Multi-Scale Damage Evolution of Soil-Rock Mixtures Under Freeze–Thaw Cycles: Revealed by Electrochemical Impedance Spectroscopy Testing and Fractal Theory

Abstract

The response of the microscopic structure and macroscopic mechanical parameters of SRM under F–T cycles is a key factor affecting the safety and stability of engineering projects in cold regions. In this study, F–T tests, EIS, and uniaxial compression tests were conducted on SRM. The construct equivalent model of different conductive paths based on EIS was constructed. A peak strength prediction model was developed using characteristic parameters derived from the equivalent models, thereby revealing the mechanism by which F–T cycles influenced both microscopic structure and macroscopic strength. The results showed that with increasing cycles, both $R_{CP}$ and $R_{CPP}$ exhibited an exponential decreasing trend, whereas C_DSRP and D_f increased exponentially. Peak strength and peak secant modulus decreased exponentially, but peak strain increased exponentially. The expansion and interconnection of pores with different radii within $CPP$ and $CP$ caused smaller pores to evolve into larger ones while generating new pores, which led to a decline in $R_{CPP}$ and $R_{CP}$. Moreover, this expansion enlarged the soil–rock contact area by connecting adjacent gas-phase pores and promoted the transformation of $CSRPP$ into $DSRPP$, enhancing the parallel-plate capacitance effect and resulting in an increase in $C_{DSRP}$. Moreover, the interconnection increased the roughness of soil–soil and soil–rock contact surfaces, leading to a rising trend in $D_f$. The combined influence of $C_{DSRP}$ and $D_f$ yielded a strength prediction model with higher correlation than a single factor, providing more accurate predictions of UCS. However, the increases in $C_{DSRP}$ and $D_f$ induced by F–T cycles also contributed to microscopic structure damage and strength deterioration, reducing the load-bearing capacity and ultimately causing a decline in UCS.

1. Introduction

Soil–rock mixtures (SRM) are loose geotechnical materials widely distributed in open-pit mine waste dumps, highway embankment slopes, and slopes in special geological settings. They are primarily composed of fine-grained soil particles, coarse rock fragments, and porosity filled with air or water [1,2,3]. Due to the unique composition of SRM, their microscopic structure and macroscopic mechanical behavior cannot be directly compared with soils or rocks. Previous studies have shown that geotechnical engineering in cold regions was subjected to complex freeze–thaw (F–T) processes, making stability analyses in such regions significantly different from those in temperate areas [4,5,6,7,8]. Repeated F–T cycles gradually damage the initial microscopic structure of soil and rock units, alter their mechanical properties, and ultimately lead to instability and failure of geotechnical structures [9,10,11,12]. Nevertheless, as a complex porous geotechnical medium, SRM are also subject to repeated freezing and thawing, which progressively alters both their microscopic structure and macroscopic parameters, ultimately affecting engineering stability [9,10,11,12]. Therefore, investigating the evolutionary patterns of SRM microscopic structure and macroscopic properties under F–T cycles, elucidating the mechanisms by which F–T processes influence these characteristics, and analyzing the intrinsic correlations between microscopic structure and macroscopic mechanical behavior are of critical significance for stability assessment and the prevention of F–T failures in geotechnical engineering of cold regions.

At present, most studies on SRM have focused on the influence of rock and moisture content on the macroscopic mechanical behavior. Xu [13,14] demonstrated that the dominant factors affecting the microscopic structure and macroscopic mechanical properties of SRM were the proportion and particle size of rock fragments. The influence of rock content on mechanical parameters exhibited a distinct threshold effect. Similarly, Yu [15] found that rock content also showed a threshold effect on the evolution of shear bands. Soil particles rotated earlier than rock fragments. In addition, shear failure of soil particles and tensile failure at soil–rock interfaces constituted the primary failure mechanisms of SRM. Zhang [16], Gao [17], and Zhao [18] conducted a study on the effects of rock and moisture content on the mechanical properties of SRM. Their findings indicated that an increase in rock content disrupted the structural integrity, leading to a decrease in uniaxial strength. However, shear strength increased with an increase in rock content. Nevertheless, the influence of moisture content on the shear strength exhibited a threshold characteristic. Wei [19] investigated the influence of varying moisture contents and rock fragment sizes on SRM mechanical properties and reported that the effect of normal stress on shear strength also showed a threshold characteristic and followed a power-law relationship. Shear strength decreased with increasing moisture content but increased with larger rock fragment sizes. With the increasing intensity of resource development in cold regions, the effects of F–T cycles on geotechnical engineering involving SRM have become increasingly prominent. Xing and Zhou [20,21,22] investigated the impact of F–T cycles on SRM with varying rock contents and found that repeated freezing and thawing weakened the bonding strength between soil and rock particles. Both the elastic modulus and shear strength decreased with increasing F–T cycles. Furthermore, F–T cycles exerted a more significant effect on cohesion, whereas rock content predominantly influenced the internal friction angle. Tang [23,24] conducted a study using Nuclear Magnetic Resonance (NMR) to investigate the effects of F–T cycles on the microscopic structure and macroscopic mechanical properties of SRM with varying rock contents. For SRM with rock contents exceeding 45%, shear strength, shear band thickness, and surface undulation of failure planes decreased with increasing cycles. Regarding failure mode evolution, low rock content SRM exhibited progressive shear failure, whereas high rock content samples displayed sudden brittle failure. NMR analysis showed that the pore structure spectra of SRM after different numbers of cycles exhibited a distinct three-peak distribution. With increasing F–T cycles, both porosity and connectivity of pores of varying radii increased. Additionally, the pore structures under different F–T cycles displayed fractal characteristics [25,26].

