Research Article  Open Access
Analysis of the ElasticPlastic Theoretical Model of the PullOut Interface between Geosynthetics and Tailings
Abstract
Aiming at the strainhardening and strainsoftening phenomena between geosynthetics and tailings during pullout tests, bilinear and trilinear shear stressdisplacement softening models were proposed. The pullout process of the hardening reinforcement was divided into the elastic stage, elastichardening transition stage, and pure hardening stage. The pullout process of the softened reinforcement was divided into the elastic stage, elasticsoftening transition stage, pure softening stage, softeningresidual transition stage, and pure residual stage. The expressions of the interface tension, shear stress, and displacement at the different stages under a pullout load were derived through the interface basic control equation. At the same time, the evolution law of the interface shear stress at different pullout stages was analysed, and the predicted results of the two elasticplastic 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 elasticplastic models for the progressive failure analysis of reinforcement at the pullout interface. During the process of pullout, 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 reinforcementtailing interfaces with different strain types and provide theoretical support for the study of the interface characteristics of geosyntheticreinforced tailings.
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 pullout curve of unidirectional geogrids tends to be strain softening, while the pullout 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 pullout test. However, in the analysis of the stability and interface interaction between the reinforcement and soil, the pullout test [4] is more suitable because the stress and displacement in the pullout test are gradually transferred from the pullout 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 pullout interface model of retractable bars based on the elasticplastic shear stressdisplacement. 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 threestage elasticplastic shear stressdisplacement 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 pullout behaviour, it fails to consider the interface progressive failure characteristics at different stages of the whole pullout process.
To truly describe the progressive failure characteristics of the reinforcementsoil interface during different stages of the pullout process, scholars worldwide have proposed some calculation models for the strainsoftening characteristics of reinforcements. For example, Hong et al. [19] used the elasticplastic theory model to study the progressive failure of the soil nailsoil interface under a pullout state. Zhu et al. [20] and Chen et al. [21] deduced the analytical expressions of the axial force and shear stress at the reinforcementsoil interface at different pullout stages through a threeparameter model. In the study of the interface characteristics of reinforcementsoil, there are few models for calculating the strainhardening 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 pullout curves between geosynthetics and tailings are investigated (hereafter referred to as strainhardening reinforcement and strainsoftening reinforcement). It is proposed that the hardening reinforcement undergoes three continuous stages during the pullout process. Softening reinforcement goes through five continuous stages during the pullout process. The distributions of the tension, shear stress, and displacement of two kinds of reinforcement at different pullout stages are deduced. Moreover, the accuracy of the proposed two elasticplastic pullout models is verified by analysing and comparing the pullout test results of two reinforcements. The evolution of the interface shear stress at different pullout stages and its influencing factors are further studied.
2. Analysis of the PullOut Test
2.1. Test Instrument
The test instrument was a strippeddown YT1200 geosynthetic direct shear and pullout system (Howard Nanjing Soil Instrument Manufacturing Company). The system mainly consisted of a test box (direct shear and pullout), vertical loading system, horizontal loading system, and data acquisition system. The test equipment is shown in Figure 1. The inner dimensions of the pullout 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 realtime monitoring of the test data for analysis or to stop the system in case of problems.
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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/cm^{3}, and the water content was 3.75%. The physical properties of the tailings were as follows: the effective particle size d_{10} = 0.10 mm, the median particle size d_{30} = 0.19 mm, and the restricted particle size d_{60} = 0.30 mm. The calculations showed that the unevenness coefficient of the tailings C_{u} was 3.5, and the curvature coefficient of the tailings C_{c} 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.
As shown in Figure 3, the geosynthetics used in the test are shortfibre needlepunched 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.

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 strength and the residual shear strength can be obtained by the following calculation:where and are the peak shear strength and residual shear strength, kPa, respectively; and are the maximum pullout force and residual pullout force, kN, respectively; and is the area of the geosynthetics embedded in the direct shear or pullout test box, which is the area of the shear plane in the test process, .
