Study of Damage Mechanism and Evolution Model of Concrete under Freeze–Thaw Cycles

Abstract

Researching the mechanical characteristics of concrete subjected to the freeze–thaw cycle is crucial for building engineering in cold climates. As a result, uniaxial compression tests were performed on concrete samples exposed to various freeze–thaw (F–T) cycles, and the measurements of the pore size distribution, porosity, and P-wave velocity of the saturated concrete samples were obtained, both before and after being exposed to the F–T cycles. Concrete’s F–T damage mechanism and damage evolution model were thoroughly examined. Using rock structure and moisture analysis test equipment to observe the T₂ spectrum, the results showed that the F–T cycles can cause the internal structure of the samples to deteriorate. Porosity and F–T cycles have a positive correlation, although P-wave velocity has a negative correlation with the F–T cycles. As the F–T cycles increased, the specimens’ peak strength and elastic modulus steadily declined, while the peak strain clearly exhibited an increasing trend. A microscopic F–T damage model that takes into account the pore size distribution was developed, based on the relative changes in the pore structure distribution (PSD), before and after the F–T cycles. The concrete sample damage evolution law under various F–T cycles was examined using the following metrics: total energy, pore size distribution, static and dynamic elastic moduli, porosity, and P-wave velocity. Uniaxial compressive strength (UCS) and peak strain tests were used to evaluate the accuracy of the pore size distribution damage model, as well as that of five other widely used damage models.

1. Introduction

In civil engineering projects such as the construction of homes, highways, bridges, slopes, tunnels, foundations, hydraulic concrete structures, petroleum engineering edifices, nuclear waste disposal sites, and enhanced geothermal systems, concrete is one of the most popular and extensively utilized building materials [1,2,3]. For an extended period, concrete is subjected to alternating cycles of seasonal and diurnal temperatures, known as freezing and thawing (F–T) cycles. This damage and deterioration of concrete is a major cause of the instability and failure of concrete materials and structures, and it plays a crucial role in the study of the durability of concrete in cold climates [4,5,6,7]. Due to the severe deterioration of concrete materials, both domestically and internationally, brought on by the F–T cycle, increased requirements are also placed on the freeze resistant durability of concrete structures in the harsh natural environment of cold areas [8,9,10,11]. The north and northeast of China, North Dakota in the United States, and Canada are examples of cold places, both domestically and internationally, where winters are long, cold, and exhibit significant day-to-night temperature differences. The F–T cycles must be repeatedly endured by concrete materials each year. The internal concrete structure will eventually deteriorate with time, causing the building’s bearing capability to be lost. Building stability and safety are seriously threatened by the F–T cycle [12,13,14,15,16,17,18,19,20]. Thus, one of the key elements influencing the stability and longevity of engineering structures is the F–T resistance of the concrete material. Studying and precisely identifying the F–T damage process of concrete is crucial for cold climate resource utilization and structural engineering [21,22,23,24,25].

Figure 1 illustrates the typical instances of F–T damage to structural elements and concrete materials in engineering. In frigid climates, a large number of dams and other hydraulic structures have experienced significant F–T degradation prior to the completion of their planned service life [26,27,28,29]. The interior pore structure of the concrete must be saturated with water infiltration for F–T degradation to occur, and the concrete must exist in a temperature environment that is subjected to both positive and negative change [30,31]. Surface denudation and the internal loose cracking of concrete are caused by the free water of the solidified and hardened concrete being subjected to the alternating action of freezing (when the temperature is below the freezing point) and thawing (when the temperature is above the freezing point) when the temperature changes [32,33,34,35]. Concrete’s internal pore structure deteriorates due to the cold weather, causing the water within to remain in an alternating state of F–T cycles for extended periods of time. This ultimately results in irreversible F–T damage to the concrete’s mechanical and physical properties [36,37,38].