In summary, NMR is currently the primary method for investigating the microscopic structure of SRM. However, it cannot detect air-filled pores within SRM, nor can it distinguish liquid-filled pores between soil particles or rock fragments. Therefore, NMR exhibits inherent limitations in the microscopic structural characterization of SRM. Electrochemical impedance spectroscopy (EIS) has recently emerged as a widely used technique for probing the microscopic structure of porous materials. By applying an electric field perturbation and analyzing the material response, EIS enables rapid, efficient, and non-destructive assessment of the microscopic structure of porous materials [27,28,29]. In recent years, many researchers have employed EIS to investigate the microscopic structure of SRM. Dong [30,31] conducted EIS tests on SRM with varying moisture and rock contents and found that, moisture content had a greater influence on EIS-detected microscopic structure than rock content. Both rock content and moisture content exhibited threshold effects on EIS responses: low levels of moisture and rock content tended to inhibit the development of mesopores, whereas higher levels promoted their formation. Additionally, a three-tier conductive path equivalent circuit model was developed based on the degree of cementation in SRM. Using characteristic parameters from impedance spectra, a mapping relationship between microscopic structure and macroscopic mechanical properties was established, enabling analysis of EIS responses under different moisture and rock content conditions [32].

Based on the aforementioned studies, current EIS investigations of SRM have only proposed a three-level conductive path equivalent model. However, derivation and calculation of characteristic parameters for the different conductive paths have not been performed, nor has the microscopic structure of SRM been characterized using these parameters. In this study, SRM samples collected from an open-pit graphite mine waste dump in Heilongjiang Province, China, were subjected to F–T cycles, EIS testing, and uniaxial compression tests. A novel equivalent model incorporating distinct conductive paths was proposed, and characteristic parameters and fractal dimensions for the different paths were derived. These parameters were used to characterize the microscopic structure, and their evolution under F–T cycles was analyzed, revealing the mechanisms by which F–T processes affect microscopic structure and macroscopic mechanical properties. The intrinsic relationships between different characteristic parameters, fractal dimensions, and mechanical properties were also explored. Furthermore, using characteristic parameters and fractal dimensions as influencing factors, a predictive model for the uniaxial compressive strength (UCS) of SRM was developed. The findings can provide guidance for understanding the intrinsic relationships between microscopic structure and macroscopic mechanical properties of SRM under F–T cycles and for safety assessment of related geotechnical engineering.

2. Materials and Method

2.1. Sample Preparation and Experimental Testing

The soil–rock threshold is an important factor determining the microscopic structure and macroscopic mechanical parameters of SRM. Based on previous studies on SRM [14,33,34,35], a threshold value of 0.05 was selected for the experiments. The calculation is expressed as follows:

$$d_{S/R}/L_s=0.05$$

(1)

