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Hygrothermal effects on complex moduli of composite laminates

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Title:
Hygrothermal effects on complex moduli of composite laminates
Creator:
Bouadi, Hacene, 1954-
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English
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xvii, 137 leaves : ill. ; 28 cm.

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Subjects / Keywords:
Composite materials ( jstor )
Damping ( jstor )
Diffusion coefficient ( jstor )
Experimental data ( jstor )
Laminates ( jstor )
Moduli of elasticity ( jstor )
Moisture content ( jstor )
Poisson ratio ( jstor )
Stiffness ( jstor )
Temperature ratio ( jstor )
Composite materials -- Testing ( lcsh )
Dissertations, Academic -- Engineering Sciences -- UF
Engineering Sciences thesis Ph.D
Hygrothermoelasticity ( lcsh )
Laminated materials -- Testing ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1988.
Bibliography:
Includes bibliographical references.
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Hacene Bouadi.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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HYGROTHERMAL EFFECTS
ON COMPLEX MODULI OF COMPOSITE LAMINATES
BY
HACENE BOUADI
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA


ACKNOWLEDGEMENTS
I would like to express my gratitude to Professor
Chang-T. Sun, the chairman of my doctoral committee, for
his guidance, time, and encouragement during this research.
Many thanks are owed to Professor Lawrence E. Malvern
and Professor Martin A. Eisenberg for their teaching and
financial support.
I also want to thank the other members of my doctoral
committee, Dr. Charles E. Taylor and Dr. Robert E.
Reed-Hill for their helpful commen ts, critique, and advic e.
In addition, I gratefully recognize the assistance of
Dr. David A. Jenkins for teaching me how to operate the
material testing equipment that was indispensable for my
work.
Finally, I appreciate Ms. Patricia Campbells help in
typing this manuscript.


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS i i
LIST OF TABLES vi
LIST OF FIGURES vii
NOMENCLATURE xiii
ABSTRACT xvi
CHAPTERS
1 INTRODUCTION 1
1.1 General Introduction 1
1.2 Moisture Diffusion 2
1.3 Hygrothermal Effects 2
1.4 Scope and Methodology 3
1.5 Dissertation Lay-Out 4
2 DIFFUSION OF MOISTURE 6
2.1 Introduction 6
2.2 Fickian Diffusion 6
2.3 Fickian Absorption in a Plate 8
2.3.1 Infinite Plate 8
2.3.2 Semi-Inf inite Plate 10
2.3.3 Experimental Measurement of
Moisture Content 11
2.3.4 Approximate Solutions of
Moisture Content 12
2.3.5 Edge Effects Corrections
in a Finite Laminated Plate 13
2.4 Diffusivity and Maximum Moisture Content . 15
3 COMPLEX MODULI OF UNIDIRECTIONAL COMPOSITES ... 21
3.1 Introduction 21
3.2 General Theory 21
3.3 Micromechanics Formulation of elastic
Moduli 22
3.4 Complex Moduli 23
i i i


4 DAMPING 29
4.1 Damping Mechanisms 29
4.1.1 Nonmaterial Damping 29
4.1.2 Material Damping 30
4.2 Characterization of Damping 30
4.2.1 Free Vibration '. 30
4.2.2 Steady State Vibration 31
4.2.3 Complex Modulus Approach 32
5 DAMPING AND STIFFNESSES OF GENERAL LAMINATES . 36
5.1 Introduction 36
5.2 Laminated Plate Theory Approach 36
5.3 Energy Method Approach 37
6 EXPERIMENTAL PROCEDURES 40
6.1 Introduction 40
6.2 Test Specimen 40
6.3 Environmental Conditioning 41
6.4 Four-Point Flexure Test Method 41
6.5 Impulse Hammer Technique 42
7 HYGR0THERMAL EXPANSION 48
7.1 Introduction 48
7.2 Coefficients of Thermal Expansion 49
7.3 Coefficients of Moisture Expansion . . 50
7.4 Experimental Data 51
7.4.1 Previous Investigations 51
7.4.2 Present Investigation 52
8 HYGROTHERMAL EFFECTS ON COMPOSITE
COMPLEX MODULI 58
8.1 Literature Survey 58
8.2 Theoretical and Experimental Assumptions. . 59
8.3 Modeling of Epoxy Properties 61
8.4 Results 63
8.4.1 Epoxy Complex Moduli 63
8.4.2 Composite Complex Moduli 65
9 HYGROTHERMAL EFFECTS ON STRESS FIELD 79
9.1 Introduction 79
9.2 Description of Study Cases 80
9.3 Numerical Results and Discussion 81
9.3.1 Glass/Epoxy 81
9.3.2 Graphi te/Epoyx 82
9.3.3 Summary 82
i v


10 HYGROTHERMAL EFFECTS ON COMPLEX STIFFNESSES
95
10.1 Introduction 95
10.2 Numerical Results and Discussion 95
10.2.1 Glass/Epoxy 95
10.2.2 Graphite/Epoxy 96
10.2.3 Summary 97
11 CONCLUSION Ill
APPENDICES
A COMPLEX STIFFNESSES OF COMPOSITES 115
A.l Elastic Stiffnesses 115
A.2 Complex Stiffnesses 117
B DEVELOPMENT OF THE FINITE ELEMENT METHOD .... 119
B.l Equilibrium Equations 119
B.2 Program Organization 122
B.3 Shape Functions, Jacobian and Strain
Ma trix 123
B.4 Elasticity Matrix 125
B.5 Element stiffness Matrix 128
B.6 Equivalent Nodal Loadings 128
B.6.1 Element Edge Loadings 128
B.6.2 Hygrothermal Loadings 129
B.7 Element Stresses 130
REFERENCES 133
BIOGRAPHICAL SKETCH 137
v


LIST OF TABLES
T abIes Page
6.1 Initial properties of Magnolia 2026 epoxy,
3M Scotchply Glass/Epoxy, and a typical
Graphite/epoxy composite 45
7.1 Coefficients of moisture and thermal
expansion of epoxy and graphite and
glassfibers 54
7.2 Properties of Glass and Graphite Fibers . 54
9.1 Description of cases in Figure 9.2 84
9.2 Typical strengths of Glass/Epoxy and
Graphite/Epoxy 84


LIST OF FIGURES
Figures Page
2.1 Plate subjected to a constant humid 18
environment on both sides.
2.2 Moisture distribution across a plate.
The numbers on the curves are the values
of (c c.)/(c c.) 18
v i y v oo i y
2.3 Semi-infinite plate in a humid
environment 19
2.4 Comparison of the exact specific moisture
concentration equation with some approximate
so 1 u t ions 19
2.5 Geometry of a plate 20
2.6 Moisture content versus square root of time.
On the curve Vt < Vt^< Vt^ and the slope
is constant for Vt" < Vt^ 20
4.1 Schematic drawing of a free-clamped beam
under free vibration and plot of its
deflection versus time 35
4.2 Schematic drawing of a free-clamped beam
under forced vibration and plots of the
deflection versus time and deflection
amplitude versus frequency 35
6.1 Schematic drawing of environmental and
testing chambers 46
6.2 Loading configuration of the 4-point
f 1 exu retest 46
v i i


6.3Schematic drawing of the impulse hammer
technique apparatus and a typical display
of the Fourier Transform 47
7.1 Transverse moisture strain of Magnolia
epoxy and 3M Scotchply Glass/Epoxy 55
7.2 Plot of the thermal expansion coefficients
in terms of fiber volume fraction of a dry
S Glassf iber/Epoxy at 20C 56
7.3 Plot of the thermal expansion coefficients
in terms of fiber volume fraction of a dry
Graphite/Epoxy at 20C 56
7.4 Plot of the moisture expansion coefficients
in terms of fiber volume fraction of a dry
S Glassf iber/Epoxy at 20C 57
7.5 Plot of the moisture expansion coefficients
in terms of fiber volume fraction of a dry
Graphite/Epoxy at 20C 57
8.1 Schematic variation of the storage modulus of
epoxy with temperature 67
8.2 Schematic variation of Poissons ratio of
epoxy with temperature 67
8.3 Schematic variation of damping of epoxy
with temperature 6S
8.4 Glass transition temperature of epoxy.
From Delasi and Whiteside [6] 68
8.5 Experimental data of the storage modulus
of epoxy as a function of temperature at
diverse constant moisture contents 69
8.6 Experimental data of the storage modulus
of epoxy as a function of moisture content
at diverse constant temperatures 69
8.7 Experimental data of the storage modulus
of epoxy as a function of normalized
temperature (T Tq)/(T Tq) 70
8.8 Experimental data of damping of epoxy as
a function of temperature at diverse constant
moisture contents 71
v i i i


8.9 Experimental data of damping of epoxy as
a function of moisture content at diverse
constant temperatures 71
8.10 Experimental data of the storage modulus
of epoxy as a function of normalized
temperature (T Tq)/(T Tq) 72
8.11 Experimental data of Poissons ratio of
epoxy in term of temperature 73
8.12 Experimental data of Poissons ratio of
epoxy in term of moisture content 73
8.13 Experimental data of Poissons ratio in
term of the normalized temperature
T = (T T )/(T T ) 74
n v o g o'
8.14 Longitudinal storage modulus (Ej^) of
Glass/Epoxy versus = (T Tq)/(T^ Tq). . 75
8.15 Transverse (E^) and shear (Gj^) storage
moduli of Glass/epoxy versus
T = (T T )/(T T ) 75
n o g o
8.16 Longitudinal transverse (rj 22^
and shear () damping of Glass/Epoxy
versus T = (T T )/(T T ) 76
n v o g o'
8.17 Poissons ratio () of Glass/Epoxy
versus T = (T T )/(T T ) 76
n o v g o
8.18 Longitudinal storage modulus (Ej^) of
Graphite/Epoxy versus = (T Tq)/(T Tq) 77
8.19 Transverse (E^) an<3 shear (G^) storage
moduli of Graphite/epoxy versus
T = (T T ) / (T T ) 77
n v o' g o'
8.20 Longitudinal (77 ^ ^ ) transverse (1722)
and shear (p^) damping of Graphite/Epoxy
versus T = (T T )/(T T ) 78
n v o' v g o'


8.21 Poissons ratio (v ^ ) of Graphite/Epoxy
versus T = (T T )/(T T ) 78
n o g o
9.1 Geometry of a laminate and finite mesh
of a 1/4 cross-section area 85
9.2 Description of the applied moisture
grad ients 86
9.3Profile of the hygrothermal stress a
across a [(90/0)^^ Glass/Epoxy laminate
at y/b = 0.472 87
9.4 Profile of the hygrothermal stress
across a [(SO/O^lg Glass/Epoxy laminate
at y/b = 0.472 88
9.5 Profile of the hygrothermal stress
across a [(90/0)2^ Glass/Epoxy laminate
at y/b = 0.472 89
9.6 Profile of the hygrothermal stress o
across a [(OO/OJ^lg Glass/Epoxy laminate
at y/b = 0.993 90
9.7 Profile of the hygrothermal stress a
across a [(OO/OJ^jg Graphite/Epoxy laminate
at y/b = 0.472 91
9.8 Profile of the hygrothermal stress
across a [^O/OJ^jg Graphite/Epoxy laminate
at y/b = 0.472 92
9.9 Profile of the hygrothermal stress
across a [(QO/O^jg Graphite/Epoxy laminate
at y/b = 0.472 93
9.10 Profile of the hygrothermal stress yX
across a [(OO/O^lg Graphite/Epoxy laminate
at y/b = 0.993 94
10.1 Line style legend of Figures 10.2-13 98
x


10.2
Complex in-plane stiffness A^ of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 99
10.3 Complex in-plane stiffness of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 100
10.4 Complex in-plane stiffness Agg of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 101
10.5 Complex bending stiffness of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 102
10.6 Complex bending stiffness of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 103
10.7 Complex bending stiffness Dgg of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 104
10.8 Complex in-plane stiffness A^ of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 105
10.9 Complex in-plane stiffness A^ of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 106
10.10 Complex in-plane stiffness Agg of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 107
10.11 Complex bending stiffness of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 108


a
10.12 Complex bending stiffness of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 109
10.13 Complex bending stiffness Dgg of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 110
B.l Organization of the F.E.M. program 131
B.2 Local axes f and r¡, Gauss point numbers
and local node numbers of an eight-node
isoparametric element 132
x i i


NOMENCLATURE
B
*
B *
B"
c
comp 1 ex
in-p1ane
stiff ne s s
comp 1 ex
coup 1ing
stiff ne s s
comp 1 ex
modu1u s
s torage
modu1u s
loss modulus
moisture concentration
c average specific moisture
cm equilibrium moisture concentration
D* .
i J
D D
X XX
[D]
E11
E22
G12
K
spec ific heat
complex bending stiffness
moisture d i ffusivities
diffusivity matrix, elasticity matrix
longitudinal Young modulus
transverse Young modulus
in-plane shear modulus
thermal conductivity
m weight of absorbed moisture
M percent moisture content
x i i i


initial percent moisture content
M .
i
equilibrium percent moisture content
Q. transformed stiffness
i J
s
t
T
v
m
w
a .
i

T7
12
0 .
J
complex transformed stiffness
specific gravity
time
temperature
fiber volume fraction
matrix volume fraction
we igh t
coefficient of thermal expansion
coefficient of moisture expansion
s t rain
damping or loss factor
major Poissons ratio
fiber orientation of j-th layer
density
stress
Subsc rip t s
1, 2, 3 principal directions of the fibers
f fiber
i initial
j layer number
x i v


L
longitudinal direction
m ma t rix
x, y, z Cartesian coordinates
00 maximum or equilibrium
Superscripts
H mo is ture
o initial
T transpose, thermal
* complex value
real part
imaginary part
xv


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillement
of the Requirements for the Degree of Doctor of Philosophy
HYGROTHERMAL EFFECTS
ON THE COMPLEX MODULI OF COMPOSITE LAMINATES
By
Hacene Bouadi
Apr i 1 1988
Chairman: Dr. Chang-T. Sun
Major Department: Engineering Sciences
The effects of absorbed moisture and temperature on
the complex moduli of composite laminates are investigated
and the mechanisms of moisture diffusion in a lamina are
also analyzed.
First, the variation of the complex moduli of epoxy
in terms of temperature and moisture content are
experimentally determined. Then, the hygrothermal effects
on the complex moduli of composites are derived by using
the complex moduli of the matrix, micromechanical formulas
and experimental data. Only the hygrothermal effects on the
complex moduli of pure epoxy need to be experimentally
determined since these effects on the fibers properties
are neg1 igib1e.
xv i


In addition, the effects of hygrothermal environment
on the stress field and material damping of general
laminated composite plates are analyzed. It is shown that
hygrothermal stresses induced directly by moisture and
indirectly by material property changes can be very high,
but the effects on damping are less pronounced.
xv i i


CHAPTER 1
INTRODUCTION
1 1 General Introduction
The introduction of advanced composites in aerospace
applications has led to an extensive study of their
mechanical behavior. The amount of experimental and
theoretical findings of composite material researchers made
during the 1960s was so vast that Broutman and Krock [1]
needed eight volumes to edit a summary of the resulting
know 1 edge.
The interest in composite materials arose from their
ideal characteristics for aerospace structures. Replacement
of the commonly used aircraft material, aluminum, by high
strength/density ratio and versatile composites can lead to
a theoretical 60% weight reduction [2, p. 22], Due to such
benefits, lower costs and better understanding of their
mechanisms, the use of composite materials has been
increasing slowly but steadily.
Exposure of aircraft structures to high temperature
and humidity in the environment and the tendency of
composites to absorb moisture gave rise to concern about
their performances under adverse operating conditions.
1


2
Therefore, considerable work has been done to understand
the effects of hygrothermal environment on the mechanical
behavior of composite materials.
1.2 Moisture Diffusion
In a 1967 study on the effects of water on glass
reinforced composites. Fried hypothesized that water can
penetrate the resin phase by two general processes, by
diffusion through the resin and by capillary or Poiseuille
type of flow through cracks and pinholes [3], But no
mathematical theory was presented. Later, investigators
established that the primary mechanism for the transfer of
moisture through composites is a diffusion process and
adapted the general theory of mass diffusion in a solid
medium to moisture diffusion in composite materials. The
transfer of moisture through cracks is a secondary
effect [4, 5], Experimental data indicate that for most
composite materials, the diffusion of moisture can be
adequately described by a concentration dependent form of
Ficks law [4-10].
1.3 Hygrothermal Effects
The degradation of mechanical properties of glass
reinforced plastics exposed to water has long been


3
recognized by marine engineers who use "wet" strengths in
the design of naval structures [3]. Requirements in
aircraft structures are more stringent. The mechanical
properties of materials used in aerospace applications must
be completely characterized. Therefore, the effects of
hygrothermal environment on the elastic, dynamic, and
viscoelastic responses of composites have been studied. To
date, the effects of moisture and/or temperature on the
following performances have been investigated: tensile
strength, shear strength, elastic moduli [3, 11-14],
fatigue behavior [15-17], creep, relaxation, viscoelastic
responses [18-20], dimensional changes [21], dynamic
behavior [22], glass transition temperature [23], etc.
Only tensile and shear strengths and elastic moduli
have been thoroughly studied by many researchers. But data
on the other properties are more limited and hence
inadequate to constitute a good design data-base.
1.4 Scope and Methodology
The present investigation is a combined theoretical
and experimental work and is concerned with predicting the
hygrothermal effects (below the glass transition
temperature) on the complex moduli of composite materials.
This program is undertaken by carrying out the
foil owing s teps:


4
i)The complex moduli of epoxy matrix in terms of
temperature and moisture concentration are obtained by#
using experimental tests and theoretical expressions.
ii)The effects of temperature and moisture on the
complex moduli of unidirectional composites can be derived
by using the complex moduli of the matrix, micromechanics
formulas, and experimental observations. In addition, we
neglect the hygrothermal effects on the fibers.
iii)The effects of hygrothermal conditions on the
stress field and the material damping of some general
laminated composite plates undergoing simple hygrothermal
loadings are analysed.
1.5 Dissertation Lay-Out
Right af ter
the
in t roduc tion
, the
mechanism
o f
moisture diffusion
i s
described in
Chap ter
2,
whe r e
the
absorption of moisture
through thin
compos ite
1aminas
i s
analyzed in detail.
The complex moduli of unidirectional composites are
defined in Chapter 3. Sections 3.3 and 3.4 give the microm
echanics formulations of the elastic and complex moduli in
terms of the constituent material properties. The damping
of composites based on the dynamic and complex modulus
approaches is characterized and the equivalence of both
approaches is proven in Chapter 4. In Chapter 5. the
damping and complex stiffnesses of general laminates are


5
derived by using the laminated plate theory, the energy
approach, and the preceding derivations. The complex
stiffnesses are completely expressed in Appendix A.
The environmental conditioning of the test specimens,
the static flexure test, and the impulse hammer techniques
are presented in Chapter 6. These experimental methods,
although simple, are very versatile and are adequate in
determining the necessary data for the purpose of
this investigation.
The theoretical and experimental results are given in
Chapters 7-10. The moisture and thermal expansions of
composites are quantified in Chapter 7. The current
experimental results and data and conclusions of previous
investigators are used in Chapter 8 to model the complex
moduli of epoxy as functions of temperature and moisture
content. In Chapters 9 and 10, the hygrothermal effects on
the stress field across laminates and on damping of
composites are investigated with the help of the results in
the preceding Chapters. The Finite Element Method (F.E.M.)
used in determining the stresses is summarized in
Append ix B.


