Study on Dynamic Modulus and Damping Characteristics of Modified Expanded Polystyrene Lightweight Soil under Cyclic Load

In recent years, expanded polystyrene (EPS) lightweight soil has been widely used as subgrade in soft soil areas because of its light weight and environmental protection. This study aimed to investigate the dynamic characteristics of sodium silicate modified lime and fly ash treated EPS lightweight soil (SLS) under cyclic loading. The effects of EPS particles on the dynamic elastic modulus (Ed) and damping ratio (λ) of SLS were determined through dynamic triaxial tests at various confining pressures (σ3), amplitudes, and cycle times. Mathematical models of the Ed of the SLS, cycle times, and σ3 were established. The results revealed that the EPS particle content played a decisive role in the Ed and λ of the SLS. The Ed of the SLS decreased with an increase in the EPS particle content (EC). The Ed decreased by 60% in the 1–1.5% range of the EC. The existing forms of lime fly ash soil and EPS particles in the SLS changed from parallel to series. With an increase in σ3 and amplitude, the Ed of the SLS gradually decreased, the λ generally decreased, and the λ variation range was within 0.5%. With an increase in the number of cycles, the Ed of the SLS decreased. The Ed value and the number of cycles satisfied the power function relationship. Additionally, it can be found from the test results that 0.5% to 1% was the best EPS content for SLS in this work. In addition, the dynamic elastic modulus prediction model established in this study can better describe the varying trend of the dynamic elastic modulus of SLS under different σ3 values and load cycles, thereby providing a theoretical reference for the application of SLS in practical road engineering.


Introduction
Soft ground is widespread in coastal areas and cities. It is known that the insufficient bearing capacity of a soft soil foundation is easily subjected to excessive settlement under the action of traffic load, and weight of subgrade and pavement structure. This eventually leads to engineering problems such as bridge head uplift and pavement cracking [1][2][3][4]. Reducing subgrade dead weight is one of the effective methods for reducing foundation settlement, so lightweight geomaterials have attracted wide interest.
Expanded polystyrene (EPS) granules, EPS blocks, and waste tire scraps are commonly used lightweight geomaterials [5][6][7]. Compared with EPS blocks, EPS particles are more compatible and can fill structures with arbitrary shapes [8]. Therefore, EPS particles are widely used in the field of civil engineering, especially as the aggregate of lightweight soil with advantages such as low density, thermal insulation, good compatibility, wide sources, and low price [9][10][11]. EPS granular lightweight soil is a new material formed was obtained from Bengbu Jingcheng Chemical Co., Ltd., Bengbu City, Anhui Province, China, with a concentration of 40% and modulus of 3.2 ( Figure 1b).

Test Instrument and Test Scheme
The dynamic triaxial instrument used in this test is produced by GDS Company. The instrument consists of five parts including the confining pressure controller, back pressure controller, cyclic load control host, data collector, and control system, as shown in Figure  2.

Test Instrument and Test Scheme
The dynamic triaxial instrument used in this test is produced by GDS Company. The instrument consists of five parts including the confining pressure controller, back pressure controller, cyclic load control host, data collector, and control system, as shown in Figure 2. was obtained from Bengbu Jingcheng Chemical Co., Ltd., Bengbu City, Anhui Province, China, with a concentration of 40% and modulus of 3.2 ( Figure 1b).

