Study of the Applicability of the Vyalov Long-Term Strength Prediction Equation under Freezing and Thawing Conditions

Abstract

In order to determine the appropriate utilization conditions of the Vyalov long-term strength prediction equation, this study selected three soil samples from the loess region. These samples were subjected to varying numbers of freeze–thaw cycles, namely 0, 4, 6, 8, 10, 50, and 100 cycles. Subsequently, post-test soil samples underwent spherical template indenter press-in tests and grain size determinations. The experimental outcomes demonstrated that the Vyalov long-term strength prediction equation accurately reflected the long-term strength variations of frozen loess after 10 freeze–thaw cycles. A further analysis revealed that the stability of the soil samples’ grain composition played a vital role in ensuring the accuracy of the prediction equation. Notably, a highly significant positive correlation was observed between the silt content of the soil samples and the prediction equation parameter β. Moreover, employing the Vyalov long-term strength prediction equation after 10 freeze–thaw cycles yielded prediction results consistent with the freeze–thaw cycles–time analogy method for durations of 10, 20, and 30 years. This study is beneficial for the construction and operation of projects in loess areas.

1. Introduction

Loess, as an environmentally sustainable building material, finds extensive application in diverse projects within loess regions. In China, loess soils encompass approximately 6.6% of the country’s total land area, while globally they constitute approximately 10% of the Earth’s land area [1]. With the deep development of Western China (key infrastructure projects on the western seasonal permafrost areas), the problem of freeze–thaw disasters of loess is highlighted [2,3,4], and the freeze–thaw effect seriously affects the construction and operation of loess areas, such as loess-area tunnels in the construction process, due to soil relaxation; poor bearing capacity leads to poor control of tunnel construction quality [5]. The structure of loess undergoes significant transformations during freeze–thaw cycles due to cryogenic actions, resulting in substantial alterations in its long-term strength and deformation characteristics. These changes possess the potential to directly impact the stability of loess regions [6].

The testing and analysis of long-term strength in frozen loess have garnered considerable attention among scholars [7]. However, research on predicting the long-term strength of frozen loess remains relatively limited. Typically, the prediction of long-term strength in frozen soil is accomplished through methods such as a time analogy and utilization of long-term strength prediction equations. Notable examples include the temperature–time analogy method [8], the Vyalov logarithmic equation [9], the Roman power equation [10], and the Wu Ziwang power equation [8]. Due to the existence of ice cementation in frozen soil, frozen soil shows obvious rheology and temperature dependence under a load [11]. In the design of structures within cold regions, the typical service life ranges from 50 to 100 years. Therefore, in engineering applications, it becomes imperative to comprehensively account for the rheological behavior of frozen soil. The short-term test results, which yield insights into the temporal decline in strength, need to be extrapolated to the expected service life of buildings to accurately predict the long-term strength of frozen soil [12]. Roman et al. [13] conducted experiments employing the spherical template indenter, leading to the establishment of the Vyalov long-term strength prediction equation, which proves to be applicable to a wide range of soil types. Notably, the accuracy of predicting equivalent cohesion using this equation is comparable to that achieved through experimental determination. Chen et al. [14] derived the Vyalov long-term strength prediction equation specifically applicable to silt by utilizing sand, silt, and clay as representative examples. Zhou et al. [15] conducted a study on frozen loess and determined that, in comparison to the Roman method and the Wu Ziwang method, the Vyalov method exhibited a superior linear fitting degree. Consequently, the researchers concluded that the Vyalov prediction equation is more suitable for accurately predicting the long-term strength of frozen loess. However, the aforementioned studies mainly focus on identifying the soil types suitable for the Vyalov long-term strength prediction equation, without providing insights into the specific usage conditions of this equation or exploring the relationship between the prediction equation parameter β and the grain composition of the soil sample. Given that a significant portion of loess is located in the seasonal permafrost zone and undergoes repeated freeze–thaw cycles, it becomes crucial to determine the usage conditions of the Vyalov long-term strength prediction equation through freeze–thaw cycle tests and analyze the relevant parameters.

