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THESIS

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MICHIGAN STATE UNIVERSITY LIBRARIES

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This is to certify that the

dissertation entitled

Effects of Fiber Waviness and Bonding
Condition on Composite Performance by
Homogenization Method

presented by

Len Xu

has been accepted towards fulfillment
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FIBER WAVINESS AND BONDING CONDITIONS ON COMPOSITE
PERFORMANCE BY HOMOGENIZATION METHOD
By

Lan Xu

A DISSERTATION
Submitted to
Michigan State University

in Partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Material Science and Mechanics

I996

ABSTRACT

EFFECTS OF FIBERWAVINESS AND BONDING CONDITION
ON COMPOSIE PERFORMANCE BY HOMOGENIZATION METHOD

BY

LAN XU

Owing to their high efficiency as reinforcing materials, long, straight fibers were
commonly used in composite materials. However, due to some manufacturing constraints,
fiber curvature might take place in composite structures. Assuming that composite
materials were organized by periodical units of sinusoidal fibers, this study looked into the
effects of fiber waviness on composite performance. In addition, since the fiber-matrix
interfaces were not always perfectly bonded, imperfect bonding conditions including both
interfacial normal and interfacial shearing conditions were included in the investigation.
To begin with, a rigorous approach named homogenization method was used to obtain
closed-form solutions of effective moduli for the composite materials made of sinusoidal
fibers with small amplitudes. By assuming linear normal separation and linear shear slip
on composite interfaces, it was also possible to investigate the effects of bonding condition
on the composite performance. Numerical results indicated that the composite stiffness
was greatly reduced by the fiber waviness even when the waviness ratio was very small. In
addition, it was concluded that the composite materials became less stiff as the fiber-

matrix bonding became poor. In order to investigate composite materials with sinusoidal

amplitudes greater than 5% of the period, a finite element method was utilized for
numerical analysis. Nonlinear stress-strain relations for fiber waviness ratio as large as
150% were examined. Stress distributions for both periodical units and overall composite
materials were presented. By incorporating the homogenized properties into analytical
equations, some preliminary investigations regarding the effects of fiber waviness on
natural vibration frequency and critical buckling loadings were performed. In addition, the
behaviors of multi-layered composite beams with wavy-fiber layers were investigated. All
studies indicated that the effects of fiber waviness on composite performance were very

significant.

To Zhenyu
and our two children:
Yang Yang

Huan Huan

ACKNOWLEDGEMENTS

The author wishes to express her sincere appreciation and gratitude to:

Dr. Dahsin Liu, her advisor, for his constant inspiration, encouragement,
interest, and supervision under whom this investigation has been conducted.

Dr. T. Pence, Dr. H. Tsai, Dr. F Zhou, and Dr. M. Ostaja, her guidance
committee members, for their interest and suggestions.

Her parents, Mr. Ruren Xu and Ms. Meixian Wang , and her mother—in-law,
Ms. Xiejun Chen for their interest, support, and understanding through the years.

Her husband, Zhenyu Liu, and two children for their encouragement,

understanding, sacrifices, and assistance which have made this thesis possible and

DCCCSSZII')’ .

TABLE OF CONTENTS

LIST OF FIGURES ............................................................................................................ ix

CHAPTER I INTRODUCTION .................................................................................... I

l . 1 Motivation .................................................................................................... 1

1.2 Literature Survey .......................................................................................... 2

1.2.] Fiber Waviness ................................................................................ 2

1.2.2 Fiber-Matrix Bonding ..................................................................... 3

1.3 Organization of Thesis ................................................................................. 5

CHAPTER 2 HOMOGENIZATION METHOD ............................................................ 7

2.1 Fundamental Equations ................................................................................ 7

2.2 Formulation ................................................................................................ l l

2.3 Case Study 1: Straight - fiber Composites ................................................. 17
CHAPTER 3 ANALYTICAL STUDY OF EFFECTIVE MODULI OF

SINUSOIDAL—FIBER COMPOSITES .................................................. 24

3.1 Statement of Problem ................................................................................. 24

3.2 Solution for xg“ ......................................................................................... 27

32.] Domain Transformation ................................................................. 27

3.2.2 Solution of xgk' for Small Amplitude 0t ........................................ 32

3.3 Effective Moduli ....................................................................................... 42

vi

3.4

3.5

CHAPTER 4

4.1

4.2

4.3

CHAPTER 5

5.1

5.2

CHAPTER 6

6.1

6.2

6.3

3.3.1 Comparison with Other Approaches ............................................. 45

3.3.2 The Effects of Fiber Waviness on Effective Moduli ..................... 45
Effects of Interfacial Conditions on Effective Moduli ............................... 46
Effects of Volume Fraction on Effective Moduli ....................................... 60

FINITE ELEMENT ANALYSIS FOR EFFECTIVE MODULI OF

SINUSOIDAL FIBER COMPOSITES .................................................. 63
Variational Formulation ............................................................................. 63
Penalty FEA ............................................................................................... 66
4.2.1 Finite Element Formulation ........................................................... 66
4.2.2 Penalty Finite Element Analysis ................................................... 69
Effective Moduli ........................................................................................ 71
4.3.1 Effective Moduli of Sinusoidal-Fiber Composites with Small

Amplitude ...................................................................................... 71
4.3.2 Effective Moduli of Composites with Isotropic Constituents ....... 77

4.3.3 Effective Moduli of Composites with Anisotropic Constituents ..83

GLOBAL-LOCAL STRESS ANALYSIS OF SIN USOIDAL-FIBER

COMPOSITES ...................................................................................... 88
Uniaxial Tension ........................................................................................ 89
Cylindrical Bending ................................................................................. l 16

PERFORMANCE OF SINUSOIDAL—FIBER COMPOSITE BEAM. 143

Introduction .............................................................................................. 143
Material Nonlinearity ............................................................................... 143
Natural Vibration Frequency .................................................................... 147

vii

6.4 Critical Buckling Analysis ....................................................................... 155

6.5 Cylindrical Bending of Composite Beams with Sinusoidal-fiber Layer .158

6.6 Effects of Material Nonlinearity .............................................................. 158
CHAPTER 7 CONCLUSIONS .................................................................................. 168
REFERENCES ................................................................................................................ 175
APPENDICE .................................................................................................................... 176

APPENDIX A Maple Procedure ........................................................... 179

APPENDIX B Stress Distributions for Casel in Section 5.1 by

ABAQUS ..................................................................... 191
APPENDIX C Pure Tension in x2 direction ......................................... 195
APPENDIX D Natural Vibration Frequency ........................................ 218
APPENDIX E Critical Buckling Load ................................................. 220

viii

LIST OF FIGURES

Figure 2.1a - A composite material is organized by periodic unit cells microscopically....8

Figure 2.1b - A unit cell ...................................................................................................... 8
Figure 2.2 — A periodic unit consisting fiber and matrix. ................................................. 18

Figure 2.3 - Effects of interfacialimperfection on moduli of composites with straight

fibers ............................................................................................................. 22
Figure 3.1 - A unit cell consists of sinusoidal-fiber and matrix. ....................................... 25
Figure 3.2 - Details of unit normal vector n and unit tangential vectort. ........................... 28

Figure 3.3 - The geometry of the unit cell of sinusoidal-fiber composites after domain
transformation. ............................................................................................. 29

Figure 3.4 - Effects of tangential coefficient on shear modulus of a straight-fiber
composite material. ....................................................................................... 47

Figure 3.5 - Effects of Fiber waviness ratio on Young's moduli of a sinusoidal-fiber

composite material. ................................................................................ . ...... 48
Figure 3.6 - Effects of fiber waviness ratio on moduli. .................................................... 49
Figure 3.7 — Effects of fiber waviness ratio on Poisson‘s ratios ......................................... 50

Figure 3.8 - Effects of interfacial coefficients on Young’s modulus of composites with a
3% waviness ratio. ........................................................................................ 52

Figure 3.9 — Effects of interfacial coefficients on transverse Young‘s modulus of
composites with a 3% waviness ratio. .......................................................... 53

Figure 3.10 - Effects of interfacial coefficients on shear modulus ofcomposites with a
3% waviness ratio. ..................................................................................... 54

Figure 3.11 - Effects of interfacial coefficients on Poisson’s ratios ofcomposites with a
3% waviness ratio. ..................................................................................... 55

Figure 3.12 - Effects of interfacial coefficients on Poisson’s ratios of composites with a
3% waviness ratio. ..................................................................................... 56

Figure 3.13 - A unit cell with an embeded layer. .............................................................. 58

Figure 3.14 — Effects of volume fraction on moduli for the composites with perfect fiber-
matrix interface. .......................................................................................... 61

Figure 3.15 - Effects of fiber volume fraction on Poisson ratios for the composites with
perfect fiber-matrix interface. ..................................................................... 62

Figure 4.1- A basic unit cell. ............................................................................................. 64

Figure 4.2 - Comparison of two appraochs for the effects of fiber waviness on
engineering constants ................................................................................... 72

Figure. 4.3 - Effects of interfacial imperfection on engineering constants of the composite
with 3% fiber waviness. ............................................................................... 73

Figure 4.4 - A cell of unidirectional-fiber composite with a rotatin angle 9. .................... 75

Figure 4.5 - Comparisons of two approach fot the effective moduli of unidirectional-
fiber composites between tensor transformation and finite element analysis.

.................................................................................................................... 76
Figure 4.6 - Effects of fiber waviness on Young’s moduli and shear modulus of
sinusoidal glass/epoxy composites. .............................................................. 78
Figure 4.7 - Effects of fiber waviness on poisson‘s ratio of sinusoidal glass/epoxy
composites ..................................................................................................... 79
Figure 4.8 - Effects of interfacial imperfection on Young’s and shear moduli of glass/
epoxy composites with 30% fiber waviness. ................................................ 80

Figure 4.9 - Effects of interfacial imperfection on Poisson's ratios of glass/epoxy
composites with 30% fiber waviness. ........................................................... 81
x

Figure 4.10 - Effects of fiber waviness on Young’s and shear moduli of carbon/epoxy
composites ................................................................................................... 82

Figure 4.1 l - Effects of fiber waviness on Poisson’s ratio of carbon/epoxy composites. .83

Figure 4.12 - Effects of tangential bonding coefficient on Young’s and shear moduli of
carbon/epoxy composites with 30% fiber waviness. ................................. 85

Figure 4.13 - Effects of tangential bonding coefficient on Poisson’s ratios of carbon

epoxy composites with 30% fiber waviness. ............................................... 86
Figure 5. la - A sinusoidal-fiber composite under pure tension ......................................... 90
Figure 5.1b - A unit cell of the sinusoidal-fiber composite given in Figure 5.1a. ............. 90

Figure 5.1c - A unit cell of the sinusoial-fiber composite expressed by another local
coordinate system 2 1- 22. ............................................................................ 91

Figure 5.2a - Deformation of a unit cell with perfect interfaceunder uniaxial tension in
y,—direction ................................................................................................. 95

Figure 5.2b - Global deformation of a unit cell with perfect interfaceunder uniaxial
tension in yl-direction. ............................................................................... 96

Figure 5.2c - Local deformation of a unit cell with perfect interface under uniaxial
tension in yl-direction. ............................................................................... 97

Figure 5.3 - Deformation of a unit cell with tangential imperfect interface(p.=lOmm/N)
under uniaxial tension in yl-direction. ....................................................... 98

Figure 5.4 - 0'” along :l-axis in a unit cell with perfect fiber-matrix interface subject to
uniaxial tension in z,—direction. ............................................................... 100

Figure 5.5 - on along :I-axis in a unit cell with perfect fiber-matrix interface subject to
uniaxial tension in zl-direction. ............................................................... 101

Figure 5.6 - on along zl-axis in a unit cell with perfect fiber-matrix interface subject to
uniaxial tension in zl-direction. ............................................................... 102

Figure 5.7 - 0’” along :z-axis in a unit cell with perfect fiber-matrix interface subject to
uniaxial tension in z.-direction. ............................................................... 103

xi

Figure 5.8 - 622 along zz-axis in a unit cell with perfect fiber-matrix interface subject to
uniaxial tension in zl-direction. ............................................................... 104

Figure 5.9 - 012 along zz-axis in a unit cell with perfect fiber-matrix interface subject to
uniaxial tension in zl—direction. ............................................................... 105

Figure 5.10 - 0'“ along zl-axis in a unit cell with tangentially imperfect (u=10mm/N)
fiber-matrix interface subject to uniaxial tension in zl-direction. ............ 106

Figure 5.1 l - 622 along zl-axis in a unit cell with tangentially imperfect (u=10mm/N)
fiber-matrix interface subject to uniaxial tension in zl-direction. ............ 107

Figure 5.12 - 012 along zl-axis in a unit cell with tangentially imperfect (u=lOmm/N)
fiber-matrix interface subject to uniaxial tension in zl-direction. ............ 108

Figure 5.13 - 0'” along zz-axis in a unit cell with tangentially imperfect (p.=10mm/N)
fiber-matrix interface subject to uniaxial tension in zl-direction. ............ 109

Figure 5.14 - 622 along zz-axis in a unit cell with tangentially imperfect (u=10mm/N)
fiber-matrix interface subject to uniaxial tension in zl-direction. ............ 110

Figure 5.15 - 612 along zz-axis in a unit cell with tangentially imperfect (u=lOmm/N)
fiber-matrix interface subject to uniaxial tension in zl-direction. ............ 1 11

Figure 5.16 - Unit normal vector “i and unit tangential vector ti. ................................... 1 13

Figure 5.17 - Tn along the fiber-matrix interface with various imperfect tangential
coefficients u ............................................................................................ 1 14

Figure 5.18 - T, along the fiber-matrix interface with various imperfect tangential
coefficients [.1 ............................................................................................ l 15

Figure 5.19 - A simply supported composite beam under cylindrical bending ............... l 17

Figure 5.20 - Distribution of 0",. of a sinusoidal-fiber composite beam under cylindrical
bending ..................................................................................................... 120

Figure 5.21 - Distribution of 6% of a sinusoidal-fiber composite beam simply supported
under cylindrical bending. ....................................................................... 121

Figure 5.22 - Distribution of 0”,: of a sinusoidal-fiber composite beam under cylindrical
bending ..................................................................................................... 122

xii

Figure 5.23 - on at the midsection and along xZ-axis of a sinusoidal-fiber composite
beam withperfect fiber-matrix interface ................................................... 125

Figure 5.24 - 022 at the midsection and along xz-axis of a sinusoidal-fiber composite
beam withperfect fiber-matrix interface ................................................... 126

Figure 5.25 - 0'12 at the midsection and along xZ—axis of a sinusoidal-fiber composite
beam withperfect fiber-matrix interface ................................................... 127

Figure 5.26 - on at midsection as a function of x2 for a sinusoidal-fiber composite with
imperfect tangential interface (u=lOmm/N). ........................................... 128

Figure 5.27 - 0'22 at midsection as a function of x2 for a sinusoidal—fiber composite with
imperfect tangential interface (u=10mm/N). ........................................... 129

Figure 5.28 - 0'12 at beam end as a function of x2 for a sinusoidal-fiber composite with
imperfect tangential interface (p.=lOmm/N). ........................................... 130

Figure 5.29 - 9'11 along the zl-axis in a unit cell near x1=10, x2=hf2 with perfect fiber-
matrix interface. ....................................................................................... 13 1

Figure 5.30 - on along the zl-axis in a unit cell near x1=I/2, x2=h/2 with perfect fiber-
matrix interface. ....................................................................................... 132

Figure 5.31 - 0’12 along the zl-axis in a unit cell near xl=0, x2=0 with perfect fiber-matrix
interface .................................................................................................... 133

Figure 5.32 - 0'” along the zz-axis in a unit cell near xlzl/Z, xzzh/Z with perfect fiber-
matrix interface. ....................................................................................... 134

Figure 5.33 - 0'22 along the zz-axis in a unit cell near xlzl/Z, xzzh/Z with perfect fiber-
matrix interface. ...................................................................................... 135

Figure 5.34 - on along the zz-axis in a unit cell near xl=0, x2=0 with perfect fiber-matrix
interface .................................................................................................... 136

Figure 5.35 — on along the Zl-aXiS in a unit cell near x|=l/2, x2=h/2 with tangentially
imperfect fiber-matrix interface (ulemm/N). ........................................ 137

Figure 5.36 - 0'22 along the zl-axis in a unit cell near xl=l/2, x2=h/2 with tangentially
imperfect fiber-matrix interface (u=10mm/N). ........................................ 138

xiii

Figure 5.37 - 612 along the zl-axis in a unit cell near x1=0, x2=0 with tangentially
imperfect fiber-matrix interface (u=10mm/N). ........................................ 139

Figure 5.38 - on along the zz-axis in a unit cell near x1=l/2, x2=h/2 with tangentially
imperfect fiber-matrix interface (u=10mm/N). ........................................ 140

Figure 5.39 - 0'22 along the zz-axis in a unit cell near x1=l/2, x2=h/2 with tangentially
imperfect fiber—matrix interface (u=10mm/N). ........................................ 141

Figure 5.40 - on along the zz-axis in a unit cell near x|=0, x2=0 with tangentially

imperfect fiber-matrix interface (p.=10mm/N). ........................................ 142
Figure 6.1a - A sinusoidal-fiber composite is subjected to uniform tensile strain. ......... 144
Figure 6.1b - The microscopic unit cell of the composite. .............................................. 144

Figure 6.2. - Stress-strain relations for sinusoidal—fiber composites (carbon/epoxy) with
25% fiber volume fraction at various fiber waviness ratios ..................... 148

Figure 6.3. - Stress—strain relations for sinusoidal-fiber composites (carbon/epoxy) with
50% fiber volume fraction at various fiber waviness ratios. .................... 149

Figure 6.4 - A simply supported composite beam ........................................................... 150

Figure 6.5 - Effects of fiber waviness on natural vibration frequency of simply
supported beams made of sinusoidal-fiber composites .............................. 154

Figure 6.6 - A simply supported composite beam subjected to uniaxial compression. .. 156
Figure 6.7 - Effects of fiber waviness ratio on critical buckling load .............................. 157

Figure 6.8 - A simply supported multilayered composite beam under cylindrical bending.

.................................................................................................................. 159
Figure 6.9 - OH in simply supported beams under cylindrical bending .......................... 163
Figure 6.10 - 6.2 in simply supported beams under cylindrical bending. ....................... 164
Figure 6.1 l - 022 in simply supported beams under cylindrical bending ........................ 165
Figure 6.12 - Effects of fiber waviness on maximum 01 1 for various beams. ................ 166

xiv

Figure 6.13 - Effects of fiber waviness on maximum 011 for various beams. ................ 167

Figure 6.14 - A simply supported beam with nonlinear materialproperty under 3-point

bending ..................................................................................................... 169
Figure 6.15 - finite element model. ................................................................................. 170
Figure 6.16 - Linear and nonlinear stress-strain relation. ................................................ 171

Figure 6.17 - Force-displacement relations based on both linear and nonlinear stress-
strain relations. .......................................................................................... 172

Figure 6.18 - 0'” versus loading P. .................................................................................. 173

Figure 8.1 - Finite element model for a sinusoidal-fiber composite beam under uniaxial
loading in xl-direction ............................................................................. 192

Figure B.2 - SI 1 for a sinusoidal-fiber composite beam under uniaxial loading in x]-
direction ................................................................................................... 193

Figure 3.3 - S12 for a sinusoidal-fiber composite beam under uniaxial loading in x,-

direction ................................................................................................... 194
Figure C. 1a - A curved fiber composite under pure tension ............................................ 197
Figure C. lb - A unit of the curved fiber composite given in Figure C. la. ...................... 197

Figure C. lc - A unit of the sinusoial-fiber composite expressed by another local
coordinate system zl-zz ............................................................................ 198

Figure C.2 - Deformation of a unit cell with perfect interface ........................................ 201

Figure C.3 - Deformation of a unit cell with imperfect perfect interface (k: =0.lmm/N)..

.................................................................................................................. 202
Figure C.4 - 0'” along :l-axis in a unit cell with perfect fiber-matrix interface. ............ 204
Figure C.5 - 0'22 along zl-axis in a unit cell with perfect fiber-matrix interface. ............ 205
Figure C.6 - 012 along :l-axis in a unit cell with perfect fiber-matrix interface. ............ 206
Figure C.7 - 0” along :z-axis in a unit cell with perfect fiber-matrix interface. ............ 207

XV

Figure C.8 - 0'22 along zz-axis in a unit cell with perfect fiber-matrix interface. ............ 208
Figure C.9 - 012 along zz-axis in a unit cell with perfect fiber-matrix interface. ............ 209
Figure C.10 - on along zl-axis in a unit cell with imperfect fiber-matrix interface. ...... 210

Figure C.11 - 0'22 along zl-axis in a unit cell with imperfect fiber-matrix interface
(k=u=0.1mm/N). ...................................................................................... 21 1

Figure C.12 - 0'12 along zl-axis in a unit cell with imperfect fiber-matrix interface
(k=u=0. 1mm/N). ...................................................................................... 212

Figure C.13 - on along zz-axis in a unit cell with imperfect fiber-matrix
interface(k=p.=0. lmm/N). ........................................................................ 213

Figure C. 14 - 0'22 along zz-axis in a unit cell with imperfect fiber-matrix
interface(k=u=0. lmm/N). ........................................................................ 214

Figure C. 15 — on along zz-axis in a unit cell with imperfect fiber-matrix
interface(k=u=0. 1mm/N). ........................................................................ 215

Figure - C. 16 Normal traction Tn on fiber-matrix interface and along yl-axis at different
bonding coefficients u, k. ......................................................................... 216

Figure - C. 17 Tangential traction TI on fiber-matrix interface and along yl-axis at
different bonding coefficients u,k. ........................................................... 217

xvi

CHAPTER 1

INTRODUCTION

1.1 Motivation

Due to their high stiffness and high strength with low density, fiber-reinforced
polymer—matrix composite materials have become very important candidate materials for
high-performance structures. Various fiber geometries have been utilized in composite
structure constructions; for example, chopped fiber, swirl mats, woven fabrics, and
unidirectional tapes. As compared to straight fibers, curved fibers provide an extra variable
- continuous change of fiber orientation - for composite materials and structural designs.
Although the use of curved fibers in composite materials and structural designs seems
promising, the current engineering composites are limited to straight fibers due to the lack
of efficient design tools and manufacturing facilities. In addition to the positive
applications, curved fiber or wavy layers created during composite manufacturing, e.g.,
thermal processing, filament winding, braiding, and stitching, may give negative effects on
composite material properties and structure performance.