The majority of researchers have conducted a significant number of theoretical investigations, as well as experimental work, to determine the impact of F–T cycles on the mechanical and physical characteristics of concrete materials [39,40,41,42]. The five-damage models listed below serve as a basic summary of the findings of this theoretical analysis. The damage models that are classified as (1) P-wave [43,44,45,46,47], (2) porosity [48,49,50,51,52,53,54], (3) static elastic modulus [55,56,57,58,59], (4) dynamic-elastic modulus [60,61,62,63], and (5) energy damage models [63,64,65,66,67,68,69]. Nearly all of the fundamental mechanical and physical characteristics form the basis of the current F–T damage models [70,71,72,73,74,75]. Nevertheless, there are few reports regarding the development of damage factor and damage models based on the distribution of microscopic pore sizes to quantitatively assess the degree of internal damage in concrete under F–T cycle circumstances [76,77,78]. Therefore, a damage model considering the microscopic pore size distribution was introduced to quantify the concrete sample damage evolution process during F–T cycles. In addition, the six-damage model is rarely considered to quantitatively compare and evaluate the degree of internal damage of concrete under F–T cycle conditions. Therefore, the damage variables were created to indicated the damage evolution process of concrete samples subjected to various F–T cycles based on wave velocity, static elastic modulus, porosity, dynamic-elastic modulus, energy, and pore size distribution.

In order to investigate how F–T cycles affect the macroscopic and microscopic mechanical characteristics of saturated concrete specimens under uniaxial compression conditions, a thorough analysis was conducted on the changes in the mechanical characteristics, porosity, P-velocity, and pore size distribution across the F–T cycles. To measure and assess the damage evolution process of concrete samples subjected to various F–T cycles, the damage factor and damage model were created based on wave velocity, static elastic modulus, porosity, dynamic-elastic modulus, energy, and pore size distribution. The ideal damage evolution prediction model was produced by comparing the predicted values with the experimental values. The six-damage model was introduced to forecast the UCS and peak strain of concrete samples during the F–T cycle process.

2. Damage Variables Reviews

2.1. Damage Factors Established Based on the P-Wave Velocity

With the development of nondestructive testing technology, the ultrasonic method has been widely used in many research fields due to its advantages, e.g., simple operation, low costs, and small influence on materials. The ultrasonic detection method is a convenient way to obtain the internal information regarding a substance through ultrasonic waves [79]. The wave velocity type usually adopted is the P-wave velocity, and the F–T damage variable of the concrete is defined as:

$$D_p=1-{\displaystyle \frac{V_P^{\prime 2}}{V_P^2}}$$

(1)

where $V_p$ and $V_P^{\prime }$ are the P-wave velocity before and after the F–T cycles experienced by the concrete.

2.2. Damage Factors Established Based on the Static Elastic Modulus

The degree of deterioration inside rock and concrete materials can be characterized by the macroscopic physical and mechanical properties of the materials based on the theory of damage mechanics. The change in the elastic modulus of rock and concrete materials exhibits a reliable consistency with its damage and deterioration processes. In addition, the elastic modulus of concrete and rock materials after experiencing the F–T cycle is easy to analyze and measure, and it can be utilized as a parameter to quantitatively assess the damage of concrete and rock materials. Therefore, the damage variable of concrete can be defined based on the change in the elastic modulus, before and after F–T cycles, as follows:

$$D_n=1-{\displaystyle \frac{E_n}{E_0}}$$

(2)

where E₀ and E_n are the elastic modulus before and after the F–T cycle treatment.

2.3. Damage Factors Established Based on the Porosity

The change in porosity is the fundamental reason for the change in macroscopic mechanical properties of rock and concrete samples. Therefore, the porosity is used to define the damage variable to reflect the degree of damage in rocks under different F–T cycles. The microscopic F–T damage variable expression is as follows:

$$D_{\varphi }=1-{\displaystyle \frac{1-{\varphi }_n}{1-{\varphi }_0}}$$

(3)

where $D_{\varphi }$ is the microscopic F–T damage variable of the rock, ${\varphi }_n$ is the porosity of the concrete sample subjected to n F–T cycles, and ${\varphi }_0$ is the initial porosity of the concrete sample.

