For the reconstituted soil with 3 different constant matric suctions, 2 different initial stress ratios and 3 different subsequent stress ratios in triaxial tests, the measured SWRC (i.e. *S*r - *s* curves), pore water volume change (i.e. *V*m - *ε*1 curves), stress-strain (i.e. *q* - *ε*1 curves), volume change (i.e. *ε*v - *ε*1 curves), effective stress path (i.e. *q* - *p'* curves) were summarised in this section. For the unsaturated samples, both the net mean stress (*p*), mean Bishop's effective stress (*p'* = *p* + *S*r*s*) and effective stress ratio at the critical state (*M'* = *q* / *p*') were employed in this paper.

## 3.1 SWRC of clay under different stress paths

The soil-water retention behaviour of unsaturated clays under various stress paths was depicted in Fig. 4. Specimens subjected to an initial stress ratio of 0.5 displayed an increase in air entry value with the increase of the subsequent stress ratio, whereas limited variation in air entry was observed for specimens at an initial stress ratio of 1.0. At a given suction level, a reduction in saturation was attained with increasing subsequent stress ratio, indicative of the diminishing water retention capacity as the subsequent stress ratio increased. Negligible variations in the SWRC were obtained for specimens at an initial stress ratio of 0.5 after undergoing loading with different subsequent stress ratios. In contrast, a more discernible distinction of the SWRC was achieved for specimens with initial stress ratio of 1.0 under different subsequent stress ratios. The test data is represented by the VG model [45], and the van Genuchten equation can be expressed as a function of the degree of saturation *S*r:

$${S_r}={\left( {\frac{1}{{1+{{(s/a)}^m}}}} \right)^{1 - \left( {{1 \mathord{\left/ {\vphantom {1 m}} \right. \kern-0pt} m}} \right)}}$$

1

where *s* is the matric suction, *a* and *m* are the fitting parameters and *a* represents the value of the air entry. The fitting parameters evolution of VG model were given in Fig. 5. For samples with initial stress ratios of 0.5 and 1, the fitting parameter *a* linearly decreased with increasing subsequent stress ratio, while the parameter *m* decreased nonlinearly. Therefore, in the range of 1 ≤ *R*2 ≤ 3, parameters *a* and *m* can be expressed as:

$$\left\{ \begin{gathered} a= - 3{R_2}+11.533{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({R_1}=0.5,{\kern 1pt} {\kern 1pt} {\kern 1pt} {R^2}=0.9996) \hfill \\ a= - 2.5{R_2}+27.667{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({R_1}=1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {R^2}=0.9868) \hfill \\ \end{gathered} \right.$$

2

$$\left\{ \begin{gathered} m= - 0.0005{R_2}^{2}+0.0045{R_2}+1.026{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({R_1}=0.5,{\kern 1pt} {\kern 1pt} {\kern 1pt} {R^2}=1) \hfill \\ m= - 0.015{R_2}^{2}+0.11{R_2}+0.98{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({R_1}=1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {R^2}=1) \hfill \\ \end{gathered} \right.$$

3

The experimental results demonstrated that stress path imposed a significant impact on the water retention behavior of the specimens. At an initial stress ratio of 0.5, the weakly anisotropic structure rendered the degree of saturation more susceptible to variations in subsequent stress ratio. With increasing subsequent stress ratio, gradual compression of voids impeded air entry into the soil interior, thereby elevating the degree of saturation. Under high matric suction, the cohesion and internal friction angle of the specimens increased due to the enhanced bonding and interparticle friction, thus increasing the shear strength. Hence, the stress path effects on the pore water gradually declined, while matric suction effects became more pronounced. However, owing to the homogeneous void distribution, similar trends were observed in volumetric water content under different subsequent stress ratios, despite slight differences in the rate of change. In specimens subjected to an initial stress ratio of 1.0, the strong anisotropy meant pore structure varied little with subsequent stress ratios. Considering the air-entry value of soil samples remains essentially consistent under varying subsequent stress ratios, this results in negligible changes in the volumetric water content under low suction conditions. However, under high suction conditions, the increase in the subsequent stress ratio gradually weakens the anisotropy of the soil sample, consequently leading to a gradual slowdown in the rate of change of the volumetric water content under higher subsequent stress ratios.

