# Mechanical Behavior of Interface between Composite Geomembrane and Permeable Cushion Material

Authors: Haimin Wu; Yiming Shu; Linjun Dai; Zhaoming Teng
Journal: Advances in Materials Science and Engineering
 (2014)

---

## Abstract

An accurate description of composite geomembrane-cushion interface behavior is of great importance for stress-deformation analysis and stability assessment of geomembrane surface barrier of rock-fill dam. A series of direct shear tests were conducted to investigate the friction behaviors of interfaces between composite geomembrane and two different permeable cushion materials (crushed stones and polyurethane mixed crushed stones). The shear stress-displacement relationships of the two interfaces show different characteristics and were described by the nonlinear-elastic model and nonlinear-elastic perfectly plastic model, respectively. Then the two models were implemented into the Fast Lagrangian Analysis of Continua in Three Dimensions (F
L
A
C
3
D) procedure correctly. By verification of a numerical example, numerical calculation results showed a good agreement with the theoretical solutions and test results.


---

## Body

## 1. Introduction

Due to the advantages of strong ability to adapt differential deformation, short construction period, low cost, field preservation, low carbon, and environment protection, geomembranes have increasingly replaced traditional barrier materials such as compacted clay, cement concrete, asphalt concrete, or grouting as watertightness of earth and rock-fill dam. By the year of 2010, there have been 167 large embankment dams that had chosen geomembranes as their barrier systems according to the International Commission on Large Dams (ICOLD) database [1]. With the rapid construction of water conservancy and hydropower engineering, more and more dams have to be constructed on the thick pervious foundation in China. Due to its strong ability to adapt differential deformation, geomembrane surface barrier systems become the preferred solution for the watertightness of high rock-fill dam on thick pervious foundation [2].Geomembrane surface barriers are commonly comprised by three layers, namely, basal cushion layer, geomembrane barrier, and protection cover layer. In Chinese design specification for rolled earth-rock-fill dam, the geomembrane barrier is usually used with composite geomembrane (CGM) compounded by two layers of needle punched nonwoven geotextiles and a layer of high density polyethylene (HDPE) or polyvinyl chloride (PVC) geomembrane [3].There are generally two kinds of traditional permeable materials that can be used as cushion layer for geomembrane surface barrier, namely, gravel materials and porous concrete. For high rock-fill dam with a steep dam slope, gravel materials cushion can hardly maintain its stability. Because of the inability to resist flexural deformation of the surface of dam slope, the brittle porous concrete may yield bending failure and develop into cracks. That would pose a threat to the safety of geomembrane under long-term loading of high water pressure.Polyurethane (PUR) mixed crushed stone is a new elastic porous material casted by polyurethane adhesive mixed crushed stones. Due to the higher bending strength, excellent toughness, and permeability, it was used as the basal cushion layer of geomembrane in surface barrier of high rock-fill dam on thick pervious foundation [4]. In actual working condition, flexural deformations of the surface of dam slope resulted from self-weight and high water pressure loads may induce shear displacement along geomembrane-cushion interface. The shear displacement can mobilize shear stress on the interface between geomembrane and cushion. And the low shearing strength on the interface between geomembrane and cushion materials may increase the tensile force of geomembrane and cause sliding failure of the whole barrier systems [5].The mechanical behavior of geomembrane-cushion interface is the critical issue for deformation analysis and stability assessment of geomembrane surfaced barrier of high rock-fill dam. For CGM-cushion interface, the essence is the frictional behavior between needle punched nonwoven geotextile and cushion materials. Many researches on interface behaviors between geotextile and different soil materials have been reported during the last decade. For example, research involving the interface between geotextile and cohesive soil can be found in several literatures [6–8]. Studies on interface behaviors between geotextile and different sands have been reported by Lee and Manjunath [9] and Anubhav and Basudhar [10]. G. Zhang and J.-M. Zhang [11] and G. Zhang et al. [12] investigated the monotonic and cyclic behaviors of interface between geotextile and gravelly soil by laboratory test and numerical modeling. It can be seen that the research works mainly focus on interfaces between geotextile and different soils and sands in reinforcement structures. The interface between a geotextile and gravel materials with a grain size larger than that of soil and sand had not been investigated thoroughly. The research on interface between CGM and permeable cushions of barriers on slope had rarely been reported. Therefore, further study should be conducted on the friction behaviors of the interfaces between CGM and cushions materials.The objectives of this study are to obtain shear strength and shear stress-displacement response of CGM-cushion interface by direct shear test and numerical modeling. The characteristics of interface shear strength and shear stress-displacement relationship observed from test results are reported. Based on the experimental observations, the nonlinear-elastic model and nonlinear-elastic perfect-plastic model are applied to describe the interface stress-displacement relationships of CGM with crushed stones and PUR mixed crushed stones, respectively. Finally, the two interface models are incorporated into the Geogrid element ofFLAC
3
D procedure to predict the mechanical behaviors of two interfaces.


