Experimental Study on the Fracture and Failure of the Locking Section of Rock Slopes Caused by Freeze–Thaw of Fracture Water

Abstract

In rock slopes with a three-section landslide, the locking section is the key control factor. This study conducted double-sided freeze–thaw tests on a scale model of a rock slope with a three-section landslide in a cold region. We monitored the changes in frost heave force, strain, and fracture during the water–ice phase change and investigated the effect of the trailing edge tensile crack length on the frost heave fracture of the locking section. A crack frost heave model was proposed based on rock and fracture mechanics to explore the mechanism of slope crack freeze–thaw weathering. According to the results, the slope shoulder froze first, with the freezing front progressing from the slope shoulder to the interior of the rock mass. The fracture failure in the three-section rock slopes was mostly caused by the frost heave of the trailing-edge tensile cracks. The largest frost heave force and locking section deformation occurred when the temperature of the top of the trailing edge tensile crack decreased from −3.5 °C to −6 °C (whereas that of the bottom of the crack dropped from 0 °C to −2.6 °C). Additionally, the results demonstrate that the frost heave force is positively correlated with the length of the trailing edge tension crack, and shear marks are virtually absent on the tensile fracture surface.

1. Introduction

A three-section landslide in a rock slope describes the deformation and failure of a slope through sliding, tensile cracking, and shear. Specifically, the lower part of the slope creeps along the outer structural plane of the gentle slope, causing tensile cracks in the trailing edge and the shearing of the locking section in the middle slope. The geological structure of a rock slope with a three-section landslide is characterized by severe tensile fractures at its trailing edge and a structural plane outside the gentle slope at its leading edge. The locking section refers to the area between the severe tensile fractures at the trailing edge and the structural plane of the gentle slope at the leading edge [1]. This type of landslide is widely distributed in China, and its stability is mainly determined by the rock strength of the locking section and the geological structure characteristics [2]. The tensile crack and frost heave fracture at the trailing edge accelerate three-section landslides in rock slopes in high-cold regions, so it is critical to study the fracture mechanism of the locking section under freeze–thaw action.

Scholars have studied the damage of fractured rock masses under combined freeze–thaw and load actions [3]. Murton [4] and Draebing [5] believed that the moisture and heat migration inside the fractured rock mass is primarily responsible for its frost heave cracking, and freeze–thaw damage must be explored at multiple scales. Dwivedi et al. [6] observed that pore ice has a substantial impact on the deformation and failure of frozen rock samples. Akagawa [7] studied the development rate of ice crystals [8] and freezing fronts. Walder et al. [9] proposed a migration rule for unfrozen water and frozen edges in frozen rocks. Huang et al. [10] proposed a mechanical model to study the frost heave strain of cracks and investigated the effects of temperature, pore water, and pore ice pressure on frost heave strain [11]. Shi et al. [12] studied the mechanical properties of rock samples with cracks of different lengths and observed that their strengths were significantly lower than those of intact rock samples. Momeni [13] conducted 300 freeze–thaw tests on hard rocks, and the number of freeze–thaw cycles was positively correlated with porosity and water absorption but negatively correlated with dry density, longitudinal wave (P-wave) velocity, and compressive and tensile strength [14].

Research has been conducted on the damage and instability behaviors of rock slopes in cold regions. Most large-scale rock mass failures originate from cracks or joints. Guglielmi [15] employed a constitutive model to predict the failure of the rock slope locking section. In cold regions, rocks freeze when the temperature drops, and their strength and resistance to deformation increase [16,17,18,19]. Li [20] conducted model experiments on joint slopes to examine their failure mechanism by monitoring temperature, deformation, and frost heave pressure in real time. Landslide risks are most likely to occur on slopes in cold regions during ice crystal melting [21]. Girard [22] employed acoustic emission testing to measure in situ frost heave damage on steep slopes, demonstrating that stress, seepage, temperature, and chemical fields all impact slope stability [23]. Additionally, the large temperature difference in cold regions has a marked impact on the stability of rock slopes. Repeated freezing and thawing of pore water leads to fatigue damage (by generating microcracks) and a decrease in the rock strength [24]. The ice and snow on the surface of the rock mass melt under sunlight and seep into the cracks, and a decrease in temperature at night freezes the water in the cracks. Cracks extend and expand [25] as a result of repeated frost heave caused by freeze–thaw due to day and night cycles, leading to the shortening of the locking section and inducing slope instability [26]. Phillips [27] conducted long-term tracking of the rock avalanches in the Alps and determined that the fracture surface was covered in ice. The repeated frost heave effect of rock fractures and the structural characteristics of the rock mass are both important factors in slope collapse. Zhou et al. [28] conducted on-site research and physical experiments on the occurrence mechanism of a large rock slope in Yigong, Tibet. Their results revealed that the controlled sliding surface of the landslide was fully saturated with glacier melting water and rainfall, and the freeze–thaw effect accelerated the occurrence of disasters.