## 3.2 Variation of pore water volume in triaxial shear test

The variation of pore water volume in soil specimens under different stress paths during shearing is shown in Fig. 6. It can be observed that the stress path has a significant influence on the water retention behavior of the soil specimens. In Fig. 6(a), under low suction condition, the specimens with an initial stress ratio of 0.5 experience an increase in pore water change with increasing subsequent stress ratio. Because of the small volume change of the specimen under the initial stress ratio (*R*1 = 0.5) loading, leaving more pore water inside the specimen. In this case, the pore water change is closely related to the axial stress. The higher the subsequent stress ratio, the greater the axial stress under the same conditions, and thus the more significant the pore water change. With the increase of subsequent stress ratio, the rate of pore water change in the soil specimen also accelerates. Under subsequent stress ratio loading at a lower level, the pore water change is slower, while under subsequent stress ratio loading at a higher level, the pore water change rapidly reaches a steady state. As shown in Fig. 6(b), for the specimen with an initial stress ratio of 1, due to the smaller pores and higher air entry value, the pore water change mainly depends on the stress path, especially under low matric suction conditions where high pore water content leads to more significant pore water change under different subsequent stress ratio loading. When the subsequent stress ratio is 1, since the stress path is the same as the initial loading, the pore structure of the soil does not change much, resulting in sufficient drainage and significant pore water change. When the subsequent stress ratio is 2, the vertical pores of the specimen are further compressed, causing difficulty in pore water drainage and less pore water change. As the subsequent stress ratio further increases, the axial strain increases rapidly, leading to difficulty in rapid drainage of pore water inside the specimen in a short time. Therefore, the pore water change is small at low axial strain. However, as the axial strain increases, the accumulated pore water is gradually drained out, accelerating the pore water change. After the accumulated pore water is drained out, the pore water change gradually stabilizes.

From Fig. 6(c) and Fig. 6(d), it can be seen that matric suction significantly affects the change in pore water within the soil specimen. For the specimen with an initial stress ratio of 0.5, due to the large internal pores, the change in pore structure is more pronounced under the influence of subsequent stress ratios. With increasing subsequent stress ratios, the vertical strain is greater than the horizontal strain, leading to smaller vertical pores than horizontal pores, which makes it difficult for pore water to dissipate. The volume change of pore water gradually decreases. In contrast, specimens subjected to an initial stress ratio of 1.0 exhibit lower initial pores, the pore structure is less affected by subsequent stress ratios. With increasing matrix suction, the internal pore water gradually decreases, further reducing the effect of subsequent stress ratios on pore water. Therefore, while the evolution of pore-fluid volumes under various stress paths is not considerably altered with rising matric suction, the differences in pore water change between different stress paths gradually decrease.

As can be seen from Fig. 6(e) and Fig. 6 (f), under high suction conditions, the soil sample gradually shifts from drainage to water absorption, and the change in pore water under different stress paths is also quite obvious. When the initial stress ratio is 0.5, due to the larger pore volume inside the soil sample, and more pore water content, the initial pore water content is less under high suction. During the shearing process, the volume of air inside the soil sample gradually decreases, while the volume of pore water gradually increases. When the subsequent stress ratio is 1, the change in pore structure and pore water content during the loading process is slow. As the subsequent stress ratio increases, the changes in the pore structure inside the soil body are faster, the internal pores gradually become smaller, on one hand, this causes the air pressure in the upper part of the soil sample to not fully enter the soil body, on the other hand, it makes it difficult for the pore water at the bottom of the soil sample to enter the soil body. Therefore, as the subsequent stress ratio continues to increase, the change in pore water during the shearing process of the soil sample continues to decrease. When the initial stress ratio equals 1, the initial pore of the soil sample is relatively small, the matrix suction has a smaller impact on the pore water of the soil sample, while the stress path has a more significant effect on the pore water. During the shearing process with a subsequent stress ratio of 1, the pore water inside the soil sample first gradually increases, as the volume of air inside the soil body gradually decreases, the volume change of the soil sample gradually exceeds the volume change of the pore water, and the pore water is gradually expelled. As the subsequent stress ratio gradually increases, the rate at which the pore volume of the soil sample decreases also increases, the process of soil body absorbing water gradually shortens, and the drainage process gradually accelerates. And the pores of the soil body become denser and denser, causing the flow of pore water to become slower and slower, therefore, as the subsequent stress ratio increases, the change in pore water gradually decreases.

