Hygrothermal effects on complex moduli of composite laminates

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

Subjects

Subjects / Keywords:
Hygrothermoelasticity   ( lcsh )
Laminated materials -- Testing   ( lcsh )
Composite materials -- Testing   ( lcsh )
Engineering Sciences thesis Ph.D
Dissertations, Academic -- Engineering Sciences -- UF
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bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

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

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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oclc - 19755822
notis - AFJ8829
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Full Text










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


1988

















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 comments, critique, and advice.

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


ACKNOWLEDGEMENTS . .

LIST OF TABLES . . .

LIST OF FIGURES . . .

NOMENCLATURE . . .

ABSTRACT . . .

CHAPTERS

1 INTRODUCTION . .

1.1 General Introduction . .
1.2 Moisture Diffusion . .
1.3 Hygrothermal Effects . .
1.4 Scope and Methodology . .
1.5 Dissertation Lay-Out . .

2 DIFFUSION OF MOISTURE . .

2.1 Introduction . .
2.2 Fickian Diffusion . .
2.3 Fickian Absorption in a Plate .
2.3.1 Infinite Plate . .
2.3.2 Semi-Infinite Plate .
2.3.3 Experimental Measurement of
Moisture Content . .
2.3.4 Approximate Solutions of
Moisture Content .
2.3.5 Edge Effects Corrections
in a Finite Laminated Plate .
2.4 Diffusivity and Maximum Moisture Content

3 COMPLEX MODULI OF UNIDIRECTIONAL COMPOSITES

3.1 Introduction . .
3.2 General Theory . .
3.3 Micromechanics Formulation of elastic
Moduli . . .
3.4 Complex Moduli . .


Page

. ii

S. vi

. vii

. xiii

S. xvi



1

1
S 2
S 2
3
S 4

S 6

S 6
S 6
8
8
10

S. 11

12

13
S. 15

S. 21

21
21

22
23


iii









4 DAMPING .


4.1 Damping Mechanisms . .
4.1.1 Nonmaterial Damping .
4.1.2 Material Damping . .
4.2 Characterization of Damping .
4.2.1 Free Vibration . .
4.2.2 Steady State Vibration .
4.2.3 Complex Modulus Approach ..

5 DAMPING AND STIFFNESSES OF GENERAL LAMINATES

5.1 Introduction . .
5.2 Laminated Plate Theory Approach .
5.3 Energy Method Approach . .

6 EXPERIMENTAL PROCEDURES . .

6.1 Introduction . .
6.2 Test Specimen . .
6.3 Environmental Conditioning .
6.4 Four-Point Flexure Test Method .
6.5 Impulse Hammer Technique .


. 29

. 29
. 29
. 30
. 30
30
. 31
. 32


7 HYGROTHERMAL 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 Graphite/Epoyx .. 82
9.3.3 Summary .... 82









10 HYGROTHERMAL EFFECTS ON COMPLEX STIFFNESSES

10.1 Introduction . .
10.2 Numerical Results and Discussion .
10.2.1 Glass/Epoxy . .
10.2.2 Graphite/Epoxy .
10.2.3 Summary . .


. 95

. 95
. 95
. 95
. 96
. 97


11 CONCLUSION .


. 111


APPENDICES


A COMPLEX STIFFNESSES OF COMPOSITES .


115

115
117


A.1 Elastic Stiffnesses
A.2 Complex Stiffnesses


B DEVELOPMENT OF THE FINITE ELEMENT METHOD

B.1 Equilibrium Equations .
B.2 Program Organization .
B.3 Shape Functions, Jacobian and Strain
Matrix . .
B.4 Elasticity Matrix . .
B.5 Element stiffness Matrix .
B.6 Equivalent Nodal Loadings .
B.6.1 Element Edge Loadings .
B.6.2 Hygrothermal Loadings .
B.7 Element Stresses . .


REFERENCES .

BIOGRAPHICAL SKETCH


. 119

. 119
. 122

. 123
. 125
. 128
. 128
. 128
. 129
. 130


. 133


137
















LIST OF TABLES


Page


Initial properties of Magnolia 2026 epoxy,
3M Scotchply Glass/Epoxy, and a typical
Graphite/epoxy composite .. 45

Coefficients of moisture and thermal
expansion of epoxy and graphite and
glass fibers. . ... 54

Properties of Glass and Graphite Fibers 54

Description of cases in Figure 9.2 .... 84

Typical strengths of Glass/Epoxy and
Graphite/Epoxy. . .. 84


Tables


6.1



7.1



7.2

9.1

9.2















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 ci)/(c.- ci). . ... 18

2.3 Semi-infinite plate in a humid
environment. . . 19

2.4 Comparison of the exact specific moisture
concentration equation with some approximate
solutions. . .. 19

2.5 Geometry of a plate. . .. 20

2.6 Moisture content versus square root of time.
On the curve v 7< VNT< vTL and the slope
1 2 L
is constant for 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
flexure test. . . 46


vii








6.3 Schematic 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 Glassfiber/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 Glassfiber/Epoxy at 20C. .. 57

