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Biomass pyrolysis using microwave technology

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8402312
Lee, Woo II
MICROWAVE CURING OF COM POSITES .'
T h e U n iv ers ity o f M i c h i g a n
University
Microfilms
International
Ph.D.
1983
300 N. Zeeb Road, Ann Arbor, Ml 48106
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
MICROWAVE CURING OF COMPOSITES
by
Woo II Lee
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Mechanical Engineering)
in The University of Michigan
1983
Doctoral Committee:
Professor
Professor
Associate
Professor
Professor
George S. Springer, Chairman
William P. Graebel
Professor Robert B. Keller
Andrew F. Nagy
Gene E. Smith
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MICROFILMED DISSERTATIONS
Microfilmed or bound copies of doctoral dissertations submitted
to The University of Michigan and made available through University Micro­
films International or The University of Michigan are open for inspection,
but they are to be used only with due regard for the rights of the author.
Extensive copying of the dissertation or publication of material in excess of
standard copyright limits, whether or not the dissertation has been copy­
righted, must have been approved by the author as well as by the Dean of
the Graduate School.
Proper credit must be given to the author if any
material from the dissertation is used in subsequent written or published
work.
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to Eun Hee,
Sun Goo,
and Kyung Goo
ii
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ACKNO W LEDGM ENTS
I would like to express deepest thanks to Professor
George S. Springer
course of this
for his support and guidance during the
research and his help in the preparation of
this thesis.
I also would like to thank my thesis committee of
Professors William P. Graebel,
F. Nagy,
Robert
B. Keller,
Andrew
and Gene E. Smith for their willingness to serve on
my doctoral committee,
and thank Professor Valdis V. Liepa
in the Department of Electrical and Computer Engineering
for
his many valuable advice and help in the experiment.
In addition,
I would like to thank Messrs.
and A. R. Allen for their assistance
experimental apparatus,
typing,
in constructing
Miss K. Bublitz
and Mr. T. Kearns
J. Wulster
the
for her help in
for his advice on the experiment.
This work was supported by the U. S. Air Force Systems
Command,
Materials Laboratory,
Base, Dayton,
Ohio with Dr.
engineer.
financial support
The
Wright-Patterson Air Force
S. W. Tsai acting as a project
received from this project
is gratefully acknowledged.
Finally,
I would like to thank my family
and friends
for their support and encouragement.
iii
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TABLE
OF
CONTENTS
D E D I C A T I O N ....................................................ii
ACKNOWLEDGMENTS
............................................
iii
LIST OF
F I G U R E S .............................................. vi
LIST OF
T A B L E S .............................................. xi
LIST OF
A P P E N D I C E S ........................................ xii
N O M E N C L A T U R E ............................................... xi i i
SECTION
I.
INTRODUCTION
11 . MODEL
..........................................
1
• . ................................................ 3
2.1 Problem Statement
..............................
2.2 Electromagnetic Model
2.2.1
..........................
3
5
Electric Field, Linearly Polarized
TEM W a v e .................................... 6
2.2.2 Reflectance and Transmittance
. . . .
2.2.3 Overall Reflection and Transmission
Coefficients ............................
17
19
2.2.4 Absorbed E n e r g y ........................... 22
2.2.5 Incident Isotropic Electromagnetic
Wave
......................................25
2.3 Thermochemical Model
.........................
28
2.4 Resin Flow M o d e l ................................. 35
2.5 Input and Output Parameters
2.5.1 Composite Properties
III.
................. 35
...................
40
N U M E R I C A L ........................................... 44
3.1 Numerical Solution, Electromagnetic
M o d e l ............................................ 44
iv
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3.2 Numerical Solution, Thermochemical
M o d e l .................................... 4 7
3.2.1 G r i d ................................ 47
3.2.2 Finite Difference Equations
.........
3.3 Numerical Solution, Resin Flow Model
3.4 Material Properties
50
. . . .
54
.........................
54
3.5 Computer C o d e ........................... 54
IV.
E X P E R I M E N T A L ...................................58
4.1 W a v e g u i d e ................................ 58
4.2 Microwave O v e n ........................... 61
4.3 Measurement of Electromagnetic
P r o p e r t i e s ................................ 64
4.3.1 Dielectric Constants
..................
4.3.2 Measurement of the Reflectance
. . . .
4.3.3 Measurement of the Transmittance
4.4 Microwave Curing
V.
..............................
EXPERIMENTAL VALIDATION OF THE MODELS
5.1 Electromagnetic Model
NUMERICAL RESULTS
69
70
70
..........
73
.......................
73
5.2 Thermochemical and Resin Flow Models
VI.
...
65
. . . .
.....................
74
85
6.1 Electromagnetic Wave-Composite Material
Interactions ..................................
85
6.2 Microwave Curing-General Considerations
94
. .
6.3 Microwave Curing-Selection of Cure
Cycles . . . . .
..............................
96
.VII. SUMMARY AND C O N C L U S I O N S .....................116
A P P E N D I C E S ................................................. 119
R E F E R E N C E S ................................................. 145
v
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LIST
OF
F IG U R E S
Figure
1.
Geometry of the Problem,
Polarized TEM Wave.
Incident Linearly
2.
Designations of the Plies and the
Interfaces.
3.
Illustration of the Incident, Reflected, and
Transmitted Electric Field Vectors at an
Interface.
Reflected Portion
of the Right Traveling wave e T .
(E I)r is
the Reflected Portion of the Left Traveling
Wave e T.
4.
Illustration of the Off-axis
axis (p-q) Coordinates.
(1-2) and On'1
5.
Illustration of the Electric Field Vectors
in the m-th Ply (Linearly Polarized TEM
W a v e ).
6.
Illustration of the Electric Field Vectors
at the Front (m=1) and Back (m=M+1)
Interfaces (Linearly Polarized TEM Wave).
7.
Illustration of the Overall Incident,
Reflected, and Transmitted Energies
(Linearly Polarized TEM Wave).
8.
18
Illustration of the Electric Field Vectors
in the m-th Ply (Linearly Polarized TEM
W a v e ).
24
Illustration of the Incident Isotropic Wave
and the "Equivalent" Linearly Polarized TEM
Waves.
29
10.
Schematic of a Typical Cure Assembly.
11.
Description of the Cure Assembly Used
Thermochemical Model.
12.
Arrangement of the Grid Points.
48
13.
Illustration of the Control Volume about the
8-th Grid Point.
49
Schematic of the Waveguide Set-up.
60
14.
in the
vi
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31
15.
15.
17.
18.
19.
20.
21.
Schematic of the Press Used during Microwave
Curing.
Illustration of the Waveguide Arrangement
during the Measurements of the Dielectric
Constants.
Reflectances of Fiberite S2/9134B Glass
Epoxy Uncured Unidirectional Composite as
Functions of the Number of
Plies. Comparisons between the Data and
Results Computed by the Model.
Data were
Generated in a Waveguide with Incident
Linearly Polarized TEM Waves (Polarization
A n g l e , 6).
63
66
75
Reflectances and Transmittances of Hercules
AS/3501-6 Graphite Epoxy Uncured
Unidirectional Composite as Functions of the
Number of Plies.
Comparisons between the
Data and Results Computed by the Model.
Data were Generated in a Waveguide with
Incident Linearly Polarized TEM Waves
(Polarization Angle, o).
76
Reflectance of Fiberite S2/9134B Glass Epoxy
Uncured Cross-ply Composite as a Function of
the Number of Plies.
Comparison between the
Data and Results Computed by the Model.
Data were Generated in a Waveguide with
Incident Linearly Polarized TEM Waves
(Polarization Angle, 6).
77
Reflectance as a Function of Polarization
Angle 6 for Hercules AS/3501-6 Graphite
Epoxy Uncured Single Ply Composite.
Comparison between the Data and Results
Computed by the
Model.
Data were Generated
in a Waveguide with Incident Linearly
Polarized TEM Waves.
78
The Change in Reflectance with the
Orientation of the Second Ply, 0, for a Twoply Hercules AS/3501-6 Graphite Epoxy
Uncured Composite.
Comparison between the
Data and Results Computed by the Model.
Data were Generated in a Waveguide with
Incident Linearly Polarized TEM Waves
(Polarization Angle, 6).
79
vi i
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22.
23.
24.
Temperature as Function of Time during
Microwave Curing of 32-ply Fiberite S2/9134B
Glass Epoxy and Hercules AS/3501-6 Graphite
Epoxy Composites.
Comparisons between the
Data and Models.
Cure Assembly Shown in
Figures 14 and 23.
Power Inputs to the
Composites were as Indicated.
3!
Mass Losses Normal (top) and Parallel
(center) to the Tool Plate, and the Total
Mass Loss (bottom) as Functions of Time
during Microwave Curing of 32-ply Hercules
AS/3501-6 Unidirectional Composites.
Comparisons between the Data and Results
Computed by the Models.
Cure Assembly is
Shown in Figures 14 and 23.
The- Power Input
W in' Cure Pressure, P q , and Bleeder
Pressure, P., are as indicated.
The Initial
Resin Content was 42%.
82
Components of the Cure Assembly Used in
Modelling the Temperature Distribution and
the Resin Flow during Microwave Cure.
Complete Cure Assembly is Shown in Figure
15.
84
25.
26.
27.
The Variation in Reflectance and
Transmittance with Polarization Angle, 5,
for Single Plies of Fiberite S2/9134B Glass
Epoxy and Hercules AS/3501-6 Graphite Epoxy
Composites Exposed to Linearly Polarized TEM
Waves.
Results of the Model.
Material
Properties Listed in Appendix G.
87
Reflectances of a Two-ply Hercules AS/3501-6
Graphite Epoxy Composite Exposed to a
Linearly Polarized TEM Wave (Polarization
Angle, 6 = 90°) and to an Isotropic Wave.
Results of the Model.
Material Properties
Listed in Appendix G.
89
Reflectances and Transmittances of
Unidirectional, Cross-ply and Q u a s i ­
isotropic Fiberite S2/9134B Glass Epoxy
Composites Exposed to Linearly Polarized TEM
Waves (Polarization Angle 0° ^ 6 ^ 90°) and
to Isotropic Waves.
Results of the Model.
Material Properties Listed in Appendix G.
90
vi i i
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28.
29.
30.
31.
32.
33.
34.
35.
Reflectances and Transmittances of Hercules
Unidirectional, Cross-ply and Q u a s i ­
isotropic AS/3501-6 Graphite Epoxy
Composites Exposed to Linearly Polarized TEM
‘ Waves and to Isotropic Waves.
Results of
the Model.
Material Properties Listed in
Appendix G.
91
The Absorbed Energies in the m-th Ply of
Fiberite S2/9134B Glass Epoxy and Hercules
A S/3501-6 Graphite Epoxy Unidirectional
Composites Exposed to Linearly Polarized TEM
Waves.
Results of the Model.
Material
Properties Listed in Appendix G.
93
Reflectances of Fully Cured 64 ply Fiberite
S2/9134B Glass Epoxy (d=0.96cm) and Hercules
AS/3501-6 Graphite Epoxy (d=0.77cm)
Unidirectional Composites Coated with a
Homogeneous Isotropic Material on the Front
(Left), Back (Middle), and Both Front and
Back (Right).
Linearly Polarized TEM Wave
Incident on the Front Surface.
Results of
the Model.
Properties of the Composites
Given in Appendix G.
The Dielectric
Constant of the Coating is e'. The
Dissipation Factor of the Coating is e'^ = 0 .
95
Illustration of the Cure Cycle Used in the
Parametric Study of Microwave Curing.
99
Manufacturer's Recommended Cure Cycle
Hercules AS/3501-6 Prepreg [27].
for
101
Temperature Distribution Across the
Composite as a Function of Time for
Different Levels of Microwave Power Input.
Results Obtained by the Models for the Cure
Cycle Shown in Figure 31.
102
Temperature Distribution Across the
Composite as a Function of Time for
Different Power Cycles.
Results Obtained by
the Models.
^2
Temperature Distribution Across the
Composite as a Function of Time for a
Composite Thermally Insulated (Left) and for
a Composite without Thermal Insulation
(Right).
Results Obtained by the Models for
the Cure Cycle Shown in Figure 31.
105
ix
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36.
Viscosity Distribution at Different Times
Inside the Composite. Results obtained by
the Models for the Cure Cycle Shown in Figure
31.
37.
The Maximum Viscosity umax Inside the
Composite as a Function or Time.
Gel is
Assumed to Occur When Viscosity Reaches 7
Pa»s.
Results Obtained by the Model for the
Cure Cycle Shown in Figure 31 at Power
Inputs of 100 W and 200 W.
38.
Gel Time as a Function of Power Input.
Result of the Models .for the Cure Cycle
Shown in Figure 31.
39.
Number of Compacted Plies as a Function of
Time for Different Microwave Power Inputs
and Different Cure Pressures.
The Results
Obtained by the Models for the Cure Cycle
Shown in Figure 31.
The Result Shown for
the Autoclave Cure Cycle is from Reference
[13].
40.
Degree of Cure Distribution Across the
Composite as a Function of Time.
Results
Obtained by the Models for the Cure Cycle
Shown in Figure 31.
41.
Minimum Degree of Cure as a Function of Time
for Two Different Power Inputs.
Results
Obtained by the Models for the Cure Cycle
Shown in Figure 31.
42.
The Time Required to Reach the Gel point
Using Microwave Curing with 100 W and 200 W
Power Inputs and Using the Autoclave Cure
Cycle Rerommended by the Prepreg
Manufacturer.
The Results for Microwave
Curing were Obtained by the Models for the
Cure Cycle Shown in Figure 31.
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LIST
OF
TABLES
Table
1.
Input Parameters Required for the
Electromagnetic Model.
2.
Input Parameters Required for the
Thermochemical Model.
3.
Input Parameters Required for the Resin Flow
Model.
4.
Output Parameters Given by the Models.
5.
The Reflectance, ^ r a n s m i t t a n c e , and Total
Absorbed Energy <
Calculated by the Closed
Form Solutions (Eqs. 96-106) , and by the
Computer Code
xi
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LIST
OF
A P P E N D IC E S
Appendix
A.
The Reflection Coefficient at
In t e r f a c e .
the m-th
120
B.
The Attenuation Tensor for the m-th Ply.
123
C.
Rate of Energy Absorption per Unit Volume by
the m-th Ply.
126
D.
Normal Incident Energy for an
Isotropic Wave.
128
E.
F.
G.
H.
Relationship between the Platen Force and Air
Bag Pressure.
131
Electric Field Strength Distribution Inside
the Microwave Oven.
134
Properties of the Composites and Surrounding
Materials.
Reflectance for the Front Face of a Single
Ply Composite.
xi i
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135
142
NOMENCLATURE
A^j
attenuation tensor
ap
amplitude of the incoming wave, V/m
Bm
constant defined in Eq.
C
(80)
c
specific heat of the composite, kJ/(kg*K)
Q
speed of light (=3.0x10
m/s)
d
total thickness of the composite, m
dm
thickness of m-th ply of the composite,
E^
electric
E?
resultant electric
field vector,
m
V/m
field vector, V/m
f
frequency,
Hz
H
amount of heat evolved by the chemical reactions,
kJ/kg
Hr
ultimate heat of reaction during cure,
H
rate of heat generation by the chemical
reactions, kJ/(kg*s)
h^
thickness of the i-th layer, m
h
total thickness of the
I
total number of grid points
j
,
K
m
= /“
thermal conductivity perpendicular to the plane
of the composite, W/(m*K)
M
total number of plies
Mf
fiber mass,
kg
M re
resin mass,
kg
m
cure assembly,
kJ/kg
$
measured mass loss of the composite,
Nj| j
tensor defined by Eq.
ng
number of compacted plies
percent
(A.7)
XI 1 1
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pressure
in the bleeder,
kPa
P1
electric
V/m
Pg
applied cure pressure,
R
reflectance
R
reflectance at the front
composite
r
reflection coefficient
r^j
reflection coefficient tensor
$
r^j
overall reflection coefficient
T
temperature,
T ^ n ^t
initial temperature,
Tr
transmittance
t
transmission coefficient
t^j
overall transmission coefficient tensor
U^j
unit matrix
u
number of layers below the
VSWR
voltage standing wave ratio
v
number of layers above the
W^n
microwave power
w
width of the waveguide,
AX*
parameter defined by Eq.
x
vertical coordinate normal to the tool
Ax
grid spacing
x 1,X2 ,Xg
field vector defined by Eq.
