# Analysis of the Elastic-Plastic Theoretical Model of the Pull-Out Interface between Geosynthetics and Tailings

Authors: Changbo Du; Fu Yi
Journal: Advances in Civil Engineering
 (2020)

---

## Abstract

Aiming at the strain-hardening and strain-softening phenomena between geosynthetics and tailings during pull-out tests, bilinear and trilinear shear stress-displacement softening models were proposed. The pull-out process of the hardening reinforcement was divided into the elastic stage, elastic-hardening transition stage, and pure hardening stage. The pull-out process of the softened reinforcement was divided into the elastic stage, elastic-softening transition stage, pure softening stage, softening-residual transition stage, and pure residual stage. The expressions of the interface tension, shear stress, and displacement at the different stages under a pull-out load were derived through the interface basic control equation. At the same time, the evolution law of the interface shear stress at different pull-out stages was analysed, and the predicted results of the two elastic-plastic models were compared with the experimental results. The results show that the predicted results are in good agreement with the experimental data, which verifies the validity of the proposed two elastic-plastic models for the progressive failure analysis of reinforcement at the pull-out interface. During the process of pull-out, the transition stage is not obvious. When the reinforcement is in the elastic stage, the nonlinearity and maximum value of the interface shear stress increase with an increase in the elastic shear stiffness, while the tensile stiffness shows the opposite trend. When the reinforcement is in the hardening or softening stage, the larger the hardening (softening) shear stiffness is, the larger the change range of shear stress is and the more obvious the hardening (softening) characteristics of the reinforcement are. The results comprehensively reflect the progressive failure of reinforcement-tailing interfaces with different strain types and provide theoretical support for the study of the interface characteristics of geosynthetic-reinforced tailings.


---

## Body

## 1. Introduction

The strain hardening and strain softening of geosynthetics are common mechanical properties of geosynthetics in reinforced engineering. This remarkable characteristic of geosynthetics has been emphasized in many experimental studies of the interface interactions between the reinforcement and soil [1, 2]. Shi et al. [3] compared the reinforcement conditions of unidirectional and bidirectional geogrids and found that the pull-out curve of unidirectional geogrids tends to be strain softening, while the pull-out curve of bidirectional geogrids tends to be strain hardening. Generally, there are two kinds of mechanical tests to evaluate the interface properties of reinforced soil: the direct shear test and the pull-out test. However, in the analysis of the stability and interface interaction between the reinforcement and soil, the pull-out test [4] is more suitable because the stress and displacement in the pull-out test are gradually transferred from the pull-out end to the free end, which can fully reflect the interaction mechanism between the reinforcement and soil.Generally, the relationship between the shear stress and displacement should be taken into account in the theoretical study of the interface characteristics between the reinforcement and soil [5–10]. Sobhi and Wu [11] proposed a pull-out interface model of retractable bars based on the elastic-plastic shear stress-displacement. Long et al. [12] used a parabolic fitting curve to describe the nonuniform shear distribution of the interface between the reinforcement and soil. Gurung et al. [13] and Misra et al. [14] used a hyperbolic model to analyse the relationship between the shear stress and displacement. Gurung [15] simplified the boundary conditions of the anchorage section on the basis of previous studies and obtained the stress and displacement solutions of the anchorage section by fitting the hyperbolic model. Esterhulzen et al. [16] proposed a hyperbolic displacement softening model before and after the peak value. Lin et al. [17] used linear simulations to simulate the plastic softening before and after the peak value and the shear stress and displacement changes in the plastic flow. Zhang et al. [18] proposed a three-stage elastic-plastic shear stress-displacement model, in which a hyperbolic model was used before the peak value, while a linear model was used to simulate the plastic softening and plastic flow after the peak value. Although the abovementioned interface model can effectively simulate pull-out behaviour, it fails to consider the interface progressive failure characteristics at different stages of the whole pull-out process.To truly describe the progressive failure characteristics of the reinforcement-soil interface during different stages of the pull-out process, scholars worldwide have proposed some calculation models for the strain-softening characteristics of reinforcements. For example, Hong et al. [19] used the elastic-plastic theory model to study the progressive failure of the soil nail-soil interface under a pull-out state. Zhu et al. [20] and Chen et al. [21] deduced the analytical expressions of the axial force and shear stress at the reinforcement-soil interface at different pull-out stages through a three-parameter model. In the study of the interface characteristics of reinforcement-soil, there are few models for calculating the strain-hardening characteristics, and there is no computational model that includes both the strain hardening and strain softening of the reinforcement.In this study, the strain hardening and strain softening of the pull-out curves between geosynthetics and tailings are investigated (hereafter referred to as strain-hardening reinforcement and strain-softening reinforcement). It is proposed that the hardening reinforcement undergoes three continuous stages during the pull-out process. Softening reinforcement goes through five continuous stages during the pull-out process. The distributions of the tension, shear stress, and displacement of two kinds of reinforcement at different pull-out stages are deduced. Moreover, the accuracy of the proposed two elastic-plastic pull-out models is verified by analysing and comparing the pull-out test results of two reinforcements. The evolution of the interface shear stress at different pull-out stages and its influencing factors are further studied.


## 2. Analysis of the Pull-Out Test

### 2.1. Test Instrument

The test instrument was a stripped-down YT1200 geosynthetic direct shear and pull-out system (Howard Nanjing Soil Instrument Manufacturing Company). The system mainly consisted of a test box (direct shear and pull-out), vertical loading system, horizontal loading system, and data acquisition system. The test equipment is shown in Figure1. The inner dimensions of the pull-out test box were 300 × 300 × 220 mm. A narrow slit with dimensions 300 × 10 mm was opened in the middle of the front and back of the test groove to extract the geosynthetics. In the vertical loading system, the cylinder with a pressure sensor applied the overburden pressure through the reaction device. The cylinder was a 30 L air compressor. On top of the pressure loading system, there was a pressure plate with dimensions 295 × 295 × 10 mm, which evenly applied overburden pressures within the range of 0–200 kPa. The tension and compression motor of the horizontal loading system with a tension sensor had a controllable rate, which can exert a constant loading speed in the range of 0–5 mm/min and measure the test force. This machine was equipped with a control panel, which can set the overlying pressure, and the right side of the control panel was connected to the horizontal loading system, which can display the test results on a screen in real time and provide real-time monitoring of the test data for analysis or to stop the system in case of problems.Figure 1
Pull-out test: (a) test device; (b) sample preparation and dimensions; (c) test setups.
(a)(b)(c)


### 2.2. Parameter Index of the Tailing Filling and Geosynthetics

The tailing filler used in the test came from the tailing reservoir of Dazhong Co., Ltd. in Baotou City, Inner Mongolia. To reduce the influence of the water content in the sand on the test results, dry tailings were used. The density of tailings was 1.83 g/cm3, and the water content was 3.75%. The physical properties of the tailings were as follows: the effective particle size d10 = 0.10 mm, the median particle size d30 = 0.19 mm, and the restricted particle size d60 = 0.30 mm. The calculations showed that the unevenness coefficient of the tailings Cu was 3.5, and the curvature coefficient of the tailings Cc was 1.2. The latter value is between 1 and 3, which indicates that the tailings were of poor gradation. The particle grading curve is shown in Figure 2.Figure 2
Gradation curve of the tailing sand.As shown in Figure3, the geosynthetics used in the test are short-fibre needle-punched geotextiles (Figure 3(a)) and TGSG35 biaxial tension plastic geogrids (Figure 3(b)). These two geosynthetics have good application effects in various reinforcement projects. The concrete parameters of the reinforcement materials are shown in Table 1.Figure 3
The geogrid used in the test: (a) geotextiles and (b) geogrid.Table 1
The technology parameters of the geogrids.
Mechanical parameters of the staple fibre needling geotextilesTechnical indexMechanical parameters of the TGSG35 geogridTechnical indexLongitudinal and transverse fracture strength (kN·m−1)30Tensile strength (kN·m−1)Radial, zonal35Elongation corresponding to the standard strength (%)40～80Nominal elongation (%)Radial15CBR breaking strength ≥ (kN)13Zonal13Longitudinal and transverse tearing strength ≥ (kN)12Tensile strength at 2% elongation ≥ (kN·m−1)Radial12Equivalent aperture (mm)0.05～0.2Zonal12Vertical permeability coefficient (cm·s−1)1.0 × 10−3Tensile strength at 5% elongation ≥ (kN·m−1)Radial24Thickness (mm)4.2Zonal24


### 2.3. Test Principle

During the tests, when the geosynthetics were going to be pulled out, it was assumed that the shear stress on the upper and lower surfaces of the reinforcement was distributed evenly and met the equilibrium condition. The interface peak shear strengthτp and the residual shear strength τt can be obtained by the following calculation:(1a)τp=Tp2AInterface,(1b)τr=Tr2AInterface,where τp and τr are the peak shear strength and residual shear strength, kPa, respectively; Tp and Tr are the maximum pull-out force and residual pull-out force, kN, respectively; and AInterface is the area of the geosynthetics embedded in the direct shear or pull-out test box, which is the area of the shear plane in the test process, AInterface=0.09m2.The pull-out tests were carried out under four different normal stresses (10 kPa, 20 kPa, 30 kPa, and 40 kPa). The interface interaction between the reinforcement and soil can be described by the pseudofriction coefficientf, which is generally calculated as the ratio of the interface friction strength to the corresponding normal stress:(2a)f=τpσn,(2b)f=τrσn,where σn=qs+γh is the normal stress, in which qs is the additional stress, γ is the bulk density of the filler, and h is the paving height of the filler (see Figure 4).Figure 4
Pull-out sketch of the reinforcement: (a) overall stress of reinforcement and (b) force analysis of the microelement of reinforcement.


### 2.4. Test Method

To reduce the dispersion of the test results, 3 groups of parallel tests were conducted in each group. During the tests, the density of the tailing sand was used to control the amount of sand in the test box, and the sand loading process was stratified-compacted to ensure that each group had the same density. We simultaneously applied lubricating oil evenly on both sides of the pull-out test box to reduce the size effect during testing. The shear rate of the pull-out test is set to 2 mm/min to eliminate the influence of the test rate on the results. The tests were performed in strict accordance with the “Test Methods of Geosynthetics for Highway Engineering (JTG E50-2012)” [22].


### 2.5. Analysis of the Test Results

The pull-out test results of the tailings and geogrids or geotextiles under different normal stresses are shown in Figure5. In the pull-out test curve of the geogrids, the pull-out force increases with an increase in the pull-out displacement, but the rate of increase slows gradually, showing the characteristics of strain hardening as a whole. However, in the pull-out test curve of the geotextiles, the pull-out force rapidly reaches its peak value with an increase in the pull-out displacement and then decreases obviously, showing strain-softening characteristics. In addition, both kinds of pull-out curves require a certain pulling force at the beginning of pull-out because the reinforcement requires a certain pulling force to resist the friction of the interface between the reinforcement and tailings.Figure 5
Pull-out test curves between geosynthetics and tailings: (a) geogrid and (b) geotextiles.
(a)(b)According to formula (2), the relationship between the pull-out pseudofriction coefficient and normal stress can be calculated, as shown in Figure 6. The pseudofriction coefficient between geogrids and geotextiles is negatively exponential to normal stress. With an increase in the normal stress, the pseudofriction coefficient decreases and the decreasing speed slows down gradually. The pseudofriction coefficient of geogrids is between 0.48 and 0.82. The pseudofriction coefficient of geotextiles in the peak state is between 0.50 and 0.88, the pseudofriction coefficient of geotextiles in the residual state is between 0.46 and 0.70, and the difference between them is approximately 30% under the same normal stress condition.Figure 6
Relationship between the pull-out pseudofriction coefficient and normal stress: (a) geogrid and (b) geotextiles.
(a)(b)


## 2.1. Test Instrument

The test instrument was a stripped-down YT1200 geosynthetic direct shear and pull-out system (Howard Nanjing Soil Instrument Manufacturing Company). The system mainly consisted of a test box (direct shear and pull-out), vertical loading system, horizontal loading system, and data acquisition system. The test equipment is shown in Figure1. The inner dimensions of the pull-out test box were 300 × 300 × 220 mm. A narrow slit with dimensions 300 × 10 mm was opened in the middle of the front and back of the test groove to extract the geosynthetics. In the vertical loading system, the cylinder with a pressure sensor applied the overburden pressure through the reaction device. The cylinder was a 30 L air compressor. On top of the pressure loading system, there was a pressure plate with dimensions 295 × 295 × 10 mm, which evenly applied overburden pressures within the range of 0–200 kPa. The tension and compression motor of the horizontal loading system with a tension sensor had a controllable rate, which can exert a constant loading speed in the range of 0–5 mm/min and measure the test force. This machine was equipped with a control panel, which can set the overlying pressure, and the right side of the control panel was connected to the horizontal loading system, which can display the test results on a screen in real time and provide real-time monitoring of the test data for analysis or to stop the system in case of problems.Figure 1
Pull-out test: (a) test device; (b) sample preparation and dimensions; (c) test setups.
(a)(b)(c)


## 2.2. Parameter Index of the Tailing Filling and Geosynthetics

The tailing filler used in the test came from the tailing reservoir of Dazhong Co., Ltd. in Baotou City, Inner Mongolia. To reduce the influence of the water content in the sand on the test results, dry tailings were used. The density of tailings was 1.83 g/cm3, and the water content was 3.75%. The physical properties of the tailings were as follows: the effective particle size d10 = 0.10 mm, the median particle size d30 = 0.19 mm, and the restricted particle size d60 = 0.30 mm. The calculations showed that the unevenness coefficient of the tailings Cu was 3.5, and the curvature coefficient of the tailings Cc was 1.2. The latter value is between 1 and 3, which indicates that the tailings were of poor gradation. The particle grading curve is shown in Figure 2.Figure 2
Gradation curve of the tailing sand.As shown in Figure3, the geosynthetics used in the test are short-fibre needle-punched geotextiles (Figure 3(a)) and TGSG35 biaxial tension plastic geogrids (Figure 3(b)). These two geosynthetics have good application effects in various reinforcement projects. The concrete parameters of the reinforcement materials are shown in Table 1.Figure 3
The geogrid used in the test: (a) geotextiles and (b) geogrid.Table 1
The technology parameters of the geogrids.
Mechanical parameters of the staple fibre needling geotextilesTechnical indexMechanical parameters of the TGSG35 geogridTechnical indexLongitudinal and transverse fracture strength (kN·m−1)30Tensile strength (kN·m−1)Radial, zonal35Elongation corresponding to the standard strength (%)40～80Nominal elongation (%)Radial15CBR breaking strength ≥ (kN)13Zonal13Longitudinal and transverse tearing strength ≥ (kN)12Tensile strength at 2% elongation ≥ (kN·m−1)Radial12Equivalent aperture (mm)0.05～0.2Zonal12Vertical permeability coefficient (cm·s−1)1.0 × 10−3Tensile strength at 5% elongation ≥ (kN·m−1)Radial24Thickness (mm)4.2Zonal24


## 2.3. Test Principle

During the tests, when the geosynthetics were going to be pulled out, it was assumed that the shear stress on the upper and lower surfaces of the reinforcement was distributed evenly and met the equilibrium condition. The interface peak shear strengthτp and the residual shear strength τt can be obtained by the following calculation:(1a)τp=Tp2AInterface,(1b)τr=Tr2AInterface,where τp and τr are the peak shear strength and residual shear strength, kPa, respectively; Tp and Tr are the maximum pull-out force and residual pull-out force, kN, respectively; and AInterface is the area of the geosynthetics embedded in the direct shear or pull-out test box, which is the area of the shear plane in the test process, AInterface=0.09m2.The pull-out tests were carried out under four different normal stresses (10 kPa, 20 kPa, 30 kPa, and 40 kPa). The interface interaction between the reinforcement and soil can be described by the pseudofriction coefficientf, which is generally calculated as the ratio of the interface friction strength to the corresponding normal stress:(2a)f=τpσn,(2b)f=τrσn,where σn=qs+γh is the normal stress, in which qs is the additional stress, γ is the bulk density of the filler, and h is the paving height of the filler (see Figure 4).Figure 4
Pull-out sketch of the reinforcement: (a) overall stress of reinforcement and (b) force analysis of the microelement of reinforcement.