## 3.3 Stress-strain characteristics of unsaturated clay

In stress path controlled triaxial testing, under certain stress ratio conditions, the deviatoric stress of the specimen continues increasing without failure. In this study, the peak shear resistance is defined as the maximum deviatoric stress within 20% axial strain. Figure 7(a)-(c) exhibit the stress-strain curves for specimens under varied initial stress ratios at 30kPa matric suction. It is evident that the peak shear resistance of the *R*1 = 0.5 specimen exceeds that of the *R*1 = 1 specimen. Under different initial stress ratios, the shear strength first decreases then increases with increasing subsequent stress ratio, and the disparity in shear strength between specimens of diverse initial stress ratios progressively amplifies. This predominantly correlates with the initial pore geometry. The higher the initial stress ratio, the denser and more saturated the specimen becomes. During loading, the *R*1 = 1 specimen yields first. When the subsequent stress ratio is low, the volumetric strain rate is diminished, enabling adequate dissipation of pore fluid, hence the discrepancy in peak shear strength between specimens of varied initial stress ratios is small. As the subsequent stress ratio increases, the volumetric strain rate heightens, and pore water cannot drain out fully in a short time. The shear strength rapidly increases in a short period then becomes stable, and the higher the subsequent stress ratio, the greater the increase in shear strength (refer to Fig. 7(b) and Fig. 7(c)). With increasing axial strain, the accumulated pore water gradually dissipates, and the strength of the specimen progressively declines.

Figure 7 (d)-(e) exhibit the stress-strain curves for specimens under varied initial stress ratios at 100kPa matric suction. When the subsequent stress ratio *R*2 is 1 and 2, the shear strength of soil specimens with initial stress ratio *R*1 = 1 is greater than those with *R*1 = 0.5, while the pattern is reversed when *R*2 is 3. Overall, with increasing subsequent stress ratio, the shear strength of soil specimens with the same initial stress ratio exhibits a trend of initial decrease followed by increase, and the difference in shear strength between soil specimens with different initial stress ratios is gradually diminishing. Analysis indicates that as matric suction increases and pore-water content decreases under initial loading and unloading, the soil becomes more compacted. The higher the initial stress ratio, the greater the compactness of the soil. Therefore, when the subsequent stress ratio *R*2 is 1 and 2, the shear strength of the soil is positively correlated with the initial stress ratio. As *R*2 gradually increases, excess pore-water pressure is generated within the soil, and the influence of matric suction on shear strength diminishes gradually, leading to a negative correlation between shear strength and initial stress ratio.

The stress-strain curves for specimens under varied initial stress ratios at 200kPa matric suction are displayed in Fig. 7(g)-(i). When the subsequent stress ratio *R*2 is 1, the shear strength of specimens with *R*1 = 1 exceeds those with *R*1 = 0.5. However, when *R*2 are 2 and 3, the opposite trend is observed. Overall, increasing *R*2 causes the shear strength of specimens with identical *R*1 to first decrease then increase, while the difference between specimens with varying *R*1 first decreases then increases. Analysis reveals that as matric suction rises further, the increased loss of pore water accelerates deviatoric stress-induced structural changes. Greater initial stress ratios increase soil density, when *R*2 is 1, shear strength positively correlates with initial stress ratio. As *R*2 gradually increases, the structure of specimens with *R*1 = 0.5 is more readily destroyed during loading, enabling their density exceed specimens with *R*1 = 1, inducing a negative correlation between shear strength and initial stress ratio.

## 3.4 Deformation characteristics of unsaturated clay

(1) Volumetric change during shearing

The volumetric strain curves for specimens under varied initial stress ratios at three matric suctions (30kPa, 100kPa and 200kPa) are exhibited in Fig. 8. According the conventions used in soil mechanics, positive volumetric strain signifies contraction, whereas negative volumetric strain indicates dilation. It can be observed that under lower matric suction conditions, an increase in initial stress ratio leads to a larger volumetric strain in the samples, with dilatant behavior occurring during the loading process. This is due to the increase in initial stress ratio reducing the initial void ratio of the specimens, thus increasing the air entry value (see Fig. 4). At lower matric suctions (30kPa and 100kPa), the specimens have higher saturation levels, which result in increased drainage during the shearing process (see Fig. 6), leading to larger volumetric strain. As the matric suction increases, the degree of saturation of all initial stress ratio specimens decreases, but the denser structure of specimens with higher initial stress ratio gives them stronger resistance against deformation and smaller volumetric change. At the matric suction of 200kPa, with the increase of subsequent stress ratio, the volume of the soil undergoes considerable change in a short period, which causes the transition of specimen volume change from compressive to dilative state.

