Micromechanical Discrete Element Modeling of FiberReinforced Polymer Composites
Ahmed Khattab,1 Mohammad J. Khattak,2 Imran M. Fadhil31Department of Industrial Technology, College of Engineering, University of Louisiana at Lafayette,Lafayette, Louisiana 70504-2972
2Department of Civil Engineering, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-2291
3Department of Petroleum Engineering, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-4690
An analytical model of mechanical behavior of carbonfiber reinforced polymer composites using an advanceddiscrete element model (DEM) coupled with imagingtechniques is presented in this article. The analysisfocuses on composite materials molded by vacuumassisted resin transfer molding. The molded compositestructure consists of eight-harness carbon fiber fabricsand a high-temperature polymer. The actual structure ofthe molded material was captured in digital imagesusing optical microscopy. DEM was developed using theimage-based-shape structural model to predict thecomposite elastic modulus, stress–strain response, andcompressive strength. An experimental case study ispresented to evaluate the accuracy of the developedanalytical model. The results indicate that the image-based DEM micromechanical model showed fairlyaccurate predictions for the elastic modulus andcompressive strength. POLYM. COMPOS., 32:1532–1540,2011. ª 2011 Society of Plastics Engineers
INTRODUCTION
The discrete element (DE) method is a numerical tech-
nique, in which Newton’s second law and a finite differ-
ence scheme are utilized to study the interaction among
discrete particles in contact. In this technique, Newton’s
law of motion is successively solved for each particle and
force-displacement law is applied for every contact. An
explicit time stepping scheme is employed to integrate
Newton’s law of motion for each particle. This results in
a set of contact forces acting on the particle which are
updated after each time step. Based on the new contact
forces, new unbalanced forces are produced and the
particles’ positions and velocities are calculated. With
these new positions, the relative displacements of each
pair of particles are updated and used [1]. The DE method
was introduced by Cundall [2] for analysis of rock
mechanics, and then applied to soils by Cundall and
Strack [3]. Today, in addition to the granular materials,
solid materials can be modeled with bonded contact mod-
els. Many computation codes have been developed and
applied to granular and solid materials simulation with
the DE method, such as the early problem Ball and
revised version of Ball, or Trubal, universal distinct
element code (UDEC), DE code in three dimensions
(3DEC), and particle flow code in two and three dimen-
sions (PFC2D/3D) [1–13]. PFC2D/3D codes have higher
computation efficiency; and the ability to model fracture
cracks within the reinforcement materials and the matrix,
as well as the interface between the reinforcement and the
matrix.
This article presents the modeling of mechanical
behavior of carbon fiber reinforced polymer composites
using an advanced discrete element model (DEM) coupled
with latest imaging techniques [11–13]. Traditionally, the
composite materials are modeled using a continuum
mechanics approach. However, the composite materials
comprise at least of two constituents: a matrix and rein-
forcement. In general, the deformation takes place in the
matrix and the microstructure of reinforcement skeleton.
In this study, the DEM approach considered the micro-
structure and the material complexity of the composite.
The fundamental properties of matrix and reinforcement
material were used in the DEM. The actual structure of
the molded polymer composite material was captured in
digital images using optical microscope. DEM was devel-
oped using the image-based-shape structural model and
fundamental properties of the molded material to predict
the composite elastic modulus, stress–strain response, and
compressive strength.
Correspondence to: Ahmed Khattab; e-mail: [email protected]
DOI 10.1002/pc.21182
Published online in Wiley Online Library (wileyonlinelibrary.com).
VVC 2011 Society of Plastics Engineers
POLYMER COMPOSITES—-2011
ANALYTICAL INVESTIGATION
Microfabric Micromechanical Discrete Element Modeling
This study utilized an extension of microfabric micro-
mechanical discrete element modeling technique [11–13]
to model the micromechanical behavior of carbon fiber
reinforced polymer composites. The matrix (polymer) and
the reinforcement (carbon fibers) materials were modeled
with clusters of very small DEs. The modeling method
maintained all the benefits of traditional DEM, such as
the ability to handle complex and changing contact
geometries, displacements, potential crack initiation
contacts, and the ability to simulate specimen assembly.
