Microstructure of loess after WD cycles
Qualitative analysis of SEM images of samples
Figure 5 presents the SEM images (×2000) of the RL after different WD cycles. From the Figure, the soil particles of the RL without WD cycles are mainly silt grains and siltclay aggregates. The contact area between the particles is small, and the contact mode is mainly point contact. The aerial pores distribution has an obvious difference with the intergranular large pores, the aerial pores are interconnective between soil particles. The skeletal structure of the loess presents a clot mosaic colloidal structure. With the increase of WD cycles, the clay films are generated on the surface of silt grains with a diameter of about 30 µm. The partially cemented siltclay aggregates gradually separate during the water migration process, the number of intergranular pores increases and some intergranular pores are gradually connected to form new water migration channels. The cementation between the particles is weakened to produce the loosened soil structure, resulting in the reduction of soil strength. After four WD cycles, the original pores continue to serve as the main water migration channel, and the weakly cemented particles lose their linkage, leading to an increase in the number of intergranular pores, and the soil structure is further reduced. However, after five WD cycles, the changes in the number of pores and soil particles are insignificant, and a new relatively stable structure was formed inside the soil, and the soil strength stabilized with the increase of WD cycles
Figure 6 shows the SEM images (×2000) of BML with different WD cycles. From the figure, the particles of BML without the WD cycle are mainly composed of cemented clay aggregates, the surface of the particles is wrapped by clay film, the contact between particles is mainly surface cementation, and the pores are uniformly distributed and are mainly small and mediumsized intergranular pores. The skeleton of the modified loess particles presents an inlaid cemented structure, so the structure of the modified loess is more stable than unmodified loess. The cementation between the soil particles decreases after the fourth WD cycle, but the contact between particles is still dominated by surface cementation, the soil structure becomes looser, and intergranular pores with larger diameters are generated. After five WD cycles, the contact mode between particles gradually transitions from surface cementation to point contact. The small and mediumsized pores continue to decrease, while the large intergranular pores gradually increase and tend to be connected. The pore wall develops smoothly as a water migration channel during WD cycles. After the tenth WD cycle, the monomer silt particles with a diameter of about 25µm in the skeleton particles increases, the cemented clay minerals are significantly reduced, the cementation between the particles is weak, and the contact mode is mainly point contact. The pores are mainly aerial pores, which leads to the weakening of the soil structure and strength.
Quantitative analysis of loess SEM images
Based on the particles (pores) and cracks analysis system (PCAS) 26, SEM images with the magnification of 1000 times were selected for microscopic quantitative analysis. Fig. 7 shows the identification and analysis process of pores and particles. Binarized images are obtained through threshold segmentation and denoising steps, and the pores and particles are automatically segmented and identified. The white and black areas represent the pores and particles, respectively. Various geometric parameters and statistical parameters of pores and particles are obtained by a builtin algorithm. The quantitative indicators of microstructure such as pore area ratio (n) and fractal dimension (Df) were selected to describe the changes of particles and pores of the samples after the WD cycles, which can be described by:
$$n=\frac{{{A_v}}}{A}$$
1
$$\log \left( C \right)=\left( {{D_f}/2} \right) \cdot \log \left( S \right)+{c_1}$$
2
where Av and A are the pore area and the total area in the statistical area; C and S are the perimeter and area of the pore; c1 is a constant.
The fractal dimension of the pore reflects the development and complexity of the soil. Generally, the larger the fractal dimension, the more complex the pore structure distribution, the scattered distribution of soil particles and the weak degree of agglomeration.
