Creep behavior of frozen soil and creep analytical model of frozen wall

Abstract

Taking the Gubei Mine as the research object, indoor uniaxial creep tests were carried out at temperatures of −5, −10, and −15℃ under loads of 0.3, 0.5, and 0.7σs to study the creep characteristics of deep soil, Based on the Burgers rheological model and the auxiliary formula of viscoelastic rheological formula, the freezing pressure under the combined action mechanism of frozen wall, outer shaft wall and polystyrene foam board was theoretically analyzed. The results show that the creep deformation of frozen clay under a certain constant stress loading changes with time. At three temperatures, the creep is attenuated creep under constant stress of 0.3 and 0.5σs; it exhibits non-attenuated creep at a constant stress level of 0.7σs, and the growth rate increases over a certain period of time, and finally it is destroyed by excessive deformation over time. The calculated value of the freezing pressure calculated by the frozen wall model is roughly the same as the development law of the measured value, and the calculated value of the theoretical value of the freezing pressure is slightly larger than the measured value, indicating that the calculated value can be an effective reference for the creep of the frozen wall and can provide a guarantee for the construction safety.

Introduction

Artificial ground freezing (AGF) is an advanced, environmentally friendly, and versatile construction technique. This method uses artificial refrigeration technology to freeze the water in rock and soil layers, forming hard ice that transforms natural rock and soil into artificially frozen ground. This process significantly increases the material’s strength and stability while reducing its permeability, thereby isolating groundwater from underground engineering activities. This enables the safe excavation and construction of shafts or other underground structures under the protection of a frozen wall. With continuous advancements in science and technology, AGF has been progressively refined and innovated to address the evolving demands of modern geotechnical engineering. This method offers significant engineering advantages, particularly in mitigating the adverse effects of water seepage on construction stability. AGF is widely used in various underground engineering fields in regions with abundant water and soft soil, such as subway construction^1,2, tunnel excavation support^3,4, underpinning⁵, impermeable walls for excavation pits⁶, mine shaft sinking^7,8, and the containment and rehabilitation of mines⁹, among others.

Over the decades of AGF development, the creep behavior of frozen soil has remained a focal point of research globally. Yao et al.¹⁰. improved the traditional Gauss-Newton algorithm by incorporating a fuzzy random iterative search and developed the steps for a fuzzy random Gauss-Newton algorithm. This enhanced algorithm was employed to optimize the parameters of the generalized Kelvin constitutive model, yielding an optimized creep model under varying temperatures and stress levels, which accurately characterizes the creep properties of frozen soil. Luo et al.¹¹. proposed a creep constitutive model based on the Nishihara model, incorporating stress-time coupling, offering a new approach for predicting the creep settlement of frozen sand. Li et al.¹². developed an improved Nishihara model for deeply frozen clay by combining the generalized Kelvin model with an enhanced viscoplastic body.Shan et al.¹³. introduced a nonlinear creep damage model tailored for studying the mechanical creep behavior of frozen rock under unloading conditions. Their findings provide a valuable reference for assessing the long-term stability of frozen rock walls in coal mine shafts. Yao et al.¹⁴. applied an improved fuzzy random particle swarm optimization algorithm to frozen silty clay under various temperature and pressure conditions, optimizing the Burgers damage creep model. Song et al.¹⁵. incorporated hardening and damage variables into the Nishihara model to account for the hardening and damage effects, effectively simulating the entire creep process of frozen soil. Zhang et al.¹⁶. utilized a Burgers model that integrates the effects of ice and damage to describe the creep behavior of pile-soil interfaces in permafrost regions. Zhelnin et al.¹⁷. established a creep model based on the Kelvin-Voigt fractional derivative (KVFD) framework through uniaxial and triaxial creep tests, achieving a clear description of creep deformation with fewer physical parameters. Additionally, they investigated the impact of frost heave and water migration in frozen soil on the formation of frozen walls and subsequent vertical shaft sinking activities. Using the modified Clausius-Clapeyron relation, they developed a constitutive model to describe the creep behavior of frozen soil, successfully applying it to simulate the artificial freezing of mud and sand layers at the Petrikov potassium mine in Belarus. Yang and Jiang¹⁸ used mathematical analysis software to evaluate the influence of various parameters on creep and focused on analyzing creep characteristics in a creep damage model. They proposed a nonlinear creep damage model to capture the behavior of freeze-thaw and bedding-damaged rocks. Shao et al.¹⁹. modified the creep constitutive formula for frozen soil by introducing a damage variable DDD, effectively modeling the creep process of frozen soil with this modified constitutive formula.

