Strength Characteristics of Unsaturated Compacted Loess Under Complex Stress Paths

Abstract

A series of (45 sets) true triaxial shear tests with controlled suction and under conditions of equal-b and equal-p are performed on unsaturated compacted loess. By incorporating the effects of matric suction in unsaturated loess and the influence of middle principal stress conditions, the characteristics of the stress–strain curve evolution, strength failure line and related strength parameters of the unsaturated compacted loess under complex stress paths are analyzed, and the applicability of different strength criteria is analyzed. The results indicated that both the matric suction and the middle principal stress condition have significant effects on the magnitude of shear stress. While matric suction exerts minimal effect on the configuration of the stress–strain curve, the middle principal stress condition markedly affects its shape. A function is established to approximately describe the correlation between the strength-related parameters and the middle principal stress coefficient. A comparison of the failure lines predicted with several strength criteria and the experimental data indicates that the experimental results of unsaturated compacted loess under a controlled suction state are consistent with the results of the predicted failure lines of the Lade–Duncan strength criterion. The findings can provide crucial theoretical support for engineering design and disaster prevention in loess regions.

1. Introduction

Loess, a natural material used for construction and filling, is an aeolian quaternary sediment that is mainly distributed in arid and semiarid areas [1,2]. Due to the influence of arid and semiarid climatic conditions, a large amount of loess is located above the groundwater level, and the majority of loess encountered in engineering applications is typically in an unsaturated state; so, the occurrence of loess engineering problems such as the failure of loess subgrade engineering, the instability of loess slope engineering and the collapse of loess foundation engineering is mostly related to its unsaturated characteristics. For example, during the dry season, the surface of shallow loess dries out due to the evaporation of water, resulting in shrinkage and cracking. In contrast, precipitation occurring during the wet season leads to water infiltration, resulting in an elevation in pore water pressure and a reduction in the shear strength of the superficial loess soil. As a result, the collapsibility of the loess may occur [3,4,5,6,7,8]. On a loess slope, a potential sliding surface may form in the unsaturated soil layer above the groundwater level, and long-term rainfall may cause changes in pore water pressure, which will then cause instability of the loess slope, resulting in landslides, collapses and other loess geological disasters [9,10,11,12]. The compacted loess backfill of a retaining structure is also basically unsaturated. When the backfill is wet by rain, it may cause lateral earth pressure and stability problems in the retaining structure [13,14]. Many geotechnical engineering challenges in loess regions, including bearing capacity, slope stability and lateral soil pressure problems, are related to the shear strength of the soil mass [15,16,17]; so, it is extremely important to study the shear strength of unsaturated loess.

Geotechnical testing methods must be selected based on the environmental conditions of the soil. For soil located below the groundwater table or in areas with significant rainfall, saturated soil testing is required [18,19]. For soil in cold environment or areas with significant temperature fluctuations, testing under low-temperature conditions or freeze–thaw cycle simulations is necessary [20,21]. For soil situated above the groundwater table or in dry environments, unsaturated soil testing should be conducted [22,23]. According to the above description, unsaturated soil testing is required for loess in this case. Scholars have examined the strength characteristics of unsaturated loess though various method. Direct shear and triaxial tests of loess with different moisture contents have been carried out to study the strength characteristics of unsaturated loess, and the influence of suction on the shear strength of unsaturated soil was indirectly considered through the relationship between water content and suction [24,25,26]. A triaxial shear test of unsaturated loess under a constant moisture content was carried out to measure the change in matric suction during the shear process [27,28,29]. To study the effect of matric suction (s = u_a − u_w) on the shear characteristics of loess is usually performed by controlling the pore air pressure (u_a) or pore water pressure (u_w) in a consolidation apparatus, direct shear apparatus or triaxial apparatus. Ng et al. [30] utilized unsaturated direct shear testing to examine the shear characteristics of compacted loess under high suction. The results indicated that while the peak strength increased with rising suction, the rate of increase decreased. Ng et al. [31] utilized suction- and temperature-controlled direct shear apparatuses to investigate the shear behavior of loess under different matric suctions and temperatures. Wang et al. [32] conducted shear tests under constant suction on intact loess via unsaturated triaxial tests. The results show that the strength of loess was highly responsive to matric suction, with the dependence of strength behavior on matric suction primarily reflected in the evolution of cohesion. Zhang et al. [33] and Wei et al. [34] utilized an unsaturated soil triaxial apparatus to perform triaxial shear tests under constant suction on both unsaturated undisturbed loess and reshaped loess. The results indicated that there were significant differences in strength between the unsaturated undisturbed loess and the remolded loess, with the related strength parameters of the undisturbed loess being greater than those of the remolded loess. Yates and Russell [27] carried out tests on loess under constant suction and moisture conditions using an unsaturated triaxial apparatus. Gao et al. [35] conducted a triaxial shear test while maintaining constant suction levels. The results indicated that the internal friction angle remained nearly constant as suction increased, whereas cohesion increased linearly with suction. These studies indicate that the key to determining the strength characteristics of unsaturated loess is to examine the matric suction effects.

