micromechanical discrete element modeling of fiber reinforced polymer composites

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.


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.


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 –


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.]


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.]


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

REFERENCES

1. Itasca Consulting Group. ‘‘PFC 2D Ver. 3.1.’’ Minneapolis

(2004a).

2. P.A. Cundall, Symposium of the International Society forRock Mechanics, Nancy, France, II-8 (1971).

3. P.A. Cundall and O.D.L. Strack, Geotechnique, 29, 1

(1979).

4. Y.-C. Chen and H.-Y. Hung, Soils Foundation, 31, 4 (1991).

5. D.W. Washington and J.N. Meegoda, Int J Numer AnalMeth Geomech, 27, 14 (2003).

6. J.G. Dickens and P.J. Walker, Struct Eng Rev, 8, 2–3,

(1996).

7. J. Idris, T. Verdel, and M. Al-Heib, Tunnel UndergroundSpace Technol, 23, 3 (2008).

8. S.S. Vardakos, M.S. Gutierrez, and N.R. Barton, TunnelUnderground Space Technol, 22, 4 (2007).

9. X.B. Zhao, J. Zhao, J.G. Cai, and A.M. Hefny, ComputGeotech, 35, 1 (2008).

10. A. Abbas, E. Masad, T. Paapagiannakis, and T. Harman, IntJ Geomech, 7, 2 (2007).

11. W.G. Buttlar and Z. You, Transportation Research Record,1757 (2001).

12. Z. You and W. G. Buttlar, Transportation Research Record,1970 (2006).

13. M.J. Khattak and C. Roussel, Transportation ResearchBoard, 2127 (2009).

14. A. Khattab and A.S. El-Gizawy, J Adv Mater, 40, 3 (2008).

FIG. 10. Stress–strain curve for one of the specimen tested experimen-

tally. [Color figure can be viewed in the online issue, which is available

at wileyonlinelibrary.com.]


micromechanical discrete element modeling of fiber reinforced polymer composites

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