COMPOSITES

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Composites Science and Technology 67 (2007) 1666–1673

SCIENCE ANDTECHNOLOGY

Finite element modeling of polymer curing in natural fiberreinforced composites

T. Behzad a, M. Sain b,*

a Department of Chemical Engineering and Applied Chemistry, University of Toronto, 200 College Street, Toronto, Ont., Canada M5S 3E5b Faculty of Forestry/Chemical Engineering, 33 Willcocks Street, University of Toronto, Ont., Canada M5S 3B3

Received 22 November 2005; received in revised form 16 June 2006; accepted 29 June 2006Available online 2 October 2006

Abstract

Plant-based fibers have been selected as suitable reinforcements for composites due to their good mechanical performances and envi-ronmental advantages. This paper describes the development of a simulation procedure to predict the temperature profile and the curingbehavior of the hemp fiber/thermoset composite during the molding process. The governing equations for the non-linear transient heattransfer and the resin cure kinetics were presented. A general purpose multiphysics finite element package was employed. The procedurewas applied to simulate one-dimensional and three-dimensional models. Experiments were carried out to verify the simulated results.Experimental data shows that the simulation procedure is numerically valid and stable, and it can provide reasonably accurate predic-tions. The numerical simulation was performed for a three-dimensional complex geometry of an automotive part to predict the temper-ature distribution and the curing behavior of the composite during the molding process.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Natural fiber composite; B. Curing; B. Modeling; C. Finite element analysis (FEM)

1. Introduction

Natural fibers such as hemp, flax, jute, wood, and sev-eral waste cellulosic products have been used as suitablealternatives to synthetic reinforcements for composites inmany applications. These fibers offer specific benefits suchas low density, low pollutant emissions, biodegradability,high specific properties, and low cost [1,2].

Many studies have been carried out to develop differentmanufacturing processes and to study the mechanical per-formances of natural fiber composites [3–7]. The successfulproduction of thermoset composite parts depends upon aproper cure cycle during the molding process that leadsto uniform curing and compaction. A few reports in the lit-erature employed numerical analysis to study curingbehavior and temperature distribution of synthetic fibercomposites during autoclave, resin transfer molding

0266-3538/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compscitech.2006.06.021

* Corresponding author. Tel.: +1 416 946 3191; fax: +1 416 978 3834.E-mail address: m.sain@utoronto.ca (M. Sain).

(RTM), and other manufacturing processes. For instance,Guo et al. [8] conducted a one-dimensional transient heattransfer analysis during the autoclave cure cycle for thickcarbon fiber/epoxy laminates using a commercial finite ele-ment (FE) software. It was found that the conventionalcuring cycles should be modified to prevent temperatureovershoot. The temperature profiles of a thick unidirec-tional glass/epoxy laminate during an autoclave vacuumbag process were predicted by Oh and Lee [9] usingthree-dimensional transient heat transfer FE analysis.Then, the viscosity profiles, degree of cure, and resin pres-sure distribution in the laminate were obtained from theresults of the heat transfer analysis. Joshi et al. [10] per-formed a transient heat transfer analysis using a generalpurpose FE software and two user programs to simulateresin cure kinetics of a thick graphite/epoxy laminate.The results showed excellent agreement with the experi-mental data. The modeling and simulation of resin flow,heat transfer, and the curing of multilayer thermoset com-posites during autoclave processing were investigated by

T. Behzad, M. Sain / Composites Science and Technology 67 (2007) 1666–1673 1667

Blest et al. [11]. The simulation was performed with varyingcomposite thicknesses and confirmed the approximatevalidity of the model. Liu et al. [12] developed a numericalprocedure for the simulation of temperature and cure pro-files for the pultrusion process. A commercial general pur-pose FE package was combined with a user program tosolve the convection heat transfer and the resin reaction.A one-dimensional non-linear cure simulation model wasdeveloped by Pantelelis et al. [13] to track the temperatureevolution in thick composites. Moreover, an optimizationalgorithm was coupled to the model to design the curecycle. Cheung et al. [14] presented a three-dimensionalthermo-chemical cure simulation employing the GalerkinFE method. Several numerical examples were depictedfor different synthetic fibers and resins in flat or curveshaped parts. Park et al. [15] studied modeling for cure sim-ulation of composite structures with an arbitrary geometryunder non-uniform autoclave temperature distribution.

