Experimental and theoretical investigations on the mechanical behavior of frozen silt

    Experimental and theoretical investigations on the mechanical behaviors of frozen silt Yang Yugui, Gao Feng, Lai Yuanming, Cheng Hongmei PII: DOI: Reference:

S0165-232X(16)30136-7 doi: 10.1016/j.coldregions.2016.07.008 COLTEC 2296

To appear in:

Cold Regions Science and Technology

Received date: Revised date: Accepted date:

7 April 2014 24 July 2016 30 July 2016

Please cite this article as: Yugui, Yang, Feng, Gao, Yuanming, Lai, Hongmei, Cheng, Experimental and theoretical investigations on the mechanical behaviors of frozen silt, Cold Regions Science and Technology (2016), doi: 10.1016/j.coldregions.2016.07.008

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ACCEPTED MANUSCRIPT Experimental and theoretical investigations on the mechanical behaviors of frozen silt

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Yang Yugui a,b, *, Gao Feng b, Lai Yuanming c, Cheng Hongmei a,b

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a. State Key Laboratory for Geomechanics and Deep Underground Engineering; China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

b. School of Mechanics and Civil Engineering; China University of Mining and Technology;

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Jiangsu, 221116, China

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c. State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Lanzhou, 730000, China

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Abstract

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The mechanical properties of frozen soil are complicated due to their complex components and

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sensitivity to temperature, water content and pressure. This study conducts triaxial compressive tests to experimentally investigate the mechanical properties of frozen silt under different confining

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pressures and at temperatures of -2.0, -4.0, -6.0 and -8.0C. Constitutive models, which are used to describe the response behaviors of natural and artificial materials under different loading and environmental conditions, were used as the bases for the description of the mechanical behavior of this frozen silt under external loads. After introducing continuous damage and statistics theories, a statistical damage constitutive model was proposed to reproduce the deformation of frozen silt. The Weibull distribution function was used in the model to describe the propagations of micro-voids and micro-cracks during loading process. The validity of this model was verified by comparing its modeling predictions with the experimental results. It was found that the predictions by this model agree well with the corresponding experimental data.

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ACCEPTED MANUSCRIPT Keywords: frozen silt; compressive test; constitutive model; artificial freezing method

1. Introduction

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The artificial freezing method has been widely used in metro engineering, tunnel construction,

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mine shaft and other underground engineering activities (Ma et al., 1999; Wang et al., 2005; Li et al., 2006). It has minimal effect on the subsidence disturbances of the ground surface and adjacent buildings, and can also be formed at any ground depth under complex geologic conditions (Cui, 1998;

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Zhang et al., 2002; Tang and Wang, 2007). Frozen soil has complicated mechanical properties due to

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its complex composition and sensitivity to temperature, water content and pressure (Ting, et al., 1983). The studies on the deformation behaviors and failure mechanisms of frozen soil are important to

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artificial freezing engineering construction. Therefore, constitutive model, which is used to describe

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the response of geomaterials under different loadings and environmental conditions, is a key issue in

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the theory development and numerical analysis for geotechnical engineering. The choice of an appropriate constitutive model, which will adequately describe the deformation

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behavior of the material, plays a significant role in the accuracy and reliability of numerical predictions. Since the early 1930s, much effort has been made towards the relationships between load and deformation of frozen soil (Tsytovich and Sumgin, 1937; Zhu and Carbee, 1987; Zhu et al., 1992; Da Re et al., 2003; Li et al., 2004; Arenson and Springman, 2005; Wang et al., 2005; Torrance et al., 2007; Qi and Ma, 2007; Lai et al., 2007, 2008; Shoop et al., 2008; Yang et al., 2010). Frozen soil is made up of solid mineral particles, ice inclusions, liquid water and gaseous inclusions (Goughnour and Andersland, 1968; Tsytovich, 1985). There are many fissures and cavities in frozen soil, as shown in Fig. 1. The distribution randomness of the micro-voids or micro-cracks in frozen soil causes the great uncertainty and randomness of the mechanical behaviors under an external