7.5 Plot of the moisture expansion coefficients
in terms of fiber volume fraction of a dry
Graphite/Epoxy at 200C. . 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 . .. 68

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 T )/(T T ). 70

8.8 Experimental data of damping of epoxy as
a function of temperature at diverse constant
moisture contents .. 71


viii








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 T )/(T T ). 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 o g o

8.14 Longitudinal storage modulus (E1 ) of
Glass/Epoxy versus T = (T T )/(T T ). 75
n o g o

8.15 Transverse (E 2) and shear (Gi2) storage
moduli of Glass/epoxy versus
T = (T T )/(T T ) . .75
n o g o

8.16 Longitudinal (R711), transverse ('722).
and shear (nG) damping of Glass/Epoxy
versus T = (T T )/(T T ) .. 76
n o g o

8.17 Poisson's ratio (v12) of Glass/Epoxy
versus T = (T T )/(T T ) .. 76
n o g o

8.18 Longitudinal storage modulus (E11) of
Graphite/Epoxy versus T = (T T )/(T T ) 77

8.19 Transverse (E22) and shear (G'2) storage
moduli of Graphite/epoxy versus
T = (T T )/(T T ) . 77
n o g o

8.20 Longitudinal (TI11), transverse (7722)'
and shear ('G) damping of Graphite/Epoxy
versus T = (T T )/(T T ) .. 78
n o g o








8.21 Poisson's ratio (v12) of Graphite/Epoxy
versus T = (T T )/(T T ) .. 78

9.1 Geometry of a laminate and finite mesh
of a 1/4 cross-section area. .. 85

9.2 Description of the applied moisture
gradients. . 86

9.3 Profile of the hygrothermal stress a
y
across a [(90/0)2]s Glass/Epoxy laminate
at y/b = 0.472 . .. 87

9.4 Profile of the hygrothermal stress a
Z
across a [(90/0)2 s Glass/Epoxy laminate
at y/b = 0.472 . .. 88

9.5 Profile of the hygrothermal stress a
X
across a [(90/0)2]s Glass/Epoxy laminate
at y/b = 0.472 . .. 89

9.6 Profile of the hygrothermal stress a
yz
across a [(90/0)2]s Glass/Epoxy laminate
at y/b = 0.993 . .. 90

9.7 Profile of the hygrothermal stress a
y
across a [(90/0)2]s Graphite/Epoxy laminate
at y/b = 0.472 . .. 91

9.8 Profile of the hygrothermal stress a
z
across a [(90/0)2]s Graphite/Epoxy laminate
at y/b = 0.472. .. . 92

9.9 Profile of the hygrothermal stress a
x
across a [(90/0)2]s Graphite/Epoxy laminate
at y/b = 0.472 . .. 93

9.10 Profile of the hygrothermal stress a
yz
across a [(90/0)21s Graphite/Epoxy laminate
at y/b = 0.993 . 94

10.1 Line style legend of Figures 10.2-13. ... 98









10.2 Complex in-plane stiffness Al of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping. . 99

10.3 Complex in-plane stiffness A2 of Glass/Epoxy.
12
a) Non-dimensional Real part;
b) corresponding damping .. 100

10.4 Complex in-plane stiffness A66 of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping. .. 101

10.5 Complex bending stiffness D1 of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping .. 102

10.6 Complex bending stiffness D12 of Glass/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping . 103

10.7 Complex bending stiffness D66 of Glass/Epoxy.
66
a) Non-dimensional Real part;
b) corresponding damping .. 104

10.8 Complex in-plane stiffness Al of Graphite/Epoxy.
11
a) Non-dimensional Real part;
b) corresponding damping .. 105

10.9 Complex in-plane stiffness A2 of Graphite/Epoxy.
12
a) Non-dimensional Real part;
b) corresponding damping ... 106

10.10 Complex in-plane stiffness A66 of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping. . 107

10.11 Complex bending stiffness D1 of Graphite/Epoxy.
11
a) Non-dimensional Real part;
b) corresponding damping. . ... 108









10.12 Complex bending stiffness D12 of Graphite/Epoxy.
12


a) Non-dimensional Real part;
b) corresponding damping. .


109


10.13 Complex bending stiffness D66 of Graphite/Epoxy.
a) Non-dimensional Real part;
b) corresponding damping .. 110


B.1 Organization of the F.E.M. program.


