The dynamic triaxial apparatus of the British GDS company was used in this test, and the dynamic loading module of the equipment was selected in this study, as shown in Fig. 1. W is the natural moisture content, W_l is the liquid limit, W_p is the plastic limit, G_s is the relative density of soil particles and F_s is the free expansion rate. GDSLSAB software was used to apply axial stress, confining pressure and back pressure to the cylindrical sample with a height of 100 mm and a diameter of 50 mm. The maximum range of the equipment confining pressure was 2 MPa, and the maximum range of the axial force was 10 kN. The measurement system can accurately measure the pore water pressure, axial stress, confining pressure, volumetric strain and other data that change with time. The measurement accuracy of the displacement sensor is 0.0001 mm.

The test soil sample was obtained from an expressway project in Nanyang, Henan Province and was grey yellow. Its physical and mechanical indices are shown in Table 1. According to the free expansion rate of expansive soil, the sample can be classified as weak expansive soil, and its main mineral components are illite and a small amount of kaolinite. Sodium chloride solutions of 0, 0.1, 0.2, 0.3 and 0.6 mol/L were prepared following the standard solution preparation procedure, and the crystal technical conditions met the relevant specifications.

According to the compaction test, the optimal moisture content of the soil sample was 17.3%, and the maximum dry density was 1.7 g/cm³. The soil sample was mixed with sodium chloride solutions and stirred evenly, and a cylindrical sample with a diameter of 50 mm and a height of 100 mm was made according to the mass ratio. The sample preparation process is shown in Fig. 2.

Before the test, the prepared test specimen is put into the vacuum saturator for pumping and saturation, and the concentration of the saturated solution used in the saturation process is the same as the concentration of sodium chloride solution used in the preparation of the specimen. GDSLSAB software was used to apply initial static stress and confining pressure on the soil samples in a linear loading mode to conduct a consolidated undrained shear test. In the dynamic load part of this test, the haversine wave loading mode with frequency f of 1 Hz is adopted [14], and its function expression is as follows:(1) σd=σs+σmsin⁡(2πft) where σ_d is the dynamic stress, σ_s is the initial static stress, σ_m is the dynamic stress amplitude, f is the loading frequency, and t is the loading time. Fig. 3 shows the dynamic loading process of the soil. The OB section is the experimental consolidation stage; the OA section is the confining pressure applied before consolidation; and the AB section is the isotropic consolidation stage by setting the corresponding confining pressure value. In the dynamic load phase, a smaller initial axial force σ_s= 10 kPa needs to be applied first, that is, the initial static stress, to prevent the specimen from decoupling from the applied force rod when the axial force is applied; this is the BC phase. The CD stage is expressed as the amplitude stage of dynamic stress applied to the dynamic load.

Considering that the subgrade soil under vertical load is divided into the upper static load and dynamic load caused by the traffic, the dynamic loads that can transfer to the soil depth are limited. In this limited soil layer, the vertical static stress caused by the overlying static load is relatively small, and a relatively small initial static stress is applied in the testing and adjusting the stiffness coefficient stage. Therefore, no additional initial static stress is set here, and only the vertical dynamic load is considered. The vertical static stress caused by the overlying static load is small, so only the vertical dynamic load is considered. Based on the existing literature and comprehensive consideration, we selected the dynamic stress amplitude σ_d in the range of 60∼150 kPa [15,16], the confining pressure σ₃ = 200 kPa [17–19], the concentration n is 0, 0.1, 0.2, 0.3, 0.6 mol/L [8,11,12], and the number of vibrations N = 10,000. The test is terminated when the axial strain reaches 5% [8–10,16,20,21], as shown in Table 2.

The degree of nonclosure of hysteretic curve ε_p reflects the residual strain of soil under cyclic loading, that is, the greater the degree of nonclosure, the greater the plastic deformation, residual strain and settlement generated when the soil is subjected to dynamic cyclic loading, and vice versa [24]. The value of ε_p can be expressed by the difference in the strain corresponding to the starting point and the ending point of loading in one cycle:

(2) εp=|εs−εe|

where ε_s is the starting point of loading in one cycle and ε_e is the ending point of loading in one cycle.