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ACCEPTED MANUSCRIPT load. The effect of the inner flaw evolution should be considered in order to study the stress-strain relationship of frozen soil. The probability and stochastic theories, which describe the concepts that

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are not certain, are used to study the variability of frozen soil. The continuum damage mechanic has

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been developed to describe the effects of the growth of the micro-voids or micro-cracks on the mechanical behavior of the materials (Murakami, 1983). Recent developments in engineering have brought about serious and enlarged demand for the reliability and safety ranging from civil and

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structural engineering to automotive engineering, which has caused more interest in continuum

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damage mechanics and its engineering applications (Murakami, 2012). With the process of loading, plastic flow of the ice and soil particles in the frozen soil occur, which will lead to the microstructure

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damage of frozen soil. In order to describe the damage deformation behavior, Lai et al. (2009)

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proposed an elastoplastic damage constitutive model for frozen sandy soil. Lai et al. (2008, 2012)

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proposed the statistical damage constitutive models to describe the stress-train relationship of warm and ice-rich frozen soil. Their results showed that the probability and stochastic theories are effective

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tools to describe the mechanical behavior of warm and ice-rich frozen soil. However, the related theoretical study on the mechanical uncertainty of artificial frozen soil, subjected to low temperature and high confining pressure, has not been found so far. The purpose of this study is to propose a statistical damage constitutive model to describe the mechanical properties of artificial frozen silt through considering the effects of confining pressure and temperature. Further, the Weibull function is used to describe the random distribution of the inner flaws. The constitutive model is then developed based on a continuous damage theory and the probability and statistics theories. The model parameters are determined by using the conventional triaxial compressive tests and model is verified by experimental data

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ACCEPTED MANUSCRIPT 2. Test conditions and results

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2.1 Experimental samples and method

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The test soil used was silt, and its particle distribution is listed in Table.1. The specimens were

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prepared as cylinders with a diameter of 6.18cm and a height of 12.5cm. The specimens had water content of 12.8% and the dry density of 1.85g/cm3. Their plastic limit and liquid limit are 15.0% and

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23.2%, respectively. The preparation for the specimens observed the following procedure: First, according to the water content of the specimen, the dry silt and the water were weighed for the

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specimens. Then, the water was added into the dry soil and mixed thoroughly. After the preparations

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of the soil-water mixtures were completed, the specimen was compacted in a split mold, and

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subsequently refrigerated at a temperature of approximately -35C for 48h. The specimen was quickly frozen in order to prevent ice lens formations caused by water transmission. At this point, the mold

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was dismantled and the specimen was subsequently coated with plastic film and covered by an epoxy resin cap to avoid moisture evaporation. The specimen was placed into the pressure cell of the

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MTS-810, whose confining and axial pressures could be controlled synchronously. The MTS-810 has a capacity of 250kN in axial force and 20MPa in confining pressure. Its loading displacement ranges from 0 to 75mm. Its accuracy is ±3% for long-term measurement and ±1% in short-term measurement. Next, the specimen was placed into the low temperature testing machine for 24h at the given temperature. Then, the confining pressure was applied to the specimen until the given pressure was reached and kept constant for 5 minutes. The triaxial shear tests began, and shear strain rate was 1.67 10 1/sec. 4

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ACCEPTED MANUSCRIPT 2.2 Experimental results and analysis

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The typical shapes of the frozen soil samples and stress–strain curves of the frozen silt at the

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confining pressures and temperatures of -2.0, -4.0, -6.0 and -8.0C are shown in Fig. 2 and Fig. 3,

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respectively. It can be seen that the stress–strain curves of the frozen silt experience approximately three stages: an initial linear elastic stage; a nonlinear strengthening stage; and a softening stage. In

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initial elastic stage, the stress increases linearly with the increase of the axial strain. With the further increase of the axial strain, the slopes of the stress-strain curves gradually decrease in the nonlinear

