Simple Test of Liner-Zone Interface Behavior

Problem Statement

Note

The project file for this example is available to be viewed/run in FLAC3D. The main data files used are shown at the end of this example. The remaining data files can be found in the project.

A liner is placed on top of a soil block and then moved in various directions. The normal and shear stress and displacement of one of the liner coupling springs is monitored and plotted to demonstrate the effects of the liner-zone interface properties on system response.

The FLAC2D model is a one cubic meter block of soil composed of 24 zones, with a concrete liner composed of 2 liner elements centered on top of the soil block, as shown in Figure 1. The soil is assigned elastic properties (\(E\) = 15 MPa, \(\nu\) = 0.3). The liner material is assigned properties corresponding with a 0.1 m thick concrete liner (\(E\) = 25 GPa, \(\nu\) = 0.15, \(t\) = 0.1 m). The liner-zone interface properties (\(k_n\), \(k_s\), \(f_t\), \(c\), \(c_r\), \(\phi\)) are assigned as follows.

The interface stiffnesses (\(k_n\) and \(k_s\)) are set equal to 8 × 10⁸ N/m to ensure small interface deformation using the liner-stiffness equation. The following interface strength properties are assigned:

tensile strength (\(f_t\))	4 MPa
cohesion (\(c\))	4 MPa
residual cohesion (\(c_r\))	2 MPa
friction angle (\(\phi\))	20 degrees

In order to demonstrate the liner-zone interface behavior, the following conditions are imposed. The velocities of all gridpoints and nodes are fully fixed. The same velocities are applied to all nodes such that the liner moves as a rigid body. These conditions prevent internal stresses from developing in both the zones and the liner material; only the liner-zone interface springs are being exercised. The normal and shear stress and displacement at the center coupling spring is monitored and plotted. In the plots of normal stress versus normal displacement, the sign convention is that compression and overlap are positive (to correspond with the figure representing normal interface behaviors). In the plots of shear stress versus shear displacement, the magnitude of both shear stress and shear displacement are plotted (to correspond with the figure representing shear interface behaviors).

Figure 1: FLAC2D model for simple test of liner-zone interface behavior.

The test is performed in six stages. During each stage, a normal or shear displacement is applied, while the other displacement component is fixed. The displacements applied during each stage are listed in Table 2, where positive/negative normal displacement indicates separation/overlap, and shear motion along the global x-direction. The testing results are shown in Figure 2 to Figure 7.

Table 2: Applied Liner Displacements

Stage	Normal (mm)	Shear (mm)
1a	0	+15
1b	0	-5
2	-10	0
3	0	+10
4	+20	0
6a	-20	0
6b	0	+6

During stage 1, a positive shear displacement of 15 mm is followed by a negative shear displacement of 5 mm. The response is shown in Figure 2, which can be compared with shear interface behavior. The slope of this plot equals the shear coupling spring stiffness per unit area, \(k_s\), and the maximum value equals the cohesion, \(c\) (because, at this stage, the interface normal stress is zero). The unloading slope is the same as the loading slope.

During stage 2, the liner is moved downward by 10 mm. The response is shown in Figure 3, which can be compared with normal interface behavior. The slope of this plot equals the normal coupling spring stiffness per unit area, \(k_n\). At this stage, the interface normal stress, \(\sigma_n\), equals 8 MPa. During stage 3, an additional shear displacement of 10 mm is applied, and the response is shown in Figure 4. The shear stress increases and reaches a maximum value of 6.91 MPa, which corresponds with the relation

\[\tau_{\rm max} = c + \sigma_n \tan\phi\]

where:	\(c\)	=	cohesion;
	\(\sigma_n\)	=	interface normal stress; and
	\(\phi\)	=	friction angle.

