Influence of Mixing Rubber Fibers on the Mechanical Properties of Expansive Clay under Freeze–Thaw Cycles

: With the growth of the transportation industry, large volumes of waste tires are being generated, which necessitates the development of effective solutions for recycling waste tires. In this study, expansive clay was mixed with rubber fibers obtained from waste tires. Triaxial tests were conducted on the rubber fiber-reinforced expansive clay after freeze–thaw cycles. The experimental results of the unreinforced expansive clay from previous studies were used to evaluate the effect of mixing rubber fibers on the mechanical properties of rubber fiber-reinforced expansive clay under freeze–thaw cycles. The results demonstrate that the mixing of rubber fibers significantly reduces the effect of freeze–thaw cycles on the shear strength and elastic modulus of expansive clay. The shear strength and elastic modulus of the unreinforced expansive clay decrease markedly as the number of freeze–thaw cycles increases, while the shear strength and elastic modulus of the rubber fiber-reinforced expansive clay do not exhibit any remarkable change. A calculation model of the deviatoric stress–axial strain curves after freeze–thaw cycles was established. The model describes the deviatoric stress–axial strain behavior of rubber fiber-reinforced expansive clay and unreinforced expansive clay under different confining pressures and different numbers of freeze–thaw cycles.


Introduction
Automobile production has grown rapidly, owing to the expansion of the transportation industry [1].Although automobiles have made our lives more convenient, they have also negatively affected the environment.One aspect of their environmental impact is the increasing amounts of waste tires, which is becoming an urgent environmental issue.According to statistical data, the annual global production of waste tires exceeds 10 million tons [2].Currently, the primary techniques for disposing waste tires are underground landfills, centralized storage, heat recovery for energy production, and pyrolysis for other products [3][4][5][6][7][8].Tire dumps require large amounts of land and are prone to catching fire, releasing significant levels of particulate matter (PM) into air during incineration [9][10][11][12][13].
Underground landfills are commonly used to dispose of waste tires.However, waste tires buried underground do not degrade even after many years and produce toxic substances that pollute underground water [14].The European Union has enacted a law banning underground landfills for waste tires [15].The pyrolysis of waste tires has been extensively studied in recent years, with various pyrolysis techniques being used to create useful products [16][17][18][19].Rubber makes up approximately half of waste tire components; therefore, recycling rubber from waste tires has significant benefits, and an appropriate recycling method is urgently needed [14].
Numerous studies have been conducted to improve soil strength using rubber obtained from waste tires.The basic approach involves grinding waste tires into rubber granules and fibers, or cutting them into rubber strips and mixing them into the soil, which makes the soil valuable for engineering applications [14].Studies on the influence of rubber granules on soil shear strength have demonstrated that mixing rubber granules with soil can increase the shear strength [20,21].The addition of rubber granules mostly improved the shear strength by changing the internal friction angle of the soil [22,23].According to Zhang et al. [24], mixing rubber strips with sand increases the internal friction angle by 68.1%.When the soil was subjected to dynamic loads, the addition of rubber enhanced its dynamic shear modulus, dynamic elastic modulus, and damping ratio [25].In summary, mixing rubber with soil under certain conditions can change soil properties and enhance its value for engineering applications.However, approximately 23% of all land on Earth is covered with permafrost or seasonally frozen soil, where freeze-thaw cycling in seasonally frozen soil significantly affects the behavior of the soil and decreases its shear strength [26][27][28].
Studies have shown that freeze-thaw cycles reduce the strength of unreinforced soils, but there is no consensus on the effect of freeze-thaw cycles on the mechanical properties of rubber fiber-reinforced expansive clay [29][30][31][32][33][34].Under dynamic loads, freeze-thaw cycles change the dynamic shear modulus of the soil and decrease the dynamic strength [35,36].Although the characteristics of soils subjected to freeze-thaw cycles have been extensively studied, the impact of freeze-thaw cycles on rubber-reinforced soil has received relatively less attention.
The improvement of expansive soil has been a hot issue in geotechnical engineering.Expansive soil is a specialized clay rich in hydrophilic minerals such as montmorillonite, which is widely distributed worldwide.Expansive soil is considered problematic because it swells in volume when it absorbs water and shrinks and cracks when it dries.It is estimated that the cost of maintaining expansive soil already exceeds the combined economic losses caused by natural disasters worldwide [37].Therefore, a feasible solution is needed to address the negative impacts of expansive soil on building structures.
This study aims to enhance expansive clay performance in freeze-thaw cycles and to find reliable methods for recycling large volumes of waste tires.Triaxial tests were performed on rubber fiber-reinforced expansive clay exposed to freeze-thaw cycles.To evaluate the effects of rubber fiber on the mechanical properties of expansive clay under freeze-thaw cycles, the triaxial test results of rubber fiber-reinforced expansive clay and unreinforced expansive clay were compared.An expansive clay stress-strain calculation model under freeze-thaw cycles was developed based on the test results.