In Equation (1), $d_{S/R}$ is the soil-rock threshold, and $L_s$ is the shear box height. According to geotechnical testing standards [36] and the requirements of the uniaxial compression test equipment, the test specimens were cubes with a side length of 100 mm. Substituting the specimen side length into Equation (1) yields a soil–rock threshold of 5 mm, defining rock fragments as particles with a radius greater than 5 mm and soil particles as those with a radius less than 5 mm. Based on geotechnical testing standards [36], the maximum radius of rock fragments should not exceed one-quarter of the specimen box side length, which is 25 mm. All particles of SRM were collected from the crest of the waste dump at an open-pit graphite mine in Heilongjiang Province. The natural density of the particles was 2.407 g/cm³, and the initial moisture content was 9.09%. For rock fragment size classes exceeding the maximum radius, equivalent mass substitutions were applied. The resulting mass fractions of soil particles and rock fragments with radii of 10 mm, 20 mm, and 25 mm were 69.33%, 8.88%, 10.20%, and 11.59%, respectively. Since the sampling occurred in July, which is outside the rainy season of the sampling area, the moisture content was adjusted to 12%. During specimen preparation, to ensure uniform contact between the aqueous solution and the soil and rock particles, SRM was sealed with plastic wrap and allowed to rest for 24 h before being filled into the specimen box. Demolding and labeling were performed sequentially. The specimens were filled in three layers, with each layer compacted 40 times. To prevent moisture loss, the labeled specimens were fully wrapped with plastic film and maintained in this state until the uniaxial compression tests were conducted.

Figure 1 shows the distribution of average minimum temperatures in SRM sampling area from 2019 to 2021. As shown, the average minimum temperatures from November to April during this period were consistently below 0 °C, with the lowest monthly average reaching −20 °C. In contrast, the average minimum temperatures in other months remained above 0 °C, indicating the presence of F–T cycles in the sampling area. Due to the specific environmental conditions of SRM, the F–T tests were conducted using air freezing and air thawing, with freezing and thawing carried out in a low-temperature chamber and a drying oven, respectively. Based on previous F–T studies of SRM, the freezing temperature and duration were set to −20 °C and 12 h, respectively, while the thawing temperature and duration were set to 20 °C and 12 h. The number of F–T cycles was determined with reference to the design service life of the waste dump, which was 13 years, and was therefore set to 15 cycles. F–T tests were conducted for 0, 5, 10, and 15 cycles. A schematic diagram of the F–T cycle is shown in Figure 2.

The EIS testing apparatus is the PARSTAT 3000A-DX electrochemical workstation manufactured by Ametek Trading (Shanghai) Co., Ltd. in Shanghai, China. The detection wiring configuration employs a four-electrode circuit, with detailed testing parameters as shown in

Table 1. EIS test parameters.

Main Parameters	Parameter Value
Start Frequency (Hz)	10,00,000
End Frequency (Hz)	0.01
Data Quality	3
Current Range (mA)	2
Voltage Range (V)	+/−6
Electrometer Mode	Differential

. The uniaxial compression testing apparatus is the multifunctional rock direct shear tester produced by Shandong Jinan Mining and Rock Testing Instrument Co., Ltd., Jinan, China. In accordance with geotechnical testing standards [36], displacement control was selected as the control method, with a loading rate set at 0.05 mm/s. Each test group comprised three specimens. The calculation formula for uniaxial compressive strength is shown in Equation (2). Here, ${\sigma }_c$ represents uniaxial compressive strength, while $F$ and $A$ denote the specimen failure load and load cross-sectional area, respectively. The detailed testing procedure is illustrated in Figure 3.

$${\sigma }_c={\displaystyle \frac{F}{A}}$$

(2)

2.2. Equivalent Circuit Modeling

The conductive path model is an equivalent circuit model constructed based on EIS results, which interprets the microscopic structure of porous geotechnical media through characteristic parameters of different conductive paths [27,28,29]. Building on existing studies of conductive path models for SRM, this study classifies the conductive paths within SRM based on particle contacts into three types: continuous pore water conductive paths ($CPP$), discontinuous soil–rock–pore water conductive paths ($DSRPP$), and continuous soil–rock–gas pore conductive paths ($CSRPP$), as illustrated in Figure 4. As shown, the conductive paths within SRM are arranged in parallel. Based on Ohm’s law [27,28,29], the impedance of different conductive path models can be calculated as follows:

$$Z=1/({\displaystyle \frac{1}{Z_{CPP}}}+{\displaystyle \frac{1}{Z_{DSRPP}}}+{\displaystyle \frac{1}{Z_{CSRPP}}})$$

(3)

In Equation (3), $Z, Z_{CPP}{, Z}_{DSRPP}$ and $Z_{CSRPP}$ represent the impedances of the total impedance, $CPP, DSRPP,$ and $CSRPP$, respectively. The resistance of $CPP$ arises from the migration of ions in the pore water solution. Therefore, the impedance calculation formula for $CPP$ is as follows:

$$Z_{CPP}=R_{CPP}={\rho }_{w}L/\left(S{\phi }_M{\eta }_{SR}\right)$$

(4)

