The hydrostatic pressure is obtained as the solution ofthe governing equation for consolidation ofa porous bed within a given time interval with three-dimensional flow and a one-dimension
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through application ofan external pressure (or vacuum) cycle on the laminate The process is influenced by the resin viscosity, which is a function of the temperature and degree ofcure in the laminate, and the applied pressure The resin viscosity decreases initially with temperature; however, as the cross-linking reactions proceed, the vis-cosity sharply increases near the gel point, at which the visvis-cosity is theoretically infinite Thus, the cure pressure cycle is designed to take advantage ofthe window during processing when the viscosity is low
Mathematically, the consolidation process is described as that ofresin flow through
a porous bed formed by the reinforcing fibers Models for the consolidation process have been presented by Springer (1982), Davè et al (1987), and Gutowski et al
(1987), among others The models by Davè et al (1987) and Gutowski et al (1987) consider the composite to contain a deformable fiber network in which resin flow in all directions is governed by Darcy’s law The fiber network also takes part in carrying the load due to the applied pressure (or vacuum) during processing The total forceσ acting on the porous fiber bed is balanced by the sum ofthe force due to the springlike behavior ofthe fiber networkp and the hydrostatic force due to the pressure of the
resin in the layup,P
The hydrostatic pressure is obtained as the solution ofthe governing equation for consolidation ofa porous bed within a given time interval with three-dimensional flow and a one-dimensional confined compression condition (no boundary motion in thex and y directions), given by (Davè et al., 1987)
1
µm v
∂
∂x
κx ∂P
∂x
∂y
κy ∂P
∂y
+ ∂
∂z
κz ∂P
∂z
=∂P
whereµ is the viscosity ofthe resin in the porous fiber bed; κx , κ y, andκz are the specific permeabilities in thex, y, and z directions, which depend on the stress level;
t is time; and m vis the coefficient of volume change, which describes the stress–strain behavior ofa body in confined compression For a porous medium, it is the ratio of change in porosity to axial (normal) stress for confined compression of the porous body with its vertical side faces constrained from any motion (Kardos, 1997)
Solution of the foregoing equation requires information on the anisotropic perme-ability tensor and its variation as the fiber bed consolidates during processing Several theoretical and experimental studies have been conducted to determine this informa-tion (Gutowski et al., 1986; Lam and Kardos, 1991; Adams and Rebenfeld, 1991a,b;
Skartsis et al., 1992; Skartsis and Kardos, 1990), which is provided predominantly in terms ofthe Carman–Kozeny relationship
κ = (1 − ϕ)ϕ3 2sk1
whereϕ is the resin volume fraction; k ois the Kozeny constant, determined empiri-cally; ands is the specific surface of the fibers and is related to the fiber radius r f as
4/r2
f The Kozeny constant is reported to remain relatively constant over a wide range
ofresin volume fractions, except near the extreme limits ofϕ For flow parallel to the
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fibers, values ofk o in the range 0.5 to 0.7 have been reported, while for transverse flow the values range from 11 to 18 (Gutowski et al., 1987; Lam and Kardos, 1988)
Viscoelastic characteristics ofthe resin, such as near the gel point, are accounted for through the use ofa pseudo-Kozeny constant,k∗
o, which is a function of both fluid
and fiber bed properties (Skartsis et al., 1992)
Resin viscosity is another important parameter that affects resin flow and transport ofvoids during the consolidation process The viscosity is a function ofthe temper-ature and the degree ofcure and is often given by an empirical correlation ofthe form
µ = µ∞exp
E
RT + λε
(17.43)
where µ∞ is a constant, E the activation energy for viscosity, and λ a constant
that is independent oftemperature, all ofwhich are determined empirically For the Hercules 3501–6 resin, Lee et al (1982) reported the values ofthe model parameters as:µ∞= 7.93 × 10−14Pa· s, E = 9.08 × 104J/mol, andλ = 14.1 ± 1.2.