Figure 8 (a) illustrates the pore area ratio (n) of RL and BML after WD cycles. It can be seen that the n of RL and BML increases with the number of WD cycles and tends to be stable after five cycles. After five WD cycles, due to the continuous migration of water in the pore channel, some small particles are washed away from the pore wall, the original small and mediumsized pores are enlarged to form larger pores, resulting in higher n. After more WD cycles, the pores are gradually dominated by aerial pores. The pore channels for the migration of water are redistributed and soil particles are rearranged. The original structure becomes a relatively uniform loose structure, and the n no longer changes. Furthermore, the n of RL is always greater than that of BML after WD cycles. The pore structure of RL is mainly composed of large intergranular pores between particles. Part of the intergranular pores of loess particles are filled by bentonite particles, which reduces the pore volume of modified loess, while the water absorption and expansion of bentonite effectively blocks the intergranular pores, thus the n of BML is lower than that of RL during the whole WD cycles.
The variation of pore fractal dimension of RL and BML with WD cycles are shown in Fig. 8 (b). The results show that the pore fractal dimension of RL and BML increases firstly, and the fractal dimension of RL reaches the peak after three WD cycles. they all tend to be stable after five WD cycles. The fractal dimension of pores of BML is smaller than that of RL before WD cycles because of the coexistence of aerial pores and intergranular pores in the pore structure of RL and the uneven distribution of pores among particles. While the filling of bentonite makes the number of large pores in BML decrease sharply, the pores are mainly medium and small, and the pore structure of BML is more complex than that of RL. Therefore, the fractal dimension of RL in the initial state is more significant than that of the improved loess. After WD cycles, the main primary water migration channel in RL is the wellconnected pores, while the primary interconnected pore channel is less in BML. Influenced by water migration, more small and complex pores are formed in the soil. The number and shape of pores in the soil gradually develop. The pore structure is more complex than that of RL, so the pore fractal dimension is larger than that of RL. After more WD cycles, due to the rearrangement of soil particles and pores, the soil structure of RL and BML changes from the original mosaic cementation structure to a uniform loose structure, the pore structure is relatively stable, and the fractal dimension of pores tends to be stable after a decreasing trend.
Analysis of pore area ratiobased damage degree
To further analyze the change of pore area ratio of RL and BML after WD cycles, a continuous variable D is introduced as the pore area ratio damage degree, as shown in Eq. (3):
$$\left\{ \begin{gathered} {D_1}{\text{=}}\frac{{\left {{n_1}  {n_{01}}} \right}}{{1  {n_{01}}}} \hfill \\ {D_2}{\text{=}}\frac{{\left {{n_2}  {n_{02}}} \right}}{{1  {n_{02}}}} \hfill \\ \end{gathered} \right.$$
3
where D1 and D2 represent the pore area ratio damage of RL and BML after WD cycles, respectively; n1 and n2 are the pore area ratio of RL and BML samples after WD cycles, respectively (%); n01 and n02 are the initial pore area ratio of RL and BML samples (%). D=0 corresponds to the initial undamaged state of the sample, and D=1 indicates that the sample is in a fully damaged state, and 0༜D༜1indicates that the sample is in a damaged state.
According to the change curve of pore area ratio in the microquantitative index of RL and BML after the WD cycles in Fig. 8 (a), it can be found that the relationship between N and n of RL and BML after different WD cycles is as follows:
$$\left\{ \begin{gathered} {n_1}(N)={a_1}N+{b_1}{N^2}+{n_0}_{1} \hfill \\ {n_2}(N)={a_2}N+{b_2}{N^2}+{n_{02}} \hfill \\ \end{gathered} \right.$$
4
where N is the number of WD cycles; a1, a2, b1, and b2 are all fitted parameters. Transform the form of Eq. (4) into the following equation:
$$\left\{ \begin{gathered} \left {{n_1}  {n_{01}}} \right=\left {N({a_1}+{b_1}N)} \right \hfill \\ \left {{n_2}  {n_{02}}} \right=\left {N({a_2}+{b_2}N)} \right \hfill \\ \end{gathered} \right.$$
5
Substituting Eq. (5) into Eq. (3) can obtain the univariate prediction model for the number of WD cycles and pore area ratio damage as follows:
$$\left\{ \begin{gathered} {D_1}=\frac{{\left {N({a_1}+{b_1}N)} \right}}{{1  {n_{01}}}} \hfill \\ {D_2}=\frac{{\left {N({a_2}+{b_2}N)} \right}}{{1  {n_{02}}}} \hfill \\ \end{gathered} \right.$$