2.4. Damage Factors Established Based on the Dynamic Elastic Modulus

The longitudinal wave velocity defined by the elastic theory is as follows:

$$V_P=\sqrt{{\displaystyle \frac{E_d(1-\mu )}{\rho (1+\mu )(1-2\mu )}}}$$

(4)

where $V_P$ is the P-wave velocity of the concrete, $E_d$ is the dynamic elastic modulus before the F–T cycles, and $\mu$ is Poisson’s ratio of the concrete; the damage variable $D_t$ is defined by the dynamic elastic modulus of the concrete as follows:

$$D_t=1-{\displaystyle \frac{E_d^{\prime }}{E_d^{ }}}$$

(5)

If the change of Poisson’s ratio in the F–T cycles is not considered, Equation (4) is substituted into Equation (5) to obtain the following equation:

$$D_t=1-{\displaystyle \frac{{\rho }^{\prime }}{\rho }}{\displaystyle \frac{V_P^{\prime 2}}{V_P^2}}=1-{\displaystyle \frac{v}{v^{\prime }}}{\displaystyle \frac{V_P^{\prime 2}}{V_P^2}}$$

(6)

where V is the unit mass volume of the concrete before F–T cycles; ${\rho }^{\prime }$, $v^{\prime }$, and $V_p^{\prime }$ are the density, unit mass volume, and P-wave velocity of the concrete after F–T cycles, respectively.

The volume of the concrete will be irreversibly expanded and deformed under the F–T cycle action. The change in the concrete volume is the result of the initiation and evolution of micropores and fissures. If the matrix deformation is not considered, the volume change in concrete, before and after the F–T cycles, can be expressed as follows:

$${\displaystyle \frac{v}{v^{\prime }}}={\displaystyle \frac{v_s+v_o}{v_s+v_o+\Delta v_o}}$$

(7)

where $v_s$ and $v_o$ are, respectively, the volume of the matrix and pore of the concrete before the F–T cycles; $\Delta v_o$ is the pore volume variation in the concrete, before and after the F–T cycles.

The porosity of the concrete before the F–T cycles can be expressed as follows:

$${\varphi }_0={\displaystyle \frac{v_o}{v_s+v_o}}$$

(8)

The porosity of the concrete after the F–T cycles can be expressed as follows:

$${\varphi }_n={\displaystyle \frac{v_o}{v_s+v_o+\Delta v_o}}$$

(9)

By substituting Equations (7)–(9) into Equation (6), we can get

$$D_t=1-{\displaystyle \frac{1-{\varphi }_n}{1-{\varphi }_0}}{\displaystyle \frac{V_P^{\prime 2}}{V_P^2}}$$

(10)

Equation (10) is the concrete F–T damage variable expressed by the concrete porosity and the P-wave velocity [80].

2.5. Damage Factors Established Based on Energy

A damage model was established to calculate the F–T damage factors, according to the changes of input energy required by the samples before and after F–T treatment, as follows:

$$D_u=1-{\displaystyle \frac{U_n}{U_0}}$$

(11)

where U₀ and U_n are the total energy before and after the F–T cycles treatment. The energy calculation method is based on that found in Ref. [73].

2.6. Damage Factors Established Based on Microscopic Pore Size Distribution

Presently, numerous scholars have established many significant damage models for quasi-brittle materials such as the concrete based on basic physical and mechanical parameters. These models include evaluating the porosity, static elastic modulus, dynamic-elastic modulus, volume, P-wave, fractal dimension, energy, etc. [81,82]. Pore size distribution exerts significant effects on rock and concrete materials [83]. The different T₂ spectral areas represent the different pore contents of corresponding sizes from the T₂ spectrum, and the damage variables, considering the different pore sizes of the concrete exposed to F–T cycle treatment, are established as follows:

$$D_s=\eta {\displaystyle \frac{d{P^{\prime }}_{micro}+e{P^{\prime }}_{meso}+f{P^{\prime }}_{macro}}{d{P}_{micro}+e{P}_{meso}+f{P}_{macro}}}-1$$

(12)

where P_micro, P_meso, P_macro, P′_micro, P′_meso, and P′_macro represent the spectral area of the micropores, the mesopores and the macropores in the T₂ spectrum for the concrete, before and after exposure to the F–T cycles via the NMR testing; P′_micro, P′_meso, and P′_macro represent the micropores, the mesopores, and the macropores after the F–T cycles. d, e, and f are the weights. the corresponding weight is calculated using Equation (14).

