vacuum assisted resin transfer molding - clemson university

COMPUTATIONAL MODELING OF THE VACUUM

ASSISTED RESIN TRANSFER MOLDING (VARTM) PROCESS

A Thesis

Presented to

the Graduate School of

Clemson University

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

Mechanical Engineering

by

Krishna Mohan Chittajallu

May 2004

Advisor: Dr. Mica Grujicic

2

CHAPTER 1

INTRODUCTION

Ever-increasing demand for rapid production rates has pressed the polymer

composite industry to replace manual lay-up processes with alternative fabrication

techniques which are amenable to automation. Among such processes is the so called

resin transfer molding (RTM) which is one of the several processes generally referred as

the liquid molding process. RTM is a fairly simple process which utilizes matched, two-

part molds made of a metal or a composite material. The fiber reinforcements which are

placed into the mold are usually preformed off line in order to shorten the RTM

production cycle time. Once the mold is close, the resin is injected into the preform at

high pressures. Resin infiltration is frequently facilitated using vacuum.

Another, perhaps the fastest growing, liquid molding process is the so called

Vacuum Assisted Resin Transfer Molding (VARTM). Within the VARTM process,

polymer composite parts are made by placing dry fiber reinforcing fabrics into a single-

part, open mold enclosing the mold into a vacuum bag and drawing a vacuum in order to

ensure a complete preform infiltration with resin. In the last stage of the process, the

mold is heated until the part is fully cured since VARTM does not require high heat or

pressure. It is associated with low tooling cost making it possible to produce

inexpensively large, complex parts in one shot. Due to the use of a single-part mold,

VARTM allows for easy visual monitoring of the resin flow inside the mold to ensure

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complete preform infiltration and thus facilitate the production of high-quality defect-free

parts. The VARTM process has been successfully used to make both thin and very thick

laminates, as well as for large parts with complex shapes and unique fiber architectures

for high structural performance. VARTM parts are generally used in marine, ground

transportation and infrastructure applications. Some typical VARTM parts are wind

turbine blades, boats, rail cars, bridge decks, etc.

Within this thesis, the following three aspects of the VARTM process are

analyzed computationally: (a) devolatilization of resin curing by-products during the

VARTM process to ensure void-free parts; (b) non-isothermal resin infusion to shorten

manufacturing cycle time through reduction in the resin viscosity; and (c) determination

of the effect of various, fabric deformation, distortion and shifting phenomena on the

effective preform permeability. A detailed analysis of these three aspects of the VARTM

process is presented in Chapters II, III and IV respectively.

The main objective of the present thesis is to make specific contributions in the

three areas discussed above so that computer modeling can be used with more faith in the

design and development of the VARTM process for manufacturing high-performance

structural components.

CHAPTER 2

OPTIMIZATION OF THE VARTM PROCESS FOR ENHANCEMENT

OF THE DEGREE OF DEVOLATILIZATION OF POLYMERIZATION

BY-PRODUCTS AND SOLVENTS

ABSTRACT

Devolatilization of the polymerization by-products and the impregnation solvent

during Vacuum Assisted Resin Transfer Molding (VARTM) of the polyimide polymers

is analyzed using a combined continuum hydrodynamics/chemical reaction one-

dimensional model. The model which consists of seven coupled partial differential

equations is solved using a finite element collocation method based on the method of

lines. The results obtained reveal that the main process parameters which give rise to

lower gas-phase contents in the VARTM-processed polymer matrix composites are the

vacuum pressure and the tool-plate heating rate. Lower tool-plate heating rates are found

to be beneficial since they promote devolatilization of the impregnation solvent at lower

temperatures at which the degree of polymerization and, hence, resin viscosity are low.

I. INTRODUCTION

Manual lay-up of pre-impregnated fibers over a mold surface followed by

introduction of the resin using a brush or roller is quite common in manufacturing of

advanced composite structures. However this process tends to be very expensive, and

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suffers from limited pre-preg shelf lives and short lay-up times. In addition, the process is

highly labor intensive and quality control is difficult since the quality of the final product

is highly dependent on the operator skills. Many of these limitations are eliminated in the

Vacuum Assisted Resin Transfer Molding (VARTM) process which generally reduces

lay-up times and makes the fabrication process more reproducible and consistent and less

dependent upon operator skills.

VARTM is an advanced fabrication process for polymer-matrix composite

structures which is used in (ground-based and marine) commercial and military

applications [1-3]. The process has been developed over the last decade and has clear

advantages over the traditional Resin Transfer Molding (RTM) process since it eliminates

the costs associated with matched-metal tooling, reduces volatiles emission and allows

the use of lower resin injection pressures [4]. The VARTM process whose schematic is

shown in Figure 2.1, typically involves the following three steps: (a) lay-up of a fiber

preform (woven carbon or glass fabric) onto a rigid tool plate surface. The tool plate is

surrounded by a formable vacuum bag; (b) impregnation of the preform with resin. The

resin is injected through either a single or multiple inlet ports (depending on the part size

and shape) and transferred into the preform by a pressure gradient (induced by the

vacuum pressure), and by gravity and capillary effects; and (c) curing of the impregnated

preform.

6

Vacuum Bag

Tool Plate

Reinforcement StackResin Distribution Fabric

Sealant Tape

Valve

Resin Vacuum

Figure 2.1 A schematic of the vacuum assisted resin transfer molding (VARTM) process

Porosity within polymer matrix composites (including the ones fabricated by the

VARTM process) has long been recognized as a major limitation to the widespread use

of these materials in many structural applications. Growth and coalescence of the pores

under load can give rise to the formation of cracks and, in turn, result in premature

failure. For epoxy matrix composites, it is generally recognized that environmentally-

absorbed water is the primary cause for the formation of voids during processing. In the

polymeric materials based on condensation polyimide systems, on the other hand, water

and ethyl alcohol are formed as by-products of the polymerization reaction. In addition,

7

significant amounts of an impregnation solvent, such as N-methyl-2-pyrolidone (NMP),

are used in the polyimide systems during processing. Thus, to prevent void formation in

these systems, the volatile polymerization by-products and the impregnation solvent must

be removed from the reacting polymer before the part fabrication is completed. To

produce void-free high-performance composites, the devolatilization process must be

fully understood and controllable by (on-line) adjustment of the process parameters.

To analyze the polymerization of polyimide systems during the VARTM

fabrication process and help identify the optimum process parameters which ensure

minimal porosity in the final product, a mathematical model for this process has been

developed and utilized in the present work. It should be noted that the polymerization

process in the polyimide systems considered in the present work is quite complex since:

(a) it takes place in a non-ideal solution of the impregnation solvent, monomers,

polymeric fragments and the polymerization by-products; (b) the solvent and the

polymerization by-products vaporize from the liquid phase; (c) solid polymer precipitate

(and perhaps crystallize) within the liquid phase giving rise to a continuous change in the

rheological and transport properties of the liquid phase; (d) heat, mass and momentum

transport all occur in a three-phase reacting system within a consolidating fiber network;

etc. In order to make modeling of the VARTM process mathematically tractable, a

number of simplifying assumptions had to be introduced. The potential consequences of

such assumptions are discussed in the paper.

The organization of the paper is as follows. In Sections II.1 and II.2, brief

descriptions are given of the polyimide system (DuPont’s Avimid K-III linear polyimide)

and the associated fiber-reinforced composite material studied in the present work. The

8

model for the VARTM process consisting of seven coupled partial differential equations

governing the behavior of the system under investigation is presented in Section II.3. A

brief overview of the finite element collocation method based on the method of lines is

given in Section II.4. The main results obtained in the present work are presented and

discussed in Section III, while the key conclusion resulting from the present work are

summarized in Section IV.

II. COMPUTATIONAL PROCEDURE

II.1 The Basics of Imide Polymerization

Avimid K-III polymers are linear polyimides produced by condensation

polymerization from an aromatic diethylester diacid (diethyl pyromellitate) and an

aromatic ether diamine (4,4 (1,1 – biphenyl) – 2,5-diyl-bis(oxy) bis(benzeneamine)) in a

solvent typically consisting of phthalic anhydride (1 wt.%), ethanol (1 wt.%) and NMP

(98 wt.%). A schematic of the polymerization (imidization) reaction is given in Figure

2.2 where it is seen that water and ethanol polymerization by-products and the

impregnation solvent (NMP) vaporize and form the gas phase.

9

O O || || C C / \ / \ ––– N AR N – AR’––– \ / \ / C C || || n O O (Avimid K-III Polymer)

H2N – AR’– NH2

(Aromatic Ether Diamine)

Heat

C5H9NOCH3CH2OH

H2O +

+ O O || ||

CH3CH2O – C C – OCH2CH3 \ /

AR / \ HO – C C – OH || || O O

(Diethyl Pyromellitate)

C5H9NO(Solvent)+

Figure 2.2 Polymerization chemistry of the Avimid K-III.

The kinetics of the imidization reaction is very complex due to the fact that the

physical state of the material (e.g. polymer-chain flexibility) changes continuously during

the polymerization as a result of solvent loss and cyclization (imidization) of the

monomers. While the exact kinetics of the Avimid K-III is not known, Differential

Scanning Calorimetry (DSC) measurements have shown that the imide formation

reaction begins at ~390K and that it is completed at ~450K [5]. One of the most

characteristic features of the imidization reaction is that its rate undergoes a sharp drop at

a certain (temperature dependent) degree of imidization.

II.2 A Representative Material Element for the Avimid K-III Thermoplastic Matrix Fiber-reinforced Composite

A two-dimensional representative material element (RME) of the fiber-reinforced

Avimid K-III composite material fabricated by the VARTM process is shown

schematically in Figure 2.3. The height of the RME is equal to the thickness of the part

while its width is generally much smaller and scales with the periodicity of the fiber-

10

preform architecture in the horizontal direction. The RME contains three phases denoted

as the solid fiber preform (S), the liquid resin (L) and the gas (G). The solid phase is

considered as inert and rigid and to be surrounded by the liquid and gas phases. The gas

phase is shown in Figure 2.3 as a continuous phase. This is strictly valid only during the

initial stages of the devolatilization process. Near the completion of part fabrication when

most of the gas has been removed, the gas phase becomes discontinuous and consists of

discrete bubbles. Hence, the model developed in the present work would have to be

modified before it could be applied to the later stages of the VARTM process.

Resin Distribution Fabric

Gas Liquid

Solid x

x=0

Tool Plate

Figure 2.3 A representative material element (RME) for the Avimid K-III produced by the VARTM process.

11

II.3 A Model for Imide Polymerization During the VARTM Process

II.3.1 Basic Assumptions

The model developed in the present work is based on the following simplifying

assumptions: (a) the solid phase is inert and its location within a RME is fixed; (b) the

change in the laminate thickness during the VARTM process is small and can be

neglected; (c) polymerization occurs by step reaction and only in the liquid phase; (d)

transport by diffusion in the liquid and the gas phases and by convection in the liquid

phase can be neglected relative to the convective transfer in the involved gas phase. The

variation of the Avimid K-III resin viscosity with temperature displayed in Figure 2.4 [6]

shows that there are three distinct temperature regions: At temperatures below ~390K,

Region I, viscosity of the un-polymerized resin decreases with an increase in temperature.

At temperatures between ~390K and ~510K, Region II, the polymerization process takes

place and, consequently, viscosity increases with an increase in temperature. At

temperatures in excess of ~510K, Region III, the polymerized resin begins to melt and,

hence, its viscosity decreases with an increase in temperature. The data shown in Figure

2.4 suggest that diffusion may become important toward the end of the VARTM process

(Region III) when the gas phase is not continuous any longer but rather consists of

discrete bubbles. However, the role of diffusion in the overall management of the

volatiles during the VARTM process is not expected to be significant; and (e) the gas

phase can be considered as thermodynamically ideal.

12

Region I Region II Region III

Figure 2.4 The effect of temperature on viscosity of the Avimid K-III resin [6].

Since the part (laminate) thickness is generally considerably smaller than its

lateral dimensions, the VARTM process is analyzed using a one-dimensional model in

which the principal (x) direction is chosen to be perpendicular to the tool plate. The

preform is assumed to be instantaneously infiltrated with the resin and, at time equal to

zero, the preform, the resin, the resin distribution fabric and the tool plate are all assumed

to be at the same temperature, To = 353K. Then the temperature of the tool plate is

increased at a constant rate, while a constant vacuum is applied at the resin distribution

side of the laminate. The subsequent evolution of the material state and of other field

quantities (temperature, pressure, gas-phase velocity, etc.) throughout the laminate

thickness can be described using the appropriate heat, mass and momentum conservation

13

equations within each of the three phases. However these equations cannot be generally

solved due to the complex (discontinuous) microstructure (morphology) of the three-

phase composite material. To overcome this problem, the composite morphology is

homogenized using the method of volume averaging [6]. This enables the point-type

equations (applicable to the composite materials with a discontinuous morphology) to be

replaced by the corresponding volume-averaged continuity equations (applicable for the

volume-averaged continuum representation of the composite material).

II.3.2 Governing Equations

Energy Conservation Equation: Under the assumption of a local thermal equilibrium at

each material point, the heat transfer between the resin and the fiber preform can be

neglected and the temperature evolution is described by the following energy continuity

equation:

),,(

)( 2

23

1,

waterethanolNMPixTkmH

xTVC

tTC

imiivapGpGGpmm

=∂∂

=∆−+∂∂

+∂∂ ∑

=

&ρρ (2.1)

where T is the temperature, t the time, x the spatial coordinate perpendicular to the tool

plate, ρ the mass density, Cp is the constant-pressure mass heat capacity, V the gas-phase

velocity, ∆Ηvap the mass heat of vaporization, the volatilization mass flux and k the

thermal conductivity. Subscripts m, G and i are used to denote volume averages of the 3-

phase (fiber, resin, gas) mixture (composite), the gas phase and the volatile components

(NMP, ethanol, water), respectively.

m&

14

The first term on the left-hand side of the Equation (2.1) represents the rate of

change of the internal energy per unit volume, the second term accounts for the

convective gas-phase heat transfer, the third term represents the energy sink associated

with the evaporation of the volatile species. The conductive heat transfer is represented

by the right-hand side of Equation (2.1).

The volume-averaged constant-pressure (volumetric) heat capacity and thermal

conductivity of the composite material are respectively defined as:

pGGGpLLLpSSSpmm CCCC ρερερερ ++= (2.2)

and

GGLLSSm kkkk εεε ++= (2.3)

where εS, εL, and εG denote volume fractions of the solid, liquid and gas phases in the

composite material, respectively. Analogous relations are used to compute the volume-

averaged effective constant-pressure heat capacity and thermal conductivity of the liquid

and the gas phases as functions of the volume fractions of their constituents. The

constituents of the liquid phase are diamine, diacid, polymer, NMP, ethanol and water

while the constituents of the gas phase are NMP, ethanol and water.

Equation (2.1) is subjected to the following initial and boundary conditions:

Initial Condition

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( ) oTtxT == 0, (2.4)

Boundary Condition (1)

∫+==t

o dtTtxT0

),0( α (2.5)


( )( )dfLx

m TtLxThtTk −==

∂∂

−=

, (2.6)

where x = 0 corresponds to the tool-plate/composite interface, L is the laminate thickness,

To the initial temperature, α the heating rate of the tool plate, h the composite/resin-

distributive-fabric heat transfer coefficient and Tdf the temperature of the resin

distribution fabric.

The initial condition given by Equation (2.4) states that, the composite is initially

at uniform temperature. The first boundary condition, Equation (2.5), is based on the

assumption of a negligible contact thermal resistance between the tool plate and the

composite and equates the composite temperature at the tool-plate/composite interface to

that of the tool plate. The second boundary condition, Equation (2.6), postulates that heat

transfer from the composite to the distribution fabric is controlled by convection.