Test Materials
The expansive clay was obtained from Henan Province, China.Based on the reference standards listed in Table 1, the maximum dry density of the test expansive clay was 1.858 g/cm 3 and the optimum moisture content was 15.71%.The expansive clay has a liquid limit of 57.9% and plastic limit of 22.3%, and is classified as high liquid limit clay (CH) according to the Unified Soil Classification System [38].The particle size of the expansive clay is shown in Figure 1, and its uniformity coefficient (C u ) is 9.3.
One of the aims of this study was to find a sustainable solution for effective waste tire management.The rubber fibers were obtained by grinding waste tires.

Sample Preparation
Akbarimehr et al. [45] measured the damping ratio of rubber-reinforced clay and suggested an optimum rubber content of 10%.When the fiber content exceeded 20% during the preparation of the specimens, the fibers clumped together and a uniform specimen could not be obtained.Similarly, Correia et al. [46] showed that specimens with high fiber content were not uniform.To avoid non-uniformity of the specimens, the rubber fiber content used in this study was 10% (ratio of fiber mass to dry expansive clay mass).Airdried expansive clay was uniformly mixed with rubber fibers, and distilled water was added to increase the moisture content to 16% of the dry expansive clay mass.The mixture was stored in a sealed bag for 24 h to an even moisture distribution.Specimens with a diameter of 39.1 mm and height of 80 mm were obtained using the standard compaction effort [39].Because it is more difficult to bring clay to saturation than sand or silt, the rubber fiber-reinforced expansive clay specimens were treated using the vacuum saturation method [47].Some rubber fiber-reinforced expansive clay specimens are shown in Figure 2.

Sample Preparation
Akbarimehr et al. [45] measured the damping ratio of rubber-reinforced clay and suggested an optimum rubber content of 10%.When the fiber content exceeded 20% during the preparation of the specimens, the fibers clumped together and a uniform specimen could not be obtained.Similarly, Correia et al. [46] showed that specimens with high fiber content were not uniform.To avoid non-uniformity of the specimens, the rubber fiber content used in this study was 10% (ratio of fiber mass to dry expansive clay mass).Air-dried expansive clay was uniformly mixed with rubber fibers, and distilled water was added to increase the moisture content to 16% of the dry expansive clay mass.The mixture was stored in a sealed bag for 24 h to an even moisture distribution.Specimens with a diameter of 39.1 mm and height of 80 mm were obtained using the standard compaction effort [39].Because it is more difficult to bring clay to saturation than sand or silt, the rubber fiber-reinforced expansive clay specimens were treated using the vacuum saturation method [47].Some rubber fiber-reinforced expansive clay specimens are shown in Figure 2.

Test Scheme
In its actual state, the expansive clay consolidation is completed after some time.Clays have limited permeability and do not release excess pore water pressure in time; hence, a consolidated undrained test was employed to simulate the actual drainage conditions of the clays.Eight strips of filter paper were uniformly applied around each specimen to accelerate the consolidation process.Confining pressures of 100, 200, and 300 kPa were applied to specimens with the same number of freeze-thaw cycles.In this study, triaxial tests were conducted on rubber fiber-reinforced expansive clay with different numbers of freeze-thaw cycles (N), which are 0, 1, 3, 6, 10, and 15.Wang et al. [48] suggested that temperatures approaching 0 °C may lead to incomplete freezing of the specimens.The specimens were frozen at −20 °C for 12 h and thawed at 20 °C for 12 h to ensure complete freezing and thawing of the specimens.

Results and Discussion
The stress-strain curves for some of the triaxial tests are shown in Figure 3 and the shear strength results are shown in Figure 4.With confining pressures of 100 kPa and 200 kPa, the shear strength changes more significantly during the first three freeze-thaw cycles.The shear strength declines after one freeze-thaw cycle.After three freeze-thaw cycles, the shear strength reaches the maximum value.After this, the shear strength gradually decreases with the increase in the number of freeze-thaw cycles.With the confining pressure of 300 kPa, the shear strength gradually decreases with the increase in the number of freeze-thaw cycles and reaches the turning point after six freeze-thaw cycles.

