# Critical Dynamic Stress and Accumulative Deformation Evolution of Embankment Silty Clay Subjected to Cyclic Freeze-Thaw

Authors: Lina Wang; Zhiyu Weng; Tianliang Wang; Qiang Liu; Guoyu Li; Yingying Zhao
Journal: Shock and Vibration
 (2021)

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## Abstract

In cold regions, the permanent settlement of embankment is mainly caused by the repeated freeze-thaw process and long-term repeated train loads. Meanwhile, the critical dynamic stress (σdcr) is an important parameter index for determining embankment stability. Therefore, the accumulative permanent deformation evolution and critical dynamic stress of embankment soil subjected to cyclic freeze-thaw were studied using dynamic triaxial tests. Firstly, a numerical model for calculating critical dynamic stress considering the repeated freeze-thaw process was proposed, which shows that the critical dynamic stress of embankment soil rapidly decreases in the first two repeated freeze-thaw cycles, whereas it tends to be stable after the subsequent freeze-thaw process. Next, based on the normalization of the critical dynamic stress, an explicit model for predicting accumulative plastic strain (εp) of embankment soil was established. The above model considers freeze-thaw times, repeated dynamic stress amplitude (σd), and loading times, in which all material parameters of Qinghai-Tibet silty clay were presented. Thus, the critical dynamic stress and accumulative plastic strain models established in this paper can be applied to judge the embankment stability and predict the embankment settlement induced by train loads in cold regions.


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## Body

## 1. Introduction

In China, the wide-latitude, high-altitude, geological topography causes a huge difference in the distribution of cold and warm zones. Permafrost is widely distributed in China, including 2.15 × 106 km2 of permafrost area and 5.14 × 106 km2 of seasonal permafrost area, which are predominantly distributed in Northeast China, Northwest China, and Qinghai-Tibet Plateau [1]. Recently, with the development of China’s strategy of western development and revitalization of the northeast old industrial base, the infrastructure construction represented by high-speed railways and expressways will rapidly develop in the cold regions such as Northeast China and Northwest China, Central Asia, and Russia. In cold regions, the surface soil as the foundation undergoes a repeated freeze-thaw process, which changes the structure connection and arrangement of soil particles, further changing the physical-mechanical property of soil. Furthermore, the long-term and repeated action of traffic load causes the mechanical properties of embankment soil to constantly evolve, which further aggravates the occurrence of a large area of embankment subsidence, slurry turning disease, slope collapse, lateral uplift, or shear crack extrusion caused by basement extrusion and seriously affects the long-term service performance of traffic engineering. Therefore, it is critical and imperative to investigate the influence of the repeated freeze and thaw process and long-term train loading on the accumulative deformation evolution of embankment soil.Currently, the research results mainly focus on the effect of repeated freeze-thaw on the static characteristic of different types of soils [2–5]. Many researchers found that resilience modulus and shear strength of soil are reduced with an increase of repeated freeze-thaw times. Meanwhile, the compressibility of soil also affects the mechanical properties subjected to freeze-thaw cycle. The cohesive force of soil exhibiting large compactness reduced with the increase of repeated freeze-thaw, and the angle of internal friction increased or underwent insignificant changes [6–8]. However, Su et al. [9] found that the cohesive forces of soil exhibiting little compactness decrease after some freeze-thaw cycles, which is because fine-grained soils are sensitive to water and temperature and unsuitable for direct use as an engineering material. There are few studies on the dynamic properties of soil subjected to cyclic freeze-thaw under dynamic loads, including earthquake, traffic, and wave loads [10–14]. Based on the nonconsolidation and nondrainage dynamic triaxial test on Changchun silty clay, Dai et al. [15] found that the dynamic strength of silty clay after three repeated freeze-thaw cycles tended to be stable and did not change with confining pressure and freeze-thaw cycle times. The dynamic strength of fly ash soil subjected to repeated freeze-thaw also has the characteristics mentioned above [16]. After repeated freezing and thawing for 3–8 times, the dynamic modulus of fly ash soil and silty soil remained stable, among which the dynamic modulus of fly ash soil was higher than that of silty soil. Zhang [17] investigated the dynamic shear modulus and damping ratio of the silty soil of Natural Mabel Creek in Alaska subjected to repeated freeze-thaw. They observed that dynamic shear modulus of silty clay after cyclic freeze-thaw increased and the damping ratio remained unchanged at the shear strain less than 0.03%, but the damping ratio of the soil after cyclic freeze-thaw increased when it was greater than 0.03%. Based on freeze-thaw cycle tests on soils with three different plasticity indexes, the mechanical parameters of compacted embankment soil subjected to cyclic freeze-thaw were studied by Wang et al. [18]. They found that repeated freeze-thaw significantly affected the strength and deformation property of the soil. Many researchers studied the static strength of melted soil by considering the freezing temperature, melting temperature, and cyclic freeze-thaw times, while the investigation of the dynamic characteristics of melted soil has just begun. Comprehensive research on the accumulative deformation evolution of embankment silty clay subjected to repeated freeze-thaw is even less common.Therefore, centering on the demand of transportation construction in cold region, the evolution law ofσdcr and εp of Qinghai-Tibet silty clay subjected to cyclic freeze-thaw as important parameters of dynamic stability of subgrade was studied using a dynamic triaxial test. A constitutive model of permanent deformation is proposed, which considers cyclic freeze-thaw times and long-term traffic load. The research results provide significant scientific and application value for clarifying the development mechanism of accumulated deformation of roadbed soil and its role in treating roadbed diseases in traffic engineering in cold regions by providing guidance to local railroad and highway construction.