16

Overall Liquid-phase Mass Conservation Equation: Under the assumption that diffusion

and convection in the liquid phase can be neglected, the liquid-phase mass conservation

is defined as a balance of the accumulation and the reaction (evaporation) terms as:

),,(3

1waterethanolNMPim

tt ii

LL

LL =−=

∂∂

+∂

∂ ∑=

&ρ

εε

ρ (2.7)

where ρL is the overall density of the liquid phase which is a function of the composition

of the liquid phase as:

),,min,,,(6

1polymerdiacidediawaterethanolNMPiiL

iiL == ∑

=

ρφρ (2.8)

and ρi and ρLi are respectively the volume fraction and the density of the liquid-phase

species i. The second term on the left-hand side of Equation (2.7) is generally small and

can be neglected. Also, since the gas phase is initially not present, the following initial

condition can be defined:

Initial Condition

( ) oSL tx εε −== 10, (2.9)

where is the fixed volume fraction of the solid phase (fiber preform). oSε

17

Mass Balance of Water and Ethanol in the Liquid Phase: As discussed earlier, water and

ethanol are formed as by-products of the imidization reaction and also evaporate during

the VARTM process. Therefore their mass balances involve the accumulation, reaction

and devolatilization terms as:

OH

OHLOHLA

OHL MW

mt

CRt

C

2

2

2

2 2&

−∂

∂−=

∂

∂ εεε (2.10)

and

OHCHCH

OHCHCHLOHCHCHLA

OHCHCHL MW

mt

CRt

C

23

23

23

23 2&

−∂

∂−=

∂

∂ εεε (2.11)

where and C are the molar concentrations of water and ethanol,

respectively, the rate of destruction of the active (monomer) groups, and is

used to denote the molecular weight. The factor 2 in front of in Equations (2.10) and

(2.11) is used to indicate that two moles of water and two moles of ethanol are formed for

each mole of the active groups consumed. Since the resin may contain environmentally

adsorbed water and ethanol is often deliberately used as a component of the impregnation

solvent, Equations (2.10) and (2.11) are respectively subject to the following initial

conditions:

OHC2 OHCHCH 23

AR MW

AR

Initial Condition (1)

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( ) OOHOH CtxC

220, == (2.12)

Initial Condition (2)

( ) OOHCHCHOHCHCH CtxC

23230, == (2.13)

Mass Balance of NMP in the Liquid Phase: NMP is the main component of the

impregnation solvent and evaporates during the VARTM process, but it is not a by-

product of the imidization reaction. Hence, its balance involves the accumulation and

devolatilization terms as:

NMP

NMPLNMP

NMPL MW

mt

Ct

C &−

∂∂

−=∂

∂ εε (2.14)

and, hence, the corresponding initial condition can be defined:

Initial Condition

( ) ONMPNMP CtxC == 0, (2.15)

where C is the initial concentration of NMP in the liquid phase. ONMP

Mass Balance of the Active Groups: If the total (molar) concentration of active groups

present in the monomer and growing polymer chains, is denoted as C , the mass balance

of the active groups and the corresponding initial condition can be defined as:

A

19

tCR

tC L

ALAA

L ∂∂

−−=∂

∂ εεε (2.16)

and

Initial Condition

( ) OAA CtxC == 0, (2.17)

where, C is the initial concentration of the diamine (diacid). OA

Overall Mass Balance of the Gas Phase: Under the assumption that diffusion in the gas

phase is negligible in comparison to the pressure-gradient driven convection, the overall

gas-phase mass conservation can be described as:

( ) ( ) ),,(3

1waterethanolNMPimV

xt iiGGGG ==

∂∂

+∂∂ ∑

=

&ρερ (2.18)

Using the Darcy’s law to describe the relationship between the gas-phase velocity, V ,

and the pressure gradient,

G

xP ∂∂ :

xPk

VG

GG ∂

∂−=

µ (2.19)

20

the ideal-gas law to define the density of the gas phase:

RTPMWG

G =ρ (2.20)

and the relation:

1=++ GLS εεε (2.21)

Equation (2.18) can be rewritten as:

∂∂

−∂∂

∂∂

+

∂∂

+

∂∂

+∂

∂+=

∂∂ ∑

=

xT

RTP

xP

RTxPk

tT

RTP

xPk

tRTPm

MWtP

RT

G

G

G

G

GL

ii

G

G

2

22

23

1

1

1

µ

εµ

εε&

(2.22)

where R is the universal gas constant, while is permeability of the resin-infiltrated

preform,

Gk

Gµ is the gas-phase viscosity and is the mean gas-phase molecular

weight.

GMW

The initial and the boundary conditions for Equation (2.22) can be defined as:

Initial Condition

( ) oPtxP == 0, (2.23)

21


( ) 0,0 ==∂∂ tx

xP (2.24)


( ) vacPtLxP == , (2.25)

where Po and Pvac are the initial pressure and the applied vacuum pressure at the resin

distribution-fabric side of the composite respectively.

II.3.3 System-dependent Constitutive Relations

Equations (2.1), (2.7), (2.10), (2.11), (2.14), (2.16) and (2.22) represent a system

of seven coupled partial differential equations with seven unknowns: T, Lε , ,

, , and P. Before these equations can be solved, additional, material-

system dependent constitutive equations are required to define the evaporation rates of

the volatile species ( ), the rate of destruction of the active groups ( ), and the

dependencies of permeability ( ) and viscosity (

OHC2

OHCHCHC23 NMPC AC

m& AR

Gk Gµ ) on the degree of polymerization.

These equations are defined below:

Evaporation Rates of the Volatile Species: Following Yang et al. [6], the evaporation rate

of the i-th volatile component can be defined as:

( ) ( PYPAKm isat

iiiiLmi −= φγ& ) (2.26)

22

where is the liquid/gas mass transfer coefficient, the liquid/gas interfacial area

per unit volume of the composite material,

mK LA

iγ , iφ and the activity coefficient, the

volume fraction and the saturation pressure of species i in the liquid phase and Y the

molar fraction of the species i in the gas phase. Procedures used to calculate the

parameters appearing on the right hand side of Equation (2.26) are discussed below.

satiP

i

The overall liquid/gas mass transfer coefficient, , is one of the key

parameters controlling the rate of devolatilization. As devolatilization proceeds, the

gas/liquid interfacial area increases, while the mass transfer coefficient decreases due

to a polymerization-induced increase of the resin viscosity. Consequently, the volumetric

mass transfer coefficient, , experiences a maximum at a temperature T . The

temperature dependence of in kg/m

Lm AK

mK

Lm AK

Lm AK

max

3/s/Pa was experimentally determined by Yoon

et al. [5] as:

max

2

max

7

2sin105.7 TTfor

TTAK Lm ≤

×= − π (2.27)

max

4

max

7

2sin105.7 TTfor

TTAK Lm ≤

×= − π (2.28)

Yoon et al. [5] also reported that T increases moderately with the (constant) heating

rate (T (α = 0.6K/min) = 403K and (T (α = 2.2K/min) = 423K).

max

max max

23

The activity coefficient iγ can be calculated using the Flory-Huggins equation [8]

as:

+

−= 211exp PP

ni x

χφφγ (2.29)

where Pφ is the combined volume fraction of the monomer and the polymer, nx the

mean number of mers in the polymer chains (including monomers) and χ a molecular

interaction parameter whose value typically falls into a range between 0.2 and 0.5.

Following Yoon et al. [5], χ = 0.35 is used for the Avimid K-III system under

consideration. The mean number of mers in the polymer chains is related to the degree of

polymerization p ( ( ) OA

OA CCC −= A ) as:

pxn −

=1

1 (2.30)

Equation (2.28) is based on an assumption that the molecular size distribution in a

polymer can be represented by a geometric probability function which is generally

accepted as a reasonably good approximation for condensation polymers such as Avimid

K-III.

The vapor pressures of the pure volatile components of the liquid (NMP,

ethanol and water) at different temperatures have been computed using the Clausius-

satiP

24

Clapeyron equation [9] and the available heat of evaporation and boiling point data for

these components as:

),,(exp .,,, waterethanolNMPiR

THTHP iboivapivapsat

i =

∆−∆−= (2.31)

where is the heat of evaporation at the boiling point TovapH ,∆ b. The temperature

dependence of the heat of evaporation has been obtained using the Watson’s correlation

[9] as:

),,(11

,

,,,, waterethanolNMPi

TTTT

HHn

iCb

iCoivapivap =

−

−∗∆=∆ (2.32)

where TC is the critical temperature and the exponent n is assigned its standard value of

0.38 [9].

Since it is generally found that evaporation flux dominates the mass balance of

the volatile species during devolatilization, the molar fraction Yi of each volatile

component i in the gas phase can be approximated as:

),,(3

1

waterethanolNMPiMWm

MWmY

iii

iii ==

∑=

&

& (2.33)

25

Rate of Destruction of the Active Groups: As stated earlier, the exact kinetics for

polymerization of the Avimid K-III polyimide system is not well established. Following

Yoon et al. [5], the rate of destruction of the active groups, RA, is approximated using a

first-order kinetic equation as:

AAA

A Ckdt

dCR =−= (2.34)

in which the reaction-rate constant kA is defined as:

( ) ( )TA

tAAk A2

10 lnln ++= (2.35)

For kA in min-1, T in K and t in min, the three constants in Equation (2.35) take on

the following values: A0 = 31.2113, A1 = -0.1888 and A2 = -13.6 × 103 for the Avimid K-

III system [5].

Variations in the Condensed-Phase Permeability and the Gas-Phase Viscosity: As the

polymerization proceeds, condensed-phase permeability kG decreases and in the absence

of a more accurate model has been assumed to be a linear function of the degree of

polymerization, p. The viscosity, µG, on the other hand, is fully reflected by the properties

of the gas phase and is assumed to be given as a volume-average temperature-dependent

viscosities of the gas-phase components.

26

II.4 Finite Element Collocation Method

The system of seven coupled partial differential equations which govern the

behavior of the Avimid K-III system during the VARTM fabrication process is solved

using a finite element collocation method based on the method of lines [10]. Within this

method, the spatial domain (the x-direction) is discretized into elements and the x-

dependent portion of the solution represented using a polynomial basis function. The

polynomial coefficients are, on the other hand, time dependent. By requiring that the

solutions to the system of partial differential equations (PDE’s) satisfy the boundary

conditions and the continuity conditions at the nodal points separating the elements, the

system of PDE’s is converted into a semi-discrete system of Ordinary Differential

Equations (ODEs) which depend only on time. These equations are then integrated using

a standard integration procedure to determine the unknown time-dependent polynomial

coefficients at a new time step in terms of the solution at the previous time step.

III. RESULTS AND DISCUSSION

All the calculations carried out in the present work pertain to an Avimid K-III

matrix composite material reinforced with 58 vol.% of carbon fiber preforms.

Consequently, the following (typical) properties are assigned to the solid phase [11]: ρS =

1940kg/m3, CpS = 750J/kg/K and kS = 10W/m/K. Properties of the liquid and the gas

components are given in Tables I and II, respectively. Typical values and ranges of the

VARTM processing parameters used in the present work are summarized in Table III.

27

Wat

er

1.0×

10 3

4.18

×10 3

0.60

3

18.0

×10 -3

18.0

2×10

-6

373.

0

2256

.0×1

0 3

647.

4

Etha

nol

0.79

×10 3

2.43

×10 3

0.19

46.1

×10 -3

58.3

9×10

-6

351.

4

854.

0×10

3

516.

3

NM

P

1.03

×103

1.67

×10 3

1.79

99.1

×10 -3

96.4

9×10

-6

477.

3

510.

5×10

3

721.

7

Dia

cid

1.24

×103

1.68

×10 3

0.25

310×

10 -3

250×

10-6

N/A

*

N/A

*

N/A

*

Dia

min

e

0.99

4×10

3

0.34

5×10

3

0.25

368.

43×1

0 -3

370.

65×1

0-6

N/A

*

N/A

*

N/A

*

Uni

t

kg/m

3

J/kg

/K

W/m

/K

kg/m

ol

m3 /m

ol

K

J/kg

K

Sym

bol

ρ Cp k MW

Vm

T b

∆Ηva

p,o

T c

Tabl

e I.

Prop

ertie

s of t

he c

ompo

nent

s of t

he li

quid

pha

se

Prop

erty

Den

sity

Hea

t Cap

acity

Ther

mal

C

ondu

ctiv

ity

Mol

ecul

ar W

eigh

t

Mol

ar V

olum

e

Boi

ling

Poin

t

Hea

t of

Vap

oriz

atio

n (T

b)

Crit

ical

Te

mpe

ratu

re

28

Table II. Properties of the components of the gas phase

Property Symbol Unit NMP Ethanol Water

Heat Capacity Cp J/kg/K 1.26×10 3 1.43×10 3 2.02×10 3

Thermal Conductivity k W/m/K 1.79 0.19 0.603

Molecular Weight MW kg/mol 99.1×10 -3 46.1×10 -3 18.0×10 -3

Viscosity Gµ Ns/m2 1.67×10-3 1.10×10-3 0.89×10-3

Table III. Typical VARTM processing parameters for Avimid K-III material composite

Parameter Symbol Units Value

Volume Fraction of Solid Sε N/A 0.58

Initial Temperature To K 333.0

Tool-plate Heating Rate α K/min 0.5 – 2.0

Laminate Thickness L m 0.0064

Tool-plate/Composite Heat Transfer Coefficient h W/m2/K 27.9

Initial Concentration of Water oOHC

2 mol/m3

0.0

Initial Concentration of Ethanol oOHCHCHC

23

mol/m3 2.59 × 102

Initial Concentration of NMP oNMPC mol/m3

2.41 × 103

Initial Concentration of Active Groups

oAC mol/m3

1.39 × 103

Condensed Phase Permeability for the Gas Phase Gk m2 3.0×10–15-2.85×10–15p

Applied Vacuum Pvac Pa 6666.7

29

III.1 Analysis of Devolatilization under Typical VARTM Processing Conditions

The model described in Section II.3 is used in this section to analyze

devolatilization of the water, ethanol and NMP during VARTM processing of Avimid K-

III matrix carbon fiber reinforced composites under typical processing conditions. In the

following, the results obtained are presented and briefly discussed.

Variations of temperature, pressure, the degree of polymerization and the volume

fraction of the gas phase throughout the laminate thickness at different times following

infiltration (assumed to occur at a time t = 0) of the carbon fiber preform with resin are

shown in Figures 2.5.1 – 2.5.4, respectively. All the results shown in Figures 2.5.1 – 2.5.4

are obtained under a constant (1K/min) heating rate of the tool-plate.

The results displayed in Figure 2.5.1 show that at any instant during VARTM

processing of the Avimid K-III matrix carbon fiber reinforced composites, the

temperature variation throughout the laminate thickness is quite small, typically not

exceeding 4K. This finding is reasonable considering the relatively small laminate

thickness (0.0064m) and the relatively small heating rate. The Biot number, which is

defined as a ratio of the resistance to heat conduction through the laminate and the

resistance to heat convection from the laminate to the resin-rich resin-distribution fabric

is found to be around 0.03. Such a small value of the Biot number justifies the observed

high uniformity in the temperature throughout the laminate thickness.

30

t=0min

t=100min

t=200min

t=300min

Figure 2.5.1 Variations of temperature throughout the laminate thickness at different times following infiltration of the preform with resin at 1K/min heating rate.

Variations of the gas-phase pressure throughout the laminate thickness at different

times following preform infiltration with the resin is displayed in Figure 2.5.2. As

expected, the pressure is the highest at the tool-plate/laminate interface and is constant

and equal to the applied vacuum pressure (6666.7 Pa) at the resin-distribution fabric end

of the laminate. It is also seen that the pressure initially increases as water and ethanol

(generated as the polymerization by-products) and NMP solvent evaporate. However, as

devolatilization of the gas phase at the laminate/distribution-fabric interface proceeds, the

gas-phase pressure begins to decrease.

31

t=0mint=200min

t=100min

t=90min

Figure 2.5.2 Variations of pressure throughout the laminate thickness at different times following infiltration of the preform with resin at 1K/min heating rate.

The results presented in Figure 2.5.3 show that the distribution of the degree of

polymerization throughout the laminate thickness is quite uniform which is consistent

with the corresponding uniform temperature fields displayed in Figure 2.5.1.

A comparison of the results presented in Figures 2.5.2 and 2.5.4 shows that the

variation of the volume fraction of the gas phase throughout the laminate thickness at

different times following preform infiltration with the resin closely matches the

corresponding results for the gas-phase pressure.

32

t=150mint=125min

t=50min

Figure 2.5.3 Variations of degree of polymerization throughout the laminate thickness at different times following infiltration of the preform with resin at 1K/min heating rate.

t=90min

t=100min

t=200min

Figure 2.5.4 Variations of volume fraction of the gas phase throughout the laminate thickness at different times following infiltration of the preform with resin at 1K/min

heating rate.