Test Scheme
In its actual state, the expansive clay consolidation is completed after some time.Clays have limited permeability and do not release excess pore water pressure in time; hence, a consolidated undrained test was employed to simulate the actual drainage conditions of the clays.Eight strips of filter paper were uniformly applied around each specimen to accelerate the consolidation process.Confining pressures of 100, 200, and 300 kPa were applied to specimens with the same number of freeze-thaw cycles.In this study, triaxial tests were conducted on rubber fiber-reinforced expansive clay with different numbers of freeze-thaw cycles (N), which are 0, 1, 3, 6, 10, and 15.Wang et al. [48] suggested that temperatures approaching 0 • C may lead to incomplete freezing of the specimens.The specimens were frozen at −20 • C for 12 h and thawed at 20 • C for 12 h to ensure complete freezing and thawing of the specimens.

Results and Discussion
The stress-strain curves for some of the triaxial tests are shown in Figure 3 and the shear strength results are shown in Figure 4.With confining pressures of 100 kPa and 200 kPa, the shear strength changes more significantly during the first three freeze-thaw cycles.The shear strength declines after one freeze-thaw cycle.After three freeze-thaw cycles, the shear strength reaches the maximum value.After this, the shear strength gradually decreases with the increase in the number of freeze-thaw cycles.With the confining pressure of 300 kPa, the shear strength gradually decreases with the increase in the number of freeze-thaw cycles and reaches the turning point after six freeze-thaw cycles.

Effect of Mixing Rubber Fibers on Shear Strength of Expansive Clay
Numerous studies have shown that the freeze-thaw cycling process significantly reduces the shear strength of unreinforced soils [29][30][31][32][33][34]. Figure 4 shows the relationship between the shear strength of rubber fiber-reinforced expansive clay and the number of freeze-thaw cycles.The shear strength decreases after 15 freeze-thaw cycles.However, the maximum decrease in shear strength was 9.6% for all confining pressures, which was significantly different from the performance of unreinforced expansive clay under freezethaw cycles.The changes in shear strength of unreinforced expansive clay under freezethaw cycles from previous studies are referenced to show how mixing rubber fibers affects the shear strength, and the test conditions are summarized in Table 2 [49,50].Liu et al. [51] indicated that the freezing or thawing temperature had almost no effect on shear strength; therefore, the magnitude of freezing or thawing temperatures was not differentiated in this study.
The normalized shear strength was obtained by dividing the shear strength after freeze-thaw cycles by the shear strength without freeze-thaw cycles.The effect of mixing rubber fibers on the shear strength under freeze-thaw cycles can be determined by comparing the normalized shear strength of the rubber fiber-reinforced expansive clay with that of the unreinforced expansive clay.The results are shown in Figure 5.

Effect of Mixing Rubber Fibers on Shear Strength of Expansive Clay
Numerous studies have shown that the freeze-thaw cycling process significantly re duces the shear strength of unreinforced soils [29][30][31][32][33][34]. Figure 4 shows the relationship between the shear strength of rubber fiber-reinforced expansive clay and the number o freeze-thaw cycles.The shear strength decreases after 15 freeze-thaw cycles.However the maximum decrease in shear strength was 9.6% for all confining pressures, which was significantly different from the performance of unreinforced expansive clay under freezethaw cycles.The changes in shear strength of unreinforced expansive clay under freezethaw cycles from previous studies are referenced to show how mixing rubber fibers affects the shear strength, and the test conditions are summarized in Table 2 [49,50].Liu et al. [51 indicated that the freezing or thawing temperature had almost no effect on shear strength therefore, the magnitude of freezing or thawing temperatures was not differentiated in this study.
The normalized shear strength was obtained by dividing the shear strength after freeze-thaw cycles by the shear strength without freeze-thaw cycles.The effect of mixing rubber fibers on the shear strength under freeze-thaw cycles can be determined by com paring the normalized shear strength of the rubber fiber-reinforced expansive clay with that of the unreinforced expansive clay.The results are shown in Figure 5.