The pullout 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 coefficient , which is generally calculated as the ratio of the interface friction strength to the corresponding normal stress:where is the normal stress, in which is the additional stress, is the bulk density of the filler, and is the paving height of the filler (see Figure 4).
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 stratifiedcompacted to ensure that each group had the same density. We simultaneously applied lubricating oil evenly on both sides of the pullout test box to reduce the size effect during testing. The shear rate of the pullout 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 E502012)” [22].
2.5. Analysis of the Test Results
The pullout test results of the tailings and geogrids or geotextiles under different normal stresses are shown in Figure 5. In the pullout test curve of the geogrids, the pullout force increases with an increase in the pullout displacement, but the rate of increase slows gradually, showing the characteristics of strain hardening as a whole. However, in the pullout test curve of the geotextiles, the pullout force rapidly reaches its peak value with an increase in the pullout displacement and then decreases obviously, showing strainsoftening characteristics. In addition, both kinds of pullout curves require a certain pulling force at the beginning of pullout because the reinforcement requires a certain pulling force to resist the friction of the interface between the reinforcement and tailings.
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According to formula (2), the relationship between the pullout 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.
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3. ElasticPlastic Model of the ReinforcementTailing Interface
3.1. StrainHardening Model of the Reinforcement
From the pullout test results between geotextiles and tailings in Figure 5(a), it can be seen that the interface shear stress and the displacement curve of the strainhardened reinforcement can be approximately expressed as an elastic relationship before reaching the peak value and then as strainhardening characteristics. In this paper, the curve form is simplified to a bilinear shear stressdisplacement 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, and are the slopes of OA and AB, which are also called the elastic interface shear stiffness and hardening shear stiffness, respectively; is the initial shear stress, and its value is the ratio of the initial pullout force to the shear area, i.e., ; is the peak shear stress of the interface; and is the corresponding pullout displacement, where . 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 [18], i.e., , which generally has a value between 0.5 and 1.0:
3.2. StrainSoftening Model of the Reinforcement
According to the test results between the geogrid and tailings shown in Figure 5(b), the shear stress and the displacement curve of the strainsoftening 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 threestage linear shear stressdisplacement 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, is the slope of OA (the elastic shear stiffness), with , where is the peak shear stress of the interface and is the corresponding displacement; is the initial shear stress, , where is the initial pullout force; and is the slope of AB (softening shear stiffness) , where is the residual shear stress of the interface and is the corresponding displacement. The analogical damage ratio defines the ratio between and as , that is, :
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 pullout test curves of bidirectional plastic geogrids and different fillers (clay and sand) are generally strain hardening. Yang et al. [25] found that the pullout 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 geotextilereinforced tailings. The main reason for strain hardening of geogridreinforced tailings is the blocking effect of the transverse rib of the geogrid. The strainsoftening phenomenon of geotextilereinforced tailings is due to the fact that when the pullout force reaches the peak value with the increase of the pullout displacement, the whole reinforcement undergoes sliding and strainsoftening characteristics appear.
4. Fundamental Equation of the ReinforcementSoil Interface
The pullout test diagram of the reinforcement is shown in Figure 4. 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 and , respectively, and the tensile modulus is . When the shear stress at x is , the microelement with length 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 thatwhere T is the tensile force per unit width of the reinforcement at x, is the deformation length of the microelement, and is the strain.
Equation (5) can be rewritten as follows:
The strain of reinforcement at x can be written as follows: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.,
From Equations (5)–(8), we can obtain the following:
In general, the actual strain in the pullout process is very small [25], which can be neglected. Therefore, Equation (9) is approximately expressed as follows:
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. PullOut Interface Analysis of the StrainHardening Reinforcement
5.1. Analysis of the Hardening PullOut Interface
Basic assumption: according to the definition of the theoretical model of the strainhardening reinforcement, it is considered that the interface between the reinforcement and tailings will undergo an elastic stage, an elastichardening transition stage, and a pure hardening stage under a pullout load, corresponding to stages I, II, and III in Figure 9, respectively. Through theoretical calculations, the expressions of the tension, shear stress, and displacement at each stage of pullout can be obtained.