In addition to fiber waviness, another parameter which significantly affects
composite material properties is the bonding condition between fiber and matrix. Perfect
bonding is excellent for load transferring between the constituents and can result in a high
version of Young’s moduli for composites. However, high rigidity usually means low
toughness, which absorbs less energy during the damage process. In order to raise the

toughness level, the bonding condition should be relatively poor. When a composite

2

structure has misoriented fibers or is exerted by misaligned loading, a tougher material is
more durable and thus desired. It then is obvious that the bonding condition also plays a

very important role in composite performance.

1.2 Literature Survey

1.2.1 Fiber Waviness

The study of fiber waviness’ effects on composite materials has received a lot of
attention in recent years. Some were associated with material properties, such as effective
moduli, and others were concerned with the structure performance, like buckling. An
idealized sinusoidal fiber waviness was considered as a microstructure model for
analytical study and its effects on Young’s moduli of composite materials were
investigated by Kuo, Takahoshi, and Chou [1]. Based on an integral scheme, they obtained
the bounds of effective moduli under uniform stress and uniform strain. Their theoretical
model revealed that the upper-bound and the lower-bound of the effective moduli were
one in the same. A similar study was also conducted by Rai, Rogers, and Crane [2]. Both
of these studies were verified by experiments. However, it should be pointed out that in
both cases only waviness of small amplitudes (less than 5%) were considered. Amplitudes
up to 20% were later examined by Takahashi and Chou [3] with a numerical method.
Nonlinear stress-strain relations were presented in their studies by virtue of an incremental
loading technique.

Another study presented to find the effective Young’s modulus of composite

materials with sinusoidal fibers was performed by Lee and Harris [4]. Instead of using the

3

integral method, their approach was based on both the energy method and Euler-
Bemoulli’s Beam Theory. The results from their studies showed that fibers with a slight
curvature could produce a noticeable nonlinear effect on the stress-strain relationship.
However, only isotropic constituents were considered in their study.

A more refined technique for studying composite materials with sinusoidal fibers
was presented by Akbarov and Guz [5]. They took the amplitude as a small parameter to
perturb the equilibrium equations of the linear elasticity theory. The stress-strain
relationship of a laminated composite with wavy layers under uniform tension was
determined. In their study, they also concluded that a small curve could have a significant
effect on the constitutive relation of the composite.

In addition to the effective modulus, the behaviors of composite structures are also
strongly affected by fiber waviness. Raouf [6] discussed the effects of layer waviness on
interlaminar stresses in thick composites. His investigation was based on the Laminate
Theory and the results showed that layer waviness could cause a significant increase in
interlaminar normal and shear stresses.

By virtue of nonlinear finite element analysis, Wisnom [7, 8] studied the effects of
fiber waviness on the compressive strength of composite beams under compressive
loading and pure bending. He pointed out that the maximum compressive stress was
mainly dependent on the fiber misalignment. With the increase of compressive loading the
shear stress resulting from the fiber misalignment could cause the fiber to rotate and,
subsequently, increase the degree of fiber misalignment. Another finite element analysis
used to study the influence of fiber waviness on stress distribution was performed by Hyer,

Mass, and Fuchs [9] for hydrostatically loaded cylinders. They found that the thermal—

 

4

stress—induced fiber curvature was able to cause a significant reduction in the stability of

composite cylinders.

1.2.2 Fiber—Matrix Bonding

In addition to fiber waviness, the behavior of composite structures is also
significantly affected by the nature of the bonding condition between fiber and matrix.
Studies on the interfacial effects on composite properties could be divided into two
groups. One is related to the bonding effect on laminate performance and the other is
associated with microscopic modeling.

A Linear Shear Slip Theory combined with Euler-Bernoulli’s Beam Theory was
introduced by Newmark, Seiss, and Viest [10] to study laminated beams with imperfect
bonding. Similar investigations to incorporate different slip theories were performed by
Goodman and Popkov [11]; Goodman [12]; Thompson, Goodman, and Vanderbilt [13];
and Vanderbilt, Goodman, and Grisnell [14]. However, it should be noted that the
transverse shear effect, which was important to the failure analysis of laminated
composites, was not considered in their studies. In addition to the Beam Theory, an
elasticity approach was performed to study the interlaminar bonding problems by Rao and
Ghosh [15], and Fazio; and Hussein, and Ha [16], though the elasticity analysis was only
limited to some special cases.

A third technique which combined a Laminate Theory and an Interfacial Shear
Slip condition was first carried out by Toledano and Murakami [17]. Based on a layer-wise
laminate theory [18] and an objective to satisfy of the continuity conditions on composite

interfaces, Lu and Liu [19] presented an Interlayer Shear Slip Theory for the delamination

5

analysis. A more general theory concerning both interlayer shear slip and interlayer
normal separation was further discussed by Liu, Xu, and Lu [20]. Their theory was valid
for composite laminates having shearing mode (mode 11) and normal mode (mode I)
delamination.

Quite a few studies were presented in microscopic modeling. An elasticity
approach was used to investigate the elastic moduli of composites with rigid sliding
inclusions by Jasiuk, Chen, and Thrope [21]. Achenbach and Zhu [22] studied composite
interfaces with both shear-slip and normal separation by the boundary element method.
Lene and Leguilion [23] investigated the interfaces with tangential slip by homogenized
constitutive law. The interface and interphase conditions have also been characterized
experimentally by Kalantar and Drzal [24]. It was concluded from all of these
investigations, both theoretical and experimental, that the interfacial condition has a

significant effect on the behavior of composite structure performance.

1.3 Organization of Thesis

The behavior of curved-fiber composites is a primary concern of this thesis. In
order to present a rigorous analysis, only sinusoidal fiber composites are investigated. In
addition, various conditions of interfacing between fiber and matrix are also of interest.
The thesis consists of seven chapters. Chapter 2 discusses a homogenization method to
undertake the analysis. The homogenization method is introduced and developed for the
composites made of two constituents with various interfacial conditions. Chapter 3 is
devoted to an analytical process for finding effective moduli of composites with small-

amplitude sinusoidal fibers. The purpose of Chapter 4 is to obtain the effective moduli of

6

sinusoidal-fiber composites with large amplitudes. A finite element method is performed
in this chapter. In order to account for continuity conditions on fiber-matrix interface and
periodic conditions on unit cell boundaries, a penalty method is required in the finite
element analysis. Chapter 5 focused on global and local stress distributions for two
loading cases; one is pure tension and the other is cylindrical bending. Chapter 6 presents
the performance of composite structures made of sinusoidal fiber composites. Nonlinear
stress-strain relationship, natural frequencies, critical buckling loading, and cylindrical

bending are of interesting. Finally, Chapter 7 gives the conclusions for the thesis.

CHAPTER 2

THE HOMOGENIZATION METHOD

The homogenization method is a rigorous mathematical approach for double-scale
analysis. It has been used to study material response and effective moduli of composites
and soils. It is especially useful for studying materials having periodic microstructures.
Combining with numerical techniques such as the finite element method, the
homogenization method has been used to investigate bounds of moduli of braided
composites [27], which have no other analytical solution. Since the homogenization
method is aiming at double-scale analysis, it divides the solution into two parts. One is the
global part which expresses the periodically heterogeneous material by “equivalent”
properties and the other is the local part which represents the disturbance due to
microscopic irregularity. The foundations of the homogenization method can be found in

articles by Babuska [25] and Benssousan, Lion, and Papanicoulau [26].

2.1 Fundamental Equations

Consider a composite material consisting of periodic unit cells which have two
constituents. The whole domain (28 of the composite material is shown in Fig. 2.1a while
a unit cell is shown in Fig. 2.1b. The unit cell is organized by a discontinuous constituent
embedded in a continuous constituent. In order to investigate the effects of the interfacial
condition on composite performance, the interfaces between the constituents are assumed

to be nonrigid.

   
 

  
  

      

 

 

     

 

 

 

 

 

 

 

           

 

 

 

 

 

 

 

 

    

. o o o o o o o o
444—; . ___._ 2 2 2
7 2 2 :— .____Q g 2 2
Tu _> / O o o o o o o o o
O o o o o o O o o
O o o o o o o o o
O o o o o o o o o
O o o o o o o o o
O o o o o o o o . 1'21.
0 o o o - - - -
Qe

 

 

 

 

 

Figure 2.1a - A composite material is organized by periodic unit cells.

 

 

 

)2
interface Y
continous constitutent .___> , . .
__ discontinous constituent
9 .1’1

 

Figure 2. lb - A unit cell.

9

From the linear elasticity theory, the following basic equations; i.e., equilibrium
equations, constitutive equations, linear strain-displacement relations. and corresponding

boundary conditions; are employed in this study:

Gij.j+fi = 0 in Q (2.1.1)
Gij = Eijklekl in Q (2.1.2)
ek, = %(“K1IL"“}C:9 in Q (2.1.3)
ui = 17,.(x) on F“ (2.1.4)
Gil-n}- = Ti(x) 0n FT (2.1.5)

In the above equations GU, eij, ui, and T,- represent stress, strain, displacement, and
traction, respectively, while 17,- and T,- are prescribed displacement and traction on the
composite boundary, respectively. The domain 9 is defined as the domain (28 without the
interfaces between the constituents and F , which is the sum of sub-boundaries Fu, and Fr
is the global boundary of the composite material. However, the local domain of the unit
cell is represented by Y, as shown in Fig. 2.1b. Since both global scale for the composite
and local scale for the‘u—nits are involved in the study, two coordinate systems xl-xz and y]-
)‘2 are required for global and local scales, respectively.

In investigating the homogenized properties of a composite material with nonrigid

fiber-matrix interfaces, i.e., imperfect interfaces, two interfacial theories, one for normal

direction and the other for shearing (or tangential) direction, are required. In this study,

10

linear interfacial theories are assumed.
When shear slip takes place on an interface, the shear traction is assumed to be

linearly proportional to the difference between the in-plane displacements of the

constituents, i.e.,

1

TI] : th = al[ut2-u!1] (2.1.6)

where T11 and T,2 are the shear tractions on the interface of the two constituents while
“:1 and u,2 are the shearing displacement components. In addition, 11 is the shear slip
coefficient. If u approaches infinite, the shear tractions will vanish and the interface will
become completely debonded in the tangential direction. On the contrary, when [1
becomes zero, the difference of shearing displacements must vanish in order to have finite
shear tractions. It then represents a perfect tangential-interface condition.

Similarly, a linear relation can also be used to correlate normal tractions with the
difference of normal separations on the interface between fiber and matrix, i.e.,

‘1‘ ' l. __ I I -’ 6:)

1‘1“)”;
1
Tnl = T112 = ETJL‘nZ—“nll (2'1'7)

where k is the normal separation coefficient which has similar features as u. i.e., k
becomes zero when the interface is perfectly rigid in normal direction while infinitely
large when there is no normal bonding on the interface. In addition, care should be
exercised during numerical analysis to avoid any physical absurdity of penetration
between the constituents on the interface.

Moreover, it should be pointed out that the normal and tangential components of

ll

displacement are defined as:

u = u-n. (2.1.8)

and

(u) = u.-—u n- (2.1.9)

T = O-n-n. (2.1.10)

(2.1.11)

2.2 Formulation

As mentioned before, a double-scale analysis is required to investigate problems
with periodic microstructure. In this study, x is chosen as the macroscopic variable while
y is chosen as the microscopic variable. The relation between the two scales can be

expressed as

(2.2.1)

mlk

where 8 has an order approximately equal to the dimension of the unit cell, which is
considered to be very small when compared to the dimensions of the composite.
In performing a double-scale analysis, the following asymptotic expansions are

asSumed:

12

ul-(x, y) = u?(x, y) + su}(x, y) +€2u,-2(x, y) + (2.2.2)
1 0 1 2

Since the domain of interest consists of periodically distributed units, both u?(x, y) and
(SE-(x, y) (n=0,l,2,3....) should be periodic functions of y. Hence, the mean values of

their differentiations of displacement and stress with respect to y vanish, i.e., _
Mia: ,1] = Mm”. ] = 0 (2.2.4)

11'. )7

The mean value of a scalar function, e.g. v(x, y) , is defined as

l
1Y1

__._—

M[v]=—Iv(x,y)dY _ , ' '° (2.2.5)
Y \ 4

It is noted that the compound function v(x, y) can be differentiated in the following

manner:
a ' x 8v 13v 1
_) ,— : ———+——— ’ = ’- —’, 2..26
afiix 8) (8x,- say)” y)|y_£ i" +8i’)’ ( )
I. E
-3
Substituting the asymptotic expansions (2.2.2) and (2.2.3) into Eqns. (2.1.2) and

(2.1.1), respectively, and following Eqn. (2.2.6) and using Eqn. (2.1.3) it yields

1 0 l 0 1 l 2

. 1" - ,'
-, | . ,3 , ‘
13*) 2 . w . , ,

/ .
1 J ‘ A

1 0 0 0 1 1 2 2
0.” = 22£ijk1(uk~"!+ up“) + Eijkl(uk, 1+ “k.y,) + EEl-jk,(uk’ l + ukm) + 0(8 ) (2.2.8)

By comparing the coefficients of the 1/82 term on both sides of Eqn. (2.2.7), the following

equation can be concluded:

0
“11. y,

= 0 (2.2.9)

Similarly, by comparing the coefficients of 1/8 term in Eqn. (2.2.8) and Eqn. (2.2.3), it

yields
0 _v 0
CU = Eijklekl(ui) (2.2.10)
where
. 0 1 0 o ‘ '
e;,(u,.) = §(uk‘),l+u,‘yk) (2.2.11)

. . . . 0 0 .
In add1t1on, the boundary conditions of 11,. and Oi}- can be expressed as follows by

substituting the asymptotic expansions into Eqns. (2.1.4) and (2.1.5)

14? = ill-(x) on F“ (2.2.12)
0 — 0 r 2213
Gil-n!- — on T ( . . )

They can be further simplified by combining Eqns. (2.2.9—2.2. 13), i.e..

u‘.’ = uf.’(x) (2.2.14)

I

0.. = 0 (2.2.15)

With the same manner as used in deriving Eqns. (2.2.9) and (2.2.10), the

coefficients associated with the We term of Eqn. (2.2.7) give rise to

l

GILL : O (2.216)

while those associated with the 80 term of Eqn. (2.2.8) produces

l O ' l
61‘} = Eijk,[ek,(ui)+e',:l(ui)] v (2.2.17)

Eqn. (2.2.17) can also be expressed in another form:
r .

M
I

.w-

» ‘ - .l
1 7-;

_. o 1 .2 _
[Eat/(”10.x”) +Eijk1()’)uk,yl(x,N11,.) = 0 (2.2.18)

. . . . . 1 .
By usmg the separation of variables technique, the solution of u, can be found to me

kl 0 ,
alum) = -x, (y)u,,.,,(x) (2.2.19)

in which x: should satisfy

kl ' ,
[Eijg;,(.V)Xg,)-h()’)]'yj = Ely/((0’)), 1 (2.2.20)

1

. . o . . ,
Moreover, from comparing the coefficients of 8 terms on both Sides of Eqn. (2.2.7), the

following equation can be obtained

1 2
91),j+91j,y,+f1 = O (22.21)

15

in which

1 kl 0
Gij = (EU/(10’) _Eijgh(Y)Xg,yh(.Y))u/(,1(x)

(2.2.22)

The associated boundary conditions can also be obtained through a similar process. They

are I

o -
ul- = Ill-(X) on F

0"": = 61(x) 0" FT
By giving the following definitions,
1
TU = M[o,-j(x, )0]

H kl
Eijkl = M[Eijk1_Eijghxg, 1).]

the equations studying homogenization problems can be summarized as
tij,j+fi=O In Q
H 0 .
T1) = Eijklekl(ui) m Q

0 —

I-n- = 6i(x) 0n FT

(2.2.23)

(2.2.24)

(2.2.25)

(2.2.26)

(2.2.27)

(2.2.28)

(2.2.29)

(2.2.30)

16

It should be noted that Eqn. (2.2.27) is, in fact, the equilibrium equation for global scale.
The constitutive equations, Eqn. (2.2.28), are based on the homogenized material properties

defined in Eqn. (2.2.26), i.e.,

H kl 1 kl ‘
E0.“ = M[Eijk,—Eijghxg,yh] = mfiEUH—EvghnghmY (2.2.31)
Y

where x?( y) should satisfy both periodic conditions and interfacial conditions. The

interfacial conditions are:

(51,-1.1 — 15,),th ,,),, = fiiag’)” — (1:25.21 (2.2.32)
(E.,-,.,—E,-,-,,.x’;f,,>,, = fii<x’;’),1-(x’;’>,2i (2.2.33)
(131,11 - 5.),th ”1,, = gulf)... — (£5.21 (2.2.34)
(E,,,., — E.,,,,x:f ,,),,, = 504’)... — (1(5),,21 (2.2.35)

where subscripts t1 and t2 are in the tangential direction for material 1 and material 2,
respectively. Similarly, subscripts n, and n2 refer to the normal direction for material 1 and
material 2, respectively.

With the use of asymptotic expansion, the homogenization method can be used to
established a double-scale analysis. The investigation of composite materials made of
periodic microstructure can be divided into global and local solutions. The local solution

kl . . . . . . .
X3 (y) should satisfy Eqn. (2.2.20), periodic conditions of Eqn. (2.2.4), and 1nterfac1al

17

conditions of Eqns. (22.32-22.35). Once the local solution is determined, the effective
moduli can then be calculated from Eqn. (2.2.26). Subsequently, the global solution for
u?(x) can be found from Eqns. (22.27-22.30). From both local and global solutions, the
displacement field u: and the stress field 0:}. can be determined from Eqns. (2.2.19) and
(2.2.22), respectively. The total solutions containing up to the first-order term of Eqns.

(2.2.2) and (2.2.3) can be obtained.

2.3 Case Study 1: Straight — fiber Composites

”Both distribution function x:[( y) and effective moduli of composite materials are
required in the homogenization analysis. As an example, a straight-fiber composite with the
microstructure of unit cell shown in Figure 2.2 is of concern. In fact, this study can be
reduced to a one-dimensional analysis and an analytical solution is possible.

With the use of Eqn. (2.2.20), and periodic and interfacial conditions, Eqns. (2.2.32-
2.2.35), the distribution function of a unit cell with imperfect bonding can be proposed.
Assuming both the constituents are orthotropic, there are four independent moduli E 1' j“,

. . . . . . k1 .
i.e., Elm, E1122, E3222, and 51212- A linear distribution function xg (y) is proposed as

follows:

'.
’ ,fe’ e
I

kl (k1 kl
X3 0') = A?) i2 +8,” (Os)?2 s c) (2.3.1)

xilm = Aim“): + 311W“ (C 5.12 S (C + 61)) (232’

where the superscript (1‘) represents for the fiber domain while the superscript (m) the matrix

domain. The distribution function x:’( y) given in Eqns. (2.3.1) and (2.3.2) satisfy the

18

.V2

 

d matrix

 

 

6 er .
.1 Y]

 

Figure 2.2 - A periodic unit consisting straight fiber and matrix.

l9

governing equation, Eqn. (2.2.20). The periodic conditions for the microstructure are:
kl k1
x8 (Y1, 0) = X8 (ch'i'd) (2.3.3)

kl kl
x8 (09 y2) = X8 (1’ )’2) (2.3.4)
By using Eqn.’(2.3.3), it yields

(m)kl

38”)“ :88, "W" +(c+d)Ag (2.3.5)

In addition, it should be noted that the components of unit normal vector on the interface
y2 = c, are:
nl = 0, n2 = 1

By using Eqns. (22.32-22.35), the four interfacial conditions are obtained as
A ., I

follows: , h, r l". . l . 2 11., .
V )4.
/
kl (f) kl (M)
(512kl“51212A1 ) ‘ (512kl‘Ei212A1 )
n kl_ Alikl kl kl
kl (f) kl (m)
(E22kl_E2222A2 ) =2(EZZkl—E2222A )

. . kl
Making use of these equations, the constants A g are found to be

20

f m I
(Did _ d(E12kl‘Eizkl)+uEizklE’1’212
A1 " f 15“" m f
1151212 1212+(651212+d51212)

 

(2.3.8)

f f m
U)“ _ d(E22kl-E,2n2kl)+kE22klE2222
A2 _ f m f (2.3.9)
[(5222252222 + (05222 ‘1' (152222)

 

 

 

f f
Atmikl _ _C(E12kl—E,Il2kl) +ILE’1nZkIE1212 2310
1 ‘ f f ( ' ' )
“5121257212+(CE’1n212'i'dE1212)
f m f
—cE ,—l~:,, +kE’" E
21‘2“)“: ( 1211 1.kl) 22k! 2222 (2.311)

f m m f
kE1212E1212 + (651212 + delZlZ)
The effective moduli can be rewritten as:

H 1 kl (f) 1 +4 kl (m)
E0.“ = m£(Eijk,-EU82A8) dy2+m£ (EW—Eijngg) (1y, (2.3.12)

By substituting Eqns. (2.3.8-2.3.11) into Eqn. (2.3.12), the effective moduli can then be

identified, i.e.,

 

 

 

f m
H _ aEllll'I'bEllll
1111 — 'c+d
’ 2 2 2 (2.3.13)
Cd(EI122-E1'1122) +kC(Ei122) Eg'222+kd(E'l"122) Eizzz
(C + d)(kE§222E’2"222 + 6521222 + dEizzz)
f m
c+d E E
8122 = ( ) 2222 2222 (2.3.14)

.-

f m m f
[‘5 2222E 2222 + 652222 + d5 2222

21

 

f m m f
E” -EH _ 65112252222+d5112252222 2315
1122- 2211- f Em m f (--)
[(152222 2222 ‘1‘ C E 2222 + d E 2222
f m
H H (C+d)E E
E1212 = E2121 = 1212 1212 (2.3.16)

 

f m m f
”5121251212 + 651212 + dElZlZ

It is noted that all other EH11“ equal to zero.