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


. 131


132


xii
















NOMENCLATURE




A complex in-plane stiffness.
ij

B.. complex coupling stiffness
ij

B complex modulus

B' storage modulus

B" loss modulus

c moisture concentration


c average specific moisture

c equilibrium moisture concentration


C specific heat
v


D.. complex bending stiffness
ij

D Dxx moisture diffusivities


[D] diffusivity matrix, elasticity matrix

E11 longitudinal Young modulus


E22 transverse Young modulus


G12 in-plane shear modulus


K thermal conductivity
x

m weight of absorbed moisture

M percent moisture content


xiii










M. initial percent moisture content
1

M equilibrium percent moisture content


Q.i transformed stiffness


Qij complex transformed stiffness


s specific gravity

t time

T temperature

vf fiber volume fraction


v matrix volume fraction
m

w weight

a. coefficient of thermal expansion
1

.i coefficient of moisture expansion

E strain

T damping or loss factor

v12 major Poisson's ratio

8. fiber orientation of j-th layer


p density

a stress


Subscripts


1. 2. 3 principal directions of the fibers

f fiber

i initial

j layer number


xiv










L

m

x, y, z

a


longitudinal direction

matrix

Cartesian coordinates

maximum or equilibrium


Superscripts


moisture

initial


transpose, thermal

complex value

real part

imaginary part
















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

April 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 negligible.


xvi









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.


xvii















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

knowledge.

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.









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









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

following steps:









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 after the introduction, the mechanism of

moisture diffusion is described in Chapter 2, where the

absorption of moisture through thin composite laminas is

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









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

Appendix B.













CHAPTER 2
DIFFUSION OF MOISTURE



2.1 Introduction



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-dimensional temperature and moisture concentration, are

respectively given by


BT 6 aT]
C OT K, K (2.1)
Pv at ax x jx


ac [ c 1
O- Dax (2.2)
where is the dens t ia x is the


where p is the density of the material, C is the
v









specific heat, T is the temperature, t and x are the time

and spatial coordinates, respectively, K is the thermal
x
conductivity, c is the moisture concentration, and D is
x
the moisture diffusivity.

The moisture diffusivity, Dx, 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



ac c
t D a2c (2.3)
at x 2



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


Oc
at = v.([D].vc) (2.4)



where the diffusivity matrix is


D D D
xx xy xz
[D].= Dyz Dyy D (2.5)
yz yy yz
D D D
zx zy zx









Expansion of Eq (2.4) results in an equation of the form


2 2 2
8c 2c a2c c
SD + D + D c + (D
at xx ax2 yy 2 zz a2 yz


2
+(D + D ) z + (D x
+(Dzx xz Ox oz xy


2
+ D ) ac
zy Oy 6z


2
+ D ) a c
yx ax oy


if the coefficients D..'s are considered to be constant.



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 = T.
c for 0 < z < h and t < 0
c = Ci

(2.7)


(2.6)






9


T = T.
ST for z = 0, z = h and t > 0
c = c J



where T. is a constant temperature, c. is the initial
1 1
moisture concentration inside the material, and c. is the

maximum moisture concentration. It is assumed that the

moisture concentration on the exposed sides of the plate

reaches c instantly.

The solution of Eq (2.3) in conjunction with the

conditions of Eqs. (2.7) is given by Jost [25]



c ce2
c c 4 V 1 1 i+l (2j+l) 2
= 1 1 }in2-1z exp 2j+1)U2D t
co ci c (2j+1) h h2 IT z
j=o
(2.8)

Equation (2.8) is plotted in Figure 2.2.

The average moisture concentration is given by



1
c = c dz (2.9)
h o



Substitution of Eq. (2.8) into Eq. (2.9) and integration

result in



c c. D t
1- = 1 exp -(j+)2 2 z
c i j=0 (2j+1) h

(2.10)

This analysis can be applied to the case of diffusion of









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 = T.]
1I for 0 < z < and t < 0
c = c.
1
(2.11)
T = T
i for z = O and t > o
c = cJ



The solution of Eq (2.3) in this case is [24, 25]



c c. e r z ]
1- 1 erf z (2.12)
c-- ci



The rate at which the total specific mass of moisture, m,

is diffusing into the half-plane is








dm [ac
dt z z=


(2.13)


Thus, the total mass of moisture entering through an area A

in time t is


m= pADz dt = 2pA(c c. )
Jo z= 0


(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

content 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, wd. from the current weight, w, of the plate, i.e.


m = w wd


(2.16)


A parameter of practical interest is the percent moisture

content defined as








w wd
M = 100 (2.17)
Wd



Since



M M. c.
1 1 M = lOOc (2.18)
M0- M. c0- c.



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.

Small 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. D t
ci _-=4 z (2.19)
cw- ci M0 M. 2



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.,











c c. D t-
1 8 2 2z_
S1 -- -ep I (2.20)
c- ci 1 2 h 2



Shen and Springer formulation. These researchers

have derived in Ref. [4] the following approximation



c C. D t 0.75
c 1 exp 7.3 (2.20)
c- ci h 2



Figure 2.4 shows a comparison of Eqs (2.19-21) with the

exact solution.



