# Pullout Characteristics and Damage Softening Model of the Geogrid-Soil Interface

Authors: Xiaoyong Liang; Jing Jin; Guangqing Yang; Xizhao Wang; Yitao Zhou
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

The mechanical properties of geogrid-soil interface is very important in design and stability analysis of reinforced soil structure. In order to study the complicated mechanism of geogrid-soil interface, a series of pullout tests of HDPE uniaxial tensile geogrids with the different transverse ribs spacing is used to investigate the interaction characteristics in the laboratory. The test results show that pullout force and displacement curves are characterized as strain softening; compared with the no-reinforced case, the case reinforced with geogrid has larger cohesion and lower friction angles. The ductility of soil is enhanced due to geogrid reinforcement. Based on the basic control equations of the interface and damage theory, trilinear shear stress-displacement damage softening model is proposed to describe the strain-softening characteristics of geogrid-soil interface. Analytical solutions of interface tension, shear stress, and displacement at different stages are derived considering strain softening based on damage, and the development of shear stress and progressive failure mode of the geogrid-soil interface at different pullout stages is revealed. Furthermore, the proposed model is verified by experimental results.


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## Body

## 1. Introduction

As an important reinforced material, geogrid has been widely used in various geotechnical engineering, such as improving the structural stability of retaining wall, improving the bearing capacity of foundation, and dealing with the differential settlement between the old and the new subgrade. The reinforcement of geogrid is realized by the interaction between geogrid and surrounding soil. Therefore, the behavior of reinforcement soil interface is an important factor affecting the reinforcement mechanism of geogrid. There are still some deficiencies in the design method of geogrid reinforced soil structure. The main problem is that the understanding of the mechanism of reinforcement soil interface is not comprehensive enough. Scholars have carried out a lot of relevant experimental research work [1], such as direct shear test [2–5], triaxial compression test [6, 7], biaxial compression test [8], and pullout test [9]. The geogrid pullout test has been regarded as a simple and direct way to investigate the geogrid-soil interface behavior among all the experimental approaches.Yang et al. [10] carried out pullout tests of geogrids with different stiffnesses in gravel and clay and discussed the influence of geogrid stiffness and soil type on the friction characteristics of reinforcement soil interface. Wang et al. [11] studied the influence of various test factors on the pullout behavior of reinforced soil interface. Xiao et al. [12] analyzed the influence of interface normal stress and cohesive water content on the interface interaction characteristics in the pullout test. Cao et al. and Zheng et al. [13, 14] took three-dimensional geogrid as the research object and, through the indoor pullout test, studied the interaction characteristics of reinforcement soil interface and the deformation of geogrid. Jin [15] studied the influence of geogrid transverse rib spacing on the pullout test results. Abdi et al. [16] analyzed the interface parameters between geogrid and various kinds of filling through the pullout test. Wang et al. [17] analyzed the relationship between maximum pullout force and displacement through the pullout test and Fourier series approximation method. Alagiyawanna et al. [18] studied the contribution and effect of transverse rib and longitudinal rib on pullout resistance through the number of longitudinal rib and transverse rib in the pullout test. The research results of Moraci et al. [19] show that the passive resistance of transverse rib bears more than 75% of the pullout force. Cardile et al. [20] proposed a method to determine the pullout force of geogrids in sand considering the interaction between transverse ribs of geogrids.At present, the pullout tests usually measure the pullout force and displacement and then the relationship between the normal stress and the relative displacement by calculation, but the progressive yield mechanism of interface is not considered. Baker et al. [21] studied the progressive mechanism of interfacial friction in the pullout test. Nakamura et al.’s research results [22] show that there is deformation of reinforcement in the drawing test, and the interface friction resistance gradually develops. Ma et al. [23] divided the exertion of interfacial friction resistance in the pullout test into initial stage, local yield stage, and complete yield stage. Current research result shows that there are less theoretical studies on the interaction mechanism between reinforcement and soil. Gurung et al. [24] studied the 1-D expression of shear stress and displacement that was capable to analyze small strain cases. Filz et al. [25] drew the softening curve of shear stress shear displacement and obtained the plastic displacement softening model and plastic work softening model by hyperbolic fitting. Zhu et al. and Zhang et al. [26, 27] used a three-parameter model to simulate the progressive failure characteristics of fibers at the interface of fiber reinforced soil. Li et al. [28] obtained the analytical solutions of the pullout force, displacement, and shear stress of the reinforcement material without considering the softening characteristics of the reinforced soil interface. Cheng et al. [29] studied the shear behavior of geogrid-soil interface and its statistical damage softening model.In this paper, the pullout characteristics of the interface between sand and geogrid are studied by the large-scale indoor pullout test. On this basis, the damage theory is introduced to establish the damage softening model which can describe the softening characteristics of geogrid-soil interface. The pullout process of strain-softening reinforcement is divided into elastic stage, softening-elastic stage, pure softening stage, residual-softening stage, and residual stage. Based on the basic control equation and damage theory of geogrid-soil interface, the expressions of interface tension, shear stress, and displacement at different stages of reinforcement of strain softening, under pullout load, are derived, respectively, which reflect the progressive failure of the pullout interface between reinforcement and soil. Finally, the proposed model is verified by the test results. The research results provide technical reference for reasonable selection of reinforcement materials in reinforced soil engineering.