In this study, the Gubei Mine project is used as the research background. Through indoor uniaxial creep tests and based on the Burgers rheological model, the interaction mechanisms among the frozen wall, the outer shaft wall, and the polystyrene foam board are analyzed. A viscoelastic rheological formula is introduced to complement the theoretical analysis of the freezing pressure on the frozen wall. This research provides a novel reference for analyzing the freezing creep behavior of frozen walls and contributes valuable data to the advancement of creep theory.

Methods

Project background

The Gubei Mine, part of the Huainan Mining Group, is a significant new coal mining project in the Panxie New District of Huainan. It is a key construction project approved by the National Planning Commission under the “Tenth Five-Year Plan.” The mine is situated approximately 25 km northwest of Fengtai County, Huainan City, 1.2 km south of Guqiao Town, and 0.8 km east of the Feng-Li Highway. The mine site lies within the Huai River alluvial plain, characterized by flat topography with ground elevations ranging from + 22.0 m to + 23.9 m. The auxiliary shaft of the Gubei Mine has a diameter of 8.4 m, with a topsoil layer depth of 462.65 m. Its construction and geological conditions are more complex compared to the main and ventilation shafts. The shaft is constructed using the freezing method, with a freezing arrangement designed in the form of an outer ring hole, main ring hole, and inner ring hole. To prevent premature excavation spalling, an anti-wall ring hole is included inside the inner ring hole. Table 1 outlines the basic parameters of the auxiliary shaft, while Table 2 provides the technical parameters of the freezing design. Figure 1 presents a plan view of the freezing scheme.

Table 1.

The main design parameters of Gubei auxiliary shaft.

Names	Parameters	Names	Parameters
Design depth of pit shaft/m	705.9	Thickness of surface soil/m	462.65
Net diameter of pit shaft/m	8.4	Thickness of bedrock weathering section/m	23.4
Surface soil section barren diameter of pit shaft/m	12.95	Full depth of pit shaft/m	705.9
Bare diameter of pit shaft bedrock section/m	9.2

Table 2.

The freezing design technical parameters auxiliary shaft.

Serial number	Project name			Unit	Design parameter
1	Depth of frost penetration			m	500
2	Inner ring freezing hole	Contained line		m	16.0/14.2
		Hole numbers		Number	12/12
		Opening spacing		m	4.141/3.675
		Depth		m	467/280
		Specification	0–300 m	mm	φ159 × 7
			Under 300 m	mm
3	Middle ring freezing hole	Ring diameter		m	21.4
		Hole numbers		number	54
		Opening spacing		m	1.244
		Depth		m	467
		Specification	0–300 m	mm	φ159 × 7
			Under 300 m	mm
4	Outer ring freezing hole	Contained line		m	27.6
		Hole numbers		number	52
		Opening spacing		m	1.666
		Depth		m	500
		Specification	0–300 m	mm	φ159 × 6
			Under 300 m	mm	φ159 × 7

Fig. 1.

Freezing scheme plan layout.

To provide fundamental data for analyzing the displacement characteristics of the frozen wall’s temperature field, predicting the safety of both the frozen wall and shaft wall, and evaluating the interaction between them, real-time monitoring of borehole wall stress is essential. Consequently, a real-time monitoring system for assessing the safety of the auxiliary shaft wall in the Huainan Mine has been developed and implemented.

Among them, The detection elements are buried in the burial orientation as shown in the (Fig. 2):

Fig. 2.

The detection elements are buried in the burial orientation.

Test method and test content

The experimental specimens were extracted from deep clay soil at a depth of 382 m below ground. The natural moisture content of the soil sample is 20.6%. Tests were conducted at three different temperature levels (−5, −10, and −15 °C), with no fewer than three frozen soil samples tested under identical conditions at each temperature level. The sample was made of undisturbed soil. Cylindrical standard specimens with dimensions of Φ50 × 100 mm were prepared.The specimen is wrapped in plastic film and stored at an appropriate temperature for testing¹⁹.The frozen soil uniaxial unconfined compression test was carried out at three temperatures.

