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A Calibration Procedure for Modeling HDPE Geomembrane Using Discrete Element Method

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Abstract

Geomembranes are geosynthetic impermeable materials used as hydraulic barriers in waste containment facilities. Continuum methods are generally used to analyze the behaviour of geomemberanes and calculate tensile and interface stresses under various loading conditions. However, it is sometimes desired to simulate the interaction behaviour between a geomembrane liner and granular soil subjected to large movements. Discrete element methods have proven to be efficient in modeling granular materials using discrete particles. Using the same procedure to model geomembranes would lead to significant reduction in calculation cost and eliminates the need to use hybrid methods, which require simultaneous use of both continuum and discontinuum modeling approaches. This study presents a procedure to calibrate a discrete element model of a HDPE geomembrane using spherical particles. A constitutive model that takes into account particle normal and shear cohesion is used. Standard index tests used to measure the properties of HDPE geomembrane including tensile and puncture tests are applied to validate the model developed. The effect of microscopic parameters on the overall response is examined and recommendations are made regarding to the optimum approach to simulate continuous geomembrane materials using discrete element method. RÉSUMÉ Les géomembranes sont des matériaux imperméables géosynthétiques utilisés comme barrières hydrauliques dans les installations de confinement des déchets. Les méthodes de Continuum sont généralement utilisés pour analyser le comportement des geomemberanes et calculer la traction et contraintes d'interface dans diverses conditions de charge. Cependant, il est parfois souhaitable de simuler le comportement d'une interaction entre une géomembrane et un sol granulaire soumis à de grands mouvements. Les méthodes d'éléments discrets se sont avérés efficaces dans la modélisation de matériaux granulaires en utilisant des particules discrètes. En utilisant la même procédure pour modéliser les géomembranes conduirait à une réduction significative des coûts de calcul et élimine la nécessité d'utiliser des méthodes hybrides qui nécessitent simultanément l'utilisation des approches de continuum et de discontinuum. Cette étude présente une procédure pour calibrer un modèle d'élément discret d'une géomembrane en HDPE en utilisant des particules sphériques. Un modèle constitutif est utilise qui tient compte des cohésions normale et de cisaillement de particule. Les tests d'index standards utilisés pour mesurer les propriétés de HDPE géomembrane, y compris des essais de traction et de perforation sont appliquées pour valider le modèle développé. L'effet des paramètres microscopiques sur la réponse globale est examiné et des recommandations sont formulées en ce qui concerne l'approche optimale pour simuler des matériaux de géomembrane en continu en utilisant la méthode des éléments discrets.
A Calibration Procedure for Modeling
HDPE Geomembrane Using Discrete Element Method
Masood Meidani 1, Mohamed A. Meguid 2& Luc E. Chouinard 2
(1) Graduate student, (2) Associate Professor
Department of Civil engineering and applied mechanics, McGill University
Montreal. QC, Canada, H3A 0C3
ABSTRACT
Geomembranes are geosynthetic impermeable materials used as hydraulic barriers in waste containment facilities.
Continuum methods are generally used to analyze the behaviour of geomemberanes and calculate tensile and interface
stresses under various loading conditions. However, it is sometimes desired to simulate the interaction behaviour between
a geomembrane liner and granular soil subjected to large movemen ts. Discrete element methods have proven to be
efficient in modeling granular materials using discrete particles. Using the same procedure to model geomembranes would
lead to significant reduction in calculation cost and eliminates the need to use hybrid methods, which require simultaneous
use of both continuum and discontinuum modeling approaches. This study presents a procedure to calibrate a discrete
element model of a HDPE geomembrane using spherical particles. A constitutive model that takes into account particle
normal and shear cohesion is used. Standard index tests used to measure the properties of HDPE geomembrane including
tensile and puncture tests are applied to validate the model developed. The effect of microscopic parameters on the overall
response is examined and recommendations are made regarding to the optimum approach to simulate continuous
geomembrane materials using discrete element method.
RÉSUMÉ
Les géomembranes sont des matériaux imperméables géosynthétiques utilisés comme barrières hydrauliques dans les
installations de confinement des déchets. Les méthodes de Continuum sont généralement utilisés pour analyser le
comportement des geomemberanes et calculer la traction et contraintes d'interface dans diverses conditions de charge.