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5.1.1. Elastic Stage (I Stage)
When , the shear stress and displacement are elastic, which satisfies the equation . The governing equation of this stage can be obtained by the simultaneous Equations (10) and (3a):where .
Solving Equation (11), the following can be obtained:where is the tension of the reinforcement at the elastic stage and and are the integral constants.
In the pullout test, when the tension at the pullout end, i.e., x = 0, is and the tension at the free end, i.e., x = L, is 0, the following boundary conditions exist:
By substituting the boundary condition (13) in Equation (12), we obtain
The tension expression of stage I can be obtained as follows:
According to Equations (10) and (3a), the corresponding shear stress and displacement relations can be obtained:
The transformation of the pullout displacement can be obtained by substituting x = 0 into Equation (17):
When and , from Equation (18), the critical tension of the elastic stage and the elastichardening transition stage, i.e., the maximum tension in the elastic stage, can be obtained as follows:
5.1.2. ElasticHardening Transition Stage (II Stage)
With increasing tension, the interface shear stress gradually transfers from the pullout end to the tail until it reaches its peak value. Then, the plastic characteristics of the pullout end begin to occur, and the strainhardening phenomenon occurs and enters the II stage. The critical point P () is defined to divide the elastic zone and the hardening zone. When , the interface is in the stage II hardening zone, and when , the interface is in the stage II elastic zone (where is the length of the hardening zone).(1)Elastic Zone ().
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 obtainwhere is the tension of the transition point P.
Considering that the interface shear stress at the transition point P is equal to the peak shear stress, the following result can be obtained:(2)Hardening Zone ().
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:where .
Solving Equation (24) can yieldwhere is the tension of reinforcement at the hardening zone of stage II and and are the integral constants.
Consider the boundary conditions as follows:
By substituting the boundary condition (26) in Equation (25), we can obtain
The expressions of the tension, shear stress, and displacement in stage II can be obtained as follows:
Since the shear stress at the transition point P between the elastic zone and the hardening zone is continuous, i.e., , the following can be obtained:
When , the critical tension between the elastichardening transition stage and the pure hardening stage can be obtained by Equation (31):
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:
The expressions of the interface tension, shear stress, and displacement at the pure hardening stage are as follows:
Let x = 0 be substituted into Equation (36) to obtain the transformation of the pullout displacement at this stage:
In the pure hardening stage, the interface pullout force and shear stress at the pullout end all increase. When the shear stress at the pullout end increases to the ultimate stress of the reinforcement, the reinforcement is destroyed. Then, , combined with , so Equation (37) can be written as follows:
In summary, closed solutions are obtained for the three stages of the pullout process of strainhardening reinforcement. The critical tension and the initial tension between the two stages are shown in Tables 2 and 3.


5.2. Verification of the Hardening Model
To verify the elasticplastic theoretical model of strain hardening, the pullout test results between geogrids and tailings mentioned above are used for the simulation, and the predicted results are shown in Figure 10. According to the graph, the prediction results are in good agreement with the pullout test results, and the model can effectively describe the progressive pullout 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 pullout behaviour of geogrids in the tailings.
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Furthermore, to better understand the five pullout stages of geogrids in the tailings, the relationship between the interface shear stiffness (elastic shear stiffness K_{s1} and hardening shear stiffness K_{s2}) and the initial shear stress with the normal stress is given, as shown in Figure 11. With an increase in the normal stress, K_{s1} and K_{s2} 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.
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5.3. Distribution of the Interface Shear Stress
To visually reflect the stress evolution law of strainhardening reinforcement at different stages of the pullout interface, the interface shear stress distribution at three stages was analysed. The model parameters selected according to the above test results are shown in Table 4. To simplify the analysis, the model parameters were normalized. The normalized reinforcement position was , and the normalized interface shear stress was .

