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Fractional derivative-based creep constitutive model of deep artificial frozen soil
Journal Pre-proof Fractional derivative-based creep constitutive model of deep artificial frozen soil
Dongwei Li, Chaochao Zhang, Guosheng Ding, Hua Zhang, Junhao Chen, Hao Cui, Wangsheng Pei, Shengfu Wang, Lingshi An, Peng Li, Chang Yuan PII:
S0165-232X(18)30590-1
DOI:
https://doi.org/10.1016/j.coldregions.2019.102942
Reference:
COLTEC 102942
To appear in:
Cold Regions Science and Technology
Received date:
20 December 2018
Revised date:
25 October 2019
Accepted date:
5 November 2019
Please cite this article as: D. Li, C. Zhang, G. Ding, et al., Fractional derivative-based creep constitutive model of deep artificial frozen soil, Cold Regions Science and Technology(2019), https://doi.org/10.1016/j.coldregions.2019.102942
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© 2019 Published by Elsevier.
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Fractional Derivative-Based Creep Constitutive Model of Deep Artificial Frozen Soil Dongwei Li1*,2 , Chaochao Zhang1 , Guosheng Ding3 , Hua Zhang3 , Junhao Chen1 , Hao Cui4 , Wangsheng Pei2 , Shengfu Wang1 , Lingshi An1 , Peng Li1 , Chang Yuan1 1 School of Civil and Architecture Engineering, East China University of Technology, Nanchang 330013, China
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2 State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences,
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Lanzhou 730000, China
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3 Fuzhou Metro Co. Ltd., Fuzhou 350004, China 4 Beijing China Coal Mine Engineering Co. Ltd, Beijing 100013, China
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*Corresponding author: Dongwei Li, School of Civil and Architecture Engineering, East China University of Technology, Nanchang 330013, China
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Abstract: A fractional derivative constitutive model of deep artific ial frozen so il was
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established based on the Nishihara model by analyzing the triaxial creep and shear test results of
deep artific ial frozen soil under different confining pressures and temperatures. Considering the
physical and mechanical properties of frozen soil, linear Newton equations were substituted with
nonlinear Newton equations, and a flexibility matrix was derived for numerical calculations. A
relevant constitutive finite element program was developed and used to simulate a deep artific ial
frozen curtain of a coal mine through a user subroutine of the ADINA commerc ial finite element
software. The theoretical calculations agreed well with the experimental results of the triaxial
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creep tests, and the numerical simulation results were consistent with the field measurement data.
This fractional derivative constitutive model was verified to describe the non-decaying creep
deformation characteristics of deep artific ial frozen soil under high confining pressures. It can
also be applied to the design and calculation of artificial freezing in deep strata.
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Keywords : fractional derivative constitutive model; triaxial creep characteristic; deep artific ial
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frozen soil; numerical simulation
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1 Introduction
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Frozen soil is a special material that contains both unfrozen water and ice, and it creeps
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with time, that is, its deformation increases with time under constant load and its stress
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decreases with time under a certain deformation (Vyalov, 2013; Цытович, 1985). Because of the
strong rheological properties of frozen soil, an artific ial frozen curtain is vulnerable to excess ive
creep deformation before it is destructed or loses its bearing capacity, which may lead to the
rupture of freezing pipes. It is important to construct a rheological constitutive equation
reflecting the mechanical properties of frozen soil for the des ign of an artificial frozen curtain.
The deformation of frozen soil generally involves instantaneous elastic deformation and plastic
deformation, and the most notable feature is that irrevers ible creep develops continuous ly with
time, which is a s ignificant part of total deformation. The creep of frozen soil is an important 2
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topic in frozen soil mechanics, and determining the creep inde x and establishing a creep
prediction model are essential to study the properties of frozen soil.
Several scholars have made great progress in understanding how frozen soil properties
change. Anders land et al. (Anders land and Akili, 1967) investigated the stress effect on creep
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rates of frozen soil and proposed a creep model for frozen soil based on the changes activation
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energy during frozen soil deformation. Ladanyi (Ladanyi, 1972; Ladanyi, 1975) presented a
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nonlinear yield criterion for frozen soil. Arenson et al. (Arenson et al., 2006) reviewed studies
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on frozen soil creep. Ma et al(Ma et al., 1994) analyzed the triaxial creep law of frozen soil and
the influence of temperature and confining pressure on the creep strength of frozen soil and
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proposed a creep strength criteria of frozen soil us ing a parabolic equation. Zhang et al
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(Changqing et al., 1995; Changqing et al., 1995) researched the creep damage characteristics of
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frozen soil with respect to its microstructure; Similarly Miao et al (Tiande et al., 1995) studied
the relationship between frozen soil creep and changes in its microstructure and proposed a
general creep damage theory for frozen soil. Yao et al (Yao et al., 2018) proposed a
one-dimens ional creep model for frozen soil taking temperature as an independent variable. Lai
et al. (Lai et al., 2009; Lai et al., 2010) carried out triaxial compression tests, and propos ed a
yield criterion and elasto-plastic damage constitutive model for frozen sandy soil. Liao et al.
(Liao et al., 2017) studied the creep damage characteristics of frozen soil with a fractional creep
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constitutive model. Zhou et al.(Zhou et al., 2016) performed several triaxial creep tests on frozen
loess and established a constitutive model. Loria et al. (Loria et al., 2017) proposed a nonlinear
model to describe the mechanical behavior of artific ial frozen soil strata. Christ et al. (Christ et
al., 2009) studied the mechanical and acoustic characteristics of frozen soil under different
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temperatures and cycles. Li et al. (Li et al., 2011; Li et al., 2017) investigated the creep
characteristics and developed a constitutive model of artificial frozen soil under different
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confining pressures. Amiri et al. (Ghoreishian Amiri et al., 2016) proposed a constitutive model
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of saturated frozen soil. Zhao et al. (Zhao et al., 2014; Zhao et al., 2014; Zhao et al., 2017)
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studied the cyclic direct shear behavior of an artific ial frozen silt-structure interface. Liu et al.
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(Liu et al., 2018) proposed an elasto-plastic model for saturated frozen soils. Dong (Dong et al.,
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2014) conducted creep experiments to analyze the creep curves and creep index.
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The current creep model theory does not present an absolute solution to the entire
rheological behavior of frozen soil under high stress in deep alluvium. Although the Nishihara
model and the generalized Nishihara model are widely used in geotechnical engineering and
their theory is relatively complete, the models are based on an ideal linear element and the three
stages of the creep curve are ignored; thus, their results differ cons iderably from the actual creep
phenomenon of frozen soil under high stress. It is found that fractional calculus can be an
excellent mathematical instrument for modeling some phenomena, espec ially memory-intens ive
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or path-dependent phenomena, and damping description may also be involved in. Fractional
order models also have the advantages of fewer parameters and s imple forms. It is known that
the mechanical behaviors of viscoelastic solids (Enelund et al., 2011), viscoelastic fluid (Hyder
Ali Muttaqi Shah et al., 2011), and hyperviscoelastic materials (Libertiaux et al., 2011), can be
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described by the fractional calculus. Zhou (Zhou et al., 2011) attempted to replace a Newtonian
dashpot in the c lassical Nishihara model (a mathematical model) with the fractional derivative
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Abel dashpot in his salt rock creep research. However, it is rarely investigated whether artific ial
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frozen soil can be modeled using fractional calculus.
This paper aims at developing a fractional derivative-based creep constitutive model of
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artificial frozen soil under high stress. Analytic al formulas are derived and then validated
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through experimentally obtained data. A relevant constitutive finite element program is
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developed and used to s imulate a deep artificial frozen curtain of a coal mine. This research has
theoretical and practical s ignificance. It can be applied in artific ial ground freezing method in
deep coal mine excavation and subway tunnel engineering.
2 Triaxial Tests on Deep Artificial Frozen Soil
2.1 Test and scheme design
The frozen soil samples are representative clay samples obtained from a mine in Huainan,
their main phys ical and mechanical properties are shown in Table 1, and the grain s ize curve is 5
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shown in Fig. 1. The soil s amples were prepared in batches us ing a standard sample preparation
method. Spec ifically, the on-site drilling samples were first crushed, dried, and sifted. Each wet
sample was remolded to meet a specific moisture content and then placed in an airtight,
non-evaporating environment for approximately 6 h to ensure uniform waterdistribution. Next,
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the wet soil was placed ins ide a special mold in layers and compacted to a specific dry bulk
dens ity under a certain consolidation pressure by the sample-making machine. The purpose of
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consolidation is to match the dry bulk dens ity of the samples to that of the natural soils at
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different depths. The consolidation confining pressure was determined according to the buried
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depth of the stratum, and the consolidation processes were completed when the dens ities met the
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requirements. The mold loaded with the wet soil was tightly sealed for approximately 24 h to
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shape the soil sample; then, the molded soil sample was cut into a standard cylindrical shape
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(diameter of 61.8 mm and height of 125 mm with a prec is ion control of 0.1 mm) . The sample
was rapidly frozen to temperatures of -5℃, -10℃, and -15℃ to avoid frost heave. After
refrigeration for 48 h, the samples were covered with a rubber sleeve to avoid moisture
evaporation before testing.
After the soil samples were prepared, a series of triaxial creep and shear tests as well as
uniaxial creep tests were carried out. To ensure the reliability of the test results, the loading
process and data acquis ition were controlled using a computer. The test scheme is shown in
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Table 2.
Table 1 Physical and mechanical properties of conventional soil samples
Liquid
Moisture Sampling
Wet Density
Proportion
Dry Density
Content
Limit Il 3
Depth (m)
3
(g/cm )
(g/cm )
IP (%)
(n/ Gs)
2.038
1.679
3.709
(%)
47.5
27.5
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34.52
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(%)
425.6
Plastic Limit
Fig. 1 Grain size distribution of the sample soil
Table 2 Scheme of triaxial test on deep artificial frozen soil Consolidation Test No.
