# Experimental Study on the Anisotropy and Non-coaxiality of Frozen Standard Sand under Different Principal Stress Directions

Authors: Dun Chen; Guoyu Li; Xiaodong Zhao; Wei Ma; Zhiwei Zhou; Yanhu Mu; Zejin Lai; Tuo Chen
Journal: Geofluids
 (2022)

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## Abstract

Owing to the limitations of the apparatus, the influence of its principal stress direction on the anisotropic behavior and non-coaxiality of frozen soil has not been fully considered in previous studies. At a temperature of -10°C, a series of hollow cylinder tests for frozen standard sand (FSS) was conducted under different directional angles of major principal stress and mean principal stresses in this study. The experimental results indicate that the stress-strain-strength anisotropy and non-coaxiality of the FSS are highly dependent on the principal stress direction. The stress components of the FSS vary linearly with increasing shear stress at different directional angles of the major principal stress and mean principal stresses. With a linear increase in shear stress, the strain components of the FSS exhibited a nonlinear increasing trend. The FSS strength gradually decreased as the directional angle of the major principal stress and the mean principal stresses in the test range increased. Under the different principal stress directions, the non-coaxiality of the FSS, non-coincidence of the direction of the principal strain increment and the principal stress direction, were observed. The directions of the principal strain increment and principal stress gradually tended to be coaxial as shear stress increased. Although the non-coaxial angle of the FSS increased gradually with an increase in the directional angles of the major principal stress, it did not change with the change in the mean principal stress. The non-coaxial angle of the FSS was observed to be as large as 35° in the early stage of shearing under different mean principal stresses.


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## Body

## 1. Introduction

The principal stress direction varies in almost all geotechnical constructions [1–4],such as earthquakes, traffic loading, and sea waves, and significantly impacts geotechnical engineering. Numerous experimental studies have been conducted to verify the anisotropic behavior and non-coaxiality of soil under different principal stress directions [3, 4]. Anisotropic behavior and non-coaxiality are two important characteristics of soil that substantially influence its mechanical behavior [5–7]. When designing infrastructure, without considering the influence of these soil characteristics, the deformation of the soil may be severely underestimated. Therefore, it is important to investigate the anisotropic behavior and non-coaxiality of frozen soil, to understand mechanical behavior under different principal stress directions.Depending on the cause, soil anisotropy can be classified as inherent anisotropy or stress-induced anisotropy [5]. Stress-induced anisotropy refers to the difference in the mechanical properties of soil in different stress directions triggered by various stress conditions under complex stress states. Several studies on the stress-strain-strength anisotropy of soil have been conducted from both micro and macro perspectives. Microscopic analysis has revealed the mechanism of soil properties in different stress directions from the fabric and arrangement of soil particles, whereas macroscopic analysis has elucidated the anisotropic behavior of soil through mechanical properties, such as modulus, stress-strain properties, and strength under different stress paths. Several scholars have investigated the anisotropic characteristics of soil at the microscopic level, comparing the fabric and particle arrangement of soil before and after test loading via the electrical conductivity method (ECM) [5], scanning electron microscopy (SEM) [6], computed tomography (CT) scanning, X-ray methods, and discrete element methods (DEM) [7]. Klein and Santamarina [5] identified anisotropic behavior in the structure of mica sheets using ECM tests. Ye et al. [6] conducted SEM tests on soft soil from the eastern China and discovered that permeability varied in the horizontal and vertical directions. In addition, seepage characteristics were verified to be closely related to the pore distribution and connectivity. Jiang et al. [7] investigated the macro and micro anisotropic behaviors of soil using numerical techniques, such as DEM, and then discovered that the macroscopic anisotropy of soil depends on the anisotropy of the arrangement of its microstructure. Several researchers [8–11] have investigated the anisotropic behavior of soil using different geotechnical tests in the macroscopic study of soil anisotropy and discovered that the stress loading directions significantly influence the mechanical properties of soil. Bodner [8] reported that the shear strength of clay is closely related to the shear failure surface based on an earlier study on the anisotropic behavior of soil through direct-shear tests. Gong [9] investigated the anisotropic behavior of soil in various specimen cutting directions using triaxial compression tests. This study identified a link between specimen cutting angle and soil strength. The anisotropic behavior of the soil was then obtained in experimental studies on sand using a true triaxial apparatus that can control stress in three orthogonal directions [10, 11]. The true triaxial test revealed that the deformation of sand in three different orthogonal directions exhibited clear anisotropic behavior. Despite extensive studies on the anisotropic behavior of soil, it does not correspond to the actual stress state was triggered by the principal stress direction in the field. The hollow cylinder apparatus (HCA) was employed to determine the anisotropy of the soil while considering the influence of principal stress rotation. Symes et al. [1] published a study on the anisotropy of soil under various principal stress directions. Several researchers [12–17] have previously investigated the anisotropic behavior of sand and clay using HCA tests, and the obtained results have suggested that the strength and stress-strain response of soil strongly depend on the change in the principal stress direction. Moreover, multiple researchers [18, 19] have investigated the anisotropic behavior of soils subjected to principal stress rotation. These investigations indicate that the variations in the principal stress direction are related to the mechanical properties of the saturated soft clay.Non-coaxiality is defined as the non-coincidence of the direction of the principal strain increment and the direction of the principal stress [20]. Non-coaxiality is the key to understanding strength behavior and establishing constitutive models of soil. Roscoe et al. [21] discovered the non-coaxiality of soil with principal stress direction inconsistent with the direction of the plastic principal strain increment as early as the 1960s using direct shear tests. The non-coaxiality of the soil was clearer at the beginning of shearing; however, as the specimen approached failure, the directions of the plastic principal strain increment and the principal stress tended to be coaxial. Wong and Arthur [22] discovered that the rotation of the principal stress axis triggers a non-coaxial behavior, with the angle between the principal stress direction and the direction of the principal strain increment reaching up to 30°. Numerous studies conducted in recent years have demonstrated the non-coaxiality of soil using HCA, owing to the advancement of test instruments. Symes et al. [1] conducted undrained tests on sand, confirmed the existence of the non-coaxiality of sand under a directional shear stress path, and discovered that the non-coaxial angle gradually decreased with an increase in shear stress, eventually tending to be coaxial. According to some researchers, the non-coaxiality of Toyoura sand is clearer under the continuous rotation of the principal stress axis than under monotonic shear [23–25]. Numerous experimental studies conducted recently have demonstrated the non-coaxial behavior of soil under directional shear stress path, with the elastic component exerting a minor influence [26–30]. However, owing to the limitations of the apparatus, the non-coaxiality characteristics of frozen soil under different principal stress directions are yet to be investigated.As summarized above, several experimental studies on anisotropy and non-coaxiality, as represented by variations in the principal stress direction, on soil behavior have been reported in the literature. However, there have been no attempts to present the anisotropy and non-coaxial behavior of frozen soil. Several researchers have investigated the mechanical properties of frozen soil using conventional geotechnical tests [31–37]. In previous studies, HCA tests have not been used as a preferred method for studying the anisotropic behaviors and non-coaxiality of frozen soil because they are limited by laboratory methods. Recently, although the strength and dynamic deformation of frozen clay have been described and analyzed by researchers using the dynamic hollow cylinder apparatus for frozen soil (FHCA-300) [2, 38, 39], the stress-strain-strength anisotropy and non-coaxiality of frozen soil that varies with the in situ loading direction cannot be simulated. Hence, the non-coaxiality phenomenon of frozen soil has not been studies, and research on the anisotropy of frozen soil remains insufficient. Therefore, in this study, we present an experimental study on the anisotropy and non-coaxiality of frozen standard sand (FSS) using the FHCA-300, which considers the influence of the principal stress direction. The test results of directional shear stress path with different directional angles of the major principal stress and mean principal stresses on the FSS were presented and analyzed. These studies can provide further insights into the influence of principal stress direction on the mechanical characteristics of frozen soil.


