Abstract

In order to study the creep characteristics of frozen soil under different confining pressures, triaxial creep tests of frozen soil under different temperatures are carried out and the creep deformation law of frozen soil under different confining pressures and different freezing temperatures is analyzed. In this paper, the unsteady Abel dashpot is constructed by the unsteady viscosity coefficient. On this basis, the fractional order of the viscosity dashpot is unsteady. Then, a time-dependent creep constitutive model for frozen soil based on the unsteady fractional order is established. The model can better describe the law of accelerated creep deformation of rock. Through the Levenberg–Marquardt algorithm, the parameters of the triaxial creep model are identified to verify the correctness of the creep model. The results show that the model can describe the creep deformation of frozen soil well and overcome the shortcoming that the traditional model cannot describe the law of accelerated deformation. At the same time, the agreement between the model curve and the test curve is also much greater than that between the traditional Nishihara model curve and the test curve. This lays the foundation for the later use of the creep model to simulate the settlement and horizontal displacement of subgrade soil under long-term loads and also provides a theoretical basis for the construction of other roadbed projects.

1. Introduction

Frozen soil is a substance with temperature less than zero degree Celsius and ice inside. It consists of solid soil particles, unfrozen water, gas inside the pores, and solid ice [1, 2]. It widely exists in most parts of northern China. The biggest difference between frozen soil and loose soil or ordinary unfrozen soil is that it contains a large amount of solid ice. Therefore, in a narrow sense, frozen soil samples containing solid ice are collectively referred to, and the bond strength of soil samples is greatly improved due to the presence of solid ice. In a broad sense, frozen soil is a kind of “rock” that internal pore water freezes into solid ice [35]. In the freezing environment below zero degree Celsius, frozen soil has similar engineering mechanical properties to rock and its strength is basically the same as that of plain concrete. The frozen soil also has obvious rheological properties, which are mainly manifested in the following two aspects [6, 7]. (1) The long-term strength of frozen soil under long-term loads is far less than its instantaneous strength. (2) Under long-term loads, the deformation of frozen soil increases with time and shows obvious viscosity. The study of the constitutive model of frozen soil is the basic and important part in the study of creep properties of frozen soil. At the same time, it is also an important part of transforming field monitoring data and test data into engineering practice [8, 9]. There are three main methods to construct the constitutive model of frozen soil creep. (1) The creep constitutive model for frozen soil is obtained by fitting the experimental data with the empirical formula. (2) The creep constitutive model for frozen soil is constructed by a series-parallel combination of viscous, elastic, and plastic elements, and the parameters in the model are determined by parameter identification with regression theory. (3) The empirical and semiempirical creep constitutive models for frozen soil are constructed by using classical endochronic theory, damage mechanic theory, and nonlinear theory. The model constructed by this method can not only describe the accelerated creep stage of frozen soil but also reflect the steady creep stage and the decay creep stage of frozen soil. This requires the creep test of indoor frozen soil to study the deformation and failure mechanism of frozen soil in the process of creep deformation. This can pave the way for the establishment of creep constitutive models.

Scholars have carried out a lot of research on the creep deformation characteristics of frozen soil [1012]. The creep deformation stage of frozen soil and the change in the internal microstructure of frozen soil during the deformation stage are mainly summarized as follows: When the frozen soil is under low-stress loads, the creep deformation of frozen soil will only appear at two stages of attenuation creep and stable creep [13, 14]. Creep deformation characteristics and creep deformation rate characteristics are basically consistent with rock creep. When the frozen soil is under high stress, the creep deformation of frozen soil will appear as accelerated creep deformation. Under the action of high stress, the soil particles inside the frozen soil are broken and the solid ice inside the pores is broken, which makes the mechanical properties of frozen soil decrease sharply. This further affects the bonding between soil particles and solid ice. Macroscopically, it is manifested as unstable deformation and failure of frozen soil [1517]. The creep deformation rate of frozen soil also increases sharply at this stage. In order to better illustrate the research status of scholars on the creep deformation characteristics of frozen soil in recent years, some main research results are listed as follows: In order to study the microstructure deterioration of frozen soil under freezing action, Miao et al. [18] introduced the damage variable into the study of creep deformation of frozen soil for the first time and established a damage model to describe the creep deformation law of frozen soil. When studying the viscoelastic-plastic deformation law of frozen soil, He et al. [19] found that the damage potential function can better describe the deterioration of mechanical properties of frozen soil. A damage evolution model was established to describe the damage-deformation law of frozen soil under freezing action. When studying the creep deformation of metal materials, Vyalov [20] found that aging theory can be applied to establish the creep damage model for frozen soil. The established model can better describe the attenuation creep deformation law of frozen soil. Based on the analysis of the triaxial creep and shear test results of deep artificial frozen soil under different confining pressures and temperatures, a fractional derivative constitutive model for deep artificial frozen soil was established [21]. The fractional derivative constitutive model can describe the nondecay creep deformation characteristics of deep artificial frozen soil under high confining pressures. Based on the spherical indentation test, uniaxial creep test, and triaxial creep test, the creep model for frozen soil based on the Kelvin–Voigt fractional derivative model (KVFD) with a few parameters and clear physical description was established [22]. The results of the SIT, uniaxial, and triaxial creep tests show that the model is in good agreement with the existing laboratory test results of frozen soil. Zhang et al. [23] carried out the uniaxial compressive strength test, triaxial shear test, and triaxial creep test on artificially frozen soil at different temperatures. Based on the creep test results, a fractional constitutive model for sand-bearing frozen silt was proposed. Compared with other frozen soil creep models, this model has higher precision and stronger stress sensitivity. Taking the Cretaceous saturated frozen soil as the research object, Li et al. [24] carried out creep mechanical tests on frozen rock under different confining pressure conditions. The strain-time curves under different loading stress states are obtained. Based on the Riemann–Liouville-type integral function, a five-element nonlinear creep damage constitutive equation with low temperature-damage-stress coupling was established. The results show that the model can well describe the whole creep process of frozen rock.