## 2. Direct Shear Tests of Interfaces

### 2.1. Direct Shear Apparatus

The size of shear box for direct shear apparatus has an effect on the shear strength at geosynthetic-soil interface. By contrasting the test results with different size of shear box, C. Hsieh and M.-W. Hsieh [13] recommended that a minimum dimension of 300 × 300 mm direct shear box should be used. The similar regulations are also required in the ASTM D5321 [14]. In this study, a large-scale direct shear machine was employed in shear tests. The schematic view of the apparatus is shown in Figure 1. The apparatus comprises a 300 mm square top box and a 300 × 350 mm rectangular lower box. It has a maximum shear displacement of 50 mm with no loss in area of shear plane during shearing test.Figure 1
Schematic diagram of interface shear apparatus.The load plate rigidity in the normal direction of upper shear box can also influence the test results. The results of the study conducted by C. Hsieh and M.-W. Hsieh [13] indicated that the use of conventional rigid load plate in direct shear test resulted in a concave pressure distribution at the shearing plane. It was observed that using flexible load plate showed a better reproducibility of test data than using rigid load plate. So in order to apply a uniform flexible normal pressure on the interface, a 2 cm thick soft silicon plate, placed at the bottom of rigid load plate, was used to form a flexible load plate in the upper shear box.The shear displacement of the lower box was controlled by a precise step-motor system. The shear displacement and shear force were monitored by a Linear Variable Differential Transducer (LVDT) and a load cell, respectively. A computerized data-acquisition system was installed to collect the displacement and shear stress data.


### 2.2. Materials

The CGM commonly used as surface barrier of rock-fill dam in China was chosen in the tests. The photograph of the CGM is shown in Figure2. It consists of a 0.8 mm thick HDPE geomembrane laminated to a 400 g/m2 polyethylene terephthalate (PET) needle punched nonwoven geotextile at both sides. Its ultimate tensile strength (ASTM D4595, 2005) in machine direction and cross-machine direction is 75.9 kN/m and 58.3 kN/m, respectively.Figure 2
Composite geomembrane (CGM).The crushed stones used in the tests are shown in Figure3. It is a type of coarse gravelly material crushed from fresh granite stones. The physical properties of crushed stones are presented in Table 1.Table 1
Physical properties of crushed stones.
Size range (mm)
d
50 (mm)
C
u
C
c
ρ
d (g/cm3)
Φ (°)
5–20
10
4.2
2.4
1.725
48Figure 3
Crushed stones.The PUR mixed crushed stones used in the tests is shown in Figure4. It was casted by polyurethane adhesive mixed above-mentioned crushed stones with a quality ratio of 2.5%. Its effective porosity and compressive modulus of elasticity are 32% and 8 GPa, respectively. By curing 2 days in standard curing box, the unconfined compressive strength and bending strength can reach 3.1 MPa and 1.6 MPa, respectively.Figure 4
PUR mixed crushed stones.


### 2.3. Test Procedure

In the preparation stage, the CGM samples were cut into rectangles of 450 × 300 mm. The specimens of PUR mixed crushed stones were casted into 300 × 300 × 10 mm blocks at least 2 days in advance.Standard Procedure B for soil and geosynthetic friction adopted from ASTM D-5321(2008) standard test method was used in this study. An iron block with a rough surface was placed in the lower box as a rigid substrate. Then CGM specimen was glued to the top face of the rigid block. And the left end of CGM was fixed on the side of lower box with steel bar and screws. Then the crushed stones or precast PUR mixed crushed stones specimen was placed in the upper box. The bottom surface of the cushion should keep a good contact state with upper surface of CGM. For cushion of crushed stones, it was compacted by tamping to a desired relative density of 70%.In each series of tests, the normal stress was maintained at 25, 50, 75, and 100 kPa, respectively. The shearing rate was kept at 1.0 mm/min. Every test was conducted until the applied shear force remains constant with increasing displacement or till shear displacement reaches 20 mm (1/15 of the sample length) if shear force continuously increases with increasing displacement.