Test Instrument and Test Scheme
The dynamic triaxial instrument used in this test is produced by GDS Company. The instrument consists of five parts including the confining pressure controller, back pressure controller, cyclic load control host, data collector, and control system, as shown in Figure  2.  According to the "Technical Guidelines for Construction of Highway Roadbases" JTG/T F20-2015 [37] and reference [38], the proportion of lime and fly ash was 1:3, the mass of lime and fly ash accounted for 20% of the mass of dry soil, the moisture content was 50%, and the masses of EPS particles were 0.5%, 1%, 1.5%, 2%, and 2.5% of the mass of dry soil. In addition, through an unconfined compression test and scanning electron microscope test, different proportions of sodium silicate were added to the EPS granular lightweight soil with lime and fly ash. The results are shown in Figure 3. It was found that the UCS reached a maximum when the sodium silicate content was 6-8%. The SEM test showed that, compared with no sodium silicate, after adding 6% sodium silicate, the internal hydration products (calcium silicate (C-S-H), calcium aluminate (C-A-H)) and gelling products (hydrated calcium aluminosilicate (sodium) (C (N)-A-S-H)) increased significantly, the porosity decreased significantly, and the strength improved [11,35]. Considering the comprehensive economy, the quality of sodium silicate was 6% of the mass of the dry soil. Based on this, dynamic triaxial tests with different confining pressures, amplitudes, and cycle times were conducted. The specific test schemes are listed in Table 4. In this test, displacement was used as the control mode to load it. The loading waveform was sinusoidal, and 50 points were collected for each cycle. This test simulated the traffic cyclic load, and the vibration frequency was 1 Hz [39]. As the axial displacement cyclic loading mode was adopted, the dynamic characteristics changed mainly at the initial stage of cyclic loading; therefore, the maximum number of cycles selected for this study was 100. According to the "Technical Guidelines for Construction of Highway Roadbases" JTG/T F20-2015 [37] and reference [38], the proportion of lime and fly ash was 1:3, the mass of lime and fly ash accounted for 20% of the mass of dry soil, the moisture content was 50%, and the masses of EPS particles were 0.5%, 1%, 1.5%, 2%, and 2.5% of the mass of dry soil. In addition, through an unconfined compression test and scanning electron microscope test, different proportions of sodium silicate were added to the EPS granular lightweight soil with lime and fly ash. The results are shown in Figure 3. It was found that the UCS reached a maximum when the sodium silicate content was 6-8%. The SEM test showed that, compared with no sodium silicate, after adding 6% sodium silicate, the internal hydration products (calcium silicate (C-S-H), calcium aluminate (C-A-H)) and gelling products (hydrated calcium aluminosilicate (sodium) (C (N)-A-S-H)) increased significantly, the porosity decreased significantly, and the strength improved [11,35]. Considering the comprehensive economy, the quality of sodium silicate was 6% of the mass of the dry soil. Based on this, dynamic triaxial tests with different confining pressures, amplitudes, and cycle times were conducted. The specific test schemes are listed in Table  4. In this test, displacement was used as the control mode to load it. The loading waveform was sinusoidal, and 50 points were collected for each cycle. This test simulated the traffic cyclic load, and the vibration frequency was 1 Hz [39]. As the axial displacement cyclic loading mode was adopted, the dynamic characteristics changed mainly at the initial stage of cyclic loading; therefore, the maximum number of cycles selected for this study was 100.

Specimen Preparation
As a large number of EPS particles were mixed in light soil, the sample preparation method of layered compaction will lead to deformation and rebound of EPS particles, which will eventually lead to soil rebound, cracking, and other problems. Therefore, the sample was prepared using manual layered vibration molding. The sample preparation process is as follows (see Figure 4): (1) The soil was dried, crushed, and sieved in a 2 mm sieve before being mechanically combined with lime and fly ash until it was uniformly dispersed. Simultaneously, water and the sodium silicate solution were mixed and stirred evenly. (2) The mixture of sodium silicate solution and water was poured into the mixture of lime, fly ash, and soil and stirred for 3 min. Finally, it was poured into weighed EPS particles and stirred until the EPS particles were evenly distributed in the soil. 3 times and vibrated 50 times after each loading. The mixture was allowed to stand for 3 h after being vibrated 3 times. A scraper was then used to smooth the surface of the sample. The bottom plates at both ends and the sample maker were removed, and the sample was demolded, then obtained. (4) The sample was placed in a standard curing room for 28 d. During curing, the temperature in the standard curing room fluctuated at 20 ± 2 • C, and the humidity was always above 90%.

Specimen Preparation
As a large number of EPS particles were mixed in light soil, the sample preparation method of layered compaction will lead to deformation and rebound of EPS particles, which will eventually lead to soil rebound, cracking, and other problems. Therefore, the sample was prepared using manual layered vibration molding. The sample preparation process is as follows (see Figure 4): (1) The soil was dried, crushed, and sieved in a 2 mm sieve before being mechanically combined with lime and fly ash until it was uniformly dispersed. Simultaneously, water and the sodium silicate solution were mixed and stirred evenly. (2) The mixture of sodium silicate solution and water was poured into the mixture of lime, fly ash, and soil and stirred for 3 min. Finally, it was poured into weighed EPS particles and stirred until the EPS particles were evenly distributed in the soil. times and vibrated 50 times after each loading. The mixture was allowed to stand for 3 h after being vibrated 3 times. A scraper was then used to smooth the surface of the sample. The bottom plates at both ends and the sample maker were removed, and the sample was demolded, then obtained. (4) The sample was placed in a standard curing room for 28 d. During curing, the temperature in the standard curing room fluctuated at 20 ± 2 °C, and the humidity was always above 90%. The weight of the sample after curing is shown in Table 5. The weight of the sample after curing is shown in Table 5.

