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A creep constitutive model for frozen soils with different contents of coarse grains
Accepted Manuscript A creep constitutive model for frozen soils with different contents of coarse grains
Feng Hou, Yuanming Lai, Enlong Liu, Huiwu Luo, Xingyan Liu PII: DOI: Reference:
S0165-232X(17)30486-X doi:10.1016/j.coldregions.2017.10.013 COLTEC 2469
To appear in:
Cold Regions Science and Technology
Received date: Revised date: Accepted date:
12 April 2016 22 June 2017 12 October 2017
Please cite this article as: Feng Hou, Yuanming Lai, Enlong Liu, Huiwu Luo, Xingyan Liu , A creep constitutive model for frozen soils with different contents of coarse grains. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Coltec(2017), doi:10.1016/j.coldregions.2017.10.013
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ACCEPTED MANUSCRIPT
A creep constitutive model for frozen soils with different contents of coarse grains Feng Hou1, Yuanming Lai2, Enlong Liu1,2* , Huiwu Luo1, Xingyan Liu2 State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resources
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1
and Hydropower, Sichuan University, Chengdu 610065, P.R.China;
State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and
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2*
Corresponding author: Enlong Liu
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*
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Engineering Institute, Chinese Academy of Sciences, Lanzhou 730000, P.R.China.
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Email: [email protected]
Abstract: The triaxial creep tests of frozen silty clay mixed with different contents of coarse grains
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were performed at 0.3 MPa confining pressure at -10 °C. Four sets of mass ratio of coarse-grained
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particles to silty clay are 0.0, 0.2, 0.4 and 0.6 used to form the frozen samples, and the test results demonstrate that: when the shear stress level is low, the tested sample exhibits an attenuation creep;
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when the shear stress level is high, the tested sample behaves as a non-attenuation creep. It is found that the initial shear modulus and yield strength become greater with the increasing mass ratio of
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coarse-grained particles to silty clay, but the long-term strength shows an opposite trend. Through the analysis of creep mechanism, both the new hardening variable and damage variable are introduced to improve the Nishihara model, and the three-dimensional formulation of the model is also formulated. According to the experimental results and analysis on the creep model, the model parameters are determined and identified. The simulation results show that the new model can describe the whole creep process of artificial frozen soils with different contents of coarse-grained particles well. Finally,
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the influences of the mass ratio of coarse-grained particles to silty clay on the parameters are also analyzed, which provides a reference for engineering application in cold regions.
Keywords: hardening; damage; mass ratio of coarse-grained particles to silty clay; creep constitutive
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mode
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1 Introduction
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In the past few decades of years, a number of geotechnical engineering problems have been arising (Lai et al., 2013) with more and more civil engineering projects constructed in the Qinghai-Tibet plateau.
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Many investigators have proved that the creep properties of frozen soils have a great influence on the
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long-term safety of structures (Yu et al., 2013; Yang et al., 2010). With the rapid development of hydraulic and hydropower engineering in China, there are some dams designed in cold regions
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undergoing great change of temperature. The materials used in the core wall of these dams are usually
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composed of mixed coarse grains and silty clays for preventing failure resulted from seepage and external loads. When these dams located in cold regions are working, the creep properties of these core
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wall materials at low temperature below zero will affect the safety of these dams. Therefore, it is necessary to investigate the creep properties and constitutive model for these frozen mixed soils of core
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wall dam materials.
Both performing laboratory tests and formulating constitutive models are conventional methods employed to study the creep properties of frozen soils. Therefore, many researchers have conducted creep tests under different shear stress levels and conditions to study the creep features of frozen soils (Yang et al., 2014; Li et al., 2011). For example, Zhou et al. (2016) conducted a series of triaxial compression and creep tests on frozen loesses under different confining pressure and temperature
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conditions in order to study the rate-dependent mechanical behaviors; and Zhao and Zhou (2013) carried out a series of uniaxial creep tests on frozen saturated clay under various thermal gradients and creep stresses by GFC (freezing with non-uniform temperature without experiencing K 0 consolidation) method to investigate the creep behaviors of frozen soil with thermal gradient. However, there are few
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studies on creep properties of frozen mixed soils, especially the influences of different components
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on the creep properties of frozen mixed soils. In addition, the constitutive models, used to reflect the
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response behavior of natural and manufactured materials under different stress and environmental conditions, are the bases of describing the mechanical behavior of materials under external load and
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usually expressed in terms of stress, strain and time (Willam, 2002). Hence, there are many creep
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models have been put forward by researchers, which can be roughly categorized as one of the following types (Liingaard et al., 2004): 1) empirical models; 2) rheological models; and 3) general
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stress–strain–time models. The rheological models consisting of a series of mechanical elements, e.g.
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Hookean spring, Newtonian dashpot and Saint Vernant’s slider, are widely accepted for its simplicity, convenience and good performance to represent the main deformation behavior of geomaterials. Four
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well-known models widely used are the Maxwell model, the Kelvin–Voigt model, the Bingham model,
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and Nishihara model. Li et al. (2011) deduced an improved Nishihara creep model for the frozen deep clay by combining a generalized Kelvin model and an improved visco-plastic body. Wang et al. (2014) proposed a simple model by combining Maxwell, Kelvin and Bingham body with a parabolic yield criterion so as to describe the settlement of underlying warm and ice-rich permafrost. Yang et al. (2014) proposed a new creep constitutive model of frozen silt by modifying the generalized Burgers model to reproduce the time-dependent deformation. Li et al. (2011) established a new creep constitutive model for describing frozen soils’ creep characteristics under high confining pressures based on the triaxial
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creep experiment of artificial frozen soil in deep alluvium. According to the theory and results of compressional tests, there are hardening effects with deformation development (Yao et al., 2009; Yao et al., 2013; Yao et al., 2015), and thus as one kind of deformation mechanism, the hardening effects should also be considered when a creep deformation occurs. What is more, the creep mechanism has
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been analyzed by many researchers with the aid of computer scanning technology, and it is found that
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there are two rival effects existing in the creep process: hardening and damage (Miao et al., 1995; Fan
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et al., 2010). When formulating the creep model for frozen soils, many attentions have been paid to the damage effect with neglecting the hardening effect(Zhu et al., 2010; Zhou et al., 2011; Li et al., 2012;
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Zhou et al., 2013; Kang et al., 2015).
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From the analysis on the existing creep models for frozen soils mentioned above, we know that few of them consider the combining effects of hardening and damage mechanism during the process of
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creep. The Nishihara model is widely used to model the creep properties of frozen soils, which can be
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revised or improved to simulate the effects of coarse-grained particles on the creep properties of frozen mixed soils and will be carried out here. In this paper, firstly, we carried out the triaxial creep tests on
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the frozen silty clay mixed with different contents of coarse grains with the same dry density and their
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creep properties were analyzed, followed by putting forward an improved Nishihara model based on the new hardening variable and damage variable based on the triaxial creep test results.. Furthermore, the parameters of the creep model have been analyzed and the new model is also verified with creep tested results of frozen mixed soils. Finally, the influences of the contents of coarse grains on the model parameters are analyzed. 2 Triaxial creep tests of frozen silty clay mixed with different contents of coarse grains 2.1. Test conditions
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The tested soils were silty clay and coarse grains with 2-4 mm particle size of quartz. Their physical parameters are shown in Table 1 and Fig. 1, respectively. We define λ as the mass ratio of coarse grains to silty clay in the sample for convenience. First, the silty clay, dried and sieved through the 1 mm screen, was prepared with water content of 16.0 %, and kept for 24 h without evaporation, so that its
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moisture was uniformly distributed. The silty clay and coarse grains were weighted according to λ
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equal to 0.0, 0.2, 0.4, and 0.6, respectively, then the prepared silty clay and coarse grains were mixed
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uniformity, and filled in a cylindrical mold to make cylindrical soil specimens under a certain suitable compression rate provided by the machine, by which the specimens were prepared as cylinders with
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diameter and height of 6.18 cm and 12.5 cm, respectively. Then, the specimens were placed in another
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mold with three same parts and saturated for over 12 h under a vacuum for 3 h. In order to avoid a large frost heave and prevent moisture migration, the soil samples submerged in distilled water were placed
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in a refrigeration unit and frozen quickly at a temperature of -30 °C. After 48 h of freezing, the molds
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were removed and the specimens were mounted with epoxy resin plates on both ends and covered with a rubber sleeve to avoid moisture evaporation. Finally, the samples were then kept in an incubator for
Table 1
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temperature.