CHAPTER 2
DIFFUSION OF MOISTURE
2.1 Introduc tion
The mechanism of moisture absorption and desorption
in most fiber reinforced composites is adequately described
by Ficks law [4]. Fick recognized that heat transfer by
conduction is analogous to the diffusion process.
Therefore, he adopted a mathematical formulation similar to
Fouriers heat equation to quantify the diffusion
process [24, 25].
2.2 Fickian Diffusion
The Fourier and Ficks equations, describing the
one-dimens iona1 temperature and moisture concentration, are
respectively given by
(2.1)
dc d ~ Sc_
3t dx x dx
(2.2)
where p is the density of the material, is the
6


7
specific heat, T is the temperature, t and x are the time
and spatial coordinates, respectively, is the thermal
conductivity, c is the moisture concentration, and is
the moisture diffusivity.
The moisture diffusivity, D and the thermal
diffusivity, Kx/(pCv), are the rate of change of the
moisture concentration and the temperature, respectively.
In general, both parameters depend on temperature and
moisture concentration. But experimental data show that,
for most composites, moisture diffusivity does not depend
strongly on moisture concentration [4], Hence, Eq (2.2)
becomes
dc
3t
D
a2
a c
x 2
dx
(2.3)
and is solved independently of Eq (2.1).
The three-dimensional diffusion in an anisotropic
medium is obtained by generalization of Eq (2.2) as follows
dc mi * \
JTT = v. ( [D] vc)
where the diffusivity
ma t rix
i s
D
D
D
XX
xy
xz
[D]
-
D
D
D
yz
yy
yz
D
D
D
zx
zy
zx
(2.4)
(2.5)


8
Expansion of Eq (2.4) results in an equation of the form
2 2 2 2
dc d c d c d c d c
§7- = D + d 2_£ + d 2_£ + (D + D ). g--
3t xxg^2 yyay2 K yz xy'ay ox
~2 .2
+ (D + D + (D + D ).g g
zx xz ox oz xy yx'ox oy
(2.6)
if the coefficients D. ,s are considered to be constant.
i J
2.3 Fickian Diffusion in a Plate
Laminated plates are widely used in the experimental
characterization of composites. Hence, being of practical
interest, the problem of moisture absorption in a plate is
thoroughly discussed in this section.
2.3.1 Infinite Plate
The case of moisture absorption through a material
bounded by two parallel planes is considered. The initial
and boundary conditions of an infinite plate exposed on
both sides to the same constant environment (Figure 2.1)
are given by
T
c
for 0 < z < h and t < 0
(2.7)


9
T = T. ]
1 f for z = 0, z = h and t > 0
C = C<* J
where T. is a constant temperature, is the initial
moisture concentration inside the material, and c is the
CO
maximum moisture concentration. It is assumed that the
moisture concentration on the exposed sides of the plate
reaches cra instantly.
The solution of Eq (2.3) in conjunction with the
conditions of Eqs. (2.7) is given by Jost [25]
c -
c
oo
c .
1
c .
1
00
4 V 1 2 ,j + 1
W 1 (2jTTTsln ~Sz exp
j=o
^iii)2!72D t
,2 z
n
(2.8)
Equation (2.8) is plotted in Figure 2.2.
The average moisture concentration is given by
c
'll
c dz
^ o
(2.9)
Substitution of Eq. (2.8) into Eq (2.9) and integration
result in
c
c
00
c .
1
c .
1
8_
ir
I
j=o
1
(2j + l)
exp
.2_2 Dzt
-(2j + 1) U s-
(2.10)
This analysis can be applied to the case of diffusion of


10
moisture into a laminated composite plate so thin that
moisture enters predominantly through the plane faces.
2.3.2 Semi-Infinite Medium
In the early stages of moisture diffusion into a
plate, there is no interaction between moisture entering
through different faces. Therefore, the solution of
moisture absorption into a semi-infinite half-plane is
applicable to a plate for short time.
The initial and boundary conditions of a semi
infinite plane (Figure 2.3) exposed to a constant moist
environment are
T
c
for 0 < z < 00 and t < 0
0 and t > o
(2.11)
The solution of Eq (2.3) in this case is [24, 25]
erf
z
2 / D t
L v z J
(2.12)
The rate at which the total specific mass of moisture, m,
is diffusing into the half-plane is


de
dz _
z =0
(2.13)
1 1
dm
d t
D
Thus, the total mass of moisture entering through an area A
in time t is
n t
m = -
pAD
dc
dz
dt = 2pA(c -
z=0
D t
z
17
(2.14)
Equation (2.14) shows that the mass of diffusing substance
is proportional to the square root of time.
2.3.3 Experimental Measurement of Moisture Content
In the case of a finite plate, the total moisture
con tent is
m = pVc
(2.15)
where V is the volume of the test piece. The total moisture
content is experimentally measured by subtracting the dry
weight, w^, from the current weight, w, of the plate, i.e.
m = w
w
(2.16)
A parameter of practical interest is the percent moisture
content defined as


12
M
100
(2.17)
Since
M M c c .
n = 1 : M = 100c (2.18)
M M. c c v
oo i co i
the experimentally measured M of Eq (2.18) can be compared
to the analytical value given by Eq (2.10).
2.3.4 Approximate Solution of Moisture Content
Approximate solutions of the specific moisture
distribution in a plate subjected to the conditions given
by Eqs (2.7) are useful, since the difficulty of dealing
with infinite series can be avoided.
Sma11 time. As discussed in section 2.3.2, Eq (2.14)
can be applied during the early stages of absorption. It
yields
c c M M.
i i
Large time. Tsai and Hahn [26, p. 338] suggest that,
for sufficiently large t, Eq. (2.10) can be approximated by
using the first term of the series, i.e.,
D t
z
n h
(2.19)


13
c
c
00
C .
1
C .
1
8
-9-exp
UZ
IT
D ti
z
(2.20)
Shen and Springer formulation. These researchers
have derived in Ref. [4] the following approximation
c
c
00
c .
1
c .
1
exp
7.3
D t
z
0.75'
1 h2 j
(2.20)
Figure 2.4 shows a comparison of Eqs (2.19-21) with the
exac t so 1u tion.
2.3.5 Edge-Effect Corrections in a Finite Laminated Plate
A plate exposed to a humid environment absorbs
moisture through all its six sides. At small time, the
interaction of moisture entering through different sides is
negligible. Therefore, Eq. (2.19) can be applied to such
cases. It yields
m = 4P(C oo_ ci)
bL /d + bh nr + hL rr rm
V z vx v y J v
(2.22)
where D D and D are the diffus ivities in the x, y, and
x y z
z directions, respectively. The geometry of the plate is
shown in Figure 2.5. Rewriting Eq. (2.22) in terms of the
percent moisture content gives


14
M
4M
Dt
> nh2
where the effective diffusivity D is
(2.23a)

D
D
1 h
1 + L
x h
_ i
y
D b
D
>
z >1
z
(2.23b)
The micromechanics formulation for diffus ivities proposed
by Shen and Springer [4] and modified by Hahn [26] for
impermeable, circular cross-section, fiber-reinforced
compos i tes is
D
L
D
m
D
T
(2.24)
where D D_ and D, are the matrix, transverse, and
m T L
longitudinal diffus ivities, respectively. Equation (2.23b)
for a unidirectional lamina with all fibers parallel to the
x-direction can be written as
D
(2.25)


15
For a general laminated plate consisting of N layers with
fiber orientations 0., the diffusivities are
J
D
z
D
T
D
x
D
y
N N
D. y h .cos^0 + D~ T h.sin^0.
L L j j TZ.J j
J = i ii
N
2",
j = l
N N
D, y h.sin^0. + D y h.cos^0.
L L j j 7 L j j
1 = 1 j=l
N
(2.26)
where h. is the thickness of
J
diffusivity of a general
substituting Eqs. (2.26) into
the j-th layer.
1amina t e is
Eq (2.23b).
The effective
obtained by
2.4 Diffusivity and Maximum Moisture Content
The diffusivity and the maximum moisture content
Mro must be experimentally determined in order to predict
the moisture content and distribution in a lamina. These
parameters are obtained by the following procedures:
- a thin, unidirectional composite plate is
completely dried and its weight is recorded,


16
- the specimen is then placed in a constant
temperature and constant relative humidity environment, and
its weight as function of time is recorded,
- the moisture content, M, versus the square root of
time, is plotted as shown in Figure 2.6.
The maximum moisture content is determined from the
plot and the diffusivity from the following equation
m2 Ml
= 4M
D
I7h
(2.27)
The subscripts 1 and 2 are defined in Figure 2.6.
The diffusivity depends only on the material and
temperature as follows
D
z
D exp
o
(2.28)
where R is the gas constant, D and E, are the
o a
pre-exponential factor and the activation energy,
respec tively.
Experimental research has shown that the maximum
moisture content depends on environment humidity content
and material. For a material exposed to humid air [4], the
equilibrium moisture content can be expressed as
M
00
(2.26)


17
where 0 is the relative humidity, a aad b are material
cons tan t s.


IS
>
>
Moisture
>
5
Fig. 2.1 Plate subjected to a constant humid environment
on both sides.
Z =* 0 is the center of the
crosssection of the plate
z/h
Fig. 2.2
Moisture distribution across
numbers on the curves are
a plate. The
the values of
(c -
c.)/(c -
J v oo
ci}


(M M,)
19
Fig. 2.3 Semi-infinite plate in a humid environment.
Exact
Oneterm
Shen and Springer
Fig. 2.4 Comparison of the exact specific moisture
concentration equation with some approximate
solutions.


Moisture content (%)
20
Fig. 2.5 Geometry of a plate
i
Fig. 2.6 Moisture content versus square root of time. On
the curve J t ^ < J t^ < J t^ and the slope is
for/T <
cons tan t


CHAPTER 3
COMPLEX MODULI OF UNIDIRECTIONAL COMPOSITES
3.1 In t roduction
Composite materials, such as Glass/Epoxy and
Graphite/Epoxy, have a polymeric matrix. Therefore, they
display viscoelastic behavior. Some of the effects of this
time-dependent phenomenon are: stress relaxation under
constant deformation, creep under constant load, damping of
dynamic response, etc.
This chapter is an introduction to the dynamic
behavior of viscoelastic composites in terms of complex
modu 1 i .
3.2 General Theory
A usual representation of the one-dimens iona1
stress-strain relation of a viscoelastic material subjected
to a harmonic strain history of the form
( t) = e1(Jt (3.1)
21


22
is given by
a(t) = B (iw)e*Wt = B (icj)G(t) (3.2)
The complex modulus B can be decomposed into its real and
imaginary parts as follows
B*(iu) = B ((i)) + i B" ( gj ) (3.3)
The terms B' and B are called the storage and loss
moduli, respectively, and the ratio of the loss over the
storage modulus
D *
V = g-r (3.4)
is referred to as either the loss factor or damping. The
loss modulus is a measure of the energy dissipated or lost
as heat per cycle of harmonic deformation.
3.3 Micromechanics Formulation of Elastic Moduli
The longitudinal modulus E^, the transverse modulus
E22 the in-plane shear modulus G^, anc* the major
Poissons ratio v 12 can be obtained by using the rule of
mixtures and the Halpin-Tsai equations, viz.,
+ v E
m m
E = v E
11 f f 11
(3.5)


23
E
22
E
m
1 + 2rijV
1 n ^ v
(3-6)
1 + n9v
G10 = G t ^-3-
12 ml- n2vf
(3.7)
wher e
The subscripts f
respectively, and v
n, =
n~ =
= VfUf12 +
V V
m m
(3.8)
(Ef22/Em>
- i
(3.9)
(Ef22/Em)
+ 2
(Gfl2/Gm>
- 1
(3.10)
(Gfl2/Gm)
+ 1
m stand
for fiber and
ma t rix,
v are the
m
volume fractions
Como lex Moduli
The micromechanics formulations of the complex moduli
are obtained by applying the e1astic-viscoe1astic
correspondence principle [27-29], i.e., by undertaking the
following steps:
i) determining the elastic moduli of composites in
terms of the constituent material properties,


ii) replacing the elastic moduli of fibers and matrix
by corresponding complex expressions.
For a viscoelastic composite. the properties of the
constituent materials are
Efll f11 + lEfll
"£22 ~ Ef22 + lEf22
G, =
G1 + iG"
E =
E' + iE
m m
(3-11)
G =
m
G1 + iG"
m m
v =
V + IB
m m
J f12 L f 12
The bulk modulus of eDoxy matrix, K is real and
m
independent of frequency [2S]. It is given by
-
%m 3V 1 2v )
v m'
(3.12)
while the viscoelastic bulk modulus is obtained from the
correspondence principle


25
E' + iE"
m m
K* =
m 3[1 2(u' + iu")]
m
m
(3.13)
Separation of the real and imaginary parts of Eq. (3.13)
yields
K
*
m
(1
2d')E
m m
3 [( 1
2u"E" + i[2E"d" + E'( 1
m m L mm m
2u')2 + 4d"2 J
m m J
2d )]
m'J
(3.14)
Since the dilatation bulk modulus is real, the imaginary
part of Eq. (3.14) is equal to zero; hence
2E"d" + E(1 2d) = 0 (3.15)
mmmv m' v '
Equation (3.15) results in
m,
d = ^(d 0.5)
m E v m
(3.16)
m
The shear modulus of the matrix is given by
G* =
m
m
3K E*
m m
2(1 + d*) 9K E*
v m' m m
(3.17a)
Separating the real and imaginary parts and neglecting the
2
terms of the form (^) yield


26
3K E'
P* nn m
m 9K E
m
1 + i
9K E"
m m
9K E' E1
m mm
Introduction of the material properties
rj = ft
m E
'f 1 1
f 1 1
E
f 11
f 22
f 22
E
f22
m
9K
m
'Gm G 1 9K E 'm
m mm
into Eq s.
(3.11)
r e su 1t s
m
i n
E f 11 ( 1 + 1T?f 11
Ef22^ + 1T?f22
G f 12 ( 1 + 1T?fl2
E'(1 + ip )
m v m'
G 1 ( 1 + i r\r )
mv Gmy
)
)
)
(3.17b)
(3.18)
(3.19)


27
*
v
m
v +
m
T7 ( v '
m m
V £ 12 ~ V £ 12 ~ v £ 12
There are no satisfactory data on the shear and transverse
damping of fibers. Fibers have damping with a magnitude
order ten times smaller than epoxy. The dampings ^fii
T7f22> and q^.^ are assumed to be equal and are replaced by
t]j. in subsequent equations. Since the fiber damping, q^. ,
is much smaller than the matrix damping, q the
m
imaginary part of the fiber Poissons ratio is neglected.
The preceding assumptions have a negligible effect on the
complex moduli of composites.
Application of the e1astic-viscoelast i c
correspondence principle to Eqs. (3.5)-(3.10) and
substituting them into Eqs (3.19) yield the following
complex material properties
E. = v.E' ,(1 + i T7r) + v E'(l + iq )
11 ffllv f mmv my
E
x
22
E ( 1 + i q )
mv m
1 + 2njVj.
*
1 nivf
(3.20)
12
Gm(1 + iTW
1 + n2Vf
- n2vf
1


whe r e
Ef22<1 + V Em( lflm>
Ef22(l + "fl + 2Em(1 iT>m>
Cfl2*1 lr|f) ~ Gm*1 1 ^Gm ^
Gf]2(l + iut) + Gm(l + lGm)
The elastic moduli given by Eqs. (3.5)-(3.8) model
experimental results with a good accuracy [2], Therefore,
they are used instead of mathematically exact
micromechanics formulas, such as those derived by
Hashin [27, 29].


CHAPTER 4
DAMPING
4.1 Damping Mechanisms
Any vibrational energy introduced in a structure
tends to decay in time. This phenomenon is called damping.
There are two types of damping mechanisms, external or
nonmaterial and internal or material.
4.1.1 Nonmaterial Damping.
Two common types of external damping are
- Accoustic damping: a vibrating structure always
interacts with the surrounding fluid medium (air, water,
etc.). This effect can lead to noise emission and even to
changes of the natural frequencies and mode shapes. Thus,
mechanical responses might be modified.
- Coulomb friction damping: two contacting surfaces
in relative motion dissipate energy through frictional
forces.
29


30
4.1.2 Material Damping
There are many damping mechanisms that dissipate
vibrational energy inside the volume of a material. Damping
phenomena include thermal effects, magnetic effects, stress
relaxation, phase processes in solid solutions [30, p. 61],
etc.
The internal damping of polymeric matrix composites,
such as Glass/Epoxy and Graphite/Epoxy, is dominated by
viscoelastic damping.
4.2 Characterization of Damping
4.2.1 Free Vibration
A cantilever under free vibration oscillates
regularly with an amplitude that decreases from one
oscillation to the next one (Figure 4.1). A measure of
damping is the logarithmic decrement defined as
6
1 n
(4.1)
whe r e
A^ = amplitude of the n-th cycle
^n + N = amPlituc*e the (n+N)-th cycle
The damping defined in Eq (4.1) is applicable to viscous


31
damping and for hysteretic damping that is represented by a
complex modulus approach.
4.2.2 Steady State Vibration
Damping also influences the dynamic equilibrium
amplitude of structures (e.g. beams) that undergo harmonic
oscillation. A resonance usually occurs (Figure 4.2). The
following measure of damping is used
V
(j.
- (jj.
CJ
(4.2)
whe r e
o
resonant frequency
"l-
w2 =
frequencies on either sides of such that
the amp 1itude
is 1/y 2 times the resonant
amp 1itude.
I n the
case
of a vibration
induced by the force
f(t) = Fsin(ut)
the response (deflection), w(t), is out of phase with f(t)
by an angle e such that
w( t)
Wsinfut + e)


32
The work done per cycle is
217/u
D
f ( t)^- dt = I7WF sin(e)
(4.3)
J o
The strain energy stored in the system at the maximum
displacement is half the product of the maximum
displacement by the corresponding value of the force, i.e.,
U = ^FW cos(e)
(4.4)
There is no damping if the work done per cycle is zero,
i.e, if sin(e) = 0.
The ratio of energy dissipated in a cycle to energy
stored at the maximum displacement is another measure of
damping. Therefore, the damping is
(4.5)
The definitions of damping given by Eqs. (4.2) and (4.5)
are equivalent [33].
4.2.3 Complex Modulus Approach
The one-dimensional stress-strain relation of a
viscoelastic material undergoing harmonic motion has been
shown to be (Eq. (3.2))


33
* ( t) = EH((o)o eUt = (E'(g)) + iE"())o eiwt (4.6)
Noting that i |u | = d/dt, Eq. (4.6) can be written as
^ 1(Jt E iutc
ct ( t ) = e + -ir ue
o co o
(4.7)
The real part is given by (after algebraic manipulation)
a(t) = E e^sinfut + e) i + rj
+ e ) -11 + r/2
(4.8)
where q = tan(e) = E"/E '
The energy dissipated during a cycle per unit volume is
D = () a d =
x
217/(0
d
x
d t
d t =
UqE'e2
o
(4.9)
The maximum energy stored is
1 2
u = E* ~
2 o
(4.10)
The r e f o r e,
V
(4.11)


34
Hence, the definitions of damping given by Eq. (3.4) and
Eq. (4.5) are equivalent. This conclusion is also valid for
general cases of structural vibration.


35
Fig. 4.1
Schematic drawing of
free vibration and
versus time.
a free-clamped beam under
plot of its deflection
4.2
Schematic drawing
forced vibration
versus time and
f requency.
of a free-clamped beam under
and plots of the deflection
deflection amplitude versus
Fig .


CHAPTER 5
DAMPING AND STIFFNESSES OF GENERAL LAMINATES
5.1 Introduction
Both the laminated plate theory and the energy method
approaches for analyzing the damping and the stiffnesses of
general laminates are presented in this chapter.
5.2 Laminate Plate Theory Approach
Four independent parameters are needed to determine
completely the damping of a unidirectional composite. But,
the analysis of the material damping of a general laminated
composite requires the use of eighteen parameters. These
quantities are the ratios of the imaginary over the real
parts of the complex in-plane stiffnesses A. ,s, the
i J
and the complex bending
The terms A. ,s, B. ,'s, and
1 J i J
and D. ,s are defined as
i J
+h/2
A. .
i J
dz
J -h/2
36


37
r+h/2
-h/2
(5.1)
0
p+h/2
J-h/2
2tt*
2 Qij
dz
where the complex transformed stiffness Q_s depend on
* *
Ell E22 G12
the 1amina t e.
12
and the orientation of each layer of
The in-plane, coupling, and flexural material damping
are defined as
1*1 j
A 7 .
i J
a: .
i j
C^ij
B7 .
-U.
b: .
i j
(5.2)
r-7?
F i J
D7 .
i i
d: .
1J
respec tively.
5.3 Energy Method Approach
The energy method can be used to
damping of laminated composite materials
determine the
under certain
loading and boundary conditions. The damping of a laminated


38
composite material in the first mode of vibration can be
defined as
N
^ ^ k^d ^ cyc.
V = ^ (5.3)
I 2n ks
k= 1
where N is the total number of layers, (. U.) is the
v k deye.
energy dissipated in the k-th layer during a cycle, and ^_Us
is the maximum energy stored in the k-th layer. The storage
and the dissipated energy are given by
k
U
s
1_
2
C .
V J J1 1
k
dV
kUd
n
e c
v J J'1 1
k
dV
(5.4)
where i and j are
C'.' are the real
J i
stiffnesses and V.
k
total" damping of
the material principal axes, Cj and
and imaginary parts of the complex
is the volume of each layer. Hence, the
an N-layered laminate is given by
V
(5.5)


39
The maximum strain vector {} can be determined by the
finite element method first. Then, the damping can be
deduced. Equation (5.5) is used to determine the damping of
a beam with variable thickness or of more general
s t rue tures.