Gutowski and co-workers (Gutowski, 1985; Gutowski et al., 1987) modeled fiber deformation by assuming that a composite is a porous, nonlinear elastic medium that
is filled with a viscous resin They modeled the deformation ofbundles offibers as beams bending between multiple contact points, and derived an expression for the stiffnessp as
p = 3π E
β4
v f /v i− 1
whereE is the bending stiffness of the fiber and v a,v f, andv iare the maximum fiber volume fraction, the instantaneous fiber volume fraction, and the initial fiber volume fraction, respectively The termβ is a geometric parameter that is related to the fiber architecture and fiber diameter For well-aligned fiber bundles, the parameters,β, v i, andv a, determined by fitting experimental measurements to the model, were given
by Gutowski and co-workers asβ = 225, v i = 0.50, and v a = 0.829 The values
increase with increasing fiber alignment
A model for diffusion-controlled void growth and dissolution in an epoxy resin system during consolidation was developed by Kardos et al (1986) The model provides the size ofa void located in an infinite isotropic medium as a function of the processing parameters and identifies conditions under which void growth can be prevented or voids can be made to collapse during the cure process Based on their model, they proposed that the resin pressure (atm) at any point within the laminate being cured must be greater than a minimum value as given below:
P ≥ Pmin Pmin= 4.962 × 103exp
−4892T
whereP is the pressure (atm) in the resin, obtained as solution ofeq (17.41), Pmin the minimum resin pressure required to prevent void growth by moisture diffusion at
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any time during cure,w othe relative humidity (%) to which the resin in the prepreg
is equilibrated prior to processing, andT the temperature (K).
Design ofthe cure process calls for selecting the optimum temperature and pres-sure cycles so as to minimize the fabrication time and simultaneously minimize the void content and processing-induced stresses Optimization ofcure cycles utilizing the physical models has been reported by Pillai et al (1994), Rai and Pitchumani (1997a,b), Diwekar and Pitchumani (1993), and Pitchumani and Diwekar (1994)
17.7.3 Processing of Thermoplastic-Matrix Composites
Unlike thermosetting resins, thermoplastic melts have significantly higher viscosity, which renders fabrication ofquality composites via impregnation ofa net-shaped fiber structure as a single step impractical Continuous fiber-reinforced thermoplastic composites are commonly fabricated in two stages, as shown schematically in Fig
17.22 In the first stage, called prepregging, thin reinforcement layers are impregnated with the thermoplastic to form prepregs (short for preimpregnated reinforcements).
Because the reinforcement layers in prepregs are usually about 150 to 200µm in thickness, a high degree ofimpregnation can be achieved under controlled conditions
Thermoplastic prepregs are commercially available in a variety ofwidths, ranging from small-width ribbons or tows to wider tapes and sheets, and have a low void con-tent with a fairly uniform fiber distribution Processes for the fabrication of prepregs are outside the scope ofthis chapter (discussed in detail in Pitchumani, 2002)
The focus of this section is on the second stage, referred to as consolidation,
in which the prepregs are stacked to the desired shape and thickness and fusion-bonded and solidified to obtain the final composite product Consolidation ofprepreg layers is achieved by a number ofprocesses, including prepreg layup with autoclave
Stage 1: Prepregging
Stage 2: Consolidation
Fiber Matrix
Prepregging
Prepreg
Prepreg
Heating + Pressure
T
t Solidification (crystallization)
Figure 17.22 Two main steps in thermoplastic composite processing
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V
z y
Prepreg supply spool Hot gas torch
F
Roller
Tool
Figure 17.23 Tow placement/tape laying process with in situ consolidation
consolidation, and pultrusion, which are similar to processes described in the context ofthermosetting matrix composites processing, with the notable difference that there
is usually a cooling step to solidify the consolidated composite Thermoplastic tape laying and automated tow placement, illustrated in Fig 17.23, are processes based
on incrementally laying down and continuously consolidating prepreg layers to build the composite product The nip point formed by the incoming tow and the substrate layer at the entry to the roller region is heated by an appropriate means (such as the hot-gas torch shown in Fig 17.23) The mechanisms leading to consolidation and bonding take place under the roller, and the fully consolidated structure emerges ready
to form the substrate for new tows added during subsequent passes of the process The rollers and torches, along with the supply spool, are mounted on a common frame,
called the tow-placement head, which is translated with a line speed V during the