## 2.4. Test Method

To reduce the dispersion of the test results, 3 groups of parallel tests were conducted in each group. During the tests, the density of the tailing sand was used to control the amount of sand in the test box, and the sand loading process was stratified-compacted to ensure that each group had the same density. We simultaneously applied lubricating oil evenly on both sides of the pull-out test box to reduce the size effect during testing. The shear rate of the pull-out test is set to 2 mm/min to eliminate the influence of the test rate on the results. The tests were performed in strict accordance with the “Test Methods of Geosynthetics for Highway Engineering (JTG E50-2012)” [22].


## 2.5. Analysis of the Test Results

The pull-out test results of the tailings and geogrids or geotextiles under different normal stresses are shown in Figure5. In the pull-out test curve of the geogrids, the pull-out force increases with an increase in the pull-out displacement, but the rate of increase slows gradually, showing the characteristics of strain hardening as a whole. However, in the pull-out test curve of the geotextiles, the pull-out force rapidly reaches its peak value with an increase in the pull-out displacement and then decreases obviously, showing strain-softening characteristics. In addition, both kinds of pull-out curves require a certain pulling force at the beginning of pull-out because the reinforcement requires a certain pulling force to resist the friction of the interface between the reinforcement and tailings.Figure 5
Pull-out test curves between geosynthetics and tailings: (a) geogrid and (b) geotextiles.
(a)(b)According to formula (2), the relationship between the pull-out pseudofriction coefficient and normal stress can be calculated, as shown in Figure 6. The pseudofriction coefficient between geogrids and geotextiles is negatively exponential to normal stress. With an increase in the normal stress, the pseudofriction coefficient decreases and the decreasing speed slows down gradually. The pseudofriction coefficient of geogrids is between 0.48 and 0.82. The pseudofriction coefficient of geotextiles in the peak state is between 0.50 and 0.88, the pseudofriction coefficient of geotextiles in the residual state is between 0.46 and 0.70, and the difference between them is approximately 30% under the same normal stress condition.Figure 6
Relationship between the pull-out pseudofriction coefficient and normal stress: (a) geogrid and (b) geotextiles.
(a)(b)


## 3. Elastic-Plastic Model of the Reinforcement-Tailing Interface

### 3.1. Strain-Hardening Model of the Reinforcement

From the pull-out test results between geotextiles and tailings in Figure5(a), it can be seen that the interface shear stress and the displacement curve of the strain-hardened reinforcement can be approximately expressed as an elastic relationship before reaching the peak value and then as strain-hardening characteristics. In this paper, the curve form is simplified to a bilinear shear stress-displacement relationship [23], as shown in Figure 7. It can be seen from the graph that the first stage (OA section) represents the relationship between the shear stress and displacement before the peak shear stress reaches a straight line, and the second stage (AB section) represents the strain hardening of reinforcement by a straight line. In Figure 7, Ks1 and Ks2 are the slopes of OA and AB, which are also called the elastic interface shear stiffness and hardening shear stiffness, respectively; τo is the initial shear stress, and its value is the ratio of the initial pull-out force to the shear area, i.e., τo=To/2A; τp is the peak shear stress of the interface; and up is the corresponding pull-out displacement, where Ks1=τp−τo/up. When the shear stress increases to the ultimate shear stress of reinforcement, the failure of the reinforcement occurs. The ratio between them is defined as the failure ratio Rf [18], i.e., Rf=τp/τult, which generally has a value between 0.5 and 1.0:(3a)τ=Ks1u+τo,0≤u≤up,(3b)τ=Ks2u−up+τp,u>up.Figure 7
Relationship between the interface shear stress and the displacement of the bilinear (τ−u).


### 3.2. Strain-Softening Model of the Reinforcement

According to the test results between the geogrid and tailings shown in Figure5(b), the shear stress and the displacement curve of the strain-softening reinforcement interface can be approximately expressed as an elastic relationship before reaching the peak value and then as plastic softening and plastic flow. In this paper, the characteristic of this curve is simplified to a three-stage linear shear stress-displacement relationship [24], as shown in Figure 8. It can be seen from the figure that in the first stage (O′A′), the relationship between the shear stress and displacement before shear stress reaches its peak value is represented by a straight line; in the second stage (A′B′), the strain softening of reinforcement is represented by a straight line; and in the third stage (B′C′), the plastic flow of reinforcement is represented by a horizontal straight line. In Figure 8, ks1 is the slope of OA (the elastic shear stiffness), with ks1=τp−τo′/up, where τp is the peak shear stress of the interface and up is the corresponding displacement; τo′ is the initial shear stress, τo′=To/2A, where To′ is the initial pull-out force; and ks2 is the slope of AB (softening shear stiffness) ks2=τr−τp/ur−up<0, where τr is the residual shear stress of the interface and ur is the corresponding displacement. The analogical damage ratio defines the ratio between τr and τp as Rf, that is, Rf=τr/τp:(4a)τ=ks1+τo′u,0≤u≤up,(4b)τ=ks2u−up+τp,up<u≤u,(4c)τ=Rfks1up,u>ur.Figure 8
Relationship between the interface shear stress and the displacement of the trilinear (τ−u).The suitability of the proposed model can be proven by referring to previous scholars’ literature to increase the number of types of samples. Shi et al. [3] consider that the pull-out test curves of bidirectional plastic geogrids and different fillers (clay and sand) are generally strain hardening. Yang et al. [25] found that the pull-out curves are softening when studying the interface friction between geotextiles and loess. The suitability of the model can also be confirmed by the analysis of two mechanisms of geogrid- and geotextile-reinforced tailings. The main reason for strain hardening of geogrid-reinforced tailings is the blocking effect of the transverse rib of the geogrid. The strain-softening phenomenon of geotextile-reinforced tailings is due to the fact that when the pull-out force reaches the peak value with the increase of the pull-out displacement, the whole reinforcement undergoes sliding and strain-softening characteristics appear.


## 3.1. Strain-Hardening Model of the Reinforcement

From the pull-out test results between geotextiles and tailings in Figure5(a), it can be seen that the interface shear stress and the displacement curve of the strain-hardened reinforcement can be approximately expressed as an elastic relationship before reaching the peak value and then as strain-hardening characteristics. In this paper, the curve form is simplified to a bilinear shear stress-displacement relationship [23], as shown in Figure 7. It can be seen from the graph that the first stage (OA section) represents the relationship between the shear stress and displacement before the peak shear stress reaches a straight line, and the second stage (AB section) represents the strain hardening of reinforcement by a straight line. In Figure 7, Ks1 and Ks2 are the slopes of OA and AB, which are also called the elastic interface shear stiffness and hardening shear stiffness, respectively; τo is the initial shear stress, and its value is the ratio of the initial pull-out force to the shear area, i.e., τo=To/2A; τp is the peak shear stress of the interface; and up is the corresponding pull-out displacement, where Ks1=τp−τo/up. When the shear stress increases to the ultimate shear stress of reinforcement, the failure of the reinforcement occurs. The ratio between them is defined as the failure ratio Rf [18], i.e., Rf=τp/τult, which generally has a value between 0.5 and 1.0:(3a)τ=Ks1u+τo,0≤u≤up,(3b)τ=Ks2u−up+τp,u>up.Figure 7
Relationship between the interface shear stress and the displacement of the bilinear (τ−u).


## 3.2. Strain-Softening Model of the Reinforcement

According to the test results between the geogrid and tailings shown in Figure5(b), the shear stress and the displacement curve of the strain-softening reinforcement interface can be approximately expressed as an elastic relationship before reaching the peak value and then as plastic softening and plastic flow. In this paper, the characteristic of this curve is simplified to a three-stage linear shear stress-displacement relationship [24], as shown in Figure 8. It can be seen from the figure that in the first stage (O′A′), the relationship between the shear stress and displacement before shear stress reaches its peak value is represented by a straight line; in the second stage (A′B′), the strain softening of reinforcement is represented by a straight line; and in the third stage (B′C′), the plastic flow of reinforcement is represented by a horizontal straight line. In Figure 8, ks1 is the slope of OA (the elastic shear stiffness), with ks1=τp−τo′/up, where τp is the peak shear stress of the interface and up is the corresponding displacement; τo′ is the initial shear stress, τo′=To/2A, where To′ is the initial pull-out force; and ks2 is the slope of AB (softening shear stiffness) ks2=τr−τp/ur−up<0, where τr is the residual shear stress of the interface and ur is the corresponding displacement. The analogical damage ratio defines the ratio between τr and τp as Rf, that is, Rf=τr/τp:(4a)τ=ks1+τo′u,0≤u≤up,(4b)τ=ks2u−up+τp,up<u≤u,(4c)τ=Rfks1up,u>ur.Figure 8
Relationship between the interface shear stress and the displacement of the trilinear (τ−u).The suitability of the proposed model can be proven by referring to previous scholars’ literature to increase the number of types of samples. Shi et al. [3] consider that the pull-out test curves of bidirectional plastic geogrids and different fillers (clay and sand) are generally strain hardening. Yang et al. [25] found that the pull-out curves are softening when studying the interface friction between geotextiles and loess. The suitability of the model can also be confirmed by the analysis of two mechanisms of geogrid- and geotextile-reinforced tailings. The main reason for strain hardening of geogrid-reinforced tailings is the blocking effect of the transverse rib of the geogrid. The strain-softening phenomenon of geotextile-reinforced tailings is due to the fact that when the pull-out force reaches the peak value with the increase of the pull-out displacement, the whole reinforcement undergoes sliding and strain-softening characteristics appear.


## 4. Fundamental Equation of the Reinforcement-Soil Interface

The pull-out test diagram of the reinforcement is shown in Figure4. In the test, the bottom of the test groove and the lateral boundary are fixed, and the vertical upward pressure can be applied on the bearing plate. The length and thickness of reinforcement are L and t, respectively, and the tensile modulus is E. When the shear stress at x is τ, the microelement with length dx is taken for the analysis. The unit width of the reinforcement is taken as the width, and the boundary effect of the reinforcement is neglected. Then, according to the force balance, it can be concluded that(5)T+dT−T+2τdx+εdx=0,where T is the tensile force per unit width of the reinforcement at x, εdx is the deformation length of the microelement, and ε is the strain.Equation (5) can be rewritten as follows:(6)dTdx+2τ1+ε=0.The strain of reinforcement atx can be written as follows:(7)ε=−dudx,where u is the relative displacement of the reinforcement at x.It is assumed that the strain is linearly correlated with the tensile force per unit width [21], i.e.,(8)ε=TEt.From Equations (5)–(8), we can obtain the following:(9)Etd2udx2+2τε−1=0.In general, the actualε strain in the pull-out process is very small [25], which can be neglected. Therefore, Equation (9) is approximately expressed as follows:(10)Etd2udx2−2τ=0.Equation (10) is the basic equation of the interface between the reinforcement and soil, and it is of great significance to the study of the friction characteristics of the interface between the reinforcement and soil.


## 5. Pull-Out Interface Analysis of the Strain-Hardening Reinforcement

### 5.1. Analysis of the Hardening Pull-Out Interface

Basic assumption: according to the definition of the theoretical model of the strain-hardening reinforcement, it is considered that the interface between the reinforcement and tailings will undergo an elastic stage, an elastic-hardening transition stage, and a pure hardening stage under a pull-out load, corresponding to stages I, II, and III in Figure9, respectively. Through theoretical calculations, the expressions of the tension, shear stress, and displacement at each stage of pull-out can be obtained.Figure 9
The progressive pull-out process of the strain-hardening reinforcement in the pull-out model analysis: (a) elastic stage; (b) elastic-hardening transition stage; (c) pure hardening stage.
(a)(b)(c)


#### 5.1.1. Elastic Stage (I Stage)

When0≤u<up, the shear stress and displacement are elastic, which satisfies the equation τ=Ks1u+τo. The governing equation of this stage can be obtained by the simultaneous Equations (10) and (3a):(11)d2Tdx2−α2T=0,where α=2Ks1/Et.Solving Equation (11), the following can be obtained:(12)Tex=C1exp−αx+C2expαx,where Tex is the tension of the reinforcement at the elastic stage and C1 and C2 are the integral constants.In the pull-out test, when the tension at the pull-out end, i.e.,x = 0, is T0 and the tension at the free end, i.e., x = L, is 0, the following boundary conditions exist:(13)Tx=0=T01,Tx=L=0.By substituting the boundary condition (13) in Equation (12), we obtain(14)C1=expαLexpαL−exp−αLT01,C2=−exp−αLexpαL−exp−αLT01.The tension expression of stage I can be obtained as follows:(15)Tex=T01sinhαL−xsinhαL.According to Equations (10) and (3a), the corresponding shear stress and displacement relations can be obtained:(16)τex=αT012⋅coshαL−xsinhαL,(17)uex=αT012Ks1⋅coshαL−xsinhαL.The transformation of the pull-out displacementue0 can be obtained by substituting x = 0 into Equation (17):(18)T01=2tanhαLαKs1ue0,Whenue0=up and τp=Ks1ue0+τo, from Equation (18), the critical tension Teh0c of the elastic stage and the elastic-hardening transition stage, i.e., the maximum tension in the elastic stage, can be obtained as follows:(19)Teh0c=2τp−τotanhαLα.