Cracks in the locking section of slopes are vital in freeze–thaw damage to rock masses in cold regions. Therefore, investigating the deterioration mechanism of crack frost heaves on the locking section is vital. However, research on the fracture failure of rock slopes with three-section landslides in cold regions is still in its infancy, and certain fundamental issues remain unknown. For instance, as one of the main factors determining slope stability, the impact of tensile cracks at the trailing edge on the crack propagation morphology under frost heave remains unclear, as does the impact of the trailing edge tensile crack length on slope fracture. Accordingly, the potential mechanisms should be further investigated. To perform freeze–thaw testing on slope models in the lab, high-performance equipment is needed. In this study, freeze–thaw testing was conducted on a scale model of rock slopes with three-section landslides using a two-way freezing box. This study considered the impacts of temperature and trailing edge tensile crack length on the fracture process of the locking section and established a crack frost heave model by correlating frost heave force, deformation, and the freeze–thaw process. Moreover, the freeze–thaw damage process of the rock slope with three-section landslides was discussed in detail. The findings can guide further studies on the stability of rock slopes with three-section landslides in cold regions.

2. Test Content

2.1. Sample Preparation

After an on-site investigation, this study took a rock slope with a three-section landslide in the Mila Mountain area as the case study. The generalized model of the slope is shown in Figure 1. The gray sandstone formations on the 20 m-high slopes are all carved to an angle of 80°. Based on geometric similarity [29], the model was made at a scale of 1:50. A scale model of rock slopes with a three-section landslide was made using boulders of gray sandstone gathered from the base of the slope, as shown in Figure 2. The trailing edge tension crack length of the model is $L_1$. The angle (${\alpha }_3$) between the leading edge shear crack ($L_3$) and the horizontal line is 20°, and the locking sections of each model are equal in length. The gray sandstone is rather firmly cemented, with different particle sizes and forms, as shown in Figure 2c. The quartz particle content is relatively high, whereas the clay mineral content is very low. Some particles have poor bonding, and natural microcracks of variable lengths exist (

Table 1. Average physical parameters of the sandstone.

Dry Density (g/cm3)	Saturation Density (g/cm3)	Saturated Water Capacity (%)	Porosity (%)	Longitudinal-Wave Velocity (m/s)
2.52	2.58	2.279	5.7429786	3020–3080

).

2.2. Experimental Design

Water and temperature are important factors in freeze–thaw weathering. Rain, snow, and water cannot accumulate on the steep slope after manual excavation. The saturation is relatively high at approximately 50 cm below the surface. Rain, snow, and water easily fill the tensile cracks at the trailing edge, resulting in frost heave. The freeze–thaw damage of the leading edge shear fracture is not considered in this study because groundwater is scarce at the gentle slope of the leading edge, where rainwater and snow water hardly penetrate and are weakly affected by temperature. Due to rainfall and snowfall, the top and surface of the slope were saturated with water. The areas that can reach critical saturation are shown in Figure 3, and Figure 4 depicts the areas affected by freeze–thaw. Based on the temperature conditions of Mount Mila, the experimental temperature was set between 20 °C and 20 °C to simulate a natural freeze–thaw environment using a self-made double-sided (top and surface) freeze–thaw device. The specific test plan is detailed in

Table 2. Freeze–thaw test plan.

Model No.	Trailing Edge Tension Crack Length/mm	Trailing Edge Tension Crack Width/mm	Test Content
L	40	13	temperature, frost heave force, crack propagation morphology
M-5	60	13
N	80	13
K-2	intact sample

.

2.3. Test Scheme

The procedures for the double-sided freeze–thaw test are as follows:

(1)

The side facade of the model crack is sealed with waterproof adhesive. Figure 5 depicts the installation of the sensors.

(2)

Saturate the slope top and surface (40 mm) of the model with water.

(3)

Paste strain gauges in the locking section area, fix the membrane pressure sensor (for testing frost heave force) and temperature sensor, and then connect the acquisition system.

(4)

Fill the trailing edge tensile crack with rainwater only to freeze and thaw.

(5)

Wrap the model in cling film and leave it to set for 24 h.

(6)

Conduct tests on slope models using a self-made double-sided (top and surface) freeze–thaw device (Figure 6). The coolant flows into the aluminum box through a refrigeration circulator. Wrap the aluminum box and model with 15 cm-thick thermal insulation cotton for subsequent heat preservation.

