Mechanical Properties and Strength Evolution Model of Sandstone Subjected to Freeze–Thaw Weathering Process: Considering the Confining Pressure Effect

Abstract

Freeze-and-thaw (F&T) weathering cycles induced by day–night and seasonal temperature changes cause a large number of rock mass engineering disasters in cold areas. Investigating the impact of F&T weathering process on the strength and deformation characteristics of frozen–thawed rocks is therefore of critical scientific importance for evaluating the stability and optimizing the design of rock mass engineering in these areas. In this research, the evolution characteristics of F&T damage were analyzed based on T₂ spectrum distribution curves of sandstone specimens before and after F&T weathering cycles. The coupling impact of the quantity of F&T weathering cycles and confining pressure on pre-peak and post-peak deformation behaviors of sandstone specimens were analyzed in detail. By introducing the confining pressure increase factor (CPIF), the impact of confining pressure on the triaxial compressive strength (TCS) of sandstone specimens after undergoing different quantities of F&T weathering cycles was further investigated. A novel strength evolution model was proposed that could effectively describe the coupling impact of the quantity of F&T weathering cycles and confining pressure on TCS of rocks after undergoing the F&T weathering process. The proposed strength evolution model was cross-verified with experimental data from the published literature and all correlation coefficients were above 0.95, which proved that the strength evolution model proposed in this paper was reasonable; in addition, this model has strong applicability.

1. Introduction

During the construction of rock mass engineering (such as mines, roads, and tunnels) in cold regions, the recurrence of F&T weathering processes induced by day–night and seasonal temperature changes causes rapid damage and the deterioration of rock masses and has initiated a large number of F&T disasters such as rock falls, landslides [1,2,3], and the cracking of rock surrounding tunnels [4,5], which has a major impact on the design, construction, and operation of rock mass engineering in cold areas. The strength and deformation behaviors of rocks are the theoretical basis for evaluating the stability and optimizing the engineering design in rock mass engineering [6]. Therefore, investigating the mechanical properties of rocks after undergoing F&T weathering cycles has great significance in evaluating the stability and optimizing the design in rock mass engineering in cold areas.

The impact of F&T weathering process on rock mechanical properties have been investigated by a considerable number of scholars. These investigation results indicated that after undergoing the F&T weathering process, the elastic modulus [7,8,9,10,11], uniaxial compressive strength (UCS) [7,8,10,11,12,13,14], Brazilian tensile strength (BTS) [15,16,17,18], point load strength (PLS) [15,16], dynamic uniaxial compressive strength (UCS_d) [7,8,10,19,20,21,22] and dynamic tensile strength (BTS_d) [19,23] of rocks such as sandstone, tuff, gneiss, granite and shale all decreased as the quantity of F&T weathering cycles increased, but to different extents. To reveal the change laws in these mechanical parameters with the quantity of F&T weathering cycles, a sequence of regression analysis models [24,25] and exponential decay models [15,16,26,27,28] were proposed based on the relationships between these mechanical parameters and the quantity of F&T weathering cycles. In addition, Liu et al. [29] regarded F&T weathering cycles as a kind of fatigue damage and thus established a fatigue damage model of rocks after undergoing F&T weathering cycles. A prediction model of UCS of rocks after undergoing F&T weathering cycles was obtained based on this fatigue damage model. Gao et al. [30] established a UCS evolution model of frozen–thawed rocks based on the energy evolution characteristics of rock failure. However, these research findings mainly focused on the uniaxial mechanical properties of rocks after undergoing F&T weathering cycles and did not consider the impact of the confining pressure on the mechanical properties of rocks after undergoing F&T weathering process. There is no doubt that actual rock mass engineering is always in a certain stress field [31,32]. To be closer to the actual engineering, it is necessary to study the triaxial mechanical properties of rocks after undergoing F&T weathering cycles.