It is concluded from Eqns. (2.3.13-2.3.16) that the tangential bonding coefficient 11
only affects the shear modulus £11,212 , i.e., the engineering shear modulus 012- Also from
these equations, the normal bonding coefficient k can affect the engineering Young’s
moduli E11 and E22, and Poisson’s ratios v12 and v2].

As a numerical example, the following orthotropic material properties are

examined:

f f f f
Eiiii = E2222 — 30061)“, 51122 = E1212 = 100010“

Einlll = 5,2222 = 300p“, Einizz = Einziz = 106])“

C
C+

 

With the volume fraction V f = = 0.25, the effects of normal bonding coefficient k

on E1 1’ E22, v12, and v31, and the effects of tangential bonding coefficient 11 on 012 are
shown graphically in Fig. 2.4. The relationships between engineering constants and

material property tensor are:

 

 

 

 

 

 

 

3.5T El l/Em

3...
2:5—

2._
1.5+

a 612/ Grn
1...
EZZ/Em

risk 1v”

 

0-4 -3 -2 -1 0 i 2 3 1
log 11 (log k)

 

Figure 2.3 - Effects of interfacial imperfection on moduli of composites
with straight fibers.

 

23

H
_ E1122
V12 - T
1"32222

H
_ E1122
V21 - T
E1111

H
E11 = E1111(1-V12V21)
H

Fig. 2.4 reveals that the effects of k on E11 and v12 are not as significant as its effects on
E22 and v21. As the coefficients 1.1 and k increase, the bonding condition becomes poor,
resulting in the decrease of E22, 612, and v21. The results also indicate a distinct transition

zone for E22 from k=10-3 to k=1 , and for Glz from 11:1 to it: 103.

CHAPTER 3

ANALYTICAL SOLUTION FOR EFFECTIVE MODULI

OF SINUSOIDAL-FIBER COMPOSITES

The global behavior of a heterogeneous material is governed by the effective
moduli of the material. This chapter is aimed at identifying the effective moduli of
composite material consisting of sinusoidal fibers. In order to obtain the effective moduli
by the homogenization method, the distribution function x]: needs to be identified first.
The effective moduli can then be calculated from the distribution function, the geometry

of the unit cell, and the material properties of the constituents.

3.1 Statement of Problem

A composite material consisting of curved fibers with a sinusoidal shape is

considered in this chapter. The material properties of both constituents are assumed to be

{Jae

isotropic. The unit cell of the microstructure is shown in Fig. 3.1. Let Yf be the fiber

domain and Ym the matrix domain, then the entire domain of the unit cell is
Y = Yf + Ym. Assuming the length of the unit cell in the yl-direction is land the heights
for the fiber and matrix in the yz-direction are c and (1, respectively, the volume fraction of

fiber becomes

 

c . .
, while that of the matrix .
+d c+d

From Eqn. (2.2.20), the partial differential equations to govern the distribution

 

. kl .
function Xg (y) in Yare:

25

 

 

 

 

Figure 3.1-A unit cell consists of sinusoidal-fiber and matrix.

26

Ellllxlifylyl+(E1122+£1212)xl2fy,y3+El2l2xiiy3yz = 0 (3-1-1)
E1212Xl2i,\',)'1+(E1122+El2lZ)XlIiy,y3+£2222X2iy3y3 = 0 (3:12)

Denote
x:'(y) = ng’kky) in Y, (3.1.3)
xglm = xémwm m Y,,, (3.1.4)

The periodic conditions which x:[( y) should satisfy are:

kl kl
xi,“ (0.35) = Xéf) (1.1'2) (3.1.5)

kl m kl
71‘!” (0.112) = x; ’ (1.1-3) (3.1.6)
xf,’"’<.v,,asin(By.>) = xf,’"’“<y,,c-+c1+ocsin<13.v,)) (3.1.7)

It is noted that the interface between fiber and matrix is
.122 = c + Otsin([3yl) (3.1.8)

. . . . . . kl . . .
In addition to the periodic conditions. x, (y) should also satisfy the 1nterfac1al

conditions, i.e.,

(m)

(f)kl (m)kl

kl (f) kl 1
(Eijkl‘EijglzX.e.1-.) "1"; = (Eljkl—El‘jgl1Xg.j-h) "1le = 2W x, M, (3.1.9)

27

(f)

_ E E kl (m)
"1’,- - ( ljkl‘ ij.8llx.€~)‘l.)

kl (f)kl (m)kl
(EU/<1 _ Eijghxg. 1'1) _

1
where n,- and t,- are components of unit normal vector 11 and unit tangential vector t. They

are shown in Fig. 3.2. The tangential angle 0 (slope) is defined by

l

)1
tanG = —'—“— = ochos(Byl) (3.1.11)
dyl

. . 21c . . .
in which [3 = 7. Accordingly, the unit tangential components are:

H
~
ll

cosB

_ (3.1.12)
t7 = Sin 9
In addition, the relations between the unit normal vector n and the unit tangential vector t are :

nl = —t2 = —sin9 (3113)
n. =1l = cosB H

- kl
3.2 Solution forxg

3.2.1 Domain Transformation

In obtaining an analytical solution, a domain transformation is performed. Let

31 = ."1

(3.2.1)

:3 = yz—(xsin(Byl) (3.2.2)

28

9 yz=c+0tsin(Byl)

  

 

 

Figure 3.2 - Details of unit normal vector n and unit tangential vector t.

29

z2

 

 

 

 

_Zl

Figure 3.3 - The geometry of the unit cell of the sinusoidal-fiber composite
after domain transformation.

30

The sinusoidal geometry in the yz-direction can be eliminated from the domain

transformation. The unit cell then becomes a rectangular geometry as depicted by 2:1 -22

coordinate system as shown in Fig. 3.3. However, it should also be noted that although the

geometry of the domain becomes much simpler after transformation, the governing

equations and the interfacial conditions become more complicated instead.

Before performing the domain transformation, some fundamentals of calculus are

recalled below. The partial derivatives between yl-yz and zl-z2 systems can be

interchanged as follows:

a _ a _a_
3'71 -- ‘a—Z-l-*(1BCOS(BZI)aZ2

 

 

 

 

.9. = .3.
ayz 822
a2 82 i . 82 282 - a
— = ——201 cos( z )—+(0t COS( z )) .+<1 81MB“-
ayi azf B B 1 32,32, B B 1 8:3 B I a“?
82 = 82
av2 dz:
8?. 2 _, 82
31,31, a ,a .‘aBCOSw‘l’az,’

(3.2.3)

(3.2.4)

(3.2.5)

(3.2.6)

(3.2.7)

The equations for distribution functions x210) , Eqns (3.1.1) and (3.1.2), can then be

rewritten as follows:

31

kl ' k1
Ellll(xl,11- 2acos(Bz1)Xif12 + (0608(1321))2Xi,122 + aBsm<BZ1>Xl~ 3) (3 2 8)

kl kl kl
2“ (51122 + E1212)(X2, 12 ‘ “005(1331)X2,22) 2‘ E121211, 22 = O

k, k, 2 kl . k1
E1212(X2.11‘2aCOS(le)X2. 12 + (acos(lel) X2, 22 +0135m(BZi)X2.3) (3 2 9)

kl kl kl
+(E1122+E1212)(X1,12_aCOS(BZl)X1,22) +£2222X2.22 = 0
where a = (113. Notice that x2,- in Eqns. (3.2.8) and (3.2.9) indicates partial

. . . kl .
differentiation of x8 on Z1» i.e.,

This notation will be used throughout this chapter. Similarly, by using 21 -23 system, the

periodical conditions become

l l
ng”(o,z Q) = x""(1, zz) (3.2.10)
x‘,“"’(0,z z21= ’“(Lza (3.2.11)
XU’“(;,,0 0:) xl""“(; l,c+cl) (3.2.12)

and the interfacial conditions become

f")
(Eijkl_ Eijghx: [11+EU-glaCOS((BZ ”X: 2))( n}

(m n) X( )k1
= (Eljkl £02th [l1+EijglaCOS((Bz 0X: 2)) 2} "inf (3'2'13)

(m)k k

= 2092f WTXR )"s’

32

kl kl (f)
(Eijkl " Elngng. l: + Eijglacos((le ”(3. 2)) "inl'

(”’x‘gf’k’n .r. (3.2.14)

[(1 kl

1 ()k1 ()kl
=l—10c2f —xg"' )Ig

. . . . . kl
Since k, l, and g are valued 1 and 2 in two-dimen51onal analy51s, there are twelve 18 (z)
to be determined, six in the fiber domain Yf and the other six in the matrix domain Ym. In

. . . kl .
order to obtain a unique solution for x8 (z), a constant needs to be a551gned [25].

Accordingly, an additional condition

x:1(z) = 0 (3.2.15)

at z] = O and 22 = O, is imposed in the analysis.

3.2.1 Solution of X; for Small Amplitude (1

Assume that x:l(y) has the form

’3 ’3

x21 = x: + ax; + a‘x; + (3.2.16)

For briefness, the superscript kl is omitted. i.e.,

0_ Okl 1_1kl 2_ 2kl
Xg-X‘Q‘Xg_Xg’Xg-Xg

By substituting Eqn. (3.2.16) into governing equations. periodic conditions. and interfacial
conditions, i.e., Eqns.(3.2.8-3.2.l4), and collecting the similar terms of (1 together, the

following equations and conditions can be concluded.

33

. . . 0

The differential equations for x8 are:
E O E E 0 E 0 —o
1111X1,11+( 1122+ 1212)7(2,12+ 1212X1,22-

51212X2.11+(E1122+51212)X1,12+52222X2,22 = 0

. . . . 0
The periodic conditions for 18 are:

x‘g”°(0. z2) - meU 22)

7((f)0(O 22) = x(f)0(l 2)
x(f)0(zl,0) — Xg'")0(z126+d)
The interfacial conditions for x: are:
0 (f) 0 (In)

(El'Zkl ‘ Elzghxg, h) = (Elzkl ‘ Ellgllxg, h)

0 (f) (2" )0

(E22k1_E22gth,ll) = ",(Xzf I )

0 (f) (f)0_ X(lm)0

(Eizkl’ ElZghngl) = film )

(3.2.17)

(3.2.18)

(3.2.19)

(3.2.20)

(3.2.21)

(3.2.22)

(3.2.23)

(3.2.24)

Similarly, the governing equations, the periodic conditions, and the interfacial conditions

for x; can be expressed by the following equations:

34

1 1 1 0
E1111X1,11+(51122 + £1212)X2. 12 + 5121276132 ‘ 2C05(13Z1>£11117C1. 12

. 0 0
+ EllllBSln(le)xl.2 ‘ (£1122 +51212)C°S(531)X2.22 = 0

1 1 1 0
51212262, 11 + (51122 + 51212)X1, 12 + 52222X2, 22 ‘ ZCOSwzi 151212X2, 12

. o 0
+ 512121351“(BZ1)X2.2 ‘ (51122 + E1212)COS(BZI)XI.22 = 0

l 1
x") (0222) = x") (I,z2>
ml m1
xf. ’(0 02:2) = x‘ ’(I,z2
1 m1
x‘f’ (2,,0> =x§2 )(zl,c+d)

l O O (f)
(—Ei2ghxg, ll + (‘ Eilkl + Elighxg,h + EiZgIXg, 2)C°S(l331 ))

- E ‘ E E O E 0 ‘ - (m)
-(‘ lzglng,h+(‘ likl+ ilghxg,h+ lzglxg.2)C05(B~1))

I o 0 (f)
(—E22ghxg. I1 + (‘ 2E12kl+ 2E12gth.h + E22ngg, 2)C05( 1331 l)

1 l m 0 mO .
= ;(x‘2f’—x ‘2 ”—(xlf) -x1 ’ )OS(1321))

(_E12ghxg [1+(E22kl_Ellkl—E22ghxg11+Ellghngl+El2glxg,2)cos(le))m

=51(X(1f)l— 911"”+(x‘f’O—x(2""0)cos(13:.))

(3.2.25)

(3.2.26)

(3.2.27)

(3.2.28)

(3.2.29)

(3.2.30)

(3.2.31)

(3.2.32)

. 2 . . . .
Moreover, the function xg should satisfy the followmg: governing equations [Eqns.

(3233—3234)], periodic conditions [Eqns. (3235-3237)], and interfacial conditions

[Eqns. (3238-3240)].

35

2 2 2 1
E1111X1,11 +(131122 + l5121221212 + £1212X1, 22 ‘ 2COS([321)Ellllxl,12
. 1 1
"’ El 1111351n(1331)X1, 2 ‘ (E1122 + 51212)C°S(531)X2, 22 (32-33)

2 0
+Ellll(COS(le)) X1,22 = 0

2 2 2 1
51212X2.11 +(151122 + 512122261, 12 + 52222X2, 22 ‘ 2‘30S(l321)1512127C2, 12
. 1 1
+ 5121255111030 ”(2. 2 ‘ (51122 + 51212)C°3(331)X1,22 (3234)

2 0
+51212(C°5(1331)) X222 = O

ng’2(02 z2)= x‘gf’2(l2 :2) (3.2.35)

(m)2

xg’""(0.z 22)— — x ((22) (3.2.36)

(m)2

xfgf)2(zl,0)- = Xg (zl,c+d) (3.2.37)

2 1 l 2 O (f)
(‘EiZgthJ1+(Eilgth,h+Ei2gIXg, 2)cos((le)-E,-181(C08(1311)) X8 3)) (..3238)
m)

2 1 l
= (_El2glixg.ll+(El'lghngz+El2glxg,2)cos((BZl)_ Eilgl(COS(le))2 Xg, 2))(

2 1
(_E22ghxg.ll+(2'El2g/ingi+E22g1xg 2)COS(BZI)+(EIIkI E22k1)(cos([3,,1))2 )(f
0
+ (_El]gth,h+E22gth h —2E]2gIXg 2)( M(COS(BZI)) (3.2.39)

=]:(X(2f)l_ x(m)1_ (X3 f)0_x(IM)O)COS(BZ1 )__ %(x(2f)0_x(2m)0) (COS(BZ1))2)

. 1 1 1 2 2 (f)
((_E22ghxg, ll + EligthJ1+ E12g1Xg. 2)C°S(1331)‘Eizgkxg, I: ‘ 2Eizkl(cos(1231 )) )
0 0 (f) 2
+(2E12812Xg,12+(E2281- Engl)Xg,2) (“251630) (3.2.40)

1 ml 0 m m 2
=“(xfi“—x ‘ ’ +(x‘2f’ —x‘2 ’01cos(Bz,>——(x‘“'—x‘2 ”>(cos(Bz.)>)

0
3.2.2.1 Solution of 13(2)

36

By inspecting the equations and conditions consisting of Eqns. (32.17-32.24) for

xg(z), it can be found that the solution form of the transformed problem is identical to

that of straight-fiber composites. Accordingly, x:(z) are linear functions of 2:2, as shown

in Section 2.3, i.e.,

0 kl kl

(3.2.41)

With the same procedures used in finding Ag for straight-fiber composites, A g for

sinusoidal-fiber composites are identified, i.e.,

A(lf)kl

A(2f)kl

(m)/<I _

(m)/<I _

2 -

With the additional condition given in Eqn. (3.2.15), Be are found to be

f f M
(“512111- ET2k1)+ “5121.151212
f m m f
”5121251212 + (651212 + d51212)

 

(1(522/(1‘ 5,221.1) + k522k15’21222

f m m f
k5222252222 + “£222 + (”52222)

 

_ f m m f
— “E12111“ 5121.1) + HE12klE1212
HEi21zE’1nz12 '*' (€31,212 "' dEi212)

 

f m m f
‘ C(E12k1 ‘ E1211) + kE22klE2222

 

k512125’1n212 + (CET212 + deizn)

(fl/<1 _
Bg — 0

(3.2.42)

(3.2.43)

(3.2.44)

(3.2.45)

(3.2.46)

37

Bg""”" = —(c +d)A;m)k1 (3.2.47)

3.2.2.2 Solution of x;(z)

By inspecting the equations and conditions for x;(z). x;(z) can be expressed as

A
x;(z,,zz) = Gg(zz)cos(le)+(Hg(zz)+—B‘§)sin(le) (3.2.48)

where A g are given by Eqns. (32.42-32.45). With this form, the periodic conditions
(3.2.27) and (3.2.28) are satisfied automatically.

Substituting Eqn. (3.2.48) into Eqns. (3.2.25) and (3.2.26) and identifying terms of
cos(le) and sin(le) , the following ordinary differential equations about Gg and 1‘1g

are obtained:

2
E11115 (51122+E1212)BH._

 

 

 

 

 

 

 

G "— G + . — 0 (3.2.49)
1 E1212 1 51212 '
2
E E +E .
Hf— 121251,’2 ( 1122 121-)BG . = 0 (3.2.50)
£2222 52222
2
E E +E —.
H2"— 1212i3 H2-( 1122 1212)BG. = 0 (3.2-51)
E2222 52222
2
E (E +5
”1"‘ 1111‘3 H _ 1122 120302. = 0 (3.2.52)

1
El212 E1212

The eigenvalue equation for the above four equations is

38

 

 

 

 

 

 

 

' 2
E E . E +E " E .
2.4- 1111+ 1212_( 1212 2222) szz+ 1111 4 = 0 (3.2.53)
E1212 E2222 E1212E2222 E2222
The four roots of the equation are:
_ -2 _
:48 —4C .
41.2. 3.4 = :8] 2 (3.2.54)
E E (E +F )2 E
where f; = 1111+ 1212_ 1212 42222 a Cl? ___ 1111
£1212 52222 5121252222 52222

It is obvious that when B > 0 and BZ — 45 2 O, the four eigenvalues are real and

G,- and H,- (i=1,2) are exponential functions. However, when 792—45 <0 the four

eigenvalues are complex and G,- and H,- (i=1,2) are combinations of exponential and

sinusoidal functions.

If the constituents are isotropic materials, the four eigenvalues are

21:22:13 23:24:43

(3.2.55)
In such cases. G422) and H2(:2) can be expressed by the following forms
H (~ ) — H eBzZ+H filth. H 13:.“ H ’8“ 3256)
g ”2 — 8' .97} Z: 5338 ”2 5246 ( - -
G ~ — G BS? G "B53 G ‘33: ~ G ‘35? 3257

By using Eqns. (3.2.50) and (3.2.52) again, the coefficients Ggi can be expressed by H‘gi,
(g =1. 2,i = l, 2, 3,4), i.e.,

39

D
011 = H22+BH23
D
012 = H21+§H24
00"Hu
G14 '- H23 (3258)
D ..
(le—HD+BHB
D
022 = ‘H11+‘B'H14
023 — ’H14
G24 = ‘H13

where

Hence, if H 8,- are found, G 81- can be identified. The periodic and interfacial conditions

provide the following equations for Gg(z»_,) and Hg(:3) (g=l.2 and i=1,2 in the following

equafions)

G (“(0) = Gg(m)(c+b) (3.2.59)

.9

, A ('j.)—A (121))
Hg('f)(0) = H2(")(c+b)—( ? l3 .2 (3.2.60)

 

(f)
g)

(‘ Ell/(1+ E A — EiZgIBHg_ EiZglAg " 512va'

ilgl g ..