The parameters , , and were calculated. According to the model parameters given, the following formulas were substituted.
5.3.1. Elastic Stage
Substitution of Equation (16) can yield , i.e., .
The substitution of Equation (19) indicates that the critical tension of the elastic stage and the elastichardening transition stage is .
5.3.2. ElasticHardening Transition Stage
(i)Elastic Zone ().
The substitution of into Equation (64) yields , i.e., .(ii)Hardening Zone ().
The substitution of into Equation (29) results in , i.e., .
From Equation (32), the critical tension of the elastichardening transition stage and the pure hardening stage is .(iii)Pure Hardening Stage.
The substitution of Equation (35) results in , i.e., .
The substitution of Equation (38) shows that the ultimate tensile force at the stage of pure hardening is .
According to the above expression of the interface shear stress, the evolution law of the shear stress in the three stages of elastic, elastichardening transition, and pure hardening of the strainhardening reinforcement can be obtained, as shown in Figure 12.
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When , the interface is in the elastic stage, and the shear stress decreases nonlinearly from the pullout end to the free end. When , the interface is in the critical stage of elastic and pure hardening, and the peak shear stress reaches 20 kPa at the pullout end.
When , the interface is in the transition stage of elastichardening, in which the shear stress at the pullout end reaches its peak value and then drives the free end to continue to increase. The shear stress decreases from the pullout end to the free end. When , the free end also reaches its peak value, and the interface will enter the stage of pure hardening.
When , the interface is in the stage of pure hardening, and the shear stress at the pullout end continues to increase until , when the shear stress at the pullout 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 (K_{s1}) 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 (L_{h}) and the hardening shear stiffness (K_{s2}). 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 .
5.4.1. Elastic Stage
The effect of the elastic shear stiffness (K_{s1}) 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 pullout end to the free end. The maximum shear stress is the same, which is the peak shear stress. The larger the K_{s1} is, the more obvious the nonlinearity of the interface shear stress curve is.
The effect of the singlewidth 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 K_{s1} ().
5.4.2. ElasticHardening Transition Stage
The influence of the hardening zone length 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 elastichardening transition stage decreases nonlinearly in both the elastic zone and the hardening zone. The maximum shear stress is at the pullout 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.
The effect of the hardening shear stiffness (K_{s2}) 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 K_{s2} 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 K_{s2} is, the larger the range of the shear stress curve in the hardening zone is.
5.4.3. Pure Hardening Stage
The effect of the hardening shear stiffness (K_{s2}) 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 K_{s2} is, the greater the sag of the interface shear stress curve in stage III is. In practice, K_{s2} is very small. In the elastichardening transition stage, the shear stress at the pullout end should be slightly larger than the peak shear stress and then slowly decrease until the shear stress at the pullout end reaches the ultimate shear stress and the reinforcement is damaged.
6. PullOut Interface Analysis of the StrainSoftening Reinforcement
6.1. Analysis of the Softening PullOut Interface
Basic assumption: the pullout process of the strainsoftening reinforcement under the pullout load is divided into five stages [24]: the elastic stage, elasticsoftening transition stage, pure softening stage, softeningresidual 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 pullout can be obtained by theoretical calculations.
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6.1.1. Elastic Stage (I Stage)
Similar to the distribution law of the strainhardening 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):
In Equation (41), x = 0 can be substituted to obtain the transformation of the pullout displacement :when and , the critical tension of the elastic and elasticsoftening transition stages can be obtained from Equation (42), that is, the maximum tension in the elastic stage,
6.1.2. ElasticSoftening Transition Stage (II Stage)
With increasing tension, the interface shear stress gradually transfers from the pullout end to the tail until it reaches its peak value. Then, the plastic characteristics of the pullout end begin to occur, and the strainhardening phenomenon occurs and enters the II stage. The critical point is defined to divide the elastic zone and the hardening zone. When , the interface is in the stage II hardening zone, and when , the interface is in the stage II elastic zone (where is the length of the hardening zone).(1)Elastic Zone ().