Confining Pressure (MPa)
Stress
Temperature (℃)
7
Test Type
rate (N/s)
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Triaxial creep test and
A-1
3.0
-5
A-2
3.5
-5
A-3
4.0
-10
A-4
5.0
-15
A-5
0
-10
B-1
4.0
-5
B-2
4.5
-10
B-3
5.0
-10
B-4
6.0
-15
B-5
0
C-1
5.0
C-2
6.0
triaxial shear test Triaxial creep test and triaxial shear test
70
70
Triaxial creep test and triaxial shear test Triaxial creep test and triaxial shear test
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Uniaxial creep test
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Triaxial creep test and triaxial shear test
70
70 70 110
C-4
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C-5
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triaxial shear test
Triaxial creep test and triaxial shear test Triaxial creep test and triaxial shear test Uniaxial creep test
110
110
110 110
Triaxial creep test and
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C-3
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Triaxial creep test and
triaxial shear test
-10
6.5
-15
7.0
-15
0
-10
Triaxial creep test and triaxial shear test Triaxial creep test and triaxial shear test
150
150
150
Triaxial creep test and triaxial shear test Uniaxial creep test
150 150
Note: Consolidation confining pressure is the confining pressure at which the consolidated sample is frozen. 2.2 Test results
(1) Triaxial shear test
Stress-strain curves of deep artific ial frozen soil under triaxial shear test were obtained 8
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through a series of measurements, as shown in Fig. 2, where σ 1 -σ 3 denotes the deviatoric stress, and ε1 denotes the vertical strain; the p-q curves were obtained by analyzing the test
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results, as shown in Fig. 3.
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Fig. 2 Triaxial Stress-strain Curves of Artificial Frozen Soil under Different Temperatures
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q/MPa
Pr
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p/MPa
11
pr
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q/MPa
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p/MPa
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Fig. 3 p-q Relation Curves of Deep Artificial Frozen Soil at the Same Temperature
3
, q 1 3 )
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revealed the following:
1 2 3
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The triaxial shear test of deep artific ial frozen soil (for which p
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1. When confining pressure is less than 7 MPa, the stress-strain curve is strain-hardening
curve. A typical characteristic of this kind of curve is the existence of yield limit. The frozen soil
stress-strain curve consists of two deformation stages, the initial elastic phase and the plastic
hardening phase. When the deviatoric stress is less than the yield limit, the frozen soil receives
less axial force, and the additional damage ins ide the soil is less. The material is in the stage of
linear elastic deformation, and the relationship between stress and strain obeys Hooke's law.
With the increase of the axial load, the deviatoric stress of the frozen soil exceeds the yield limit,
and the joint between the soil partic les and the ice crystal begins to appear in the soil, which 12
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reduces the ability of frozen soil to resist compressive deformation. When the frozen soil enters
the stage of plastic hardening, the trend of stress increas ing with strain begins to s low down, and
the stress maintains an increasing state, and finally stabilizes.
2. When the confining pressure is greater than or equal to 7 MPa, the trend of the
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stress-strain curve develops toward the strain softening curve. Confining pressure has two
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oppos ite effects on the strength of frozen soil. On the one hand, with the increase of confining
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pressure, the defects (micro-cracks, voids, etc.) in frozen soil gradually heal, and the
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cementation force, structural force, friction force and cohes ion of soil particles gradually
increase. The frozen soil is continuous ly strengthened, and the plastic properties of frozen soil
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are enhanced. On the other hand, with the continuous increase of confining pressure, the ice in
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frozen soil melts continuous ly, and the weak mineral partic les are crushed continuous ly. The
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frozen soil is weakened, and the strength tends to decrease.
3. At same confining pressure, the shape of the stress-strain curve of frozen soil is similar at
different temperatures, but the initial slope of the stress-strain curve is different. The hardening
rate of the stress-strain curve is different at different temperatures. The lower the temperature is,
the faster the hardening rate of the stress-strain curve at the initial stage is. On the contrary, the
slower the hardening rate is. This is because with the decrease of temperature, the content of
unfrozen water in frozen soil decreases gradually, and the cementation between ice and soil
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increases, so that the lower the temperature of soil, the greater the modulus of deformation. At
the same time, the yield limit increases with the decrease of temperature.
4. At same temperature, the initial s lope of stress-strain curve of frozen soil increases
with the increase of confining pressure, which mainly shows that the hardening rate of
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stress-strain curve is different under different confining pressure. The hardening rate of
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frozen soil at the initial stage of stress-strain curve increases with the increase of confining
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pressure. This is because confining pressure increases the compactness of soil and improves
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the resistance of soil to initial compressive deformation. At the same time, the yield limit
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(2) Triaxial creep test
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increases with the increase of confining pressure.
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Creep curves of artific ial frozen soil under different confining pressures and freezing
temperatures were obtained through triaxial creep test of deep artific ial frozen soil, as shown
in Fig. 4, and typical isochronous stress-strain curves of artificial frozen soil were obtained
by analyzing the creep curves, as shown in Fig. 5.
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MPa
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Fig. 4 Typical Creep Test Curves of Artificial Frozen Soil at Different Temperatures
ε1 /% Fig. 5 Isochronous Stress-strain Curves at a Freezing Temperature of -10℃
The triaxial creep test of deep artificial frozen soil revealed the following:
(1) The test curves of triaxial creep and uniaxial creep are s imilar, demonstrating the
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theoretical basis of the transition from uniaxial creep to triaxial creep;
(2) We can see from the data presented above that isochronous creep curves include four
segments and represent a linear distribution at low axial pressure, showing that the
Drucker-Prager yield criterion can also be used to describe the yield of frozen soil;
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(3) The creep properties of frozen soil are c losely related to time and stress levels. In the
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creep tests, when the stress level is lower than the yield stress of the frozen soil, the frozen soil
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will experience the first and second creep stages, namely the decay and steady-state creep stage,
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but no acceleration creep stage, and mainly manifested as viscoelasticity.
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(4) During the creep tests, when the creep stress exceeds a critical value σ s ( yield stress ),
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the third creep stage occurs. The frozen soil w ill undergo three stages of decay, steady state and
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acceleration. The inside of the frozen soil will be damaged and gradually accumulated until the
accelerated creep of the frozen soil occurs. Acceleration creep stage lasts only for a short time
before damage occurs. Because frozen soil in deep alluvium is subjected to a significant ground
pressure and enters the third creep stage, the accelerated creep stage of frozen soil must be
considered.
(5) At the same temperature, the creep value of frozen soil increases with the increase of the value σ 1 -σ 3 , and at the same value σ 1 -σ 3 , the creep value of frozen soil decreases with the
decrease of the temperature. 17
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(6) During tests, frozen soil shows transient elastic strain, but less than 10% creep
deformation;
The Drucker–Prager yield criterion can be used for the triaxial compress ion and creep tests
of frozen soil, providing the basis for fractional derivative model fitting.
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2.3 Validation of yield criterion for deep artificial frozen soil The measured creep yield criterion of frozen soil is described here. Tests reveal the creep
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J
0.5
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as presented in Fig. 6 and Fig. 7.
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strength envelope and Drucker-Prager yield criterion of deep artificial frozen soil with the results
σ m/MPa Fig. 6 Drucker-Prager Yield Criterion of Artificial Frozen Soil
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σ/MPa
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Pa
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Fig. 7 Creep Envelope of Deep Artificial Frozen Soil
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Based on the data shown in Fig. 6 and Fig. 7, the following conclusions can be obtained from
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the tested pressure range: the p-q relation of frozen soil is a curve and is linear at low stress, and
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the creep envelope of frozen soil is nearly straight. We can conclude that the transient deformation
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of frozen soil complies with the property of Drucker-Prager material; therefore the yield of the
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frozen soil can be approximately described using the Drucker-Prager yield criterion.
3 Fractional Derivative-Based Creep Constitutive Model of Deep Artificial Frozen Soil
3.1 The fractional derivative constitutive equation Since the traditional integral order calculus constitutive model has some limitations, fractional calculus is applied to calculate the creep of soft soil. The fractional derivative-based rheological model uses Abel theorems to replace the Newton equations in classical Nishihara theory. At the same time, in classical Nishihara model, the yield limit σ s is a one-dimensional constant，and When σ is larger than σ s, the model enters the plastic yield stage. Under the
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three-dimensional stress state, the fractional derivative-based rheological model replaces σ s with a three-dimensional function group F， considering the anisotropy of the mechanical properties of the material and the changes under the influence of temperature. When the value under various factors exceeds the limit of this function, the soil sample will enter the plastic yield stage, which can more accurately describe the plasticity yield stage of frozen soil. The traditional Nishihara
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model is shown in Fig. 8.
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Fig. 8 Traditional Nishihara Model The traditional Nishihara model consists of Hooker body (H), viscoelastic body (N/H) and
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viscoplastic body (N/St V). In Fig 8, where E0 denotes the elastic modulus; E1 denotes the
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viscoelastic modulus; η1 , η2 denote the viscous coefficients of the kettle.
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The fractional derivative-based creep model is shown in Fig. 9.
Elastic stage
Elastic-visco stage
Visco-plastic stage
Fig. 9 Fractional Derivative-Based Creep Model The fractional derivative-based visco-elastic-plastic constitutive model is explained as follows:
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(1)
1 d t x d 1 r dt 0 t r
(2)
D r f t can be solved by Laplace transform:
m r L Dr f t L D m D f t
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m 1 mr mr S m L D f t S mk 1D f t k 0 t 0
m 1
S r F s S mk 1D k mr f 0 （m – 1 < r < m, m = 1, 2, 3…）
(3)
pr
k 0
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wherein, s and m are the Laplace transform parameters.