, (m)
2Ag _ [31‘2ngng —_ E‘ZglAg _ EiZgZG g)

1

(3.2.61)
= (‘ Eilkl + E

ilg

4O

(Eiz‘ngGg —E,.‘,_1,42H'g)‘f ’ = (EizngGg — Eizng'g)""’ (3.2.62)

I ( )
(_E1122BH1_ 522220 2 + 2151212141 — 25121(1)]

 

 

= Hog“ — G§""—(A‘,“—A‘{"’)c— (An—3‘1“)» (32.63)
(15.1221301 — E2222H'2)(f) = k[H(2f) — H‘z’") + (Ag) yam”) (3.2.64)
(‘512123H2‘512126'1+ E1122A2_E2222A2+£22k1—Ellkl)(f) (3 2 65)

= u(G‘,“-0‘1"”+(A‘Zf’—A‘2'"’)c+(B‘2fl—B‘2m’» ' °
(51212302—51212H'1)(fl = l1[H(1f)-H(1M)+(ADE-Agni”) (32-66)

With the additional condition (3.2.15) which provides four more equations, a total of
sixteen equations are formed for finding the sixteen unknowns of H(f)g,- and H(m)g,-

where g: l .2 and i: l .234.

3.2.2.3 Solution of x;(z)
The solution for x:(z) is assumed to be
1.2(21’32) = Pg<32>+ngz>COS<2an+Sg(:2)sin(213:,), g = 1.2 (3.2.67)

This solution satisfies the periodic conditions, i.e., Eqns. (3.2.35) and (3.2.36). For

effective moduli, only the functions P423) are of concerns. since the second and third

4l

terms in Eqn. (3.2.67) have no contribution to the effective moduli.

Substituting Eqn. (3.2.67) into Enqs. (3.2.33) and (3.2.34), it yields

P (37):

g - 2‘3”}; (Z2)+Uggz2+Q (3.2.68)

where U 8 and Q g are constants.

The periodic conditions (3.2.37) gives:

1H
ZBH

1

2—-BH'W(C c+ d) + U“ ’(c + d) + Q‘f’" (3.2.69)

.éf)(0)+Q(f)_

The following four equations for constants U 8 and Q g can be obtained from interfacial

conditions, i.e., Eqns. (32.38-32.40),

(f)
(— 2512,;ng +(Ei2gl+E11g2)Gg. +13E Hg)

"3 1 (3.2.70)
(M)

=(- ZEiZgZPg +(Ei2gl+Eilg2)Gg +I3Eilgly .2)

1 (f)
+(§(Ell:—E22kl+(E2222+E1122)A2)) (3'2'71)

' n l I l n l I
= k(P(gj)—P(2l)—-(G(1f)—G(ln))—Z(Agf)-A(2'))Vf—&(B§f)—Bl_,")))

2
' 1 I ' (f)
(—E1212Pl +§l3(51111“15221015“45(152211‘15222.2.“'l":1212)02)
(f) 3272
+(‘ E22kl+E1212Al) (" )

‘11(Plf) _P(lm)_ gm?)—C(2"'))—i(A(1f)—A‘,"”))

42
With the additional conditions Eqn. (3.2.15), two more equations can be obtained.

(f)

Consequently, there are eight equations for eight unknowns, i.e., U (f ) , Ugm), Qg , and
Qém) , where g=l,2.
3.3 Effective Moduli
Recalling the formula for finding effective moduli, i.e., Eqn. (2.2.31)
H l kl (f)
Eijkl 1' mJ-gj‘;(Eijk1—Eijghx g,h(zl,Z2)) szldzz (33 I)

l

+d k] (m)
+l(c+d)oJ: (Eil'kl-Eijghx 8'h(zl’z2)) sz‘dZ3

 

where J is the Jacobian due to coordinate transformation. If the properties of the

constituents are given as follows:

E1f111 = 52222 = 3000120, 511322 = 51f212 = 1000120 (332)

m

E1111 = 521222 = 300m), Ein122 = 571212 = 1001)“

and a volume fraction of Vf = 0.25 is chosen, the following numerical equations can be

obtained:

(171)” (111)“

EH11”=97.5—25A2(‘f)“—7.5A2 +a2(—25U2(f)11—7.5U2 )

+a2(ll9.3H12(f)” + 54.4H22‘f’ll + 149.8H32‘f"'—54.7H42‘f“') (3.3.3)

(m)ll (m)ll (m)ll

+ (120.09.5le + 4.1H22("'m + 392.6H32 - 2.6H42 )

43

" 1—22.5A,""’” + a2(—75 112"” 1-—22.5 Uz‘m’”)

+ a2(—l19.3H12(f)11—54.4H22(f)11- 73.8H32(f "1 + 20.1H42(f ’1‘) (3.3.4)

EH1122 = 32.5—75.4;f

+ a2(—209.5Hl2(m)1l—4.1sz(m)11 — 259.3H32(m)” — 0.05H42(m)”)

(m)22 m)22

EH2222 = 97.5 - 75A,”)22 — 22.5.42

+ a2(—75 112‘f ’22—225 U2( )
+ a2(-l19.3H12(f)22—-54.4H22(f)22 — 73.8H32‘f’22 + 20.1 Hum”) (335)
+ a2(—209.5H12(m)22—4. 1 Hum” — 259.3H32(m)22 — 0.05H42(m)22)

11 . .
where Hi2”) , Ham)”, Hi2(f)22, and Ham)22 are linear functions of p. and

quadratic functions of k, i.e., H12=flk1k2,l~l, 0191118), while U2(f)11, U2(m)ll

’

9 . . - ' '
U2(f)22, and U2(m)2' are linear functions of u and cubic functions of k, i.e., U2:

g(k, k2, k3, 11, 11k, uk2,11k3) .

Moreover,

E”1212 = 32.5—25A,‘f"2—7.5.«i,‘"‘)12 + a2(—25U,‘f"2—7.5UIW”)

+ 612(—l19.3HlIll-)n—54.4H21(f)12 — 35.8H31(f)12 + 2.7mm”) (3.3.6)

(m)l2 (m)12 (m)12 (m)l2

+a2(—209.5HH —4.1H21 —192.6H31 —1.4H41

)

12 12 . . . .
where H11”) and H ”(m are linear functions of k and quadratic functions of 1.1,

i.e.,Hl-1(f)12=f(k, u, ku, kuz) , while UIUHZ, and Ulm)12 are linear functions of k and

UIUHZ

cubic functions of 1.1, i.e.. =g(k, it, ku, kuz, ku3) . It should be noted that all other

. . H
components of effective moduli E ijkl = 0.

. . kl ' kl 1-1 kl .
Since the coeffic1ents H12”) , Ham) , U2”) , and U30") are independent

44

variables of three effective moduli EH1 111 , EH1122, and EH2222 from Eqns. (3.33-3.35),
it is concluded that these three moduli are also dependent on 11 linearly, and on k cubically.
Similarly, EHmz depends on 11 cubically. and on k linearly since it is a function of the
coefficients Hum“, HUM)“, U10)“ , and U1('"W as shown in Eqn. (3.3.6). It then is
concluded that the normal bonding coefficient k affects the effective moduli EH11”,
EH1122, and EH2222 more significantly than the tangential bonding coefficient 11. On the
contrary, the tangential bonding coefficient H has more significant effects on effective
moduli EH1212 than the normal bonding coefficient k. Once the effective moduli (in tensor

form) are identified, the engineering constants can be computed. They are:

 

<
J
H

 

In this study, a symbolic software MAPLE is also used to verify the analytical
solution and subsequent numerical applications. The procedures of MAPLE analysis are

briefly summarized in Appendix A.

45

3.3.1 Comparison with Other Approaches

Fig. 3.4 shows the comparison of the present study with Hashin’s results [28]
about the effects of tangential bonding coefficient 11 on the shear modulus G12 of
straight-fiber composites. In Ref. [28], Hashin used elasticity method to calculate the
thermoelastic properties of fiber composites with imperfect interface. An unidirectionally
reinforced cylindrical specimen was subjected to uniform shear strain in his study. The
shear stress can then be found by elasticity. The averaged shear stress over uniform shear
strain gave the effective shear moduli. In Fig. 3.3, Gm is the shear modulus of matrix
material. The results indicate that both studies have the same tendency and the shear
modulus changes significantly from [1:1 to p.=lOO.

Fig. 3.5 shows the comparison of the present study with that by Rai [2], and Chou
[3] about the effects of the fiber waviness ratio 01/1 on Young’s modulus E”. Kai and
Chou used the mechanics of materials approach. They took a unit sinusoidal cell as a unit.
A uniform stress was applied in the longitudinal direction. An average strain was
calculated by integrating the strain at each infinitesimal element. The effective modulus in
longitudinal direction was then obtained by comparing the applied uniform stress with the

average strain. Both curves decrease monotonously as the waviness ratio increases.

3.3.2 The Effects of Fiber Waviness on Effective Moduli

Figure 3.6 gives more results of the present study. The effects of the fiber waviness
ratio on Young’s moduli and shear modulus of a sinusoidal-fiber composite material are

investigated. It is concluded from the results that the waviness ratio affects the Young’s

46

modulus in the longitudinal direction, E 11’ much more than that in the transverse
direction, E 22, and that in the shearing direction, 012' The increase in the waviness ratio
results in reduction of stiffness in the longitudinal direction. However, the effects of the
waviness ratio on the stiffnesses in both transverse and shearing directions are
insignificant in small waviness ratios.

Figure 3.7 shows the effects of fiber waviness on Poisson’s ratios. It is obvious that

Poisson’s ratios increase as the fiber waviness increases even the waviness ratio is small.

3.4 Effects of Interfacial Conditions on Effective Moduli

In studying the effects of interfacial conditions on effective moduli, the fiber
waviness ratio of 3% is chosen in the following investigations. When k = 0, fiber and
matrix are perfectly bonded in the normal direction on the interface. When pi = 0, they
are perfectly bonded in the tangential direction on the interface.

Fig. 3.8 shows the effects of interfacial coefficients k and u on the Young’s
modulus in the longitudinal direction, i.e.. E”. The diagram gives relations of Ell/Em
versus tangential bonding coefficients it at three values of k. The doted sections of the
lines indicate that k is greater than it, i.e., the bonding condition in the tangential
direction is better than that in the normal direction. In contrast to the straight-fiber
composite case in which only k affects the value of E 119 both k and it have significant
effects in the sinusoidal-fiber composite.

Fig. 3.9 shows the effects of bonding coefficients k and it on Young’s modulus in
the transverse direction, i.e., E22. The diagram gives relations of 522/5112 versus tangential

coefficients 1.1 at three values of k. The dotted lines also emphasize that k is greater than

47

 

 

 

 

 

1.5?
— present
---- Hashin
1._
0.5“
0-2 -'1 0 4

 

 

Figure 3.4 - Effects of tangential coefficient on shear modulus of a straight-
fiber composite material.

 

48

 

 

Ell/Em
3.5-

— present
---- Rai and Chou

 

 

 

2.5 0 0.61 0.62 . 0.63 0.64 0.65

waviness ratio 01/!

 

Figure 3.5 - Effects of Fiber waviness ratio on Young's modulus of a
sinuosoidal-fiber composite material.

 

49

 

 

 

 

 

 

 

3.5-—
Ell/Em

3...
2.51-

24-
15.- E22/Em

GIZ/Gm
10 0.61 0.62 . 0.63 0.64 0.65

waviness ratio 01/]

 

Figure 3.6 - Effects of fiber waviness ratio on Young’s moduli and shear
modulus.

 

50

 

 

 

 

 

 

0.41-
V12 //
0.3--
0.2-L
V21
0.10 0.01 0.62 . 0.63 0.64 0.65

waviness ratio Ot/l

 

Figure 3.7 - Effects of fiber waviness ratio on Poisson’s ratios.

 

51

it. It can be seen that E22 changes significantly with the change of k from 0 to 1. However,
between k=1 and k=lO, E22 does not change very much as it changes.

Fig. 3.10 shows the effects of bonding coefficients k and u on shear modulus 012-
The diagram gives the relations between shear modulus and tangential coefficient 11 at
two values of k. It can be concluded that k has no significant effect on shear modulus.
However, the tangential bonding coefficient affects the shear modulus significantly. This is
because the shear modulus is associated with [1 in a cubic order, while with k it is
associated in a linear order.

Figs. 3.11, and 3.12 show the effect of bonding coefficients k and it on the
Poisson’s ratios v12 and v2]. If only the region k is smaller than ii is considered, the
bonding coefficients have little effect on v12. However, k seems to affect the Poisson’s

ratio v21 very significantly.

3.4.] Notes on Bonding Coefficients

In pervious sections. it has been found that Poisson’s ratios can be negative in the
regions where bonding in normal direction is poorer than that in tangential direction. Since
Poisson’s ratio is usually positive for engineering materials, the negative Poisson’s ratio
deserves further investigation. especially in both the physical sense and the mathematical
manipulations.

Based on equilibrium equations, the traction should always be continuous across
the interfaces. However, if the interface is not perfect, there will be a displacement jump
between the constituents. Linear relations are usually introduced to correlate the traction

and displacement jump, due to their simplicity.

52

 

Ell/Em
3.1~

p-

3.051

2 .95 i
k=0

 

 

2.93 k=1

 

k=10

 

 

i—b l-
”.9
in

2.85-5 34 33 52 31 6

 

 

Figure 3.8 - Effects of interfacial coefficients on Young’s modulus of
composites with a 3% waviness ratio.

 

53

 

 

Ezz/Em

1.41 k=0

I

1.2:

I

0.8:

0.6::

I

0.4:

I

0.2:

 

 

------------------------------------------------------------------ . : ;k=10

...............................................
----;l"'---l-1-1Ii aaaaaaaaaa

 

Figure 3.9 - Effects of interfacial coefficients on transverse Young’s
modulus of composites with a 3% waviness ratio.

 

54

 

 

GIZ/Gm
1.4

I

 

I

1.2:

1.4..

I

0.8“

I

0.6:

I

0.4“

0.2‘

I

 

 

: k=10

 

 

 

Figure 3.10 - Effects of interfacial coefficients on shear modulus of

composites with a 3% waviness ratio.

 

55

 

 

 

 

 

 

V12
0.57" k=0
0.11. k:]
-0.S-t ................................ .
k=10
-1—~ .
-1.5“
-2-5 -‘4 :3 :2 :1 o 1 2 '
logu

 

Figure 3.1 l - Effects of interfacial coefficients on Poisson’s ratios of
composites with a 3% waviness ratio.

 

56

 

 

 

 

 

 

 

 

V21
0.21-
/ k=O
0.153-
0.13-
0.05-
k=1
0+ _ _ - ----------------- 1:10
-0-05-5 54 -3 -2 Si 0 i 2 3

 

Figure 3.12 - Effects of interfacial coefficients on Poisson’s ratios of
composites with a 3% waviness ratio.

 

57

Most of the papers available in literature only address tangential bonding
conditions. Only very few papers such as Ref. [22] and [30] discuss bonding conditions in
both tangential and normal directions. Even so, only the results of the cases where the
tangential coefficient [.1 is great than the normal coefficient k are investigated. They seem
to imply that this type of bonding condition can closely represent the real bonding
phenomenon.

Homogenization method has the advantage of decoupling the macroscopic and
microscopic problems. This feature brings in mathematical simplicity in dealing with
problems. However, it also increases the difficulty in solving the bonding problems. As
mentioned in previous sections, finding effective moduli is the center of the microsc0pic
problem, while finding stresses is the center of the macroscopic problem, which is
dependent on boundary conditions and applied loading. Since solving for the effective
moduli should be prior to obtaining the stress distributions, the bonding coefficients may
not be correct to avoid unreal physical phenomenon- such as penetration between the
constituents on the interface.

Consider the straight fiber composite as shown in Fig. 3.13, which has a unit cell with
an interphase between the fiber and the matrix. Suppose that no penetration occurs when
an interphase exists between the constituents. Assume the thickness of the interphase layer
e is very small and the material is isotropic with Young’s modulus and shear modulus E,
and Ge, which are also very small. With the same analytical procedures as performed

before. the effective moduli for this case are:

58

 

Matrix

 

 

 

e, Ee, Ge

3’1

c Fiber

Figure 3.13 - A unit cell with an embeded layer.

59

(c + d + e)E2222E2222

 

 

 

 

           

 

 

 

 

 

£2222 = e f f (3.4.2)
E 2222E 2222 + C E 2222 ‘l' d E 2222
2222
Ef dE f eE___1_122Ef E
C 112252222 + 112252222+ 2222 2222
H H 2222
E1122 = E2211 = e f m f (3'43)
5 2222E 2222 + C E 2222 ‘l’ d E 2222
2222
H (C+d+e)512125m212
E1212 = E2121 = (3'4'4)
e f f
_EnnEun+CEnn+dEnn
1212
aEf + bE’" eE"
EH = 1111 1111 2222
““ c+d+e c+d+e
“((51122 E1122)2 +— —C(51122) sz 1122) 2£222
2222 2222
d _Ef Em Em dEf
(C + + e) 2222 2222 + C 2222 + 2222 (3 4 5)
2222 . .
( )
651122 615,2."2229151122151122)+ dE2222(251n122 ‘ E1,122)
52222 f ’"
(C+d+€) e £2222E2322+CE2222+dE2222
k E2222 /

Notice that when e approaches to zero,

 

eve also approaches zero. Comparing

Eqn. (3.4.2) with Eqn. (2.3.14); Eqn. (3.4.3) with Eqn. (2.3.15); Eqn. (3.4.4) with Eqn.

(2.3.16); and Eqn. (3.4.5) with Eqn. (2.3.13), respectively, it is obvious that k is equal to

e(ll-ve2 )/Ee and it is equal to e/Ge. If V1» is to be greater than -1, k must be less than 11.

60

3.5 Effects of Volume Fraction on Effective Moduli

Fig. 3.14 shows the effects of fiber volume fraction on Young’s moduli E11 and
E22, and shear modulus 012- The results reveal that increasing the fiber volume fraction
results in the increases in all of the three moduli of the composite. .In addition, Fig. 3.15
shows the effects of fiber volume fraction on Poisson’s ratios. Increasing the fiber volume
fraction results in the increase of Poisson’s ratio v12 but not v21 . Compared to all other
volume fractions, a 50% fiber volume fraction gives the smallest Poisson’s ratio v21 for

the fiber waviness ratio up to 5%.

61

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\ V o 65
62- f:
5“ \me
4«- \
vf=035
3L E22/Em
v,=0.65
2 V .5
F035
v .65
1 fi5 i
012/6m F035
00 0.61 0.62 0.63 0.64 (Yes
waviness ratio 01/]

 

 

Figure 3.14 - Effects of volume fraction on moduli for the composites with

perfect fiber-matrix interface.

62

 

 

 

 

 

 

 

0.41-
V .65
0:35
0.3»
0.2~L
V .35
F065
V21 1,; VFO‘S
0.10 0.01 0.02 0.03 0.04 0.05

waviness ratio a/l

 

Figure 3.15 - Effects of fiber volume fraction on Poisson’s ratios for
composites with perfect fiber—matrix interface.

 

CHAPTER 4

FIN ITE ELEMENT ANALYSIS FOR EFFECTIVE MODULI OF

SINUSOIDAL-FIBER COMPOSITES

Finite element methods have been proved to be powerful techniques for boundary
value problems which have no analytical solutions. Since the distribution function x: is
the key to the solution of homogenization analysis and only sinusoidal-fiber composites
with small waviness ratios have analytical solutions, it is necessary to find a numerical
solution for the distribution function x: with large waviness ratios. In this chapter, a finite
element method is established to find solutions for sinusoidal-fiber composites with large

waviness ratios.

4.1 Variational Formulation

Consider an arbitrary unit domain as shown in Fig. 4.1. Define

kl
Fiji, = 5.9-MU) - E,-_,-,5.,,(y)xg, ”(y) (4.1.1)

The governing equations, Eqn. (2.2.20), becomes

Fijk1(y)~.\‘l, = 0 (4.1.2)
The periodic conditions are:
kl kl
Xgi. = Xgls (4.1.3)
“.13 DC

63

 

 

 

 

3’2
A Y
D C
interface F

Figure 4.1- A unit cell.

65

kl kl
xngBC : ngSAD (414)

and the interfacial conditions on the interface F are:

( ) ( ) ( )kl ( )kl
(Fl-j“) f "1"} = (Ft-j“) m "1"} = [1(8(X f —Xgm )ng (4.1.5)
In kl kl
(Fijk,)(f)nitj = (Fljk,)(n ’ = 110‘”) (gm) )zg (4.1.6)

For an arbitrary periodic function vi, the variational form of Eqn. (4.1.2) is

J'Fijk,(y),ylvidY = 0 (4.1.7)
Y

By using Gauss’ Theorem, Eqn. (4.1.7) can be rewritten as

J‘F ijkl(y yi)vn nj(IS‘J(Fjjk1(y))Vj‘de = 0 (4.1.8)
5 Y

By imposing interfacial conditions and periodic conditions. the first term in Eqn. (4.1.8)

becomes

1 "I m ' "1
—1[(k—u)(x(€f)k1-xg( W)’Q’1,+"(X(f)kl- X1 )u)](vitf)_vi( ))dS (4.1.9)

Then final variational formulation of the governing equation. Eqn. (4.1.2), is

66

kl
Y

1 1 1 (4.1.10)
k )kl m kl kl m)kl ) (m)
+IJ:[(;-fi)(ng fit; ) )ngni+fi(x:f) —x§ )](vi(f ‘Vi )d5 = 0

It can be seen from Eqn. (4.1.10) that in the perfect bonding condition, i.e., 11:0 and k=0,

my)"l — Xémfld) have to be zero in order to have the finite value for the product.