No significant work has been reported concerning thesimulation of heat transfer and cure kinetics of natural fiberthermoset polymer composites during the molding process.Rouison et al. [16] presented only a one-dimensional modelusing finite difference (FD) method to predict the tempera-ture distribution and cure behavior of natural fiber compos-ites in the RTM process. Most of the commercially usedfinite element software are not applicable for temperatureand cure behavior simulation of thermoset composites dur-ing the molding process. In this case, either some user pro-grams should be combined with the commercially availablesoftware or a non-linear transient heat transfer finite ele-ment model should be coded. In the present study, a non-linear transient heat transfer finite element model is intro-duced to simulate the curing behavior of natural fiber ther-moset polymer composites. The model is defined for oneand three-dimensional analysis using only a commerciallyavailable multiphysics software (COMSOL 3.2) withoututilizing user written programs. It is assumed that no resin

Fig. 1. Schematic diagram of

flow or thickness reduction occurs during the curing pro-cess. The temperature and degree of cure profiles insidethe composite can be evaluated by solving the non-linearanisotropic heat conduction equation including the internalheat consumption as a result of the chemical reaction. Theinternal heat consumption can be expressed in terms of thecure rate. The minimum curing time required to completelysolidify the composite can also be predicted. Hemp fiber/acrylic composites were considered and thermo-physicalproperties and the equation for the cure rate were studied.

2. Experimental

An environmentally friendly acrylic polymer wasobtained from BASF Company; Ontario, Canada. Hempfibers were supplied by Hempline Company, Ontario, Can-ada. In order to prepare prepreg mats, hemp fibers wererandomly oriented on a perforated screen and the acrylicresin solution was circulated to impregnate the fibers withthe solution. Vacuum filtration was applied to remove theexcess solution. After circulation of the resin solution, thewet mat was displaced on a polyester sheet and then keptin the oven at 50 �C for 48 h to remove all the moisturecontent and to complete the thickening process [17,18].Heat compression molding experiments were conductedto verify the validation of the simulation. A metallic blockmold with dimension of 15 · 15 · 1.2 cm3 was sprayed withthe released agent. Total of ten prepreg plies were placedinside the mold to completely compact the composite. Inorder to verify the numerical modeling, temperature pro-files at five different points inside the composite block wererecorded through the experiment and compared with thenumerical results. Fig. 1 illustrates the schematic diagramof the experimental set-up used to measure the temperatureevolution during the molding process. The thermocoupleswere placed at different locations inside the compositeand connected to a data acquisition system to monitor

the experimental set-up.

Table 1Resin kinetic parameters used in modeling

Total heatof reaction(J/kg)

Frequencyfactor (1/s)

Activationenergy(J/mol)

Order ofreaction

491,000 18,033 46,700 1.4

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the temperature versus time. The mold temperature waskept at 175 �C under the constant pressure for 25 min.

The evolution of the heat capacity with temperature forhemp fibers was measured using a differential scanning cal-orimeter (DSC) in a TA instruments DSC Q 1000. Thenon-cured resin’s heat capacity was obtained by a similarexperiment.

3. Analysis

3.1. Governing equations

In compression molding, the reinforcement was alreadysaturated with the resin and the resin was uniformly distrib-uted through the mat. Therefore, the convective heat trans-fer effect caused by the resin flow is negligible. Moreover, itwas assumed that the geometry, the thickness, and the resinmass remain constant during the molding process. It wasalso assumed that the curing reaction takes place after clos-ing the mold. Moreover, the thermo-physical properties ofthe resin and composite were considered to be independentof the degree of cure. With these assumptions, the transientnon-isothermal heat transfer is governed by the followingequation in the Cartesian system [19]:

qcCpc

oTot¼ o

oxKx

oTox

� �þ o

oyKy

oToy

� �þ o

ozKz

oToz

� �� oQ

ot

ð1Þwhere qc, Cpc, and Ki (i = x,y,z) are density, specific heat,and thermal conductivity of the material in three orthogo-nal directions, respectively.

The endothermic effect of the resin cure reaction repre-

sents by the internal heat consumption sink termoQot

� �.

Ignoring the effect of resin flow in the material, the term

can be directly related to the rate of cureoaot

� �by the fol-

lowing equation:

oQot¼ qrV rHR

oaot

ð2Þ

where, qr is the density of resin, Vr is the resin volume frac-tion of composite, and HR is the total heat of reaction, anda is the degree of cure.