In Equation (4), ${\rho }_w$ and $L$ represent the resistivity of pore water solution ($\Omega \mathrm{cm}$) and the thickness of the soil-rock mixture parallel to the applied electric field direction ($\mathrm{cm}$), respectively, while $S$ denotes the cross-sectional area of SRM perpendicular to the electric field intensity (cm²). ${\eta }_{SR}$ denotes the porosity of SRM (%), while ${\phi }_M$ represents the proportion of $CPP$ pore volume to the total porosity (%). Based on published electrochemical equivalent models for porous geomaterials, $DSRPP$ can be further divided into a continuous pore water path ($CP$) and a discontinuous soil-rock conductive path ($DSRP$). Furthermore, according to the principles of impedance detection [27,28], $DSRP$ can be regarded as a double-parallel-plate capacitor. Thus, the impedance calculation expression for $DSRP$ is as follows:

$$\left.\begin{array}{c}{R}_{CP}={\displaystyle \frac{{L^{\prime }\rho }_w}{S{\eta }_{SR}{\phi }_M^{\prime }}}\\ Z_{DSRP}={\displaystyle \frac{1}{j\mathsf{\omega }{C}_{DSRP}}}\\ C_{DSRP}={\displaystyle \frac{{\phi }_M^{\prime }{\eta }_{SR}S{\epsilon }_A{\epsilon }_S}{d}}\end{array}\right\}$$

(5)

$$Z_{DSRPP}=R_{CP}+Z_{DSRP}={\displaystyle \frac{{L^{\prime }\rho }_w}{S{\eta }_{SR}{\phi }_M^{\prime }}}+d/j\mathsf{\omega }\left[{\phi }_M^{\prime }{\eta }_{SR}S{\epsilon }_A{\epsilon }_S\right]$$

(6)

In Equations (5) and (6), $Z_{DSRPP}, R_{CP}$, and $Z_{DSRP}$ represent the impedance of $DSRPP$, the resistance of $CP$ within $DSRPP$, and the impedance of $DSRPP$ within the $DSRPPP$, respectively. $C_{DSRP}$ denotes the double parallel-plate capacitance of $DSRPP$. $L^{\prime }$ and $d$ denote the thickness of $CP$ parallel to the electric field (cm) and the cumulative thickness of the discontinuous points in $DSRP$(cm), respectively. ${\phi }_M^{\prime }$, ${\epsilon }_A$, and ${\epsilon }_S$ represent the porosity of $CP$(%), the vacuum permittivity (8.85 × 10⁻¹⁴ F/cm), and the relative permittivity of the soil (F/cm), respectively. $\mathsf{\omega }$ and $j$ denote angular frequency (rad/s) and the imaginary unit ($j^2=-1$). Similarly, based on the principle of resistive impedance detection [26,28], the soil-rock matrix within $CSRPP$ can also be regarded as a double-parallel-plate capacitor. Here, the finer-grained soil acts as the dielectric. Due to the capacitive charging and discharging effect, the alternating current applied by the electrochemical workstation can penetrate the soil-rock matrix of $CSRPP$. The impedance of $CSRPP$ is calculated as follows:

$$\left.\begin{array}{c}{Z}_{CSRPP}={\displaystyle \frac{1}{j\mathsf{\omega }{C}_{CSRPP}}}\\ C_{CSRPP}={\displaystyle \frac{S{\epsilon }_A{\epsilon }_S}{L}}\end{array}\right\}$$

(7)

In Equation (7), $Z_{CSRPP}$ and $C_{CSRPP}$ represent the impedance and parallel plate capacitance of $CSRPP$, respectively. Substituting Equations (4), (6) and (7) into Equation (3) yields the total impedance calculation formula for different conductive paths, as shown in Equation (8).

$$Z=1/\left\{j\mathsf{\omega }{\displaystyle \frac{S{\epsilon }_A{\epsilon }_S}{L}}+\left(S{\phi }_M{\eta }_{SR}\right)/{\rho }_{w}L+1/[{\displaystyle \frac{{L^{\prime }\rho }_w}{S{\eta }_{SR}{\phi }_M^{\prime }}}+1/(j\mathsf{\omega }{\displaystyle \frac{{\phi }_M^{\prime }{\eta }_{SR}S{\epsilon }_A{\epsilon }_S}{d}}\left)\right]\right\}$$

(8)