## 2.1. Direct Shear Apparatus

The size of shear box for direct shear apparatus has an effect on the shear strength at geosynthetic-soil interface. By contrasting the test results with different size of shear box, C. Hsieh and M.-W. Hsieh [13] recommended that a minimum dimension of 300 × 300 mm direct shear box should be used. The similar regulations are also required in the ASTM D5321 [14]. In this study, a large-scale direct shear machine was employed in shear tests. The schematic view of the apparatus is shown in Figure 1. The apparatus comprises a 300 mm square top box and a 300 × 350 mm rectangular lower box. It has a maximum shear displacement of 50 mm with no loss in area of shear plane during shearing test.Figure 1
Schematic diagram of interface shear apparatus.The load plate rigidity in the normal direction of upper shear box can also influence the test results. The results of the study conducted by C. Hsieh and M.-W. Hsieh [13] indicated that the use of conventional rigid load plate in direct shear test resulted in a concave pressure distribution at the shearing plane. It was observed that using flexible load plate showed a better reproducibility of test data than using rigid load plate. So in order to apply a uniform flexible normal pressure on the interface, a 2 cm thick soft silicon plate, placed at the bottom of rigid load plate, was used to form a flexible load plate in the upper shear box.The shear displacement of the lower box was controlled by a precise step-motor system. The shear displacement and shear force were monitored by a Linear Variable Differential Transducer (LVDT) and a load cell, respectively. A computerized data-acquisition system was installed to collect the displacement and shear stress data.


## 2.2. Materials

The CGM commonly used as surface barrier of rock-fill dam in China was chosen in the tests. The photograph of the CGM is shown in Figure2. It consists of a 0.8 mm thick HDPE geomembrane laminated to a 400 g/m2 polyethylene terephthalate (PET) needle punched nonwoven geotextile at both sides. Its ultimate tensile strength (ASTM D4595, 2005) in machine direction and cross-machine direction is 75.9 kN/m and 58.3 kN/m, respectively.Figure 2
Composite geomembrane (CGM).The crushed stones used in the tests are shown in Figure3. It is a type of coarse gravelly material crushed from fresh granite stones. The physical properties of crushed stones are presented in Table 1.Table 1
Physical properties of crushed stones.
Size range (mm)
d
50 (mm)
C
u
C
c
ρ
d (g/cm3)
Φ (°)
5–20
10
4.2
2.4
1.725
48Figure 3
Crushed stones.The PUR mixed crushed stones used in the tests is shown in Figure4. It was casted by polyurethane adhesive mixed above-mentioned crushed stones with a quality ratio of 2.5%. Its effective porosity and compressive modulus of elasticity are 32% and 8 GPa, respectively. By curing 2 days in standard curing box, the unconfined compressive strength and bending strength can reach 3.1 MPa and 1.6 MPa, respectively.Figure 4
PUR mixed crushed stones.


## 2.3. Test Procedure

In the preparation stage, the CGM samples were cut into rectangles of 450 × 300 mm. The specimens of PUR mixed crushed stones were casted into 300 × 300 × 10 mm blocks at least 2 days in advance.Standard Procedure B for soil and geosynthetic friction adopted from ASTM D-5321(2008) standard test method was used in this study. An iron block with a rough surface was placed in the lower box as a rigid substrate. Then CGM specimen was glued to the top face of the rigid block. And the left end of CGM was fixed on the side of lower box with steel bar and screws. Then the crushed stones or precast PUR mixed crushed stones specimen was placed in the upper box. The bottom surface of the cushion should keep a good contact state with upper surface of CGM. For cushion of crushed stones, it was compacted by tamping to a desired relative density of 70%.In each series of tests, the normal stress was maintained at 25, 50, 75, and 100 kPa, respectively. The shearing rate was kept at 1.0 mm/min. Every test was conducted until the applied shear force remains constant with increasing displacement or till shear displacement reaches 20 mm (1/15 of the sample length) if shear force continuously increases with increasing displacement.


