained very good results in using the GaussAmp peak function to describe the unit volume energy absorption curve of the frozen soil. A dynamic constitutive model for the frozen soil was developed based on the results obtained. The model can describe the stress–strain responses of frozen soil under the impact loading conditions well. Also, the parameters used in the proposed constitutive model can be determined easily. (4) The model developed in this study can be used to numerically predict the mechanical behavior of frozen soil under impact loading. The results also provide a theoretical basis for the design and safety assessment of construction in the frozen soil of cold regions. Acknowledgement Financial supports by the Project of National Natural Science Foundation of China (No. 11172251), the Open Fund of State Key Laboratory of Frozen Soil Engineering (No. SKLFSE201001), the Opening Project of State Key Laboratory of Explosion Science and Technology (Beijing

Institute of Technology) (No. KFJJ13-10 M), and the Project of Sichuan Provincial Youth Science and Technology Innovation Team, China (2013TD0004) are appreciated. References Chen, B.S., Hu, S.S.,Ma, Q.Y., Tu, Z.Y., 2005. Experimental Research of Dynamic Mechanical Behaviors of Frozen Soil [J]. Chin. J. Theor. Appl. Mech. 37 (6), 724–728. French, H.M., 2007. The periglacial environment [M]. Wiley. com. Johnson, G.R., Cook, W.H., 1983. A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures [C]. Proceedings of the 7th International Symposium on Ballistics., 21. International Ballistics Committee, The Hague, Netherlands, pp. 541–547. Lee, M.Y.,Fossum, A., Costin, L.S., et al., 2002. Frozen soil material testing and constitutive modeling [R]. Report No. SAND2002-0524Sandia National Laboratory, USA. Li, W., Xu, J., 2009. Impact characterization of basalt fiber reinforced geopolymeric concrete using a 100-mm-diameter split Hopkinson pressure bar [J]. Mater. Sci. Eng. A 513, 145–153. Li, S.,Lai, Y.,Zhang, M., et al., 2009. Seismic analysis of embankment of Qinghai–Tibet railway [J]. Cold Reg. Sci. Technol. 55 (1), 151–159. Ma, Q.Y., 2005. Study on the Dynamic Mechanical Properties of Frozen Soil under Impact Loading [D]. Beijing University of Science and Technology. Ma, Q.Y., 2009. Research status of dynamic properties of artificial frozen soil and its significance [J]. Rock Soil Mech. 30 (supp), 10–14. Ma, Q.Y., 2010. Experimental analysis of dynamic mechanical properties for artificially frozen clay by the split Hopkinson pressure bar [J]. J. Appl. Mech. Tech. Phys. 51 (3), 448–452. Mukai, T.,Kanahashi, H.,Miyoshi, T., et al., 1999. Experimental study of energy absorption in a close-celled aluminum foam under dynamic loading [J]. Scr. Mater. 40 (8), 921–927. Qi, J.L., Ma, W., 2010. State-of-art of research on mechanical properties of frozen soils [J]. Rock Soil Mech. 31 (1), 133–143. Steinberg, D.J.,Cochran, S.G.,Guinan, M.W., 1980. A constitutive model for metals applicable at high‐strain rate [J]. J. Appl. Phys. 51 (3), 1498–1504. T S ytovich, N.A., 1975. The mechanics of frozen ground [M]. Scripta Book Co. Wang, L.L., 2007. Foundations of stress Waves [M]. Elsevier, Amsterdam. Wang, L.L., Shi, S.Q., 2000. ZWT nonlinear viscoelastic constitutive relation of thermal research and application (in Chinese) [J]. J. Ningbo Univ. 13 (B12), 141–149. Zhang, H., Zhu, Z., Song, S., et al., 2013. Dynamic behavior of frozen soil under uniaxial strain and stress conditions [J]. Appl. Math. Mech. 34, 229–238. Zhu, Z.,Ling, X.,Chen, S., et al., 2010. Experimental investigation on the train-induced subsidence prediction model of Beiluhe permafrost subgrade along the Qinghai–Tibet Railway in China [J]. Cold Reg. Sci. Technol. 62 (1), 67–75.