Based on previous studies of EIS [26,28], it is observed that the impedance response of different conductive-path equivalent models for SRM (Figure 5a) typically exhibited a two-semicircle arc pattern (Figure 5b). However, this response can also be represented by conventional electrochemical equivalent circuit models (Figure 5c). In this study, the impedance expressions of both the conductive-path equivalent models of SRM and the traditional electrochemical circuit models are derived. Through simplification and comparison, the characteristic parameters corresponding to the different conductive paths in the electrochemical equivalent model are obtained, with their calculation expressions given as follows:

$$\left.\begin{array}{c}{R}_{CP}=(R_1+R_2)R_1{/R}_2\\ R_{CPP}=R_1+R_2\end{array}\right\}$$

(9)

$$\left.\begin{array}{c}{C}_{DSRP}=(C_1+C_2){[{R_2/(R}_1+R_2)]}^2\\ C_{CSRPP}=C_1{C}_2/(C_1+C_2)\end{array}\right\}$$

(10)

$$C_{DSRP}=C_2\times {[{R_2/(R}_1+R_2)]}^2$$

(11)

In Equations (9) and (10), $C_1$, $C_2$, $R_1$, and $R_2$ are fundamental parameters of the traditional equivalent impedance model. However, in laboratory measurements, due to testing limitations of experimental equipment, the impedance test results yield a single circular arc curve (Figure 5d), whose equivalent circuit model is shown in Figure 5e. Furthermore, since the thickness ($L$) of $CSRPP$ parallel to the electric field direction is significantly greater than the cumulative thickness ($d$) of the discontinuous points in $DSRP$, according to Equations (5) and (7), $C_{DSRP}$ is substantially larger than $C_{CSRPP}$. Neglecting the influence of $C_{CSRPP}$, Equation (10) can be simplified and transformed into Equation (11). Moreover, the equivalent circuit path model of SRM in Figure 5a can be represented as the equivalent model shown in Figure 5f. Due to the complex distribution of soil and rock particles with different radii within SRM, the constant phase angle equivalent model was selected for circuit fitting (Figure 5g) to enhance the accuracy of calculating characteristic parameters for different paths in the equivalent model. The fitting software used in the experiment was Zview2, with fitting data comprising electrochemical impedance spectroscopy data covering frequencies from 1000 Hz to 1,000,000 Hz. The impedance calculation formula for the constant-phase-angle equivalent model is as follows:

$$Z_{CPE}=1/[CPE1\_T\times {\left(j\mathsf{\omega }\right)}^{CPE1\_P}]$$

(12)

In Equation (12), $CPE1\_P$ is the dimensionless fitting index of $CPE$, with a fitting range from 0 to 1. $CPE1\_T$ represents the amplitude coefficient of $CPE$($F·s^{CPE1\_P-1}$). Comparing Equations (7) and (12) reveals a one-to-one correspondence between $CPE1\_T$ and $C_2$. Therefore, $CPE1\_T$ can be used as a substitute for $C_2$.

2.3. Calculation of Fractal Dimension for Contact Surfaces of Soil-Rock Agglomerates

Based on existing studies on EIS of geotechnical materials and Ohm’s law [27,28,29], the impedance of the discontinuous soil–rock–pore water solution channels—modeled as a double parallel-plate capacitor formed by discontinuous soil–rock pathways—can be expressed as shown in Equation (12). In this formulation, the fitting range of $CPE1\_P$ is 0 to 1. When $CPE1\_P$ approaches 0, the cemented soil–rock conduction path within SRM behaves as an ideal resistor. Conversely, when $CPE1\_P$ approaches 1, the cemented conduction path acts as an ideal parallel-plate capacitor. Therefore, $CPE1\_P$ can be employed as an impedance-based characteristic parameter to construct the fractal dimension describing the roughness of soil–rock cemented interfaces. According to published research on the fractal dimension of soil–rock contact surfaces [37,38], the corresponding calculation expression is given in Equation (13), where $D_f$ denotes the fractal dimension.

$$D_f=(1+CPE1\_P)/CPE1\_P$$

(13)

3. Results Analysis

3.1. Evolution Patterns of Characteristic Parameters in Equivalent Models of Different Conductive Pathways Under F–T Cycles