## 3. Test Results

### 3.1. Shear Stress-Shear Displacement Responses of Interfaces

The shear stress-shear displacement responses on the interfaces between CGM and two cushion materials under different normal loads are presented and discussed in this section.The shear stress versus shear displacement curves of CGM-crushed stones interface are shown in Figure5(a). The shear stresses increase rapidly as soon as shear displacements start and continue to increase to comparatively large shear displacement values. The test results show that no well-defined peak shear strength is observed for the CGM-crushed stones interface. An obvious stain-hardening characteristic of the shear stress-shear displacement response can be seen from the test results. It is different from geotextile-gravelly soil interface that exhibited evident strain-softening characteristics in literature [11]. The main reason could be the difference of surface texture and elasticity between CGM and geotextile.Shear stress versus shear displacement of CGM-cushion interfaces.
(a)
CGM-crushed stones interface
(b)
CGM-PUR mixed crushed stones interfaceThe ultimate state of CGM surface after testing is shown in Figure6. It can be seen that the filament structures on the surface were destroyed by the angular crushed stones. Many broken particles of the crushed stones had embedded in the surface of CGM. It is indicated that the interlocking exists between the angular-shaped gravels and geotextile at the upper side of CGM. It can also be observed that the increasing normal load results in increased interlocking. That may be the reason why the CGM-crushed stones interface shown significant hardening characteristics. A nonlinear hyperbolic relationship between the shear stress and shear displacement can be found.Figure 6
Ultimate surface state of CGM after testing.The shear stress versus shear displacement curves of CGM-PUR mixed crushed stones interface under different normal loads are shown in Figure5(b). A different shear stress-displacement response can be observed compared with the GM-crushed stones interface. An initial increase in shear stress with increasing displacement can be seen in the shear stress-displacement curves. When the shear stress reaches peak shear strength, the plastic failure begins with the increasing of shear displacement. The interface shows a failure mode of elastic perfect-plastic and sliding along the shear plane.The ultimate surface state of CGM and PUR mixed crushed stones after testing is shown in Figure7. It can be seen that the filament structure of CGM surface realigned along the shearing direction through frictional interaction. Some filament had dropped off from the surface of CGM and embedded into the slits of PUR mixed crushed stones. It is indicated that a slippage failure had occurred on the interface for large shear displacement. It also can be found that a nonlinear-elastic perfect-plastic relationship can be used to express the complete shear stress and shear displacement relationship from the curves in Figure 5(b).Ultimate surface state of CGM and PUR mixed crushed stones after testing.
(a)
(b)


### 3.2. Peak Shear Strength of Interfaces

The variations of the peak shear stress with normal stress for the two interfaces are shown in Figure8. The results show an apparent adhesion on the peak shear strength of the two interfaces. It is observed that the peak shear stresses increase with increase of normal stress. The shear strength of CGM-crushed stones interface is always higher than that of CGM-PUR mixed crushed stones interface under different normal loads. The shear strength of the two interfaces can be expressed as a function of normal stress by Mohr-Coulomb criteria.Figure 8
Shear strength envelops of two interfaces.For CGM-crushed stones interface, a friction angle of 35.2° and an adhesion intercepts of 6.93 kPa can be found from the linear regression fits of the test results. And lower shear strength with friction angle of 32.3° and adhesion intercepts of 5.38 kPa can be found for the CGM-PUR mixed crushed stones interface. It is supposed that the adhesive film wrapping the surface of gravels had decreased the roughness of the crushed stones. Besides, the measured shear strength of CGM-crushed stones interface is composed of the friction between gravel and geotextile surface and some interlocking resistance between gravel particles. Therefore, the shear strength for CGM-crushed stones interface is larger than that for CGM-PUR mixed crushed stones interface.


## 3.1. Shear Stress-Shear Displacement Responses of Interfaces

The shear stress-shear displacement responses on the interfaces between CGM and two cushion materials under different normal loads are presented and discussed in this section.The shear stress versus shear displacement curves of CGM-crushed stones interface are shown in Figure5(a). The shear stresses increase rapidly as soon as shear displacements start and continue to increase to comparatively large shear displacement values. The test results show that no well-defined peak shear strength is observed for the CGM-crushed stones interface. An obvious stain-hardening characteristic of the shear stress-shear displacement response can be seen from the test results. It is different from geotextile-gravelly soil interface that exhibited evident strain-softening characteristics in literature [11]. The main reason could be the difference of surface texture and elasticity between CGM and geotextile.Shear stress versus shear displacement of CGM-cushion interfaces.
(a)
CGM-crushed stones interface
(b)
CGM-PUR mixed crushed stones interfaceThe ultimate state of CGM surface after testing is shown in Figure6. It can be seen that the filament structures on the surface were destroyed by the angular crushed stones. Many broken particles of the crushed stones had embedded in the surface of CGM. It is indicated that the interlocking exists between the angular-shaped gravels and geotextile at the upper side of CGM. It can also be observed that the increasing normal load results in increased interlocking. That may be the reason why the CGM-crushed stones interface shown significant hardening characteristics. A nonlinear hyperbolic relationship between the shear stress and shear displacement can be found.Figure 6
Ultimate surface state of CGM after testing.The shear stress versus shear displacement curves of CGM-PUR mixed crushed stones interface under different normal loads are shown in Figure5(b). A different shear stress-displacement response can be observed compared with the GM-crushed stones interface. An initial increase in shear stress with increasing displacement can be seen in the shear stress-displacement curves. When the shear stress reaches peak shear strength, the plastic failure begins with the increasing of shear displacement. The interface shows a failure mode of elastic perfect-plastic and sliding along the shear plane.The ultimate surface state of CGM and PUR mixed crushed stones after testing is shown in Figure7. It can be seen that the filament structure of CGM surface realigned along the shearing direction through frictional interaction. Some filament had dropped off from the surface of CGM and embedded into the slits of PUR mixed crushed stones. It is indicated that a slippage failure had occurred on the interface for large shear displacement. It also can be found that a nonlinear-elastic perfect-plastic relationship can be used to express the complete shear stress and shear displacement relationship from the curves in Figure 5(b).Ultimate surface state of CGM and PUR mixed crushed stones after testing.
(a)
(b)