The experiment was conducted using the WDT-100 type frozen soil testing machine, as shown in (Fig. 3). It is equipped with a cooling system that ensures the creep test can be maintained at the required subzero temperature conditions.According to the rules of MT/T 593.6–2011. The specimens were subjected to loading at 0.3σs, 0.5σs, and 0.7σs. Here, σs represents the uniaxial compressive strength peak value of the permafrost, which is obtained from the uniaxial compressive strength test of the frozen soil. Each specimen was tested under a single load and temperature condition, with a test duration of 8 h per specimen. Throughout the test, the strain of each specimen was continuously monitored to prevent significant deformation of the displacement meter due to external factors, which could lead to premature termination of the test before completion. Upon completion of the tests, the collected data were analyzed, and a strain-time relationship curve was plotted. To account for variability among specimens, the data were averaged across multiple trials, ensuring accurate determination of the necessary parameters for the final analysis.

Fig. 3.

The WDT-100 type frozen soil testing machine.

Results

According to the test results, the relationship between uniaxial creep and time for frozen soil under different freezing temperatures and loading levels is illustrated as shown in (Fig. 4).

Fig. 4.

The relationship between uniaxial creep and time of frozen soil at different freezing temperatures and loading loads.

The uniaxial creep curves indicate that the creep deformation of frozen clay under constant stress loading changes with time. At three temperatures, the creep is decaying under constant stress levels of 0.3 and 0.5σs, demonstrating that creep deformation increases over time and gradually approaches a stable value; However, at a constant stress level of 0.7σs, the creep becomes non-decaying, increasing continuously with time. Beyond a certain period, the rate of deformation accelerates, ultimately leading to failure due to excessive deformation over time.

Freezing wall model and viscoelastic formula analysis

Freeze wall mechanic models

The excavation of soil is a primary cause of deformation in the frozen wall, due to the release of the stress field during the unloading process. Therefore, when analyzing the stress-strain state of a frozen wall in practical projects, it is unnecessary to account for the original stress field; instead, only the effects after the stress field has been relieved are considered. Once the relieved stress field is determined, the relieving stress on the inner surface of the frozen wall can be represented as an equivalent stress applied to the outer boundary of the surrounding infinite soil. For a plane strain problem, the equivalent load can be calculated using the following Eq. (1):

1

.

Where represents the equivalent load of the soil body; P_eq represents the original horizontal stress of the soil; represents the Poisson’s ratio of the soil.

Basic assumptions

To facilitate the study of the rheology and deformation pressures of the frozen wall, the following assumptions are made:

1. At a constant mean freezing temperature of frozen wall, the frozen wall can be considered as a homogeneous, elasto-viscous thick-walled cylinder and the freezing expansion force of the frozen wall can be neglected;

2. The bulk strain in the freezing wall is assumed to be zero, so that the interaction between the outer shaft wall, the foam board and the freezing wall can be regarded as an axisymmetric plane problem;

3. The ratio of the outer diameter to the inner diameter of the frozen wall is greater than three.

Calculation model

Building upon Huang Daoliang’s²⁰ research on the creep properties of polystyrene foam boards:

The interaction between the outer shaft wall, the foam board, and the frozen wall is illustrated in (Fig. 5). In this diagram, the support force exerted by the outer shaft wall is referred to as the freezing stress of the frozen wall on the outer shaft wall.

Fig. 5.

The diagram of the interaction between the outer shaft wall, the foam board and the freezing wall.

Where P represents the support force of the outer shaft wall on freezing wall, which is equal to the freezing pressure exerted by the frozen wall on the outer shaft wall due to the principle of action and reaction; represents the pressure on the exterior surface of the freezing wall. To simplify the calculations, is taken as.

Viscoelastic rheological formula

The rheological constitutive model of frozen soil serves as a critical foundation for analyzing the rheological behavior of the frozen wall and its interaction with the outer shaft wall. Therefore, selecting an appropriate model is a key prerequisite for conducting rheological analysis of the frozen wall.In this study, the creep characteristics of permafrost are comprehensively considered. Taking into account the actual engineering conditions, the Burgers constitutive model is adopted to describe the rheological properties of the frozen soil²¹. Based on this model, an analysis of the frozen wall is performed, leading to the derivation of an analytical equation for the freezing pressure, as shown in (Fig. 6).

Fig. 6.

Burgers constitutive model.

According to the series connection of the model, the equations are obtained: ,.The following equations are the creep formula of the Burgers constitutive models.

2

3

4

5

6

The Burgers model can be rewritten as:

7

In the equation: and represent the stress and the strain in the Burgers model; and represent the viscous coefficient and elastic modulus in the Maxwell model; and represent the viscous coefficient and elastic modulus in the Kelvin model.