Figure 6 presents the influence of F–T cycles on the characteristic parameters of different conductive-path equivalent models of SRM. As shown in Figure 6a, when the number of cycles increases from 0 to 15, the value of $R_{CP}$ decreases from 386.0148 Ω to 224.4813 Ω, corresponding to a reduction of 41.85%. Moreover, $R_{CP}$ exhibits a clear exponential relationship with the number of cycles, with a fitting coefficient of 0.951. Figure 6b illustrates the evolution of $R_{CPP}$ under different F–T cycles. With increasing cycles, $R_{CPP}$ decreases from 12,120.1 Ω to 7837.8 Ω, representing a reduction of 64.67%. Similarly, an exponential relationship is observed, with a fitting coefficient of 0.9936. According to Figure 6c, as the number of F–T cycles increases from 0 to 15, $C_{DSPR}$ increases from 3.7005×10⁻¹⁰ F to 9.1189×10⁻⁹ F, showing a growth factor of 23.6. The relationship between $C_{DSPR}$ and F–T cycles also follows a clear exponential trend, with an excellent fitting coefficient of 0.9999. Figure 6d shows the effect of F–T cycles on the fractal dimension of the soil–rock cementation interface. As the number of cycles increases from 0 to 15, the fractal dimension rises from 2.0442 to 2.2702, corresponding to an increase of 11.06%. This parameter also exhibits an exponential relationship with the number of cycles, with a fitting coefficient of 0.9982.

3.2. Evolution of Uniaxial Mechanical Properties of SRM Under F–T Cycles

Figure 7a presents the stress–strain curves of SRM under varying numbers of F–T cycles. The curves exhibit four distinct stages: micropore compaction (OA), elastic deformation (AB), plastic deformation (BC), and post-peak softening (After point C). With an increase in cycles from 0 to 15, the compaction stage becomes more pronounced, while both peak stress and peak strain show contrasting trends, with peak stress decreasing and peak strain increasing. In addition, the elastic deformation stage becomes progressively less distinct. To facilitate the quantification of SRM deformability, the peak secant modulus was adopted as the representative parameter. Figure 7b illustrates the variation in UCS with increasing cycles. The UCS decreases from 0.2797 MPa at 0 cycles to 0.0753 MPa at 15 cycles, corresponding to a reduction of 373.08%. The relationship follows a clear exponential decay, with a fitting correlation coefficient of 0.9908. Figure 7c shows the evolution of the peak secant modulus under repeated F–T cycles. As the number of cycles increases from 0 to 15, the modulus decreases from 0.0051 GPa to 8.22 × 10⁻² GPa, representing an 83.83% reduction. The relationship also follows an exponential law, with an excellent fitting correlation coefficient of 0.9968. Figure 7d depicts the evolution of peak strain with increasing F–T cycles. The peak strain increases from 5.512% to 9.156% as the number of cycles increases from 0 to 15, representing an increase of 66.11%. This trend also follows an exponential relationship, with a correlation coefficient of 0.9997.

4. Discussion

4.1. Mechanism of F–T Cycle Effects on the Microstructure of SRM

Figure 8 illustrates the mechanism by which F–T cycles influence the internal microscopic structure of SRM. As shown in the schematic, when the experimental temperature drops below 0 °C, the pore water in the $CPP$ and $DSRPP$ pathways undergoes a liquid–ice phase transition. The volume expansion associated with this phase change generates frost heave forces acting on the soil–soil/soil–rock cementation structures (pore walls) along different conductive paths. When the frost heave force exceeds the peak cementation strength of the pore walls, the initial pores within the mixture expand and interconnect, transforming small-radius pores into larger ones and generating new pores. According to Equation (4), the expansion and interconnection of pores with different radii in $CPP$ increase the porosity (${\phi }_M/{\eta }_{SR}$), ultimately leading to a reduction in $R_{CPP}$ with increasing cycles. Similarly, as indicated by Equation (5), the pore expansion within $CP$ under F–T cycles increases its porosity, thereby decreasing $R_{CP}$ as the number of cycles increases. In contrast, $DSRPP$ contains both liquid-phase and gas-phase pores. When the frost heave force exceeds the cementation strength of the pore walls in $CP$, the expansion and interconnection of liquid-phase pores of various radii increase the soil–rock contact area ($S$). Furthermore, these expanded pores also connect to the surrounding gas-phase pores, promoting the transformation of $CSRPP$ into $DSRPP$ and thereby enhancing the dual parallel-plate capacitance effect. Consequently, based on Equation (5), when the number of F–T cycles increases from 0 to 15, both the enhanced capacitance effect and the enlarged soil–rock contact area contribute to an increase in $C_{DSRP}$.