## 2. Test Scheme

### 2.1. Soil Properties

The particle distribution curve of the material used in this paper is shown in Figure1, which was taken from the Qinghai-Tibet Railway in China. The physical characteristics of the soil are summarized in Table 1, such as the maximum dry density, plasticity index, and optimum water content.Figure 1
Particle analysis curve of Qinghai-Tibet silty clay.Table 1
Physical properties of Qinghai-Tibet silty clay.
Maximum dry densityLiquid limit (%)Plastic limit (%)Optimum water content (%)Saturated water content (%)Plasticity index1.89 g/cm326.117.312.3517.458.8


### 2.2. Test Equipment

In this paper, the MTS-810 closed-loop servohydraulic material testing machine, equipped with automatic numerical control and data acquisition system, was used for the dynamic triaxial tests. The various indexes of the triaxial test machine are shown in [19]. A refrigerator equipped with an automatic temperature control system is used to freeze Qinghai-Tibet silty clay. The freezing temperature of the refrigerator is up to 30 °C, and the temperature accuracy is 0.1 °C.


### 2.3. Test Programme

According to the requirements of Railway Engineering Geotechnical Test Code (TB10102-2010), the silty clay samples were prepared in batches and the experimental procedures involved the following five steps. (1) Cylindrical soil specimens with a compressed dry density of 1.89 g/cm3, a diameter of 61.8 mm, and a height of 120 mm were prepared by the artificial freezing method. (2) To ensure uniform freezing of soil samples, the prepared soil specimen with rubber sleeve was placed in the refrigerator for 24 hours at −11°C. (3) The specimen was melted at a maximum temperature of 20°C for 18 h. The time of freezing and thawing was enough which had been proved by experiments [20]. (4) After 0, 1, 2, 4, and 6 repeated freeze-thaw cycles, the cylindrical soil specimen was placed in the triaxial test instrument chamber. (5) The tests were carried out by the undrained and unconsolidated loading method. The repeated cyclic loads of sinusoidal waveform with vibration frequency of 2 Hz were applied in the test, as shown in Figure 2. Figure 2 also shows definition of accumulative plastic strain using dynamic stress-strain relationship. It can be seen from the figure that the line of plastic strain of soil in each load cycle is the curve of εp changing with the number of load vibration. Test termination criteria are 10,000 vibrations or 8% dynamic axial strain. Table 2 shows dynamic triaxial experiment conditions of each sample.Figure 2
Axial strain of sample under long‐term repeated cyclic loading.Table 2
Dynamic triaxial test condition of thawed soil.
Water content (%)Times of cyclic freeze-thaw (Ncyc)Dynamic stress amplitude (kPa) (σd)12.350300, 360, 400, 440, 5001200, 240, 300, 4002150, 200, 240, 300, 4004150, 200, 240, 3006150, 180, 200, 240