33

The evolution of the pressure and degree of polymerization in the course VARTM

processing of the Avimid K-III base carbon-fiber reinforced composites can be further

understood by analyzing the results displayed in Figures 2.6.1 and 2.6.2. The results

displayed in Figure 2.6.1 show the variation of pressure at the tool-plate laminate

interface as a function of tool-plate temperature under a constant (1K/min) heating rate of

the tool plate. As discussed earlier, evaporation of water, ethanol and NMP give rise to an

increase in the gas-phase pressure at lower tool-plate temperatures. However, as

devolatilization proceeds, the amount of volatiles in the liquid phase decreases and so

does the gas-phase pressure. The results displayed in Figure 2.6.2 show that the

polymerization of the resin in contact with the tool-plate is complete by ~470K. In

addition a comparison of the results displayed in Figures 2.6.1 and 2.6.2 shows that at the

completion of polymerization, the gas phase pressure is still quite high (~10,000MPa)

and that, due to high viscosity of the fully-polymerized resin, the rate of pressure

reduction by devolatilization has been substantially decreased.

34

Figure 2.6.1 Variations of pressure as a function of temperature at the tool-plate/laminate interface at a heating rate of 1K/min.

Figure 2.6.2 Variations of degree of polymerization as a function of temperature at the tool-plate/laminate interface at a heating rate of 1K/min.

35

Variations of the accumulated mass fluxes of water-vapor, ethanol, NMP and the

(total) gas phase at the resin-distribution fabric end of the laminate as a function of

temperature of the tool plate at three different heating rates are shown in Figures 2.7.1 –

2.7.4, respectively. The accumulated mass flux of the three gas-phase components at the

resin-distribution fabric end of the composite material is given by the following equation:

Q (2.36) ),,(0

waterethanolNMPidtVt

GiGi == ∫ ρ

where Giρ is the density of i-th component of the gas phase. The total accumulated flux

of the gas phase is defined as a sum of the accumulated mass fluxes of the three gas-

phase components.

The results displayed in Figures 2.7.1 – 2.7.4 show that the lower is the heating

rate of the tool plate the larger is the accumulation flux of each of the three gas-phase

components and, thus, the lower are the amounts of volatiles left in the composite at

completion of the VARTM process. Hence, the use of lower heating rates is preferred

from the standpoint of achieving a more complete removal of the gas-phase in VARTM-

processed composites. On the other hand, the benefits of lower heating rates have to be

balanced against the resulting longer part-manufacturing times and the associated higher

manufacturing cost.

36

0.5K/min

1K/min

2K/min

Figure 2.7.1 Variations of the accumulated fluxes for water vapor as a function of temperature at the resin distribution fabric end at three different heating rates.

2K/min1K/min

0.5K/min

Figure 2.7.2 Variations of the accumulated fluxes for ethanol as a function of temperature at the resin distribution fabric end at three different heating rates.

37

2K/min

1K/min

0.5K/min

Figure 2.7.3 Variations of the accumulated fluxes for NMP as a function of temperature at the resin distribution fabric end at three different heating rates.

2K/min

1K/min

0.5K/min

Figure 2.7.4 Variations of the accumulated fluxes for the gas phase as a function of temperature at the resin distribution fabric end at three different heating rates.

38

The effect of vacuum pressure Pvac is also studied in the present work but the

results are not shown for brevity. The main finding of this portion of the calculations is

that, at a given tool-plate heating rate, the fraction of the gas phase left in the composite

at the completion of the VARTM process decreases linearly with the vacuum pressure

Pvac.

III.2 Optimization of the VARTM Process

As discussed in the introduction, Section I, the main objective of the present work

is to develop a mathematical model of the VARTM process which can be used to

optimize this process with respect to achieving the highest extent of devolatilization.

From the findings and considerations presented in the previous section, it is clear that to

achieve this goal, the extent of devolatilization of the volatile species (NMP in particular)

should be maximized at lower temperature, while the degree of polymerization and,

hence, the resin viscosity are low and the mass transfer coefficient (Km) is high. As

discussed earlier, to lower the resin viscosity and, thus, help ensure a complete preform

infiltration with the resin, NMP solvent is mixed with diamine and diacid. Using the

experimental measurements of the temperature dependence of viscosity of the Avimid K-

III reported by Yoon [7], and the NMP viscosity value of 0.0016kg/m/s, the following

relation is obtained between the liquid-phase viscosity, Lµ , in kg/m/s, the temperature

and the initial molar concentrations of the active groups and NMP in mol/m3: Diamine

( ) ( )( ) m

NMPoNMP

mDiacod

moA

mDiacod

moA

L VCVVCVVC

TT++

++−=

Diamine

Diamine20318.09868.691.431µ (2.37)

39

where V is used to denote the molar volume in mol/mm 3.

The functional relationship defined by Equation (2.37) is used to generate the

liquid-phase viscosity contour plot shown in Figure 2.8.1 for C = 1.39 × 103 mol/moA

3.

As expected, liquid-phase viscosity is seen to decrease with an increase in temperature

and an increase in the concentration of NMP.

40

50

5060

60

60

70

70

7080

90

Temperature, K

NM

PC

once

ntra

tion,

mol

e/m

3

340 350 360 370

1500

2000

2500

3000

3500

4000

4500

Figure 2.8.1 Variations of viscosity in kg/m/s of the liquid phase with temperature and molar concentration of NMP.

Using the experimental data for temperature dependences of the liquid-phase

viscosity, the overall liquid/gas mass transfer coefficient, KmAL, and the volume fraction

and the mean diameter of gas-phase bubbles in the resin, as reported by Yoon [7], the

40

following relationship is derived between the mass transfer coefficient, Km, in kg/m5/Pa/s

and the liquid-phase viscosity kg/m/s:

2161310 101187.0101184.0101342.0 LLmK µµ −−− ×+×−×= (2.38)

The functional dependence between the mass transfer coefficient in kg/m5/Pa/s,

temperature and the NMP molar concentration is displayed using a contour plot in Figure

2.8.2. As expected, higher temperatures and higher NMP concentrations are seen to

promote evaporation of the polymerization by-products and the NMP solvent by

increasing the mass transfer coefficient.

1.26E-111.27E-11

1.28E-11

1.29E-11

1.29E-11

Temperature, K

NM

PC

once

ntra

tion,

mol

e/m

3

340 350 360 370

1500

2000

2500

3000

3500

4000

4500

Figure 2.8.2 Variations of mass transfer coefficient in kg/m5/Pa/s with temperature and molar concentration of NMP.

41

Since, according to Equation (2.38), the mass transfer coefficient, Km, increases

with a decrease in the liquid-phase viscosity, an increased concentration of NMP in the

liquid phase is preferred. However, an increase in the concentration of NMP in the liquid

phase also means that a larger amount of the gas-phase will have to be removed from the

composite through devolatilization. This implies that there is and optimum NMP

concentration in the liquid phase which, at a given heating rate, gives rise to a maximum

degree of gas-phase removal from the composite material.

The notion of the optimal concentration of NMP can be further understood by

analyzing the contour plot shown in Figure 2.9. In this figure, the fraction of volatiles left

in the composites at the highest temperature attained during the VARTM process (taken

to be 623K for Avimid K-III matrix carbon fiber reinforced composites) is plotted as a

function of the tool-plate heating rate and the molar concentration of NMP. It is seen that

at each constant level of the fraction of volatiles left in the composite, there is an

optimum concentration of NMP which is associated with the highest allowable tool-plate

heating rate. A dashed line is used to connect the optimum NMP concentrations/heating

rate points at different contour lines and, thus, to show that both the optimal NMP

concentration and the corresponding highest heating rate decrease as the fraction of the

volatiles left in the composites decreases. However, the results displayed in Figure 2.9

show that the increase in the tool-plate heating rate associated with the use of the

optimum NMP concentration is quite small. Hence, the optimum concentration of NMP

in the resin would be, in general, governed by the NMP’s role in lowering resin viscosity

and, thus, in promoting a complete preform infiltration with the resin rather than by the

devolatilization aspects of the VARTM process.

42

0.001

0.01

08

0.0206

0.03

04

0.0402

0.05

Tool-plate Heating Rate, K/min

NM

PC

once

ntra

tion,

mol

e/m

3

0 0.5 1 1.5 21500

2000

2500

3000

3500

Figure 2.9 Contour plot of the volume fraction of gas phase left in the composite at the completion of the VARTM process as a function of the tool-plate heating rate and the

initial molar fraction of NMP in the resin.

IV. CONCLUSIONS

Based on the results obtained in the present work, the following main conclusions

can be drawn:

1. Adequate modeling of devolatilization during the VARTM process requires consideration of both chemical effects associated with polymerization of the resin and hydrodynamic effects associated with the transport of volatiles through the resin.

2. Lower tool-plate heating rates promote devolatilization of the volatiles (in

particular, of the NMP solvent) at lower temperatures at which, due to a low degree of polymerization, resin viscosity is low. This results in a more complete removal of the volatiles and a lower gas-phase content in VARTM-

43

processed fiber reinforced polymer matrix composites. However, lower tool-plate heating rates are generally associated with higher manufacturing costs.

3. From the standpoint of achieving a high degree of gas-phase removal at a

highest possible tool-plate heating rate, there is, in general, an optimum concentration of the solvent. However, the benefits of using the optimum NMP concentration relatively to increasing the tool-plate heating rate and, thus, in reducing the VARTM processing time, are relatively limited.

V. REFERENCES

1. Lewit, S. M., and J. C. Jakubowski, SAMPE International Symposium, 42, 1173 (1997).

2. Nquyen, L. B., T. Juska and S. J. Mayes, AIAA/ASME/ASCE/AHS/ASC

Structures, Structural Dynamics and Materials Conference, 38, 992 (1997).

3. Lazaras, P., SAMPE International Symposium, 41 1447 (1996).

4. Sayre, J. R., PhD Thesis, Virginia Polytechnic Institute and State University,

Blacksburg, VA (2000).

5. Yoon, I. S., Y. Yang, M. P. Dudukovic and J. L. Kardos, Polymer Composites, 15, 184 (1994).

6. Yang, Y. B., J. L. Kardos, I. S. Yoon, S. J. Choi and M. P. Dudukovic, Proc.

Amer. Soc. Compos., 4th Tech. Conf., Western Hemisphere Publ., Blacksburg, VA, p. 36-45 (1989).

7. Yoon, I. S., “Experimental Investigation of the Devolatilization in Polyimide

Fiber Composites,” DSc. Thesis, Washington University, St. Louis, MO (1990).

8. Biesenberger I. A., and D. H. Sebastian, Principles of Polymerization

Engineering, John Wiley and Sons, New York (1983).

9. Reid, R. C., J. M. Prausnitz and B. E. Poling, “The Properties of Gases and Liquids,” 4th Edition, Mc Graw Hill, New York (1987).

10. Sincovec, R. F., and N. K. Madsen, Software for Nonlinear Partial Differential

Equations, ACM-TOMS, Vol.1, No.3, pp. 232-260 (1975).

11. Cambridge Engineering Selector, Version 3.1, Granta Design Ltd, Cambridge, UK (2000).

CHAPTER 3

NON-ISOTHERMAL PREFORM INFILTRATION DURING THE

VACUUM ASSISTED RESIN TRANSFER MOLDING (VARTM) PROCESS

ABSTRACT

A control-volume finite-element model is developed to analyze the infiltration of

the fiber preform with resin under non-isothermal conditions within a high-permeability

resin-distribution medium based Vacuum Assisted Resin Transfer Molding (VARTM)

process. Due to the exposure to high temperatures during preform infiltration, the resin

first undergoes thermal-thinning which decreases its viscosity. Subsequently, however,

the resin begins to gel and its viscosity increases as the degree of polymerization

increases. Therefore the analysis of preform infiltration with the resin entails the

simultaneous solution of a continuity equation, an energy conservation equation and an

evolution equation for the degree of polymerization. The model is applied to simulate the

infiltration of a rectangular carbon-fiber based preform with the NBV-800 epoxy resin

and to optimize the VARTM process with respect to minimizing the preform infiltration

time. The results obtained suggest that by proper selection of the ramp/hold thermal

history of the tool plate, one can reduce the preform infiltration time relative to the room-

temperature infiltration time. This infiltration time reduction is the result of the thermal-

thinning induced decrease in viscosity of the ungelled resin.

45

I. INTRODUCTION

Vacuum-Assisted Resin Transfer Molding (VARTM) is an open-mold polymer-

matrix composite manufacturing process widely used in a variety of commercial

applications (e.g. boats, refrigerated cargo boxes, etc.). In addition, the VARTM process

is being considered for different automotive, aerospace and military applications [1-3].

The process is based on the use of a single rigid mold (tool plate) which is laid up with

fiber-reinforcement preforms and enclosed in an air-impervious vacuum bag. The

preform is next infiltrated with resin using vacuum pressure, Figure 2.1. Lastly, the rigid

mold is heated to a resin-dependent temperature and held at that temperature for a

sufficient amount of time to ensure a complete curing of the resin-impregnated preform.

The VARTM process offers several advantages over the competing polymer-matrix

composites manufacturing processes such as: (a) a low tooling cost; (b) a low emission of

volatile organic chemicals; (c) processing flexibility; (d) a low void-content in the

fabricated parts; and (e) a potential for fabrication of the relatively large (surface area

~150-200m2) and thick (0.1-0.15m) composite parts, containing a large content (75-

80wt.%) of the reinforcing fiber-preform phase.

There are several versions of the VARTM process which differ mainly with

respect to the type of resin distribution system used. Among these, two are most

frequently used: (a) a VARTM process based on the use of a high-permeability medium,

Figure 3.1.1 and (b) a VARTM process based on the use of grooves located within a low-

density core of the fiber preform, Figure 3.1.2.

46

High Permeability Medium

Fiber Preform

Fiber Preform

Peel Ply

Vacuum Bag

Core

Tool Plate

Figure 3.1.1 A high-permeability medium based resin distribution system used in VARTM process.

Vacuum Bag

Fiber Preform

CoreGrooves

Fiber Preform

Tool Plate

Figure 3.1.2 Grooves based resin distribution system used in VARTM process.

While the VARTM process has been commercialized for more than a decade, its

application to manufacture of complex composite structures is based almost entirely on

experience and on a trial–and–error approach. One of the most critical steps during the

VARTM process is the resin infiltration stage. Ideally one would want a complete

infiltration of the preform (mold filling) with the resin, in a shortest period of time in

order to minimize the production time and, thus, the manufacturing cost. In addition, one

must ensure that the resin completely wets the fiber preform in order to avoid formation

of “dry spots”. Due to a lack of polymer/fiber bonding, dry spots can act as crack

47

nucleation sites, when VARTM-fabricated structures are subjected to loads while in

service. In general, polymerization (gelation) of the resin should not take place during the

mold-filling stage, since the resulting increase in resin viscosity may lead to incomplete

preform infiltration. Therefore, computer models which can provide a better

understanding of the fiber-preform infiltration process and enable an accurate prediction

of the infiltration time, should help the manufacturers of composite structures to design

and optimize the VARTM process so that, through the selection of the tool-plate heating

profile and the resin composition (e.g. the concentrations of infiltration solvent, initiator,

etc.), a complete mold filling is attained in a shortest period of time. In a series of papers,

Lee and co-workers [4-8] developed a finite-element control-volume based model for

isothermal, preform filling for both high-permeability medium and grooves–based

VARTM processes. The models developed by Lee and co-workers [4-8] enable the

prediction of the filling time and the flow pattern and were validated by comparing the

model predictions with the (room-temperature) flow-visualization experimental results

for preform infiltration with several oils of different viscosity. In a typical VARTM

process, heating of the tool-plate is started only after the mold-filling stage of the process

is completed. Under such conditions, the models developed by Lee and co-workers [4-8]

can be used to analyze the mold filling process. However, one may identify at least two

potential benefits that heating of the tool plate during the resin infiltration stage of the

VARTM process can have: (a) moisture absorbed onto the fiber-preform surface could be

removed to a greater extent promoting a better-polymer/fiber bonding and (b)

temperature-induced lowering of the viscosity of the “ungelled” resin can facilitate a

more complete mold filling and give rise to shorter infiltration times. Due to the

48

isothermal nature of the models developed by Lee and co-workers [4-8], they cannot be

used to analyze the preform infiltration process under the conditions when the tool plate

is being heated and a non-uniform, time-dependent temperature field is developed in the

mold. To overcome these limitations, the (isothermal) two-dimensional finite-element

control-volume mold-filling model of Lee and co-workers [4] for the VARTM process

based on the use of a high-permeability medium, is extended to account for the thermal

effects and the associated changes in the degree of polymerization and, in turn, in the

resin viscosity.