process Unlike the autoclave process, which involves batch consolidation oftow layers, tow placement and tape-laying processes involve consolidation in situ and offer the potential for rapid fabrication These processes can be used to produce part geometries with intricate features and are particularly suited for fabrication
of large structures such as aircraft wing skins and the fuselage (Lamontia et al.,
1992, 1995) Tape laying and tow placement differ principally in the size of the prepregs used; while wider tapes are used in tape laying, the prepreg tows used in tow placement are smaller (on the order ofabout 0.25 in wide) In this process, a thermoplastic impregnated tow is passed under a lay-down roller onto a substrate formed of previously deposited and consolidated tows on a cylindrical or flat tool
Fabrication of composites from prepregs using the aforementioned processes is
based on the principle of fusion bonding, which fundamentally consists of application
ofheat and pressure at the interface oftwo layers ofthermoplastic prepregs in contact,
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Interface softening
Pressure
Tow compaction/
void reduction
Healing Bonded material
Figure 17.24 Mechanisms involved in the fusion bonding process
and subsequently, cooling down ofthe interface to obtain a bonded product The mechanisms involved in fusion bonding of two thermoplastic surfaces are illustrated schematically in Fig 17.24 The applied high temperatures cause the interface to soften, and the simultaneous application of pressure serves to flatten the surface
asperities and establish an area contact at the interface, referred to as the intimate
contact process Further, the elevated temperatures cause an interdiffusion of polymer
molecules, termed autohesion or healing, across the interfacial areas in intimate
contact, resulting in the development ofbond strength in the laminate Polymer
degradation refers to the cumulative effect of exposure of the polymer matrix to
high temperatures during the process The cooling down ofthe bonded layers causes
solidification ofthe molten thermoplastic matrix, which in the case ofsemicrystalline
thermoplastics, influences the crystalline morphology ofthe polymer matrix in the composite product
The dominant transport mechanisms involved in the process and their relation-ship to the pressure and temperature cycles applied are shown in Fig 17.25 As the
material is heated, when its temperature reaches a certain value known as the glass
transition temperature T g, the crystal structure ofsemicrystalline thermoplastics be-gins to break, and material softening takes place At temperatures exceedingT g, the crystalline structure disintegrates progressively until the material melting pointTmp
is reached, whereupon all crystallinity is lost and the polymer is fully molten For amorphous thermoplastics, owing to the absence ofany significant crystalline struc-ture, the glass transition temperature and the melting point are nearly identical As seen in Fig 17.25, the material temperature exceeding the glass transition temperature
is a prerequisite for most ofthe mechanisms offusion bonding, while heat transfer and polymer degradation occur throughout the process Healing takes place as long
as the temperature is above the melting point, and polymer crystallization
accompa-nying solidification occurs when the material is cooled down from the melting point
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Temperature Pressure
Time, t
T mp
T c
Heat transfer 1
2
3
4 5
Void dynamics/tow compaction
Polymer crystallization (Solidification) Polymer degradation
Interfacial bonding
Healing ( >T T mp)
Intralaminar void growth ( = 0; > )p T T g
Intimate contact ( > 0; > )p T T g
Intralaminar void reduction tow compaction ( > 0; > )p T T g
Figure 17.25 Dominant mechanisms during thermoplastic composites processing and their relationship to the temperature and pressure cycles
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past the crystallization temperature (T c ) to the glass transition point Intimate contact,
intralaminar void reduction (i.e., reduction ofvoids present within prepregs), and tow compaction processes prevail during the application ofpressure, whereas intralami-nar void growth occurs in the absence ofapplied pressure and while the temperature
is greater than the glass transition point The mechanisms are discussed in detail next
by considering the tape-laying and tow-placement processes
Heat Transfer Heat transfer during thermoplastic composites processing has been studied extensively by many investigators [see Pitchumani (2002) for a detailed list]
in the context ofautoclave, filament winding, tape/tow placement, and pultrusion processes The goal ofheat transfer analysis is to predict the transient temperature field within the tow layers, which is utilized in the analysis ofthe other mechanisms involved in the process The thermal model formulation consists of the energy equa-tion in an appropriate coordinate system with a source term corresponding to the heat ofcrystallization The material domain modeled is usually two-dimensional along the fiber length (they direction) and through the thickness ofthe tow layers (the z