Test Results and Discussion
E d and λ mostly represent the dynamic properties of soil [40,41]. A series of hysteresis loops can be obtained under the action of axial cyclic loads [42]. Figure 5 shows a typical hysteresis loop [43,44], where E d is the ratio of the maximum stress difference (Equation (1) to the maximum strain difference (Equation (2)) under the action of a cyclic load [45,46]. The calculation formula is shown in Equation (3).
where σ i,max and σ i,min represent the maximum and minimum dynamic stresses corresponding to the current hysteresis loop, respectively (kPa). ε i,max and ε i,min represent the maximum and minimum dynamic strains corresponding to the current hysteresis loop, respectively (%). E d is the dynamic modulus of elasticity (MPa). ∆σ represents the dynamic stress (kPa), and ∆ε is the dynamic strain (%).

Test Results and Discussion
Ed and λ mostly represent the dynamic properties of soil [40,41]. A series of hysteresis loops can be obtained under the action of axial cyclic loads [42]. Figure 5 shows a typical hysteresis loop [43,44], where Ed is the ratio of the maximum stress difference (Equation (1)) to the maximum strain difference (Equation (2)) under the action of a cyclic load [45,46]. The calculation formula is shown in Equation (3).  The damping of soil reflects the characteristics of the dissipation of deformation energy with vibration due to retardation under dynamic loads. The commonly used damping ratio λ indicates its size [47,48]. The larger the λ, the larger the soil mass is, and the greater is its ability to resist vibration attenuation. Equation (4) is the formula for λ: where λ is the damping ratio, A0 is the area enclosed by the dynamic stress and dynamic strain hysteretic loops, and Ar is the area of the triangular AOB, as shown in Figure 5. The damping of soil reflects the characteristics of the dissipation of deformation energy with vibration due to retardation under dynamic loads. The commonly used damping ratio λ indicates its size [47,48]. The larger the λ, the larger the soil mass is, and the greater is its ability to resist vibration attenuation. Equation (4) is the formula for λ: where λ is the damping ratio, A 0 is the area enclosed by the dynamic stress and dynamic strain hysteretic loops, and A r is the area of the triangular AOB, as shown in Figure 5.  Figure 6 shows the effect of EC on E d . It can be observed that the E d of the SLS had the same change rule under different constraining pressures; that is, E d decreased in a "Z" shape with an increase in EC, indicating that EC plays a decisive role in the E d of the SLS. This was because the elastic modulus of EPS particles was smaller than that of sodium silicate solution modified lime fly ash soil (LFS). With an increase in EC, the overall elastic modulus decreased. After cyclic loading, EPS particles and LFS were unable to deform, so they separated along the interface and reduced E d [49] When EPS particles were mixed into the LFS, the LFS produced pores. The larger the number of EPS particles, the larger the number of pores [50]. The E d decline mechanism was analyzed in three stages. Figure 6 shows the effect of EC on Ed. It can be observed that the Ed of the SLS had the same change rule under different constraining pressures; that is, Ed decreased in a "Z" shape with an increase in EC, indicating that EC plays a decisive role in the Ed of the SLS. This was because the elastic modulus of EPS particles was smaller than that of sodium silicate solution modified lime fly ash soil (LFS). With an increase in EC, the overall elastic modulus decreased. After cyclic loading, EPS particles and LFS were unable to deform, so they separated along the interface and reduced Ed [49] When EPS particles were mixed into the LFS, the LFS produced pores. The larger the number of EPS particles, the larger the number of pores [50]. The Ed decline mechanism was analyzed in three stages. (1) Slow-descent stage When EC < 1%, there were relatively few EPS particles in the SLS and more consolidated soil in the SLS of the same volume. Lime fly ash and sodium silicate underwent hydration, ion exchange, crystallization, and other reactions during maintenance in the soil. The generated cement filled the pores and refined the pore size, thereby making the overall skeleton more compact [45][46][47][51][52][53]. Under the action of cyclic loading, the damage to the overall skeleton was not obvious; therefore, the Ed of the SLS decreased slowly with an increase in EC.