Recently, scholars have increasingly studied the triaxial mechanical properties of rocks after undergoing F&T weathering cycles. Tan et al. [9] used granite as the research object to investigate the impact of the quantity of F&T weathering cycles on uniaxial and triaxial mechanical properties according to uniaxial compression tests (UCTs) and triaxial compression tests (TCTs) after granite specimens being underwent F&T weathering process. They found that both the UCS and TCS decreased exponentially as the quantity of F&T weathering cycles increased, as did the elastic modulus and cohesion. The relationships between these mechanical properties and the quantity of F&T weathering cycles were built according to an exponential function. Wang et al. [31] and Hosseini and Khodayari [33] also carried out similar studies but took sandstone as the research object; the research results for both were in accordance with those of Tan et al. [9]. In addition, Hosseini and Khodayari [33] also found that the rate of reduction in the TCS was less than that of the UCS at the equivalent quantity of F&T weathering cycles and that the higher the confining pressure was, the lower the rate of reduction. It was evident that the confining pressure had a significant impact on the TCS of rocks after undergoing F&T weathering cycles. To this end, Fu et al. [34] operated a sequences of TCTs after transversely isotropic rocks underwent an F&T weathering process and found that the quantity of F&T weathering cycles, bedding plane orientation, and confining pressure had a significant impact on the TCS of slate. A TCS prediction model for transversely isotropic rocks after undergoing an F&T weathering process was proposed based on the single discontinuity theory and the functional relationships between cohesion and internal friction angle and the quantity of F&T weathering cycles. Seyed Mousavi et al. [35] also carried out similar studies taking calc-schist rock specimens as the research objects. Finally, an empirical expression among the TCS, the quantity of F&T weathering cycles, and the confining pressure was obtained according to experimental results and the prediction model suggested by Fu et al. [34]. Although the prediction model proposed by Fu et al. [34] could effectively reflect the coupling impact of the quantity of F&T weathering cycles and confining pressure on TCS of rocks after undergoing an F&T weathering process, this prediction model was based on transversely isotropic rocks and its applicability was not strong. Therefore, it is necessary to establish a more applicable strength evolution model to reveal the coupling impact of the quantity of F&T weathering cycles and confining pressure on TCS of rocks after undergoing an F&T weathering process. In addition, the above-mentioned studies mainly focused on investigating the strength-deterioration characteristics of rocks after undergoing an F&T weathering process and did not deeply investigate the deformation behaviors of rocks, especially the coupling impact of the quantity of F&T weathering cycles and confining pressure on deformation behaviors of rocks during the entire loading procedure.

In this study, saturated sandstone specimens were first subjected to different quantities of F&T weathering cycles, then change laws in the T₂ spectrum distribution curves of sandstone specimens before and after the F&T weathering cycles were investigated. UCTs and TCTs of sandstone specimens after undergoing different quantities of F&T weathering cycles were conducted to obtain the UCS and TCS and the corresponding stress–strain curves. The coupling impact of the quantity of F&T weathering cycles and confining pressure on the deformation behaviors of rocks were investigated. A novel strength evolution model that considered the coupling impact of the quantity of F&T weathering cycles and confining pressure on the TCS of rocks after undergoing the F&T weathering process was established; the proposed strength evolution model was cross-verified with experimental data from the published literature.

2. Experimental Materials and Methods

2.1. Rock Specimen Preparation

In this paper, rock specimens of sandstone were taken from the Jiama open pit copper mine located in the Tibet Autonomous Region of China. The sampling site is shown in Figure 1a. Figure 1b shows the distribution map for the frozen soil of China. It can be seen in Figure 1b that the sampling site was on the boundary between the permafrost regions and seasonal frozen regions; therefore, the rock was subjected to repeated F&T weathering cycles.

Table 1. Mineral composition of sandstone specimens.

Rock Type	Mineral Composition
	Quartz (%)	Kaolinite (%)	Feldspar (%)	Mica (%)
Sandstone	88.14	7.44	3.05	1.37

shows the mineral composition of sandstone obtained via an X-ray diffraction (XRD) technique.

Based on the method suggested by the International Society for Rock Mechanics (ISRM), all rock specimens were processed into a cylinder with a diameter of 50 mm; we ensured that the flatness of the end surfaces was less than 0.05 mm [36]. Careful preparations ensured that the maximum deviations of the diameter and height were less than 0.30 mm and the vertical variance was less than 0.25° [36]. In this study, 15 and 60 rock specimens were used in UCTs and TCTs, respectively, and the length/diameter ratio of the rock specimens was 2.0 [36]. These specimens were divided into five groups (labeled A, B, C, D, and E); each group comprised 15 rock specimens (labeled 1, 2 … 14, 15). The sandstone specimens from groups A, B, C, D, and E were treated in 0 cycles, 10 cycles, 20 cycles, 30 cycles, and 40 cycles, respectively. As shown in Figure 2a, the sandstone specimens labeled 1–3 in each group were used in the UCTs. The confining pressures were 3, 6, 9, and 12 MPa in the TCTs, corresponding to the sandstone specimens labeled 4–6, 7–9, 10–12, and 13–15 in each group, as shown in Figure 2b.