By modeling the carbon fiber and polymer with a mesh
of small DEs called ‘‘microfabric,’’ it was possible to
model complex shape and contours and mechanical char-
acteristics of the composite during a strength test simula-
tions. The constitutive behavior of the composite was
simulated using the software called, Particle Flow Code
2D (PFC2D) [1].
Micromechanical Behavior
In the DEM approach, the complex behavior of a mate-
rial can be simulated by combining simple contact constitu-
tive models with complex geometrical features. Shear
and normal stiffness, static and sliding friction, and inter-
particle bonds (cohesion/adhesion) are three of the simpler
contact models which can be employed. The stiffness
model provides an elastic relationship between the contact
force and relative displacement between particles [1].
Figure 1 illustrates two particles A and B in contact, where
normal stiffnesses are shown to have magnitudes KnA and
KnB, and shear stiffnesses are shown with magnitudes KsA
and KsB. For a linear contact model, the contact stiffness is
computed by assuming that the stiffnesses of the two con-
tacting entities act in series. The force-displacement law of
the two particles in contact in a contact-stiffness model can
be expressed using the following relationships.
Kn ¼ KnAKn
B
KnA þ Kn
Bð1Þ
Ks ¼ KsAKs
B
KsA þ Ks
Bð2Þ
Fn ¼ n:Kn:Un ð3ÞDFs ¼ �Ks:DUs ð4Þ
Equation 3 relates the total normal force (Fn) to the total
normal displacement (Un) where n is the total number of
contacts. Equation 4 relates the incremental shear force
(DFs) to the incremental shear displacement (DUs). The
normal and shear strength between two contacting balls
can be simulated using simple contact-bond models,
which are applied at the contact point (Figs. 1 and 2).
When the tensile and/or shear stress at a contact exceeds
the strength of the bond (Sn and/or Ss), the bond breaks,
and separation and/or frictional sliding can occur. The
friction force Fr is given by Fr ¼ lFn, where l is coeffi-
cient of friction between the contacting bodies.
Figure 2 illustrates the microstrength bond, including
cohesive strength in the fiber and the polymer elements
and adhesive strength between fiber and polymer ele-
ments. When stress exceeds the bond strength, a bond
will break. This will represent the microcrack initiation
point. The microstrength bond will be converted to the
macro strength and compared with the experimentally
measured strength. Constitutive behavior of a finite-sized
piece of cementatious material deposited between two
balls can be modeled using the parallel-bond. These
bonds establish an elastic interaction between particles
that acts in parallel with the slip or contact-bond models
described above. A parallel bond can be envisioned as a
set of elastic springs with constant normal and shear
stiffnesses, uniformly distributed over either a circular or
rectangular cross-section lying on the contact plane and
centered at the contact point. The force and moment
acting on the two bonded particles can be related to maxi-
mum normal and shear stresses acting within the bond
FIG. 1. Schematics of contact model [1].
DOI 10.1002/pc POLYMER COMPOSITES—-2011 1533
material at the bond periphery. If either of these maxi-
mum stresses exceeds its corresponding bond strength, the
parallel bond breaks [1, 11–13].
As shown in Fig. 2, there are three types of contacts
representing three different interactions within the
carbon fiber reinforced polymer composite material: con-
tacts within polymer, contacts within carbon fiber, and
contacts between polymer and carbon fiber. All three con-
tacts were considered as a pure elastic material and the
spring element was employed to represent the constitutive
mechanical behavior. At a contact, the linear contact
model was employed as discussed earlier. In summary,
four corresponding constitutive models, at contacts, were
built to characterize these contact behaviors; (1) stiffness
models at contacts within the polymer, (2) stiffness
models at contacts within the carbon fiber, (3) stiffness
models at contacts between the carbon fiber and polymer,
and (4) slip and bonding models within the composite
material. The slip model characterized with friction and a
bonding model (contact bond or parallel bond) defined
with tensile and shear strengths are applied at contacts as
shown in Fig. 2.