Under different concentrations and dynamic stress amplitudes, the variation law of the nonclosure degree ε_p of the hysteretic curve of the test soil sample after cyclic dynamic loading is shown in Figs. 5 and 6.

As shown in Fig. 5, at the initial stage of cyclic loading, the nonclosure degree of the hysteretic curve decreases sharply. When the number of cycles N reaches 2,000, the nonclosure degree decreases, and ε_p gradually stabilizes. As the concentration of sodium chloride solutions increased, the relation curve of ε_p-N showed an obvious downwards trend. The nonclosure degree of the hysteretic curve of the plain soil sample is greater than that of the soil sample mixed with sodium chloride solutions. Yao et al. [12] showed that salt solution concentrations were inversely proportional to the thickness of the double electric layer, as shown in Eq. (3):(3) 1K=(ε0Dk0T2n0e2v2)12 where 1/K is the thickness of the double electric layer; ε₀ is the dielectric constant; k₀ is the Boltzmann constant; T is the absolute temperature; n₀ is the ion concentration; e is the electron charge; and v is the number of ions.

Eq. (3) indicates that the increase in concentration can significantly reduce the thickness of the double electric layer of soil particles in expansive soil. Since the free Na⁺ ions in sodium chloride solutions are positively charged, they tend to be attracted by the negatively charged soil particles, achieving electrical equilibrium to a certain extent, and the polar water molecules adsorbed in the diffusion layer between soil particles are correspondingly reduced [25–28]. The subsequent accumulation of dynamic load is actually a process of compaction of soil, the water molecules between soil particles are extruded, and the spacing between particles is constantly reduced. For the sample mixed with sodium chloride solutions, especially the one with a higher concentration of sodium chloride solutions, the reaction is more obvious. At the microscale level, the distance between soil particles has been reduced, and in the process of subsequent cumulative load application, it naturally shows smaller plastic deformation at the macroscale level.

Fig. 6 shows that the soil hysteresis curve ε_p changes significantly under different dynamic stress amplitudes under the same concentration. For σ_d = 60 and 90 kPa, the soil hysteresis curve ε_p-N decreases sharply at first and then tends to be stable, showing a basically closed state. When it increases from σ_d = 90 to 120 and 150 kPa, the ε_p value of the soil hysteretic curve obviously increases at the initial stage, and the overall trend of the curve moves upwards with the increase in the dynamic stress amplitude. In the initial stage of cyclic loading, the larger dynamic stress amplitude will cause more serious damage to the test soil. According to the above phenomena, when the dynamic stress amplitude of the improved expansive soil foundation exceeds 90 kPa, the settlement of the foundation should be given close attention, and the monitoring of the section with the heavy traffic load should be appropriately increased.

The hysteresis curve saturation α can be expressed by the ratio of the short and long axes of ellipses of similar morphological size, i.e., the width of the hysteresis curve, as shown in Eq. (4). The width of the hysteresis curve also reflects the cohesiveness of the soil. The wider the curve is, the greater the viscosity of the soil, and the slower the response to dynamic load changes in practical engineering [29–33].(4) α=EFBC=ab where “a” is the short axis of the hysteretic curve and “b” is the long axis of the hysteretic curve.

Figs. 7 and 8 show the variation rule of the saturation α of the hysteretic curve of the test soil samples under different concentrations and dynamic stress amplitudes after cyclic loading.

From Fig. 7, it is clear that the α value decreased with increasing concentration because the pores in the soil were compressed under the confining pressure in the consolidation stage before the dynamic load test. The degree of pore compression is directly proportional to the concentration of the sodium chloride solutions. With increasing concentration, the double electric layer between soil particles becomes thinner, resulting in a decrease in the distance between particles and an increase in the van der Waals force between molecules and Coulomb force between charged particles. At the same time, embedding and friction between particles is more significant [25–27]. Macroscopically, the elastic deformation of the soil skeleton accounts for a larger proportion of the overall deformation when it is subjected to an external load, and the response to the load is more rapid; that is, the soil viscosity is reduced. At the same concentration, with the increase in N, α decreases until it stabilizes. When the number of cyclic vibrations is less than 300, the saturation of the hysteresis curve changes greatly; when the number of vibrations is more than 300, the curve’s variation amplitude decreases and gradually tends to be stable.