2.2. Test Procedures and Experimental Apparatus

All sandstone specimens were divided into five groups and dried in an oven for 48 h at 65 °C. The sandstone specimens that did not undergo F&T weathering cycles were used directly in the UCTs and TCTs. The other sandstone specimens were placed in a vacuum pump at a pressure of 0.1 MPa for 4 h and then soaked in distilled water for 24 h. The saturated specimens then went through the specified quantity of F&T weathering cycles in a TDS-300 automatic F&T testing machine (as shown in Figure 3a). When the quantity of F&T weathering cycles reached the specified quantities, the corresponding sandstone specimens were removed and UCTs and TCTs were conducted using an MTS815 electrohydraulic servo-controlled rock-testing machine (as shown in Figure 3c).

2.3. F&T Weathering Cycle Tests

Designed based on the local climate of the sampling site, one F&T weathering cycle in our tests included freezing the saturated rock specimens at −20 °C for four hours and then thawing in water at +20 °C for four hours. The temperature variation curve of the F&T weathering process is shown in Figure 4. In this study, four groups of sandstone specimens were subjected to F&T weathering tests corresponding to 10 cycles, 20 cycles, 30 cycles and 40 cycles. To reveal the evolution characteristics of the F&T damage, the MesoMR23-060H-I NMR system (as shown in Figure 3b) was used to conduct nuclear magnetic resonance (NMR) tests to obtain T₂ spectrum distribution curves before and after the F&T weathering cycles [37,38,39,40,41,42,43].

Figure 5 displays T₂ spectrum distribution curves of the sandstone specimens before and after 20 (as shown in Figure 5a) and 40 (as shown in Figure 5b) F&T weathering cycles. In the T₂ spectrum distribution curves, the T₂ relaxation time (horizontal axis) is a measurement of the internal pore sizes and the porosity component (vertical axis) is the proportion of the corresponding pore sizes [20]. As can be seen in Figure 5a, after the sandstone specimens underwent 20 cycles, the peak values of the T₂ spectrum distribution curve increased, indicating that the sizes of the internal pores increased. As shown in Figure 5b, when the sandstone specimens underwent 40 cycles, the increases in the peak values were even more significant. In addition, the expansion of the curve to the left suggested that the sizes of some small pores increased. This demonstrated that in the early stage of the F&T weathering cycles, the original internal pores and microcracks were constantly developing. With the increase in the quantity of F&T weathering cycles, in the later stage of the cycles, in addition to the expansion of original internal pores and the constant extension of microcracks, new pores and microcracks were generated; i.e., with the increase in the quantity of the F&T weathering cycles, the accumulated damage to the inner sandstone specimens constantly increased.

2.4. Uniaxial and Triaxial Compression Tests

UCTs and TCTs were conducted on an MTS815 electrohydraulic servo-controlled rock-testing machine; axial and circumferential extensometers were used in our experiment to measure the axial and lateral strains. In our experiment, the displacement-control loading mode was used at a loading rate of 0.1 mm/min. The measured UCSs and TCSs of the sandstone specimens are listed in

Table 2. Uniaxial and triaxial compression test results for sandstone specimens after undergoing different quantities of F&T weathering cycles.