## 2.1. Soil Properties

The particle distribution curve of the material used in this paper is shown in Figure1, which was taken from the Qinghai-Tibet Railway in China. The physical characteristics of the soil are summarized in Table 1, such as the maximum dry density, plasticity index, and optimum water content.Figure 1
Particle analysis curve of Qinghai-Tibet silty clay.Table 1
Physical properties of Qinghai-Tibet silty clay.
Maximum dry densityLiquid limit (%)Plastic limit (%)Optimum water content (%)Saturated water content (%)Plasticity index1.89 g/cm326.117.312.3517.458.8


## 2.2. Test Equipment

In this paper, the MTS-810 closed-loop servohydraulic material testing machine, equipped with automatic numerical control and data acquisition system, was used for the dynamic triaxial tests. The various indexes of the triaxial test machine are shown in [19]. A refrigerator equipped with an automatic temperature control system is used to freeze Qinghai-Tibet silty clay. The freezing temperature of the refrigerator is up to 30 °C, and the temperature accuracy is 0.1 °C.


## 2.3. Test Programme

According to the requirements of Railway Engineering Geotechnical Test Code (TB10102-2010), the silty clay samples were prepared in batches and the experimental procedures involved the following five steps. (1) Cylindrical soil specimens with a compressed dry density of 1.89 g/cm3, a diameter of 61.8 mm, and a height of 120 mm were prepared by the artificial freezing method. (2) To ensure uniform freezing of soil samples, the prepared soil specimen with rubber sleeve was placed in the refrigerator for 24 hours at −11°C. (3) The specimen was melted at a maximum temperature of 20°C for 18 h. The time of freezing and thawing was enough which had been proved by experiments [20]. (4) After 0, 1, 2, 4, and 6 repeated freeze-thaw cycles, the cylindrical soil specimen was placed in the triaxial test instrument chamber. (5) The tests were carried out by the undrained and unconsolidated loading method. The repeated cyclic loads of sinusoidal waveform with vibration frequency of 2 Hz were applied in the test, as shown in Figure 2. Figure 2 also shows definition of accumulative plastic strain using dynamic stress-strain relationship. It can be seen from the figure that the line of plastic strain of soil in each load cycle is the curve of εp changing with the number of load vibration. Test termination criteria are 10,000 vibrations or 8% dynamic axial strain. Table 2 shows dynamic triaxial experiment conditions of each sample.Figure 2
Axial strain of sample under long‐term repeated cyclic loading.Table 2
Dynamic triaxial test condition of thawed soil.
Water content (%)Times of cyclic freeze-thaw (Ncyc)Dynamic stress amplitude (kPa) (σd)12.350300, 360, 400, 440, 5001200, 240, 300, 4002150, 200, 240, 300, 4004150, 200, 240, 3006150, 180, 200, 240


## 3. Results and Discussion

### 3.1. Effect of the Dynamic Stress Amplitude

Figure3 illustrates the influence of σd on εp of Qinghai-Tibet silty clay subjected to different number of repeated freeze-thaw cycles. It is clear that εp increases with increasing vibration numbers (N). For all the tests, there is similar changing tendency of accumulative strain including initial creep stage, steady creep stage, and tertiary stage. During the initial creep stage, the accumulative strain increases at a rapid rate, while the strain rate gradually decreases until it reaches a minimum value. During the stable creep stage, εp develops at a constant rate. As expected, the accumulative plastic strain increased with the increase of σd. For the subgrade fill, there is a threshold value of dynamic stress. When the dynamic stress is greater than this value, the accumulative plastic strain of the soil will increase significantly until failure, which is called failure type. When the dynamic stress is less than this value, the accumulative plastic strain of soil is small and tends to be stable, and the soil sample is elastic, which is called stable type.Figure 3
Effect of dynamic stress amplitude on the accumulative plastic strain after different number of freeze-thaw cycles (Ncyc). (a) Ncyc = 0. (b) Ncyc = 1. (c) Ncyc = 2. (d) Ncyc = 4. (e) Ncyc = 6.
(a)(b)(c)(d)(e)Figure3(b) shows that when σd is less than 200 kPa, the accumulative axial strain of silty clay gradually increases during the initial stage of vibration and tends to be stable with the increase of vibration numbers, which shows a stable type. When σd is greater than 400 kPa, εp of silty clay increases slowly at the initial stage of vibration, then starts to increase sharply, and quickly reaches failure which indicated a failure‐type curve. After the turning point, εp of silty clay increases sharply and rapidly reaches the failure which indicated a failure type curve. Figure 3(b) also shows the relationship between εp of silty clay and σd under different vibration numbers. From the results obtained, when σd is 240 kPa, εp of silty clay linearly increases with σd at the vibration number of 1000, indicating that the sample underwent stable deformation within the range of dynamic stress level and vibration numbers. When σd is greater than 240 kPa, the slope of the relationship curve between accumulative plastic strain and σd increases suddenly, indicating that the sample is damaged.