CHAPTER 6
EXPERIMENTAL PROCEDURES
6 1 Introduction
A description of the test specimens and the
experimental procedures of the present investigation is
given in this chapter.
6.2 Test Specimens
The test specimens used to determine the complex
moduli of epoxy and of composite materials are thin strips
of approximate dimensions, 150mm by 25mm by 2mm. The only
materials tested are Magnolia 2026 laminating epoxy and 3M
Scotchply Glass/Epoxy. The curing temperatures of the epoxy
and the Glass/Epoxy are 175C and 170C, respectively. The
initial properties of these materials (at 20 C and without
moisture), as well as those of a typical Graphite/Epoxy,
are given in Table 6.1.
40


41
6.3 Environmental Conditioning
The specimens are conditioned in a Thermotron
environment chamber at a constant temperature and constant
relative humidity. The weight gain of the test pieces as a
function of time is monitored. Right after moisture
equilibrium is reached, the specimens undergo all tests at
diverse temperatures inside a testing chamber connected to
the environment chamber (Figure 6.1). The range of
temperature achieved inside the environment chamber is 4C
to 90C and the range of relative humidity is 4% to 99% for
temperatures below 75C. As temperature increases, the
highest relative humidity that can be obtained decreases
steadily to 75% at 90C.
6.4 Four-Point Flexure Test Method
The Youngs modulus and the Poissons ratio can be
determined with the four-point flexure test method. The
loading configuration of this test is shown in Figure 6.2.
The elastic flexural analysis yields [31]
E
PI'
Sbh^w
(6.1)
where E is the effective modulus, P is the applied load,
1 is the length of the specimen, b is the specimen width, h


42
is the thickness, and w is the deflection at quarter-point.
Poissons ratio is expressed as
(6.2)
where the transverse strain is measured
y
transverse strain gage cemented in the middle
specimen.
with a
of the
6.5 Impulse Hammer Technique
The material damping and the storage modulus of a
one-dimensional thin beam are determined with the impulse
hammer technique. This technique was pioneered by Halvorsen
and Brown [32]. The equipment set-up is shown in
Figure 6.3. The specimen is clamped inside the testing
chamber. A force impulse is applied to the test piece by a
force transducer. The end displacement of the specimen is
recorded with a non-contacting motion transducer. Both
responses from the force and motion transducers go through
signal conditioning equipments (filters, amplifiers). These
responses are digitized in a Fast Fourier Transform
analyzer (FFT) to obtain the transfer function in terms of
the frequency. The transfer function is defined as the
ratio of the Fourier Transform of the output (displacement


43
v(t)) over the Fourier Transform of the input (force
impulse u(t)); that is.
H(f)
_ mi
~ U(f)
(6.3)
where
t = time
f = f requency
V(f) = Fourier Transform of v(t)
U(f) = Fourier Transform of u(t)
The real and imaginary parts of H(f) are displayed on the
FFT analyser CRT (Figure 6.3). The material damping defined
by Eq. (4.11) is experimentally obtained by the following
expr e s sion
(fa/fb>2 1
2 + 1
(6.4)
where the frequencies f and f, are defined in Figure 6.3.
a b
The storage modulus is expressed as [33, p.464]
2 1 ^
E' = 38.32 f p^r (6.5)
r li
where f is the resonant frequency in Hz. p, is the
material density, 1 is the length of the specimen and h is
the thickness of the specimen. Equation (6.5) is valid only


44
for the case of the first mode free vibration of a
clamped-free beam. A complete description and analysis of
the impulse hammer technique are presented in Lees
dissertation [34].


45
Tab le 6.1
Initial properties of Magnolia 2026 epoxy,
3M Scotchply Glass/Epoxy, and a typical
Graphite/Epoxy composite.
Properties
Epoxy
G1as s/Epoxy
Graphite/Epoxy
Vf
0.50
0.70
p (g/cm3)
1.25
1.93
1.6
En (GPa )
4.0
37.00
155.23
E22 (GPa)
4.0
11.54
10.81
G12 (GPa.)
1.52
3.46
4.35
U 1 2
0.32
0.285
0.217
*11
0.018
0.0023
0.0019
V22
0.018
0.015
0.0078


46
Fig. 6.1
Schematic drawing of environmental and testing
chamber s.
Loading configuration of
f1exure test.
Fig. 6.2
the
4-poin t


47
Fourier Transform
Real part
Im. part
Schematic drawing
technique apparatus
the Transfer Fourier
of the impulse hammer
and a typical display of
Trans form.
Fig. 6.3


CHAPTER 7
HYGROTHERMAL EXPANSION
7.1 Introduction
When a metallic or composite structure is subjected
to a change of temperature, there are dimensional
variations and there may be stress development. For a
one-dimensional case, it is assumed that the thermal strain
is given by
T = a.(T T ) = a.AT
i i o 1
where
= coefficient of thermal expansion
T = actual temperature
Tq = reference temperature.
A polymer matrix composite exposed to a humid
environment absorbs moisture. Hence, it increases in weight
and dimensions. This situation produces a moisture strain
that varies linearly with moisture concentration [26]. In
the one-dimensional case, the hygros train is given by
(7.1)
48


49
H = p (c c ) = P.Ac (7.2)
i iv o' l v
where c is the initial moisture concentration and 6. is
o 1
the coefficient of moisture expansion.
7.2 Coefficients of Thermal Expansion
In the case of laminated composite plates, three
coefficients of thermal expansion are used in determining
the thermal strains. These parameters can be written in
terms of fiber and matrix properties. The micromechanics
formulas for a unidirectional orthotropic lamina are given
by (see Refs. [35, p. 24], and [36, p. 405] for a detailed
de riva tion)
l =
vParEr + v a E
f f f m m m
En
a2 =
vfaf +
v a
m m
vfufaf
v v a -
m m m
U12al
(7.3)
where the subscripts 1 and 2 represent the fiber and the
transverse directions. The thermal expansion coefficients
of an orthotropic lamina whose fibers make an angle 0
with the x-direction (Figure 2.5) are given by


50
2 2
a = a, cos 0 + o' sin 0
x 1 2
2 2
a = a, sin 0 + a_ cos 0
y 1 2
(7.4)
a = 2(a, a~) cos 0 sin 0
xy v 1 2'
7.3 Coefficients of Moisture Expansion
Similarly, the coefficients of moisture expansions of
an orthotropic lamina with impermeable fibers can be
expressed as [36, p. 406]
sE
p = £p
1 s E, m
m 11
P2 = f-d + vjpm v12px (7.5)
m
*12 =
where s and s^ are the specific gravities of the composite
ma terial and the ma trix. The moisture expansion
coefficients expressed in an axis system such that the
x-direction makes an angle 0 with the fibers are given by
Eqs. (7.4) after replacement of a's ^7 P^'s-


51
7.4 Experimental Data
7.4.1 Previous Investigations
Hahn and coworkers investigations [21, 26, 37] of
swelling of composites are outlined in this section. Some
of the typical results of the transverse strain versus
percent moisture gain are obtained by conducting the
following tests: absorption is conducted in moisture
saturated air such that Eqs (2.2) and (2.7) are satisfied;
while desorption takes place in vacuum at the same
temperature. Their data show a hysteretic nature of
swelling in this case.
But, when swelling of composites is given in terms of
moisture concentration, the average behavior of
S2-G1ass/Epoxy, Kevlar 49/Epoxy and Graphite/Epoxy can be
approximated by
0.43c = P2c (7.6)
Since the data presented in their publications display a
wide scatter, Hahn et al. suggest that Eq. (7.6) can be
used to estimate the moisture strains for most composite
materials.


52
7.4.2 Present Investigation
Epoxy and Glass/Epoxy specimens are conditioned at a
constant relative humidity until the absorbed moisture
reaches equilibrium. The changes in transverse dimensions
are measured. This procedure is repeated at diverse values
of relative humidity. The results are plotted in
Figure 7.1. The longitudinal swelling strains could not be
measured since the micrometer calipers
used
were not
suf ficiently
accurate. These data yield
the
foil ow i ng
expe rimen ta1
values
Pm(epoxy) = 0.25
(7.7)
P2(G1ass/Epoxy) = 0.48 (7.8)
Substitution of Eq. (7.7) and the parameters given in
Table 6.1 into Eqs. (7.5) yields the following empirical
values
j3 = 0.042 (7.9a)
P2 = 0.47 (7.9b)
for the 3M Glass/Epoxy composite. The experimental and
empirical values of P^ are practically equal. Hence, the


53
present results differ slightly from the approximation
given by Eq. (7.6).
The above coefficients and the typical coefficients
of expansion of graphite are quantified in Table 7.1, while
the storage moduli and the density of glass and graphite
fibers are listed in Table 7.2. These properties are used
to plot the thermal and moisture expansion coefficients
versus the fiber volume fraction of Glass/Epoxy and
Graphite/Epoxy in Figures 7.2 through 7.5.
The values in these plots are valid for dry
composites at room temperature. Since the storage modulus
of epoxy varies with temperature and moisture content, this
additional effect is investigated in Chapter 8.
In general, the thermal expansion coefficients are
functions of temperature, but this temperature effect is
negligible below 100C. Therefore, in the subsequent
chapters, the thermal expansion coefficients are assumed to
be independent of temperature.


54
Table 7.1 Coefficients of moisture and thermal expansion
of epoxy and graphite and glass fibers.
Epoxy
Glass
Graphite
a (pm/m)/K
54.0
5.0
0.2
P
0.25
0.0
0.0
Table 7.2
Proper ties
of Glass and
graphite Fibers.
Glass
Graphite
En(GPa)
70.0
220.0
E-22(Gpa)
70.0
16.6
Gf12(GPa)
28.7
8.27
71 f
0.0015
0.0015
V£ 12
0.22
0.16
p (g/cm3)
2.60
1.75


55
Moisture concentration (%)
Fig. 7.1
Transverse moisture strain of Magnolia
and 3M Scotchply Glass/epoxy.
epoxy


56
Longitudinal
transverse
Fig. 7.2 Plot of the thermal expansion coefficients in
terms of fiber volume fraction of a dry
S Glassfiber/Epoxy at 20C.
Longitudinal
Transverse
Fiber volume fraction
Fig. 7.3 Plot of the thermal expansion coefficients in
terms of fiber volume fraction of a dry
Graphite/Epoxy at 20C.


Coefficient of moisture expansion ^ Coefficient of moisture expansion
57
Longitudinal
Transverse
Plot of the moisture expansion coefficients in
terms of fiber volume fraction of a dry
S G1assfiber/Epoxy at 20C.
Fiber volume fraction
Longitudinal
Transverse
Plot of the moisture expansion coefficients in
terms of fiber volume fraction of a dry
Graphite/Epoxy at 20C.
Fig. 7.5


CHAPTER 8
HYGROTHERMAL EFFECTS ON COMPOSITE COMPLEX MODULI
S.1 Literature Survey
The storage moduli (real parts of Eqs. (3.11)) of
composites are usually determined by dynamic testings, such
as the technique described in section 6.5. They can be
approximated by using static tests [38].
Shen and Springer [12] investigated the environmental
effects on the elastic moduli of a Graphite/Epoxy composite
and made a survey of existing data showing the effects of
temperature and moisture on the elastic modulus of several
composites. Their conclusions are listed below.
i) The hygrothermal effects on the 0 fiber
direction laminates are negligible.
ii) For 90 fiber direction laminates, the
hygrothermal effects on the modulus are insignificant in
the 200K to 300K temperature range. But, between 300K and
450K, the hygrothermal effects on the modulus are
impor tan t.
Putter et al. [38] investigated the influence of
frequency and environmental conditions on the dynamic
58


59
behavior of Graphite/Epoxy composites. Their overall
conclusions are
i)The effects of frequency on the modulus and
damping are quite small in all cases.
ii)The effects of frequency on the modulus and
damping are relatively greater for matrix-controlled
laminates at higher frequencies (above 400 Hz.).
iii)At the same temperature, damping increases with
moisture saturation. But for dry laminates, damping
decreases slightly as temperature increases.
From all these experimental works, a general summary
can be drawn: the influence of hygrothermal conditions on
the elastic modulus, dynamic modulus and damping of
composites is matrix dominated.
8.2 Theoretical and Experimental Assumptions
Since the hygrothermal influence on composite
properties is matrix controlled [12, 38], the fiber
properties are assumed to be constant at any temperature
below the glass transition temperature and at any moisture
content. Therefore, to obtain the values of the complex
moduli of composites, it is sufficient to know how
temperature and moisture affect the complex moduli of the
epoxy matrix, and then use the micromechanics formulations
given by Eqs. (3.20). Thus, only the following functions


60
E '
m
E 1
m
(T.c)
u = u (T c ) (8.1)
mm v '
t] = r) (T c )
m mv '
need to be experimentally evaluated. The constant fiber
properties are given in Table 7.2. The effects of frequency
are negligible below 400 Hz. The results of this
investigation are not accurate for higher frequencies since
their effects have not been taken into account.
The qualitative influence of temperature only on the
storage modulus, real part of Poissons ratio, and damping
of epoxy is illustrated in Figures 8.1-8.3. There are three
distinct regions. At room temperature (in the glassy
region), the storage modulus, Poissons ratio, and damping
of epoxy are equal to about 4.0 GPa, 0.35, and 0.018,
respectively. In the glassy region, the storage modulus
decreases slowly, while Poissons ratio and the damping
increase as temperature increases. In the next region
(transition region), the storage modulus decreases rapidly,
and both Poissons ratio and damping reach their maximum
values. The last region is the rubbery region where the
modulus takes a very low value, and all three parameters
stay relatively constant.
Typical values of the modulus in the rubbery region
-2
could be 10 times the glassy modulus or lower. The damping


61
can
reach a
value of 1 or even 2
in the
transition region
[30,
p. 90].
Poissons ratio reaches the
1 imiting
value o f
0.5,
wh i ch
is approximated by
incompressible
rubber s
[39,
p. 293]
The position of the transition region depends on the
moisture concentration. The effects of moisture on the
glass transition temperature, T^_, of six epoxy resins have
been determined by Delasi and Whiteside [6]. These results
are plotted in Figure 8.4. They are compatible with the
data of McKague [40] and satisfy the theoretical relation
derived in Ref. [41, p. 69].
8.3 Modeling of Epoxy Properties
The observations of the preceding section are used
for modeling the material properties of epoxy that are
given by Eq. (8.1).
The glass transition region of epoxy resin is not
broad [6], therefore, a glass transition temperature is
used instead. The temperature T^ is usually obtained by
measuring the expansion of a specimen as function of
temperature. The point where the epoxy stops expanding as
temperature increases corresponds to the first deviation
from the glassy state and is termed T .
According to the experimental data plotted in
Figure 8.4, T^ is stongly dependent on absorbed moisture.
These results show that, as the moisture content of epoxy


62
increases, the transition temperature moves to the left in
Figures S. 1-8.3. Hence, the abrupt change of the material
properties starts at a lower temperature as the moisture
content increases. This fact and the conclusions reached by
previous investigators [12] suggest that the following
modelings of E, v', and n' are appropriate,
mm m
(8.2)
(8.3)
(8.4)
where the temperature Tq is equal to 273K. The moisture
concentration appears implicitly in T The glass
transition temperature is represented by
Tg = 210 exp(- 9c) (C) (8.5)
where c is the moisture concentration.
This modeling has been chosen so that it does not
represent the material properties beyond T since the
study of epoxy in the rubbery stage is not within the scope
of this research. Equations (8.2)-(8.4) are valid only for


63
the continuous parts of the curves plotted in
Figures 8.1-8.3. ,
8.4 Results
All test specimens are conditioned in a constant
relative humidity environment until moisture equilibrium is
reached. Then, the test pieces undergo the impulse hammer
technique and the four-point flexure tests to determine the
storage moduli, the material damping, and Poissons ratio
at several temperature and moisture contents.
8.4.1 Complex Moduli of Epoxy
Storage modulus. The experimental data on the storage
modulus of epoxy in term of temperature at three different
equilibrium moisture concentrations are plotted in
Figure 8.5.
It
can be concluded
tha t
increase in either
temperature
or
moisture
content
or
both
results in a
decrease in
the
s t orage
modu1u s.
P1 o 11ing
these data in
terms of moisture content in Figure 8.6 does not lead to
any additional insight. But, representing these results in
term of the following normalized non-dimensional
temperature
(8.5)


64
in Figure 8.7 shows a clear trend. Experimental studies
have shown that the modulus of polymer is very low at the
glass transition temperature, therefore, adding the value
E =0 for T = T to the data yields the following modeling
m g
E = 4.0(1 T ) (GPa)
m v nJ y '
(8.6)
Material damping. Similarly, the experimental data of
the hygrothermal effects on the damping of epoxy are
plotted in three Figures (8.8-8.10). There is very little
change in damping for all the considered conditions.
Therefore, it is proposed to let
T7 = 0.018 (8.7)
for temperatures up to 80C and moisture contents up to 5%.
The conclusion that the hygrothermal effects on the damping
of epoxy is negligible is qualitatively corroborated by
Putter et al. [38]. A quantitative comparison cannot be
made since these researchers have not included in their
publication the values of the fiber volume fraction and
moisture content of the test specimens.
Poissons ratio. The experimental values of the
Poissons ratio in terms of temperature at two different


65
moisture contents. are plotted in Figure 8.11. These
results show that Poissons ratio increases at a negligible
rate as temperature varies from 0 to SOC. Representing the
same data in terms of the moisture content up to M = 4.5%
(Figure 8.12) shows that the moisture effect is also
negligible. Therefore,
v = 0.32 (8.8)
m v
for temperatures up to 80C and moisture contents up to 5%.
Since v equals 0.5 at the glass transition temperature
m
(T = 0), the plot of Poissons ratio versus the normalized
temperature has been extrapolated as shown in Figure 8.13.
The extrapolation displays a qualitative trend only.
8.4.2 Complex Moduli of Composites
The complex moduli of Glass/Epoxy and Graphite/Epoxy
in terms of moisture content and temperature can be
determined by using the fibers properties given in
Table 7.2, Eqs (8.6) through (8.8) and the micromechanics
formulas (Eqs. (3.20)).
This procedure is illustrated by determining the
storage moduli and the damping of a Glass/Epoxy lamina with
a fiber volume fraction of 0.5 and a Graphite/Epoxy lamina
with a fiber volume fraction of 0.7.


66
G 1 a s s/Epoxy The parameters E^, E 22' ^12
'll
T]22 and v 12 versus the normalized temperature are
plotted in Figures 8.14-17. The experimental data
substantiate the theoretical results.
h.
Graph i te/Epoxv Similarly, E^, ^22 *^12
11
T,22 and v'i2 versus the normalized temperature of
Graphite/Epoxy are plotted in Figures 8.18-21.
For both Glass/Epoxy and Graphite/Epoxy, the results
show that the matrix-dominated parameters (E^ and G^) are
strongly affected by moisture and temperature, while the
fiber-dominated parameters (Ej^, s tay practically
cons tan t.


67
Fig. 8.1 Schematic variation of the storage modulus of
epoxy with temperature.
Fig. 8.2 Schematic variation of Poissons ratio of epoxy
with temperature.


68
Temperature
Fig. 8.3 Schematic variation of damping of epoxy with
temperature.
O
o
Fig. 8.4
Glass
De lasi
transition temperature of epoxy. From
and Whiteside [6].


69
2.5
2 L_ i i I i i i 1 i
0 20 40 60 30 100
Temperature (C)
M
M
M
0.0%
2.90%
3.70%
Fig. 8.5 Experimental data of the storage modulus of
epoxy as a function of temperature at diverse
constant moisture contents.
20 C
50 C
70 C
Fig. 8.6 Experimental data of the storage modulus of
epoxy as a function of moisture content at
diverse constant temperatures.


Storage modulus (GPa)
TO
Normalized temperature
Fig. 8.7
Experimental data of the storage
epoxy as a function of normalized
(T -
T ) / (T
o g
" T )
o
modulus of
temperature


o u
3.703
a.
E
a
a
0.015
I
0.01 -
0.005 j-
o r . i I >
0 20 40 SO 30 100
Temperature (C)
Fig. 8.8 Experimental data of damping of epoxy as a
function of temperature at diverse constant
moisture contents.
* T = 20 C
O T = 50 C
T = 70 C
Fig. 8.9 Experimental data of damping of epoxy as a
function of moisture content at diverse
constant temperatures.