The coefficient of variation of the spectral peak area corresponding to the pores of each size inside the concrete samples subjected to different F–T cycles was obtained using the Equation (13), as follows:

$${C.V.}_i=1-{\displaystyle \frac{{\beta }_i}{\overline{{\theta }_i}}}$$

(13)

where $C.V{.}_i$ is the coefficient of variation; ${\beta }_i$ and $\overline{{\theta }_i}$ are the standard deviation and average of the peak area index.

The coefficients of variation of the peak areas of the micropores, the mesopores, and the macropores were 0.213, 0.256, and 0.536, respectively. The corresponding weight is calculated using Equation (14). The weights d, e, and f of the micropores, the mesopores, and the macropores were 0.215, 0.292, and 0.503, respectively.

$$W_i={\displaystyle \frac{C.V{.}_i}{\overline{{\displaystyle {\sum }_{i}C.V{.}_i}}}}$$

(14)

3. Experimental Scheme

3.1. Sample Preparation

The concrete samples were created by combining water, sand, and regular Portland cement. The sand was a mixture of 60% fine (0.1~0.25 mm) and 40% medium (0.25~0.40 mm) sands, the water is regular tap water, and the cement is 42.5 strength grade P.I. Portland cement [84]. To create the concrete samples, the cement, sand, and water were combined in a blender at a weight ratio of 2:1:0.4 [84]. Subsequently, the newly mixed concrete was transferred into a 50 mm diameter and 100 mm high cylindrical mold. Using a vibrating table, the mold containing the fresh concrete was vibrated for 10 min. It was then maintained at room temperature (20 °C) for 28 days. The concrete specimen test scheme under different conditions is shown as

Table 1. Concrete samples test scheme under different conditions.

Freeze–Thaw Cycles Test	F–T Cycles	Samples
Concrete samples	0	F–T-0
	10	F–T-10
	20	F–T-20
	30	F–T-30
	40	F–T-40
	50	F–T-50

.

3.2. Experimental Process and Equipment

The specific test process includes the P-wave velocities test, the drying test, the vacuum saturation test, the F–T cycles test, the PSD test, and the UCS test. The test procedure is illustrated in Figure 2. The specific steps are as follows:

(1)

The six samples with similar P-wave velocities were selected via the P-wave velocities test at the YS4A tester.

(2)

The samples were dried at 105 °C for 24 h in the vacuum drying oven. The samples were vacuumed at 0.1 MPa pressure for 6 h in a vacuum saturated facility. Then, the internal pores were completely saturated with water at a pressure of 5 MPa for 24 h at the HYS-4 tester.

(3)

The pore structure distribution of saturated samples before the F–T cycles was measured using the rock structure and moisture analysis test equipment.

(4)

The samples were maintained at the HDM-10C tester for the F–T cycles test. One cycle of the F–T experimental process is shown in Figure 3. The F–T cycles consisted of 0, 10, 20, 30, 40 and 50, respectively [35]. Following the F–T cycles test, the above steps were repeated by monitoring the P-wave velocity and the PSD.

(5)

Finally, the UCS tests were performed at a rate of 0.05 MPa/min using the MTS-815 loading tests system for the samples that had been exposed to different numbers of F–T cycles at room temperature (25 °C).

4. Experimental Results and Analysis

4.1. Pore Size Distribution

The change in the T₂ spectrum can reflect the change in the pore structure in the rock. The rock structure and moisture analysis test equipmentdetects the relaxation response of the 1H proton via the CPMG pulse sequence test in the samples, and then the amplitude of the attenuation signal of the spin echo string is obtained. The set of all attenuation amplitude results forms the T₂ spectrum [85]. Due to the porous media properties of the concrete, there is no unified criterion for the classification of pore size at present. As can be observed in Figure 4, the T₂ distribution of the samples can be mainly divided into three peaks, corresponding to the transverse relaxation time T₂ spectrum of the micropores (0.01~4 ms), mesopores (4~100 ms), and macropores (100~10,000 ms), as shown from left to right [86,87,88,89].