(7)

Based on the relatively high cooling rate observed during on-site monitoring, the experiment adopted a cooling rate of 10 °C per hour. Reduce the temperature from 20 °C to −20 °C at a rate of 10 °C/h. Automatically freeze the model for 10 h before allowing it to thaw naturally.

(8)

Conduct crack propagation measurement and record.

(9)

Repeat freeze–thaw cycles until fracture failure occurs.

(10)

Export the temperature, frost heave force, and strain results of the entire freeze–thaw process and statistically analyze the measured data.

The frost heave force does not need to be measured in the complete slope model, and the other steps are the same as those above.

3. Results of Freeze–Thaw Test on Intact Rock Slopes

To acquire strain gauge deformation results at 60 mm below the slope top, a complete gray sandstone slope model was tested under double-sided freeze–thaw (from −20 °C to 20 °C). Figure 7 displays the maximum values of chilling shrink and frost heave deformation for the intact sample K-2. The maximum value of frost heave no longer exceeded 70 με, whereas the maximum value of chilling shrink was no larger than 50 με. Noticeably, the sample was very slightly deformed after 16 freeze–thaw cycles. According to literature analysis, intact rock slopes can cause rock degradation and geological disasters during long-term freeze–thaw cycles (including annual and century cycles) [30,31,32].

4. Freeze–Thaw Test Results for the Locking Section of the Rock Slope with Three-Section Landslide

4.1. Effect of Freeze–Thaw Cycles on the Damage of the Locking Section

This study conducted double-sided freeze–thaw tests to monitor the deformation of rock slopes with a three-section landslide and analyze the stability of the dangerous rock mass. The most vulnerable region to freeze–thaw failure is the exposed rock cracks. Abrupt changes in the deformation curve of the locking section and the connection of tensile cracks at the trailing edge of the rock mass indicate that a landslide is going to occur. Model M-5 was utilized to analyze the freeze–thaw fracture process of rock slopes with a three-section landslide, and the evolution of freeze–thaw damage in the locking section is illustrated in Figure 8. A temperature drop causes changes in the water phase, resulting in wedge-shaped ice and local stress inside the tensile cracks at the trailing edge. Some increases in local tension trigger fracture propagation at the crack tip and stress relaxation.

In early stage I, the temperature dropped, but the deformation remained constant. In late stage I, the temperature in sensor 8 continued to drop from 6.5 °C, while strain gauges 1, 2, and 3 had variable degrees of chilling shrink deformation (with negative strain values). The highest chilling shrink deformation among them was observed in strain gauge 1, whereas the smallest was in strain gauge 3. This is due to the thermal shrinking of the rock matrix, which reduces the strain of the gray sandstone locking section as the temperature decreases. Fracture water remained liquid at stage I.

In early stage II, the open end of the crack was frozen and sealed, and crack water gradually froze from the top down, corresponding to temperature variations. The frost heave force was rather minimal at this time, with the strain gauges 1, 2, and 3 progressively shifting from chilling shrink to frost heave state. Frost heave occurred initially in gauge 1 and finally in gauge 3. In the early stages of freezing, the adhesive force was relatively weak between the ice wedge at the top of the crack and the crack wall, which was not tightly restricted. The frost heave force squeezed the ice wedge upward, causing a decrease in the frost heave force.

In early stage III, the temperature in sensor 8 dropped from 0 °C to −6 °C, and all of the crack water gradually turned into ice wedges. The volume of ice increased while the water in the crack rapidly shrank. Unfrozen water migrated toward the ice wedge and froze. As the ice wedge expanded, the frost heave force rapidly increased. However, the frost heave force decreased due to the compression of ice wedges and the opening of fractures, partially reducing the frost heave force. During the first freeze–thaw cycle, strain gauge 1 readings significantly increased, strain gauge 2 readings sharply increased, and strain gauge 3 readings indicated minimal deformation. Moreover, only a moderate change in the lengthening and a small increase in the openness of microcracks occurred after the first cycle. The frost heave deformation of strain gauges 1, 2, and 3 gradually increased with the number of freeze–thaw cycles. Strain gauge 1 showed the greatest rise in frost heave deformation, followed by strain gauge 2. Due to the frost heave of ice wedges during the eighth freeze–thaw cycle, strain gauges 1, 2, and 3 were damaged sequentially within 12 min. The reading in temperature sensor 10 dropped from 0 °C to −2.6 °C, whereas that in temperature sensor 8 dropped from −3.5 °C to −6 °C. At this stage, frost heave damage was the most significant, which is in line with earlier research findings (Table 3). The temperature range of effective freezing damage may vary slightly due to rock properties.

