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

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.


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 di erential settlement between the old and the new subgrade. e reinforcement of geogrid is realized by the interaction between geogrid and surrounding soil. erefore, the behavior of reinforcement soil interface is an important factor a ecting the reinforcement mechanism of geogrid.
ere are still some de ciencies in the design method of geogrid reinforced soil structure. e 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][3][4][5], triaxial compression test [6,7], biaxial compression test [8], and pullout test [9]. e 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 di erent sti nesses in gravel and clay and discussed the in uence of geogrid sti ness and soil type on the friction characteristics of reinforcement soil interface. Wang et al. [11] studied the in uence of various test factors on the pullout behavior of reinforced soil interface. Xiao et al. [12] analyzed the in uence 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. e 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.
e 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.
e research results provide technical reference for reasonable selection of reinforcement materials in reinforced soil engineering.

Test Materials.
e 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 Figure 1.
e spacing between transverse ribs of geogrid used in this test is 35 cm and 50 cm, respectively. e technical indexes of geogrid are shown in Table 1. e 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.

Test Scheme.
e pullout equipment was mainly composed of four parts: pullout box, normal loading system, horizontal loading system, and data acquisition system. e 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 Figure 3. e test filling height is 50 cm, which is compacted in four layers, and compaction degree is taken as 90%. e 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. e 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.

Test Results.
e relationship curve of pullout force and displacement with different transverse rib spacing under different normal stress is shown in Figure 4. 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. e larger the normal stress, the larger the displacement corresponding to the maximum drawing force. e 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 Figure 5. It can be seen that the maximum shear stress at the interface increases linearly with the increase of normal stress. e 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 geogridsoil 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. e reinforcement of geogrid improves the cohesion of soil, but the friction angle decreases.

Mechanical Analysis of Geogrid-Soil Interface
During the pullout test, the geogrid is placed in the middle of the upper and lower test chambers. e top of the upper box is applied with vertical compressive stress, and geogrid is pulled out slowly. e schematic diagram of the pullout  Mass percentage of soil less than a certain particle size (%) 0   Advances in Materials Science and Engineering 3 process is shown in Figure 6. 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. e microelement with length of x is taken along the length direction of geogrid; according to the force balance and physical and geometric equations, where x � arbitrary coordinate as shown in Figure 6; � tensile strain at point x; τ(x) � interface shear stress at an arbitrary point x; E r � tensile stiffness, E r � Et; E is elastic modulus of geogrid; and t is thickness of geogrid. According to (1) and (2), the basic equation of load transfer can be expressed as In the pullout test of geogrid, the pullout force F at pulling end can be measured by the sensor, and the force at free end can be ignored. e boundary condition of the pullout force at both ends of the geogrid can be expressed as In the process of pullout, it is necessary to establish the constitutive model of the geogrid-soil interface to solve u(x) and τ(x) in (3).

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 Figure 7. 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. In order to consider the strain-softening constitutive model of geogrid-soil interface with damage, according to Rabotnov damage factor D, the geogrid-soil interface under pullout force is assumed to be composed of damaged and undamaged parts, as shown in Figure 7. e damage factor D can be expressed as the ratio of the damage length to the original length, which can be expressed as where d(x) � initial original length of geogrid-soil interface; d ′ (x) � undamaged part length; d ″ (x) � damaged part length; τ(x) � 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 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 Figure 8 that the shear strength of geogrid-soil interface reaches the peak shear stress τ 1 ; afterward, the strength decreases until it reaches the stable residual strength τ 2 . In the damage model, the shear stress of damaged length is not zero and still bear the shear stress of τ 2 , that is τ ″ � τ 2 ; formula (7) can be expressed as F (x)+dF (x) Figure 6: Schematic illustration of pullout mechanism of geogrid in soil.

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For the undamaged part, the shear stress τ ′ (x) still obeys the linear part of the initial interface shear stress-displacement, and formula (8) can be expressed as 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 ≤ u 1 , the effect of damage is ignored; for u 1 ≤ u ≤ u 2 , the damage factor variable satisfies 0 < D < 1; for u ≥ u 2 , the damage factor variable satisfies D � 1.
For trilinear shear stress-displacement softening model, relationship between shear stress τ(x) and shear displacement u(x) can be expressed by where G � shear stiffness at the geogrid-soil interface that should be determined experimentally and u 1 and u 2 � shear displacement corresponding to the peak shear resistance τ 1 and the residual shear resistance τ 2 , respectively. Considering that u 1 � τ 1 /G and u 2 � (2τ 1 -τ 2 )/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 Figure 9. ese 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 P 1 , as shown in Figure 9(b), is introduced here to divide the elastic and softening zones; (III) pure softening phase; (IV) softeningresidual 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 P 2 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.

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 where α � ����� 2G/E r and α is the interface influence coefficient which reflects the influence of geogrid on the tensile force distribution.
A general solution of (11) is where C 1 and C 2 � integration constants, which can be determined with specific boundary conditions. e solution of the boundary-value problem defined by (4) and (12) is e corresponding interfacial shear stress τ e (x) and pullout force F e (x) are then calculated using (10) and (2), respectively, which can be expressed by

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∼ l s ) from the pulling end enters the softening stage, while the geogrid within (l s ∼ 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 (l s ∼ l) are similar to those in the pure elastic phase. erefore, the following equations are easily obtained:   where l hs � 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 In the softening zone (0∼l s ), the relationship between the interfacial shear stress and the shear displacement is defined by (10). Combining (3) and (10), we get A general solution of (17) is Accordingly, the solutions for the softening zone are

Softening Stage.
Similar to the analysis of the softening zone in the softening-elastic phase, (17) is still applicable in the pure softening phase. e boundary conditions are e solutions for the pure softening phase are

Residual-Softening Stage.
As the pullout force decreases, the residual state initiates at the pulling end and extends towards the free end. e residual and softening zones are divided by the transition point P 2 . 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-l r ), l by (l -l r ), and F by F'.

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Because the interfacial shear stress at P 2 equals the residual shear resistance, F′ in equations (25)- (27) can be rewritten as As for the residual zone, the interfacial shear stress is equal to the residual shear resistance. (1) leads to Combining it with the boundary conditions, e distribution of tensile force is obtained by integrating (28): Substituting (30) into (2) and considering P 2 , e shear displacement in the residual zone can be expressed by where C 5 � the constant term which can be obtained according to (31).

Residual Stage.
In residual stage, the pullout force of geogrid does not change any more. e interfacial shear stress of the whole area of geogrid is τ 2 , and the pullout force of geogrid is linear distribution: Pullout displacement of geogrid at this stage is u 0 r ; therefore, the distribution of tensile force is given as

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/E r , and β � ������� � 2 DG/E r , where G � the shear stiffness of geogrid-soil interface (it represents the slope of the straight line in the shear stress-displacement curve); E r is the tensile modulus of geogrid, E r � 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, where m � the shape parameter of Weibull function, which reflects the shape of the function, and u 0 � 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 e parameters u 0 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 u 1 and the stress reaches the peak point, the stress increment at this point is zero, which can be expressed as where τ 1 is peak shear stress at geogrid-soil interface and u 1 is the interface displacement corresponding to τ 1 . Combining (37) and (38), the parameters m and u 0 are

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 Table 4. 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 u 0 and m basically do not change the basic shape of the model curve, but with the increase of parameters u 0 and m, the curve moves upward, that is, the peak value of interfacial shear stress and residual shear stress increases. By 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 Figure 10. It can be seen that the calculated results of the model are in good Advances in Materials Science and Engineering 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.

Conclusions
e 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. e 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. e mechanism of geogridsoil 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.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.