#### 5.1.2. Elastic-Hardening Transition Stage (II Stage)

With increasing tension, the interface shear stress gradually transfers from the pull-out end to the tail until it reaches its peak value. Then, the plastic characteristics of the pull-out end begin to occur, and the strain-hardening phenomenon occurs and enters the II stage. The critical pointP (x=Lh) is defined to divide the elastic zone and the hardening zone. When 0≤x<Lh, the interface is in the stage II hardening zone, and when Lh<x≤L, the interface is in the stage II elastic zone (where Lh is the length of the hardening zone).(1)
Elastic Zone (Lh≤x≤L).The distributions of the interface tension, shear stress, and displacement in the stage II elastic zone are similar to those in the elastic stage; thus, we can obtain(20)Tex=T02sinhαL−xsinhαL−Lh,(21)τex=αT022⋅coshαL−xsinhαL−Lh,(22)uex=αT022Ks1⋅coshαL−xsinhαL−Lh,where T02 is the tension of the transition point P.Considering that the interface shear stress at the transition pointP is equal to the peak shear stress, the following result can be obtained:(23)T02=2τptanhαL−Lhα.(2)
Hardening Zone (0≤x≤Lh).The relationship between the interface shear stress and displacement in the stage II hardening zone is defined by Equation (3b), and the simultaneous Equations (10) and (3b) can be obtained as follows:(24)d2Tdx2−β2T=0,where β=2Ks2/Et.Solving Equation (24) can yield(25)Thx=C3exp−βx+C4expβx,where Thx is the tension of reinforcement at the hardening zone of stage II and C1 and C2 are the integral constants.Consider the boundary conditions as follows:(26)Thx=0=T03,Thx=Lh=Tex=Lh.By substituting the boundary condition (26) in Equation (25), we can obtain(27)C3=−T03exp−βLhexpβLh−exp−βLh+2τptanhαL−LhαexpβLh−exp−βLh,C4=T03expβLhexpβLh−exp−βLh−2τptanhαL−LhαexpβLh−exp−βLh.The expressions of the tension, shear stress, and displacement in stage II can be obtained as follows:(28)Thx=T03sinhβLh−xsinhβLh+2τptanhαL−Lhα⋅sinhβxsinhβLh,(29)τhx=βT032⋅coshβLh−xsinhβLh−βτptanhαL−Lhα⋅coshβxsinhβLh,(30)uhx=βT032Ks2⋅coshβLh−xsinhβLh−βτptanhαL−LhαKs2⋅coshβxsinhβLh−τpKs2+up.Since the shear stress at the transition pointP between the elastic zone and the hardening zone is continuous, i.e., τex=Lh=τhx=Lh, the following can be obtained:(31)T03=2τpsinhβLhβ+2τptanhαL−Lhα⋅coshβLh.WhenLh=L, the critical tension Th0c between the elastic-hardening transition stage and the pure hardening stage can be obtained by Equation (31):(32)Th0c=2τpsinhβLβ.


#### 5.1.3. Pure Hardening Stage (III Stage)

Similar to the analysis of the hardening zone of stage II, Equation (25) is still applicable to the pure hardening stage, and the boundary conditions are as follows:(33)Tsx=0=T04,Tsx=L=0.The expressions of the interface tension, shear stress, and displacement at the pure hardening stage are as follows:(34)Thx=T04⋅sinhβL−xsinhβL,(35)τhx=βT042⋅coshβL−xsinhβL,(36)uhx=βT042Ks2⋅coshβL−xsinhβL−τpKs2+up.Letx = 0 be substituted into Equation (36) to obtain the transformation of the pull-out displacement uh0 at this stage:(37)T04=2tanhβLβKs2uh0−up+τp.In the pure hardening stage, the interface pull-out force and shear stress at the pull-out end all increase. When the shear stress at the pull-out end increases to the ultimate stressτult of the reinforcement, the reinforcement is destroyed. Then, τult=Ks2uh0−up+τp, combined with Rf=τp/τult, so Equation (37) can be written as follows:(38)T04=2τptanhβLRfβ.In summary, closed solutions are obtained for the three stages of the pull-out process of strain-hardening reinforcement. The critical tension and the initial tension between the two stages are shown in Tables2 and 3.Table 2
Critical tensile force between the two stages of the hardening pull-out model.
Pull-out stageI-IIII-IIICritical tensile forceTeh0c=2τptanhαL/αTh0c=2τpsinhβL/βTable 3
Initial tensile forces at different stages of the hardening pull-out model.
Pull-out stageIII (elastic zone, hardening zone)IIIInitial tensile forcesT01=2tanhαL/αKs1ue0T02=2τptanhαL−Lh/α,T03=2τpsinhβLh/β+2τptanhαL−Lh/α⋅coshβLhT04=2τptanhβL/Rfβ


### 5.2. Verification of the Hardening Model

To verify the elastic-plastic theoretical model of strain hardening, the pull-out test results between geogrids and tailings mentioned above are used for the simulation, and the predicted results are shown in Figure10. According to the graph, the prediction results are in good agreement with the pull-out test results, and the model can effectively describe the progressive pull-out behaviour of geogrids in the tailings. Moreover, the displacement of stage II is relatively small compared with that of stages I and III, which is due to the large elastic modulus and the small length of reinforcement, resulting in the progressive failure of the reinforcement not obvious. This transitional stage can be neglected in the analysis of the pull-out behaviour of geogrids in the tailings.Figure 10
Comparison between the pull-out test results and model-predicted results of geogrids under different normal stresses: (a) 10 kPa; (b) 20 kPa; (c) 30 kPa; and (d) 40 kPa.
(a)(b)(c)(d)Furthermore, to better understand the five pull-out stages of geogrids in the tailings, the relationship between the interface shear stiffness (elastic shear stiffnessKs1 and hardening shear stiffness Ks2) and the initial shear stress with the normal stress is given, as shown in Figure 11. With an increase in the normal stress, Ks1 and Ks2 increase linearly. The initial shear stress is also linearly related to the normal stress, and the initial shear strength index can be obtained from the Mohr–Coulomb strength criterion.Figure 11
The relationship between the normal stress and shear stiffness of the hardening model: (a) shear stiffness; (b) initial shear stress.
(a)(b)


### 5.3. Distribution of the Interface Shear Stress

To visually reflect the stress evolution law of strain-hardening reinforcement at different stages of the pull-out interface, the interface shear stress distribution at three stages was analysed. The model parameters selected according to the above test results are shown in Table4. To simplify the analysis, the model parameters were normalized. The normalized reinforcement position was X=x/L, and the normalized interface shear stress was ρ=τ/τp.Table 4
Hardening model parameters.
σn (kPa)L (m)t (mm)E (GPa)Ks1 (MPa/m)Ks2 (MPa/m)fRf300.3211.4490.0660.5260.8The parametersτp=σnf=15.78kPa, α=2Ks1/Et=1.20, and β=2Ks2/Et=0.26 were calculated. According to the model parameters given, the following formulas were substituted.


#### 5.3.1. Elastic Stage

Substitution of Equation (16) can yield τex=τpcoshαL−x/coshαL, i.e., ρ=coshα1−X/coshαL.The substitution of Equation (19) indicates that the critical tension of the elastic stage and the elastic-hardening transition stage is Teh0c=2τptanhαL/α=9.08kN/m.


#### 5.3.2. Elastic-Hardening Transition Stage

(i)
Elastic Zone (0.5≤X≤1).The substitution ofLh=L/2 into Equation (64) yields τex=τptanhαL/2coshαL−x/sinhαL/2, i.e., ρ=coshαL1−X/coshαL/2.(ii)
Hardening Zone (0≤X≤0.5).The substitution ofLh=L/2 into Equation (29) results in τhx=τp⋅coshβL/2−x+τpβtanhαL/2/αsinhβL/2⋅coshβL/2⋅coshβL/2−x−coshβx, i.e., ρ=coshβL0.5−X+βtanhαL/2/αsinhβL/2⋅coshβL/2⋅coshβL0.5−X−coshβLX.From Equation (32), the critical tension of the elastic-hardening transition stage and the pure hardening stage is Th0c=2τpsinhβL/β=9.48kN/m.(iii)
Pure Hardening Stage.The substitution of Equation (35) results in τhx=τpcoshβL−x/RfcoshβL, i.e., ρ=coshβL1−X/RfcoshβL.The substitution of Equation (38) shows that the ultimate tensile force at the stage of pure hardening is T04=2τptanhβL/Rfβ=11.81kN/m.According to the above expression of the interface shear stress, the evolution law of the shear stress in the three stages of elastic, elastic-hardening transition, and pure hardening of the strain-hardening reinforcement can be obtained, as shown in Figure12.Figure 12
Evolution of the interfacial shear stress in different stages of the pull-out process of the reinforced tailings (strain-hardening reinforcement): (a) elastic stage; (b) elastic-hardening transition stage; (c) pure hardening stage.
(a)(b)(c)WhenT≤9.08kN/m, the interface is in the elastic stage, and the shear stress decreases nonlinearly from the pull-out end to the free end. When T=9.08kN/m, the interface is in the critical stage of elastic and pure hardening, and the peak shear stress reaches 20 kPa at the pull-out end.When9.08kN/m≤T≤9.48kN/m, the interface is in the transition stage of elastic-hardening, in which the shear stress at the pull-out end reaches its peak value and then drives the free end to continue to increase. The shear stress decreases from the pull-out end to the free end. When T=9.48kN/m, the free end also reaches its peak value, and the interface will enter the stage of pure hardening.When9.48kN/m≤T≤11.81kN/m, the interface is in the stage of pure hardening, and the shear stress at the pull-out end continues to increase until T=11.81kN/m, when the shear stress at the pull-out end reaches the ultimate stress and the reinforcement is damaged.


### 5.4. Parametric Impact Analysis

From the above analysis, it can be seen that the interface shear stress is mainly affected by the elastic shear stiffness (Ks1) and the tensile stiffness (Et) in the elastic stage, and the distribution of shear stress in the hardening stage is related to the length of the hardening zone (Lh) and the hardening shear stiffness (Ks2). The calculated parameters are consistent with those in the previous section, in which the ratio of the hardening zone length to the reinforcement length is defined as η=Lh/L.


#### 5.4.1. Elastic Stage

The effect of the elastic shear stiffness (Ks1) on the interface shear stress at different locations of the reinforcement in stage I is shown in Figure 13. The distribution of the interface shear stress in the elastic stage decreases nonlinearly from the pull-out end to the free end. The maximum shear stress is the same, which is the peak shear stress. The larger the Ks1 is, the more obvious the nonlinearity of the interface shear stress curve is.Figure 13
Effect of the interface frictional resistance at stage I onKs1.The effect of the single-width tensile stiffness (Et) on the interface shear stress at different locations of the reinforcement in stage I is shown in Figure 14. It can be seen from the graph that the distribution of the interface shear stress decreases nonlinearly at this stage, but the smaller is the Et, the more obvious is the nonlinearity of the curve, which is due to the negative correlation between Et and Ks1 (α=2Ks1/Et).Figure 14
Effect of the interface frictional resistance at stage I onEt.


#### 5.4.2. Elastic-Hardening Transition Stage

The influence of the hardening zone lengthLpη on the interface shear stress at different locations of the reinforcement in stage II is shown in Figure 15. It can be found from the graph that the interface shear stress in the elastic-hardening transition stage decreases nonlinearly in both the elastic zone and the hardening zone. The maximum shear stress is at the pull-out end, and the minimum shear stress is at the free end. The critical point of the two points is the transition point. The nonlinearity of the shear stress curve in the elastic zone is more obvious than that in the hardening zone.Figure 15
Effect of the interface frictional resistance at stage II onLp.The effect of the hardening shear stiffness (Ks2) on the interface shear stress at different locations of the reinforcement in stage II is shown in Figure 16. It can be seen from the graph that Ks2 only affects the change in the shear stress in the hardening zone of stage II but has no effect on the change in the shear stress in the elastic zone. The larger the Ks2 is, the larger the range of the shear stress curve in the hardening zone is.Figure 16
Effect of the interfacial frictional resistance at stage II onKs2.


#### 5.4.3. Pure Hardening Stage

The effect of the hardening shear stiffness (Ks2) on the interface shear stress at different locations of the reinforcement in stage III is shown in Figure 17. It can be found from the graph that the larger the Ks2 is, the greater the sag of the interface shear stress curve in stage III is. In practice, Ks2 is very small. In the elastic-hardening transition stage, the shear stress at the pull-out end should be slightly larger than the peak shear stress and then slowly decrease until the shear stress at the pull-out end reaches the ultimate shear stress and the reinforcement is damaged.Figure 17
Effect of the interfacial frictional resistance at stage III onKs2.


## 5.1. Analysis of the Hardening Pull-Out Interface

Basic assumption: according to the definition of the theoretical model of the strain-hardening reinforcement, it is considered that the interface between the reinforcement and tailings will undergo an elastic stage, an elastic-hardening transition stage, and a pure hardening stage under a pull-out load, corresponding to stages I, II, and III in Figure9, respectively. Through theoretical calculations, the expressions of the tension, shear stress, and displacement at each stage of pull-out can be obtained.Figure 9
The progressive pull-out process of the strain-hardening reinforcement in the pull-out model analysis: (a) elastic stage; (b) elastic-hardening transition stage; (c) pure hardening stage.
(a)(b)(c)


### 5.1.1. Elastic Stage (I Stage)

When0≤u<up, the shear stress and displacement are elastic, which satisfies the equation τ=Ks1u+τo. The governing equation of this stage can be obtained by the simultaneous Equations (10) and (3a):(11)d2Tdx2−α2T=0,where α=2Ks1/Et.Solving Equation (11), the following can be obtained:(12)Tex=C1exp−αx+C2expαx,where Tex is the tension of the reinforcement at the elastic stage and C1 and C2 are the integral constants.In the pull-out test, when the tension at the pull-out end, i.e.,x = 0, is T0 and the tension at the free end, i.e., x = L, is 0, the following boundary conditions exist:(13)Tx=0=T01,Tx=L=0.By substituting the boundary condition (13) in Equation (12), we obtain(14)C1=expαLexpαL−exp−αLT01,C2=−exp−αLexpαL−exp−αLT01.The tension expression of stage I can be obtained as follows:(15)Tex=T01sinhαL−xsinhαL.According to Equations (10) and (3a), the corresponding shear stress and displacement relations can be obtained:(16)τex=αT012⋅coshαL−xsinhαL,(17)uex=αT012Ks1⋅coshαL−xsinhαL.The transformation of the pull-out displacementue0 can be obtained by substituting x = 0 into Equation (17):(18)T01=2tanhαLαKs1ue0,Whenue0=up and τp=Ks1ue0+τo, from Equation (18), the critical tension Teh0c of the elastic stage and the elastic-hardening transition stage, i.e., the maximum tension in the elastic stage, can be obtained as follows:(19)Teh0c=2τp−τotanhαLα.