In practice, loess projects located above the water table are often subjected to three-dimensional stress gradients [36,37] caused by changes in net stress (σ_ij − u_aδ_ij) and matric suction (u_a − u_w)δ_ij. The abovementioned studies on the matric suction characteristics of unsaturated loess were mostly limited to static, one-dimensional and axisymmetric stress loading modes. Most of them adopted triaxial tests, and the stress conditions that can be simulated in triaxial tests are axisymmetric. However, in engineering practice, the soil is complicated, and the three-dimensional force is often asymmetric. It is challenged to reflect the real complex stress state of rock and soil under triaxial conditions. A reasonable solution for geotechnical engineering problems in practical engineering largely depends on understanding the mechanical nature of loess under different stress paths in practical engineering. Some scholars have used true triaxial testing to examine the effects of different stress paths in the study of loess mechanics [38,39,40,41,42,43]. However, due to the limitations of test conditions, there are few studies on the influence of matric suction on the strength characteristics of unsaturated loess under complex stress paths, and further research is needed. To more accurately reflect the complex stress state of unsaturated loess in practical engineering, this paper uses a true triaxial apparatus to carry out constant-suction true triaxial shear tests of unsaturated remolded loess with different stress paths; this approach can consider the effects of the matric suction and stress conditions on the strength characteristics of the unsaturated loess. This study examines the changes in strength parameters in relation to matric suction of unsaturated loess and various middle principal stress conditions. A quantitative relationship is established between the strength parameters, matric suction of unsaturated loess, and parameters of middle principal stress. Additionally, the evolution of the strength failure line on the meridian plane and π-plane is analyzed to characterize the three-dimensional strength behavior of unsaturated loess. The findings of research are helpful for understanding the mechanical behavior of unsaturated loess under various stress paths in practical engineering and provide a theoretical and experimental basis for solving geotechnical engineering problems in practical loess engineering.

2. Experimental Overview

2.1. Sample Preparation

The loess used for the test was taken from a subway station construction site located in the eastern suburb of Xi’an city, Shaanxi Province, China. Xi’an is situated in the Guanzhong Basin and experiences a relatively humid climate. The natural moisture content of the Xi’an loess ranges from 15% to 20%. Vertical joints are well-developed, with high porosity (40% to 50%), a dry density between 1.3 and 1.5 g/cm³, and a slightly elevated clay content. The primary minerals present are quartz and feldspar. The sampling site had a flat terrain, and the loess sampling was conducted at depths ranging between 4 and 5 m. Based on the stratum investigation conducted at the site, it was determined that the soil sample obtained from the sampling point was Q₃ loess (Figure 1). The manual shaft excavation method was employed for soil collection, where a soil block was chiseled from the pit’s sidewall. Subsequently, the soil sample was carefully sealed using black plastic bags and tape to minimize any disturbance during transportation to the laboratory. The fundamental physical properties of the loess samples determined by laboratory tests are presented in

Table 1. Fundamental physical properties of the loess samples.