This study took Fuping loess, Haidong loess, and Lanzhou loess with high silt content and strong frost susceptibility as the objects of study. After 0, 4, 6, 8, 10, 50, and 100 freeze–thaw cycles, the long-term strength changes were tested using a spherical template indenter. The variations of the prediction results of the Vyalov long-term strength prediction equation were studied under a different number of freeze–thaw cycles. Then, through grain size determination and a correlation analysis, the reason for improving the accuracy of the prediction equation and the impact of grain composition on the parameter β of the prediction equation were further analyzed. Additionally, the long-term strength of loess was predicted for different time periods, specifically 10, 20, and 30 years. Two different methods were employed for this prediction after subjecting the samples to 10 freeze–thaw cycles: the freeze–thaw cycles–time analogy method [16] and the Vyalov long-term strength prediction equation. To verify the accuracy of the Vyalov long-term strength prediction equation after 10 freeze–thaw cycles, a comparative analysis was conducted by comparing the results obtained from both methods. This study has important theoretical significance for the comparative study of the mechanical behavior of frozen loess under freeze–thaw actions. It also has practical value for the prediction and analysis of engineering stability in cold areas.

2. Theoretical Basis

2.1. Spherical Template Indenter Press-In Test

The working principle of the spherical template indenter is similar to that of the Brinell hardness tester [17], which is based on the analysis method of the plasticity theory first proposed by Russian scientist Isherinsky.

Previous research results have proved that the shear strength of frozen soil consists of two parts; one is friction and the other is the cohesion between soil grains [18].

$${\tau }_f=c\left(\theta ,w,t\right)+{\sigma }^{\prime }\tan \phi \left(\theta ,w,t\right)$$

(1)

where ${\tau }_f$ is the shear strength of the soil, kPa; $c\left(\theta ,w,t\right)$ is the cohesion of the soil, kPa; ${\sigma }^{\prime }$ is the normal stress in the shear plane, kPa; $\phi \left(\theta ,w,t\right)$ is the angle of internal friction of the soil, °; $\theta$ is the temperature, °C; $w$ is the water (ice) content; and $t$ is the time, s.

According to previous research, it has been found that the internal friction of silty soil is nearly zero under freezing conditions [19]. Then, the above equation was abbreviated as:

$${\tau }_f=c\left(\theta ,w,t\right)$$

(2)

That is, in theory, when the shear strength of frozen loess was tested with a spherical template indenter, its shear strength ${\tau }_f$ was equivalent to the cohesive force $c\left(\theta ,w,t\right)$. However, in practical operation, there may be other factors that affect it, resulting in the actual cohesive force value being greater than the shear strength ${\tau }_f$. Therefore, according to the suggestion of Tsytovich, in order to accurately reflect the shear strength ${\tau }_f$, the actual cohesive force value can be regarded as the equivalent cohesive force $C_{equ}$, and the equivalent cohesive force $C_{equ}$ can be used as the strength index of frozen soil, which is expressed as ${\tau }_f=C_{equ}$ [20]. If according to the derivation of the deformation equation of incomplete elastic and viscoplastic objects, the formula of cohesion can be expressed as Equation (3) [21,22]:

$${\tau }_f=C_{equ}=K\frac{P}{\mathrm{\pi }d{S}_t}$$

(3)

where $C_{equ}$ is the equivalent cohesive force per unit area over time, MPa; $P$ is the vertical load acting on the spherical template indenter, N; $K$ is the correction factor, indicating the hardness ratio of the platen to the soil, which is taken as 0.18 for plastic materials; $d$ is the diameter of the spherical template indenter, mm, and this study selects $d$ = 22 mm for testing according to the recommendations given in the Russian national standard; and $S_t$ is the depth of the spherical template indenter pressed into the soil over time, mm.

The actual cohesion values measured in this study were all the equivalent cohesive force values $C_{equ}$. The process of pressing the spherical template indenter into the soil is shown in Figure 1, and according to the test, the law of the depth of pressing the spherical template indenter with time could be obtained, and then the equivalent cohesive force value of the frozen soil was calculated with Equation (3).