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 c ) bL /Dz + bh /DJx+ hL /JDy/T7i

(2.22)

where Dx. D and D are the diffusivities in the x, y, and

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











Dt
M = 4MJ D-2 (2.23a)
2h



where the effective diffusivity D is




D = D 1 + h + h (2.23b)



The micromechanics formulation for diffusivities proposed

by Shen and Springer [4] and modified by Hahn [26] for

impermeable, circular cross-section, fiber-reinforced

composites is



DL = Dm
DL m
(2.24)


DT = 1 2 D
T ~ v



where D D and D are the matrix, transverse, and
m T L
longitudinal diffusivities, respectively. Equation (2.23b)

for a unidirectional lamina with all fibers parallel to the

x-direction can be written as




D = D 1 + + m (2.25)
z +
1 2 _f









For a general laminated plate consisting of N layers with

fiber orientations 8., the diffusivities are
J


Dz = DT


N N
D, h.cos2. + DT h.sin 2
DL Z j 2j jT Z J
D= Nj=N j=l (2.26)

h.
j=l
N N
2 2
DL 2 h.sin 9. + DT h cos 28
L ZJ J T J
D j=l j=l
y N
hI
j=l


where h. is the thickness of the j-th layer. The effective
3
diffusivity of a general laminate is obtained by

substituting Eqs. (2.26) into Eq (2.23b).



2.4 Diffusivity and Maximum Moisture Content



The diffusivity D and the maximum moisture content

M 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,








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, q/ 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 M D
= 4M (2.27)
t/2 --./ t Hh2


The subscripts 1 and 2 are defined in Figure 2.6.

The diffusivity depends only on the material and

temperature as follows



D =D exp (2.28)
z 0 RT



where R is the gas constant, Do and Ed are the

pre-exponential factor and the activation energy,

respectively.

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 = ab


(2.26)






17





where is the relative humidity, a aad b are material

constants.
















Moisture
---


h

^-s^~


Moisture
c----


4


Fig. 2.1


Plate subjected to a constant humid environment
on both sides.


1
I



8
. 0.8

0.4
I


01 0.2 .3 0.4 0~J


z = 0 is the center of the
cross-section of the plate


z/h


Fig. 2.2


Moisture distribution across
numbers on the curves are
(c ci)/(cm ci).


a plate.
the values


The
of















x







Moisture
-Dow



Dow)


Fig. 2.3











0.8








0.
0


Semi-infinite plate in a humid environment.









Exact
-- One-term
S* -* Shen and Springer


Vh2


Fig. 2.4


Comparison of the exact specific moisture
concentration equation with some approximate
solutions.















Fiber direction


Fig. 2.5


Geometry of a plate


0 M1





77 f Square root of time


Fig. 2.6


Moisture content versus square root of time. On
the curve /i < / < V and the slope is

constant for,/T < /t.


I
















CHAPTER 3
COMPLEX MODULI OF UNIDIRECTIONAL COMPOSITES




3.1 Introduction



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

moduli.



3.2 General Theory



A usual representation of the one-dimensional

stress-strain relation of a viscoelastic material subjected

to a harmonic strain history of the form



E(t) = e0e (3.1)









is given by



a(t) = B(i)EOeit = B (ie)E(t) (3.2)



The complex modulus B can be decomposed into its real and

imaginary parts as follows



B (iw) = B'(w) + iB"(w) (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


B"
S B' (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 E11, 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.,


E11 = fE + v mEm
11 f f1l m m


(3.5)












E = E
22 m



G = G
12 m


1 + 2nlvf
1 n1vf


1 + n2vf
1 n2vf


12 = Vf f12 + Vm m






(Ef22/E) 1
S= (Ef22/Em) + 2



(Gf12/G) 1
2 = (Gf12/Gm) + 1


The subscripts f and

respectively, and vf and


m stand for fiber and matrix,

v are the volume fractions.
m


3.4 Complex Moduli



The micromechanics formulations of the complex moduli

are obtained by applying the elastic-viscoelastic

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.


(3.6)




(3.7)



(3.8)


where


(3.9)




(3.10)









ii) replacing the elastic moduli of fibers and matrix

by corresponding coDplex expressions.

For a viscoelastic composite, the properties of the

constituent materials are


= E 11 + iE l
11E1


E E' + iE"
E22 = f22 22



G = G + iG"
f f



E = E' + iE"
m* m



G = G" + iG"
m* m



D = U' + iv"
1m m a



Vf 12 = fl2


(3.11)


The bulk modulus of epoxy

independent of frequency [28].


matrix. K is real and
m
It is given by


E
K -
m 3(1 2v )


(3.12)


while the viscoelastic bulk modulus is obtained from the

correspondence principle











E' + iE"
n m m
m 3[1 2(v' + iv")] (3.13)
m m



Separation of the real and imaginary parts of Eq. (3.13)

yields



S(1 2v)E 2E"m + i[2E"v" + E'(1 2v')]
m m mm mm m m
S 3[(1 2v)2 + 4v2 J
m m
(3.14)

Since the dilatation bulk modulus is real, the imaginary

part of Eq. (3.14) is equal to zero; hence



2E"v" + E"(l 2v') = 0 (3.15)
mm m m



Equation (3.15) results in



E"
v" (v' 0.5) (3.16)
m



The shear modulus of the matrix is given by



E 3K E
n m mm
G = m (3.17a)
S2(1 + v) 9K -
m m m



Separating the real and imaginary parts and neglecting the
2
terms of the form (E") yield
m