In this paper, the physical properties of clay from roadbed are analyzed through indoor physical tests and the soil is rerolled and improved according to the test requirements to obtain a standard cylindrical sample. The triaxial compression test and the triaxial creep test of soil samples under different conditions are carried out.

2. Physical Properties of Clay

The physical properties of frozen soil along the Donggang-Heda highway subgrade are determined by laboratory tests to determine the type of soil. Sampling was performed in areas without interference at the beginning of construction. When sampling, the upper humus soil, grass roots, tree roots, and other soil that affect the test should be removed. The soil depth is positioned below 1 meter. The soil samples are taken back to the laboratory in the test bag, and the soil samples are tested according to the requirements of the geotechnical test method standard. The specific physical and mechanical parameters of the soil samples are shown in Table 1.

The soil samples were placed in a drying oven for drying, and the dried soil samples were weighed 300 g. Finally, the soil samples were crushed for the screening test, and the percentage of the particle size of the soil sample is shown in Table 2.

It can be seen in Table 2 that the particle size in the range of 0.075 mm∼1.0 mm accounts for more than 90%. In other particle size ranges, the content of the soil sample is less than 10%. Therefore, the main particles of clay are fine particles.

3. Study on Triaxial Deformation of Seasonal Viscous Frozen Soil

The soil sample is made into a marked specimen with a height of 125 mm and a diameter of 68.1 mm by using the three-lobe saturator of the test instrument. Before the final frozen soil sample, the specimen is saturated by vacuum immersion. The photographs of the clay samples are shown in Figure 1(a). The frozen soil sample is a remolded soil sample. The radial strain gauge and the axial strain gauge of the instrument are fixed in the radial direction of the specimen. The strain changes in the soil samples in the axial and radial directions during the loading process are measured, as shown in Figure 1(b).

The soil samples are tested by the loading method, as shown in Figure 1. The axial strain of the soil samples under different freezing temperatures is measured. In this paper, it is assumed that the height of the soil sample is l, its diameter is d, and its external load is P. Under the action of stress, the compressive deformation of the soil sample in the axial direction is Δl and the compressive deformation in the radial direction is Δd. It can be obtained that the axial and radial deformations of the soil sample under external loads arewhere εx is the axial strain and εy is radial strain.

The stress applied in the axial direction of the soil sample iswhere σ is the stress.

The soil sample is placed in the freezing equipment for freezing after demoulding and curing. The freezing temperatures are −5°C, −10°C, −15°C, and −20°C. The freezing time is 48 hours. The soil sample is taken out from the cooling equipment for the conventional triaxial compression test. The confining pressures of the triaxial compression test are 0.5 MPa, 1.0 MPa, 2.0 MPa, and 3.0 MPa.

The specific loading steps are as follows: (1) It is necessary to apply the confining pressure value to the predetermined value and keep the confining pressure value unchanged. The frozen soil sample is consolidated under the condition of the fixed confining pressure. The condition of consolidation termination is that the deformation of the soil sample is less than 0.005 mm/h. (2) The triaxial compression test is carried out at a rate of 0.002 mm/s. (3) When the soil sample is damaged, the axial strain rate is continuously increased for a period of time, and then, loading is stopped. (4) After the test, the damaged frozen soil sample is taken out. It should be noted that during the loading test, the test temperature of the soil sample remains unchanged under the action of the refrigeration system.