According to the definition of creep, when the stress is a constant value, we can obtain formula:, substituting this formula into Eq. (7),The Burgers creep model is derived as follows:

8

Equation (8) can be rewritten as:

9

In the equation.

Each parameter corresponds to its respective value.

The Burgers model is used to perform a nonlinear curve fitting of the test data based on the Origin software, in order to obtain the parameters at different temperatures.

As is shown in Table 3.

Table 3.

The parameters of the model at different temperatures.

Temperature/℃	Stress/MPa	/MPa	/MPa	/MPa·h	/MPa·h
−5	0.408	3.091	0.156	24.286	0.248
	0.680	5.155	0.260	42.288	0.358
	0.952	7.217	0.364	62.189	0.501
−10	0.720	6.235	0.122	15.280	0.210
	1.200	10.391	0.203	25.467	0.350
	1.680	14.548	0.284	35.653	0.490
−15	0.972	13.081	0.123	12.253	0.178
	1.620	16.893	0.205	20.242	0.297
	2.268	20.235	0.287	28.593	0.416

According to the definition of creep, when the stress is a constant value, we can obtain formula: , the creep formula can be obtained as:

10

In practical engineering, the support pressure on the frozen wall, influenced by the interaction between the two walls, changes over time, leading to variations in the stresses acting on the frozen wall.

The deformation of the frozen wall is a rheological deformation that varies with changes in stress, rather than a true creep process. Therefore, the deformation of the frozen wall must be analyzed using a rheological constitutive model.

The frozen wall is in a three-dimensional rheological state, while the above equation represents a one-dimensional rheological constitutive model. Based on the fundamental principles of elastic mechanics, the three-dimensional stress at any point can be decomposed into deviatoric stress and spherical stress. Combining this with the previous assumptions that the volume strain of the frozen wall is zero, the three-dimensional rheological behavior is simplified to shear rheology under deviatoric stress.

Based on the above analysis, the three-dimensional rheological constitutive equation of the Burgers model can be derived as follow:

11

In the expression above:

In the equation: represents the deviatoric stress tensor; represents the deviatortic strain tensor; represents the Poisson’s ratio of the Hookean body; represents the Poisson’s ratio of Kelvin body.

Based on the previous hypothesis, disregarding the volumetric rheology of the frozen wall, the problem is simplified into a plane stress problem. As a result, the above equation is converted into a tangential rheological formula:

12

Substituting ,,, into Eq. (12),Where is the frozen wall load, we can get:

13

Thus, the Eq. (14) can be derived as follows:

14

In the Eq. (14):

Substituting into Eq. (14), we can obtain the rheological formula for the surface of the frozen wall:

15

Equation (15) represents the relationship between the internal displacement of the frozen wall and the supporting stress acting on it. Both the displacement and the supporting stress are time-dependent unknowns, which must be solved using the auxiliary formula.

3 Auxiliary Formula of Viscoelastic Rheological formula.

Auxiliary formula of viscoelastic rheological formula

Based on the previous analysis of the compressive properties of polystyrene foam boards²⁰, this paper derives the analytical expression for the freezing pressure acting on the wellbore wall under the combined effect of the frozen wall, outer wellbore wall, and polystyrene foam board:

(1)In the first stage of foam board compression, the compression rate is less than 75%.

During this period, the compression stiffness of the foam board is denoted as , The Interaction mode between the support pressure and the deformation at the inner surface of the frozen wall is expressed as follows:

16

In the Eq. (16): represents instantaneous displacement of the inner surface of the frozen wall in the excavation, where is .

(2) In the second stage of foam board compression, the compression rate ranges from 75 to 95%.

During this period, the compressive stiffness of the foam board is denoted as , The relationship between the support stress and the displacement at the inner surface of the frozen wall is expressed as follows:

17

In the Eq. (17):

represents the average compressive stiffness of the foam board at this stage;

represents the radial compressive stiffness of the outer shaft wall;

represents the stress when the foam board reaches the maximum compression deformation in the first stage.

(3)In the third stage of foam board compression, the compression rate exceeds 95%.

During this period, the foam plate is essentially compacted, and the supporting stiffness is roughly equal to the radial compression stiffness of the outer wall. At this stage, the interaction between the supporting pressure and the deformation of the inner surface of the frozen wall is expressed as follows:

18

In the Eq. (18): , represents the stress of the foam board at the maximum compression deformation in the second stage.

Viscoelastic freezing pressure formula of frozen wall

The relationship between the freezing stress and the inner displacement of the frozen wall has been presented and analyzed in the previous section.By combining this with the auxiliary formulas for the compression stiffness of the foam board under different conditions, expressions for the freezing pressure at different stages can be derived.