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over 24 h at the test target temperature of -10 °C such that the specimens adopted a uniform
The physical parameters of the silty clay and coarse grains Specific gravity
Liquid limit
Plastic limit
Water content
(%)
(%)
(%)
Silty clay
2.72
27.58
19.37
1.35
coarse grain
2.66
--
--
0.0
5
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Fig. 1. Grain distribution of the silty clay
The test equipment used in this study is a cryogenic triaxial apparatus modified from the MTS-810
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material test machine, which contains the pressure chamber to apply the confining pressure and axial
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pressure, whose schematics was illustrated by Lai et al. (2012). After the samples mentioned above were prepared, they were placed into the pressure cell of the MTS-810 material test machine and a
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series of the creep tests were performed. First, the temperature in the pressure chamber was set at
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-10 °C with a precision of ±0.1 °C. Before the creep testing, the prepared samples were consolidated under confining pressure of 0.3 MPa for 30 min prior to the axial load, respectively. The triaxial creep
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tests on samples were conducted under different shear stress levels. In the initial loading, the axial load
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was applied to the tested samples from zero to the stress level value of the creep test within 60 s. During the process of testing, the axial pressure on the specimens was kept constant with a precision of ±10 kPa.
2.2. Test results and analysis The curves of the axial strain-time obtained by the triaxial creep tests of frozen mixed soil samples under different shear stress levels are shown in Fig. 2. As can be seen from Fig. 2, there are two types of creep, i.e. an attenuate creep and a non-attenuate creep. When the shear stress is lower than the
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long-term strength, the deformation with time is an attenuate creep. When the shear stress is higher than the long-term strength, it turns to be a non-attenuation creep and can be divided into three stages, i.e. the unsteady stage, steady stage and accelerating stage. The tested sample quickly fails once it enters the accelerating stage. For λ=0 (pure silty clay), the creep curves exhibit three stages at higher
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shear stress level as shown in Fig 2 (a). But with the increasing of λ (contents of coarse-grained
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particles), the creep curves mainly behave as two stages at higher stress level, in which the last stage of
q=σ1-σ3=7.75MPa q=7.34MPa
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accelerating stage is not obvious (see Fig. 2 (d) for λ=0.6 ).
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Axial strain ε1( %)
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q=6.54MPa
q=5.71MPa
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q=4.08MPa
Time (h) (a) λ=0.0
q=5.85MPa
Axial strain ε1( %)
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q=6.69MPa q=6.27MPa
q=5.43MPa q=5.01MPa q=3.33MPa
Time (h) (b) λ=0.2
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q=5.75MPa
q=5.39MPa
Axial strain ε1( %)
q=5.03MPa
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q=4.31MPa
q=5.84MPa
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Time (h) (c) λ=0.4
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q=3.59MPa
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Axial strain ε1( %)
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q=5.08MPa
q=4.36MPa q=3.63MPa
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q=4.71MPa
(d)Time λ=0.6(h)
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Fig. 2. The creep test results of frozen mixed samples
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2.2.1 The initial shear modulus By the triaxial creep tests, we can obtain the different initial stains under different shear stress levels as shown in Fig. 3. In general, the initial stain is deemed to the elastic stain, which can be computed by Eq. (1), without considering the volumetric change. So we can get the initial shear modulus for λ=0.0, 0.2, 0.4, and 0.6 are 0.37 MPa, 0.91MPa, 1.21MPa, and 1.97 MPa, respectively. It is found that the initial shear modulus increases with increasing λ, which means that the higher the coarse-grained content of the sample, the greater the initial shear modulus is.
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q , 3G0
(1)
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where 0 is the initial stain and G0 is the initial shear modulus.
Fig. 3. The initial shear modulus under different λ
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2.2.2. The yield stress s
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The yield stress s can be obtained by the stress-stain curve of triaxial tests as shown in Fig. 4, in which we take the result of frozen mixed sample with λ=0.0 as an example. Following the method
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proposed by Ma and Wang (2014), it is easy to obtain the yield stress s =2.25 MPa when λ=0.0. So,
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in the same way, we can get the yield stress of frozen samples with λ= 0.2, 0.4, and 0.6 are 3.76 MPa, 3.90 MPa and 4.16 MPa, respectively. The yield stress s becomes greater with the increase of λ, and
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the higher the content of coarse grains is, the larger the elastic stage is.
The yield stress σs
Fig. 4. The yield stress s of frozen sample with λ=0.0 9
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finally behave an accelerating failure stage once it enters the steady creep stage with a constant creep
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rate, and in this stage, the greater the shear stress is, the larger the creep rate is. So based on the creep
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rate of the steady creep stage, a method is put forward here to determine the long-term creep strength. Firstly, the relationship between the creep rate of steady creep stage and the shear stress is obtained as
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shown in Fig. 5. Then, an equation is proposed to fit the relationship between the rate and the shear
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stress as follows:
v q k ln 1 , v0
(2)
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where k is a material constant, whose unit is MPa; v is the creep rate of steady creep stage; and v0 is the
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reference of creep rate, and can be taken as 1.0.
Fig. 5. The long-term strength under different λ So, according to Eq. (2), we can obtain the long-term strength of frozen samples with λ=0.0, 0.2, 0.4,
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and 0.60 are 6.32 MPa, 5.32 MPa, 4.85 MPa, and 4.57 MPa, respectively, as shown in Fig.5. The long-term strength decreases with increasing λ, and the higher the content of coarse grains is, the lower the damage stress threshold is. 3 A creep constitutive model based on hardening and damage
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Hardening comes from the compression of micro cracks, the emergence of the new connection between
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soil particles and the recombination of mineral grains, which make the strength higher during the
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process of creep deformation. Usually, there is an initial damage in the frozen sample within the initial state, resulted by the micro cracks. During the loading process of creep,
there are two deformation
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mechanisms of hardening and damage existing within the frozen soils simultaneously. When the shear
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stress level is lower, the sample exhibits a non-attenuation creep, which shows the characteristics of viscoelasticity, and this is because under the low stress level, the initial micro cracks of the sample
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tends to be closed, and the sample behaves hardening; when the shear stress level is higher, the initial
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micro cracks continue to develop and accumulate, resulting in destruction of the continuity of frozen sample and the sample damages gradually.
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3.1 The hardening parameter H
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The hardening will not occur until the plastic deformation has happened, which shows that the frozen samples become stronger with the growth of the plastic deformation. So, the hardening parameter is introduced here to reflect the change of the viscous coefficient of visco-plastic element, which has the following properties: H 0 when t 0, and H a certain constant when t . We can define H A 1 e t ,
(3)
where H is the hardening parameter; and A denotes the level of the hardening effect, and the bigger its value is, the greater the strengthening effect is, with A≥0.