di-rection) Considering a tape laying or tow-placement process for rectangular product geometries, the governing equations for the transient temperature field in the com-posite may be written as follows:
∂
∂t (ρcT ) +
∂
∂y (ρcV T ) =
∂
∂y
k y ∂T
∂y
+ ∂
∂z
k z ∂T
∂z
+ ρm (1 − v f ) ∆H c d ˆc
dt
(17.46)
whereρ and c are the density and specific heat ofthe composite medium, T the
temperature,t the time, V refers to the tow velocity in the direction of travel, and v f
refers to the fiber volume fraction in the composite The anisotropic conductivities of the composite medium,k yandk z, are evaluated as the conductivities ofan equivalent
homogeneous medium The longitudinal conductivity in the fiber direction(k y ) is
determined using the rule of mixtures, which is simply a volume average ofthe fiber
and the matrix conductivities, while the transverse conductivity(k z ) is obtained by
any ofa number ofanalytical models available [see Hashin (1983), Han and Cosner (1981), and Pitchumani (1999) and references therein]
The last term in eq (17.46) denotes the source/sink effects due to the crystalliza-tion, in which∆H cis the heat ofcrystallization,ρm the matrix density, andd ˆc/dt
the crystallization rate Generally, the magnitude ofthe crystallization source term is very small in comparison to the other terms in the energy equation, during the heat-ing process, and it has been customary to neglect the source term in the heat transfer analysis ofthe process during the heating stage Crystallization, however, plays a significant role in the cooling-down step, which is discussed later in the section on solidification The governing equations are subject to initial conditions on the tem-perature and appropriate boundary conditions, specific to the process and the product geometry under consideration For example, in the tape-laying or tow-placement pro-cesses (Pitchumani et al., 1996), the bottom surface of the prepreg layers in contact
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with a tool surface is subject to a prescribed temperature either with perfect contact or with a finite contact resistance The top surface condition is, in general, a combination ofconvection to the ambient, exposure to the heat source, and contact with the roller
or tool/die surfaces The heat transfer coefficients and contact resistance values used
in the problem formulation are generally determined from experiments, or assumed empirically The equations are solved numerically using a finite difference or finite element method
Void Dynamics The application ofa pressure cycle to a prepreg tow heated to
above its glass transition temperature causes deformation, referred to as tow
com-paction, due to flow ofthe softened material, whereby the tow thickness decreases
accompanied by a corresponding increase in the width During the process, owing
to the high viscosity ofthermoplastics, the fibers in the prepregs move along with the thermoplastic resin rather than relative to the resin (Pitchumani et al., 1996; Ran-ganathan et al., 1995) Therefore, the process may best be described as a squeeze flow instead ofa Darcy flow The fiber–resin–voids mixture is modeled as an equivalent homogeneous fluid with the rheological properties ofthe continuum dependent on the temperature, the fiber volume fraction, and the void content
The compaction process is accompanied by void reduction, and the regions outside the compaction zone subject to high temperature are areas ofvoid growth The void reduction and void growth mechanisms are collectively referred to in this discussion
as void dynamics Several mechanisms contribute to void dynamics, including void
migration, void compression and expansion, void coalescence, gas diffusion from the void to the melt, and void bubbling (Ranganathan et al., 1995) The dominant consolidation-related void dynamics mechanisms in thermoplastics processing are those ofvoid migration along with resin and void compression due to the effects of cooling, and compaction under the applied pressure The diffusion of gases across the void–tow melt interface may be assumed to be negligible in the analysis (Pitchumani
et al., 1996; Ranganathan et al., 1995), owing to the poor solubility ofthe gases in the
thermoplastic melt The effect of void migration is accounted for in a macroscopic flow model, while the void compression effects are considered in a microscopic
void dynamics model These two models are coupled and necessitate a simultaneous
solution for the void fraction in the composite The macroscopic flow model further
yields the pressure field in the consolidation region, which governs the intimate contact process discussed in a later subsection
Pressure Field in the Consolidation Region (Macroscopic Model) Pitchumani
