The calcareous clay (CC) used in the test was sourced from a deep layer (420–430 m) of a coal mine in the Lianghuai mining area, northern Anhui Province, China. It was transported to the laboratory in sealed packaging. The basic physical and mechanical properties of the undisturbed soil samples are shown in Table 1. Following standard provisions (China Planning Press, 2019), the undisturbed soil was dried naturally. After drying, the soil was placed in an oven at 105° C for 12 h A crusher. The dried and processed lumpy soil is crushed once by a crusher. However, the particle size of the crushed soil did not meet the test requirements, necessitating secondary crushing. After the second crushing, the soil was sieved through a 0.5 mm sieve to obtain the desired particle size for the experiment. The crushing process is shown in Figure 1. The particle size distribution was measured using a laser particle size analyser (BT-2001) (He et al., 2024), with results presented in Figure 2. The analysis revealed that the particle size distribution of the calcareous clay primarily ranged from 0.5 to 450 microns, with a significant concentration between 25 and 180 microns. The cumulative content curve of the particle size distribution indicates a relatively stable slope, demonstrating a uniform texture and consistent particle size range within the calcareous clay. This uniformity is beneficial to ensure that the prepared sample has a consistent texture and is isotropic.

The stress-strain curves of frozen calcareous clays at various temperatures are depicted in Figure 7 (In the figure σ 1 − σ 3 is the deviatoric stress and ε 1 is the axial strain). These curves exhibit strain-hardening characteristics across different interaction levels (Wu et al., 2017). Specifically, at −5°C and −10°C, the curves demonstrate weak hardening behavior, while at −15°C and −20°C, they exhibit general hardening behavior. The transition from weak to general hardening with decreasing temperature can be attributed to the gradual transformation of liquid water in the soil into solid ice crystal particles. This transformation enhances both the strength and strain-hardening characteristics of the soil body due to the increased cementing effect of the ice crystal particles (Shi et al., 2020).

Figure 8 schematically illustrates the microstructure and mechanical behavior of frozen calcareous clay (In the figure σ 3 is the confining pressure). Under axial and confining pressure, frozen calcareous clay experiences extrusion and friction between soil particles. During the freezing process, the transformation of liquid water into ice diminishes its lubricating effect. The formation of cementing ice tightly binds adjacent soil particles together, enhancing cohesion. Additionally, the presence of pore ice acts similarly to the addition of aggregates to cement, significantly increasing the strength of the calcareous clay (Wu et al., 2024; Wu et al., 2022).

The deviatoric stress-axial strain curves of frozen calcareous clay exhibit distinct elastic and elastoplastic phases, which can be elucidated through the following analysis: Initially, under the applied hydrostatic pressure for 30 min, the pore space within the sample undergoes compaction. Concurrently, the presence of cemented ice fosters a dense structure among soil particles. During this phase, the specimen’s deformation modulus is substantial, leading to a rapid increase in preload with axial strain, characterized by a linear relationship. This behavior, which may originate from the presence of intact cemented ice and pore ice, reflects elastic properties. As axial strain continues to increase, the preload and axial strain exhibit non-linear growth. The slope of the curve, representing the deformation modulus, gradually diminishes and stabilizes, without exhibiting a distinct peak throughout the curve. At this stage, the internal cemented ice and pore ice within the specimen undergo compression or even disintegrate, leading to the gradual formation and expansion of internal cracks (Wu et al., 2020). Consequently, the specimen becomes damaged, the deformation modulus gradually decreases, and its resistance to deformation weakens (Zhang et al., 2024).

Figure 7 reveals that during the elastic phase, both the slope (modulus of elasticity) and stress value at the inflection point (Junction of elastic and elastoplastic phases) demonstrate an increasing trend with decreasing temperature and increasing confining pressure. This phenomenon may be attributed to several factors: Firstly, the decrease in temperature reduces the content of unfrozen water, thereby increasing the presence of cemented ice and pore ice. Consequently, the effects of cementation and “replacement” are amplified, leading to an augmentation in the specimen’s deformation modulus. Secondly, the rise in confining pressure intensifies the external load on the specimen (Zhang et al., 2023). Microscopically, the pore space experiences further compaction due to the elevated confining pressure, resulting in closer contact between soil particles and increased compression and friction. Macroscopically, the specimens endure greater confining pressures, enhancing confinement and thereby augmenting their ability to resist deformation.