Parameter	Value	Unit
Specific gravity	2.70	-
Liquid limit	34.2	%
Plastic limit	18.6	%
Plastic index	15.6	-
Dry density	1.40	g/cm3
Initial pore ratio	0.93	-

, and the particle distribution curves of the soil samples are shown in Figure 2, in which the proportions of clay, silt and sand are 15.91%, 81.69% and 2.40%, respectively.

A ring knife sample with a diameter of 6.18 cm, a height of 2.0 cm and a dry density of 1.40 g·cm⁻³ was prepared, and the soil–water characteristic curve of the sample was determined (Figure 3). Two model were used to analyze and fit the experimental data, and both models demonstrated a good fit to the test results.

Before preparing the soil sample, the soil materials cut from the original sample were subjected to air-drying for several days to reduce its moisture content. Following air-drying, the soil was pulverized and subsequently sieved using a sieve with a 2 mm aperture. Water addition was performed according to the moisture content corresponding to the required matrix suction for the test. The wetted loose soil was put into a plastic bag, sealed, and left for 48 h to ensure that the water distribution in the loose soil was uniform. The sample dimensions utilized were 70 mm by 70 mm by 140 mm, with a target dry density established at ρ_d = 1.40g·cm⁻³. The true triaxial remolded samples were compacted in seven layers. For each layer, the surface of the previous layer of compacted soil was shaved to ensure that the latter layer of compacted soil was well bonded with the previous layer of soil. The process of soil sample compaction is illustrated in Figure 4. The prepared remolded samples were positioned within a moisturizing cylinder and left for a duration of 48 h.

2.2. Test Equipment

A true triaxial apparatus for unsaturated soil (Figure 5) was utilized to conduct the test. The testing device employs a mixed boundary loading mode, combining rigid loading in one direction and flexible loading in two directions. The vertical major principal stress (σ₁) is exerted via a rigid plate, while the lateral middle principal stress (σ₂) and minor principal stress (σ₃) are applied through hydraulic rubber bags. A ceramic plate designed for high air intake is strategically positioned at the base of the pressure chamber, enabling the effective separation of pore air from pore water. Additionally, this apparatus is equipped with a standalone measurement and control system for both pore air and pore water, allowing for a wide range of tests related to suction.

2.3. Test Scheme

The remolded loess samples that were prepared underwent a true triaxial shear test while maintaining controlled suction conditions. The true triaxial shear testing under controlled suction was categorized into three phases: the suction equilibrium phase, isotropic consolidation phase, and true triaxial shear phase. As shown in Figure 6, the AB section was the suction equilibrium phase, in which the net mean stress (p, p = (σ₁ + σ₂ + σ₃)/3 − u_a) was consistently held at a fixed value of 5 kPa. The BC section was the isotropic consolidation phase, in which the matric suction (s) was kept constant, and the major principal stress (σ₁), middle principal stress (σ₂), and minor principal stress (σ₃) were all equal to the net mean stress (p), i.e., σ₁ = σ₂ = σ₃ = p. The CD section represented the true triaxial shear phase under controlled suction, during which equal-b and equal-p shear tests were performed. In this phase, both s, and p were maintained at constant levels, while the value of middle principal stress coefficient (b) was set to values of 0, 0.25, 0.5, 0.75, and 1. The specific test scheme of is shown in

Table 2. Test scheme.

Pore Air Pressure, ua (kPa)	Pore Water Pressure, uw (kPa)	Matric Suction, s (kPa)	Net Mean Stress, p (kPa)	Middle Principal Stress Coefficient, b	Number of Specimens
50	0	50	100/200/300	0/0.25/0.5/0.75/1	15
100	0	100	100/200/300	0/0.25/0.5/0.75/1	15
200	0	200	100/200/300	0/0.25/0.5/0.75/1	15

.

