Upload
others
View
6
Download
0
Embed Size (px)
Citation preview
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
3
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
5
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
15
( ) oTtxT == 0, (2.4)
Boundary Condition (1)
∫+==t
o dtTtxT0
),0( α (2.5)
Boundary Condition (2)
( )( )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)
18
( ) 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
Boundary Condition (1)
( ) 0,0 ==∂∂ tx
xP (2.24)
Boundary Condition (2)
( ) 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. COMPUTATIONAL PROCEDURE
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
Based on the results obtained in the present work, the following main 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. COMPUTATIONAL PROCEDURE
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
Based on the results obtained in the present work, the following main 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.