2.2. Vyalov Logarithmic Equation

Based on the long-term strength destructive equation and creep equation, Vyalov obtained the long-term strength prediction equation for isothermal conditions [9]:

$$C_{equ}=\frac{\beta }{\ln \frac{t+t*}{B}}$$

(4)

where:

$C_{equ}$—the equivalent cohesive force, MPa;

$t$—time of soil experience, s;

$t*$ is a very small value of time, equal to the loading time corresponding to the instantaneous strength;

$\beta$, $B$ are the parameters obtained from the experiment, $\beta =C_0\ln \frac{1}{B}$;

$C_0$ is the equivalent cohesive force of the soil when $t$ equals 0 s.

Taking the inverse of both sides of Equation (4) at the same time and turning it into a linear equation gives:

$$\frac{1}{C_{equ}}=\frac{1}{\beta }\ln (t+t*)-\frac{1}{\beta }\ln B$$

(5)

where $y=\frac{1}{C_{equ}}$, $x=\ln (t+1)$, $\tan \phi =\frac{1}{\beta }$, and $\alpha =-\frac{1}{\beta }\ln B$; then, Equation (5) could be rewritten as a linear equation like $y=x\tan \phi +\alpha$. Therefore, the parameters in Equation (4) could then be determined according to the slope of the line on the $\frac{1}{C_{equ}}-\ln t$ coordinate system and the intercept of the vertical coordinate for $\beta$ and $B$, respectively. The schematic is shown in Figure 2.

2.3. Freeze–Thaw Cycles–Time Analogy Method

The time analogy method serves as a means to establish a relationship between time and long-term strength, shedding light on the interplay between stress, strain, and time by considering the influential factors that govern the deformation process [12]. That is to say, the influence of these factors on stress and strain is similar to that of time and can be transformed into each other. During the test, the strain, when the parameter value is high (such as more freeze–thaw cycles or a higher negative temperature), can reach the same strain by prolonging the time, so these parameter values can be strengthened, and the deformation in the design service time of buildings can be predicted through the experimental data. A comprehensive explanation of the derivation process of the time analogy method can be found in the book Frozen Soil Mechanics [12], where frozen peat soil is employed as an illustrative example to demonstrate the applicability and effectiveness of the time analogy method in determining the strength of frozen soil.

In this study, the number of freeze–thaw cycles was taken as an important influencing factor, and the number of freeze–thaw cycles was converted into the time to experience the freeze–thaw, i.e., T (Times)—t (time). The conversion formula is shown in Equation (6). The normalized long-term strength equation could be obtained by normalizing the curve, which could be used to predict the long-term strength of frozen soil. Figure 3 shows the schematic diagram of the freeze–thaw cycles–time analogy method.

$$t_i=t{N}_i$$

(6)

where t is the loading time required for deformation stability of the soil sample, N_i is the number of freeze–thaw cycles, and t_i is the physical time after the conversion of the ith freezing and thawing times.

2.4. Evaluation Indicators of Forecast Results

To evaluate the validity of the long-term intensity forecast equation, PRECISION and root mean square error (RMSE) were used to define the error of each forecast curve with respect to the measured curve [23].

$$\mathrm{PRECISION}=\frac{1}{n}{\displaystyle \sum _{i=1}^n(1-abs(\frac{T_i-P_i}{T_i})}\times 100\%)$$

(7)

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}\times {\displaystyle \sum _{i=1}^n{(T_i-P_i)}^2}}$$

(8)

where T_i, P_i are the measured and predicted values of the ith validation sample; n is the number of samples.

3. Sample Preparation and Testing

3.1. Soil Sample Preparation

Lanzhou loess, Haidong loess, and Fuping loess were selected for the experiment. The sampling location is shown in Figure 4. The preparation of soil samples met the standard preparation requirements. The air-dried soil samples were crushed and passed through a 2 mm sieve. Then, distilled water was sprayed evenly on the soil samples, they were mixed thoroughly and sealed with cling film, and they were left for 24 h to make the moisture content uniform. The soil samples were configured according to the parameters in

Table 1. Basic physical properties of loess.