6
where D1 and D2 respectively represent the pore area ratio damage of RL and BML after WD cycles; N is the number of WD cycles; n01 and n02 are the initial pore area ratio of RL and BML samples, the values are 29.6% and 16.57%; a1, a2, b1, and b2 are all fitted parameters, with values of 0.0377, 0.0651, 0.0023, and 0.0042 respectively.
In Eq. (6), D1/D2 obtains Eq. (7), and Eq. (7) can reflect the relationship between the pore area ratio damage of RL and 15% BML during the WD cycles:
$$\frac{{{D_1}}}{{{D_2}}}=\left {\frac{{{a_1}+{b_1}N}}{{{a_2}+{b_2}N}}} \right * \frac{{1  {n_{02}}}}{{1  {n_{01}}}}$$
7
Figure 9 (a) presents the variation of the pore area ratio damage degree of RL and BML with the number of WD cycles. The result shows that the pore area ratio damage of RL and BML increase during the previous five WD cycles, stabilizing from the fifth WD cycle. The pore area ratio damage of the BML samples is greater than that of RL during the development of WD cycles. Although the pore area ratio of RL is always higher than that of BML during the whole WD cycles, the pore area ratio of BML varies within a large range after the impact of the WD cycle. Because the fine bentonite particles swell and fill in the large pores of the loess grains skeleton before the WD cycle. During the reciprocating WD cycle, the montmorillonite continuously undergoes the processes of waterswellable and shrinkage. After several WD cycles, the damage of the expanded montmorillonite in the process of water shrinkage is irreversible, resulting in a more significant change in the pore structure of the BML than before. Therefore, the pore area ratio damage also shows a larger trend than that of the RL. Figure 9 (b) compares the calculated value and the tested value of the univariate damage prediction model based on the pore area ratio. From the figure that most of the data points are distributed on both sides of the straightline y=x, indicating that the predicted value of the damage model is similar to the experimental value, the model can better predict the RL and BML pore area ratio damage rules under the action of WD cycles.
Analysis of shear strength of loess under WD cycles
The curves between shear stress and shear displacement
Figure 10 and Fig. 11 show the relationship between shear stress and shear displacement of RL and BML under different vertical pressures during WD cycles. The relationship between shear stress and shear displacement of RL shows strain softening under the action of low vertical pressure (p = 30kPa), the curves of RL gradually transits to strain hardening with the vertical pressure increase. The shear stress of samples increases with the growth of shear displacement after different WD cycles. However, the relationship between shear stress and shear displacement of BML is strainsoftening without the influence of WD cycles, while the relationship between shear stress and shear displacement is strain hardening when it experiences the WD cycles, and the shear stress increases with the growth of shear displacement.
Strength parameters
Figure 12(a) shows the variation of the cohesion of RL and BML with the number of WD cycles, the cohesion of both loess decreased gradually with the increase of the WD cycles, and tended to be stable after five WD cycles. The cohesion of BML was higher than that of RL in the initial state, it was consistently lower than that of RL in the first to fourth WD cycles, and when it is stable after five cycles, it is higher than that of RL. Because the cohesion is related to the cementation between soil particles, doublelayer water and the attraction between molecules. Without the effect of the WD cycle, due to montmorillonite and other minerals in bentonite, water swelling fills the pores between loess particles, and ion exchange forms a diffuse double layer, which enhances the bonding force and electrostatic attraction between soil particles. The interaction force between modified loess particles is more substantial than RL particles, so the modified loess cohesion is greater than RL. During the WD cycles, the cement structure and diffusion electric double layer are destroyed due to water migration, and the cohesion decreases sharply. Finally, when the cohesion tends to be stable, the existence of some clay aggregates makes the cohesion of modified still higher than that of RL after multiple WD cycles.