In later stage III, temperature reading in sensor 8 continued to decrease from −6 °C, with a small amount of pore water in the side wall of the crack migrating toward the ice wedge. As the number of freeze–thaw cycles increased, the deformation reading of strain gauge 1 did not increase, whereas that of strain gauges 2 and 3 gradually increased under the cyclic frost heave effect. The locking section underwent frost heave deformation under the action of ice wedges, and the frost heave force gradually decreased after deformation. In the early stage of crack water freezing, the initial freezing at the open end of the crack served as a seal, and the ice wedge at the top of the crack was squeezed upward under the piston action of the frost heave force. The fracture walls of gray sandstone have weak permeability, making the interfacial water film between ice and rock the only pipeline leading to the surface. The water film has high hydraulic impedance due to its thin and special structure [33]. Because the bonding force between the crack wall and the ice wedge is higher than the strength of the ice wedge, the ice wedge at the top of the crack is pulled apart from the middle and squeezed out slightly toward the upper part, as shown in Figure 9a. After the 8th frost heave fracture of the locking section (after the frost heave fracture of the strain gauge), the frost heave force did not immediately dissipate, but gradually decreased.

Table 3. Temperature range for effective freezing damage [9,34,35,36].

Lithology	Porosity (%)	$\mathbf{Upper} \mathbf{Limit} {\mathit{T}}_{\mathbf{H}}$ (°C)	$\mathbf{Lower} \mathbf{Limit} {\mathit{T}}_{\mathbf{L}}$ (°C)	Indicators	Water Supply	References
Low porosity (<5%)
Granite and marble	1–3	−4	−15	theory	open	Walder [9]
Medium porosity (5–20%)
Berea sandstone	20	−3	−6	length change	open	Hallet [34]
High porosity (>20%)
Ohya tuff	38	−1.4	−5	ice crystal formation	open	Akagawa [35]
Brézé chalk	47	−0.2	−2	length change	open	Murton [36]

Table 3. Temperature range for effective freezing damage [9,34,35,36].

Lithology	Porosity (%)	$\mathbf{Upper} \mathbf{Limit} {\mathit{T}}_{\mathbf{H}}$ (°C)	$\mathbf{Lower} \mathbf{Limit} {\mathit{T}}_{\mathbf{L}}$ (°C)	Indicators	Water Supply	References
Low porosity (<5%)
Granite and marble	1–3	−4	−15	theory	open	Walder [9]
Medium porosity (5–20%)
Berea sandstone	20	−3	−6	length change	open	Hallet [34]
High porosity (>20%)
Ohya tuff	38	−1.4	−5	ice crystal formation	open	Akagawa [35]
Brézé chalk	47	−0.2	−2	length change	open	Murton [36]

In stage IV, the top of the crack melts first when the wedge-shaped ice begins to heat up from the top and bottom of the crack (Figure 9b). Meltwater migrates to the ice wedge in the middle of the crack and freezes again, resulting in a modest increase in frost heave force. As the temperature rises, the amount of meltwater steadily increases, the frost heave force falls to zero, and the deformation in the strain gauges gradually diminishes. The frost heave effect vanishes at the end of this stage. This suggests that the temperature of crack water has an impact on the frost heave force, and the frost heave force (wedge-shaped ice) leads to the opening and deformation of cracks.

In stage V, with the accelerating melting rate of ice, the amount of liquid water increased as the volume of the ice wedge shrinks. The ice wedge is frozen in a filamentous shape from the top of the crack down as it gradually separates from the crack (Figure 9c).

In stage VI, until the crack ice was entirely melted, some floating ice remained at the top of the crack. Figure 9d shows that many bubbles are present in the upper ice wedge, but the lower portion has few bubbles. A reasonable explanation is that bubbles in the crack water rise to the surface before freezing and will be squeezed during the freezing process of the ice wedge, causing some stress relaxation.

No evident surface peeling or slag drop on the model occurred after eight freeze–thaw cycles. The freeze–thaw failure rate of the trailing edge tensile crack exceeds the freeze–thaw spalling of particles on the slope surface on site. Frost heave force is a regulating factor in the freeze–thaw failure of rock slopes with a three-section landslide, as well as a major factor in the deformation fracture of the locking section.

4.2. Impact of Crack Depth on Frost Heave Damage

The freeze–thaw damage process of the trailing edge tensile fractures in a “three-stage” slope is related to the temperature variation trend of the fracture water. The key variations are as follows: the volumes of crack water and ice wedges generated increase with the depth of the tensile crack at the trailing edge. The migration of unfrozen water toward the ice wedge is more significant the greater the temperature difference between the top and bottom of the fracture. Figure 10 demonstrates the maximum frost heave force of a three-section slope model with different crack depths. Noticeably, the frost heave force increases with the depth of the crack. The maximum frost heave force increases by 233 N as the tensile crack at the trailing edge increases from 40 mm to 60 mm. In contrast, the maximum frost heave force only increases by 107 N as the tensile crack at the trailing edge increased from 60 mm to 80 mm. The tensile cracks at the 40 mm, 60 mm, and 80 mm trailing edges were fractured and destroyed in the locking section after 14, 8, and 4 freeze–thaw cycles, respectively. Noticeably, the fracture failure of the locking section accelerates as the tensile crack length increases at the trailing edge of the slope.