Quantity of F&T Weathering Cycles	Confining Pressure (MPa)	Specimen ID	Diameter (mm)	Height (mm)	Peak Compressive Strength (Mpa)
					Tested Value	Average Value
0	0	A1	48.54	100.26	26.94	27.50
		A2	49.12	100.28	27.51
		A3	48.96	99.88	28.05
	3	A4	49.06	100.02	48.78	49.34
		A5	49.14	99.84	47.99
		A6	49.02	100.16	51.26
	6	A7	48.94	100.10	62.87	63.28
		A8	49.06	100.14	64.75
		A9	49.00	100.28	62.23
	9	A10	48.74	100.22	72.12	74.61
		A11	48.88	100.08	75.46
		A12	48.96	100.20	76.25
	12	A13	49.48	99.84	83.76	85.08
		A14	48.94	99.88	85.24
		A15	48.04	100.16	86.25
10	0	B1	49.12	100.02	25.60	24.38
		B2	49.46	100.02	24.11
		B3	48.96	99.84	23.44
	3	B4	48.86	100.04	44.88	44.58
		B5	48.74	100.28	45.58
		B6	48.56	100.06	43.28
	6	B7	48.52	100.12	56.86	57.57
		B8	49.10	100.12	57.22
		B9	49.04	100.66	58.64
	9	B10	49.02	100.18	70.33	68.99
		B11	48.98	99.94	67.40
		B12	48.96	100.24	69.24
	12	B13	49.14	99.94	81.88	80.13
		B14	48.82	99.72	79.55
		B15	48.84	100.16	78.96
20	0	C1	49.04	100.02	19.68	20.79
		C2	48.88	100.10	20.62
		C3	49.18	99.90	22.06
	3	C4	48.96	100.24	39.12	40.11
		C5	49.04	100.10	42.89
		C6	49.04	100.12	38.33
	6	C7	49.08	100.18	52.11	53.25
		C8	48.76	99.82	54.48
		C9	48.52	100.22	53.15
	9	C10	48.74	99.98	65.40	64.24
		C11	48.70	99.86	64.06
		C12	49.12	100.14	63.25
	12	C13	49.20	100.20	74.59	74.28
		C14	49.78	99.86	72.66
		C15	48.88	99.96	75.58
30	0	D1	49.52	100.16	19.04	18.17
		D2	48.84	99.88	18.2
		D3	49.04	100.02	17.26
	3	D4	48.36	100.02	35.18	36.47
		D5	49.12	99.78	38.97
		D6	48.76	100.06	35.26
	6	D7	48.82	100.16	48.80	49.29
		D8	49.06	99.78	50.81
		D9	49.92	100.04	48.25
	9	D10	48.80	99.86	58.28	59.41
		D11	49.06	100.02	60.16
		D12	49.82	99.90	59.78
	12	D13	49.02	100.22	66.62	67.26
		D14	49.28	100.16	65.37
		D15	49.12	99.94	69.78
40	0	E1	48.84	99.84	15.88	16.39
		E2	48.84	100.04	16.41
		E3	48.60	100.20	16.88
	3	E4	49.02	100.18	33.06	32.84
		E5	48.86	99.88	31.78
		E6	48.68	100.30	33.68
	6	E7	48.76	99.84	45.82	43.94
		E8	48.98	100.08	42.75
		E9	48.68	100.14	43.25
	9	E10	48.96	10.14	52.73	52.29
		E11	48.88	100.02	49.9
		E12	49.08	100.14	54.25
	12	E13	48.92	100.18	61.77	61.13
		E14	49.16	99.92	59.38
		E15	48.90	100.20	62.25

.

3. Experimental Results and Analysis

3.1. Uniaxial Mechanical Properties Variation Characteristics for Sandstone after Undergoing F&T Weathering Cycles

3.1.1. Stress–Strain Curve

Figure 6 displays the stress–strain curves for the sandstone specimens after undergoing different quantities of F&T weathering cycles under uniaxial compression conditions. All of the stress–strain curves had the same variation patterns and could be separated into five stages during the entire loading process; that is, the compaction stage, elastic deformation stage, yield stage, failure stage, and strain softening stage. As the quantity of F&T weathering cycles increased, the stress–strain curves showed three obvious features: (1) the compaction stage became longer; (2) the slope at the linear deformation stage decreased, as did the UCS; and (3) the stress-dropping rate of the post-peak decreased. The main reason was that the length of the compaction stage and the slope of the linear deformation stage were proportional to the number of microdefects inside the rock [44]. Under the F&T weathering process, water migration and transformation from water to ice caused microdefects to gradually develop; the number of microdefects increased with the increase in the quantity of F&T weathering cycles [45,46]. The main reason for the reduction in the stress-dropping rate of the post-peak was that the F&T weathering cycles caused the cohesion between the particles to gradually decrease, which caused the sandstone specimens to become soft and the plasticity to increase.