4.2. The Variation in Porosity

The porosity of a saturated concrete samples is positively proportional to the NMR signal of its pore water pressure [90]. Hence, the porosity of the concrete samples can be obtained using Equation (15), as follows:

$$\varphi ={\displaystyle \frac{P(0)}{P_{100\%}(0)}}={\displaystyle \frac{\sum P_{0i}}{P_{100\%}(0)}}=\sum {\displaystyle \frac{\sum P_{0i}}{P_{100\%}(0)}}=\sum {\varphi }_i$$

(15)

where $\varphi$ and ${\varphi }_i$ are the total porosity and the porosity of the class i dimension pores (%), respectively. P(0), P_100%(0), and P_0i represent the total signal intensity, the signal intensity produced, and the signal intensity of pores of the class i dimension, respectively.

The test results and the calculation of porosity are shown in Figure 5. With the increase in the number of F–T cycles, the change in the porosity of the samples was positively correlated with the F–T cycles. The porosity of the samples after 10 F–T cycles increased by 10.15% compared with that of the samples without exposure to the F–T cycles. The porosity of the samples after 20 F–T cycles increased by 12.01% compared with that after 10 F–T cycles; the porosity of the samples after 30 F–T cycles increased by 14.21% compared with that after 20 F–T cycles; and the porosity of the samples after 40 F–T cycles increased by 15.50% compared with that after 30 F–T cycles. The porosity of the samples after 50 F–T cycles increased by 16.08% compared with that after 40 F–T cycles. This indicates that the pore structure damage of the samples after numerous F–T cycles was significant more serious than that causes by the previous 10 F–T cycles. This is because the first 10 F–T cycles have little influence on the pore structure of the samples, but the F–T cycles caused a certain auxiliary effect on the development of the pore structure of the sample. With the increase in the number of F–T cycles, the number of newly generated pores continues to increase, gradually developing into mesopores and macropores, which develop continuously, expanding and connecting to form microcracks, causing a significant increasing in the porosity of the samples.

4.3. The Variation in P-Velocity

The changes in P-wave velocity, before and after F–T cycles, are presented in the Figure 6. The P-wave velocity of the samples decreases as the F–T cycles increase. The internal pores of the sample developed after 10 F–T cycles, and the P-wave velocity decreased from 4.61 km/s to 4.44 km/s, a decrease of 3.68%. After 20 F–T cycles, the P-wave velocity of the samples decreased from 4.5 km/s to 4.15 km/s, a decrease of 7.77%. After 30 F–T cycles, the P-wave velocity decreased from 4.37 km/s to 3.81 km/s, a decrease of 12.81%. With the increase in the number of F–T cycles, the pores continued to develop and began to connect, and the total number increased significantly. After 40 F–T cycles, the P-wave velocity of the samples decreased from 4.29 km/s to 3.44 km/s, a decrease of 19.81%, After 50 F–T cycles, the wave velocity of the samples decreased from 4.2 km/s to 3.19 km/s, a decrease of 24.04%.

This is due to the fact that the F–T cycle effect creates a huge number of pores in the concrete samples, altering the internal structure and ultrasonic wave propagation mode and path. Constantly forming new pores, the existing pores also keep growing and spreading as the F–T cycle times rise. The sample’s mechanical strength and P-wave velocity significantly decrease as a result of the initial pores continuing to expand with each cycle due to the irreversible breakdown of the pore structure within the sample. The changes in both the porosity and the P-wave velocity were in good agreement, suggesting that material deterioration and damage buildup occur as the frequency of F–T cycles increases.