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3.2 The damage variable D The experimental investigation has verified that the damage effect has a threshold value [Li et al., 2012; Sidoroff, 1981; Ju, 1989; Martin and Chandler, 1994; Aubertin and Simon, 1997; Eberhardt et al., 1997]. Only if the loading stress exceeds the threshold value, the microscopic cracks initiate, propagate
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and evolve such that the creep damage accumulates and the accelerating creep occurs. The stain rate of
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the frozen sample increases non-linearly in this process, and failure will take place once the strain
damage variable as E , t E0
E0 E , t E0 exp Ct
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and
,
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D , t 1
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reaches a critical value (Yang et al., 2014). Therefore, based on damage mechanics, we can define the
0
0
(4)
,
(5)
where D , t is the damage variable; E , t is the elastic modulus at time t; E0 is the initial
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elastic modulus; and C is the material constant to reflect the influence of damage.
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Substituting Eq. (5) into Eq. (4), we have
0
0 D , t Ct 1 e
0
.
(6)
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Based on the definition of the effective stress [Kachanov, 1986], it can be defined as follows:
1 D
,
(7)
where is the effective stress; and is the Cauchy stress. So, we can obtain
Ct e
0 0
.
(8)
3.3 A creep constitutive model based on hardening and damage The Nishihara model can reflect the creep properties of elastic, viscoelastic and viscoplastic, 12
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recognized by a greatnumber of researchers (Li et al., 2011). In this paper, the hardening parameter and
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damage variable are introduced to improve Nishihara model, as shown in Fig. 6.
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According to a series rules, the following relation holds:
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Fig. 6. The improved Nishihara model
e ve vp , and E vp ,
(9)
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where E and e are the stress and strain of the elastic element, with E E0 e ; and ve are the stress and strain of Abel dashpot; vp and vp are the stress and strain of visco-plastic element; and
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and are the total stress and strain.
(1) When s , according to the elasto-plastic theory, vp =0, then, the model is degenerated to be
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the Kelvin model, there is no hardening and damage effects during the deformation. So, we can get
E0
E1 1 exp t , E1 1
(10)
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e ve
where E1 and 1 are the parameters of the viscoelastic element. (2) When s , the hardening effect occurs with the growing plastic deformation, but the damage does not work. So, the total strain can be expressed as
e ve vp
E0
E1 s t , 1 exp t E1 1 2 H
(11)
where 2 H is the viscous coefficient of the visco-plastic element with the hardening effect considered, which can be expressed as follows by use of hardening parameter H defined in Eq. (3): 13
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(12)
where 20 is the initial viscous coefficient of the visco-plastic element. (3) When , both hardening effect and damage effect works with the development of the plastic deformation. Hence, the damage effect should be considered. There is no damage effect in both
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the elastic element and viscoelastic element when the stress and strain grow. But, for the visco-plastic
E0
E1 s t. 1 exp t E1 1 2 H
(13)
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e ve vp
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element should be the effective one . Then we can obtain
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element, it will damage because of the plastic deformation, and the total stress of the visco-plastic
So, in summary, the constitutive equation of the improved Nishihara model in one dimension is
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expressed as follows:
D
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e ve vp
E 1 exp 1 t E0 E1 1 E s 1 exp 1 t t 1 2 H E0 E1 1 exp E1 t s t E0 E1 1 2 H
s s .
(14)
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3.4. Three-dimensional formulation of the creep constitutive model
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In general, the frozen soil is in the three-dimensional stress state, and the shear deformation is the main source of creep. Thus, in this study, the volume deformation is not considered, and we assume that the material is isotropic.
In three-dimensional stress state, the total strain of the model can be easily obtained and expressed as:
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q s
F Q G1 1 t 1 exp t 1 2 H F0 ij
s q
F Q G1 1 t 1 exp t H F0 ij 1 2
q
,
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s s ij ij 2G0 2G1 sij sij 2G0 2G1 sij sij 2G0 2G1
(15)
where sij is the Cauchy deviatoric tensor of stress; G0 is the shear modulus of the elastic element;
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G1 is the shear modulus of the viscoelastic element; q is the shear stress; F is the yield function;
F0
is
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the initial reference of the yield function, and can be taken as 1.0; Q is the plastic potential function,
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and the tested material is assumed to be associated flow rule with F Q ; and is the form of the power function, and its exponent sign is taken as 1.0; and ij and ij are Cauchy stress tensor and
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effective stress tensor, respectively.
According to Yu et al., (1985), the effect of the mean stress on creep of frozen soils can be neglected,
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and the deviator stress plays a main role in the creep. So, the yield function can be chosen as:
F J2
s 3
,
(16)
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where J 2 is the second diviatoric stress tensor invariant, which should be the effective second diviatoric stress tensor invariant when q .
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Under the triaxial creep test, 2 3 , and thus we have
m
1 1 2 1 2 3 , s11 1 m 1 3 , J 2 1 3 . 3 3 3
(17)
Thus, Eq. (15) can be rewritten as:
G1 3 3 1 1 1 exp t 3G1 3G0 1 G s 1 1 3 1 3 1 exp 1 t 1 3 t 3G1 32 H 1 3G0 1 3 1 3 1 exp G1 t 1 3 s t 3G 3G1 32 H 0 1 15
q s
s q , q
(18)
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1 1 D
1eCt ; and 3
3 1 D
3eCt .
4. The determination of model parameters and model verification 4.1. The determination of parameters 4.1.1 Parameters of the visco- elastic element G1 and 1
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Under the same content of coarse grains, we can obtain the varying rules of G1 and 1 with the
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shear stress when q s as follow:
1 e q ,
and
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G1 e q ,
(20) (21)
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where , , , and are materials patameters, which can be determined by test results.
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4.1.2 Parameters of the visco-plastic element 2
The visco-plastic parameter 2 contains two parameters 20 and A, and the first does not change with
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the varying stress level as the initial viscous coefficient 20 is constant. The parameter A is to reflect
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assumed as follows:
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the hardening level of material, which decreases with the increasing deviator stress q, and thus, it is
A e
q s
s
,
(22)
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where and are parameters of material. 4.1.3 Damage parameter C As we know, the greater the shear stress is, the higher level the damage is. According to the test results and the above analysis, the relationship between C and shear stress is assumed as follows: m
q C k , where k and m are parameters of material. 4.2. Model verification 16
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From the triaxial creep test results under the four contents of coarse grains, the proposed model parameters are given in Table 2, and the calculated results of creep curves are shown in Fig. 7. From Fig. 7, it is found that the calculated results of the proposed model are in good agreement with the tested results under the different values of λ. The proposed model can not only accurately reproduce the
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unsteady stage when the shear stress is lower than the long-term strength, but also can predict the
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steady stage and progressive stage when the shear stress is greater than the long-term strength.
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Table 2 The proposed model parameters η1(MPa)
G1(MPa) α
θ -1
γ -1
(MPa )
(MPa)
(MPa )
280 763 780 2600
0.26 0.62 0.72 0.93
13981 1245 571 458
0.94 0.74 0.69 0.54
1000 500 50 13
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(MPa)
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0.0 0.2 0.4 0.6
β
20 (MPa)
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λ
(a) λ=0.0.
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μ
20.34 20.09 14.66 10.00
A
C χ
k
0.81 1.81 1.90 1.93
19.05 5.95 1.20 0.20
m
1.57 1.47 1.40 1.39
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D
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(b) λ=0.2.
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(c) λ=0.4.
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(d) λ=0.6.