Considering the strain-hardening nature of the stress-strain curves in this test, the deviator stress at an axial strain of 15% serves as the failure stress for analysis (China Planning Press, 2019). Figure 9A compares the results at different temperatures, indicating a gradual increase in the specimen’s failure stress as temperature decreases, a trend that is pronounced. Under identical temperature conditions, the failure stress of the specimen exhibits a positive correlation with confining pressure. Notably, at 5°C, the peak deviatoric stress of A-2 surpasses that of A-3. Analyzing A-2 and A-3 specimens reveals that A-2 has a dry density of 1.86g/cm3 and a moisture content of 17.5%, while A-3 has a dry density of 1.76g/cm3 and a moisture content of 20%. Although A-2 exhibits higher dry density compared to A-3, its moisture content is lower. The discrepancy in failure stress between A-2 and A-3 may originate from the fact that at 5°C, some free water within the specimen remains in liquid form despite freezing. It should be indicated that liquid water, while possessing a lubricating effect, weakens the cementation and “replacement” effect of ice, consequently diminishing the specimen’s strength. However, the increase in dry density increases the number of soil particles in the specimen, reduces porosity, enhances effective contact area between particles, and intensifies extrusion and friction, thereby significantly improving compressive strength. Hence, at this temperature, the impact of water content on sample compressive strength is minimal compared to dry density.

In Figure 9B, at a moisture content of 17.5%, the failure stress of C-1 surpasses that of B-3, despite B-3 having higher confining pressure and dry density. This may be attributed to C-1 being at −15°C, where nearly all free water freezes and the process of bonded water freezing initiates. At this temperature, the lubricating effect of liquid water diminishes significantly, while the cementation and “replacement” effects of ice are greatly enhanced. This substantial enhancement in ice cementation and “replacement” significantly increases the compressive strength of the specimen. This observation aligns with the general hardening nature of the stress-strain curve at −15°C, underscoring the significant impact of temperature on failure stress magnitude.

Analysis of Figures 9B,C reveals a disordered distribution of failure stress under constant dry density and moisture content conditions. This suggests that while dry density and moisture content exert some influence on failure stress, their impact is almost negligible compared to temperature and confining pressure.

The test data acquired from orthogonal experiments (Jiang et al., 2021) are typically subjected to sensitivity analysis for data processing. This involves averaging the extreme differences in the target value influenced by each factor, thereby identifying the primary factors affecting the target value through the difference between the maximum and minimum test values (Deng et al., 2023). The sensitivity analysis results, presented in Table 3, indicate that temperature exhibits the most substantial effect on failure stress, with an extreme difference value of 2.99 MPa, while moisture content has the least impact, registering an extreme difference value of 0.78 MPa. The sensitivity ranking of each influencing factor on failure stress is as follows: temperature > pressure > dry density > moisture content. This hierarchy underscores that the influence of temperature and pressure on failure stress far surpasses that of moisture content and dry density. Thus, the pronounced effect of temperature and pressure on failure stress aligns with the earlier analysis.

Alterations in the unfrozen water content of permafrost, representing the phase transition between solid and liquid water directly affect the soil’s mechanical properties (Pardo Lara et al., 2021; Wen et al., 2012). To investigate this phenomenon, an NMR test system was employed in this study to measure the unfrozen water content of the samples under the specified orthogonal test conditions. It was ensured that the temperature had reached the designated level and had stabilized for a minimum of 2 hours before conducting the measurements. The NMR test system setup is illustrated in Figure 10.

Figure 11 depicts the evolution of unfrozen water content concerning temperature, delineated into three distinct phases: (1) From 0°C to −2.7°C: Sharp Decrease Phase - As the temperature drops below the freezing point of the calcareous clay, the free water within the soil undergoes a rapid phase transition and rapid freezing. This abrupt change manifests as a sharp decline in unfrozen water content. (2) From −2.7°C to −11.5°C: Rapid Decrease Phase - Within this temperature range, any remaining free water in the soil continues its freezing process, leading to a significant reduction in unfrozen water content. (3) From −11.5°C to −25°C: Slow Reduction Stage - At this juncture, the free water in the soil is predominantly frozen, while the phase transition of bound water commences. Bound water, tightly adhering to soil particles due to capillary action, electrostatic adsorption, and surface tension, encounters difficulty in freezing (Wen et al., 2012). Consequently, the unfrozen water content diminishes gradually at a slower pace. In summary, the progressive decrease in temperature prompts the gradual freezing of free water within the soil, while the freezing of bound water occurs with greater resistance, resulting in the observed evolution of unfrozen water content across the described phases.