(8) and
(9),
kPa
face of a single ply
tensor
C
C
composite
(Figure
11)
composite
(Figure
11)
input, W
m
(109)
coordinates defined in Figure
Z
impedance,
z
parameter defined by Eq.
plate
1
ohm
(110)
xiv
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GREEK SYMBOLS
a
degree of cure
a •„
min
minimum degree of cure
a
specified- degree of cure
y
propagation constant as defined by Eq.
6
polarization angle of linearly polarized T EM
wave, degree
^
(4)
—
£
energy absorbed rate per unit volume,
<5^nc
incident energy flux, W/m^
£
absorbed energy per unit area per unit time
the m-th ply, W/m
m
3
W/m
in
r e f l e c t e d e n e r g y oe r u ni t a r e a D er u n i t t im e ,
W/m
/_
total absorbed energy per unit area per
time, W/m
<; .
c
transmitted energy per unit area per unit time,
w/m
e
complex dielectric constant
e'
dielectric constant
e"
dissipation
n
a tensor defined by Eq.
0
angle of ply orientation,
X
wavelength, m
Xq
wavelength in free space, m
y
resin viscosity,
Umax
maximum viscosity
unit
factor
(39)
degree
Pa*s
inside the composite,
Pa*s
complex permeability
Vf
fiber volume fraction of the composite
£
a tensor defined by Eq.
(38)
xv
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p
density,
0
3
electrical conductivity of the composite,
time,
t
t
T
kg/m
g
s
cure time,
c
e
i
co
mho/m
gel time,
s
s
angular velocity,
rad/s
SUBSCRIPTS
A
null position with the sample,
B
null position without the sample,
f
fiber
1
components of
inc
incident electromagnetic wave
j
components of
a vector in 1,
k
components of
a vector
L
lower surface
of the cure assembly
1
layers of cure assembly
m
m-th ply or interface
p
direction parallel to the fibers
q
direction perpendicular to the fibers
r
reflected electromagnetic waves
re
resin
T
total
t
transmitted electromagnetic waves
U
upper
wg
waveguide
0
free space
surface
a vector
m
m
in 1, 2, and 3
directions
2 and 3directions
in 1, 2 ,and 3
directions
of the cure assembly
xvi
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1,2,3
1,2,3-directions
(Figure
g
g-th grid or grid spacing
6
polarization angle 6
1)
SUPERSCRIPTS
c
cured composite
i
isotropic wave
q
time step
u
uncured composite
+
electromagnetic wave travelling
right (Figure 5)
electromagnetic wave travelling
left (Figure 5)
from left to
from right to
xvi i
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S E C T IO N
I
INTRODUCTION
Parts and structures constructed
from fiber reinforced
matrix composites are manufactured by arranging the uncured
fiber-resin mixture
into the desired shape and then curing
the material at elevated temperatures and pressures.
Generally,
part
the curing process
in an autoclave,
is accomplished by placing the
with temperature and pressure
inside
the autoclave being maintained at prescribed levels.
Autoclave curing
parts made of thin,
is suitable
for individually cured
uniform laminates.
less suitable when the parts are large,
uneven dimensions,
simultaneously.
gradients
Autoclave curing
thick,
or have
or when several different parts are cured
In these cases,
the unavoidable temperature
inside the autoclave and the thermal
the autoclave make
is
it difficult
inertia of
to ensure that the parts are
cured uniformly and completely.
Microwave curing offers the possibility of uniform,
complete,
the part.
and economical cure regardless of the geometry of
In order to utilize the
microwave curing,
full potential of
the incident electromagnetic
radiation and
the a pplied pressure must be related to the thermal,
chemical,
and physical processes occurring
in the composite
1
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2
during cure.
However,
to date, microwave curing has only
been studied by experimental methods
these empirical
studies do not shed
[1-7],
The results of
light on the importance
of the various parameters which affect the cure.
The shortcoming of the empirical approach could be
overcome by use of analytical models;
unfortunately,
no
analytical model exists that could describe the behavior of
composite materials during microwave curing.
Therefore,
the
first and major objective of this study was to develop
models which characterize the response of continuous
reinforced thermosetting matrix composites
and relates
fiber
to
electromagnetic
radiation,
electromagnetic
radiation to the relevant cure process
variables such as temperature,
viscosity,
resin content,
the incident
degree of cure,
resin
and cure time.
The second objective was to generate
can be used to validate the models.
test data which
The third and
final
objective was to determine the response of composite
materials to electromagnetic waves and to demonstrate
usefulness,
as well as the limitations,
The emphasis
the
of microwave cure.
in this investigation was on microwave
curing of composites.
Therefore,
experimental results presented are
the analytical and
for linearly polarized,
and for isotropic electromagnetic waves at the microwave
frequency of 2.45 GHz.
Nevertheless,
the models and the
numerical procedures developed here are general,
and are
applicable over the entire frequency spectrum.
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S E C T IO N
I I
MODEL
2.1 Problem Statement
Consider a composite
laminate constructed of M plies,
each ply consisting of unidirectional
organic matrix.
x 1 axis
fibers embedded
The fiber orientation with respect to the
is denoted by the angle
6 (Figure
1).
The composite
is exposed on one side to a known electromagnetic
The incoming electromagnetic
plane
field.
field may be constrained to a
(linearly polarized transverse electromagnetic
TEM wave)
or may be isotropic.
polarized TEM wave,
the direction
field and the x 1 axis
polarization angle)
wave,
In case of a linearly
perpendicular to the composite.
electric
in an
of the propagation must be
The angle between the
(referred to as the
is denoted by the symbol
6.
It
is desired fo find the following parameters:
a)
the ratio of the reflected to the
incident energy
(r e f l e c t a n c e ) ;
b)
the ratio of the transmitted to the incident energy
(transmittance);
c ) the total absorbed energy;
d)
the absorbed energy as a function of position;
e ) the temperature T as a function of time and
3
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
COMPOSITE
FIBERS
INCIDENT LINEARLY
POLARIZED TEM WAVE
Figure 1
Geometry of the Problem,
Incident Linearly Polarized TEM Wave.
5
position;
and
f) the degree of cure of the resin as a
function of
time and position.
In addition,
for incoming linearly polarized TEM waves,
is desired to find the overall
it
reflection a n d transmission
coefficients.
A model suitable for calculating
is developed below in two parts.
model,
referred
overall
to
first part of the
as the electromagnetic model,
reflection coefficient,
transmission coefficient,
absorbed energy.
The
the above parameters
the reflectance,
the transmittance,
The second part,
yields the
the overall
and the
referred to as a
thermochemical model, gives the temperature and the degree
of cure.
In addition,
it is discussed how the information
generated by these models can be used to calculate the
viscosity and the resin flow.
In order to emphasize the concepts and the solution
methods,
the model
is presented
for a composite in which the
properties vary only across the thickness
problem,
(one-dimensional
flat plate geometry).
2.2 Electromagnetic Model
The electromagnetic model
is developed
linearly polarized TEM waves and for
separately for
isotropic waves.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6
2.2.1 Electric Field,
Linearly Polarized TEM Waves.
A linearly polarized T EM wave of known amplitude ag and
known wavelength X
impinges on a composite material.
The
direction of propagation is perpendicular to the composite,
i.e., the wave propagates
in the x^ direction
The x 1 a nd x2 components of the vector
this electric
field can be expressed as
(£,)
[8]
e*P (jM,r ~ yXi)
where ag is the amplitude,
1).
representing
= Q 0 cos<£ exp C j w T - 7X3)
(E*) = CU
the time, x^
(Figure
(1)
<2>
5 is the polarization angle,
t is
is the coordinate perpendicular to the plane of
the composite with the origin
the laminate,
and
j
(Xg=0) at the front surface of
has the common meaning,
j=/-T.
The
parameter u is the angular velocity
60
and
y
=
zirc/ Z
(3)
is the propagation constant
c is the speed of light,
e
and u
are the complex
dielectric constant and the complex permeability of the
medium through which the wave propagates.
materials,
^ * s=1.
For non-magnetic
The parameter e* may be expressed as
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7
where e'
is dielectric constant and e" is the dissipation
fa ctor;
It is desired to find the electric
inside of the material.
field distribution
A procedure suitable for performing
the calculations is described below.
To analyze the
problem,
we focus our attention on the m-th ply of the
laminate
(Figure 2).
Electromagnetic waves.arrive at each interface from the
right and from the left directions
(Figure 3).
Right and
left traveling waves are represented by the vectors,
e
T, respectively.
The subscript
E^ and
i has the values of i=1 or
2.
At the interface,
a portion of the arriving wave
transmitted through the interface
is reflected
and a portion of it
T he reflected portion of the right
traveling wave is denoted by
(E { ) r / and the reflected
portion of the left traveling wave
tangential
is
components
[9].
(ET)r .
The
of the vectors representing the
electromagnetic waves entering
must be equal
is denoted by
and leaving
This condition
theinterface
requires that the
following equalities be satisfied at the interface
E f = (El)± - ( E J r
<6>
E c = ( £ - ) t - ( E L )r
n)
We define now the following two parameters
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B A C K
8
2
•
tn
<D
0
(0
4-t
M
<D
+J
C
H
0)
X
•
-P
•
nd
c
(0
•
+
tn
Q)
•H
f-H
Cu
E
E
d)
XX
+>
•
•
0
•
tn
c
m
0
•H
+j
(0
c
CP
•H
tn
<u
Q
CM
OJ
>_l
CL
—
H
2
O
UJ
o
g
h2
q:
Q)
P
3
Cn
•H
u.
LU
cc
9 IxJ
Ll I
I—
2
o
>
2:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
INTERFACE
Ef
(Et),
pf
Pf <
(ET)r
E7
(E D .
IJ
Figure 3
Illustration of the Incident, Reflected, and Transmitted Electric
Field Vectors at an Interface.
(Ei)r is the reflected portion of the
right travelling v/ave
. (Ei)r is the reflected portion of the left
travelling w a v e E f .
vo
P: = (t;;r T
The reflected and
incident electric
field vectors are
related by the expressions
( E ^
=
(A jK E -)
<1°>
CEL)r - - (rcj)(bj )
'where r^j
(1 1 )
is the reflection coefficient tensor at the
interface.
The subscripts
components
(i,j=1,2).
i and j represent the
Here,
and in all subsequent analyses,
the Einstein summation convention
subscripts
is used,
i.e.,
repeated
imply summations.
For the
interface between the m and m-1 ply
the reflection coefficient tensor r^j
H j
where N^j
and
is
(Figure 3),
(Appendix A)
-
d 2)
is defined as
CoS0 - S i n 6 \ , / i - f
j
'■5iVi©
coS&A
O \/fc S 0
( ] 3)
l°S&)
o
t
The subscripts p and q refer to the directions parallel and
perpendicular to the fibers,
p and q and
and 0 is the angle between the
1 and 2 coordinate axes
By substituting Eqs.
(8)-( 11)
(Figure 4).
into Eqs.
(6) and
(7), we
obtain
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11
q
Figure 4
Illustration of the Off-axis
On-axis (p-q) Coordinates.
(1-2 ) and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
12
P-+
=
Pr =
U^j
( U;j +
r y ) ( E j ) - ( Tcj) ( E j )
(P 5)(eJ) +
(14)
(Ucj -r£j- ) CEJ)
os)
represents the unit matrix.
The electric field traveling through the material
attenuated.
Thus,
the electric
is
field arriving at an
interface is related to the electric field vector leaving
the adjacent interface by the expressions
(see Figure 5 and
Appendix B)
(E;+)m = (Aij)„<Pi)m
(16>
(Ei)m 1
A^.
is the attenuation coefficient
tensor in the m-th ply
(Appendix B ) .
, a > s e - s i n 6 s , e rJm 0
~
The parameter
y
' Sin0
ooSB
coefficient tensor
tensor
S'Vife-1
Q >iiy\rSind
' 0
was defined in Eq.
thickness of the m-th ply.
\,M 8
(4), and dm is the
The components of the reflection
(r^j) and the attenuation coefficient
(A^j) depend on the dielectric properties of the
composite,
wavelength.
the fiber orientation
Thus,
in each ply, and the
for a given material,
incoming electric field,
laminate layup, and
the components of the r^j and A^j
tensors are known.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•r
fl
UJ f
I
<
ii
PLY
i—
+LlT
-TH
CL
'oT
3
+q T
<*
n
it
UJ
i
UJ
1
i •UJ
Figure
5
£
I
~1
Illustration of the Electric Field Vectors
Ply (Linearly Polarized TEM W a v e ) .
in
the
m-th
13
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14
By utilizing Eqs.
Eqs.
(14) and
(15),
(16) and
(17), together with
the following expressions are obtained
,19)
(P; )m_," (Pj)m ^Ajk)m.|( )rn-i
+ [ U £ 3 - < ^ ) B3(AjK)w (TS')nl
The subscripts m and m-1
respectively.
coefficient
m-th ply.
refer to the m-th and
The parameter
<20)
(m-1)-th ply,
(fjj)m is the reflection
tensor at the interface between the
The subscript k has the values of
(m-1)-th and
1 and 2
(k=1,2).
At the front
material,
we have
interface
(m=1),
where the wave enters the
(Figure 6)
( Ec* )0 "
(E
c )(
=
The components of the
and
Eqs.
(21>
( A t j ),
(22)
),
(22)
incident electric
(E2 )0 , are given by Eqs.
(21) and
( fj
into Eqs.
(1) and
(14) and
field vector,
(2).
(15),
(E^q
By substituting
for m=1,
we
obtain
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Figure
6
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Illustration of the Electric Field Vectors at the Front
(m=l) and Back
(rr.=M+l) Interfaces
(Linearly Polarized TEM
M+1
15
16
(E-)0
( F f ; , = C Ucj +
- ( rcj)I (.& j k ) ,( P* )0
(p a
= ( r a a )
At the back interface
material,
(m=M+1),
where no wave enters the
the following equations hold
(Figure 6)
(E*)M = (Acj)M (Pj+JM
<25>
f E-' )
(26)
=
By substituting Eqs.
for m=M+1,
0
(25) and
(26)
into Eqs.
F’c )|v,
t
Uij
+ c 'pj
(15),
i (Ajk)„(
(24),
<27)
-
(28)
For a laminate consisting of M plies,
(23),
(14) and
we obtain
< ' P a . , =
(
(23)
(27), and
(28)
represent
Eqs.
(19),
2M+2 equations
(20),
for the
2M+2 unknowns which are the pT and PT vectors.
Note that
the equations and unknowns are in vector
Therefore,
form.
solutions must be obtained by expressing the equations
component
in
form, and by solving for the x 1 and x ^ components
of P* and P^.
A procedure for determining
the unknowns
outlined in Section 3.1.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is
17
2.2.2
Reflectance and Transmittance
The reflectance
is the ratio of the reflected to the
incident energy flux, and the transmittance
is the ratio of
the transmitted to the incident energy flux
[10].
incident energy flux is (Figure 7)
The
[8]
(29)
’o
where Z q is the impedance
120ir ohms
[11].
in free space and has the value of
The reflected energy flux, (5r is [8]
\(P: ) J
(30)
0
Thus the reflectance
is given by
Z
<fr
KRD0I
The transmitted energy f l u x , £ t is
(31 )
[8]
(32)
0
Accordingly,
the transmittance
is
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m=1
m= M + 1
INCIDENT
ENERGY
inc
2Z
TRANSMITTED
ENERGY
2
LAMINATE
.
K pD m
2Z
h
I
REFLECTED
ENERGY
.
c - K flk J 1
'
2Zn
' /I
FRONT
Figure 7
BACK
Illustration of the Overall Incident, Reflected, and
Transmitted Energies (Linearly Polarized TEM W a v e ) .
19
^
T_
<ft
—
=
=
- \- (- P
£lnc
- w f
1
, 4
(33)
I C E D J 2
In the above equations,
(P^)g anc^
are
e ^e c t r ^c
field vectors entering and leaving the front interface,
respectively
the back
(Figure 6).
The electric
interface is (Pj^M+l
field vector leaving
(Figure 6).
These parameters
can be calculated using the analysis developed in the
previous
2.2.3
section.
Overall Reflection and-Transmission Coefficients
$
The overall reflection coefficient tensor r^j and the
overall
transmission coefficient tensor t^j are defined by
the expressions
[10,29]
( P t" ) a =
(r j(e p 4
(34)
3 (t;p (E p 0
The electric field vectors (P- )n and
1 U
(3 5 )
(P.+ )u ,. are to be
1 M+ I
calculated by the method described previously
in Section
2.2.1.
is
However,
a knowledge of these vectors
insufficient to readily calculate the r^j and t^j tensors.
A procedure suitable
for determining these tensors
is
described below.
At the back interface
equations apply
(m=M+1),
the following two
(Section 2.2.1)
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20
( F£Ih,
=
( ^C.