### 3.2. Effect of the Number of Cyclic Freeze-Thaw

Figure4 gives the relationship between εp and vibration times of the Qinghai‐Tibet Railway silty clay under different freeze‐thaw times. Figure 4(b) shows that when σd was 300 kPa, εp of the silty clay is small for unfrozen soil, but the relationship curve between εp of soil subjected to two repeated freeze-thaw cycles and vibration numbers appeared as a turning point, sharply increasing and rapidly reaching failure. Figure 5 shows the relationship between Ncyc and εp of Qinghai-Tibet silty clay. εp of soil induced by train load is significantly affected by the repeated freeze-thaw process. When σd was 200 kPa, the accumulative plastic strain of the embankment soil exhibited an obvious increasing trend with the increase of cyclic freeze-thaw. When σd was increased to 300 kPa, the accumulative plastic strain showed an increasing trend during the first two times of freeze-thaw cycles, and then the accumulative plastic strain changed slightly with the increase in the subsequent freeze-thaw cycles, indicating that the failure form of silty clay gradually evolved from brittle failure to plastic failure.Figure 4
Relationship between the accumulative plastic strain and vibration numbers under different number of freeze-thaw cycles. (a)σd = 200 kPa. (b) σd = 300 kPa.
(a)(b)Figure 5
Relationship between the accumulative plastic strain and (N)cyc. (a) σd = 200 kPa. (b) σd = 300 kPa.
(a)(b)


### 3.3. Effect of Freeze-Thaw Cycles on Critical Dynamic Stress

One of important parameters for determining embankment stability is the critical dynamic stress (σdcr). According to the fatigue test of cohesive soil, Heath et al. [21] defined the critical dynamic stress, which means that repeated dynamic stress greater than it will cause large permanent deformation of subgrade soil, and permanent deformation is small and terminated when repeated applied load is less than it, as shown in Figure 6.Figure 6
Schematic diagram of accumulative plastic strain development patterns.Consequently, Figure3(c) shows that σdcr of silt clay at freeze-thaw cycle Ncyc = 2 was 200 kPa under certain test conditions. Based on this, σdcr of silty clay is 200 kPa subjected to two freeze-thaw cycles. Figure 7 shows the influence of cyclic freeze-thaw on σdcr of Qinghai-Tibet silty clay. The x and y axes represent Ncyc and σd, respectively, and each point obtained from the above test is plotted in this coordinate. From Figure 6, the critical dynamic stress decreased with an increase in the number of freeze-thaw cycles, suggesting that Ncyc significantly influenced σdcr. Through regression analysis of σdcr subjected to repeated freeze-thaw, the fitting formula of σdcr of Qinghai-Tibet silty clay is obtained:(1)σdcr=213.59+183.63×exp−Ncyc0.474R2=0.98,where Ncyc and σdcr represent the number of freeze-thaw cycles and the critical dynamic stress, respectively.Figure 7
Relationship between the number of freeze-thaw cycles and threshold dynamic stress.