## 2. Pullout Test of Geogrid

### 2.1. Test Materials

The reinforcement material is unidirectional tensile HDPE plastic geogrid, and width and thickness of the geogrid sample are 30 cm and 6 mm, respectively, as shown in Figure1. The spacing between transverse ribs of geogrid used in this test is 35 cm and 50 cm, respectively. The technical indexes of geogrid are shown in Table 1. The physical and mechanical indexes of sand selected for filling are obtained through the particle screening test, relative density test, and direct shear test, as shown in Table 2. Coarse sand with good grading is used for filling, and the grain grading curve is shown in Figure 2.Figure 1
Uniaxial geogrids.Table 1
The technical indexes of geogrid.
Transverse rib spacing (cm)Tensile strength (kN/m)Tensile strength at 2% strain (kN/m)Tensile strength at 5% strain (kN/m)Peak strain (%)35183.266.76136.917.2450175.058.63109.868.93Table 2
The physical and mechanical indexes of soil.
Specific gravityExperimental relationship (g/cm3)Maximum dry density (g/cm3)Void ratioNonuniform coefficient (Cu)Curvature coefficient (Cc)Cohesion (kPa)Internal friction angle (o)2.651.421.830.745.311.42126.5Figure 2
The soil grading curve.


### 2.2. Test Scheme

The pullout equipment was mainly composed of four parts: pullout box, normal loading system, horizontal loading system, and data acquisition system. The pullout box was made of 10 mm thick steel plate and was 600 mm in length, 400 mm in width, and 500 mm in height in internal dimensions, as shown in Figure3. The test filling height is 50 cm, which is compacted in four layers, and compaction degree is taken as 90%. The normal stresses, 50 kPa, 100 kPa, 150 kPa, and 200 kPa, are applied to the test chamber filled with coarse sand. According to the Test Methods of Geosynthetics for Highway Engineering (JTG E50-2006) [30] and considering the large mesh size of geogrid, the pullout rate during the test is 1 mm/min, so as to ensure that the geogrid will not produce torsion during the test. When the pullout force is no longer increased or continues to decrease, stop the test. In the test, lubricant was applied on the inner surface of the four walls of the box to reduce the friction between the box wall and the soil. The width of the sample was controlled to ensure a certain distance between the geogrid and the side wall of the box, so as to minimize the influence of boundary effect and size effect.Figure 3
The pullout test equipment.


### 2.3. Test Results

The relationship curve of pullout force and displacement with different transverse rib spacing under different normal stress is shown in Figure4. It can be seen that with the increase of pullout displacement, pullout curve shows strain-softening trend after reaching the peak value. In the softening process, with the continuous increase of pullout displacement, pullout force gradually tends to a stable value, that is to say, the residual strength value. The larger the normal stress, the larger the displacement corresponding to the maximum drawing force.Figure 4
The curve between pullout displacement and force of different transverse rib spacings. (a)s = 35 cm. (b) s = 50 cm.
(a)(b)The reason is that the interface progressive yield mechanism gradually plays a role with the increase of pullout displacement. With the increase of normal stress, the stronger the restraint effect of geogrid-soil interface is, the slower the progressive exertion process of geogrid-soil interface load is, and more displacement is required when the pullout force reaches the peak value.According to the Mohr–Coulomb strength criterion, the maximum shear stress and normal stress of geogrid-soil interface are fitted, as shown in Figure5. It can be seen that the maximum shear stress at the interface increases linearly with the increase of normal stress. The parameters of peak shear strength at the geogrid-soil interface are obtained from the fitting results, as shown in Table 3. For the transverse rib spacing of 35 cm, the cohesion and friction angle of geogrid-soil interface are 13.55 kPa and 4.23°, respectively. For the transverse rib spacing of 50 cm, the cohesion and friction angle of geogrid-soil interface are 10.35 kPa and 4.11°, respectively. However, cohesion and friction angle of pure sand are 1 kPa and 26.5°, respectively. The reinforcement of geogrid improves the cohesion of soil, but the friction angle decreases.Figure 5
Relationship between interfacial shear strength and normal stress.Table 3
Strength parameters of geogrid-soil interface.
Transverse rib spacing s (cm)Cohesionc (kPa)Friction angleφ (°)3513.554.235010.354.11