Fig. 7. Comparison between tested results and calculated ones
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In order to verify the applicatity of model proposed here for other experimental results by different
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scholars, we used the experimental resulus for artificial frozen silt samples at -8 °C by Yang et al.(2014). The computational parameters obtained from the test results are as follows: σ s=9.2 MPa, G0=66700 MPa, α=2500, β=0.38, θ=8500, γ=0.47,
20 =14 MPa, μ=15, χ=0.9, κ=0.45
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σ∞=9.43MPa,
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and m=1.4. The comparsions of computated and testes results are shown in Fig. 8, which demonstrates that the new creep model propsoed here can also predict the creep properties of artificial frozen silt
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samples.
Fig. 8. Comparisons between tested results (Yang et al., 2014) and calculated ones
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5 Discussions In the following, the influence of coarse-grained content is discussed firstly. According to the test results shown in Table 2, we can obtain the relationships between the parameters (G1, η1, A and C) and
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G1 (MPa)
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λ, as shown in Figs. 9-12.
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q (MPa)
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η1(MPa)
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Fig. 9. The relationship between G1 and q under different λ
q (MPa)
Fig. 10. The relationship between 1 and q under different λ
20
PT
A
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(q-σs)/σs
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D
C
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Fig. 11. The relationship between A and (q-σs)/σs under different λ
(q-σ∞)/σ∞
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Fig. 12. The relationship between C and (q-σ∞)/σ∞ under different λ
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Fig. 9 shows that the initial shear modulus decreases with the increasing of shear stress under the same content of coarse grains, and the higher of the coarse grains content is, the faster the initial shear modulus reduces with the increasing shear stress. From the Figs. 10 and 11, we know that 1 and A have the same properties, which decrease with the increasing of shear stress, and increase with the increasing of coarse-grained content. But from the Fig. 12, we know that damage parameter C increases with the increasing of shear stress, and decrease with the increasing of coarse-grained content, which means that the larger the coarse-grained content is, the more difficult the sample fails.
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In the new creep model proposed, the non-attenuation creep at lower shear stress level and attenuation creep at higher shear stress level can be modeled simultaneously. For considering these effects, we introduce two competing physical mechanisms, including hardening and damage, during the creep process. For the same confining pressure, when the shear stress level is lower, the frozen soil
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sample will exhibit attenuation creep, which is mainly dominated by hardening due to the closeness of
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micro cracks; while when the shear stress level is higher, the frozen soil sample will exhibit
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non-attenuation creep, which is mainly dominated by gradually damage due to the propagation and development of micro cracks. In addition to the creep model properties considered above, the
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influences of mass ratio on the creep behavior can also be taken into account by varying the model
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parameters.
In order to verify the applicability of the existing creep model to the experimental data, we use both
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the Nishihara model and the propsed model here to predict the tested results of the forzen soils with
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mass ratio of λ=0.2. The computed parameters are the same as those shown in Table 2. The comparisons between results predicted by Nishihara model (the old model) and proposed model (the
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new model) are shown in Fig. 13, which demonstrates that the Nishihara model (the old model) can
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model the attenuation creep at lower shear stress level well, but with the increasing shear stress level, the Nishihara model (the old model) will give incresing errors between the predicted and tested results, and for the non-attenuation creep at higher shear stress level, the Nishihara model (the old model) can not give the accelerating stage. 6 Conclusions Based on the analysis of a series of the triaxial creep tests results of frozen silty clay mixed with coarse grains at 0.3 MPa confining pressure and -10 °C, an improved Nishihara model is formulated to
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predict the creep deformation behavior of frozen soil. Some conclusions can be drawn as follows:
Fig. 13. Comparisons between results predicted by Nishihara model (the old model) and proposed
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model (the new model)
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(1) The test results show that the sample exhibits an attenuate creep when the shear stress is lower than the long-term strength, and an accelerating creep failing in accelerating stage when the shear stress
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is greater than the long-term strength.
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(2) The hardening parameter and the damage variable are introduced to the Nishihara model to describe the hardening and damage effects in the process of creep. Though validating the proposed
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model by the test data of frozen silty clay mixed with coarse grains under different coarse- grained
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contents, it is found that the proposed model can describe the creep behavior of frozen silty clay mixed with coarse grains well. (3) The initial shear modulus G1 , the viscous coefficient of elastic element 1 , and the hardening parameter A decrease with the increase of shear stress, and increase with the increasing content of coarse grains. But the damage factor C increase with the increase of shear stress, and decrease with the increasing of coarse-grained content. References
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Aubertin M., Simon R., 1997. A damage initiation criterion for low porosity rocks. International Journal of Rock Mechanics and Mining Sciences 34 (3-4), 017. Eberhardt E., Stead D., Stimpson B., Read R.S., 1997. Changes in acoustic event properties with progressive fracture damage. International Journal of Rock Mechanics and Mining Sciences 34 (3-4),
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Fan Q.Y., Yang K.Q., Wang W.M., 2010. Study of creep mechanism of argillaceous soft rocks.
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Chinese Journal of Rock Mechanics and Engineering, 29(8), 1555-1561. (in Chinese) Ju J.W., 1989. On energy-based coupled elastoplastic damage theories: constitutive modeling and
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simulation for warm ice-rich frozen silt. Cold Reg. Sci. Technol.71, 102–110. Lai Y.M., Xu X.T, Dong Y.H., Li S.Y., 2013. Present situation and prospect of mechanical research on frozen soils in China. Cold Reg. Sci. Technol. 87, 6–18. Li D.W., Fan J.H., Wang R.H., 2011. Research on visco-elastic-plastic creep model of artificially frozen soil under high confining pressures. Cold Reg. Sci. Technol. 65, 219–225. Liingaard M., Augustesen A., Lade
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behavior of soils. Int. J. Geomech. 4(3), 157-177.
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Wang S.H., Qi J.L., Yin Z.Y., Zhang J.M., Ma W., 2014. A simple rheological element based creep model for frozen soils. Cold Reg. Sci. Technol. 106–107, 47–54.
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Willam K.J., 2002, Constitutive models for engineering materials. Encyclopedia of Physical Science
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and Technology 3, 603–633. Yang W.D., Zhang Q.Y., Li S.C., 2014. Time-dependent behavior of diabase and a nonlinear creep model. Rock Mech. Rock Eng.47, 1211–1224. Yang Y.G., Gao F., Cheng H.M., Lai Y.M., and Zhang X.X., 2014. Researches on the constitutive models of artificial frozen silt in underground engineering. Advances in Materials Science and Engineering 902164. Yang Y., Gao F., Cheng H., Lai Y., Zhang X., 2014. Researches on the constitutive models of artificial
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frozen silt in underground engineering. Advances in Materials Science and Engineering 902164. Yang Y.G, Lai Y.M., Chang X.X., 2010. Experimental and theoretical studies on the creep behavior of warm ice-rich frozen sand. Cold Reg. Sci. Technol. 63, 61–67. Yao Y.P., Hou W., Zhou A.N., 2009. UH model: three-dimensional unified hardening model for
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clays. Geotechnique 63 (15), 1328–1345.
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Yao Y.P., Zhou A.N., 2013. Non-isothermal unified hardening model: a thermo-elastoplastic model for
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Yu Q.H., 1985. Rheological failure process of rock and finite element analysis. Journal of Hydraulic Engineering 6(1), 55–61.(in Chinese)
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Zhou H.W., Wang C.P., Han B.B., Duan Z.Q., 2011. A creep constitutive model for salt rock based on
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fractional derivatives. International Journal of Rock Mechanics and Mining Sciences 48, 116–121. Zhou H.W., Wang C.P., Mishnaevsky L., Duan Z.Q., and Ding J.Y., 2013. A fractional derivative approach to full creep regions in salt rock. Mech. Time-Depend Mater. 17, 413-425. Zhao X.D., Zhou G.Q., 2013. Experimental study on the creep behavior of frozen clay with thermal gradient. Cold Reg. Sci. Technol. 86, 127–132. Zhou Z.W., Ma W., Zhang S.J., Du H.M., Mu Y.H., Li G.Y., 2016. Multiaxial creep of frozen loess. Mechanics of Materials 95, 172–191.