The stress-strain relationship of soil under different freezing temperatures and confining pressures is shown in Figure 2.

It can be seen in Figure 2 that the stress-strain curve trend of frozen soil samples is basically the same under different freezing temperatures and confining pressures. The strain increases with an increase in deviatoric stress. When the strain reaches a certain value, the curve gradually becomes flat. Under the same freezing temperature, as the confining pressure increases, the strength of the stabilized soil sample becomes larger. This shows that an increase in the confining pressure improves the strength of frozen soil to a certain extent. Under the same confining pressure, with a decrease in temperature, the strength value of the soil sample tends to be more stable. This shows that the bearing capacity of soil samples after freezing shows a significant increase. The pore water inside the original soil sample freezes into ice at low temperature. The volume of the soil sample expands. At the same time, the strength of frozen ice is far greater than that of water. The frozen ice filling in the pore makes the overall structural improvement of the soil sample.

4. Study on Creep Deformation Characteristics of Seasonal Viscous Frozen Soil

The confining pressure is 3 MPa, and the freezing temperature are −5°C, −10°C, −15°C, and −20°C. The creep characteristics of frozen soil at different freezing temperatures are tested. The frozen soil creep test equipment adopts the W3Z-200 frozen soil low temperature triaxial test system. The equipment comes with a low temperature control system. It can ensure that the initial freezing temperature remains unchanged when the frozen soil is creep-tested at a specific temperature.

The specific loading scheme is as follows.

(1) The confining pressure is loaded to a predetermined value at a rate of 200 N/s. (2) The soil sample is consolidated under the condition of the fixed confining pressure. The condition of consolidation termination is that the deformation of the soil sample is less than or equal to 0.005 mm/h. (3) The soil samples after consolidation are frozen according to the freezing temperatures of −5°C, −10°C, −15°C, and −20°C, and the freezing time is 48 h. (4) The axial pressure is loaded at a predetermined stress level of 0.002 mm/s. In order to ensure that the confining pressure remains constant during the loading process, the confining pressure value must be supplemented to the predetermined value during the loading process. (5) When the specimen has been damaged, it must immediately stop the test. When each level of stress is loaded to be stable, it enters the next level of the creep-loading stage until the creep failure of frozen soil occurs. Finally, the test data need to be exported and saved. (6) After unloading the confining pressure and axial stress, the damaged soil samples are taken out and preserved. (7) The above steps need to be repeated according to the test plan.

The axial strain-time creep curve is shown in Figure 3.

It can be seen in Figure 3 that the axial creep-time curve of frozen soil has the following characteristics under different freezing temperatures. When a load is applied to the frozen soil at the same temperature, a large instantaneous strain will be generated at the initial time of loading. With an increase in stress, the instantaneous strain increases. Frozen soil has only decay creep under low stress. Under higher stress, the creep deformation of the frozen soil samples consists of decay creep and stable creep. After the stress is applied to a certain value, frozen soil will produce obvious accelerated creep. This is because the internal defects of the soil sample have developed to a certain extent after the first two creep stages. After the stress increases, it begins to gradually penetrate from microcracks into macrocracks, making the soil sample creep damage. As the temperature decreases, the creep failure time gradually decreases. When the freezing temperatures are −5°C, −10°C, −15°C, and −20°C, the last creep failure times are 33.84 h, 46.66 h, 48.01 h, and 52.82 h, respectively. It shows that a decrease in the freezing temperature can effectively enhance the strength of soil samples and delay the creep failure time. This is because the lower the freezing temperature, the more the volume of the pore water inside the soil sample becomes frozen ice (the strength of ice is much greater than that of water). The pore water is removed, and the soil sample damage caused by the volume expansion of the frozen ice is removed. The lower freezing temperature on the whole enhances the bearing capacity of the frozen soil.

The relationship between the freezing temperature and creep rupture time is shown in Figure 4.

According to Figure 4, the relationship between the freezing temperature and creep failure time shows a nonlinear increasing trend. The correlation coefficient is 0.977.

The isochronous stress-strain curve is a strain curve under different stress levels at the same time [25]. Before the divergence point, the deformation of the soil sample is considered to be in the linear elastic stage of loading. After the divergence point, the deformation of the soil sample is considered to be a nonlinear deformation. Therefore, the stress corresponding to the divergence point can be used as the long-term strength value of the soil sample. According to the creep test results, the isochronous stress-strain curves of frozen soil under different freezing temperatures are shown in Figure 5.