The freeze–thaw failure morphology of the three-section slope model with different crack depths is shown in Figure 11. Figure 12 depicts the propagation of cracks from the fracture at the bottom of the cracks. The fracture cracks extend downward along the direction of tensile cracks at the trailing edge until they intersect with shear cracks at the leading edge. The flatness and lack of shear marks on the fracture surface can be attributed to the tensile failure induced by frost heave. The direction of tensile cracks at the trailing edge is consistent with that of frost heave cracks based on the analysis of the on-site research findings of rock slopes with a three-section landslide. The larger the length and width of tensile cracks at the trailing edge, the stronger the water catchment and conductivity of the cracks, and the more severe the impact of freeze–thaw weathering.

Microscopic imaging was performed on the fracture cracks of Model M-5 after eight freeze–thaw failures, as shown in Figure 13. The crack in sections ① and ② is the widest, ranging from 0.17 mm to 0.23 mm. A 30-fold enlargement of the two cracks in section ① reveals that the main crack has a large local curvature, accompanied by small dendritic cracks (0.03 mm) on both sides. The crack morphology mainly comprises an intergranular fracture and rarely a transgranular fracture, which is consistent with previously described tensile failure characteristics (tensile failure mainly manifests as an intergranular fracture). The crack width in section ③ is similar to that in ① and ②, and that in section ④ is the smallest. As the number of freeze–thaw cycles increases, the deterioration length of the trailing edge tensile crack increases and the effective length of the locking section decreases, as does the integrity and stability of the slope.

4.3. Theoretical Analysis of Frost Heave Deformation in Fractured Rock Slopes

Using Model M-5 as an example, this study established a theoretical model under double-sided freeze–thaw conditions. The tensile crack at the trailing edge of the model is 13 mm ($W_1$) wide, 150 mm long, and 60 mm ($L_1$) deep. The maximum frost heave force measured is 9.96 MPa. Temperature affects the status of crack water, and the extrusion output of ice during the freezing process is 2.5% (1.5 mm). According to the experimental results, the downhill propagation speed of the ice front along the crack is positively connected with the freezing time.

Figure 14 depicts a phase transition squeezing model of the fracture water. The initial water level is only 2 mm below the top, and freezing has developed to $vw$, with a distance of $l$ from the opening end of the crack. Considering the freezing front moves to $v^{\prime }{w}^{\prime }$, the pressure in the water increases when water freezes and expands, forcing the crack walls to open. As shown in Figure 14a, the ice seal will only consist of contacts above $v$ and $w$. In fact, ice above $vw$ is under elastic pressure (water pressure) and tension (ice pressure). When the crack wall is forced to open by increased water pressure and ice pressure, the compression of $vw$ is slightly relaxed, but it does not break, thus maintaining the seal. The opening in the middle of the crack ice above $tu$ can be attributed to the same reason. The distance that the freezing front advances from $vw$ to $v^{\prime }{w}^{\prime }$ is denoted as $\mathrm{d}s$.

For simplicity, the bottom of the crack is considered flat and parallel to $vw$, and the crack wall is bent inward by an angle $\alpha$, as illustrated in Figure 14b. The squeezing of the ice jam is thus ignored at this point, and the angle $\alpha$ is calculated based on mass conservation. The ice mass in length $\mathrm{d}s$ plus the water mass in the expansion volume below $vw$ equals the water mass in length $\mathrm{d}s$. For the unit length perpendicular to the graph, which is ${\rho }_w\alpha (L_1-l)\mathrm{d}s$, the following equation can be obtained:

$${\rho }_{\mathrm{w}}{W}_1\mathrm{d}s={\rho }_{\mathrm{i}}{W}_1\mathrm{d}s+{\rho }_{\mathrm{w}}\alpha (L_1-l)\mathrm{d}s$$

(1)

where ${\rho }_{\mathrm{w}}$ and ${\rho }_{\mathrm{i}}$ refer to the density of water and ice, respectively, and $W_1$ is the length of $vw$. Therefore, the following can be obtained:

$$\alpha =(1-{\displaystyle \frac{{\rho }_{\mathrm{i}}}{{\rho }_{\mathrm{w}}}}){\displaystyle \frac{W_1}{L_1-l}}$$

(2)

Substituting $W_1$ = 13 mm, $L_1-l$, and ${\rho }_{\mathrm{i}}/{\rho }_{\mathrm{w}}$ = 0.91 into Equation (2), $\alpha$ can be identified as being very small. Therefore, the downward-moving freezing front forms inclined, wedge-shaped ice (to house the volume expansion generated by freezing). As $L_1-l$ decreases to 0, the required $\alpha$ becomes very large, indicating cracking at the bottom of the cracks.