Soil	Density (g/cm3)	Water Content (%)	Liquid Limit (%)	Plastic Limit (%)	Plasticity Index
Fuping loess	1.89	22	29	18	11
Haidong loess	1.43	11.81	25.64	14.77	10.87
Lanzhou loess	2.02	10	24.8	15.5	9.3

.

3.2. Test Method

The experimental design is shown in Figure 5 below. The soil samples were all water-saturated. During the freeze–thaw cycle, the soil samples were in a closed system and did not drain or replenish water. The prepared samples were placed into the freeze–thaw test device for 0, 4, 6, 8, 10, 50, and 100 freeze–thaw cycles. Before the test, repeated freezing and thawing tests were carried out on a single sample at the set freezing and thawing temperatures. It was found that when the temperature was set to ±20 °C and the freezing and thawing time was set to 2 h, the sample could be fully frozen and thawed. Therefore, the test freezing environment temperature was set to −20 °C, the thawing environment temperature was set to 20 °C, and the freezing and thawing time of the sample was set to 2 h. Three parallel samples were taken from each type of soil and subjected to spherical template indenter press-in tests and grain size determinations under the same conditions. The test devices are shown in Figure 6.

4. Results

4.1. Equivalent Cohesion Test Results

The test results after 0, 4, 6, 8, 10, 50, and 100 freeze–thaw cycles are presented in Figure 7, where C represents the equivalent cohesive force and the subsequent numbers indicate the press-in time. For example, C₂₄ represents the equivalent cohesion of a spherical template indenter after 24 h of pressing into the soil. The equivalent cohesion decreased with increasing time, while the pressing depth of the spherical template indenter increased. This phenomenon is indicative of the rheological behavior of frozen soil. As the spherical indenter sunk deeper into the soil, the stress gradually decreased, thereby promoting the stability of deformation. From Figure 7, it is observed that the rate of cohesion reduction for the three types of loess was initially high and then gradually stabilized within the time period of 17 h to 24 h. The cohesion observed during this period represents the long-term strength of the loess.

4.2. Long-Term Strength Change Rules

Figure 8 shows the long-term strength of frozen loess measured at 24 h (1440 min) after different freeze–thaw cycles. Long-term strength is a comprehensive characterization of the internal structure, mineral composition, and physical properties of frozen soil. This study observed that under repeated freezing and thawing, the long-term strength exhibited an unstable pattern during the first 10 cycles, characterized by a damping shape change. In the first 10 freeze–thaw cycles, the long-term strength of Fuping loess and Lanzhou loess was the maximum under natural sample conditions, 0.17 Mpa and 0.29 Mpa, respectively, and both reached the minimum after 10 freeze–thaw cycles, 0.07 Mpa and 0.17 Mpa, respectively. However, the long-term strength of Haidong loess was the minimum under natural sample conditions, 0.05 Mpa, and the maximum after 10 freeze–thaw cycles, 0.12 Mpa. Beyond 10 cycles, the long-term strength gradually declined and approached stability, indicating that the soil properties began to reach an equilibrium state.

4.3. Long-Term Strength Prediction Results

The long-term strength prediction of three kinds of loess was carried out for 24 h using the Vyalov long-term strength prediction equation of Equation (4), and then the measured and predicted values were compared. The parameters employed in the prediction equation are provided in

Table 2. Parameter values of the Vyalov long-term strength prediction equation.

Soil	Freeze–Thaw Cycles	β	B
Fuping loess	0	4.879	2.380 × 10−7
	4	1.244	40.993
	6	0.841	8.551
	8	1.005	30.111
	10	0.913	0.456
	50	0.385	0.087
	100	0.541	0.087
Haidong loess	0	0.363	333.416
	4	1.191	0.005
	6	17.455	1.018 × 10−68
	8	2.056	1.497 × 10−8
	10	6.068	2.976 × 10−18
	50	1.237	5.641 × 10−7
	100	0.676	0.000198
Lanzhou loess	0	11.874	1.806 × 10−13
	4	2.849	1.286
	6	23.132	4.848 × 10−54
	8	28.098	5.450 × 10−40
	10	2.095	0.307
	50	7.914	5.012 × 10−42
	100	13.604	2.715 × 10−82

below.