## 2. Research Significance

Anisotropy and non-coaxiality, two important characteristics of soil, significantly impact on the mechanical behavior of soil [1, 40]. Varying strengths and stress-strain relationships of frozen soil were observed with different stress paths, particularly under different principal stress directions, exhibiting significant anisotropic characteristics of frozen soil [38, 39]. The stress-strain-strength behaviors of frozen clay are substantially affected by the fixed direction of principal stress, which has an evident anisotropic behavior of frozen clay [2, 38]. However, anisotropic behavior, especially considering the effect of the various directions of principal stress, has rarely been reported for the FSS. Therefore, the anisotropic behavior of the FSS under different principal stress directions must be investigated.Conventional elastic-plastic constitutive models of frozen soil were used to predict the deformation and strength under different stress states. The coaxiality assumption (the consistency between the directions of the principal strain increment and principal stress) has been implied in the constitutive models of frozen soil established in the past [31–37], which cannot reflect non-coaxiality, thereby resulting in a serious underestimation of the deformation of the frozen soil. Infrastructure design that does not consider the effect of non-coaxiality may be unsafe in permafrost regions. The evolution law of non-coaxiality characteristics in frozen soil is unclear, especially under the stress paths involved in the principal stress direction. Therefore, it is important to consider the non-coaxiality of frozen soil in the mechanical behavior and constitutive model of frozen soil under different principal stress directions. The primary purpose of this study is to use the FHCA-300 to investigate the anisotropy and non-coaxiality of the FSS under different principal stress directions.


## 3. Materials and Laboratory Tests

### 3.1. Test Apparatus

The experiments in this study were performed using the FHCA-300. Chen et al. [38] provided a detailed description of apparatus functions. Unlike other geotechnical test apparatus, the FHCA-300 can perform principal stress rotations at different temperatures [38, 39]. The HCA tests are extremely useful for studying the mechanical behavior of frozen soil under complex loading conditions. Therefore, complex geotechnical tests with multiple stress paths (such as directional shear stress paths) can be performed.By applying controlled loads (W, MT, pi, and po), the four stress components (σz, σr, σθ, and τzθ) of the hollow cylindrical soil specimens can be controlled. Hight et al. [18] provided the equations for calculating the stress and strain components, as presented in Table 1. The non-coaxial angle is adopted to quantify the degree of non-coaxiality and is defined as the angle between the directions of the principal strain increment and principal stress [18]. The non-coaxial angle can be calculated as
(1)α=12arctan2τzθσz−σθ.(2)αds=12arctandγzθdεz−dεθ.(3)β=αds−α.Table 1
Equations for data interpretation [39].
Stress
Strain
Axial
σ
z
=
W
/
π
r
o
2
−
r
i
2
+
p
o
r
o
2
−
r
p
2
−
p
i
r
i
2
/
r
o
2
−
r
i
2
ε
z
=
Δ
h
/
H
o
Circumferential normal
σ
θ
=
p
o
r
o
−
p
i
r
i
/
r
o
−
r
i
ε
θ
=
−
u
o
+
u
i
/
r
o
+
r
i
Radial
σ
r
=
p
o
r
o
+
p
i
r
i
/
r
o
+
r
i
ε
r
=
−
u
o
−
u
i
/
r
o
−
r
i
Circumferential shear
τ
z
θ
=
τ
θ
z
=
M
T
r
o
+
r
i
/
π
r
o
4
−
r
i
4
γ
z
θ
=
γ
θ
z
=
2
Δ
θ
r
o
3
−
r
i
3
/
3
H
o
r
o
2
−
r
i
2
Major principal
σ
1
=
σ
z
+
σ
θ
/
2
+
σ
z
−
σ
θ
/
2
2
+
τ
z
θ
2
ε
1
=
ε
z
+
ε
θ
/
2
+
ε
z
−
ε
θ
/
2
2
+
γ
z
θ
2
Intermediate principal
σ
2
=
σ
r
ε
2
=
ε
r
Minor principal
σ
3
=
σ
z
+
σ
θ
/
2
−
σ
z
−
σ
θ
/
2
2
+
τ
z
θ
2
ε
3
=
ε
z
+
ε
θ
/
2
−
ε
z
−
ε
θ
/
2
2
+
γ
z
θ
2


### 3.2. Specimen Preparation

#### 3.2.1. Material Selection

In a series of experiments, China’s ISO standard sand was adopted to circumvent differences caused by the inherent anisotropy of frozen soil (refer to Figure1). The specific gravity and particle size of standard sand were determined as 2.643 and 0.5–1.0 mm, respectively. Their physical properties are presented in Table 2.Figure 1
China’s ISO standard sand.Table 2
Physical properties of standard sand.
Specific gravityGs
Nonuniformity coefficientCU
Curvature coefficientCC
Particle sizeD
Dry densityg/cm3
2.643
1.16
1.06
0.5–1.0
1.53


#### 3.2.2. Specimen Preparation Method

Compared with the specimen preparation of frozen clay [38], the hollow cylindrical specimen of the FSS was prepared on the FHCA-300, with dimensions of 100 mm/60 mm/200 mm (OD/ID/height). In this study, the hollow cylindrical specimen of the FSS was prepared using the following procedure (as shown in Figure 2):
(a)
Installation preparation. The inner membrane was inserted into the bottom of the base pedestal of the FHCA-300 by the “O” shaped rubber ring, as illustrated in Figure 2(a). The outer membrane and mold (refer to Figure 2(b)) were then assembled on the base pedestal of the FHCA-300.(b)
Specimen preparation. The pluviation method was adopted to create a hollow cylindrical specimen of the FSS. In addition, the prepared dry sand was sprinkled evenly along the gap between the inner and outer membranes using a funnel (Figure 2(c)). The top surface of the hollow cylindrical specimen was leveled with a brush, and the upper indenter was installed for sealing, as illustrated in Figures 2(d) and 2(e).(c)
Complete installation. First, a pressure rod was employed to fix the upper part of the specimen. Second, after slowly lowering and tightening the cell chamber of the FHCA-300, the outer and inner cells were filled with aviation oil. Third, as illustrated in Figure 2(f), the pressure tank was wrapped with thermal insulation foam to protect it from external heat exchange.(d)
Specimen freezing. Under vacuum conditions, the hollow cylindrical specimens of the FSS were suctioned in distilled water until they were fully saturated. The FSS specimens were then quickly frozen at −30°C for 48 h. The HCA tests of the FSS began after the specimen temperature was increased to −10°C and maintained for 12 h.Figure 2
Preparation procedures of hollow cylindrical specimen of the FSS.
(a)
(b)
(c)
(d)
(e)
(f)


### 3.3. Program of Laboratory Tests

The stress path of directional shear was selected in this study to investigate the anisotropic behavior and non-coaxiality of the FSS under different principal stress directions. The hollow cylindrical specimens of the FSS were subjected to a series of HCA tests atα=0°,15°,and30° with p=2,4.5,and6MPa. Meanwhile, as illustrated in Table 3, all tests in this study were performed keeping T constant at −10°C and b at 0.5.Table 3
Summay of directional shear tests.
Test Nos.
T
/
°
C
α
/
°
p
/
MPa
b
T1
−10
0
4.5
0.5
T2
−10
15
4.5
0.5
T3
−10
30
4.5
0.5
T4
−10
30
2
0.5
T5
−10
30
6
0.5The four parameters (p, α, b, and T) remained constant during the stress path loading step, while qs was increased at a rate of 30 kPa/min until σz reached 20% or γzθ reached 30% [41], i.e., the shear stress increased along the different parameters (α and p) until failure occurred. A flowchart of the testing program is presented in Figure 3.Figure 3
Flowchart of testing program.