### 3.4. Prediction Model for Accumulative Deformation of Thawed Soil

This study adopted the Monismith model, which is simple and practical and conforms to equation (2). Based on this model, the evolution law of accumulated plastic strain of silty soil shown in Figure 3 was described by regression analysis. Parameters A and B of the Monismith model were obtained, respectively, as shown in Table 3.(2)εp=ANB,where εp is the accumulative plastic strain, N represents the vibration numbers, and A and B are the parameters. The correlation coefficient R2 under each test condition shown in Table 3 was mostly above 0.95, and the fitting effect was good.Table 3
Fitting parameters of the empirical model of dynamic permanent deformation of embankment soil under long-term train load.
Ncycσd (kPa)ScrABR203000.750.00230.15140.9953600.900.00260.2010.9974001.000.00310.1560.9934401.100.00420.1710.9955001.250.00430.1590.99712000.830.00240.1720.9972401.000.00310.1370.9893001.250.00450.2340.9894001.670.00530.3300.98121500.750.00190.1990.9982001.000.00290.1860.9922401.200.00290.3040.9973001.500.00370.3510.9754002.000.00450.4060.92641500.630.00220.1150.9952000.830.00350.1820.9822401.000.00400.2520.9913001.250.00380.3410.98861500.750.00260.1730.9801800.900.00320.1780.9642001.000.00320.2980.9962401.200.00280.3610.995To reduce the discretization of data points, the dynamic stress amplitude of each level was divided by the critical dynamic stress in the corresponding physical state. After normalization treatment, the dimensionless dynamic stress ratio Scr was used to describe the magnitude of each level of dynamic stress. Equation (3) shows the stress ratio formula.(3)Scr=σdσdcr=qdσdcr.Figure8 shows that with an increase in the dynamic stress ratio, model parameters A and B also increased. Parameter A linearly increased with the stress ratio Scr, and the fitting formula can be expressed using equation (4). The exponent B of load times linearly increased with stress ratio Scr, and the fitting formula can be expressed using equation (5).(4)A=0.00103+0.00211ScrR2=0.78,(5)B=0.00825+0.20837ScrR2=0.77.Figure 8
Parameters A and B vs. dynamic stress ratioScr.The above analysis shows that the permanent deformation prediction formula of Qinghai-Tibet silty clay after cyclic freeze-thaw was as follows:(6)εp=ANB=0.00103+0.00211ScrN0.00825+0.20837Scr.The above formula indicates that the accumulative plastic strain prediction model considers the comprehensive action of the freeze-thaw cycle and long-term train load.Figure9 compares the calculated and measured results of the accumulative plastic strain prediction model of Qinghai-Tibet silty clay considering repeated freeze-thaw cycles. The predicted permanent strain agrees well with the test results. The above model can be used for predicting the permanent deformation induced by the long-term action of traffic load. The model has important guiding significance for predicting the accumulative permanent deformation of permafrost embankment soil under freeze-thaw cycles and long-term train loads.Figure 9
Comparison of predicted and experimental values of accumulative plastic strain.


## 3.1. Effect of the Dynamic Stress Amplitude

Figure3 illustrates the influence of σd on εp of Qinghai-Tibet silty clay subjected to different number of repeated freeze-thaw cycles. It is clear that εp increases with increasing vibration numbers (N). For all the tests, there is similar changing tendency of accumulative strain including initial creep stage, steady creep stage, and tertiary stage. During the initial creep stage, the accumulative strain increases at a rapid rate, while the strain rate gradually decreases until it reaches a minimum value. During the stable creep stage, εp develops at a constant rate. As expected, the accumulative plastic strain increased with the increase of σd. For the subgrade fill, there is a threshold value of dynamic stress. When the dynamic stress is greater than this value, the accumulative plastic strain of the soil will increase significantly until failure, which is called failure type. When the dynamic stress is less than this value, the accumulative plastic strain of soil is small and tends to be stable, and the soil sample is elastic, which is called stable type.Figure 3
Effect of dynamic stress amplitude on the accumulative plastic strain after different number of freeze-thaw cycles (Ncyc). (a) Ncyc = 0. (b) Ncyc = 1. (c) Ncyc = 2. (d) Ncyc = 4. (e) Ncyc = 6.
(a)(b)(c)(d)(e)Figure3(b) shows that when σd is less than 200 kPa, the accumulative axial strain of silty clay gradually increases during the initial stage of vibration and tends to be stable with the increase of vibration numbers, which shows a stable type. When σd is greater than 400 kPa, εp of silty clay increases slowly at the initial stage of vibration, then starts to increase sharply, and quickly reaches failure which indicated a failure‐type curve. After the turning point, εp of silty clay increases sharply and rapidly reaches the failure which indicated a failure type curve. Figure 3(b) also shows the relationship between εp of silty clay and σd under different vibration numbers. From the results obtained, when σd is 240 kPa, εp of silty clay linearly increases with σd at the vibration number of 1000, indicating that the sample underwent stable deformation within the range of dynamic stress level and vibration numbers. When σd is greater than 240 kPa, the slope of the relationship curve between accumulative plastic strain and σd increases suddenly, indicating that the sample is damaged.