### 5.1.2. Elastic-Hardening Transition Stage (II Stage)

With increasing tension, the interface shear stress gradually transfers from the pull-out end to the tail until it reaches its peak value. Then, the plastic characteristics of the pull-out end begin to occur, and the strain-hardening phenomenon occurs and enters the II stage. The critical pointP (x=Lh) is defined to divide the elastic zone and the hardening zone. When 0≤x<Lh, the interface is in the stage II hardening zone, and when Lh<x≤L, the interface is in the stage II elastic zone (where Lh is the length of the hardening zone).(1)
Elastic Zone (Lh≤x≤L).The distributions of the interface tension, shear stress, and displacement in the stage II elastic zone are similar to those in the elastic stage; thus, we can obtain(20)Tex=T02sinhαL−xsinhαL−Lh,(21)τex=αT022⋅coshαL−xsinhαL−Lh,(22)uex=αT022Ks1⋅coshαL−xsinhαL−Lh,where T02 is the tension of the transition point P.Considering that the interface shear stress at the transition pointP is equal to the peak shear stress, the following result can be obtained:(23)T02=2τptanhαL−Lhα.(2)
Hardening Zone (0≤x≤Lh).The relationship between the interface shear stress and displacement in the stage II hardening zone is defined by Equation (3b), and the simultaneous Equations (10) and (3b) can be obtained as follows:(24)d2Tdx2−β2T=0,where β=2Ks2/Et.Solving Equation (24) can yield(25)Thx=C3exp−βx+C4expβx,where Thx is the tension of reinforcement at the hardening zone of stage II and C1 and C2 are the integral constants.Consider the boundary conditions as follows:(26)Thx=0=T03,Thx=Lh=Tex=Lh.By substituting the boundary condition (26) in Equation (25), we can obtain(27)C3=−T03exp−βLhexpβLh−exp−βLh+2τptanhαL−LhαexpβLh−exp−βLh,C4=T03expβLhexpβLh−exp−βLh−2τptanhαL−LhαexpβLh−exp−βLh.The expressions of the tension, shear stress, and displacement in stage II can be obtained as follows:(28)Thx=T03sinhβLh−xsinhβLh+2τptanhαL−Lhα⋅sinhβxsinhβLh,(29)τhx=βT032⋅coshβLh−xsinhβLh−βτptanhαL−Lhα⋅coshβxsinhβLh,(30)uhx=βT032Ks2⋅coshβLh−xsinhβLh−βτptanhαL−LhαKs2⋅coshβxsinhβLh−τpKs2+up.Since the shear stress at the transition pointP between the elastic zone and the hardening zone is continuous, i.e., τex=Lh=τhx=Lh, the following can be obtained:(31)T03=2τpsinhβLhβ+2τptanhαL−Lhα⋅coshβLh.WhenLh=L, the critical tension Th0c between the elastic-hardening transition stage and the pure hardening stage can be obtained by Equation (31):(32)Th0c=2τpsinhβLβ.


### 5.1.3. Pure Hardening Stage (III Stage)

Similar to the analysis of the hardening zone of stage II, Equation (25) is still applicable to the pure hardening stage, and the boundary conditions are as follows:(33)Tsx=0=T04,Tsx=L=0.The expressions of the interface tension, shear stress, and displacement at the pure hardening stage are as follows:(34)Thx=T04⋅sinhβL−xsinhβL,(35)τhx=βT042⋅coshβL−xsinhβL,(36)uhx=βT042Ks2⋅coshβL−xsinhβL−τpKs2+up.Letx = 0 be substituted into Equation (36) to obtain the transformation of the pull-out displacement uh0 at this stage:(37)T04=2tanhβLβKs2uh0−up+τp.In the pure hardening stage, the interface pull-out force and shear stress at the pull-out end all increase. When the shear stress at the pull-out end increases to the ultimate stressτult of the reinforcement, the reinforcement is destroyed. Then, τult=Ks2uh0−up+τp, combined with Rf=τp/τult, so Equation (37) can be written as follows:(38)T04=2τptanhβLRfβ.In summary, closed solutions are obtained for the three stages of the pull-out process of strain-hardening reinforcement. The critical tension and the initial tension between the two stages are shown in Tables2 and 3.Table 2
Critical tensile force between the two stages of the hardening pull-out model.
Pull-out stageI-IIII-IIICritical tensile forceTeh0c=2τptanhαL/αTh0c=2τpsinhβL/βTable 3
Initial tensile forces at different stages of the hardening pull-out model.
Pull-out stageIII (elastic zone, hardening zone)IIIInitial tensile forcesT01=2tanhαL/αKs1ue0T02=2τptanhαL−Lh/α,T03=2τpsinhβLh/β+2τptanhαL−Lh/α⋅coshβLhT04=2τptanhβL/Rfβ


## 5.1.1. Elastic Stage (I Stage)

When0≤u<up, the shear stress and displacement are elastic, which satisfies the equation τ=Ks1u+τo. The governing equation of this stage can be obtained by the simultaneous Equations (10) and (3a):(11)d2Tdx2−α2T=0,where α=2Ks1/Et.Solving Equation (11), the following can be obtained:(12)Tex=C1exp−αx+C2expαx,where Tex is the tension of the reinforcement at the elastic stage and C1 and C2 are the integral constants.In the pull-out test, when the tension at the pull-out end, i.e.,x = 0, is T0 and the tension at the free end, i.e., x = L, is 0, the following boundary conditions exist:(13)Tx=0=T01,Tx=L=0.By substituting the boundary condition (13) in Equation (12), we obtain(14)C1=expαLexpαL−exp−αLT01,C2=−exp−αLexpαL−exp−αLT01.The tension expression of stage I can be obtained as follows:(15)Tex=T01sinhαL−xsinhαL.According to Equations (10) and (3a), the corresponding shear stress and displacement relations can be obtained:(16)τex=αT012⋅coshαL−xsinhαL,(17)uex=αT012Ks1⋅coshαL−xsinhαL.The transformation of the pull-out displacementue0 can be obtained by substituting x = 0 into Equation (17):(18)T01=2tanhαLαKs1ue0,Whenue0=up and τp=Ks1ue0+τo, from Equation (18), the critical tension Teh0c of the elastic stage and the elastic-hardening transition stage, i.e., the maximum tension in the elastic stage, can be obtained as follows:(19)Teh0c=2τp−τotanhαLα.


## 5.1.2. Elastic-Hardening Transition Stage (II Stage)

With increasing tension, the interface shear stress gradually transfers from the pull-out end to the tail until it reaches its peak value. Then, the plastic characteristics of the pull-out end begin to occur, and the strain-hardening phenomenon occurs and enters the II stage. The critical pointP (x=Lh) is defined to divide the elastic zone and the hardening zone. When 0≤x<Lh, the interface is in the stage II hardening zone, and when Lh<x≤L, the interface is in the stage II elastic zone (where Lh is the length of the hardening zone).(1)
Elastic Zone (Lh≤x≤L).The distributions of the interface tension, shear stress, and displacement in the stage II elastic zone are similar to those in the elastic stage; thus, we can obtain(20)Tex=T02sinhαL−xsinhαL−Lh,(21)τex=αT022⋅coshαL−xsinhαL−Lh,(22)uex=αT022Ks1⋅coshαL−xsinhαL−Lh,where T02 is the tension of the transition point P.Considering that the interface shear stress at the transition pointP is equal to the peak shear stress, the following result can be obtained:(23)T02=2τptanhαL−Lhα.(2)
Hardening Zone (0≤x≤Lh).The relationship between the interface shear stress and displacement in the stage II hardening zone is defined by Equation (3b), and the simultaneous Equations (10) and (3b) can be obtained as follows:(24)d2Tdx2−β2T=0,where β=2Ks2/Et.Solving Equation (24) can yield(25)Thx=C3exp−βx+C4expβx,where Thx is the tension of reinforcement at the hardening zone of stage II and C1 and C2 are the integral constants.Consider the boundary conditions as follows:(26)Thx=0=T03,Thx=Lh=Tex=Lh.By substituting the boundary condition (26) in Equation (25), we can obtain(27)C3=−T03exp−βLhexpβLh−exp−βLh+2τptanhαL−LhαexpβLh−exp−βLh,C4=T03expβLhexpβLh−exp−βLh−2τptanhαL−LhαexpβLh−exp−βLh.The expressions of the tension, shear stress, and displacement in stage II can be obtained as follows:(28)Thx=T03sinhβLh−xsinhβLh+2τptanhαL−Lhα⋅sinhβxsinhβLh,(29)τhx=βT032⋅coshβLh−xsinhβLh−βτptanhαL−Lhα⋅coshβxsinhβLh,(30)uhx=βT032Ks2⋅coshβLh−xsinhβLh−βτptanhαL−LhαKs2⋅coshβxsinhβLh−τpKs2+up.Since the shear stress at the transition pointP between the elastic zone and the hardening zone is continuous, i.e., τex=Lh=τhx=Lh, the following can be obtained:(31)T03=2τpsinhβLhβ+2τptanhαL−Lhα⋅coshβLh.WhenLh=L, the critical tension Th0c between the elastic-hardening transition stage and the pure hardening stage can be obtained by Equation (31):(32)Th0c=2τpsinhβLβ.


## 5.1.3. Pure Hardening Stage (III Stage)

Similar to the analysis of the hardening zone of stage II, Equation (25) is still applicable to the pure hardening stage, and the boundary conditions are as follows:(33)Tsx=0=T04,Tsx=L=0.The expressions of the interface tension, shear stress, and displacement at the pure hardening stage are as follows:(34)Thx=T04⋅sinhβL−xsinhβL,(35)τhx=βT042⋅coshβL−xsinhβL,(36)uhx=βT042Ks2⋅coshβL−xsinhβL−τpKs2+up.Letx = 0 be substituted into Equation (36) to obtain the transformation of the pull-out displacement uh0 at this stage:(37)T04=2tanhβLβKs2uh0−up+τp.In the pure hardening stage, the interface pull-out force and shear stress at the pull-out end all increase. When the shear stress at the pull-out end increases to the ultimate stressτult of the reinforcement, the reinforcement is destroyed. Then, τult=Ks2uh0−up+τp, combined with Rf=τp/τult, so Equation (37) can be written as follows:(38)T04=2τptanhβLRfβ.In summary, closed solutions are obtained for the three stages of the pull-out process of strain-hardening reinforcement. The critical tension and the initial tension between the two stages are shown in Tables2 and 3.Table 2
Critical tensile force between the two stages of the hardening pull-out model.
Pull-out stageI-IIII-IIICritical tensile forceTeh0c=2τptanhαL/αTh0c=2τpsinhβL/βTable 3
Initial tensile forces at different stages of the hardening pull-out model.
Pull-out stageIII (elastic zone, hardening zone)IIIInitial tensile forcesT01=2tanhαL/αKs1ue0T02=2τptanhαL−Lh/α,T03=2τpsinhβLh/β+2τptanhαL−Lh/α⋅coshβLhT04=2τptanhβL/Rfβ


## 5.2. Verification of the Hardening Model

To verify the elastic-plastic theoretical model of strain hardening, the pull-out test results between geogrids and tailings mentioned above are used for the simulation, and the predicted results are shown in Figure10. According to the graph, the prediction results are in good agreement with the pull-out test results, and the model can effectively describe the progressive pull-out behaviour of geogrids in the tailings. Moreover, the displacement of stage II is relatively small compared with that of stages I and III, which is due to the large elastic modulus and the small length of reinforcement, resulting in the progressive failure of the reinforcement not obvious. This transitional stage can be neglected in the analysis of the pull-out behaviour of geogrids in the tailings.Figure 10
Comparison between the pull-out test results and model-predicted results of geogrids under different normal stresses: (a) 10 kPa; (b) 20 kPa; (c) 30 kPa; and (d) 40 kPa.
(a)(b)(c)(d)Furthermore, to better understand the five pull-out stages of geogrids in the tailings, the relationship between the interface shear stiffness (elastic shear stiffnessKs1 and hardening shear stiffness Ks2) and the initial shear stress with the normal stress is given, as shown in Figure 11. With an increase in the normal stress, Ks1 and Ks2 increase linearly. The initial shear stress is also linearly related to the normal stress, and the initial shear strength index can be obtained from the Mohr–Coulomb strength criterion.Figure 11
The relationship between the normal stress and shear stiffness of the hardening model: (a) shear stiffness; (b) initial shear stress.
(a)(b)


## 5.3. Distribution of the Interface Shear Stress

To visually reflect the stress evolution law of strain-hardening reinforcement at different stages of the pull-out interface, the interface shear stress distribution at three stages was analysed. The model parameters selected according to the above test results are shown in Table4. To simplify the analysis, the model parameters were normalized. The normalized reinforcement position was X=x/L, and the normalized interface shear stress was ρ=τ/τp.Table 4
Hardening model parameters.
σn (kPa)L (m)t (mm)E (GPa)Ks1 (MPa/m)Ks2 (MPa/m)fRf300.3211.4490.0660.5260.8The parametersτp=σnf=15.78kPa, α=2Ks1/Et=1.20, and β=2Ks2/Et=0.26 were calculated. According to the model parameters given, the following formulas were substituted.


### 5.3.1. Elastic Stage

Substitution of Equation (16) can yield τex=τpcoshαL−x/coshαL, i.e., ρ=coshα1−X/coshαL.The substitution of Equation (19) indicates that the critical tension of the elastic stage and the elastic-hardening transition stage is Teh0c=2τptanhαL/α=9.08kN/m.


### 5.3.2. Elastic-Hardening Transition Stage

(i)
Elastic Zone (0.5≤X≤1).The substitution ofLh=L/2 into Equation (64) yields τex=τptanhαL/2coshαL−x/sinhαL/2, i.e., ρ=coshαL1−X/coshαL/2.(ii)
Hardening Zone (0≤X≤0.5).The substitution ofLh=L/2 into Equation (29) results in τhx=τp⋅coshβL/2−x+τpβtanhαL/2/αsinhβL/2⋅coshβL/2⋅coshβL/2−x−coshβx, i.e., ρ=coshβL0.5−X+βtanhαL/2/αsinhβL/2⋅coshβL/2⋅coshβL0.5−X−coshβLX.From Equation (32), the critical tension of the elastic-hardening transition stage and the pure hardening stage is Th0c=2τpsinhβL/β=9.48kN/m.(iii)
Pure Hardening Stage.The substitution of Equation (35) results in τhx=τpcoshβL−x/RfcoshβL, i.e., ρ=coshβL1−X/RfcoshβL.The substitution of Equation (38) shows that the ultimate tensile force at the stage of pure hardening is T04=2τptanhβL/Rfβ=11.81kN/m.According to the above expression of the interface shear stress, the evolution law of the shear stress in the three stages of elastic, elastic-hardening transition, and pure hardening of the strain-hardening reinforcement can be obtained, as shown in Figure12.Figure 12
Evolution of the interfacial shear stress in different stages of the pull-out process of the reinforced tailings (strain-hardening reinforcement): (a) elastic stage; (b) elastic-hardening transition stage; (c) pure hardening stage.
(a)(b)(c)WhenT≤9.08kN/m, the interface is in the elastic stage, and the shear stress decreases nonlinearly from the pull-out end to the free end. When T=9.08kN/m, the interface is in the critical stage of elastic and pure hardening, and the peak shear stress reaches 20 kPa at the pull-out end.When9.08kN/m≤T≤9.48kN/m, the interface is in the transition stage of elastic-hardening, in which the shear stress at the pull-out end reaches its peak value and then drives the free end to continue to increase. The shear stress decreases from the pull-out end to the free end. When T=9.48kN/m, the free end also reaches its peak value, and the interface will enter the stage of pure hardening.When9.48kN/m≤T≤11.81kN/m, the interface is in the stage of pure hardening, and the shear stress at the pull-out end continues to increase until T=11.81kN/m, when the shear stress at the pull-out end reaches the ultimate stress and the reinforcement is damaged.


## 5.3.1. Elastic Stage

Substitution of Equation (16) can yield τex=τpcoshαL−x/coshαL, i.e., ρ=coshα1−X/coshαL.The substitution of Equation (19) indicates that the critical tension of the elastic stage and the elastic-hardening transition stage is Teh0c=2τptanhαL/α=9.08kN/m.