Influence of EPS
(2) Rapid descent stage When 1% < EC < 1.5%, the LFS in the SLS was relatively small. Under the action of a cyclic load, the overall skeleton was gradually damaged and loosened, and then gradually deformed. Therefore, an increase in EC led to a rapid decrease in the Ed of the SLS, and the Ed decreased by 60%.
(3) Slow decline to stabilize stage When EC > 1.5%, more EPS particles were present in the SLS. Under cyclic loading, the entire skeleton was more likely to be damaged, resulting in less constraint of the skeleton on the EPS and easier EPS deformation. The elastic properties of EPS particles played a role; thus, the Ed of the SLS gradually decreased and stabilized at the modulus of the EPS particles with an increase in EC. (1) Slow-descent stage When EC < 1%, there were relatively few EPS particles in the SLS and more consolidated soil in the SLS of the same volume. Lime fly ash and sodium silicate underwent hydration, ion exchange, crystallization, and other reactions during maintenance in the soil. The generated cement filled the pores and refined the pore size, thereby making the overall skeleton more compact [45][46][47][51][52][53]. Under the action of cyclic loading, the damage to the overall skeleton was not obvious; therefore, the E d of the SLS decreased slowly with an increase in EC.
(2) Rapid descent stage When 1% < EC < 1.5%, the LFS in the SLS was relatively small. Under the action of a cyclic load, the overall skeleton was gradually damaged and loosened, and then gradually deformed. Therefore, an increase in EC led to a rapid decrease in the E d of the SLS, and the E d decreased by 60%.
(3) Slow decline to stabilize stage When EC > 1.5%, more EPS particles were present in the SLS. Under cyclic loading, the entire skeleton was more likely to be damaged, resulting in less constraint of the skeleton on the EPS and easier EPS deformation. The elastic properties of EPS particles played a role; thus, the E d of the SLS gradually decreased and stabilized at the modulus of the EPS particles with an increase in EC.

Series Parallel Model for E d of SLS
EPS particles and LFS are two kinds of materials with different physical properties. The elastic modulus E1 of EPS particles and E2 of LFS are not equal, and both resist deformation together. In addition, in general physical phenomena series and parallel are common forms of material arrangement. For example, the effective thermal conductivity system of porous material composed of any two phases is determined by the thermal conductivity of the two phases and their distribution. Therefore, when studying the E d of SLS, the arrangement form of EPS particles and LFS is regarded as a linear combination of series and parallel [54,55]. Assuming that the SLS sample was made up of a series of The elastic modulus E1 of EPS particles and E2 of LFS are not equal, and both resist deformation together. In addition, in general physical phenomena series and parallel are common forms of material arrangement. For example, the effective thermal conductivity system of porous material composed of any two phases is determined by the thermal conductivity of the two phases and their distribution. Therefore, when studying the Ed of SLS, the arrangement form of EPS particles and LFS is regarded as a linear combination of series and parallel [54,55]. Assuming that the SLS sample was made up of a series of springs, the sample could contain series and parallel forms of LFS and EPS particles, as shown in Figure 7. As shown in Figure 7b, the EPS particles in the SLS were connected in series with the LFS, and their elastic modulus is expressed in Equation (5).
where Es is the elastic modulus in the series, is the strain of the EPS particles, and is the strain of the LFS. E1 is the elastic modulus of the EPS particles (generally, 2.5 MPa) [50,56], and E2 is the elastic modulus of the LFS. The elastic modulus was 311.5 MPa based on the preliminary test. Based on Equation (5), Es = 2.48 MPa.
As shown in Figure 7c, the EPS particles in the SLS were connected in parallel with the LFS, and their elastic modulus is expressed in Equation (6).
where Ep is the elastic modulus in parallel mode and is the strain of the SLS sample. Based on Equation (6), Ep = 314 MPa.
In the actual test, both series and parallel modes co-existed in the SLS sample, as shown in Figure 7a. Assuming that they existed in the form of a linear superposition, the elastic modulus of the SLS is expressed in Equation (7).
where η denotes the proportion of the series mode. When η = 1, the particles of the LFS and EPS in the SLS existed in series, and their elastic moduli were small. When η = 0, the LFS and EPS particles in SLS existed in parallel, and their elastic moduli were large. The As shown in Figure 7b, the EPS particles in the SLS were connected in series with the LFS, and their elastic modulus is expressed in Equation (5).
where E s is the elastic modulus in the series, ε 1 is the strain of the EPS particles, and ε 2 is the strain of the LFS. E 1 is the elastic modulus of the EPS particles (generally, 2.5 MPa) [50,56], and E 2 is the elastic modulus of the LFS. The elastic modulus was 311.5 MPa based on the preliminary test. Based on Equation (5), E s = 2.48 MPa. As shown in Figure 7c, the EPS particles in the SLS were connected in parallel with the LFS, and their elastic modulus is expressed in Equation (6).
where E p is the elastic modulus in parallel mode and ε is the strain of the SLS sample. Based on Equation (6), E p = 314 MPa.
In the actual test, both series and parallel modes co-existed in the SLS sample, as shown in Figure 7a. Assuming that they existed in the form of a linear superposition, the elastic modulus of the SLS is expressed in Equation (7).
where η denotes the proportion of the series mode. When η = 1, the particles of the LFS and EPS in the SLS existed in series, and their elastic moduli were small. When η = 0, the LFS and EPS particles in SLS existed in parallel, and their elastic moduli were large. The value of η is related to the EC and load type and was calculated using the experimental data. The value of η was obtained using Equation (7). Under different EC conditions, the proportions of SLS samples in series and parallel were obtained. The relationship between η and EC is illustrated in Figure 8. It can be observed that when EC was less than 1%, the value of η was less than 0.23, and the change was slight. The EPS and LFS in the sample mostly existed in parallel. When the EC increased from 1% to 1.5%, the η value increased rapidly from 0.23 to 0.65, and the LFS and EPS particles in the sample changed mainly from being in parallel to being in series. When the EC value was greater than 1.5%, the η value increased gradually to 0.74 with an increase in EPS content. The LFS and EPS particles mainly existed in series mode in the sample. The series and parallel models reasonably explained how the E d in Figure 6 decreased in three stages of the "Z" shape with an increase in EC.
proportions of SLS samples in series and parallel were obtained. The relationship between η and EC is illustrated in Figure 8. It can be observed that when EC was less than 1%, the value of η was less than 0.23, and the change was slight. The EPS and LFS in the sample mostly existed in parallel. When the EC increased from 1% to 1.5%, the η value increased rapidly from 0.23 to 0.65, and the LFS and EPS particles in the sample changed mainly from being in parallel to being in series. When the EC value was greater than 1.5%, the η value increased gradually to 0.74 with an increase in EPS content. The LFS and EPS particles mainly existed in series mode in the sample. The series and parallel models reasonably explained how the Ed in Figure 6 decreased in three stages of the "Z" shape with an increase in EC.  Figure 9 shows the impact of EPS particles on the λ value of the SLS. It can be observed that for SLS under different σ3 environments, the changing trend of λ is an "N" shape with the content of EPS particles. This shows that EC plays an important role in the λ of the SLS. The change mechanisms were analyzed in three stages.