72
Normalized temperature
* Experimental data
Fig. 8.10
Experimental data of the storage
epoxy as a function of normalized
(T T )/(T
o; g
T )
o
modulus of
temperature


73
QZ
4.17S
Fig. S.ll Experimental data of Poissons ratio of epoxy-
in term of temperature
* t = 20^:
O T = 50 C
a T => 75C
Fig. 8.12 Experimental data of Poissons ratio of epoxy
in term of moisture content.


Poissons ratio
74
Fig
^ Experimental data
Fit to data
Extrapolation
8.13 Experimental
term of the
(T T )/(T
o' g
data of Poisson's ratio in
normalized temperature =


75
Theoretical
* Experimental data
Fig. 8.14 Longitudinal
s to rage
modu1u s
Glass/Epoxy versus = (T Tq)/(T
(Eil)
- v-
o f
Theoretical E
22
^ Experimental EI,2
Theoretical G^2
Fig. 8.15 Transverse (E^) an<^ shear (G^) storage moduli
of Glass/epoxy versus T = (T T )/(T T ).
n o g o'


76
Theoretical *1,,
^ Experimental 7) n
Theoretical T)^
O Experimental 7)^
Theoretical 71
s
Fig. 8.16
Long itudi na 1
shear
(le1
(Dll).
damping
n
= (T -
T ) / (T T ) .
o g o
transverse (^22
of Glass/Epoxy
Theoretical
* Experimental
Fig. 8.17
Poisson s ra t i o
T = (T T ) / (T
n o g
(i2)
- v-
of Glass/Epoxy
) and
versus
versus


180
a
CL
a
160
0)
3
3
T3
O
140
QJ
cr
a
o
-*-<
tn
120
0 0.2 0.4 0.6 0.8 1
Normalized Temperature
Fig. 8.IS Longitudinal storage modulus (E^) of
Graphite/Epoxy versus T = (T T )/(T T ).
n o g o
Transversa
Shear
0 0.2 0.4 0.6 0.8 1
Normalized Temperature
(Gj2) storage
versus T =
n
Fig. 8.19
T ransverse
modu 1 i of
(T T )/(T
o' g
(^2) anc* shear
Graphite/epoxy
- T ).
o


Poissons ratio ^ Damping
7 S
0.02
0.015 -
0.01 -
0.005 -
Longitudinal damping
Transverse damping
Shear damping
Normalized temperature
(nn)
ig. 8.20 Longitudinal
shear (nG)
T = (T T )/(T T ) .
n v o' g o
transverse
damping of Graphite/Epoxy versus
(ti22), and
0.4
Normalized temperature
Theoretical
Pois sons ratio
("2>
o f
versus T = (T T )/(T T ).
n o g o
Fig. 8.21
Graphite/Epoxy


CHAPTER 9
HYGROTHERMAL EFFECTS ON STRESS FIELD
9.1 Introduction
The hygrothermal effects on the stress field are
investigated by considering an infinitely long, finite
width and symmetric composite laminate undergoing
hygrothermal loadings. The Finite Element Method is used in
order to estimate the magnitude of hygrothermal stresses in
laminated composites (see Appendix B). The geometry of a
laminate and the finite mesh of a quarter cross-section are
shown in Figure 9.1 and the boundary conditions are given
by
v = 0 for (y,z) = (0,z)
(9.1)
w = 0 f or (y,z) = (y 0)
where v and w are the displacements in the y and z
directions, respectively. The grid consists of 24 eight
node isoparametric elements and 93 nodes. Only 24 elements
79


80
are used since increasing the number of elements to 48
results in a relatively small change in the stress
magnitudes. The material properties in terms of temperature
and moisture content have been derived in the preceding
chapter. The constitutive equations are given by Eq. (B.12)
and can be written in matrix form as
{a} = [Q]({} {a}AT (P}c) (9.2)
where {a} and {/3} are the vectors of thermal and moisture
expansion coefficients.
9.2 Description of Study Cases
The considered stacking sequence is the [^O/O^jg
lay-up. The cross-ply laminate is preferred over other
laminate since hygrothermal loadings induce very high
stresses in this case. The volume fiber fractions of the
Glass/epoxy and the Graphite/Epoxy are 0.5 and 0.7,
respectively. The thickness and the width of the laminates
are assumed to be 2 mm and 20 mm, respectively.
Three cases of moisture gradients are applied. They
are described in Figure 9.2 and Table 9.1. Cases A and C
correspond to the dry and moisture saturated states,
respectively. While the non-uniform moisture gradient
(case B) corresponds to a moisture profile as derived in
section 2.3. Two uniform temperatures (20C and 80C) are


81
used. All laminates are assumed to be initially (dry at
20C) free of stress. Hence, residual stresses are not
taken into account. The elastic moduli used in computing
the stresses are approximated by the real parts of the
complex moduli. Therefore, the hygrothermal effects on the
elastic properties can be deduced from the results given
in Chapter 8
9.3 Numerical Results and Discussion
For all considered cases, the following remarks can
be drawn: at z/h = constant, the stresses away from the
free edge stay constant and the shear stress (CTyz)
zero, but, as y/b approaches 1,
non-zero values and there are small variations in the
a takes significant
yz
values of the other stresses. Hence, the stresses a a
y z
and a are plotted across the section of the laminate at
x
y/b = 0.472 and the shear stress a is plotted across
yz
the section at y/b = 0.993 (close to the free edge).
The stresses are compared to typical strengths of
Glass/Epoxy and Graphite/Epoxy that are provided in
Table 9.2.
9.3.1 G1 as s/Epoxv
The equilibrium moisture concentration, c of the
Glass/Epoxy material is 0.025.


82
The stress a is plotted in Figure 9.3. It reaches
y
a maximum magnitude of 166 MPa. for case C at 20C. It is
compressive for the 0 layer and tensile for the 90 layer.
The stress a is shown in Figure 9.4. It is
compressive everywhere and reach a magnitude of 288 MPa.
for the case C at 20C. The stress is also compressive
(Figure 9.5) and reaches a maximum of 245 MPa. The free
edge shear stress yz (Figure 9.6) is very significant
since its maximum magnitude is about 80 MPa..
9.3.2 Graphite/Epoxy
The equilibrium moisture concentration c^ for these
cases is 0.015. The stresses a a o and a are
y z x yz
plotted in Figures 9.7-10. These results show the same
trend as for the Glass/Epoxy cases. However, since the
moisture concentration is lower and graphite fibers have
stiffer moduli and lower coefficient of thermal expansion,
the magnitude of the stresses is smaller.
9.3.3 Summary
The hygrothermal conditions used in the preceding
sections are practically achieved only under very adverse
conditions. Hence, the induced stresses can be considered


Full Text

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HYGROTHERMAL EFFECTS
ON COMPLEX MODULI OF COMPOSITE LAMINATES
BY
HACENE BOUADI
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA

ACKNOWLEDGEMENTS
I would like to express my gratitude to Professor
Chang-T. Sun, the chairman of my doctoral committee, for
his guidance, time, and encouragement during this research.
Many thanks are owed to Professor Lawrence E. Malvern
and Professor Martin A. Eisenberg for their teaching and
financial support.
I also want to thank the other members of my doctoral
committee, Dr. Charles E. Taylor and Dr. Robert E.
Reed-Hill for their helpful commen ts, critique, and advic e.
In addition, I gratefully recognize the assistance of
Dr. David A. Jenkins for teaching me how to operate the
material testing equipment that was indispensable for my
work.
Finally, I appreciate Ms. Patricia Campbell’s help in
typing this manuscript.

TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS i i
LIST OF TABLES vi
LIST OF FIGURES vii
NOMENCLATURE xiii
ABSTRACT xvi
CHAPTERS
1 INTRODUCTION 1
1.1 General Introduction 1
1.2 Moisture Diffusion 2
1.3 Hygrothermal Effects 2
1.4 Scope and Methodology 3
1.5 Dissertation Lay-Out 4
2 DIFFUSION OF MOISTURE 6
2.1 Introduction 6
2.2 Fickian Diffusion 6
2.3 Fickian Absorption in a Plate 8
2.3.1 Infinite Plate 8
2.3.2 Semi-Inf inite Plate 10
2.3.3 Experimental Measurement of
Moisture Content 11
2.3.4 Approximate Solutions of
Moisture Content 12
2.3.5 Edge Effects Corrections
in a Finite Laminated Plate 13
2.4 Diffusivity and Maximum Moisture Content . . 15
3 COMPLEX MODULI OF UNIDIRECTIONAL COMPOSITES ... 21
3.1 Introduction 21
3.2 General Theory 21
3.3 Micromechanics Formulation of elastic
Moduli 22
3.4 Complex Moduli 23
i i i

4 DAMPING 29
4.1 Damping Mechanisms 29
4.1.1 Nonmaterial Damping 29
4.1.2 Material Damping 30
4.2 Characterization of Damping 30
4.2.1 Free Vibration '. 30
4.2.2 Steady State Vibration 31
4.2.3 Complex Modulus Approach 32
5 DAMPING AND STIFFNESSES OF GENERAL LAMINATES . . 36
5.1 Introduction 36
5.2 Laminated Plate Theory Approach 36
5.3 Energy Method Approach 37
6 EXPERIMENTAL PROCEDURES 40
6.1 Introduction 40
6.2 Test Specimen 40
6.3 Environmental Conditioning 41
6.4 Four-Point Flexure Test Method 41
6.5 Impulse Hammer Technique 42
7 HYGR0THERMAL EXPANSION 48
7.1 Introduction 48
7.2 Coefficients of Thermal Expansion 49
7.3 Coefficients of Moisture Expansion . . . . . 50
7.4 Experimental Data 51
7.4.1 Previous Investigations 51
7.4.2 Present Investigation 52
8 HYGROTHERMAL EFFECTS ON COMPOSITE
COMPLEX MODULI 58
8.1 Literature Survey 58
8.2 Theoretical and Experimental Assumptions. . . 59
8.3 Modeling of Epoxy Properties 61
8.4 Results 63
8.4.1 Epoxy Complex Moduli 63
8.4.2 Composite Complex Moduli 65
9 HYGROTHERMAL EFFECTS ON STRESS FIELD 79
9.1 Introduction 79
9.2 Description of Study Cases 80
9.3 Numerical Results and Discussion 81
9.3.1 Glass/Epoxy 81
9.3.2 Graphi te/Epoyx 82
9.3.3 Summary 82
i v

10 HYGROTHERMAL EFFECTS ON COMPLEX STIFFNESSES
95
10.1 Introduction 95
10.2 Numerical Results and Discussion 95
10.2.1 Glass/Epoxy 95
10.2.2 Graphite/Epoxy 96
10.2.3 Summary 97
11 CONCLUSION Ill
APPENDICES
A COMPLEX STIFFNESSES OF COMPOSITES 115
A.l Elastic Stiffnesses 115
A.2 Complex Stiffnesses 117
B DEVELOPMENT OF THE FINITE ELEMENT METHOD .... 119
B.l Equilibrium Equations 119
B.2 Program Organization 122
B.3 Shape Functions, Jacobian and Strain
Ma trix 123
B.4 Elasticity Matrix 125
B.5 Element stiffness Matrix 128
B.6 Equivalent Nodal Loadings 128
B.6.1 Element Edge Loadings 128
B.6.2 Hygrothermal Loadings 129
B.7 Element Stresses 130
REFERENCES 133
BIOGRAPHICAL SKETCH 137
v

LIST OF TABLES
T abIes Page
6.1 Initial properties of Magnolia 2026 epoxy,
3M Scotchply Glass/Epoxy, and a typical
Graphite/epoxy composite 45
7.1 Coefficients of moisture and thermal
expansion of epoxy and graphite and
glassfibers 54
7.2 Properties of Glass and Graphite Fibers . . . 54
9.1 Description of cases in Figure 9.2 84
9.2 Typical strengths of Glass/Epoxy and
Graphite/Epoxy 84

LIST OF FIGURES
Figures Page
2.1 Plate subjected to a constant humid 18
environment on both sides.
2.2 Moisture distribution across a plate.
The numbers on the curves are the values
of (c - c.)/(c - c.) 18
v i y v oo i y
2.3 Semi-infinite plate in a humid
environment 19
2.4 Comparison of the exact specific moisture
concentration equation with some approximate
so 1 u t ions 19
2.5 Geometry of a plate 20
2.6 Moisture content versus square root of time.
On the curve Vt < Vt^< Vt^ and the slope
is constant for Vt" < Vt^ 20
4.1 Schematic drawing of a free-clamped beam
under free vibration and plot of its
deflection versus time 35
4.2 Schematic drawing of a free-clamped beam
under forced vibration and plots of the
deflection versus time and deflection
amplitude versus frequency 35
6.1 Schematic drawing of environmental and
testing chambers 46
6.2 Loading configuration of the 4-point
f 1 exu retest 46
v i i

6.3Schematic drawing of the impulse hammer
technique apparatus and a typical display
of the Fourier Transform 47
7.1 Transverse moisture strain of Magnolia
epoxy and 3M Scotchply Glass/Epoxy 55
7.2 Plot of the thermal expansion coefficients
in terms of fiber volume fraction of a dry
S Glassf iber/Epoxy at 20°C 56
7.3 Plot of the thermal expansion coefficients
in terms of fiber volume fraction of a dry
Graphite/Epoxy at 20°C 56
7.4 Plot of the moisture expansion coefficients
in terms of fiber volume fraction of a dry
S Glassf iber/Epoxy at 20°C 57
7.5 Plot of the moisture expansion coefficients
in terms of fiber volume fraction of a dry
Graphite/Epoxy at 20°C 57
8.1 Schematic variation of the storage modulus of
epoxy with temperature 67
8.2 Schematic variation of Poisson’s ratio of
epoxy with temperature 67
8.3 Schematic variation of damping of epoxy
with temperature 6S
8.4 Glass transition temperature of epoxy.
From Delasi and Whiteside [6] 68
8.5 Experimental data of the storage modulus
of epoxy as a function of temperature at
diverse constant moisture contents 69
8.6 Experimental data of the storage modulus
of epoxy as a function of moisture content
at diverse constant temperatures 69
8.7 Experimental data of the storage modulus
of epoxy as a function of normalized
temperature (T - Tq)/(T - Tq) 70
8.8 Experimental data of damping of epoxy as
a function of temperature at diverse constant
moisture contents 71
v i i i

8.9 Experimental data of damping of epoxy as
a function of moisture content at diverse
constant temperatures 71
8.10 Experimental data of the storage modulus
of epoxy as a function of normalized
temperature (T - Tq)/(T - Tq) 72
8.11 Experimental data of Poisson’s ratio of
epoxy in term of temperature 73
8.12 Experimental data of Poisson’s ratio of
epoxy in term of moisture content 73
8.13 Experimental data of Poisson's ratio in
term of the normalized temperature
T = (T - T )/(T - T ) 74
n v o g o
8.14 Longitudinal storage modulus (Ej^) of
Glass/Epoxy versus = (T - Tq)/(T^ - Tq). . . 75
8.15 Transverse (E^) and shear (Gj^) storage
moduli of Glass/epoxy versus
T = (T - T )/(T - T ) 75
n o g o
8.16 Longitudinal transverse ( rj22^’
and shear () damping of Glass/Epoxy
versus T = (T - T )/(T - T ) 76
n v o g o'
8.17 Poisson’s ratio () of Glass/Epoxy
versus T = (T - T )/(T - T ) 76
n o v g o'
8.18 Longitudinal storage modulus (Ej^) of
Graphite/Epoxy versus = (T - Tq)/(T - Tq) 77
8.19 Transverse (E^) an<3 shear (Gjg) storage
moduli of Graphite/epoxy versus
T = (T - T ) / (T - T ) 77
n v o' g o'
8.20 Longitudinal (77 ^ ^ ) , transverse (^22^ ’
and shear (^q) damping of Graphite/Epoxy
versus T = (T - T )/(T - T ) 78
n v o' v g o'

8.21 Poisson’s ratio (v ’ ^ ) of Graphite/Epoxy
versus T = (T - T )/(T - T ) 78
n o g o
9.1 Geometry of a laminate and finite mesh
of a 1/4 cross-section area 85
9.2 Description of the applied moisture
grad ients 86
9.3Profile of the hygrothermal stress a
across a [(90/0)^^ Glass/Epoxy laminate
at y/b = 0.472 87
9.4 Profile of the hygrothermal stress
across a [(OO/O^lg Glass/Epoxy laminate
at y/b = 0.472 88
9.5 Profile of the hygrothermal stress
across a [(90/0)2^ Glass/Epoxy laminate
at y/b = 0.472 89
9.6 Profile of the hygrothermal stress a
across a [(OO/OJ^lg Glass/Epoxy laminate
at y/b = 0.993 90
9.7 Profile of the hygrothermal stress a
across a [(OO/OJ^jg Graphite/Epoxy laminate
at y/b = 0.472 91
9.8 Profile of the hygrothermal stress
across a [^O/OJ^jg Graphite/Epoxy laminate
at y/b = 0.472 92
9.9 Profile of the hygrothermal stress
across a [(QO/O^jg Graphite/Epoxy laminate
at y/b = 0.472 93
9.10 Profile of the hygrothermal stress °yX
across a [(OO/O^lg Graphite/Epoxy laminate
at y/b = 0.993 94
10.1 Line style legend of Figures 10.2-13 98
x

10.2
Complex in-plane stiffness A^ of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 99
10.3 Complex in-plane stiffness of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 100
10.4 Complex in-plane stiffness Agg of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 101
10.5 Complex bending stiffness of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 102
10.6 Complex bending stiffness of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 103
10.7 Complex bending stiffness Dgg of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 104
10.8 Complex in-plane stiffness A^ of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 105
10.9 Complex in-plane stiffness A^ of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 106
10.10 Complex in-plane stiffness Agg of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 107
10.11 Complex bending stiffness of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 108

a
10.12 Complex bending stiffness of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 109
10.13 Complex bending stiffness Dgg of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping 110
B.l Organization of the F.E.M. program 131
B.2 Local axes f and , Gauss point numbers
and local node numbers of an eight-node
isoparametric element 132
x i i

NOMENCLATURE
B
*
B *
B"
c
comp 1 ex
in-p1ane
stiff ne s s
comp 1 ex
coup 1ing
stiff ne s s
comp 1 ex
modu1u s
s torage
modu1u s
loss modulus
moisture concentration
c average specific moisture
cm equilibrium moisture concentration
D* .
i J
D , D
X XX
[D]
E11
E22
G12
K
spec ific heat
complex bending stiffness
moisture d i ffusivities
diffusivity matrix, elasticity matrix
longitudinal Young modulus
transverse Young modulus
in-plane shear modulus
thermal conductivity
m weight of absorbed moisture
M percent moisture content
x i i i

initial percent moisture content
M .
i
equilibrium percent moisture content
Q. . transformed stiffness
i J
s
t
T
v
m
w
a .
i
€
V
12
e .
j
complex transformed stiffness
specific gravity
time
temperature
fiber volume fraction
matrix volume fraction
we igh t
coefficient of thermal expansion
coefficient of moisture expansion
s t rain
damping or loss factor
major Poisson’s ratio
fiber orientation of j-th layer
density
stress
Subsc rip t s
1, 2, 3 principal directions of the fibers
f fiber
i initial
j layer number
x i v

L
longitudinal direction
m ma t rix
x, y, z Cartesian coordinates
00 maximum or equilibrium
Superscripts
H mo is ture
o initial
T transpose, thermal
* complex value
real part
imaginary part
xv

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillement
of the Requirements for the Degree of Doctor of Philosophy
HYGROTHERMAL EFFECTS
ON THE COMPLEX MODULI OF COMPOSITE LAMINATES
By
Hacene Bouadi
Apr i 1 1988
Chairman: Dr. Chang-T. Sun
Major Department: Engineering Sciences
The effects of absorbed moisture and temperature on
the complex moduli of composite laminates are investigated
and the mechanisms of moisture diffusion in a lamina are
also analyzed.
First, the variation of the complex moduli of epoxy
in terms of temperature and moisture content are
experimentally determined. Then, the hygrothermal effects
on the complex moduli of composites are derived by using
the complex moduli of the matrix, micromechanical formulas
and experimental data. Only the hygrothermal effects on the
complex moduli of pure epoxy need to be experimentally
determined since these effects on the fibers’ properties
are neg1 igib1e.
xv i