## 2.1. Test Materials

The reinforcement material is unidirectional tensile HDPE plastic geogrid, and width and thickness of the geogrid sample are 30 cm and 6 mm, respectively, as shown in Figure1. The spacing between transverse ribs of geogrid used in this test is 35 cm and 50 cm, respectively. The technical indexes of geogrid are shown in Table 1. The physical and mechanical indexes of sand selected for filling are obtained through the particle screening test, relative density test, and direct shear test, as shown in Table 2. Coarse sand with good grading is used for filling, and the grain grading curve is shown in Figure 2.Figure 1
Uniaxial geogrids.Table 1
The technical indexes of geogrid.
Transverse rib spacing (cm)Tensile strength (kN/m)Tensile strength at 2% strain (kN/m)Tensile strength at 5% strain (kN/m)Peak strain (%)35183.266.76136.917.2450175.058.63109.868.93Table 2
The physical and mechanical indexes of soil.
Specific gravityExperimental relationship (g/cm3)Maximum dry density (g/cm3)Void ratioNonuniform coefficient (Cu)Curvature coefficient (Cc)Cohesion (kPa)Internal friction angle (o)2.651.421.830.745.311.42126.5Figure 2
The soil grading curve.


## 2.2. Test Scheme

The pullout equipment was mainly composed of four parts: pullout box, normal loading system, horizontal loading system, and data acquisition system. The pullout box was made of 10 mm thick steel plate and was 600 mm in length, 400 mm in width, and 500 mm in height in internal dimensions, as shown in Figure3. The test filling height is 50 cm, which is compacted in four layers, and compaction degree is taken as 90%. The normal stresses, 50 kPa, 100 kPa, 150 kPa, and 200 kPa, are applied to the test chamber filled with coarse sand. According to the Test Methods of Geosynthetics for Highway Engineering (JTG E50-2006) [30] and considering the large mesh size of geogrid, the pullout rate during the test is 1 mm/min, so as to ensure that the geogrid will not produce torsion during the test. When the pullout force is no longer increased or continues to decrease, stop the test. In the test, lubricant was applied on the inner surface of the four walls of the box to reduce the friction between the box wall and the soil. The width of the sample was controlled to ensure a certain distance between the geogrid and the side wall of the box, so as to minimize the influence of boundary effect and size effect.Figure 3
The pullout test equipment.


## 2.3. Test Results

The relationship curve of pullout force and displacement with different transverse rib spacing under different normal stress is shown in Figure4. It can be seen that with the increase of pullout displacement, pullout curve shows strain-softening trend after reaching the peak value. In the softening process, with the continuous increase of pullout displacement, pullout force gradually tends to a stable value, that is to say, the residual strength value. The larger the normal stress, the larger the displacement corresponding to the maximum drawing force.Figure 4
The curve between pullout displacement and force of different transverse rib spacings. (a)s = 35 cm. (b) s = 50 cm.
(a)(b)The reason is that the interface progressive yield mechanism gradually plays a role with the increase of pullout displacement. With the increase of normal stress, the stronger the restraint effect of geogrid-soil interface is, the slower the progressive exertion process of geogrid-soil interface load is, and more displacement is required when the pullout force reaches the peak value.According to the Mohr–Coulomb strength criterion, the maximum shear stress and normal stress of geogrid-soil interface are fitted, as shown in Figure5. It can be seen that the maximum shear stress at the interface increases linearly with the increase of normal stress. The parameters of peak shear strength at the geogrid-soil interface are obtained from the fitting results, as shown in Table 3. For the transverse rib spacing of 35 cm, the cohesion and friction angle of geogrid-soil interface are 13.55 kPa and 4.23°, respectively. For the transverse rib spacing of 50 cm, the cohesion and friction angle of geogrid-soil interface are 10.35 kPa and 4.11°, respectively. However, cohesion and friction angle of pure sand are 1 kPa and 26.5°, respectively. The reinforcement of geogrid improves the cohesion of soil, but the friction angle decreases.Figure 5
Relationship between interfacial shear strength and normal stress.Table 3
Strength parameters of geogrid-soil interface.
Transverse rib spacing s (cm)Cohesionc (kPa)Friction angleφ (°)3513.554.235010.354.11