Microscale Model Parameters
The constitutive models within the polymer composite
material are microscale mechanical models. The microscale
model parameters are usually difficult to be measured in
the laboratory. To determine the model parameters,
macroscale properties, which can be obtained directly from
measurements performed on laboratory specimens, are
needed. In laboratory, elastic modulus (E), poisons ratio
(u), tensile strengths (St) of polymer and carbon fibers can
be determined.
In DE elastic modeling, the behavior of a contact
between two DEs is related to that of an elastic beam
with its ends at the centers of the two DEs, and the beam
is loaded at its ends by the corresponding force and
moment vectors acting at each DE center [1]. The follow-
ing are the conversions from macroscale properties to
microscale model parameters which were applied for the
2D DE modeling in this study:
1. Stiffness model parameters between polymer or carbon
fiber elements—The macroscale elastic properties of
polymer and carbon fiber such as E and l can be
measured in the laboratory and the corresponding
microscale model parameters (knA, ks
A, knB, ks
B) can be
expressed with the macroscale properties. In the nor-
mal direction, the following equation was derived as:
F ¼ KnDLF ¼ EeA ¼ EADL
L
)Kn ¼ EA
L¼ Et
ð5Þ
where F ¼ axial loading; Kn ¼ contact stiffness
at a contact point and represents the stiffness;
E ¼ Young’s modulus; DL ¼ increment of rela-
tive displacement; e ¼ strain under the axial load-
ing condition; L ¼ length; t ¼ thickness/height;
and A ¼ Lt ¼ cross-section area.
FIG. 2. Microstrength bonds and stiffness models illustration. [Color figure can be viewed in the online issue,
which is available at wileyonlinelibrary.com.]
1534 POLYMER COMPOSITES—-2011 DOI 10.1002/pc
In shear direction, a similar equation was obtained as:
Ks ¼ 12IG
L3¼ Gt ð6Þ
where Ks ¼ shear stiffness at the contact point;I ¼ L3
12t ¼ moment of inertia;
G ¼ shear modulus can be expressed with E and m as:
E ¼ 2Gð1þ mÞ ð7Þ
Since the stiffness of the two balls at a contact in
Fig. 2 have the same value. Therefore, according to
Eqs. 1, 2, 5, and 6 the microscale model parameters
can be expressed as:
knA ¼ kn
B ¼ 2Et ksA ¼ ks
B ¼ 2Gt ð8Þ
2. Slip and bonding model parameters—The friction l of
the slip model at a contact is determined by the maxi-
mum value between the frictions of the two contact
balls. If the material tensile and shear strengths at a
contact are termed with rc, and sc, then the contact
bonds force in normal (fn) and shear (fs) directions
can be expressed as:
/n ¼ rcA ¼ rc Lt /s ¼ scA ¼ scLt ð9Þ
The microscale model parameters for parallel
bonds are normal and shear stiffnesses (kApn, kAps,
kBpn, kBps), normal and sheer strength (rpc, spc) and
radius of the parallel bond (Rp).
kpn;ps ¼ kn;sAp
¼ kn;sLpt
¼ kn;s2Rpt
¼ kn;s
2ðnRÞt ð10Þ
where, R ¼ mean radius of two contact elements;
Lp ¼ length of the parallel bond; and n ¼ radius
multiplier, a maximum value of 1 indicates that
the parallel bond (cementatious effect) extends to
the mean diameter of the two contact elements.
IMAGE-BASED MODEL
Fiber Fabric Structure
To model mechanical properties of carbon fiber fabric
composites, the geometry of the fabric was first analyzed
on the microscopic level. Figure 3 shows a schematic of a
top view of eight-harness (8H) weave. This number (8H)
refers to the number of yarns that are passed over by one
yarn. In an eight-harness satin weave, yarns are weaved
by passing over seven and under one yarn before the
pattern repeats itself. The fiber yarns which run in
x-direction are called the fill yarns, and those running in
the y-direction are called the warp yarns, with the same
fibers count in both directions.