The distributions of the interface tension, shear stress, and displacement in the stage II elastic zone are similar to those in the elastic stage:where is the pullout force of the transition point .
Considering that the interface shear stress of the transition point Q_{1} is equal to the peak shear stress, the following can be obtained:(2)Softening Zone ().
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:where .
Solving Equation (48) can yieldwhere is the tension of the reinforcement at the elastic stage and and are the integral constants.
Consider the following boundary conditions:
The substitution of Equation (47) into Equation (46) results in
Since the shear stress at the transition point Q_{1} between the elastic zone and the softening zone is continuous, i.e., , it can be determined that is as follows:
The expressions of the interface tension, shear stress, and displacement in stage II can be obtained:when , the critical pullout force of the elasticsoftening transition stage and the pure softening stage can be obtained from Equation (52):
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:
The solution of the pure softening stage is as follows:
Let x = 0 be substituted into Equation (60) to obtain the transformation of the pullout displacement at this stage:
In the pure softening stage, the interface tension and shear stress decrease. When and , the critical pullout force of the elasticsoftening transition stage and the pure softening stage can be obtained from Equation (61); that is, the minimum pullout force of the pure softening stage is
6.1.4. SofteningResidual Transition Stage (IV Stage)
When the pullout force decreases to , the pullout end of the reinforcement begins to enter the residual state and gradually extends when the end of reinforcement enters stage IV. The transition point is defined to divide the softening zone and the residual zone. When , the interface is in the residual zone of stage II, and when , the interface is in the softening zone of stage II ( is the residual zone length).(1)Softening Zone ().
The distributions of the tension, shear stress, and displacement in the pure softening stage are similar to those in the softeningresidual transition stage (, , and ). According to Equations (58)–(60), the relationship of the tension, shear stress, and displacement in the elastic zone of stage IV can be obtained:
Since the interface shear stress at the transition point Q_{2} is equal to the residual shear stress , it can be obtained according to Equation (64):(2)Residual Zone ().
The interface shear stress in the residual zone is equal to the residual shear strength, which can be obtained by Equations (10) and (4c):
Consider the boundary conditions:
It can be obtained as follows:where
The critical tension of the softeningresidual transition stage and the pure residual stage is obtained at , and the result of the substitution of into Equation (69) is as follows:
6.1.5. Pure Residual Stage (V Stage)
The critical shear displacement at the pullout end of the reinforcement is obtained by substituting and into Equation (71):
Assuming that the displacement of the reinforcement at the pullout end is , the distribution of the shear displacement can be derived as follows:
At this stage, the tension and the interface shear stress remain constant. Therefore, the distribution of tension at different locations of the reinforcement is as follows:
In summary, closed solutions are obtained for the five stages of the pullout process of strainhardening reinforcement. The critical tension and the initial tension between the two stages are shown in Tables 5 and 6.


6.2. Verification of the Softening Model
To verify the elasticplastic theoretical model of strain softening, the pullout test results between geotextiles and tailings mentioned above are used for simulation, and the predicted results are shown in Figure 19. The predicted results are in good agreement with the pullout test results, and the model can effectively describe the pullout 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 pullout behaviour of the geotextiles in the tailings.
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In addition, to better understand the five pullout stages of the geotextiles in the tailings, the relationship between the interface shear stiffness (elastic shear stiffness k_{s1} and softening shear stiffness k_{s2}) and the initial shear stress with the normal stress is given, as shown in Figure 20. The graph shows that k_{s1} and k_{s2} 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.
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6.3. Distribution of the Interface Shear Stress Values
To visually reflect the stress evolution law of strainsoftening reinforcement at different stages of the pullout interface, the interface shear stress distribution at three stages was analysed. The model parameters selected according to the above test results are shown in Table 7. To simplify the analysis, the model parameters were normalized. The normalized reinforcement position is , and the normalized interface shear stress is .