## 3.1. Test Apparatus

The experiments in this study were performed using the FHCA-300. Chen et al. [38] provided a detailed description of apparatus functions. Unlike other geotechnical test apparatus, the FHCA-300 can perform principal stress rotations at different temperatures [38, 39]. The HCA tests are extremely useful for studying the mechanical behavior of frozen soil under complex loading conditions. Therefore, complex geotechnical tests with multiple stress paths (such as directional shear stress paths) can be performed.By applying controlled loads (W, MT, pi, and po), the four stress components (σz, σr, σθ, and τzθ) of the hollow cylindrical soil specimens can be controlled. Hight et al. [18] provided the equations for calculating the stress and strain components, as presented in Table 1. The non-coaxial angle is adopted to quantify the degree of non-coaxiality and is defined as the angle between the directions of the principal strain increment and principal stress [18]. The non-coaxial angle can be calculated as
(1)α=12arctan2τzθσz−σθ.(2)αds=12arctandγzθdεz−dεθ.(3)β=αds−α.Table 1
Equations for data interpretation [39].
Stress
Strain
Axial
σ
z
=
W
/
π
r
o
2
−
r
i
2
+
p
o
r
o
2
−
r
p
2
−
p
i
r
i
2
/
r
o
2
−
r
i
2
ε
z
=
Δ
h
/
H
o
Circumferential normal
σ
θ
=
p
o
r
o
−
p
i
r
i
/
r
o
−
r
i
ε
θ
=
−
u
o
+
u
i
/
r
o
+
r
i
Radial
σ
r
=
p
o
r
o
+
p
i
r
i
/
r
o
+
r
i
ε
r
=
−
u
o
−
u
i
/
r
o
−
r
i
Circumferential shear
τ
z
θ
=
τ
θ
z
=
M
T
r
o
+
r
i
/
π
r
o
4
−
r
i
4
γ
z
θ
=
γ
θ
z
=
2
Δ
θ
r
o
3
−
r
i
3
/
3
H
o
r
o
2
−
r
i
2
Major principal
σ
1
=
σ
z
+
σ
θ
/
2
+
σ
z
−
σ
θ
/
2
2
+
τ
z
θ
2
ε
1
=
ε
z
+
ε
θ
/
2
+
ε
z
−
ε
θ
/
2
2
+
γ
z
θ
2
Intermediate principal
σ
2
=
σ
r
ε
2
=
ε
r
Minor principal
σ
3
=
σ
z
+
σ
θ
/
2
−
σ
z
−
σ
θ
/
2
2
+
τ
z
θ
2
ε
3
=
ε
z
+
ε
θ
/
2
−
ε
z
−
ε
θ
/
2
2
+
γ
z
θ
2


## 3.2. Specimen Preparation

### 3.2.1. Material Selection

In a series of experiments, China’s ISO standard sand was adopted to circumvent differences caused by the inherent anisotropy of frozen soil (refer to Figure1). The specific gravity and particle size of standard sand were determined as 2.643 and 0.5–1.0 mm, respectively. Their physical properties are presented in Table 2.Figure 1
China’s ISO standard sand.Table 2
Physical properties of standard sand.
Specific gravityGs
Nonuniformity coefficientCU
Curvature coefficientCC
Particle sizeD
Dry densityg/cm3
2.643
1.16
1.06
0.5–1.0
1.53


### 3.2.2. Specimen Preparation Method

Compared with the specimen preparation of frozen clay [38], the hollow cylindrical specimen of the FSS was prepared on the FHCA-300, with dimensions of 100 mm/60 mm/200 mm (OD/ID/height). In this study, the hollow cylindrical specimen of the FSS was prepared using the following procedure (as shown in Figure 2):
(a)
Installation preparation. The inner membrane was inserted into the bottom of the base pedestal of the FHCA-300 by the “O” shaped rubber ring, as illustrated in Figure 2(a). The outer membrane and mold (refer to Figure 2(b)) were then assembled on the base pedestal of the FHCA-300.(b)
Specimen preparation. The pluviation method was adopted to create a hollow cylindrical specimen of the FSS. In addition, the prepared dry sand was sprinkled evenly along the gap between the inner and outer membranes using a funnel (Figure 2(c)). The top surface of the hollow cylindrical specimen was leveled with a brush, and the upper indenter was installed for sealing, as illustrated in Figures 2(d) and 2(e).(c)
Complete installation. First, a pressure rod was employed to fix the upper part of the specimen. Second, after slowly lowering and tightening the cell chamber of the FHCA-300, the outer and inner cells were filled with aviation oil. Third, as illustrated in Figure 2(f), the pressure tank was wrapped with thermal insulation foam to protect it from external heat exchange.(d)
Specimen freezing. Under vacuum conditions, the hollow cylindrical specimens of the FSS were suctioned in distilled water until they were fully saturated. The FSS specimens were then quickly frozen at −30°C for 48 h. The HCA tests of the FSS began after the specimen temperature was increased to −10°C and maintained for 12 h.Figure 2
Preparation procedures of hollow cylindrical specimen of the FSS.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.2.1. Material Selection

In a series of experiments, China’s ISO standard sand was adopted to circumvent differences caused by the inherent anisotropy of frozen soil (refer to Figure1). The specific gravity and particle size of standard sand were determined as 2.643 and 0.5–1.0 mm, respectively. Their physical properties are presented in Table 2.Figure 1
China’s ISO standard sand.Table 2
Physical properties of standard sand.
Specific gravityGs
Nonuniformity coefficientCU
Curvature coefficientCC
Particle sizeD
Dry densityg/cm3
2.643
1.16
1.06
0.5–1.0
1.53