Cependant, il est parfois souhaitable de simuler le comportement d'une interaction entre une géomembrane et un sol
granulaire soumis à de grands mouvements. Les méthodes d'éléments discrets se sont avérés efficaces dans la
modélisation de matériaux granulaires en utilisant des particules discrètes. En utilisant la même procédure pour modéliser
les géomembranes conduirait à une réduction significative des coûts de calcul et élimine la nécessité d'utiliser des
méthodes hybrides qui nécessitent simultanément l'utilisation des approches de continuum et de discontinuum. Cette
étude présente une procédure pour calibrer un modèle d'élément discret d'une géomembrane en HDPE en utilisant des
particules sphériques. Un modèle constitutif est utilise qui tient compte des cohésions normale et de cisaillement de
particule. Les tests d'index standards utilisés pour mesurer les propriétés de HDPE géomembrane, y compris des essais
de traction et de perforation sont appliquées pour valider le modèle développé. L'effet des paramètres microscopiques sur
la réponse globale est examiné et des recommandations sont formulées en ce qui concerne l'approche optimale pour
simuler des matériaux de géomembrane en continu en utilisant la méthode des éléments discrets.
1 INTRODUCTION
In the field of solid waste landfill engineering, the use and
acceptance of geosynthetics and high density polyethylene
(HDPE) geomembrane (GM) has increased over the past
few years. HDPE geomembrane is usually used as a
hydraulic barrier in waste containment applications
including municipal solid waste facilities.
One of the greatest risk of damage in geomembranes
is associated with stress concentrations from direct contact
with coarse soil particles (e.g., gravel or stones), which can
occur from an underlying soil subgrade or an overlying
granular soil layer (Nosko and Touze-Foltz 2000; Giroud
and Touze-Foltz 2003). Extensive research has been
conducted on granular soil-geomembrane interaction using
experimental and numerical methods (Reddy et al. 1996a;
Koerner et al. 2010; Hornsey and Wishaw 2012; Brachman
and Sabir 2013). Among the different numerical methods
that have been developed by researchers to study this
interaction, the discrete element method (DEM) has proven
to be efficient in modeling granular materials involving large
deformations. Also, using the same approach to model
geomembrane leads to significant reduction in calculation
cost in comparison with other methods such as hybrid
procedure that requires simultaneous use of both
continuum and discontinuum modeling approaches.
The discrete element method (DEM) has gained
popularity in the past few decades among geotechnical
engineers and researchers involved in granular soil-
structure interaction problems. The method was first
proposed by Cundall and Strack (1979) and has been used
to analyse geotechnical engineering problems. Laboratory
tests such as triaxial and direct shear have been modelled
using DEM to investigate the microscopic behaviour of soil
samples (Cui and O'Sullivan 2006). Also, several
researchers applied this method to model soil-
geosynthetics problems including elements such as
textiles, grids and membranes (McDowell et al. 2006;
Effeindzourou et al. 2016). In most of these studies a
membrane is modelled using a set of spherical particles
bonded together. These bonded particles can simulate the
membrane behaviour correctly if the input parameters are
chosen precisely.
In this work, a calibration procedure is proposed which
takes into account the role of each parameter in the
macroscopic behaviour. Two index tests, namely, tensile
and puncture tests are numerically simulated to determine
the microscopic parameters of the bonded HDPE
geomembrane particles.
2 DISCRETE ELEMENT MODELING
2.1 General formulation
The discrete element method (DEM) treats the interaction
between particles as a dynamic process that reaches static
equilibrium when the internal and external forces are
balanced. This dynamic process is usually modeled using
a time-step algorithm based on an explicit time-difference
scheme. Displacement and rotation of each particle are
then determined using Newton’s and Euler’s equations.
The DEM simulation in this study are performed using the
open source discrete element code YADE (Kozichi and
Donze, 2008; Smilauer et al., 2010).
The contact law between particles is briefly described
below:
After collision of particles A and B with radii rAand rB,
contact penetration depth is defined as
∆= + [1]
Where is the distance between centers of particles A
and B. Interaction between the two particles is represented
by the force vector F. This vector can be decomposed into
normal and tangential forces (Fig. 1)
= . [2]
= . [3]
Where is the normal force; is the incremental
tangential force; and are the normal and tangential
stiffnesses at the contact point; is the normal
penetration between the two particles and is the
incremental tangential displacement between the two
particles.
The normal stiffness between particle A and B at
contact point is defined by
=.[4]
Where and are the particles normal stiffnessess
calculated using particle radius and the particle material
modulus .
= 2 = 2 [5]
So the normal stiffness at contact point can be written
as:
=.[6]
The interaction tangential stiffness is defined as a
ratio of the computed as = .