In addition, the effects of hygrothermal environment
on the stress field and material damping of general
laminated composite plates are analyzed. It is shown that
hygrothermal stresses induced directly by moisture and
indirectly by material property changes can be very high,
but the effects on damping are less pronounced.
xv i i

CHAPTER 1
INTRODUCTION
1 . 1 General Introduction
The introduction of advanced composites in aerospace
applications has led to an extensive study of their
mechanical behavior. The amount of experimental and
theoretical findings of composite material researchers made
during the 1960’s was so vast that Broutman and Krock [1]
needed eight volumes to edit a summary of the resulting
know 1 edge.
The interest in composite materials arose from their
ideal characteristics for aerospace structures. Replacement
of the commonly used aircraft material, aluminum, by high
strength/density ratio and versatile composites can lead to
a theoretical 60% weight reduction [2, p. 22]. Due to such
benefits, lower costs and better understanding of their
mechanisms, the use of composite materials has been
increasing slowly but steadily.
Exposure of aircraft structures to high temperature
and humidity in the environment and the tendency of
composites to absorb moisture gave rise to concern about
their performances under adverse operating conditions.
1

2
Therefore, considerable work has been done to understand
the effects of hygrothermal environment on the mechanical
behavior of composite materials.
1.2 Moisture Diffusion
In a 1967 study on the effects of water on glass
reinforced composites. Fried hypothesized that water can
penetrate the resin phase by two general processes, by
diffusion through the resin and by capillary or Poiseuille
type of flow through cracks and pinholes [3], But no
mathematical theory was presented. Later, investigators
established that the primary mechanism for the transfer of
moisture through composites is a diffusion process and
adapted the general theory of mass diffusion in a solid
medium to moisture diffusion in composite materials. The
transfer of moisture through cracks is a secondary
effect [4, 5], Experimental data indicate that for most
composite materials, the diffusion of moisture can be
adequately described by a concentration dependent form of
Fick’s law [4-10].
1.3 Hygrothermal Effects
The degradation of mechanical properties of glass
reinforced plastics exposed to water has long been

3
recognized by marine engineers who use "wet" strengths in
the design of naval structures [3]. Requirements in
aircraft structures are more stringent. The mechanical
properties of materials used in aerospace applications must
be completely characterized. Therefore, the effects of
hygrothermal environment on the elastic, dynamic, and
viscoelastic responses of composites have been studied. To
date, the effects of moisture and/or temperature on the
following performances have been investigated: tensile
strength, shear strength, elastic moduli [3, 11-14],
fatigue behavior [15-17], creep, relaxation, viscoelastic
responses [18-20], dimensional changes [21], dynamic
behavior [22], glass transition temperature [23], etc.
Only tensile and shear strengths and elastic moduli
have been thoroughly studied by many researchers. But data
on the other properties are more limited and hence
inadequate to constitute a good design data-base.
1.4 Scope and Methodology
The present investigation is a combined theoretical
and experimental work and is concerned with predicting the
hygrothermal effects (below the glass transition
temperature) on the complex moduli of composite materials.
This program is undertaken by carrying out the
foil owing s teps:

4
i)The complex moduli of epoxy matrix in terms of
temperature and moisture concentration are obtained by#
using experimental tests and theoretical expressions.
ii)The effects of temperature and moisture on the
complex moduli of unidirectional composites can be derived
by using the complex moduli of the matrix, micromechanics
formulas, and experimental observations. In addition, we
neglect the hygrothermal effects on the fibers.
iii)The effects of hygrothermal conditions on the
stress field and the material damping of some general
laminated composite plates undergoing simple hygrothermal
loadings are analysed.
1.5 Dissertation Lay-Out
Right af ter
the
in t roduc tion
, the
mechanism
o f
moisture diffusion
i s
described in
Chap ter
2,
whe r e
the
absorption of moisture
through thin
compos ite
1aminas
i s
analyzed in detail.
The complex moduli of unidirectional composites are
defined in Chapter 3. Sections 3.3 and 3.4 give the microm¬
echanics formulations of the elastic and complex moduli in
terms of the constituent material properties. The damping
of composites based on the dynamic and complex modulus
approaches is characterized and the equivalence of both
approaches is proven in Chapter 4. In Chapter 5. the
damping and complex stiffnesses of general laminates are

5
derived by using the laminated plate theory, the energy
approach, and the preceding derivations. The complex
stiffnesses are completely expressed in Appendix A.
The environmental conditioning of the test specimens,
the static flexure test, and the impulse hammer techniques
are presented in Chapter 6. These experimental methods,
although simple, are very versatile and are adequate in
determining the necessary data for the purpose of
this investigation.
The theoretical and experimental results are given in
Chapters 7-10. The moisture and thermal expansions of
composites are quantified in Chapter 7. The current
experimental results and data and conclusions of previous
investigators are used in Chapter 8 to model the complex
moduli of epoxy as functions of temperature and moisture
content. In Chapters 9 and 10, the hygrothermal effects on
the stress field across laminates and on damping of
composites are investigated with the help of the results in
the preceding Chapters. The Finite Element Method (F.E.M.)
used in determining the stresses is summarized in
Append ix B.

CHAPTER 2
DIFFUSION OF MOISTURE
2.1 Introduc tion
The mechanism of moisture absorption and desorption
in most fiber reinforced composites is adequately described
by Fick’s law [4]. Fick recognized that heat transfer by
conduction is analogous to the diffusion process.
Therefore, he adopted a mathematical formulation similar to
Fourier’s heat equation to quantify the diffusion
process [24, 25].
2.2 Fickian Diffusion
The Fourier and Fick’s equations, describing the
one-dimens iona1 temperature and moisture concentration, are
respectively given by
(2.1)
3c _ 3 ~ 3c
3t - 3x x dx
(2.2)
where p is the density of the material, is the
6

7
specific heat, T is the temperature, t and x are the time
and apatial coordinates, respectively, is the thermal
conductivity, c is the moisture concentration, and is
the moisture diffusivity.
The moisture diffusivity, D , and the thermal
diffusivity, Kx/(pCv), are the rate of change of the
moisture concentration and the temperature, respectively.
In general, both parameters depend on temperature and
moisture concentration. But experimental data show that,
for most composites, moisture diffusivity does not depend
strongly on moisture concentration [4], Hence, Eq (2.2)
becomes
3c
3 t
D
a2
3 c
x _ 2
dx
(2.3)
and is solved independently of Eq (2.1).
The three-dimensional diffusion in an anisotropic
medium is obtained by generalization of Eq (2.2) as follows
3c , mi —* \
7TT = V. ( [D] . Vc)
where the diffusivity
ma t rix
i s
D
D
D
XX
xy
xz
[D]
-
D
D
D
yz
yy
yz
D
D
D
zx
zy
zx
(2.4)
(2.5)

8
Expansion of Eq (2.4) results in an equation of the form
2 2 2 2
dc d c d c d c d c
§7- = D + D 2_£ + d 2_£ + (D + D )_ ■ g--
3t xxg^2 yyay2 xx^^2 K yz xy'ay ox
R2 R2
+ (D + D + (D + D K° §
zx xz ox oz xy yx'ox oy
(2.6)
if the coefficients D. ,’s are considered to be constant.
i J
2.3 Fickian Diffusion in a Plate
Laminated plates are widely used in the experimental
characterization of composites. Hence, being of practical
interest, the problem of moisture absorption in a plate is
thoroughly discussed in this section.
2.3.1 Infinite Plate
The case of moisture absorption through a material
bounded by two parallel planes is considered. The initial
and boundary conditions of an infinite plate exposed on
both sides to the same constant environment (Figure 2.1)
are given by
T
c
for 0 < z < h and t < 0
(2.7)

9
T = T. j
1 f for z = 0, z = h and t > 0
C = C03 J
where T. is a constant temperature, is the initial
moisture concentration inside the material, and c is the
CO
maximum moisture concentration. It is assumed that the
moisture concentration on the exposed sides of the plate
reaches cra instantly.
The solution of Eq (2.3) in conjunction with the
conditions of Eqs. (2.7) is given by Jost [25]
c -
c
oo
c .
1
c .
1
oo
4 V 1 . 2 ,j + 1 „
W 1 (2jTTTsln ~Sz exp
j=o
^iii)2!72D t
v 2 z
n
(2.8)
Equation (2.8) is plotted in Figure 2.2.
The average moisture concentration is given by
c
â– 'h
c dz
^ o
(2.9)
Substitution of Eq. (2.8) into Eq . (2.9) and integration
result in
c
c
00
c .
1
c .
1
8_
ir
I
j=o
1
( 2 j + 1 )'
exp
.2_2 Dzt
-(2j + 1) U s-
(2.10)
This analysis can be applied to the case of diffusion of

10
moisture into a laminated composite plate so thin that
moisture enters predominantly through the plane faces.
2.3.2 Semi-Infinite Medium
In the early stages of moisture diffusion into a
plate, there is no interaction between moisture entering
through different faces. Therefore, the solution of
moisture absorption into a semi-infinite half-plane is
applicable to a plate for short time.
The initial and boundary conditions of a semi¬
infinite plane (Figure 2.3) exposed to a constant moist
environment are
T
c
for 0 < z < 00 and t < 0
0 and t > o
(2.11)
The solution of Eq (2.3) in this case is [24, 25]
erf
z
2 / D t
L v z J
(2.12)
The rate at which the total specific mass of moisture, m,
is diffusing into the half-plane is

de
dz _
z =0
(2.13)
1 1
dm
d t
D
Thus, the total mass of moisture entering through an area A
in time t is
n t
m = -
pAD
dc
dz
dt = 2pA(c -
z=0
D t
z
17
(2.14)
Equation (2.14) shows that the mass of diffusing substance
is proportional to the square root of time.
2.3.3 Experimental Measurement of Moisture Content
In the case of a finite plate, the total moisture
con tent is
m = pVc
(2.15)
where V is the volume of the test piece. The total moisture
content is experimentally measured by subtracting the dry
weight, w^, from the current weight, w, of the plate, i.e.
m = w
w
(2.16)
A parameter of practical interest is the percent moisture
content defined as

12
M
100
(2.17)
Since
M - M . c - c .
n 7r~ = 1 : M = 100c (2.18)
M - M. c - c . v ’
00 } CO i
the experimentally measured M of Eq (2.18) can be compared
to the analytical value given by Eq (2.10).
2.3.4 Approximate Solution of Moisture Content
Approximate solutions of the specific moisture
distribution in a plate subjected to the conditions given
by Eqs (2.7) are useful, since the difficulty of dealing
with infinite series can be avoided.
Sma11 time. As discussed in section 2.3.2, Eq (2.14)
can be applied during the early stages of absorption. It
yields
c - c . M - M.
i i
Large time. Tsai and Hahn [26, p. 338] suggest that,
for sufficiently large t, Eq. (2.10) can be approximated by
using the first term of the series, i.e.,
D t
z
TJh
(2.19)

13
c
c
00
C .
1
c .
1
8
-9-exp
UZ
IT
D ti
z
(2.20)
Shen and Springer formulation. These researchers
have derived in Ref. [4] the following approximation
c
c
00
c .
1
c .
1
exp
7.3
D t
z
0.75'
1 h2 j
(2.20)
Figure 2.4 shows a comparison of Eqs (2.19-21) with the
exac t so 1u tion.
2.3.5 Edge-Effect Corrections in a Finite Laminated Plate
A plate exposed to a humid environment absorbs
moisture through all its six sides. At small time, the
interaction of moisture entering through different sides is
negligible. Therefore, Eq. (2.19) can be applied to such
cases. It yields
m = 4P(C oo_ ci)
bL /d + bh nr + hL rr rm
V z vx v y J v
(2.22)
where D , D , and D are the diffus ivities in the x, y, and
x y z
z directions, respectively. The geometry of the plate is
shown in Figure 2.5. Rewriting Eq. (2.22) in terms of the
percent moisture content gives

14
M
4M
Dt
' nh2
where the effective diffusivity D is
(2.23a)
»
D
D
1 h
1 + L
x h
_ i
y
D b
D
>
Z N
z
(2.23b)
The micromechanics formulation for diffus ivities proposed
by Shen and Springer [4] and modified by Hahn [26] for
impermeable, circular cross-section, fiber-reinforced
compos i tes is
D
L
D
m
D
T
(2.24)
where D , D_ and D, are the matrix, transverse, and
m T L
longitudinal diffus ivities, respectively. Equation (2.23b)
for a unidirectional lamina with all fibers parallel to the
x-direction can be written as
D
(2.25)

15
For a general laminated plate consisting of N layers with
fiber orientations 0., the diffusivit i es are
J
D
z
D
T
D
x
D
y
N N
D. y h .cos^0 . + D~ T h.sin^0.
L L j j TZ.J j
J = i iüi
N
2",
j = l
N N
D, y h.sin^0. + D„ y h.cos^0.
L L j j 7 L j j
â– 1 = 1 j=l
N
(2.26)
where h. is the thickness of
J
diffusivity of a general
substituting Eqs. (2.26) into
the j-th layer.
1amina t e is
Eq (2.23b).
The effective
obtained by
2.4 Diffusivity and Maximum Moisture Content
The diffusivity and the maximum moisture content
Mra must be experimentally determined in order to predict
the moisture content and distribution in a lamina. These
parameters are obtained by the following procedures:
- a thin, unidirectional composite plate is
completely dried and its weight is recorded,

16
- the specimen is then placed in a constant
temperature and constant relative humidity environment, and
its weight as function of time is recorded,
- the moisture content, M, versus the square root of
time, , is plotted as shown in Figure 2.6.
The maximum moisture content is determined from the
plot and the diffusivity from the following equation
M2 - Ml
= 4M
D
I7h
(2.27)
The subscripts 1 and 2 are defined in Figure 2.6.
The diffusivity depends only on the material and
temperature as follows
D
z
D exp
o
(2.28)
where R is the gas constant, D and E, are the
o a
pre-exponential factor and the activation energy,
respec tively.
Experimental research has shown that the maximum
moisture content depends on environment humidity content
and material. For a material exposed to humid air [4], the
equilibrium moisture content can be expressed as
M
00
(2.26)

17
where 0 is the relative humidity, a aad b are material
cons tan t s.

IS
>
>
Moisture
>
5»
Fig. 2.1 Plate subjected to a constant humid environment
on both sides.
z =* 0 is the center of the
cross—section of the plate
z/h
Fig. 2.2 Moisture distribution across a plate. The
numbers on the curves are the values of
(c -
c.)/(c -
Í J v co
ci).

(M - M,)
19
Fig. 2.3 Semi-infinite plate in a humid environment.
Exact
One—term
Shen and Springer
Fig. 2.4 Comparison of the exact specific moisture
concentration equation with some approximate
solutions.

Moisture content (%)
20
Fig. 2.5 Geometry of a plate
i
Fig. 2.6 Moisture content versus square root of time. On
the curve J t ^ < J t^ < y t^ and the slope is
f°r/T < yr^.
cons tan t

CHAPTER 3
COMPLEX MODULI OF UNIDIRECTIONAL COMPOSITES
3.1 In t roduction
Composite materials, such as Glass/Epoxy and
Graphite/Epoxy, have a polymeric matrix. Therefore, they
display viscoelastic behavior. Some of the effects of this
time-dependent phenomenon are: stress relaxation under
constant deformation, creep under constant load, damping of
dynamic response, etc.
This chapter is an introduction to the dynamic
behavior of viscoelastic composites in terms of complex
modu 1 i .
3.2 General Theory
A usual representation of the one-dimens iona1
stress-strain relation of a viscoelastic material subjected
to a harmonic strain history of the form
€( t) = €°e1CJt (3.1)
21

22
is given by
a(t) = B (iw)€°e^Wt = B (icj)G(t) (3.2)
The complex modulus B can be decomposed into its real and
imaginary parts as follows
B*(iu) = B ' ((i)) + i B" ( gj ) (3.3)
The terms B' and B” are called the storage and loss
moduli, respectively, and the ratio of the loss over the
storage modulus
D »*
V = g-r (3.4)
is referred to as either the loss factor or damping. The
loss modulus is a measure of the energy dissipated or lost
as heat per cycle of harmonic deformation.
3.3 Micromechanics Formulation of Elastic Moduli
The longitudinal modulus E^, the transverse modulus
E22• the in-plane shear modulus G^, and the major
Poisson’s ratio v12 can be obtained by using the rule of
mixtures and the Halpin-Tsai equations, viz.,
+ v E
m m
E = v E
11 f f 11
(3.5)

23
E
22
E
m
1 + 2rijV
1 - n ^ v
(3-6)
1 + n9v
G10 = G t ^-3-
12 ml- n2vf
(3.7)
wher e
The subscripts f
respectively, and v
n, =
n~ =
= VfUf12 +
V V
m m
(3.8)
(Ef22/Em>
- i
(3.9)
(Ef22/Em)
+ 2
(Gfl2/Gm>
- 1
(3.10)
(Gfl2/Gm)
+ 1
m stand
for fiber and
ma t rix,
v are the
m
volume fractions
Como lex Moduli
The micromechanics formulations of the complex moduli
are obtained by applying the e1astic-viscoe1astic
correspondence principle [27-29], i.e., by undertaking the
following steps:
i) determining the elastic moduli of composites in
terms of the constituent material properties,

ii) replacing the elastic moduli of fibers and matrix
by corresponding complex expressions.
For a viscoelastic composite. the properties of the
constituent materials are
Efll ' “fll + lEfll
^f22 ~ Ef22 + lEf22
G, =
G1 + iG"
E =
E' + iE”
m m
(3-11)
G =
m
G1 + iG"
m m
v =
V + IB
m m
J f12 _ L f 12
The bulk modulus of epoxy matrix, K , is real and
m
independent of frequency [2S]. It is given by
-
•%n _ 3V 1 - 2v )
v m'
(3.12)
while the viscoelastic bulk modulus is obtained from the
correspondence principle

25
E' + iE"
m m
K* =
m 3[1 - 2(u' + iu")]
m
m •
(3.13)
Separation of the real and imaginary parts of Eq. (3.13)
yields
K
*
m
(1
2d ')E’
m m
3 1(1
2u"E" + i[2E"d" + E'( 1
m m L mm m
2u')2 + 4d"2 J
m ’ m J
2d ' )]
m'J
(3.14)
Since the dilatation bulk modulus is real, the imaginary
part of Eq. (3.14) is equal to zero; hence
2E"d" + E”(1 - 2d‘) = 0 (3.15)
mmmv m' v '
Equation (3.15) results in
m,
d" = =^(d * - 0.5)
m E v m ’
(3.16)
m
The shear modulus of the matrix is given by
G* =
m
m
3K E*
m m
2(1 + d*) 9K - E*
v m' mm
(3.17a)
Separating the real and imaginary parts and neglecting the
2
terms of the form (E”) yield

26
3K E'
P* nn m
m “ 9K - E
m
1 + i
9K E"
m m
9K - E’ E*
m mm
Introduction of the material properties
p = ft
m E
'f 1 1
f 1 1
E ’
f 11
f 22
f 22
E ’
f22
m
9K
m
'Gm " G 1 “ 9K - E’ 'm
m mm
into Eq s.
(3.11)
r e su 1t s
i n
E f 11 ( 1 + 1T7f 11
E f 22 ^ 1 + 1T?f 22
G f 12 ( 1 + 1T?fl2
E'(1 + ip )
m v m'
G 1 ( 1 + ipp )
mv Gmy
)
)
)
(3.17b)
(3.18)
(3.19)

27
*
v
m
v ' +
m
ÍT7 ( v '
m m
U f 12 “ U f 12 “ u f 1 2
There are no satisfactory data on the shear and transverse
damping of fibers. Fibers have damping with a magnitude
order ten times smaller than epoxy. The dampings ^fii’
T7f22> and q^.^ are assumed to be equal and are replaced by
t]j. in subsequent equations. Since the fiber damping, q^. ,
is much smaller than the matrix damping, q , the
m
imaginary part of the fiber Poisson’s ratio is neglected.
The preceding assumptions have a negligible effect on the
complex moduli of composites.
Application of the e1astic-viscoelastic
correspondence principle to Eqs. (3.5)-(3.10) and
substituting them into Eqs (3.19) yield the following
complex material properties
E.. = V.E' ,(1 + iqr) + v E'(l + iq )
11 ffllv f’ mmv my
E
x
22
E'(1 + iq )
mv m
l + 2njVj.
' *
1 - nivf
(3.20)
12
= G'n + iTw)
m
, *
1 + n2Vf
Gm; . ^
1 ' n2Vf

whe r e
Ef22< 1 + ‘V ~ Em(1 * lT|m>
Ef22(I + ‘"fl + 2Em(1 * iT>m>
Cf 12^ 1 * lr|f ) ~ Gm*1 * 1’1Cni*
Gf]2(l + iuf) + Gm(l + mGm)
The elastic moduli given by Eqs. (3.5)-(3.8) model
experimental results with a good accuracy [2], Therefore,
they are used instead of mathematically exact
micromechanics formulas, such as those derived by
Hashin [27. 29],