According to the stress value corresponding to the divergence point in Figure 6, the long-term strength of the soil sample at a freezing temperature of −20°C is 5.30 MPa. Before the divergence point, the broken line segment almost overlaps into a straight line, and after the divergence point, the divergence of the broken line segment also increases.

5. Establishment of the Damage Creep Model for Frozen Clay

The description of the change in the fractional function has many forms, and the Riemann–Liouville calculus method is most commonly used to define the fractional order. For any real number γ, the Riemann–Liouville fractional function can be expressed as [2629]

As shown in Figure 6, the constitutive equation of the Abel clay pot iswhere η is the viscosity coefficient.

When the stress is kept constant, equation (4) is integrated to obtain equation (5).where σ is the stress.

In the creep process, each creep parameter is not fixed. The damage variable D is introduced to describe the damage process of frozen soil, and the deterioration of the viscosity coefficient η is described by defining the damage variable D. Therefore, the expression of the viscosity coefficient η after deterioration is

The damage evolution equation can be expressed aswhere β1 is the damage coefficient under the influence of temperature and β2 is the damage coefficient under the influence of the creep load.

Equation (9) is obtained by combining equations (6) and (8).

Considering only the influence of the load time, the relationship between parameters and time and temperature satisfieswhere X represents the creep parameters in the creep model of frozen soil.

The variable fractional order can be obtained by further expanding the fractional order. The fractional order can be regarded as a time-dependent variable parameter so that the fractional order has time-dependent characteristics. Therefore, it can be considered that the fractional order is not only affected by time but also by the applied stress when describing the creep deformation of rock. Under the influence of both time and stress, the fractional order γ is a function of the stress and time.where β1 and β2 are the damage degree coefficient affecting the fractional order.

Generally, the creep of frozen soil can be divided into three stages. Creep includes the attenuation creep stage, stable creep stage, and accelerated creep stage. Under most conditions, the Nishihara model (Figure 7(a)) can well describe the creep characteristics of the first two stages of frozen soil, but it is difficult to describe the characteristics of accelerated creep. Fractional calculus is the study of any order differential and integral theory. It is also a generalization of integer-order calculus to any order. It can convert the linear Newtonian dashpot in the Nishihara body into the linear Newtonian dashpot to better simulate the creep process of actual frozen soil. In this paper, the constant coefficient sticky pot is transformed into an unsteady sticky pot body through the above discussion and all other creep parameters are transformed into unsteady parameters. Then, the original linear elements in the Nishihara model are replaced in turn to obtain the unsteady and fractional creep model, as shown in Figure 7(b).

According to rheological model theory, the total strain ε of the fractional-order model satisfies the following conditions:where εe is the elastic strain, εve is the viscoelastic strain, and εvp is the viscoplastic strain.

For the elastic Hooke element, the parameter E0 deteriorates under different stress levels. The elastic strain εe of the Hooke element iswhere E0 is Hooke element’ elastic modulus and β10 and β20 are the damage influence coefficient of the elastomer.

By substituting the viscoelastic model parameters into equation (10), the viscoelastic model parameters considering the influence of temperature and time can be obtained.where β11, β21, , and are, respectively, the damage influence coefficients of the viscoelastic body.

The rheological equation of the modified viscoelastic body is

Equation (14) can be obtained by solving equation (13).

It can be assumed that the damage degree of the model corresponding to the elastic modulus E1 and the viscosity coefficient η1 is the same [30]. When the initial value is t = 0, εve = 0 is substituted into equation (14), and the Laplace transform is used to calculate equation (15).

The constitutive relation of the viscoelastic body is obtained.where η1 is the Abel dashpot viscosity coefficient and E1 is the elastic modulus of viscoelasticity. The viscoplastic strain of the viscoplastic body can be divided into the following two parts [31].

When σ < σs, the friction block does not work.when σ ≥ σs, the friction block does starts.where η2 is the viscosity coefficient of the viscoplastic body and β12 and β22 are the damage influence coefficient of the viscoplastic body.

Under uniaxial stress, the constitutive equation of the creep model is as follows.

When σ < σs, we get

When σ ≥ σs, we get

The three-dimensional unsteady fractional creep model for frozen soil can be derived by the analogy method.

In the three-dimensional stress state, the total strain εij satisfieswhere is the elastic deformation tensor, is the viscoelastic deformation tensor, and is the viscoplastic deformation tensor.