The quantity of unfrozen water content within permafrost significantly influences its strength, making the investigation of unfrozen water content essential (Chen et al., 2021; Zhang et al., 2020). To estimate the unfrozen water content, this study employs the empirical formula introduced by Xu et al. (2001). The amount of unfrozen water content can be calculated using the Equation 3: ω u = ω T f b T − b (3) where ω u is the unfrozen water content at -T°C, %; ω is the initial water content, %; T f is the absolute value of the freezing temperature of the sample, °C; T is the absolute value of temperature, °C; b is the test constant.

According to the NMR test results, the starting freezing temperature of calcareous clay was identified as −0.8°C. The measured data were subjected to fitting, and the fit is illustrated in Figure 12. The fit appears to be satisfactory, with a correlation coefficient of R ² = 0.97687. This high value suggests that the formula can effectively depict the relationship between the unfrozen water content of calcareous clay and the temperature and initial water content. Consequently, it can provide more accurate predictions of the unfrozen water content of calcareous clay. ω u = 0 . 8 0.54945 ω T − 0.54945 = 0.8846 ω T − 0.54945 (4)

The Duncan-Chang model (Dong et al., 2023) employs hyperbolic equations to describe the stress-strain behavior of soil. The model can be mathematically expressed as follows: σ 1 − σ 3 = ε 1 λ + v ε 1 (5) where λ and v are experimental constants. Through coordinate transformation, Equation 5 can be rewritten in the form below: ε 1 σ 1 − σ 3 = λ + v ε 1 (6)

Equation 6 can be considered the primary function of ε 1 / σ 1 − σ 3 and ε 1 . Using this functional form, the orthogonal test data can be analyzed and sorted in a secondary analysis to derive the relevant parameters λ and v . The sorting results are presented in Figure 13.

Figure 13 reveals that the parameters’ linear fit correlation coefficients R ² exceed 0.9, indicating a significant linear fit correlation. In Equation 6, the parameter v represents the slope of the straight line and λ is the intercept of the straight line on the vertical axis. The values of these parameters obtained from the fit are collated and presented in Table 4.

Table 4 indicates that the parameters v and λ vary with changes in confining pressure and temperature. The temperature affects the unfrozen water content within the sample, so it is assumed that the relationship between the parameters v and λ , the confining pressure, and the unfrozen water content can be expressed as Equations 7, 8: v = f σ 3 , ω u (7) λ = g σ 3 , ω u (8) where f σ 3 , ω u and g σ 3 , ω u are functional relations related to confining pressure σ 3 and unfrozen water content ω u .

The specific values of the unfrozen water content of the samples at four temperatures were calculated using Equation 4. These values were then fitted to the parameters v and λ by combining the data in Table 4. The results of this fitting process are illustrated in Figure 14.

It is observed that the binary nonlinear fitting correlation coefficients of parameters v and λ exceed 0.9, indicating a significant nonlinear fitting correlation. The fitted expressions are as follows: v = 0.5039 − 0.04621 σ 3 − 0.33883 ω u + 0.03993 ω u 2 + 0.12461 σ 3 ω u 1 + 0.38433 σ 3 − 0.92555 ω u + 0.18341 σ 3 2 + 0.10091 ω u 2 + 0.44398 σ 3 ω u (9) λ = 5.54723 9.73223 − 2.7894 ω u + 28.35811 σ 3 − 2.05817 σ 3 ω u (10)

According to Equation 4, ω u is a function of initial water content ω and temperature T. Accordingly, the relationship between parameters v , λ and confining pressure, temperature, initial water content can be expressed as Equations 11, 12: v = F σ 3 , T , ω (11) λ = G σ 3 , T , ω (12) where F σ 3 , T , ω and G σ 3 , T , ω are functional relations that are related to the confining pressure σ 3 , temperature T, and initial water content ω .

Introducing Equation 4 into Equation 9 and Equation 10, yields the following expression: v = 0.5039 − 0.04621 σ 3 − 0.29973 ω T − 0.54945 + 0.03125 ω 2 T − 1.0989 + 0.11023 σ 3 ω T − 0.54945 1 + 0.38433 σ 3 − 0.81874 ω T − 0.54945 + 0.18341 σ 3 2 + 0.07896 ω 2 T − 1.0989 + 0.39274 σ 3 ω T − 0.54945 (13) λ = 5.54723 9.73223 − 2.4675 ω T − 0.54945 + 28.35811 σ 3 − 1.82066 σ 3 ω T − 0.54945 (14)