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strengthening stage. The slope of the stress-strain curve of the frozen silt gradually becomes negative with the increase of the axial strain in the softening stage. The stress-strain curve presents a strain

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softening phenomenon during the shearing process under the low confining pressures. Then, the strain softening phenomenon decreases with the increasing of the confining pressure, and even presents a

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strain hardening phenomenon under the high confining pressure. The corresponding peak strengths of the frozen silt under different confining pressures are shown

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in Fig. 4. This figure shows that the confining pressure and temperature greatly affect the strength of the frozen silt. Under low confining pressure  =MPa, the strength increases from 4.7MPa to 9.0MPa as the temperature decreases from -2.0C to -8.0C. Under the confining pressure of MPa, the strength increases from 11.7MPa to 16.3MPa. At a temperature of -2.0C, the strength increases from 4.7MPa to 11.7MPa, with the increase of confining pressure from 1.0MPa to 8.0MPa.

3. Statistical damage constitutive model for frozen soil The concept of an effective stress proposed by Kachanov (1958) has been used to formulate

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ACCEPTED MANUSCRIPT constitutive equations for damaged materials (Lemaitre and Chaboche, 1990). The effective stress theory postulates that the material damage is caused mainly by the decrease in the load-carrying

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effective area, due to the growth of micro-voids or micro-cracks. Suppose that A0 is the initial

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cross-sectional area of the undamaged specimen and A is the cross-sectional area of the damaged specimen. The damage parameter D is defined as the ratio of the voids to the total cross-sectional area of the specimen D  A0  A .

A0

 is the nominal stress applied to the damaged specimen,   F

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and ~ is the effective stress for the damaged specimen.

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A  A0 (1 D)

F  A 1 D

~  ~ 

A0

(1) (2)

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According to Hooke's law, the following equation can be given as

1  E1  (2  3 )

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E is the elastic modulus; and 

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where

(3)

is the Poisson's ratio.

Substituting Eq. (2) into Eq. (3) gets

1

  E  ( 1 D 1 D 2



3

) 1 D

(4)

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By combining this result with the damage variable D defined above, Eq. (4) can be rewritten in the following form,

  E(1 D)  (  ) 1

1

2

3

(5)

In order to formulate the evolution equation in the compression process, a comprehensive understanding of the evolution law for the damage variable D should be the first step. For this purpose, we herein rewrite Eq. (5) into the following form,

D  E   (  ) E 1

1

2

3

(6)

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Eq. (6) expresses the evolution of damage variable with stress, strain and mechanical parameters.

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ACCEPTED MANUSCRIPT That is to say that D can be evaluated when the values of 1, 2, 3 and 1, along with E and

 are

known. The values of 1 and the corresponding strain of 1 at different confining pressures 3, are

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chosen using the experimental stress-strain curves for the frozen soil specimens loaded at a certain

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strain-rate. The Young's modulus can be calculated from the initial slopes of the stress-strain curves, and the Poisson's ratio is estimated as 0.35 (He et al., 1999).

Now, according to Eq. (6), we can determine the damage evolution by means of a group of the

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experimental data. The damage variable D versus axial strain 1 is illustrated in Fig.5. It can be clearly

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seen that the damage variables increase as the axial strain 1 increases. The variation of damage D, under various confining pressures, is invariably limited within the range of 0 to 1. The damage

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initiates once an applied load is given, and continues to develop until the full damage state or D

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approaches to 1. In the case where the axial strain 1 reaches the same value, the damage variables D

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are slightly different under the different confining pressures, but the tendencies are roughly the same with the increase of axial strain.

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Various investigations have been implemented in order to characterize the damage evolution process. When considering the randomness of the distribution and the growth of micro-voids or micro-cracks in the geomaterials, a statistical method provides a powerful tool. The statistical damage approach, which can be quite an attractive tool for investigating the deformation process and failure mechanism in the mechanics of geomaterials, has been successfully applied to deal with the deformation response of soil materials (Li et al., 2012). In statistical damage mechanics, the state of the damage is characterized by a measure of the voids in a cross-section with a ratio. When N denotes the number of all mesoscopic elements, and N() denotes the number of all the failed mesoscopic elements, the damage variable D can be directly

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ACCEPTED MANUSCRIPT defined as

D  N( ) N

(7)

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to 1, corresponding to the intact and damaged states, respectively.