The parameters , , , and 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 , i.e., .
The substitution of Equation (43) shows that the critical tension of the elastic stage and the elasticsoftening transition stage is .
6.3.2. ElasticSoftening Transition Stage
Elastic Zone ().
Let in Equation (45) produce , i.e., . Softening Zone ().
The substitution of into Equation (54) results in , i.e., .
From Equation (56), the critical tension of the elasticsoftening transition stage and the pure softening stage is .
6.3.3. Pure Softening Stage
The substitution of Equation (59) results in , i.e., .
The critical tension of the pure softening stage and the softeningresidual transition stage obtained by substitution of Equation (62) is .
6.3.4. SofteningResidual Transition Stage
Softening Zone ().
The substitution of into (64) results in , i.e., . Residual Zone ().
The interface shear stress entering the residual zone is , i.e., .
From Equation (70), the critical tension of the softeningresidual transition stage and the pure residual stage can be calculated to be .
6.3.5. Pure Residual Stage
In the pure residual stage, the interface shear stress is , i.e., .
According to the above expressions, the evolution laws of the pullout interface shear stress in the elastic stage, the elastichardening transition stage, and the pure hardening stage of strainsoftening reinforcement can be obtained, as shown in Figure 21. The graph shows the following.
(a)
(b)
(c)
(d)
(e)
When , the interface is in the elastic stage, and the shear stress decreases nonlinearly from the pullout end to the free end. At that time, the interface is in the critical stage of the elastic and elasticsoftening transition, and the pullout end reaches the peak shear stress of 16.8 kPa.
When , the interface is in the transition stage of elasticsoftening, 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 , the free end shear stress also reaches the peak value, and the interface will enter the stage of pure softening.
When , the interface is in the pure softening stage, and the shear stress increases nonlinearly from the pullout end to the free end. When , the shear stress at the pullout end reaches the residual stress, and the interface will enter the softeningresidual transition stage.
When , the interface is in the softeningresidual transition stage. In this stage, the pullout end begins to enter the residual state and gradually transits to the free end. When , the free end also reaches the residual stress, and the interface will enter the pure residual stage.
When , 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 (k_{s1}) and the singlewidth 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 (L_{s}), the softening shear stiffness (k_{s2}), the residual interval length (L_{r}), the elastic shear stiffness (k_{s2}), and the failure ratio R_{f} 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 , and the ratio of residual zone length to the reinforcement length is defined as .
6.4.1. Elastic Stage
The variation in the interface shear stress of the strainsoftening reinforcement in the elastic stage is consistent with that of the strainhardening reinforcement at this stage, which has not been thoroughly described here.
6.4.2. ElasticSoftening Transition Stage
The influence of the softening zone length 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 pullout 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 , the peak value of the interface shear stress gradually transfers to the free end.
As shown in Figure 23, the effect of the softening shear stiffness (k_{s2}) on the interface shear stress at different locations of the reinforcement in stage II is studied. It can be seen from the graph that k_{s2} 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 k_{s2} is, the larger the range of the interface shear stress changes is, which indicates that the softening characteristics of the reinforcement are more obvious.
6.4.3. Pure Softening Stage
As shown in Figure 24, the effect of the softening shear stiffness (k_{s2}) 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 pullout end to the free end. The shear stress at the pullout end is the smallest, and the shear stress at the free end is the largest. At this time, the shear stress at the pullout end has been reduced to the residual stress.
6.4.4. SofteningResidual Transition Stage and Pure Residual Stage
As shown in Figure 25, the effect of the softening shear stiffness (k_{s2}) 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 k_{s2} 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 k_{s2} is, the greater the nonlinear increase in the interface shear stress is.
As shown in Figure 26, the effect of the residual zone length 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 softeningresidual 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.