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In the case of m = 1, and with the initial condition set as 0,
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1 r L Dr f t S r F s D f 0 S r F s
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The constitutive equation of the fractional derivative-based Nishihara model can be solved by
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Laplace transform:
s E s q1S r s
(4)
wherein, q1 indicates the creep modulus of fractional derivative-based Nishihara model. Therefore, for σ1 -σ3 <σs , the constitutive equation of the fractional derivative-based Nishihara model is as follows:
t
1 Er t ，t 0 ， E1
t / Er t 1 1 1 rn n 1
wherein,
0
n
rn
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Therefore:
t / t 1 1 E1 n 1 1 rn 0
rn
n
(5)
In the case of σ 1 -σ 3 ≥σ s , the constitutive equation of the fractional derivative-based Nishihara model is as follows:
t / F t t 1 1 E1 n 1 1 rn 2 0
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(6)
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3.2 Viscoelastic modulus of artificial frozen soil
The visco-elastic-plastic stress can be calculated using forward integration, and the stress
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increment is given by equation (7)
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i( k ) Dije (ej e( k ) eepj ( k ) )
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Wherein:
e( k )
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e j
t
1
e
E1
1
(7)
t
(8)
Therefore, the viscoelastic modulus of artificial frozen soil can be expressed as follows:
t / Er t 1 1 1 rn n 1
n
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(9)
3.3 Visco-elastic-plastic flexibility matrix It is assumed that the creep of frozen soil satisfies the classical theory of plastic mechanics: (1) the rheological deformation results from deviatoric stress and is not related to the hydrostatic pressure; and (2) the principal axis of stress coincides with the principal axis of strain. Thus, the
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viscoplastic increment can be expressed as follows:
m cm em
(10)
s j cl (elj eepj )
(11)
wherein:
Et 1 2
cl
Et 1
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Partial derivatives of formula (10) and (11) are as follows
(12)
f
cm
(13)
pr
m cm [1 1 1 0 0 0]T { } 3
(14)
Pr
e-
si el (eiep ) cl ( i ) e j e j e j
wherein:
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4 2 elj 1 2 e j 6 0 0 0
(ej p ) ei
2 2 0 0 0 4 2 0 0 0 2 4 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3
t t F
2
F { }
(15)
(16)
According to the above tests, the Drucker–Prager yield criterion can be used to describe the creep yield of frozen soil.
F
6sin 6c cos m 3J 2 (3 sin ) (3 sin )
(17)
Wherein: φ is the internal friction angle of frozen soil, in degrees; c is the cohesive force of 23
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frozen soil, in MPa; and σ m and J 2 are the first stress tensor invariant and the second deviatoric stress tensor invariant, respectively.
F 6sin m 3J 2 { } (3 sin ) { } { }
J2 1 Sx { } 2 J 2
Sy
2 yz
Sz
(18)
2 zx
2 xy
T
1 1 1 [ P] 0 0 0
0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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Pr
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1 1
1 1 0
0
1 0
0
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2 1 1 [ R] 0 0 0
pr
wherein: [P] and [R] are coefficient matrixes;
2
(20)
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F 6sin [ P] 3[ R] ( ){ } { } (3 sin ) 9 m 6 J 2
(19)
1 2
0
0
0
0
6
0
0
0
0
6
0
0
0
0
0 0 0 0 0 6
(21)
(22)
The relationship between the deviatoric stress tensor invariant and the components of stress is as follows:
i si m
i = 1, 2, 3
(23)
i si
i = 4, 5, 6
(24)
Therefore, the visco-elastic-plastic matrix components can be expressed as follows
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si m e j e j
DijVP
Assuming a 2
t t F
2
si e j
i = 1, 2, 3
(25)
i = 4, 5, 6
(26)
2sin and 3 m (3 sin )
b2
t t F
2
3 , the 6 J2
components of the viscoplastic flexibility matrix can be expressed as follows: (27)
1 1 d12 d13 d 23 cm cl a +b 3 3
(28)
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1 2 d11 d 22 d33 cm cl a 2b 3 3
(29)
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1 d 44 d55 d66 cl 6b 2
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The flexibility matrix is a symmetric matrix and the remaining components are 0.
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of Constitutive Model
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4 ADINA Finite Element Program-Based Secondary Development and Validation
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ADINA 8.1 version supports the secondary development of material constitutive relations, unit algorithm, element failure criterion, structure rupture criterion, crack extension rule, and boundary conditions.
4.1 Verification of fractional derivative model of artificial frozen soil FORTRAN language was used to develop the constitutive model for artificial frozen soil, and the element type, integration method, and nonlinearity of finite elements were considered.
Uniaxial and triaxial creep tests were carried out on artificial frozen clay samples under different temperatures and stresses, and regression analysis was conducted to assess the constitutive relation of artificial frozen soil under complex stress. The rheological parameters (as
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shown in Table 3) were obtained by uniaxial creep test on deep artificial frozen soil using cylindrical samples with a diameter of 50 mm and height of 100 mm.
Table 3 Parameters for the Numerical Calculation of the Fractional Derivative Model of Deep Artificial Frozen Soil
Test
E0
E1
η1
η2
c
φ
μ MPa
MPa
MPa·min
-5℃
76
0.26
140
2358
-10℃
138
0.22
250
3250
-15℃
204
0.18
375
4670
10 MPa·min
MPa
°
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Temperature
m 3
1.5
5.48
1.4
5856
3.1
6.08
1.4
8750
4.8
6.68
1.4
e-
pr
3762
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al
Pr
The creep curves obtained by numerical simulation are shown in Fig. 10.
Time/h Fig. 10 ADINA-Simulated Triaxial Creep Test Curves of Cylindrical Samples
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Pr
1 /%
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Time/h
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1 /%
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Time/h
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Time/h
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1 /%
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%
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Fig. 11 Measured and Simulated Triaxial Creep Test Curves for Cylindrical Samples
Time/h Fig. 12 Error between Triaxial Creep Test and Simulation versus Time As can be seen from the data presented in Fig. 11 and 12, the values obtained from the triaxial compression creep test and the simulation changed similarity, with error within 0.15%, further demonstrating that the fractional derivative-based creep model can accurately describe the constitutive relation of frozen soil under high stress. The fractional derivative-based creep constitutive model was added and the ADINA material library was expanded with the powerful secondary development platform of ADINA. The uniaxial and triaxial compression creep test results show that this model can accurately describe creep under high ground pressure. 28
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(1) Under a low constant force, frozen oil experiences both unstable and stable creep, i.e., the first and second creep stages, and the theoretical value agrees well with the measured value. (2) When the creep stress exceeds the critical value σ s , the third creep stage occurs, but lasts only for a short time before damage occurs. Because frozen soil in deep alluvium withstands significant ground pressure and enters the third creep stage, the accelerated creep stage of frozen
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soil must be considered. According to the test results, there are differences between the values
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obtained from theoretical calculation and tests early in the accelerated creep stage, but these
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values are consistent in the damage stage.
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Therefore, it is feasible to use this fractional derivative-based creep constitutive model for
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theoretical calculations of frozen soil under high stress such as deep alluvium and foundation of high-rise buildings.
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4.2 Numerical simulation of frozen curtain by artificial freezing method For the excavation of a coal mine with the artificial ground freezing method, the freezing
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depth is 500 m, and it freezes with three circles of freezing tubes. The net diameter of the shaft is 6 m, and the designed thickness of the frozen wall is 9 m. Considering the axial symmetry of the cylindrical frozen wall, one-fourth of the physical model is simulated, as shown in Fig. 13. In the calculated physical model, the upper surface is set as the force boundary, bearing the dead weight of the overlying topsoil, its value is 6 MPa; the right boundary is set the lateral pressure calculated by the theoretical formula of heavy liquid, its value is 4 MPa; and the front and rear boundaries are symmetric boundaries with constrained tangential displacement in the circumference direction; the temperature of frozen wall is -15 ℃, and the excavation height is 20 m. To verify the calculated results, a measured displacement site was set at 0.8 m depth from top to bottom. The main 29
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freezing technical parameters of the construction are shown in Table 4.
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Fig. 13 Calculation Model for Shaft Excavation
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Table 4 Main freezing technical parameters
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Item
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Freezing depth (m)
500 6
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Net diameter of the shaft (m)
Parameter
9
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Freezing wall thickness (m) Average temperature (℃)
-15
Number of freezing holes
40
Circulation temperature of freezing brine (℃)
-26～-28
Temperature of side wall (℃)
-6～-8
The distribution rule of the sidewall displacement with the depth is obtained by the numerical simulation, as shown in Fig. 14. As shown in the graph, for the radial creep displacement of the frozen wall in the exposed section at a section height and different exposure time, the displacement at the middle and lower parts are higher than the displacement at the upper part, and
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the curves form the shape of a fish belly.
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Depth/m
Displacement/mm
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Fig. 14 Distribution Rule of Sidewall Displacement with Depth Constrained by the construction status and conditions, it is usually impossible to obtain all
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the necessary displacement data of the sidewall of the frozen curtain. The measuring points at the
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sidewall can be arranged only after they are excavated. In addition, the measurement may often be
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limited by construction conditions. For instance, a concrete wall may be built after a working of the section height of the shaft is finished, and the displacement measuring must be stopped. Therefore, to evaluate the difference between the measured displacement of the sidewall of the frozen curtain and the actual displacement of the frozen curtain, the measured results and ADINA-based numerical simulation data were compared and analyzed, as shown in Fig. 15.
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Time/h
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Displacement/mm
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Fig. 15 Comparison of Sidewall Displacement Measured at Measuring Points and Obtained
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from Numerical Simulation
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Fig. 15 presents the comparison of the measured displacement data of the frozen curtain of
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the sidewall at 0.8 m from the top to bottom and the numerical simulation calculations of the
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corresponding points during shaft excavation. It can be seen from the graph that, at the instant of
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excavation of the section height, the lower measuring points of the displacement working face experience advanced displacement (approximately 3.5 mm). With the excavation of the working face, there is a gradual increase in the displacement of the measuring points, and the displacement of the sidewall reaches 22 mm when the measuring point is exposed. At this time, the measuring point was established 8 h after the excavation of the section height. The comparison of the measured displacement curves and numerical simulation curves shows that these curves are similar in shape. Limited by the supporting work, it is impossible to obtain all the displacement data of the excavated section height by measurement. A comparison of measured data and maximum numerical simulation data shows that the measured data only accounts for 16/53 of the
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actual values under the engineering conditions, approximately 30%.