## 5.3.2. Elastic-Hardening Transition Stage

(i)
Elastic Zone (0.5≤X≤1).The substitution ofLh=L/2 into Equation (64) yields τex=τptanhαL/2coshαL−x/sinhαL/2, i.e., ρ=coshαL1−X/coshαL/2.(ii)
Hardening Zone (0≤X≤0.5).The substitution ofLh=L/2 into Equation (29) results in τhx=τp⋅coshβL/2−x+τpβtanhαL/2/αsinhβL/2⋅coshβL/2⋅coshβL/2−x−coshβx, i.e., ρ=coshβL0.5−X+βtanhαL/2/αsinhβL/2⋅coshβL/2⋅coshβL0.5−X−coshβLX.From Equation (32), the critical tension of the elastic-hardening transition stage and the pure hardening stage is Th0c=2τpsinhβL/β=9.48kN/m.(iii)
Pure Hardening Stage.The substitution of Equation (35) results in τhx=τpcoshβL−x/RfcoshβL, i.e., ρ=coshβL1−X/RfcoshβL.The substitution of Equation (38) shows that the ultimate tensile force at the stage of pure hardening is T04=2τptanhβL/Rfβ=11.81kN/m.According to the above expression of the interface shear stress, the evolution law of the shear stress in the three stages of elastic, elastic-hardening transition, and pure hardening of the strain-hardening reinforcement can be obtained, as shown in Figure12.Figure 12
Evolution of the interfacial shear stress in different stages of the pull-out process of the reinforced tailings (strain-hardening reinforcement): (a) elastic stage; (b) elastic-hardening transition stage; (c) pure hardening stage.
(a)(b)(c)WhenT≤9.08kN/m, the interface is in the elastic stage, and the shear stress decreases nonlinearly from the pull-out end to the free end. When T=9.08kN/m, the interface is in the critical stage of elastic and pure hardening, and the peak shear stress reaches 20 kPa at the pull-out end.When9.08kN/m≤T≤9.48kN/m, the interface is in the transition stage of elastic-hardening, in which the shear stress at the pull-out end reaches its peak value and then drives the free end to continue to increase. The shear stress decreases from the pull-out end to the free end. When T=9.48kN/m, the free end also reaches its peak value, and the interface will enter the stage of pure hardening.When9.48kN/m≤T≤11.81kN/m, the interface is in the stage of pure hardening, and the shear stress at the pull-out end continues to increase until T=11.81kN/m, when the shear stress at the pull-out end reaches the ultimate stress and the reinforcement is damaged.


## 5.4. Parametric Impact Analysis

From the above analysis, it can be seen that the interface shear stress is mainly affected by the elastic shear stiffness (Ks1) and the tensile stiffness (Et) in the elastic stage, and the distribution of shear stress in the hardening stage is related to the length of the hardening zone (Lh) and the hardening shear stiffness (Ks2). The calculated parameters are consistent with those in the previous section, in which the ratio of the hardening zone length to the reinforcement length is defined as η=Lh/L.


### 5.4.1. Elastic Stage

The effect of the elastic shear stiffness (Ks1) on the interface shear stress at different locations of the reinforcement in stage I is shown in Figure 13. The distribution of the interface shear stress in the elastic stage decreases nonlinearly from the pull-out end to the free end. The maximum shear stress is the same, which is the peak shear stress. The larger the Ks1 is, the more obvious the nonlinearity of the interface shear stress curve is.Figure 13
Effect of the interface frictional resistance at stage I onKs1.The effect of the single-width tensile stiffness (Et) on the interface shear stress at different locations of the reinforcement in stage I is shown in Figure 14. It can be seen from the graph that the distribution of the interface shear stress decreases nonlinearly at this stage, but the smaller is the Et, the more obvious is the nonlinearity of the curve, which is due to the negative correlation between Et and Ks1 (α=2Ks1/Et).Figure 14
Effect of the interface frictional resistance at stage I onEt.


### 5.4.2. Elastic-Hardening Transition Stage

The influence of the hardening zone lengthLpη on the interface shear stress at different locations of the reinforcement in stage II is shown in Figure 15. It can be found from the graph that the interface shear stress in the elastic-hardening transition stage decreases nonlinearly in both the elastic zone and the hardening zone. The maximum shear stress is at the pull-out end, and the minimum shear stress is at the free end. The critical point of the two points is the transition point. The nonlinearity of the shear stress curve in the elastic zone is more obvious than that in the hardening zone.Figure 15
Effect of the interface frictional resistance at stage II onLp.The effect of the hardening shear stiffness (Ks2) on the interface shear stress at different locations of the reinforcement in stage II is shown in Figure 16. It can be seen from the graph that Ks2 only affects the change in the shear stress in the hardening zone of stage II but has no effect on the change in the shear stress in the elastic zone. The larger the Ks2 is, the larger the range of the shear stress curve in the hardening zone is.Figure 16
Effect of the interfacial frictional resistance at stage II onKs2.


### 5.4.3. Pure Hardening Stage

The effect of the hardening shear stiffness (Ks2) on the interface shear stress at different locations of the reinforcement in stage III is shown in Figure 17. It can be found from the graph that the larger the Ks2 is, the greater the sag of the interface shear stress curve in stage III is. In practice, Ks2 is very small. In the elastic-hardening transition stage, the shear stress at the pull-out end should be slightly larger than the peak shear stress and then slowly decrease until the shear stress at the pull-out end reaches the ultimate shear stress and the reinforcement is damaged.Figure 17
Effect of the interfacial frictional resistance at stage III onKs2.


## 5.4.1. Elastic Stage

The effect of the elastic shear stiffness (Ks1) on the interface shear stress at different locations of the reinforcement in stage I is shown in Figure 13. The distribution of the interface shear stress in the elastic stage decreases nonlinearly from the pull-out end to the free end. The maximum shear stress is the same, which is the peak shear stress. The larger the Ks1 is, the more obvious the nonlinearity of the interface shear stress curve is.Figure 13
Effect of the interface frictional resistance at stage I onKs1.The effect of the single-width tensile stiffness (Et) on the interface shear stress at different locations of the reinforcement in stage I is shown in Figure 14. It can be seen from the graph that the distribution of the interface shear stress decreases nonlinearly at this stage, but the smaller is the Et, the more obvious is the nonlinearity of the curve, which is due to the negative correlation between Et and Ks1 (α=2Ks1/Et).Figure 14
Effect of the interface frictional resistance at stage I onEt.


## 5.4.2. Elastic-Hardening Transition Stage

The influence of the hardening zone lengthLpη on the interface shear stress at different locations of the reinforcement in stage II is shown in Figure 15. It can be found from the graph that the interface shear stress in the elastic-hardening transition stage decreases nonlinearly in both the elastic zone and the hardening zone. The maximum shear stress is at the pull-out end, and the minimum shear stress is at the free end. The critical point of the two points is the transition point. The nonlinearity of the shear stress curve in the elastic zone is more obvious than that in the hardening zone.Figure 15
Effect of the interface frictional resistance at stage II onLp.The effect of the hardening shear stiffness (Ks2) on the interface shear stress at different locations of the reinforcement in stage II is shown in Figure 16. It can be seen from the graph that Ks2 only affects the change in the shear stress in the hardening zone of stage II but has no effect on the change in the shear stress in the elastic zone. The larger the Ks2 is, the larger the range of the shear stress curve in the hardening zone is.Figure 16
Effect of the interfacial frictional resistance at stage II onKs2.


## 5.4.3. Pure Hardening Stage

The effect of the hardening shear stiffness (Ks2) on the interface shear stress at different locations of the reinforcement in stage III is shown in Figure 17. It can be found from the graph that the larger the Ks2 is, the greater the sag of the interface shear stress curve in stage III is. In practice, Ks2 is very small. In the elastic-hardening transition stage, the shear stress at the pull-out end should be slightly larger than the peak shear stress and then slowly decrease until the shear stress at the pull-out end reaches the ultimate shear stress and the reinforcement is damaged.Figure 17
Effect of the interfacial frictional resistance at stage III onKs2.


## 6. Pull-Out Interface Analysis of the Strain-Softening Reinforcement

### 6.1. Analysis of the Softening Pull-Out Interface

Basic assumption: the pull-out process of the strain-softening reinforcement under the pull-out load is divided into five stages [24]: the elastic stage, elastic-softening transition stage, pure softening stage, softening-residual transition stage, and pure residual stage, corresponding to stages I, II, III, IV, and V in Figure 18, respectively. The expressions of the interfacial tension, shear stress, and displacement at the different stages of pull-out can be obtained by theoretical calculations.Figure 18
The progressive pull-out process of strain-softening reinforcement in the pull-out model analysis: (a) elastic stage; (b) elastic-softening transition stage; (c) pure softening stage; (d) softening-residual transition stage; and (e) pure residual stage.
(a)(b)(c)(d)(e)


#### 6.1.1. Elastic Stage (I Stage)

Similar to the distribution law of the strain-hardening reinforcement in the elastic stage, the expressions of the tension, shear stress, and displacement at this stage can be obtained by Equations (10) and (4a):(39)Tex=T01⋅sinhαL−xsinhαL,(40)τex=αT012⋅coshαL−xsinhαL,(41)uex=αT012ks1⋅coshαL−xsinhαL.In Equation (41), x = 0 can be substituted to obtain the transformation of the pull-out displacement ue0:(42)T01=2tanhαLαks1ue0,when ue0=up and τp=ks1ue0+τo, the critical tension Teh0c of the elastic and elastic-softening transition stages can be obtained from Equation (42), that is, the maximum tension in the elastic stage,(43)Tes0c=2τp−τotanhαLα.


#### 6.1.2. Elastic-Softening Transition Stage (II Stage)

With increasing tension, the interface shear stress gradually transfers from the pull-out end to the tail until it reaches its peak value. Then, the plastic characteristics of the pull-out end begin to occur, and the strain-hardening phenomenon occurs and enters the II stage. The critical pointPx=Lh is defined to divide the elastic zone and the hardening zone. When 0≤x<Lh, the interface is in the stage II hardening zone, and when Lh<x≤L, the interface is in the stage II elastic zone (where Lh is the length of the hardening zone).(1)
Elastic Zone (Ls≤x≤L).The distributions of the interface tension, shear stress, and displacement in the stage II elastic zone are similar to those in the elastic stage:(44)Tex=T02⋅sinhαL−xsinhαL−Ls,(45)τex=αT022⋅coshαL−xsinhαL−Ls,(46)uex=αT022ks1⋅coshαL−xsinhαL−Ls,where T02 is the pull-out force of the transition point P.Considering that the interface shear stress of the transition pointQ1 is equal to the peak shear stress, the following can be obtained:(47)T02=2τptanhαL−Lsα.(2)
Softening Zone (0≤x≤Ls).The relationship between the interface shear stress and the displacement in the softening zone is defined by Equation (4b). The simultaneous Equations (10) and (4b) can be obtained as follows:(48)d2Tdx2+β2T=0,where β=−2ks2/Et>0.Solving Equation (48) can yield(49)Tsx=C3cosβx+C4sinβx,where Tsx is the tension of the reinforcement at the elastic stage and C3 and C4 are the integral constants.Consider the following boundary conditions:(50)Tsx=0=T03,Tsx=Ls=Tex=Ls.The substitution of Equation (47) into Equation (46) results in(51)C3=T03,C4=2τptanhαL−LsαsinβLs−T03cotβLs.Since the shear stress at the transition pointQ1 between the elastic zone and the softening zone is continuous, i.e., τex=Ls=τsx=Ls, it can be determined that T03 is as follows:(52)T03=2τpsinβLsβ+tanhαL−LsαcosβLs.The expressions of the interface tension, shear stress, and displacement in stage II can be obtained:(53)Tsx=2τpsinβLs−xβ+tanhαL−LsαcosβLs−x,(54)τsx=τpcosβLs−x−βtanhαL−LsαsinβLs−x,(55)usx=τpks2cosβLs−x−βtanhαL−LsαsinβLs−x−τpks2+up,when Ls=L, the critical pull-out force Ts0c of the elastic-softening transition stage and the pure softening stage can be obtained from Equation (52):(56)Ts0c=2τpsinβLβ.


#### 6.1.3. Pure Softening Stage (III Stage)

Similar to the analysis of the stage II softening zone, Equation (50) is still applicable to the pure softening stage with the following boundary conditions:(57)Tsx=0=T04,Tsx=L=0.The solution of the pure softening stage is as follows:(58)Tsx=T04cosβx−cotβLsinβx,(59)τsx=βT042sinβx+cotβLcosβx,(60)usx=βT042ks2sinβx+cotβLcosβx−τpks2+up.Letx = 0 be substituted into Equation (60) to obtain the transformation of the pull-out displacement us0 at this stage:(61)T04=2tanβLβks2us0−up+τp.In the pure softening stage, the interface tension and shear stress decrease. Whenus0=ur and τr=ks2us0−up+τp, the critical pull-out force Tsr0c of the elastic-softening transition stage and the pure softening stage can be obtained from Equation (61); that is, the minimum pull-out force of the pure softening stage is(62)Tsr0c=2τrtanβLβ.


#### 6.1.4. Softening-Residual Transition Stage (IV Stage)

When the pull-out force decreases toTsr0c, the pull-out end of the reinforcement begins to enter the residual state and gradually extends when the end of reinforcement enters stage IV. The transition point Q2x=Lr is defined to divide the softening zone and the residual zone. When 0≤x<Lr, the interface is in the residual zone of stage II, and when Lr<x≤L, the interface is in the softening zone of stage II (Lr is the residual zone length).(1)
Softening Zone (Ls≤x≤L).The distributions of the tension, shear stress, and displacement in the pure softening stage are similar to those in the softening-residual transition stage (x⟶x−Lr, L⟶L−Lr, and T04⟶T05). According to Equations (58)–(60), the relationship of the tension, shear stress, and displacement in the elastic zone of stage IV can be obtained:(63)Tsx=T05cosβx−Lr−cotβL−Lrsinβx−Lr,(64)τsx=βT052sinβx−Lr+cotβL−Lrcosβx−Lr,(65)usx=βT052ks2sinβx−Lr+cotβL−Lrcosβx−Lr−τpks2+up.Since the interface shear stress at the transition pointQ2 is equal to the residual shear stress τsx−Lr=τr, it can be obtained according to Equation (64):(66)T05=2τrtanβL−Lrβ.(2)
Residual Zone (0≤x≤Lr).The interface shear stress in the residual zone is equal to the residual shear strength, which can be obtained by Equations (10) and (4c):(67)dTdx=−2τr.Consider the boundary conditions:(68)Trx=0=T06,Trx=Lr=Tsx=Lr.It can be obtained as follows:(69)T06=2τrtanβL−Lrβ+2τrLr,(70)Trx=2τrLr−x+2τrtanβL−Lrβ,(71)urx=τrEtx2−2τrEtLrx+tanβL−Lrβx+C5,where(72)C5=τrLr2Et−2τrLrβEttanβL−Lr+τr−τpks2+up.The critical tension of the softening-residual transition stage and the pure residual stage is obtained atx=0, and the result of the substitution of Lr=L into Equation (69) is as follows:(73)Ts0c=2τrL.