Effect of Mixing Rubber Fibers on Shear Strength of Expansive Clay
Numerous studies have shown that the freeze-thaw cycling process significantly reduces the shear strength of unreinforced soils [29][30][31][32][33][34]. Figure 4 shows the relationship between the shear strength of rubber fiber-reinforced expansive clay and the number of freeze-thaw cycles.The shear strength decreases after 15 freeze-thaw cycles.However, the maximum decrease in shear strength was 9.6% for all confining pressures, which was significantly different from the performance of unreinforced expansive clay under freeze-thaw cycles.The changes in shear strength of unreinforced expansive clay under freeze-thaw cycles from previous studies are referenced to show how mixing rubber fibers affects the shear strength, and the test conditions are summarized in Table 2 [49,50].Liu et al. [51] indicated that the freezing or thawing temperature had almost no effect on shear strength; therefore, the magnitude of freezing or thawing temperatures was not differentiated in this study.
The normalized shear strength was obtained by dividing the shear strength after freezethaw cycles by the shear strength without freeze-thaw cycles.The effect of mixing rubber fibers on the shear strength under freeze-thaw cycles can be determined by comparing the normalized shear strength of the rubber fiber-reinforced expansive clay with that of the unreinforced expansive clay.The results are shown in Figure 5.   Figure 5 shows the variation in the normalized shear strength with the number of freeze-thaw cycles, and the different curves in the same color represent different confining pressures.The normalized shear strength of rubber fiber-reinforced expansive clay changed the least after 15 freeze-thaw cycles.However, the shear strength of unreinforced expansive clay decreased by approximately 34% (average) after 9 freeze-thaw cycles in Tang et al. [49], and by approximately 27% (average) after 10 freeze-thaw cycles in Zhao et al. [50].Unreinforced expansive clay generates cracks under freeze-thaw cycles and the cracks develop with the increasing number of freeze-thaw cycles [52][53][54].The cracks reduce the shear strength of the expansive clay [53].When cracks are generated in the expansive clay, the rubber fibers produce a tensile force to weaken the development of the cracks, thus reducing the effect of freeze-thaw cycles on the shear strength of the expansive clay [55,56].It can be observed from the above discussion that mixing rubber fibers enhances the ability of the expansive clay to resist the effects of freeze-thaw cycles and thus the service life of expansive clay engineering in seasonal frozen regions is prolonged

Effect of Mixing Rubber Fibers on Elastic Modulus of Expansive Clay
Elastic modulus is an important mechanical parameter for soil materials.The stressstrain curves shown in Figure 3 are not straight lines, indicating that the secant elastic modulus is different at different strain levels.Some scholars have defined the elastic modulus of soil as the secant elastic modulus at 1% axial strain [34,49].In this study, the elastic modulus was calculated from the secant elastic modulus at 2% axial strain based on the stress-strain curves of the specimens, as expressed in Equation ( 1), and the results are shown in Figure 6.
( ) where 2% E is the elastic modulus, ( ) is the deviatoric stress at 2% axial strain Figure 5 shows the variation in the normalized shear strength with the number of freeze-thaw cycles, and the different curves in the same color represent different confining pressures.The normalized shear strength of rubber fiber-reinforced expansive clay changed the least after 15 freeze-thaw cycles.However, the shear strength of unreinforced expansive clay decreased by approximately 34% (average) after 9 freeze-thaw cycles in Tang et al. [49], and by approximately 27% (average) after 10 freeze-thaw cycles in Zhao et al. [50].Unreinforced expansive clay generates cracks under freeze-thaw cycles and the cracks develop with the increasing number of freeze-thaw cycles [52][53][54].The cracks reduce the shear strength of the expansive clay [53].When cracks are generated in the expansive clay, the rubber fibers produce a tensile force to weaken the development of the cracks, thus reducing the effect of freeze-thaw cycles on the shear strength of the expansive clay [55,56].It can be observed from the above discussion that mixing rubber fibers enhances the ability of the expansive clay to resist the effects of freeze-thaw cycles and thus the service life of expansive clay engineering in seasonal frozen regions is prolonged.