3.2. Cure kinetics

One of the most widely accepted methods to study thecure kinetics of a thermoset resin system is differential scan-ning calorimetry (DSC) [20,21]. The reaction rate, ra, andthe degree of cure are related as follows:

oaot¼ ra ð3Þ

The rate of the curing reaction is often described accord-ing to the following equation:

ra ¼ kð1� aÞn ¼ Ae�Ea=RT ð1� aÞn ð4Þ

where k is the reaction constant and is usually assumed tobe in the Arrhenius form, A is frequency factor, Ea is theactivation energy, R is the universal gas constant, T isthe absolute temperature, and n is order of the reaction.This expression can be written in the logarithmic form:

lnoaot

� �¼ ln A� Ea

RTþ n lnð1� aÞ ð5Þ

A multilinear regression can be performed to calculatethe values of A, Ea, and n. In a previous study, the totalheat of reaction and the kinetic parameters of the resinhave been investigated [22]. The kinetic parameters of theresin are shown in Table 1.

3.3. Thermo-physical properties

To predict the curing behavior of the composite accu-rately, the thermo-physical properties of the compositehave to be well-known. The densities of the non-curedand cured resin, qr, were measured 1.53 g/cm3 and 1.45g/cm3, respectively. The density of the hemp fiber, qf, wasassumed to be 1.48 g/cm3. The density of the composite,qc, was calculated using the rule of mixture as follows:

qc ¼ tfqf þ ð1� tfÞqr ð6Þwhere tf is the volume fraction of fiber.

The variations of the fiber’s heat capacity, Cpf, could befitted quite well by a polynomial of the second order as fol-low in J/g K [22]:

Cpf ¼ 2� 10�5T 2 � 0:0008T þ 1:0812 ð7Þwhere T is the absolute temperature.

The heat capacity of the non-cured resin increased line-arly with temperature in the range of 35–190 �C [22]. Thesevariations could not be neglected in the model and the fol-lowing linear relationship was used as the resin’s heatcapacity, Cpr, in J/g K:

Cpr ¼ 0:0149T þ 0:459 ð8ÞTo predict the temperature variations in the composite

during the molding process, the heat capacity of the com-posite materials is required. Therefore, the heat capacityof the composite, Cpc, was evaluated using the rule of mix-ture as follows:

CpcðT Þ ¼ CpfðT Þ � tf þ CprðT Þ � ð1� tfÞ ð9Þwhere tf is the volume fraction of the fiber.

The transverse conductivity of the composite, Kt, (z-direction in Fig. 1) was calculated based on the thermal-electrical analogy technique for elliptical filaments and a

T. Behzad, M. Sain / Composites Science and Technology 67 (2007) 1666–1673 1669

square packing array unit cell model (E–S model) [23]. Itcan be expressed as follows:

K t

Km

¼1�1=cþp=2d�c=dffiffiffiffiffiffiffiffiffiffiffiffiffiffid2�c2

pln

dþffiffiffiffiffiffiffiffiffiffiffiffiffiffid2�c2p

c

���������� ð10Þ

where Km is the thermal conductivity of matrix, c ¼ffiffiffiffiffiffiffiffiffiffiffiffipl=tf

p=2, d = l(1/b � 1), and b = Kf/Km. Where l is the

geometry ratio of the filler (l = a/b), where a and b arethe axial lengths of the ellipse along the x-axis and y-axis,respectively, tf is fiber volume fraction of the composite,and Kf is the conductivity of fiber. When l = 1 (i.e.,a = b), the present model can be simplified as the cylindri-cal filaments in a square packing array unit cell model.

The in-plane longitudinal conductivity of the composite,Kl (x- and y-directions in Fig. 1) was experimentally mea-sured at different volume fractions of fiber. Experimentalprocedure and results were reported in detail elsewhere[24]. The longitudinal thermal conductivity of the compos-ite versus the volume fraction of fibers can be expressed bythe following linear relationship:

K l ¼ 0:724tf þ 0:4033 ð11Þ

where tf is the volume fraction of fiber.

0

20

40

60

80

100

120

140

160

180

0 100 200 300 400 500 600

Time (second)

Tem

pera

ture

(˚C

)

FDM FEM

Fig. 2. Temperature profiles at the center of the composite versus timeobtained by different numerical methods.