et al (1996) and Ranganathan et al (1995) presented models for tow compaction for consolidation under a roller as encountered in tow/tape placement, and filament winding processes Figure 17.26 shows an enlarged view ofthe modeling domain, which is the region under the compaction roller A tow ofa given height h i and widthw i enters the region under the roller with a specified line speedV , and in the
compaction process, its height reduces toh f while its width increases tow f Since
the tow dimension in they direction is much larger than the x and z dimensions,
flow in they direction may be neglected Further, owing to the high viscosity ofthe
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z
Fiber-resin mixture Voids
Section A-A ( )a
( )b
( )c
h1
h f
y
z L c A
Consolidation region
T y, z( )
y x
x
Compaction roller
D r A
Figure 17.26 Region under the consolidation roller illustrating the tow compaction process
matrix resin, the inertial effects may be neglected and the consolidation process may
be treated as a creeping flow problem
The fluid motion under the compaction roller is governed by the continuity and momentum equations in Cartesian coordinates, which may be simplified by utilizing the fact that the tow thicknesses (typically about 0.006 in.) are small relative to the width and length Further, because the quantity ofinterest in the consolidation region is the pressure field under the rollers and not the actual velocity profiles, the governing equations may be cast in an integral form, as described in Pitchumani et
al (1996) The resulting integral equation is given by eq (17.47), which constitutes
the macroscopic governing equation for determining the pressure distribution under
the rollers
h ∂ρ∗
∂τ + ρ∗
dh
dτ +
∂
∂x
ρ∗ h 0
v x (0) + dp
dx
z 0
ξ
µdξ
+ C1(x)
z 0
1
µdξ
dz
= 0
(17.47)
where the process is assumed to be at steady state,ξ is a dummy variable ofinte-gration,C1(x) is a constant ofintegration, ρ∗ is the density ofthe fiber–resin–voids mixture scaled with respect to the density ofthe mixture in the absence ofvoids, and
τ is the Lagrangian time, which is related to the line speed V and the location under
the roller (measured from the entrance to the roller region)y, as τ = y/V The term h
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is the instantaneous thickness ofthe tow under the compaction roller, which is related
to the under-roller distancey, based on geometric considerations (Pitchumani et al.,
1996), andv x (0) is the width-wise velocity component at the tow–substrate interface
(z = 0 in Fig 17.26).
The boundary conditions on the pressurep correspond to the tow being
uncon-strained along its width, which implies thatp(x = ±w/2) = patm, the atmospheric pressure Furthermore, the unknownsC1(x) and v x (0), which result from integration
ofthe continuity equation across the instantaneous tow thickness, are determined us-ing the no-slip condition at the tow–substrate interface and a partial slip condition at the tow–roller interface (Pitchumani et al., 1996)
Void Compression (Microscopic Model) The time derivative ofthe dimension-less density,∂ρ∗/∂τ, appearing in eq (17.47) is evaluated from a microscopic model
which accounts for void compression (one of the void reduction mechanisms) during the consolidation process A typical void at any location may be approximated by a sphere ofradiusR and is assumed to be surrounded by a concentric spherical resin
shell ofouter radiusS The ratio of R and S is determined by the void fraction at
the location under consideration Void growth or collapse is governed by a balance between the pressure inside and outside the void and the surface tensionσ and resin viscosityµ Using conservation ofthe incompressible resin mass, and the ideal gas
law for the gas within the void, the microscopic void dynamics equation may be
writ-ten as (Pitchumani et al., 1996)
4µ
R∗3
S∗3
o + R∗3− 1 − 1
dR∗
dτ +
p go
R∗3
T
T o − p f
R∗
2σ
µR o = 0 (17.48) where the bubble and outer shell radii are scaled with respect to the initial radius of the void,R o, which is determined based on the void fraction in the tow at the entrance
to the consolidation roller In eq (17.48)p gandp goare the instantaneous and initial pressures inside the voids, respectively, and the fluid pressure surrounding the void
is labeledp f Further, for the concentric spherical shell description, the rate of change ofρ∗with respect to time can be expressed in terms ofthe rate ofchange ofthe nondimensional radius ofvoid as (Ranganathan et al., 1995)
∂ρ∗
∂τ =
−3R∗2
S∗3
o − 1
S∗3
o − 1 + R∗32
dR∗
For a set ofgiven initial conditions on the void radii, density, pressure, and temper-ature, the rate ofchange ofthe nondimensional density with respect to time may be obtained using eqs (17.48) and (17.49) This expression may be used in eq (17.47)
to compute the pressure distribution in the consolidation region Equation (7.14) can then be used to compute the change in radius ofthe voids and update the void radius at various locations in the domain Similarly, eq (17.49) may be integrated numerically
to obtain the local densities in the tow as a function ofx and τ.