The organization of the paper is as follows. In Sections II.1, II.2, and II.3, brief

descriptions are given for the governing equations, the finite-element control-volume

method and its two-dimensional implementation, respectively. Temperature and time

effects on the degree of polymerization and, in turn, on the viscosity of the NBV-800, a

two-component, toughened, 400K-curing, epoxy-type VARTM resin [9], are discussed in

Section III. The main flow-front mold-filling and the filling-time results obtained in the

present work are presented and discussed in Section IV, while the key conclusions

resulting from the present work are summarized in Section V.


II.1 Formulation of the Model

II.1.1 General Consideration

The mold-filling model developed in the present work is based on the following

assumptions and simplifications:

(a) fiber preform placed into the mold cavity prior to its infiltration with resin does not undergo any rigid-body motion during mold filling but can undergo

49

reversible deformations due to a pressure difference across the vacuum-bag walls which affects preform permeability;

(b) due to a low value of the Reynolds number of the resin flow, inertia effects are

negligible;

(c) the effects of surface tension are negligible in comparison with the viscous-force effects;

(d) the size of the mold cavity is much larger than the average fiber-preform pore

size so that the momentum conservation equation can be replaced by the Darcy’s law for fluid flow through a porous medium; and

(e) the resin can be considered as an incompressible fluid.

The steady-state resin flow is governed by the following incompressible-fluid

continuity equation:

0=⋅∇ ϑ (3.1)

where ∇ denotes a divergence operator, and ⋅ ϑ the resin velocity vector.

Integration of Equation (3.1) over a control volume, VCV, gives:

0=⋅∇∫ dVCVV

ϑ (3.2)

where V denotes the volume and, through the use of the Divergence theorem, Equation

(3.2) can be transformed into:

0=⋅∫ dSnCVS

ϑ (3.3)

50

where S denotes the surface, SCV the surface of a control volume, n is the outward unit

vector normal to the surface of the control volume and the raised dot is used to represent

a scalar product of two vectors.

The Darcy’s law for flow through a porous medium can be expressed as:

PK∇−=

µϑ (3.4)

where K is a second-order permeability tensor, µ resin viscosity and ∇ denotes a

gradient operator.

Substitution of Equation (3.4) into Equation (3.3) yields:

01=∇⋅∫ dSPKn

CVS µ (3.5)

While Equation (3.5) is derived starting from the steady-state continuity equation,

Equation (3.1), it is used in the present study to analyze the transient fluid flow during

mold filling. This is justified by the fact that mold filling takes place at a relatively low

rate and can be considered as a quasi steady-state process in which a steady-state

condition can be assumed to hold over each small time step.

The control-volume formulation developed up to this point is applicable only for

an isothermal mold-filling process. To include the effect of temperature, in addition to the

continuity equation, Equation (3.1), one must also consider the energy conservation

equation. Since, in general, carbon-based fiber preforms have good thermal conductivity,

51

thermal conduction is expected to be an important mechanism for transfer of the heat

from the tool-plate, through the fiber preform to the resin. In addition, the transfer of heat

by the moving resin is also expected to play a significant role. To simplify the resulting

energy conservation equation, a homogenization approach is used which eliminates a

separate treatment of the fiber preform and resin in a control volume and, instead, uses

effective (volume-averaged) gravimetric and thermal properties of the materials in a

control volume. Under such conditions, the energy conservation equation is defined as:

( ) sPP QTkTCtTC =∇−⋅∇+∇⋅+

∂∂ ϑρρ (3.6)

where ρ , Cp and k denote the effective density, heat capacity and thermal conductivity,

respectively, T is the temperature, and Qs a heat sink or a heat source term.

II.1.2 Two-dimensional Formulation

In the following analysis, preform length is assumed to be aligned in the x-

direction, preform thickness in the y-direction and preform width in the z-direction. In

many VARTM applications, the width of the fiber preform does not vary along the length

of the preform and, hence, the mold geometry and its filling can be simplified using a

two-dimensional representation. When the preform is curved along its length, a local

coordinate system is used whose z-axis is still aligned in the preform width direction.

Constant pressure is assumed to exist in the z-direction and, hence, the pressure is

assumed to be only a function of the local x and y coordinates. However, since resin

viscosity (due to gradients of the degree of polymerization and temperature in the width

52

direction) and differences in the preform permeability may vary in the width direction, a

width averaged resin velocity is defined, within a three-dimensional Cartesian coordinate

system, as:

∫=zh

z

dzzyxh

yx0

),,(1),( ϑϑ (3.7)

where is the local preform width and an overbar is used to denote an average quantity.

Substitution of Equation (3.7) into Equation (3.5) yields for a two-dimensional case:

zh

[ ] 0=

∂∂∂∂

=∇⋅∫ ∫ dl

yPxP

SSSS

nnhdlPSnhCV CVC C yyyx

xyxxyxzz (3.8)

where

∫=zh

z

dzKh

S0

'1µ

(3.9)

is a 2×2 flow-coefficient matrix defined in terms of the width-averaged resin viscosity

and the width-averaged preform permeability matrix, 'K . Assuming that the preform

width is uniformed over the surface area of a single control volume, the surface integral

in Equation (3.8) is replaced by a product of the preform width at the location of a given

control volume and the corresponding line integral.

53

II.2 Finite Element Implementation

In this section, Equation (3.8) and a finite-element control-volume method are

used to develop a model for simulation of a two-dimensional mold filling process.

Toward that end, the entire flow field is divided into four-node quadrilateral elements, as

schematically shown in Figure 3.2. Next centroids of the four adjacent elements are

connected with straight lines to form quadrilateral control volumes (more precisely

control areas in the present two-dimensional formulation).

Control Volume

Finite Element

Node

Figure 3.2 Discretization of the computational domain into four-node quadrilateral finite elements and quadrilateral control volumes.

As shown in Figure 3.3, each control area is composed of four sub-regions each

associated with a different finite element (element numbers are encircled in Figure 3.3).

54

4 3

2 1

n(b)

C B

n(a)ay/2

2 1 Aax/2

Control Volume

3 4

Figure 3.3 A control volume is composed of four quadrilateral segments each residing in a different finite element (element number is encircled). Outward control-volume surface

normals are denoted by n.

Next, following the standard finite element formulation, the pressure at each point

within an element is defined in terms of the pressures at the four nodes, (i=1-4), as: iP

∑=

=4

1iii NPP (3.10)

55

where the four bi-linear shape functions, (i=1-4), are defined in terms of a control-

volume based – coordinate system whose origin is located at the lower-left node of

the element as:

iN

'x 'y

( )( ) ( )

( )A

yxaN

AyxN

Ayax

NA

yaxaN

x

yyx

''''

''''

43

21

−==

−=

−−=

(3.11)

where and are the width and the height of an element, respectively, and the

element surface area .

xa ya

yx aaA ≡

Partial derivatives of the shape functions 'dxdNii ≡α and 'dydNii ≡β are then

computed as:

( )

( )

( )

( )A

xx

Axx

Ayy

Ay

y

Axx

Ax

x

Ayy

Ay

y

'a)'(

')'(

')'(

'a)'(

')'(

'a)'(

')'(

'a)'(

x4

2

4

y2

3

x1

3

y1

−=

−=

−=

−=

=

−−=

=

−−=

β

β

α

α

β

β

α

α

(3.12)

56

Substitution of Equation (3.12) into Equation (3.8) yields:

0=⋅∫CVC

z SBPdlnh (3.13)

where

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

=

''''''''

4321

4321

xxxxyyyy

Bββββαααα

(3.14)

and

[ TPPPPP 4321= ] (3.15)

and the superscript T is used to denote a transpose.

As seen in Figure 3.3, the control volume boundary within each element consists

of two straight segments and, hence, the contour integral given in Equation (3.13) can be

rewritten as:

∑ ∫∫=

=

⋅+⋅

4

1

)()()()()()()()( 0i C

bbi

bii

bi

C

aai

aii

aiz

CVCV

dlPBSndlPBSnh (3.16)

57

where super scripts (a) and (b) are used to denote the two line segments within a given

element and subscript i to indicate the finite elements associated with a given control

volume.

Upon integration, Equation (3.16) for each control volume becomes a linear

algebraic equation of the pressures associated with the node coinciding with the centroid

of the control volume in question and with the surrounding eight nodes. After Equation

(3.16) is written for all control volumes within the flow field and the appropriate

boundary conditions for pressure are applied, a system of linear algebraic equations for

the unknown pressures is obtained. The size of the system increases as the mold filling

proceeds until the filling is complete.

The two-dimensional finite-element control volume model presented thus far is

strictly valid only for planar geometries in which the local coordinate system coincides

with the global coordinate system. In addition, the z-axis is generally aligned with the

preform width. However, for curved thin three-dimensional preforms, the local

coordinate system whose orientation varies along the surface of the preform generally

differs from the global coordinate system. In such cases, the local z-axis is still aligned

with the preform width. However, before the current formulation can be applied, all the

quantities of a control volume defined with respect to the global coordinal system must

be transformed into the local coordinate system. This procedure is briefly discussed in the

Appendix.

The solution of the energy conservation equation, Equation (3.6), is carried out

using the same type of four-node quadrilateral finite elements, Equation (3.10), and the

same bi-linear shape functions, Equation (3.11). The solution to this problem can be

58

found in many finite element books [e.g. Equation (8.13), page 354 in Ref. 10] and,

hence, is not repeated here.

It should be noted that since the temperature-dependent resin viscosity appears in

the continuity equation, Equation (3.5), and pressure-dependent resin velocities appear in

the energy conservation equation, Equation (3.6), the two equations are coupled. The

procedure used to handle coupling of the two equations is described in the next section.

II.3 A Two-dimensional Mold Filling Model

The model developed in the previous section is used here to analyze two-

dimensional mold filling. For simplicity, the preform is assumed to be rectangular in

shape with its horizontal and vertical dimensions set to 0.1m and 0.01m, respectively.

The thicknesses of the high-permeability medium, the peel ply and the fiber preform are

set to 0.002m, 0.0005m, and 0.0075m, respectively. The in-plane permeabilities for the

high-permeability medium, the peel ply and the fiber preform in the absence of a pressure

difference across the wall of the vacuum bag are assigned typical values of 2800darcy

(1darcy = 1.0×10-12m2), 60darcy and 60darcy, respectively. The corresponding through-

the-thickness permeabilities are also set to their typical values of 2800darcy, 10darcy and

10darcy, respectively. To account for the effect of such pressure difference, the model

and the model parameters proposed by Johnson and Pitchumani [11] are used. Viscosity

of the resin and its variation with the degree of polymerization and temperature are

described in next section. The remaining thermal properties of the fiber preform and the

NBV-800 resin are listed in Table IV.

59

Peel

Pl

y

2.0×

103

0.6×

10 3

15

0.6 60

10

N/A

N/A

Hig

h Pe

rmea

bilit

y M

ediu

m

2.0×

103

0.6×

10 3

15

0.4

2800

2800

N/A

N/A

Car

bon-

fiber

Pr

efor

m

2.0×

103

0.75

×10 3

10

0.7 60

10

N/A

N/A

NB

V-8

00

Res

in

1.2×

103

0.7×

10 3

0.5

bala

nce

N/A

N/A

300

10,0

00

Uni

t

kg/m

3

J/kg

/K

W/m

/K

N/A

darc

y

darc

y

cps

cps

Sym

bol

ρ Cp k f V

K

K η η

Tabl

e IV

. The

rmal

pro

perti

es o

f the

NB

V-8

00 e

poxy

resi

n an

d th

e ca

rbon

fibe

r pre

form

.

Prop

erty

Den

sity

Hea

t Cap

acity

Ther

mal

Con

duct

ivity

Vol

ume

Frac

tion

In-p

lane

Per

mea

bilit

y

Thro

ugh-

the-

thic

knes

s Pe

rmea

bilit

y

Roo

m-te

mpe

ratu

re N

eat-

resi

n V

isco

sity

Roo

m-te

mpe

ratu

re F

ully

-cu

red

Res

in V

isco

sity

60

The computational domain is discretized into four-node quadrilateral elements of

the size 0.0005m × 0.0005m. This yielded the following numbers of the finite elements in

the thickness direction in the high-permeability medium (4), peel ply (1) and the fiber

preform (15).

Mold filling is carried out under the following boundary conditions:

1. Resin is allowed to enter the computational domain at the left-hand edge of the computational domain;

2. The pressure at the inlet is specified and set equal to the atmospheric pressure

(Pin = 101,325.33 Pa);

3. The pressure at the flow front is specified and set equal to the absolute vacuum pressure (Pvac = 0 Pa);

4. A filled-fraction parameter (0 ≤ f ≤ 1) is used to denote the volume fraction of

the resin in a control volume at the flow front;

5. The control volumes associated with the inlet are assumed to be filled (f = 1) at the onset of a mold-filling simulation run;

6. Pressure at the centroid of partially-filled (0 < f < 1) control volumes at the

flow front are set to Pvac;

7. Since there is no resin flow at the mold wall in the direction normal to the mold wall, the first derivative of the pressure in this direction is set to zero, in accordance with the Darcy’s law;

8. The temperature of the resin at the inlet is assumed to be equal to the room

temperature;

9. The temperature of the fiber preform and the resin in contact with the tool plate is set equal to the temperature of the tool plate. In other words, a zero resistance to the heat transfer from the tool plate to the preform and the resin is assumed;

10. The temperature at the top surface of the high-permeability medium is taken

to be governed by a convective flux to the environment. Following the procedure described in our recent work [15], a heat transfer coefficient is assessed as h = 30W/m2/K, while the film temperature is set to Tfilm = 295K;

61

11. The thermal flux at the right hand side of the computational domain is set to zero; and

12. The degree of polymerization of the resin at the inlet is set to an initial value

which was determined using the procedure described in Section III.

At the beginning of each new time step, the pressure, velocity, fill-fraction

parameter, permeability, temperature and degree of polymerization fields are known.

Specifically, at the beginning of a simulation run, the only control volumes filled with

resin are the ones associated with the inlet (the left hand side of the computational

domain). From the known pressure (Pin) at the centroid of these control volumes and the

known pressure (Pvac) at the centroid of the surrounding control volumes at the flow

front, and using the known initial permeability, temperature and degree of polymerization

fields and the Darcy’s law, Equation (3.4), the flow velocities at the flow front are

calculated. These velocities are assumed to remain constant over a small time step. As

discussed earlier, mold filling is treated as a quasi steady-state process in which the

steady-state condition is assumed to hold over a small time step. To ensure stability of

such an approach, the time increment associated with a given computational step is set

equal to the minimum time needed to completely fill one of the previously partially filled

control volume at the flow front. In some cases, however, more than one flow-front

control volume becomes simultaneously filled within the selected time increment.

The velocities obtained above are next used to solve the energy conservation

equation and to obtain the temperature field at the end of the time step. Next, the

temperature is assumed to vary linearly over the time step and the resulting degree of

polymerization field computed. This was done in two steps: (a) First a change in the

degree of polymerization due to the resin exposure to elevated temperatures over the

62

given time step is calculated and the corresponding change in the viscosity of the resin

within each control volume calculated (details of the calculation are presented in the next

section); (b) Next, the change in the resin viscosity within all fully or partially filled cells

due to the resin flow into and out of the control volumes is calculated. The average

temperature and the average degree of polymerization values for the given time

increment are used to recalculate the pressure and velocity fields and to recalculate the

minimal time increment needed to completely fill one of the previously unfilled control

volumes at the flow front. The resulting velocity fields are used to recalculate the

temperature field. This procedure is repeated until a preset convergence limit is reached

with respect to the minimal time increment for filling one flow-front control volume.

Once the convergence is attained, mold-filling simulation is continued over the

next time step. Toward that end, the pressures at the centroid of the filled control volumes

are declared as unknowns and the system of linear algebraic equations, Equation (3.16),

reassembled and solved.