3.4. Finite element modeling of heat transfer

The temperature distribution in the composite duringthe curing process can be obtained using a non-linear tran-sient heat transfer analysis including the internal heat con-sumption (Eq. (1)). The model is defined based on finiteelement method (FEM) using only a commercially avail-able software with multiphysics environment (COMSOL3.2) and without adding any user written programs. First,the transient heat transfer model is defined in the heattransfer module of the software to obtain the temperature.Then, a general form equation is considered separately inthe partial differential equation (PDE) module to evaluatethe cure kinetics and degree of cure reached in each ele-ment. The above mentioned governing equations can besolved considering the initial and boundary conditions.At t = 0, T = T0 and a = a0 (a0 = 0), where T0 and a0 arethe initial temperature and degree of cure of the material,respectively. For the heat transfer model, the temperatureof boundaries is set to the temperature of mold. The Neu-mann boundary conditions are considered for the kineticmodel. The rule of mixture is used to calculate the densityand heat capacity of the composite (Eqs. (6) and (9)). Thethermal conductivity of the composite at different direc-tions is obtained using different models (Eqs. (10) and (11)).

The energy and kinetic equations are coupled where therate of cure reaction is a function of temperature whichleads to a non-linear system. The rates of internal heat sinkare calculated from the rates of cure at each time step forall the nodal elements. These are then applied as heat sinksat the respective nodal points for the finite element tran-sient heat transfer analysis in the next time step. The execu-

tion of the finite element analysis is repeated until thecompletion of the cure cycle.

4. Results and discussion

4.1. One-dimensional cure modeling

The one-dimensional cure modeling of the compositewas carried out considering the transient heat transfer anal-ysis combined with cure kinetics. The effect of number ofelements and time steps on the temperature distribution ofthe composite was investigated. Little variation with respectto the mesh density and time step confirmed the stabilityand convergence of the procedure. It means that the resultsare not sensitive to either the mesh density or time step.

To validate the present procedure (FEM) it was com-pared to the one-dimensional result obtained from finitedifference method (FDM). For the finite difference method,a code in MATLAB was developed to simulate the heattransfer and cure kinetic for a one-dimensional model.The code employed a finite difference solution techniqueto predict the temperature and degree of cure profiles forthin flat composites. Detail descriptions of the FDM proce-dure and results were published elsewhere [22]. A one-dimensional model with 1.2 cm thickness with the samenumber of elements for both procedures was considered.A time step of 1 s was selected to achieve the desired accu-racy. The temperature results for the central point of thecomposite by FDM and the present procedure (FEM) werecompared; see Fig. 2. A very good agreement was observedduring the first and the last part of the curing cycle. A fewdegree discrepancies in the intermediate stage of the curingcycle can be observed.

The degree of cure at the center of the composite versustime is presented in Fig. 3. The FDM technique predictsthe degree of cure with slower rate compared to FEMwhich can be due to differences in the estimated tempera-tures. In FDM, the Crank–Nicholson technique wasapplied followed by a dichotomy method to obtain the

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600

Time (second)

Deg

ree

of c

ure

FDM FEM

Fig. 3. Degree of cure profiles at the center of the composite versus timeobtained by different numerical methods.

0

20

40

60

80

100

120

140

160

180

0 200 400 600 800 1000

Time (second)

Tem

pera

ture

(˚C

)

1D 3D

Fig. 5. Temperature profiles at the center of the composite in 1D and 3Dmodel.

1670 T. Behzad, M. Sain / Composites Science and Technology 67 (2007) 1666–1673

temperature and degree of cure at each time step. Whereas,in FEM, the Newton–Raphson method was employedwhich results in a different temperature and degree of cureat each time step due to different numerical errors.

4.2. Three-dimensional cure modeling for a block

In three-dimensional heat transfer analysis, due toanisotropic nature of the material, it is necessary to providethermal conductivity of the material in three orthogonaldirections. For the three-dimensional analysis, a simpleblock (15 · 15 · 1.2 cm3) was used which composed of600 solid elements and the time step was selected to be10 s. The temperature of all outer surfaces was set to themold temperature. Fig. 4 represents the meshed finite ele-ment model for the analysis.

The model was simulated in both 1D and 3D heat trans-fer models. Fig. 5 compares one-dimensional and three-dimensional temperature simulations at the center of theblock. It shows that there is good agreement between 1Dand 3D temperature profiles from heat transfer models.This result indicates that for the center of block, a 1D heattransfer model can precisely predict the temperature pro-file. As one moves from the center to corner of the block,discrepancies between the temperature obtained in 1D

Fig. 4. Finite element model of a block of the composite (all units are inmeters).

and 3D heat transfer model will arise due to boundary con-ditions. Fig. 6 shows the temperature evolution versus timeat the center and corner of the block using the 3D heattransfer model. As it can be seen, the simple 1D heat trans-fer model is not applicable and 3D model has to be consid-ered to predict the results accurately.