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where N ( ) is the number of damaged elements; N denotes all of the elements; and D varies from 0

In statistical constitutive modeling, the probabilistic distribution of the microscopic element failure strain in frozen soil is a critical piece of information. This is conventionally described by

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various empirical distributions, among which the Weibull distribution is extensively used. The

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Weibull distribution has been widely applied to model the damage behavior of geomaterials (Wang et al., 2007; Lai et al., 2008; Li et al., 2009; Deng and Gu, 2011; Li and Zhou, 2013). In this study, the

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failure strain of the frozen soil mesoscopic element is chosen as the random variable, and the

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probability density function of the Weibull distribution can be written as

where

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   th  1     th  f ( )  s ( ) exp    s s   s 

th

is the location parameter,

s

s

  

th  0 for frozen soil in this study; s

(8)

is the shape parameter;

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and s is the scale parameter.

When the strain reaches a certain value, the number of damaged elements N ( ) can be obtained as

      s  th N( )  0 Nf ( )dt  N 1 exp          s   

(9)

Substituting Eq. (9) into Eq. (7), the cumulative distribution function is

     s  th D( )  1 exp       s   Therefore, the statistical damage model can be obtained from Eq. (5) and Eq. (10) is as

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(10)

ACCEPTED MANUSCRIPT 

s



1  E exp    th     (2 3 )

(11)

  s  

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In order to describe the influences of the confining pressure and temperature, the damage variable

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D in Eq. (11) is regarded as a function of confining pressure and temperature, and the stress-strain

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relationship for conventional triaxial compression is 

s (3 ,T )

  th     s (3 , T ) 

1  E(3,T ) exp 

    (2 3 ) 

(12)



s

and

s

can be calculated using a linear regression method, from the

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The parameters

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where E(3,T) , s (3 ,T ) , s (3 ,T ) are the material parameters of the frozen soil.

experimental results of the triaxial compression. They can be obtained from Eq. (13) to Eq. (17) as

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follows

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     (  )  exp        E   s  1

2

(13)

s

3

1





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obtained as

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By taking the natural logarithm on both sides of the Eq. (13), the following expression can be

Y  X M

(14)

s



where X  ln ; Y  ln  ln



1  ( 2  3 )  ; and  exp(M / )  E1  s

s

Consider a set of experimental data points, ( 11 , 11 ), ( 12 , 12 ),…, ( 1n , 1n ) and corresponding ( X1 , Y1 ),( X2 , Y2 ),…, ( Xn , Yn ); in this study, Xi is an independent variable and

Yi is a dependent

variable. The model function has the form g ( x,  ) , where the adjustable parameters are held in the vector   (s , M ) . The least squares method finds its optimum when the sum Q of the squared residuals is a minimum as follows

Q   d   Y g(x , ) n

i 1

2

i

n

i 1

i

i

9

2

(15)

ACCEPTED MANUSCRIPT The minimum is determined by calculating the partial derivatives of Q(s , M ) with respect to

s

and M , and setting them to zero,

1

s

2

2

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s and s can be written as follows

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 XY  n  X Y ,  1  X  n  X 

(16)

1 n 1 n  Xi  Yi   ns  i 1  n i1 

s  exp 

(17)

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By solving Eq. (16), the expressions

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Q Q 0, 0 s M

4. Model verification and analysis

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The stress-strain curves are adopted again under the confining pressures of 1.0, 3.0, 5.0 and 8.0MPa,

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as illustrated in Fig. 3. The proposed statistical damage model is now used to simulate some typical

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laboratory tests to check its validity. The entire deformation process of the frozen silt is calculated using the parameters in Table 2. Fig.6 shows the stress-strain curves obtained by both the computation

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and experiment. In order to verify the adaptability of the model for other frozen soil, comparisons of the stress-strain curves between the test results and predicted results of the frozen sand have also been

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given in Fig.7. Fig.7 shows that the model gives relatively good predictions of the measured stress-strain curve under different confining pressures. The influences of confining pressure and temperature on the deformation behavior of frozen soil are found to be correctly described by the proposed model.