## 3. Mechanical Analysis of Geogrid-Soil Interface

During the pullout test, the geogrid is placed in the middle of the upper and lower test chambers. The top of the upper box is applied with vertical compressive stress, and geogrid is pulled out slowly. The schematic diagram of the pullout process is shown in Figure6. Geogrid is assumed to be an axially loaded tension member; when a pullout force F is applied, shear stresses are mobilized at the geogrid-soil interface to resist the pullout force. The microelement with length of x is taken along the length direction of geogrid; according to the force balance and physical and geometric equations,(1)dFx+2τxdx=0,(2)Fx=Erεx=−Erduxdx,where x = arbitrary coordinate as shown in Figure 6; F(x) = pullout force at point x; ux  = displacement at point x; εx  = tensile strain at point x; τx  = interface shear stress at an arbitrary point x; Er  = tensile stiffness, Er=Et; E is elastic modulus of geogrid; and t is thickness of geogrid.Figure 6
Schematic illustration of pullout mechanism of geogrid in soil.According to (1) and (2), the basic equation of load transfer can be expressed as(3)Erd2uxdx2−2τxdx=0.In the pullout test of geogrid, the pullout forceF at pulling end can be measured by the sensor, and the force at free end can be ignored. The boundary condition of the pullout force at both ends of the geogrid can be expressed as(4)F0=−Erduxdx|x=0=F,Fl=−Erduxdx|x=l=0.In the process of pullout, it is necessary to establish the constitutive model of the geogrid-soil interface to solveux and τx in (3).


## 4. Damage Softening Model of Geogrid-Soil Interface under Pullout Load

A trilinear softening model is adopted to quantify the shear stress-displacement relationship of the geogrid-soil interface, as depicted in Figure7. In this model, the geogrid-soil interface first behaves elastically, which is characterized by an ascending branch up to the peak shear resistance. Afterward, stress softening emerges at the of geogrid-soil interface. Once the interfacial shear stress decreases to the residual shear resistance, the shear stress remains constant.Figure 7
Strain-softening model of geogrid-soil interface. (a) Curve relationship of shear stress-displacement. (b) Trilinear softening model.
(a)(b)In order to consider the strain-softening constitutive model of geogrid-soil interface with damage, according to Rabotnov damage factorD, the geogrid-soil interface under pullout force is assumed to be composed of damaged and undamaged parts, as shown in Figure 7. The damage factor D can be expressed as the ratio of the damage length to the original length, which can be expressed as(5)D=d″xdx,(6)τxdx=τ′xd′x+τ″xd″x,where dx  = initial original length of geogrid-soil interface; d′x  = undamaged part length; d″x  = damaged part length; τx  = apparent shearing stress at point x; τ′x  = shear stress of undamaged length; and τ″x  = shear stress of damaged length.According to (5) and (6), expression can be expressed as(7)τx=τ′x1−D+τ″xD.It can be seen from formula (7) that the damage factor variable D represents the damage degree of the geogrid-soil interface in the process of pullout, and the damage factor D satisfies 0≤D≤1.It can be seen from Figure8 that the shear strength of geogrid-soil interface reaches the peak shear stress τ1; afterward, the strength decreases until it reaches the stable residual strength τ2. In the damage model, the shear stress of damaged length is not zero and still bear the shear stress of τ2, that is τ″=τ2; formula (7) can be expressed as(8)τ=τ′1−D+τ2D.Figure 8
Micro-stress analysis of geogrid-soil interface.For the undamaged part, the shear stressτ′x still obeys the linear part of the initial interface shear stress-displacement, and formula (8) can be expressed as(9)τ=Gu1−D+τ2D.According to Lemaitre's strain equivalent principle [31], the constitutive relation of damage state is the same as undamaged state. Under the effect of damage, the relationship of shear stress-displacement is obtained by multiplying the damage factor variable D. For 0 ≤ u ≤ u1, the effect of damage is ignored; for u1 ≤ u ≤ u2, the damage factor variable satisfies 0 < D < 1; for u ≥ u2, the damage factor variable satisfies D = 1.For trilinear shear stress-displacement softening model, relationship between shear stressτ(x) and shear displacement u(x) can be expressed by(10)τx=Gu,0≤u<u1,a,D2τ1−Gux,u1≤u≤u2,b,τ2,u≥u2,c,where G = shear stiffness at the geogrid-soil interface that should be determined experimentally and u1 and u2 = shear displacement corresponding to the peak shear resistance τ1 and the residual shear resistance τ2, respectively. Considering that u1 = τ1/G and u2 = (2τ1-τ2)/G, in order to simplify the calculation, it is assumed that the slopes of the ascending and descending sections in the interface softening model are equal, and both of them are taken as G.Based on the above assumptions, the progressive pullout behavior of geogrid in soil can be divided into five consecutive phases, as shown in Figure9. These five phases are described as follows: (I) initial pure elastic phase—the relationship between shear stress and displacement is linear; (II) softening-elastic phase—a transition point P1, as shown in Figure 9(b), is introduced here to divide the elastic and softening zones; (III) pure softening phase; (IV) softening-residual phase—similar to Phase II, once the interfacial shear stress at the pullout end decreases to the residual shear resistance, the geogrid turns into the softening-residual transition state; again, a transition point P2 is introduced here to divide the softening and residual zones, as shown in Figure 9(d); in this phase, both the pullout force and the interfacial shear stress decrease slightly; (V) final pure residual phase—the final stage starts when the interfacial shear stress at the free end decreases to the residual shear resistance. As shown in Figure 9(e), the residual zone now occupies the entire geogrid. In this stage, the pullout force remains constant, whereas the pullout displacement increases continuously.Figure 9
Progressive pullout process of geogrid in softening model. (a) Elastic phase. (b) Softening-elastic phase. (c) Softening phase. (d) Residual-softening phase. (e) Residual phase.
(a)(b)(c)(d)(e)