It is assumed that the crack wall below $vw$ is allowed to somewhat bend and that the acute angles at $v$ and $w$ are reduced to make the model more realistic. $\alpha \mathrm{d}s$ is the average opening of cracks below $vw$. Ice jams are allowed to squeeze, as 2.5% of the ice was extruded from the cracks during freezing, which is insufficient (9%) to explain the general volume change. Assume the ice jam advances upwards by $f\mathrm{d}s$ as the freezing front moves downwards (Figure 14a), where $f$ is a fraction between 0 (no compression) and 0.09 (maximum compression). Equation (2) for calculating $\alpha$ can be further described as

$$\alpha ={\displaystyle \frac{k{W}_1}{L_1-l}}$$

(3)

where the expansion fraction after extrusion can be calculated using the formula $k=({\rho }_{\mathrm{w}}-{\rho }_{\mathrm{i}}-f{\rho }_{\mathrm{i}})/{\rho }_{\mathrm{w}}$; if $f$ = 0 (no compression), $k$ = 0.09; if $f$ = 0.09 (maximum compression), $k$ ≈ 0.

Specifically, when allowing for compression deformation in the model, it is necessary to consider that the crack walls above $vw$ are not parallel. Therefore, they should be bent inward at a certain angle, as shown in Figure 14c. When the freezing front moves from $vw$ to $v^{\prime }{w}^{\prime }$, the ice particles at $v$ and $w$ move upward through $f\mathrm{d}s$ and then outward to $v^{″}$ and $w^{″}$ through $\mathrm{d}a$. Therefore, the opening of $2\mathrm{d}a$ is the result of the wedging effect when the conical ice jam is pressed upward. When the ice wedge at position $vw$ moves upward along the crack, the pressure on it decreases. Therefore, it undergoes elastic relaxation strain $\mathrm{d}\epsilon$ (expansion), which can be obtained from the following equation:

$$W_1\mathrm{d}\epsilon =2\mathrm{d}a-2f\beta \mathrm{d}s$$

(4)

The normal pressure on both sides of the ice jam decreases from $v$, and the hydrostatic pressure in the water is zero from $v$ to the top of the crack, as illustrated in Figure 14a). As the ice water boundary advances a distance of $\mathrm{d}s$, and the zero-pressure line advances from $tu$ to $t^{\prime }{u}^{\prime }$. $t{t}^{\prime }$ and $v{v}^{\prime }$ are 10 s of times larger than $v{v}^{″}$. Therefore, when evaluating the pressure reduction on the ice wedge $vw$ in the later stage of the freezing process, the movement can be ignored.

Because both water pressure and the length of the ice–rock interface ($tv$) increase approximately linearly with crack ice volume [37], the pressure above $vw$ decreases to $p(\mathrm{d}{s}^{\prime }/c)$, where $\mathrm{d}{s}^{\prime }=t{t}^{\prime }$ and $c=tv$. The corresponding strain is as follows:

$$\mathrm{d}\epsilon ={\displaystyle \frac{p\mathrm{d}{s}^{\prime }}{Ec}}$$

(5)

where $E$ is the elastic modulus of ice. The ratio of the first and third terms in Equation (4) can be represented by Equation (6):

$${\displaystyle \frac{W_1\mathrm{d}\epsilon }{2f\beta \mathrm{d}s}}={\displaystyle \frac{W_{1}p}{2Ecf\beta }}\cdot {\displaystyle \frac{\mathrm{d}{s}^{\prime }}{\mathrm{d}s}}$$

(6)

According to the experimental results, $p$ = 9.96 MPa, $\mathrm{d}{s}^{\prime }/\mathrm{d}s$ = 0.55, $c$ = 60 mm, $f$ = 0.025, $E$ = 6 × ${10}^3$ MPa, and Equation (6) yields 2.6 × ${10}^{-3}$/$\beta$. Therefore, if $\beta$ is much greater than 2.6 × ${10}^{-3}$, the ratio is very small. This corresponds to a small angle in the length of the crack, whereas the angle caused by artificial cracks is much larger. Therefore, the third term in Equation (4) will definitely be much larger than the first term due to the elasticity of ice. Equation (4) can be further written as:

$$\mathrm{d}a=f\beta \mathrm{d}s$$

(7)

Therefore, ice wedges in the late stage of freezing are considered solid bodies. $\alpha$ can be calculated using Equation (3), and the ratio of the crack opening above $vw$ (2$\mathrm{d}a$) to the average opening below ($\alpha \mathrm{d}s$) is as follows:

$${\displaystyle \frac{2\mathrm{d}a}{\alpha \mathrm{d}s}}={\displaystyle \frac{2f\beta (L_1-l)}{k{W}_1}}=0.416\beta$$

(8)

The ratio is relatively modest because $\beta$ is much smaller than 1/0.416 in the experiment. This suggests that the crack opening happens primarily in the unfrozen water area before the crack water is fully frozen, which is consistent with the experimental findings.