Optical Microscope
The cross-section of some samples produced by vacuum
assisted resin transfer molding (VARTM) were prepared,
polished, and analyzed under an optical microscope.
Figure 4b shows the shape of a cross-section through the
thickness of the molded sample. All the dimensions of the
fill and warp yarn were obtained from the observation under
the microscope and then the cross-section was redrawn in
CAD system as shown in Fig. 4a. These observations
FIG. 3. Schematic of a top view of eight-harness carbon fiber fabric.
FIG. 4. Geometry of a cross-section of eight-harness carbon fiber fabric, (a) a CAD drawing (b) a picture
obtained by optical microscope, at 403, for sample produced by VARIM.
DOI 10.1002/pc POLYMER COMPOSITES—-2011 1535
showed that the warp yarn, which crosses over the fill yarn,
resemble an ellipse. Due to the compression in VARTM
process, the following seven yarns, which cross under
the fill yarn, regroup together to act as a one bundle.
Measurements were made, under the microscope, of the
wavelength of a yarn, yarn thickness, elliptical shape of the
warp yarn, and some other necessary dimensions to draw
the cross-section in CAD system. The area of the elliptical
shape of the warp yarn, and the total fiber bundle cross-
section area were used to calculate the bundle fiber volume
fraction.
The analysis is performed on a unit cell which repre-
sents the building block of the entire fabric. The geometry
of eight-harness fabric repeats itself every eight yarns in
both x- and y-directions. The unit cell chosen for the anal-
ysis should contain all the patterns present in the fabric.
So, the complete unit cell for eight-harness fabric consists
of eight rows. Figure 5 shows the geometry of eight-
harness carbon fiber fabric as was seen under the micro-
scope. Each section represents a cross-section of the fabric
along the y-direction, as shown in Fig. 3. The section
length (l) in Fig. 5 represents the length of a unit cell.
The unit cell was color coded to interpret the polymer,
longitudinal carbon fiber, and transverse carbon fiber
elements (Fig. 6a). The color coded image was processed
through a routine developed in Matlab to establish a nu-
merical logical matrix of the image. The logical matrix
was used to establish three computer files of pixels
representing the polymer, longitudinal carbon fiber, and
transverse carbon fiber. These files were further processed
to establish x- and y-coordinates and create consistent
coding for PFC2D with appropriate ball radii, heights,
and widths. Various computer files were generated and
processed using PFC2D to produce synthetic composite
material specimen for compression test simulations. Fig-
ure 6b–d shows three typical composite material synthetic
specimens modeled in PFC2D. Some portions of the
images were also zoomed in for details of particles.
Discrete Element Model Simulation
Discrete element model simulations of uniaxial com-
pressive tests were conducted using the digital synthetic
specimens in PFC2D as shown in Fig. 6. Model parame-
ters for DEM simulations were determined using the
procedure discussed previously in this article. The macro-
mechanical properties obtained from the venders used to
determine the micromechanical properties are listed in
Table 1. Compressive loading in the y-direction was
applied to the top loading plate of the synthetic specimen
in PFC2D, while the bottom loading plate of this specimen
was fixed in all directions. The simulated specimens
contained up to 5,500 disk-shaped particles with a radius
of 0.164 mm. Figure 6b–d illustrates the micromechanical
models, where the elements for the microstructure of
polymer and carbon fiber are shown. It should be noted
that contact bonds were applied for carbon fibers in longi-
tudinal directions while parallel bonds were assigned
between carbon fibers and polymer to simulate the cemen-
tation effect of the polymer. After a certain continuous
incremental uniaxial compressive loading, the response of
each carbon fiber element and each piece of the polymer
element can be monitored.
Figure 7 illustrates the compressive (black) and tensile
(red) contact force chains for Section 3 as the result
of the compressive test simulation in PFC2D. Figure 7b
demonstrates the contact force chains for the enlarged top
portion of Fig. 7a. Even in a compressive test, due to the
FIG. 5. Geometry of eight-harness carbon fiber fabric, each section represents a cross-section of the fabric
along the y-direction.