#### 6.1.5. Pure Residual Stage (V Stage)

The critical shear displacement at the pull-out end of the reinforcement is obtained by substitutingx=0 and Lr=L into Equation (71):(74)ur0c=τrL2Et+τr−τpks2+up.Assuming that the displacement of the reinforcement at the pull-out end isu0′, the distribution of the shear displacement can be derived as follows:(75)urx=τrEtx2−L2−2τrLEtx−L+τr−τpks2+up+u0′−ur0c=u0′+2τrLEtx2−2Lx.At this stage, the tensionTr0c and the interface shear stress τr remain constant. Therefore, the distribution of tension at different locations of the reinforcement is as follows:(76)Trx=2τrL−x.In summary, closed solutions are obtained for the five stages of the pull-out process of strain-hardening reinforcement. The critical tension and the initial tension between the two stages are shown in Tables5 and 6.Table 5
Critical tensile force between the two stages of the softening pull-out model.
Pull-out stageI-IIII-IIIIII-IVIV-VCritical tensile forceTes0c=2τp−τotanhαL/αTs0c=2τpsinβL/βTsr0c=2τrtanβL/βTs0c=2τrLTable 6
Initial tensile forces at different stages of the softening pull-out model.
Pull-out stageInitial tensile forcesIElastic stage:T01=2tanhαL/αks1ue0IIElastic zone:T02=2τptanhαL−Ls/α; softening zone: T03=2τpsinβLs/β+tanhαL−Ls/αcosβLsIIISoftening stage:T04=2tanβL/βks2us0−up+τpIVSoftening zone:T05=2τrtanβL−Lr/β; residual zone: T06=2τrtanβL−Lr/β+2τrLrV—


### 6.2. Verification of the Softening Model

To verify the elastic-plastic theoretical model of strain softening, the pull-out test results between geotextiles and tailings mentioned above are used for simulation, and the predicted results are shown in Figure19. The predicted results are in good agreement with the pull-out test results, and the model can effectively describe the pull-out behaviour of geotextiles in the tailings. Moreover, the displacements of the II and IV stages are relatively smaller than those of the I, III, and V stages, so the two transitional stages can be neglected in the analysis of the pull-out behaviour of the geotextiles in the tailings.Figure 19
Comparison between the pull-out test results and model-predicted results of geotextiles under different normal stresses.
(a)(b)(c)(d)In addition, to better understand the five pull-out stages of the geotextiles in the tailings, the relationship between the interface shear stiffness (elastic shear stiffnessks1 and softening shear stiffness ks2) and the initial shear stress with the normal stress is given, as shown in Figure 20. The graph shows that ks1 and ks2 increase linearly with an increase in the normal stress, and the initial shear stress is linearly related to the normal stress. The initial shear strength index can be obtained by the Mohr–Coulomb strength criterion.Figure 20
The relationship between the normal stress and the shear stiffness of the softening model: (a) shear stiffness and (b) initial shear stress.
(a)(b)


### 6.3. Distribution of the Interface Shear Stress Values

To visually reflect the stress evolution law of strain-softening reinforcement at different stages of the pull-out interface, the interface shear stress distribution at three stages was analysed. The model parameters selected according to the above test results are shown in Table7. To simplify the analysis, the model parameters were normalized. The normalized reinforcement position is X=x/L, and the normalized interface shear stress is ρ=τ/τp.Table 7
Softening model parameters.
σn (kPa)L (m)t (mm)E (GPa)ks1 (MPa/m)ks2 (MPa/m)fRf300.3212.190−0.4500.5600.7The parametersτp=σnf=16.80kPa, τr=Rfτp=11.76kPa, α=2ks1/Et=1.48, and β=−2ks2/Et=0.67 are calculated. According to the model parameters given, the following formulas are substituted:


#### 6.3.1. Elastic Stage

The substitution of Equation (40) can yield τex=τpcoshαL−x/coshαL, i.e., ρ=coshα1−X/coshαL.The substitution of Equation (43) shows that the critical tension of the elastic stage and the elastic-softening transition stage is Tes0c=2τptanhαL/α=9.47kN/m.


#### 6.3.2. Elastic-Softening Transition Stage

Elastic Zone (0.5≤X≤1).LetLs=L/2 in Equation (45) produce τex=τpcoshαL−x/coshαL/2, i.e., ρ=coshαL1−X/coshαL/2.Softening Zone (0≤X≤0.5).The substitution ofLs=L/2 into Equation (54) results in τsx=τpcosβL/2−x−βtanhαL/2/αsinβL/2−x, i.e., ρ=cosβL0.5−X−βtanhαL/2/αsinβL0.5−X.From Equation (56), the critical tension of the elastic-softening transition stage and the pure softening stage is Ts0c=2τpsinβL/β=10.03kN/m.


#### 6.3.3. Pure Softening Stage

The substitution of Equation (59) results in τsx=RfτptanβLsinβx+cosβx, i.e., ρ=RftanβLsinβLX+cosβLX.The critical tension of the pure softening stage and the softening-residual transition stage obtained by substitution of Equation (62) is Tsr0c=2τptanβL/β=7.16kN/m.


#### 6.3.4. Softening-Residual Transition Stage

Softening Zone (0.5≤X≤1).The substitution ofLr=L/2 into (64) results in τsx=τrtanβL/2sinβx−L/2+cosβx−L/2, i.e., ρ=RftanβL/2sinβLX−0.5+cosβLX−0.5.Residual Zone (0≤X≤0.5).The interface shear stress entering the residual zone isτrx=Rfτp, i.e., ρ=Rf=0.7.From Equation (70), the critical tension of the softening-residual transition stage and the pure residual stage can be calculated to be Ts0c=2τrL=7.06kN/m.


#### 6.3.5. Pure Residual Stage

In the pure residual stage, the interface shear stress isτrx=Rfτp, i.e., ρ=Rf=0.7.According to the above expressions, the evolution laws of the pull-out interface shear stress in the elastic stage, the elastic-hardening transition stage, and the pure hardening stage of strain-softening reinforcement can be obtained, as shown in Figure21. The graph shows the following.Figure 21
Evolution of the interfacial shear stress in the different stages of the pull-out process of the reinforced tailings (strain-softening reinforcement): (a) elastic stage; (b) elastic-softening transition stage; (c) pure softening stage; (d) softening-residual transition stage; and (e) pure residual stage.
(a)(b)(c)(d)(e)WhenT≤9.47kN/m, the interface is in the elastic stage, and the shear stress decreases nonlinearly from the pull-out end to the free end. At that time, the interface is in the critical stage of the elastic and elastic-softening transition, and the pull-out end reaches the peak shear stress of 16.8 kPa.When9.47kN/m≤T≤10.03kN/m, the interface is in the transition stage of elastic-softening, the shear stress first increases and then decreases, and the peak point is at the junction of the elastic zone and the softening zone. With increasing tension, the softening zone length increases, and the shear stress at the free end approaches the peak gradually. When T=7.16kN/m, the free end shear stress also reaches the peak value, and the interface will enter the stage of pure softening.When7.16kN/m≤T≤10.03kN/m, the interface is in the pure softening stage, and the shear stress increases nonlinearly from the pull-out end to the free end. When T=7.16kN/m, the shear stress at the pull-out end reaches the residual stress, and the interface will enter the softening-residual transition stage.When7.06kN/m≤T≤7.16kN/m, the interface is in the softening-residual transition stage. In this stage, the pull-out end begins to enter the residual state and gradually transits to the free end. When T=7.06kN/m, the free end also reaches the residual stress, and the interface will enter the pure residual stage.WhenT=7.06kN/m, the shear stress remains unchanged when the interface is in the pure residual stage.


### 6.4. Parametric Impact Analysis

The results show that the interface shear stress is mainly affected by the elastic shear stiffness (ks1) and the single-width tensile stiffness (Et) in the elastic stage. The distribution of the shear stress in the softening stage and the residual stage is related to the length of the softening zone (Ls), the softening shear stiffness (ks2), the residual interval length (Lr), the elastic shear stiffness (ks2), and the failure ratio Rf of the reinforcement. The calculated parameters are consistent with the previous section. The ratio of the softening zone length to reinforcement length is defined as η=Ls/L, and the ratio of residual zone length to the reinforcement length is defined as ξ=Lr/L.


#### 6.4.1. Elastic Stage

The variation in the interface shear stress of the strain-softening reinforcement in the elastic stage is consistent with that of the strain-hardening reinforcement at this stage, which has not been thoroughly described here.


#### 6.4.2. Elastic-Softening Transition Stage

The influence of the softening zone lengthLsη on the interface shear stress at different locations of the reinforcement in stage II is shown in Figure 22. It can be found from the graph that the interface shear stress in stage II increases first and then decreases from the pull-out end to the free end, and the peak value is at the critical point of the elastic zone and the softening zone in stage II. With an increase in η, the peak value of the interface shear stress gradually transfers to the free end.Figure 22
Effect of the interface frictional resistance at stage II onLs.As shown in Figure23, the effect of the softening shear stiffness (ks2) on the interface shear stress at different locations of the reinforcement in stage II is studied. It can be seen from the graph that ks2 only affects the change in the shear stress of the softening zone in stage II but has no effect on the shear stress of the elastic zone. The larger the ks2 is, the larger the range of the interface shear stress changes is, which indicates that the softening characteristics of the reinforcement are more obvious.Figure 23
Effect of the interface frictional resistance at stage II onks2.


#### 6.4.3. Pure Softening Stage

As shown in Figure24, the effect of the softening shear stiffness (ks2) on the interface shear stress at different locations of the steel bars in stage III was studied. It can be found from the graph that the interface shear stress increases nonlinearly from the pull-out end to the free end. The shear stress at the pull-out end is the smallest, and the shear stress at the free end is the largest. At this time, the shear stress at the pull-out end has been reduced to the residual stress.Figure 24
Effect of the interface frictional resistance at stage III onks2.


#### 6.4.4. Softening-Residual Transition Stage and Pure Residual Stage

As shown in Figure25, the effect of the softening shear stiffness (ks2) on the interface shear stress at different locations of the steel bars in stage IV is studied. It can be seen from the figure that ks2 only affects the change in the shear stress of the residual zone in stage IV, in which the softening zone decreases to the residual stress. The larger the ks2 is, the greater the nonlinear increase in the interface shear stress is.Figure 25
Effect of the interface frictional resistance at stage IV onks2.As shown in Figure26, the effect of the residual zone length Lrξ on the interface shear stress at different locations of the reinforcement in stage IV was studied. It can be seen from the graph that the interface shear stress is the smallest at the critical point between the softening zone and the residual zone in stage IV; at this time, the shear stress is the residual stress. The shear stress at the free end is the largest, which indicates that the interface friction of the reinforcement in the softening zone can more easily play a role, and the shear stress at the free end increases. With an increase in the residual zone length, the interface gradually transitions from the softening-residual stage to the pure residual stage. The distribution of the shear stress tends to be constant. The interface completely enters the residual stage, and the shear stress presents a horizontal distribution. That is, when the interface shear stress decreases to the residual stress, it will not change and will be in a plastic flow state, which is in accordance with the assumption.Figure 26
Effect of the interface frictional resistance at stage IV onLr.


## 6.1. Analysis of the Softening Pull-Out Interface

Basic assumption: the pull-out process of the strain-softening reinforcement under the pull-out load is divided into five stages [24]: the elastic stage, elastic-softening transition stage, pure softening stage, softening-residual transition stage, and pure residual stage, corresponding to stages I, II, III, IV, and V in Figure 18, respectively. The expressions of the interfacial tension, shear stress, and displacement at the different stages of pull-out can be obtained by theoretical calculations.Figure 18
The progressive pull-out process of strain-softening reinforcement in the pull-out model analysis: (a) elastic stage; (b) elastic-softening transition stage; (c) pure softening stage; (d) softening-residual transition stage; and (e) pure residual stage.
(a)(b)(c)(d)(e)


### 6.1.1. Elastic Stage (I Stage)

Similar to the distribution law of the strain-hardening reinforcement in the elastic stage, the expressions of the tension, shear stress, and displacement at this stage can be obtained by Equations (10) and (4a):(39)Tex=T01⋅sinhαL−xsinhαL,(40)τex=αT012⋅coshαL−xsinhαL,(41)uex=αT012ks1⋅coshαL−xsinhαL.In Equation (41), x = 0 can be substituted to obtain the transformation of the pull-out displacement ue0:(42)T01=2tanhαLαks1ue0,when ue0=up and τp=ks1ue0+τo, the critical tension Teh0c of the elastic and elastic-softening transition stages can be obtained from Equation (42), that is, the maximum tension in the elastic stage,(43)Tes0c=2τp−τotanhαLα.


### 6.1.2. Elastic-Softening Transition Stage (II Stage)

With increasing tension, the interface shear stress gradually transfers from the pull-out end to the tail until it reaches its peak value. Then, the plastic characteristics of the pull-out end begin to occur, and the strain-hardening phenomenon occurs and enters the II stage. The critical pointPx=Lh is defined to divide the elastic zone and the hardening zone. When 0≤x<Lh, the interface is in the stage II hardening zone, and when Lh<x≤L, the interface is in the stage II elastic zone (where Lh is the length of the hardening zone).(1)
Elastic Zone (Ls≤x≤L).The distributions of the interface tension, shear stress, and displacement in the stage II elastic zone are similar to those in the elastic stage:(44)Tex=T02⋅sinhαL−xsinhαL−Ls,(45)τex=αT022⋅coshαL−xsinhαL−Ls,(46)uex=αT022ks1⋅coshαL−xsinhαL−Ls,where T02 is the pull-out force of the transition point P.Considering that the interface shear stress of the transition pointQ1 is equal to the peak shear stress, the following can be obtained:(47)T02=2τptanhαL−Lsα.(2)
Softening Zone (0≤x≤Ls).The relationship between the interface shear stress and the displacement in the softening zone is defined by Equation (4b). The simultaneous Equations (10) and (4b) can be obtained as follows:(48)d2Tdx2+β2T=0,where β=−2ks2/Et>0.Solving Equation (48) can yield(49)Tsx=C3cosβx+C4sinβx,where Tsx is the tension of the reinforcement at the elastic stage and C3 and C4 are the integral constants.Consider the following boundary conditions:(50)Tsx=0=T03,Tsx=Ls=Tex=Ls.The substitution of Equation (47) into Equation (46) results in(51)C3=T03,C4=2τptanhαL−LsαsinβLs−T03cotβLs.Since the shear stress at the transition pointQ1 between the elastic zone and the softening zone is continuous, i.e., τex=Ls=τsx=Ls, it can be determined that T03 is as follows:(52)T03=2τpsinβLsβ+tanhαL−LsαcosβLs.The expressions of the interface tension, shear stress, and displacement in stage II can be obtained:(53)Tsx=2τpsinβLs−xβ+tanhαL−LsαcosβLs−x,(54)τsx=τpcosβLs−x−βtanhαL−LsαsinβLs−x,(55)usx=τpks2cosβLs−x−βtanhαL−LsαsinβLs−x−τpks2+up,when Ls=L, the critical pull-out force Ts0c of the elastic-softening transition stage and the pure softening stage can be obtained from Equation (52):(56)Ts0c=2τpsinβLβ.