Based on the numerical simulation results, the displacement of the floor heave of the working
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Pr
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face is shown in Fig. 16.
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Fig. 16 Distribution Rule of Working Face Displacement with Shaft Center The displacement of the floor heave of the working face of shaft has the following distinct features: 1. The displacement curve is a curve of more than cubic power, and is inflected near the sidewall. 2. The maximum displacement occurs at the center of shaft. 3. The displacement is one-fourth times larger than that of the sidewall. 4. The deformation of the floor heave is the main contributor of the advanced displacement of the frozen wall below the working face. 5. The deformation of the floor heave increases with the increase in the excavated section height and the extension of the excavation time.
After the excavation of the section height for 4 h, significant displacement takes place at the
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floor heave of the working face of the shaft, reaching 22 mm and above. As the working face is excavated deeper and deeper, the displacement of the floor heave gradually increases to reach a maximum of 32 mm after 8 h and 47 mm after 12 h. When the entire section height is accomplished, the maximum displacement of the floor heave is 66 mm. A comparison of the measured displacement of the floor heave of the excavated working face and the simulated values
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was performed and is shown in Table 5.
Working Face
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Table 5 Comparison of the Displacement of the Sidewall of the Frozen Curtain and the
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Numerical Simulation Numerical Simulation Calculation Method
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Maximum Displacement of
24.5
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Sidewall/mm Maximum Uplift Displacement
(16 h)
Measurement (8 h)
49.8
16
66
21
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(8 h)
Field
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32
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of Working Face/mm
We can see from Table 5, because the displacement of the floor heave of the working face can only be measured at the excavated section height and in the supporting process, there are limited field measurements of the displacement. Only 8 h were available for the measurement, and the maximum displacement occurred at the center of the shaft, at approximately 21.0 mm. The maximum displacement determined from numerical simulation is 32 mm in 8 h. Therefore, the measured value is only 60% of the simulated value.
5 Conclusions A self-developed W3Z-200 testing system was used for triaxial compression and creep tests
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on frozen soil, and the test results were analyzed. Linear Newton equations in the Nishihara model were replaced with nonlinear Newton equations, and a linear spring was replaced with a Drucker-Prager yield function. In this manner, the Nishihara model was improved, resulting in the fractional derivative-based
creep
constitutive model for
artificial frozen
soil under
three-dimensional stress. In addition, the flexibility matrix of application and numerical
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calculation was derived with the Drucker-Prager yield criterion. The fractional derivative
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constitutive model of artificial frozen soil requires few parameters and simple calculations .
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(1) When stress is low, deep artificial frozen soil shows a stable creep. When σ1 -σ3 >σs , the
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soil shows non-decaying creep for a short time before the damage occurs, indicating that the
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fractional derivative constitutive model is suitable to describe the creep of the frozen soil. (2) The constitutive model is added into a large nonlinear program ADINA by secondary
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development of a finite element program, and uniaxial and triaxial creeps of frozen soil were
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tested and simulated. The results show some errors between the stress-strain curves and measured
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curves of frozen soil before the stress reaches its limits, and the theoretical values coincide with the measured values in the final creep stage at maximum stress, demonstrating that this model can describe the overall non-decaying creep process of the frozen soil under high stress. (3) This model can be used for the mechanical analys is of a frozen curtain in a deep mine. The maximum displacement of the sidewall of the frozen curtain obtained from the numerical simulation within 16 h was 48.9 mm, while the measured maximum displacement is 16.8 mm at the same section height. The floor heave of the working face was distributed as measured on field, and is especially large near the center of the shaft with an ADINA-based simulated value of 66.0 mm. The floor heave measured on site was 40.0 mm, 60% of the simulated value.
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Acknowledgements This study is supported by the National Natural Science Foundation of China (Grant No. 41672278, No. 41271071, No. 41977236, and No. 51504070), the Fujian Provincial Natural Science Foundation Projects (Grant No. 2017Y4001), the Jiangxi Provincial Natural Science Foundation Projects (Grant No. 20192ACBL20002), and the State Key Laboratory Program of
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Frozen Soil Engineering (Grant No. SKLFSE201204), we gratefully acknowledge these supports.
References:
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Andersland, O. and Akili, W., 1967. Stress effect on creep rates of a frozen clay soil. Geotechnique, 17(1): 27-39.
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Arenson, L.U., Springman, S.M. and Sego, D.C., 2006. The rheology of fro zen soils. Applied Rheology, 17(1): 12147.
Changqing, Z., Tiande, M., Jiacheng, W., Lina, M. and Yafeng, L., 1995. An Analysis on Creep Loess by Electronmicroscope. JOURNA L OF GLACIOLOGY AND
GEOCRYOLOGY, 17(S): 54-59.
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Damage of Fro zen
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Changqing, Z., Xuexia, W. and Tiande, M., 1995. Microstructure Damage Behaviour and Change Characteristics in the Frozen Soil Creep Process. JOURNA L OF GLA CIOLOGY AND GEOCRYOLOGY, 17(S1): 60-65.
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Christ, M., Kim, Y. and Park, J., 2009. The influence of temperature and cycle s on acoustic and mechanical properties of frozen soils. KSCE Journal of Civil Engineering, 13(3): 153-159. Dong, L., Zhang, G., Zhao, S. and Pan, W., 2014. Experimental study of the creep indexes of frozen soil. Journal of Glaciology and Geocryology, 36(1): 130-136. Enelund, M., Mahler, L., Runesson, K. and Josefson, BL., 1999. Formu lation and integration of the standard linear viscoelastic solid with fractional o rder rate laws. International Journal of Solids And Structuresv, 36(16):2417– 2442. Ghoreishian Amiri, S., Grimstad, G., Kad ivar, M. and Nordal, S., 2016. Constitutive model for rate-independent behavior of saturated frozen soils. Canadian Geotechnical Journal, 53(10): 1646-1657. Hyder Ali Muttaqi Shah, S., Qi, H., 2010. Starting solutions for a vis coelastic flu id with fractional burgers’ model in an annular pipe. Nonlinear Analysis: Real World Applications, 11(1):547–554. Lai, Y., Jin, L. and Chang, X., 2009. Yield criterion and elasto -plastic damage constitutive model for frozen sandy soil. International journal of plasticity, 25(6): 1177-1205. Lai, Y., Yang, Y., Chang, X. and Li, S., 2010. Strength criterion and elastoplastic constitutive model of frozen silt in generalized plastic mechanics. International Journal of Plasticity, 26(10): 1461-1484. Li, D., Fan, J. and Wang, R., 2011. Research on visco-elastic-plastic creep model of artificially frozen 36
Journal Pre-proof soil under high confining pressures. Cold Regions Science and Technology, 65(2): 219-225. Li, D., Yang, X. and Chen, J., 2017. A study of triaxial creep test and yield criterion of artificial frozen soil under unloading stress paths. Cold Regions Science and Technology, 141: 163-170. Liao, M., Lai, Y., Liu, E. and Wan, X., 2017. A fractional order creep constitutive model of warm frozen silt. Acta Geotechnica, 12(2): 377-389. Libertiau x, V., Pascon, F., 2010. Differential versus integral formu lation of fractional hyperviscoelastic constitutive laws for brain tissue modelling. Journal of Co mputational and Applied Mathematics; 234(7):2029–2035. Liu, E., Lai, Y., Wong, H. and Feng, J., 2018. An elastoplastic model for saturated freezing soils based on thermo-poromechanics. International Journal of Plasticity, 107: 246-285. Loria, A.R., Frigo, B. and Chiaia, B., 2017. A non -linear constitutive model for describing the mechanical behaviour of frozen g round and permafrost. Co ld regions science and technology, 133:
f
63-69.
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Ma, W., Wu, Z.W. and Sheng, Y., 1994. Creep and creep strength of fro zen soil. Journal of Glaciology and Geocryology, 16(2): 113-118. frozen soil. Science in China(Series B)(03): 309-317.
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Tiande, M., Xuexia, W. and Changqing, Z., 1995. M icrostructure damage theory of creep process in Vyalov, S.S., 2013. Rheological fundamentals of soil mechanics. Elsevier.
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Yao, X., Qi, J., Zhang, J. and Yu, F., 2018. A one-dimensional creep model for frozen soils taking temperature as an independent variable. Soils and foundations, 58(3): 627-640.
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Zhao, L., Yang, P., Wang, J. and Zhang, L., 2014. Cyclic direct shear behaviors of frozen soil– structure interface under constant normal stiffness condition. Co ld Regions Science and Technology, 102: 52-62.
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Zhao, L., Yang, P., Wang, J. and Zhang, L., 2014. Impacts of surface roughness and loading conditions on cyclic direct shear behaviors of an artificial fro zen silt–structure interface. Cold Reg ions Science
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and Technology, 106: 183-193.
Zhao, L., Yang, P., Zhang, L. and Wang, J., 2017. Cyclic d irect shear behaviors of an artificial frozen
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soil-structure interface under constant normal stress and sub -zero temperature. Co ld Regions Science and Technology, 133: 70-81.
Zhou, Z. et al., 2016. Multiaxial creep of frozen loess. Mechanics of Materials, 95: 172-191. Цытович, Н.А., 1985. Frozen soil mechanics. Science press, Beijing. Zhou, H.W., Wang, C.P., Han, B.B., and Duan, Z.Q., 2011 A creep constitutive model for salt rock based on fractional derivatives. International Journal Of Rock Mechanics And Mining Sciences, 48(1):116–121.
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The authors declare that there is no conflicts of interest regarding the publication of this paper.
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Dongwei Li, Chaochao Zhang, Guosheng Ding, Hua Zhang, Junhao Chen, Hao Cui, Wangsheng Pei, Shengfu Wang, Lingshi An, Peng Li, Chang Yuan
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Highlights 1, A fractional derivative constitutive model of deep artificial frozen soil was established; 2, Nonlinear Newton equations were used to consider the physical and mechanical properties of frozen soil; 3, The fractional derivative constitutive model can be used to describe frozen soil under high
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Pr
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confining pressures.