As shown in Figure 6d, the fractal dimension (D_f) of the soil–rock cementation surface of $DSRP$ increases with the number of F–T cycles from 0 to 15. Combined with the mechanism illustrated in Figure 8, when the temperature falls below 0 °C, repeated freezing and thawing induce water–ice phase transitions, resulting in volumetric expansion. This expansion generates frost-heaving forces acting on the pore walls formed by soil–soil and soil–rock cementation within $CP$. Once the frost-heaving force exceeds the ultimate bonding strength of the pore walls, liquid-phase pores of different radii expand and interconnect, thereby increasing the surface roughness of the soil–soil and soil–rock contact interfaces. Consequently, D_f of the soil–rock cementation surface exhibits a progressive increase with the accumulation of F–T cycles.

4.2. Predicting Model for Freeze–Thaw UCS of SRM

Previous studies have demonstrated that the microscopic structural parameters of porous geo-materials were closely related to their macroscopic mechanical properties [5,6,7,8]. To establish the intrinsic relationship between the microscopic structural parameters and the macroscopic strength of SRM, this study employed the characteristic parameters of different conductive paths in EIS equivalent model as influencing factors to construct a predictive model for UCS of SRM under F–T cycles. Figure 9a–c present the correlation coefficients between the characteristic parameters of different equivalent model paths and UCS. The results indicate that R_CP and R_CPP are positively correlated with UCS, whereas C_DSRP exhibits a negative correlation. Among these parameters, C_DSRP shows the strongest correlation with UCS, with a correlation coefficient of 0.9588. Figure 9d shows the correlation between the fractal dimension of the soil–rock cementation surfaces and UCS. Analysis reveals a negative correlation between D_f and UCS, with a correlation coefficient of 0.8476.

Based on the above analysis, among the equivalent model characteristic parameters, C_DSRP exhibits the strongest correlation with UCS of SRM. Therefore, this study selected C_DSRP and D_f as influencing factors to construct a predictive model for the UCS of SRM under F–T cycles.

Table 2. Calculation results of UCS regression model of influencing factors for C_DSRP under F–T cycles.

Regression Parameters	Regression Coefficient	p	F	R2
$c_1$	0.260942484	8.86 × 10−11 (**)	8.74689 × 10−7 (**)	0.9193
CDSPR	−20,825,862.1	8.75 × 10−7 (**)

(**): the regression model is statistically significant at the 95% confidence level.

and Figure 10a present the linear regression model results for UCS with C_DSRP. The model is expressed by Equation (14), where $\alpha$ and $c_1$ represent the coefficient of the predictor and the model intercept, respectively. The results indicate that both the regression coefficient and intercept are significant ($p$< 0.05), and the model itself demonstrates statistical significance with a correlation coefficient of 0.9193. Similarly,

Table 3. Calculation results of UCS regression model of influencing factors for D_f under F–T cycles.

Regression Parameters	Regression Coefficient	p	F	R2
$c_2$	1.520750596	0.000398 (**)	0.001047 (**)	0.6749
$D_f$	−0.61370175	0.001047 (**)

(**): the regression model is statistically significant at the 95% confidence level.

and Figure 10b show the linear regression results for UCS with D_f, expressed by Equation (15), where $\beta$ and $c_2$ are the regression coefficient and intercept. The regression results indicate significant influence ($p$< 0.05), and the model correlation coefficient is also 0.9193. To account for the combined influence of C_DSRP and D_f on UCS, a multiple linear regression model was developed. Figure 10c shows the correlation coefficient between C_DSRP and D_f, which is 0.6789 (<0.7), indicating that multicollinearity can be neglected.

Table 4. Calculation results of UCS regression model of influencing factors for C_DSRP and D_f under F–T cycles.

Regression Parameters	Regression Coefficient	p	F	R2
$c_3$	0.75759538	0.000112 (**)	8.34657 × 10−8 (**)	0.9733
CDSPR	−16,158,088	3.51 × 10−6 (**)
$D_f$	−0.2364532	0.002097 (**)

(**): the regression model is statistically significant at the 95% confidence level.

and Figure 10d present the regression results for UCS considering both C_DSRP and D_f, expressed by Equation (16), where ${\alpha }_1$ and ${\beta }_1$ are the regression coefficients and $c_3$ is the intercept. Both coefficients and the model intercept are significant ($p$< 0.05), and the model exhibits high correlation ($R^2$= 0.9733). This demonstrates that the predictive capability of the model considering both C_DSRP and D_f is superior to that of models based on single predictors. Furthermore, both C_DSRP and D_f are negatively correlated with UCS under F–T cycles, leading to the final predictive model expression given in Equation (17).