1. In the first stage of foam board compression.

Substituting Eq. (16) into Eq. (15), we can obtain:

19

Based on the findings, we can conclude that:

20

The solution to the above equation was

21

In the equation,

Under the initial conditions, ,,.

• (2)In the second stage of foam board compression.

Substituting Eq. (17) into Eq. (15), we can obtain:

22

The solution to the above equation was

23

In the equation:

• (3)In the third stage of foam board compression.

Substituting Eq. (18) into Eq. (15), we can get:

24

The solution to the above equation was

25

In the equation,

By applying the constitutive formula of the Burgers model with viscoelasticity, the rheological relationship of the calculation model is derived from the constitutive relationships of each component. In conjunction with the compression stiffness of the polyethylene foam board at different compression stages, the viscoelastic freezing pressure formula for the corresponding stage is derived.

Calculation and analysis of frozen wall model

Determination of rheological model parameters

The parameters of the Burgers model at different temperatures are presented in (Table 4).

Table 4.

The parameters of the burgers model at different temperatures.

Temperature (℃)	Parameters
	/MPa	/MPa	/MPa·h	/MPa·h
−5	5.155	0.260	42.288	0.358
−10	10.391	0.203	25.467	0.350
−15	16.893	0.205	20.242	0.297

Based on real-time monitoring of the freezing temperature at the project site, the measured average temperature of the frozen wall is approximately −3.6℃. To better align with the actual conditions, the model parameters for − 5 °C were selected.

The Poisson’s ratio  = 0.278 was determined through the indoor freezing test.The linear fitting relationship is expressed as follow : =0.30667 + 0.008T, R² = 0.95918.

Figure 7 shows the measured Poisson’s ratio data and corresponding fitting curve.

Fig. 7.

The Poisson’s ratio measured data and fitting curve.

By substituting the parameters above into Eq. 11, the values were obtained: ,

, , .

Substituting , , , into Eq. (14),we can obtain:

, , , .

According to the design parameters of the auxiliary shaft wall, the inner diameter and outer diameter of outer shaft wall were obtained as follows: , , , ,According to Eq. (1), the external load on the frozen wall at this level is approximately 3.57 MPa.

According to Huang Daoliang ‘s indoor foam board test¹⁰, the support stiffness of the foam board at each compression stage is obtained, as shown in (Table 5).

Table 5.

The support stiffness at each compression stage of the foam board.

Stage number	Stage character	Parameter description
1	Elastic compression	According to Hooke law, the average support stiffness of foam board can be obtained:K1 = 0.00265 MPa/mm.
2	Curve compression	The radial compressive stiffness of the outer shaft wall can be obtained:Kr=1.3538 MPa/mm, The average compressive stiffness of the foam board at this stage is Ks=0.1239 MPa/mm, The support stiffness of foam board is Kr=(Ks·Kr)/(Ks+Kr) = 0.1135 MPa/mm.
3	Complete compaction	The foam board has been basically compacted, and the support stiffness is approximately the radial compressive stiffness of the outer wall, we get K3 = Ks=1.3538 MPa/mm.

From the compression curve of the foam board, it can be determined that the boundary points are 73 and 96%,we can obtain : , , Thus, the time of the demarcation point is determined as follows:, .

By substituting A, B, D, and E from the previous text into , for the foam board in the first stage, we obtained the following result:

Similarly, In the second stage, we can obtain: , ;

In the third stage, we can obtain: , .

The calculated value of viscoelastic frozen wall pressure can be obtained by substituting each parameter into Eqs. (22), (23),and (25) for different stages.

Comparison of theoretical calculation value and measured value of freezing pressure

The curve of the theoretically calculated freezing pressure is plotted using Origin software and compared with the measured freezing pressure values from the project. Figure 8 shows the measured freezing pressure.Here, D1,D2,D3,D4,D5, and D6 represent the six measurement points shown in (Fig. 2). Figure 9 shows the calculated average freezing pressure; Fig. 10. presents the measured average freezing pressure; Fig. 11 presents a comparison between the measured and calculated freezing pressures.

Fig. 8.

The measured value of freezing pressure.

Fig. 9.

The calculating average of freezing pressure.

Fig. 10.

The measured average of freezing pressure.

Fig. 11.

The comparison diagram of measured and calculated freezing pressure.

The development trend of the theoretically calculated freezing pressure closely corresponds to the field-monitored data, which can be divided into three stages: rapid growth, slow growth, and stable stages.