1536 POLYMER COMPOSITES—-2011 DOI 10.1002/pc
heterogeneity of the material, local tensile stresses
occurred, which caused bond breakage between particles
and resulted in crack initiation. The crack propagation
mechanism has been shown in Fig. 7b–d. At strain level
of 0.475% and stress of 305 Mpa just before failure stress
and strain, the bonds between polymer and longitudinal car-
bon fiber broke at contact points in the synthetic specimen.
This resulted in numerous microcracks throughout the spec-
imen. Once the stress reached the strength of the material,
successive bond breakage occurred for adjacent particles
which caused the formation of macrocracks as can be seen
on the upper right corner of the Fig. 7c. Finally, the macro
crack grew and ended in complete failure and splitting of
the specimen at the interface of longitudinal carbon fibers
and polymer as shown in Fig. 7d.
FIG. 6. (a) Color coded digital image, (b) DEM synthetic model of Section 1, (c) DEM synthetic model of
Section 2, and (d) DEM synthetic model of Section 3. [Color figure can be viewed in the online issue, which
is available at wileyonlinelibrary.com.]
TABLE 1. Macro-mechanical properties used to calculate micro-
mechanical properties.
Materials
Elastic
modulus (GPa)
Tensile
strength (MPa)
Polymer 4.6 104
Carbon fibers—Longitudinal direction 228 4,300
Carbon fibers—Transverse direction 60 –
DOI 10.1002/pc POLYMER COMPOSITES—-2011 1537
The aforementioned micromechanical model using
imaged-based DEM approach provided a reasonable physi-
cal portrayal of the force chains developed in the carbon
fibers and polymer skeleton, which is known to be a critical
aspect of mechanical modeling. These simulations pre-
dicted the stress–strain response of the composite material
which was then used to determine the elastic modulus and
compressive strength of the composite material. Figure 8
shows the stress–strain curve under compressive loading
obtained as a result of DEM PFC2D simulations for the
three sections shown in Fig. 6. Several simulations were
conducted for various sections of the unit cell. Using the
stress–strain curve, the elastic modulus and compressive
strength of the composite material was determined and plot-
ted as function of the length of unit cell, as shown in Fig. 9.
The modulus and strength distributions reveal that the mini-
mum values occur due to the polymer rich area and the
fibers undulation. Nevertheless, the modulus and strength
functions shown in Figure 9 were integrated and averaged
along the length of the unit cell to obtain an effective elastic
modulus (Ex) and an effective compressive strength (Scx) ofthe composite material, respectively.
Ex ¼ 1
lx
Zlx0
ExðxÞdx ð11Þ
FIG. 7. Distribution of contact force chains and crack propagation in Section 3 (Black: compressive contact
forces, Red: tensile contact forces). [Color figure can be viewed in the online issue, which is available at
wileyonlinelibrary.com.]
1538 POLYMER COMPOSITES—-2011 DOI 10.1002/pc
Scx ¼1
lx
Zlx0
ScxðxÞdx ð12Þ
Based on the results of the DEM simulation, the effective
elastic modulus and effective compressive strength was
determined as 75 GPa and 643 MPa, respectively.
Experimental Study
The presented study of a laminate with six-layer fab-
rics was investigated experimentally to determine the
elastic modulus and compressive strength. The laminate
was molded by VARTM process. The materials used are
AS4-8H carbon fiber fabrics, produced by Hexcel, and a
high-temperature polymer called Cycom 5250-4-RTM,
produced by Cytec Engineered Materials. The carbon
fiber fabric has a fiber areal weight of 373 g/m2 and a
fabric thickness of 0.42 mm. Molding process parameters
were set at a maximum cure temperature of 1888C and a
heating rate of 1.678C/min. Full description of the experi-
mental setup was reported in a previous publication by
one of the authors [14].