Effect of Mixing Rubber Fibers on Elastic Modulus of Expansive Clay
Elastic modulus is an important mechanical parameter for soil materials.The stressstrain curves shown in Figure 3 are not straight lines, indicating that the secant elastic modulus is different at different strain levels.Some scholars have defined the elastic modulus of soil as the secant elastic modulus at 1% axial strain [34,49].In this study, the elastic modulus was calculated from the secant elastic modulus at 2% axial strain based on the stress-strain curves of the specimens, as expressed in Equation ( 1), and the results are shown in Figure 6.
where E 2% is the elastic modulus, (σ 1 − σ 3 ) 2% is the deviatoric stress at 2% axial strain, and ε 1, 2% is 2% axial strain.The results shown in Figure 6 indicate that the elastic modulus of rubber fiber-reinforced expansive clay decreases slowly from an overall point of view at the confining pressures of 100 kPa and 200 kPa.However, at the confining pressure of 300 kPa, the decrease in the elastic modulus of the rubber fiber-reinforced expansive soil is more significant.The results of the elastic modulus of unreinforced expansive clay under freeze-thaw conditions from previous studies were referred to compare the effect of mixing rubber fibers on the elastic modulus of expansive soil [49,50].The results of the normalized elastic modulus are shown in Figure 7 and the test conditions are presented in Table 2. Figure 7 shows the variation in the normalized elastic modulus with the number of freeze-thaw cycles, and the different curves in the same color represent different confining pressures.The normalized elastic modulus of rubber fiber-reinforced expansive clay showed a decreasing trend within 15 freeze-thaw cycles.The elastic modulus of rubber fiber-reinforced expansive clay decreased by approximately 10% (average) after 15 freezethaw cycles.However, the elastic modulus of unreinforced expansive clay decreased by approximately 36% (average) after 9 freeze-thaw cycles in Tang et al. [49], and by approximately 37% (average) after 10 freeze-thaw cycles in Zhao et al. [50].The rubber fibers The results shown in Figure 6 indicate that the elastic modulus of rubber fiberreinforced expansive clay decreases slowly from an overall point of view at the confining pressures of 100 kPa and 200 kPa.However, at the confining pressure of 300 kPa, the decrease in the elastic modulus of the rubber fiber-reinforced expansive soil is more significant.The results of the elastic modulus of unreinforced expansive clay under freeze-thaw conditions from previous studies were referred to compare the effect of mixing rubber fibers on the elastic modulus of expansive soil [49,50].The results of the normalized elastic modulus are shown in Figure 7 and the test conditions are presented in Table 2.The results shown in Figure 6 indicate that the elastic modulus of rubber fiber-reinforced expansive clay decreases slowly from an overall point of view at the confining pressures of 100 kPa and 200 kPa.However, at the confining pressure of 300 kPa, the decrease in the elastic modulus of the rubber fiber-reinforced expansive soil is more significant.The results of the elastic modulus of unreinforced expansive clay under freeze-thaw conditions from previous studies were referred to compare the effect of mixing rubber fibers on the elastic modulus of expansive soil [49,50].The results of the normalized elastic modulus are shown in Figure 7 and the test conditions are presented in Table 2. Figure 7 shows the variation in the normalized elastic modulus with the number of freeze-thaw cycles, and the different curves in the same color represent different confining pressures.The normalized elastic modulus of rubber fiber-reinforced expansive clay showed a decreasing trend within 15 freeze-thaw cycles.The elastic modulus of rubber fiber-reinforced expansive clay decreased by approximately 10% (average) after 15 freezethaw cycles.However, the elastic modulus of unreinforced expansive clay decreased by approximately 36% (average) after 9 freeze-thaw cycles in Tang et al. [49], and by approximately 37% (average) after 10 freeze-thaw cycles in Zhao et al. [50].The rubber fibers produce tensile forces that weaken the development of cracks (caused by freeze-thaw cy- Figure 7 shows the variation in the normalized elastic modulus with the number of freeze-thaw cycles, and the different curves in the same color represent different confining pressures.The normalized elastic modulus of rubber fiber-reinforced expansive clay showed a decreasing trend within 15 freeze-thaw cycles.The elastic modulus of rubber fiber-reinforced expansive clay decreased by approximately 10% (average) after 15 freezethaw cycles.However, the elastic modulus of unreinforced expansive clay decreased by approximately 36% (average) after 9 freeze-thaw cycles in Tang et al. [49], and by approximately 37% (average) after 10 freeze-thaw cycles in Zhao et al. [50].The rubber fibers produce tensile forces that weaken the development of cracks (caused by freezethaw cycles) in the expansive clay, thus reducing the effect of freeze-thaw cycles on the elastic modulus of the expansive clay [53,55,56].Based on the above discussion, the mixing of rubber fibers lowers the effect of freeze-thaw cycles on the elastic modulus and thus improves the engineering performance of expansive clay in seasonal frozen regions.