CHAPTER 4
DAMPING
4.1 Damping Mechanisms
Any vibrational energy introduced in a structure
tends to decay in time. This phenomenon is called damping.
There are two types of damping mechanisms, external or
nonmaterial and internal or material.
4.1.1 Nonmaterial Damping.
Two common types of external damping are
- Accoustic damping: a vibrating structure always
interacts with the surrounding fluid medium (air, water,
etc.). This effect can lead to noise emission and even to
changes of the natural frequencies and mode shapes. Thus,
mechanical responses might be modified.
- Coulomb friction damping: two contacting surfaces
in relative motion dissipate energy through frictional
forces.
29

30
4.1.2 Material Damping
There are many damping mechanisms that dissipate
vibrational energy inside the volume of a material. Damping
phenomena include thermal effects, magnetic effects, stress
relaxation, phase processes in solid solutions [30, p. 61],
etc.
The internal damping of polymeric matrix composites,
such as Glass/Epoxy and Graphite/Epoxy, is dominated by
viscoelastic damping.
4.2 Characterization of Damping
4.2.1 Free Vibration
A cantilever under free vibration oscillates
regularly with an amplitude that decreases from one
oscillation to the next one (Figure 4.1). A measure of
damping is the logarithmic decrement defined as
5
1 n
(4.1)
whe r e
A^ = amplitude of the n-th cycle
^n + N = amPlituc*e the (n+N)-th cycle
The damping defined in Eq (4.1) is applicable to viscous

31
damping and for hysteretic damping that is represented by a
complex modulus approach.
4.2.2 Steady State Vibration
Damping also influences the dynamic equilibrium
amplitude of structures (e.g. beams) that undergo harmonic
oscillation. A resonance usually occurs (Figure 4.2). The
following measure of damping is used
V
(j.
- (jj.
(i>
o
(4.2)
whe r e
o
resonant frequency
"l-
w2 =
frequencies on either sides of such that
the amp 1itude
is 1/y 2 times the resonant
amp 1itude.
I n the
case
of a vibration
induced by the force
f(t) = Fsin(ut)
the response (deflection), w(t), is out of phase with f(t)
by an angle e such that
w( t)
Wsinfut + e)

32
The work done per cycle is
â– 217/u
D
f ( t)^- dt = I7WF sin(e)
(4.3)
J o
The strain energy stored in the system at the maximum
displacement is half the product of the maximum
displacement by the corresponding value of the force, i.e.,
U = ^FW cos(e)
(4.4)
There is no damping if the work done per cycle is zero,
i.e, if sin(e) = 0.
The ratio of energy dissipated in a cycle to energy
stored at the maximum displacement is another measure of
damping. Therefore, the damping is
(4.5)
The definitions of damping given by Eqs. (4.2) and (4.5)
are equivalent [33].
4.2.3 Complex Modulus Approach
The one-dimensional stress-strain relation of a
viscoelastic material undergoing harmonic motion has been
shown to be (Eq. (3.2))

33
°* ( t) = EH((o)€o ei(Jt = (E ' ((J) + iE” (<>;)) €q eiwt (4.6)
Noting that i |u | € = d€/dt, Eq. (4.6) can be written as
^ 1(Jt . E . l (J t ^
a (t) = € e + -i—r íue €
o co o
(4.7)
The real part is given by (after algebraic manipulation)
a(t) = E' e^sinfut + t)'! i + rj
+ e ) -11 + r/2
(4.8)
where 77 = tan(e) = E"/E'
The energy dissipated during a cycle per unit volume is
D = () a d€ =
x
â– 217/(0
d€
x
d t
d t =
UqE'e2
o
(4.9)
The maximum energy stored is
1 2
u = E* €~
2 o
(4.10)
The r e f o r e,
V
(4.11)

34
Hence, the definitions of damping given by Eq. (3.4) and
Eq. (4.5) are equivalent. This conclusion is also valid for
general cases of structural vibration.

35
Fig. 4.1 Schematic drawing of a free-clamped beam under
free vibration and plot of its deflection
versus time.
4.2
Schematic drawing
forced vibration
versus time and
f requency.
of a free-clamped beam under
and plots of the deflection
deflection amplitude versus
Fig .

CHAPTER 5
DAMPING AND STIFFNESSES OF GENERAL LAMINATES
5.1 Introduction
Both the laminated plate theory and the energy method
approaches for analyzing the damping and the stiffnesses of
general laminates are presented in this chapter.
5.2 Laminate Plate Theory Approach
Four independent parameters are needed to determine
completely the damping of a unidirectional composite. But,
the analysis of the material damping of a general laminated
composite requires the use of eighteen parameters. These
quantities are the ratios of the imaginary over the real
parts of the complex in-plane stiffnesses A. ,’s, the
i J
and the complex bending
The terms A. ,’s, B. ,'s, and
1 J i J
and D. ,’s are defined as
i J
â– +h/2
A. .
i J
dz
J -h/2
36

37
r+h/2
-h/2
(5.1)
0
P+h/2
J-h/2
2tt*
2 Qij
dz
where the complex transformed stiffness Q_’s depend on
* * *
Ell’ E22’ G12’
the 1amina t e.
12
and the orientation of each layer of
The in-plane, coupling, and flexural material damping
are defined as
I77! j
A 7 .
i J
a: .
i j
C^ij
B7 .
-U.
b: .
i j
(5.2)
r-7? • •
F i J
D7 .
i i
d: .
1J
respec tively.
5.3 Energy Method Approach
The energy method can be used to
damping of laminated composite materials
determine the
under certain
loading and boundary conditions. The damping of a laminated

38
composite material in the first mode of vibration can be
defined as
N
^ ^ k^d ^ cyc.
V = ^ (5.3)
I 2n k“s
k= 1
where N is the total number of layers, U.) is the
vk d'cyc.
energy dissipated in the k-th layer during a cycle, and ^_Us
is the maximum energy stored in the k-th layer. The storage
and the dissipated energy are given by
k
U
s
2
€ . C '. . € .
V J J1 1
k
dV
kUd
U
€ C" €
V J J'1 1
k
dV
(5.4)
where i and j are
C'.' . are the real
J i
stiffnesses and V.
k
’’total” damping of
the material principal axes, Cj . and
and imaginary parts of the complex
is the volume of each layer. Hence, the
an N-layered laminate is given by
(5.5)

39
The maximum strain vector {€} can be determined by the
finite element method first. Then, the damping can be
deduced. Equation (5.5) is used to determine the damping of
a beam with variable thickness or of more general
s t rue tures.

CHAPTER 6
EXPERIMENTAL PROCEDURES
6 . 1 Introduction
A description of the test specimens and the
experimental procedures of the present investigation is
given in this chapter.
6.2 Test Specimens
The test specimens used to determine the complex
moduli of epoxy and of composite materials are thin strips
of approximate dimensions, 150mm by 25mm by 2mm. The only
materials tested are Magnolia 2026 laminating epoxy and 3M
Scotchply Glass/Epoxy. The curing temperatures of the epoxy
and the Glass/Epoxy are 175°C and 170°C, respectively. The
initial properties of these materials (at 20° C and without
moisture), as well as those of a typical Graphite/Epoxy,
are given in Table 6.1.
40

41
6.3 Environmental Conditioning
The specimens are conditioned in a Thermotron
environment chamber at a constant temperature and constant
relative humidity. The weight gain of the test pieces as a
function of time is monitored. Right after moisture
equilibrium is reached, the specimens undergo all tests at
diverse temperatures inside a testing chamber connected to
the environment chamber (Figure 6.1). The range of
temperature achieved inside the environment chamber is 4°C
to 90°C and the range of relative humidity is 4% to 99% for
temperatures below 75°C. As temperature increases, the
highest relative humidity that can be obtained decreases
steadily to 75% at 90°C.
6.4 Four-Point Flexure Test Method
The Young’s modulus and the Poisson’s ratio can be
determined with the four-point flexure test method. The
loading configuration of this test is shown in Figure 6.2.
The elastic flexural analysis yields [31]
E
PI'
Sbh^w
(6.1)
where E is the effective modulus, P is the applied load,
1 is the length of the specimen, b is the specimen width, h

42
is the thickness, and w is the deflection at quarter-point.
Poisson’s ratio is expressed as
(6.2)
where the transverse strain € is measured
y
transverse strain gage cemented in the middle
specimen.
with a
of the
6.5 Impulse Hammer Technique
The material damping and the storage modulus of a
one-dimensional thin beam are determined with the impulse
hammer technique. This technique was pioneered by Halvorsen
and Brown [32]. The equipment set-up is shown in
Figure 6.3. The specimen is clamped inside the testing
chamber. A force impulse is applied to the test piece by a
force transducer. The end displacement of the specimen is
recorded with a non-contacting motion transducer. Both
responses from the force and motion transducers go through
signal conditioning equipments (filters, amplifiers). These
responses are digitized in a Fast Fourier Transform
analyzer (FFT) to obtain the transfer function in terms of
the frequency. The transfer function is defined as the
ratio of the Fourier Transform of the output (displacement

43
v(t)) over the Fourier Transform of the input (force
impulse u(t)); that is.
H(f)
_ mi
~ U(f)
(6.3)
where
t = time
f = f requency
V(f) = Fourier Transform of v(t)
U(f) = Fourier Transform of u(t)
The real and imaginary parts of H(f) are displayed on the
FFT analyser CRT (Figure 6.3). The material damping defined
by Eq. (4.11) is experimentally obtained by the following
expr e s sion
T7
(fa/fb>2 - 1
2 + 1
(6.4)
where the frequencies f and f, are defined in Figure 6.3.
a b
The storage modulus is expressed as [33, p.464]
2 1 ^
E' = 38.32 f p^r (6.5)
r li
where f is the resonant frequency in Hz. , p, is the
material density, 1 is the length of the specimen and h is
the thickness of the specimen. Equation (6.5) is valid only

44
for the case of the first mode free vibration of a
clamped-free beam. A complete description and analysis of
the impulse hammer technique are presented in Lee’s
dissertation [34].

45
Tab le 6.1
Initial properties of Magnolia 2026 epoxy,
3M Scotchply Glass/Epoxy, and a typical
Graphite/Epoxy composite.
Proper ties
Epoxy
G1as s/Epoxy
Graphite/Epoxy
Vf
0.50
0.70
p (g/cm3)
1.25
1.93
1.6
En (GPa . )
4.0
37.00
155.23
E22 (GPa‘)
4.0
11.54
10.81
G12 (GPa.)
1.52
3.46
4.35
U 1 2
0.32
0.285
0.217
*11
0.018
0.0023
0.0019
*22
0.018
0.015
0.0078

46
Fig. 6.1
Schematic drawing of environmental and testing
chamber s.
Loading configuration of
f1exure test.
Fig. 6.2
the
4-poin t

47
Fourier Transform
Real part
Im. part
Schematic drawing of the impulse hammer
technique apparatus and a typical display of
the Transfer Fourier Transform.
Fig. 6.3

CHAPTER 7
HYGROTHERMAL EXPANSION
7.1 Introduction
When a metallic or composite structure is subjected
to a change of temperature, there are dimensional
variations and there may be stress development. For a
one-dimensional case, it is assumed that the thermal strain
is given by
€T = a.(T - T ) = a.AT
i i o 1
where
= coefficient of thermal expansion
T = actual temperature
Tq = reference temperature.
A polymer matrix composite exposed to a humid
environment absorbs moisture. Hence, it increases in weight
and dimensions. This situation produces a moisture strain
that varies linearly with moisture concentration [26]. In
the one-dimensional case, the hygros train is given by
(7.1)
48

49
€H = /3 . (c - c ) = 0 . Ac (7.2)
i iv o' i v ’
where c is the initial moisture concentration and 6. is
o 1
the coefficient of moisture expansion.
7.2 Coefficients of Thermal Expansion
In the case of laminated composite plates, three
coefficients of thermal expansion are used in determining
the thermal strains. These parameters can be written in
terms of fiber and matrix properties. The micromechanics
formulas for a unidirectional orthotropic lamina are given
by (see Refs. [35, p. 24], and [36, p. 405] for a detailed
de riva tion)
“l =
v Pa PE P + v a E
iff m m m
En
a2 =
vfaf +
v a
m m
vfufaf
v v a -
m m m
U12al
(7.3)
where the subscripts 1 and 2 represent the fiber and the
transverse directions. The thermal expansion coefficients
of an orthotropic lamina whose fibers make an angle 0
with the x-direction (Figure 2.5) are given by

50
2 2
a = a, cos 0 + o' sin 0
x 1 2
2 2
a = a, sin 0 + a_ cos 0
y 1 2
(7.4)
a = 2(a, - a~) cos 0 sin 0
xy v 1 2'
7.3 Coefficients of Moisture Expansion
Similarly, the coefficients of moisture expansions of
an orthotropic lamina with impermeable fibers can be
expressed as [36, p. 406]
sE
p 1 = —£—p
1 s E, , m
m 11
P2 = f-O + vm)Pm - vl2Px (7.5)
m
* 12 = °
where s and s^ are the specific gravities of the composite
ma terial and the ma trix. The moisture expansion
coefficients expressed in an axis system such that the
x-direction makes an angle 0 with the fibers are given by
Eqs. (7.4) after replacement of a¿'s ^7 P^'s-

51
7.4 Experimental Data
7.4.1 Previous Investigations
Hahn and coworkers’ investigations [21, 26, 37] of
swelling of composites are outlined in this section. Some
of the typical results of the transverse strain versus
percent moisture gain are obtained by conducting the
following tests: absorption is conducted in moisture
saturated air such that Eqs (2.2) and (2.7) are satisfied;
while desorption takes place in vacuum at the same
temperature. Their data show a hysteretic nature of
swelling in this case.
But, when swelling of composites is given in terms of
moisture concentration, the average behavior of
S2-G1ass/Epoxy, Kevlar 49/Epoxy and Graphite/Epoxy can be
approximated by
0.43c = P2c (7.6)
Since the data presented in their publications display a
wide scatter, Hahn et al. suggest that Eq. (7.6) can be
used to estimate the moisture strains for most composite
materials.

52
7.4.2 Present Investigation
Epoxy and Glass/Epoxy specimens are conditioned at a
constant relative humidity until the absorbed moisture
reaches equilibrium. The changes in transverse dimensions
are measured. This procedure is repeated at diverse values
of relative humidity. The results are plotted in
Figure 7.1. The longitudinal swelling strains could not be
measured since the micrometer calipers
used
were not
suf ficiently
accurate. These data yield
the
foil ow i ng
expe rimen ta1
values
Pm(epoxy) = 0.25
(7.7)
^2(G1ass/Epoxy) = 0.48 (7.8)
Substitution of Eq. (7.7) and the parameters given in
Table 6.1 into Eqs. (7.5) yields the following empirical
values
j3 = 0.042 (7.9a)
P2 = 0.47 (7.9b)
for the 3M Glass/Epoxy composite. The experimental and
empirical values of 02 are practically equal. Hence, the

53
present results differ slightly from the approximation
given by Eq. (7.6).
The above coefficients and the typical coefficients
of expansion of graphite are quantified in Table 7.1, while
the storage moduli and the density of glass and graphite
fibers are listed in Table 7.2. These properties are used
to plot the thermal and moisture expansion coefficients
versus the fiber volume fraction of Glass/Epoxy and
Graphite/Epoxy in Figures 7.2 through 7.5.
The values in these plots are valid for dry
composites at room temperature. Since the storage modulus
of epoxy varies with temperature and moisture content, this
additional effect is investigated in Chapter 8.
In general, the thermal expansion coefficients are
functions of temperature, but this temperature effect is
negligible below 100°C. Therefore, in the subsequent
chapters, the thermal expansion coefficients are assumed to
be independent of temperature.

54
Table 7.1 Coefficients of moisture and thermal expansion
of epoxy and graphite and glass fibers.
Epoxy
Glass
Graphite
a (pm/m)/K
54.0
5.0
0.2
P
0.25
0.0
0.0
Table 7.2
Proper ties
of Glass and
graphite Fibers.
Glass
Graphite
Efii(GPa)
70.0
220.0
E f 22 ^ GPa)
70.0
16.6
Gf12(GPa)
28.7
8.27
71 f
0.0015
0.0015
V{ 12
0.22
0.16
p (g/cm3)
2.60
1.75

55
Moisture concentration (%)
Fig. 7.1
Transverse moisture strain of Magnolia
and 3M Scotchply Glass/epoxy.
epoxy

Thermal expansion coefficient (/im/rn)/K ^ Thermal expansion coefficient Oim/m)/K
56
Longitudinal
transverse
2 Plot of the thermal expansion coefficients in
terms of fiber volume fraction of a dry
S Glassfiber/Epoxy at 20°C.
Longitudinal
Transverse
0 0.2 0.4 0.6 0.3 1
Fiber volume fraction
Fig. 7.3 Plot of the thermal expansion coefficients in
terms of fiber volume fraction of a dry
Graphite/Epoxy at 20°C.