## 3.2. Peak Shear Strength of Interfaces

The variations of the peak shear stress with normal stress for the two interfaces are shown in Figure8. The results show an apparent adhesion on the peak shear strength of the two interfaces. It is observed that the peak shear stresses increase with increase of normal stress. The shear strength of CGM-crushed stones interface is always higher than that of CGM-PUR mixed crushed stones interface under different normal loads. The shear strength of the two interfaces can be expressed as a function of normal stress by Mohr-Coulomb criteria.Figure 8
Shear strength envelops of two interfaces.For CGM-crushed stones interface, a friction angle of 35.2° and an adhesion intercepts of 6.93 kPa can be found from the linear regression fits of the test results. And lower shear strength with friction angle of 32.3° and adhesion intercepts of 5.38 kPa can be found for the CGM-PUR mixed crushed stones interface. It is supposed that the adhesive film wrapping the surface of gravels had decreased the roughness of the crushed stones. Besides, the measured shear strength of CGM-crushed stones interface is composed of the friction between gravel and geotextile surface and some interlocking resistance between gravel particles. Therefore, the shear strength for CGM-crushed stones interface is larger than that for CGM-PUR mixed crushed stones interface.


## 4. Numerical Modeling of Interface Behavior

Based on the results of direct shear tests, the mechanical behavior of the CGM-crushed stones interface can be described by a nonlinear hyperbolic model [15], namely, the nonlinear-elastic stage of the nonlinear-elastic perfectly plastic interface model (Figure 9). And for the CGM-PUR mixed crushed stones interface, a nonlinear-elastic perfectly plastic interface constitutive model that combines the nonlinear hyperbolic model (nonlinear-elastic stage) with the Mohr-Coulomb plastic failure envelope (perfectly plastic stage) can be used to describe the mechanical behavior.Figure 9
Nonlinear-elastic perfectly plastic model of interface.


### 4.1. Nonlinear-Elastic Perfectly Plastic Constitutive Model of Interface

#### 4.1.1. Nonlinear-Elastic Stage

Before shear stress reaches peak shear strength, the relationship between shear stress and shear displacement of the interface can typically be modeled by a hyperbolic equation:(1)
τ
=
u
a
+
b
u
,
where u is shear displacement of interface.The parametersa and b can be expressed as
(2)
a
=
1
k
1
γ
w
(
σ
n
/
Pa
)
n
,
b
=
R
f
σ
n
tan
ϕ
+
c
,
where γ
w is the unit weight of water; σ
n is the normal effective stress of the interface; Pa is the atmospheric pressure; τ is the shear stress; c is the cohesion of the interface; ϕ is the friction angle of interface; k
1, n, and R
f are the nonlinear parameters that can be derived from interface direct shear tests.By combining the three equations above, the tangent modulus of the shear stress-displacement curve of the interface (shear stiffness)k
s can be expressed as follows [16]:
(3)
k
s
=
K
1
γ
w
(
σ
n
Pa
)
n
[
1
-
R
f
τ
c
+
σ
n
tan
ϕ
]
2
.


#### 4.1.2. Perfectly Plastic Stage

When shear stress reaches the peak shear strength of interface, the plastic failure occurs with the increase of shear displacement. The shear strength envelope in the postpeak stage is expressed by Mohr-Coulomb failure criteria:(4)
τ
=
τ
max
⁡
=
c
+
σ
n
tan
ϕ
,
where τ
max
⁡ is the shear strength of interface; c is the cohesion of interface; ϕ is the friction angle of interface.