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Dongwei Li, Chaochao Zhang, Guosheng Ding, Hua Zhang, Junhao Chen, Hao Cui, Wangsheng Pei, Shengfu Wang, Lingshi An, Peng Li, Chang Yuan PII:
S0165-232X(18)30590-1
DOI:
https://doi.org/10.1016/j.coldregions.2019.102942
Reference:
COLTEC 102942
To appear in:
Cold Regions Science and Technology
Received date:
20 December 2018
Revised date:
25 October 2019
Accepted date:
5 November 2019
Please cite this article as: D. Li, C. Zhang, G. Ding, et al., Fractional derivative-based creep constitutive model of deep artificial frozen soil, Cold Regions Science and Technology(2019), https://doi.org/10.1016/j.coldregions.2019.102942
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© 2019 Published by Elsevier.
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Fractional Derivative-Based Creep Constitutive Model of Deep Artificial Frozen Soil Dongwei Li1*,2 , Chaochao Zhang1 , Guosheng Ding3 , Hua Zhang3 , Junhao Chen1 , Hao Cui4 , Wangsheng Pei2 , Shengfu Wang1 , Lingshi An1 , Peng Li1 , Chang Yuan1 1 School of Civil and Architecture Engineering, East China University of Technology, Nanchang 330013, China
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2 State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences,
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Lanzhou 730000, China
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3 Fuzhou Metro Co. Ltd., Fuzhou 350004, China 4 Beijing China Coal Mine Engineering Co. Ltd, Beijing 100013, China
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*Corresponding author: Dongwei Li, School of Civil and Architecture Engineering, East China University of Technology, Nanchang 330013, China
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Abstract: A fractional derivative constitutive model of deep artific ial frozen so il was
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established based on the Nishihara model by analyzing the triaxial creep and shear test results of
deep artific ial frozen soil under different confining pressures and temperatures. Considering the
physical and mechanical properties of frozen soil, linear Newton equations were substituted with
nonlinear Newton equations, and a flexibility matrix was derived for numerical calculations. A
relevant constitutive finite element program was developed and used to simulate a deep artific ial
frozen curtain of a coal mine through a user subroutine of the ADINA commerc ial finite element
software. The theoretical calculations agreed well with the experimental results of the triaxial
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creep tests, and the numerical simulation results were consistent with the field measurement data.
This fractional derivative constitutive model was verified to describe the non-decaying creep
deformation characteristics of deep artific ial frozen soil under high confining pressures. It can
also be applied to the design and calculation of artificial freezing in deep strata.
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Keywords : fractional derivative constitutive model; triaxial creep characteristic; deep artific ial
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frozen soil; numerical simulation
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1 Introduction
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Frozen soil is a special material that contains both unfrozen water and ice, and it creeps
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with time, that is, its deformation increases with time under constant load and its stress
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decreases with time under a certain deformation (Vyalov, 2013; Цытович, 1985). Because of the
strong rheological properties of frozen soil, an artific ial frozen curtain is vulnerable to excess ive
creep deformation before it is destructed or loses its bearing capacity, which may lead to the
rupture of freezing pipes. It is important to construct a rheological constitutive equation
reflecting the mechanical properties of frozen soil for the des ign of an artificial frozen curtain.
The deformation of frozen soil generally involves instantaneous elastic deformation and plastic
deformation, and the most notable feature is that irrevers ible creep develops continuous ly with
time, which is a s ignificant part of total deformation. The creep of frozen soil is an important 2
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topic in frozen soil mechanics, and determining the creep inde x and establishing a creep
prediction model are essential to study the properties of frozen soil.
Several scholars have made great progress in understanding how frozen soil properties
change. Anders land et al. (Anders land and Akili, 1967) investigated the stress effect on creep
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rates of frozen soil and proposed a creep model for frozen soil based on the changes activation
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energy during frozen soil deformation. Ladanyi (Ladanyi, 1972; Ladanyi, 1975) presented a
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nonlinear yield criterion for frozen soil. Arenson et al. (Arenson et al., 2006) reviewed studies
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on frozen soil creep. Ma et al(Ma et al., 1994) analyzed the triaxial creep law of frozen soil and
the influence of temperature and confining pressure on the creep strength of frozen soil and
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proposed a creep strength criteria of frozen soil us ing a parabolic equation. Zhang et al
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(Changqing et al., 1995; Changqing et al., 1995) researched the creep damage characteristics of
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frozen soil with respect to its microstructure; Similarly Miao et al (Tiande et al., 1995) studied
the relationship between frozen soil creep and changes in its microstructure and proposed a
general creep damage theory for frozen soil. Yao et al (Yao et al., 2018) proposed a
one-dimens ional creep model for frozen soil taking temperature as an independent variable. Lai
et al. (Lai et al., 2009; Lai et al., 2010) carried out triaxial compression tests, and propos ed a
yield criterion and elasto-plastic damage constitutive model for frozen sandy soil. Liao et al.
(Liao et al., 2017) studied the creep damage characteristics of frozen soil with a fractional creep
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constitutive model. Zhou et al.(Zhou et al., 2016) performed several triaxial creep tests on frozen
loess and established a constitutive model. Loria et al. (Loria et al., 2017) proposed a nonlinear
model to describe the mechanical behavior of artific ial frozen soil strata. Christ et al. (Christ et
al., 2009) studied the mechanical and acoustic characteristics of frozen soil under different
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temperatures and cycles. Li et al. (Li et al., 2011; Li et al., 2017) investigated the creep
characteristics and developed a constitutive model of artificial frozen soil under different
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confining pressures. Amiri et al. (Ghoreishian Amiri et al., 2016) proposed a constitutive model
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of saturated frozen soil. Zhao et al. (Zhao et al., 2014; Zhao et al., 2014; Zhao et al., 2017)
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studied the cyclic direct shear behavior of an artific ial frozen silt-structure interface. Liu et al.
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(Liu et al., 2018) proposed an elasto-plastic model for saturated frozen soils. Dong (Dong et al.,
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2014) conducted creep experiments to analyze the creep curves and creep index.
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The current creep model theory does not present an absolute solution to the entire
rheological behavior of frozen soil under high stress in deep alluvium. Although the Nishihara
model and the generalized Nishihara model are widely used in geotechnical engineering and
their theory is relatively complete, the models are based on an ideal linear element and the three
stages of the creep curve are ignored; thus, their results differ cons iderably from the actual creep
phenomenon of frozen soil under high stress. It is found that fractional calculus can be an
excellent mathematical instrument for modeling some phenomena, espec ially memory-intens ive
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or path-dependent phenomena, and damping description may also be involved in. Fractional
order models also have the advantages of fewer parameters and s imple forms. It is known that
the mechanical behaviors of viscoelastic solids (Enelund et al., 2011), viscoelastic fluid (Hyder
Ali Muttaqi Shah et al., 2011), and hyperviscoelastic materials (Libertiaux et al., 2011), can be
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described by the fractional calculus. Zhou (Zhou et al., 2011) attempted to replace a Newtonian
dashpot in the c lassical Nishihara model (a mathematical model) with the fractional derivative
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Abel dashpot in his salt rock creep research. However, it is rarely investigated whether artific ial
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frozen soil can be modeled using fractional calculus.
This paper aims at developing a fractional derivative-based creep constitutive model of
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artificial frozen soil under high stress. Analytic al formulas are derived and then validated
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through experimentally obtained data. A relevant constitutive finite element program is
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developed and used to s imulate a deep artificial frozen curtain of a coal mine. This research has
theoretical and practical s ignificance. It can be applied in artific ial ground freezing method in
deep coal mine excavation and subway tunnel engineering.
2 Triaxial Tests on Deep Artificial Frozen Soil
2.1 Test and scheme design
The frozen soil samples are representative clay samples obtained from a mine in Huainan,
their main phys ical and mechanical properties are shown in Table 1, and the grain s ize curve is 5
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shown in Fig. 1. The soil s amples were prepared in batches us ing a standard sample preparation
method. Spec ifically, the on-site drilling samples were first crushed, dried, and sifted. Each wet
sample was remolded to meet a specific moisture content and then placed in an airtight,
non-evaporating environment for approximately 6 h to ensure uniform waterdistribution. Next,
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the wet soil was placed ins ide a special mold in layers and compacted to a specific dry bulk
dens ity under a certain consolidation pressure by the sample-making machine. The purpose of
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consolidation is to match the dry bulk dens ity of the samples to that of the natural soils at
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different depths. The consolidation confining pressure was determined according to the buried
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depth of the stratum, and the consolidation processes were completed when the dens ities met the
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requirements. The mold loaded with the wet soil was tightly sealed for approximately 24 h to
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shape the soil sample; then, the molded soil sample was cut into a standard cylindrical shape
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(diameter of 61.8 mm and height of 125 mm with a prec is ion control of 0.1 mm) . The sample
was rapidly frozen to temperatures of -5℃, -10℃, and -15℃ to avoid frost heave. After
refrigeration for 48 h, the samples were covered with a rubber sleeve to avoid moisture
evaporation before testing.
After the soil samples were prepared, a series of triaxial creep and shear tests as well as
uniaxial creep tests were carried out. To ensure the reliability of the test results, the loading
process and data acquis ition were controlled using a computer. The test scheme is shown in
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Table 2.
Table 1 Physical and mechanical properties of conventional soil samples
Liquid
Moisture Sampling
Wet Density
Proportion
Dry Density
Content
Limit Il 3
Depth (m)
3
(g/cm )
(g/cm )
IP (%)
(n/ Gs)
2.038
1.679
3.709
(%)
47.5
27.5
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Pr
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pr
34.52
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(%)
425.6
Plastic Limit
Fig. 1 Grain size distribution of the sample soil
Table 2 Scheme of triaxial test on deep artificial frozen soil Consolidation Test No.