4.2 Penalty Finite Element Method
4.2.1 Finite Element Formulation

A finite element formulation for a two-dimensional, four-node element is
considered here. Accordingly, x]: can be expressed in terms of nodal values, i.e.,

4
kl kl .
Xg : z ngWj J = 1, 29 39 4 (4.2.1)

j=1

kl . . . .
where xg are the values at the nodes 1n the element and w]. are linear interpolation
functions for the four-node, rectangular elements.

Substituting Eqn. (4.2.1) into the variational formulation Eqn. (4.1.10), the finite

element equations can be given by the following matrix form
[K]{X} = {F} (4.2.2)

where

[K] = 2 [K“”1 (4.2.3)

e=1

67

and n is number of elements. For composites with perfect interfaces, the stiffness matrix of

the individual element is

k11 kiz k13 k14 le ki6 k17 k13
k21 k22 k23 1‘24 k25 k26 k27 k28
k3i k32 k33 k34 k35 k36 k37 k38
(e) k4i k42 k43 k44 k45 k46 k47 k48
[K ] = (4.2.4)

k5! k52 k53 k54 kSS k56 k57 k58
k61 k62 k63 k64 k65 k66 k67 k68

k71 k72 k73 k74 k75 k76 k77 k78

 

 

k81 k82 k83 k84 k85 k86 k87 k88

where
k2i—l,2j-l = 1(Eiiiiwi,i‘l’j,1+Eiiiz‘l’i.in,2+Eiiiz\Vi.2‘l’j.i+Ei:12“’i.2‘l’j,2)dY
Y.
k2i,2j-—l = 1(E1122‘l’i.2‘l’j,1+Eiiiz‘l’i.i\l’j.i+52221Wi.2‘l’j,2+51212‘1’i.1Wj,2)dY
Y.

I‘m-1,21 = 1(51122‘1’1.i‘l’j.2+Eiiiz‘l’i.i‘Vj.i+52221W1.2Wj.2+51212‘l’i.2‘l/j,ildY
Y

I

[(21.21 = 1(51212Wi.1‘l’j. i +E‘MI‘V1’.le.2+E2221wi.2wj.l +52222Wi.2‘l’j.2)dY

Y,

in which i=l,2 andj=l,2.

. . . . . . . . . kl
Vector {X} is the combination of all nodal variables of distribution function Xe .

68

The right hand side vector is defined as

{F} = 21F“) (4.2.5)
and
{F‘e’} = {f1, f2, f3, f4. f5, f6. f7. fg}T (4.2.6)
where
fZi—l = 91(5111-1‘1’1‘4 + EizkiWi.2)dY (4-2-7)
f2i = )J:(E12klwi. l + Ezzki‘l’i. zldY (4-2-8)

(

i=1,2,3,4, k, [212

If the composite material has imperfect interface. the integral terms with k and it
coefficients in the variational formulation Eqn. (4.1.10) should be considered. Define a
bonding element that it has two nodes on fiber side, and the other two nodes on the matrix
side and the height of the bonding element close to zero. By doing so. only the second
term, i.e., the linear intergal term in Eqn. (4. 1.10), has contributed to the element stiffness
matrix, since the bonding element area is close to zero. In addition, it should be noted that

the interfacial variables also appeared in the same equation.

69

For the bonding elements, the stiffness matrix is

 

51 5, 5, 5, 5, 5, 5, 5,
"6 "3' '6 ’6 6 6 6 6
5, 5, 5, 5, 5, 5, 5, 5,
'75“? ‘6-6' 616 6”?
5, 5, 5, 5, 5, 5, 5, 5,
66-6"7?—6 6?? 6“6
5, 5, 5, 5, 5, 5, 5, 5,
g, '6”7§ 66—6 676 6“6
[K ] = (4.2.9)
51 62 51 52 5l 52 61 52
616 6”? '6"7§ 66—6
5, 5, 5, 5, 5, 5, 5, 5,
616 6”? ’6"7§ 66—6
61 52 51 52 5l 52 61 82
6 6 6 6 ‘6 ”6 '6 ’6
5, 5, 5, 5, 5, 5, 5, 5,
L 6 6 6 6 _6 —6 "3— "3—

 

, and

1 l 1 (e) 1 1 (6’)
5, = ((k_fi)n]nl_t-JAI , 52 = (E—fijnlnzAl , 53 = 5

_ l_l -1 (“1
84—(1k “)"3’12 u)Al

In the above equations. Aim is the length of the element.

IQ

4.2.2 Penalty Finite Element Analysis
Physical problems usually come with some constraints which play very important

roles in determining the nature of the solution. In imposing the constraints, both the

70

Lagrangian Method and the Penalty Method are commonly used. Lagrangian Method
requires introducing Lagrangian Multipliers {K}. They are considered as additional
unknowns. The number of k,- is equal to the number of constrained equations. Combining

the constrained equations [A ] { X} = {b} , the finite element formulation becomes

11411543

In the penalty method, if the aforementioned constrained equations are also assumed, i.e.,
[A]{X} = {1)} (4.2.11)
a transpose matrix is usually employed to make the stiffness matrix symmetric. i.e.,
[Afr/111m = 141%} (4.2.12)
By denoting
1K,,i = [AlTlAl {F,,1=141T{b} (4.2.13)
the penalty finite element model gives the following linear system:

[K]{X} +01[Kp]{X} = {F} +a{FP} (4.2.14)

where Cl is called penalty parameter.
. . . . kl . . .
To solve for the distribution function x8 in this study, the procedures are different

from solving common elasticity problems. The boundary conditions for the distribution

71

function are neither prescribed displacement nor traction, but are periodic conditions and
interfacial conditions. The number of constrained equations depends on the number of
elements and nodes used in the finite elements method. For example, if the structure is
meshed in such a way that there is n nodes in yl-direction and m nodes in yZ-direction,
then the total number of unknowns for x? is 6(m+l)(n+l ), and the number of constrained
equations is 6(2n+m+2). In order to save computation time, penalty finite element analysis
is used in this study. Compared to the Lagrangian Method, the penalty finite element
method has the advantage of introducing no new variables. Penalty parameter Ct
represents the stiffness of constraint elements. The constraints can then be satisfied by
assigning a high modulus to the constraint elements which are already in the program. As
or increases, the constrained equations are better satisfied. However, in order to avoid
numerical error, 0t cannot be too large. As suggested in Ref. [32], 01 of 1,000,000 is used

in the study.

4.3 Effective Moduli

4.3.1 Effective Moduli of Sinusoidal-Fiber Composites with Small Waviness Ratios

To evaluate the program, the effective moduli of the composites with small
amplitude are examined. The comparisons of finite element analysis with exact solutions
are shown in Figs. 4.2 and 4.3. In these studies, the fiber volume fraction is assumed to be
25% and the material constants are identical to those used in Chapter 3.

Fig. 4.2 shows the effective moduli versus amplitude for perfect interface. The

solid lines are the results from Chapter 3. The open circles represent the results from finite

72

 

 

 

 

 

 

 

 

 

 

 

Exact
oooo FEA
3.5« ,Ei “Em
3.”- W
2.5--
2.-
1.54- 522/5"! If}...
1* VGlz/Gm
0.5" - V12 -
: ; V21 i I
0 0.01 0.02 - 0.03 0.04 0.05

 

waviness ratio a/l

 

Figure 4.2 - Comparison of two approaches for the effects of fiber waviness

on engineering constants.

 

73

 

 

 

 

 

 

 

 

 

— Exact
3.5" Ell/Em oooo PEA
3+-
2.5"
2.-
v.12 -
0 -4 -3 -2 -'1 0 1 2 3

 

Fig. 4.3 - Effects of interfacial imperfection on engineering constants ofthe
composite with 3% fiber waviness.

 

74

element analysis. The results from those two methods are very close.

Fig. 4.3 shows the effects of bonding coefficient [.1 on the engineering constants.
The composite considered has a 3% fiber waviness ratio. Once again, the solid lines are
the results from Chapter 3 and the open circles are from the finite element analysis. It is
also shown that the results from these two methods agree very well.

Another example used to justify the finite element program is to identify the
effective moduli of the unidirectional-fiber composite structure shown in Figure 4.4. When
0 equals to zero, the effective moduli are given by Eqns. (2313-2316). The moduli at

various 0 can be obtained from tensor transfonnation, i.e.,

O 4 O 2 . 2 O . 4 O 2 . 2
Elm = EHHcos 0+2E1122cos Gsm 0+E22225m 0+4E1212cos 03m 0

0 4 O O 2 . 2 O . 4 0 2 . 2
E”22 = Ellzzcos 9+(Eml +E2222)c05 03m 0+E11223m 0—4E1212cos 05m 0
(4.3.2)

0 0 0 2 . 2 0 2 . 2 3
E1212 = (Ellll +£2232—251122fl'03 03m 9+E1212(COS G—sm 9)

Fig. 4.5 shows the effective moduli of the composite with 25% fiber volume
fraction at various 0. The material used in the studies is glass/epoxy with moduli
E = 72.08GPa and v = 0.22 for glass fiber. and E = 3.4SGpa and v = 0.35 for
epoxy matrix. It can be concluded from Fig. 4.5 that when 0 is less than 85 degree, the

results from finite element analysis and tensor transformation agree very well.

75

.1’2

 

 

   

) 0

Figure 4.4 - A cell of unidirectional-fiber composite with a rotation angle 0.

76

 

 

60"

404-

30+

201-

—— Tensor transformation rule
444+ FEA

 
   

E1212*10

 

 

 

 

1 : l 1El122 l l :7 j ’ l
0 f0 20 3O 40 50 60 70 80 9O
angle 0

 

Figure 4.5 - Comparisons of the effective moduli of unidirectional—fiber

composites between tensor transformation and finite element analysis.

 

77

4.3.2 Effective Moduli of Composites with Isotropic Constituents

The effective moduli of a sinusoidal-fiber composite with 25% fiber volume
fraction is of concern in this section. The isotropic glass/epoxy given in the previous
section is used in this investigation.

Fig. 4.6 shows the effects of fiber waviness on the Young’s and shear moduli. It is
clear that E 11 decreases very quickly at a low waviness ratio up to about 20% and then
becomes stable. In contrast, E22 changes slowly at lower waviness and then increases
quickly. Shear modulus, however, increases first and decreases when waviness ratio
reaches 40%. Fig. 4.7 shows the effects of fiber waviness on Poisson’s ratios. It reveals
that Poisson’s ratios increase at first, then decrease as the fiber waviness ratio increase.

Figs. 4.8 and 4.9 present the effects of tangential bonding coefficient [.1 on
engineering constants. The composite of interest is of 30% fiber waviness ratio. Fig. 4.8
indicates that all Young’s moduli and shear modulus decrease with the increasing of it (i.
e., the bonding becomes poor). In contrast to the composites made of straight fibers, the
tangential bonding coefficient not only affects the shear modulus but also Young’s moduli
of the sinusoidal-fiber composites significantly. However, it can be concluded from Fig.
4.9 that the tangential bonding coefficient it affects Poisson’s ratios v12 and v21 in different
ways. When it increases, v12 increases but v2, decreases. Both Fig. 4.8 and Fig. 4.9 also
show that in the range between 0.0001 and 0 of u, the effective moduli are affected by the

tangential bonding coefficient significantly.

4.3.3 Effective Moduli of Composites with AnisotrOpic Constituents

78

 

 

 

 

 

 

61
5,.
El l/Em EZZ/Em
4...
3,.
GlZ/Gm
2.. l
0 0'5 waviness ratio a/I 1 1'5

 

Figure 4.6 - Effects of fiber waviness on Young’s moduli and shear
modulus of sinusoidal glass/epoxy composites.

 

79

 

 

 

 

0.5+
V 2
0.4» ,2]
0.3~-
0.24-
0.1-~
0 0i5 i 13

waviness ratio a/l

 

Figure 4.7 - Effects of fiber waviness on Poisson’s ratio of sinusoidal glass/
epoxy composite.

 

80

 

 

2.81
2.61
2.4‘
2.2‘

1.8‘
1.61
1.4“

1.2*” m/Em

I

I

I

GlZ/Gm

 

I

Ell/Em

 

 

hr

-5 -4 3 3310125
logu

 

Figure 4.8 - Effects of interfacial imperfection on Young’s and shear
moduli of sinusoidal glass/epoxy composites with 30% fiber waviness.

 

81

 

 

I

0.46'

V12

 

0.44“

\

0.42‘

\

U

0.4-

0.38‘

l

 

036*

 

 

Figure 4.9 - Effects of interfacial imperfection on Poisson’s ratios of
sinusoidal glass/epoxy composites with 30% fiber waviness.

 

82

 

 

 

 

 

 

0 03 13

waviness ratio a/l 1

 

Figure 4.10 - Effects of fiber waviness on Young’s and shear modulus of
sinusoidal carbon/epoxy composite.

 

83

 

 

 

 

0.5“"
V21
0.4"
0.34-
0.21”
V12
0.1-
0 0:5 1 is

waviness ratio a/l

 

Figure 4.1 l - Effects of fiber waviness on Poisson’s ratio of sinusoidal
carbon/epoxy composite.

 

84

The effective moduli of carbon/epoxy composite with 25% fiber volume fraction
are investigeted in this section. An orthotropic material carbon/epoxy with material
properties Ell = 226.99GPa,E22 = 20.720Pa, G12 = 41.16GPa,and v12 = 0.3 for
carbon fibers, and E = 3.4SGPa and v = 0.35 for epoxy matrix are used in the
investigation.

Figs. 4.10 and 4.11 present the effects of tangential bonding coefficient 11 on
engineering constants. As expected, E 1 , decreases very quickly at low waviness ratio up to
20% and then becomes stable as the fiber waviness ratio increases. In contrast, E22
changes slowly first and then increases fast. Shear modulus 012, however, increases first
and then decreases gradually. Fig. 4.11 shows the changes of Poisson’s ratios as a function
of waviness ratio.

Fig. 4.12 and Fig. 4.13 present the effects of tangential bonding coefficient [.1 on
engineering constants. The composite in these studies is of 30% fiber waviness ratio. Fig.
4.12 shows that all the Young’s moduli and shear modulus decrease with the increase of u
(i.e., the bonding becomes poor). It can be concluded from Fig. 4.9 that the bonding
coefficient it affects the Poisson’s ratios v12 and v21 in different ways. When it increases.
v12 increases but v21 decreases. Both Fig. 4.8 and Fig. 4.9 also indicate that in the range
between 0.0001 and 0, the tangential bonding coefficient affects the effective moduli
significantly.

In comparison with the composites made of isotropic constituents, E 11 decreases
much faster while E22 increases much slower in the sinusoidal-fiber composites. For
example, Fig. 4.6 and Fig. 4.10 show that E 1 1 drops about 75% for glass/epoxy composite

but 90% for carbon/epoxy composite at 30% fiber waviness ratio. At 150% fiber waviness

 

 

 

 

85
3 .5 ‘-
3" G,,/G,,,
2.5“
2“
15.. El l/Em
522/15,,

 

 

 

 

Figure 4.12 - Effects of tangential bonding coefficient on Young’s and
shear moduli of carbon/epoxy composites with 30% fiber waviness.

 

 

86

 

 

0:46“

\Hz

 

0444*

I

0:421

I

0.4‘

I

0.381

v21

 

 

I

0.361 . . . . . . . . .
-5 -4 -3 -2 -1 0 1 2 3
logil

 

Figure 4.13 - Effects of tangential bonding coefficient on Poisson’s ratios
of carbon/epoxy composites with 30% fiber waviness.

 

 

 

 

 

87

ratio, E22 rises about 90% for glass/epoxy composite, though only 50% for carbon/
epoxycomposite. It can be seen from all these diagrams that the interfacial bonding

coefficient affects the engineering constants of the two composites in a similar way.

CHAPTER 5

GLOBAL-LOCAL STRESS ANALYSIS OF SINUSOIDAL-FIBER COMPOSITES

Three cases of stress analysis for sinusoidal-fiber composites are investigated in this
chapter. Case 1 studies the stress distribution of a sinusoidal-fiber composite under pure
tension. In case 2, a tensile force is exerted on the transverse direction of the sinusoidal-
fibcr composite. Results of case 2 are given in Appendix C. In case 3, the cylindrical bending
of the sinusoidal-fiber composite is of interest. In addition to the fiber waviness, the effects
of the interfacial bonding condition on the stress distributions are also of concern.

As mentioned in Chapter 2, two decoupled steps have to be considered in each case.

The first step is to find the distribution function xgl. Subsequently, the effective moduli E 5k 1

with given geometry and properties of the constituents can be identified. The second step

. . . 0 0 . .
is to look for global displacements and stresses, i.e.. Hg, 91',‘ by solv1ng the equ1valent
. . . . H .
homogenized problem With the effective moduli Ell/<1“ Then the local displacements and
. . . 0 o . .
stresses can be obtained by us1ng global solutions Hg and 8,]- , and the local material properties

and distribution functions, i.e. Elfjkl, E2?“ , and 1:]. Finally, the total deformation and stress
distributions can be obtained by combining the displacements and stresses of global and
local components.

The composite investigated in this chapter has sinusoidal fibers and matrix. In order
to have analytical solutions, the fiber waviness ratio is chosen to have a low value. i.e.. 3%,

though the properties of the constituents are identical to those given in the previous chapters,

88

89

i.e.,

E1111 = 1“32222 = 30001)“, E1122 = E1212 = 1006’)“,

E,,,, = E,,,, = 3OGPa, ET,,, = ET,” = IOGPa

and the fiber volume fraction is assumed to be 50%.

5.1 Uniaxial Tension

Consider a sinusoidal-fiber composite loaded in tension as shown in Fig. 5. la. The
unit cell of the sinusoidal-fiber composite is given in Fig. 5.1b and the transformed local

coordinate system 2 1 -z2 for depicting stress distributions given in Fig. 5. 1c. The relationships

between the transformed and the original coordinate system are 2, = y1 and
zz = yz—asin([3y,) as indicated by Eqns. (3.2.1) and (3.2.2). Fig. 5.1b and Fig. 5.1c

show that 0 S 22 < 0.5 represents for the fiber domain while 0.5 < :3 S 1 matrix domain.

The interface between the fiber and the matrix is given by 23:05

. . . . kl . . H . . .
The distribution function x g and the effective moduli E 1 j k I for various fiber wavmess

ratios, interfacial conditions, and volume fractions have been obtained in Chapter 3. The

effective moduli E21,, for 50% fiber volume fraction are shown in Figs. 3.14 and 3.15. Since

the homogenized material is of an orthotropic material, the global solutions for this

homogenization problem as obtained in Chapter 2 are:

90

 

Figure 5.1a - A curved fiber composite under pure tension.

 

12:0

Figure 5.1b - A unit cell of the sinusoidal—fiber composite given in Figure
5.1a.

91

Matrix

 

Figure 5.1c - A unit cell of the sinusoidal-fiber composite expressed by
another local coordinate system (zl-zz).

O 0
011—0
0

O
012:0

From the orthotropic constitutive equations

0 H 0 H 0
911: Ellllell+51122£22

0 H 0 H 0
022 = E1122“311+l’:2222822

0 H 0
912 = 2151212512

the strains can be expressed in terms of stresses

0 152222 0
811 = “—0 911
H
0 E1122 0
822 ‘ —D 11
0

where D = 5111111331222 - (E7122)- .The global displacements are:

(5.1.1)

(5.1.2)

(5.1.3)

(5.1.4)

(5.1.5)

(5.1.6)

(5.1.7)

15.1.8)

(5.1.9)

93

0 0

“I = 811x]
0 0
“2 = 822412

with the assumptions that 141:0 when x1=0, and u2=0 when x2=0.

By using Eqn. (2.2.19), the local solutions are:

i_ klO_ 110 220
“g — *Xg €11 - “Xg sir-7C3 822

1 kl 0
Cl} = (EU/(10’)“Eijg},(Y)Xg,h(.Y))8/(1

The total solutions for the problem then become

(g = 1, 2 and there is no sum forg in egg) or

1 kl 0
91} = 91') = (Eijkl_Eijghxg, rile/<1

ll , 11 0 22
= (Eijll6Eijllxl.l_Eij2212.2)811+(Eijll—Eijllxl.l_E

Accordingly, the deformed coordinates can then be defined as

x]: Xi‘H‘i

x7 = X7+Uq

(5.1.10)

(5.1.11)

(5.1.12)

(5.1.13)

(5.1.14)

(5.1.15)

22 0
1,22X2. 2)522

(5.1.16)

(5.1.17)

94

x1

y, = _ (5.1.18)
8
”‘2

where X I and X 2 are undeformed coordinates while x, and x2 are deformed coordinates.

The deformed and undeformed cells for both perfect bonding and imperfect bonding

(11:10mm/N) are shown in Figs. 5.2 and 5.3, respectively. The uniaixl tension 0'0 is chosen
as 4 GPa in this study. The undeformed cell is shown in Fig 5.1b. The dimensions of the
unit cell are 2le by hE where hg is the height of the unit cell. The dotted lines in the diagrams
represent the undeformed cell, while the solid lines represent the deformed cell. It is
concluded that from these two diagrams that both global and local deformations contribute

to the deformation of the unit cell. In fact, the global deformation introduces a uniform

stretching in the xl-direction and a uniform shrinkage in the xz-direction while the local

deformation gives nonuniform reduction in the sinusoidal amplitude.