Compressive properties were determined according to
ASTM standard D6641 using a combined loading com-
pression (CLC) test fixture. Specimens, used in the test,
are 140-mm long, 12.5-mm wide, and on average 2.1-mm
thick. The machine which was used in the evaluation is
Series 812 Materials Test System from MTS Systems
Corporation. The compression test was performed using a
constant head speed of 1.27 mm/min. Three tests were
conducted. A data acquisition system based on LAB-
VEIW was used to collect the test data. Figure 10 shows
a typical stress–strain curve obtained from one of the test.
The compressive strength was determined as the maxi-
mum stress at failure.
The statistical analysis of the results shows an average
modulus of 80 GPa, a standard deviation of 6.5 and a
coefficient of variation of 8.1%. The value of modulus
obtained experimentally, 80 GPa, in this study is 6.25%
higher than the value obtained by the DE model, 75 GPa.
On the other hand, the compressive strength value
obtained from DEM PFC2D simulation is 8.6% higher
than the average experimental value of 592 MPa which
has a standard deviation of 58 MPa and coefficient of
variation of 9.8%. The results obtained from DEM
simulations are within the coefficient of variation of the
experimental values. The results indicate that the image-
based DEM micromechanical model showed fairly accu-
rate predictions for the elastic modulus and compressive
FIG. 9. Distribution of elastic modulus and compressive strength along the length of the unit cell. [Color
figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
FIG. 8. Typical stress–strain curve using PFC2D simulations.
[Color figure can be viewed in the online issue, which is available at
wileyonlinelibrary.com.]
DOI 10.1002/pc POLYMER COMPOSITES—-2011 1539
strength of the molded carbon fiber reinforced polymer
composite.
CONCLUSIONS
This article presents an analytical modeling of mechan-
ical behavior of carbon fiber reinforced polymer compo-
sites using an advanced DEM coupled with latest imaging
techniques. A micromechanical DE model is developed to
determine the compression modulus and strength of car-
bon fiber reinforced polymer composite. Three constitu-
tive laws are used to represent the interactions at contacts
of DEs within the carbon fibers, within the polymer, and
between polymer and carbon fibers. Each of these consti-
tutive laws consists of: a stiffness model, a slip model,
and a bonding model. A 2D synthetic, heterogeneous
microstructure of the composite material is reconstructed
using digital image from an optical microscope. Bulk me-
chanical characteristics are used to determine the micro-
characteristics of the polymer and carbon fibers to con-
duct virtual compressive test simulations. An experimental
case study is also presented to evaluate the accuracy of
the developed analytical model. The following conclu-
sions can be drawn from the findings of this study:
1. The micromechanical DEM, developed in this study,
using the image-based-shape structural model provides
the capability for prediction of the composite material
elastic modulus, stress–strain response, and compres-
sive strength.
2. The developed micromechanical DEM showed fairly
accurate predictions with 6.25% and 8.6% differences
from the experimental results for the elastic modulus
and compressive strength, respectively. The results
obtained from DEM simulations are within the coeffi-
cient of variation of the experimental values.
3. The micromechanical DEM simulation exhibits a
reasonable physical portrayal of the compressive and
tensile force chains developed in the carbon fibers and
polymer, which can assist in determination of stress–
strain distribution at critical locations of composite
materials.
4. The developed model may assist in the fundamental
understanding of the micromechanical behavior of the
composite materials in relation to microcrack initiation
and propagation thus, leading to complete failure.
ACKNOWLEDGMENTS
The authors acknowledge the support of the University
of Louisiana at Lafayette. Also, the authors acknowledge
the support of the Airtech Advanced Materials Group,
Hexcel Corporation, and Cytec Engineered Materials.
ABBREVIATIONS
CLC combined loading compression
DE discrete element
DEM discrete element model
UDEC universal distinct element code
VARTM vacuum assisted resin transfer molding
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tally. [Color figure can be viewed in the online issue, which is available
at wileyonlinelibrary.com.]
1540 POLYMER COMPOSITES—-2011 DOI 10.1002/pc