Stress-Strain Curves after Freeze-Thaw Cycles of Expansive Clay
The shape of the stress-strain curves shown in Figure 3 is very similar to that of the nonlinear elastic model proposed by Duncan and Chang [57].The relationship between the deviatoric stress and axial strain is expressed as Equation ( 2): where σ 1 − σ 3 is the deviatoric stress, ε 1 is the axial strain, and a and b are the two model parameters.
The a and b in Equation ( 2) correspond to the initial elastic modulus and ultimate strength in the deviatoric stress-axial strain curve, respectively, and this correspondence is expressed in Equation ( 3): where E i is the initial elastic modulus, and (σ 1 − σ 3 ) ult is the ultimate strength.Equation ( 2) converted to Equation ( 4) is expressed as follows: Equation ( 4) shows that there is a linear relationship between ε 1 /(σ 1 − σ 3 ) and ε 1 .The ε 1 /(σ 1 − σ 3 )-ε 1 curves are drawn according to the test results of rubber fiber-reinforced expansive clay and the results of unreinforced expansive clay conducted by Tang et al. [49] and Zhao et al. [50].The partial results are displayed in Figure 8.The intercept a and slope b were obtained by linear fitting of the ε 1 /(σ 1 − σ 3 )-ε 1 curves, and the initial elastic modulus and ultimate strength after different numbers of freeze-thaw cycles at different confining pressures were obtained by substituting the correspondence expressed in Equation ( 3).The results are summarized in Table 3.
Appl.Sci.2024, 14, x FOR PEER REVIEW 8 of 15 rubber fibers lowers the effect of freeze-thaw cycles on the elastic modulus and thus improves the engineering performance of expansive clay in seasonal frozen regions.

Stress-Strain Curves after Freeze-Thaw Cycles of Expansive Clay
The shape of the stress-strain curves shown in Figure 3 is very similar to that of the nonlinear elastic model proposed by Duncan and Chang [57].The relationship between the deviatoric stress and axial strain is expressed as Equation ( 2): ( ) where 1 3 σ σ − is the deviatoric stress, 1 ε is the axial strain, and a and b are the two model parameters.
The a and b in Equation ( 2) correspond to the initial elastic modulus and ultimate strength in the deviatoric stress-axial strain curve, respectively, and this correspondence is expressed in Equation ( 3): where i E is the initial elastic modulus, and ( ) Equation ( 2) converted to Equation ( 4) is expressed as follows: ( ) Equation (4) shows that there is a linear relationship between ( )  3).The results are summarized in Table 3.   Table 3 shows the results for ultimate strength and initial elastic modulus.R 2 are all above 0.99, which indicates that ε 1 /(σ 1 − σ 3 ) and ε 1 exhibit a highly linear relationship.Figure 9 shows the relationship between the ultimate strength and initial elastic modulus.
Appl.Sci.2024, 14, 5437 10 of 15 Analysis of the data shows that there is a proportional relationship between the initial elastic modulus and ultimate strength, and the correspondence is expressed as Equation ( 5): where M is the coefficient between the initial elastic modulus and ultimate strength.
Table 3 shows the results for ultimate strength and initial elastic modulus.R 2 are all above 0.99, which indicates that ( ) ε exhibit a highly linear relationship.
Figure 9 shows the relationship between the ultimate strength and initial elastic modulus.Analysis of the data shows that there is a proportional relationship between the initial elastic modulus and ultimate strength, and the correspondence is expressed as Equation ( 5): ( ) where M is the coefficient between the initial elastic modulus and ultimate strength.Analysis of the data reveals a certain regularity in the variation in the ultimate strength.The normalized ultimate strength is defined as the ratio of the ultimate strength after freeze-thaw cycles over the ultimate strength without freeze-thaw cycles, expressed as Equation ( 6): ) where R is the normalized ultimate strength, ( ) is the ultimate strength after N freeze-thaw cycles, and ( ) is the ultimate strength without freeze-thaw cycles.
The normalized ultimate strengths were averaged (average values for different confining pressures) and are shown in Figure 10.The relationship between the average value of the normalized ultimate strength and the number of freeze-thaw cycles was established and shown as Equation (7), where the parameter λ is a deterioration parameter.