## 3.2.2. Specimen Preparation Method

Compared with the specimen preparation of frozen clay [38], the hollow cylindrical specimen of the FSS was prepared on the FHCA-300, with dimensions of 100 mm/60 mm/200 mm (OD/ID/height). In this study, the hollow cylindrical specimen of the FSS was prepared using the following procedure (as shown in Figure 2):
(a)
Installation preparation. The inner membrane was inserted into the bottom of the base pedestal of the FHCA-300 by the “O” shaped rubber ring, as illustrated in Figure 2(a). The outer membrane and mold (refer to Figure 2(b)) were then assembled on the base pedestal of the FHCA-300.(b)
Specimen preparation. The pluviation method was adopted to create a hollow cylindrical specimen of the FSS. In addition, the prepared dry sand was sprinkled evenly along the gap between the inner and outer membranes using a funnel (Figure 2(c)). The top surface of the hollow cylindrical specimen was leveled with a brush, and the upper indenter was installed for sealing, as illustrated in Figures 2(d) and 2(e).(c)
Complete installation. First, a pressure rod was employed to fix the upper part of the specimen. Second, after slowly lowering and tightening the cell chamber of the FHCA-300, the outer and inner cells were filled with aviation oil. Third, as illustrated in Figure 2(f), the pressure tank was wrapped with thermal insulation foam to protect it from external heat exchange.(d)
Specimen freezing. Under vacuum conditions, the hollow cylindrical specimens of the FSS were suctioned in distilled water until they were fully saturated. The FSS specimens were then quickly frozen at −30°C for 48 h. The HCA tests of the FSS began after the specimen temperature was increased to −10°C and maintained for 12 h.Figure 2
Preparation procedures of hollow cylindrical specimen of the FSS.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.3. Program of Laboratory Tests

The stress path of directional shear was selected in this study to investigate the anisotropic behavior and non-coaxiality of the FSS under different principal stress directions. The hollow cylindrical specimens of the FSS were subjected to a series of HCA tests atα=0°,15°,and30° with p=2,4.5,and6MPa. Meanwhile, as illustrated in Table 3, all tests in this study were performed keeping T constant at −10°C and b at 0.5.Table 3
Summay of directional shear tests.
Test Nos.
T
/
°
C
α
/
°
p
/
MPa
b
T1
−10
0
4.5
0.5
T2
−10
15
4.5
0.5
T3
−10
30
4.5
0.5
T4
−10
30
2
0.5
T5
−10
30
6
0.5The four parameters (p, α, b, and T) remained constant during the stress path loading step, while qs was increased at a rate of 30 kPa/min until σz reached 20% or γzθ reached 30% [41], i.e., the shear stress increased along the different parameters (α and p) until failure occurred. A flowchart of the testing program is presented in Figure 3.Figure 3
Flowchart of testing program.


## 4. Results and Discussion

### 4.1. Realization of Directional Shear Stress Path

In this study, the stress path of the directional shear was followed. A series of HCA tests were performed on frozen soil at differentα and p values. As illustrated in Figure 4, the stress path of the directional shear was determined by increasing qs until failure, while the α value was fixed at 0°, 15°, and 30°. The angle between σz−σθ/2 and τzθ was twice the directional angle of the major principal stress as given by Equation (1). In the deviatoric stress space, α value remained constant, and the hollow cylindrical specimen of the FSS was along the loading stress path until the specimen failed, as illustrated in Figure 5. The corresponding values of the mean principal stresses when α=30° are 2 MPa, 4.5 MPa, and 6 MPa (as shown in Figures 6 and 7), respectively. In Figure 6, the stress paths overlap because the directional angles of the major principal stresses are all 30° in the deviatoric stress space, thereby confirming the accuracy of the loading paths. In summary, the actual stress paths (scatter points) closely match the theoretical curves (solid lines).Figure 4
Stress path under different principal stress directions.Figure 5
Relationships between directional angle of the major principal stress and shear stress under different principal stress directions.Figure 6
Stress path under different mean principal stresses.Figure 7
Relationships between mean principal stress and shear stress under different mean principal stresses.


### 4.2. Stress and Strain Component Characteristic of the FSS under the Directional Shear Stress Path

This section presents and discusses the results obtained from a series of the FSS tests with varyingp and α value. Based on Table 1, the following equations can be used to calculate the corresponding relationship between the generalized stresses (qs, p, α, and b) and stress components (σz, σr, σθ, and τzθ) [42]:
(4)σz=p−2bqs−q/23+qscos2α,σr=p+22bqs−qs3,σθ=p−2bqs−qs3−qscos2α,τzθ=qssin2α.


#### 4.2.1. Stress Characteristics of the FSS under Different Directional Angles of Major Principal Stress

Figure8 presents the relationships between qs and the stress components of the FSS under different α values at T=−10°C. The following features can be clearly identified: (a) the σz of the FSS increases linearly with increasing qs under the directional shear stress path; however, the increased amplitude of the axial stress varies with the α value; (b) when α=0°, the hollow cylindrical specimen of the FSS is only subjected to an axial load, and τzθ is always zero throughout the loading procedure; (c) the σr of the FSS is always constant with the linear increase in qs, and the value is equal to the mean principal stress (p=4.5MPa); (d) the σθ of the FSS decreases with increasing qs; and (e) when α=15°and30°, the axial stress and circumferential shear stress of the FSS increases simultaneously owing to the coupling effect of axial load and torque, and the hollow cylindrical specimen of the FSS exhibits both compression and torsional shears.Figure 8
Relationships between shear stress and stress components of the FSS.
(a)
α
=
0
°
(b)
α
=
15
°
(c)
α
=
30
°


#### 4.2.2. Strain Characteristics of the FSS under Different Directional Angles of Major Principal Stress

Figure9 presents the variations in qs and the strain components of the FSS under different α values. The development of the strain component curves of the FSS exhibits significant differences with the nonlinear increase in shear stress under the directional shear stress path. Only the axial strain of the FSS increases nonlinearly with an increase in qs at α=0°, as illustrated in Figure 9(a); γzθ, εr, and εθ are always maintained at zero. The strain component exhibits a nonlinear growth trend at α=15°and30°. And the hollow cylindrical specimen of the FSS exhibits the axial compression and radial expansion states owing to the coupling effects of the axial load and torque. Therefore, the radial strain of the FSS exhibits a negative growth trend at α=15°and30°. Simultaneously, the strain components exhibit a slow growth trend in the early stage and then a rapid growth trend as they approach failure.Figure 9
Relationships between shear stress and strain components of the FSS.
(a)
α
=
0
°
(b)
α
=
15
°
(c)
α
=
30
°


#### 4.2.3. Stress Characteristics of the FSS under Different Mean Principal Stresses

The stress components are plotted against the shear stress in Figures10(a) and 10(b) at p=2and6MPa, respectively. Figure 8(c) depicts the relationship curves between the stress components and shear stress with p=4.5MPa under α=30° at T=−10°C. The different stress components change linearly with an increase in qs during the directional shear tests of the FSS. As qs, σz, and τzθ increase linearly, σθ decreases linearly, while the σr simultaneously remains constant, with its value being equal to the mean principal stress.Figure 10
Relationships between shear stress and stress components of the FSS.
(a)
p
=
2
MPa
(b)
p
=
6
MPa


#### 4.2.4. Strain Characteristics of the FSS under Different Mean Principal Stresses

Figures11(a), 9(c), and 11(b) present the strain components versus shear stress obtained from experiments under p=2,4.5,and6MPa at T=−10°C, respectively. During the shearing process, the strain component curves of the FSS exhibit a nonlinear increase with a linear increase in qs, thus indicating that the strain component initially increases slowly and then rapidly it approached failure. The radial strain and circumferential normal strain increase negatively as qs increases; however, the variation in the radial strain is greater than that in the circumferential normal strain of the FSS.Figure 11
Relationships between shear stress and strain components of the FSS.
(a)
p
=
2
MPa
(b)
p
=
6
MPa