Theoretical insights suggest that the simplified model in Figure 14 (including some squeezing and related wedging effects, as well as the smoothing of the edges and corners of the crack wall) can accurately represent the experimental crack opening process. The results indicate that the crack ice above $tu$ opens from the middle. This is mainly because water pressure pushes the ice wedge upward in the early stage of freezing, but the ice wedge adheres more tightly to the crack wall, forcing the crack ice to widen and fracture in the middle.

Whether to ice or to the interface, the plastic friction law $\tau$ = $k_0$ (constant) can be applied, where $\tau$ refers to the average shear traction force on the wall. As the freezing depth increases, the ice wedge undergoes plastic deformation. The water pressure on the ice wedge (upwards) is balanced with the friction force from the wall (downward), which can be described using the following formula:

$$p{W}_1=2k_{0}c,p={\displaystyle \frac{2k_{0}c}{W_1}}$$

(9)

where $c$ is generally proportional to the quantity of crack water that freezes, with which the static water pressure grows roughly linearly, as does $p$. The amount of crack propagation is nearly proportional to the depth of frost heave penetration [38]. Therefore, the experimental results are extremely consistent with the plastic friction law ($p$ increases almost linearly). The obtained value of $k_0$ is 1.079 MPa. Although the pressure is substantially lower than the expected expansion volume, ice wedges nonetheless form near ice fronts in water-filled fractures at temperatures below 0 °C [39].

The high roughness of on-site cracks increases the contact area between ice and rock mass. The ice wedge only requires a small amount of frozen ice to maintain a water seal due to the geometry of the crack. Water pressure and ice wedges both influence how cracks expand as freezing progresses. The crack propagation morphology has two states: (1) the ice plug is being squeezed out or at the extrusion point and (2) ice jams are essentially fixed and grow in place.

In case (1), the irregularity of the cracks leads to plastic deformation of the ice. Equation (9) can be used in this situation, where $c$ is the result of subtracting the distance from the top of the crack to the freezing front from the length of the ice opening crack. Because $2k_0$ is larger than 2 MPa, and the ratio of crack depth to crack width ($c/W_1$) is larger than 4, the value of $P$ will exceed 9.2 MPa, which is sufficient to propagate the crack. Therefore, the pressure can be calculated solely based on the yield strength of the ice in case (1). In case (2), the pressure generated depends on the elastic and strength properties of the rock, which determine cracking. As $c/W_1$ increases, a point is reached where the ice stops squeezing out, at which point case (2) can be used to determine $P$. In addition, sealing is necessary because natural sealing is formed when the exposed rock surface freezes [37].

5. Discussion

Freeze–Thaw Damage Mechanism of Rock Slope with Three-Section Landslide

The freeze–thaw weathering of gray sandstone pores is quite slow at suitable temperatures (below 0 °C). On-site analysis revealed that the surface particles of rock slopes tend to peel off first, indicating a progressive deepening trend from the outside to the core. This is directly related to the physical properties of the rock, such as porosity, cohesiveness, and permeability. The frequency of freeze–thaw cycles is mainly determined by the environment. Because crack frost heave damage is substantially greater than pore freeze–thaw damage, rock slopes with a three-section landslide generate not only weak pore freeze–thaw damage but also more severe freeze–thaw weathering at the cracks. This study focuses on the fracture failure brought on by the frost heave of tensile cracks at the trailing edge, as slope collapse is mostly influenced by crack frost heave.

This study generalized the microstructure into four parts, namely, pores, cracks, particle microelements, and damage propagation microelements [40], and proposed a static equilibrium relationship between the microelements to precisely characterize the propagation mechanism of tensile cracks at the trailing edge of the rock slope with a three-section landslide. Figure 15 illustrates the freeze–thaw mechanism of tensile cracks at the trailing edge of rock slopes with a three-section landslide. Figure 16 shows a schematic of the freeze–thaw mechanism of tensile cracks at the trailing edge (local and non-proportional expansion to depict the microelements).