Smm I
m 9K E
m


Introduction of


9K E"
m i
9K E' E'
m m m


the material properties


E"
m
7m E'
m


E'
fl1
4f 11 ~ E
ti=t


E"
f22
f22 E22



G" 9K
?m m
Gm G-' 9K E' m
m m m



into Eqs. (3.11) results in


= E11 ( + if 1)



= E 22(1 + i" 22)


E
fll1



E
f22



G
f12


E =
m


= G 12(1 +


'fl12)


E'(1 + iTm)
m- im


G = G'(1 + iT )
m m Gm


(3.17b)


(3.18)


(3.19)








n 1
v = U' + iTm (v' -
m m m m 2


Vfl2 = 'fl2 = f12



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 rf11'

"f22o and nfl2 are assumed to be equal and are replaced by
Tf in subsequent equations. Since the fiber damping, 7f,

is much smaller than the matrix damping, m, the

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 elastic-viscoelastic

correspondence principle to Eqs. (3.5)-(3.10) and

substituting them into Eqs (3.19) yield the following

complex material properties



E11 = fE11(l + inf) + vmE'(1 + inm)



1 + 2n vf
E22 E(1 + inm) f
22 m m n
1 nlvf
1 (3.20)



1 + n2vf
G12 = G(1 + inGm) 2
1 n2vf








1 1
v12 = (Vfvfl2 + v v') + im (v )v
12 fmm m m m


where


SEf22(1 + i-qf) Em(1 + im)
n1
E 22( + f) + 2E'(1 + im)




n2
Gfl2(l + i) + Gm(1 + iGm)
2 = Gj1(l + inf) + G(l + in )


The elastic

experimental

they are

micromechanic

Hashin [27. 2


moduli given

results with a

used instead

s formulas,


by

good

of

such


Eqs. (3.5)-(3.8) model

accuracy [2]. Therefore,

mathematically exact

as those derived by

















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.









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 N In[A (4.1)
n+N



where

A = amplitude of the n-th cycle

An+N = amplitude of the (n+N)-th cycle

The damping defined in Eq (4.1) is applicable to viscous









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



"2 "1
n 2 (4.2)
0
o



where

W = resonant frequency
o
1., W2 = frequencies on either sides of o such that

the amplitude is 1//"2 times the resonant

amplitude.

In the case of a vibration induced by the force



f(t) = Fsin(wt)



the response (deflection), w(t), is out of phase with f(t)

by an angle e such that


w(t) = Wsin(wt + e)








The work done per cycle is



dw
D = f(t) dt = TWF sin(e) (4.3)




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= 1-FW cos(a) (4.4)



There is no damping if the work done per cycle is zero,

i.e, if sin(a) = 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



2IU = tan(e) (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))











iWt iwt
S(t) = E(w)E e0 = (E'(() + iE"(w))E et (4.6)



Noting that i[il = dE/dt, Eq. (4.6) can be written as


, i t E" i t
a (t) = E e + i e E
a 0Fo


(4.7)


The real part is given by (after algebraic manipulation)


a(t) = E'E sin(wt + e)1J + n2


(4.8)


where n = tan(e) = E"/E'

The energy dissipated during a cycle per unit volume is


dE
x 2
D = 0 d = dE dt = lT1E'E
x fdt o


(4.9)


The maximum energy stored is


1 2
U = E' E
2 o


(4.10)


Therefore,


E" D
7 E' = 2- U


(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.






















Vibrating beam


Fig. 4.1


Envelope Equation
w(t) = e- t


Response


Schematic drawing of a free-clamped beam under
free vibration and plot of its deflection
versus time.


Bem under forced vibration
Beam under forced vibration


'A
aI
crI/


Deflection versus time


aA
Amplitude versus frequency


Fig. 4.2


Schematic drawing of a free-clamped beam under
forced vibration and plots of the deflection
versus time and deflection amplitude versus
frequency.


MIM_














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

complex coupling stiffnesses B..'s, and the complex bending

stiffnesses D. 's (see Appendix A).

The terms A .'s, B .'s, and D .'s are defined as


+h/2
A. = Q. dz
ij ij
-h/2








.h/2

ij -h/2

h/2


Dh/2
D. = h/
1 -h/2


zQi. dz
1J


(5.1)


z2 Q dz
1j


where the complex transformed stiffness Q..'s depend on

E1, E22, G12. v12 and the orientation of each layer of

the laminate.

The in-plane, coupling, and flexural material damping

are defined as


Al.
I ij A
ij


B"
ij


(5.2)


D".
Fij D:.
ij


respectively.