(
) where ave R is the average normalized ultimate strength, λ is the deterioration parame- ter of the ultimate strength under freeze-thaw cycles, and N is the number of freeze-thaw cycles.
In Equation (7), λ reflects the soil ultimate strength as affected by freeze-thaw cycles.The value of λ is between 0 and 1.The ultimate strength is less affected by freeze-thaw cycles when the value of λ is close to 0. The ultimate strength is more affected by freezethaw cycles when the value of L is close to 1.The ultimate strength is (1-λ) times the ultimate strength without freeze-thaw cycles if the expansive clay is subjected to an infinite Analysis of the data reveals a certain regularity in the variation in the ultimate strength.The normalized ultimate strength is defined as the ratio of the ultimate strength after freeze-thaw cycles over the ultimate strength without freeze-thaw cycles, expressed as Equation ( 6): where R is the normalized ultimate strength, (σ 1 − σ 3 ) ult, N is the ultimate strength after N freeze-thaw cycles, and (σ 1 − σ 3 ) ult, 0 is the ultimate strength without freeze-thaw cycles.The normalized ultimate strengths were averaged (average values for different confining pressures) and are shown in Figure 10.The relationship between the average value of the normalized ultimate strength and the number of freeze-thaw cycles was established and shown as Equation (7), where the parameter λ is a deterioration parameter.
where R ave is the average normalized ultimate strength, λ is the deterioration parameter of the ultimate strength under freeze-thaw cycles, and N is the number of freeze-thaw cycles.
In Equation (7), λ reflects the soil ultimate strength as affected by freeze-thaw cycles.The value of λ is between 0 and 1.The ultimate strength is less affected by freeze-thaw cycles when the value of λ is close to 0. The ultimate strength is more affected by freezethaw cycles when the value of L is close to 1.The ultimate strength is (1 − λ) times the ultimate strength without freeze-thaw cycles if the expansive clay is subjected to an infinite number of freeze-thaw cycles, according to Equation (7). Figure 10 shows that λ = 0.336 for unconsolidated expansive clay in Tang et al. [49] and λ = 0.275 in Zhao et al. [50].However, the ultimate strength of rubber fiber-reinforced expansive clay is almost unaffected by freeze-thaw cycles, with λ = 0. number of freeze-thaw cycles, according to Equation (7). Figure 10 shows that λ = 0.336 for unconsolidated expansive clay in Tang et al. [49] and λ = 0.275 in Zhao et al. [50].However, the ultimate strength of rubber fiber-reinforced expansive clay is almost unaffected by freeze-thaw cycles, with λ = 0.The ultimate strength after freeze-thaw cycling is obtained through the deterioration parameters and the ultimate strength without freeze-thaw cycles, expressed as Equation ( 8): Equation ( 9) shows the Mohr-Coulomb strength theory for soil, in which the shear strength of the soil can be divided into cohesive and frictional components.
where f τ denotes the shear strength, c denotes the cohesive force, φ denotes the internal friction angle, and σ denotes the normal stress on the shear surface.
The ultimate strength can be obtained from the cohesion force and internal friction angle according to the geometrical relationship of Mohr's circle, as expressed in Equation ( 10): ( ) By substituting Equations ( 3), ( 5), ( 7), (8), and (10) into Equation ( 2), the formula for the deviatoric stress after different numbers of freeze-thaw cycles at different confining pressures can be obtained, and is expressed as Equation (11): Equation (11) demonstrates that the deviatoric stress is related to the number of freeze-thaw cycles, confining pressure, and axial strain.The deviatoric stress-axial strain curve after any number of freeze-thaw cycles at any confining pressure can be obtained according to Equation (11).The cohesive force and internal friction angle were determined using conventional triaxial tests.The parameters needed for Equation (11) are listed in Table 4.The deviatoric stress-axial strain curves were calculated for different confining The ultimate strength after freeze-thaw cycling is obtained through the deterioration parameters and the ultimate strength without freeze-thaw cycles, expressed as Equation ( 8): Equation ( 9) shows the Mohr-Coulomb strength theory for soil, in which the shear strength of the soil can be divided into cohesive and frictional components.
where τ f denotes the shear strength, c denotes the cohesive force, φ denotes the internal friction angle, and σ denotes the normal stress on the shear surface.
The ultimate strength can be obtained from the cohesion force and internal friction angle according to the geometrical relationship of Mohr's circle, as expressed in Equation ( 10): By substituting Equations (3), ( 5), ( 7), (8), and (10) into Equation (2), the formula for the deviatoric stress after different numbers of freeze-thaw cycles at different confining pressures can be obtained, and is expressed as Equation (11): Equation (11) demonstrates that the deviatoric stress is related to the number of freezethaw cycles, confining pressure, and axial strain.The deviatoric stress-axial strain curve after any number of freeze-thaw cycles at any confining pressure can be obtained according to Equation (11).The cohesive force and internal friction angle were determined using conventional triaxial tests.The parameters needed for Equation (11) are listed in Table 4.The deviatoric stress-axial strain curves were calculated for different confining pressures after different numbers of freeze-thaw cycles, and the partial results shown in Figure 11 indicate that Equation (11) yielded satisfactory results.