5. Summaries and Conclusions In order to understand the mechanical behaviors of frozen soil, the mechanical characteristics of frozen silt were investigated through a series of experimental data studies under different confining pressures and at temperatures of -2.0, -4.0, -6.0 and -8C, respectively. The results show that the mechanical characteristics of frozen silt are greatly affected by confining 10

ACCEPTED MANUSCRIPT pressure and temperature. The stress-strain curve presents strain softening during the shearing process under low confining pressure. However, with increase of confining pressure, the strain softening

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phenomenon decreases, and the stress-strain curve presents strain hardening under high confining

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pressure. Under the low confining pressure (1.0MPa), as temperature decreases from -2.0C to -8.0C, the strength increases from 4.7MPa to 11.7MPa (~250%). Under the high confining pressure (8.0MPa), the strength increases from 9.0MPa to 16.3MPa (~180%).

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To reproduce the deformation behavior of the frozen soil, a statistical damage constitutive model,

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in which the Weibull function was employed to describe the random distribution of inner flaw, was proposed based on the continuous damage theory and probability and statistics theories. All of the

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parameters of the model were determined by the experimental data. It was found that the predictions

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Acknowledgements

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by this model agree well with the corresponding experimental data.

We would like to kindly thank our anonymous reviewers and Editors, whose constructive

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comments were helpful for the revision of this study; and J. Wang for his assistance during this work. This research is supported by National Natural Science Foundation of China (51574219), the Fundamental Research Funds for the Central Universities (2015QNA61).

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ACCEPTED MANUSCRIPT situation of the rock surrounding the tunnel in cold regions under the conditions of different construction seasons, initial temperatures and insulations. Tunneling and Underground Space

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Zhu, Y.L., Carbee, D., 1987. Creep and strength behaviour of frozen silt in uniaxial compression. U.S.A. CRREL report, p. 8710.

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compression. Journal of Glaciology and Geocryology. 14(3), 210-217.

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ACCEPTED MANUSCRIPT List of Figure Captions Fig. 1 Electron microscope image of frozen soil(×350)

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Fig. 2 Shapes of frozen soil samples

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Fig. 3 Stress-strain curves of frozen silt at different temperatures and confining pressures Fig. 4 Strength of frozen silt at different confining pressures Fig. 5 Relationships between damage and axial strain

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Fig. 6 Comparisons of stress-strain curves between test results and predicted results of frozen silt

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Fig. 7 Comparisons of stress-strain curves between test results and predicted results of frozen sand (The experimental data are cited from Yang et al. (2010)).

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Fig. 1 Electron microscope image of frozen soil(×350)

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(a) Before test (b) After test Fig. 2 Shapes of frozen soil samples

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T

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SC R

 (MPa)

12

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T= -2.0C 1.0 MPa 3.0 MPa 5.0 MPa 8.0 MPa

9

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6

0 0

2

4

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3

6

8

10

12

14

16

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12

14

16

18

1 

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TE

D

（a）T=-2.0C

 (MPa)

AC

15

12

T= -4.0C 1.0 MPa 3.0 MPa 5.0 MPa 8.0 MPa

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6

3

0 0

2

4

6

8

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1 

（b）T=-4.0C

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T= -6.0C 1.0 MPa 3.0 MPa 5.0 MPa 8.0 MPa

T

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 (MPa)

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4

0 2

4

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8

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0

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1 

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（c）T=-6.0C

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 (MPa)

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T= -8.0C 1.0 MPa 3.0 MPa 5.0 MPa 8.0 MPa

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8

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0 0

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1 

（d）T=-8.0C Fig. 3 Stress-strain curves of frozen silt at different temperatures and confining pressures