5.3 Energy Method Approach



The energy method can be used to determine the

damping of laminated composite materials under certain

loading and boundary conditions. The damping of a laminated








composite material in the first mode of vibration can be

defined as


N

I kUdcyc.
k=l
S= N (5.3)

21 kUs
k=l


where N is the total number of layers. (kUd)cyc. is the

energy dissipated in the k-th layer during a cycle, and kU
ks
is the maximum energy stored in the k-th layer. The storage

and the dissipated energy are given by



kUs = E .C'.. dV
k 2 j j1 1
k
(5.4)

kUd = E.C".. dV
k d JV k ji I
k


where i and j are the material principal axes, C. and
Ji
C". are the real and imaginary parts of the complex
ji
stiffnesses and Vk is the volume of each layer. Hence, the

"total" damping of an N-layered laminate is given by


N
I fV {}T[C"]{} dV
k = k (5.5)

kV {E}T[C']{} dV
k=l k





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

structures.
















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 1750C and 1700C. respectively. The

initial properties of these materials (at 200 C and without

moisture), as well as those of a typical Graphite/Epoxy,

are given in Table 6.1.









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 900C.



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]



3
S= P3 (6.1)
8bh w


where E is the effective modulus, P is fhe applied load,

1 is the length of the specimen, b is the specimen width, h









is the thickness, and w is the deflection at quarter-point.

Poisson's ratio is expressed as


E 2
S = E (6.2)
6hw y
x



where the transverse strain E is measured with a
y
transverse strain gage cemented in the middle of the

specimen.



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 equipment (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









v(t)) over the Fourier Transform of the input (force

impulse u(t)); that is,



H(f) = (f) (6.3)
U(f)


where

t = time

f = frequency

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

expression



(fa/fb)2 1
S= 2 (6.4)
(f/f b) + 1



where the frequencies fa and fb are defined in Figure 6.3.

The storage modulus is expressed as [33, p.464]



2 1
E' = 38.32 fr p (6.5)



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].
















Table 6.1


Initial properties of Magnolia 2026 epoxy,
3M Scotchply Glass/Epoxy, and a typical
Graphite/Epoxy composite.


Properties Epoxy Glass/Epoxy Graphite/Epoxy


vf 0.50 0.70


p (g/cm3) 1.25 1.93 1.6

E11 (GPa.) 4.0 37.00 155.23

E22 (GPa.) 4.0 11.54 10.81

G12 (GPa.) 1.52 3.46 4.35

v12 0.32 0.285 0.217

11 0.018 0.0023 0.0019

T22 0.018 0.015 0.0078














I- I-


Testing chamber

/Environn
/


mental chamber


Fig. 6.1


Schematic drawing of environmental and testing
chambers.


Fig. 6.2


Loading configuration
flexure test.


of the 4-point


MTS Ic
frame




































Fourier Transform

Real part







Im. part


Motion transducer

Testing chamber


fr Frequency


Fig. 6.3


Schematic drawing of the impulse hammer
technique apparatus and a typical display of
the Transfer Fourier Transform.
















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 (7.1)
1 1 o 1



where

a. = coefficient of thermal expansion

T = actual temperature

T = 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 hygrostrain is given by








H
EH = p.(c c ) = 1.Ac (7.2)
1 1 0o 1



where c is the initial moisture concentration and 3. is

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

derivation)


VfafEf + vaE
a EE



a2 fa + mam + fa + vm am- 12al (7.3)



"12 = 0



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 6

with the x-direction (Figure 2.5) are given by








2 2
a = a1 cos 8 + a2 sin 8


2 2
a = a sin 8 + a2 cos 8 (7.4)



axy = 2(al a2) cos 0 sin 8



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
m
1 sE m
m 11



2 = s(1 + V)m v12 P (7.5)
m



12 =



where s and s are the specific gravities of the composite
m
material and the matrix. The moisture expansion

coefficients expressed in an axis system such that the

x-direction makes an angle 8 with the fibers are given by

Eqs. (7.4) after replacement of a.'s by Pi's.









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-Glass/Epoxy, Kevlar 49/Epoxy and Graphite/Epoxy can be

approximated by



EH 0.43c = Pfc (7.6)
2 .2


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.








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

sufficiently accurate. These data yield the following

experimental values



Pm(epoxy) = 0.25 (7.7)



P2(Glass/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



pi = 0.042 (7.9a)



p2 = 0.47 (7.9b)



for the 3M Glass/Epoxy composite. The experimental and

empirical values of P2 are practically equal. Hence, the









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 1000C. Therefore, in the subsequent

chapters, the thermal expansion coefficients are assumed to

be independent of temperature.













Table 7.1


a (Wm/m)/K

P


Table 7.2


Ef (GPa)

E 22(GPa)

G 12(GPa)

If


vf12


p (g/cm )


Coefficients of moisture and thermal expansion
of epoxy and graphite and glass fibers.


Epoxy


54.0

0.25


Glass


5.0

0.0


Graphite


0.2

0.0


Properties of Glass and graphite Fibers.


Glass

70.0

70.0

28.7

0.0015

0.22


2.60


Graphite

220.0

16.6

8.27

0.0015

0.16


1.75

























-- Glass/Epoxy

- 4- Epoxy


Fig. 7.1


Moisture concentration (%)














Transverse moisture strain of Magnolia epoxy
and 3M Scotchply Glass/epoxy.