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Zhu Z.W., Ning J.G., Song S.C., 2010. Finite-element simulations of a road embankment based on a constitutive model for frozen soil with the incorporation of damage. Cold Reg. Sci. Technol. 62,
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ACCEPTED MANUSCRIPT List of Tables and Figures Table 1. The physical parameters of the silty clay and quartz sand. Table 2. The proposed model parameters. Figure1. Grain distribution of the silty clay.
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Figure 2. The creep test results of frozen samples: (a) λ=0.0. (b) λ=0.2. (c) λ=0.4. (d) λ=0.6.
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Figure 4.The yield stress s of frozen sample with λ=0.0. Figure 5. The long-term strength under different λ.
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Figure 6. The creep constitutive model.
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Figure 3.The initial shear modulus under different λ.
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Figure 7. Comparison between tested results and calculated ones: (a)λ=0.0. (b)λ=0.2 (c)λ=0.4. (d)λ=0.6. Figure 8. Comparisons between tested results (Yang et al., 2014) and calculated ones.
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Figure 9. The relationship between G1 and q under different λ.
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Figure 10. The relationship between 1 and q under different λ. Figure 11. The relationship between A and (q-σs)/σs under different λ.
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Figure 12. The relationship between C and (q-σ∞)/σ∞ under different λ.
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Figure 13. Comparisons between results predicted by Nishihara model (the old model) and proposed model (the new model).
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1. The coarse-grain contents have great influence on creep properties of frozen
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mixed soils.
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2. Hardening and damage mechanism are considered in the new creep model.
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3. A new creep model based on Nishihara model is proposed.
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Feng Hou, Yuanming Lai, Enlong Liu, Huiwu Luo, Xingyan Liu PII: DOI: Reference:
S0165-232X(17)30486-X doi:10.1016/j.coldregions.2017.10.013 COLTEC 2469
To appear in:
Cold Regions Science and Technology
Received date: Revised date: Accepted date:
12 April 2016 22 June 2017 12 October 2017
Please cite this article as: Feng Hou, Yuanming Lai, Enlong Liu, Huiwu Luo, Xingyan Liu , A creep constitutive model for frozen soils with different contents of coarse grains. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Coltec(2017), doi:10.1016/j.coldregions.2017.10.013
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A creep constitutive model for frozen soils with different contents of coarse grains Feng Hou1, Yuanming Lai2, Enlong Liu1,2* , Huiwu Luo1, Xingyan Liu2 State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resources
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1
and Hydropower, Sichuan University, Chengdu 610065, P.R.China;
State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and
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2*
Corresponding author: Enlong Liu
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*
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Engineering Institute, Chinese Academy of Sciences, Lanzhou 730000, P.R.China.
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Email: [email protected]
Abstract: The triaxial creep tests of frozen silty clay mixed with different contents of coarse grains
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were performed at 0.3 MPa confining pressure at -10 °C. Four sets of mass ratio of coarse-grained
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particles to silty clay are 0.0, 0.2, 0.4 and 0.6 used to form the frozen samples, and the test results demonstrate that: when the shear stress level is low, the tested sample exhibits an attenuation creep;
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when the shear stress level is high, the tested sample behaves as a non-attenuation creep. It is found that the initial shear modulus and yield strength become greater with the increasing mass ratio of
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coarse-grained particles to silty clay, but the long-term strength shows an opposite trend. Through the analysis of creep mechanism, both the new hardening variable and damage variable are introduced to improve the Nishihara model, and the three-dimensional formulation of the model is also formulated. According to the experimental results and analysis on the creep model, the model parameters are determined and identified. The simulation results show that the new model can describe the whole creep process of artificial frozen soils with different contents of coarse-grained particles well. Finally,
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the influences of the mass ratio of coarse-grained particles to silty clay on the parameters are also analyzed, which provides a reference for engineering application in cold regions.
Keywords: hardening; damage; mass ratio of coarse-grained particles to silty clay; creep constitutive
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mode
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1 Introduction
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In the past few decades of years, a number of geotechnical engineering problems have been arising (Lai et al., 2013) with more and more civil engineering projects constructed in the Qinghai-Tibet plateau.
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Many investigators have proved that the creep properties of frozen soils have a great influence on the
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long-term safety of structures (Yu et al., 2013; Yang et al., 2010). With the rapid development of hydraulic and hydropower engineering in China, there are some dams designed in cold regions
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undergoing great change of temperature. The materials used in the core wall of these dams are usually
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composed of mixed coarse grains and silty clays for preventing failure resulted from seepage and external loads. When these dams located in cold regions are working, the creep properties of these core
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wall materials at low temperature below zero will affect the safety of these dams. Therefore, it is necessary to investigate the creep properties and constitutive model for these frozen mixed soils of core
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wall dam materials.
Both performing laboratory tests and formulating constitutive models are conventional methods employed to study the creep properties of frozen soils. Therefore, many researchers have conducted creep tests under different shear stress levels and conditions to study the creep features of frozen soils (Yang et al., 2014; Li et al., 2011). For example, Zhou et al. (2016) conducted a series of triaxial compression and creep tests on frozen loesses under different confining pressure and temperature
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conditions in order to study the rate-dependent mechanical behaviors; and Zhao and Zhou (2013) carried out a series of uniaxial creep tests on frozen saturated clay under various thermal gradients and creep stresses by GFC (freezing with non-uniform temperature without experiencing K 0 consolidation) method to investigate the creep behaviors of frozen soil with thermal gradient. However, there are few
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studies on creep properties of frozen mixed soils, especially the influences of different components
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on the creep properties of frozen mixed soils. In addition, the constitutive models, used to reflect the
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response behavior of natural and manufactured materials under different stress and environmental conditions, are the bases of describing the mechanical behavior of materials under external load and
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usually expressed in terms of stress, strain and time (Willam, 2002). Hence, there are many creep
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models have been put forward by researchers, which can be roughly categorized as one of the following types (Liingaard et al., 2004): 1) empirical models; 2) rheological models; and 3) general
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stress–strain–time models. The rheological models consisting of a series of mechanical elements, e.g.
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Hookean spring, Newtonian dashpot and Saint Vernant’s slider, are widely accepted for its simplicity, convenience and good performance to represent the main deformation behavior of geomaterials. Four
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well-known models widely used are the Maxwell model, the Kelvin–Voigt model, the Bingham model,
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and Nishihara model. Li et al. (2011) deduced an improved Nishihara creep model for the frozen deep clay by combining a generalized Kelvin model and an improved visco-plastic body. Wang et al. (2014) proposed a simple model by combining Maxwell, Kelvin and Bingham body with a parabolic yield criterion so as to describe the settlement of underlying warm and ice-rich permafrost. Yang et al. (2014) proposed a new creep constitutive model of frozen silt by modifying the generalized Burgers model to reproduce the time-dependent deformation. Li et al. (2011) established a new creep constitutive model for describing frozen soils’ creep characteristics under high confining pressures based on the triaxial
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creep experiment of artificial frozen soil in deep alluvium. According to the theory and results of compressional tests, there are hardening effects with deformation development (Yao et al., 2009; Yao et al., 2013; Yao et al., 2015), and thus as one kind of deformation mechanism, the hardening effects should also be considered when a creep deformation occurs. What is more, the creep mechanism has
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been analyzed by many researchers with the aid of computer scanning technology, and it is found that
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there are two rival effects existing in the creep process: hardening and damage (Miao et al., 1995; Fan
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et al., 2010). When formulating the creep model for frozen soils, many attentions have been paid to the damage effect with neglecting the hardening effect(Zhu et al., 2010; Zhou et al., 2011; Li et al., 2012;
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Zhou et al., 2013; Kang et al., 2015).