3.1.2. UCS

Figure 7 shows the changes in the UCS and its reduction ratio for different quantities of F&T weathering cycles. The reduction ratio for UCS is defined as follows:

$$\eta =\frac{{\sigma }_0-{\sigma }_{\mathrm{N}}}{{\sigma }_0}\times 100\%$$

(1)

where ${\sigma }_0$ is the UCS of sandstone specimens without any F&T weathering cycles, ${\sigma }_{\mathrm{N}}$ is the UCS of sandstone specimens after undergoing N quantity of F&T weathering cycles, and $\eta$ is the reduction ratio of the UCS.

As shown in Figure 7, compared with the original average UCS (27.50 MPa), the reduction ratios were 11.33% (24.38 MPa), 24.41% (20.79 MPa), 33.94% (18.17 MPa), and 40.40% (16.39 MPa), corresponding to 10 cycles, 20 cycles, 30 cycles, and 40 cycles, respectively. The reason can be explained as follows: the water migration and transformation from water to ice under the F&T weathering process caused microdefects to gradually develop and the sandstone specimens to become more fragmented. It was noticeable that the UCS exponentially decayed as the quantity of F&T weathering cycles increased, similar to laws found in the literature [16,26,28]. The experimental data were fitted by the decay model suggested by Mutlutürk et al. [26]; this decay model is defined as follows:

$$I_{\mathrm{N}}=I_0{e}^{-\lambda \mathrm{N}}$$

(2)

where I is the rock integrity, $\lambda$ is the decay coefficient, and N is the quantity of F&T weathering cycles.

In this paper, the UCS was regarded as the rock integrity; therefore, the decay model became as follows:

$${\sigma }_{\mathrm{N}}={\sigma }_0{e}^{-\lambda \mathrm{N}}$$

(3)

The fitting curves of our test are shown in Figure 7. The model fit well with the experimental data: the fitting coefficient of determination (R²) was greater than 0.99.

3.1.3. Failure Modes

Figure 8 shows the failure modes of the sandstone specimens after undergoing different quantities of F&T weathering cycles in uniaxial compression conditions. The failure mode was single inclined plane shear failure when the sandstone specimens did not undergo F&T weathering cycles. The failure modes became tension-shear comprehensive failure and splitting failure as the quantity of F&T weathering cycles increased. Tension-shear comprehensive failure occurred in the sandstone specimens after undergoing 10 cycles and 20 cycles and the sandstone specimens were fragmented. Splitting failure occurred in the sandstone specimens when the quantity of F&T weathering cycles was 30 and 40. The increase in the number of macroscopic cracks on the surfaces of sandstone specimens caused them to become more fragmented. The reason why the failure modes changed with the quantity of F&T weathering cycles was that the F&T weathering process causes microdefects to gradually develop inside the sandstone specimens, which caused cracks to be more likely to expand in the axial direction under uniaxial compression conditions.

3.2. Triaxial Mechanical Properties’ Variation Characteristics for Sandstone after Undergoing F&T Weathering Cycles

3.2.1. Stress–Strain Curve

Figure 9 displays the stress–strain curves for the sandstone specimens after undergoing different quantities of F&T weathering cycles under different confining pressures. Compared with the stress–strain curves of sandstone specimens after undergoing different quantities of F&T weathering cycles under uniaxial compression conditions, all of the stress–strain curves had obvious residual strength characteristics. In addition, the variation characteristics of the stress–strain curves were similar under different confining pressures as the quantity of F&T weathering cycles increased. All of the stress–strain curves could be separated into six stages: compaction stage, elastic deformation stage, yield stage, failure stage, strain softening stage, and residual strength stage. When the confining pressure was constant, with the increase in the quantity of the F&T weathering cycles, except for a gradual decrease in the residual strength, the other variation characteristics were similar to those in uniaxial compression conditions: (1) the compaction stage became longer; (2) the slope at the linear deformation stage decreased, as did the TCS; and (3) the stress-drop rate in the post-peak stress decreased. The main reason for these characteristics was the same as that under uniaxial compression conditions.