Effect of EPS Particle Content on λ
(1) When EC < 1%, λ first increased with an increase in EC. This was because, as the cyclic load progressed, the overall skeleton loosened, and λ increased because the energy dissipated by friction increased. (2) When 1% < EC < 1.5%, λ decreased with an increase in EC. This was because, as the cyclic load progressed and when the EC was approximately 1.5%, the EPS particles in SLS were completely compacted. The slip and dislocation of particles in the SLS were reduced, and λ decreased owing to the reduction in energy dissipated by friction and elastic deformation of EPS particles. (3) When EC > 1.5%, λ increased with increasing EPS particle content. This was caused by the increase in the EPS particle content, which replaced the LFS. As the pore structure in the SLS increased, the LFS volume decreased, and more energy was lost in the propagation of stress waves in the SLS, which contained a large proportion of EPS particles. As the cyclic loading continued, there were still many EPS particles that were not compacted. This portion of the EPS particles produced elastic deformation and consumed energy; thus, λ increased again.  Figure 9 shows the impact of EPS particles on the λ value of the SLS. It can be observed that for SLS under different σ 3 environments, the changing trend of λ is an "N" shape with the content of EPS particles. This shows that EC plays an important role in the λ of the SLS. The change mechanisms were analyzed in three stages.  Figure 10 shows the effect of σ3 on Ed. When EC was the same, Ed follows the σ3 raise and lower. This was caused by the fact that increasing the confining pressure decreases the elastic characteristics of EPS particles, and that cyclic loading causes an increase in irreversible plastic deformation. The deformation of the SLS increased, resulting in a decrease in Ed. However, when EC was greater than 2%, the decrease in Ed with an increase in the confining pressure was not obvious. This was because the EC was too high and the increase in the confining pressure could not effectively inhibit the elastic properties of the EPS particles. There were still some EPS particles in the SLS that could provide elastic properties; therefore, the reduction in Ed was not apparent. (1) When EC < 1%, λ first increased with an increase in EC. This was because, as the cyclic load progressed, the overall skeleton loosened, and λ increased because the energy dissipated by friction increased. (2) When 1% < EC < 1.5%, λ decreased with an increase in EC. This was because, as the cyclic load progressed and when the EC was approximately 1.5%, the EPS particles in SLS were completely compacted. The slip and dislocation of particles in the SLS were reduced, and λ decreased owing to the reduction in energy dissipated by friction and elastic deformation of EPS particles.