(D
c 1



'o
0)
11





L.
I-













- Longitudinal

- transverse


0 0.2 0.4 0.6 0.8
Fiber volume fraction


Fig. 7.2


Plot of the thermal expansion coefficients in
terms of fiber volume fraction of a dry
-S Glassfiber/Epoxy at 200C.


- Longitudinal

- Transverse


0 0.2 0.4 0.6 0.8 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 200C.













- Longitudinal

- -. Transverse


0.4 0.6 0.8
Fiber volume fraction


Fig. 7.4


Plot of the moisture expansion coefficients in
terms of fiber volume fraction of a dry
S Glassfiber/Epoxy at 20C.


./






3 0.2 0.4 0.6 0.8
Fiber volume fraction


- Longitudinal

- Transverse


Fig. 7.5


Plot of the moisture expansion coefficients in
terms of fiber volume fraction of a dry
Graphite/Epoxy at 200C.


0.8


0.6


0 0.2


1


0.8


0.6


0.4


0.2


0


......,....I.........















CHAPTER 8
HYGROTHERMAL EFFECTS ON COMPOSITE COMPLEX MODULI



8.1 Literature Survey



The storage moduli (real parts of Eqs. (3.11)) of

composites are usually determined by dynamic testing, 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 00 fiber

direction laminates are negligible.

ii) For 900 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

important.

Putter et al. [38] investigated the influence of

frequency and environmental conditions on the dynamic









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









E' = E'(T,c)
m m



v' = v'(T,c) (8.1)
m m



nm = 7m(T,c)



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









can reach a value of 1 or even 2 in the transition region

[30, p. 90]. Poisson's ratio reaches the limiting value of

0.5, which is approximated by incompressible rubbers

[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
g
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
g
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 strongly dependent on absorbed moisture.
These results show that, as the moisture content of epoxy
These results show that. as the moisture content of epoxy









increases, the transition temperature moves to the left in

Figures 8.1-8.3. 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 77 are appropriate,
m m m



E' E' f o (8.2)
m o T T 0
g o



v = g[ o] (8.3)
m o T -T
g o



T = i o h T T (8.4)
g o


where the temperature T is equal to 273K. The moisture

concentration appears implicitly in T The glass
g
transition temperature is represented by



T = 210 exp(- 9c) (C) (8.5)
g


where c is the moisture concentration.

This modeling has been chosen so that it does not

represent the material properties beyond T since the
g
study of epoxy in the rubbery stage is not within the scope

of this research. Equations (8.2)-(8.4) are valid only for









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 that increase in either

temperature or moisture content or both results in a

decrease in the storage modulus. Plotting 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


T T
T T T
g o











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) (8.6)
m n



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



Tm = 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









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



for temperatures up to 80 C and moisture contents up to 5%.

Since v' equals 0.5 at the glass transition temperature
m
(Tn = 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.








Glass/Epoxy. The parameters E'll E'22' G12' 11

T22' -G, and v12 versus the normalized temperature are
plotted in Figures 8.14-17. The experimental data

substantiate the theoretical results.

Graphite/Epoxy. Similarly, E'22, GI2 711

T22' T), and vu2 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'2 and G12) are

strongly affected by moisture and temperature, while the

fiber-dominated parameters (E11' v12) stay practically

constant.

























Temperature


Fig. 8.1


Schematic variation of the storage modulus of
epoxy with temperature.


--_-J----------



Transition
Glassy region region Rubbery region
I I


Temperature


Fig. 8.2


Schematic variation of Poisson's ratio of epoxy
with temperature.


Transition
region

Glassy region I Rubbery region
I "s T


1


I I m


r















S ---I-----


Temperature


Fig. 8.3





o
250


S200


E 150


100

50
: 50
c,
03


Fig. 8.4


Schematic variation of damping of epoxy with
temperature.


L
0


2 4 6 8


Moisture concentration x 100 (%)




Glass transition temperature of epoxy. From
Delasi and Whiteside [6].











5


O4.5

a 4
0

0 3.5


O3
0
o,


2.5


2


SM = 0.0%

* M = 2.90%

0 M = 3.70%


Fig. 8.5


Experimental data of the storage modulus of
epoxy as a function of temperature at diverse
constant moisture contents.


5


a0


i3
--!
0
E 3.5
So
0'
. 3
o


2.5


2


)


* T = 20 oC

0 T = 50 C

O T = 70 C


1 2 3 4
Moisture content (%)


Fig. 8.6


Experimental data of the storage
epoxy as a function of moisture
diverse constant temperatures.


modulus of
content at


0











0 20 40 60 80 1
Temperature (oC)


. *





-8
0*
B I21


















o 4
C0




E


03


1




0 0.2 0.4 0.6 0.8
Normalized temperature


Fig. 8.7


SExperimental data

- Fit to data


Experimental data of the storage modulus of
epoxy as a function of normalized temperature
(T T )/(T T )
o g o

















- *
8



A
-







0 20 40 60 80 1
Temperature (oC)


A M 0.