Figure 10 displays the stress–strain curves of the sandstone specimens after undergoing 20 F&T weathering cycles at different confining pressures. With the increase in confining pressure, the stress–strain curves showed four obvious features: (1) the compaction stage became shorter; (2) the slope at the linear deformation stage increased, as did the TCS; (3) the stress-dropping ratio of the post-peak decreased; and (4) the residual strength increased. The main reason was that the length of the compaction stage and the slope of the linear deformation stage are proportional to the number of microdefects in the rock. Under the confining pressure, microdefects inside the rock caused by the F&T weathering process were pre-compression, which caused a decrease in the number of microdefects; the higher the confining pressure was, the larger the number of microdefects that the pre-compression had. The main reason for the decrease in the stress-dropping ratio of the post-peak and the increase in residual strength was that the confining pressure could make the post-peak deformation behavior of the rocks transition from brittleness to plasticity [47,48,49,50].

These variation characteristics indicated that the impact of the quantity of F&T weathering cycles on the pre-peak deformation behavior and residual strength was the opposite of the confining pressure while the post-peak deformation behaviors were similar to the confining pressure.

3.2.2. TCS

Figure 11 displays the relationship between the TCS and its reduction ratio and the quantity of F&T weathering cycles at different confining pressures. It can be seen that the TCS decreased with an increase in the quantity of F&T weathering cycles when the confining pressure was constant, which was similar to laws in the uniaxial compression conditions. Compared with the original average TCS, the reduction ratios were: 9.55%, 18.71%, 26.09%, and 33.45% for 10 cycles, 20 cycles, 30 cycles, and 40 cycles, respectively, when the confining pressure was 3 MPa; 9.02%, 15.86%, 22.12%, and 30.57% for 10 cycles, 20 cycles, 30 cycles, and 40 cycles, respectively, when the confining pressure was 6 MPa; 7.53%, 13.90%, 20.38%, and 29.91% for 10 cycles, 20 cycles, 30 cycles, and 40 cycles, respectively, when the confining pressure was 9 MPa; and 5.82%, 12.70%, 20.95%, and 30.57% for 10 cycles, 20 cycles, 30 cycles, and 40 cycles, respectively, when the confining pressure was 12 MPa. At different confining pressures, the variation characteristics between the TCS of the sandstone specimens and the quantity of F&T cycles were similar to the UCS. Therefore, the experimental data could be used to fit the model suggested by Mutlutürk et al. [26]; the fitting results of our tests are shown in Figure 11. The model fit well with experimental data: the fitting coefficient of determination (R²) was greater than 0.98. The decay coefficient was 0.01018, 0.00881, 0.00819, and 0.00778 for 3, 6, 9, and 12 MPa, respectively. The results indicated that the decay coefficient decreased as the confining pressure increased and that the higher the confining pressure was, the larger the reduction in the decay coefficient. Therefore, the TCS of the sandstone after undergoing F&T weathering process was impacted by the confining pressure. Further study of the impact of confining pressure on the TCS of sandstone after undergoing the F&T weathering process referred to the definition of the dynamic increase factor [51,52]. The confining pressure increase factor (CPIF) could be defined as TCS/UCS.

Figure 12 displays the CPIF curves of the sandstone specimens after undergoing different quantities of F&T weathering cycles in different confining pressures. As shown in Figure 12, the variation characteristics of the CPIF curves were similar under different confining pressures as the quantity of F&T weathering cycles increased; that is, the CPIF increased as the quantity of F&T weathering cycles increased. However, the higher the confining pressure was, the larger the increased amplitude of the CPIF. For example, when the confining pressure was 3 MPa, the value of the CPIF was 1.79 without F&T weathering cycles; the CPIF is 2.31 after the sandstone underwent F&T weathering cycles, an increase ratio of 28.97%; when the confining pressure was 12 MPa, the value of the CPIF was 3.09 without F&T weathering cycles and the CPIF was 4.83 after the sandstone underwent F&T weathering cycles, an increase ratio of 56.27%. This indicated that sandstone specimens were more sensitive to the confining pressure after undergoing more F&T weathering cycles.