Effect of Confining Pressure on Ed
(3) When EC > 1.5%, λ increased with increasing EPS particle content. This was caused by the increase in the EPS particle content, which replaced the LFS. As the pore structure in the SLS increased, the LFS volume decreased, and more energy was lost in the propagation of stress waves in the SLS, which contained a large proportion of EPS particles. As the cyclic loading continued, there were still many EPS particles that were not compacted. This portion of the EPS particles produced elastic deformation and consumed energy; thus, λ increased again.  Figure 10 shows the effect of σ 3 on E d . When EC was the same, E d follows the σ 3 raise and lower. This was caused by the fact that increasing the confining pressure decreases the elastic characteristics of EPS particles, and that cyclic loading causes an increase in irreversible plastic deformation. The deformation of the SLS increased, resulting in a decrease in E d . However, when EC was greater than 2%, the decrease in E d with an increase in the confining pressure was not obvious. This was because the EC was too high and the increase in the confining pressure could not effectively inhibit the elastic properties of the EPS particles. There were still some EPS particles in the SLS that could provide elastic properties; therefore, the reduction in E d was not apparent.  Figure 10 shows the effect of σ3 on Ed. When EC was the same, Ed follows the σ3 raise and lower. This was caused by the fact that increasing the confining pressure decreases the elastic characteristics of EPS particles, and that cyclic loading causes an increase in irreversible plastic deformation. The deformation of the SLS increased, resulting in a decrease in Ed. However, when EC was greater than 2%, the decrease in Ed with an increase in the confining pressure was not obvious. This was because the EC was too high and the increase in the confining pressure could not effectively inhibit the elastic properties of the EPS particles. There were still some EPS particles in the SLS that could provide elastic properties; therefore, the reduction in Ed was not apparent.   Figure 11 shows the effect of σ 3 on λ. When EC remained constant, the λ value of the SLS decreased as σ 3 increased; however, the total variation range of λ with σ 3 was minimal, falling within 0.5%. This was consistent with the test results of Lu [3]. This was because the interior of the SLS was more tightly packed due to the increase in σ 3 . Thus, there was less sliding and dislocation between them and less friction between the various particles. The value of λ decreased because there was less energy lost by the stress wave during its propagation. Figure 11 shows the effect of σ3 on λ. When EC remained constant, the λ value of the SLS decreased as σ3 increased; however, the total variation range of λ with σ3 was minimal, falling within 0.5%. This was consistent with the test results of Lu [3]. This was because the interior of the SLS was more tightly packed due to the increase in σ3. Thus, there was less sliding and dislocation between them and less friction between the various particles. The value of λ decreased because there was less energy lost by the stress wave during its propagation.  Figure 12 shows the effect of the amplitude on Ed. It can be observed that when EC was the same, Ed decreased with an increase in amplitude. This was because the increase in amplitude (although small amplitude was not sufficient to destroy the SLS) loosened the overall skeleton of the SLS and made the EPS particles easier to compact. The greater the amplitude, the greater the deformation of SLS, resulting in a decrease in Ed. The Ed of SLS with high EC decreased relatively slowly, with an increase in amplitude owed to excessive EC. Under cyclic loading, the increase in amplitude did not effectively inhibit the elastic potential energy of the EPS particles. The deformation of the SLS decreased slowly, indicating that Ed decreased relatively slowly with a decrease in amplitude.  Figure 12 shows the effect of the amplitude on E d . It can be observed that when EC was the same, E d decreased with an increase in amplitude. This was because the increase in amplitude (although small amplitude was not sufficient to destroy the SLS) loosened the overall skeleton of the SLS and made the EPS particles easier to compact. The greater the amplitude, the greater the deformation of SLS, resulting in a decrease in E d . The E d of SLS with high EC decreased relatively slowly, with an increase in amplitude owed to excessive EC. Under cyclic loading, the increase in amplitude did not effectively inhibit the elastic potential energy of the EPS particles. The deformation of the SLS decreased slowly, indicating that E d decreased relatively slowly with a decrease in amplitude.  Figure 13 shows the effect of amplitude on λ. As can be observed, the λ of the SLS decreased as the amplitude increased, and the variation range was very limited. This was caused by the increase in the amplitude. During cyclic loading, the interior of the SLS was gradually compacted, and the slip and dislocation of particles in the SLS were reduced, reducing the energy consumed by friction. When EC was 0.5%, the λ value of the SLS increased rapidly with an increase in amplitude. This was because when EC was small  Figure 13 shows the effect of amplitude on λ. As can be observed, the λ of the SLS decreased as the amplitude increased, and the variation range was very limited. This was caused by the increase in the amplitude. During cyclic loading, the interior of the SLS was gradually compacted, and the slip and dislocation of particles in the SLS were reduced, reducing the energy consumed by friction. When EC was 0.5%, the λ value of the SLS increased rapidly with an increase in amplitude. This was because when EC was small there was more LFS in the same volume of the SLS. With an increase in amplitude, under the action of a cyclic load, the overall skeleton loosens. Because of the increase in energy dissipated by friction, more energy was lost during the transmission process of the stress wave, resulting in an increase in λ. Amplitude (mm) Figure 12. Relationship between amplitude and Ed. Figure 13 shows the effect of amplitude on λ. As can be observed, the λ of the SLS decreased as the amplitude increased, and the variation range was very limited. This was caused by the increase in the amplitude. During cyclic loading, the interior of the SLS was gradually compacted, and the slip and dislocation of particles in the SLS were reduced, reducing the energy consumed by friction. When EC was 0.5%, the λ value of the SLS increased rapidly with an increase in amplitude. This was because when EC was small there was more LFS in the same volume of the SLS. With an increase in amplitude, under the action of a cyclic load, the overall skeleton loosens. Because of the increase in energy dissipated by friction, more energy was lost during the transmission process of the stress wave, resulting in an increase in λ.  Figure 13. Relationship between amplitude and λ. Figure 13. Relationship between amplitude and λ. Figure 14 shows hysteretic curves under different confining pressures and different cycles. Figure 14a shows the stress-strain curves (hysteretic loops) of the cyclic load when the EC was 1.5% under different confining pressures. The dip angle of the hysteresis ring can represent the relaxation degree of soil under the action of axial dynamic stress, and can reflect the change of soil stiffness under the action of dynamic load and the cumulative damage degree in the sample [57]. That is, the change in the inclination of the hysteresis loop can reflect the E d change of the SLS under cyclic loading. The smaller the inclination, the smaller the E d . As shown in Figure 14a, with an increase in σ 3 , the hysteresis loop gradually moves downward and the inclination decreases; therefore, the E d of the SLS decreased with an increase in σ 3 . This conclusion is consistent with that in the previous text. Figure 15b shows the hysteretic loops of the SLS with the number of cycles applied by the cyclic load when σ 3 = 100 kPa. It is clear that as the cyclic load was applied with more cycles, the inclination of the hysteretic loop gradually decreased and its E d diminished.