Coefficient of moisture expansion ^ Coefficient of moisture expansion
57
Longitudinal
Transverse
Plot of the moisture expansion coefficients in
terms of fiber volume fraction of a dry
S G1assfiber/Epoxy at 20°C.
Fiber volume fraction
Longitudinal
Transverse
Plot of the moisture expansion coefficients in
terms of fiber volume fraction of a dry
Graphite/Epoxy at 20°C.
Fig. 7.5

CHAPTER 8
HYGROTHERMAL EFFECTS ON COMPOSITE COMPLEX MODULI
S.1 Literature Survey
The storage moduli (real parts of Eqs. (3.11)) of
composites are usually determined by dynamic testings, such
as the technique described in section 6.5. They can be
approximated by using static tests [38].
Shen and Springer [12] investigated the environmental
effects on the elastic moduli of a Graphite/Epoxy composite
and made a survey of existing data showing the effects of
temperature and moisture on the elastic modulus of several
composites. Their conclusions are listed below.
i) The hygrothermal effects on the 0° fiber
direction laminates are negligible.
ii) For 90° fiber direction laminates, the
hygrothermal effects on the modulus are insignificant in
the 200K to 300K temperature range. But, between 300K and
450K, the hygrothermal effects on the modulus are
impor tan t.
Putter et al. [38] investigated the influence of
frequency and environmental conditions on the dynamic
58

59
behavior of Graphite/Epoxy composites. Their overall
conclusions are
i)The effects of frequency on the modulus and
damping are quite small in all cases.
ii)The effects of frequency on the modulus and
damping are relatively greater for matrix-controlled
laminates at higher frequencies (above 400 Hz.).
iii)At the same temperature, damping increases with
moisture saturation. But for dry laminates, damping
decreases slightly as temperature increases.
From all these experimental works, a general summary
can be drawn: the influence of hygrothermal conditions on
the elastic modulus, dynamic modulus and damping of
composites is matrix dominated.
8.2 Theoretical and Experimental Assumptions
Since the hygrothermal influence on composite
properties is matrix controlled [12, 38], the fiber
properties are assumed to be constant at any temperature
below the glass transition temperature and at any moisture
content. Therefore, to obtain the values of the complex
moduli of composites, it is sufficient to know how
temperature and moisture affect the complex moduli of the
epoxy matrix, and then use the micromechanics formulations
given by Eqs. (3.20). Thus, only the following functions

60
E 1
m
E 1
m
(T.c)
u ' = u ’ (T . c ) (8.1)
mm v ’
t] = r) (T , c )
m mv '
need to be experimentally evaluated. The constant fiber
properties are given in Table 7.2. The effects of frequency
are negligible below 400 Hz. The results of this
investigation are not accurate for higher frequencies since
their effects have not been taken into account.
The qualitative influence of temperature only on the
storage modulus, real part of Poisson’s ratio, and damping
of epoxy is illustrated in Figures 8.1-8.3. There are three
distinct regions. At room temperature (in the glassy
region), the storage modulus, Poisson’s ratio, and damping
of epoxy are equal to about 4.0 GPa, 0.35, and 0.018,
respectively. In the glassy region, the storage modulus
decreases slowly, while Poisson’s ratio and the damping
increase as temperature increases. In the next region
(transition region), the storage modulus decreases rapidly,
and both Poisson’s ratio and damping reach their maximum
values. The last region is the rubbery region where the
modulus takes a very low value, and all three parameters
stay relatively constant.
Typical values of the modulus in the rubbery region
-2
could be 10 times the glassy modulus or lower. The damping

61
can
reach a
value of 1 or even 2
in the
transition region
[30,
p. 90].
Poisson’s ratio reaches the
1 imiting
value o f
0.5,
wh i ch
is approximated by
incompressible
rubber s
[39,
p. 293]
The position of the transition region depends on the
moisture concentration. The effects of moisture on the
glass transition temperature, T^_, of six epoxy resins have
been determined by Delasi and Whiteside [6]. These results
are plotted in Figure 8.4. They are compatible with the
data of McKague [40] and satisfy the theoretical relation
derived in Ref. [41, p. 69].
8.3 Modeling of Epoxy Properties
The observations of the preceding section are used
for modeling the material properties of epoxy that are
given by Eq. (8.1).
The glass transition region of epoxy resin is not
broad [6], therefore, a glass transition temperature is
used instead. The temperature T^ is usually obtained by
measuring the expansion of a specimen as function of
temperature. The point where the epoxy stops expanding as
temperature increases corresponds to the first deviation
from the glassy state and is termed T .
According to the experimental data plotted in
Figure 8.4, T^ is stongly dependent on absorbed moisture.
These results show that, as the moisture content of epoxy

62
increases, the transition temperature moves to the left in
Figures S.1-8.3. Hence, the abrupt change of the material
properties starts at a lower temperature as the moisture
content increases. This fact and the conclusions reached by
previous investigators [12] suggest that the following
modelings of E', v', and n' are appropriate,
mm m
(8.2)
(8.3)
(8.4)
where the temperature Tq is equal to 273K. The moisture
concentration appears implicitly in T . The glass
transition temperature is represented by
Tg = 210 exp(- 9c) (°C) (8.5)
where c is the moisture concentration.
This modeling has been chosen so that it does not
represent the material properties beyond T , since the
study of epoxy in the rubbery stage is not within the scope
of this research. Equations (8.2)-(8.4) are valid only for

63
the continuous parts of the curves plotted in
Figures 8.1-8.3. ,
8.4 Results
All test specimens are conditioned in a constant
relative humidity environment until moisture equilibrium is
reached. Then, the test pieces undergo the impulse hammer
technique and the four-point flexure tests to determine the
storage moduli, the material damping, and Poisson’s ratio
at several temperature and moisture contents.
8.4.1 Complex Moduli of Epoxy
Storage modulus. The experimental data on the storage
modulus of epoxy in term of temperature at three different
equilibrium moisture concentrations are plotted in
Figure 8.5.
It
can be concluded
tha t
increase in either
temperature
or
moisture
content
or
both
results in a
decrease in
the
s t orage
modu1u s.
P1 o 11ing
these data in
terms of moisture content in Figure 8.6 does not lead to
any additional insight. But, representing these results in
term of the following normalized non-dimensional
temperature
(8.5)

64
in Figure 8.7 shows a clear trend. Experimental studies
have shown that the modulus of polymer is very low at the
glass transition temperature, therefore, adding the value
E =0 for T = T to the data yields the following modeling
m g
E’ = 4.0(1 - T ) (GPa)
m v nJ y '
(8.6)
Material damping. Similarly, the experimental data of
the hygrothermal effects on the damping of epoxy are
plotted in three Figures (8.8-8.10). There is very little
change in damping for all the considered conditions.
Therefore, it is proposed to let
r, = 0.018 (8.7)
for temperatures up to 80°C and moisture contents up to 5%.
The conclusion that the hygrothermal effects on the damping
of epoxy is negligible is qualitatively corroborated by
Putter et al. [38]. A quantitative comparison cannot be
made since these researchers have not included in their
publication the values of the fiber volume fraction and
moisture content of the test specimens.
Poisson’s ratio. The experimental values of the
Poisson’s ratio in terms of temperature at two different

65
moisture contents. are plotted in Figure 8.11. These
results show that Poisson’s ratio increases at a negligible
rate as temperature varies from 0 to 80°C. Representing the
same data in terms of the moisture content up to M = 4.5%
(Figure 8.12) shows that the moisture effect is also
negligible. Therefore,
v’ = 0.32 (8.8)
m v ’
for temperatures up to 80°C and moisture contents up to 5%.
Since u ’ equals 0.5 at the glass transition temperature
m
(T = 0), the plot of Poisson’s ratio versus the normalized
temperature has been extrapolated as shown in Figure 8.13.
The extrapolation displays a qualitative trend only.
8.4.2 Complex Moduli of Composites
The complex moduli of Glass/Epoxy and Graphite/Epoxy
in terms of moisture content and temperature can be
determined by using the fibers’ properties given in
Table 7.2, Eqs . (8.6) through (8.8) and the micromechanics
formulas (Eqs. (3.20)).
This procedure is illustrated by determining the
storage moduli and the damping of a Glass/Epoxy lamina with
a fiber volume fraction of 0.5 and a Graphite/Epoxy lamina
with a fiber volume fraction of 0.7.

66
G 1 a s s/Epoxy . The parameters E’^, E 22' ^12
'll 1
T]22• t7q . and v [2 versus the normalized temperature are
plotted in Figures 8.14-17. The experimental data
substantiate the theoretical results.
Tli
Graphi te/Epoxv . Similarly, E’^, ^’22’ *^12
11 ’
T,22’ , and ujg versus the normalized temperature of
Graphite/Epoxy are plotted in Figures 8.18-21.
For both Glass/Epoxy and Graphite/Epoxy, the results
show that the matrix-dominated parameters (E^ and G’g) are
strongly affected by moisture and temperature, while the
fiber-dominated parameters (E^, stay practically
cons tan t.

67
Fig. 8.1 Schematic variation of the storage modulus of
epoxy with temperature.
Fig. 8.2 Schematic variation of Poisson’s ratio of epoxy
with temperature.

Glass transition temperature, Tg (°C) ^ Damping
68
Tg
Temperature
.3 Schematic variation of damping of epoxy with
temperature.
250
Fig. 8.4
Glass
De lasi
transition temperature of epoxy. From
and Whiteside [6].

69
0.0%
2.90%
3.70%
Fig. 8.5 Experimental data of the storage modulus of
epoxy as a function of temperature at diverse
constant moisture contents.
20 °C
50 °C
70 °C
Fig. 8.6 Experimental data of the storage modulus of
epoxy as a function of moisture content at
diverse constant temperatures.

Storage modulus (GPa)
TO
Normalized temperature
Fig. 8.7
Experimental data of the storage
epoxy as a function of normalized
(T -
T ) / (T
o g
" T )
o
modulus of
temperature

o u
3.703
a.
E
a
a
0.015
I
0.01 -
0.005 j-
o r . . . ■ ■ ■ I ■ • >
0 20 40 SO 30 100
Temperature (°C)
Fig. 8.8 Experimental data of damping of epoxy as a
function of temperature at diverse constant
moisture contents.
* T = 20 °C
O T = 50 °C
□ T = 70 °C
Fig. 8.9 Experimental data of damping of epoxy as a
function of moisture content at diverse
constant temperatures.

72
Normalized temperature
* Experimental data
Fig. 8.10
Experimental data of the storage
epoxy as a function of normalized
(T - T )/(T
o; g
T )
o
modulus of
temperature

73
QZ
4.17S
Fig. S.ll Experimental data of Poisson’s ratio of epoxy-
in term of temperature
* T =* 20^
O T = 50 °C
d T => 75°C
Fig. 8.12 Experimental data of Poisson’s ratio of epoxy
in term of moisture content.

Poisson’s ratio
74
Fig
^ Experimental data
Fit to data
• • • • Extrapolation
8.13 Experimental
term of the
(T - T )/(T
o' g
data of Poisson's ratio in
normalized temperature =

75
Theoretical
* Experimental data
Fig. 8.14 Longitudinal
s to rage
modu1u s
Glass/Epoxy versus = (T - Tq)/(T
(Eil)
- v-
o f
Theoretical E’„
22
+ Experimental EI,2
Theoretical G^2
Fig. 8.15 Transverse (E^) an<^ shear (G^) storage moduli
of Glass/epoxy versus T = (T - T )/(T - T ).
n o g oJ

76
Theoretical *1,,
^ Experimental 7) n
• • • • Theoretical l]^
O Experimental 7)^
Theoretical 71
6
Fig. 8.16
Long itudi na 1
shear
(vG)
(Dll).
damping
n
= (T -
T ) / (T - T ) .
o g o
transverse (^22
of Glass/Epoxy
Theoretical
* Experimental
Fig. 8.17
Poisson ' s ra t i o
T = (T - T )/(T
n o g
(°i2)
- v-
of Glass/Epoxy
) , and
versus
versus

180
a
CL
a
160
0)
3
3
TJ
O
140
QJ
O'
a
o
-*-<
tn
120
0 0.2 0.4 0.6 0.8 1
Normalized Temperature
Fig. 8.IS Longitudinal storage modulus (E^) of
Graphite/Epoxy versus T = (T - T )/(T - T ).
n o g o
Transversa
Shear
0 0.2 0.4 0.6 0.8 1
Normalized Temperature
(Gj2) storage
versus T =
n
Fig. 8.19
T ransverse
modu 1 i of
(T - T )/(T
oJ g
(^¿2) anc* shear
Graphite/epoxy
- T )â– 
o

Poisson’s ratio ^ Damping
0.02
— Longitudinal damping
— - Transverse damping
0.015 -
0.01 -
0.0C5 -
0 L
0 0.2 0.4 0.6 0.3 1
Normalized temperature
Shear damping
ig. 8.20
Longitudinal ( p ^ ^ , transverse (^22
shear (p^) damping of Graphite/Epoxy
T = (T - T )/(T - T ) .
n v o' g o’
0.4
0.3 -
0.2 =
0.1 -
0 ■ • • > ‘
0 0.2 0.4 0.6 0.3 1
Theoretical
Normalized temperature
Poisson’s
versus T
n
ratio (uj2)
= (T - T )/(T -
o' g
of
T
o
)•
) , and
versus
Fig. 8.21
Graphite/Epoxy

CHAPTER 9
HYGROTHERMAL EFFECTS ON STRESS FIELD
9.1 Introduction
The hygrothermal effects on the stress field are
investigated by considering an infinitely long, finite
width and symmetric composite laminate undergoing
hygrothermal loadings. The Finite Element Method is used in
order to estimate the magnitude of hygrothermal stresses in
laminated composites (see Appendix B). The geometry of a
laminate and the finite mesh of a quarter cross-section are
shown in Figure 9.1 and the boundary conditions are given
by
v = 0 for (y,z) = (0,z)
(9.1)
w = 0 f or (y,z) = (y . 0)
where v and w are the displacements in the y and z
directions, respectively. The grid consists of 24 eight
node isoparametric elements and 93 nodes. Only 24 elements
79

80
are used since increasing the number of elements to 48
results in a relatively small change in the stress
magnitudes. The material properties in terms of temperature
and moisture content have been derived in the preceding
chapter. The constitutive equations are given by Eq. (B.12)
and can be written in matrix form as
{a} = [Q]({€} - {a}AT - (P}c) (9.2)
where {a} and {/3} are the vectors of thermal and moisture
expansion coefficients.
9.2 Description of Study Cases
The considered stacking sequence is the [^O/OJgjg
lay-up. The cross-ply laminate is preferred over other
laminate since hygrothermal loadings induce very high
stresses in this case. The volume fiber fractions of the
Glass/epoxy and the Graphite/Epoxy are 0.5 and 0.7,
respectively. The thickness and the width of the laminates
are assumed to be 2 mm and 20 mm, respectively.
Three cases of moisture gradients are applied. They
are described in Figure 9.2 and Table 9.1. Cases A and C
correspond to the dry and moisture saturated states,
respectively. While the non-uniform moisture gradient
(case B) corresponds to a moisture profile as derived in
section 2.3. Two uniform temperatures (20°C and 80°C) are

81
used. All laminates are assumed to be initially (dry at
20°C) free of stress. Hence, residual stresses are not
taken into account. The elastic moduli used in computing
the stresses are approximated by the real parts of the
complex moduli. Therefore, the hygrothermal effects on the
elastic properties can be deduced from the results given
in Chapter 8
9.3 Numerical Results and Discussion
For all considered cases, the following remarks can
be drawn: at z/h = constant, the stresses away from the
free edge stay constant and the shear stress (CTyz)
zero, but, as y/b approaches 1,
non-zero values and there are small variations in the
a takes significant
yz
values of the other stresses. Hence, the stresses a , a
y z
and a are plotted across the section of the laminate at
x
y/b = 0.472 and the shear stress a is plotted across
yz
the section at y/b = 0.993 (close to the free edge).
The stresses are compared to typical strengths of
Glass/Epoxy and Graphite/Epoxy that are provided in
Table 9.2.
9.3.1 G1 as s/Epoxv
The equilibrium moisture concentration, c , of the
Glass/Epoxy material is 0.025.

82
The stress a is plotted in Figure 9.3. It reaches
y
a maximum magnitude of 166 MPa. for case C at 20°C. It is
compressive for the 0° layer and tensile for the 90° layer.
The stress
a
is shown in Figure 9.4. It is
compressive everywhere and reach a magnitude of 2S8 MPa.
for the case C at 20°C. The stress ct^ is also compressive
(Figure 9.5) and reaches a maximum of 245 MPa. . The free
edge shear stress °yz (Figure 9.6) is very significant
since its maximum magnitude is about 80 MPa..
9.3.2 Graphite/Epoxy
The equilibrium moisture concentration cm for these
cases is 0.015. The stresses o , o , a , and a are
y z x yz
plotted in Figures 9.7-10. These results show the same
trend as for the Glass/Epoxy cases. However, since the
moisture concentration is lower and graphite fibers have
stiffer moduli and lower coefficient of thermal expansion,
the magnitude of the stresses is smaller.
9.3.3 Summary
The hygrothermal conditions used in the preceding
sections are practically achieved only under very adverse
conditions. Hence, the induced stresses can be considered

S3
an upper bound for hygrothermal stresses. The results yield
the following observations:
1) The stresses induced by temperature only (dry at
80°C) are much smaller than those induced by high moisture
content.
2) The stresses due to a non-uniform moisture
gradient can be as high as those induced by the saturated
mo isture case.
3) Since the hygrothermal conditions degrade the
modulus of the epoxy matrix, the stresses caused by the
most severe hygrothermal condition (moisture case C at
80°C) are lower than for some of the other cases.
4) The hygrothermal stresses of the cross-ply
laminates are very significant since their magnitude is of
the same order of those of the strengths given in
Table 9.2.

S4
Table 9.1 Description of cases in Figure 9.2.
Case
Descrip tion
A
Dry
B
(M
- M )/(Ho- M.) = 0.5
(absorption cycle)
C
(M
- M.)/(Mffl- M.) = 1.0
(fully saturated)
Table 9.2
Typical strengths
Graphite/Epoxy
of Glass/Epoxy
S t reng ths
G1as s/Epoxy
Graphite/Epoxy
(MPa.)
Xt
1000.0
1200.0
X
1000.0
700.0
C
Y
t
30.0
o
o
Y
140.0
70.0
c
S
40.0
70.0
and
X (^c) = Longitudinal strength in tension (compression)
Y^ (Yc) = Transverse strength in tension (compression)
S
Shear strength.

S5
Finite element mesh of shaded area
Fig. 9.1 Geometry of a laminate and finite mesh of a 1/4
cross-section area.

86
Moisture case A
Moisture case B
Moisture case C
Fig. 9.2
Description of the applied moisture gradients.

Stress (MPa.)
87
z/h
—*— A at 80 °C
--O- B at 20 °C
•••Q” B at 80 °C
— ¿r- C at 20 °C
—■— C at 80 °C
Profile of the hygrothermal stress a across
y
a [(90/0)2] Glass/Epoxy laminate at
y/b = 0.472.
Fig. 9.3

Stress (MPa.)
ss
—A at 80 °C
- -O- - B at 20 °C
• • -a- • B at 80 °C
—C at 20 °C
—•— C at 80 °C
Fig. 9.4 Profile of the hygrothermal stress across
a [(90/0)2]s Glass/Epoxy laminate at
y/b = 0.472.

89
.. .0..
—A—
A at 80 °C
B at 20 °C
B at 80 °C
C at 20 °C
C at 80 °C
Fig. 9.5 Profile of the hygrothermal stress across
a [(90/0)2] Glass/Epoxy laminate at
y/b = 0.472.

90
—*♦— A at 80 °C
--o-- B at 20 °C
B at 80 °C
—A— C at 20 °C
—•— C at 80 °C
Fig. 9.6 Profile of the hygrothermal stress a across
yz
a [(90/0)2]s Glass/Epoxy laminate at
y/b = 0.993.

91
A at 80 C
--o-- B at 20 C
•••&•• B at 80 °C
—A— C at 20 °C
C at 80 °C
Fig. 9.7 Profile of the hygrothermal stress aacross
a [(90/0)2ls Graphite/Epoxy laminate at
y/b = 0.472.

92
—+— A at 80 °C
--o-- B at 20 °C
•••a-- B at 80 °C
-A- C at 20 C
C at 80 C
Fig. 9.8 Profile of the hygrothermal stress a across
z
a [(90/0)<-,]s Graphite/Epoxy laminate at
y/b = 0.472.

93
--O--
—A—
A at 80
B at 20
B at 80
C at 20
C at 80
Fig. 9.9 Profile of the hygrothermal stress across
a [(90/0)2] Graphite/Epoxy laminate at
y/b = 0.472.

94
- -O- -
. . .Q. .
—A—
Fig. 9.10 Profile of the hygrothermal stress
a [(90/0)2^ Graphi te/Epoxy
y/b = 0.993.
A at 80 °C
B at 20 °C
B at 80 °C
C at 20 °C
C at 80 °C
a across
yz
laminate at

CHAPTER 10
HYGROTHERMAL EFFECTS ON COMPLEX STIFFNESSES
10.1 In t roduction
The hygrothermal effects on the in-plane (A_) and
the bending (D_) complex stiffnesses of Glass/Epoxy and
Graphite/Epoxy angle-ply laminates are
investigated. The applied moisture gradients are the cases
A, B, and C that are given in section 9.2 and the uniform
applied temperatures are 20°C and 80°C. The material
properties in terms of moisture content and temperature
have been determined in Chapter 8. The fibers properties
are given in Table 7.2. The theoretical expressions of A
and D. . in terms of the fibers and matrix properties have
ij
been developped in Chapter 5 and Appendix A.
10.2 Numerical Results and Discussion
10.2.1 G1 a s s/Epoxy
The fiber volume fraction of the Glass/Epoxy is 0.5
and the equilibrium moisture content cm is 0.025. Since
95

96
the fibers do not absorb any moisture, the equilibrium
moisture concentration of the matrix is 0.05. The thickness
x x
of the laminate is 2.0 mm and A. . and D. . are normalized
i J i J
with respect to 75.86 x 10 N/m and 25.29 N.m,
respect ively.
The real part of the longitudinal in-plane complex
stiffness, Aj^, and its corresponding damping, are
plotted in Figure 10.2. As the moisture gradients and the
temperature change, the relative changes of Ajj vary from
6% (for ±0 = 0°) to 33% (for ±0 = 90°). The line style
legend of the figures of this chapter is defined in
Figure 10.1.
Similarly, A^ and 1^12 are Plotted in Figure 10.3
and A ’ and
DO
lV66
in Figure 10.4. All these cases show
that the hygrothermal effects on A. . is matrix dominated.
1 J
The real part of the bending stiffnesses, D ^ ^ . and
their corresponding damping, -17. ., are plotted in Figures
r 1J
10.5-7. These results yield similar conclusions to those
of A* . .
1 J
10.2.2 Graphite/Epoxy
The volume fraction of the Graphite/Epoxy laminate is
0.7 and its equilibrium moisture concentration cm is
0.015. Since all the moisture is absorbed by the matrix ,
cro of the epoxy is 0.050. The laminate is 2.0 mm thick.