7. Conclusions
(1)The pullout test curves of the geogrids and tailings show strainhardening characteristics, and the pullout test curves of the geotextiles and tailings show strainsoftening characteristics. In addition, both kinds of pullout curves require a certain pullout force at the beginning of pullout because the reinforcement requires a certain pullout force to resist the friction of the interface between the reinforcement and tailings.(2)Considering that the initial pullout force is required in the initial pullout process, a bilinear shear stressdisplacement hardening model and a trilinear shear stressdisplacement softening model are proposed. The pullout process of the hardening reinforcement is divided into the elastic stage, the elastichardening transition stage, and the pure hardening stage. The pullout process of softening reinforcement is divided into the elastic stage, elasticsoftening transition stage, pure softening stage, softeningresidual 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 pullout load are derived, which satisfactorily reflect the progressive failure of the reinforcementtailing interface.(3)The predicted results of the two models are basically in agreement with the pullout test data, which verifies the validity of the proposed two elasticplastic models for the progressive failure analysis of the reinforcement at the pullout interface; in the pullout process, the general transition stage is not obvious. The new linear shear stressdisplacement hardening and softening model is simple, easy to calculate, and has good applicability and can be used in the study of the characteristics of pullout interfaces.(4)In the pullout 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.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The research was supported by the National Natural Science Foundation of China (51774163) and the Scientific Research Project of Angang Group (2018, KeA19).
References
 G. L. S. Babu, A. Sridharan, and K. K. Babu, “Composite reinforcement for reinforced soil applications,” Journal of the Japanese Geotechnical Society of Soils and Foundations, vol. 43, no. 2, pp. 123–128, 2003. View at: Publisher Site  Google Scholar
 T. Nakamura, T. Mitachi, and I. Ikeura, “Estimating method for the insoil deformation behavior of geogrid Based on the results of direct box shear test,” Journal of the Japanese Geotechnical Society of Soils and Foundations, vol. 43, no. 1, pp. 47–57, 2003. View at: Publisher Site  Google Scholar
 D. D. Shi, W. B. Liu, and W. H. Shui, “Comparative experimental studies of interface characteristics between uniaxial/biaxial plastic geogrids and different soils,” Rock and Soil Mechanics, vol. 30, no. 8, pp. 2237–2244, 2009. View at: Google Scholar
 G. Q. Yang, G. X. Li, and B. J. Zhang, “Experimental studies on interface friction characteristics of geogrids,” Chinese Journal of Geotechnical Engineering, vol. 28, no. 8, pp. 948–952, 2006. View at: Google Scholar
 A. Sawicki, “Modelling of geosynthetic reinforcement in soil retaining walls,” Geosynthetics International, vol. 5, no. 3, pp. 327–345, 1998. View at: Publisher Site  Google Scholar
 M. Abramento and A. J. Whittle, “Analysis of pullout tests for planar reinforcements in soil,” Journal of Geotechnical Engineering, vol. 121, no. 6, pp. 476–485, 1995. View at: Publisher Site  Google Scholar
 Z. Yuan, “Pullout response of geosynthetic in soil—theoretical analysis,” in Proceedings of the GeoFrontiers Congress, Dallas, TX, USA, March 2011. View at: Publisher Site  Google Scholar
 N. Gurung, “1D analytical solution for extensible and inextensible soil/rock reinforcement in pullout tests,” Geotextiles and Geomembranes, vol. 19, no. 4, pp. 195–212, 2001. View at: Publisher Site  Google Scholar