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T= -6.0C 12

T

T= -8.0C

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 (MPa)

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T= -2.0C 9

0

2

4

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6

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 (MPa)

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Fig. 4 Strengths of frozen silt at different temperatures and confining pressures

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0.8

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T= -2.0C 1.0 MPa 3.0 MPa 5.0 MPa 8.0 MPa

0.6

0.4

0.2

3

6

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0

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Damage parameter D

1.0

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Strain ()

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（a）T=-2.0C

D TE

0.8

T= -4.0C 1.0 MPa 3.0 MPa 5.0 MPa 8.0 MPa

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CE P

Damage parameter D

1.0

0.6

0.4

0.2

0

3

6

9

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Strain ()

（b）T=-4.0C

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0.4

0.2

3

6

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0

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T= -6.0C 1.0 MPa 3.0 MPa 5.0 MPa 8.0 MPa

0.6

T

0.8

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Damage parameter D

1.0

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15

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15

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Strain (%)

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（c）T=-6.0C

TE

0.8

0.6

T= -8.0C 1.0 MPa 3.0 MPa 5.0 MPa 8.0 MPa

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Damage parameter D

D

1.0

0.4

0.2

0

3

6

9

12

Strain (1)

（d）T=-8.0C Fig. 5 Relationships between damage and axial strain

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15

 (MPa)

12

T

T= -2.0C 1.0 MPa 3.0 MPa 5.0 MPa 8.0 MPa

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9

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6

3

0

2

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6

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0 8

10

12

14

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1 

D

（a）T=-2.0C

15

TE

T= -4.0C 1.0 MPa 3.0 MPa 5.0 MPa 8.0 MPa

CE P

 (MPa)

12

9

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6

3

0 0

2

4

6

8

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1 

（b）T=-4.0C

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12

14

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T= -6.0C 1.0 MPa 3.0 MPa 5.0 MPa 8.0 MPa

T

12

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 (MPa)

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8

0 0

2

4

6

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4

8

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1 

D

（c）T=-6.0C

TE

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CE P

 (MPa)

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T= -8.0C 1.0 MPa 3.0 MPa 5.0 MPa 8.0 MPa

12

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8

4

0 0

2

4

6

8

10

12

14

16

1 

（d）T=-8.0C Fig. 6 Comparisons of stress-strain curve between test results and predicted results

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0.5 MPa 1.0 MPa 3.0 MPa 10.0 MPa

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6

4

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 (MPa)

8

0 4

8

12

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0

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2

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1 

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Fig. 7 Comparisons of stress-strain curves between test results and predicted results of frozen sand (The experimental data are cited from Yang et al. (2010)).

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Table 1. Basic physical parameters of silt

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Composition of particle diameters （%）

T

Tables

0.50~0.25mm

0.25~0.10mm

0.10~0.075mm

0.075~0.05mm

0.05~0.005mm

0

0.792

22.798

13.50

17.15

35.93

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CE P

TE

D

MA

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>0.50mm

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<0.005mm 9.83

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0.0661

0.703

0.0534

3MPa

0.622

0.0607

0.594

0.0498

5MPa

0.553

0.0565

0.563

0.0495

8MPa

0.523

0.0465

0.489

0.0311

0.758

0.582

0.0381

0.582

0.0381

0.502

0.0351

0.502

0.0351

0.468

0.0301

0.468

0.0301

0.758

MA D TE CE P AC

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0.0474

SC R

0.86

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1MPa

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Table 2. Principal parameters determined by fitting analysis at different temperatures Confining -2.0C -4.0C -6.0C -8.0C pressures s s s s s s s s 0.0474

ACCEPTED MANUSCRIPT Highlights

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1. The compressive tests of frozen silt have been conducted at -2,-4,-6,-8C;

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2. The nonlinear characteristics of the stress-strain curves were analyzed; 3. The evolution process of damage has been investigated;

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4. The statistical damage constitutive model for frozen silt is given.

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