### 4.3. Anisotropy Behavior of the FSS

#### 4.3.1. Strength Characteristics of the FSS under Different Directional Angles of Major Principal Stress

A series of FSS tests were performed using the FHCA-300 to determine the effects of different principal stress directions on deformation and strength. From Figure12, the strain components develop with stress components with p=4.5MPa under α=0°,15°,and30° at T=−10°Candb=0.5.Figure 12
Stress-strain curves of the FSS under different principal stress directions.
(a)
Axial stress-strain curves
(b)
Circumferential shear stress-strain curves
(c)
Generalized shear stress-strain curvesFigure12(a) illustrates the relationship between the axial stress and strain of the FSS. Under different principal stress directions, the axial stress-strain curves of the FSS all exhibit the strain-hardening phenomena. The axial strength of the FSS gradually decreases as α increases. The anisotropy of the FSS was induced by the principal stress direction. Figure 12(b) depicts the circumferential shear stress-strain behavior of the FSS at various values of α. The torsional shear strength of the FSS increases with increasing α value in the directional shear tests. The principal stress direction had a significant impact on the torsional shear strength of the FSS. When α=0°, the hollow cylindrical specimens of the FSS were only subjected to an axial load, thereby resulting in approximately zero of circumferential shear strain. However, the circumferential shear stress-strain curves exhibit strain hardening at α=15°and30°. Figure 12(c) illustrates the relationship between the generalized shear stress-strain curves obtained from the directional shear tests of the FSS under different principal stress directions. The generalized shear stress-strain curves of the FSS are primarily strain-hardening curves. However, the generalized shear stress-strain curves of the FSS exhibit a weak hardening tendency at α=15°. The strength of the FSS gradually decreases as α value increases. The axial component dominates the generalized shear stress-strain curves of the FSS more than the circumferential shear component.In this study, theq value corresponding to γg=5% was considered as the failure point of the FSS. Figure 13 illustrates the changes in the strength of the FSS and frozen clay at various α values. Chen et al. [38] described the test data for frozen clay with the same stress path as that of the FSS. As observed in Figure 13, the strengths of the FSS and frozen clay differ in two ways: on the one hand, the strength of the FSS is greater than that of the frozen clay under the same directional angle of major principal stress; the strength deviation between the FSS and frozen clay can be as large as 4.797 MPa and 3.057 MPa at α=15°and30°, respectively; on the other hand, with the variation in α value, the strength of the FSS varies more abruptly than that of the frozen clay. Based on the preceding analysis and discussion, it can be inferred that the anisotropic behavior of frozen soil is caused by the principal stress direction; frozen soil exhibits stress-strain-strength variations depending on the principal stress directions.Figure 13
Strength variations of the FSS and frozen clay with different principal stress directions.


#### 4.3.2. Strength Characteristics of the FSS under Different Mean Principal Stresses

Figure14 presents the variations in the FSS with stress and strain under p=2,4.5,and6MPa at T=−10°C. The axial stress-strain curves of the FSS exhibit strain-hardening characteristics under different mean principal stresses. The strength gradually increases as the p value increases. The hollow cylindrical specimen of the FSS was always compressed. Figures 14(b) and 14(c) present the circumferential shear and generalized shear stress-strain curves of the FSS, respectively. The strength of the FSS gradually increases as the p value increases within the test range. The strength of the FSS at p=6MPa was significantly higher than those at p=2and4.5MPa.Figure 14
Stress-strain curves of the FSS under different mean principal stresses.
(a)
Axial stress-strain curves
(b)
Circumferential shear stress-strain curves
(c)
Generalized shear stress-strain curves


### 4.4. Non-coaxiality of the FSS

#### 4.4.1. Non-coaxiality of the FSS under Different Directional Angles of Major Principal Stress

The relationship betweenβ and qs was calculated based on the test data of the FSS under different fixed principal stress directions. Figure 15 presents the variation in the direction of the principal strain increment with qs. The principal strain increment directions of the FSS and stress path do not coincide at α=15°and30°; however, when α=0°, the non-coaxial behavior of the FSS remains unclear. When α=15°and30°, the non-coaxial angle of the FSS (the non-coaxial angle was approximately 20° and 40° in the early stage, respectively) decreases as qs increases, and the degree of non-coaxiality of the FSS decreases as the hollow cylindrical specimen approaches failure. Initially, the axes of the principal stress and principal strain increment are non-coincident. The non-coaxial angle of the FSS at α=30° is inferred to be greater than that at α=15°. Namely, for variations in the non-coaxiality of the FSS under different fixed principal stress directions, the non-coaxial angle fluctuation increases as the directional angle of the major principal stress increases.Figure 15
Variations of non-coaxial angle with shear stress under different principal stress directions.As illustrated in Figure16, the stress envelope surface is represented by the σz−σθ/2 versus the τzθ stress space. Meanwhile, various directions of the principal stress and principal strain increment are presented. According to Equation (1), the angle is formed by σz−σθ/2 and the stress path is twice the directional angle of the major principal stress. At α=15°, the direction of the principal strain increment of the FSS deviates slightly from that of the principal stress. In tests with α=30°, there are relatively large deviations between the direction of principal stress and principal strain increment.Figure 16
Direction of principal strain increment along different principal stress directions.


#### 4.4.2. Non-coaxiality of the FSS under Different Mean Principal Stresses

Figure17 depicts the non-coaxial behavior of the FSS between the axes of the principal stresses and principal strain increments under different mean principal stresses. The non-coaxiality of the FSS with shear stress follows the same change rule as that of the FSS under α=15°and30°. When qs is small, the non-coaxial angle of the FSS is larger, and as qs increases, the non-coaxial angle of the FSS gradually decreases. When the hollow cylindrical specimen of the FSS is nearing failure, the non-coaxial angle reaches its minimum value, thus indicating that as the shear stress increases, the direction of the principal strain increment and the direction of the principal stress tend to be coaxial. In other words, the deviation of the non-coaxial angle of the FSS can be as large as 35° in the early stage of shearing. Subsequently, the deviation decreases as the shear strain increases, and the non-coaxiality behavior of the FSS is almost coaxiality at failure. The evolution law of the non-coaxial angle of the FSS does not change as the mean principal stress changes and remains within a certain range.Figure 17
Variations of non-coaxial angle with shear stress under mean principal stresses.


## 4.1. Realization of Directional Shear Stress Path

In this study, the stress path of the directional shear was followed. A series of HCA tests were performed on frozen soil at differentα and p values. As illustrated in Figure 4, the stress path of the directional shear was determined by increasing qs until failure, while the α value was fixed at 0°, 15°, and 30°. The angle between σz−σθ/2 and τzθ was twice the directional angle of the major principal stress as given by Equation (1). In the deviatoric stress space, α value remained constant, and the hollow cylindrical specimen of the FSS was along the loading stress path until the specimen failed, as illustrated in Figure 5. The corresponding values of the mean principal stresses when α=30° are 2 MPa, 4.5 MPa, and 6 MPa (as shown in Figures 6 and 7), respectively. In Figure 6, the stress paths overlap because the directional angles of the major principal stresses are all 30° in the deviatoric stress space, thereby confirming the accuracy of the loading paths. In summary, the actual stress paths (scatter points) closely match the theoretical curves (solid lines).Figure 4
Stress path under different principal stress directions.Figure 5
Relationships between directional angle of the major principal stress and shear stress under different principal stress directions.Figure 6
Stress path under different mean principal stresses.Figure 7
Relationships between mean principal stress and shear stress under different mean principal stresses.