The procedure described above is repeated until the entire computational domain

is filled with the resin (if the objective of the simulation is to determine the filling time)

or up to a certain time shorter than the filling time (if the objective of the simulation is the

analysis of the flow-front shape).

III. RHEOLOGY OF THE NBV-800 EPOXY VARTM RESIN

NBV-800 is a two-component, toughened, epoxy-based resin which is frequently

used in VARTM applications. Due to its low room-temperature viscosity (~300 cps, 1cps

= 10-3kg/m/s), NBV-800 is recommended for room temperature preform infiltration.

63

Upon infiltration, the following curing cycle is recommended: Heating from the room

temperature to ~400K at a heating rate ~1.67K/min followed by holding at 400K for 2

hrs.

The gelation temperature vs. time curve for the NBV-800 is displayed in Figure

3.4 [9]. To model the kinetics of polymerization of this resin, it is assumed that, at each

temperature, the onset of gelation corresponds to a same value of the degree of

polymerization, p. The degree of polymerization at the onset of gelation was determined

by requiring that after the recommended curing cycle given above, the NBV-800 is

practically fully-polymerized (p = 0.999). This procedure yielded p = 0.17 at the onset on

gelation.

ExperimentFitted

Figure 3.4 Isothermal temperature-time gelation curve for the NBV-800 epoxy resin.

64

Assuming first-order reaction kinetics for the isothermal polymerization process,

the degree of polymerization is taken to evolve with time at a constant temperature as:

( )tTkp −−= exp1 (3.17)

where the temperature-dependent reaction rate constant k(T) is defined as:

RTQ

ATk−

= exp)( (3.18)

By setting p = 0.17 and using the resin gelation data from Figure 3.4, the two

Arrhenius kinetic parameters are determined via a least-squares fitting procedure as A =

89.61min-1, and Q = 11,600J/mole. A comparison of the fitting function (the solid line)

and the experimental data (solid circles) in Figure 3.4 shows that the assumption

regarding the first-order reaction kinetics for polymerization of the NBV-800 is justified.

The variation of the degree of polymerization in the NBV-800 during holding time at

various temperatures is shown in Figure 3.5. It is seen that isothermal curing at 400K

gives rise to a practically complete polymerization of the NBV-800 which is consistent

with the recommended holding stage of the curing cycle discussed above.

65

0.1

0.1

0.2

0.2

03

0.3

0.3

04

0.4

0.4

0.4

0.5

0.50.5

0.6

0.6

0.6

0.7

0.7

0.70.8

0.8

0.80.9

0.9

0.9

Temperature, K

Tim

e,m

in

370 380 390 400 410 4200

10

20

30

40

50

60

70

80

90

100

110

120

Figure 3.5 Variation of the degree of polymerization in the NBV-800 epoxy resin with time during isothermal holdings at different temperatures.

The evolution of the degree of polymerization of a material point in the resin

subjected to a non-uniform temperature history during preform infiltration is obtained by

integrating the following differential equation:

tT

Tp

tp

dtdp

∂∂

∂∂

+∂∂

= (3.19)

to obtain:

66

)))'))'(/(exp(exp()1(1( ∫∆+

∆+ −−−−=tt

tttt dttRTQApp

(3.20)

where subscripts t and t+∆t are used to denote the value of a quantity at the beginning and

at the end of a time step with the duration ∆t and the integral can be readily evaluated

using numerical integration.

Next, following our recent work, the resin viscosity at each temperature is taken

to be a single power-law function of the degree of polymerization [9] and that there is a

degree of polymerization invariant thermal-thinning effect. Consequently, the resin

viscosity is defined as:

( )[ ]

−−

=== ⋅−+= RTTTRTQ

ppp ep11

3.11010

*

ηηηη (3.21)

where 0=pη (=300cps) and 1=pη (=10,000cps) are the room-temperature viscosity of fully

un-polymerized and fully polymerized resin, Q (=2,600J/mole) [9] is a thermal-thinning

activation energy and T

*

RT (=295K) is the room temperature. The effect of degree of

polymerization and the temperature on the logarithm of viscosity of the NBV-800 is

shown in Figure 3.6. It is seen that the degree of polymerization has a substantially larger

effect on viscosity of the NBV-800 than the temperature and that the effect of

temperature on the relative change in viscosity of the NBV-800 is quite similar at

different levels of the degree of polymerization.

67

2.6

2.7

2.8

2.9

3

3.2

3.3 3.4

3.5

3.6

3.7

3.8

Degree of Polymerization

Tem

pera

ture

,K

0 0.25 0.5 0.75 1370

380

390

400

410

420

2-hour Curing Line

Figure 3.6 Variation of the logarithm of viscosity in NBV-800 epoxy resin with the degree of polymerization and temperature.

IV. RESULTS AND DISCUSSION

IV.1 Room-temperature Mold Filling Simulations

The control-volume finite-element method developed in Sections II.1-II.3 is

utilized in this section to analyze preform infiltration with the NBV-800 epoxy resin at

room temperature. The details regarding the preform dimensions including the

thicknesses of the high-permeability medium, the peel-ply and the fiber-preform as well

as the values of the planar and the through-the-thickness permeabilities of these layers are

given in Section II.3 and in Table IV. The value for room-temperature viscosity of the

neat NBV-800 resin is given in Table IV.

68

Since, no experimental investigation is carried out as part of this work, the present

model is validated using the experimental flow visualization results of Lee and co-

workers [7] for room-temperature infiltrations of a carbon preform with three different

mineral oils: DOP oil (room-temperature viscosity 43 cps), Mobil Extra Heavy Oil

(room-temperature viscosity 320 cps), Mobil BB oil (room-temperature viscosity 530

cps). Both the computed shapes of the flow front and the infiltration times are found to

agree quite well with their experimental counterparts.

Figure 3.7 shows the NBV-resin flow-fronts at infiltration times of 5.2s, 42.9s,

111.8s, 211.7s and 346.8s, respectively. As expected, it is seen that in the preform length

direction, the resin flows primarily through the high-permeability medium. Infiltration of

the peel-ply and the fiber-preform, on the other hand, is primarily the result of resin flow

from the fully-infiltrated high-permeability medium into the peel-ply and the fiber-

preform in the through-the-thickness direction. Consequently, the flow-front in the fiber

preform lags behind the flow front in the high-permeability medium. Short-time

simulation results of the evolution of the resin flow front (not included here for brevity)

showed that that the lag distance, ll, reaches a nearly constant, steady-state value of

~27mm, approximately 4.4 seconds after the start of the mold filling process. This value

of the lag distance is in a reasonably good agreement with the corresponding value (25

mm) obtained using the following analytically-derived equation proposed by Ni et al.

[12]:

69

x

z

FP

HPM

x

z

FP

HPM

x

z

FP

HPM

x

z

FPl

KK

hh

KK

hh

KK

hh

KK

hl

2

112

π

π+

−+

−

= (3.20)

where hFP and hHPM are the thicknesses of the fiber preform and the high-permeability

medium, respectively and Kx and Kz are the fiber-preform in-plane and through-the-

thickness permeabilities, respectively.

((ee))

((aa))

((bb))

((cc))

((dd))

0.01m

Figure 3.7 Resin flow fronts during room-temperature infiltration at the filling times: (a) 5.2s; (b) 42.9s; (c) 111.8s; (d) 211.7s; and (e) 346.8s.

70

The isothermal mold-filling simulation yielded a time of 471s for a complete

filling of the mold. This value, as well as the flow-front profiles displayed in Figure 3.7,

is found not to be significantly affected when the mesh size is reduced by a factor of 2.

Consequently all the remaining calculations reported here were carried out using the

mesh size reported in Section II.3.

IV.2 Non-Isothermal Mold Filling Simulations

The control-volume finite-element method developed in Sections II.1-II.3 is

utilized in this section to analyze preform infiltration with the NBV-800 epoxy resin

under different thermal histories of the tool plate. The details regarding the preform

dimensions including the thicknesses of the high-permeability medium, the peel-ply and

the fiber-preform as well as the values of the planar and the through-the-thickness

permeabilities of these layers are given in Section II.3 and in Table IV. The change of the

viscosity of the NBV-800 resin with the degree of polymerization and with temperature is

discussed in Section III.

The mold-filling simulations carried out in this section involved ramping of the

tool-plate temperature, from the onset of preform infiltration, at a constant heating rate of

0.03K/s until a desired (holding) temperature is reached and holding the temperature

constant thereafter until the completion of mold filling. The effect of the holding

temperature on the time required filling a half of the mold (the half-filling time) and the

time for complete filling of the mold (the complete-filling time) are displayed in Figures

3.8.1 and 3.8.2, respectively. It is seen that the minimum half-filling time is attained for

71

the holding temperature of ~334K, while any heating of the tool plate increases the

complete-filling time. This finding can be readily explained by considering the direct

(thermal-thinning) effect of the temperature on the resin viscosity and its indirect effect

via an increase in the degree of polymerization in the resin. The thermal-thinning effect is

dominant at shorter infiltration times when the degree of polymerization in the resin is

close to zero. Consequently, the resulting lower viscosity of the resin can give rise to a

decrease in the infiltration time, Figure 3.8.1. Contrary, at longer infiltration times, a

viscosity increase due to the associated increase in the degree of polymerization of the

resin becomes dominant and, consequently, the infiltration times are increased, Figure

3.8.2.

Figure 3.8.1 The effect of the tool-plate holding temperature on the mold half-filling time. The tool-plate heating rate from the room temperature to the holding temperature is

fixed at 0.3K/s.

72

Figure 3.8.2 The effect of the tool-plate holding temperature on the complete-filling time. The tool-plate heating rate from the room temperature to the holding temperature is fixed

at 0.3K/s.

To demonstrate the effect of thermal-thinning at short infiltration times, the flow-

front shape and the temperature and the viscosity contour plots at fill fraction of the mold

of ~13% for the case of the tool-plate heating from the onset of infiltration at a rate of

5K/s are shown in Figures 3.9.1 – 3.9.3, respectively. A comparison of the results

displayed in Figure 3.9.1 and Figure 3.7 shows that, due to a lower viscosity of the resin

in the vicinity of the tool-plate, there is a measurable contribution of the resin flow in the

longitudinal direction which changes the shape of the flow front. Specifically, the largest

lag distance in the preform is not at the tool-plate surface but somewhat removed from it.

A comparison of the results displayed in Figures 3.9.2 and 3.9.3 shows a direct

73

correlation between the temperature and the resin viscosity at shorter infiltration times

when the direct thermal-thinning effect of the temperature on resin viscosity is prevalent.

Figure 3.9.1 The flow front for the case of preform infiltration in which the tool plate is heated, from the onset of infiltration at a rate of 5K/s for ~5s.

Resin

298K

304K310K

316K322K

Figure 3.9.2 The temperature contour plot for the case of preform infiltration in which the tool plate is heated, from the onset of infiltration at a rate of 5K/s for ~5s.

74

290cps

190cps

270cps250cps

230cps210cps

Figure 3.9.3 The resin viscosity contour plot for the case of preform infiltration in which the tool plate is heated, from the onset of infiltration at a rate of 5K/s for ~5s.

IV.3 Optimization of the Mold Filling Process

The model developed in the present work enables determination of an optimum

ramping/holding thermal history of the tool plate which can give rise to a minimum

complete-filling time. An example of such an optimization procedure is presented in this

section.

In general, optimization of the preform-infiltration process under non-isothermal

conditions can be done by decomposing the time-temperature profile of the tool plate into

a number of constant heating-rate ramping steps and a number of constant-temperature

holding steps. Then the ramping heating rates and holding temperatures and times can be

used as optimization parameters. An optimization procedure, such as the simplex method

[13] or the genetic algorithm [14], can then be used to determine the optimum thermal

history of the tool plate which minimizes the complete-filling time. Such an optimum

analysis will be used in our future work. In this paper, however, we report the results of a

75

simple two-parameter optimization analysis which does not entail the use of an

optimization algorithm.

Based on the results presented in the previous section, it was concluded that

heating of the tool plate should start only after a substantial fraction of the mold has been

infiltrated with the resin at the room temperature. Otherwise, the associated prolonged

heating of the resin during preform infiltration gives rise to an increase in the degree of

polymerization and, in turn, to an increase in the resin viscosity causing the rate of

infiltration to decrease. Toward that end, a two-parameter optimization analysis is carried

out in which the fraction of the mold filled with the resin at the room temperature and the

heating rate at which the tool plate is subsequently heated are used as the optimization

parameters. The first parameter is varied from 0.5 to 1.00 in increments of 0.1, while the

second parameter is varied from 0.5 to 4K/s in increments of 0.25K/s. The results of this

optimization analysis are presented using a complete-filling time contour plot shown in

Figure 3.10. It is seen that relative to the case of complete mold-filling at the room

temperature, ~80% preform infiltration at the room temperature followed by heating of

the tool plate at a rate of ~3.2K/s can reduce the complete-filling time by ~5%, from 471s

to 447s. While this level of complete-filling time reduction does not appear very

significant, one can generally expect more significant complete-filling time reductions in

the VARTM preforms with more complex shapes.

76

Figure 3.10 Variation of the complete-filling time with the room-temperature fill fraction and the tool-plate heating rate. The solid white circle is used to denote the minimum

complete-filling time.

V. CONCLUSIONS


can be drawn:

1. By adding to the incompressible-fluid mass conservation equation, an energy conservation equation and an equation for the time and temperature evolution of the degree of polymerization of the resin, the control-volume finite-element method originally proposed by Lee and co-workers [4-8] has been extended to analyze preform infiltration stage of a high-permeability medium based VARTM process.

2. Simulations of the preform infiltration process under non-isothermal

conditions showed that, at short infiltration times, the effect of tool-plate heating can be beneficial and can lead to an increase in the rate of infiltration.

Room-temperature Fill Fraction

Tool

-pla

teH

eatii

ngTe

mpe

raur

e,K/

s

0.5 0.6 0.7 0.8 0.9 1

1

2

3

4

490s 500s480s

470s

510s 520s

460s470s

450s

77

This effect has been attributed to a thermal-thinning based reduction in the resin viscosity.

3. An optimization analysis of the VARTM preform infiltration process showed

that, in order to take a full advantage of tool-plate heating, ~70-80% of the mold should be filled with the resin at the room temperature before heating of the tool-plate is initiated. For the simple rectangular geometry of the fiber preform used in the present work, ~80% room-temperature preform infiltration followed by a tool-plate heating at a rate of ~3.2K/s, can reduce the infiltration time by ~5% relative to the room-temperature complete-infiltration time.

VI. REFERENCES

1. Lewit, S. M., and J. C. Jakubowski, “Low Cost VARTM Process for Commercial and Military Applications,” SAMPE International Symposium, 42, 1173 (1997).

2. Nquyen, L. B., T. Juska and S. J. Mayes, “Evaluation of Low Cost Manufacturing

Technologies for Large Scale Composite Ship Structures,” AIAA/ASME/ASCE/AHS/ ASC Structures, Structural Dynamics and Materials Conference, 38, 992 (1997).

3. Lazarus, P., “Resin Infusion of Marine Composites,” SAMPE International

Symposium, 41, 1447 (1996).

4. Young, W. B., K. Han, L. H. Fong, L. J. Lee and M. J. Liou, “Flow Simulation in Molds with Preplaced Fiber Mats,” Polymer Composites, 12, 391 (1991).

5. Young, W. B., K. Rupel, K. Han, L. J. Lee and M. J. Liou, “Analysis of Resin

Injection Molding in Molds With Preplaced Fiber Mats I: Permeability and Compressibility Measurements,” Polymer Composites, 12, 30 (1991).

6. Young, W. B., K. Rupel, K. Han, L. J. Lee and M. J. Liou, “Analysis of Resin

Injection Molding in Molds With Preplaced Fiber Mats II: Numerical Simulation and Experiments of Mold Filling,” Polymer Composites, 12, 20 (1991).

7. Sun, X. D., S. Li and L. J. Lee, “Mold Filling Analysis in Vacuum-Assisted Resin

Transfer Molding, Part I: SCRIMP Based on a High-Permeable Medium,” Polymer Composites, 19, 807 (1998).

8. Sun, X. D., S. Li and L. J. Lee, “Mold Filling Analysis in Vacuum-Assisted Resin

Transfer Molding, Part II: SCRIMP Based on Grooves,” Polymer Composites, 19, 818 (1998).