Fig. 8 shows that the amplitude has a significant influence on the saturation α of the hysteretic curve of the test soil sample, and the α-N relationship curve rises gradually with the increase in the amplitude of dynamic stress. As shown in Fig. 8b, when the dynamic stress amplitude σ_d increases from 60 to 150 kPa, the corresponding average variation in α increases from 0.05 to 0.1. In the whole process of cumulative cyclic loading, the internal pores of the soil are constantly compressed with the increase in the number of cyclic loadings, which makes the soil particles bond more closely, and the viscosity improves accordingly. The macroscopic performance results in the hysteresis curve α increasing with increasing amplitude of the dynamic stress.

The slope of the hysteresis curve k reflects the elastic properties and stiffness of the soil. Its value can be expressed by the slope of the line connecting the two endpoints of the hysteresis curve, i.e., the ratio of stress difference to strain difference in the long axis BC, as in Eq. (5).(5) k=tan⁡θ=σC−σBεC−εB where σ_C, σ_B, ε_B and ε_C correspond to the maximum dynamic stress, the minimum dynamic stress, the maximum dynamic strain, and the minimum dynamic strain in a period, respectively. A larger difference indicates a greater elastic property and stiffness of the soil.

Under different concentrations and dynamic stress amplitudes, the slope k of the hysteretic curve of the test soil samples after cyclic dynamic loading changes, as shown in Figs. 9 and 10.

Fig. 9 shows the k-N relationship curves of the hysteresis curve for soil samples with the same dynamic stress amplitude and different concentrations under cyclic loading. The figure shows that the curve gradually shifts upwards as the concentration of sodium chloride solutions increases. For the first 300 times, k tends to decrease as the number of vibrations increases. However, when the number of vibrations exceeds 300, k tends to increase gradually. This indicates that strain hardening of the soil occurred at this stage. This is because there are still many pores in the soil in the early stage of loading, and under the premise that no structural failure occurs in the soil skeleton, there is more allowance for the plastic deformation. With the increase in the vibration times, the pores in the soil body are continuously compressed, the spacing between particles is reduced, and a denser load-bearing structure is formed. The overall strain gradually transitioned from mainly plastic deformation to mainly elastic deformation. Macroscopically, the curve of the single k-N relationship first decreases and then increases during the cyclic loading process [34,35].

Fig. 10 shows the k-N relationship curve of the soil samples at the same concentration and different dynamic stress amplitudes. The k-N curve moves downwards as the dynamic stress amplitudes increase. The above results indicate that in highway engineering, both the influence of the sodium chloride solution concentration on the soil in the pore environment and the influence of highway traffic load on the soil strength and stiffness should be considered to prevent subgrade settlement.

The hysteretic curve area S reflects the energy dissipation and seismic performance of soil due to the damping action. A larger S means that more energy is consumed by the soil. Due to the differences in the forms of the hysteretic curves of the test soil under different small and large dynamic stress amplitude conditions, two methods are adopted to calculate the area of the hysteretic curve, S₁ and S₂, respectively, as shown in formulas (6) and (7):

(6) S1=πab4

(7) S2=∑inSi=−12(|ε1dσ1dε2dσ2d|+|ε2dσ2dε3dσ3d|+⋅⋅⋅+|ε(n−1)dσ(n−1)dεndσnd|+|εndσndε1dσ1d|)

where a is the short axis of the hysteresis curve, b is the long axis of the hysteresis curve, ε_id is the dynamic strain, σ_id is the dynamic stress, and i is the 1-n point of the hysteresis curve.

Under different concentrations and dynamic stress amplitudes, the variation rule of the hysteretic curve area S of the test soil samples after cyclic dynamic loading is shown in Figs. 11 and 12.