The cracks have sufficient length and aspect ratio [33]. Water is prone to freezing in large cracks. Therefore, water volume plays a crucial role in frost heave cracking. A water-to-ice expansion of 9% is undeniable, but rare rocks in nature can approach saturation (which is larger than 91%). After the cracks are filled with water for a period, the crack wall can reach high saturation within a certain depth, as shown in Figure 16a. When the cracks and pores reach critical saturation and the temperature drops below the freezing point, the water in the cracks and pores freezes in situ, and it is difficult for water to flow around. The volume expansion (about 9%) generated by this water–ice transition leads to the extension of microcracks. Bound water and capillary water migrate to the freezing front under a temperature gradient, thus gradually increasing the size of ice wedges and leading to the extension of microcracks [34,41].

A temperature gradient is produced by the difference in temperature between day and night, and a thermal molecular pressure gradient is produced at a sufficient temperature (below 0 °C). The temperature at the top of the crack first drops below the freezing point and freezes. Due to the thermal conductivity rate of water in the crack being lower than that of gray sandstone, the ice body grows from the crack wall toward the middle (Figure 16b). As the temperature of the rock mass decreases, ice crystals form in the trailing edge tensile cracks, whereas smaller cracks and pores undergo surface energy effects, inhibiting the growth of ice crystals and keeping water in a liquid state. When water freezes, the ice–water interface approaches the water–rock interface. As the temperature continues to decrease, the upward crack opening is frozen, generating a closed environment, accompanied by a volume expansion of 9% in the ice wedge. The bonding force between the crack wall and ice is higher than the tensile strength of the ice wedge, so the ice wedge at the top of the crack is stretched and cracked from the middle under the compression of the frost heave force (Figure 16c). Squeezing or uneven deformation will gradually occur inside the crack, resulting in tensile or shearing stress, and the crack will produce small tensile deformation. The intensity of frost heave force determines the depth of rock mass fracture.

At a certain depth, the cooling rate becomes relatively slow. Under the adsorption effect and ice wedging, water is sucked in from the surrounding pores due to the free-energy gradient, which both promote the slow expansion of the ice wedge. The increasing frost heave force and water pressure generated by the ice wedge promote crack propagation, leading to the formation of microcracks at the pre-existing crack tips (Figure 16d). The in situ tensile crack at the trailing edge is not a fully sealed rigid body, and its frost heave force will not fully act on the crack wall. Moreover, the frost heave force and deformation caused by the freezing of crack water need to be investigated based on different constraint conditions. Due to water supply considerations, the volume of ice wedges will slow down after rapid growth. Until all liquid water freezes, microcracks at the tip continue to expand (Figure 16e).

Rock masses are generally poor conductors of heat, as they are made up of mineral particles, cementitious components, pores, and cracks, among other compounds. The considerable temperature range in the surrounding environment leads to uneven heating and heat dissipation of the rock mass. Differential deformation (thermal expansion and contraction) occurs in each part of the rock mass, and the degree of deformation is controlled by the intensity of temperature changes. Stress concentration is more likely to occur at the crack tip, and long-term differential deformation results in fatigue damage, which, along with ice wedges, accelerates crack propagation. The longer the cooling period to the proper low temperature, the more adequate the water supply, and the sooner frost heave and cracking occur.

A tiny portion of the ice wedge at the crack top will melt under the impact of freeze–thaw over day and night cycles in the sunny area of the slope [42,43]. The ice wedge at the crack bottom maintains a subzero temperature, and the meltwater on the surface migrates and freezes again (Figure 16f). The fractured ice wedge produces greater expansion, accelerating the ice-splitting fracture [44,45]. As the temperature continues to rise and the ice at the bottom of the crack melts, water will flow into the newly expanded crack tips and pores [46], leading to further erosion of the crack water (Figure 16g). The pre-existing cracks continue to widen and deepen under the impact of ice wedges, and microcracks eventually expand into macrocracks under the freeze–thaw cycles (Figure 16h). Finally, the locking section of the “three-stage” rock slope fails.

6. Conclusions

(1)

The tensile crack frost heave at the trailing edge dominates the fracture failure of the rock slope with a three-section landslide, with substantially less freeze–thaw damage occurring in pores than in cracks.

(2)

The largest frost heave force and locking section deformation occur when the temperature of the top of the trailing edge tensile crack is lowered from −3.5 °C to −6 °C (while that of the bottom of the crack dropped from 0 °C to −2.6 °C).

(3)

The frost heave force is positively correlated with the trailing edge tension crack length, and the fracture crack extends from the base of the crack to its intersection with the shear crack at the leading edge. Additionally, there are almost no shear marks on the tensile fracture surface.

(4)

Based on rock mechanics and fracture mechanics, a theoretical failure model of crack frost heave was established in this study, which can be used to ascertain whether frost heave fracture occurs. The size effect directly affects the temperature response. In the next step, detailed field tests should be conducted to verify the model and theoretical results.