Since the data from measuring points 5 and 6 do not align with the actual conditions.The reason may be that the pressure boxes at these two measurement points became loosened or displaced due to vibration during the concrete pouring process, causing them to malfunction and fail to provide accurate readings.There, the average value of measuring points 1 to 4 is used for comparison between the measured and calculated freezing pressures. The measured freezing pressure reaches its peak value of 3.865 MPa after approximately 75 days.;The period from 1 to 5 days corresponds to the rapid growth stage, with the freezing pressure measured at 5 days being 2.96 MPa, approximately 76.6% of the value at 75 days. The period from 6 to 36 days represents the slow growth stage, with the freezing pressure at 36 days reaching 3.71 MPa, about 96% of the value at 75 days. After 36 days, the freezing pressure enters a stable stage, with an increase of no more than 4%.

In the theoretical freezing pressure values, the maximum value occurs around 8 days, reaching 4.569 MPa. The calculated freezing pressure from 1 to 4 days corresponds to the rapid growth stage, with the freezing pressure at 4 days being 4.05 MPa, approximately 88.6% of the measured freezing pressure at 7 days. The period from 5 to 7 days represents the slow growth stage, with the calculated freezing pressure at 7 days reaching 4.5 MPa, about 98.5% of the measured value. After 8 days, the freezing pressure enters a stable stage, with an increase of no more than 2%.

The large difference in the time required to reach the maximum value between the two may be attributed to the following reasons:

The freezing pressure is influenced by the deformation of frozen wall, soil properties, depth, the frost heave deformation of the thawing soil behind the wall, the water absorption and expansion deformation of the soil layer, the temperature of frozen soil behind the wall and construction technology^3,22,23.

The theoretical calculation values are based on the indoor test results, which may not fully account for the influence of frozen soil under complex geological conditions and field environment, and the uniaxial and creep tests of frozen soil are idealized, and the calculated values do not consider the effect of the refreezing of frozen wall caused by small heat of hydration^24,25 in the later stage of the outer shaft wall pouring.

The theoretical calculated freezing pressure is marginally higher than the measured value, indicating that the calculated value can serve as an effective reference and provide a guarantee for construction safety.

Conclusion

Through a series of indoor tests on deep clay in Gubei Mine, the parameters required for the rheological formula were obtained. Using the Burgers rheological model as a starting point, the interaction mechanism between the polystyrene foam board, the frozen wall, and the outer shaft wall is analyzed. A viscoelastic rheological formula is introduced to calculate the freezing pressure of the frozen wall, which is then compared with the field-measured data.

Results show that:

1. At three temperatures, the creep deformation increases with time and gradually approaches a stable value under constant pressures of 0.3σ_s and 0.5σs.At the constant stress level of 0.7σs, the creep deformation also increases with time, but the rate of increase accelerates after a certain period, eventually leading to failure due to excessive deformation over time.

2. Through the analysis of experimental data, the properties of the elastic modulus and Poisson’s ratio of frozen soil were obtained: the elastic modulus increases as the freezing temperature decreases and can be expressed by the linear equation A; Poisson’s ratio decreases with the decrease in freezing temperature and can be expressed by the linear equation B, though the decrease is relatively small.

3. The rheological relationship of the calculation model is derived from the constitutive relationships of each component using the Burgers viscoelastic model. By combining this with the compression stiffness of polyethylene foam board at different compression stages, the viscoelastic freezing pressure formula for each corresponding stage is derived.

4. The development of the theoretical calculated freezing pressure closely matches the field-measured data, which exhibit three stages: rapid growth, slow growth, and stable phases. The theoretical freezing stress calculated value is slightly higher than the measured value, indicating that the calculated value can serve as an effective reference and provide assurance for construction safety.

This study suggests that the proposed model may offer an innovative method for estimating the creep deformation characteristics of the freezing wall.

Author contributions

Conceptualization, Z.C. and H, H,; Methodology, Z.C. and H.H.; Validation, Z.C.; Formal analysis; Z.C.; Investigation, H.H.; Resources, B.L.; Datacuration, Z.C. and H.H.; Writing—original draft preparation, Z.C. and H.H.; Writing—review, Z.C.; Visualization, Z.C.; Supervision, Z.C.; Project administration, B.L.; Funding acquisition, B.L. All authors have read and agreed to the published version of the manuscript.

Data availability

Data is provided within the manuscript.

Declarations

Competing interests

The authors declare no competing interests.