As shown in Fig. 11, the S–N relationship curve presents an overall downwards trend with increasing concentration. When the dynamic stress amplitude is small, the S–N relation curve shows a converging state. When the dynamic stress amplitude is large, the S–N relation curve presents a discrete state. Fig. 12 shows the S–N relation curve of the hysteretic curve of the soil samples with the same concentration and different dynamic stress amplitudes under dynamic cyclic loading. With the increase in concentration, the S–N relation curve of the soil samples with different dynamic stress amplitudes gradually draws closer. As shown in Fig. 12c, the S–N relation decreases nonlinearly in the first 300 cycles of cyclic vibration, and when it exceeds 300 cycles, the hysteresis curve area S changes slowly, and the energy loss changes less.

Both concentration and dynamic stress amplitude have an effect on the hysteretic curve area S. The effect of concentration is significant at larger values; at smaller values, the effect of concentration on the soil is relatively weaker. This is because free Na⁺ ions in sodium chloride solutions weakened the expansive soil particles electronegativity and reduced the thickness of the electric double layer. In the consolidation stage, the pores with a high concentration of sodium chloride solutions, compared with those in the lower concentration solution, have been compressed to a greater extent, and the deformable margin has been reduced accordingly. The energy dissipated through deformation under primary cyclic loading can be reflected by the hysteretic curve area S. The reduction of the deformation allowance leads to the weakening of the ability of the soil sample to dissipate energy through deformation, and the load energy can be transmitted through the soil skeleton structure more effectively. From this point of view, under the same dynamic stress amplitude, the increase in sodium chloride solution concentration weakens the energy dissipation performance of the sample, which corresponds to a lower S–N relation curve and reduces the influence of amplitude on the energy dissipation capacity of the sample. With the increase in the dynamic stress amplitude, the total deformation of the sample under cyclic loading increases, and the influence of concentration on the energy dissipation capacity is more obvious, which corresponds to the more discrete S–N relationship curve under different concentrations of sodium chloride solutions.

The stress–strain curve with N < 5,000 times dynamic cyclic loading, namely, the hysteresis curve, was selected. As the number of dynamic cyclic loadings increased, its shape changed from wide fat to narrow, as shown in Fig. 13. At the initial stage of dynamic cyclic loading, the hysteretic curve has a large opening, and it is not closed. With the increase in the number of dynamic cyclic loadings, the test sample becomes increasingly dense in the later loading stage, and the plastic deformation ratio decreases, indicating that cyclic strain hardening occurs on the whole sample.

The above curve rulessuggest that concentration has a great influence on the four morphological characteristics of the hysteretic curve: the degree of nonclosure ε_p, saturation α, slope k and area S. Because all the characteristic parameters of the hysteretic curve meet the development trend of the exponential function, an exponential function is used for the curve fitting analysis of ε_p, α, k and S. To study the development law of the four morphological characteristics of the concentration and hysteresis curve with increasing vibration times, the data of the dynamic stress amplitude σd = 150 kPa are selected and processed accordingly, and the relationship curves ε_p-N, α-N, k-N and S–N can be obtained. It is fitted by exponential functions (8)–(11). The fitting curve is shown in Fig. 14, and the fitting parameters are shown in Table 3.

(8) εp=φe−Nβ+nγ

(9) α=φeNβ+γeNζ+nη

(10) k=φN+βγN+nη

(11) S=φeNβ+γeNζ+nη

In the equations in Table 3, n is the concentration of salt solutions, φ, β, γ, ζ and η are the fitting parameters, and N is the number of vibrations [36–41].

The fitting prediction model shows that the unclosed degree ε_p of the sample changes gently during the cyclic loading process, and the three-parameter exponential prediction model can be used to predict and analyse it well. However, the saturation α and area S of the hysteretic curve are the coupling results of a variety of factors, and the curve changes more sharply during the whole cumulative cyclic loading process. Therefore, the five-parameter exponential prediction model is used to predict and analyse the curve. For the prediction of slope K, the curve shows a pair hook shape that falls first and then rises. Therefore, the shah function, which can describe the trend of the pair hook, should be used to perform the fit. Figs. 14a–14d show that the correlation index function can well reflect the relationship between the concentration and morphological characteristics ε_p, α, k and S of the hysteretic curve under cyclic dynamic loading. The fitting degree R² is above 0.99. These results show that the prediction model and the related fitting parameters can provide a reference for practical engineering [42–45].