Figure 12(b) presents the variation curve of the internal friction angle of RL and BML with the number of WD cycles. As shown in the figure, the internal friction angle of RL samples fluctuates with the increase of the WD cycles, while the internal friction angle of BML shows a trend of continuous attenuation with the increase of the WD cycles and the attenuation amplitude gradually decreases. The internal friction angle of the sample depends on the friction and occlusion between the particles. The addition of bentonite in the loess acts as the clay aggregates between the soil particles, and more small particles adsorb on the surface of particles, causing the friction between the soil particles to increase. With the development of WD cycles, the friction between particles weakens, and the internal friction angle between particles decreases due to the lubrication of water film.
Analysis of shear strength
Figure 13 illustrates the variation of shear strength of RL and BML under different WD cycles. Under different vertical pressures, the shear strength of RL and BML first decreases and then tends to be stable with the increase of the number of WD cycles, which is consistent with the variation law of cohesion under WD cycles. Before the WD cycles, the clay minerals in the BML fill in the aerial pores and large intergranular voids, the soil particles are mainly surface cementation, the BML particles structural stability is higher than that of the RL, so the shear strength of the BML is higher than that of the RL. The shear strength of BML is lower than that of RL after the WD cycles, but the shear strength of BML and RL tends to be stable after five WD cycles, and the shear strength of BML is higher than that of RL. In the previous WD cycles, as the scouring effect of water migration on soil particles, the clay film on the surface of modified loess particles is continuously dissolved, the large agglomerated particles are separated. The cementation between particles is weakened, the contact area is continuously reduced, the soil structure changes significantly, and the strength change of BML is more extensive than RL. With the increasing of WD cycles, the influence of water migration on soil structure is weakened, and the soil particles and pores are rearranged, the particles are mainly pointtopoint contact. However, due to the clay minerals in bentonite, the bond strength between the modified loess particles is greater than that of RL, so the shear strength of BML is higher than that of RL.
Analysis of cohesion damage degree
The influence of WD cycles on the internal friction angle of RL and BML is smaller than that of the cohesion. Therefore, based on the cohesion to further analyze the deterioration rule during the WD cycles. A continuous variable λ was introduced, named cohesion damage degree:
$$\left\{ \begin{gathered} {\lambda _1}=\frac{{\left {{c_1}  {c_{01}}} \right}}{{{c_{01}}}} \hfill \\ {\lambda _2}=\frac{{\left {{c_2}  {c_{02}}} \right}}{{{c_{02}}}} \hfill \\ \end{gathered} \right.$$
8
where: λ = 0 corresponds to the nondamaged state, λ = 1 corresponds to the completely damaged state, 0 < λ < 1 corresponds to the different degree of cohesive damage state; c01 and c02 represent the cohesion of RL and BML samples in the initial state, respectively (kPa); c1 and c2 respectively represent the cohesion of RL and BML after WD cycles (kPa).
According to the curve of cohesion of RL and BML under WD cycles in Fig. 12(a), it can be found that there are quadratic polynomial relationships and the exponential relationship between the cohesion c of RL and BML after experiencing different WD cycles N:
$$\left\{ \begin{gathered} {c_1}={c_{01}}+{a_1}N+{b_1}{N^2} \hfill \\ {c_2}={a_2} * {N^{{b_2}}} \hfill \\ \end{gathered} \right.$$
9
Substituting Eq. (9) into Eq. (8), the relationship between the cohesion of RL and BML and the number of WD cycles are obtained, respectively:
$$\left\{ \begin{gathered} {\lambda _1}=\frac{{\left {N({a_1}+{b_1}N)} \right}}{{{c_{01}}}} \hfill \\ {\lambda _2}=\left {1  \frac{{{a_2} * {N^{{b_2}}}}}{{{c_{02}}}}} \right \hfill \\ \end{gathered} \right.$$
10
where λ1 and λ2 are the cohesion damage degree of RL and BML after WD cycles, respectively; N is the number of WD cycles; c01 and c02 are the initial cohesion of RL and BML samples, respectively, with values of 58.68 kPa and 87.33 kPa; a1, a2, b1 and b2 are the fitted parameters, with values of 10.3979, 0.6000, 46.0906 and 0.3422, respectively.
In Eq. (10), λ1 /λ2 can be obtained as Eq. (11), which reflects the quantitative relationship between the cohesion damage degree of RL and that of BML in the process of WD cycles:
$$\frac{{{\lambda _1}}}{{{\lambda _2}}}=\left {\frac{{N\left( {{a_1}+{b_1}N} \right)}}{{{{\text{c}}_{{\text{02}}}}  {a_2} * {N^{{b_2}}}}}} \right * \frac{{{c_{02}}}}{{{c_{01}}}}$$
11
Figure 14 (a) presents the curves of cohesionbased damage degree of RL and BML after different WD cycles. The cohesionbased damage degree of RL and BML increases with the increase of WD cycles and tends to be stable after five cycles. In the process of WD cycles, the damage degree of BML is greater than that of RL. The reason is that the water migration in the process of WD cycles constantly destroys the surface electric double layer of clay minerals attached to the surface of soil particles, the intermolecular force decreases continuously, and the cementation between particles continued to decrease, the cohesion after the WD cycles decreases more significant than the initial state, so the BML cohesion damage degree under the WD cycles is more significant than that of RL.
The cohesion damage degree of RL and BML after WD cycles was calculated by equation (8), and compared with the experimental values of cohesion damage degree. The comparison results are shown in Fig. 14 (b). It can be seen from the figure that the data points are distributed on both sides of the straightline y=x, which indicates that the predicted value of the cohesive damage model is similar to the experimental value and that the model can better predict the damage regularity of the cohesion of RL and BML under the action of WD cycles.
Correlation analysis of shear strength and microstructure of loess
Multivariate linear stepwise regression method
The mechanical properties of loess and modified loess are not only related to their mineral composition, but also the degree of association of pores and particles. The expansion and development of surface cracks of loess under WD cycles are closely related to the connection of internal particle pore structure. Therefore, to investigate the correlation between the strength characteristics of loess and microscopic particle pores, the regression equation between the shear strength parameters of loess and microscopic pore structure parameters was established by the multivariate linear stepwise regression analysis. The stepwise method involves introducing the independent variables into the model individually, conducting an Ftest after each new independent variable is introduced, and performing a ttest on the selected independent variables singly, and removing the variables with insignificant changes from the equation individually. The expression of the model is as follows27,28:
$$Y={\beta _0}+{\beta _1}{X_1}+{\beta _2}{X_2}+ \cdots ++{\beta _i}{X_i}$$
12
where: β0 is the regression constant; β1 is the partial regression coefficient of the independent variable X1; β2 is the partial regression coefficient of the independent variable X2; βi is the partial regression coefficient of the independent variable Xi; i is the number of independent variables. The results of partial regression coefficients and regression constants were obtained by data analysis.
In the multivariate linear stepwise regression analysis, the shear strength parameters were selected as the cohesion (c) and the internal friction angle (ϕ). The parameters of the micropore structure of loess were selected as equivalent diameter (De), oblate degree (Od), pore area ratio (n), average form factor (F), probability entropy (H), fractal dimension (Df)26,29. The De is the average diameter of an equivalent circle equal to the pore area of the soil. The Od is the ratio of the short axis and the long axis of the pore, the value is less than 1. The smaller the value, the pore tends to be elongated, otherwise, the closer to the round shape. The n indicates the ratio of the pore area to the total area within the same crosssection. The F characterizes the pore boundary roundness, and the value is in the range of (0,1), the pore shape is closer to round with the increase of F. The H is used to describe the directionality of the pore arrangement. The value of H varies between (0,1) and the smaller the H, the more pronounced the directionality of the pores. The Df reflects the variation law of pore complexity with its area, and the larger the Df reflects the more complex pore structure and the more dispersed pore distribution of the soil. The experimental results and the obtained parameters are shown in Table 3.
Table 3
Summary of strength and micropore structure parameters of loess
Loess