## 4.2. Stress and Strain Component Characteristic of the FSS under the Directional Shear Stress Path

This section presents and discusses the results obtained from a series of the FSS tests with varyingp and α value. Based on Table 1, the following equations can be used to calculate the corresponding relationship between the generalized stresses (qs, p, α, and b) and stress components (σz, σr, σθ, and τzθ) [42]:
(4)σz=p−2bqs−q/23+qscos2α,σr=p+22bqs−qs3,σθ=p−2bqs−qs3−qscos2α,τzθ=qssin2α.


### 4.2.1. Stress Characteristics of the FSS under Different Directional Angles of Major Principal Stress

Figure8 presents the relationships between qs and the stress components of the FSS under different α values at T=−10°C. The following features can be clearly identified: (a) the σz of the FSS increases linearly with increasing qs under the directional shear stress path; however, the increased amplitude of the axial stress varies with the α value; (b) when α=0°, the hollow cylindrical specimen of the FSS is only subjected to an axial load, and τzθ is always zero throughout the loading procedure; (c) the σr of the FSS is always constant with the linear increase in qs, and the value is equal to the mean principal stress (p=4.5MPa); (d) the σθ of the FSS decreases with increasing qs; and (e) when α=15°and30°, the axial stress and circumferential shear stress of the FSS increases simultaneously owing to the coupling effect of axial load and torque, and the hollow cylindrical specimen of the FSS exhibits both compression and torsional shears.Figure 8
Relationships between shear stress and stress components of the FSS.
(a)
α
=
0
°
(b)
α
=
15
°
(c)
α
=
30
°


### 4.2.2. Strain Characteristics of the FSS under Different Directional Angles of Major Principal Stress

Figure9 presents the variations in qs and the strain components of the FSS under different α values. The development of the strain component curves of the FSS exhibits significant differences with the nonlinear increase in shear stress under the directional shear stress path. Only the axial strain of the FSS increases nonlinearly with an increase in qs at α=0°, as illustrated in Figure 9(a); γzθ, εr, and εθ are always maintained at zero. The strain component exhibits a nonlinear growth trend at α=15°and30°. And the hollow cylindrical specimen of the FSS exhibits the axial compression and radial expansion states owing to the coupling effects of the axial load and torque. Therefore, the radial strain of the FSS exhibits a negative growth trend at α=15°and30°. Simultaneously, the strain components exhibit a slow growth trend in the early stage and then a rapid growth trend as they approach failure.Figure 9
Relationships between shear stress and strain components of the FSS.
(a)
α
=
0
°
(b)
α
=
15
°
(c)
α
=
30
°


### 4.2.3. Stress Characteristics of the FSS under Different Mean Principal Stresses

The stress components are plotted against the shear stress in Figures10(a) and 10(b) at p=2and6MPa, respectively. Figure 8(c) depicts the relationship curves between the stress components and shear stress with p=4.5MPa under α=30° at T=−10°C. The different stress components change linearly with an increase in qs during the directional shear tests of the FSS. As qs, σz, and τzθ increase linearly, σθ decreases linearly, while the σr simultaneously remains constant, with its value being equal to the mean principal stress.Figure 10
Relationships between shear stress and stress components of the FSS.
(a)
p
=
2
MPa
(b)
p
=
6
MPa


### 4.2.4. Strain Characteristics of the FSS under Different Mean Principal Stresses

Figures11(a), 9(c), and 11(b) present the strain components versus shear stress obtained from experiments under p=2,4.5,and6MPa at T=−10°C, respectively. During the shearing process, the strain component curves of the FSS exhibit a nonlinear increase with a linear increase in qs, thus indicating that the strain component initially increases slowly and then rapidly it approached failure. The radial strain and circumferential normal strain increase negatively as qs increases; however, the variation in the radial strain is greater than that in the circumferential normal strain of the FSS.Figure 11
Relationships between shear stress and strain components of the FSS.
(a)
p
=
2
MPa
(b)
p
=
6
MPa


## 4.2.1. Stress Characteristics of the FSS under Different Directional Angles of Major Principal Stress

Figure8 presents the relationships between qs and the stress components of the FSS under different α values at T=−10°C. The following features can be clearly identified: (a) the σz of the FSS increases linearly with increasing qs under the directional shear stress path; however, the increased amplitude of the axial stress varies with the α value; (b) when α=0°, the hollow cylindrical specimen of the FSS is only subjected to an axial load, and τzθ is always zero throughout the loading procedure; (c) the σr of the FSS is always constant with the linear increase in qs, and the value is equal to the mean principal stress (p=4.5MPa); (d) the σθ of the FSS decreases with increasing qs; and (e) when α=15°and30°, the axial stress and circumferential shear stress of the FSS increases simultaneously owing to the coupling effect of axial load and torque, and the hollow cylindrical specimen of the FSS exhibits both compression and torsional shears.Figure 8
Relationships between shear stress and stress components of the FSS.
(a)
α
=
0
°
(b)
α
=
15
°
(c)
α
=
30
°


## 4.2.2. Strain Characteristics of the FSS under Different Directional Angles of Major Principal Stress

Figure9 presents the variations in qs and the strain components of the FSS under different α values. The development of the strain component curves of the FSS exhibits significant differences with the nonlinear increase in shear stress under the directional shear stress path. Only the axial strain of the FSS increases nonlinearly with an increase in qs at α=0°, as illustrated in Figure 9(a); γzθ, εr, and εθ are always maintained at zero. The strain component exhibits a nonlinear growth trend at α=15°and30°. And the hollow cylindrical specimen of the FSS exhibits the axial compression and radial expansion states owing to the coupling effects of the axial load and torque. Therefore, the radial strain of the FSS exhibits a negative growth trend at α=15°and30°. Simultaneously, the strain components exhibit a slow growth trend in the early stage and then a rapid growth trend as they approach failure.Figure 9
Relationships between shear stress and strain components of the FSS.
(a)
α
=
0
°
(b)
α
=
15
°
(c)
α
=
30
°


## 4.2.3. Stress Characteristics of the FSS under Different Mean Principal Stresses

The stress components are plotted against the shear stress in Figures10(a) and 10(b) at p=2and6MPa, respectively. Figure 8(c) depicts the relationship curves between the stress components and shear stress with p=4.5MPa under α=30° at T=−10°C. The different stress components change linearly with an increase in qs during the directional shear tests of the FSS. As qs, σz, and τzθ increase linearly, σθ decreases linearly, while the σr simultaneously remains constant, with its value being equal to the mean principal stress.Figure 10
Relationships between shear stress and stress components of the FSS.
(a)
p
=
2
MPa
(b)
p
=
6
MPa