Influence of Cycle Times on E d and λ
Taking an EPS content of 1.5% as an example, the E d of each hysteretic loop in a process of 100 cycles was calculated, and the change in the E d of the SLS with the number of cycles under different σ 3 was obtained, as shown in Figure 15. At the initial stage of the cyclic loading, the E d of the SLS under different σ 3 values decreased rapidly with an increase in the number of cycles and then tended to stabilize gradually. This is because at the initial stage of the cyclic loading, after the dynamic load was applied the unrecoverable plastic deformation of the SLS increased, the deformation of EPS particles continued to increase, and Ed decreased rapidly. With the increase in the number of cycles and the limitation of the amplitude (0.1 mm), the deformation of the EPS particles did not change and tended to stabilize gradually. The same phenomenon has also appeared in other studies. The researchers believe that this phenomenon is due to the rapid deformation of EPS during the initial stage of strain cycling, resulting in a rapid change in elastic modulus. As the strain cycle continues, the deformation phase difference of EPS particles causes the deformation of EPS particles to not occur simultaneously with the cyclic strain, so it tends to stabilize [43].
can reflect the change of soil stiffness under the action of dynamic load and the cumulative damage degree in the sample [57]. That is, the change in the inclination of the hysteresis loop can reflect the Ed change of the SLS under cyclic loading. The smaller the inclination, the smaller the Ed. As shown in Figure 14a, with an increase in σ3, the hysteresis loop gradually moves downward and the inclination decreases; therefore, the Ed of the SLS decreased with an increase in σ3. This conclusion is consistent with that in the previous text. Figure 15b shows the hysteretic loops of the SLS with the number of cycles applied by the cyclic load when σ3 = 100 kPa. It is clear that as the cyclic load was applied with more cycles, the inclination of the hysteretic loop gradually decreased and its Ed diminished.   can reflect the change of soil stiffness under the action of dynamic load and the cumulative damage degree in the sample [57]. That is, the change in the inclination of the hysteresis loop can reflect the Ed change of the SLS under cyclic loading. The smaller the inclination, the smaller the Ed. As shown in Figure 14a, with an increase in σ3, the hysteresis loop gradually moves downward and the inclination decreases; therefore, the Ed of the SLS decreased with an increase in σ3. This conclusion is consistent with that in the previous text. Figure 15b shows the hysteretic loops of the SLS with the number of cycles applied by the cyclic load when σ3 = 100 kPa. It is clear that as the cyclic load was applied with more cycles, the inclination of the hysteretic loop gradually decreased and its Ed diminished.   Based on the previous analysis, the E d of the SLS first decreased rapidly, then tended to stabilize with an increase in the number of loading cycles, and decreased with an increase in σ 3 . Based on the above rules [58], the E d and cycle times under various σ 3 values satisfied the power function relationship, as shown in Equation (8).
where a and b are related to the confining pressure. Based on the relationship between the confining pressure and E d shown in Figure 15, the functional relationships between the parameters a and b and the confining pressure were determined, as shown in Equations (9) and (10), respectively. a = −0.2189p + 120.32, R 2 = 0.96 Equations (9) and (10) were substituted into Equation (8) to obtain Equation (11).
E d = (−0.2189p + 120.32)N −0.0003p 2 +0.0167p−1.6282 (11) where N is the number of cycles and p is the confining pressure. The predicted value of the dynamic elastic modulus calculated using Equation (11) was compared with the actual value of the dynamic elastic modulus obtained from the tests. The comparison results are shown in Figure 16. It can be observed that the predicted value of E d was consistent with the experimental value, with an error within 5%. The model predicted the varying trend of the E d of the SLS with different confining pressures and cycle times. The predicted value of the dynamic elastic modulus calculated using Equation (9) was compared with the actual value of the dynamic elastic modulus obtained from the tests. The comparison results are shown in Figure 16. It can be observed that the predicted value of Ed was consistent with the experimental value, with an error within 5%. The model predicted the varying trend of the Ed of the SLS with different confining pressures and cycle times. The varying pattern of the Ed was obtained using Equation (9), as shown in Figure 17. It was found that with an increase in confining pressure and cycle times, the Ed of the SLS gradually decreased, and the effect of the confining pressure was relatively obvious. In the yellow area in Figure 17, the Ed of the SLS tended to be minimal. When the number of cycles continued to increase, the Ed tended to stabilize.  The varying pattern of the E d was obtained using Equation (11), as shown in Figure 17. It was found that with an increase in confining pressure and cycle times, the E d of the SLS gradually decreased, and the effect of the confining pressure was relatively obvious. In the yellow area in Figure 17, the E d of the SLS tended to be minimal. When the number of cycles continued to increase, the E d tended to stabilize. The varying pattern of the Ed was obtained using Equation (11), as shown in Figure  17. It was found that with an increase in confining pressure and cycle times, the Ed of the SLS gradually decreased, and the effect of the confining pressure was relatively obvious. In the yellow area in Figure 17, the Ed of the SLS tended to be minimal. When the number of cycles continued to increase, the Ed tended to stabilize.