Stress distributions along zl-axis for a unit cell with a perfect interfacial bonding

condition between fiber and matrix are shown in Fig. 5.4 through Fig. 5.6. Results reveal

that the normal stresses 0'” and 0'22 reach their extreme values, maximum or minimum, at
the peak and trough points of the sinusoidal curve, i.e., :, = 0.5 , and z, = 1.5. On the
contrary. the shear stress 0'12 reaches its maximum and minimum values at the middle point

and the ends. 0'” and 0'12 in the fiber domain are greater than those in the matrix domain

while stresses C522 show no difference between the fiber and the matrix domains for the pure

95

 

 

yz/hE interface on matrix

1.21

I

---- undeformed shape

— deformed shape

 

 

 

 

 

 

 

 

 

0.2"
n m: - 6H-"
00 0.5 1 ... . 2
-0.2"
.Vl/hs

 

Figure 5.2a - Deformation of a unit cell with perfect interface under
uniaxial tension in yl-direction.

 

96

 

 

Yz/he
1.21

---- undeformed shape

— deformed shape

 

 

 

 

0.22

I

 

 

0 ois
-02

 

 

I

 

Figure 5.2b - Global deformation of a unit cell with perfect interface under
uniaxial tension in yrdirection.

 

97

 

 

---- undeformed shape

H

.

M ‘
‘1’;

—- deformed shape

08

I

I

0.6‘

 

 

 

I

0.4‘

0.2l

I

 

r
.‘C
N
§
('5

 

-0.2‘

 

Figure 5.2c - Local deformation of a unit cell with perfect interface under
uniaxial tension in yl-direction.

 

98

 

 

yZ/he .... interface on matrix
1.21L ---- undeformed shape
— deformed shape

 

 

 

0.2‘

 

--_... -._ no... 0... “m-.. .-

 

 

 

 

-0.2‘~

 

Figure 5.3 - Deformation of a unit cell with tangential imperfect
interface(u=10mm/N) under uniaxial tension in yl-direction.

 

99

tension case. The stress distributions along the zz-axis are shown in Figs. 5.7-5.9. The results
indicate that on and 012 are discontinuous across the interface, i.e., 22:0.5. However, 0'22
is continuous across the interface. This is because that 0'22 is equal to the normal traction Tn if
the zeroth-order and the first-order terms of Eqn. (3.2.16) are retained. However, since T, is
the combination of the three stress components, the continuity of T, cannot be concluded
from the stress distribution.

The stress distributions for a sinusoidal-fiber composite with tangentially imperfect

interface (e.g., u=10mm/N) are shown in Fig. 5.10 through Fig. 5.15. Figs. 5.10-5.12 depict

the stress distributions along zl-axis. The shapes of the stress distributions are no longer

symmetrical with respect to the midpoint when compared with the cases of perfect interface.

The extreme values of on and on increase while those of 0'22 decrease. The stress
distributions along the zz-axis are depicted in Figs. 5.13-5. 15. These results further indicate
that the difference across the fiber-matrix interface decreases in 0'11 but increases in 0'12,

The normal and shearing tractions can be obtained by the following equations:

T" = Gijninj (5.1.20)

T=o

!

The unit normal vector n, and the unit tangential vector t, are shown in Fig. 5.16. Since the

curve is given by yzzasin(B_v1), the tangential angle 0, i.e., slope, is

tan0 = chos(By,) (5.1.22)

100

 

 

 

 

 

0 5555 global stress
0 lo
2; — stresses in fiber domain
4 + + + stresses in matrix domain
2.51" 22:05
2..
IDS-IL.
22:0
14>06¢990930°OOOOOOCQCOQOOOOCOCOCCO$3°€9399¢0¢99°¢O
0 .5" 22:1 22:05
..ggifiizz:ZIIIE5353g;...gg§§EEEIIIIZIISEEEEng.
0 0.5 1/ 1.5 2
Z2 he

22:0.25

 

 

Figure 5.4 - on along the zl-axis in a unit cell with perfect fiber-matrix

interface subject to uniaxial tension in z. direction.

101

 

 

 

0 0000 global stress
0 /o
0 032 —— stresses in fiber domain
' 1 ++++ stresses in matrix domain
0.0 1*-
. 22:0
0 i 1 1.5 2:
ZI/h8
-0 .01 " Z2:05
-0.02 *

 

 

Figure 5.5 - 0'22 along the zl-axis in a unit cell with perfect fiber—matrix

interface subject to uniaxial tension in z, direction.

 

102

 

 

012/00

I

0.11

0.05“

-0.05"

 

I
O
p...

1
I

O
O
0 Q
o 0 o .

0000 global stress
stresses in fiber domain

4 + 4 + stresses in matrix domain

 

22:0.5,0

:mfio. OH": 025
i . Z = .

0 ° : z o ..... Z%=O.75

..... z2=0.5,1

 

Figure 5.6 - 012 along the zl-axis in a unit cell with perfect fiber-matrix

interface subject to uniaxial tension in z, direction.

 

103

 

 

 

 

 

611/00
3“ — stresses in fiber domain
\ 1 5 +++ + stresses in matrix domain
2.54- Zl" °
2f
21:0, 1
1.51-
1»- 21:0.5
0.5 *
21:05
3333555522525umum-m-uussssszziiE22535 z1=0, 1
0 0’2 0’4 . 016 0:8 121:1.5
ZZ/hg

 

Figure 5.7 - on along the zz-axis in a unit cell with perfect fiber-matrix
interface subject to uniaxial tension in z] direction.

 

104

 

 

 

 

022/60 --— stresses in fiber domain
++ ++ stresses in matrix domain
0 .02“
1.
0.011
4. Z1=O.5
Z]: ,1
0. Zz/he 0T6 0.8 i
4 21:1.5
-0 .01 ‘
-0.02 "

 

 

Figure 5.8 - 0’22 along the zz-axis in a unit cell with perfect fiber-matrix
interface subject to uniaxial tension in z, direction.

 

105

 

 

— stresses in fiber domain

0 1, ++++ stresses in matrix domain

 

 

 

 

Figure 5.9 4 on along the zz-axis in a unit cell with perfect fiber-matrix
interface subject to uniaxial tension in 21 direction.

 

106

 

 

011/00 °°°° global stress
3.5“

— stresses in fiber domain

I

4 4 4 4 stresses in matrix domain

22:05

 

22:0.25

 

22:0

      

 

 

1‘ 000096000 0°9¢°0°°¢°°°¢°9 °¢°¢99°¢9
. 22—1 2230.5
1"”...{6260-‘75 ”"1182: ...... 1 ..... :in
....... ,,, .. °' ' '8: 3" '
00 0.5 ' . 1 15 2
ZI/he

 

 

Figure 5.10 - 0'” along the zl-axis in a unit cell with tangentially imperfect
(u=10mm/N) fiber-matrix interface subject to uniaxial tension in z]
direction.

107

 

 

(522/00 0000 global stress

0 01, — stresses in fiber domain
' 4 444 stresses in matrix domain

I

   

0.005- 22:025. z2=0.75

-0.005‘

I

 

-0.011

 

Figure 5.1 1 - 032 along the zl-axis in a unit cell with tangentially imperfect
(p.=|0mmN) fiber-matrix interface subject to uniaxial tension in z,-
direction.

 

108

 

 

 

(512/00 0000 global stress
stresses in fiber domain
0.1+ 4444 stresses in matrix domain
22:05
22:0
0.05"
....... . . . . =0 5
”2 “3" 2222.075
0 [1 ....... I“ , ° ° ....... 222- 23
22:
-0.05"
-0.1*

 

 

 

Figure 5.12 - 0'12 along the zl-axis in a unit cell with tangentially imperfect
(11:10mm/N) fiber-matrix interface subject to uniaxial tension in z]
direction.

109

 

 

 

 

 

0 . .
(511/0 —— stresses in fiber domain
35+ 4444 stresses in matrix domain

Zj—1.5
3.-
2.5+
2.-
21:0,1
1.51-
1.
Z1=0.5
0.5-”-
III-=05
{ % 6666666666 § 0090000oooooo:::o:E£E:EEI'..§;ii§;::ofiu 4'l=0,1
0 0 0.2 0.4 t'“~~0~.6°~"°°“’" 0.8 "'1.-..=1.5
ZZ/he

 

Figure 5.13 - 0'” along the zz-axis in a unit cell with tangentially imperfect
(1t=10mm/N) fiber-matrix interface subject to uniaxial tension in z;

direction.

 

110

 

 

 

stressesin fiber domain
4 4 4 4 stresses in matrix domain

z,=0.5

21:0,1

L 1 a l
r v *7 fiw

Z2/h9.6 0.8 g 1

o
......
000000

 

 

 

Figure 5.14 - 0’22 along the zz-axis in a unit cell with tangentially imperfect
(u=10mm/N) fiber-matrix interface subject to uniaxial tension in z,
direction.

 

 

111

 

 

—- stresses in fiber domain
0. 11' 444 4 stresses in matrix domain

 

 

 

 

Figure 5.15 - 0'12 along the zz-axis in a unit cell with tangentially imperfect
(uzlomm/N) fiber-matrix interface subject to uniaxial tension in z.
direction.

 

 

112

235

where B = l

. With this angle, the components of unit tangential vector become

t, = cos0 t2 = sin0 (5.1.23)
The relations between the unit normal vector 11, and the unit tangential vector t, are:
721 = --t2 = ——sin9 n2 = t1 = C089 (5.1.24)

In the case of small amplitude, i.e., only the zeroth-order, first-order, and second-order terms

of 01 are retained, cos0 and sin0 can be expressed as

c080 = i 1 = i(1—%(0t[3cos(13y,))2) (5.1.25)

«11 -1-(tan0)2

 

sin0 = cosGtanG = iothos(By,) (5.1.26)
Since n2 is positive on the interface. the positive sign is chosen. The unit normal then becomes

n] = —01[3cos([3_vl) (5.1.27)

n2 = 1-%((XBCOS(B_V]))2 (5.1.28)

Figs. 5.17 and 5.18 show the distributions of normal traction T,, and tangential traction
T, along zl-axis at the fiber-matrix interface for various conditions of tangential bonding
coefficient 11. Results show that the extreme values of both T" and T, decrease as the bonding

coefficient 11 increases (i.e.. the bonding becomes poor). In addition. it should be noted that

 

113

.V2

0 Y2=0151n(13)’1)
t K

v y]

 

 

Figure 5.16 - Unit normal vector 11, and unit tangential vector ‘1-

 

114

 

 

 

T,,/o0
0.024L
“:0 ”Flo—l
0.01-- "
11:10
00 0 5
-0.011--
-0.02*~

 

 

Figure - 5.17 T“ along the fiber—matrix interface with various imperfect
tangential coefficients p.

 

 

 

115

 

 

 

-0.01*

-0.02‘

 

 

-0.03*

 

Figure - 5.18 T, along the fiber-matrix interface with various imperfect
tangential coefficients [.1

 

 

 

116

both Tn and T, are continuous across the interface of both perfect and imperfect conditions.

5.2 Cylindrical Bending

In this section, a beam made of sinusoidal-fiber composite is simply supported at
both ends and subjected to cylindrical bending. In order to achieve cylindrical bending, a
sinusoidal loading is exerted on the top surface of the composite beam, as shown in Fig.
5.19. Following the technique given by Pagano [33], the cylindrical bending problem can
be summarized in the following manner.

In order to satisfy the equilibrium equations, i.e.,

0 0

<3,,,,+<5,2‘2 = 0 (5.2.1)
0 0

0,2, , +022“, = 0 (5.2.2)

the following stress fields are proposed

0

0,, = f"(x2)sin(px,) (5.2.3)
0'23 = —p2f(x2)sin(px,) (5.2.4)
6?, = —pf'(x2)cos(px,) (5.2.5)

where p =

NIH

. In order to obtain a compatible displacement field. the stresses should also

satisfy the compatibility equation, i.e.,

117

.132
A C1051)

 

 

 

 

7
7
i/

 

Figure 5.19 - A simply supported composite beam under cylindrical
bending.

118

0 0 0 0
81). 1-1 + 311,1)“ elk, jl ‘ ejl, ik = 0 (52-6)

Eqn. (5.2.6) together with the stress-strain relations, Eqns. (5.1.4-5.1.6), give the fact that

f (x2) satisfies a fourth-order linear ordinary differential equation. Thus, f (x2) can be

written as

mgr,

f(x,) = A ,e'""3 + A,e'"2"’ + A ,em’“ + A ,e (5.2.7)

"1le ”12x2 ’II3X3 In x2 . o
where e , e , e , and e 4 are the base solutions of a fourth—order linear

ordinary differential equation. All m ,5 which are the eigenvalues of a linear ordinary

. . . . . . H
differential equation can be determmed from the homogenized material constants E ,1.“

and the beam length l . The four coefficients of A j can be determined by imposing four

boundary conditions, i.e.,

h
62,(.x,, 2) = q(x,) (5.2.8)
632(x1,—g) = 0 (5.2.9)
6120123) = 0 (5.2.10)

6?,(x,, 123) = 0 (5.2.11)

 

119

TCX‘

In this study, q(x,) is chosen as qosin(-§,—l). Using the same stress-strain relationship

given in Eqns. (514-516), the global strain components 82, can be found. The stress

field can then be obtained by

i kl 0
91} = 0,, = (Eijkl—Ei'jghxg, 111511 (52-12)

Figs. 5.20-5.22 show the global stress distributions over the beam span. Fig. 5.20 reveals
that (5011 has an asymmetric pattern along xz-axis but symmetric with respect to the
midsection of the beam (x,=l/2). More specifically, it reaches its maximum tension at the
point x1=l/2 and x2=h/2, and maximum compression at the point x1=l/2 and X2=-lI/2. These
results seem to imply that those two points have the highest possibility of having structural

tensile or compressive failures. In addition, 0'0” vanishes on the midsurface of the beam
(x2=0) and two ends (x1=0, x121). Fig. 5.21 shows that 0022 increases through the
thickness of the beam. Fig. 5.22 gives 0013 which have a symmetric pattern along .1‘3-axis

but asymmetric about the midsection of the beam (xlzl/2). In addition, 0012 reaches its
maximum at the points xlzl. x220, and x1=0, x3=0 and vanishes on the top. bottom, and
midsection of the beam. These results imply that these two points have the highest
possibility of shearing failure. Among the three stresses, 0'0“ is the dominant one,

indicating that the beam is subjected to more loading along the fiber direction than the

transverse direction.

From Fig. 5.23 to Fig. 5.28, the stress distributions of the beam in the thickness

120

 

 

 

. . _\ . .
4
\ 4
. '7’ N" 4 —
.. .-C . .-\' . .
, , \ _. _-'
. . ‘ . . . \
. i \ ___
4 . 3.1“
w": 4 4‘ .v'
' K, ‘-. - («"2
‘ ‘6 _\-_""'.T I » ___.r'
. . , fl .\
H~¢tx " ‘ ”x \
- ~ —. ~ . w
20 ., . \\—"' \ \ M'K
.\

 

0.4

 

0.2
leh ‘0-2 -0.4

 

 

Figure 5.20 - Distribution of 6011 of a sinusoidal-fiber composite beam
under cylindrical bending.

 

121

 

 

0
0 22/90

0.8
0.6 l
0.4
0.2

 

0.4

 

 

 

Figure 5.21 - Distribution of 6022 of a sinusoidal-fiber composite beam
simply supported under cylindrical bending.

 

122

 

 

_ P-"v'

 

 

 

0
02
Wk .04 0 2 4

 

Figure 5.22 - Distribution of 0012 of a sinusoidal-fiber composite beam
under cylindrical bending.

 

123

direction, i.e., xz-direction, are shown. The solid line indicates the maximum stresses in

the fiber domain, the line with cross symbols gives the minimum stresses in the matrix
domain, while the line with open circles reveal the global stresses. Figs. 5.23-5.25 show
the stresses of the composite beam with perfect fiber-matrix interface. The results seem to
indicate that the maximum stresses are much greater than the global stresses for both
normal and tangential components. They seem to imply that the global stresses are not
enough in describing the composite stress distribution and the local details are necessary
for giving a complete stress description.

Figs. 5.26-5.28 show the stresses for the composites with imperfect tangential
interface (11:10mm/N). Apparently, as the bonding coefficient increases (the interface
becomes poor) the maximum values of 0'“, 0'22, and 0'12 also increase while the minimum
values decrease. These results indicate that the poorer the interface, the easier the
composite to fail. They also show that the shapes of 0', land 0'22 distributions are distinctly
different.

Figs. 5.29-5.34 show the global-local stress distributions in a unit cell near the
point ofxlzl/2. x3=hl2 for o, 1, 0'22 and near the point .r,=0. x2=0 for 0'12. Figs. 5.29-5.31
show the stress distributions along zl-axis for the composites with perfect interface.

Obviously, three stresses reach their maximum or minimum values at the peak and trough

points of the sinusoidal curve, i.e., :, = 0.5, and :l = 1.5. Because of the sinusoidal-
fiber, even 0'” can have great difference between its maximum and minimum values in the
same unit. on in the fiber domain is not necessarily greater than those in the matrix

domain as obtained in the pure tension case. However. 032 and 0'12 show no difference in

124

both fiber and matrix domains in the cylindrical bending case. It should be noted that the

shape of on in bending is different from that in tension.

Figs. 5.32-5.34 show the stress distributions along the zz-axis for the composites
with perfect bonding. These diagrams depict that the on is discontinuous on the fiber-
matrix interface, i.e., 22:0.5, while 0’22 and 0'12 continues across the interface. As shown
in the tension case, the continuity of T” across the interface can be expressed by 0'22 in a
unit near the point of xl=l/2, xzzh/2. The continuity of T, can be revealed by 012 in a unit
near the point of x1=0, x2=0 because T, is equal to 0'12 in bending case if the zeroth-order

and first-order terms of Eqn. (3.2.16) are retained.

Fig. 5.35 through Fig. 5.40 show the stress distributions for the composites with
imperfect tangential interface. Figs. 5.35-5.37 show the stress distributions along zl-axis.
Apparently, the shapes of the stress distribution are no longer symmetric. The extreme

values of on and 612 increase. Figs. 5.38-5.40 show the stress distribution along the zz-

axis for the composite with imperfect tangential interface (p.=10mm/N). These results

indicate that the difference across the fiber-matrix interface increases in 0“,,

125

 

 

all/C10

oooo global stress

   
  
  
    

— stresses in fiber domain 150
4444 stresses in matrix domain

100

50

0

°‘ ‘01'1' ' 052' : 0'3 0’4 ois-xz/h

-015 20".4' :0’3 2012; 3051;

O
.9

 

Figure 5.23 - on at the midsection and along xz-axis of a sinusoidal—fiber
composite beam with perfect fiber-matrix interface.

 

126

 

 

 

 

 

0'22/(10

4....

b

 

;0 01T0f20i3 0’4

00 o o global stress

stresses ,in fiber domain
4 4 4 4 stresses in matrix domain

 

     
 

oooo¢°°°°°‘:

, x
0.5

2/h

 

Figure 5.24 «3'22 at the midsection and

along xz-axis of a sinusoidal-fiber

composite beam with perfect fiber-matrix interface.

 

127

 

 

O.12/(10 ° °° ° global stress
— stresses in fiber domain

 

4 444 stresses in matrix domain

 

 

0.5 0.4 0.3 0.2 0.1 00 Oil 0.2 .03 0’4 0f5x2/h

 

 

Figure 5.25 - on at the midsection and along xz-axis of a sinusoidal-fiber
composite beam with perfect fiber-matrix interface.

128

 

 

o ooo global stress

— stresses in fiber domain
4444 stresses in matrix domain

011/90

200‘,

1003’

 

 

-05. r064 ~03 £1023°90

o:
0
0°

-100-"

-2003

.0 0.102 03 04 05

T

 

x2/h

 

Figure 5. 26- on at midsection as a function of Q for a sinusoidal- fiber

composite with imperfect tangential interface (11:10mm/N).

 

 

129

 

 

0’ /
22 C10 0000 global stress

41." —— stresses in fiber domain

4 4 4 4 4 stresses in matrix domain

)-

4

o v

0 0°
0.°o¢°o°
c0

"6

A
I U

 

. 03 0’4 015x2/h

     

 

 

Figure 5.27 - 0'22 at midsection as a function of x2 for a sinusoidal-fiber
composite with imperfect tangential interface (ulemm/N).

 

130

 

 

512/90 . ... global stress
— stresses in fiber domain

    

4 44 4 stresses in matrix domain

..................
.l 9 .
' A

-0.5 -014 0.3-0120.100 0102 0.3 014 0.5x2/h

 

 

 

Figure 5.28 - on at beam end as a function of x2 for a sinusoidal-fiber
composite with imperfect tangential interface (11:10mm/N).

 

 

 

 

131

 

 

 

01 1/40
200‘-
150“ 22:0.5
1001

V0

0 o
10°C?9099000900°°¢09200090¢090c°99¢090°9909¢990°9

 

 

 

50.-
;;;;; 3220-5
00 05 ‘1‘? ..1...5. 2
Z 1/ he, 22:0 75 22:1
-50 -
-100*~
Figure 5.29 - on along the zj-axis in a unit cell near x,=l/2, x2: lz/2 with

perfect fiber—matrix interface.

 

 

 

132

 

 

o o . lobal stress
0 / ° 8 . .
22 (10 — stresses in fiber domain
41- 4444 stresses in matrix domain

   

2.2:025, 0.75

 

 

 

Figure 5.30 - 022 along the :l-axis in a unit cell near x,=l/2, x3: 11/2 with
perfect fiber-matrix interface.