78

9. Grujicic, M., and K. M. Chittajallu, “Kinetics of Polymerization of NBV-800

Two-Component Epoxy-based Resin,” Journal of Materials Science, submitted for publication, November 2003.

10. Huebner, K. H., Donald L. Dewhirst, Douglas E. Smith and Ted G. Byrom, “The

Finite Element Method for Engineers,” Fourth Edition, Wiley-Interscience (2001).

11. Johnson, R. J., and R. Pitchumani, “Enhancement of Flow in VARTM using

Localized Induction Heating,” Composites Science and Technology, 63, 2201 (2003).

12. Ni, J., Y. Zhao, L. J. Lee, and S. Nakamura, “Analysis of Two-Regional Flow in

Liquid Composite Molding,” Polymer Composites, 18, 254 (1997).

13. Grujicic, M., Y. Hu, and G. M. Fadel, “Optimization of the LENS™ Rapid Fabrication Process for In-Flight Melting of the Feed Powder,” Journal of Materials Synthesis and Processing, 9, 223 (2002).

14. Grujicic, M., G. Cao and B. Gersten, “Optimization of the Chemical Vapor

Deposition Process for Carbon Nanotubes Fabrication,” Applied Surface Science, 191, 223 (2002).

15. Grujicic, M., K. M. Chittajallu and S. Walsh, “Optimization of the VARTM

Process for Enhancement of the Degree of Devolatilization of Polymerization By-products and Solvents,” Journal of Materials Science, 38, 1729 (2003).

CHAPTER 4

EFFECT OF SHEAR, COMPACTION AND NESTING ON PERMEABILITY OF THE

ORTHOGONAL PLAIN-WEAVE FABRIC PREFORMS

ABSTRACT

Permeability of fabric preforms and its changes due to various modes of the fabric

distortion or deformation as well due to fabric-layers shifting and compacting is one of

the key factors controlling infiltration of the preforms with resin within the common

polymer-matrix composite liquid-molding fabrication processes. While direct

measurements of the fabric permeability generally yield the most reliable results, a large

number of the fabric architectures used and numerous deformation and layers

rearrangement modes necessitate the development and the use of computational models

for the prediction of preform permeability. One such model, the so-called lubrication

model is adapted in the present work to study the effect of the mold walls, the

compaction pressure, the fabric-tows shearing and the fabric-layers shifting on

permeability of the preforms based on orthogonal balanced plain-weave fabrics. The

model predictions are compared with their respective experimental counterparts available

in the literature and a reasonably good agreement is found between the corresponding

sets of results.

80

NOMENCLATURE

η - Resin viscosity (Ns/m2)

f - Fiber volume fraction

h - Fabric thickness (m)

φ - Relative dimensionless shift of the adjacent fabric layers

K - Permeability tensor of the fabric (m2)

L - Length of the quarter unit cell (m)

p - Pressure (Pa)

θ - Shear angle (deg.)

r - Fiber radius (m)

s - Relative shift of the adjacent fabric layers (m)

u - x-component of the resin velocity (m/s)

U - In-plane resin velocity magnitude (m/s)

v - y-component of the resin velocity (m/s)

w - z-component of the resin velocity (m/s)

W - Transverse resin velocity magnitude (m/s)

Subscripts

bot - Quantity associated with the bottom surface of the fabric

corr - Quantity corrected for the effect of shear on the fiber volume fraction

low - Quantity associated with the lower mold surface

o - Quantity associated with un-sheared fabric preform

top - Quantity associated with the top surface of the fabric

θ - Quantity associated with sheared fabric preform

81

upp - Quantity associated with the upper mold surface

Superscripts

B - Quantity associated with the bottom channel

F - Quantity associated with the fabric

T - Quantity associated with the top channel

I. INTRODUCTION

Over the last two decades, processing of high-performance polymer-matrix

composites via the use of modern resin-injection technologies has made major advances

and expanded from its aerospace roots to military and diverse civil applications. At the

same time, processing science has become an integral part of the composite-

manufacturing technology so that empiricism and semi-empiricism have given way to

greater use of computer modeling and simulations of the fabrication processes. Among

the modern polymer-matrix composite manufacturing techniques, liquid molding

processes such as Resin Transfer Molding (RTM), Vacuum Assisted Resin Transfer

Molding (VARTM) and Structural Reaction Injection Molding (SRIM) have a prominent

place. A detailed review of the major liquid molding processes can be found in the recent

work of Lee [1]. One common feature to all these composite fabrication processes is the

use of low-pressure infiltration of the porous fabric preforms with a viscous fluid (resin).

Conformation of the fabric preforms to the ridges and recesses in the mold and the

applied pressure can induce significant distortions and deformations in the fabric as well

as give rise to shifting of the individual fabric layers and, in turn, cause significant

change in local permeability of the preform. Since the infiltrating fluid follows the path

82

of least resistance, local changes in the fabric permeability can have a great influence on

the mold filling process influencing the filling time, the filling completeness and the

formation of pores and dry spots. Hence, understanding the changes in fabric

permeability caused by various local distortions, shearing and shifting of its tows is

critical for proper design of the liquid molding fabrication processes.

Permeability of a porous medium is one of the most important parameters

controlling the flow of a fluid through such medium. In simple terms, permeability can be

defined as a (tensorial) quantity which relates the local velocity vector of the fluid flow

with the associated pressure gradient. In polymer-matrix composite liquid-molding

manufacturing processes (e.g. in the RTM and the VARTM processes), the porous

medium consists of woven- or weaved-fabric preforms placed in the mold and the fluid

flow of interest involves preform infiltration with resin. Complete infiltration of the

preform with resin is critical for obtaining high-integrity, high-quality composite

structures. The knowledge of the preform permeability and its changes due to fabric

bending, shearing, compression, shifting, etc. is crucial in the design of a composite

fabrication process (e.g. in the design of the tool plate, or for placement of the resin

injection ports). In general, the most accurate value of permeability of a porous medium

is obtained by direct experimental measurements. However, the number of fabric

architectures can be quite large and fabric distortion modes numerous making

permeability determinations via the purely experimental means not a very appealing

alternative. In addition, sometimes the experimentally-determined permeability values

reported by different researchers for the apparently identical fabric architectures can

differ significantly. Consequently, development of the computational models for

83

prediction of the preform permeability to complement experimental measurements has

become a standard practice.

For the computational modeling approach to be successful in predicting

permeability of the fabric preforms, it must include, in a correct way, both the actual

architecture of the fabric and the basic physics of the flow through it. A schematic of the

relatively-simple orthogonal plain-weave one-layer fabric architecture is shown in Figure

4.1.

(b)

(a)

Weft

PoreWarp

Warp Weft

Figure 4.1 A schematic of: (a) the top view and (b) the edge view of a one-layer orthogonal plain-weave fabric preform.

84

As seen in Figure 4.1, the fabric consists of orthogonal (warp and weft) fiber

yarns, which are woven together to form an interconnected network. Each yarn, on the

other hand, represents a bundle of the individual fibers held together with thread. In

addition, the fabric involves a network of empty pores and channels. When a fabric like

the one shown in Figure 4.1 is being infiltrated, the resin flows mainly through the pores

and the channels. However, since the fabric tows are porous (pores and channel on a finer

length scale exist between the fibers in tows), the resin also flows within the yarn. Thus

when predicting the effective permeability of a fabric, the computational model must

account for both components of the resin flow.

Prediction of the permeability of porous medium has been the subject of intense

research for at least last two decades. Due to space limitations in this paper, it is not

possible to discuss all the models proposed over this period of time. Nevertheless, one

can attempt to classify the models. One such classification involves the following main

types of models for permeability prediction in the porous media: (a) the

phenomenological models based on the use of well established physical concepts such as

the capillary flow [e.g. 2,3] or the lubrication flow [e.g. 4]. These models generally

perform well within isotropic porous media with a simple architecture; (b) the numerical

models which are based on numerical solutions of the governing differential equations.

These models generally attempt to realistically represent the architecture of the fiber

preform but, due to limitations in the computer speed and the memory size, are ultimately

forced to the introduction of a number of major simplifications [e.g. 5,6]; and (c) the

models which are based on a balance of the fabric-architecture and the flow physics

simplifications, enabling physically-based predictions of the preform permeability within

85

reasonably realistic fabric architectures. One of such models is the one proposed by

Simacek and Advani [7]. The model of Simacek and Advani [7] also includes the effect

of important factors such as: (a) the flow within the fiber yarn; (b) nesting in multi-layer

fabric; and (c) distortion and deformation of the fabric. In the present work, the model of

Simacek and Advani [7] is extended to include the effect of shear of the fabric tows on

the effective volume fraction of fibers.

The organization of the paper is as follows: A brief overview of the model

proposed by Simacek and Advani [7] and its modifications are presented in Section II.

The application of their model to reveal the role of various fabric distortion and layers-

compaction phenomena is presented and discussed in Section III. The main conclusions

resulted from the present work are summarized in Section IV.


II.1 Fabric Architecture

In this work, only (un-sheared and sheared) balanced orthogonal plain-weave

fabric is considered. Due to the in-plane periodicity, the fabric architecture can be

represented using a unit cell. The entire orthogonal plain-weave fabric can then be

obtained by repeating the unit cell in the in-plane (x- and y-) directions. A schematic of

one quarter of a plain-weave unit cell with the appropriate denotation for the system

dimensions are shown in Figure 4.2. In a typical plain-weave fabric, the fabric thickness

(h) to the quarter cell in-plane dimension (L) ratio, h/L, is small (0.01-0.1), while the tow

cross section is nearly elliptical in shape with a large (width-to-height) aspect ratio (5 or

larger). The geometry of the tows within the cell can be described using various

86

mathematical expressions [e.g. 8] for the top, ( )yxztop , , and the bottom, ( )yxzbot , ,

surfaces of the fabric, respectively. In the present work, the following sinusoidal

functions originally proposed by Ito and Chou [9] are used:

( )

+= y

Lx

Lhyxztop

ππ 2sin2sin2

, (4.1)

( )

+−= y

Lx

Lhyxzbot

ππ 2sin2sin2

, (4.2)

y z

L2

L1 ~ L2 ~ LWarp Tow

Weft Tow Top Channel

x

Weft Tow

Top Mold Wall

Bottom Mold Wall

Bottom Channel

h

L1

Figure 4.2 Schematic of one quarter of the unit cell for a one-layer balanced orthogonal plain-weave fabric.

87

As pointed out earlier, fiber tows have typically a near elliptical cross section and

hence Equations (4.1) and (4.2) only approximate the actual tow cross-section shape.

Nevertheless, they are used in the present work since they greatly simplify permeability

calculations in the distorted fabric and are generally considered as a good approximation

for the actual tow cross-section shape.

A simple examination of Figure 4.2 shows that within a single-layer orthogonal

plain-weave fabric unit cell, one can identify three distinct domains:

The top channel, Region T: 2hzztop <<

The fabric, Region F: topbot zzz <<

The bottom Channel, Region B: botzzh<<

2−

Regions T and B contain only the resin, while region F contains both the fiber

tows and the resin. The resin flow through a unit cell is analyzed in the present paper by

first considering the flow within the three regions separately and then utilizing the

matching boundary conditions which ensure continuity in the pressure and the velocity

components across the contact surfaces of the adjacent regions. The resin is considered as

a Newtonian (constant density) fluid. The flows within the top and the bottom channels

are assumed to be of a creeping nature (i.e. the inertial effects are neglected) while the

flow within the fabric is assumed to be governed by the Darcy’s law (a velocity vs.

88

pressure gradient relation which eliminates the need for use of the momentum

conservation equations).

II.2 Governing Equations

II.2.1 Flow Within the Top and the Bottom Channels

Under typical fabric infiltration conditions, the resin flow within the regions T and

B can be considered as a creeping flow in which inertial effects are negligibly small in

comparison to the viscous effects. Under such conditions, at constant temperature, the

resin flow can be described by the Stokes equations as:

0222

2

2

2

2

2

2

=

∂∂

∂+

∂∂∂

+∂∂

+∂∂

+∂∂

⋅+∂∂

−zx

wyxv

zu

yu

xu

xp η (4.3)

0222

2

2

2

2

2

2

=

∂∂

∂+

∂∂∂

+∂∂

+∂∂

+∂∂

⋅+∂∂

−zy

wyx

uzv

xv

yv

yp η (4.4)

0222

2

2

2

2

2

2

=

∂∂

∂+

∂∂∂

+∂∂

+∂∂

+∂∂

⋅+∂∂

−zyv

xzu

yw

xw

zw

zp η (4.5)

0=∂∂

+∂∂

+∂∂

zw

yv

xu (4.6)

where p is the pressure, u, v and w are respectively the x-, y- and z- components of the

resin velocity and η is the resin viscosity.

89

Following the procedure of Simacek and Advani [7], which involves non-

dimensionalization of the governing equations, and the use of the conditions: h/L << 1

and ( ) ( ) 1≈UhWL , (U and W are (mean) in-plane and transverse resin velocity

magnitudes, respectively), Equations (4.3) – (4.6) can be simplified to yield:

02

2

=∂∂

+∂∂

−zu

xp η (4.3')

02

2

=∂∂

+∂∂

−zv

yp η (4.4')

0=∂∂

zp (4.5')

0=∂∂

+∂∂

+∂∂

zw

yv

xu (4.6')

Equations (4.3') – (4.6') are generally referred to as “two-dimensional lubrication-

flow equations” in which the pressure variation in the z-direction is negligibly small.

However, in contrast to the traditional lubrication models, the transverse velocity w (the

velocity in the z-direction) is generally not zero (or constant) in the present case and,

consequently, the last term on the left hand side of the continuity equation, Equation

(6.6'), does not vanish. Nevertheless, this term can be eliminated by integrating Equation

(6.6') in the z direction to yield:

90

0=−+

∂∂

+∂∂

∫upp

low lowupp

z

z zzwwdz

yv

xu (4.7)

where and are mathematical expressions for the upper and the lower

surfaces of the channels and the associated transverse velocities,

),( yxzupp ),( yxzlow

uppzw and

lowzw , are

given by the appropriate boundary conditions discussed later.

II.2.2 Flow Within the Fiber Tows

The resin flow through the fabric is described in the present work using the

Darcy’s law for an anisotropic porous medium as:

∂∂

+∂∂

+∂∂

−=zpK

ypK

xpKu xzxyxxη

1 (4.8)

∂∂

+∂∂

+∂∂

−=zpK

ypK

xpKv yzyyyxη

1 (4.9)

∂∂

+∂∂

+∂∂

−=zpK

ypK

xpKw zzzyzxη

1 (4.10)

0=∂∂

+∂∂

+∂∂

zw

yv

xu (4.11)

91

where , , , xxK yyK zzK yxxy KK = , zxxz KK = and zyyz KK = are the components of the

symmetric tow permeability tensor.

Equations (4.8) – (4.11) can be simplified under the following assumptions: (a)

and (b)zzyyxx KKK == 0== zxyz KK . The first assumption is not typically fully

justified since the longitudinal components of the permeability ( and ), are

generally larger (up to an order of magnitude) than the transverse component ( ) of the

permeability. However, this assumption greatly simplifies the computational procedure

and, for simple fabric geometries, it is found, in the present work, that the results are

different by only 1-2% relative to their more accurately determined counterparts

corresponding to

xxK yyK

zzK

10== zzyy KKzzxx KK . The second assumption, on the other hand, is

generally expected to be valid for at least two reasons: (a) for the orthogonal plain-weave

architecture of the fabric, the material transverse principal direction is expected to be

essentially coincident with the global z-axis; and (b) the second assumption is valid

whenever the first assumption is valid. Again following the procedure of Simacek and

Advani [7], which involves non-dimensionalization of the governing equations, and the

use of the conditions: h/L << 1 and ( ) ( ) 1≈UhWL , Equations (4.8) – (4.11) become:

0=u (4.8')

0=v (4.9')

zpK

w zz

∂∂

−=η

(4.10')

92

0=∂∂

zw (4.11')

Equations (4.8') – (4.11') indicate that the only non-zero component of the resin

velocity within the fabric is the one in the z-direction and that, at given values of the in-

plane x- and y- coordinates, this component of the velocity does not vary in the z-

direction.