”
C U j t c r y )M+il C A j O m
'(Pk+ )M
£ ^ ) M+j (Ajk)|V| (
By rearranging these equations,
(27)
|
(28)
we obtain
(36)
( P : \ = {<AcpM ru j k H f jk ) J ] (
-11 . r^+,
< PT) m = ( < rCj;M/ u jk+ ( rjk )Mti] j ( Pk ; Mw
w .
These equations are now represented symbolically as
.
M'tl
+■
<38)
(P i)M =
The tensors
tensors
(39)
<39)
and
(see Eqs.
into Eqs.
contain only the known r^j and A^j
36 and 37).
(19)
and
(20),
vectors at the M - 1 interface
Symbolically,
j.
By substituting Eqs.
we obtain the electric
(38) and
field
in terms of
the result can be expressed as
^M
r**s
,40)
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
21
where the tensors
and
tensors A^j and r^j.
The procedure is repeated step-by-step
until the m=2 interface
(P-),
(
(23)
is reached
-
Pi ^1 ~
( 7ij
As the final step,
Eqs.
again contain only the known
Eqs.
^
+l
(42) and
(43) are substituted
=
the
is obtained
Jm+ i
C,?Cj) ( P p M +i
(44)
(45>
(44) can be inverted to yield
( 5:j)
where
into
and (24), and after algebraic manipulations,
following expression
Equation
(43)
((i|j)
CEj)c
1 is the inverse matrix of
(46)
Equations
(45)
and (46) give
(PDo
It
-
is emphasized again
«7
( Sj'k) ' ( Efe )o
that the
and n]j tensors depend
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22
only on the known reflection coefficient and attenuation
coefficient tensors,
expressions for
r^'s
and A^j's.
However,
and njj cannot be derived.
closed form
These
tensors must be evaluated by a numerical method.
Comparisons of Eqs.
(46) and
(47) with Eqs.
(34) and
(35) yield the overall reflection and transmission
coefficients
r c*
=
(48)
<Sy>"
"
2.2.4
Absorbed Energy
The rate of absorbed energy per unit volume in each ply
is taken to be uniform and constant across the ply.
this approximation,
volume
the rate of absorbed energy per unit
in the m-th ply
= ^
With
is (Appendix C)
| ( E
+
^
e
m + c E , V ;" M
( E * ) c o j 6 m - ( E 4 )M S r n 0 m |
(50)
L
where Op and Og are the electrical conductivities
in the
%
e
directions parallel and perpendicular to the fibers.
E° and
E^ are the x 1 and x2 components of the resultant electric
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
23
field vector E? passing through the ply
(Figure 8).
The
vector E? is approximated by the expression
CEL
s)m =
j
(51)
The energy absorbed by the m-th ply per unit area per unit
time
is
(52)
~
where dm
(Em
^ rw
is the thickness of the m - t h ply.
absorbed by the entire laminate
unit area per unit time)
The energy
(total absorbed energy per
is
M
& T1
Alternately,
=
Z (£ „)
(53)
ro-i
the total absorbed energy per unit area per
unit time may be calculated by the law of conservation of
energy.
This law requires that the
following equality be
satisfied
/lncident\
/ReflectedX
/ Transmitted\
/ AbsorbedX
1 Energy j =
Energy
I + [ Energy
+ f Energy
\ Flux /
\ Flux
/
\
Flux
\
Flux
J
Equations
(30) and
er -
w h e r e i s
(32)
together with Eq.
<5 inc • ( I - R
given by Eq.
J
(54)
(54) yield
- T r )
(29).
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(55)
24
m - 1
K
m
U
,
"
m
►
. (p i+) m
M EI U
(AijUPPm
-«■
(PD m
I
i[(P D m +(Aij )m(P7)m]
Figure 8
Illustration of the Electric Field Vectors in
the m-th Ply (Linearly Polarized TEM W a v e ) .
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25
2.2.5 Incident
Isotropic Electromagnetic Wave
When an isotropic electromagnetic wave impinges on the
surface,
the wavelength X, and the total energy content
are known.
waves.
The superscript
i is used to denote
The total energy flux
normal
one half of the total energy flux
6 j
=
£ l/
isotropic
to the surface
is
(Appendix D)
z
<5S)
In analyzing the problem,
we replace
the isotropic wave
with a single unpolarized plane wave h a ving a total energy
content (S^f and traveling in the di r e c t i o n perpendicular to
the surface
plane wave
(x^ direction,
Figure 9).
This unpolarized
is considered to be made up of
plane waves,
the vectors representing these waves being
evenly distributed in a plane parallel
plane).
linearly polarized
to the surface
Each linearly polarized wave has
the same
wavelength X and the same energy content
Since
linearly polarized waves are evenly distributed,
incident energy flux at any polarization angle
(fjnJj
=
=
£Y
For each linearly polarized TEM wave
reflectance
(R)g,
the transmittance
absorbed energy per unit volume
each ply
4
6 is
ir
(Tr)^,
(5 7 )
the
the rate of
the energy absorbed by
(Sm )g, and the total absorbed energy
can be
calculated by the method described in Sections 2.2.2 and
2.2.4.
the
the
in the 6 plane,
(£)g,
( x 1~ x 2
Once these parameters are known for each
incident
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
LINEARLY
POLARIZED TEM WAVE
WAVELENGTH: X
NORMAL INCIDENT
ENERGY FLUX OF
ONE WAVE :
WAVELENGTH: X
TOTAL INCIDENT
]/
/
/
/
/
= £ / 4 7T
/
/
/
/
/
/
v
/
/
ISOTROPIC WAVE
Figure 9
DIRECTION OF
PROROGATION OF
UNPOLARIZED
TEM WAVE
(SUM OF LINEARLY
POLARIZED TEM
WAVES)
LINEARLY POLARIZED TEM WAVE
Illustration of the Incident Isotropic Wave and the
"Equivalent" Linearly Polarized TEM Waves.
/
/
/
/
/
/
/
/
/
/
/
z'
Xi
to
CTv
27
linearly polarized TEM waves,
for the
incident
they can then be calculated
isotropic wave by the procedure described
below.
The
isotropic reflectance
R ‘
3
is defined as
(see Eq.31)
/ ei
tr
(58)
<£*, the total reflected energy flux normal to the surface,
is the sum of the reflected normal energy fluxes of the
linearly polarized TEM waves
£=$'(£,)***
The reflectance
for a single polarized TE M wave
(R)s
By combining Eqs.
expression
=
(57)-(60),
reflectance
2T
~ I
2.7T o
An expression for the
I
=
—
The result
is
ar
f (TrLJS
The rate of absorbed energy per unit volume
is taken
(6,)
isotropic transmittance can be
in a completely analogous manner.
-JV
(60)
we obtain the following
for the isotropic
R‘ =
is
( £ f)s/ ( & • * ) £
I
developed
,59)
(62)
in the m-th ply
to be the sum of the rates of absorbed energy per
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
28
unit volume due to each linearly polarized TEM wave
2t
(63)
The absorbed energy by the m-th ply per unit area per unit
time
is
(64)
The total energy absorbed by the laminate per unit area per
unit time is
(65)
m-i
Equations
(1)— (65) complete the electromagnetic model.
The input parameters required for the solution and the
output parameters are specified in Section 2.5.
The
numerical method used to generate solutions is described
in
Section 3.
2.3 Thermochemical Model
During microwave curing,
the material to be cured is
generally surrounded by various layers.
Typically,
composite will be placed on a solid plate.
composite
is a bleeder;
on top of
Above
the bleeder,
an air breather and another solid plate.
the
the
there may be
Sheets of porous
or non-porous teflon release clothes are placed between the
various layers
(Figure
10).
A microwave of known energy content and wavelength
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is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
MICROWAVE
\ i /
PRESSURE
I
T E MP: T ,
|
|
TOOL
1
1
1
1
PLATE
AIR BREATHER
NON- POROUS
^ /T E F L O N
BLEEDER
^ /P O R O U S TEFLON
COMPOSITE
NON-POROUS
/TEFLON
TOOL PLATE
TEMP: T l
t t t t t t t PRESSURE
/t\
MICROWAVE
Figure 10
Schematic of a Typical Cure Assembly.
to
VO
30
incident on both the top and bottom surfaces of this
assembly.
In addition,
bottom surfaces,
the temperatures at the top and the
and
calculate the temperature,
, are specified.
It is desired to
the degree of cure,
and the
viscosity of the composite as functions of position and
time.
In order to generalize the problem,
cure assembly
illustrated in Figure
11.
we consider the
This assembly
consists of an arbitrary number of layers above and below
the composite.
Each of these layers are taken to be
homogeneous and isotropic with specified thicknesses and
known material properties.
Microwave energy absorbed by the
layers surrounding the composite
is taken to be negligible.
By considering energy transfer only in the direction
perpendicular to the plane of the composite
(x direction),
and by assuming perfect thermal contact between each layer,
the temperature distribution at any position
inside the
assembly can be calculated by the following form of the
energy equation
t
is the time,
[12]
x is the coordinate normal to the plane of
the assembly with its origin on the bottom surface of the
lowest layer.
T is the temperature,
and specific heat,
conductivity
respectively.
p and C are the density
K is the thermal
in the x direction, (- .and H are the rates of
heat generated by the absorbed microwave energy and by the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
31
INCIDENT MICROWAVE
Tu
LAYER: i =u+v+1
u+v+1
NUMBER OF
LAYERS: v
LAYER: I = u + 2
COMPOSITE
(Layer: 1 = u + l}
h u+ 2
u+1
/•
Layer: 1 = u
NUMBER OF
LAYERS:u
<
Layer: I = 2
Layer:
1
T
l
/ t\
NCIDENT MICROWAVE
Figure 11
Description of the Cure Assembly Used
in- the Thermochemical Model.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
32
chemical reactions,
the composite,
respectively.
In the layers surrounding
both £ and H are zero. For the composite, <£
as a function of position x may be found by the procedure
described in Section 2.2.4.
H is defined by the expression
[13]
clod
•
H -=
Hr
h r
is the total or ultimate heat of reaction during cure,
and a is the degree of cure defined as [13]
H(r)
A
(6 8 )
-
HR
where H(t)
is the heat evolved from the beginning of the
reaction to some intermediate time,
material
a=0,and
approaches unity.
neglected,
t
.
For an uncured
for a completely cured material
a
If diffusion of chemical species is
the degree of cure at each point
inside the
material can be calculated once the cure rate
is known
in
the following way
d o/ , ^
{gg)
- - f 0 < •& )
In order to complete the model,
the dependence of the
cure rate on the composite and on the degree of cure must be
known.
This dependency may be expressed symbolically as
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
33
(70)
The functional relationship in Eq.
(70), along with the
value of the heat of reaction Hp for the prepreg material
under consideration,
can be determined experimentally by the
procedures described
in Reference
Solutions to Eqs.
(66) and
[13].
(67)— (70) can be obtained
once the initial and boundary conditions are specified.
The
initial conditions require that the temperature inside the
assembly and the degree of cure
given before the start of cure
inside the composite be
(time
t < 0).
The boundary
conditions require that the temperatures on the top and
bottom surfaces of the assembly
and the incident microwave
energy be known as functions of time during cure
Accordingly,
(time
the initial and boundary conditions
corresponding to Eqs.
(66) and (70) are
Initial conditions:
t <0
T init
temperature
in the assembly.
Boundary conditions:
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t ^O).
34
where
and TL are the temperatures on the top and bottom
surfaces of the assembly,
respectively
addition to the temperatures
waves
and T^,
10).
In
the electromagnetic
incident on the top and bottom surfaces must be
specified.
aQ,
(Figure
For linearly polarized TEM waves,
the amplitude
the w a velength X, and the polarization angle 6 must be
given.
For
isotropic waves,
the wavelength X and the total
energy content of the wave £.1 must be specified.
Solutions to Eqs.
the degree of cure
(66)— (72) provide the temperature and
in the composite as functions of position
and time.
Once these parameters are known,
can be calculated,
the resin viscosity
provided a suitable expression
relating
resin viscosity to its temperature and degree of cure
available.
If the resin viscosity
is
is assumed to be
independent of shear rate, then the relationship between
viscosity,
temperature,
and degree of cure can be
represented in the form:
jji
The manner
2
< T , ° < )
'
(73)
in which the relationship between viscosity,
temperature,
described
=
and degree of cure can be established
in Reference
Equations
is
[13],
(66)— (73) complete the thermochemical model.
A numerical procedure suitable
for generating solutions
presented
input parameters required for
in Section 3.2.
The
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is
35
the solutions are discussed subsequently
(Section 2.5).
2.4 Resin Flow Model
During the cure, a pressure is applied to the cure
assembly
(Figure
composite,
10) to squeeze the excess resin out of the
to consolidate the plies,
void content.
and to minimize the
It is desired to estimate the resin flow out
of the composite and the amount of resin
functions of position and time.
in the composite as
These parameters can be
calculated by the method proposed by Loos and Springer
The detailed steps of the method are not given here,
be found in Reference
[14].
but can
[14].
The equations proposed by Loos and Springer were
incorporated into the numerical solutions
(Section
3)
to
enable us to calculate the resin flow during microwave
curing of composites.
2.5 Input and Output
Parameters
Solutions to the electromagnetic(Eqs. 1-65)
thermochemical(Eqs. 66-73), and to the resin flow models
(Eqs.
10-33 in Reference
parameters be specified.
[14])
require that the input
The parameters needed for the
solutions are summarized in Tables
1-3.
Solutions to the electromagnetic,
thermochemical,
resin flow models provide the information
listed
and
in Table 4.
The methods used in obtaining solutions to the appropriate
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36
Table
1
Input Parameters Required for the Electromagnetic Model
A. Geometry
1) Thickness of each ply
2) Number of plies
3)
Ply orientation
B. Composite Material Properties
4) Dielectric constants parallel
and perpendicular
the fibers for the uncured composite
to
5) Dielectric
constants
parallel
and perpendicular
the fibers for the cured composite
to
C. Incident Wave Properties^
6) Wavelength
7) Amplitude
8)
Polarization angle
9)
Incident Energy' Flux
1. For incident linearly polarized T E M waves, only items
6,
7,
and 8 are needed.
For incident isotropic waves, only
items 6 and 9 are needed.
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37
Table 2
Input Parameters Required for the Thermochemical Model
These parameters are in addition to those required in the
electromagnetic model (Table 1).
A. Geometry
1) Length of the composite
2) Width of the composite
3) Number of layers above and below the composite
4) Thickness of each layer above and below the composite
5) Temperature at the upper surface of the cure assembly
as a function of time
6) Temperature at the lower surface of the cure assembly
as a function of time
B. Composite Properties
7) Initial thickness of one ply
8) Initial resin mass fraction of one ply
9) Resin content of one compacted ply
C. Resin Properties
10) Density
1 1) Specific heat
12) Thermal conductivity
13) Heat of reaction
14) Relationship between the cure rate,
degree of cure
temperature,
D. Fiber Properties
15) Density
16) Specific heat
17) Thermal conductivity
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and
38
Table 2. (cont.)
E. Layer Properties
18) Density of each layer
19) Specific heat of each layer
20) Thermal conductivity of each layer
F.
Initial and Boundary Conditions
21) Initial temperature distribution in the composite and
in the layers above and below the compos ite
22) Initial degree of cure of the resin
in the composite
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39
Table 3
Input Parameters Required for the Resin Flow Model
These parameters are in addition to those required in the
electromagnetic and thermochemical models (Tables 1 and 2)
A. Composite Material Properties
1) Apparent permeability of the prepreg
plane of the composite
2) Flow
coefficient
fibers
of
the
prepreg
normal
parallel
to
the
to the
B. Resin Properties
3) Relationship between the viscosity,
degree of cure
temperature,
C. Bleeder Properties
4) Apparent permeability
5) Porosity
D.
Initial and Boundary Conditions
6) Cure pressure as a function of time
7) Pressure
in the bleeder
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and
40
equations are described in Section 3.
2.5.1
Composite Properties
Solutions to the electromagnetic and
thermochemical
models described in Sections 2.2 - 2.4 require
that the
complex dielectric constant e , density p, specific heat C,
heat of reaction H, and thermal conductivity normal to the
fibers K of the composite,
be known.
These properties
depend on the local resin and fiber contents and on the
degree of cure of each ply.
By assuming that the complex dielectric constant is
directly proportional to the resin content of the composite,
the changes
in the complex dielectric constants in the m-th
ply during cure may be approximated by the expressions
c
jy*
f
=W
*
'JC C
'Jt U
Mfio “
Mft.