SM U 2.90X

0 M 3.702


Fig. 8.8


Experimental data of damping of epoxy as a
function of temperature at diverse constant
moisture contents.


0.03

g, 0.025

S0.025
0.015

0.015




0.005


0


* T 20 OC

0 T 50 C

0 T 70 C


0 1 2 3 4
Moisture content (X)


Fig. 8.9


Experimental data of damping of
function of moisture content
constant temperatures.


epoxy as a
at diverse


0.03

o 0.025
c

S0.02

0.015

0.01

0.005

0


*







- I.






















0.03


0.025


0.02


0.015


0.01


0.005


0 I
0 0.2 0.4 0.6 0.8 1
Normalized temperature


* Experimental data


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



























0 20 40 60 80
Temperature (OC)


* U 0%

0 M 4.17%


Fig. 8.11 Experimental data of Poisson's ratio of epoxy
in term of temperature


0.2 -


0 1 2 3 4
Moisture content (%)


* T = 20C

0 T = 50'C

ST = 75C


Fig. 8.12 Experimental data of Poisson's ratio of epoxy
in term of moisture content.































: *
*











0 0.2 0.4 0.6 0.8
Normalized temperature


* Experimental data

- Fit to data

*... Extrapolation


Fig. 8.13 Experimental
term of the


data of
normalized


(T T )/(Tg T ).
o vg o


Poisson's ratio in
temperature T =
n
















m0 40
:3
a- 35
0
E
S3

25
0
U2
20


- Theoretical

* Experimental data


Fig. 8.14 Longitudinal storage modulus (Ei1)
Glass/Epoxy versus T = (T T )/(T T ).
n o g o


0 0.2 0.4 0.6 0.8
Normalized Temperature


--Theoretical E'2

* Experimental E22

.... Theoretical G;2


Fig. 8.15 Transverse (E22) and shear (G12) storage moduli
of Glass/epoxy versus T = (T T )/(T T ).
n o g o


-
* **


0 0.2 0.4 0.6 o0.
Normalized Temperature


t-


















0


Theoretical 31,

Experimental 11,

Theoretical 1122

Experimental 122

Theoretical a
G


0 0.2 0.4 0.6 0.8
Normalized temperature


Longitudinal
shear (inG)
T = (T T )/
n 0


(711), transverse (722), and
damping of Glass/Epoxy versus
(T T ).
g o


-- Theoretical

SExperimental


0.6 0.8
temperature


Fig. 8.17 Poisson's ratio (v12)
T = (T T )/(T T ).
n o g o


of Glass/Epoxy versus


0.03


0.025 -


0.02

0.015

0.01

0.005

0


:0 0
0



.
*-s *0



' !... i,, i^ i*, ,


Fig. 8.16


* *
*

*


I .


0 0.2 0.4
Normalized


0


























0 0.2 0.4 0.6 0.8

Normalized Temperature


Fig. 8.18 Longitudinal storage modulus (E11) of
Graphite/Epoxy versus T = (T T )/(T T ).
n o g o


-Transverie

-- Shear


0 0.2 0.4 0.6 0.8

Normalized Temperature


Fig. 8.19 Transverse (E22) and shear
moduli of Graphite/epoxy
(T T )/(Tg T ).
1 ovg o'


(G12)
versus


storage
T =
n











0.02



. 0.015
a0
E

0.01



0.005


0


0 0.2 0.4 0.6 0.8
Normalized temperature


Fig. 8.20


- Longitudinal damping
- Transverse damping

.... Shear damping


Longitudinal (11 ), transverse (T22), and
shear (7G) damping of Graphite/Epoxy versus
T = (T T )/(T T ).
n o g o


0 0.2 0.4

Normalized


0.6 0.8

temperature


- Theoretical


Fig. 8.21


Poisson's ratio (v12) of Graphite/Epoxy
versus T = (T T )/(T T ).
n o g o


..
r-- ~'
f
*
















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) = (Oz)

(9.1)

w = 0 for (y,z) = (y,O)



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









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



{o} = [Q]({E} {a}AT {j}c) (9.2)



where {a} and {(} are the vectors of thermal and moisture

expansion coefficients.



9.2 Description of Study Cases



The considered stacking sequence is the [(90/0)2]S

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 (200C and 800C) are









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 (ayz) is

zero, but, as y/b approaches 1, a takes significant
yz
non-zero values and there are small variations in the

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 Glass/Epoxv



The equilibrium moisture concentration, cW, of the

Glass/Epoxy material is 0.025.









The stress a is plotted in Figure 9.3. It reaches

a maximum magnitude of 166 MPa. for case C at 20 C. It is

compressive for the 00 layer and tensile for the 900 layer.

The stress a is shown in Figure 9.4. It is
Z
compressive everywhere and reach a magnitude of 288 MPa.

for the case C at 200C. The stress a is also compressive

(Figure 9.5) and reaches a maximum of 245 MPa.. The free

edge shear stress a (Figure 9.6) is very significant
yz
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 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









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

moisture 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.