97
The real parts of the complex stiffnesses (A^. an<^
Djj) and their corresponding damping (j^ij an<^ F^ij^
are
plotted in Figures 10.8-13. The terms A! . and D; . are
i j i j
0
normalized with respect to 311.3 x 10 N/m and 103.77 N.m,
respectively. These results show the same tendency as those
of Glass/Epoxy. Since the volume fraction and the
longitudinal modulus of the graphite fibers are higher, the
hygrothermal effects are less pronounced.
10.2.3 Summary
The effects of moisture on the stiffnesses A.,’s and
i J
D. ,’s of composites at room temperature are negligible for
all the considered cases. But, as temperature increases,
the combined influence induces significant changes in the
complex stiffnesses especially for the matrix
dominated terms.

9S
Fig. 10.1 Line
Moisture gradient
Moisture gradient
Moisture gradient
Moisture gradient
Moisture gradient
Moisture gradient
style 1 egend o f
case
A
at
o
o
o
CNj
case
B
at
20 °C
case
C
at
20°C
case
A
at
80 °C
case
B
at
80 °C
case
C
at
80 °C
Figures 10.2-13.

Damping Real pQr¿ Qf complex stiffness
99
(a)
(b)
Fig. 10.2 Complex in-plane stiffness of Glass/Epoxy.
a) Non-dimensional Real part ; b) corresponding
damping.

Real part of complex stiffness
100
(a)
Fig. 10.3 Complex in-plane stiffness of
a) Non-dimensional Real part; b)
damping,
Glass/Epoxy.
corresponding

part of complex stiffness
101
(a)
(b)
10.4 Complex in-plane stiffness Agg of
a) Non-dimensional Real part; b)
damping.
Glass/Epoxy.
co r r e sponding
Fig .

Real part of complex stiffness
102
(a)
0.02
cn 0.015
c
'cl
a 0.01
a
0.005
0
0 15 30 45 60 75 90
(b)
Fig. 10.5 Complex bending stiffness of
a) Non-dimensional Real part; b)
damping.
Glass/Epoxy.
c o r r e sponding

Real part of complex stiffness
103
(a)
(b)
Fig. 10.6 Complex bending stiffness of
a) Non-dimensional Real part; b)
damping.
Glass/Epoxy.
corresponding

Real part of complex stiffness
104
(a)
±0
(b)
Fig. 10.7 Complex bending stiffness D* of
66
a) Non-dimensional Real part; b)
damping.
Glass/Epoxy.
corresponding

Real part of complex stiffness
105
(a)
(b)
Fig. 10.8 Complex in-plane stiffness
Graphite/Epoxy, a) Non-dimensional Real
b) corresponding damping.
of
par t ;

Real part of complex stiffness
106
(a)
Fig. 10.9 Complex in-plane stiffness of
Graphite/Epoxy, a) Non-dimensional Real part;
b) corresponding damping.

Real part of complex stiffness
107
(a)
(b)
Fig. 10.10 Complex in-plane stiffness of
bo
Graphite/Epoxy, a) Non-dimensional Real part;
b) corresponding damping.

Real part of complex stiffness
10S
(a)
(b)
Fig. 10.11 Complex bending stiffness D*j of
Graphite/Epoxy, a) Non-dimensional Real part;
b) corresponding damping.

Real part of complex stiffness
109
(a)
(b)
Fig. 10.12 Complex bending stiffness
Graphite/Epoxy, a) Non-dimensional Real
b) corresponding damping.
o f
par t;

Real part of complex stiffness
110
(a)
(b)
Fig. 10.13 Complex bending stiffness D*
bo
Graphite/Epoxy, a) Non-dimensional Real
b) corresponding damping.
o f
par t ;

CHAPTER 11
CONCLUSION
Theoretical and experimental methods have been
incorporated in order to determine the effects of
temperature and absorbed moisture on the complex moduli of
composite materials. The effects of hygrothermal loadings
and of the changes of the complex moduli on the stress
field and on the structural damping of composite laminates
are also analyzed.
The Fickian theory of mass diffusion has been used
for analyzing the diffusion of moisture. The Fickian
diffusion is adequate for the experimental determination of
moisture diffusion through unstressed test specimens.
Therefore, theories that incorporate the coupling of
moisture diffusion with stress, viscoelastic relaxation,
entropy inequality, etc. [43-45], have not been included.
The complex moduli of unidirectional composites are
expressed in terms of the constituent material proper-ties
in Chapter 3 by using the following steps:
- first, the elastic moduli of unidirectional
composites in terms of the fibers and epoxy properties are

112
obtained by using the rule of mixture and the Halpin-Tsai
equa tions,
- then, the correspondence principle is applied to
determine the complex moduli.
Equations (3.20) show that in order to determine the
hygrothermal effects on the complex moduli of composites,
the effects on eight distinct parameters need to be
assessed. But, these effects on the fibers properties are
negligible. Hence, only three terms, E^(T,c), tj^CT.c) and
u^(T,c), have to be experimentally measured. The necessary
test procedures are descibed in Chapter 6.
The complex moduli of epoxy in terms of temperature
and moisture content are presented in Chapter 8. The
storage moduli, E^, is strongly dependent on temperature
and moisture content. But, n and v' stay constant up to
a moisture content M = 4.5% and a temperature T = 80°C.
More severe hygrothermal conditions could not be reached in
the environmental chamber due to operating temperature
limitations of the chamber and of the motion and force
transducers used in the impulse hammer technique.
The experimental results of the epoxy properties in
combination with the micromechanics formulas of Chapter 3
are used to determine the hygrothermal effects on the
complex moduli of unidirectional Glass/Epoxy and
Graphite/Epoxy laminates. It is shown that only the matrix

113
dominated terms (^-¿2 anc* ^12^ are strongly affected by
temperature and/or moisture.
Hygrothermal conditions and changes in the complex
moduli influence the stress field and the structural
damping of laminated composites.
The stresses induced by hygrothermal conditions in
Glass/Epoxy and Graphite/Epoxy cross-ply laminates are
illustrated in Chapter 9. The [^O/O^jg lay-ups are chosen
since they display the highest hygroscopic stresses. The
induced stresses in the 90° layers are of the same order of
magnitude as the strengths. Hence, high loadings induced by
temperature and moisture content can lead to failure.
The complex stiffnesses and structural damping of
[(iGJ^jg Glass/Epoxy and Graphite/Epoxy laminates are
analyzed in Chapter 10. These results display the same
general trend as those of the complex moduli of
unidirectional composites. That is, only the matrix
dominated terms are strongly affected by moisture and/or
temperature.
The important conclusions of this investigations are:
• hygrothermal effects are very significant for the
matrix dominated properties only;
• for certain laminates, severe hygrothermal stresses
alone can lead to failure. Therefore, hygrothermal effects
need to be taken into account when designing composite
structures that are subjected to moisture and/or
tempera ture.

114
Recommendations and additional remarks that arose
from this investigation are listed below.
• The results and conclusions could vary among
different material systems. Hence, it might be necessary to
repeat the tests and the methodology for different
materials.
• The experimental results are valid up to a 80°C
temperature and a 4.5% moisture content. In order to
determine the properties of epoxy through a wider range of
temperature (from - 50°C to 200°C) and moisture content, an
environmental chamber and transducers that can operate
under more severe hygrothermal conditions need to be used.

APPENDIX A
COMPLEX STIFFNESSES OF COMPOSITES
A.1 Elastic Stiffnesses
The in-plane, coupling and bending stiffnesses of a
general laminated elastic composite plate (Figure A.l) are
given by
A. .
i J
B. .
i J
D. .
1 J
w-h/2
Q. . dz
i J
-h/2
p+h/2
zQ. . dz
i J
-h/2
P+h/2
z^Q. . dz
1 J
J -h/2
(A.l)
where the components of the transformed matrix
[Q]
ill
^12
^16
Q12
^22
5.26
Q16
Q26
Q66
(A.2)
115

116
are given by
Q11 = Qllm + 2^Q12 + 2Q66^m n + Q22n
Q12 = ^Qll + Q22 ~ 4Q66^m n + Q12^m + n ^
Q22 = Qnn4 + 2(Q12 + 2Q66)m2n2 + Q^m4
(A.3)
Q16 = CQx x ~ Q12)m3n + (Q12 - Q22)mn3 + 2Q66(m2- n2)mn
Q26 = (^n “ Q12)mn3 + (Q12 - Q22)m3n + 2Qg6(m2 - n2)mn
Q66 = ^Qll + Q22 ~ 2Q12 ~ 2Q66^m n +-Q66^m ” n ^
and
m = cos 0
n = sin 0
The angle 0 represents the fibers orientation of the
lamina under consideration.
The parameters Q.^.’s can be expressed in terms of the
longitudinal (E^) and transverse (E22) Young modulus, the
shear modulus (Gj2) and the major Poisson’s ratio (u^2)

117
Q11 = Ell/‘1 “ u12"21>
Q12 - "l2Ell/(1 ' “12U21>
Q22 = E22/^1 - u12y21^ (A-4)
“21 ” v12E22/E11
The parameters E^, ^22’ *^12 an<^ v 12 are obtained as
functions of the properties of the constituent materials
(fibers and matrix) by using Eqs. (3.5)-(3.10).
A.2 Complex Stiffnesses
The laminate complex stiffnesses are determined by
carrying the following steps
i)The elastic-viscoelastic correspondence
principle is applied to Eqs. (3.5)-(3.10)
ii)The resulting complex values replace their
corresponding elastic modulus in Eqs. (A.4)
iii)Consequently, the complex stiffnesses Q_’s are
obtained and substituted for their correspondig elastic
stiffnesses in Eqs. (A.3)

118
Qu
Th i s
iv) Finally, the complex transformed s
are substituted in Eqs. (5.1).
procedure is easily executed with a FORTRAN
tiff ne s se s
program.

APPENDIX B
DEVELOPMENT OF THE FINITE ELEMENT METHOD
The finite element displacement method used to derive
the stress field in the cases of Chapter 9 is presented in
this Appendix. The F.E.M presented below is used only to
give a first approximation of the hygrothermal stress
magnitudes in laminated composites. In order to obtain more
accurate results, another method that takes into account
the zero-stress boundary conditions on the free surfaces
should be applied.
B. 1 Equilibrium Equations
The stress-strain relation of a linear elastic solid
continuum undergoing any type of loading can be written in
the f orm
W = [D]({€} - {€°}) + {a°} (B.l)
where [D] is the elasticity matrix, {€} is the strain
vector, {€°} is the hygrothermal and/or initial strain
vector, {ct°} is any initial stress vector.
119

120
In this method, it is assumed that the displacements
have unknown values only at the nodal points. The
displacements are
wher e
n
{6} = [N]{66} = l [N.]{6.}
i = 1
( B . 2 )
[N] = [N x, N2. ,Nn]
[Ni] = N.[I]
and n is the number of variables per node and [I] is the
identity matrix.
The strains in terms of the displacements are given
by the following expression
tr
dv
y
dy
J
dvt
Z
dz
nr
dv dw
yz
dz + dy
n
= E B ] { <5e } = l [B.]{6.} ( B . 3 )
i = 1
where the strain matrix is defined as

121
[Bi]
dN .
i
3y
0
dN .
i
dz
0
dN
dz
dN
3 y
(B.4)
The set of functions are called the shape functions and
are subsequently defined.
The continuum is subdivided into a finite number of
elements. Consider an element that is acted upon by nodal
forces {F } and body forces {p}. Then, application of the
virtual work principle to an element e yields
[de]T{Fe} +
[6]T{p}dV =
J V
e
[€]T{a}dV
J V
e
(B.5)
Substitution of Eqs . (B.1)-(B.4) into Eq. (B.5) results in
{ F e } +
[N]T{P}dV =
[B]T[D][B]dV
V
e
. V
V
e
o r
[B]T[D]{€°}dV +
[B]T{a°}dV (B.6)
{Fe> + {Fp + {Fe£Q} + {F^o} = [Ke ] {6e} (B.7a)
whe r e

122
[Ke]
[B]T[D][B]dV
J V
e
element stffness matrix
{ Fe } =
P
[N]‘{p}dV
= equivalent nodal body force
{FGo>
[B]T[D]{e°}dV
J V
e
hygrothermal or initial
strain 1oading
[B]T{a}dV
J V
e
= initial stress loading
These equations are valid for one element only. For the
complete structure, Eq. (B.6) should be summed over all
elements. It is noted that if an element is subjected to
surface traction forces, {t}, then the additional
equivalent nodal force
{F*} =
[N]‘{t}dS
(B.7b)
should be added to the left hand side of Eq. (B.7a)
B.2 Program Organization
Hinton and Owen [42] have developed a detailed F.E.M.
code to solve isotropic beam, plane stress/strain and plate
bending problems. Their procedure for the plane
stress/strain has been modified so that it is adapted to

123
composite laminates that have nonconstant material
properties and undergo thermal and moisture induced
expansion.
The F.E.M. operations are performed by modular
subroutines. The general organization of these programs is
shown in Figure B.l.
B.3 Shape Functions. Jacobian and Strain Matrix
Eight node isoparametric elements are used in the
F.E.M. code. In this case, the shape functions are used to
approximate both the geometry and the displacement field.
The coordinates of any point of the element shown in
Figure B.2 are
8
y(£ .t?) = ^ N . (f .T7)y .
i = 1
(B.8)
8
z(f . T?) = J N . (f ,T7)z .
i = l
where (y. , z. ) are the coordinates of the node i and the
w i i '
quadratic shape functions are defined as
(1 - f)(l - T7) ( 1 + f + T7)/4

124
N2 = (1 ~f2)(l - i7)/2
N3 = (1 + f) ( 1 - v)(S - n - l)/4
N4 = (1 + f)(l - T72)/2 (B . 9)
Ng = (1 + f) ( 1 + T7 )(f + T7 - l)/4
Ng = (1 - f2)(l + tj)/2
N? = (1 - f)(l + T7) (- f + tí - l)/4
Ng = (1 - f)(l - T72)/2
The direction of the local curvilinear coordinates f and 17
are given in Figure B.2. The displacements at any point are
expressed as
8
v(f.ri) = ^ N.(f,T7)v.
i = 1
(B.10)
8
w(f . T7) = ^ N. (f , T7)w.
i = 1
The Jacobian matrix [J(f.h)] is necessary to derive
the elemental volume dV, and area dS. It is defined as
foilows

125
5N .
i
[J] =
dy
d£
dz
as
00
II
as yi
dy
dz
i = 1
3N .
[ dr/
a T)
an yi
3N .
3T*.
<3N .
dr) Z i
(B.11)
Then the elemental area dS is given by
dS = dy dz = det[J] df dp
Once the shape functions are chosen, the strains can be
written in terms of and 6^ (Eq. (B.3)).
B.4 Elasticity Matrix
The stress-strain relation of an orthotropic lamina
is given by
â–  *
a
X
'Qll
^12
°13
0
0
^16
e
X
a AT -
X
¡3 c
X
a
y
Q12
^22
^23
0
0
^26
e
y
a AT -
y
P c
y
o
z
« —
Q13
Q23
Q33
0
0
Q36
â– 
€
z
a AT -
z
/3 c ?
z
CT
yz
0
0
0
q44
^45
0
nr
yz
a
xz
0
0
0
Q45
Q55
0
or
xz
a
xy
«16
^26
^36
0
0
^66-
7
xy
- a AT
xy
- /3 c
xy J
(B•12)
where the transformed stiffnesses that have not been
defined in Appendix A are expressed as

126
Q13
= Q13m2 + Q23n2
G23
= Q13n2 + Q23m2
G33
= Q33 G36
= (Q13 ' Q23)mn
q44
= G23m2 + G12n2
G45
= (G12 - G23)mn
G55
= G12m + G23n
The stiffnesses Q. ,’s are
i J
Q13 =
u21(1 + u12)E11/Q
Q23 =
u12 ^1 + U21^E22/Q
Q33 =
t1 " U12U21)E22/Q
Q =
1 " 2v12v21^1 + 2v23^ ~ v23
wher e

127
r v,
’23
V . + T) . V
f 4 m
f 12
m
+ ^4 G“
m
^4 =
3 - 4u + G /G10
m m 12
4(1 - v )
m'
v
23
v
m
For the case of Chapter 9, Eq. (B.12) can be reduced to
a
y
Q22
Q23
0
* *
e
y
T &
z
» =
^23
^33
0
<
e ^
z
a
L yzJ
0
0
Q44
a
L yzJ
Q12
Q22
Q23
Q26
fa AT +
X
a AT +
P c
X
Pc
Ql3
^23
^33
^36
<
y
a AT +
z
y
pzc
0
0
0
0
a AT
1 xy
+ p c
xy J
or
(B.15)
W = [D1]{€> - [D2]{€°}
The last equation is a reduced form of Eq. (B.l) with the
initial stresses left out.
The through the thickness stress is given by
a = Q (- a AT - j3 c) + Q 10(€ - a AT - 0 c)
x 1 1 v x 'x' 1 2 v y y Y
+ Q13(€z - azAT - p„c) + Q1R(- a_,AT - /3_c) (B.16)
16v xy xy

1 28
B.5 Element Stiffness Matrix
0
A submatrix of the element stiffness matrix [K ]
linking nodes i and j is evaluated from the expression
â– >+1 p+ 1
J-lJ-1
[B.]T[D][B.] t de t[J ] df dp
(B.17)
where t is the thickness of the element under
consideration. The integration is done by using a 3-point
Gauss integration rule.
B . 6 Equivalent Nodal Loadings
B.6.1 Element Edge loadings
An element edge might have both tangential and normal
distributed load per unit length. Every edge of the
isoparametric element has three nodal points. The values of
the normal and tangential loads at each nodal point are
called fp ). and (p ).. Then, the distributed loads at any
point along the edge are given by
*
Pn
3
'(pn>i'
= ) N-
>
Pt.
L* 1
i = 1
.(Pn}i.
(B.18)

129
It can be shown that the equivalent nodal forces are
expressed as
yi
N . (p - p |f-)df
iVit *n b
(B.19)
z .
i
N. (p ||- + p |r)df
i v *n df *^t 1
if the loads are applied on an edge parallel to the
curvilinear coordinate f. The Gaussian numerical
integration is used to derive Eqs. (B.19) which are a form
of Eq. (B.7b).
B.6.2 Hygrothermal Loadings
The equivalent nodal loadings due to hygrothermal
strains are given in matrix form as
[Bi]T([D ]{€°})dv
J y 1 ^
e
(B.20)
where the matrices
[D2] and
{€°} are defined in
Eq. (B.15).

130
B. 7 Element Displacements and Stresses
The global stiffness matrix as well as the equivalent
nodal loading matrix are assembled. Then, the displacements
are computed by a frontal solution method. Once the
displacements are known, the strains and the stresses can
be deduced

131
Input the geometric data, boundary conditions,
material properties, moisture contents and
temperatures at the nodal points. Generate the
sampling Gauss points.
1
Calculate all the element stiffnesses
Fig. B.l Organization of the F.E.M. program.

132
Fig. B.2 Local axes f and 77, Gauss point numbers and
local node numbers of an eight-point
isoparametric element.

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BIOGRAPHICAL SKETCH
Hacene Bouadi was born in Algeria in August 12, 1954.
After graduating from high school in 1972, he attended the
Ecole Po1ytechni que d’Alger (Polytechnic Institute of
Algiers). He obtained a bachelor’s degree in mechanical
engineering in 1977.
He was awarded a graduate scholarship by the Algerian
Government which allowed him to obtain a Master of Science
degree in 1979 and the Degree of Engineer in 1982 from the
Aeronautics and Astronautics Department of Stanford
University. Thereafter, he continued his studies toward the
doctorate degree at the University of Florida. He completed
his Ph.D. in December 1987 in the field of aerospace
enginee ring.
137

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Charles E. Taylor x
Professor of Engineering Sciences
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Robert E. Reed-Hill
Professor Emeritus of Material
Sciences and Engineering
This dissertation was submitted to the Graduate
Faculty of the College of Engineering and to the Graduate
School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
Apr i 1 1988
Dean, Graduate School

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Chang-T. Sun, Chairman
Professor of Engineering Sciences
I certify that
opinion it conforms
presentation and is
as a dissertation for
I have read this study and that in my
to acceptable standards of scholarly
fully adequate, in scope and quality,
the degree of Doctor of Philosophy.
Lawrence E. Malvern
Professor of Engineering Sciences
I certify tha t
opinion it conforms
presentation and is
as a dissertation for
I have read this study and that in my
to acceptable standards of scholarly
fully adequate, in scope and quality,
the degree of Doctor of Philosophy.
Martin A. Eisenberg
Professor of Engineering Sciences

UNIVERSITY OF FLORIDA
3 1262 08556 7880





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