M ^ - M re
yi C
+l€f}
1
™
{Ui
(75)
Mr.
- M r
’
*
The subscripts p and q represent the direction parallel and
perpendicular
to the fibers,
respectively.
u and c denote uncured and cured materials,
re
and
re
composites,
are the resin masses
The superscripts
respectively.
in the uncured and cured
and M rg is the resin mass at any intermediate
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4'1
Table 4
Output Parameters Given by the Models
Electromagnetic Model^
1) Overall reflection coefficient
2) Reflectance
3) Overall transmission coefficient
4) Transmittance
5) Absorbed energy
in each ply
6) Total absorbed energy
7)
Isotropic reflectance
8)
Isotropic transmittance
Thermochemical Model
9) Temperature distribution as a function of time and
position
10) Degree of cure of resin as a function of time and
position
11) Viscosity of the resin as a function of time and
position
Resin Flow Model
12)
Number of compacted plies as
a function of time
13)
Amount of resin flow normal to theplane of the
composite as a function of time
14) Amount of resin flow parallel to the plane of the
composite as a function of time
15) Total
time required for the cure
1. For incident linearly polarized TEM waves, only items 1
through 6 are calculated.
For incident isotropic waves,
only items 5 through 8 are calculated.
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42
time
t
.
The parameters
(ep)U f
^eq ^ U ' ^£p^C ' and
^eq ^ C are
determined from the measurements of the dielectric constants
(see Section 4.3.1).
The parameter M re can be calculated by
the resin flow model.
The variations
in p, C, H, and K with the degree of
cure are generally unknown.
Therefore,
the change
parameters with the degree of cure are neglected
in these
in this
study, and only changes due to the resin content are taken
into account.
Accordingly,
p, C, and H are calculated by
the expressions developed in Reference
[14]
(76)
(77)
(78)
where subscripts re and f refer to resin and fiber,
respectively.
M T is the total mass of the m-th ply,
and M f are the masses of the resin and the fiber.
the heat of reaction per unit mass for the resin
thermal conductivity normal to the fibers
and M rg
(H R ^ re is
only.
(x direction,
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The
43
Figure
11)
K may be estimated from the expression
Km
Krc^'-i
fn>
[16]
(79)
+
B/
Z, /
where
B
=
2
-
i )
(80)
K c
Kj and K re are the thermal conductivities of the fiber and
the resin,
ply
(vf )m
respectively.
is given by
The fiber volume fraction of m-th
[16]
(81 )
w/re.
As noted previously,
M re w ^t^1 position and time
the variation
in the resin mass
is given by the numerical
solution described in Section 3.
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SECTION III
NUMERICAL
Solutions to the electromagnetic,
resin
The
thermochemical,
and
flow models must be obtained by numerical methods.
numerical procedures used to obtain solutions and the
associated computer code are described below.
3.1
Numerical Solution,
Electromagnetic Model
The starting point of the solution
(39)
is Eqs.
(38) and
in Section 2.2.3
M+i
• +
(38)
(39)
The components of the tensors
Then,
as was discussed
an^
in Section 2.2.3,
are known.
the calculations
proceed from ply to ply toward the front
(40)
(41)
0
44
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45
j
CPc^m
At each step,
calculated.
reached,
^
(
F
f
)
' M+!
U
Thus, once the front
(82)
,
the components of tensors
tensors
composite
(44),
=
/ V 1J
and n^j are
interface
(m=1)
and n^j are known throughout the
from m=1
to m=M+1.
Then,
using Eqs.
the overall transmission coefficient t ^ j
parameter
is
(35) and
and the
(p i)M + i are calculated
,*
,
■tj:
=
(
1 s-l
fcj )
<49»
(P-+)
= (' - £ * )' ( y E J^ )<0
*■ 'm+I
'
r
C
From the known values of
of tensors
and r^jf
calculated at every
(82) and
(83).
vectors,
e
T and
(p T ) + , and the known values
the parameters pt and PT are
interface using Eqs.
Once P^ and P^ are known,
e
(35>
‘J
7, at each
(38)
- (41),
the electric
(45),
field
interface are calculated using
E q s .(16) and (17).
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46
The reflectance,
transmittance,
are calculated from Eqs.
In case the wave
composite
(as,
(31),
(33), and
(50) -(53).
impinges on the two sides of the
for example,
absorbed energy
and absorbed energies
in a microwave oven)
the
is taken to be the sum of the absorbed
energies of the waves entering from both sides.
For isotropic waves, calculation of the isotropic
reflectance,
transmittance,
intergration of
(R)gf
angle 5 (see Section 2.2.5,
integration
and absorbed energy requires an
an<^ ^ ’V 5 over
polarization
E q s . 61-65).
the
interval was divided
Here,
in tt/6 segments and the
integration was performed numerically by taking the
parameters
(R)g>
^T r ^5
radian segments.
calculated
Thus,
^5
t0
be constant over
the isotropic
n/6
reflectance was
by the expression
I
R‘ = i f
j2?r
2ir °
The isotropic
volume were
anc^ ^
I 11
( R )sdS -
C -Z(R ;
W
^
rk
(84:
^ j
transmittance and absorbed energy rate per unit
calculated
in similar manners.
It is noted that the foregoing calculation procedures
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47
may also be applied to problems
outside the composite.
in which layers are placed
These layers may then be treated as
"plies" with known thicknesses and with known dielectric
properties.
3.2 Numerical Solution, Thermochemical Model
3.2.1 Grid
In order to calculate the temperature distribution
inside the composite and in the layers surrounding the
composite,
the cure assembly
one dimensional grids
(Figure
(Figure
11).
10) was divided into
Inside the composite,
grid points were located at the interface of each adjacent
ply, and on the lower and upper surfaces of the composite.
The layers surrounding the composite were also divided into
grids.
The distance between any two grid points
as Ax„ (Figure
p
composite,
12).
is denoted
In the layers surrounding the
the grid spacing Ax„
is constant.
In the
p
composite,
the grid spacings vary with time depending on the
resin content of the ply.
Each grid point is indicated by subscript g and time is
designated by the superscript q.
A control volume is placed around each grid point,
shown
in Figure
13.
The material properties are the same
between any two grid points
between
as
(i.e., between g-1 and
g, and
g and g+1), but are different across the control
volume.
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48
t
♦ 0-1
LAVER
COMPOSITE
PLIES
— - — ■
LAYER
a X/3
♦
r
LAYER
t ^ =2
Kj3*\
// / / / ////// \ /////; s ; s ; y / r
Figure 12
Arrangement of the Grid Points
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49
I
oa.
oa.
X
X
<J
<u
x
t
+j
•P
3
O
XI
(d
iH
O
>
pH
0
V)
44
+
G
O
U
dJ
-C
OQ.
---
4-1
44 4-i
O G
•H
c O
o cu
•H
+J XI
<d •H
p >4
4-> a
ui
3 -G
r-H 4-1
H I
H CQ
n
aj
_J
o IxJ
0c
V4
3
Cd
•H
Pm
I-
oo $
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50
3.2.2
Finite Difference Equations
Equation
(66) together with Eqs.
following expression
(67) — <70) give the
for the energy equation
In order to proceed with the solution,
we focus our
attention to the control volume surrounding the (3-th grid
point
(Figure
13).
control volume,
By integrating Eq.
(87) across the,
we obtain
S^<fCT)Jx
-- K g ]
+
(8e>
a.
Solution to the above equation
finite difference method
in a form suitable
[17].
is obtained by an
Equation
(88)
for numerical calculations.
the left-hand side of the Eq.
(88)
implicit
is expressed
The term on
is approximated by
where A t is the time step equal to the time between
T1^.
Here,
on T refers
and in the subsequent analyses,
to the (3-th grid point;
[18]
p
+ 1 and
the subscript (3
on all other parameters,
it refers to the region between the 8-1 and 8“th grid
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51
points..
The term on the right-hand side of Eq.
(88)
is
approximated by
a
"
(3
u
(5-i
where Axf is the distance between the grid points
8 and 8+1
p
at time
and A x ^ _ 1 is the distance between the grid
points 8 and 8_1 at time t^.
right-hand side
«
The second integral on the
is expressed as
r
fljn>H‘ * £i
By combining Eqs.
(89)
- (91),
/z
we obtain the following
algebraic expression
r
( K ]f + ( A f
Ui'p
L W ?-|
+
1 (£94$
J?
z
_ /J< \l J-H jK
Mxjj-/ P'1
1
°^
A* JL
+
z
J )
/>
Is*'
|
( /£ V s > W
£
-rf-"
Z
+
^ -yb
f
(f HriK 1- £
7
^ ,
-2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(92)
52
Equation
(92) represents 1-1 linear equations corresponding
to the 1-1
"interior" grid points
(Figure
equations,
called a tridiagonal system,
12).
This set of
is solved for the
temperature at each grid point at the new time T ^ +1 using
p
the Gaussian elimination method.
An algorithm for the
solution of
the above tridiagonal system is given by
Carnahan et
al. [19].
The temperatures at the lower
(T^+ 1 ) and upper
(T^+j)
boundary grid points are specified by the boundary
conditions.
The boundary temperatures may vary with time in
an arbitrary manner.
In the calculations,
the boundary
temperatures were .assumed to be constant.
It is noted that the numerical procedure outlined
previously
is an implicit method of solution.
The implicit
method does
not impose a limit on the size of the time step
(At ), which
can be used with a desired grid spacing
ensure stability of the numerical solution.
stability criterion
procedure,
is not required
(Ax)
Therefore,
to
a
for the above numerical
and the size of time step can be chosen
ar b i t r a r i l y .
Once the temperature
(Tf+ 1 ) is known at each grid
p
point,
the resin degree of cure can be determined at each
grid point
inside the composite.
relationship between cure rate,
cure,
The
functional
temperature,
and degree of
is given by
=
-f C T , e O
(70)
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53
By approximating the cure
rate by the difference equation
d°( ^ <?\s“
^
dz )n
A
”
and substituting Eq.
(93)
into Eq.
(70), we obtain the
following expression
for the degree of cure at the new time
T * +1
£
n
The viscosity of the
grid point and at time
V
+ -FCTp
t^+
'
%
A
(54:
resin can be determined at each
1 once the temperature
(Tf+ 1 ) and
p
the degree of cure
(a?+ 1 ) are known.
p
between viscosity,
temperature,
The exact relationship
and degree of cure, depends
on the resin system being studied
(Eq.
73).
A symbolic
relationship between the viscosity at each grid point
(u?+ ^), the temperature
(T^+ 1 ), and the degree of cure
p
p
(a?+ 1 ), can be expressed as
p
(95)
The rate of microwave energy absorption per unit volume
inside the composite
is calculated from the solution of
the electromagnetic model
described in Section 3.1.
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54
3.3 Numerical Solution,
Resin Flow Model
A Numerical method of solution
was developed by Loos and Springer
described in detail
in Reference
for the resin flow model
[14].
This method,
[14], was adopted in this
investi g a t i o n .
3.4 Material Properties
Solution of the finite difference equation
requires the complex dielectric constant
(Eq.92)
$
e , density p,
specific heat C, heat of reaction H, and thermal
conductivity normal to the fibers K be known at each grid
point
inside the composite.
Once the mass of the resin and
the total volume of the ply surrounding the grid point are
determined,
from Eqs.
the aforementioned properties can be calculated
(74) - (81).
The resin mass and the ply volume
are obtained from the resin flow model.
3.5 Computer Code
A computer code
(designated as "EMWAVE")
was developed
to implement the foregoing numerical procedures.
To test
the accuracy of the computer code pertaining to the
electromagnetic model,
results were obtained to problems for
which known analytical
solution exists.
The parts of the
code pertaining to the thermochemical and resin
were not tested because
flow models
the accuracies of these portions of
the code were evaluated by Loos and Springer
[15].
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55
In order
pertaining
to assess
the accuracy of the computer code
to the electromagnetic model,
the transmittance,
the reflectance,
and the total absorbed energy were
c alculated with the computer code
for unidirectional
32 ply
graphite epoxy and glass epoxy composites exposed to a
po l arized TEM wave
(6=90°).
AS/3501-6 graphite
epoxy and Fiberite S2/9134B glass epoxy
composites were
used
are
listed
and
(28), closed
The properties of Hercules
in the calculations.
in Appendix G.
By using Eqs.
form solutions
transmittance can be obtained.
are
These properties
(23),
(24),
(27)
for the reflectance and
The
resulting expressions
[20]
R =
E
u>sho( +■F sinho( - Q-Cosfl
+• H sinfi
(96)
(rtf it)
Tr -
(97)
E £<>5h o( + P S i n h o t ~ O '
+ H ?
where
4'fTkd
(98)
P
4-iTnd
and
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OQ )
56
k + 1) - 4 n
A = c
B = ( n z+ k * - 1 ) + 4 k
E = CnVkV// t
F = 4-n ( n h k V i )
( n l+fc1-/ A 4 k
Q
H = 4 k ( n + k L- i )
(100)
( 101)
( 102)
(103)
(104)
~
(105)
where n and -k are the real and imaginary parts of
respectively.
The total energy absorbed may be calculated
for the known value of R and Tr
(see Eqs.
- R
1
“
54 and 55)
- Tr
doe)
'
C ? .
^
(Y i <
The reflectance,
the transmittance,
and total absorbed
energy calculated by the computer code and by the closed
form expressions
results
(Eqs.
96-106) are given in Table 5.
The
in this table show good agreement between the
computer and the analytical solutions.
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57
TABLE 5
The Reflectance R, Transmittance Tr, and Total Absorbed
Energy
Calculated by Closed Form Solutions
(Eqs. 96 - 106) and by the Computer Code
S 2 / 9 13 4 B ( c u r e d , (0]3 2 , 6=90°)
Closed Form Solution
Tr
Computer Code
0.181
0.181
0.750
0.750
0.069
0.069
AS/350 1-6(c u r e d , [0]3 2 , 6 = 90°)
Closed Form Solution
Computer Code
0.735
0.181
Tr
0.014
0.014
(**
O rp
0.251
0.251
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SECTION IV
EXPERIMENTAL
In this chapter,
the experimental apparatus and
procedures are described which were used to measure
a) the dielectric properties of the material,
b) the reflection and the transmission of linearly
polarized T E M waves by the material,
c) the temperature distribution
inside the material
during microwave heating, and
d) the resin flow of the composite during microwave
curing.
Two types of apparatus were used
waveguide and a microwave oven.
in the tests: a
A waveguide was used
for
measuring the dielectric constants and the reflected and
transmitted energies.
A microwave oven was utilized
temperature and resin flow measurements.
apparatus is described below.
description
test
Each of
in the
the two
In addition, a brief
is given about the methods used to construct the
specimens.
4.1 Waveguide
In the experiments employing a waveguide,
2.45 GHz
microwaves were generated by a microwave oscillator
(General
58
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59
Radio Type
1360-B).
The microwave oscillator was connected
by coaxial cables to a waveguide adaptor
S281A) via a frequency meter
isolator
(Hewlett-Packard
(Hewlett-Packard 536A) and an
(Huggins Lab HC-7082)
(Figure
14).
The adaptor
was clamped to the waveguide.
The waveguide
section
consited of two parts:
a "slotted"
(Hewlett-Packard S810A) and a "solid" section.
material to be tested was placed between
solid sections.
clamps.
The
the slotted and the
The two sections were fastened together by
The waveguide was for S-band microwaves.
Accordingly,
dimensions
the cross-section was rectangular,
of 7.214 cm x 3.607 cm [21].
having
The length of the
slotted section was 32.4 cm.
Three different types of solid sections were used:
for measuring dielectric constants,
reflectances,
one
one for measuring
and one for measuring transmittances of the
material.
For measuring
the dielectric constants,
the solid
section was a waveguide with an adjustable
short
Packard S920A).
inside the
The postion of the short
(Hewlett-
waveguide could be adjusted by a screw mechanism.
measuring the reflectances,
termination
a resistor
the solid section was a
(Hewlett-Packard S910A).
This termination had
inside to minimize back reflection.
transmittance measurements,
an attenuator
In
For the
the solid section consisted of
(Polytechnic Research Development Type
with an adaptor
(Hewlett-Packard S281A)
171)
attached to the far
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
STANDING
WAVE
INDICATOR
MOVABLE
PROBE
o
0*1
MICROWAVE
OSCILLATOR
ADAPTOR
FREQUENCY
METER
ISOLATOR
SLOTTED
SECTION
WAVEGUIDE-
Figure 14
Schematic of the Waveguide Set-up.
SOLID
SECTION
61
side.
A probe measuring the electric
into the slotted section.
standing wave
position
indicator
of the probe
field strength was built
The probe was connected to a
(Hewlett-Packard 415B).