The bulk modulus and the shear modulus satisfy the following conditions:where G0 is the shear modulus, K0 is the bulk modulus, and is Poisson’s ratio.

The elastic strain satisfies

The viscoelastic strain satisfies

The analogy method is suitable and feasible for deriving fractional elastic and viscoelastic deformation [32, 33]. But for viscoplastic deformation, it also involves the yield function F and plastic potential function Q. The stress tensor Sij cannot be simply used to replace the stress σ in the original one-dimensional model. Therefore, the viscoplastic strain is [34]where F0 is the initial reference value of the yield function of frozen soil and n is a material constant; usually, n is 1.

It can be assumed that the initial yield function of frozen soil is F0 = 1. According to the flow rule in plasticity theory, equation (22) can be transformed into

The yield function is

The creep model for frozen soil under the three-dimensional stress state is as follows.

When σ < σs, we get

When σ ≥ σs, we get

6. Validation of the Unsteady Fractional Creep Model

The Levenberg–Marquardt algorithm is used to fit the creep test curve and identify the parameters. In the test, the instantaneous deformation includes not only elastic deformation but also plastic deformation. For the convenience of analysis, the instantaneous deformation is assumed to be elastic deformation. Through the instantaneous volumetric strain value of the test curve, the initial bulk modulus under various stress states can be determined to be

In this paper, taking a freezing temperature of −20°C as an example, according to the axial creep curve, the inversion model parameters are shown in Table 3.

According to the unsteady creep equation established above, the parameters obtained by inverting the creep test curves of a freezing temperature of −20°C are substituted into the creep equation, and the creep deformation and time are plotted, as shown in Figure 8(a). Similarly, the relationship between the creep test curves and the model curves at other freezing temperatures can be obtained, as shown in Figures 8(b)8(d).

In order to better verify the correctness of the model, the traditional Nishihara model creep model will be used for comparative analysis, and the comparison curve is shown in Figure 9.

It can be seen in Figures 8 and 9 that the established creep model curve can better describe the whole creep process curve of frozen soil under different temperatures and stress loads. In particular, it can make up for the shortcomings of the traditional Nishihara model which cannot describe accelerated creep deformation. At the same time, the agreement between the model curve and the test curve is also much greater than that between the traditional Nishihara model curve and the test curve. This lays the foundation for the later use of the creep model to simulate the settlement and horizontal displacement of subgrade soil under long-term loads and also provides a theoretical basis for the construction of other roadbed projects.

7. Parameter Sensitivity Analysis

Taking a freezing temperature of −20°C and a stress of 127 kPa as an example, the parameter sensitivity analysis is carried out. The variation of creep deformation and time under different parameter values is shown in Figure 10.

It can be seen in Figure 10 that the order of the fractional order affects the size and rate of creep. The higher the fractional order, the greater the influence on the creep rate. Unsteady parameters β12 and β22 also have a significant effect on creep. The parameter β22 has a great influence on the creep rate, especially on the accelerated creep rate. The influence of the parameter β12 is relatively small. Its influence is mainly in the accelerated creep stage. With an increase in parameters β12 and β22, the creep rate increases gradually.

8. Conclusions

In the process of creep, each creep parameter is not fixed. The rheology of frozen soil is a process of continuous accumulation of frozen soil damage and continuous deterioration of mechanical parameters. Therefore, the creep parameters are transformed into time-dependent variables, and the time-dependent creep model can better analyze the creep characteristics of frozen soil.

The derived unsteady fractional creep model conforms to the creep deformation law under various freezing temperatures. From the above comparison of frozen soil creep test curves and model curves under different freezing temperatures and different stresses, it can be fully explained that the unsteady model is suitable and feasible to reflect the whole process deformation law of frozen soil creep. It not only accurately reflects the creep characteristics of attenuation and stable creep stage but also overcomes the shortcomings of the traditional Nishihara model which is difficult to describe accelerated creep. In general, the model has a higher fitting degree and has a good predictive analysis of triaxial creep test data.

Under the influence of time and stress, the fractional order is regarded as a function of stress and time. The three-dimensionaltime-dependent creep constitutive model for unsteady fractional frozen soil well describes the accelerated creep deformation law of rock.

Data Availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Hongmiao Lv was responsible for data curation, conceptualization, writing the original draft, methodology, reviewing and editing the manuscript, supervision, and funding acquisition. Lei Chen was responsible for investigation, project administration, writing, reviewing, and editing the manuscript, supervision, data curation, and validation.

Acknowledgments

This work was supported by the Eastern Liaoning University Doctoral Research Start-Up Foundation (Grant no. 2021BS011).