Figure 1. Interaction between two DE particles
Rolling resistance between two particles A and B is
determined using a rolling angular vector . This vector is
calculated by summing the angular vector of the
incremental rolling (Smilauer et al., 2010)
=[7]
A resistant moment is calculated by
= . [8]
Where is the rolling stiffness of the interaction and is
defined as
= . ( ) . [9]
Elastic limits can be defined for Eqs. (2) and (3) using
shear ( )and tensile strength ( ).
× [10]
tan + × [11]
Where is the microscopic friction angle between
particles and =is the reference surface area ( is
the reference radius of the contact, = min ( and )).
Note that normal force is only limited in traction and it is
assumed that compression at contact is always elastic.
2.2 Discrete element modeling of a flexible membrane
The developed HDPE geomembrane model consists of
an array of bonded spherical particles which are arranged
hexagonally. The bonds are defined by shear and normal
tensile strength, set high enough that the membrane does
not split. Also rotation of particles and the transmission of
moments are restricted to ensure membrane flexibility (De
Bono et al. 2012). The main properties of the spherical
particles which are needed for the calibration procedure are
listed in Table 1. Among these parameters, the value of the
micro-friction angle () is assigned to zero based on
the findings of De Bono et al. (2012) and Bourrier et al.
(2013). All four remaining parameters need to be extracted
using the calibration method described in the next section.
2.2.1 Tensile test specimen
The tensile test specimen is created based on ASTM
D6693 (standard test method for determining tensile
properties of flexible geomembranes). The specimen has a
dog bone shape and its dimensions are illustrated in Fig. 2.
Table 1- Parameters of the contact model used in the
modeling of HDPE geomembrane
Properties
Particle material modulus ( )
Density
Micro friction angle ()
=
Tensile strength ( )
Shear strength ( )
The test procedure is described below
1- Measuring the width and thickness of the sample
(W=6 mm, t=1.5 mm)
2- Placing the specimen in the grips of the test
apparatus (to prevent slippage of the specimen).
Grip dimension is 25 mm on each side.
3- Installing the strain gage on the specimen (gage
initial length=33 mm).
4- Applying the load at a rate of 50 mm/min on the
right side while the left grip is fixed. Then, recording
the load-displacement data.
Figure 2. Tensile test specimen dimensions
The diameter of the particles in the discrete element
model is chosen considering a balance between simulation
time and the geomembrane flexibility. Based on these
criteria, spherical particles with diameter of 0.3 mm are
created and arranged in hexagonal pattern. Two
specimens with different thicknesses are created. First
sample with thickness of 0.3 mm consists of 28564
particles arranged in one row. The other sample includes 6
rows of the first specimen with 171,384 particles and final
thickness of 1.5 mm. Most of the particles located between
the grips do not have interactions with other particles as
they have a zero or constant velocity under the specified
test condition. Hence, to increase the simulation speed,
only 2.5 mm of each grip is modeled. Figure 3 illustrates
the final discrete element samples of the tensile test and a
close view of the specimen.
2.2.2 Puncture test specimen
The puncture test specimen is created based on ASTM
D4833 (standard test method for index puncture resistance
of geomembranes). The specimen has a circular shape
and its dimensions are illustrated in Fig. 4.
Figure 3. a) Top view of the first test specimen with
thickness of 0.3 mm and a partial view of the specimen to
illustrate the hexagonal arrangement of particles, b) A 3D
view of specimen with a thickness of 1.5 mm.
To perform the puncture test, geomembrane needs to
be fixed among an O-ring plate with outer diameter of 100
mm and an open internal diameter of 45 mm. Then a solid
steel rod (test probe) is pushed downward with a speed of
300 mm/min towards the center. Probe load (puncture
resistance) is recorded until the steel rod completely
ruptures the test specimen.
The diameter of the particles in the discrete element
model and their arrangement are chosen the same method
as the tensile test specimens; and two samples with
different thickness are created as well. Particles in the fixed
part of the sample don’t have any effects on the outcome
force. Hence, to decrease the number of particles and
duration of the simulation, only 2.5 mm of the fixed part is
created. Thus, the diameter of the DE sample is 50 mm.
Two samples with thicknesses of 0.3 mm and 1.5 mm and
total number of particles of 25,198 and 151,188,
respectively, are created. The discrete element model of
the puncture test and a partial view of the sample are
presented in Fig. 5.