These results demonstrated that the UCS of the sandstone after undergoing F&T weathering process was mainly controlled by the quantity of F&T weathering cycles under uniaxial compression conditions. The TCS of the sandstone after undergoing the F&T weathering process was impacted by the quantity of F&T weathering cycles and confining pressure under triaxial compression conditions; these two factors had opposite impacts. The main reason was that the water–ice phase and water migration under the F&T weathering cycles caused microdefects to gradually develop inside the sandstone specimens, which induced F&T damage. However, under confining pressure conditions, some microdefects inside the sandstone specimens were closed, which lessened the damage induced by the F&T weathering process. Therefore, for the slope engineering of open-pit mines in cold areas, the application of anchor reinforcement technology to provide pre-stress could lessen the damage induced by the F&T weathering process and improve the stability of the slope.

3.2.3. Failure Modes

Under different quantities of F&T weathering cycles, the failure modes of the sandstone specimens had the same evolution characteristics with the change in confining pressure. Therefore, Figure 13 only shows the failure modes under different confining pressures after the sandstone specimens underwent 20 F&T weathering cycles. As shown in Figure 13, under different confining pressures, the failure modes of the sandstone specimens all were single incline plane shear failure; however, the length of the shear failure plane became shorter with the increase in the confining pressure. The main reason was that the lateral deformation was limited under the confining pressure, so the sandstone specimens only exhibited single inclined plane shear failure. In addition, the higher the confining pressure was, the more severe the limiting effect, so the length of the shear failure plane became shorter with the increase in the confining pressure.

4. Strength Evolution Model of Rock Specimens Considering the Freeze–Thaw Weathering Process and Confining Pressure

The experimental results demonstrated that the TCS of rocks after undergoing the F&T weathering process was impacted by the quantity of F&T weathering cycles and the confining pressure under triaxial compression conditions. The model suggested by Mutlutürk et al. [26] could only describe the change in the peak compressive strength of rocks after undergoing the F&T weathering process at a specific confining pressure, which did not consider the coupling impact of the quantity of F&T weathering cycles and the confining pressure. Therefore, it was necessary to establish a novel model that could reflect the evolution laws of the TCS of rocks after undergoing the F&T weathering process. The impact of the confining pressure on the TCS could be described by the rock-strength criterion. The Hoek–Brown strength criterion proposed by Hoek–Brown could describe the failure of the broken rock mass. The expression of the Hoek–Brown strength criterion is as follows [53,54]:

$${\sigma }_1={\sigma }_3+\sqrt{m_i{\sigma }_{ci}{\sigma }_3+{\sigma }_{ci}^2}$$

(4)

where ${\sigma }_{ci}$ is the UCS, ${\sigma }_1$ is the TCS, $m_i$ is the material constant, and ${\sigma }_3$ is the confining pressure.

Under the F&T weathering process, the water migration and transformation from water to ice caused microdefects to gradually develop, which caused the rock specimens to become more fragmented. Therefore, the Hoek–Brown strength criterion could be adopted to describe the impact of the confining pressure on the TCS of rocks after undergoing the F&T weathering process. According to the expression of the Hoek–Brown strength criterion, the relationship between the TCS ${\sigma }_1$ and the confining pressure ${\sigma }_3$ could be simplified into a quadratic function expression. Based on this simplified relationship and the model suggested by Mutlutürk et al. [26], this paper proposed a novel strength evolution model to describe the coupling impact of the quantity of F&T weathering cycles and the confining pressure on the TCS. The expression of the strength evolution model is as follows:

$${\sigma }_1=\left(a+b{\sigma }_3+c{\sigma }_3^2\right)\exp \left[\left(d+e{\sigma }_3+f{\sigma }_3^2\right)\mathrm{N}\right]$$

(5)

where a, b, c, d, e, and f are fitting parameters determined by the properties of rocks.

Using Equation (5), MATLAB was adopted to fit the experimental data of the TCS of the sandstone specimens after undergoing different quantities of F&T weathering cycles and different confining pressures; the fitting surface is shown in Figure 14 and the fitting parameters are shown in

Table 3. Fitting parameters of strength evolution model of sandstone specimens.

Fitting Parameters	a	b	c	d	e	f	R2	RMSE
Value	27.358	7.412	−0.216	1.101 × 10−2	3.257 × 10−4	−6.043 × 10−6	0.992	1.731

. As shown in Figure 14 and Table 3, the fitting surface agreed well with the experimental data: the correlation coefficient was up to 0.992, which indicated that the proposed model could effectively describe the coupling impact of the quantity of F&T weathering cycles and the confining pressure on the TCS of the sandstone after undergoing the F&T weathering process.