The
in the slotted section could be
varied along the slot.
During the transmittance measurements,
the electric
field strength was m e a s u r e d with a fixed probe located
the adaptor attached to the solid section.
attached to a standing wave
indicator
in
The probe was
(Hewlett-Packard
415B).
The procedures used to determine the dielectric
constants and the reflectances and transmittances are
described
in Section 4.3.
4.2 Microwave Oven
A commercially available
Model
700 watt,
2.45 GHz
(Litton
1290) microwave oven was used in the curing tests.
There were two controls built
into this oven:
one was a
timer which controlled the length of the periods during
which the power was on and off;
which controlled the total
the other one was a timer
length of time during which the
oven was operating.
The temperatures
inside the microwave oven were
measured either by type T
(c o p p e r - c o n s t a n t a n ) or by type J
(iron-co n s t a n t a n ) thermocouples.
shielded with either
These thermocouples were
stainless steel or inconel overbraid,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
respectively.
The thermocouples were fed through a hole on
the back of the oven, and the shield was grounded
[22].
The
output of each thermocouple was measured by a digital
voltmeter.
During cure,
the pressure to the composite was applied,
using the fixture shown
in Figure
constructed of polypropylene
dissipation
teflon
15.
(dielectric constant
factor e " = 0.0009 [ 2 3 ] )
(e'=2.3
,
e"=0.002
The fixture was
[23]).
e'=2.2,
and glass reinforced
These materials were used
to prevent significant energy absorption and corresponding
temperature
The
rise of the fixture materials.
fixture consisted of three 25.4 cm x 25.4 cm square
polypropylene plates.
The plates were separated by four 2.54
cm diameter ploypropylene rods.
thick) a nd the top plate
The bottom plate
(5.08 cm
(2.54 cm thick) were attached to
the rods and were prevented from moving by polypropylene
bolts.
The middle plate
up and down
(2.45 cm thick)
could freely slide
the rods.
A pancake shaped rubber bag
(25.4 cm diameter)
enclosed
in glassfiber cloth was placed in between the top and the
middle plates.
The bag was pressurized by compressed air.
A 20.3-cm x 20.3 cm
(2.54 cm thick)
glass reinforced
teflon plate was placed on the bottom plate.
to be cured was placed on this teflon plate.
x 15 cm,
Dams
(2.5 cm
1.5 cm thick) made of glass reinforced teflon were
placed around the composite.
thick)
The composite
glass
A 10.2 .cm x 10.2 cm (3.2 cm
reinforced teflon plate was placed on the top
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
POLYPROPYLENE
PLATE
AIR BAG
SUPPORT RODS
GLASS
REINFORCED
TEFLON
GLASS
REINFORCED
TEFLON
DAM
COMPOSITE
Ch
GLASS
REINFORCED
TEFLON
POLYPROPYLENE
PLATE
Figure 15
Schematic of the Press Used during Microwave Curing.
co
64
of the composite.
By inflating the air bag, pressure could be exerted on
the composite placed between the middle and the bottom
plates.
The relationship between the air pressure and the
pressure applied to the composite was determined by
replacing the composite with a load cell, and by measuring
the load at different bag pressures.
procedure
is described
further
This calibration
in Appendix E.
The uniformity of the power distribution
inside the
oven was evaluated by the method proposed in Reference
[24].
Nine 50 ml glass cups of water were placed inside the oven
at different locations.
The temperature rise of the water
was measured by thermocouples during a 60 second time
interval.
These results are given
maximum variation
percent.
The
in the power across the entire oven was 46
This maximum variation of electric field occured
across a distance of
test specimen was
variation
in Appendix F.
13.5 cm.
10.1 cm.
in electric
Maximum dimension of the
Across this distance,
the
field strength should be less than
about 35 percent.
4.3. Measurement of the Electromagnetic Properties
In measuring the electromagnetic properties,
the
composite material was placed between the slotted and the
solid sections of the waveguide.
vectors of
The electric
incoming waves were always
direction(Figure
16).
field
in the x^
Tests were performed with fibers
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
oriented in different directions with respect to the
electric
field.
4.3.1 Dielectric Constants
In order to measure the dielectric constants of the
material,
the composite of thickness d was placed between
the slotted and the solid section of the waveguide
16).
The position of the adjustable short
adjusted so that it was at a distance of
back surface of the material.
(Figure
(plunger)
was
1/4 X, „ from the
wg
The parameter
Xwg„
is given
J
by
(107)
where Xg is the wavelength in free space,
of the waveguide.
manner,
and w is the width
By positioning the plunger
in this
it was assured that the electromagnetic wave had a
maximum amplitude at the back surface of the material
(x.j=d).
This
is referred to as the "open position."
The wave generator was then turned on.
The probe
in
the slotted section was moved until the location of the
first minimum electric
field strength
(null position,
*A ),
and the location where the electric field had the maximum
amplitude were found.
section waveguide,
From a scale mounted on the slotted
the null position xA was recorded.
the standing wave indicator,
From
the strength of the electric
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ELECTRIC
FIELD
FIRST NULL
POSITION
COMPOSITE
MATERIAL
PLUNGER
MAXIMUM
AMPLITUDE
Figure 16
Illustration of the Waveguide Arrangement during the
Measurements of the Dielectric Constants.
67
field at the maximum and min i m u m
(E
max
, E
• ).
min
locations were
From the latter measurements,
standing wave ratio
(VSWR)
was c a lculated
recorded
the voltage
=
[21]
E f/)ax
VSWR
=
<108>
Emin
The composite material
was
then
taken out of the w a v eguide
and was replaced by an a l u m i n u m plate.
of the slotted section was again
parameter
AX
found
The null position
(position Xg).
A
was calculated
to?
=
Knowing the value of AX
and VSWR,
found from the Smith chart
[21].
impedance at the front surface of
calculated by the expression
the
impedance
Once z was
ratio z was
known,
the material
Z(0)
the
was
[21]
Z(0)
Z
=
(110)
where
7
)
{ L "fo
Under
rests
12.0 If
~.T ... . :
=
/
,
-
^
(111)
f
the "open p o s ition" condition employed
(see above),
the
impedance
material can be exDressed as
at the
front
face
in the
of the
[23]
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68
(1 1 2 )
where
/20V
(113)
( 1 14)
( 115)
From Eqs.
(110)
the dissipation
- (115),
the dielectric constant e' and
factor e" could readily be calculated by a
trial and error procedure.
The dielectric constants thus
measured are the values parallel to the direction of the
incident electric
fibers parallel
field vector.
Thus,
by aligning the
or perpendicular to the incoming wave,
the
dielectric constants of the composite normal or parallel
to
the fiber direction could be determined.
The measured values of the dielectric constants are
listed
in Appendix G for Hercules AS/3501-6 graphite epoxy
unidirectional composites and for Fiberit(e S2/9134B glass
epoxy unidirectional composites.
Graphite
is a good electrical conductor.
Therefore,
the values of e' and e" measured for graphite epoxy
composites along the fiber direction must be used with
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
caution.
For example,
for Hercules AS/3501-6 graphite epoxy
composites, measurements gave e T= 1 and e"=:25000
(Table G.2
in Appendix G).
The high e"/e'
absorption rate,
and consequently a high heat-up rate by
this material.
However,
because the material
ratio implies a high energy
high absorption rates do not occur
is also a good reflector
by the high value of e"),
(as manifested
and hence only a small
the electromagnetic wave penetrates
fraction of
into the material.
4.3.2 Measurement of the Reflectances
The reflectances were measured by placing the material
between the slotted section and the solid section of the
waveguide.
With the wave generator turned on, VSWR was
determined in the manner described in the previous section.
The ratio of the incoming and the reflected electric
fields (reflection coefficient)
expression
was calculated by the
[25]
VSWR - I
V5 WR
+
/
The ratio of the incoming energy
energy flux
(reflectance)
( 116)
flux and the reflected
is [10]
2
r
( 117)
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70
4.3.3 Measurement of the Transmittances
The material was placed between the slotted and the
solid section of the waveguide.
turned on, the electric
With the wave generator
field intensity was measured with a
probe positioned in the solid' section.
The signal from the
probe was displayed on the standing wave
indicator
decibels
(dB).
material
in the waveguide.
The measurement was repeated without the
the incoming electric
coefficient)
in
The ratio of the transmitted to
field strength
(transmission
was calculated by the expression
,
t
-
[25]
in-J-o(dZi-d e <)
10
(118)
where dB^ and d B 2 are the dB readings with and without the
material
in the waveguide,
respectively.
The ratio of the transmitted and the incoming energy
flux
(transmittance)
T
=
is given by [10]
t
(119)
4 .4 Microwave Curing
Two types of experiments were performed with the
microwave oven.
One was to measure the temperature
distributions during the cure of graphite epoxy and glass
epoxy
laminates;
the other one was to measure the resin flow
from laminates made of unidirectional graphite epoxy prepreg
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
71
tapes.
During the temperature measurements,
constantan)
type T (copper-
or type J (ir o n - constantan) thermocouples were
imbedded into the specimens at different locations.
thermocouple wires were shielded with either
or inconel overbraids.
The
stainless steel
During the temprature measurements,
pressure was applied to the graphite epoxy samples,
pressure was applied to the glass epoxy samples.
while no
The
microwave oven was turned on and the temperatures as
functions of time were recorded.
The amounts of resin flow were measured using
unidirectional graphite epoxy specimens.
Aluminum foils
extending approximately 5 mm from the edges were wrapped
around each edge of the graphite epoxy specimens.
The
purpose of the foil was to minimize the buildup of the
electric
field concentrated around the edges.
foils were perforated to permit resin
foils.
A thermocouple was also
Before each test,
The aluminum
flow through the
imbedded in the specimen.
the composite
layups and the bleeders
were weighed on a Mettler analytical balance.
The
composite specimen was then placed in the press.
Bleeders
were placed on the top of the specimen and along the edges
perpendicular to the fibers.
Cork dams were placed along
the edges parallel to the fibers.
The air bag was then
pressurized to the desired level and the press was inserted
into the microwave oven.
Next to the press,
a small cup of
water was placed in the oven to prevent possible buildup of
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72
the electromagnetic
field.
After the timer was set,
During cure,
the composite shrinks in height;
calibration tests showed that
changes
the oven was turned on.
in the force
however,
there were no appreciable
(pressure) applied to the composite,
due to the motion of the middle plate caused by the
shrinkage.
After the desired cure time was reached,
removed from the oven,
the press was
the pressure was released,
and the
composite-bleeder assembly was cooled to room temperature.
The weights of the composite and the bleeder were then
measured.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
SECTION V
EXPERIMENTAL VALIDATION OF THE MODELS
Experiments were performed
to generate data which can
be used to evaluate the accuracies of the electromagnetic,
thermochemical,
and resin
flow models.
Fiberite S2/9134B
glass epoxy and Hercules AS/3501-6 graphite epoxy prepregs
were used
in the tests.
The material properties used in the
numerical calculations are listed in Appendix G.
5.1
Electromagnetic Model
The reflectances of uncured glass epoxy and graphite
epoxy composites were measured using a' waveguide.
M e a s urements were performed with composites consisting of
different
and
number of plies having different ply orientations,
for different values of the polarization angle,
ranging from 0° to 90°.
In addition,
functions of thicknesses
(number of plies)
6,
the transmittances as
were measured for
graphite epoxy composites.
The data are given
the
in Figures
17-21.
In these figures,
results of the electromagnetic model are also included
and are represented by solid lines.
In the calculations,
the values of the complex dielectric constants e* specified
for
free space and for the materials
(Appendix G) had to be
73
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74
m odified to account for the fact that the waves
waveguide
travel
in a confined space.
dielectric constant
in the
The complex
in the waveguide eW g is related to the
complex dielectric constant
in free space e* by the
expression
*
*
=
6
(e*g = 0.28),
(120’
in free space
(Xg = 12.24 cm for
and W is the width of the waveguide
= 7.214 cm for S band waveguide
were calculated
2o \
" f e i
where Xg is the wavelength
2.45 GHz microwave),
/
[21]).
(w
The values of e*„
wg
for air in front of and behind the material
for the Fiberite S2/9134B glass epoxy prepreg
in the direction parallel
r
(e„„ = 4.7 - j0.42) and
wg
J
*$•
perpendicular to the fibers (eWg = 4.5 - j0.40), and for the
Hercules AS/3501-6 graphite epoxy prepreg in the directions
parallel and perpendicular to the fibers
and' 32.3 - j53.3).
(e*g = 0.3 - j25000
The results of the model
in Figures
17 -
•
•
21 were obtained with these e,,„ values.
wg
As can be seen from the Figures
17 - 21,
the agreements
between the data and the results of the model are excellent.
This creates confidence
in the accuracy of the
electromagnetic model.
5,2
Thermochemical and Resin Flow Models
The thermochemical and resin
flow models had already
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75
0.2
S 2 / 9 I 3 4 B (UNCURED)
REFLECTANCE,
oc
O DATA
-M O D E L
8=90
0
Figure 17
2
4
NUMBER OF PLIES, M
Reflectances of Fiberite S2/9134B
Glass Epoxy Uncured Unidirectional Composite
as Functions of the Number of Plies.
Com­
parisons between the Data and Results Com­
puted by the Model.
Data were Generated in a
Waveguide with Incident Linearly Polarized
TEM Waves (Polarization Angle, o).
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0
1.0
8 =0
0.8
0.8
UJ
LlJ
H 06
8=90
o DATA
MODEL
0.2
O
Figure 18
A S /3 5 0 1 -6
(UNCURED)
[0 ]M
2
8
10
4
6
NUMBER OF PLIES, M
12
0.6
o
2
£
h-
0.4
cn
2
<
oc
0.2
14
Reflectances and Transmittances of Hercules AS/3501-6
Graphite Epoxy Uncured Unidirectional Composite as Functions
of the Number of Plies.
Comparisons between the Data and
Results Computed by the Model.
Data were Generated in a Wave­
guide with Incident Linearly Polarized TEM Waves (Polarization
A n g l e , ,<5) .
ar\
0.2
S 2 /9 1 3 4 B (UNCURED)
[0 /9 0 ]
REFLECTANCE,
8=90
DATA
MODEL
0.1
-
6
8
10
NUMBER OF PLIES, M
Figure 19
Reflectance of Fiberite S2/9134B Glass Epoxy
Uncured Cross-ply Composite as a Function of the
Number of Plies.
Comparison between the Data
and Results Computed by the Model.
Data were
Generated in a Waveguide with Incident Linearly
Polarized TEM Waves (Polarization Angle, 5).
R eproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
78
A S / 3 5 0 1 -6
(UNCURED)
0.8
[0]
REFLECTANCE
a:
0.6
0.4
O DATA
— MODEL
0.2
30
60
POLARIZATION ANGLE , 8 (degree)
Figure 20
90
Reflectance as a Function of Polarization
Angle 6 for Hercules AS/3501-6 Graphite Epoxy
Uncured Single Ply Composite. Comparison between
the Data and Results Computed by the Model.
Data were Generated in a Waveguide with Incident
Linearly Polarized TEM Waves.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A S /3 5 0 1 -6
(UNCURED)
8=90°
o 0.6
[ 0 /0 ]
<-> 0.4
o Data
— Model
*
0.2
0
VO
30
60
ORIENTATION OF SECOND P L Y ,
Figure 21
90
6 { deg re e )
The Change in Reflectance with the Orientation of the Second Ply,
o, for a Two-ply Hercules AS/3501-6 Graphite Epoxy Uncured Composite.
Comparison between the Data and Resulus Computed by the Model.
Data were Generated in a Waveguide with Incident Linearly Polarized
TEM Waves (Polarization Angle, 6).
80
been tested by Loos and Springer
were performed here
[15].
Additional tests
to evaluate these models when coupled
with the electromagnetic model and when applied to microwave
curing.
The tests were performed in a microwave oven.
unidirectional composites,
Using
temperature distributions and
resin flows were measured parallel and perpendicular to the
tool plate.
In the experiments with glass epoxy composites,
pressure was not applied and the resin flow was not
measured.
In the tests with graphite epoxy composites,
pressure was applied at the beginning of the cure and was
kept constant at 446 k P a ( a b s )(64.7 psia) during the cure.
The pressure
pressure
in the bleeder was m a intained at ambient
(101
kPa or
14.7 psia).
The temperatures as functions of time measured at two
different
locations
inside 32 ply composites are shown
in
Figure 22.
The results of the resin flow measurements are
presented
in Figure
on the abscissa.
23.