Confining Pressure (MPa)
Stress
Temperature (℃)
7
Test Type
rate (N/s)
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Triaxial creep test and
A-1
3.0
-5
A-2
3.5
-5
A-3
4.0
-10
A-4
5.0
-15
A-5
0
-10
B-1
4.0
-5
B-2
4.5
-10
B-3
5.0
-10
B-4
6.0
-15
B-5
0
C-1
5.0
C-2
6.0
triaxial shear test Triaxial creep test and triaxial shear test
70
70
Triaxial creep test and triaxial shear test Triaxial creep test and triaxial shear test
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Uniaxial creep test
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Triaxial creep test and triaxial shear test
70
70 70 110
C-4
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C-5
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triaxial shear test
Triaxial creep test and triaxial shear test Triaxial creep test and triaxial shear test Uniaxial creep test
110
110
110 110
Triaxial creep test and
-5
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C-3
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Triaxial creep test and
triaxial shear test
-10
6.5
-15
7.0
-15
0
-10
Triaxial creep test and triaxial shear test Triaxial creep test and triaxial shear test
150
150
150
Triaxial creep test and triaxial shear test Uniaxial creep test
150 150
Note: Consolidation confining pressure is the confining pressure at which the consolidated sample is frozen. 2.2 Test results
(1) Triaxial shear test
Stress-strain curves of deep artific ial frozen soil under triaxial shear test were obtained 8
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through a series of measurements, as shown in Fig. 2, where σ 1 -σ 3 denotes the deviatoric stress, and ε1 denotes the vertical strain; the p-q curves were obtained by analyzing the test
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results, as shown in Fig. 3.
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Fig. 2 Triaxial Stress-strain Curves of Artificial Frozen Soil under Different Temperatures
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q/MPa
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p/MPa
11
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q/MPa
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p/MPa
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Fig. 3 p-q Relation Curves of Deep Artificial Frozen Soil at the Same Temperature
3
, q 1 3 )
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revealed the following:
1 2 3
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The triaxial shear test of deep artific ial frozen soil (for which p
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1. When confining pressure is less than 7 MPa, the stress-strain curve is strain-hardening
curve. A typical characteristic of this kind of curve is the existence of yield limit. The frozen soil
stress-strain curve consists of two deformation stages, the initial elastic phase and the plastic
hardening phase. When the deviatoric stress is less than the yield limit, the frozen soil receives
less axial force, and the additional damage ins ide the soil is less. The material is in the stage of
linear elastic deformation, and the relationship between stress and strain obeys Hooke's law.
With the increase of the axial load, the deviatoric stress of the frozen soil exceeds the yield limit,
and the joint between the soil partic les and the ice crystal begins to appear in the soil, which 12
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reduces the ability of frozen soil to resist compressive deformation. When the frozen soil enters
the stage of plastic hardening, the trend of stress increas ing with strain begins to s low down, and
the stress maintains an increasing state, and finally stabilizes.
2. When the confining pressure is greater than or equal to 7 MPa, the trend of the
oo
f
stress-strain curve develops toward the strain softening curve. Confining pressure has two
pr
oppos ite effects on the strength of frozen soil. On the one hand, with the increase of confining
e-
pressure, the defects (micro-cracks, voids, etc.) in frozen soil gradually heal, and the
Pr
cementation force, structural force, friction force and cohes ion of soil particles gradually
increase. The frozen soil is continuous ly strengthened, and the plastic properties of frozen soil
al
are enhanced. On the other hand, with the continuous increase of confining pressure, the ice in
rn
frozen soil melts continuous ly, and the weak mineral partic les are crushed continuous ly. The
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frozen soil is weakened, and the strength tends to decrease.
3. At same confining pressure, the shape of the stress-strain curve of frozen soil is similar at
different temperatures, but the initial slope of the stress-strain curve is different. The hardening
rate of the stress-strain curve is different at different temperatures. The lower the temperature is,
the faster the hardening rate of the stress-strain curve at the initial stage is. On the contrary, the
slower the hardening rate is. This is because with the decrease of temperature, the content of
unfrozen water in frozen soil decreases gradually, and the cementation between ice and soil
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increases, so that the lower the temperature of soil, the greater the modulus of deformation. At
the same time, the yield limit increases with the decrease of temperature.
4. At same temperature, the initial s lope of stress-strain curve of frozen soil increases
with the increase of confining pressure, which mainly shows that the hardening rate of
oo
f
stress-strain curve is different under different confining pressure. The hardening rate of
pr
frozen soil at the initial stage of stress-strain curve increases with the increase of confining
e-
pressure. This is because confining pressure increases the compactness of soil and improves
Pr
the resistance of soil to initial compressive deformation. At the same time, the yield limit
rn
(2) Triaxial creep test
al
increases with the increase of confining pressure.
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Creep curves of artific ial frozen soil under different confining pressures and freezing
temperatures were obtained through triaxial creep test of deep artific ial frozen soil, as shown
in Fig. 4, and typical isochronous stress-strain curves of artificial frozen soil were obtained
by analyzing the creep curves, as shown in Fig. 5.
14
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rn
al
Pr
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pr
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f
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15
pr
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f
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al rn Jo u
MPa
Pr
e-
Fig. 4 Typical Creep Test Curves of Artificial Frozen Soil at Different Temperatures
ε1 /% Fig. 5 Isochronous Stress-strain Curves at a Freezing Temperature of -10℃
The triaxial creep test of deep artificial frozen soil revealed the following:
(1) The test curves of triaxial creep and uniaxial creep are s imilar, demonstrating the
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theoretical basis of the transition from uniaxial creep to triaxial creep;
(2) We can see from the data presented above that isochronous creep curves include four
segments and represent a linear distribution at low axial pressure, showing that the
Drucker-Prager yield criterion can also be used to describe the yield of frozen soil;
oo
f
(3) The creep properties of frozen soil are c losely related to time and stress levels. In the
pr
creep tests, when the stress level is lower than the yield stress of the frozen soil, the frozen soil
e-
will experience the first and second creep stages, namely the decay and steady-state creep stage,
Pr
but no acceleration creep stage, and mainly manifested as viscoelasticity.
al
(4) During the creep tests, when the creep stress exceeds a critical value σ s ( yield stress ),
rn
the third creep stage occurs. The frozen soil w ill undergo three stages of decay, steady state and
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acceleration. The inside of the frozen soil will be damaged and gradually accumulated until the
accelerated creep of the frozen soil occurs. Acceleration creep stage lasts only for a short time
before damage occurs. Because frozen soil in deep alluvium is subjected to a significant ground
pressure and enters the third creep stage, the accelerated creep stage of frozen soil must be
considered.
(5) At the same temperature, the creep value of frozen soil increases with the increase of the value σ 1 -σ 3 , and at the same value σ 1 -σ 3 , the creep value of frozen soil decreases with the
decrease of the temperature. 17
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(6) During tests, frozen soil shows transient elastic strain, but less than 10% creep
deformation;
The Drucker–Prager yield criterion can be used for the triaxial compress ion and creep tests
of frozen soil, providing the basis for fractional derivative model fitting.
oo
f
2.3 Validation of yield criterion for deep artificial frozen soil The measured creep yield criterion of frozen soil is described here. Tests reveal the creep
e-
al rn
Jo u
J
0.5
Pr
as presented in Fig. 6 and Fig. 7.
pr
strength envelope and Drucker-Prager yield criterion of deep artificial frozen soil with the results
σ m/MPa Fig. 6 Drucker-Prager Yield Criterion of Artificial Frozen Soil
18
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σ/MPa
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f
Pa
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Fig. 7 Creep Envelope of Deep Artificial Frozen Soil
e-
Based on the data shown in Fig. 6 and Fig. 7, the following conclusions can be obtained from
Pr
the tested pressure range: the p-q relation of frozen soil is a curve and is linear at low stress, and
al
the creep envelope of frozen soil is nearly straight. We can conclude that the transient deformation
rn
of frozen soil complies with the property of Drucker-Prager material; therefore the yield of the
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frozen soil can be approximately described using the Drucker-Prager yield criterion.
3 Fractional Derivative-Based Creep Constitutive Model of Deep Artificial Frozen Soil
3.1 The fractional derivative constitutive equation Since the traditional integral order calculus constitutive model has some limitations, fractional calculus is applied to calculate the creep of soft soil. The fractional derivative-based rheological model uses Abel theorems to replace the Newton equations in classical Nishihara theory. At the same time, in classical Nishihara model, the yield limit σ s is a one-dimensional constant，and When σ is larger than σ s, the model enters the plastic yield stage. Under the
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three-dimensional stress state, the fractional derivative-based rheological model replaces σ s with a three-dimensional function group F， considering the anisotropy of the mechanical properties of the material and the changes under the influence of temperature. When the value under various factors exceeds the limit of this function, the soil sample will enter the plastic yield stage, which can more accurately describe the plasticity yield stage of frozen soil. The traditional Nishihara
e-
pr
oo
f
model is shown in Fig. 8.
Pr
Fig. 8 Traditional Nishihara Model The traditional Nishihara model consists of Hooker body (H), viscoelastic body (N/H) and
al
viscoplastic body (N/St V). In Fig 8, where E0 denotes the elastic modulus; E1 denotes the
rn
viscoelastic modulus; η1 , η2 denote the viscous coefficients of the kettle.
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The fractional derivative-based creep model is shown in Fig. 9.
Elastic stage
Elastic-visco stage
Visco-plastic stage
Fig. 9 Fractional Derivative-Based Creep Model The fractional derivative-based visco-elastic-plastic constitutive model is explained as follows:
20
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(1)
1 d t x d 1 r dt 0 t r
(2)
D r f t can be solved by Laplace transform:
m r L Dr f t L D m D f t
oo
f
m 1 mr mr S m L D f t S mk 1D f t k 0 t 0
m 1
S r F s S mk 1D k mr f 0 （m – 1 < r < m, m = 1, 2, 3…）
(3)
pr
k 0
e-
wherein, s and m are the Laplace transform parameters.