## 5. The Pullout Process of Damage Softening Process

### 5.1. Elastic Stage

In this phase, the relationship between the interfacial shear stress and the shear displacement can be expressed by (10). By combining (3) and (10), we get(11)d2uexdx2−α2uxe=0,where α=2G/Er and α is the interface influence coefficient which reflects the influence of geogrid on the tensile force distribution.A general solution of (11) is(12)uex=C1eαx+C2e−αx,where C1 and C2 = integration constants, which can be determined with specific boundary conditions.The solution of the boundary-value problem defined by (4) and (12) is(13)uex=FEr⋅αcoshαl−xsinhαl.The corresponding interfacial shear stressτe(x) and pullout force Fe(x) are then calculated using (10) and (2), respectively, which can be expressed by(14)τex=GFErαcoshαl−xsinhαl,Fex=Fαsinhαl−xsinhαl.


### 5.2. Softening-Elastic Stage

When the pullout force continues to increase, the pulling end begins to soften, that is, the geogrid within the range of (0∼ls) from the pulling end enters the softening stage, while the geogrid within (ls ∼ l) is still in elastic state, as shown in Figure 9(b).In the elastic zone, the distributions of tensile force, interfacial shear stress, and shear displacement in the elastic zone (ls ∼ l) are similar to those in the pure elastic phase. Therefore, the following equations are easily obtained:(15)uex=F′Er⋅αcoshαl−ls−xsinhαl−ls,τex=GF′Erαcoshαl−ls−xsinhαl−ls,Fex=F′αsinhαl−ls−xsinhαl−ls,where lhs = length of the softening zone and F′  = tensile force at the transition point P. Considering that the interfacial shear stress at P equals the peak shear resistance, we get(16)F′=Erατ1Gcoshαl−2lssinhαl−ls.In the softening zone (0∼ls), the relationship between the interfacial shear stress and the shear displacement is defined by (10). Combining (3) and (10), we get(17)d2usxdx2+β2usx=4Dτ1Er.A general solution of (17) is(18)usx=C3cosβx+C4sinβx+4Dτ1Er.Accordingly, the solutions for the softening zone are(19)usx=τ1sinn2βl−ls2Gsinβls⋅sinh2βl−ls−FErβcotβls⋅cosβx−FErβsinβx+4Dτ1Er,Fsx=τ1βErsinn2βl−ls2Gsinβls⋅sinh2βl−ls−Fcotβls⋅sinβx+Fcosβx,τsx=2Dτ1−Dτ1sinn2βl−ls2Gsinβls⋅sinh2βl−ls−DFErβcotβls⋅cosβx+DFErβsinβx−4Dτ1Er.