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From the analysis on the existing creep models for frozen soils mentioned above, we know that few of them consider the combining effects of hardening and damage mechanism during the process of
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creep. The Nishihara model is widely used to model the creep properties of frozen soils, which can be
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revised or improved to simulate the effects of coarse-grained particles on the creep properties of frozen mixed soils and will be carried out here. In this paper, firstly, we carried out the triaxial creep tests on
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the frozen silty clay mixed with different contents of coarse grains with the same dry density and their
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creep properties were analyzed, followed by putting forward an improved Nishihara model based on the new hardening variable and damage variable based on the triaxial creep test results.. Furthermore, the parameters of the creep model have been analyzed and the new model is also verified with creep tested results of frozen mixed soils. Finally, the influences of the contents of coarse grains on the model parameters are analyzed. 2 Triaxial creep tests of frozen silty clay mixed with different contents of coarse grains 2.1. Test conditions
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The tested soils were silty clay and coarse grains with 2-4 mm particle size of quartz. Their physical parameters are shown in Table 1 and Fig. 1, respectively. We define λ as the mass ratio of coarse grains to silty clay in the sample for convenience. First, the silty clay, dried and sieved through the 1 mm screen, was prepared with water content of 16.0 %, and kept for 24 h without evaporation, so that its
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moisture was uniformly distributed. The silty clay and coarse grains were weighted according to λ
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equal to 0.0, 0.2, 0.4, and 0.6, respectively, then the prepared silty clay and coarse grains were mixed
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uniformity, and filled in a cylindrical mold to make cylindrical soil specimens under a certain suitable compression rate provided by the machine, by which the specimens were prepared as cylinders with
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diameter and height of 6.18 cm and 12.5 cm, respectively. Then, the specimens were placed in another
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mold with three same parts and saturated for over 12 h under a vacuum for 3 h. In order to avoid a large frost heave and prevent moisture migration, the soil samples submerged in distilled water were placed
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in a refrigeration unit and frozen quickly at a temperature of -30 °C. After 48 h of freezing, the molds
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were removed and the specimens were mounted with epoxy resin plates on both ends and covered with a rubber sleeve to avoid moisture evaporation. Finally, the samples were then kept in an incubator for
Table 1
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temperature.
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over 24 h at the test target temperature of -10 °C such that the specimens adopted a uniform
The physical parameters of the silty clay and coarse grains Specific gravity
Liquid limit
Plastic limit
Water content
(%)
(%)
(%)
Silty clay
2.72
27.58
19.37
1.35
coarse grain
2.66
--
--
0.0
5
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Fig. 1. Grain distribution of the silty clay
The test equipment used in this study is a cryogenic triaxial apparatus modified from the MTS-810
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material test machine, which contains the pressure chamber to apply the confining pressure and axial
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pressure, whose schematics was illustrated by Lai et al. (2012). After the samples mentioned above were prepared, they were placed into the pressure cell of the MTS-810 material test machine and a
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series of the creep tests were performed. First, the temperature in the pressure chamber was set at
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-10 °C with a precision of ±0.1 °C. Before the creep testing, the prepared samples were consolidated under confining pressure of 0.3 MPa for 30 min prior to the axial load, respectively. The triaxial creep
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tests on samples were conducted under different shear stress levels. In the initial loading, the axial load
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was applied to the tested samples from zero to the stress level value of the creep test within 60 s. During the process of testing, the axial pressure on the specimens was kept constant with a precision of ±10 kPa.
2.2. Test results and analysis The curves of the axial strain-time obtained by the triaxial creep tests of frozen mixed soil samples under different shear stress levels are shown in Fig. 2. As can be seen from Fig. 2, there are two types of creep, i.e. an attenuate creep and a non-attenuate creep. When the shear stress is lower than the
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long-term strength, the deformation with time is an attenuate creep. When the shear stress is higher than the long-term strength, it turns to be a non-attenuation creep and can be divided into three stages, i.e. the unsteady stage, steady stage and accelerating stage. The tested sample quickly fails once it enters the accelerating stage. For λ=0 (pure silty clay), the creep curves exhibit three stages at higher
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shear stress level as shown in Fig 2 (a). But with the increasing of λ (contents of coarse-grained
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particles), the creep curves mainly behave as two stages at higher stress level, in which the last stage of
q=σ1-σ3=7.75MPa q=7.34MPa
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accelerating stage is not obvious (see Fig. 2 (d) for λ=0.6 ).
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Axial strain ε1( %)
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q=6.54MPa
q=5.71MPa
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q=4.08MPa
Time (h) (a) λ=0.0
q=5.85MPa
Axial strain ε1( %)
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q=6.69MPa q=6.27MPa
q=5.43MPa q=5.01MPa q=3.33MPa
Time (h) (b) λ=0.2
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q=5.75MPa
q=5.39MPa
Axial strain ε1( %)
q=5.03MPa
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q=4.31MPa
q=5.84MPa
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Time (h) (c) λ=0.4
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q=3.59MPa
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Axial strain ε1( %)
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q=5.08MPa
q=4.36MPa q=3.63MPa
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q=4.71MPa
(d)Time λ=0.6(h)
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Fig. 2. The creep test results of frozen mixed samples
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2.2.1 The initial shear modulus By the triaxial creep tests, we can obtain the different initial stains under different shear stress levels as shown in Fig. 3. In general, the initial stain is deemed to the elastic stain, which can be computed by Eq. (1), without considering the volumetric change. So we can get the initial shear modulus for λ=0.0, 0.2, 0.4, and 0.6 are 0.37 MPa, 0.91MPa, 1.21MPa, and 1.97 MPa, respectively. It is found that the initial shear modulus increases with increasing λ, which means that the higher the coarse-grained content of the sample, the greater the initial shear modulus is.
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q , 3G0
(1)
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where 0 is the initial stain and G0 is the initial shear modulus.
Fig. 3. The initial shear modulus under different λ
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2.2.2. The yield stress s
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The yield stress s can be obtained by the stress-stain curve of triaxial tests as shown in Fig. 4, in which we take the result of frozen mixed sample with λ=0.0 as an example. Following the method
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proposed by Ma and Wang (2014), it is easy to obtain the yield stress s =2.25 MPa when λ=0.0. So,
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in the same way, we can get the yield stress of frozen samples with λ= 0.2, 0.4, and 0.6 are 3.76 MPa, 3.90 MPa and 4.16 MPa, respectively. The yield stress s becomes greater with the increase of λ, and
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the higher the content of coarse grains is, the larger the elastic stage is.
The yield stress σs
Fig. 4. The yield stress s of frozen sample with λ=0.0 9
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finally behave an accelerating failure stage once it enters the steady creep stage with a constant creep
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rate, and in this stage, the greater the shear stress is, the larger the creep rate is. So based on the creep
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rate of the steady creep stage, a method is put forward here to determine the long-term creep strength. Firstly, the relationship between the creep rate of steady creep stage and the shear stress is obtained as
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shown in Fig. 5. Then, an equation is proposed to fit the relationship between the rate and the shear
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stress as follows:
v q k ln 1 , v0
(2)
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where k is a material constant, whose unit is MPa; v is the creep rate of steady creep stage; and v0 is the
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reference of creep rate, and can be taken as 1.0.