Pr
In the case of m = 1, and with the initial condition set as 0,
al
1 r L Dr f t S r F s D f 0 S r F s
rn
The constitutive equation of the fractional derivative-based Nishihara model can be solved by
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Laplace transform:
s E s q1S r s
(4)
wherein, q1 indicates the creep modulus of fractional derivative-based Nishihara model. Therefore, for σ1 -σ3 <σs , the constitutive equation of the fractional derivative-based Nishihara model is as follows:
t
1 Er t ，t 0 ， E1
t / Er t 1 1 1 rn n 1
wherein,
0
n
rn
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Therefore:
t / t 1 1 E1 n 1 1 rn 0
rn
n
(5)
In the case of σ 1 -σ 3 ≥σ s , the constitutive equation of the fractional derivative-based Nishihara model is as follows:
t / F t t 1 1 E1 n 1 1 rn 2 0
rn
(6)
oo
f
n
pr
3.2 Viscoelastic modulus of artificial frozen soil
The visco-elastic-plastic stress can be calculated using forward integration, and the stress
e-
increment is given by equation (7)
Pr
i( k ) Dije (ej e( k ) eepj ( k ) )
al
Wherein:
e( k )
Jo u
rn
e j
t
1
e
E1
1
(7)
t
(8)
Therefore, the viscoelastic modulus of artificial frozen soil can be expressed as follows:
t / Er t 1 1 1 rn n 1
n
rn
(9)
3.3 Visco-elastic-plastic flexibility matrix It is assumed that the creep of frozen soil satisfies the classical theory of plastic mechanics: (1) the rheological deformation results from deviatoric stress and is not related to the hydrostatic pressure; and (2) the principal axis of stress coincides with the principal axis of strain. Thus, the
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viscoplastic increment can be expressed as follows:
m cm em
(10)
s j cl (elj eepj )
(11)
wherein:
Et 1 2
cl
Et 1
oo
Partial derivatives of formula (10) and (11) are as follows
(12)
f
cm
(13)
pr
m cm [1 1 1 0 0 0]T { } 3
(14)
Pr
e-
si el (eiep ) cl ( i ) e j e j e j
wherein:
Jo u
rn
al
4 2 elj 1 2 e j 6 0 0 0
(ej p ) ei
2 2 0 0 0 4 2 0 0 0 2 4 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3
t t F
2
F { }
(15)
(16)
According to the above tests, the Drucker–Prager yield criterion can be used to describe the creep yield of frozen soil.
F
6sin 6c cos m 3J 2 (3 sin ) (3 sin )
(17)
Wherein: φ is the internal friction angle of frozen soil, in degrees; c is the cohesive force of 23
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frozen soil, in MPa; and σ m and J 2 are the first stress tensor invariant and the second deviatoric stress tensor invariant, respectively.
F 6sin m 3J 2 { } (3 sin ) { } { }
J2 1 Sx { } 2 J 2
Sy
2 yz
Sz
(18)
2 zx
2 xy
T
1 1 1 [ P] 0 0 0
0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
al
Pr
e-
1 1
1 1 0
0
1 0
0
Jo u
rn
2 1 1 [ R] 0 0 0
pr
wherein: [P] and [R] are coefficient matrixes;
2
(20)
oo
f
F 6sin [ P] 3[ R] ( ){ } { } (3 sin ) 9 m 6 J 2
(19)
1 2
0
0
0
0
6
0
0
0
0
6
0
0
0
0
0 0 0 0 0 6
(21)
(22)
The relationship between the deviatoric stress tensor invariant and the components of stress is as follows:
i si m
i = 1, 2, 3
(23)
i si
i = 4, 5, 6
(24)
Therefore, the visco-elastic-plastic matrix components can be expressed as follows
24
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si m e j e j
DijVP
Assuming a 2
t t F
2
si e j
i = 1, 2, 3
(25)
i = 4, 5, 6
(26)
2sin and 3 m (3 sin )
b2
t t F
2
3 , the 6 J2
components of the viscoplastic flexibility matrix can be expressed as follows: (27)
1 1 d12 d13 d 23 cm cl a +b 3 3
(28)
pr
oo
f
1 2 d11 d 22 d33 cm cl a 2b 3 3
(29)
e-
1 d 44 d55 d66 cl 6b 2
Pr
The flexibility matrix is a symmetric matrix and the remaining components are 0.
rn
of Constitutive Model
al
4 ADINA Finite Element Program-Based Secondary Development and Validation
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ADINA 8.1 version supports the secondary development of material constitutive relations, unit algorithm, element failure criterion, structure rupture criterion, crack extension rule, and boundary conditions.
4.1 Verification of fractional derivative model of artificial frozen soil FORTRAN language was used to develop the constitutive model for artificial frozen soil, and the element type, integration method, and nonlinearity of finite elements were considered.
Uniaxial and triaxial creep tests were carried out on artificial frozen clay samples under different temperatures and stresses, and regression analysis was conducted to assess the constitutive relation of artificial frozen soil under complex stress. The rheological parameters (as
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shown in Table 3) were obtained by uniaxial creep test on deep artificial frozen soil using cylindrical samples with a diameter of 50 mm and height of 100 mm.
Table 3 Parameters for the Numerical Calculation of the Fractional Derivative Model of Deep Artificial Frozen Soil
Test
E0
E1
η1
η2
c
φ
μ MPa
MPa
MPa·min
-5℃
76
0.26
140
2358
-10℃
138
0.22
250
3250
-15℃
204
0.18
375
4670
10 MPa·min
MPa
°
oo
f
Temperature
m 3
1.5
5.48
1.4
5856
3.1
6.08
1.4
8750
4.8
6.68
1.4
e-
pr
3762
rn Jo u
al
Pr
The creep curves obtained by numerical simulation are shown in Fig. 10.
Time/h Fig. 10 ADINA-Simulated Triaxial Creep Test Curves of Cylindrical Samples
26
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rn
al
Pr
1 /%
e-
pr
Time/h
oo
f
1 /%
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Time/h
27
Time/h
oo
f
1 /%
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rn
al
%
Pr
e-
pr
Fig. 11 Measured and Simulated Triaxial Creep Test Curves for Cylindrical Samples
Time/h Fig. 12 Error between Triaxial Creep Test and Simulation versus Time As can be seen from the data presented in Fig. 11 and 12, the values obtained from the triaxial compression creep test and the simulation changed similarity, with error within 0.15%, further demonstrating that the fractional derivative-based creep model can accurately describe the constitutive relation of frozen soil under high stress. The fractional derivative-based creep constitutive model was added and the ADINA material library was expanded with the powerful secondary development platform of ADINA. The uniaxial and triaxial compression creep test results show that this model can accurately describe creep under high ground pressure. 28
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(1) Under a low constant force, frozen oil experiences both unstable and stable creep, i.e., the first and second creep stages, and the theoretical value agrees well with the measured value. (2) When the creep stress exceeds the critical value σ s , the third creep stage occurs, but lasts only for a short time before damage occurs. Because frozen soil in deep alluvium withstands significant ground pressure and enters the third creep stage, the accelerated creep stage of frozen
f
soil must be considered. According to the test results, there are differences between the values
oo
obtained from theoretical calculation and tests early in the accelerated creep stage, but these
pr
values are consistent in the damage stage.
e-
Therefore, it is feasible to use this fractional derivative-based creep constitutive model for
Pr
theoretical calculations of frozen soil under high stress such as deep alluvium and foundation of high-rise buildings.
rn
al
4.2 Numerical simulation of frozen curtain by artificial freezing method For the excavation of a coal mine with the artificial ground freezing method, the freezing
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depth is 500 m, and it freezes with three circles of freezing tubes. The net diameter of the shaft is 6 m, and the designed thickness of the frozen wall is 9 m. Considering the axial symmetry of the cylindrical frozen wall, one-fourth of the physical model is simulated, as shown in Fig. 13. In the calculated physical model, the upper surface is set as the force boundary, bearing the dead weight of the overlying topsoil, its value is 6 MPa; the right boundary is set the lateral pressure calculated by the theoretical formula of heavy liquid, its value is 4 MPa; and the front and rear boundaries are symmetric boundaries with constrained tangential displacement in the circumference direction; the temperature of frozen wall is -15 ℃, and the excavation height is 20 m. To verify the calculated results, a measured displacement site was set at 0.8 m depth from top to bottom. The main 29
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f
freezing technical parameters of the construction are shown in Table 4.
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Fig. 13 Calculation Model for Shaft Excavation
e-
Table 4 Main freezing technical parameters
Pr
Item
al
Freezing depth (m)
500 6
rn
Net diameter of the shaft (m)
Parameter
9
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Freezing wall thickness (m) Average temperature (℃)
-15
Number of freezing holes
40
Circulation temperature of freezing brine (℃)
-26～-28
Temperature of side wall (℃)
-6～-8
The distribution rule of the sidewall displacement with the depth is obtained by the numerical simulation, as shown in Fig. 14. As shown in the graph, for the radial creep displacement of the frozen wall in the exposed section at a section height and different exposure time, the displacement at the middle and lower parts are higher than the displacement at the upper part, and
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the curves form the shape of a fish belly.
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pr
oo
f
Depth/m
Displacement/mm
Pr
Fig. 14 Distribution Rule of Sidewall Displacement with Depth Constrained by the construction status and conditions, it is usually impossible to obtain all
al
the necessary displacement data of the sidewall of the frozen curtain. The measuring points at the
rn
sidewall can be arranged only after they are excavated. In addition, the measurement may often be
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limited by construction conditions. For instance, a concrete wall may be built after a working of the section height of the shaft is finished, and the displacement measuring must be stopped. Therefore, to evaluate the difference between the measured displacement of the sidewall of the frozen curtain and the actual displacement of the frozen curtain, the measured results and ADINA-based numerical simulation data were compared and analyzed, as shown in Fig. 15.
31
Time/h
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f
Displacement/mm
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Fig. 15 Comparison of Sidewall Displacement Measured at Measuring Points and Obtained
e-
from Numerical Simulation
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Fig. 15 presents the comparison of the measured displacement data of the frozen curtain of
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the sidewall at 0.8 m from the top to bottom and the numerical simulation calculations of the
rn
corresponding points during shaft excavation. It can be seen from the graph that, at the instant of
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excavation of the section height, the lower measuring points of the displacement working face experience advanced displacement (approximately 3.5 mm). With the excavation of the working face, there is a gradual increase in the displacement of the measuring points, and the displacement of the sidewall reaches 22 mm when the measuring point is exposed. At this time, the measuring point was established 8 h after the excavation of the section height. The comparison of the measured displacement curves and numerical simulation curves shows that these curves are similar in shape. Limited by the supporting work, it is impossible to obtain all the displacement data of the excavated section height by measurement. A comparison of measured data and maximum numerical simulation data shows that the measured data only accounts for 16/53 of the
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actual values under the engineering conditions, approximately 30%.