### 5.3. Softening Stage

Similar to the analysis of the softening zone in the softening-elastic phase, (17) is still applicable in the pure softening phase. The boundary conditions are(20)Fsx|x=0=F,Fsx|x=l=0.The solutions for the pure softening phase are(21)Fsx=Fcotαlαsinβx+FErcosβx,(22)usx=FcotβlErβ2cosβx−FErβsinβx+4Dτ1Er,(23)τsx=2Dτ1−G⋅D⋅FcotβlErβ2cosαx+G⋅D⋅FErβsinβx−4G⋅D⋅τ1Er.


### 5.4. Residual-Softening Stage

As the pullout force decreases, the residual state initiates at the pulling end and extends towards the free end. The residual and softening zones are divided by the transition pointP2. In the softening zone, the distributions of tensile force, interfacial shear stress, and shear displacement are given by (22)–(24), respectively, with x replaced by (x-lr), l by (l -lr), and F by F'.(24)Fsx=F′cotβl−lrβsinβx−lr+F′Ercosβx−lr,(25)usx=F′cotβl−lrErβ2cosβx−lr−F′Erβsinβx−lr+4Dτ1Er,(26)τsx=2Dτ1−G⋅D⋅F′cotβl−lrErβ2cosβx−lr+G⋅D⋅F′Erβsinβx−lr−4G⋅D⋅τ1Er.Because the interfacial shear stress atP2 equals the residual shear resistance, F′ in equations (25)–(27) can be rewritten as(27)F′=Erβ2G2τ1−τ2D−4τ1β2tanβl.As for the residual zone, the interfacial shear stress is equal to the residual shear resistance. (1) leads to(28)dFrx=−2τ2dx.Combining it with the boundary conditions,(29)Frx|x=0=F,Frx|x=lr=Fsx|x=lr.The distribution of tensile force is obtained by integrating (28):(30)Frx=−2τ2x+β2G2τ1−τ2D−4τ1β2Ertanβl+2τ2lr.Substituting (30) into (2) and considering P2,(31)urx|x=lr=usx|x=lr.The shear displacement in the residual zone can be expressed by(32)urx=−τ2x2Er−β2G2τ1−τ2D−4τ1β2Ertanβl+2τ2lrx+C5,where C5 = the constant term which can be obtained according to (31).


### 5.5. Residual Stage

In residual stage, the pullout force of geogrid does not change any more. The interfacial shear stress of the whole area of geogrid isτ2, and the pullout force of geogrid is linear distribution:(33)Frx=−2τ2l−x.Pullout displacement of geogrid at this stage isur0; therefore, the distribution of tensile force is given as(34)Frx=τ2l−x2Er+ur0.


## 5.1. Elastic Stage

In this phase, the relationship between the interfacial shear stress and the shear displacement can be expressed by (10). By combining (3) and (10), we get(11)d2uexdx2−α2uxe=0,where α=2G/Er and α is the interface influence coefficient which reflects the influence of geogrid on the tensile force distribution.A general solution of (11) is(12)uex=C1eαx+C2e−αx,where C1 and C2 = integration constants, which can be determined with specific boundary conditions.The solution of the boundary-value problem defined by (4) and (12) is(13)uex=FEr⋅αcoshαl−xsinhαl.The corresponding interfacial shear stressτe(x) and pullout force Fe(x) are then calculated using (10) and (2), respectively, which can be expressed by(14)τex=GFErαcoshαl−xsinhαl,Fex=Fαsinhαl−xsinhαl.