Conclusions
Adding EPS particles into subgrade filling material can effectively reduce the density of filling material, improve the stability of the original subgrade, and reduce the uneven

Conclusions
Adding EPS particles into subgrade filling material can effectively reduce the density of filling material, improve the stability of the original subgrade, and reduce the uneven settlement of soft subgrade. In this study, the effects of EPS particle content, confining pressure, amplitude, and cycle times on the E d and λ of SLS were evaluated by using a dynamic triaxial test, which provides theoretical basis and data support for the application of SLS in road engineering. The main conclusions are as follows: (1) The EPS particle content played a decisive role in the E d and λ of the SLS, and the optimal dosage of EPS particles is between 0.5% and 1%. When σ 3 and amplitude are constant, η increases with the increase of EC, and the layout of EPS and LFS in SLS is transferred from parallel to series. When EC increased from 1% to1.5%, E d decreases by 60%, resulting in a Z-shaped decline in E d of SLS. The λ of SLS is N-shaped with the increase in EPS particle content. (2) When EC was constant, the E d of the SLS decreased with an increase in σ 3 and amplitude; however, the effect of change in confining pressure and amplitude on E d was not obvious under high EC values. This was because the number of EPS particles was too high and some EPS particles in the SLS provided elastic properties; thus, the reduction in E d was not obvious. The λ of the SLS generally decreased with an increase in the confining pressure and amplitude, and the variation range was small (within 0.5%).