### 4.2. Parameters of CGM-Cushion Interface Model

The fitting curves of shear stress versus shear displacement from the test results of the two interfaces using nonlinear-elastic perfectly plastic interface model are shown in Figure10. The fitting parameters for the two interface models are given in Table 2. The comparisons between theoretical solutions calculated by the model and test results show good agreement under different normal loads.Table 2
Fitting parameters of two interfaces by nonlinear-elastic perfectly plastic model.
Parameters
K
n
R
f
c (Pa)
φ (°)
GM-crushed stones interface
1624
1.12
0.75
6930
35.2
GM-PUR mixed crushed stones interface
2871
0.185
0.893
5380
32.3Comparison of test data with fitting curves of interface model.
(a)
CGM-crushed stones interface
(b)
CGM-PUR mixed crushed stones interface


### 4.3. Numerical Implementation of Interface Constitutive Model

In order to simulate the mechanical behaviors of CGM-cushion interfaces, the nonlinear-elastic perfectly plastic interface model was incorporated into the built-in Geogrid element ofFLAC
3
D procedure by user-defined FISH program. The Geogrid element in FLAC
3
D provides interface model that is characterized by Coulomb sliding and shear bonding. Interfaces of Geogrid element have the properties of friction, cohesion, shear stiffness, and shear bond strength [17]. But the Geogrid element in FLAC
3
D can only simulate the interface with linear elastic characteristics. It cannot be used to simulate geosynthetic-soil interfaces which are characterized by the nonlinear behaviors. So the Geogrid element in FLAC
3
D should be improved to simulate the CGM-cushion interface behaviors by user-defined fish program.The detailed program flow chart of implementation of CGM-cushion interface model inFLAC
3
D procedure is shown in Figure 11.Figure 11
Program flow chart of user-defined interface model of Geogrid element.As shown in Figure11, at every calculation step, the program first reads the normal effective stress and shear stress of the three interface nodes for every Geogrid element. The resultant interface normal stress and shear stress of the Geogrid element are calculated according to the node variables. Then the stiffness of the interface was calculated according to the resultant interface variables by (3). The shear stiffness is inputted into the Geogrid element using fish functions. Then, the calculated new parameters for each Geogrid element were inputted to carry out the calculation of the next step. In this way, the program continuously circulated until the cycle for all Geogrid elements has been completed.


### 4.4. Verification of Interface Constitutive Model

In order to verify the effectiveness of the imbedded interface model, a simple numerical example is chosen to model the direct shear test results of CGM-cushion interfaces. As shown in Figure12, the numerical model of the test is composed of two parts. The upper part is a shear box with cushion materials in it; the lower one is a rigid block where the geomembrane is glued at the top. The numerical model of cushion is a 300 × 300 × 100 mm cuboid which is simulated using hexahedral brick elements. The numerical model of rigid block is a 300 × 350 × 100 mm cuboid which is also simulated using hexahedral brick elements. The Geogrid element is used to simulate the GCM. The newly imbedded interface model is used to simulate the CGM-cushion interface behaviors. The boundary conditions correspond to roller boundaries on both sides (y-direction andx-direction) and to fixed displacements in they- andz-direction at the model base.Figure 12
Grid of numerical model for direct shear test.In order to compare the numerical results with theoretical solutions and test results, a linear elastic model was employed for the cushions in the upper box as well as the rigid block in the lower box. Gravity forces of cushions and rigid block were not considered during the numerical modeling. Parameters in Table2 were used in numerical model. According to the typical procedures for direct shear test, a constant normal pressure was applied on the top surface of the cushion. Then, the displacements and velocities of all elements were reset to zero. A fixed shear velocity of 1 × 10−5 m/(time step) was applied to all the elements of the lower box to simulate the actual shearing rate of 1 mm/min. That led to a shear displacement on the interface between the upper cushion and the lower CGM. Four numerical direct shear tests were conducted with constant normal pressures of 25 kPa, 50 kPa, 75 kPa, and 100 kPa, respectively.The numerical computed results of the average shear stress versus shear displacement curves of two interfaces and the comparison with test results and theoretical solution are shown in Figure13. It is obvious that the numerical results are very close to the theoretical solutions calculated by (1)–(4) using the same parameters. Both the numerical and theoretical solutions show good agreements with test results for the two interfaces. Figure 13 illustrates that the implemented interface model in Geogrid element of FLAC
3
D is capable of modeling the nonlinear-elastic and perfectly plastic behavior of interface between CGM and two permeable cushions.Comparison of numerical results with test results and theoretical solution of interfaces.
(a)
CGM-crushed stones interface
(b)
CGM-PUR mixed crushed stones interface