II.3 Boundary Conditions

The following boundary conditions are used for the resin flow problem in the

three regions:

• No slip (u = v = w = 0) at the mold walls, 2hz ±= ;

• At the fabric/channels contact surfaces, and , the velocities and the

pressure continuity are assumed, i.e.: topz botz

( ) ( )toptop zz

FinTin φφ = (4.12)

( ) ( )botbot zz

FinBin φφ = (4.13)

where φ = p, u, v, or w.

93

It should also be noted that, as established in the previous section, u (in F) = v (in

F) = 0. In addition, the definition of the (x- and y-) in-plane boundary conditions is

deferred until the final system of equations is derived (the next section).

II.4 The Final System of Equations

The resin velocities in the two channels can be obtained by integrating twice

Equations (4.3') and (4.4'), and using the boundary conditions given by Equations (4.12)

and (4.13) to determine the integration constants. This procedure yields:

( )

( )

−⋅

−⋅

∂∂=

−⋅

−⋅

∂∂=

top

top

zzhzypv

zzhzxp

u

2

2

η

η in Region T (4.14)

( )

( )

−⋅

+⋅

∂∂=

−⋅

+⋅

∂∂=

bot

bot

zzhzypv

zzhzxp

u

2

2

η

η in Region B (4.15)

The subsequent equations can be simplified by introducing the following

expressions: ( ) topT zhyxh −= 2, , ( ) 2, hzyxh bot

B −= , ( ) bottopF zzyxh −=, which

denote the height fields of the top channel and the bottom channels and the thickness

field of the fabric, respectively.

94

Substitution of Equations (4.14) and (4.15) into the integrated form of the

continuity equation, Equation (4.7), for the two channels yields:

( ) ( )0

61

33

2=

∂

∂

∂⋅∂

+∂

∂

∂⋅∂

⋅⋅

−−== y

yph

xx

phww

TT

TT

zz

T

hz

T

top η (4.16)

( ) ( )0

61

33

2=

∂

∂

∂⋅∂

+∂

∂

∂⋅∂

⋅⋅

−−−== y

yph

xx

phww

BB

BB

hz

B

zz

B

bot η (4.17)

where superscripts T and B are used to denote the quantities pertaining to the top and the

bottom channels.

The first two terms on the left hand side of Equations (4.16) and (4.17) are

defined by the boundary conditions discussed earlier as:

02

==hz

Tw (4.18)

( )F

TBzz

zz

T

hppK

wtop ⋅

−=

= η (4.19)

02

=−= hz

Bw (4.20)

95

( )F

TBzz

zz

B

hppK

wbot ⋅

−=

= η (4.21)

Consequently Equations (4.16) and (4.17) can be rewritten as:

( ) ( ) ( )0

61

33

=

∂

∂

∂⋅∂

+∂

∂

∂⋅∂

⋅⋅

−⋅

−−

yy

ph

xx

ph

hppK

TT

TT

F

TBzz

ηη (4.22)

( ) ( ) ( )0

61

33

=

∂

∂

∂⋅∂

+∂

∂

∂⋅∂

⋅⋅

−⋅

−y

yph

xx

ph

hppK

BB

BB

F

TBzz

ηη (4.23)

Equations (4.22) and (4.23) represent the final system of equations consisting of

two coupled linear elliptic partial differential equations with the pressures and as

dependent variables. To solve these equations, boundary conditions along the (x-y) in-

plane boundaries of the unit cell must be prescribed. For the un-sheared balanced plain-

weave fabric architecture in which the unit cell boundaries are the lines of geometrical

symmetry, a fixed pressure gradient can be enforced in one principal direction while

requiring periodicity in the pressure distribution in the direction normal to the direction in

which the pressure gradient is prescribed. This type of boundary conditions is generally

used since it enables determination of the off-diagonal ( , and ) components

Tp

xz

Bp

xyK yzK K

96

of the effective preform permeability tensor. A more detailed discussion of the in-plane

boundary conditions is given later in the context of the effect of fabric shearing on the

choice of in-plane boundary conditions.

The final system of partial differential equations, Equations (4.22) and (4.23),

contains the thickness fields: ( )yxhT , , ( )yxh B , and ( )yxh F , . These fields are defined in

the present work using the analytical expressions for the top and the bottom surfaces of

the fabric preform, Equations (4.1) and (4.2). These expressions are generally considered

as reasonably good approximations of the actual orthogonal plain-weave fabric

architecture with a near elliptical cross-section area. It should be noted, however, that

over-simplification of the fabric architecture (e.g. using square or circularly shaped tows)

may lead to erroneous results and must be avoided. In general, the thickness fields can be

constructed using direct experimental measurements such as quantitative metallographic

analysis of consolidated and sectioned parts [e.g. 10] and through the use of

computerized image analysis of the fabric surface [e.g. 11]. The second of these two

methods is quite appealing since the image conversion procedure can be directly coupled

with the solution scheme for Equations (4.22) and (4.23).

Due to complexity in the ( )yxT , ,h ( )yxh B , and ( )yxh F , functions, Equations

(4.22) and (4.23), cannot be solved analytically. However, finding the numerical solution

to Equations (4.22) and (4.23) is relatively straightforward. In the present work,

MATLAB general-purpose mathematical package [12] and a finite difference method are

used to solve Equations (4.22) and (4.23).

Once Equations (4.22) and (4.23) are solved, the resulting pressure fields can be

used, in conjunction with Equations (4.14) and (4.15), to compute the corresponding in-

97

plane velocity fields in the two channels. Integration of these velocity fields over the side

boundaries of the quarter unit cell then enables determine of the total resin flow rate, Q =

[Qx Qy], through the quarter unit cell in the two principal direction. The components of

the effective in-plane preform permeability, and are then computed using

the two-dimensional Darcy’s law and the known imposed values of the pressure gradient.

effyy

effxx KK , eff

xyK

II.5 Application of the Model to the Multi-Layer Fabric

The model developed thus far pertains to a single-layer fabric preform. In typical

RTM and VARTM processes, the preforms may contain several fabric layers. In such

multi-layer preforms, nesting and compaction generally have a significant effect and must

be included when predicting preform permeability. Numerous experiments [e.g. 13,14]

confirmed that permeability varies with a number of layers.

The single-layer model developed in the previous section can be readily extended

to a multi-layer preform. A schematic of two types of two-layer plain-weave fabric

preforms is given in Figure 4.3.1 and 4.3.2. The two types are generally referred to as

“in-phase” and “out-of-phase” fabric architectures or laminates. In the case of an n-layer

fabric preform, if the channels are labeled using consecutive integers (with the bottom

channel being denoted as channel “1”), the analytical procedure for a single-layer fabric

preform used in the previous section yields (n+1) coupled elliptical partial differential

equations with n+1 unknown pressures ( )1p , ( )2p , … ( )1+np as:

( ) ( )( ) ( )( ) ( )( ) 06

1 131121

=∇∇⋅

−−⋅

− phpph

KF

zz

ηη (4.24)

98

( ) ( )( ) ( ) ( )( ) ( )( ) ( )( )ni

phpph

Kpp

hK iiii

Fzzii

Fzz

ii

,,3,2

06

1 311

1

K=

=∇∇⋅

−−⋅

−−⋅

++

− ηηη (4.25)

( ) ( )( ) ( )( ) ( )( ) 06

1 1311 =∇∇⋅

−−⋅

+++ nnnnF

zz phpph

Kn ηη

(4.26)

where denotes the thicknesses of the i-th fabric layer (numbered starting

from the bottom of the mold) and

( nih iF ,,1K= )

( ) ( )1,,1 += nii Kh are the heights of the inter-fabric or

tool/fabric resin channels (also numbered starting from the bottom of the mold). The

system of equations defined by Equations (4.24) – (4.26) is solved using the same

computational procedure used for the one-layer fabric preform.

Top Mold Wall

Weft Tow Top Channel

Weft Tow Warp TowMiddle Channel

Weft TowBottom Channel

Weft Tow Warp Tow

z Bottom Mold Wall

x

Figure 4.3.1 x-z section of a quarter of the unit cell for an in-phase two-layer orthogonal plain-weave fabric.

99

Top Mold Wall

Weft Tow

Weft Tow

Middle ChannelWeft Tow

Bottom Channel

Top Channel

Warp Tow

Warp Tow

Bottom Mold Wall z

x

Figure 4.3.2 x-z section of a quarter of the unit cell for an out-of-phase two-layer orthogonal plain-weave fabric.

II.6 Shear-induced Fiber Volume Fraction Correction for Permeability

When the fabric is sheared in the x-direction, as shown in Figure 4.7.2, weft tows

are rotated but remain stress free. Consequently, the dimension of the fabric-preform unit

cell in the y-direction is altered causing a change in the effective fiber volume fraction in

the unit cell. This change in the fiber volume fraction can have a significant effect on

preform permeability at large shear angles and, hence, must be taken into account. The

procedure described below is used to correct permeabilities in the sheared fabrics

obtained using the original model of Simacek and Advani [7] as reviewed in Sections II.4

and II.5.

100

To quantify the permeability correction described above, the Kozeny-Carman

relation [e.g. 15] for permeability of the porous media with a fibrous architecture is used.

According to this relation, permeability of such media is given by:

2

32 )1(cf

frK −= (4.27)

where r and f are the fiber radius and the fiber volume fraction, respectively, while c is a

fibrous-medium architecture-dependent constant.

When the fabric preform is sheared in the x-direction by an angle θ , the fiber

volume fraction in fabric tows changes as:

)90sin( θθ −= of

f (4.28)

where the angle θ is given in degrees and the subscripts o and θ are used to denote the

value of a respective quantity in the un-sheared fabric and in the fabric sheared by an

angle θ , respectively.

To account for a shear-induced change in the fiber volume fraction, the

permeability values for sheared fabric preforms obtained using the models described in

Sections II.4 and II.5, should be multiplied by the following correction factor:

23

32

)1()1(

θ

θ

ffff

Ko

ocorr −

−= (4.29)

101

III. RESULTS AND DISCUSSION

III.1 Un-Sheared Single-Layer Plain-Weave Fabric Preforms

The model developed in Section II.4 is used in this section to analyze the pressure

distribution within the un-sheared single-layer balanced orthogonal plain-weave quarter

unit cell. Due to the symmetry of the unit cell with respect to the z = 0 plane, the pressure

distributions within the top and the bottom resin channels are identical and, hence, there

is no transverse flow of the resin through the fabric preform. Also, as established in

Section II.4, there is no variation of the pressure in the z-direction within the channels.

The variation of the top- and bottom-channel heights and of the fabric thickness in the x-y

plane within a quarter unit cell, used as input in the present analysis, are shown in Figures

4.4.1 and 4.4.2, respectively.

0.00045m

0.00035m

0.00005m

0.00025m

0.00015

Figure 4.4.1 Resin channels height in an un-sheared one-layer orthogonal plain-weave fabric preform.

m

102

0.0001m

0.0009m

0.0003m

0.0007m

0.0005

Figure 4.4.2 Fabric thickness field in an un-sheared one-layer orthogonal plain-weave fabric preform.

m

The variation of the pressure in the x-y plane within the resin channels of a quarter

unit cell for the fixed pressure drop of 1.0×105 in the x-direction is shown in Figure 4.5.

The pressure distribution (or more precisely its gradient) at a given x-y location correlates

inversely with the local height of the resin channel in order to satisfy the continuity

equation. It should be also noted that due to the symmetry of the fabric geometry with

respect to the quarter unit cell boundaries normal to the y-direction, zero-flux (i.e. zero

pressure gradient) conditions are found in the y-direction at these boundaries.

103

Distance in x-direction, m

Dis

tanc

ein

y-di

rect

ion,

m

0 0.002 0.004 0.006 0.008 0.010

0.002

0.004

0.006

0.008

0.01

0.7

atm

0.9

atm

0.3

atm

0.1

atm

1.0

atm

0.5

atm

0.0

atm

0.2

atm

0.4

atm

0.8

atm

0.6

atm

Figure 4.5 Pressure distribution in the x-y plane within a resin channel in the case of an un-sheared single-layer balanced orthogonal plain-weave fabric preform.

III.2 Effect of the Number of Layers in Un-sheared Plain-Weave Fabric Preforms

The model developed in Section II.5 is used in the present section to predict

permeability of the balanced un-sheared single- and multi-layer orthogonal plain-weave

fabric architectures. In all the calculations carried out in this section, as well as in the

calculations carried out in the previous section, the following unit cell parameter and one-

layer fabric thickness values are used: L1 = L2 = L = 0.01m and h = 0.001m. Also the

transverse permeability of the fiber tows is set to a typical (fixed) value, Kzz = 1×10-10m2.

To determine the effect of the number of fabric layers on the effective

permeability, the model developed in the previous section is used for the cases of 1-, 2-,

104

3-, 5-, 10- and 20-layer in-phase orthogonal balanced plain-weave fabric preforms in the

absence of layer nesting. The results of this calculation are presented in Figure 4.6. These

results show that as the number of layers increases, the permeability rises but at an ever

decreasing rate so that in fabric preforms with 10 or more layers, the effect of the number

of layers on permeability becomes insignificant. This finding can be easily rationalized

by recognizing that as the number of layers in the fabric increases, the effect of the

bottom and the top resin channels which are more restrictive to the fluid flow (and thus

reduce effective preform permeability) decreases.

Number of Fabric Layers in the Preform

Effe

ctiv

ePr

efor

mPe

rmea

bilit

y,m

2

0 3 6 9 12 15 18 215E-10

1E-09

1.5E-09

2E-09

2.5E-09

3E-09

Figure 4.6 The effect of the number of fabric layers on the effective permeability of an un-sheared balanced plain-weave fabric preform.

105

III.3 Effect of Fabric Shear on Permeability

As pointed out earlier, when the fabric preform is forced to conform to the ridges

and recesses of a mold, it may locally undergo shear deformation. Such deformation can

significantly affect local permeability of the preform. As shown in Figures 4.7.1 and

4.7.2, when a balanced square-cell plain-weave fabric is sheared, two important factors

must be considered: (a) the unit cell size increases and to make the calculations of

preform permeability manageable, the shear angle ( )nm1tan −=α is generally allowed to

take only the values corresponding to relatively small integers m and n; and (b) the

boundaries of the unit cell, unlike the case of the initial square-shape unit cell, are no

longer the lines of symmetry of the fabric structure. Consequently, the boundary

conditions imposed along the boundaries of the unit cell have to be modified relative to

those used in the case of the un-sheared unit cell. For instance, if a fixed pressure drop is

applied in the x-direction, the symmetry conditions along the unit cell boundaries normal

to the y-coordinate require that zero pressure-gradient boundary conditions be applied in

the y-direction. In the case of a sheared fabric preform, on the other hand, the unit cell

boundaries normal to the y-direction are not any longer the lines of symmetry of the

fabric architecture and, hence, only the periodic boundary condition (the corresponding

pressure values along the two unit-cell boundaries normal to the y-direction are identical)

can be applied.

106

y

x

Figure 4.7.1 Quarter unit cell (denoted using heavy dashed lines) in balanced plain-weave un-sheared fabric architectures.

y

x

Figure 4.7.2 Effect of fabric shearing on the size of the quarter unit cell (denoted using heavy dashed lines) in balanced plain-weave sheared fabric architectures by an angle

( )31tan 1−=α in the x-direction.

The effect of shear deformation (measured by the magnitude of the shear angle α)

on the effective permeability of single-layer plain-weave fabric preforms is displayed in

107

Figure 4.8. An example of the variation of the top- and bottom-channel heights and of the

fabric thickness in the x-y plane within a quarter unit cell, used as input in the present

analysis, are shown in Figures 4.9.1 and 4.9.2, respectively. For comparison, the

experimental values of preform permeabilities obtained in Ref. [16] are also shown in

Figure 4.8. While the agreement between the corresponding computational and the

experimental values is only fair, the effect of shear deformation on preform permeability

appears to be quite well predicted by the model. In addition, the corresponding computed

values of the in-plane off-diagonal (Kxy and Kyx) elements of the effective permeability

are very close as required by symmetry of the orthogonal plain-weave fabric architecture.

Shear Angle, deg

Pref

orm

Perm

eabi

lity

Com

pone

nts,

m2

0 5 10 15 20 25 300

2E-10

4E-10

6E-10

8E-10

1E-09

Kxx [21]

Kyy

Kxy [21]

Kyx

Kyy

Kxx

Figure 4.8 The effect of shear on permeability of a single-layer balanced orthogonal plain-weave fabric preform.