### 6.1.3. Pure Softening Stage (III Stage)

Similar to the analysis of the stage II softening zone, Equation (50) is still applicable to the pure softening stage with the following boundary conditions:(57)Tsx=0=T04,Tsx=L=0.The solution of the pure softening stage is as follows:(58)Tsx=T04cosβx−cotβLsinβx,(59)τsx=βT042sinβx+cotβLcosβx,(60)usx=βT042ks2sinβx+cotβLcosβx−τpks2+up.Letx = 0 be substituted into Equation (60) to obtain the transformation of the pull-out displacement us0 at this stage:(61)T04=2tanβLβks2us0−up+τp.In the pure softening stage, the interface tension and shear stress decrease. Whenus0=ur and τr=ks2us0−up+τp, the critical pull-out force Tsr0c of the elastic-softening transition stage and the pure softening stage can be obtained from Equation (61); that is, the minimum pull-out force of the pure softening stage is(62)Tsr0c=2τrtanβLβ.


### 6.1.4. Softening-Residual Transition Stage (IV Stage)

When the pull-out force decreases toTsr0c, the pull-out end of the reinforcement begins to enter the residual state and gradually extends when the end of reinforcement enters stage IV. The transition point Q2x=Lr is defined to divide the softening zone and the residual zone. When 0≤x<Lr, the interface is in the residual zone of stage II, and when Lr<x≤L, the interface is in the softening zone of stage II (Lr is the residual zone length).(1)
Softening Zone (Ls≤x≤L).The distributions of the tension, shear stress, and displacement in the pure softening stage are similar to those in the softening-residual transition stage (x⟶x−Lr, L⟶L−Lr, and T04⟶T05). According to Equations (58)–(60), the relationship of the tension, shear stress, and displacement in the elastic zone of stage IV can be obtained:(63)Tsx=T05cosβx−Lr−cotβL−Lrsinβx−Lr,(64)τsx=βT052sinβx−Lr+cotβL−Lrcosβx−Lr,(65)usx=βT052ks2sinβx−Lr+cotβL−Lrcosβx−Lr−τpks2+up.Since the interface shear stress at the transition pointQ2 is equal to the residual shear stress τsx−Lr=τr, it can be obtained according to Equation (64):(66)T05=2τrtanβL−Lrβ.(2)
Residual Zone (0≤x≤Lr).The interface shear stress in the residual zone is equal to the residual shear strength, which can be obtained by Equations (10) and (4c):(67)dTdx=−2τr.Consider the boundary conditions:(68)Trx=0=T06,Trx=Lr=Tsx=Lr.It can be obtained as follows:(69)T06=2τrtanβL−Lrβ+2τrLr,(70)Trx=2τrLr−x+2τrtanβL−Lrβ,(71)urx=τrEtx2−2τrEtLrx+tanβL−Lrβx+C5,where(72)C5=τrLr2Et−2τrLrβEttanβL−Lr+τr−τpks2+up.The critical tension of the softening-residual transition stage and the pure residual stage is obtained atx=0, and the result of the substitution of Lr=L into Equation (69) is as follows:(73)Ts0c=2τrL.


### 6.1.5. Pure Residual Stage (V Stage)

The critical shear displacement at the pull-out end of the reinforcement is obtained by substitutingx=0 and Lr=L into Equation (71):(74)ur0c=τrL2Et+τr−τpks2+up.Assuming that the displacement of the reinforcement at the pull-out end isu0′, the distribution of the shear displacement can be derived as follows:(75)urx=τrEtx2−L2−2τrLEtx−L+τr−τpks2+up+u0′−ur0c=u0′+2τrLEtx2−2Lx.At this stage, the tensionTr0c and the interface shear stress τr remain constant. Therefore, the distribution of tension at different locations of the reinforcement is as follows:(76)Trx=2τrL−x.In summary, closed solutions are obtained for the five stages of the pull-out process of strain-hardening reinforcement. The critical tension and the initial tension between the two stages are shown in Tables5 and 6.Table 5
Critical tensile force between the two stages of the softening pull-out model.
Pull-out stageI-IIII-IIIIII-IVIV-VCritical tensile forceTes0c=2τp−τotanhαL/αTs0c=2τpsinβL/βTsr0c=2τrtanβL/βTs0c=2τrLTable 6
Initial tensile forces at different stages of the softening pull-out model.
Pull-out stageInitial tensile forcesIElastic stage:T01=2tanhαL/αks1ue0IIElastic zone:T02=2τptanhαL−Ls/α; softening zone: T03=2τpsinβLs/β+tanhαL−Ls/αcosβLsIIISoftening stage:T04=2tanβL/βks2us0−up+τpIVSoftening zone:T05=2τrtanβL−Lr/β; residual zone: T06=2τrtanβL−Lr/β+2τrLrV—


## 6.1.1. Elastic Stage (I Stage)

Similar to the distribution law of the strain-hardening reinforcement in the elastic stage, the expressions of the tension, shear stress, and displacement at this stage can be obtained by Equations (10) and (4a):(39)Tex=T01⋅sinhαL−xsinhαL,(40)τex=αT012⋅coshαL−xsinhαL,(41)uex=αT012ks1⋅coshαL−xsinhαL.In Equation (41), x = 0 can be substituted to obtain the transformation of the pull-out displacement ue0:(42)T01=2tanhαLαks1ue0,when ue0=up and τp=ks1ue0+τo, the critical tension Teh0c of the elastic and elastic-softening transition stages can be obtained from Equation (42), that is, the maximum tension in the elastic stage,(43)Tes0c=2τp−τotanhαLα.


## 6.1.2. Elastic-Softening Transition Stage (II Stage)

With increasing tension, the interface shear stress gradually transfers from the pull-out end to the tail until it reaches its peak value. Then, the plastic characteristics of the pull-out end begin to occur, and the strain-hardening phenomenon occurs and enters the II stage. The critical pointPx=Lh is defined to divide the elastic zone and the hardening zone. When 0≤x<Lh, the interface is in the stage II hardening zone, and when Lh<x≤L, the interface is in the stage II elastic zone (where Lh is the length of the hardening zone).(1)
Elastic Zone (Ls≤x≤L).The distributions of the interface tension, shear stress, and displacement in the stage II elastic zone are similar to those in the elastic stage:(44)Tex=T02⋅sinhαL−xsinhαL−Ls,(45)τex=αT022⋅coshαL−xsinhαL−Ls,(46)uex=αT022ks1⋅coshαL−xsinhαL−Ls,where T02 is the pull-out force of the transition point P.Considering that the interface shear stress of the transition pointQ1 is equal to the peak shear stress, the following can be obtained:(47)T02=2τptanhαL−Lsα.(2)
Softening Zone (0≤x≤Ls).The relationship between the interface shear stress and the displacement in the softening zone is defined by Equation (4b). The simultaneous Equations (10) and (4b) can be obtained as follows:(48)d2Tdx2+β2T=0,where β=−2ks2/Et>0.Solving Equation (48) can yield(49)Tsx=C3cosβx+C4sinβx,where Tsx is the tension of the reinforcement at the elastic stage and C3 and C4 are the integral constants.Consider the following boundary conditions:(50)Tsx=0=T03,Tsx=Ls=Tex=Ls.The substitution of Equation (47) into Equation (46) results in(51)C3=T03,C4=2τptanhαL−LsαsinβLs−T03cotβLs.Since the shear stress at the transition pointQ1 between the elastic zone and the softening zone is continuous, i.e., τex=Ls=τsx=Ls, it can be determined that T03 is as follows:(52)T03=2τpsinβLsβ+tanhαL−LsαcosβLs.The expressions of the interface tension, shear stress, and displacement in stage II can be obtained:(53)Tsx=2τpsinβLs−xβ+tanhαL−LsαcosβLs−x,(54)τsx=τpcosβLs−x−βtanhαL−LsαsinβLs−x,(55)usx=τpks2cosβLs−x−βtanhαL−LsαsinβLs−x−τpks2+up,when Ls=L, the critical pull-out force Ts0c of the elastic-softening transition stage and the pure softening stage can be obtained from Equation (52):(56)Ts0c=2τpsinβLβ.


## 6.1.3. Pure Softening Stage (III Stage)

Similar to the analysis of the stage II softening zone, Equation (50) is still applicable to the pure softening stage with the following boundary conditions:(57)Tsx=0=T04,Tsx=L=0.The solution of the pure softening stage is as follows:(58)Tsx=T04cosβx−cotβLsinβx,(59)τsx=βT042sinβx+cotβLcosβx,(60)usx=βT042ks2sinβx+cotβLcosβx−τpks2+up.Letx = 0 be substituted into Equation (60) to obtain the transformation of the pull-out displacement us0 at this stage:(61)T04=2tanβLβks2us0−up+τp.In the pure softening stage, the interface tension and shear stress decrease. Whenus0=ur and τr=ks2us0−up+τp, the critical pull-out force Tsr0c of the elastic-softening transition stage and the pure softening stage can be obtained from Equation (61); that is, the minimum pull-out force of the pure softening stage is(62)Tsr0c=2τrtanβLβ.


## 6.1.4. Softening-Residual Transition Stage (IV Stage)

When the pull-out force decreases toTsr0c, the pull-out end of the reinforcement begins to enter the residual state and gradually extends when the end of reinforcement enters stage IV. The transition point Q2x=Lr is defined to divide the softening zone and the residual zone. When 0≤x<Lr, the interface is in the residual zone of stage II, and when Lr<x≤L, the interface is in the softening zone of stage II (Lr is the residual zone length).(1)
Softening Zone (Ls≤x≤L).The distributions of the tension, shear stress, and displacement in the pure softening stage are similar to those in the softening-residual transition stage (x⟶x−Lr, L⟶L−Lr, and T04⟶T05). According to Equations (58)–(60), the relationship of the tension, shear stress, and displacement in the elastic zone of stage IV can be obtained:(63)Tsx=T05cosβx−Lr−cotβL−Lrsinβx−Lr,(64)τsx=βT052sinβx−Lr+cotβL−Lrcosβx−Lr,(65)usx=βT052ks2sinβx−Lr+cotβL−Lrcosβx−Lr−τpks2+up.Since the interface shear stress at the transition pointQ2 is equal to the residual shear stress τsx−Lr=τr, it can be obtained according to Equation (64):(66)T05=2τrtanβL−Lrβ.(2)
Residual Zone (0≤x≤Lr).The interface shear stress in the residual zone is equal to the residual shear strength, which can be obtained by Equations (10) and (4c):(67)dTdx=−2τr.Consider the boundary conditions:(68)Trx=0=T06,Trx=Lr=Tsx=Lr.It can be obtained as follows:(69)T06=2τrtanβL−Lrβ+2τrLr,(70)Trx=2τrLr−x+2τrtanβL−Lrβ,(71)urx=τrEtx2−2τrEtLrx+tanβL−Lrβx+C5,where(72)C5=τrLr2Et−2τrLrβEttanβL−Lr+τr−τpks2+up.The critical tension of the softening-residual transition stage and the pure residual stage is obtained atx=0, and the result of the substitution of Lr=L into Equation (69) is as follows:(73)Ts0c=2τrL.


## 6.1.5. Pure Residual Stage (V Stage)

The critical shear displacement at the pull-out end of the reinforcement is obtained by substitutingx=0 and Lr=L into Equation (71):(74)ur0c=τrL2Et+τr−τpks2+up.Assuming that the displacement of the reinforcement at the pull-out end isu0′, the distribution of the shear displacement can be derived as follows:(75)urx=τrEtx2−L2−2τrLEtx−L+τr−τpks2+up+u0′−ur0c=u0′+2τrLEtx2−2Lx.At this stage, the tensionTr0c and the interface shear stress τr remain constant. Therefore, the distribution of tension at different locations of the reinforcement is as follows:(76)Trx=2τrL−x.In summary, closed solutions are obtained for the five stages of the pull-out process of strain-hardening reinforcement. The critical tension and the initial tension between the two stages are shown in Tables5 and 6.Table 5
Critical tensile force between the two stages of the softening pull-out model.
Pull-out stageI-IIII-IIIIII-IVIV-VCritical tensile forceTes0c=2τp−τotanhαL/αTs0c=2τpsinβL/βTsr0c=2τrtanβL/βTs0c=2τrLTable 6
Initial tensile forces at different stages of the softening pull-out model.
Pull-out stageInitial tensile forcesIElastic stage:T01=2tanhαL/αks1ue0IIElastic zone:T02=2τptanhαL−Ls/α; softening zone: T03=2τpsinβLs/β+tanhαL−Ls/αcosβLsIIISoftening stage:T04=2tanβL/βks2us0−up+τpIVSoftening zone:T05=2τrtanβL−Lr/β; residual zone: T06=2τrtanβL−Lr/β+2τrLrV—


## 6.2. Verification of the Softening Model

To verify the elastic-plastic theoretical model of strain softening, the pull-out test results between geotextiles and tailings mentioned above are used for simulation, and the predicted results are shown in Figure19. The predicted results are in good agreement with the pull-out test results, and the model can effectively describe the pull-out behaviour of geotextiles in the tailings. Moreover, the displacements of the II and IV stages are relatively smaller than those of the I, III, and V stages, so the two transitional stages can be neglected in the analysis of the pull-out behaviour of the geotextiles in the tailings.Figure 19
Comparison between the pull-out test results and model-predicted results of geotextiles under different normal stresses.
(a)(b)(c)(d)In addition, to better understand the five pull-out stages of the geotextiles in the tailings, the relationship between the interface shear stiffness (elastic shear stiffnessks1 and softening shear stiffness ks2) and the initial shear stress with the normal stress is given, as shown in Figure 20. The graph shows that ks1 and ks2 increase linearly with an increase in the normal stress, and the initial shear stress is linearly related to the normal stress. The initial shear strength index can be obtained by the Mohr–Coulomb strength criterion.Figure 20
The relationship between the normal stress and the shear stiffness of the softening model: (a) shear stiffness and (b) initial shear stress.
(a)(b)


## 6.3. Distribution of the Interface Shear Stress Values

To visually reflect the stress evolution law of strain-softening reinforcement at different stages of the pull-out interface, the interface shear stress distribution at three stages was analysed. The model parameters selected according to the above test results are shown in Table7. To simplify the analysis, the model parameters were normalized. The normalized reinforcement position is X=x/L, and the normalized interface shear stress is ρ=τ/τp.Table 7
Softening model parameters.
σn (kPa)L (m)t (mm)E (GPa)ks1 (MPa/m)ks2 (MPa/m)fRf300.3212.190−0.4500.5600.7The parametersτp=σnf=16.80kPa, τr=Rfτp=11.76kPa, α=2ks1/Et=1.48, and β=−2ks2/Et=0.67 are calculated. According to the model parameters given, the following formulas are substituted:


### 6.3.1. Elastic Stage

The substitution of Equation (40) can yield τex=τpcoshαL−x/coshαL, i.e., ρ=coshα1−X/coshαL.The substitution of Equation (43) shows that the critical tension of the elastic stage and the elastic-softening transition stage is Tes0c=2τptanhαL/α=9.47kN/m.


### 6.3.2. Elastic-Softening Transition Stage

Elastic Zone (0.5≤X≤1).LetLs=L/2 in Equation (45) produce τex=τpcoshαL−x/coshαL/2, i.e., ρ=coshαL1−X/coshαL/2.Softening Zone (0≤X≤0.5).The substitution ofLs=L/2 into Equation (54) results in τsx=τpcosβL/2−x−βtanhαL/2/αsinβL/2−x, i.e., ρ=cosβL0.5−X−βtanhαL/2/αsinβL0.5−X.From Equation (56), the critical tension of the elastic-softening transition stage and the pure softening stage is Ts0c=2τpsinβL/β=10.03kN/m.