Figure 4. Puncture test details
Figure 5. a) Top view of first puncture test specimen with
thickness of 0.3 mm and a partial view of the specimen to
illustrate the hexagonal arrangement of particles, b) A 3D
view of specimen with thickness of 1.5 mm.
3 CALIBRATION OF THE LOCAL PARAMETERS
The calibration of the materialproperties with respect to the
real geomembrane is performed by comparing a simulated
and a laboratory test results. Once calibrated, the
predictive capabilities of the numerical model is checked
and validated by simulating the puncture test. For the
calibration step, the selected local parameters include,
particle material modulus ( ), Tensile strength ( ), Shear
strength ( ) and the ratio between tangential and normal
stiffness ( ). The choice of these parameters should allow
for the correct macroscopic values (Young’s modulus ,
tensile strength at the yield point, tensile elongation at yield
and puncture resistance) to be reproduced. To achieve this
objective, the impact of each local parameter on the
macroscopic response needs to be identified. Based on the
previous studies (Calvetti et al. 2003, Sibille et al. 2006,
Plassiard et al. 2009) it was found that elastic parameters
( and ) and rupture parameters ( , ) can be
calibrated separately.
The particle modulus ( ) is known to play an important
role in the elastic response whereas, the ratio between
tangential and normal stiffness ( ) has nosignificant impact
on material Young’s modulus . Therefore, will be used
first to calibrate the macroscopic elastic behavior. The
value of is set to 0.3 based on that reported by
Effeindzourou et al. (2016) in modeling a deformable
structure using DEM. Using this value for , is set such
that the target Young’s modulus based on the tensile test
results is obtained. As presented in Fig. 6, as the particle
modulus ( ) increases, the Young’s modulus of the
geomembrane increases.
Figure 6. Dependency of Young’s modulus on particles
material modulus ( )
Once the elastic parameters are set, the values of the
rupture parameters ( , ) can be determined. Changing
these two parameters separately was found to lead to
divergence in the results. Equal values for the two
parameters were considered in consistency with De Bono
et al. (2012), Bourrier et al. (2013) and Effeindzourou et al.
(2016). These two parameters were found to affect the
peak stresses with little to no effect on Young’s modulus as
illustrated in Figure 7.
Figure 7. Dependency of peak stress on particles tensile
and shear strength
4 APPLICATION OF THE PROPOSED METHOD TO
SIMULATE GEOMEMBRANE RESPONSE TO
LOADING
The selected HDPE GM was manufactured by Layfield
Corp. (USA and Canada). Geomembrane specimen has a
thickness of 1.5 mm with blown-film texturing on both sides.
The GM material properties are given in Table 2.
Following the calibration procedure described in section
2, tensile test is modeled using DEM. At first a specimen
with thickness of 0.3 mm is created and the input
parameters are determined using the calibration method
(see Table 3). To validate these parameters a second
0
10
20
30
40
50
60
0 1 2 3 4 5 6 7 8
Tensile strength (kN/m)
Strain (%)
Ei=3e9 (Pa)
Ei=4e9 (Pa)
Ei=5e9 (pa)
0
5
10
15
20
25
30
0 3 6 9 12 15
Tensile strength (kN/m)
Strain (%)
Cn=Ct= 8e8 (Pa)
Cn=Ct= 1e9 (Pa)
Cn=Ct= 8e9 (Pa)
tensile test specimen with a thickness of 1.5 mm is created
and the micro-parameters are assigned to the particles.
Results are summarized in Table 4 which show
consistency between the calculated results and the
experimental data.
Table 2- Material properties of the selected HDPE
geomembrane
Properties
Value
Thickness
1.5 mm
Density
0.94 g/cc
Tensile strength at yield
22 kN/m
Tensile elongation at yield
12 %
Puncture resistance
480 N
The effect of the applied tension force can be further
examined by inspecting the contact force distribution within
the geomembrane specimen. Figure 8 shows the contact
force network in the geomembrane during the test = 6%).
Most of the contact forces are directed parallel to the
applied external load. In addition, the magnitude of the
forces is larger for the narrow section as compared to the
rest of sample.