 

 

 

 

 

 

 

133
0 we global stress
012/C10 stresses in fiber domain
61’ 4 44 4 stresses in matrix domain
z =0.5
5 .5“ 2
51*
2 =0.25. 0.75
4.5‘“
4“ 22:0, 1
3.5' 4 4 4 4 Z I/hg
0 0.5 1 1.5 2

 

 

 

Figure 5.31 - 0,2 along the zl-axis in a unit cell near x1=0, x3: 0 with
perfect fiber-matrix interface.

134

 

 

 

 

 

91 1/q0
150» 21:15 — stresses in fiber domain
4444 stresses in matrix domain
Zl=0, 1
1001*
21:05
50.-
Zl=0, 1
0 0 0i2 0.4 0:5 03 122/12,,
° z,=1.5
-504

 

 

Figure 5.32 - 0'” along the :Z-axis in a unit cell near xl=l/2, x2: h/2 with
perfect fiber-matrix interface.

 

 

135

 

 

 

 

 

O’zz/QO 4444 stresses in matrix domain
41- — stresses in fiber domain ;
3.-

Z1=1.5
21-
217-0, 1
1 ................................................
1 4 . 1Z2/h8
00 02 04 06 08 1
. Zl=0.5
-14
-21

 

Figure 5.33 - 0'23 along the 2.2-axis in a unit cell near x,=l/2, .12: lz/2 with
perfect fiber-matrix interface.

 

136

 

 

— stresses in fiber domain

4 4 4 4 stresses in matrix domain

 

000000000000000000000000000000000000000000000000

 

 

 

 

Figure 5.34 - on along the zz-axis in a unit cell near x1=0, x2: 0 with
perfect fiber-matrix interface.

 

137

 

 

 

 

 

 

0' / oooo global stress
11 C10 stresses in fiber domain
250“ 444 4 stresses in matrix domain
2210.5
200"
=0.25
1501 Z2
100" \ 22:0
504-
:::;::z°;;; z2=0.5
.0831! l ‘ ‘5‘80 :.o,.‘!,::£:&!::2”..oo:22:0.75
0 0 0.5 1 1.5 212"1
-50 - 31/112
-100*

 

 

Figure 5.35 - on along the zl-axis in a unit cell near x1=l/2, x2: h/2 with

tangentially imperfect fiber-matrix interface (11:10mm/N).

 

138

 

 

oooo global stress

41, —- stresses in fiber domain
4444 stresses in matrix domain

\
oa°°°°°°°°°°°°°°°°°° °¢°°°coooooooooocoooooc
‘ J

 

 

 

 

Figure 5.36 - 0'22 along the :l-axis in a unit cell near x1=l/2, x2: h/2 with
tangentially imperfect fiber-matrix interface (11:10mm/N).

 

139

 

 

 

O.12/C10 a global stress

1211- — stresses in fiber domain
4 4 44 stresses in matrix domain
22:0.5
10" 32:0, 1
32:0.25

84-

6.-

4.-

2.-

, 1 § ‘ ZI/hg

0 0 0.5 1 1.5 2
-23-
_4--

 

 

Figure 5.37 - on along the zl-axis in a unit cell near xl=0, x2: 0 with

tangentially imperfect fiber-matrix interface (11:10mm/N).

 

 

 

140

 

 

 

 

 

 

011/910
250"
— stresses in fiber domain
4444 stresses in matrix domain
2006 Z2=1.5
150"
1001-
50-
22:05
............................................... 22:0, 1
1 1 :::::””’4””:3””””””f”””””’”’2: 22:15
0 0 0.2 0 '6 0:8 1

.4 . 0.
Zj/h8

 

Figure 5.38 - 0'” along the zo-axis in a unit cell near x1=l/2, x5: h/2 with

tangentially imperfecf fiber-matrix interface (11:10mmlN).

 

 

141

 

 

4 4 4 4 stresses in matrix domain

 

 

 

 

622/(10 stresses in fiber domain
41-
63; Z1=1.5
3" ’-
2‘ . ,°
1 /\ /\ /\ ....... ...... 21:0, 1
1 1 .5 1 9, Z h
00 . . . 0.6 0.8 ,» 1 2/ 8
-14
21:0.5
-24 ..........

 

 

Figure 5.39 - on along the zz-axis in a unit cell near x1=l/2, x2: h/2 with
tangentially imperfect fiber-matrix interface (11:10mm/N).

 

 

142

 

 

012/90
121

 

 

00000

— stresses in fiber domain

4 4 4 4 stresses in matrix domain

9
o
o
00’
ooooooooooooooooooooo
..00
o

0000000000000000000000000000000000000000000

 

 

 

 

Figure 5.40 - on along the zz-axis in a unit cell near x,=0, x2: 0 with
tangentially imperfect fiber-matrix interface (11:10mm/N).

 

 

 

CHAPTER 6

PERFORMANCE OF SINUSOIDAL-FIBER COMPOSITE BEAMS

6.1 Introduction

In Chapter 4, the effective moduli of sinusoidal-fiber composites are calculated
from a finite element method. In order to further investigate the effects of the sinusoidal
fibers on composite structural performance, several composite beams are examined. For
example, the material nonlinearity due to sinusoidal fibers and the effects of fiber
curvature on the natural vibration frequency and critical buckling load are of concern. In

addition, the effects of sinusoidal fibers on beam bending are also of interest.

6.2 Material Nonlinearity

Figs. 6.1a and 6.1b show the macroscopic and microscopic structures of the
sinusoidal-fiber composite, respectively. In studying the nonlinear response of sinusoidal-

fiber composite along xl-direction, a uniaxial tensile strain is applied to the composite

along that direction. A stepwise incremental method is employed in the analysis of the

stress-strain relation.

Assume that the initial cell configuration can be described by the function

(0) 10) ,(0)

E (y, ,12 ) = 0.The two-dimensional cell structurein Fig. 6.1bcan be expressed as

(0) 2n (1))

_ . 0 .-
212 -,\‘2-Otsm(7;\, )‘3 0, OS\(1)SI,OS)’ZSC‘+(1

143

 

Eii

 

Figure 6.1a - A sinusoidal-fiber composite is subjected to a uniform tensile
strain.

C+d

 

Figure 6.1b - The microscopic unit cell of the composite.

 

145

where the superscript (0) represents the initial state and 1‘12 refers to the initial height of

0 . . . . - 0
13(2). For example, if the fiber—matrix interface is expressed as yé) = c and

0 _ 271 0 . . . .
1112 ) }—x(20) — asin(— l 1’, U = 0. In this study, the material properties for fiber and matrix

(f)

are designated as E ,1, , and E, 1k, , respectively.

To begin with, an increment of tensile strain A8,, is exerted on the virgin

composite. Using Eqns. (2.2.2) and (2.2.19), the displacements can be expressed by

u,- = u? — exfleg, (6.2.1)
where u? are global displacements. As indicated in Eqn. (5.4), they are: (6.2.2)
11? = 8?,x, = A8,,x, (6.2.3)
11‘, = 9,2,4, = 0' (6.2.4)

Hence, the displacements after the first increment A8,, are:

(1)u0101(1)
u, = u—, 8x, eu- — AE,,-\‘,— ex, A8,, (6.2.5)
14‘," = 113—ex? sf, = -3 Ex” ”A3,, (6.2.6)

where superscript (1) refers to the state after the first increment. Consequently. after :1

increments. the displacements become

146

1“,") = A8, ,x,—ex’,‘“")Ae,, (6.2.7)

u, = —ex2 "”212“ (6.2.8)

Making use of relations between x and y in Eqn. (5.8), it yields

I—1 l
2’1") = (1 ‘l’Aeiilyi2 )—Xil(n)A511 (6.2-9)

52(2),) = 1'12"_l)—x;1(")A8,, (6.2.10)

These equations stand for the deformed cell after n increments of strain A8,, and can be

used for the next incremental analysis. By using this deformed geometry, i.e.,

F (">01"), y(2n)) = 0 Which can not be expressed explicitly, the material properties of the

. . . . . kl .
constituents E55,), and E32,) , and the nth distribution function Xén) , the nth effective

. H . . .
moduli Emir) can be obtained. The stress in the deformed cell can then be written as

n H I
Aofi,’ = E,,‘,',’Ae,, (6.2.11)

- l
61’," = 61’; )+Ao(,';) (6.2.12)

In performing the aforementioned analysis, a finite element program is established.

. . . . . . 0 0 0
The initial finite element mesh is based on the configuration of F1 )(,1'(, ), )‘(7 )) = 0. After

each incremental step, the program automatically remeshes the domain of the new

 

 

147

(n)kl
g

11(n)

configuration, and consequently, gives the new computed values of x , E ,jk, , and

(n)
0,, .

Fig. 6.2 and Fig. 6.3 show the stress—strain curves for a carbon/epoxy composite
with various fiber waviness ratios. The moduli of the constituents are

E,, = 226.990Pa,E,, = 20.726Pa, 0,, = 41.16GPa,v,, = 0.31 for fibers and

E = 3.4SGPa, v = 0.35 for matrix.
Ari increment of A8,, equal to 0.01 is used in the study. Fig. 6.2 shows the results

based on 25% fiber volume fraction while Fig. 6.3 shows the results based on 50%. It can
be found from the diagrams that both low and high waviness ratios give nonlinear stress-
strain relations. The nonlinearity seems to be the most distinct at the lowest waviness ratio
examined, i.e., a/l=5%. This is because of the fact that the lower the fiber waviness ratio,

the faster the Young’s modulus E ,, drops as shown in Figure 4.10.

6.3 Natural Vibration Frequency

The natural vibration frequency of sinusoidal-fiber composites is of interesting in
this study. As shown in Fig. 6.4, simply supported beams are investigated. However, it
should be mentioned that the following study is based on global analysis with the
governing equations, the constitutive equations. and the linear strain-displacement

relations given below:

<5,,,,+o,:,_;2 = p11,.” (6.3.1)

 

 

 

 

148
C
8+
a/l=0%
6..-
all=5%
a/l=10%
4* a/l=20%
a/l=30%
a/l=50%
2_- a/l=100%
1 1* 8 (%)
00 2 4 6 8 10

 

 

 

Figure 6.2. - Stress-strain relations for sinusoidal-fiber composites (carbon/
epoxy) with 25% fiber volume fraction at various fiber waviness ratios.

 

 

 

 

 

149
O'
151'
a/l=0%
= o
10" a/l=10%
a/l=20%
a/l=30%
5 a/l=50%
all=100%
-0 1 -..:4--—-#' . 0.05 ofi 81%)
-54
-10«-
-15‘

 

 

 

Figure 6.3. - Stress-strain relations for sinusoidal-fiber composites (carbon/
epoxy) with 50% fiber volume fraction at various fiber waviness ratios.

 

150

x2

 

 

 

 

8 , ,\\

 

Figure 6.4 - A simply supported composite beam.

x1

151
02,; , +0222 = pa,” (6.3.2)
011 = E1111511‘1151122522 (63-3)
922 = 52211811+Ez222€22 (63-4)
6,, = 25,,,,e,, (6.3.5)
811 = “1,1 (6.3.6)
822 = 1122 (6.3.7)
8,2 = $01,,2 + 141,) (6.3.8)

In order to find the natural vibration frequency a), the following displacement field is

assumed:

a, = g(x3)cos(1,1—t.r,)ciw (6.3.9)
. "TE 1'0)! ,
u2 = f(x2)sm(—l—x,)e (11.3.10)

Obviously, the boundary conditions of the simply supported beam are automatically

safisfied.Leq

112(0, x2) 2 112(1, x3) = 0 (6.3.11)

 

152
By substituting Eqns. (6.3.9) and (6.3.10) into equilibrium Eqns. (6.3. 1) and (6.3.2) and by

using Eqns. (6.3.3) to (6.3.8), it yields:

l1
0

ag"+bg+cf' (6.3.12)

11
O

df"+ef'—cg (6.3.13)

where the constants a, b, c, d, and e are functions of material properties, beam length l, and

frequency (1) . i.e.,

 

 

E
a = '2” (6.3.14)
2

b — 2 "7‘ 2 6315
”Pm—751111 (..)

E

5 215’
e = par—("T“) 12212 (6.3.18)

Since Eqns. (6.3. 12) and (6.3.13) are linear ordinary differential equations. they have four

_ , A y Ly luv 24 v .
solutions, i.e., b,e ’ , bze ’ , b3e ‘ and [2,6 " . where X, are functions of a, b, c, d, and

In addition to the boundary conditions given in Eqn. (6.3.1 1), four more boundary

 

153

conditions for the top and bottom surfaces of the beam need to be satisfied. They are:

6,2(x,,ig) = 0 _ (6.3.19)

o,,(.x,, :3) = 0 (6.3.20)

These four conditions form four equations and can be used to determine the four

coefficients [9, , b2 , b3 and b4. These four equations can be written in matrix form, i.e.,

[A]{b} = {0} (6.3.21)
In order to find a nontrivial solution for {b} , the determinant of [A] must be zero, i.e.,
detlA] = 0 (6.3.22)

The natural frequency (1) can then be found from Eqn. (6.3.22).

Fig. 6.5 shows the effects of fiber waviness on the fundamental mode of natural
vibration frequency of simply supported beams made of a sinusoidal-fiber composite.
Notice that the curves labeled as classical plate theory are calculated from Lekhnittski’s
formulae [35]. They are given in Appendix D. 5 is a normalized frequency and its data is

also given in Appendix D. The material used in the numerical study is glass/epoxy with
25% fiber volume fraction. It has E = 72.080Pa and v = 0.22 for glass fiber, and
E = 3.4SGPa and v = 0.35 for epoxy matrix. From Fig. 6.5, it can be seen that the

natural frequency decreases as the fiber waviness increases. It is also revealed from this

. diagram that if the aspect ratio of the beam, i.e., Mr. is large enough, the results from the

 

154

 

   

lz/l=0.5, classical plate theory

 

li/l=0.5, elasticy analysis

'/

1. h/l=0.l, classical plate theory h/l=0.l, ela ticy analysis
V Z

 

 

 

7

0:5 1 13

waviness ratio 01/1

 

Figure 6.5 - Effects of fiber waviness on natural vibration frequency of
simply supported beams made of sinusoidal-fiber composites.

 

 

155

classical plate theory and those from elasticity become very close.

6.4 Critical Buckling Analysis

A simply supported composite beam made of sinusoidal fibers is considered again.
The beam is subjected to unidirectional compression as shown in Fig. 6.6. By using the
governing equation (2), given in Ref. [34], the basic equations for critical buckling

analysis can be expressed as follows:

911,1+912.2 = P0“1.11 (94-1)

C5211145222 = Pol‘z. 11 (6-4-2)

where PO is the applied load.

The constitutive equations and the strain-displacement relations are identical to
those given in Eqns. (6.33-6.38), so are the boundary conditions at the top and bottom
surfaces of the beam. The procedures to identify the critical buckling load are almost
identical to those for natural vibration frequency. The only difference lies on the right-
hand side of Eqns. (6.4.1) and (6.4.2). Accordingly, the solutions to the goxeming

equations are assumed to be as follows:

it, = g(x2)cos(fl,l[.r,) (6.4.3)

~
K

J
11

114-2) sin (”711'“) (6.4.4)

 

156

3‘2

 

 

<—
—u 4- Po
h P0 —hb ———————————————— <———> x,
—u <—
<—

 

 

7
7/

 

Figure 6.6 - A simply supported composite beam subjected to uniaxial
compression.

 

157

 

 

l/h=2

l/h=10

 

 

 

0 0:5 waviness ratio 61/11 115

 

Figure 6.7 - Effects of fiber waviness ratio on critical buckling load.

 

158

By following the same procedures exercised in Section 6.3, the relations between
the critical buckling load and the fiber waviness ratio are shown in Fig. 6.7. It should be
noted that the material used in this analysis is also of glass/epoxy with 25% fiber volume

fraction. Its material properties are given in Section 6.3. P0 is normalized loading and its

data is given in Appendix B. As shown in Fig. 6.7, the critical buckling loading decreases
as the fiber waviness ratios increases, i.e., composites with higher waviness ratio are easier
to be buckled. It also can be seen from the diagram that the smaller the aspect ratio l/h, the

greater the critical buckling load.

6.5 Cylindrical Bending of Composite Beams

A multilayered composite beam simply supported at both ends is shown in Fig.
6.8. In formulating the bending analysis, equilibrium equations, stress—strain relations, and
the strain-displacement relations are the same as those given in Eqns. (6.3.1—6.3.8). The

boundary conditions on the top and bottom surfaces of the beam are:

624.1,, {31) = q(x,) (6.5.1)
0 h
0 h
0,2(x,, 3) = 0 (6.5.3)

6?,(x,, 31) = 0 (6.5.4)

 

159

I,

 

Figure 6.8 - A simply supported multilayered composite beam under
cylindrical bending.

160

TEX

where q(x,) is chosen as quin(-Tl) in order to have a closed—form solution.

Consequently, the composite beam experiences cylindrical bending. In addition, the

interlayer continuity conditions. i.e., interlayer normal stress 0'22, interlayer shear stress
0,2, and displacement components u, and uz on the layer interfaces should be continuous.

The interlayer continuity conditions can be expressed as follows:

C(22(.r,, xgi) = 6g; 1)(x,, X?) (6.5.5)
Gfi'é)(-r,, x9) = 0'1? 1)(x,, x?) (6.5.6)
u(,i)(x,, x?) = il(,i + 1)(x,, x5”) (6.5.7)
ugi)(x,, XS”) = 149+ I)(x,,x(2i)) (6.5.8)

where (i) and (i+l) represent for layer numbers and x9) is the thickness coordinate of the

ith layer.

By following the Pagano’s [33] solution technique, the stress field in the ith layer

can be assumed as follows:

o(,',) = f"“’(xg)sin(pxi) (6 5 9)

033 = —p2f(i)(x2)sin(px,) (6.5.10)

 

 

161

61",) = —pf""(x,)cos(px,) (6.5.11)

where p = 7. This solution technique is also given in Section 5.2. However, instead of

only one layer examined in Section 5.2, beams made of multiple layers are of concern in

the present problem. Since each layer may have different material properties, many more

f'(i)(x2) need to be solved. Recall f1i)(x2) which has the following form

. 1')
(i) "1‘3”.1-3 (i) mjx,

. . H)_ , ti)
f(l)(x’)) = Ame,"l 1’ + A, e + A(')1em3x2 +A e (6.5.12)
.. 1 .- 3 4

where my) are the functions of the material constants and beam length. The number of

A?) is proportional to the number of f (1) (x2), which increases as the layer number

increases. For example, if the composite beam has 11 layers, there will be 4n unknowns, so

are 4n A?) where j=l, 2, 3, 4. In order to identify the 4n unknowns, 4n equations are

required. Among the 4n equations. 4(n-l) will be from the interlayer continuity
conditions, i.e., Eqns, (6.55-6.58), and the remaining four equations will be given by
boundary conditions, expressed in Eqns. (65.1-65.4).

In studying the effects of fiber waviness on the composite laminate performance,
three composite beams are investigated. The first one has no wavy layer. Its fibers are

straight and aligned along the beam axis, i.e., [0] laminate. The second one is a two-layer

beam with one 0 degree-layer and one sinusoidal layer, namely [0/w]. The third one is a

three-layer beam with the sinusoidal-layer sandwiched by 0 degree layers, i.e., [0/w/0].

162

Fig. 6.9 shows the distribution of 0,, through the beam thickness, at the midspan
of the beam, i.e., x,=l/2. Results for all three types of beam are placed in the same diagram
for comparison. The material used for this study is again the glass/epoxy composite with

25% fiber volume fraction as given in Section 6.3. The sinusoidal layers have a waviness

ratio of 150%. Because of the low anisotropy of glass/epoxy, the discrepancies of 0,, on

the layer interface are not distinct as shown in Fig. 6.9.

Fig. 6.10 shows the distribution of transverse shear stress 0,2 through the beam
thickness at the beam end. i.e., x,=0, while Fig. 6.11 the distribution of transverse normal
stress 022 through the beam thickness at the midspan of the beam. The numerical results
reveal that 0,, is not continuous across the layer interfaces but 0,2 and 022 are. In
addition, no distinct kink has been observed on the layer interfaces in both 0,2 and 022. It

is believed that the smooth curves are again due to the low anisotropy of glass/epoxy.

Figs. 6.12 and 6.13 give the effects of fiber waviness on the maximum 0,, and 0,2
for one-layer, two-layer, and three-layer composite beams. In both figures, w represents
the sinusoidal layers which have waviness ratios ranging from 0% to 150%. The
maximum value of 0,, is located at .r,=l/2 and x2=lil2 for all three types of composite
beams. Similarly. the maximum value of 0,2 is located at x,=0 and 113:0 for the one-layer
and three-layer beams while that of the two-layer changes with the fiber waviness ratio.
Hence, in determining the maximum stress value for the two-layer beam, the location has
to be identified before the value can be calculated.

Concluded from Fig. 6.12. the effects of fiber waviness on 0, , are significant when

the waviness ratio is less than 30%. Fig. 6.12 also reveals that the two-layer beam has the

163

 

 

01 1/C10
80

[0/w]

I

_.«-”;_[0/w/0]

I

60‘
[0]

40:

 

-0lr.5 0 oisxz/h

1.
1‘9
83.