3. Experimental Results

3.1. Stress–Strain Relationship Curve

Figure 7 presents the correlation between shear stress (q) and shear strain (ε_d) across a range of matric suctions (s) levels and varying net mean stresses (p) while considering different middle principal stress coefficients (b). The curves of stress–strain collectively demonstrate strain hardening behavior, and the shear stress develops with increasing shear strain.

Influence of the net mean stress (p) on the shear stress–strain curve: For the same matric suction and the same b, the net mean stress exerts minimal impact on the configuration of the stress–strain curve, but it substantially affects the magnitude of the soil’s shear stress. Comparing the results for the three matric suctions and five b-values, it is evident that p significantly influences the stress–strain curve. As p increases, q also increases.

A comparison of the stress–strain curves under varying matric suctions, while maintaining consistent values for b and p, reveals that matric suction has a minimal impact on the morphology of the curve. However, it significantly enhances the shear stress of the soil. Comparing the results for the three net mean stresses and five b-values, the influence of s on the q − ε_d curve exhibits a consistent trend: as s increases, q also increases.

The influence of the middle principal stress on the stress–strain curve is illustrated through a longitudinal comparison of the q − ε_d curves, which are presented for five different b values in Figure 7a–e. As the value of b increases, the morphology of the stress–strain curve undergoes significant changes. The curve transitions from exhibiting strain hardening characteristics to resembling an ideal elastic-plastic behavior curve. The effect of parameter b on the stress–strain curve exhibits a consistent pattern: as b increases, the shear stress decreases while the shear strain increases. Consequently, the morphology of the q − ε_d curve progressively shifts from exhibiting strain-hardening behavior to resembling an ideal elastic-plastic response.

3.2. Failure Stress

Figure 8 illustrates the relationship between failure stress (q_f), the parameter b, and matric suction. Both matric suction and b significantly influence the failure stress (q_f). Matric suction can enhance the failure stress (q_f) of the sample, and with increasing b, the failure stress (q_f) of the sample decreases.

The experimental results can be simply concluded in

Table 3. Response results of different factors.

Item	Response Results of Different Factors
	p	s	b
q~εd	minimal impact on the morphology of curve	minimal impact on the morphology of curve	morphology of curve significant changes
q	p increases, q increases	s increases, q increases	b increases, q decreases
qf	p increases, qf increases	s increases, qf increases	b increases, qf decreases

.

4. Analysis and Discussion

4.1. Strength Failure Line on the Meridian Plane

The failure stresses at different net mean stresses can be expressed linearly in Equation (1), and this straight line is the strength failure line. Figure 9a–c show the strength failure lines after fitting the test data points through Equation (1) for different b values and matric suctions.

$$q_f=M_{f}p+{\mu }_f$$

(1)

where M_f is the slope of the strength failure line, and μ_f is the longitudinal intercept of the strength failure line.

Figure 10 shows the M_f and μ_f of the strength failure line with changing parameter (b). The slope continues to decrease with increasing b. The longitudinal intercept does not show a more pronounced pattern of change with increasing b. Figure 11 illustrates the changes in both M_f and μ_f in relation to matric suction. The slope varies little with increasing substrate suction, while the longitudinal intercept increases significantly with increasing matric suction.

From the above analysis of the correlation between M_f, b and s, it is determined that matric suction is mainly affected by b; so, the effect of matric suction can be neglected; thus, Equation (2) can be constructed to represent the variation in M_f with respect to b:

$$M_f(b,h)=g(b,h)M_c$$

(2)

where M_c is the slope of the critical state line under axisymmetric compression, and g(b,h) is a dimensionless function that must satisfy g(b = 0,h) = 1 at b = 0. The expression for g(b,h) is

$$g(b,h)={\displaystyle \frac{2}{1+h-(h-1){(1-b)}^2}}$$

(3)

where h is the fitting parameter; when b = 1,

$$g(b,h)={\displaystyle \frac{2}{1+h}}$$

(4)

The established function g(b,h) was fitted to the experimental results for various middle principal stress coefficients (b). Figure 12 illustrates the fitted curves corresponding to various matric suctions (s), demonstrating satisfactory fitting results.