Table 3- Input parameters of the contact model obtained
from the selected HDPE geomembrane using the
proposed calibration method
Properties
Value (Unit)
Particle material modulus (Ei)
3.0E9 (Pa)
Density
0.94 (g/cc)
Micro friction angle ()
0 (Degree)
=
0.3
Tensile strength ( )
1.0E9 (Pa)
Shear strength ( )
1.0E9 (Pa)
Table 4- Comparison between calculated and measured
tensile strength and Young’s modulus of the
geomembrane
Test method
Thickness (mm)
0.3
1.5
Experiment
22
22
Numerical
23
25
Experiment
12
12
Numerical
12.2
12.5
Figure 8. Contact force network in tensile test simulation
The puncture test was also simulated using two
specimens of different thicknesses and the input
parameters are assigned to the used particles. Numerical
results were found to be in agreement with the
experimental data as shown in Table 5. The contact force
network distribution during the puncture test before and
after failure are illustrated in Figures 9 and 10. As
presented in Fig. 9 contact forces are higher near the edge
and under the test probe in comparison with the rest of the
sample. Also, the specimen failure mode is found to be
similar to that observed in the experiment. The above
results confirm that the proposed DEM based method is
acceptable in modeling the response of geomembrane
material.
Table 5- Comparison between calculated and measured
puncture resistance of the geomembrane
Thickness
(mm)
Puncture resistance (kN)
Experiment
Numerical
1.5
480
505
0.3
96
101
Figure 9. Contact force network in puncture test before
the failure
Figure 10. Top view of the puncture test simulation at
failure state
5 CONCLUSION
A DEM model has been created that can simulates tensile
and puncture tests performed on HDPE geomembrane.
Bonded spherical particles are used to create a flexible
membrane material allowing for the correct deformation
pattern to develop. A calibration procedure is proposed
which attempts to consider the respective roles of each
local parameter on the macroscopic behaviour of the
material.
Numerical simulations areperformed to simulate tensile
and puncture tests conducted on a specific HDPE
geomembrane to evaluate the applicability of the proposed
method. An acceptable agreement between the numerical
and experimental results is obtained. In spite of the
simplicity of the suggested calibration method, the
numerical model was able to reproduce the main features
of the tensile and puncture tests up to the yielding point.
The calibration method presented in this study and the
ability of creating a flexible membrane using DEM, shows
that discontinuous methods are promising in modeling the
interaction between granular soil and geomembrane
material.
6 ACKNOWLEDGEMENT
This research is supported by the Natural Sciences and
Engineering Research Council of Canada (NSERC).
Financial support provided by McGill Engineering Doctoral
Award (MEDA) to the first author is appreciated.
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FOREWORD: Every four years, an International Conference on Geosynthetics provides an opportunity to bring together, from all over the world, a variety of people involved in this field. Thus, in September 2002, 1200 participants attended the 7 th International Conference on Geosynthetics held in Nice, France. Among several innovative features, this conference included ‘special sessions’ on key subjects selected by the conference Scientific Committee. For each special session, two session leaders, known for their expertise on the subject, were invited to plan the session several months before the conference, and, in particular, to select topics for discussion. The special session on ‘Geomembranes in landfills’ was especially significant, as it was organized with the cooperation of Geosynthetics International. This cooperation between a conference and a technical journal was proposed by the conference Scientific Committee as an original experiment aimed at promoting the diffiusion of valuable technical information. The following paper reports in great detail the discussion that took place at the special session on ‘Geomembranes in landfills’. We must say that this special session, with more than 200 attendees, and the resulting paper fully matched our highest expectations thanks to the leadership provided during the session by J. P. Giroud and N. Touze-Foltz, and their careful editing of the discussion transcript. We should also acknowledge the contribution of the fifteen attendees who took part in the discussion and agreed to review the edited transcript. It is no secret that discussion transcripts require extensive editing prior to publication. The two session leaders should be commended for producing an edited version that remains very lively and retains the favour of heated debates between researchers and practitioners. The readers will also appreciate that the paper reports the discussion of the three important topics selected by the session leaders in a well-organized manner. Based on the reported discussion, it is clear that there are significant discrepancies between the state of the art and the state of practice, as well as discrepancies between the states of practice in different countries. The paper that follows is a first step towards a reduction of these discrepancies and will, therefore, benefit practitioners in many countries. Certainly, one of the most interesting features of the discussion at the special session on ‘Geomembranes in landfills’ is the active participation of practitioners, whereas too often at international conferences practitioners essentially listen to researchers. It would have been a great loss had this discussion not been reported, as is unfortunately the case of most discussions at most conferences. Therefore, the readers of this paper will certainly agree with us that this experiment of cooperation with a technical journal should be repeated at future International Conferences on Geosynthetics. by J. P. Gourc, H. Girard and P. Delmas *