### 6.3.3. Pure Softening Stage

The substitution of Equation (59) results in τsx=RfτptanβLsinβx+cosβx, i.e., ρ=RftanβLsinβLX+cosβLX.The critical tension of the pure softening stage and the softening-residual transition stage obtained by substitution of Equation (62) is Tsr0c=2τptanβL/β=7.16kN/m.


### 6.3.4. Softening-Residual Transition Stage

Softening Zone (0.5≤X≤1).The substitution ofLr=L/2 into (64) results in τsx=τrtanβL/2sinβx−L/2+cosβx−L/2, i.e., ρ=RftanβL/2sinβLX−0.5+cosβLX−0.5.Residual Zone (0≤X≤0.5).The interface shear stress entering the residual zone isτrx=Rfτp, i.e., ρ=Rf=0.7.From Equation (70), the critical tension of the softening-residual transition stage and the pure residual stage can be calculated to be Ts0c=2τrL=7.06kN/m.


### 6.3.5. Pure Residual Stage

In the pure residual stage, the interface shear stress isτrx=Rfτp, i.e., ρ=Rf=0.7.According to the above expressions, the evolution laws of the pull-out interface shear stress in the elastic stage, the elastic-hardening transition stage, and the pure hardening stage of strain-softening reinforcement can be obtained, as shown in Figure21. The graph shows the following.Figure 21
Evolution of the interfacial shear stress in the different stages of the pull-out process of the reinforced tailings (strain-softening reinforcement): (a) elastic stage; (b) elastic-softening transition stage; (c) pure softening stage; (d) softening-residual transition stage; and (e) pure residual stage.
(a)(b)(c)(d)(e)WhenT≤9.47kN/m, the interface is in the elastic stage, and the shear stress decreases nonlinearly from the pull-out end to the free end. At that time, the interface is in the critical stage of the elastic and elastic-softening transition, and the pull-out end reaches the peak shear stress of 16.8 kPa.When9.47kN/m≤T≤10.03kN/m, the interface is in the transition stage of elastic-softening, the shear stress first increases and then decreases, and the peak point is at the junction of the elastic zone and the softening zone. With increasing tension, the softening zone length increases, and the shear stress at the free end approaches the peak gradually. When T=7.16kN/m, the free end shear stress also reaches the peak value, and the interface will enter the stage of pure softening.When7.16kN/m≤T≤10.03kN/m, the interface is in the pure softening stage, and the shear stress increases nonlinearly from the pull-out end to the free end. When T=7.16kN/m, the shear stress at the pull-out end reaches the residual stress, and the interface will enter the softening-residual transition stage.When7.06kN/m≤T≤7.16kN/m, the interface is in the softening-residual transition stage. In this stage, the pull-out end begins to enter the residual state and gradually transits to the free end. When T=7.06kN/m, the free end also reaches the residual stress, and the interface will enter the pure residual stage.WhenT=7.06kN/m, the shear stress remains unchanged when the interface is in the pure residual stage.


## 6.3.1. Elastic Stage

The substitution of Equation (40) can yield τex=τpcoshαL−x/coshαL, i.e., ρ=coshα1−X/coshαL.The substitution of Equation (43) shows that the critical tension of the elastic stage and the elastic-softening transition stage is Tes0c=2τptanhαL/α=9.47kN/m.


## 6.3.2. Elastic-Softening Transition Stage

Elastic Zone (0.5≤X≤1).LetLs=L/2 in Equation (45) produce τex=τpcoshαL−x/coshαL/2, i.e., ρ=coshαL1−X/coshαL/2.Softening Zone (0≤X≤0.5).The substitution ofLs=L/2 into Equation (54) results in τsx=τpcosβL/2−x−βtanhαL/2/αsinβL/2−x, i.e., ρ=cosβL0.5−X−βtanhαL/2/αsinβL0.5−X.From Equation (56), the critical tension of the elastic-softening transition stage and the pure softening stage is Ts0c=2τpsinβL/β=10.03kN/m.


## 6.3.3. Pure Softening Stage

The substitution of Equation (59) results in τsx=RfτptanβLsinβx+cosβx, i.e., ρ=RftanβLsinβLX+cosβLX.The critical tension of the pure softening stage and the softening-residual transition stage obtained by substitution of Equation (62) is Tsr0c=2τptanβL/β=7.16kN/m.


## 6.3.4. Softening-Residual Transition Stage

Softening Zone (0.5≤X≤1).The substitution ofLr=L/2 into (64) results in τsx=τrtanβL/2sinβx−L/2+cosβx−L/2, i.e., ρ=RftanβL/2sinβLX−0.5+cosβLX−0.5.Residual Zone (0≤X≤0.5).The interface shear stress entering the residual zone isτrx=Rfτp, i.e., ρ=Rf=0.7.From Equation (70), the critical tension of the softening-residual transition stage and the pure residual stage can be calculated to be Ts0c=2τrL=7.06kN/m.


## 6.3.5. Pure Residual Stage

In the pure residual stage, the interface shear stress isτrx=Rfτp, i.e., ρ=Rf=0.7.According to the above expressions, the evolution laws of the pull-out interface shear stress in the elastic stage, the elastic-hardening transition stage, and the pure hardening stage of strain-softening reinforcement can be obtained, as shown in Figure21. The graph shows the following.Figure 21
Evolution of the interfacial shear stress in the different stages of the pull-out process of the reinforced tailings (strain-softening reinforcement): (a) elastic stage; (b) elastic-softening transition stage; (c) pure softening stage; (d) softening-residual transition stage; and (e) pure residual stage.
(a)(b)(c)(d)(e)WhenT≤9.47kN/m, the interface is in the elastic stage, and the shear stress decreases nonlinearly from the pull-out end to the free end. At that time, the interface is in the critical stage of the elastic and elastic-softening transition, and the pull-out end reaches the peak shear stress of 16.8 kPa.When9.47kN/m≤T≤10.03kN/m, the interface is in the transition stage of elastic-softening, the shear stress first increases and then decreases, and the peak point is at the junction of the elastic zone and the softening zone. With increasing tension, the softening zone length increases, and the shear stress at the free end approaches the peak gradually. When T=7.16kN/m, the free end shear stress also reaches the peak value, and the interface will enter the stage of pure softening.When7.16kN/m≤T≤10.03kN/m, the interface is in the pure softening stage, and the shear stress increases nonlinearly from the pull-out end to the free end. When T=7.16kN/m, the shear stress at the pull-out end reaches the residual stress, and the interface will enter the softening-residual transition stage.When7.06kN/m≤T≤7.16kN/m, the interface is in the softening-residual transition stage. In this stage, the pull-out end begins to enter the residual state and gradually transits to the free end. When T=7.06kN/m, the free end also reaches the residual stress, and the interface will enter the pure residual stage.WhenT=7.06kN/m, the shear stress remains unchanged when the interface is in the pure residual stage.


## 6.4. Parametric Impact Analysis

The results show that the interface shear stress is mainly affected by the elastic shear stiffness (ks1) and the single-width tensile stiffness (Et) in the elastic stage. The distribution of the shear stress in the softening stage and the residual stage is related to the length of the softening zone (Ls), the softening shear stiffness (ks2), the residual interval length (Lr), the elastic shear stiffness (ks2), and the failure ratio Rf of the reinforcement. The calculated parameters are consistent with the previous section. The ratio of the softening zone length to reinforcement length is defined as η=Ls/L, and the ratio of residual zone length to the reinforcement length is defined as ξ=Lr/L.


### 6.4.1. Elastic Stage

The variation in the interface shear stress of the strain-softening reinforcement in the elastic stage is consistent with that of the strain-hardening reinforcement at this stage, which has not been thoroughly described here.


### 6.4.2. Elastic-Softening Transition Stage

The influence of the softening zone lengthLsη on the interface shear stress at different locations of the reinforcement in stage II is shown in Figure 22. It can be found from the graph that the interface shear stress in stage II increases first and then decreases from the pull-out end to the free end, and the peak value is at the critical point of the elastic zone and the softening zone in stage II. With an increase in η, the peak value of the interface shear stress gradually transfers to the free end.Figure 22
Effect of the interface frictional resistance at stage II onLs.As shown in Figure23, the effect of the softening shear stiffness (ks2) on the interface shear stress at different locations of the reinforcement in stage II is studied. It can be seen from the graph that ks2 only affects the change in the shear stress of the softening zone in stage II but has no effect on the shear stress of the elastic zone. The larger the ks2 is, the larger the range of the interface shear stress changes is, which indicates that the softening characteristics of the reinforcement are more obvious.Figure 23
Effect of the interface frictional resistance at stage II onks2.


### 6.4.3. Pure Softening Stage

As shown in Figure24, the effect of the softening shear stiffness (ks2) on the interface shear stress at different locations of the steel bars in stage III was studied. It can be found from the graph that the interface shear stress increases nonlinearly from the pull-out end to the free end. The shear stress at the pull-out end is the smallest, and the shear stress at the free end is the largest. At this time, the shear stress at the pull-out end has been reduced to the residual stress.Figure 24
Effect of the interface frictional resistance at stage III onks2.


### 6.4.4. Softening-Residual Transition Stage and Pure Residual Stage

As shown in Figure25, the effect of the softening shear stiffness (ks2) on the interface shear stress at different locations of the steel bars in stage IV is studied. It can be seen from the figure that ks2 only affects the change in the shear stress of the residual zone in stage IV, in which the softening zone decreases to the residual stress. The larger the ks2 is, the greater the nonlinear increase in the interface shear stress is.Figure 25
Effect of the interface frictional resistance at stage IV onks2.As shown in Figure26, the effect of the residual zone length Lrξ on the interface shear stress at different locations of the reinforcement in stage IV was studied. It can be seen from the graph that the interface shear stress is the smallest at the critical point between the softening zone and the residual zone in stage IV; at this time, the shear stress is the residual stress. The shear stress at the free end is the largest, which indicates that the interface friction of the reinforcement in the softening zone can more easily play a role, and the shear stress at the free end increases. With an increase in the residual zone length, the interface gradually transitions from the softening-residual stage to the pure residual stage. The distribution of the shear stress tends to be constant. The interface completely enters the residual stage, and the shear stress presents a horizontal distribution. That is, when the interface shear stress decreases to the residual stress, it will not change and will be in a plastic flow state, which is in accordance with the assumption.Figure 26
Effect of the interface frictional resistance at stage IV onLr.


## 6.4.1. Elastic Stage

The variation in the interface shear stress of the strain-softening reinforcement in the elastic stage is consistent with that of the strain-hardening reinforcement at this stage, which has not been thoroughly described here.


## 6.4.2. Elastic-Softening Transition Stage

The influence of the softening zone lengthLsη on the interface shear stress at different locations of the reinforcement in stage II is shown in Figure 22. It can be found from the graph that the interface shear stress in stage II increases first and then decreases from the pull-out end to the free end, and the peak value is at the critical point of the elastic zone and the softening zone in stage II. With an increase in η, the peak value of the interface shear stress gradually transfers to the free end.Figure 22
Effect of the interface frictional resistance at stage II onLs.As shown in Figure23, the effect of the softening shear stiffness (ks2) on the interface shear stress at different locations of the reinforcement in stage II is studied. It can be seen from the graph that ks2 only affects the change in the shear stress of the softening zone in stage II but has no effect on the shear stress of the elastic zone. The larger the ks2 is, the larger the range of the interface shear stress changes is, which indicates that the softening characteristics of the reinforcement are more obvious.Figure 23
Effect of the interface frictional resistance at stage II onks2.


## 6.4.3. Pure Softening Stage

As shown in Figure24, the effect of the softening shear stiffness (ks2) on the interface shear stress at different locations of the steel bars in stage III was studied. It can be found from the graph that the interface shear stress increases nonlinearly from the pull-out end to the free end. The shear stress at the pull-out end is the smallest, and the shear stress at the free end is the largest. At this time, the shear stress at the pull-out end has been reduced to the residual stress.Figure 24
Effect of the interface frictional resistance at stage III onks2.


## 6.4.4. Softening-Residual Transition Stage and Pure Residual Stage

As shown in Figure25, the effect of the softening shear stiffness (ks2) on the interface shear stress at different locations of the steel bars in stage IV is studied. It can be seen from the figure that ks2 only affects the change in the shear stress of the residual zone in stage IV, in which the softening zone decreases to the residual stress. The larger the ks2 is, the greater the nonlinear increase in the interface shear stress is.Figure 25
Effect of the interface frictional resistance at stage IV onks2.As shown in Figure26, the effect of the residual zone length Lrξ on the interface shear stress at different locations of the reinforcement in stage IV was studied. It can be seen from the graph that the interface shear stress is the smallest at the critical point between the softening zone and the residual zone in stage IV; at this time, the shear stress is the residual stress. The shear stress at the free end is the largest, which indicates that the interface friction of the reinforcement in the softening zone can more easily play a role, and the shear stress at the free end increases. With an increase in the residual zone length, the interface gradually transitions from the softening-residual stage to the pure residual stage. The distribution of the shear stress tends to be constant. The interface completely enters the residual stage, and the shear stress presents a horizontal distribution. That is, when the interface shear stress decreases to the residual stress, it will not change and will be in a plastic flow state, which is in accordance with the assumption.Figure 26
Effect of the interface frictional resistance at stage IV onLr.


## 7. Conclusions

(1)
The pull-out test curves of the geogrids and tailings show strain-hardening characteristics, and the pull-out test curves of the geotextiles and tailings show strain-softening characteristics. In addition, both kinds of pull-out curves require a certain pull-out force at the beginning of pull-out because the reinforcement requires a certain pull-out force to resist the friction of the interface between the reinforcement and tailings.(2)
Considering that the initial pull-out force is required in the initial pull-out process, a bilinear shear stress-displacement hardening model and a trilinear shear stress-displacement softening model are proposed. The pull-out process of the hardening reinforcement is divided into the elastic stage, the elastic-hardening transition stage, and the pure hardening stage. The pull-out process of softening reinforcement is divided into the elastic stage, elastic-softening transition stage, pure softening stage, softening-residual transition stage, and pure residual stage. Based on the interface basic governing equation, the analytical solutions of the interface tension, shear stress, and displacement at different stages under the pull-out load are derived, which satisfactorily reflect the progressive failure of the reinforcement-tailing interface.(3)
The predicted results of the two models are basically in agreement with the pull-out test data, which verifies the validity of the proposed two elastic-plastic models for the progressive failure analysis of the reinforcement at the pull-out interface; in the pull-out process, the general transition stage is not obvious. The new linear shear stress-displacement hardening and softening model is simple, easy to calculate, and has good applicability and can be used in the study of the characteristics of pull-out interfaces.(4)
In the pull-out process of the reinforcement, when the reinforcement is in the elastic stage, the nonlinearity and maximum value of the interface shear stress increase with an increase in the elastic shear stiffness, while the tensile stiffness is the opposite. When the reinforcement is in the hardening stage or the softening stage, the larger the hardening (softening) shear stiffness is, the larger the change in the range of the shear stress is and the more obvious the hardening (softening) characteristics of the reinforcement are.



---

*Source: 5680521-2020-06-13.xml*