## 4.1. Nonlinear-Elastic Perfectly Plastic Constitutive Model of Interface

### 4.1.1. Nonlinear-Elastic Stage

Before shear stress reaches peak shear strength, the relationship between shear stress and shear displacement of the interface can typically be modeled by a hyperbolic equation:(1)
τ
=
u
a
+
b
u
,
where u is shear displacement of interface.The parametersa and b can be expressed as
(2)
a
=
1
k
1
γ
w
(
σ
n
/
Pa
)
n
,
b
=
R
f
σ
n
tan
ϕ
+
c
,
where γ
w is the unit weight of water; σ
n is the normal effective stress of the interface; Pa is the atmospheric pressure; τ is the shear stress; c is the cohesion of the interface; ϕ is the friction angle of interface; k
1, n, and R
f are the nonlinear parameters that can be derived from interface direct shear tests.By combining the three equations above, the tangent modulus of the shear stress-displacement curve of the interface (shear stiffness)k
s can be expressed as follows [16]:
(3)
k
s
=
K
1
γ
w
(
σ
n
Pa
)
n
[
1
-
R
f
τ
c
+
σ
n
tan
ϕ
]
2
.


### 4.1.2. Perfectly Plastic Stage

When shear stress reaches the peak shear strength of interface, the plastic failure occurs with the increase of shear displacement. The shear strength envelope in the postpeak stage is expressed by Mohr-Coulomb failure criteria:(4)
τ
=
τ
max
⁡
=
c
+
σ
n
tan
ϕ
,
where τ
max
⁡ is the shear strength of interface; c is the cohesion of interface; ϕ is the friction angle of interface.


## 4.1.1. Nonlinear-Elastic Stage

Before shear stress reaches peak shear strength, the relationship between shear stress and shear displacement of the interface can typically be modeled by a hyperbolic equation:(1)
τ
=
u
a
+
b
u
,
where u is shear displacement of interface.The parametersa and b can be expressed as
(2)
a
=
1
k
1
γ
w
(
σ
n
/
Pa
)
n
,
b
=
R
f
σ
n
tan
ϕ
+
c
,
where γ
w is the unit weight of water; σ
n is the normal effective stress of the interface; Pa is the atmospheric pressure; τ is the shear stress; c is the cohesion of the interface; ϕ is the friction angle of interface; k
1, n, and R
f are the nonlinear parameters that can be derived from interface direct shear tests.By combining the three equations above, the tangent modulus of the shear stress-displacement curve of the interface (shear stiffness)k
s can be expressed as follows [16]:
(3)
k
s
=
K
1
γ
w
(
σ
n
Pa
)
n
[
1
-
R
f
τ
c
+
σ
n
tan
ϕ
]
2
.


## 4.1.2. Perfectly Plastic Stage

When shear stress reaches the peak shear strength of interface, the plastic failure occurs with the increase of shear displacement. The shear strength envelope in the postpeak stage is expressed by Mohr-Coulomb failure criteria:(4)
τ
=
τ
max
⁡
=
c
+
σ
n
tan
ϕ
,
where τ
max
⁡ is the shear strength of interface; c is the cohesion of interface; ϕ is the friction angle of interface.


## 4.2. Parameters of CGM-Cushion Interface Model

The fitting curves of shear stress versus shear displacement from the test results of the two interfaces using nonlinear-elastic perfectly plastic interface model are shown in Figure10. The fitting parameters for the two interface models are given in Table 2. The comparisons between theoretical solutions calculated by the model and test results show good agreement under different normal loads.Table 2
Fitting parameters of two interfaces by nonlinear-elastic perfectly plastic model.
Parameters
K
n
R
f
c (Pa)
φ (°)
GM-crushed stones interface
1624
1.12
0.75
6930
35.2
GM-PUR mixed crushed stones interface
2871
0.185
0.893
5380
32.3Comparison of test data with fitting curves of interface model.
(a)
CGM-crushed stones interface
(b)
CGM-PUR mixed crushed stones interface