(2) Variation of axial strain with time

The axial strain variation over time of soil specimens with an initial stress ratio (*R*1) of 0.5 under different subsequent stress ratios (*R*2) are shown in Fig. 9(a)-(c). The axial strain rate data underscores the role of matric suction in modulating the mechanical response of the specimens across varying stress path regimes. At identical applied stress ratios, elevating suction induces a marked decline in the axial strain rate. This reflects an enhancement in the specimen’s propensity to resist axial deformation owing to suction-mediated improvements in fabric stability and shear resistance. Conversely, raising the subsequent stress ratio intensifies axial strain rates for a given suction condition, signalling an escalation in deformation and structural damage.

Fundamentally, the axial strain rate signifies the temporal evolution of axial strain and provides insight into the specimen’s intrinsic load-bearing capacity. Lower strain rates denote superior resistance to axial loading, whereas higher strain rates convey accelerated structural deterioration. At *R*2 = 1, the sample drains adequately, and axial strain changes slowly. Minimal differences exist between the low-suction samples’ strain rates, while more significant divergences occur under high suction. When *R*2 is 2, the suction has a significant effect on the soil sample because the friction between soil particles gradually increases the shear strength of the soil sample. As *R*2 is 3, the effect of suction on the soil sample gradually weakens. Although high suction can improve the initial structural strength of the soil sample, the initial structure of the soil sample is destroyed with the continuous increase of shear stress, making the structural characteristics of the soil samples under different suction conditions tend to be consistent, resulting in a smaller difference in the axial strain rate of the soil sample under high suction.

The axial strain variation over time of soil specimens with an initial stress ratio (*R*1) of 1 under different subsequent stress ratios (*R*2) is depicted in Fig. 9(d)-(e). It is discernible that for a given stress path, the axial strain rate of the specimen diminishes with escalating matrix suction. Additionally, at a constant matrix suction, the specimen axial strain rate intensified with elevating subsequent stress ratio. Relative to Fig. 9(a)-(c), the specimen axial strain rate is markedly attenuated. The structure of the specimen with *R*1 = 1 is more compact, and the suction has a greater impact on its structural strength and shear resistance. At *R*2 = 1, a pronounced discrepancy in axial strain rate manifests between specimens of disparate suctions. When *R*2 increases to 2 and 3, the axial strain rate under low suction states surges drastically, signalling expeditious damage to the specimen structure. However, under elevated suction conditions, larger internal friction retards axial strain evolution, with suction-dependencies in axial strain rate declining.

## 3.5 Effective stress path of unsaturated clay

By summarizing the stress-strain relationships of unsaturated clay specimens under various matric suctions, the critical state parameter *M'* (effective stress at critical state) was obtained for unsaturated clay under different stress paths. Numerous studies [1, 12, 39, 53] indicate that that the critical state stress ratio *M'* of unsaturated clay remains unchanged under different matric suctions. Figure 10 depicts the critical state line (labeled CSL on the graph) of saturated clay in the *p'* - *q* plane. For specimens with initial stress ratios of 0.5 and 1, *M'* increases with the increasing subsequent stress ratio, and *M'* represents the sliding friction coefficient when the soil reaches the critical state. Since the specimen with an initial stress ratio of 0.5 exhibit a relatively lower density, the specimen’s volumetric strain gradually increases as the subsequent stress ratio increases, and *M'* will also increase significantly. In contrast, when initial stress ratio is 1, it can be inferred that the specimen has already undergone shearing. Despite a higher density, the yield strength is notably reduced, making it easier to reach the critical state. Moreover, as observed in Fig. 8, under the same suction, the increase in volume change and density increase of the specimen after shearing is smaller with the increase of subsquent stress ratio, hence the parameter *M'* representing the interparticle sliding friction demonstrates a smaller magnitude of increase.

Furthermore, when the subsequent stress ratio is 1(see Fig. 10(a) and Fig. 10(d)), the specimen with an initial stress ratio of 0.5 is deemed to have reached the critical state when the axial strain reaches 20%, but the specimen is actually still undergoing continuous shrinkage (see Fig. 6(g)). Conversely, for the specimen with an initial stress ratio of 1, its volume tends to stabilize when the axial strain reaches 20%. At this point, the specimen’s void ratio is relatively small, and the soil is denser, resulting in a larger critical state parameter *M'*. As the subsequent stress ratio increases, although the initial void ratio of the specimen with an initial stress ratio of 1 is smaller, the specimen exhibits a shear dilation tendency during loading. Compared with the specimen with an initial stress ratio of 0.5, the sliding friction between particles gradually decreases, and the critical state parameter *M'* exhibits a trend of initially increasing and then decreasing.