The correlations between the u_f and s for different b values can be fitted by Equation (5). Additionally, the outcomes of the fitting analysis are illustrated in Figure 13.

$$u_f={\displaystyle \frac{s}{A_f+B_{f}s}}$$

(5)

4.2. Strength Parameters

Fredlund and Morgenstern [21] proposed defining the stress state of unsaturated soil by using the net normal stress and matrix suction as stress state variables. Then, they established a shear strength formula based on these two stress state variables, as shown in Equation (6).

$${\tau }_f=c^{\prime }+(\sigma -u_a)\tan {\phi }^{\prime }+(u_a-u_w)\tan {\phi }^b$$

(6)

where τ_f is the shear strength at failure, c′ is the effective cohesion, φ′ is the effective friction angle, and φ^b represents the friction angle associated with matric suction.

Equation (6) may be alternatively expressed as follows:

$${\tau }_f=c+(\sigma -u_a)\tan {\phi }^{\prime }$$

(7)

where c is the total cohesion.

According to the relationship between the strength failure line on the p-q surface and the Mohr’s circle,

$$M_f={\displaystyle \frac{3\sin {\phi }^{\prime }}{\sqrt{3}\cos {\theta }_{\sigma }+\sin {\theta }_{\sigma }\sin {\phi }^{\prime }}}$$

(8)

$${\mu }_f={\displaystyle \frac{3c\cos {\phi }^{\prime }}{\sqrt{3}\cos {\theta }_{\sigma }+\sin {\theta }_{\sigma }\sin {\phi }^{\prime }}}$$

(9)

Based on Equations (8) and (9), the φ′ and c can be obtained as follows:

$$\sin {\phi }^{\prime }={\displaystyle \frac{\sqrt{3}{M}_f\cos {\theta }_{\sigma }}{3-M_f\sin {\theta }_{\sigma }}}$$

(10)

$$c={\displaystyle \frac{{\mu }_f(\sqrt{3}\cos {\theta }_{\sigma }+\sin {\theta }_{\sigma }\sin {\phi }^{\prime })}{3\cos {\phi }^{\prime }}}$$

(11)

where θ_σ is the stress Lode angle, and the relationship between θ_σ and the b is as follows:

$$\tan {\theta }_{\sigma }={\displaystyle \frac{2b-1}{\sqrt{3}}}$$

(12)

Based on Equations (10)–(12), the results for the effective friction angle and total cohesion are derived for various b and s. Figure 14 illustrates the relationship between φ′ and c in relation to parameter b. The φ′ angle initially rises and subsequently declines as the parameter b increases, while the c does not show an obvious variation with increasing middle principal stress coefficient, and the total cohesion fluctuates within a certain range. Figure 15 shows the change pattern in the φ′ and c with the matric suction. The effective friction angle exhibits minimal variation with an increase in s, whereas the c demonstrates a substantial increase as s rises.

From the above analysis, it can be concluded that, the φ′and c are mainly affected by the parameter b, and the influence of s can be ignored. According to Equations (2), (3), (10) and (12), the relationship between the effective friction angle and the slope of the critical state line under axisymmetric compression (M_c) and the middle principal stress coefficient is as follows:

$$\sin {\phi }^{\prime }={\displaystyle \frac{\sqrt{3}g(b,h)M_c\cos (\arctan \frac{2b-1}{\sqrt{3}})}{3-g(b,h)M_c\sin (\arctan \frac{2b-1}{\sqrt{3}})}}$$

(13)

The predicted results of φ′ with b are obtained by using Equation (13) in Figure 16.

The relationship between c and s of unsaturated loess for various b is fitted with Equation (14). The results of fitting are presented in Figure 17. Similar results were also obtained for other types of unsaturated soils under the conventional direct shear and triaxial test used to control suction [35,44,45,46], However, these results are obtained under conventional experimental conditions and do not take into account the influence of the middle principal stress conditions.

$$c={\displaystyle \frac{s}{A_c+B_{c}s}}$$

(14)

4.3. Applicability of the Strength Criteria

The strength criterion of soil represents the damage threshold of the stress state of soil. The strength criterion is generally expressed via Equation (15):

$$F({\sigma }_{ij},k_f) = 0$$

(15)

where σ_ij represents the stress tensor and k_f denotes the soil characteristic parameter.