108

0.00045m

0.00005m0.00015m

0.00025m

0.0003

Figure 4.9.1 Resin channels height in a one-layer orthogonal plain-weave fabric preform subjected to shear in the x-direction by an angle of . )3/1(tan 1−=α

5m

0.0001m

0.0009m

0.0007m0.0005m

0.0003m

Figure 4.9.2 Fabric thickness field in a one-layer orthogonal plain-weave fabric preform subjected to shear in the x-direction by an angle of . )3/1(tan 1−=α

109

III.4 Effect of Preform Compaction on Permeability

When the fabric is subjected to compression during mold closing in the RTM

process or during evacuation of the vacuum bag in the VARTM process, it undergoes a

number of changes such as: the cross-section of the fiber tow flattens, the pores and gaps

between the fibers inside tows as well as between individual tows are reduced, the tows

undergo elastic deformation, inter-layers shifting (nesting), etc. A typical compression-

pressure vs. preform thickness curve for a woven fabric is depicted in Figure 4.10 [17].

The curve shown in Figure 4.10 has three distinct parts: two linear and one nonlinear. In

the low-pressure linear and the nonlinear portions of the pressure vs. thickness curve,

preform compaction is dominated by a reduction of the pore and the gap sizes between

the fibers in tows. In the high-pressure linear region of the pressure vs. thickness curve,

on the other hand, preform compaction involves mainly tow bending and nesting. Typical

liquid molding processes such as RTM or VARTM involve pressures which correspond

to the high-pressure linear pressures vs. thickness region. Hence, the effect of fabric

compaction on permeability of the fabric preform associated with the high-pressure linear

compaction regime is investigated in this section.

110

LinearFa

bric

Thi

ckne

ss

Non-linear

Linear

Compaction PressureFigure 4.10 A typical compaction-pressure vs. preform-thickness curve for a plain-weave

fabric architecture.

To quantify the effect of preform compaction (in the high-pressure linear region)

on permeability of the balanced orthogonal plain-weave fabric, the beam-bending based

micro-mechanical model developed in a series of papers by Chen and Chou [18-20] is

utilized in the present work. The model of Chen and Chou [18-20] is based on a number

of well-justified assumptions such as: (a) the fabric is considered to extend indefinitely in

the x-y plane and, hence, can be represented using the unit cells such as the one shown in

Figure 4.2; (b) tows in the fabric are treated as a transversely isotropic solid material; (c)

the fabric is subjected to the compaction pressure only in the through-the-thickness

111

direction, and can freely adjust its shape in the x-y plane; (d) since the compaction

analyzed corresponds to the high-pressure linear region, no voids or gaps are assumed to

exist between the fibers in tows or between the tows; (e) during fabric compaction, the

cross-section area of the tows is assumed to remain unchanged but the shape of the cross-

section undergoes a change; and (f) as compaction proceeds, the deformation of the tows

leads to an increase in the effective volume fraction of the fibers in the fabric and, in the

limit of complete compaction of the tows, the volume fraction of the fibers in the fabric

becomes equal to that in the individual tows.

In order to derive a relationship between the reduction in the fabric thickness, the

effective volume fraction of the fibers and various distributions and magnitudes of the

applied compaction pressure, Chen and Chou [18-20] applied a simple procedure from

the solid-mechanics beam theory. Toward that end, the one-quarter unit cell shown in

Figure 4.2 is first simplified by replacing the two warp and the two weft tows with four

beams. Next based on the symmetry of the simplified model, it is shown that the problem

can be further simplified using a single beam and the appropriate distribution of the

applied and contacting pressures, Figure 4.11. The model of Chen and Chou [18-20] is

utilized in the present work to compute the effect of the compaction pressure on the

channel heights ( and ( yxhT , ) ( )yxh B , ) and on the fabric thickness, h fields.

These fields are, in turn, used in the lubrication model presented in the Section II.4 to

quantify the effect of fabric compaction on the effective permeability of a one-layer

orthogonal plain-weave fabric.

( yxF , )

112

Applied Pressure Distribution

Contact Pressure Distribution

Contact Pressure Distribution

Applied Pressure Distribution

Figure 4.11 A schematic of the pressure distribution on a curved beam used in the calculation of permeability of un-sheared one-layer orthogonal plain-weave fabrics.

The effect of the compaction force applied to the upper and the lower (rigid and

flat) molds on the effective permeability of a one-layer orthogonal plain-weave fabric is

displayed in Figure 4.12. In these calculations, the Young’s modulus is assigned a value

of 22GPa and a sinusoidal distribution of the applied and the contacting pressures is

assumed [20]. An example of the variation of the top- and the bottom-channel heights

and of the fabric thickness in the x-y plane within a quarter unit cell, used as input in the

present analysis, are shown in Figures 4.13.1 and 4.13.2, respectively. For comparison,

the experimental results reported by Sozer et al. [21] are also shown in Figure 4.12. It is

113

seen that a reasonably good agreement exists between both the magnitude of the

predicted preform permeability and its change with the applied compaction force.

Compaction Force, N

Effe

ctiv

ePr

efor

mPe

rmea

bilit

y,m

2

0 2 4 6 8 10 12 14 165E-10

5.5E-10

6E-10

6.5E-10

7E-10

7.5E-10

8E-10

Ref. [19]

This

Figure 4.12 Effect of compaction (represented by the magnitude of the compaction force) on permeability of a one-layer un-sheared orthogonal plain-weave fabric preform.

114

0.00035m

0.00025m

0.00005m

0.

Figure 4.13.1 Resin channels height in an un-sheared one-layer orthogonal plain-weave fabric preform subject to a total compressive force of 10.3N via rigid, flat upper and

lower tool surfaces.

00015m

0.00035m

0.00025m

0.00005m

0.00015m

Figure 4.13.2 Fabric thickness field in an un-sheared one-layer orthogonal plain-weave fabric preform subject to a total compressive force of 10.3N via rigid, flat upper and

lower tool surfaces.

115

III.5 Effect of Layer Nesting

As mentioned earlier shifting of fabric layers followed by their more compact

packing (the phenomenon generally referred to as layers “nesting”) can have a major

effect on the effective fiber density in the preform and, hence, on permeability of the

preform. Nesting of the fabric layers can particularly take place under high applied

pressures which are sufficient to overcome inter-tow friction. The thickness reduction in

balanced orthogonal plain-weave fabrics whose geometry is represented by Equations

(4.1) and (4.2), due to layers nesting has been analyzed by Ito and Chou [9] who derived

the following relation for the fabric thickness reduction caused by nesting:

≤≤≤≤−−

≤≤≤−−

≤≤≤−−

≤≤−−

=∆

πφππφπφφ

πφππφφφ

πφπφπφφ

πφπφφφ

yxyx

yxyx

yxyx

yxyx

nesting hh

2,

2,

2sin

2sin2

2,

2,

2sin

2cos2

2,

2,

2cos

2sin2

2,

2,

2cos

2cos2

(4.30)

where xx sLπφ 2

= and yy sLπφ 2

= are dimensionless while and are the

dimensional relative shifts of the adjacent layers in the x- and y- directions, respectively.

xs ys

Two non-nesting cases associated with zero nesting, reduction in the fabric

thickness can be identified: (a) 0== yx φφ which corresponds to the iso-phase laminate

case and (b) 2/πφφ ±== yx corresponding to the out-of-phase laminate case.

116

The relations given in Equation (4.30) are used in the present work to examine the

effect of layers nesting on fabric permeability. While, in general, fabric compaction

during the high-pressure linear compaction stage can involve both elastic distortions (tow

bending) and layers nesting, the two modes of fabric compaction are generally considered

as decoupled and can be considered separately.

The effect of nesting (quantified by the magnitudes of the dimensionless layer

shifts in the x- and the y-directions, xφ and yφ , respectively in a two-layer orthogonal

plain-weave fabric is shown in Figure 4.14. The values displayed in Figure 4.14 pertain

to the ratio of fabric permeability at the given values of xφ and yφ and fabric

permeability at 0== yx φφ . As expected, fabric nesting gives rise to the reduction in

fabric permeability. Furthermore, for the case of a out-of-phase laminate fabric

( 2/πφφ ±== yx ), fabric permeability is only about 30% of its value in the in-phase

laminate fabric. This finding is in excellent agreement with its experimentally counterpart

reported by Sozer et al [21].

117

Dimensionless Shift in the x-Direction

Dim

ensi

onle

ssSh

iftin

the

y-D

irect

ion

0 0.5 1 1.50

0.5

1

1.5

1.0

0.70

0.74

0.51

0.48

0.36

0.42 0.34 0.29

Figure 4.14 The effect of nesting on the ratio of fabric permeability at the given values of layer shifts in the x- and the y-directions and fabric permeability of an un-nested in-phase

laminate fabric.

IV. CONCLUSIONS


can be drawn:

1. Effective permeability of the orthogonal plain-weave fabric preforms can be determined computationally by combining a lubrication model for the resin flow through tool-surface/fabric-tow and tow/tow channels with the Darcy’s law for the resin flow through the fabric tows.

2. The computational approach presented in this work enables assessment of the

contribution that various phenomena such as the mold walls, fabric shearing, interlayer shifting and restacking as well as fabric compaction due to the infiltration pressure make to orthogonal plain-weave fabric-preform permeability.

118

3. While no comprehensive set of experimental data is available to fully test validity of the present model, the agreement of the model predictions with selected experimental results can be generally qualified as reasonable.

V. REFERENCES

1. Lee, L. J., “Liquid Composite Molding,” In: T. G. Gutowski, editor., Advanced Composites Manufacturing, New York: John Wiley & Sons, pp. 393-456 (1997).

2. Lam, R. C., and J. L. Kardos, in Proc. Third Tech. Conf., American Society for

Composites (1988).

3. Gutowski, T. G., in SAMPE Quart., 4 (1985).

4. Gebart, B. R., “Permeability of Unidirectional Reinforcement for RTM,” Journal of Composite Materials, 26, 1100 (1992).

5. Ranganathan, S., F. Phelan and S. G. Advani, Polymer Composites, 17, 222

(1996).

6. Ranganathan, S., G. M. Wise, F. R. Phelan, R. S. Parnas and S. G. Advani, “A Numerical and Experimental Study of the Permeability of Fiber Preforms,” in Proc. Tenth ASM/ESD Advanced Composites Conf., Oct. (1994).

7. Simacek, C., and S. G. Advani, “Permeability Model for a Woven Fabric,”

Polymer Composites, 17, 887 (1996).

8. Dungan, F. D., M. T. Senoguz, A. M. Sastry and D. A. Faillaci, “Simulations and Experiments on Low-Pressure Permeation of Fabrics: Part I - 3D Modeling of Unbalanced Fabric,” Journal of Composite Materials, 35, 1250 (2001).

9. Ito, M., and T. W. Chou, “An Analytical and Experimental Study of Strength and

Failure Behavior of Plain Weave Composites,” Journal of Composite Materials, 32, 2 (1998).

10. Falzon, P., and V. M. Karbhari, “Effects of Compaction on the Stiffness and

Strength of Plain Woven Composites,” draft of paper.

11. Heitzmann, K. F., “Determination of In-Plane Permeability of Woven and Non-Woven Fabrics,” Master’s thesis, University of Illinois at Urbana-Champaign (1994).

119

12. MATLAB, 6th Edition, “The Language of Technical Computing,” The MathWorks Inc., 24 Prime Park Way, Natick, MA, 01760-1500 (2000).

13. Pearce, N., and J. Summerscales, “The Compressibility of a Reinforcement

Fabric,” Composites Manufacturing, 6, 15 (1995).

14. Saunders, R. A., C. Lekakou and M. G. Bader, “Compression and Microstructure of Fiber Plain Woven Cloths in the Processing of Polymer Composites,” Composites Part A, 29A, 443 (1998).

15. Dungan, F. D., M. T. Senoguz, A. M. Sastry and D. A. Faillaci, “On the Use of

Darcy Permeability in Sheared Fabrics,” Journal of Reinforced Plastics and Composites, 18, 472 (1999).

16. Dungan, F. D., M. T. Senoguz, A. M. Sastry, and D. A. Faillaci, “Simulations and

Experiments on Low-Pressure Permeation Fabrics: Part I – 3D Modeling of Unbalanced Fabric,” Journal of Composite Materials, 35, 1250 (2001).

17. Hu, J., and A. Newton, “Low-load Lateral-Compression Behavior of Woven

Fabrics,” J Text Inst Part I, 88, 242 (1997).

18. Chen, B., and T.W. Chou, “Compaction of Woven-Fabric Preforms in Liquid Composite Molding Processes: Single-Layer Deformation,” Composites Science and Technology, 59, 1519 (1999).

19. Chen, B., and T.W. Chou, “Compaction of Woven-Fabric Preforms: Nesting and

Multi-Layer Deformation,” Composites Science and Technology, 60, 2223 (2000).

20. Chen, B., E. J. Lang and T. W. Chou, “Experimental and Theoretical Studies of

Fabric Compaction Behavior in Resin Transfer Molding,” Material Science and Engineering, A317, 188 (2001).

21. E. M. Sozer, B. Chen, P. J. Graham, S. Bickerton, T. W. Chou and S. G. Advani,

“Characterization and Prediction of Compaction Force and Preform Permeability of Woven Fabrics During the Resin Transfer Molding Process,” Proceedings of the Fifth International Conference on Flow Processes in Composite Materials, Plymouth, U.K., pp. 25-36 (1999).

CHAPTER 5

CONCLUSIONS

Based on the results obtained in the present work, the following general main

conclusions can be drawn:

1. Devolatilization of the resin-curing gaseous by-products, a controlled use of heat during the preform infiltration stage and the knowledge of the effect of various fabric deformation and sliding phenomena on the preform permeability all play important roles in affecting the cycle time and production cost of the VARTM process as well as in affecting the quality of the resulting polymer matrix composite parts.

2. Adequate modeling of devolatilization during the VARTM process requires

consideration of both chemical effects associated with polymerization of the resin and hydrodynamic effects associated with the transport of volatiles through the resin. Lower tool-plate heating rates are found to promote devolatilization of the volatiles at lower temperatures at which, due to a low degree of polymerization, resin viscosity is low. This results in a more complete removal of the volatiles and a lower gas-phase content in VARTM-processed fiber reinforced polymer matrix composites. However, lower tool-plate heating rates are generally associated with longer cycle times and, hence, with higher manufacturing costs. From the standpoint of achieving a high degree of gas-phase removal at a highest possible tool-plate heating rate, there is, in general, an optimum concentration of the solvent. However, the benefits of using the optimum concentration of the solvent to increasing the tool-plate heating rate and, thus, in reducing the VARTM processing time, are relatively limited.

3. Preform infiltration stage of a high-permeability medium based VARTM

process can be modeled by combining an incompressible-fluid mass conservation equation, an energy conservation equation and an equation for the time and temperature evolution of the degree of polymerization of the resin and utilizing a control-volume finite-element method. Such modeling of the preform infiltration process under non-isothermal conditions showed that, at short infiltration times, the effect of tool-plate heating can be beneficial and can lead to an increase in the rate of infiltration. This effect has been attributed to a thermal-thinning based reduction in the resin viscosity. An optimization analysis of the VARTM preform infiltration process showed that, in order to take a full advantage of tool-plate heating, ~70-80% of the mold

121

should be filled with the resin at the room temperature before heating of the tool-plate is initiated. For the simple rectangular geometry of the fiber preform analyzed, ~80% room-temperature preform infiltration followed by a tool-plate heating at a rate of ~3.2K/s, can reduce the infiltration time by ~5% relative to the room-temperature complete-infiltration time.

4. Effective permeability of the orthogonal plain-weave fabric preforms can be

determined computationally by combining a lubrication model for the resin flow through tool-surface/fabric-tow and tow/tow channels with the Darcy’s law for the resin flow through the fabric tows. Such a computational approach enables the assessment of the contribution that various phenomena such as the mold walls, fabric shearing, interlayer shifting and restacking as well as fabric compaction due to the infiltration pressure make to orthogonal plain-weave fabric-preform permeability. While no comprehensive set of experimental data is available to fully test validity of the present computational approach, the agreement between the computed results and their experimental counterparts is generally found to be reasonable.

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