Wettingdrying cycles

Strength parameters

Micropore structure
parameters

c / kPa

ϕ / °

De /µm

Od

n

F

H

Df

Remolded loess

0

58.68

21.42

4.5734

0.5850

29.60%

0.3728

0.9836

1.1971

1

47.86

22.00

4.3102

0.6142

29.13%

0.3337

0.9917

1.2007

2

42.00

20.86

4.7598

0.6074

36.65%

0.3529

0.9893

1.2377

3

37.69

20.98

4.4649

0.6185

38.07%

0.3479

0.9920

1.2427

4

29.17

24.45

4.5932

0.6102

40.48%

0.3412

0.9892

1.2328

5

17.07

20.66

4.0995

0.6167

40.43%

0.3349

0.9947

1.2256

10

16.00

20.75

4.7722

0.6236

42.53%

0.3298

0.9935

1.2195

Bentonite modified loess

0

87.33

31.50

3.7825

0.5909

16.57%

0.3842

0.9876

1.1626

1

46.82

28.80

4.1319

0.584

20.08%

0.3916

0.9907

1.2211

2

36.43

25.90

4.2751

0.5750

26.28%

0.3019

0.9913

1.2448

3

30.83

20.20

4.3752

0.5865

30.86%

0.2982

0.9895

1.2557

4

27.30

19.23

4.8088

0.5871

35.13%

0.3100

0.9838

1.2813

5

25.81

18.00

4.5036

0.5919

38.07%

0.2880

0.9915

1.2727

10

23.33

17.22

4.9375

0.6105

38.56%

0.3741

0.9894

1.2477

Results of regression analysis
The regression models of shear strength and micropore structure parameters were calculated by multivariate linear stepwise regression analysis through SPSS software and obtained as shown in Table 4. The cohesion of loess is negatively linearly correlated with the n, i.e., the cohesion decreases gradually with the increase of n. The n, H, and Df can jointly explain 84% of the loess cohesion, indicating that the n, H, and Df have a strong influence on the loess cohesion. The c is negatively correlated with n. The larger the pore area is, the easier for the particles to slide, and the bonding effect between the soil particles is weakened, leading to the reduction of the loess cohesion. The c is negatively correlated with H. With the increase of H, the pore orientation weakened, the pore direction distribution tended to be random, and the loess cohesion gradually decreased. The c is negatively correlated with Df. With the increase of Df, the complexity of pore distribution of different pore sizes and shapes in the plane increases, the more complex the pore structure of the soil, the more dispersed the pore distribution is, the easier the skeleton of soil particles will be destroyed, and the loess cohesion will be smaller.
The ϕ is linearly correlated with the n, and the correlation coefficient of the regression equation is 0.6. The ϕ is negatively correlated with the n, i.e., the ϕ decreases with the increase of the n. The larger n is, the more pores in the unit area, the bonding between soil particles is weakened, the easier the sliding between particles, and the internal friction angle decreases. It can be seen that the n is one of the factors affecting the ϕ of loess.
Table 4
Regression analysis of strength and micropore structure parameters of loess
Regression model

R2

F

P

c

Model 1

c = 102.061  195.169 n

0.65

25.238

<0.001

Model 2

c = 394.346  142.838 n  251.362 Df

0.77

22.257

<0.001

Model 3

c = 2087.196  115.357 n  285.656 Df  1676.723 H

0.84

22.976

<0.001

ϕ

ϕ = 35.657  40.486 n

0.60

20.368

<0.001

Figures 15 (a)(c) show the comparison between the fitted results of the stepwise regression equations and the measured values of the cohesion of the remold loess and modified loess. From the figures, as the independent variables in the regression equation increase, the range of error bands gradually decreases and the correlation coefficients of the regression models increases subsequently. When the regression model contains n, H, and Df, the fitted correlation coefficient R2 is 0.84, and the fitted value of cohesion has a high linear correlation with the measured value, indicating that the fitted regression model 3 can be used to predict the cohesion of loess. Figure 15 (d) shows the comparison between the measured and fitted values of the internal friction angle. The fitted values of the internal friction angle fit well with the measured, and both have similar decay laws with the increase of wettingdrying cycles. It is indicated that the regression equation between macromechanical parameters and microstructural parameters established by multivariate linear stepwise regression method can well reflect the influence law of loess microstructure on macromechanical strength.