 C. Huang, H. Hsieh, and Y. Hsieh, “Hyperbolic models for a 2D backfill and reinforcement pullout,” Geosynthetics Internationa, vol. 21, no. 3, pp. 168–178, 2014. View at: Publisher Site  Google Scholar
 E. M. Palmeira, “Bearing force mobilisation in pullout tests on geogrids,” Geotextiles & Geomembranes, vol. 22, no. 6, pp. 481–509, 2004. View at: Publisher Site  Google Scholar
 S. Sobhi and J. T. H. Wu, “An interface pullout formula for extensible sheet reinforcement,” Geosynthetics International, vol. 3, no. 5, pp. 565–582, 1996. View at: Publisher Site  Google Scholar
 P. V. Long, D. T. Bergado, A. S. Balasubramaniam et al., “Interaction between soil and geotextile reinforcement,” American Society of Civil Engineers, vol. 69, Geotechnical Special Publication, 1997. View at: Google Scholar
 N. Gurung, Y. Iwao, and M. R. Madhav, “Pullout test model for extensible reinforcement,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 23, no. 12, pp. 1337–1348, 1999. View at: Publisher Site  Google Scholar
 A. Misra, C. H. Chen, R. Oberoi et al., “Simplified analysis method for micropile pullout behavior,” Journal of Geotechnical and Geoenvironmental Engineering, vol. 130, no. 10, pp. 1024–1033, 2004. View at: Publisher Site  Google Scholar
 N. Gurung, “A theoretical model for anchored geosynthetics in pullout tests,” Geosynthetics International, vol. 7, no. 3, pp. 269–284, 2000. View at: Publisher Site  Google Scholar
 J. J. B. Esterhuizen, G. M. Filz, and J. M. Duncan, “Constitutive behavior of geosynthetic interfaces,” Journal of Geotechnical and Geoenvironmental Engineering, vol. 127, no. 10, pp. 834–840, 2001. View at: Publisher Site  Google Scholar
 W. A. Lin, B. Zhu, Y. M. Chen et al., “Tension analysis of geomembrane in landfill slope considering interface strainsoftening,” Rock and Soil Mechanics, vol. 29, no. 8, pp. 2063–2069, 2008. View at: Google Scholar
 P. Zhang, J. H. Wang, and J. J. Chen, “Mechanical behavior of the interface between reinforcement and soil in pullout tests on geotextiles,” Journal of Shanghai Jiaotong University, vol. 38, no. 6, pp. 999–1002, 2004. View at: Google Scholar
 C. Y. Hong, J. H. Yin, W. H. Zhou, and H.F. Pei, “Analytical study on progressive pullout behavior of a soil nail,” Journal of Geotechnical and Geoenvironmental Engineering, vol. 138, no. 4, pp. 500–507, 2012. View at: Publisher Site  Google Scholar
 H. H. Zhu, C. C. Zhang, C. S. Tang et al., “Modeling the pullout behavior of short fiber in reinforced soil,” Geotextiles and Geomembranes, vol. 42, no. 4, pp. 329–338, 2014. View at: Publisher Site  Google Scholar
 J. Chen, S. Saydam, and P. C. Hagan, “An analytical model of the load transfer behavior of fully grouted cable bolts,” Construction and Building Materials, vol. 101, no. 1, pp. 1006–1015, 2015. View at: Publisher Site  Google Scholar
 Ministry of Transport of the People’s Republic of China, “Test models of geosynthetics for highway engineering, JTG E502006,” Communication Press, Beijing, China, 2006. View at: Google Scholar
 M. R. Madhav, N. Gurung, and Y. Iwao, “A theoretical model for pullout response of extensible reinforcements,” Geosynthetics International, vol. 5, no. 4, pp. 399–424, 1998. View at: Publisher Site  Google Scholar
 F. W. Lai, L. P. Li, and F. Q. Chen, “Elasticexponential softening model for behavior of interface between geogrid reinforcement and soil against pullout,” Journal of Engineering Geology, vol. 26, no. 4, pp. 852–860, 2018. View at: Google Scholar
 M. Yang, N. Li, X. X. Liu et al., “Experimental research on interface frictional behaviors of the geotextilereinforced soil,” Journal of Xi’an University of Technology, vol. 32, no. 1, pp. 46–51, 2016. View at: Google Scholar
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Copyright © 2020 Changbo Du and Fu Yi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.