During stage 4, the liner is moved upward by 20 mm. The response is shown in Figure 5. When the interface normal stress reaches the tensile strength, \(f_t\), of 4 MPa, the spring breaks, causing the interface normal stress to drop to zero. When the liner fails in tension, the effective cohesion becomes \(c_r\), and the tensile strength is set to zero. The total relative normal displacement continues to be tracked such that compressive normal stress will again develop when the gap closes. During stage 5, an additional shear displacement of 2 mm is applied, and the response is shown in Figure 6. The peak shear stress drops from 6.91 to 2 MPa, because the interface normal stress is zero, and the residual cohesion is now being used. During stage 6, the liner is moved downward by 20 mm, and then an additional shear displacement of 6 mm is applied. The response is shown in Figure 7. The gap has closed, and the interface normal stress is again 8 MPa. The peak shear stress is 4.91 MPa, in accord with the residual cohesion.

Figure 2: Shear stress versus shear displacement (after stage 1).

Figure 3: Normal stress versus normal displacement (after stage 2).

Figure 4: Shear stress versus shear displacement (after stage 3).

Figure 5: Normal stress versus normal displacement (after stage 4).

Figure 6: Shear stress versus shear displacement (after stage 5).

Figure 7: Shear stress versus shear displacement (after stage 6).

Data File

LinerZoneInterfaceTest.dat

; Liner example application
; Simple test of Liner-Zone Interface Behavior
model new
model large-strain off
model title 'Simple Test of Liner-Zone Interface Behavior'

; Create the soil.
zone create quadrilateral size 6,4 point 0 (0.0,0.0) point 1 (1.0,0.0) ...
                             point 2 (0,1.0) point 3 (1.0,1.0)
zone cmodel assign elastic
zone property bulk=12.5e6 shear=5.77e6  ; soil
; Create the concrete liner.
structure liner create by-zone-face range position (0.33,1) (0.66,1)

structure liner property young 25.0e9 poisson 0.15 cross-sectional-area 0.01 moi 8.33e-6; concrete
structure liner property coupling-stiffness-normal=8e8 ...
             coupling-stiffness-shear=8e8 coupling-yield-normal=4e6 ...
             coupling-cohesion-shear=4e6 ...
             coupling-cohesion-shear-residual=2e6 coupling-friction-shear=20.0
;
zone gridpoint fix velocity ; fully fix all zones
structure node fix velocity    ; fully fix all nodes

structure liner history name 'NormalStress' coupling-stress-normal component-id 1 end 2
structure liner history name 'NormalDisp' coupling-displacement-normal component-id 1 end 2
structure liner history name 'ShearStress' coupling-stress-shear component-id 1 end 2
structure liner history name 'ShearDisp' coupling-displacement-shear component-id 1 end 2

; Apply six sets of normal/shear displacements to liner.
structure mechanical damping combined-local
structure node initialize velocity (1.0e-5,0) local; norm disp= 0 mm, shr disp= +15 mm
model cycle 1500                                   
structure node initialize velocity (-1.0e-5,0) local ; norm disp= 0 mm, shr disp= -5 mm
model cycle 500                                   
model save 'stage1'
structure node initialize velocity (0,-1.0e-5) local ; norm disp= -10 mm, shr disp=0 mm
model cycle 1000                                
model save 'stage2'
structure node initialize velocity (1.0e-5,0.0) local ; norm disp=0 mm, shr  disp= +10 mm
model cycle 1000                                            
model save 'stage3'

structure node initialize velocity (0.0,1.0e-5) local ; norm disp= +20 mm, shr  disp= 0 mm
model cycle 2000                                 
model save 'stage4'

structure node initialize velocity (1.0e-5,0) local ; norm disp=   0 mm
model cycle 200                                             ; shr  disp=  +2 mm
model save 'stage5'

structure node initialize velocity (0.0,-1.0e-5) local ; norm disp= -20 mm, shr  disp=   0 mm
model cycle 2000                                  ; 
structure node initialize velocity (1.0e-5,0) local ; norm disp=0 mm, shr disp=  +6 mm

model cycle 600 
model save 'stage6'