Based on the numerical simulation results, the displacement of the floor heave of the working
rn
al
Pr
e-
pr
oo
f
face is shown in Fig. 16.
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Fig. 16 Distribution Rule of Working Face Displacement with Shaft Center The displacement of the floor heave of the working face of shaft has the following distinct features: 1. The displacement curve is a curve of more than cubic power, and is inflected near the sidewall. 2. The maximum displacement occurs at the center of shaft. 3. The displacement is one-fourth times larger than that of the sidewall. 4. The deformation of the floor heave is the main contributor of the advanced displacement of the frozen wall below the working face. 5. The deformation of the floor heave increases with the increase in the excavated section height and the extension of the excavation time.
After the excavation of the section height for 4 h, significant displacement takes place at the
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floor heave of the working face of the shaft, reaching 22 mm and above. As the working face is excavated deeper and deeper, the displacement of the floor heave gradually increases to reach a maximum of 32 mm after 8 h and 47 mm after 12 h. When the entire section height is accomplished, the maximum displacement of the floor heave is 66 mm. A comparison of the measured displacement of the floor heave of the excavated working face and the simulated values
f
was performed and is shown in Table 5.
Working Face
oo
Table 5 Comparison of the Displacement of the Sidewall of the Frozen Curtain and the
pr
Numerical Simulation Numerical Simulation Calculation Method
Pr
Maximum Displacement of
24.5
al
Sidewall/mm Maximum Uplift Displacement
(16 h)
Measurement (8 h)
49.8
16
66
21
e-
(8 h)
Field
rn
32
Jo u
of Working Face/mm
We can see from Table 5, because the displacement of the floor heave of the working face can only be measured at the excavated section height and in the supporting process, there are limited field measurements of the displacement. Only 8 h were available for the measurement, and the maximum displacement occurred at the center of the shaft, at approximately 21.0 mm. The maximum displacement determined from numerical simulation is 32 mm in 8 h. Therefore, the measured value is only 60% of the simulated value.
5 Conclusions A self-developed W3Z-200 testing system was used for triaxial compression and creep tests
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on frozen soil, and the test results were analyzed. Linear Newton equations in the Nishihara model were replaced with nonlinear Newton equations, and a linear spring was replaced with a Drucker-Prager yield function. In this manner, the Nishihara model was improved, resulting in the fractional derivative-based
creep
constitutive model for
artificial frozen
soil under
three-dimensional stress. In addition, the flexibility matrix of application and numerical
f
calculation was derived with the Drucker-Prager yield criterion. The fractional derivative
oo
constitutive model of artificial frozen soil requires few parameters and simple calculations .
pr
(1) When stress is low, deep artificial frozen soil shows a stable creep. When σ1 -σ3 >σs , the
e-
soil shows non-decaying creep for a short time before the damage occurs, indicating that the
Pr
fractional derivative constitutive model is suitable to describe the creep of the frozen soil. (2) The constitutive model is added into a large nonlinear program ADINA by secondary
al
development of a finite element program, and uniaxial and triaxial creeps of frozen soil were
rn
tested and simulated. The results show some errors between the stress-strain curves and measured
Jo u
curves of frozen soil before the stress reaches its limits, and the theoretical values coincide with the measured values in the final creep stage at maximum stress, demonstrating that this model can describe the overall non-decaying creep process of the frozen soil under high stress. (3) This model can be used for the mechanical analys is of a frozen curtain in a deep mine. The maximum displacement of the sidewall of the frozen curtain obtained from the numerical simulation within 16 h was 48.9 mm, while the measured maximum displacement is 16.8 mm at the same section height. The floor heave of the working face was distributed as measured on field, and is especially large near the center of the shaft with an ADINA-based simulated value of 66.0 mm. The floor heave measured on site was 40.0 mm, 60% of the simulated value.
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Acknowledgements This study is supported by the National Natural Science Foundation of China (Grant No. 41672278, No. 41271071, No. 41977236, and No. 51504070), the Fujian Provincial Natural Science Foundation Projects (Grant No. 2017Y4001), the Jiangxi Provincial Natural Science Foundation Projects (Grant No. 20192ACBL20002), and the State Key Laboratory Program of
pr
oo
f
Frozen Soil Engineering (Grant No. SKLFSE201204), we gratefully acknowledge these supports.
References:
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Andersland, O. and Akili, W., 1967. Stress effect on creep rates of a frozen clay soil. Geotechnique, 17(1): 27-39.
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Arenson, L.U., Springman, S.M. and Sego, D.C., 2006. The rheology of fro zen soils. Applied Rheology, 17(1): 12147.
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GEOCRYOLOGY, 17(S): 54-59.
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rn
Changqing, Z., Xuexia, W. and Tiande, M., 1995. Microstructure Damage Behaviour and Change Characteristics in the Frozen Soil Creep Process. JOURNA L OF GLA CIOLOGY AND GEOCRYOLOGY, 17(S1): 60-65.
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Christ, M., Kim, Y. and Park, J., 2009. The influence of temperature and cycle s on acoustic and mechanical properties of frozen soils. KSCE Journal of Civil Engineering, 13(3): 153-159. Dong, L., Zhang, G., Zhao, S. and Pan, W., 2014. Experimental study of the creep indexes of frozen soil. Journal of Glaciology and Geocryology, 36(1): 130-136. Enelund, M., Mahler, L., Runesson, K. and Josefson, BL., 1999. Formu lation and integration of the standard linear viscoelastic solid with fractional o rder rate laws. International Journal of Solids And Structuresv, 36(16):2417– 2442. Ghoreishian Amiri, S., Grimstad, G., Kad ivar, M. and Nordal, S., 2016. Constitutive model for rate-independent behavior of saturated frozen soils. Canadian Geotechnical Journal, 53(10): 1646-1657. Hyder Ali Muttaqi Shah, S., Qi, H., 2010. Starting solutions for a vis coelastic flu id with fractional burgers’ model in an annular pipe. Nonlinear Analysis: Real World Applications, 11(1):547–554. Lai, Y., Jin, L. and Chang, X., 2009. Yield criterion and elasto -plastic damage constitutive model for frozen sandy soil. International journal of plasticity, 25(6): 1177-1205. Lai, Y., Yang, Y., Chang, X. and Li, S., 2010. Strength criterion and elastoplastic constitutive model of frozen silt in generalized plastic mechanics. International Journal of Plasticity, 26(10): 1461-1484. Li, D., Fan, J. and Wang, R., 2011. Research on visco-elastic-plastic creep model of artificially frozen 36
Journal Pre-proof soil under high confining pressures. Cold Regions Science and Technology, 65(2): 219-225. Li, D., Yang, X. and Chen, J., 2017. A study of triaxial creep test and yield criterion of artificial frozen soil under unloading stress paths. Cold Regions Science and Technology, 141: 163-170. Liao, M., Lai, Y., Liu, E. and Wan, X., 2017. A fractional order creep constitutive model of warm frozen silt. Acta Geotechnica, 12(2): 377-389. Libertiau x, V., Pascon, F., 2010. Differential versus integral formu lation of fractional hyperviscoelastic constitutive laws for brain tissue modelling. Journal of Co mputational and Applied Mathematics; 234(7):2029–2035. Liu, E., Lai, Y., Wong, H. and Feng, J., 2018. An elastoplastic model for saturated freezing soils based on thermo-poromechanics. International Journal of Plasticity, 107: 246-285. Loria, A.R., Frigo, B. and Chiaia, B., 2017. A non -linear constitutive model for describing the mechanical behaviour of frozen g round and permafrost. Co ld regions science and technology, 133:
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63-69.
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Ma, W., Wu, Z.W. and Sheng, Y., 1994. Creep and creep strength of fro zen soil. Journal of Glaciology and Geocryology, 16(2): 113-118. frozen soil. Science in China(Series B)(03): 309-317.
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Tiande, M., Xuexia, W. and Changqing, Z., 1995. M icrostructure damage theory of creep process in Vyalov, S.S., 2013. Rheological fundamentals of soil mechanics. Elsevier.
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Yao, X., Qi, J., Zhang, J. and Yu, F., 2018. A one-dimensional creep model for frozen soils taking temperature as an independent variable. Soils and foundations, 58(3): 627-640.
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Zhao, L., Yang, P., Wang, J. and Zhang, L., 2014. Cyclic direct shear behaviors of frozen soil– structure interface under constant normal stiffness condition. Co ld Regions Science and Technology, 102: 52-62.
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soil-structure interface under constant normal stress and sub -zero temperature. Co ld Regions Science and Technology, 133: 70-81.
Zhou, Z. et al., 2016. Multiaxial creep of frozen loess. Mechanics of Materials, 95: 172-191. Цытович, Н.А., 1985. Frozen soil mechanics. Science press, Beijing. Zhou, H.W., Wang, C.P., Han, B.B., and Duan, Z.Q., 2011 A creep constitutive model for salt rock based on fractional derivatives. International Journal Of Rock Mechanics And Mining Sciences, 48(1):116–121.
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The authors declare that there is no conflicts of interest regarding the publication of this paper.
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Dongwei Li, Chaochao Zhang, Guosheng Ding, Hua Zhang, Junhao Chen, Hao Cui, Wangsheng Pei, Shengfu Wang, Lingshi An, Peng Li, Chang Yuan
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Highlights 1, A fractional derivative constitutive model of deep artificial frozen soil was established; 2, Nonlinear Newton equations were used to consider the physical and mechanical properties of frozen soil; 3, The fractional derivative constitutive model can be used to describe frozen soil under high
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confining pressures.
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