Fig. 5. The long-term strength under different λ So, according to Eq. (2), we can obtain the long-term strength of frozen samples with λ=0.0, 0.2, 0.4,
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and 0.60 are 6.32 MPa, 5.32 MPa, 4.85 MPa, and 4.57 MPa, respectively, as shown in Fig.5. The long-term strength decreases with increasing λ, and the higher the content of coarse grains is, the lower the damage stress threshold is. 3 A creep constitutive model based on hardening and damage
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Hardening comes from the compression of micro cracks, the emergence of the new connection between
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soil particles and the recombination of mineral grains, which make the strength higher during the
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process of creep deformation. Usually, there is an initial damage in the frozen sample within the initial state, resulted by the micro cracks. During the loading process of creep,
there are two deformation
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mechanisms of hardening and damage existing within the frozen soils simultaneously. When the shear
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stress level is lower, the sample exhibits a non-attenuation creep, which shows the characteristics of viscoelasticity, and this is because under the low stress level, the initial micro cracks of the sample
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tends to be closed, and the sample behaves hardening; when the shear stress level is higher, the initial
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micro cracks continue to develop and accumulate, resulting in destruction of the continuity of frozen sample and the sample damages gradually.
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3.1 The hardening parameter H
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The hardening will not occur until the plastic deformation has happened, which shows that the frozen samples become stronger with the growth of the plastic deformation. So, the hardening parameter is introduced here to reflect the change of the viscous coefficient of visco-plastic element, which has the following properties: H 0 when t 0, and H a certain constant when t . We can define H A 1 e t ,
(3)
where H is the hardening parameter; and A denotes the level of the hardening effect, and the bigger its value is, the greater the strengthening effect is, with A≥0.
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3.2 The damage variable D The experimental investigation has verified that the damage effect has a threshold value [Li et al., 2012; Sidoroff, 1981; Ju, 1989; Martin and Chandler, 1994; Aubertin and Simon, 1997; Eberhardt et al., 1997]. Only if the loading stress exceeds the threshold value, the microscopic cracks initiate, propagate
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and evolve such that the creep damage accumulates and the accelerating creep occurs. The stain rate of
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the frozen sample increases non-linearly in this process, and failure will take place once the strain
damage variable as E , t E0
E0 E , t E0 exp Ct
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and
,
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D , t 1
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reaches a critical value (Yang et al., 2014). Therefore, based on damage mechanics, we can define the
0
0
(4)
,
(5)
where D , t is the damage variable; E , t is the elastic modulus at time t; E0 is the initial
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elastic modulus; and C is the material constant to reflect the influence of damage.
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Substituting Eq. (5) into Eq. (4), we have
0
0 D , t Ct 1 e
0
.
(6)
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Based on the definition of the effective stress [Kachanov, 1986], it can be defined as follows:
1 D
,
(7)
where is the effective stress; and is the Cauchy stress. So, we can obtain
Ct e
0 0
.
(8)
3.3 A creep constitutive model based on hardening and damage The Nishihara model can reflect the creep properties of elastic, viscoelastic and viscoplastic, 12
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recognized by a greatnumber of researchers (Li et al., 2011). In this paper, the hardening parameter and
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damage variable are introduced to improve Nishihara model, as shown in Fig. 6.
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According to a series rules, the following relation holds:
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Fig. 6. The improved Nishihara model
e ve vp , and E vp ,
(9)
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where E and e are the stress and strain of the elastic element, with E E0 e ; and ve are the stress and strain of Abel dashpot; vp and vp are the stress and strain of visco-plastic element; and
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and are the total stress and strain.
(1) When s , according to the elasto-plastic theory, vp =0, then, the model is degenerated to be
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the Kelvin model, there is no hardening and damage effects during the deformation. So, we can get
E0
E1 1 exp t , E1 1
(10)
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e ve
where E1 and 1 are the parameters of the viscoelastic element. (2) When s , the hardening effect occurs with the growing plastic deformation, but the damage does not work. So, the total strain can be expressed as
e ve vp
E0
E1 s t , 1 exp t E1 1 2 H
(11)
where 2 H is the viscous coefficient of the visco-plastic element with the hardening effect considered, which can be expressed as follows by use of hardening parameter H defined in Eq. (3): 13
ACCEPTED MANUSCRIPT 2 H 20 1+A 1 et ,
(12)
where 20 is the initial viscous coefficient of the visco-plastic element. (3) When , both hardening effect and damage effect works with the development of the plastic deformation. Hence, the damage effect should be considered. There is no damage effect in both
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the elastic element and viscoelastic element when the stress and strain grow. But, for the visco-plastic
E0
E1 s t. 1 exp t E1 1 2 H
(13)
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e ve vp
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element should be the effective one . Then we can obtain
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element, it will damage because of the plastic deformation, and the total stress of the visco-plastic
So, in summary, the constitutive equation of the improved Nishihara model in one dimension is
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expressed as follows:
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e ve vp
E 1 exp 1 t E0 E1 1 E s 1 exp 1 t t 1 2 H E0 E1 1 exp E1 t s t E0 E1 1 2 H
s s .
(14)
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3.4. Three-dimensional formulation of the creep constitutive model
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In general, the frozen soil is in the three-dimensional stress state, and the shear deformation is the main source of creep. Thus, in this study, the volume deformation is not considered, and we assume that the material is isotropic.
In three-dimensional stress state, the total strain of the model can be easily obtained and expressed as:
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ACCEPTED MANUSCRIPT ij ij e ij ve ij vp G1 1 exp t 1
q s
F Q G1 1 t 1 exp t 1 2 H F0 ij
s q
F Q G1 1 t 1 exp t H F0 ij 1 2
q
,
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s s ij ij 2G0 2G1 sij sij 2G0 2G1 sij sij 2G0 2G1
(15)
where sij is the Cauchy deviatoric tensor of stress; G0 is the shear modulus of the elastic element;
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G1 is the shear modulus of the viscoelastic element; q is the shear stress; F is the yield function;
F0
is
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the initial reference of the yield function, and can be taken as 1.0; Q is the plastic potential function,
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and the tested material is assumed to be associated flow rule with F Q ; and is the form of the power function, and its exponent sign is taken as 1.0; and ij and ij are Cauchy stress tensor and
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effective stress tensor, respectively.
According to Yu et al., (1985), the effect of the mean stress on creep of frozen soils can be neglected,
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and the deviator stress plays a main role in the creep. So, the yield function can be chosen as:
F J2
s 3
,
(16)
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where J 2 is the second diviatoric stress tensor invariant, which should be the effective second diviatoric stress tensor invariant when q .
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Under the triaxial creep test, 2 3 , and thus we have
m
1 1 2 1 2 3 , s11 1 m 1 3 , J 2 1 3 . 3 3 3
(17)
Thus, Eq. (15) can be rewritten as:
G1 3 3 1 1 1 exp t 3G1 3G0 1 G s 1 1 3 1 3 1 exp 1 t 1 3 t 3G1 32 H 1 3G0 1 3 1 3 1 exp G1 t 1 3 s t 3G 3G1 32 H 0 1 15
q s
s q , q
(18)
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1 1 D
1eCt ; and 3
3 1 D
3eCt .
4. The determination of model parameters and model verification 4.1. The determination of parameters 4.1.1 Parameters of the visco- elastic element G1 and 1
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Under the same content of coarse grains, we can obtain the varying rules of G1 and 1 with the
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shear stress when q s as follow:
1 e q ,
and
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G1 e q ,
(20) (21)
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where , , , and are materials patameters, which can be determined by test results.