In order to verify the three-dimensional modeling, thetemperature profiles at five points inside a block of com-posite with 15 · 15 · 1.2 cm3 dimensions were measuredexperimentally and compared with the numerical results.Fig. 7 shows the temperature profiles at two different loca-tions; center and corner of the block obtained by the sim-ulation and experiment. The predicted temperatureprofiles were in good agreement with the experimentalresults. There are a few degree discrepancies which canbe due to the assumptions considered for the modelingsuch as constant thickness, constant density, and ignoringthe effect of degree of cure on heat capacity and thermalconductivity. In addition, the thermocouples could be dis-placed from their initial location during the experiment incompression molding.

0

20

40

60

80

100

120

140

160

180

200

0 200 400 600 800 1000

Time (Second)

Tem

pera

ture

(˚C

)

CornerCenter

Fig. 6. Temperature profiles at the center and corner of the composite in3D model.

020406080

100120140160180200

0 200 400 600 800 1000Time (second)

Tem

pera

ture

(ºC)

020406080

100120140160180200

0 200 400 600 800 1000Time (second)

Tem

pera

ture

(ºC)

Experiment Model

Experiment Model

a

b

Fig. 7. Comparison of experimental and predicted temperature profiles ofthe hemp fiber acrylic composites at two different locations (a) center and(b) corner.

Fig. 8. Finite element mesh for a segment of an automotive mirror case(all units are in meters).

Fig. 9. Temperature distributions (�C) at different curing times in variousslides of the structure: (a) curing time = 50 s and (b) curing time = 500 s(units of axis are in meters).

T. Behzad, M. Sain / Composites Science and Technology 67 (2007) 1666–1673 1671

4.3. Three-dimensional cure modeling for an automotivemirror case

To demonstrate the ability of the three-dimensionalmodel, the numerical simulation was performed for a com-plex three-dimensional geometry of an automobile part topredict the temperature distribution and cure behavior ofthe composite during the molding process. A transient heattransfer three-dimensional simulation was conducted for areal automotive mirror case geometry. The mold tempera-ture was selected to be 185 �C. Due to the symmetry of thepart, to save the computation time only a segment of thecorner of the structure was considered. The thickness ofthe part is different at the corner, edge, and top sectionsof the model. The governing equation is solved subjectedto the mold-wall temperature conditions and the isother-mal condition on symmetry boundaries:

T ¼ T m on mold surfaces ð12Þ

� koTon¼ 0 on symmetry surfaces ð13Þ

Fig. 8 shows the finite element model of the geometry atdifferent views. The numerical model was performed at dif-ferent number of tetrahedral meshes (7296,11,269,15,112)and time steps (100, 50,10). Although there was no signifi-cant difference in temperature profiles, but applying 15,112

elements and 10 s for time step, the model shows more sta-ble and smoother profiles. Figs. 9 and 10 show the temper-ature distribution and degree of cure of the structure atdifferent slides inside the composite after 50 and 500 s.

Fig. 10. Degree of cure distributions at different curing times in variousslides of the structure: (a) curing time = 50 s and (b) curing time = 500 s(units of axis are in meters).

1672 T. Behzad, M. Sain / Composites Science and Technology 67 (2007) 1666–1673

As it can be seen, corners are the last sections whichreach to the final temperature of curing due to their greaterthickness. Consequently, the corners are the last areas tocompletely solidified and cured. From the simulationresults it can be estimated that the molding time to achieve100% cure for the entire structure is approximately 500 s.Hence, the simulation model is capable to display the curebehavior of any complex geometry.

5. Conclusion

A non-linear transient heat transfer analysis combinedwith a cure kinetic model based on finite element proce-dures was developed. In order to accurately predict theresult, the thermo-physical properties of the compositewere investigated. The temperature and degree of curedistribution for hemp fiber/acrylic composites duringthe molding process were developed for a one-dimen-sional model and compared with previous results froma finite difference method. A three-dimensional modelwas developed for a simple block of the composite andcompared with experimental results. Experimental datashows that the simulation procedure is numerically validand stable, and provides reasonably accurate predictions.To demonstrate the simulation model for a complexgeometry, the model was carried out for a segment of

an automotive mirror case. The temperature and degreeof cure profiles were predicted for the structure. The curebehavior of a structure with different geometries, curva-tures, and thicknesses can be simulated using the multi-physics software.

Acknowledgments

The authors would like to thank Network of Centre ofExcellence-Auto 21 Canada for their financial support.

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