In this figure,
time
t
is plotted
The ordinates represent either the total
mass loss of the composite or the mass losses due to resin
flow in the directions normal and parallel to the tool plate
in time
t
.
The mass losses shown
in Figure 23 represent
the
mass loss with respect to the initial mass of the composite
( 121 )
initial
noass
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
81
150
AS/3501-6
[0]32
x/d =0.5
x/d=0.875
50
105
TIME (min)
150
S2/9I34B
32
o
x/d = 0.5
Ui
(T 100
D
x/d =0.875
£
tr
UJ
a.
2
tu 5 0
I-
o Data
230
— Model
TIME (min)
Figure 22
Temperature as Function- of Time during Microwave
Curing of 32-ply Fiberite S2/9134B Glass Epoxy and
Hercules AS/3501-6 Graphite Epoxy Composites.
Compar­
isons between the Data and Models.
Cure Assembly Shown
in Figures 15 and 24. Power Inputs to the Composites
were as Indicated.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
82
10
0
Q.
A S /3 5 0 1 -6
£
32
20
o
—
DATA
MODEL
105
TIME (min)
Figure 23
Mass Losses Normal (top) and Parallel (center) to the
Tool Plate, and the Total Mass Loss (bottom) as Functions
of Time during Microwave Curing of 3 2-ply Hercules
AS/3501-6 Unidirectional Composites.
Comparisons be­
tween the Data and Results Computed by the Models.
Cure
Assembly is Shown in Figures 15 and 24.
The Power
Input W i n , Cure Pressure, P Q , and the' Bleeder Pressure,
Pjj, are as Indicated.
The initial Resin Content was 42%.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
83
The temperature
in the composite and the resin
flow were
calculated by the models.
The microwave power transmitted into the material was
less than the power output of the oven and was unknown.
The
power transmitted into the composite was determined by
matching the results of the model to the first three
temperature data points.
This power level was then used
in
all subsequent calculations.
In calculating the temperature
a)
inside the composite,
the outside surfaces of the top and bottom teflon
plates were taken to be at ambient
temperature
(Figure 24),
b)
the isotropic microwaves were taken to impinge
equally on both sides of the cure assembly,
c)
and
microwave energy absorption by all the layers
surrounding the composite was neglected.
In addition,
for glass epoxy composites,
energy release by
chemical reactions was assumed to be negligible.
The results of the models are represented by solid
lines in Figures 22-23.
. The calculated and measured
temperature distributions and resin flows agree well.
These
agreements tend to confirm the validities of the combined
electromagnetic-thermochemical-resin
flow models and to
demonstrate the usefulness of these models
in simulating
microwave curing of composites.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
AMBIENT TEMPERATURE
*
PRESSURE, P0
I
W
I
I
I
3 2 mm
GLASS REINFORCED
TEFLON PLATE
BLEEDER
COMPOSITE
GLASS
REINFORCED
TEFLON PLATE
2 5 mm
AMBIENT
Figure 24
TEMPERATURE
Components of the Cure Assembly Used in Modelling the Temperature
Distribution and the Resin Flow during Microwave Cure.
Complete
Cure Assembly is Shown in Figure 15.
00
S E C T IO N
VI
NUMERICAL RESULTS
The models were used to generate
illustrate the major
composite material
results which
features of electromagnetic wave-
interactions and of microwave
fiber-reinforced organic matrix composites.
curing of
Calculations
were performed for Fiberite S2/9134B glass epoxy and
Hercules AS/3501-6 graphite epoxy composites.
The material
properties listed in Appendix G were used in the
calculations.
All the calculations were performed
for
electromagnetic waves having a frequency of 2.45 GHz.
6. 1 Electromagnetic Wave-Composite Material
When an electromagnetic wave
material,
a fraction of the wave
absorbed by the material,
through the material.
Interactions
impinges on a composite
is reflected,
and the remainder
The reflectance,
a fraction
is
is transmitted
transmittance,
and
amount of absorbed energy depend on the complex dielectric
constants of the material,
orientations,
the number of plies,
the ply
and the characteristics of incident
electromagnetic wave.
The reflectance,
the transmittance,
and the amount of absorbed energy must be calculated by the
model
(Sections 2 and 3).
Results,
obtained for sample
85
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86
problems,
are presented subsequently.
First,
reflectance of a single ply is examined.
however,
The results
the
for a
single ply aid in understanding the phenomena which occur
when electromagnetic waves
impinge on multilayered
composites.
The reflectance at the front face of a single ply
composite exposed to a linearly polarized T EM wave
is
(Appendix H)
(122)
As indicated by the above equation,
the reflectance R g
depends on the complex dielectric constants parallel and
perpendicular to the fibers,
e
$
Cr
$
and e , and on the
M
polarization angle 5.
When the complex dielectric constants
nearly the same,
$
$
£p and Eg are
the reflectance becomes insensitive to the
polarization angle 6.
This
is the case,
for example,
Fiberite S2/9134B glass epoxy composites(Figure
for
25).
When the complex dielectric constants differ
considerably
the fibers,
in the directions parallel and perpendicular
the reflectance becomes a strong function of
the polarization angle.
This
of
is the case for the Hercules
AS/3501-6 graphite epoxy composite,
one hand,
to
considered here.
for polarization angle 6 = 90°,
low and a large fraction of the wave
On the
the reflectance
is transmitted across
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is
67
. 1.0
1.0
S 2 /9 1 3 4 B
cl:
2.45 GHz
Ll I
u 0.6
0.6
[0 ]
2
2
y
0.8
(CURED)
0.4
0.4
ul
LU
K 0.2
-
d
a .3
i—
i■
0.2
0
0.8
0.8-
cr
0.6
0 0.6
2
<
0.4
0.4
01 0.2
0.2
uj
_i
u.
UJ
30
60
TRANSMITTANCE, Tr
0.8
TRANSMITTANCE . Tr
U
90
POLARIZATION ANGLE, 8 (degree)
Figure 25
The Variation in Reflectance and Transmittance
with Polarization Angle, 6, for Single Plies of
Fiberite S2/9134B Glass Epoxy and Hercules AS/3501-6
Graphite Epoxy Composites Exposed to Linearly
Polarized TEM Waves.
Results of the Model.
Material
Properties Listed in Appendix G.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
88
the interface
(Figure 25);
on the other hand,
the reflectance is nearly unity,
wave is reflected.
Thus,
for 6 = 0°,
and practically the entire
significant amounts of
electromagnetic energy can be transmitted into
unidirectional graphite epoxy composites only when
polarization angle
8
electromagnetic wave
is 90°.
Most, or all,
is reflected
the
of the
from graphite epoxy
composites when the polarization angle is different than 90°
for any given ply.
Therefore,
most of the electromagnetic
wave is reflected when the composite is composed of
multidirectional plies,
as illustrated in Figure
26 and
28.
The reflectance and transmittance of multilayer glass
epoxy laminates are shown in Figure 27.
Neither the ply
orientation nor the polarization angle significantly affects
either the reflectance or transmittance.
above,
the reason for this
As discussed
is that the values of the complex
dielectric constants parallel and perpendicular
fibers are nearly the same.
electromagnetic waves,
Thus,
to the
from the point of view of
the material behaves
in a q u a s i ­
isotropic manner.
As shown
in Figure
27,
for glass epoxy composites,
the reflectance' increases and the transmittance decreases
with thickness
(number of plies).
These
trends are valid
for laminates consisting of less than 80 plies.
laminates consisting of more than 80 plies,
reversal
in these trends;
transmittance
increases
there
For
is a
the reflectance decreases and the
slightly with the thickness.
This
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
89
0.8
ISOTROPIC
REFLECTANCE
cc
0.6
8 =9 0
0.4
A S /3 5 0 1 -6
(CURED)
2 .4 5 G Hz
0.2
[ 0 /0 ]
30
60
ORIENTATION OF SECOND PLY,
0
Figure 26
9
90
(degree)
Reflectances of a Two-ply Hercules AS/3501-6
Graphite Epoxy Composite Exposed to a Linearly
Polarized TEM Wave (Polarization Angle, 6 = 90°)
and to an Isotropic Wave.
Results of the Model.
Material Properties Listed in Appendix G.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0
_S2/9!34B
(CURED)
2.45 GHz
CO1
o
0 °< S < 9 0 °
AND ISOTROPIC
UJ~0.5
(j
2
<
P-
\-
2 1.0
cn
2
<
Lu
0.5
{£ 0.5
[ 0 /9 0 ] AND
[ 0 / ± 4 5 / 90)
0
Figure 27
50
100
NUMBER OF PLIES, M
0
50
100
NUMBER OF PLIES, M
Reflectances and Transmittances of Unidirectional, Cross-ply and
Quasi-isotropic Fiberite S2/9134B Glass Epoxy Composites Exposed
to Linearly Polarized TEM Waves (Polarization Angle 0° <_ 6 <_ 90°)
and to Isotropic Waves.
Results of the Model.
Material Properties
Listed in Appendix G.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0 ^
I
0.8—
^
0.6 _
/ \ y
cr
f
ixJ 0.4 o
~
/
0.8
><00
4 5 ° AND
ISOTROPIC
/
?
0.2
o
8=90°
r1 0 !
i_
UJ
uj 1.0
cr
0.8
I
i
A S /3501-6
(CURED)
2 .4 5 GHz
[0 ]
—
. 1
1
0.4
AND
l
[o /9 Q ]
4 5 ° AND ISOTROPIC
t 0.2
co
k
0 °£ 8 < 90 °
AND ISOTROPIC
I----- 1
0.6
8 = 90
------------------------------
—
Figure 28
£
0.2
ID
0 °< 8 < 9 0 °
AND ISOTROPIC
0.1
[ 0 /± 4 5 / 9 0 ]
I
I
20
40
NUMBER OF PLIES, M
I
60
0
40
NUMBER OF P LIES , M
20
60
Reflectances and Transmittances of Hercules Unidirectional, Cross—ply,
and Quasi-isotropic AS/3501-6 Graphite Epoxy Composites Exposed to
Linearly Polarized TEM Waves and to Isotropic Waves.
Results of the
Model.
Material Properties Listed in Appendix G.
92
reversal
in the trends
transmittance
reversal
in the reflectance and in the
increases slightly with the thickness.
in the trends
This
in the reflectance and in the
transmittance are caused by the interactions between the
reflected and transmitted waves inside the material.
For multilayered graphite epoxy composites,
reflectance
is very high
(R
=
1), except
the
for unidirectional
composites exposed to linearly polarized TEM waves having
polarization angle,
laminates,
6=90°
(Figure 28).
Even for such
the reflectance reaches 0.7 for laminates
consisting of more than 32 plies.
graphite epoxy composite
both reflectance
The transmittance of the
is low because,
for this material,
(Figure 28) and absorption
(Figure 29) are
high.
The absorbed energies as function of position across
16, 32, and 64 ply unidirectional uncured glass epoxy and
graphite epoxy composites are shown in Figure
interest
It is of
to note that the amount of energy absorbed per unit
area per unit time may
increase or may decrease across the
thickness from the front towards the back.
energy
29.
The absorbed
increases across the laminate when absorption across
the laminate
is low,
so that a significant
wave reaches the back surface.
A portion of this wave
reflected back into the material.
contributes
fraction of the
is
This reflected wave
to the amount of absorbed energy,
causing the
higher absorption rate near the back surface.
The absorbed energy decreases across the laminate when
absorption
is so high that most of the wave
is absorbed
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[0 ]
S2/9I34 B
(UNCURED)
0.01
o
tr
LxJ
32
Ld
AS/3501-6
(UNCURED)
2.45 GHz
8 = 90°
64
S 0.02
CD
c
r
o
if)
WAVE
GO
<
m= 1 m = M
0.01
30 0
10
PLY NUMBER , m
Figure 29
20
30
40
50
60
The Absorbed Energies in the m-th Ply of Fiberite S2/9134B Glass
Epoxy and Hercules AS/3501-6 Graphite Epoxy Unidirectional Composites
Exposed to Linearly Polarized TEM Waves. Results of the Model.
Material
Properties Listed in Appendix G.
94
before it reaches the back surface.
reflection from the back
In this case,
surface is insignificant,
and the
absorption is mainly due to the wave traveling from the
front towards the back.
absorption
Under this condition,
the
decreases nearly exponentially with thickness.
The results discussed in the foregoing and illustrated
in Figures 25-29,
must be born
are of
importance
in microwave curing,
and
in mind when applying microwave to the curing
of composites.
Calculations were also performed to illustrate the
effects of coatings
(niade of homogeneous,
dielectric materials)
on the
sides of the composite.
covered by coatings are
isotropic
front, on the back,
or on both
The reflectances of composites
illustrated in Figure
30.
The
reflectances change significantly with the values of the
dielectric constant e^, and the thickness of the coating H.
With appropriate choices of
and H, the reflectance may be
reduced to practically zero or may be increased to a value
which is higher than the reflectance of the composite
without coating.
course,
An
increase in reflectance results,
of
in less energy being transmitted into and absorbed
by the material.
Reduced reflectance results
in more energy
being transmitted
into and absorbed by the material.
6.2 Microwave Curing - General Considerations
On the basis of the results presented
Section
in the previous
(Section 6.1), the following major conclusions can
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
.S 2 /9 I3 4 B
2.45 GHz
8 =90°
WAVE
WAVE
Hd H
:H ,
WAVE
.A S /3 5 0 1 -6
1.0 - 1°]64
UD
<J1
-€> 0
0.8
^
UJ
=4
o
z 0.6
o
y 0.4
Ll
UJ
0.2
i.
0
Figure 30
0.5
1.0
1.5 2.0 0
.5 2.0 0 0.5
COATING THICKNESS. H (cm)
1.0
1.5
2.0
Reflectances of fully cured 64 ply Fiberite S2/9134B Glass Epoxy (d=0.96 cm)
Hercules AS/3501-6 Graphite Epoxy (d=0.77 cm) Unidirectional Composites
Covered with a Homogeneous Isotropic Material on the Front (Left), Back
(Middle), and Both Front and Back (Right). Linearly Polarized TEM Wave
Incident on the Front Surface.
Results of the Model.
Properties of
the Composites Given in Appendix G.
The Dielectric Constant of the
The Dissipation Factor of the Cover is e " = 0.
Cover is
96
be made regarding microwave
a)
Glass
curing of composites:
fiber reinforced epoxy matrix composites may
be cured effectively by microwaves regardless of
ply orientation and polarization angle.
b)
Unidirectional graphite epoxy composites may also
be cured by microwaves.
c)
The curing efficiency
depends strongly
on the polarization angle 6.Most
effective curing
is achieved with 6 = 90°.
Graphite epoxy composites consisting of
multidirectional
laminate cannot be cured
effectively by microwaves.
d)
The energy absorbed
the material and
Correspondingly,
across the laminate depends on
on the thickness.
the induced temperature
distribution across the laminate may be non-uniform
during microwave cure,
and may either
increase or
decrease along the thickness.
e)
The amount of microwave
energy absorbed by the
material can be increased or decreased considerably
by coatings.
6.3 Microwave Curinq-Selection
When curing a composite,
selected in such a way that
of the Cure Cycle
the cure cycle should be
the following
requirements are
satisfied:
a) The temperature
is nearly uniform across the
material and does not exceed a prescribed limit at
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
97
any position during the cure.
b) The magnitude of cure pressure
is sufficiently high
so that all of the excess resin
is squeezed out from
every ply of the composite.
c)
The resin
is cured uniformly,
and the
degree
of cure
is above a specified limit throughout the composite
at the end of the cure.
d) The composite
is cured
in the shortest time.
The models and the corresponding computer code were
used to
generate results
which
illustrate
the effect ofthe
microwave power level on the curing process for a given
material.
The calculations were performed for Hercules
AS/3501-6 unidirectional 32 ply graphite epoxy composites
bounded by teflon tool plates on both sides
(Figure 24).
The material properties used in the calculations are given
in Appendix G.
Resin
flow only
in the direction normal
to
the tool plate was considered.
The purposes of the numerical calculations were to show
the main features of microwave curing,
and to demonstrate
the procedures that can be used to select the proper cure
cycle.
The procedure used to select an appropriate cure cycle
during autoclave cure was described by Loos and Springer
[14],
The same general steps may be used in selecting the
microwave cure cycle.
selection
indicated,
Therefore,
is not given in detail;
each step of cure cycle
only the major steps are
with emphasis on those features which are unique
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
98
to the microwave curing process.
The microwave impinging on the composite was taken to
be isotropic with the frequency of 2.45 GHz.
is used most commonly
This
frequency
in microwave heating.
The total amount of microwave power absorbed by the
composite depends on the geometries of the oven and the
composite,
and on the properties of the composite.