Figure 11 illustrates the relationship between C_DSRP, D_f, and the mechanical properties of SRM. Combining this schematic with the above regression analysis reveals that increases in C_DSRP and D_f due to F–T cycles degrade the UCS of SRM. When the number of cycles increases from 0 to 15, repeated freezing and thawing cause the expansion and interconnection of liquid-phase pores of different radii within $DSRPP$, as well as an increase in the soil–rock cementation contact area and enhancement of the double-layer capacitance effect, resulting in an increase in C_DSRP.

During the process by which F–T cycles affect the fractal dimension of soil–rock mixtures, the pore water solution inside the mixture begins to undergo a water–ice phase transition when the experimental temperature falls below 0 °C. The volumetric expansion caused by this phase transition exerts frost heaving forces on the soil–rock interface of DSPR. When these forces exceed the ultimate bearing capacity of the interface, expansion and penetration occur. Additionally, pore expansion and interconnection along the conductive paths increase the roughness of the soil–rock contact surfaces, thereby promoting the growth of the fractal dimension. At the same time, the expansion of the soil–rock interface damages the initial microscopic structure of the mixture and weakens its load-bearing strength. Consequently, the uniaxial compressive strength decreases as the fractal dimension increases.

In summary, the simultaneous increases in C_DSRP and D_f disrupt the initial microscopic structure of SRM, reducing its load-bearing capacity. Consequently, the UCS of SRM exhibits a decreasing trend with increasing C_DSRP and D_f.

$${\sigma }_c=\alpha \times C_{DSPR}+c_1$$

(14)

$${\sigma }_c=\beta \times D_f+c_2$$

(15)

$${\sigma }_c={\alpha }_1\times C_{DSPR}+{\beta }_1\times D_f+c_3$$

(16)

$${\sigma }_c=-16158088\times C_{DSPR}-0.2364532\times D_f+0.75759538$$

(17)

5. Conclusions

The study conducted F–T cycle tests, EIS measurements, and uniaxial compression tests on SRM. Equivalent models of different conductive pathways were established to analyze the response characteristics of macroscopic structures and macroscopic mechanical parameters under F–T cycles. Furthermore, the mechanisms governing the effects of F–T action on the characteristic parameters of the equivalent models, fractal dimension, and mechanical parameters were investigated. A prediction model for UCS under F–T cycles was developed by incorporating C_DSRP and D_f as influencing factors, thereby revealing the intrinsic link between meso-structural evolution and macroscopic mechanical properties. The findings provided a basis and reference for subsequent analyses of the relationship between the microscopic structure and macroscopic mechanical parameters of soil–rock mixtures under freeze–thaw cycles, as well as for the study and prevention of freeze–thaw disasters in geotechnical engineering involving soil–rock mixtures in cold regions. The main conclusions are as follows:

(1) With an increasing number of F–T cycles, both $R_{CP}$ and $R_{CPP}$ of SRM decreased, whereas $C_{DSPR}$ and $D_f$ exhibited an increasing trend. In addition, various microscopic parameters showed a clear exponential relationship with the number of cycles.

(2) The stress–strain curve of SRM comprised four stages: micropore compaction, elastic deformation, plastic deformation, and post-peak behavior. As the number of F–T cycles increased, the compaction stage became more pronounced, while the elastic stage gradually diminished. Meanwhile, peak strength and peak secant modulus decreased with F–T cycles, whereas peak strain increased. Moreover, different macroscopic mechanical parameters displayed significant exponential correlations with the number of cycles.

(3) Repeated freezing and thawing induced water–ice phase transitions in the pore solutions of $CPP$ and $CP$. The associated volumetric expansion caused initial pores to expand and interconnect, transforming small-radius pores into larger ones and creating new voids. This porosity increase resulted in decreasing $R_{CPP}$ and $R_{CP}$. Furthermore, pore expansion and interconnection enlarged the soil–rock contact surface area. Expanded pores also interconnected with nearby gaseous pores, promoting the transformation of $CSRPP$ into $DSRPP$, which enhanced the double-parallel-plate capacitance effect, thereby increasing $DSRPP$. In addition, the development and interconnection of pores of varying radii roughened soil–soil/soil–rock interfaces, leading to an increase in $D_f$ with F–T cycles.

(4) The predictive model of UCS incorporating both $C_{DSRP}$ and $D_f$ demonstrated higher correlation and accuracy than models based on single factors, enabling reliable prediction of SRM UCS after different numbers of F–T cycles.

(5) The increase in $C_{DSRP}$ and D_f caused by F–T cycles exerted a deteriorative effect on UCS of SRM. Their growth led to microscopic structural damage and weakened load-bearing capacity, thereby causing UCS to decrease with increasing $C_{DSRP}$ and D_f.