As depicted in Figure 9 and

Table 3. Accuracy of prediction equations for Fuping loess under different number of freeze–thaw cycles.

Freeze–Thaw Cycles	PRECISION	RMSE
0	96.8158%	4.264 × 10−3
4	95.2307%	9.631 × 10−3
6	95.6434%	6.371 × 10−3
8	97.3463%	3.693 × 10−3
10	99.6369%	3.314 × 10−4
50	99.7612%	9.654 × 10−5
100	99.6722%	9.700 × 10−5

, it can be observed that for Fuping loess, the prediction curve of the Vyalov long-term strength prediction equation gradually approached the measured value curve from the 4th to the 10th freeze–thaw cycles. The prediction accuracy exhibited an upward trend, with the 10th freeze–thaw cycle showing the most significant improvement. During this cycle, the PRECISION increased by 2.35% and the RMSE index decreased by 91.03%. After 10 freeze–thaw cycles, the PRECISION exceeded 99.6% and all RMSE indicators were below 3.314 × 10⁻⁴. These findings indicate that the prediction curve demonstrates a favorable fitting effect with the equivalent cohesion curve, and the predicted values were in close proximity to the measured values.

As depicted in Figure 10, it is evident that during the first four freeze–thaw cycles, the prediction curve of the Vyalov long-term strength prediction equation exhibits a considerable deviation. At natural sample conditions and the initial stage of the spherical template indenter, the maximum relative error reached 28.9%. As the spherical template indenter pressed into the soil, the pore structure was disrupted, leading to significant changes in the initial strength. Consequently, the predicted strength value deviated considerably from the measured value.

From

Table 4. Accuracy of prediction equations for Haidong loess under different number of freeze–thaw cycles.

Freeze–Thaw Cycles	PRECISION	RMSE
0	86.7535%	8.635 × 10−3
4	98.4653%	1.125 × 10−3
6	99.9211%	6.961 × 10−5
8	99.1985%	7.053 × 10−4
10	99.8185%	3.194 × 10−4
50	99.4757%	3.835 × 10−4
100	99.8282%	7.584 × 10−5

, it is evident that after the sixth freeze–thaw cycle, the PRECISION of the Vyalov long-term strength prediction equation consistently exceeded 99.19%, while the RMSE remained below 7.053 × 10⁻⁴. However, when considering Figure 10 in conjunction with the results, it becomes apparent that the prediction curve demonstrates better agreement with the measurement curve after the 10th freeze–thaw cycle.

It can be seen from Figure 11 and

Table 5. Accuracy of prediction equations for Lanzhou loess under different number of freeze–thaw cycles.

Freeze–Thaw Cycles	PRECISION	RMSE
0	99.4004%	2.503 × 10−3
4	96.7055%	9.606 × 10−3
6	99.8818%	3.678 × 10−4
8	99.4717%	9.551 × 10−4
10	99.9045%	2.243 × 10−4
50	99.9915%	6.380 × 10−6
100	99.9927%	5.949 × 10−6

that for Lanzhou loess, RMSE was the best after 10 freeze–thaw cycles, with the highest reduction of 99.94% and the lowest reduction of 63.98%. And it had obvious advantage in PRECISION, which was above 99.9%. These results highlight the accuracy and reliability of the Vyalov long-term strength prediction equation after 10 freeze–thaw cycles.

5. Discussion and Analysis

Based on the previous findings, it was observed that the long-term strength of the soil gradually decreased and reached a stabilized state after undergoing 10 freeze–thaw cycles. To gain further insights into the improved accuracy of the Vyalov long-term strength prediction equation after these cycles, it was necessary to analyze the potential influence of grain composition changes. Therefore, this study conducted grain size determinations under different freeze–thaw cycles and investigated the relationship between the parameter β of the prediction equation and the grain composition. Additionally, in order to validate the accuracy of the Vyalov long-term strength prediction equation after 10 freeze–thaw cycles, the predicted results for 10, 20, and 30 years were compared with those obtained through the freeze–thaw cycles–time analogy method.