## 4.2.4. Strain Characteristics of the FSS under Different Mean Principal Stresses

Figures11(a), 9(c), and 11(b) present the strain components versus shear stress obtained from experiments under p=2,4.5,and6MPa at T=−10°C, respectively. During the shearing process, the strain component curves of the FSS exhibit a nonlinear increase with a linear increase in qs, thus indicating that the strain component initially increases slowly and then rapidly it approached failure. The radial strain and circumferential normal strain increase negatively as qs increases; however, the variation in the radial strain is greater than that in the circumferential normal strain of the FSS.Figure 11
Relationships between shear stress and strain components of the FSS.
(a)
p
=
2
MPa
(b)
p
=
6
MPa


## 4.3. Anisotropy Behavior of the FSS

### 4.3.1. Strength Characteristics of the FSS under Different Directional Angles of Major Principal Stress

A series of FSS tests were performed using the FHCA-300 to determine the effects of different principal stress directions on deformation and strength. From Figure12, the strain components develop with stress components with p=4.5MPa under α=0°,15°,and30° at T=−10°Candb=0.5.Figure 12
Stress-strain curves of the FSS under different principal stress directions.
(a)
Axial stress-strain curves
(b)
Circumferential shear stress-strain curves
(c)
Generalized shear stress-strain curvesFigure12(a) illustrates the relationship between the axial stress and strain of the FSS. Under different principal stress directions, the axial stress-strain curves of the FSS all exhibit the strain-hardening phenomena. The axial strength of the FSS gradually decreases as α increases. The anisotropy of the FSS was induced by the principal stress direction. Figure 12(b) depicts the circumferential shear stress-strain behavior of the FSS at various values of α. The torsional shear strength of the FSS increases with increasing α value in the directional shear tests. The principal stress direction had a significant impact on the torsional shear strength of the FSS. When α=0°, the hollow cylindrical specimens of the FSS were only subjected to an axial load, thereby resulting in approximately zero of circumferential shear strain. However, the circumferential shear stress-strain curves exhibit strain hardening at α=15°and30°. Figure 12(c) illustrates the relationship between the generalized shear stress-strain curves obtained from the directional shear tests of the FSS under different principal stress directions. The generalized shear stress-strain curves of the FSS are primarily strain-hardening curves. However, the generalized shear stress-strain curves of the FSS exhibit a weak hardening tendency at α=15°. The strength of the FSS gradually decreases as α value increases. The axial component dominates the generalized shear stress-strain curves of the FSS more than the circumferential shear component.In this study, theq value corresponding to γg=5% was considered as the failure point of the FSS. Figure 13 illustrates the changes in the strength of the FSS and frozen clay at various α values. Chen et al. [38] described the test data for frozen clay with the same stress path as that of the FSS. As observed in Figure 13, the strengths of the FSS and frozen clay differ in two ways: on the one hand, the strength of the FSS is greater than that of the frozen clay under the same directional angle of major principal stress; the strength deviation between the FSS and frozen clay can be as large as 4.797 MPa and 3.057 MPa at α=15°and30°, respectively; on the other hand, with the variation in α value, the strength of the FSS varies more abruptly than that of the frozen clay. Based on the preceding analysis and discussion, it can be inferred that the anisotropic behavior of frozen soil is caused by the principal stress direction; frozen soil exhibits stress-strain-strength variations depending on the principal stress directions.Figure 13
Strength variations of the FSS and frozen clay with different principal stress directions.


### 4.3.2. Strength Characteristics of the FSS under Different Mean Principal Stresses

Figure14 presents the variations in the FSS with stress and strain under p=2,4.5,and6MPa at T=−10°C. The axial stress-strain curves of the FSS exhibit strain-hardening characteristics under different mean principal stresses. The strength gradually increases as the p value increases. The hollow cylindrical specimen of the FSS was always compressed. Figures 14(b) and 14(c) present the circumferential shear and generalized shear stress-strain curves of the FSS, respectively. The strength of the FSS gradually increases as the p value increases within the test range. The strength of the FSS at p=6MPa was significantly higher than those at p=2and4.5MPa.Figure 14
Stress-strain curves of the FSS under different mean principal stresses.
(a)
Axial stress-strain curves
(b)
Circumferential shear stress-strain curves
(c)
Generalized shear stress-strain curves


## 4.3.1. Strength Characteristics of the FSS under Different Directional Angles of Major Principal Stress

A series of FSS tests were performed using the FHCA-300 to determine the effects of different principal stress directions on deformation and strength. From Figure12, the strain components develop with stress components with p=4.5MPa under α=0°,15°,and30° at T=−10°Candb=0.5.Figure 12
Stress-strain curves of the FSS under different principal stress directions.
(a)
Axial stress-strain curves
(b)
Circumferential shear stress-strain curves
(c)
Generalized shear stress-strain curvesFigure12(a) illustrates the relationship between the axial stress and strain of the FSS. Under different principal stress directions, the axial stress-strain curves of the FSS all exhibit the strain-hardening phenomena. The axial strength of the FSS gradually decreases as α increases. The anisotropy of the FSS was induced by the principal stress direction. Figure 12(b) depicts the circumferential shear stress-strain behavior of the FSS at various values of α. The torsional shear strength of the FSS increases with increasing α value in the directional shear tests. The principal stress direction had a significant impact on the torsional shear strength of the FSS. When α=0°, the hollow cylindrical specimens of the FSS were only subjected to an axial load, thereby resulting in approximately zero of circumferential shear strain. However, the circumferential shear stress-strain curves exhibit strain hardening at α=15°and30°. Figure 12(c) illustrates the relationship between the generalized shear stress-strain curves obtained from the directional shear tests of the FSS under different principal stress directions. The generalized shear stress-strain curves of the FSS are primarily strain-hardening curves. However, the generalized shear stress-strain curves of the FSS exhibit a weak hardening tendency at α=15°. The strength of the FSS gradually decreases as α value increases. The axial component dominates the generalized shear stress-strain curves of the FSS more than the circumferential shear component.In this study, theq value corresponding to γg=5% was considered as the failure point of the FSS. Figure 13 illustrates the changes in the strength of the FSS and frozen clay at various α values. Chen et al. [38] described the test data for frozen clay with the same stress path as that of the FSS. As observed in Figure 13, the strengths of the FSS and frozen clay differ in two ways: on the one hand, the strength of the FSS is greater than that of the frozen clay under the same directional angle of major principal stress; the strength deviation between the FSS and frozen clay can be as large as 4.797 MPa and 3.057 MPa at α=15°and30°, respectively; on the other hand, with the variation in α value, the strength of the FSS varies more abruptly than that of the frozen clay. Based on the preceding analysis and discussion, it can be inferred that the anisotropic behavior of frozen soil is caused by the principal stress direction; frozen soil exhibits stress-strain-strength variations depending on the principal stress directions.Figure 13
Strength variations of the FSS and frozen clay with different principal stress directions.