## 5.2. Softening-Elastic Stage

When the pullout force continues to increase, the pulling end begins to soften, that is, the geogrid within the range of (0∼ls) from the pulling end enters the softening stage, while the geogrid within (ls ∼ l) is still in elastic state, as shown in Figure 9(b).In the elastic zone, the distributions of tensile force, interfacial shear stress, and shear displacement in the elastic zone (ls ∼ l) are similar to those in the pure elastic phase. Therefore, the following equations are easily obtained:(15)uex=F′Er⋅αcoshαl−ls−xsinhαl−ls,τex=GF′Erαcoshαl−ls−xsinhαl−ls,Fex=F′αsinhαl−ls−xsinhαl−ls,where lhs = length of the softening zone and F′  = tensile force at the transition point P. Considering that the interfacial shear stress at P equals the peak shear resistance, we get(16)F′=Erατ1Gcoshαl−2lssinhαl−ls.In the softening zone (0∼ls), the relationship between the interfacial shear stress and the shear displacement is defined by (10). Combining (3) and (10), we get(17)d2usxdx2+β2usx=4Dτ1Er.A general solution of (17) is(18)usx=C3cosβx+C4sinβx+4Dτ1Er.Accordingly, the solutions for the softening zone are(19)usx=τ1sinn2βl−ls2Gsinβls⋅sinh2βl−ls−FErβcotβls⋅cosβx−FErβsinβx+4Dτ1Er,Fsx=τ1βErsinn2βl−ls2Gsinβls⋅sinh2βl−ls−Fcotβls⋅sinβx+Fcosβx,τsx=2Dτ1−Dτ1sinn2βl−ls2Gsinβls⋅sinh2βl−ls−DFErβcotβls⋅cosβx+DFErβsinβx−4Dτ1Er.


## 5.3. Softening Stage

Similar to the analysis of the softening zone in the softening-elastic phase, (17) is still applicable in the pure softening phase. The boundary conditions are(20)Fsx|x=0=F,Fsx|x=l=0.The solutions for the pure softening phase are(21)Fsx=Fcotαlαsinβx+FErcosβx,(22)usx=FcotβlErβ2cosβx−FErβsinβx+4Dτ1Er,(23)τsx=2Dτ1−G⋅D⋅FcotβlErβ2cosαx+G⋅D⋅FErβsinβx−4G⋅D⋅τ1Er.


## 5.4. Residual-Softening Stage

As the pullout force decreases, the residual state initiates at the pulling end and extends towards the free end. The residual and softening zones are divided by the transition pointP2. In the softening zone, the distributions of tensile force, interfacial shear stress, and shear displacement are given by (22)–(24), respectively, with x replaced by (x-lr), l by (l -lr), and F by F'.(24)Fsx=F′cotβl−lrβsinβx−lr+F′Ercosβx−lr,(25)usx=F′cotβl−lrErβ2cosβx−lr−F′Erβsinβx−lr+4Dτ1Er,(26)τsx=2Dτ1−G⋅D⋅F′cotβl−lrErβ2cosβx−lr+G⋅D⋅F′Erβsinβx−lr−4G⋅D⋅τ1Er.Because the interfacial shear stress atP2 equals the residual shear resistance, F′ in equations (25)–(27) can be rewritten as(27)F′=Erβ2G2τ1−τ2D−4τ1β2tanβl.As for the residual zone, the interfacial shear stress is equal to the residual shear resistance. (1) leads to(28)dFrx=−2τ2dx.Combining it with the boundary conditions,(29)Frx|x=0=F,Frx|x=lr=Fsx|x=lr.The distribution of tensile force is obtained by integrating (28):(30)Frx=−2τ2x+β2G2τ1−τ2D−4τ1β2Ertanβl+2τ2lr.Substituting (30) into (2) and considering P2,(31)urx|x=lr=usx|x=lr.The shear displacement in the residual zone can be expressed by(32)urx=−τ2x2Er−β2G2τ1−τ2D−4τ1β2Ertanβl+2τ2lrx+C5,where C5 = the constant term which can be obtained according to (31).


## 5.5. Residual Stage

In residual stage, the pullout force of geogrid does not change any more. The interfacial shear stress of the whole area of geogrid isτ2, and the pullout force of geogrid is linear distribution:(33)Frx=−2τ2l−x.Pullout displacement of geogrid at this stage isur0; therefore, the distribution of tensile force is given as(34)Frx=τ2l−x2Er+ur0.