5.1. Grain Composition Change Rules

The grain size distribution curves are presented in Figure 12. To accurately characterize the variation in grain composition, the widely adopted grain group division scheme in China was utilized. Specifically, the sand grain group was defined as particles larger than 0.075 mm, the silt grain group ranged from 0.005 mm to 0.075 mm, and the clay grain group encompassed particles smaller than 0.005 mm. The distribution curves of these three grain groups in the frozen loess are illustrated in Figure 13. It can be observed that during the initial 10 freeze–thaw cycles, the content of the three grain groups underwent continuous changes. In the case of Fuping and Lanzhou loess, the content of clay grains (0–0.005 mm) exhibited a gradual increase during the first six freeze–thaw cycles, while the content of sand grains (>0.075 mm) demonstrated a decreasing trend. Conversely, Haidong Yellow soil displayed an opposite trend. After 50 freeze–thaw cycles, the clay content in all three types of loess gradually decreased steadily, while the silt content tended to increase.

The experimental results demonstrate that the changes occurring in frozen loess during the initial 10 freeze–thaw cycles are highly unstable, with frequent occurrences of grain aggregation and fracturing [24,25]. However, after 10 freeze–thaw cycles, as the number of cycles increases, the grain composition tends to stabilize, and the grain size exhibits minimal changes. At this stage, a new skeleton structure begins to form in the soil [26]. The Vyalov long-term strength prediction equation is derived from the perspectives of both the dynamic theory and deformation hardening theory. The onset of transformation is considered as the failure time, which is significantly influenced by both the strength of shear stress and the average normal stress. Therefore, prior to 10 freeze–thaw cycles, the soil with an unstable structure contributes to larger prediction errors.

5.2. Correlation Analysis

From the correlations of the clay grain content, silt grain content, and sand grain content with the parameter β of the long-term strength prediction equation in Figure 14, the following is shown: a real negative correlation with the clay grain, a highly significant positive correlation with the silt grain, and a slightly negative correlation with the sand grain. Consequently, after 50 freeze–thaw cycles, the impact of silt grains in loess on the parameter β of the prediction equation is significantly more influential compared to the clay and sand grains.

5.3. Freeze–Thaw Cycles–Time Analogy Method

From the previous text, it is known that the long-term strength changes under the first 10 freeze–thaw times were very drastic, so the long-term strength data after the 10th time were used for fitting. The fitting results are shown in Figure 15.

The presence of deviations in the fitting curve introduces potential errors in the prediction results. In order to assess the accuracy of the Vyalov long-term strength prediction equation, Figure 16 presents the long-term strength predictions for the three types of loess after 10, 20, and 30 years using both the Vyalov equation and the freeze–thaw cycles–time analogy method. It can be observed that, within an acceptable range of error, the results obtained from both methods exhibit a relatively close agreement. This further validates the accuracy of the Vyalov long-term strength prediction equation after 10 freeze–thaw cycles.

6. Conclusions

(1) In the first 10 cycles of freezing and thawing, the long-term strength change curve of frozen loess was in a damping form. After 10 freeze–thaw cycles, a new equilibrium structure was formed in the soil due to repeated freeze–thaw action, so the long-term strength gradually decreased, and particularly after 50 freeze–thaw cycles, the long-term strength tended to be stable.

(2) For loess, it is recommended to use the Vyalov long-term strength prediction equation to predict its long-term strength changes after undergoing 10 freeze–thaw cycles.

(3) The parameter β was an important parameter of Vyalov’s long-term strength prediction equation, and the silt content in loess played a decisive role in the value of the equation parameter β. Therefore, the prediction equation can be evaluated and analyzed based on the silt content in loess.

(4) The comparison of the prediction results between the freeze–thaw cycles–time analogy method and the long-term strength prediction equation showed that the two methods had similar prediction results in long-term forecasts such as 10, 20, and 30 years, further proving the feasibility of using the long-term strength prediction equation after 10 freeze–thaw cycles.