4.4. Changes in Mechanical Properties

Figure 7 depicts the alterations in the fundamental mechanical characteristics of concrete samples following various F–T cycles. Concrete samples that were not treated with F–T cycles exhibit clear signs of brittleness in their stress–strain curves. The stress–strain curves of the concrete samples exhibit a clear transition from brittleness to ductility as the number of F–T cycles increases. This is particularly evident for the treatment with 50 F–T cycles, which seems to indicate a yield platform. Furthermore, the samples’ compaction deformation stage lengthens as the number of F–T cycles increases during the initial loading stage. This is because there is a large number of F–T cycles creating new pores in the samples, and those pores must be compressed during the initial loading stage.

A noticeable stress drop occurs in the elastic deformation stage as the number of F–T cycles increases, and as the number of F–T cycles increases, so does the stress needed to generate the stress reduction. This stems from the fact that the samples at this stage produce macrocracks, and the actions of the F–T cycles reduce the tension necessary for the production of macrocracks. As the number of F–T cycles increases, a clear yield plateau forms at the yield failure stage (pre-peak and post-peak stages). This phenomena is explained by the fact that the F–T cycles clearly damage the samples, causing them to transition from a brittle state to ductile failure. With the increase in the number of F–T cycles, the peak strength and elastic modulus of the samples decreased, while the peak strain increased. This is because the F–T cycles cause the original pores and cracks inside the samples to spread and generate new pore cracks, resulting in damage inside the samples, ultimately reducing the bearing capacity of the samples and their ability to resist deformation. The peak strain increased, but the peak strength and elastic modulus of the samples failed as the number of F–T cycles increased. This is due to the fact that the F–T cycles caused the pores and cracks inherent in the samples to propagate and create new pore cracks, causing internal damage that eventually reduced the bearing capacity and the resistance to deformation.

5. Discussion

5.1. Comparison of the Damage Variables under F–T Cycles

In order to further quantify the influence of the F–T cycles on the concrete samples, based on the above damage variable reviews, in this paper, the damage variables were compared and analyzed. The relationship between the F–T cycles and the porosity, total energy, static and dynamic elastic moduli, P-wave velocity, and pore size distribution is shown in Figure 8.

The damage variable of the six-damage model shows a similar variation trend, as does the damage variable of the concrete samples, which increases as the number of F–T cycles increases. Furthermore, the PSD damage model demonstrated a greater sensitivity to the F–T cycles. Compared to the five-damage models, the PSD damage model performed better and showed smaller increases. For the six-damage models, the damage variables have an exponential connection with n, and the correlation coefficients are greater than 0.9159. The six-model damage evolution equations are as follows:

The damage variable of the concrete samples increases with the increase in the number of F–T cycles, and the damage variable of the six-damage models show a similar variation trend. In addition, the PSD damage model was more sensitive to the F–T cycles. The PSD damage model exhibits smaller increases than does the five-damage model. The damage variables were exponentially correlated with n for the six-damage model, and the correlation coefficients are greater than 0.9159. The damage evolution equation of the six models are as follows:

$$\begin{array}{ll}{D}_p=1-1.0717\exp 0.012n^{ }& R^2=0.9159\\ D_n=1-1.0684\exp 0.14n^{ }& R^2=0.9502\\ D_{\varphi }=1-1.0227\exp 0.013n^{ }& R^2=0.9566\\ D_t=1-0.9601\exp 0.013n^{ }& R^2=0.9819\\ D_u=1-0.998\exp 0.016n^{ }& R^2=0.9968\\ D_s=1-1.0254\exp 0.013n^{ }& R^2=0.9396\end{array}$$

5.2. Comparison of the Damage Model under the F–T Cycles

By using parameter analysis and data fitting, some prediction models have been presented to investigated the law of concrete and rock strength loss under F–T cycles by means of parameter analysis and data fitting. M. Mutluturk et al. [91] first proposed a first-order model for predicting the integrity of rock F–T damage, and it has been widely applied. The integrity index of rock mass after the F–T cycles can be expressed as:

$$I_N=I_0{e}^{-\lambda N}$$

(16)

where $I_0$ and $I_N$ are the integrity indexes of rocks before and after the F–T cycles; $\lambda$ is the decay constant that can be calculated from the half-decay period of the rock.