Conclusions
In this study, the mechanical properties of rubber fiber-reinforced expansive clay under freeze-thaw cycles were investigated and compared with those of unreinforced expansive clay.The main conclusions are as follows.
(1) The mixing of rubber fibers reduced the effect of freeze-thaw cycles on the shear strength and elastic modulus of expansive clay.The shear strength and elastic modulus of rubber fiber-reinforced expansive clay decreased slightly with an increase in the number of freeze-thaw cycles.However, the shear strength and elastic modulus of the unreinforced expansive clay decreased rapidly with an increase in the number of freeze-thaw cycles.

Conclusions
In this study, the mechanical properties of rubber fiber-reinforced expansive clay under freeze-thaw cycles were investigated and compared with those of unreinforced expansive clay.The main conclusions are as follows.
(1) The mixing of rubber fibers reduced the effect of freeze-thaw cycles on the shear strength and elastic modulus of expansive clay.The shear strength and elastic modulus of rubber fiber-reinforced expansive clay decreased slightly with an increase in the number of freeze-thaw cycles.However, the shear strength and elastic modulus of the unreinforced expansive clay decreased rapidly with an increase in the number of freeze-thaw cycles.
(2) The relationship between the number of freeze-thaw cycles and the ultimate strength was established, which can describe the change in ultimate strength under freezethaw cycles.The ultimate strength of rubber fiber-reinforced expansive clay does not change markedly as the number of freeze-thaw cycles increases, and the deterioration parameter λ is 0. In contrast, the ultimate strength of the unreinforced expansive clay significantly decreased after freeze-thaw cycles, with the deterioration parameters as 0.336 and 0.275.(3) A calculation model of the deviatoric stress-axial strain curves after freeze-thaw cycles was established, which can describe the deviatoric stress-axial strain behavior of rubber fiber-reinforced expansive clay and unreinforced expansive clay under different confining pressures and numbers of freeze-thaw cycles.

Figure 1 .
Figure 1.Particle size distribution curve of the expansive clay.

Figure 1 .
Figure 1.Particle size distribution curve of the expansive clay.

Figure 2 .
Figure 2. Rubber fiber-reinforced expansive clay specimen (a) before test and (b) after test.

Figure 2 .
Figure 2. Rubber fiber-reinforced expansive clay specimen (a) before test and (b) after test.

Figure 4 .
Figure 4. Shear strength with different number of freeze-thaw cycles.

Figure 4 .
Figure 4. Shear strength with different number of freeze-thaw cycles.

Figure 4 .
Figure 4. Shear strength with different number of freeze-thaw cycles.

Figure 6 .
Figure 6.Elastic modulus with different number of freeze-thaw cycles.

Figure 6 .
Figure 6.Elastic modulus with different number of freeze-thaw cycles.

Figure 6 .
Figure 6.Elastic modulus with different number of freeze-thaw cycles.

1 ε 1 ε
curves are drawn according to the test results of rubber fiber-re- inforced expansive clay and the results of unreinforced expansive clay conducted by Tang et al.[49] and Zhao et al.[50].The partial results are displayed in Figure8.The intercept a and slope b were obtained by linear fitting of the curves, and the ini- tial elastic modulus and ultimate strength after different numbers of freeze-thaw cycles at different confining pressures were obtained by substituting the correspondence expressed in Equation (

Table 1 .
Characteristic parameters for the expansive clay.

Table 3 .
Results for initial elastic modulus and ultimate strength.

Table 4 .
Parameters.Appl.Sci.2024, 14, x FOR PEER REVIEW 12 of 15 pressures after different numbers of freeze-thaw cycles, and the partial results shown in Figure 11 indicate that Equation (11) yielded satisfactory results.