Several commonly used strength criteria are selected for analysis and comparison.

(1) Extended von Mises criterion

According to the von Mises strength criterion, soil failure occurs when the deviatoric stress J₂ attains a certain limit, as in Equation (16):

$$F(J_2) = J_2-k_{\mathrm{M}}=0$$

(16)

To reflect the influence of the first stress invariant (I₁) on the shear strength of soil, Drucker and Prager proposed the generalized extended von Mises criterion, as in Equation (17):

$$F(J_2,I_1) =\sqrt{J_2}-\alpha I_1-k_{\mathrm{M}}=0$$

(17)

where α and k_M are strength parameters.

(2) Mohr–Coulomb criterion

According to the Mohr–Coulomb strength criterion, soil failure transpires when the stress ratio (τ/σ)_max reaches a certain limit, the soil is destroyed. When the cohesive force is reached, the Mohr–Coulomb strength criterion can be articulated as presented in Equation (18):

$$\tan \phi ={\displaystyle \frac{{\sigma }_1-{\sigma }_3}{2\sqrt{{\sigma }_1{\sigma }_3}}}$$

(18)

where φ is the internal frictional angle and σ₁ and σ₃ are the major and minor principal stresses, respectively.

(3) Lade–Ducan criterion

The Lade–Duncan strength criterion was formulated based on the true triaxial testing of sand, as in Equation (19):

$$F(I_1,I_2) = I_1^3-k_{\mathrm{L}}{I}_1^3=0$$

(19)

where k_L is the strength parameter.

Figure 18, Figure 19 and Figure 20 show the comparison of the predicted failure lines of different strength criteria and experimental data on the π-plane. The experimental findings exhibit a greater alignment with the anticipated failure boundaries as delineated by the Lade–Duncan strength criterion. The failure trajectory of the unsaturated SP-SC soil in the true triaxial test on the π plane is also consistent with the Lade-Duncan failure criterion [47]. However, the findings obtained from the true triaxial with controlled suction tests conducted on unsaturated sandy soil deviate from the predictions of the Lade-Duncan failure criterion [48]. It can be seen that the applicability of different strength criteria is related to the type of soil.

5. Conclusions

(1) The shear stress (q) of unsaturated loess exhibits an upward trend in response to increases in net mean stress (p) and matric suction (s), while it demonstrates a downward trend with respect to increases in the middle principal stress coefficient (b). The p and s exert minimal impact on the morphology of the stress–strain curve. However, as the b increases, the stress–strain curve progressively shifts from exhibiting strain hardening characteristics to demonstrating ideal elastic-plastic behavior.

(2) The slope of the strength failure line (M_f) is predominantly influenced by the middle principal stress condition, while the effect of s can be ignored. The relationship $M_f(b,h)=g(b,h)M_c$ is established to approximately describe the relationship between M_f and b.

(3) The longitudinal intercept (μ_f) does not demonstrate a significant trend of variation as the b increases. The relationship between the μ_f and s under different values of b can be described by ${\mu }_f= s/(A_f+B_{f}s)$.

(4) The friction angle (φ′) is predominantly affected by the middle principal stress condition, but the influence of s can be ignored. A relationship between φ′ and b and the relevant parameters of the axial compression state is established.

(5) The total cohesion (c) does not exhibit a clear trend as b increases, while the c increases significantly with s. And the correlation between total cohesion (c) of unsaturated loess for different values of b and s can be described by $c = s/(A_c+B_{c}s)$.

(6) After comparing the failure lines predicted with three different strength criteria and experimental data in the π-plane, the experimental results of unsaturated reshaped loess under complex stress paths are consistent with the results of the failure lines predicted by the Lade–Duncan strength criterion.