The
fraction of the total energy absorbed by the composite can
be determined experimentally,
as was discussed
In the subsequent calculations,
absorbed by the composite
assumed to be known.
in Section 5.
the total amount of power
(designated as
"power
Unless noted otherwise,
input")
was
the
calculations were performed with microwave power set at a
constant
level until the temperature inside the material
reached the prescribed m a x i m u m value T
*
then cycled on and off
in a manner
, .
max
The power was
r
such as to keep the
temperature constant across the laminate at the prescribed
Tmax v a ^ue
(Figure 31).
If the power were to be kept at a
constant level after Tm „„ was
max
reached,
the temperature
r
inside the composite w o u l d 'increase beyond the allowed
maximum.
In the present calculations,
at
177C.
The outer
the value of T
was set
surfaces of the lower teflon plate
plate) and the upper teflon plate were taken
ambient temperature of 22C
to be at the
(Figure 24).
The cure and the bleeder pressures used
calculations are shown
(tool
in Figure 31.
in the
The cure pressure P q
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
99
Tmax = 177 C
IS REACHED
TIME (m in)
a
Q_
CURE PRESSURE: P0
2C
UJ
ir
D
CO
BLEEDER PRESSURE
Pb= 16.7k Pa ( 5 in. H g)
ir
a.
TIME (min)
Figure 31
Illustration of the Cure Cycle Used in the
Parametric Study of Microwave Curing.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
was applied at the beginning of cure process
remained constant
(x=0) and
for the duration of the cure.
magnitude of cure pressure is indicated
showing the results.
The pressure
taken to be constant at
16.7 kPa.
The
in each figure,
in the bleeder
This value
was
is typical of
the bleeder pressure used in vacuum bagging procedures.
Calculations were also performed using the cure cycle
recommended by manufacturer
for Hercules AS/3501-6 graphite
epoxy prepregs.
is shown in Figure 32.
This cycle
Temperature Inside the Composite.
During the cure,
it is required that the temperature distribution across the
composite be reasonably uniform,
and that the temperature
does not exceed the prescribed maximum value at any point.
This condition can be satisfied by selecting the appropriate
power
input level and the appropriate power cycling after
Tmax *s reac^e<^*
As
input levels result
distributions.
result
illustrated in Figure 33, higher power
in less uniform temperature
However, higher power
input levels also
in a faster temperature rise and, hence,
cure time.
Thus,
the power
input level
as to ensure a reasonably uniform
in a shorter
should be chosen so
temperature distribution
and a reasonably short cure time.
It is interesting to note that the temperature
distribution and the cure time depend mainly on the
microwave energy
input level
input,
(Watt)
and are insensitive to the power
(Figure 34).
Thus,
nearly the same
temperature distributions are achieved by employing a lower
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
101
400
A S /3 5 0 1 -6
300
200
50
100
TEMPERATURE (F )
TEMPERATURE(C)
200
mmm
800
400
PRESSURE
600
(psi)
100
—
50
ABSOLUTE
ABSOLUTE
PRESSURE(kPa)
CURE PRESSURE
200
BLEEDER PRESSURE
0
■---------- 1---------- L
0
50
100
Figure
32
. I ..... 1.......... 1
150
200
150
TIME (min)
300
Manufacturer's Recommended Cure Cycle for
Hercules AS/3501-6 Prepreg [27].
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A S /3 5 0 1 -6 [ 0 ]
Tmax= <77C
200 ~Win=50W
P0 = 1479 kPa
Pb= 16.7 k Pa
"Win = <00 W
20
60
O
I—
UJ
150
cr
30
ID
<C
cr
20
UJ
CL
100
r = 1 min
T= 10 min
T= 2
0.5
Figure 33
1.0 0
0.5
POSITION, x /d
1.0 0
0.5
Temperature Distribution Across the Composite as a Function of
Time for Different Levels of Microwave Power Input.
Results
Obtained by the Models for the Cure Cycle Shown in Figure 31.
1.0
200
W
*t
♦♦♦
200l
\dTm m
-
1001
s
20
P0 = 1479 kPa
W
Pb = 16.7 kPa
400T
W
lo l*
44 s
20
2 6s
20
o
10
\-
10
10
T = 2 min
T = 2 min
r 150
tr
z>
LU
<i
oc
5
LlI
Q.
UJ
103
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A S /3 5 0 1 -6
Tmax= (77 C
100
T = 2 min
I
i
i
i
0
i
1 i
0.5
i
i
i
J
1.0 0
I L
J
I
0.5
I
J
L
1.0 0
I
I
L
J
0.5
I
I
L
1.0
POSITION , x /d
Figure 34
Temperature Distribution Across the Composite as a Function of
Time for Different Power Cycles.
Results Obtained by the Models.
104
power
input maintained at a constant level,
or higher power
input, cycled on and off.
Heat generated by the absorbed microwave energy is
transferred out of the material.
The amount of heat
transferred can be reduced considerably by placing thermal
insulators on both sides of the composite.
Thermal
insulation results in a more rapid temperature
(Figure 35).
increase
In turn, a rapid temperature rise results in a
reduced cure time.
Gel Po i n t .
Excess resin must be squeezed out of
every ply before the gel point of the resin
any point inside the composite.
is reached at
The computer code can be
used to generate the viscosity distribution
composite
(Figure 36).
inside the
From this information,
viscosity at any point inside the composite,
can be determined,
the maximum
at any time,
and a plot of maximum viscosity ym
lil_v
dA
versus time can be constructed as illustrated in Figure 37.
The gel point of the resin is assumed to occur when the
viscosity of the resin reaches a certain value.
Thus,
knowing the viscosity corresponding to the gel point,
by
the
time when gel occurs can be determined from the maximum
viscosity umax versus time curve, as shown in Figure 37.
For the present calculations,
gel was taken to occur when
the degree of cure a reaches 0.5 [13].
this degree of cure is about 7 Pa*s at
viscosity,
The viscosity at
177C
[13].
For this
Figure 37 indicates gel times of 21 and 13
minutes for 100 and 200 Watt power inputs,
respectively.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
200
Thermally
Insulated
20
o
H 150
L
u
l
tr
105
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
Not Insulated
P0 =1479 kPa
Pu =16.7 kPa
A S /3 5 0 1 -6
T"= 5 min
CL
§ 100
0
Figure 35
T = 1min
0.5
POSITION, x /d
1.0 0
0.5
POSITION, x /d
1.0
Temperature Distribution Across the Composite as a Function of Time for
a Composite Thermally Insulated (Left) and for a Composite without Thermal
Insulation (Right). Results Obtained by the Models for the Cure Cycle
Shown in Figure 31.
106
Gel times were calculated for different power
inputs,
the
results of which are given in Figure 38.
Resin Flow.
Excess resin must be squeezed out from
every ply of the composite before resin
This
resin flow results
in any ply gels.
in the compaction of plies
[14],
The number of compacted plies n g as a function of time is
shown
in Figure 39.
In this figure,
the number of compacted
plies which results from the autoclave cure cycle
recommended by the prepreg manufacturer
is also indicated.
As can be s e e n ,'microwave curing results
in much faster
compaction than autoclave curing with the manufacturer's
cure cycle.
Furthermore,
the manufacturer
the cure pressure recommended by
(Figure 32)
is insufficient to compact
every ply in a 32 ply composite.
kPa
(160 psia)
is required to compact all
before gel time is reached
Degree of Cure.
process,
the 32 plies
(Figure 39).
At the completion of the curing
and the degree of cure should exceed a prescribed
throughout the composite.
The degree of cure will
be uniform as long as the temperature distribution
the composite is uniform
Conversely,
inside
(Figures 33 and 40).
if the temperature distribution
composite becomes nonuniform,
expected to be nonuniform.
result
1100
the resin in the composite should be cured
uniformly,
value a
A minimum pressure of
inside the
the degree of cure can also be
Therefore,
cure cycles which
in uniform temperature distributions also result
composites that are cured uniformly.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
in
i 07
A S /3 5 0 1 - 6
20
V)
8.
io ° -
5.
-
*»
>
hcn
oo
T =10 min
to
>
P0 = 1479 kPa
Pu = 16.7 kPa
0 .2
0 .4
0 .6
0.8
POSITION, x/d
Figure 36
Viscosity Distribution at Different Times Inside
the Composite. Result Obtained by the Models
for the Cure Cycle Shown in Figure 31.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
106
A S /3 5 0 1 -6
J 32
s)
max
= 177 C
MAXIMUM
VISCOSITY, U ma)( (Pa
W- = 100 W
_________________ |_______
0
10
u_
20
TIME, T(m in)
Figure 37
The Maximum Viscosity
Inside the Composite as
a Function of Time. Gel is Assumed to Occur When
Viscosity Reaches 7 Pa*s. Results Obtained by the
Model for the Cure Cycle Shown in Figure 31 at Power
Inputs of 100 W and 200 W.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[O]-
c
E
®
20
o»
l±J
109
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A S /3 5 0 1 -6
30
_1
LU
CD
10
t
0
7max = 177 C
P0 = 1479 kPa
Pb = 16.7 kPa
1
100
200
POWER INPUT, Wjn (W)
Figure 38
Gel Time as a Function of Power Input.
Result of the
Models for the Cure Cycle Shown in Figure 31.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A S /3 5 0 1 -6
32
« 30
Tgei
Win = 100W (P0 = 1100 k Pa)
Tgel
_i
a_
Wjn=100W (P0 = 687kPa)
20
o
MANUFACTURER'S CURE CYCLE
(AUTOCLAVE CURE:P0 = 687 kPa =85psig)
(P0 = 687 kPa)
Tmax= 177 C
P K = 16.7 kPa
LlJ
00
0
50
100
49
150
TIME , T (min)
Figure 39
Number of Compacted Plies as a Function of Time for Different
Microwave Power Inputs and Different Cure Pressures.
The Results
Obtained by the Models for the Cure Cycle Shown in Figure 31.
The
Result Shown for the Autoclave Cure Cycle is from Reference [13].
111
0.8
AS/3501-6
P0 =1479 kPa
Pb =16.7 kPa
’ win =100 W
|
l i t
0.6
20
W
ac
o 0.4
15
LU
LU
CC
o
T=10 min
y 0-2
—
—
i
—
i
—
i
i
1
_L_
.
1
0.5
1
1
l-.O
POSITION, x/d
Figure 40
Degree of Cure Distribution Across the
Composite as a Function of Time.
Results
Obtained by the Models for the Cure Cycle
Shown in Figure 31.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
112
The computer code can be used to generate results such
as those shown
in Figure 40.
From these results,
of cure as a function of position
be determined.
the degree
inside the composite can
Once this information
is known,
it can be
determined if the composite was cured uniformly,
and if the
degree of cure exceeds the prescribed value of a
throughout
the composite.
Cure Time.
When curing a composite,
that the curing process be completed
of time.
it is desirable
in the shortest amount
The cure is considered complete when the degree of
•
cure reaches a specified value a
composite.
at every point
,
m
the
The time required to reach this value can be
established by first plotting the degree of cure as a
function of position and time, as shown in Figure 41.
this curve,
From
the lowest value of the degree of cure am -n in
the composite at each time is determined,
and a plot of the
lowest value of the degree of cure as a function of time is
constructed.
The cure
is considered complete when a
reaches the specified value a
shown in Figure 41,
process
.
From a plot such as that
the time required to complete the curing
can be determined.
For example,
for a 32 ply
unidirectional AS/3501-6 graphite epoxy composite,
is reached
in
in 21 minutes with a power
13 minutes with a power
input of
input of 200 Watt
After the gel time is reached,
a
= 0.9
100 Watt,
(Figure 42).
the composite usually is
removed from the oven, and is postcured without pressure
being applied.
and
It is interesting to compare the time
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1.0
Tr
= 4 7 min
a =0.9
c
E
a
0.8
T r = 52 min
LU
tr
z>
0.6
o
Win = 100W
Win = 2 00 W
Ll
O
LU
IU
tr
o
0.4
A S /3 5 0 1 - 6
IU
32
o
z>
2
0.2
Tmax = 177 C
P0 = 1479 kPa
PK= 16.7 kPa
2
0
20
40
60
TIME, T ( min)
Figure 41
Minimum Degree of Cure as a Function of Time for Two Different
Power Inputs. Results Obtained by the Models for the Cure Cycle
Shown in Figure 31.
114
150
AS/3501-6
[°]32
P0 = 6 87 k Pa ( 85 p s ig )
Pb = 16.7 kPa
Tmax= 177 C
c
100
i
a>
o»
M AN U FAC TU R ER /
RECOMMENDED
AUTOCLAVE
CURE CYCLE
UJ
2
h-
50
UJ
o
Win = 100 W
Win = 200 W
[7
A
Figure 42
The Time Required to Reach the Gel point
Using Microwave Curing with 100 W and 200 W
Power Inputs and Using the Autoclave Cure
Cycle Recommended by the Prepreg Manufacturer
The Results for Microwave Curing were Obtained
by the Models for the Cure Cycle Shown in
Figure 31.
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required to reach the gel point by microwave curing and by
autoclave curing,
using the cure cycle recommended by the
prepreg manufacturer.
A nearly
10-fold decrease
can be achieved by microwave curing.
This
in gel time
suggests that
considerable reductions in cure time are offered by
microwave curing.
Finally,
it
is
noted
that,
in addition to promoting
internal heat generation,
microwaves
structure
resin matrix.
of the polymer
considered in this
may
also
affect
the
This effect was not
investigation.
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S E C T IO N
V II
SUMMARY AND CONCLUSIONS
The following major tasks were completed during the
course of this investigation:
1) Models were developed to describe the interaction of
linearly polarized transverse electromagnetic waves
and isotropic electromagnetic waves with continuous
fiber reinforced organic matrix composites.
model provides the reflectance,
absorbed energy,
The
transmittance,
total
and absorbed energy distribution
inside the composite.
■2) Models were developed which simulate microwave
curing of fiber reinforced thermosetting resin
matrix composites.
power
The models relate the microwave
input and the cure pressure to the
thermochemical and physical processes occurring in
the composite during cure.
Specifically,
provide the following information
the models
for flat-plate
composites cured by a specified cure cycle:
a) the temperature T inside the composite as a
function of position and time?
b) the degree of cure of the resin a as a function
of position and time;
116
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117
c) the resin viscosity
-\i as
a function of position
and time;
d) the number of compacted plies n g as a function
of time;
e) the amount of resin in the bleeder as a
function of time; and
f) the thickness and the mass of the composite as
functions of time.
3) On the basis of the models,
a computer code was
developed which can be used to generate results.
4) Experiments were performed in a waveguide and in a
microwave oven with Fiberite S2/9134B glass epoxy
and Hercules AS/3501-6 graphite epoxy composites.
a) Using a waveguide,
the reflectances and
transmittances were measured for composites
having different ply orientations and different
thicknesses exposed to linearly polarized TEM
waves with different polarization angles.
b) Using a microwave oven,
temperature
distribution across glass epoxy and graphite
epoxy composites,
and resin
flow out of
graphite epoxy composites were measured during
microwave cure.
5) Calculations were performed with the computer code
for the conditions employed in the experiments.
calculated results-were compared with the
experimental data.
These comparisons showed that
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The
118
the models adequately describe the reflectance and
transmittance during exposure to linearly polarized
TEM waves, and the temperature distributions
the resin flow out of flat plate,
in and
unidirectional
composites during microwave cure.
6) A parametric study was performed to illustrate:
a) how the reflectance,
transmittance,
and
absorbed energy depend on the characteristics
of incident electromagnetic waves,
geometry
and the
(thickness and ply orientation of the
composite,
thickness of coating), and on the
material properties of the composite;
and
b) how the models and the associated computer code
can be used to determine the appropriate power
level during microwave curing which results
a composite that
in
is cured uniformly in the
shortest amount of time.
In this investigation,
results were generated only for
electromagnetic waves at 2.45 GHz.
the computer code are general,
entire
However,
the model and
and can be used over the
frequency range.
In the present form,
the computer code can be applied
to flat plate geometry only.
It could be readily extended
to composites made in different shapes.
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A P P E N D IC E S
119
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A P P E N D IX
A
THE REFLECTION COEFFICIENT AT THE M-TH INTERFACE
The reflection coefficient tensor
expression
(Eq.
r^-
is defined by the
10 in Section 2.2.1)
(Ec)r
"
C y H E j)
u.i>
where the notation
is same as in Section 2.2.1.
magnetic materials
$
(-p, =1),
the electric
For non-
field vectors
directions parallel and perpendicular to the fibers
directions,
E? =
hr
and e
(p and q
Figure 4) are related to the components of the
magnetic field vectors by the expressions
e
in the
[11]
HP
(A.2)
^
(A.3)
are the complex dielectric constants in the p and
q directions.