It can be seen that the UCS of rock after experiencing F–T cycles is related to the initial strength of rock and the degree of F–T cycle damage. The nonlinear prediction equation of the UCS of concrete was proposed, based on Equation (16), as follows [91]:

$${\sigma }_n={\sigma }_0(1-D)$$

(17)

where ${\sigma }_0$ and ${\sigma }_n$ are the UCS of the concrete samples before and after the F–T cycles. The damage variable $D$ can be expressed as follows:

$$D=\left\{\begin{array}{l}{D}_p=1-{\displaystyle \frac{V_P^{\prime 2}}{V_P^2}}\\ D_n=1-{\displaystyle \frac{E_n}{E_0}}\\ D_{\varphi }=1-{\displaystyle \frac{1-{\varphi }_n}{1-{\varphi }_0}}\\ D_t=1-{\displaystyle \frac{1-{\varphi }_n}{1-{\varphi }_0}}{\displaystyle \frac{V_P^{\prime 2}}{V_P^2}}\\ D_u=1-{\displaystyle \frac{U_n}{U_0}}\\ D_s={\displaystyle \frac{d{P^{\prime }}_{micro}+e{P^{\prime }}_{meso}+f{P^{\prime }}_{macro}}{d{P}_{micro}+e{P}_{meso}+f{P}_{macro}}}-1\end{array}\right.$$

(18)

Figure 9 displays the experimental and predicted UCS values of the concrete samples subjected to various F–T cycles. As the number of the F–T cycles increased, the predicted values of the wave velocity model, the static and dynamic elastic modulus model, the porosity model, and the energy model were all greater than the experimental values. The largest discrepancies between the predicted and the experimental values after 30 cycles were 25.6%, 20.82%, 17.24%, 11.43%, and 16.03%. The experimental values of concrete prior to 40 F–T cycles were most closely aligned with the anticipated values of the pore size distribution damage model; the errors were 2.24%, 1.57%, 5.14%, and 2.70%, respectively. the pore size distribution model’s projected value and the experimental value showed the least amount of error after 50 cycles, and the error was 0.73%. Therefore, the pore size distribution damage model can accurately predict the mechanical strength of concrete samples after F–T cycles.

Based on the change curve of the peak strain versus the number of F–T cycles, Gao et al. [66] defined the relative strain variables as follows:

$${\displaystyle \frac{{\epsilon }_n-{\epsilon }_0}{{\epsilon }_0}}=M_{n}D$$

(19)

Figure 10 show that the values of the peak strains predicted based on the pore size distribution damage model exhibit minimal errors and were stable when compared with the test results.

6. Conclusions

In this study, we adopted to the ultrasonic tester and nuclear magnetic resonance (NMR) to investigating the effects of F–T cycles on the microscopic and macroscopic mechanical properties of saturated concrete specimens under uniaxial compression conditions. The main research results are summarized as follows:

The F–T cycles can lead to the deterioration of the internal structure of the samples based on the observation of the T₂ spectrum using the rock structure and moisture analysis test equipment.

P-wave velocity is negatively correlated with the F–T cycles, while the porosity is positive correlated with the F–T cycles. The peak strength and elastic modulus of the specimens decrease gradually as the F–T cycles increase, and the peak strain shows an obvious increasing trend.

A microscopic F–T damage model, considering the pore size distribution, was established. The damage evolution law of concrete samples under different F–T cycles was analyzed based on the P-wave velocity, porosity, static elastic modulus, dynamic elastic modulus, total energy, and pore size distribution. The accuracy of the pore size distribution damage model and five other common damage models was tested using the uniaxial compressive strength (UCS) and the peak strain.

The newly proposed damage factor and damage model, considering the definition of pore structure changes before and after the freeze and thaw cycle, can effectively evaluate the freeze–thaw cycle properties of related concrete materials in cold areas. Without mechanical tests, it can evaluate the evolution process of the freeze–thaw damage of concrete materials, providing a reference for related projects.