## 4.3.2. Strength Characteristics of the FSS under Different Mean Principal Stresses

Figure14 presents the variations in the FSS with stress and strain under p=2,4.5,and6MPa at T=−10°C. The axial stress-strain curves of the FSS exhibit strain-hardening characteristics under different mean principal stresses. The strength gradually increases as the p value increases. The hollow cylindrical specimen of the FSS was always compressed. Figures 14(b) and 14(c) present the circumferential shear and generalized shear stress-strain curves of the FSS, respectively. The strength of the FSS gradually increases as the p value increases within the test range. The strength of the FSS at p=6MPa was significantly higher than those at p=2and4.5MPa.Figure 14
Stress-strain curves of the FSS under different mean principal stresses.
(a)
Axial stress-strain curves
(b)
Circumferential shear stress-strain curves
(c)
Generalized shear stress-strain curves


## 4.4. Non-coaxiality of the FSS

### 4.4.1. Non-coaxiality of the FSS under Different Directional Angles of Major Principal Stress

The relationship betweenβ and qs was calculated based on the test data of the FSS under different fixed principal stress directions. Figure 15 presents the variation in the direction of the principal strain increment with qs. The principal strain increment directions of the FSS and stress path do not coincide at α=15°and30°; however, when α=0°, the non-coaxial behavior of the FSS remains unclear. When α=15°and30°, the non-coaxial angle of the FSS (the non-coaxial angle was approximately 20° and 40° in the early stage, respectively) decreases as qs increases, and the degree of non-coaxiality of the FSS decreases as the hollow cylindrical specimen approaches failure. Initially, the axes of the principal stress and principal strain increment are non-coincident. The non-coaxial angle of the FSS at α=30° is inferred to be greater than that at α=15°. Namely, for variations in the non-coaxiality of the FSS under different fixed principal stress directions, the non-coaxial angle fluctuation increases as the directional angle of the major principal stress increases.Figure 15
Variations of non-coaxial angle with shear stress under different principal stress directions.As illustrated in Figure16, the stress envelope surface is represented by the σz−σθ/2 versus the τzθ stress space. Meanwhile, various directions of the principal stress and principal strain increment are presented. According to Equation (1), the angle is formed by σz−σθ/2 and the stress path is twice the directional angle of the major principal stress. At α=15°, the direction of the principal strain increment of the FSS deviates slightly from that of the principal stress. In tests with α=30°, there are relatively large deviations between the direction of principal stress and principal strain increment.Figure 16
Direction of principal strain increment along different principal stress directions.


### 4.4.2. Non-coaxiality of the FSS under Different Mean Principal Stresses

Figure17 depicts the non-coaxial behavior of the FSS between the axes of the principal stresses and principal strain increments under different mean principal stresses. The non-coaxiality of the FSS with shear stress follows the same change rule as that of the FSS under α=15°and30°. When qs is small, the non-coaxial angle of the FSS is larger, and as qs increases, the non-coaxial angle of the FSS gradually decreases. When the hollow cylindrical specimen of the FSS is nearing failure, the non-coaxial angle reaches its minimum value, thus indicating that as the shear stress increases, the direction of the principal strain increment and the direction of the principal stress tend to be coaxial. In other words, the deviation of the non-coaxial angle of the FSS can be as large as 35° in the early stage of shearing. Subsequently, the deviation decreases as the shear strain increases, and the non-coaxiality behavior of the FSS is almost coaxiality at failure. The evolution law of the non-coaxial angle of the FSS does not change as the mean principal stress changes and remains within a certain range.Figure 17
Variations of non-coaxial angle with shear stress under mean principal stresses.


## 4.4.1. Non-coaxiality of the FSS under Different Directional Angles of Major Principal Stress

The relationship betweenβ and qs was calculated based on the test data of the FSS under different fixed principal stress directions. Figure 15 presents the variation in the direction of the principal strain increment with qs. The principal strain increment directions of the FSS and stress path do not coincide at α=15°and30°; however, when α=0°, the non-coaxial behavior of the FSS remains unclear. When α=15°and30°, the non-coaxial angle of the FSS (the non-coaxial angle was approximately 20° and 40° in the early stage, respectively) decreases as qs increases, and the degree of non-coaxiality of the FSS decreases as the hollow cylindrical specimen approaches failure. Initially, the axes of the principal stress and principal strain increment are non-coincident. The non-coaxial angle of the FSS at α=30° is inferred to be greater than that at α=15°. Namely, for variations in the non-coaxiality of the FSS under different fixed principal stress directions, the non-coaxial angle fluctuation increases as the directional angle of the major principal stress increases.Figure 15
Variations of non-coaxial angle with shear stress under different principal stress directions.As illustrated in Figure16, the stress envelope surface is represented by the σz−σθ/2 versus the τzθ stress space. Meanwhile, various directions of the principal stress and principal strain increment are presented. According to Equation (1), the angle is formed by σz−σθ/2 and the stress path is twice the directional angle of the major principal stress. At α=15°, the direction of the principal strain increment of the FSS deviates slightly from that of the principal stress. In tests with α=30°, there are relatively large deviations between the direction of principal stress and principal strain increment.Figure 16
Direction of principal strain increment along different principal stress directions.


## 4.4.2. Non-coaxiality of the FSS under Different Mean Principal Stresses

Figure17 depicts the non-coaxial behavior of the FSS between the axes of the principal stresses and principal strain increments under different mean principal stresses. The non-coaxiality of the FSS with shear stress follows the same change rule as that of the FSS under α=15°and30°. When qs is small, the non-coaxial angle of the FSS is larger, and as qs increases, the non-coaxial angle of the FSS gradually decreases. When the hollow cylindrical specimen of the FSS is nearing failure, the non-coaxial angle reaches its minimum value, thus indicating that as the shear stress increases, the direction of the principal strain increment and the direction of the principal stress tend to be coaxial. In other words, the deviation of the non-coaxial angle of the FSS can be as large as 35° in the early stage of shearing. Subsequently, the deviation decreases as the shear strain increases, and the non-coaxiality behavior of the FSS is almost coaxiality at failure. The evolution law of the non-coaxial angle of the FSS does not change as the mean principal stress changes and remains within a certain range.Figure 17
Variations of non-coaxial angle with shear stress under mean principal stresses.


## 5. Conclusions

A series of hollow cylinder tests of frozen soil were conducted at various directional angles of major principal stress and mean principal stresses, to investigate the anisotropic behavior and non-coaxiality of the FSS under the influence of the principal stress direction. The evolution of the anisotropic behavior and non-coaxiality of the FSS was analyzed under different principal stress directions. Based on this analysis, the following conclusions were drawn:(1)
The linear variation of the stress components (σz, σr, σθ, and τzθ) of the FSS as qs increases for various directional angles of major principal stress and mean principal stresses. The relationships between shear stress and strain components (εz, γzθ, εr, and εθ) of the FSS with different directional angles of major principal stress and mean principal stresses exhibited a nonlinear increasing trend with a linear increase in shear stress, thus indicating that the growth in the strain components slowed in the early stage and then accelerated near failure.(2)
The stress-strain-strength anisotropy of frozen soil was determined to be strongly dependent on the principal stress direction. The FSS strength gradually decreased as the directional angle of the major principal stress increased. However, as the mean principal stress increased, the FSS strength decreased. The FSS strength was greater than that of frozen clay under the same directional angle of the major principal stress.(3)
The non-coaxiality of the FSS was highly dependent on the direction of principal stress. The non-coaxial angle of the FSS increased gradually with an increase in the directional angles of the major principal stress under the directional shear stress path; however, the non-coaxial angle of the FSS did not change with a change in the mean principal stress. The non-coaxial angle of the FSS was observed to be as large as 35° in the early stage of shearing under different mean principal stresses. As the shear stress increased, the direction of the principal strain increment and principal stress tended to become coaxial.



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*Source: 1585324-2022-05-14.xml*