To further validate the rationality of the proposed model in this paper, Equation (5) was adopted to fit the tested value from the published literature [32,33,34,35]. The fitting surfaces are shown in Figure 15 and the fitting parameters are shown in

Table 4. Fitting parameters of the strength evolution model of rock specimens from the published literature.

Data Source	Fitting Parameters
	a	b	c	d	e	f	R2	RMSE
[33]	32.242	11.493	−0.580	9.190 × 10−3	1.570 × 10−3	−1.309 × 10−4	0.988	2.249
[34]	34.444	4.333	−0.042	2.314 × 10−2	3.788 × 10−4	−2.546 × 10−7	0.977	4.341
[32]	4.387	5.155	−0.313	8.880 × 10−3	1.500 × 10−3	−9.298 × 10−5	0.992	0.688
[35]	31.680	5.907	0.299	−2.328 × 10−2	1.810 × 10−3	−8.714 × 10−5	0.965	5.817

. As shown in Figure 15 and Table 4, the fitting surfaces agreed well with the tested value and all fitting correlation coefficients were above 0.95, which proved that the strength evolution model proposed in this paper was reasonable; in addition, this model has strong applicability.

5. Conclusions

In this paper, the evolution characteristics of the F&T damage of sandstone were analyzed based on NMR techniques. Uniaxial and triaxial compression tests of sandstone after undergoing different quantities of F&T weathering cycles were conducted to investigate the coupling impact of the quantities of F&T weathering cycles and the confining pressure on the mechanical properties and failure modes. A novel strength evolution model was proposed to describe the coupling impact of the quantity of F&T weathering cycles and the confining pressure on the TCS of rocks after undergoing the F&T weathering process. The following main conclusions could be drawn from this research:

(a).

In the early stage of the F&T weathering cycles, original internal pores and microcracks were constantly developing. As the quantity of F&T weathering cycles increased, in the later stage of the F&T weathering cycles, in addition to the expansion of the original internal pores and the constant extension of microcracks, new pores and microcracks were generated; i.e., the accumulated damage inside the sandstone specimens constantly increased as the quantity of F&T weathering cycles increased.

(b).

The impact of the quantity of F&T weathering cycles on the pre-peak deformation behaviors, peak compressive strength, and residual strength was the opposite of the confining pressure while the post-peak deformation behaviors were similar to confining pressure. When the confining pressure was constant, with an increase in the quantity of F&T weathering cycles, the compaction stage became longer while the slope at the linear deformation stage, the peak compressive strength, the residual strength, and the stress-dropping rate of the post-peak all decreased. When the quantity of the F&T weathering cycles was constant, with an increase in the confining pressure, the compaction stage became shorter and the slope at the linear deformation stage, the peak compressive strength, and the residual strength all increased, but the stress-dropping rate of the post-peak decreased.

(c).

In the uniaxial compression tests, the failure mode of the sandstone specimens changed as the quantity of F&T cycles increased. The failure mode was a single inclined plane shear failure when the sandstone specimens did not undergo F&T weathering cycles. In the early and later stages of the F&T weathering cycles, the failure modes became tension–shear comprehensive failure and splitting failure, respectively. In the triaxial compression tests, the failure mode of the sandstone specimens under different confining pressures was a single inclined plane shear failure regardless of the quantity of F&T weathering cycles experienced. However, the length of the shear failure plane became shorter as the confining pressure increased.

(d).

The variation characteristics of the CPIF curves were similar under different confining pressures as the quantity of F&T weathering cycles increased; that is, the CPIF increased as the quantity of F&T weathering cycles increased. However, the higher the confining pressure was, the larger the increased amplitude of the CPIF. This indicated that the sandstone specimens were more sensitive to the confining pressure after undergoing more F&T weathering cycles.

(e).

A novel strength evolution model that could describe the coupling impact of the quantity of F&T weathering cycles and the confining pressure on the TCS of rocks after undergoing the F&T weathering process was proposed. The proposed model was cross-verified with tested values from the published literature. Fitting surfaces agreed well with the tested value and all fitting correlation coefficients were above 0.95, which proved that the strength evolution model proposed in this paper was reasonable; in addition, this model has strong applicability.