Coordinate transformation gives the
relationship between the vector components
directions and in the
/Ep\
_
/
1 and 2 directions
in the p and q
[29]
(Figure 4)
/ E,
“ * & ) \ E ZJ
<A' 4)
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
121
h
U
p
H|/
Equations
e
s i " 6 \ / H ^
\-5l'n0 60S $ / \
(A.2) - (A.5) yield
/E|W^6 - S i n 0 \ / > / ^
Ez* ISiV»d
0
\/C0S$
SM0\/Hi
(A.6)
£°*d/\ Hz
By defining the complex refraction coefficient tensor
_
\Si'r\d
The electric
0 \/OoSd sind\
0 yejjy(-£/>)0 coS0/
field vector may be expressed as
E-l = Nij Hj
For a nonmagnetic
electric
(A.7)
(u =1) homogeneous
(A.8)
isotropic material,
field vector is related to magnetic
the expression
the
field vector by
[11]
= N Ht-
(a.9)
where
(\j -
y ^
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(A.10)
122
At the interface of two different homogeneous
materials
(material m-1 and material m),
isotropic
the reflection
coefficient is [9]
-I
r
By analogy
=
( Nm-I
+-
s.
Nm) ( Nlm-i" N m )
(see Eqs. A . 9 - A . 11), at the interface
separating two anisotropic materials,
the reflection
coefficient tensor is
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(A. 11)
123
A P P E N D IX
B
THE ATTENUATION TENSOR FOR THE M-TH PLY
In the following,
tensor A^j
is derived.
an expression for the attenuation
For the m-th ply of the composite,
the attenuation tensor A^j
is defined as
(see Eq.
16,
Section 2.2.1)
Bi
(B. 1 )
= ( A ijK f p
The notations are the same as used in the text
(Section
2 .2 .1 ).
Inside a homogeneous
isotropic material,
the attenuated
electric
field vector E^ can be expressed in terms of the
electric
field vector P^ entering the material
E. - Pc exp (
-ydm)
where dm is the thickness of the ply,
main text
composite,
(Eq. 4).
[28]
By analogy,
(B.2)
y was defined in the
for the m-th ply of the
the components of attenuated electric
vector parallel and perpendicular
to the fibers
field
(Figure 4)
can be expressed as
(B .3 )
E?
=
P|. e * p
Jm)
(B .4 )
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124
Pp and Pg are
related to P^,
the coordinate transformation
/ Pr \ _
\pj
Similarly,
Pj components of this vector by
[29]
(Figure 4)
I^
s,'nA / P-'l
l-sfnacoseA
fj
<B-5)
Ep and Eg can be expressed in terms of the
components of attenuated electric
field vector
in the
1 and
2 directions
BF\ / c o s e
I £
s.V>0\/E.\
(B6)
/
I
We
introduce now the following two parameters
AP s
=
Equations
(b.7)
C*p(-y<j.d m )
(B-8)
(B.3) - (B.8)
E|\
to '
give
CoS0 Si r)d\j Pi
,S\r\b tosfy/l o A^A'Sfnfi
\
"Sin6\/Ap
A comparison of Eq.
expression
e>p(-ypdn)
(B.1)
0 \j
with Eq.
(B. 9)
(B.9) yield the desired
for the attenuation coefficient
tensor
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125
CoSQ -Sind
A l j
=
Sfnft
LoSft
Af> o
\ /
*'n8
(B. 10)
0
Af-'\-Sin()
9
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126
A P P E N D IX
C
RATE OF ENERGY ABSORPTTON PER UNIT VOLUME BY THE M-TH PLY
The rate of energy absorption per unit volume by a
homogeneous
where E^
and
a
isotropic material
is the electric
is [11]
field vector
is the electrical conductivity
o'
- (
f is the frequency,
material.
inside the material,
[11]
z'irf to) £
(c.2 )
and e" is the dissipation factor of the
Eg is a constant
£o =
Analogously,
* I0‘Z F a r a d / m
(c.3 )
we express the rate of energy absorption per
unit volume by the m-th ply as
(C .4 )
where
ap
and
are the electrical conductivities parallel
and perpendicular to the
fibers,
respectively.
are the components of resultant electric
E^ and E^
field vector
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in the
127
p and q directions.
(E®) and (E®) can be related to the
tr
resultant electric
H
field vector components in the
directions by the coordinate transformation
-
E,
0
“Ej
The
+•
(Figure 4)
Si/10
(C • 5)
l~2.
1 and 2 components of the electric
were found in Section 2.2.4
[29]
1 and 2
(c.6)
field vector
(Eq. 51)
E,s= -j ( P,+ +■Aij Rj) + j-(
E* =
(E®)
\ ( ?l + Azi p/ ^ + J (
) (c-7>
+ Az3
^
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(c-
8)
128
A P P E N D IX
D
NORMAL INCIDENT ENERGY FOR AN ISOTROPIC WAVE
[30]
When an isotropic electromagnetic wave impinges on a
surface,
the total
£
incident energy flux ~'1 is
‘
-
<d . i >
where dft is the solid angle shown in Figure D.1, and
is the intensity,
i.e.,
angle dfi.
isotropic wave,
For an
distributed,
and
(<S-1)^
the incident energy flux in solid
the energy
is uniformly
independent of the direction of
incidence.
A ccord ingly,
( C L)
-
(D. 2)
1 0 'a
Z r
The energy flux incident normal to the surface
in solid
angle dft is
d£i
J
The total energy
=
( £ L) CoS0 dJ2
Si
(D.3)
incident normal to the surface is
r
Et
= JI Z ir
3
cose
dfi
(D-«)
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129
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
cos0
Figure D.l
The Geometry Used in Calculating the Normal
Incident Energy Flux for Isotropic Electro­
magnetic Waves.
130
The solid angle d& is
(D.5)
where angles
Eqs.
(D.4)
0 and <{> are shown
and
in Figure D.1.
By combining
(D.5), we obtain
2* %
0
Integration of Eq.
r^L
S'
J0 ZTf
CoS $
st"n &
( D - 6)
(D.6) yields
<f‘
(D.7)
z
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131
A P P E N D IX
E
RELATIONSHIP BETWEEN PLATEN FORCE AND AIR BAG PRESSURE
The relationship between the
force generated by the
press platens and the indicated air bag pressure was
determined,
using a calibrated load cell,
.
as described in
Section 4.2.
The load cell was calibrated by connecting a Baldwin
(Model SR-4) 8.9 kN
Ellis Associates
(2000 l b f ) strain gage load cell to an
(Model B A M - 1) bridge amplifier and meter.
The output of
the Ellis bridge amplifier was
Fluke digital
voltmeter.
within ± 0.25% over
recorded on a
Since the load cell
its operating
range
is linear
(0-2000
to
lbf), only
two points are necessary to construct a calibration curve
[31].
The
first calibration point was obtained by balancing
the amplifier
and bridge circuits of the Ellis bridge
amplifier at the no load
cell).
point
(no force applied to the
These circuits were adjusted so that
load
the output
voltage of the Ellis bridge amplifier was zero at the no
load point.
The second point o'" une r-’libration curve was
obtained by loading the cell
universal
testing machine.
to the cell,
„o 2000 lbf
on
^n Xnjtron
With a force of 2000 lbf applied
the gain of the Ellis bridge amplifier circuit
was adjusted to give a measured output voltage of 2.0 V.
Thus,
when the applied force on the load cell
ranged from 0
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132
lbf to 2000 lbf,
amplifier
the output voltage of the Ellis
ranged from 0 V to 2.0 V.
Therefore,
the load cell could be determined directly
bridge
the force on
from the output
voltage measurements of the Ellis bridge amplifier.
The relationship between the platen force and the air
bag pressure was determined for three different
distance between the top and middle plates
ha ).
The results are given
microwave curing experiment,
mm.
After the cure,
height
(air bag height,
in Figure E.l.
as the plies compacted,
The results
In each
the initial height was h_ = 21
increased by about 2 mm, at the most
composites.
values of
in Figure E.1
the air bag
, for 32 ply
suggest
that a 2 mm
change
in air bag height would not cause a significant
change
in the platen
force.
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133
AIR BAG PRESSURE ( psig)
0
10
20
30
40
1000
(lbf)
6
FORCE
-1500
■nrr
4
500
2
0
0
100
200
PLATEN
PLATEN
FORCE (kN)
8
300
AIR BAG PRESSURE (kPag)
Figure E.l
Force Generated by the Air Bag on the Press
Platens as a Function of Air Bag Pressure for
Different Air Bag Heights (ha ) .
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
134
A P P E N D IX
F
ELECTRIC FIELD STRENGTH DISTRIBUTION
INSIDE THE MICROWAVE OVEN
The distribution of the electric
field strength inside
the microwave oven was determined by placing nine glass cups
in the oven,
each cup being filled with 50 ml of water.
temperature of the water
in each cup were measured after 60
seconds of heating at full power
(700 Watt).
uniformities
in the temperatures are
uniformities
in the electric
Non­
indicative of the non­
field strength.
temperature rise in each cup after 60 seconds
The
is shown
below.
K:--------------
40.6 c m --------------
IT
41.3 cm
The
28 ,0C
19. 1C
22. 5C
19.6C
18.6C
16.4C
18.9C
20.3C
17.7C
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135
A P P E N D IX
G
PROPERTIES OF THE COMPOSITE AND SURROUNDING MATERIALS
This appendix contains the properties of Fiberite
S2/9134B glass epoxy and Hercules AS/3501-6 graphite epoxy
cured and uncured prepregs.
The properties of Mochburg
CW1850 bleeder cloth and glass reinforced teflon plate are
also given.
The properties are tabulated
in Tables G.1
G.4.
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to
136
Table G.1
Properties of Fiberite S2/9134B Glass Epoxy Prepreg
Dielectric constant
(uncured and c u r e d ) 1
parallel to the fibers
5.46
perpendicular to the fibers
5.21
Dissipation
factor
(uncured and c u r e d ) 1
parallel to the fibers
0.42
perpendicular to the fibers
0.40
Initial (uncured)
fraction
prepreg resin mass
Initial (uncured)
prepreg
thickness of the
Resin density
0.30
1 .50x10-4 m
3
1 .2 6 x 103 k g / m 3
Specific heat of the resin
3
1.26 kJ/(kg*K)
3
Thermal conductivity of the resin
4
Fiber density
Specific heat of the fiber
0.17 W/(m«K)
2.7 x 103 k g / m 3
4
0.83 kJ/(kg*K)
Thermal conductivity of the fiber4
0.76 W / ( m * K )
1. Measured using uncured prepreg.
The dielectric
constants and dissipation factors for the cured
and uncured resins are nearly the same [2],
2. Specified by the manufacturer
prepreg.
3. From Reference
[14].
4. From Reference
[32],
for each shipment of
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137
Table G.2
Properties of Hercules AS/3501-1
Graphite Epoxy P r e p r e g 1
2
Dielectric constant
(uncured)
1
parallel to the fibers
perpendicular to the fibers
2
Dissipation factor (uncured)
parallel to the fibers
33.0
~
perpendicular to the fibers
Dielectric constant
(cured)
25,000
53.3
2
parallel to the fibers
1
perpendicular to the fibers
2
Dissipation factor (cured)
parallel to the fibers
perpendicular to the fibers
Initial (uncured)
fraction
14.5
~
25,000
75.8
prepreg resin mass
0.42
Initial uncured thickness of the prepreg
1.651x10 ^ m
Thickness of one compacted ply
1.19 4 x 10-4 m
Apparent permeability of the prepreg
normal to the plane of the composite
5.8x10-16 m 2
Flow coefficient
to the fibers
of the prepreg parallel
170
Resin density
1. 26x 103 k g / m 3
Specific heat of the resin
1.26 kJ/(kg«K)
Thermal conductivity of the resin
Heat of reaction of the resin
Fiber density
Specific heat of
0.167 W/(m*K)
473.6 kJ/kg
1 .79x 103 k g / m 3
the fiber
i .712 kJ/(kg *K)
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138
Table G.2 (cont.)
Thermal conductivity of the fiber
26.0 W/(m«K)
Relationship between the cure rate,
temperature, and degree of cure
see Table G.3
Relationship between viscosity,
temperature, and degree of cure
see Table G.3
1. Unless otherwise stated,
[14].
all values are from Reference
2. Measured using a waveguide.
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Table G.3
Degree of Cure and Viscosity of Hercules
3501-6 Epoxy Resin [13]
The relationship between the cure rate,
temperature,
and
degree of cure is
aX
(K
+ K 2*)(I-c<)(B-°<)
= K3(i-tf)
<0.
cX><3.3
where
K 1 = A lex p ( - A E 1/RT)
K 2 = A 2exp(-AE2/RT)
K 3 = A 3exp(-AE3/RT)
and
A 1 = 2. 101x10^ min
1
A 2 = -2.014x10^ min
1
A 1 = 1.9 6 0 x 10 5 m i n -1
A E 1 = 8.07x104 J/mol
A E 2 = 7.78x104 J/mol
A E 3 = 5.6 6 x 104 J/mol
R = 8.31434 J/mol
B = 0.47
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140
Table G.3 (cont.)
The relationship between
the viscosity,
temperature,
and
degree of cure is
ju -
exp ( U / R T
+ Ko<)
where
1^*
00 = 7 .93x
10“ 14 Pa *s
U = 9.08 x 104 J/mol
K = 14.1
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141
Table G.4
Properties of Mochburg CW1850 Bleeder Cloth
and Glass Reinforced Teflon
Mochburg Bleeder Cloth
Apparent p e r m eability1
5 . 6 x 1 0 " 11 m 2
Porosity1
Density
0.57
2
0. 3 8 x 10 3 kg/m3
Specific heat
2
0.83 kJ/kg*K
2
Thermal conductivity
5.0 6 x 10 ~ 2 W/m-K
Glass Reinforced Teflon
Dielectric constant
Dissipation
Density
factor
3
2.35
3
0.0035
4
2. 2 x 10 3 kg/ m 3
4
Specific heat
1.01
kJ/(kg*m)
5
Thermal conductivity
1. From Reference
0.40 W/(m*K)
[14].
2. Estimated using the rule of mixture with the properties
of polypropylene [33], and the porosity value given
above.
3. From Reference
[23]
for the frequency of 3.0 GHz.
4. Estimated using the rule of mixture with the properties
of glass [32] and teflon [33], and the glass mass
fraction of 20 %.
5. From Reference
[34].
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142
A P P E N D IX
H
REFLECTANCE FOR THE FRONT FACE OF A SINGLE PLY COMPOSITE
The reflectance R is defined by the expression
(Eq. 31
in Section 2.2.2)
KRj,0
R =
The notation
of the
and
is
(H. 1 )
I(E D
the same as in Section 2.2.
incident electric
The components
field vectors are given
by Eqs.(1)
(2)
(E*)0 - a0cosS expijwr-y*3)
=
a t SinS e/p (jwt ~y*i)
The components of the incident electric
(«.2)
(H.3)
field vectors
parallel and perpendicular to the fibers are given by the
coordinate transformations
<EfV
[29]
CoS0
-S inQ
Si' nd
C.°sd
+\ ->
(E R0
t
(H.4)
For a single ply of unidirectional composite with the fibers
aligned in the
1 direction
(0 = 0°)
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143
(Ep) = ( E ^ o = a D ccs£ c * p ( j u > r - y t j )
ih.si
(E?) =(El) = a0 sm$ c*p (ju>r'I*?,)
(H.
Similarly,
the following equations hold for the components
of the reflected electric
(p ;)„
e
=
field vectors
tensor
(H.7)
(H. 8)
( P i)
1 (free space value)
coefficient
(6 = 0°)
ip;\
-
(p,i ■
For
6)
(Eq.
and
for
0
8 = 0 , the reflection
A . 12) becomes
0
r.. ' ‘J
(K.9)
/€%
0
1 4
Also,
from Appendix A
(Eq. A.1),
+
we have
( P J, - ( Oj ) ( Ej )
By combining Eqs.
< pr>; =
(H.5)
+■
to (H.10),
(H.10)
we obtain
(epI irr7a
(*s<k (
r T'i^
ief + c
(H.11)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
144
(P ^
From Eq.
(H.12),
^>
(H.1),
the
7
^
-y^'
7
together with Eqs.
following expression
(H.5).
(H.6),
.1 2 )
(H.11)^ and
is obtained
.13)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
REFERENCES
14 5
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146
REFERENCES
1.
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147
and Sons
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Arpaci, V. S.,
(1966) .
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R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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