## 3.2. Effect of the Number of Cyclic Freeze-Thaw

Figure4 gives the relationship between εp and vibration times of the Qinghai‐Tibet Railway silty clay under different freeze‐thaw times. Figure 4(b) shows that when σd was 300 kPa, εp of the silty clay is small for unfrozen soil, but the relationship curve between εp of soil subjected to two repeated freeze-thaw cycles and vibration numbers appeared as a turning point, sharply increasing and rapidly reaching failure. Figure 5 shows the relationship between Ncyc and εp of Qinghai-Tibet silty clay. εp of soil induced by train load is significantly affected by the repeated freeze-thaw process. When σd was 200 kPa, the accumulative plastic strain of the embankment soil exhibited an obvious increasing trend with the increase of cyclic freeze-thaw. When σd was increased to 300 kPa, the accumulative plastic strain showed an increasing trend during the first two times of freeze-thaw cycles, and then the accumulative plastic strain changed slightly with the increase in the subsequent freeze-thaw cycles, indicating that the failure form of silty clay gradually evolved from brittle failure to plastic failure.Figure 4
Relationship between the accumulative plastic strain and vibration numbers under different number of freeze-thaw cycles. (a)σd = 200 kPa. (b) σd = 300 kPa.
(a)(b)Figure 5
Relationship between the accumulative plastic strain and (N)cyc. (a) σd = 200 kPa. (b) σd = 300 kPa.
(a)(b)


## 3.3. Effect of Freeze-Thaw Cycles on Critical Dynamic Stress

One of important parameters for determining embankment stability is the critical dynamic stress (σdcr). According to the fatigue test of cohesive soil, Heath et al. [21] defined the critical dynamic stress, which means that repeated dynamic stress greater than it will cause large permanent deformation of subgrade soil, and permanent deformation is small and terminated when repeated applied load is less than it, as shown in Figure 6.Figure 6
Schematic diagram of accumulative plastic strain development patterns.Consequently, Figure3(c) shows that σdcr of silt clay at freeze-thaw cycle Ncyc = 2 was 200 kPa under certain test conditions. Based on this, σdcr of silty clay is 200 kPa subjected to two freeze-thaw cycles. Figure 7 shows the influence of cyclic freeze-thaw on σdcr of Qinghai-Tibet silty clay. The x and y axes represent Ncyc and σd, respectively, and each point obtained from the above test is plotted in this coordinate. From Figure 6, the critical dynamic stress decreased with an increase in the number of freeze-thaw cycles, suggesting that Ncyc significantly influenced σdcr. Through regression analysis of σdcr subjected to repeated freeze-thaw, the fitting formula of σdcr of Qinghai-Tibet silty clay is obtained:(1)σdcr=213.59+183.63×exp−Ncyc0.474R2=0.98,where Ncyc and σdcr represent the number of freeze-thaw cycles and the critical dynamic stress, respectively.Figure 7
Relationship between the number of freeze-thaw cycles and threshold dynamic stress.