## 4.3. Numerical Implementation of Interface Constitutive Model

In order to simulate the mechanical behaviors of CGM-cushion interfaces, the nonlinear-elastic perfectly plastic interface model was incorporated into the built-in Geogrid element ofFLAC
3
D procedure by user-defined FISH program. The Geogrid element in FLAC
3
D provides interface model that is characterized by Coulomb sliding and shear bonding. Interfaces of Geogrid element have the properties of friction, cohesion, shear stiffness, and shear bond strength [17]. But the Geogrid element in FLAC
3
D can only simulate the interface with linear elastic characteristics. It cannot be used to simulate geosynthetic-soil interfaces which are characterized by the nonlinear behaviors. So the Geogrid element in FLAC
3
D should be improved to simulate the CGM-cushion interface behaviors by user-defined fish program.The detailed program flow chart of implementation of CGM-cushion interface model inFLAC
3
D procedure is shown in Figure 11.Figure 11
Program flow chart of user-defined interface model of Geogrid element.As shown in Figure11, at every calculation step, the program first reads the normal effective stress and shear stress of the three interface nodes for every Geogrid element. The resultant interface normal stress and shear stress of the Geogrid element are calculated according to the node variables. Then the stiffness of the interface was calculated according to the resultant interface variables by (3). The shear stiffness is inputted into the Geogrid element using fish functions. Then, the calculated new parameters for each Geogrid element were inputted to carry out the calculation of the next step. In this way, the program continuously circulated until the cycle for all Geogrid elements has been completed.


## 4.4. Verification of Interface Constitutive Model

In order to verify the effectiveness of the imbedded interface model, a simple numerical example is chosen to model the direct shear test results of CGM-cushion interfaces. As shown in Figure12, the numerical model of the test is composed of two parts. The upper part is a shear box with cushion materials in it; the lower one is a rigid block where the geomembrane is glued at the top. The numerical model of cushion is a 300 × 300 × 100 mm cuboid which is simulated using hexahedral brick elements. The numerical model of rigid block is a 300 × 350 × 100 mm cuboid which is also simulated using hexahedral brick elements. The Geogrid element is used to simulate the GCM. The newly imbedded interface model is used to simulate the CGM-cushion interface behaviors. The boundary conditions correspond to roller boundaries on both sides (y-direction andx-direction) and to fixed displacements in they- andz-direction at the model base.Figure 12
Grid of numerical model for direct shear test.In order to compare the numerical results with theoretical solutions and test results, a linear elastic model was employed for the cushions in the upper box as well as the rigid block in the lower box. Gravity forces of cushions and rigid block were not considered during the numerical modeling. Parameters in Table2 were used in numerical model. According to the typical procedures for direct shear test, a constant normal pressure was applied on the top surface of the cushion. Then, the displacements and velocities of all elements were reset to zero. A fixed shear velocity of 1 × 10−5 m/(time step) was applied to all the elements of the lower box to simulate the actual shearing rate of 1 mm/min. That led to a shear displacement on the interface between the upper cushion and the lower CGM. Four numerical direct shear tests were conducted with constant normal pressures of 25 kPa, 50 kPa, 75 kPa, and 100 kPa, respectively.The numerical computed results of the average shear stress versus shear displacement curves of two interfaces and the comparison with test results and theoretical solution are shown in Figure13. It is obvious that the numerical results are very close to the theoretical solutions calculated by (1)–(4) using the same parameters. Both the numerical and theoretical solutions show good agreements with test results for the two interfaces. Figure 13 illustrates that the implemented interface model in Geogrid element of FLAC
3
D is capable of modeling the nonlinear-elastic and perfectly plastic behavior of interface between CGM and two permeable cushions.Comparison of numerical results with test results and theoretical solution of interfaces.
(a)
CGM-crushed stones interface
(b)
CGM-PUR mixed crushed stones interface


## 5. Conclusions

Based on the present study, the following conclusions can be drawn for the interface behaviors of CGM and two different permeable cushions.(i)
Shear stress versus shear displacement curves of interfaces between CGM and two cushions have shown different features. The nonlinear and strain hardening characteristics can be found for CGM-crushed stones interface. Nonlinear and perfectly plastic failure characteristics were found for CGM-PUR mixed crushed stones interface.
(ii)
The peak shear strength of the two interfaces can be expressed as a function of normal stress by Mohr-Coulomb criteria. The peak shear strength of CGM-crushed stones interface is always higher than CGM-PUR mixed crushed stones interface under different normal loads. Both lower friction angle and adhesion intercepts were observed for the CGM-PUR mixed crushed stones interface.
(iii)
By fitting test data, the nonlinear hyperbolic model and nonlinear-elastic perfectly plastic interface model can be used to predict the mechanical behaviors of the two interfaces, respectively. The interface model of CGM-cushions was implemented into the Geogrid element ofFLAC
3
D. By simulation of the interface direct shear tests, the incorporated interface model was proved correct and cable of modeling the mechanical behaviors of CGM-permeable cushion interfaces.



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*Source: 184359-2014-10-22.xml*