## 6. Determination of Model Parameters

In the trilinear shear stress-displacement damage softening model of geogrid-soil interface,α and β are the interface pullout influence coefficients, α=2G/Er, and β=2DG/Er, where G = the shear stiffness of geogrid-soil interface (it represents the slope of the straight line in the shear stress-displacement curve); Er is the tensile modulus of geogrid, Er = Et, where E is the elastic modulus and t is the thickness of geogrid; and D is the damage factor variable; assuming that it is a function of interface displacement, the damage evolution law of geogrid-soil interface is expressed by two-parameter Weibull distribution function [29, 32], that is,(35)D=1−exp−uu0m,where m = the shape parameter of Weibull function, which reflects the shape of the function, and u0 = the scale parameter of Weibull function, which can enlarge and reduce the abscissa scale of curve, but does not affect the shape of curve, and it is a parameter related to the average value of all parameters.Combining (9) and (35), the interface shear stress of the model is(36)τ=Guexp−uu0m+τ21−exp−uu0m.The parametersu0 and m are determined, according to the characteristics of the τ-u curve of geogrid-soil interface. It can be seen from Figure 7 that the τ-u curve has obvious characteristics of strain softening, with peak value and residual. When the displacement reaches u1 and the stress reaches the peak point, the stress increment at this point is zero, which can be expressed as(37)∂τ∂u|u=u1=0,(38)τ1=Gu1exp−u1u0m+τ21−exp−u1u0m,where τ1 is peak shear stress at geogrid-soil interface and u1 is the interface displacement corresponding to τ1. Combining (37) and (38), the parameters m and u0 are(39)m=Gu1Gu1−τ2lnτ1−τ2/Gu1−τ2,u0=mGu1−τ2u1m−1G1/m.


## 7. Verification of Interface Constitutive Model

According to the pullout test results of geogrids with different transverse rib spacings, the parameters of the trilinear shear stress-displacement damage softening model of the geogrid-soil interface are obtained, as shown in Table4. It can be seen that the shear stiffness G decreases with the increase of transverse rib spacing, but increases with the increase of normal stress. In Weibull distribution, parameters u0 and m basically do not change the basic shape of the model curve, but with the increase of parameters u0 and m, the curve moves upward, that is, the peak value of interfacial shear stress and residual shear stress increases.Table 4
Model parameters of geogrid-soil interface.
s (cm)σ (kPa)Er (kN·m−1)G (kPa·mm−1)u0 (mm)m35500.56 × 1034.231.0632.9541000.56 × 1035.561.1843.5191500.56 × 1036.451.3954.6142000.56 × 1038.251.5125.48550500.45 × 1032.361.0062.3291000.45 × 1033.351.0542.9531500.45 × 1034.761.1473.2582000.45 × 1035.731.3244.124By substituting the model parameters into the expressions of drawing force, displacement, and interfacial shear stress at different drawing stages, the theoretical calculation results of the trilinear shear stress-displacement damage softening model are obtained, as shown in Figure10. It can be seen that the calculated results of the model are in good agreement with the laboratory test curves, which can correctly reflect the strain-hardening characteristics of the geogrid-soil interface and the strain damage softening characteristics considering the damage effects during the pullout process, better reflect the trilinear damage softening model, and can accurately analyze the interaction mechanism of geogrid-soil interface.Figure 10
Comparison of test results and theoretical results. (a)s = 35 cm. (b) s = 50 cm.
(a)(b)


## 8. Conclusions

The main conclusions of the present work can be summarized as follows:(1)
In the pullout test, the strain-softening phenomenon appears after the pullout force and displacement curve reach the peak value. The shear strength of the geogrid-soil interface increases with the increase of normal stress, which shows that the reinforcement of geogrid can improve the cohesion of soil, but the internal friction angle decreases. Based on the damage mechanics, the damage softening model is established to describe the softening characteristics of geogrid-soil interface. The mechanism of geogrid-soil interface considering damage is revealed theoretically.(2)
In order to analyze the strain-softening behaviors of geogrid-soil interface, the trilinear shear stress-displacement damage softening model is used to describe the interaction mechanism of the geogrid-soil interface. Based on the damage softening model, the whole pullout process is divided into five stages, namely, pure elastic stage, softening-elastic stage, softening stage, residual-softening stage, and residual stage.(3)
The analytical solutions of interface tension, shear stress, and displacement at different stages of damage softening characteristics under the pullout load are derived through the basic interface control equation. Meanwhile, the evolution law of interface shear stress in different pullout stages is analyzed, which reflects the progressive failure of geogrid-soil interface.(4)
Considering the influence of residual strength, the damage softening constitutive model can analyze the whole pullout process of geogrid-soil interface. Compared with pullout results, prediction results of the model are basically consistent with the pullout test, which verifies the effectiveness and rationality of model for progressive failure analysis in the pullout interface.



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*Source: 8047519-2022-05-09.xml*