## 3.4. Prediction Model for Accumulative Deformation of Thawed Soil

This study adopted the Monismith model, which is simple and practical and conforms to equation (2). Based on this model, the evolution law of accumulated plastic strain of silty soil shown in Figure 3 was described by regression analysis. Parameters A and B of the Monismith model were obtained, respectively, as shown in Table 3.(2)εp=ANB,where εp is the accumulative plastic strain, N represents the vibration numbers, and A and B are the parameters. The correlation coefficient R2 under each test condition shown in Table 3 was mostly above 0.95, and the fitting effect was good.Table 3
Fitting parameters of the empirical model of dynamic permanent deformation of embankment soil under long-term train load.
Ncycσd (kPa)ScrABR203000.750.00230.15140.9953600.900.00260.2010.9974001.000.00310.1560.9934401.100.00420.1710.9955001.250.00430.1590.99712000.830.00240.1720.9972401.000.00310.1370.9893001.250.00450.2340.9894001.670.00530.3300.98121500.750.00190.1990.9982001.000.00290.1860.9922401.200.00290.3040.9973001.500.00370.3510.9754002.000.00450.4060.92641500.630.00220.1150.9952000.830.00350.1820.9822401.000.00400.2520.9913001.250.00380.3410.98861500.750.00260.1730.9801800.900.00320.1780.9642001.000.00320.2980.9962401.200.00280.3610.995To reduce the discretization of data points, the dynamic stress amplitude of each level was divided by the critical dynamic stress in the corresponding physical state. After normalization treatment, the dimensionless dynamic stress ratio Scr was used to describe the magnitude of each level of dynamic stress. Equation (3) shows the stress ratio formula.(3)Scr=σdσdcr=qdσdcr.Figure8 shows that with an increase in the dynamic stress ratio, model parameters A and B also increased. Parameter A linearly increased with the stress ratio Scr, and the fitting formula can be expressed using equation (4). The exponent B of load times linearly increased with stress ratio Scr, and the fitting formula can be expressed using equation (5).(4)A=0.00103+0.00211ScrR2=0.78,(5)B=0.00825+0.20837ScrR2=0.77.Figure 8
Parameters A and B vs. dynamic stress ratioScr.The above analysis shows that the permanent deformation prediction formula of Qinghai-Tibet silty clay after cyclic freeze-thaw was as follows:(6)εp=ANB=0.00103+0.00211ScrN0.00825+0.20837Scr.The above formula indicates that the accumulative plastic strain prediction model considers the comprehensive action of the freeze-thaw cycle and long-term train load.Figure9 compares the calculated and measured results of the accumulative plastic strain prediction model of Qinghai-Tibet silty clay considering repeated freeze-thaw cycles. The predicted permanent strain agrees well with the test results. The above model can be used for predicting the permanent deformation induced by the long-term action of traffic load. The model has important guiding significance for predicting the accumulative permanent deformation of permafrost embankment soil under freeze-thaw cycles and long-term train loads.Figure 9
Comparison of predicted and experimental values of accumulative plastic strain.


## 4. Conclusions

The effect of cyclic freeze-thaw, train loads, and vibration numbers on the permanent settlement of silty clay widely distributed along the Qinghai-Tibet railroad was investigated. Prediction models for the critical dynamic stress and permanent deformation were presented by considering the freeze-thaw cycle and long-term train load. The main findings are summarized as follows:(1)
The repeated freeze-thaw significantly influenced the accumulation of permanent deformation evolution and critical dynamic stress of Qinghai-Tibet silty clay. WhenNcyc was less than two, εp of soil sharply increased with the vibrations, reaching a brittle failure. However, when Ncyc was six, the accumulative plastic strain of the silty clay gradually increased with the number of vibrations until it was destroyed, thus showing plastic failure. The failure form of the silty clay gradually evolved from brittle to plastic failure with an increase in the number of cyclic freeze-thaw.(2)
A model for predicting the critical dynamic stress of silty clay subjected to cyclic freeze-thaw was established by using the multiple nonlinear regression method. The critical dynamic stress of soil after two repeated freeze-thaw cycles rapidly decreases and then changes little after the following freeze-thaw process.(3)
Using the Monismith exponential model and normalizing the critical dynamic stress, a model for predicting the accumulative plastic strain of Qinghai-Tibet silty clay was established, which considersNcyc and the long-term repeated cyclic load. The model has clear parameters and considers the influence of cyclic freeze-thaw, train loads, and vibration numbers on the accumulative plastic strain.



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*Source: 9068254-2021-09-23.xml*