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4.1.2 Parameters of the visco-plastic element 2
The visco-plastic parameter 2 contains two parameters 20 and A, and the first does not change with
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the varying stress level as the initial viscous coefficient 20 is constant. The parameter A is to reflect
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assumed as follows:
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the hardening level of material, which decreases with the increasing deviator stress q, and thus, it is
A e
q s
s
,
(22)
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where and are parameters of material. 4.1.3 Damage parameter C As we know, the greater the shear stress is, the higher level the damage is. According to the test results and the above analysis, the relationship between C and shear stress is assumed as follows: m
q C k , where k and m are parameters of material. 4.2. Model verification 16
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From the triaxial creep test results under the four contents of coarse grains, the proposed model parameters are given in Table 2, and the calculated results of creep curves are shown in Fig. 7. From Fig. 7, it is found that the calculated results of the proposed model are in good agreement with the tested results under the different values of λ. The proposed model can not only accurately reproduce the
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unsteady stage when the shear stress is lower than the long-term strength, but also can predict the
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steady stage and progressive stage when the shear stress is greater than the long-term strength.
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Table 2 The proposed model parameters η1(MPa)
G1(MPa) α
θ -1
γ -1
(MPa )
(MPa)
(MPa )
280 763 780 2600
0.26 0.62 0.72 0.93
13981 1245 571 458
0.94 0.74 0.69 0.54
1000 500 50 13
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(MPa)
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0.0 0.2 0.4 0.6
β
20 (MPa)
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λ
(a) λ=0.0.
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μ
20.34 20.09 14.66 10.00
A
C χ
k
0.81 1.81 1.90 1.93
19.05 5.95 1.20 0.20
m
1.57 1.47 1.40 1.39
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(b) λ=0.2.
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(c) λ=0.4.
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(d) λ=0.6.
Fig. 7. Comparison between tested results and calculated ones
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In order to verify the applicatity of model proposed here for other experimental results by different
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scholars, we used the experimental resulus for artificial frozen silt samples at -8 °C by Yang et al.(2014). The computational parameters obtained from the test results are as follows: σ s=9.2 MPa, G0=66700 MPa, α=2500, β=0.38, θ=8500, γ=0.47,
20 =14 MPa, μ=15, χ=0.9, κ=0.45
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σ∞=9.43MPa,
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and m=1.4. The comparsions of computated and testes results are shown in Fig. 8, which demonstrates that the new creep model propsoed here can also predict the creep properties of artificial frozen silt
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samples.
Fig. 8. Comparisons between tested results (Yang et al., 2014) and calculated ones
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5 Discussions In the following, the influence of coarse-grained content is discussed firstly. According to the test results shown in Table 2, we can obtain the relationships between the parameters (G1, η1, A and C) and
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G1 (MPa)
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λ, as shown in Figs. 9-12.
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q (MPa)
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η1(MPa)
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Fig. 9. The relationship between G1 and q under different λ
q (MPa)
Fig. 10. The relationship between 1 and q under different λ
20
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A
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(q-σs)/σs
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C
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Fig. 11. The relationship between A and (q-σs)/σs under different λ
(q-σ∞)/σ∞
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Fig. 12. The relationship between C and (q-σ∞)/σ∞ under different λ
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Fig. 9 shows that the initial shear modulus decreases with the increasing of shear stress under the same content of coarse grains, and the higher of the coarse grains content is, the faster the initial shear modulus reduces with the increasing shear stress. From the Figs. 10 and 11, we know that 1 and A have the same properties, which decrease with the increasing of shear stress, and increase with the increasing of coarse-grained content. But from the Fig. 12, we know that damage parameter C increases with the increasing of shear stress, and decrease with the increasing of coarse-grained content, which means that the larger the coarse-grained content is, the more difficult the sample fails.
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In the new creep model proposed, the non-attenuation creep at lower shear stress level and attenuation creep at higher shear stress level can be modeled simultaneously. For considering these effects, we introduce two competing physical mechanisms, including hardening and damage, during the creep process. For the same confining pressure, when the shear stress level is lower, the frozen soil
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sample will exhibit attenuation creep, which is mainly dominated by hardening due to the closeness of
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micro cracks; while when the shear stress level is higher, the frozen soil sample will exhibit
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non-attenuation creep, which is mainly dominated by gradually damage due to the propagation and development of micro cracks. In addition to the creep model properties considered above, the
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influences of mass ratio on the creep behavior can also be taken into account by varying the model
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parameters.
In order to verify the applicability of the existing creep model to the experimental data, we use both
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the Nishihara model and the propsed model here to predict the tested results of the forzen soils with
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mass ratio of λ=0.2. The computed parameters are the same as those shown in Table 2. The comparisons between results predicted by Nishihara model (the old model) and proposed model (the
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new model) are shown in Fig. 13, which demonstrates that the Nishihara model (the old model) can
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model the attenuation creep at lower shear stress level well, but with the increasing shear stress level, the Nishihara model (the old model) will give incresing errors between the predicted and tested results, and for the non-attenuation creep at higher shear stress level, the Nishihara model (the old model) can not give the accelerating stage. 6 Conclusions Based on the analysis of a series of the triaxial creep tests results of frozen silty clay mixed with coarse grains at 0.3 MPa confining pressure and -10 °C, an improved Nishihara model is formulated to
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predict the creep deformation behavior of frozen soil. Some conclusions can be drawn as follows:
Fig. 13. Comparisons between results predicted by Nishihara model (the old model) and proposed
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model (the new model)
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(1) The test results show that the sample exhibits an attenuate creep when the shear stress is lower than the long-term strength, and an accelerating creep failing in accelerating stage when the shear stress
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is greater than the long-term strength.
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(2) The hardening parameter and the damage variable are introduced to the Nishihara model to describe the hardening and damage effects in the process of creep. Though validating the proposed
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model by the test data of frozen silty clay mixed with coarse grains under different coarse- grained
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contents, it is found that the proposed model can describe the creep behavior of frozen silty clay mixed with coarse grains well. (3) The initial shear modulus G1 , the viscous coefficient of elastic element 1 , and the hardening parameter A decrease with the increase of shear stress, and increase with the increasing content of coarse grains. But the damage factor C increase with the increase of shear stress, and decrease with the increasing of coarse-grained content. References
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Chinese Journal of Rock Mechanics and Engineering, 29(8), 1555-1561. (in Chinese) Ju J.W., 1989. On energy-based coupled elastoplastic damage theories: constitutive modeling and
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ACCEPTED MANUSCRIPT List of Tables and Figures Table 1. The physical parameters of the silty clay and quartz sand. Table 2. The proposed model parameters. Figure1. Grain distribution of the silty clay.
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Figure 2. The creep test results of frozen samples: (a) λ=0.0. (b) λ=0.2. (c) λ=0.4. (d) λ=0.6.
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Figure 4.The yield stress s of frozen sample with λ=0.0. Figure 5. The long-term strength under different λ.
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Figure 6. The creep constitutive model.
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Figure 3.The initial shear modulus under different λ.
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Figure 7. Comparison between tested results and calculated ones: (a)λ=0.0. (b)λ=0.2 (c)λ=0.4. (d)λ=0.6. Figure 8. Comparisons between tested results (Yang et al., 2014) and calculated ones.
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Figure 9. The relationship between G1 and q under different λ.
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Figure 10. The relationship between 1 and q under different λ. Figure 11. The relationship between A and (q-σs)/σs under different λ.
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Figure 12. The relationship between C and (q-σ∞)/σ∞ under different λ.
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Figure 13. Comparisons between results predicted by Nishihara model (the old model) and proposed model (the new model).
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ACCEPTED MANUSCRIPT Highlights
1. The coarse-grain contents have great influence on creep properties of frozen
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mixed soils.
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2. Hardening and damage mechanism are considered in the new creep model.
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3. A new creep model based on Nishihara model is proposed.
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