1"
.-1

l
.33
9

l
O\
O.

 

 

Figure 6.9 - 0, , in simply supported beams under cylindrical bending.

 

 

164

 

 

[O/w]

[O/w/Elf" . _

    

 

 

0 0 0.5
xZ/h

 

Figure 6.10 - 0'13 in simply supported beams under cylindrical bending.

 

 

 

I65

 

 

Ozz/qO

   
 

[O/w/O]

 

 

-o‘.5 09h (is

 

Figure 6.1 l - on in simply supported beams under cylindrical bending.

 

166

 

 

 

 

 

 

 

0'1 1/ C10
[O/w] _._

80* _

7 5--

[0/w/O]
70*
6 ST
[W]
0 is i 13
waviness ratio 61/]

 

 

 

Figure 6.12 - Effects of fiber waviness on maximum 0'] I for various beams.

 

 

 

 

 

 

 

 

167
all/Clo
[O/W]
5.._
4.8“ [w]
4.6‘t
O/ /O
4.4~~ [ w 1
0 0:5 1 1.5

waviness ratio a/l

 

Figure 6.13 - Effects of fiber waviness on maximum 012 for various beams.

 

 

 

168

highest maximum on among the three types of beams examined, while the one-layer
beam has the lowest one, due to its smallest stiffness in the xl-direction. Fig. 6.13 shows
that the effects of fiber waviness on 612 are also significant when the waviness ratio is less
than 30%. It also reveals that the two-layer beam has the highest maximum 0'12 among the

three types of laminates, while the three-layer beam has the lowest maximum 0'12.

6.6 Effects of Material Nonlinearity

In Section 6.1, the material nonlinearity of sinusoidal-fiber composites is
recognized. In order to further understand the effects of fiber waviness on structural
performance, the material nonlinearity is employed in structural analysis.

In conducting the material nonlinear analysis, a simply supported beam subjected
to three-point bending, as shown in Fig. 6.15, is considered. A commercial software called
ABAQUS is used to perform this study. In preparing the ABAQUS input data file, a finite
element model, material properties, boundary conditions, and a calculation algorithm are
required. A mesh of 100 x 20 elements as shown in Fig. 6.16 with the element type of
rectangular shell element is used for this study. Newton’s method and nonlinear iterative
algorithm with automatic step control are chosen for solving nonlinear equilibrium
equations.

In order to understand how the characteristics of material nonlinearity affects the
structural performance, both linear and nonlinear cases are investigated. The composite
material is of carbon/epoxy with 50% fiber volume fraction and 20% fiber waviness ratio.

Fig. 6.16 shows the stress-strain relations of both linear and nonlinear materials.

169

1‘2

 

 

 

 

& 1 s\

 

Figure 6.14 - A simply supported beam with nonlinear material prOperties
under 3-point bending.

170

 

Figure 6.15 - Finite element model.

171

 

 

-0.1

0' (GPa)
84

 

nonlinear
------- linear

I

-0fos 0' 0.65

 

oft

 

Figure 6.16 - Linear and nonlinear stress-strain relations.

 

172

 

 

0 2 ‘3' £5 3 1.0 P(KN)

 

nonlinear
------- linear

 

 

Figure 6.17 - Force-displacement relations based on linear and nonlinear
stress-strain relations.

 

173

 

 

0'11 (GPO)
1.64"

 

nonlinear

 

------- linear

L

II)

P( KN)

 

 

Figure 6.18 - 0'“ versus loading P.

174

Fig. 6.17 shows the transverse displacement at x1=ll2 and x2=—h/2 as a function of loading
P. Fig. 6.18 shows the maximum stress 0'11 at the point x1=ll2 and xzz-h/Z due to loading

P. It can be found from this diagram that the nonlinear one has larger deformation and

higher stress.

CHAPTER 7

CONCLUSIONS

The main issue undertaken in this thesis was to study a particular class of
composite which has sinusoidal fibers with a periodical microstructure. In addition,
interfacial bonding conditions was also considered in the study. Based on a double-scale
analysis, the Homogenization Method was used to study the effects of both fiber waviness
and interfacial condition on composite properties and performance.

Linear and nonlinear elastic composites with small deformation are investigated in
this study. This thesis presents a Homogenization Method which produces two different
levels of boundary value problems, one is for global analysis and the other microstructure
analysis. Solutions to these analyses lead to finding the equivalent properties of the
homogenized materials. It is concluded that the composite properties are strongly affected
by fiber waviness, even if the waviness ratio is very small. In addition, the composite
response is also strongly affected by the interfacial bonding condition. Numerical results
seem to point out that reasonable results can be obtained if the bonding quality normal to
the fiber-matrix interface is not worse than that parallel to the interface. In addition to the
analytical solution for small fiber waviness, a penalty finite element program is established
to investigate composites with relatively large fiber waviness. It is concluded that in the
case of large amplitude. the tangential bonding imperfection affects not only the Shear
Modulus but also Young's Modulus considerably. Moreover, the global and local stress
analysis was performed analytically. The homogenized method presents a powerful tool to

look into the detailed performance of composites and stress with sinusoidal fibers.

175

 

LIST OF REFERENCES

176
LIST OF REFERENCES

l. Kou, C-M, Takahashi, K., Chou, T-W, “Effect of Fiber Waviness on the Nonlinear

Elastic Behavior of Flexible Composites”, J. Composite Materials, Vol. 22, 1004-1025,
1988.

2. Rai, H. C., Rogers, C. W., and Crane, D. A., “Mechanics of Curved Fiber Composites”,
J. of Reinforced Plastics and Composites, Vol. 11, 552-566, 1992.

3. Chou, T-W, and Takahashi, K., “Non-linear Elastic Behavior of Flexible Fibre
Composites”, Composites, Vol 18, 25-34, 1987.

4. Lee, J ., and Harris, C. E., “A Micromechanics Model for the Effective Young’s Modulus
of a Piecewise-Isotropic Laminate with Wavy Patterns”, J. Composite Materials, Vol. 22,
717-741, 1988.

5. Akbarov, S. D., and Guz, A. N ., “Statics of Laminated and Fibrous Composites with
Curved Structures”, Appl. Mech. Rev., Vol. 45, l7-34, 1992.

6. Raouf, R. A.,“Effects of Layer Waviness on Interlaminar Stresses in Thick Composite
Plates”, American Society for Composite, Eighth Technical Conference, 602-611, 1993.

7. Wisnom, M. R., “The Effect of Fibre Waviness on the Relationship between
Compressive and Flexural Strengths of Unidirectional Composites”, J. Composite
Materials, Vol. 28, 66-76, 1994.

8. Wisnom, M. R., “Analysis of Shear Instability in Compression Due to Fiber Waviness”,
J. of Reinforced Plastics and Composites. Vol. 14. 45-52, 1995.

9. Telegadas, H. K., and Hyer, M. W. “The Influence of Layer Waviness on the Stress State
in Hydrostatically Loaded Cylinders: Failure Predictions”, J. of Reinforced Plastics and
Composites, Vol. 11, 127-145, 1992.

10., Newmark, N.M., Seiss. CR, and Viest, I.M., “Tests and Analysis of Composite Beams
with Incomplete Interaction", Proc. Society for Experimental Stress Analysis, Vol. 9, 73-
79, 1951.

11. Goodman, J.R. and Popkov. E.P., “Layered Beam Systems with Interlayer Slip”, J.
Structural Division, ASCE, Vol. 24, 2535-2547, 1968.

12. Goodman. J.R., “Layered Wood System with Interlayer Slip”, Wood Science, Vol. 1,
148-160, 1969.

13. Thompson, E.G., Goodman, J.R., and Vanderbilt, M.D., “Finite Element Analysis of

Layered Wood System”, J. Structural Division, ASCE, Vol. 101, 2659-2672, 1975.

14. Vanderbilt, M.D., Goodman, J .R., Criswell, W.E., “Service and Overload Behavior of
wood Joist Floor System”, J. Structural Division, ASCE, Vol. 100, 11-29, 1974.

15. Rao, KM. and Ghosh, B.G., Imperfectly Bonded Unsymmetrical Laminated Beam, J.
Engineering Mechanics Division, ASCE, Vol. 106, 685-597, 1980.

16. Fazio, P., Hussein, R., and Ha, K.H., “Beam-columns with Interlayer Slip”, J.
Engineering Mechanics Division, Vol. 108, 354-366, 1982.

17. Toledano, A., Murakami, H., “Shear-deformable Two-layer Plate Theory with
Interlayer Slip”, J. Engineering Mechanics Division, ASCE, Vol. 144, 605-623, 1988.

18. Lee, C.Y., Liu, D., and Lu, X., “Static and Vibration Analysis of Laminated Composite
Beams with an Interlaminar Shear Stress Continuity Theory”, Int. J. Numerical Methods
in Engineering, Vol. 33, 409-424, 1991.

19. Lu, X. and Liu, D., “Interlayer Shear Slip Theory for Cross-Ply Laminates with
Nonrigid Interfaces”, AIAA J ., Vol. 30, 1063-1073, 1992.

20. Liu, D., Xu, L., and Lu, X., in press. “Stress Analysis of Imperfect Composite
Laminates with an Interlaminar Bonding Theory”, Int. J. for Numerical Methods in
Engineering. Vol. 36, 2819-2839,1994.

21. Jasiuk, 1., J. Chen, and M.F. Thorpe, ”Elastic Modulus of Composites with Rigid
Sliding Inclusion,” J. Mechanics and Physics of Solids, Vol. 90, 373-391, 1992.

22. Achenbach, JD. and H. Zhu,”Effect of interphases on micro and macromechanical
behavior of hexagonal-array fiber composites,” J. Applied Mechanics, Vol. 57. 956-963,
1990.

23.'Lene, F. and D. Leguillon,”Homogenized constitutive law for a partially cohesive
composite material,” International J. Solids Structures, Vol. 18, 443-458, 1982.

24. Kalantar, J. and LT. Drzal, ”The bonding mechanism of aramid fibers to epoxy
matrices, part 11 an experimental investigation,” J. Material Science, Vol. 25. 4194-4202,
1990.

25. Babuska, 1., “Homogenization and its application. Mathematical and Computational
Problems. Numerical Solution of Partial Differential Equations”, Academic Press, New
York, 1976.

26. Benssousan, A., Lions. J. L., and Papanicoulau. G..“Asymptotic Analysis for a
periodic Structures”, NorthHolland, Amsterdam, 1978.

177

 

 

178

\/27. Guedes, J. M. and Kikuchi, N., “Preprocessing and Postprocessing for Materials Based
on the Homogenization Method with Adaptive Finite Element Methods”, Computer
Method in Applied Mechanics and Engineering, Vol. 83, 143-198, 1990.

28. Hashin, Z., “Thermoelestic Properties of Fiber Composites with Imperfect Interface”,
Mechanices of Materials, Vol. 8, 333-348, 1990.

29. Hashin, Z., “The Spherical Inclusion with Imperfect Interface”, J. Applied Mechanics,
Vol. 58, 444-449, 1991.

30. Hashin, Z., “Extreme Principles for Elastic Heterogeneous Media with Imperfect
interfaces and Their Applications to Bounding of Effective Moduli”, J. Mechanics Physics
Solids, Vol. 767-781, 1993.

31. Petit, P. H., and Waddoups, M. E., “A Method of Predicting the Nonlinear Behavior of
Laminated Composites”, J. Composite Materials, Vol.3, 2-19, 1969.

32. Reddy, N. J., “Introduction to the Finite Element Method”, McGraw-Hill Book
Company, 1984.

33. Pagano, N. J ., “Exact Solutions for Composite Laminates in Cylindrical Bending”, Int.
J. Solids structures, Vol. 3, 398-441, 1969.

34. Srinivas, S., and Rao, A. K., “Bending, Vibration, and Buckling of Simply Supported
Thick Orthotropic Rectangular Plates and Laminates”, Int. J. Solids Structures, Vol. 6,
1463-1481, 1970.

35. Lekhnitskin. S. G.. “Anisotropic Plates”, Gordon and Breach Science Publishers,
1987.

APPENDICES

179

Appendix A

Maple Procedure

As well known. Maple is a symbolic software used for mathematical derivation
and numerical calculation. In this section, the procedures of how Maple was used to obtain
linear system for unknowns and then what Maple can give for the effective moduli are
shown.

The expressions for the functions, H3612) , Gg(:3), Pg(:3) . and x:’(z) , which are
given in Eqn. (3.2.49) and (3.2.68) are as follows:

hfI(:2):=CfI I *exp(b*:,2)+Cf3I *exp(b *22)*:2+exp(-b *:2) *Cf2]+Cf4I *exp(-b*:2)*22;

hm I ( :2 ): =C ml I *e.rp( I) *32 )+ C m3] *exp( b *32 ) *:2+e.rp(-I) *:,2)*C))121 +Cm4l *e.rp( -

b *32 ) *32 .'

I1I2(:,2 ).' = C f I 2 *vxm I) *z2 )+ C I3 2 *exp( I) *32) *32 +e.t'p( -I) *32) *C I22 + C I42 *e.t'p( -I) *3.” ) *32;
Ii))12(:2 )3 = Cm I2 *arm I) ‘*:2 )+ C1113 2 *exp( I) *32 ) *32 +e.rp( -I) *32 ) *C "122 + C M42 *wim -

b*:2)*:2;

If] (:2 l: =( I) *u *CI'I 2 *arm I) *z2)+l) *u "‘Cf32 *c.rp( I) *32 ) ’*‘:2 -u "C [3 2 *cxp( I) *32 )-I) *u *epr -
I) *32) ”C I22 -I) *u C I42 *‘(ht’pI -I) “:2 ) 2*:2 -u C I42 ’*‘arp( -I) “:2 )- C II 2 *I) *e.\'p( I) *:2 )-
C f3 2 *I) ”‘6pr I) ‘*‘:2 ) *z2 - C /3 2 *c.rp( I) *:,2 ) +1) ‘l‘exm -I) *32) *Cj22 + CI1I2 *I) ‘l‘e.\'p( -I) *32) *32 -

C I42 "‘e.rp( -I) *:2 ))/I)/( - I +11);

gm [(32): =(b *u *Cm 12 *e.rp(b *:2)+b *u *an32 *exp(b *32) *12-u *Cm32 *exp(b *:2)-
b *u *exp( -b *12) *CmZZ-b *u *Cm42 *exp( -b *32 ) *zZ-u *Cm42 *epr -I) *32 )-
Cm 12 *1) *exp( I) *32 )- C m32 *b *exp( b *:,2 ) *12 - Cm32 *exp( b *32)+b *e.rp( -

b *22) *Cn122 + C ))142 *b *exp( -b *12) *12- an42 *exp( -I) *22) )/b/( - I +1) I;

gI2(32):=(CfI I *b *exp(b*:2)+Cf3l *b *exp(b *32)*:2-Cf3l *exp(b*:2)-b*exp(-
b*:2)*Cf21-Cf4l *I) *exp(-b *32) *zZ-Cf4l *exp(-I)*:2)-b*u *CfI I *exp(b *32)-b*u
*Cf3l *exp(b *:2) *:2-u *Cf3l *exp(b *32)+b *u *exp(-b *:2) *Cf21+b *u *Cf4] *exp(-

b*:2 ) *:2 -u *C f4 I *exp( -b*:2 ) )/b/( -I +u ),'

pr(:2 ): = I/Z/b *( -b *C f4 I *e.rp(-b *22 ) *22 +C f4 I *exp(-b *32)-b *e.rp( -
b*:2)*Cf2]+CfI I *I)*e.rp(I)*:2)+b*Cf3I *exp(b*:2)*:2+Cf3I *exp(b*:2))+Cpr I+Cpf21

*-2.

pm I( :2 ): = I /2/I) *( -I) "‘C m4 I *exp( -I) *32) *32 +C m4 I *arp(-I) *12)-I) “fer/)1 -

I) *:2 ) *Cm}? I + Cm I I *I) “arm I) *:2 )+I) *C m3 I *exm I) *:2 ) *32 + C n13 I *e.rp( I) *:2 ) )+ C pm I I +

C me I *32;

pf2(:2 ): = I /2/I) “*‘( C I3 2 “‘I) *exp( I) *32 ) *32 + C f3 2 *c.\'p( I) “:2 ) + C f I 2 *I) ”fur/)1 I) *:2 )-

C I42 ““I) *(’.\'])I -I) “:2 ) “:2 + C I42 *(’.\‘[)I -I) ‘*:2 )-I) *C’.\'])( -I) *z2) *CI22 ) + C pfl 2 + C [)I22 “:2.

meI:.2 I: = I /2/I) *( Cn132 *I) *e.rp( I) *:2) *32 + C1113 2 *e.rp( I) *:2 )+ Cm I2 *I) *expl I) *32 )-

180

18 1
07142 *b *exp( -I) *:2 ) *:2 + C )))42 *exp( -I) *:2)-b *exp( -I) *:2 ) *Cm22 ) + C pm I 2 + C meZ *:2;

kxfI(:I.:2):=afI *:2+I)fI+a *(gfl(:2)*cos(b *:I)+(I)fI(:2)+afI/

b)*sin(b*zI)+ifI)+a"2 *pr(:2);

kme(zI.:2):=amI *:2+bml+a*(gml(22)*cos(b*:I)+(I)mI(z,2)+am I/

I))*sin(b*:I,l+imI)+a"2 *meIZZI:

kxf2(:l,:2): =uf2 *:2+bf2+a *(g/2(:2) *cos(b *:I )+(I)I2(:2)+af2/

b) *sin(b *zI)+ if2)+a"2 *pf2(22);

chm2(:l.:2).'=a))12 *:2 +me +0 *( gm2( :2 ) *cos( b *:I )+( IimZ( :2 )+a)))2/

b) *5in(b *:I )+i)n2 )+a"2 *me(:2 ),'

The letterfin the expressions indicates that the expression is valid in fiber domain. and if

it is m then valid in matrix domain. For example. III/(:2) is Iz,(:2) in fiber domain.

AI

_. X are expressed (with Liz/.2) as:
(jg/I g' .‘II .

EijkI_E

dfl I: =ef1 Ikl-chfiI/uffl: I.:2 ).: / )-

nf’l‘di III k_rI2( : I . :2 ). :2 Ha *COSI I) *: I ) “‘(IIIIIkx'I'I (: I.:2 I. :2 ).°

(1122.-=(31221.14{fan/my)(zI.:2).:/)-

(IIIII k-\jI2 ( : I.:2 ). :2 )+u *nf*c05( I) *:I ) *(IiMkthI ( : I , :2 ). :2 ).'

182

(If/2:=ef12kI-uf*dzjf(lcrf2(:1,:2 1:1)-

ufldijfl krfI ( : I . :2 ),zZ)+a *ztf*cos( I) *:I ) *difi? krf2( : I .22 ), :2 ).'

dmI I:=emlIkI-r*difi(kxmI(z1,22),z1)-

r*nm *difi'(Icr)))2(:I.:2).:2)+r*a*cos(b*:I)*difl(kxml(:1,:2).32);

dm22:=em22kI-r*nm *dzfl(krml(zl,22),zl )-

r*di}_‘f(krm2(:I.:2).:2)+r*a *nm *cos(b *:I)*difl(kxml(:l,:2),:2);

dm12:=em IZkl-r*um *dzfl(k.xm2(zI.Z2).ZI)-

r*um *dzjflkrm I (: I .:2 ).:2 )+r*a *um *cos( b *:I ) *difi‘(IocmZ(:I ,:2 ).:2 ):

where )=0.I, and other parameters are dimensionless material constants which are the

 

 

 

(f) Etf)

. , , . . If) Ilkl . 1122
material constants dwided by EH”. For example. efIIkI= (I) . n]: (I) . and
EU) E12111 Ellll

. 1212

HI: _ .etc..
‘ E”)
1111

The expressions for preparing the conditions can be first written as:

com/pifl .‘ =u ““6031 I) *3, I ) *(Ifl I -df12.‘

( vmlpi 2: =0 *msI I) “5‘: I ) “"(III 2 -cII22 .'

183

condpim I: = -a *cos( I) *: I ) *dm I I +dm I 2;

c0ndpim2: = -a *cos( b *:I ) *dm 12 +dm22 .‘

nI: =-a *cos(b *:I); n2: = I-(a *cos(b *:I))"2/2;

c0ndpi3:=dfl I *n I *n I +df22 *)12 *))2 +2 *dfl 2 *n I *)12-kk/300.0*((Icrf1 (21,:2 )-

kme(:I,:2))*nI+(kxf2(zI.:2)-Ictm2(zl,:2))*nZ);
c0ndpi4:=(dfl I-df22) *nI *n2+df12 *(n2 *nZ-nI *nI)-km/300.0*((kxfl(zl,:2)-
kme(:I.:2))*n2-(kxf2(:l,z2)-kxm2(:1,:2))*nI);
where kk and km refer to bonding coefficients I/k and Up. respectively. Then the condition
equations can be expressed as
condi I .' =condifl +condim I .'
c0mIi2 .° =crmc/tf2 +c0mIimZ .'
c0ndi3: = mud/7131):

concIi-I: =cvnqui4I):

krflI:/.0):=k.rm I(:I, I):

184
Iocf2(:1,0): =kxm2IzI. II;

kxfI(0,0):=0; kxfI(0,0):=0

By substituting :2=vl into the above equations, the equations for the interfacial
conditions can be obtained. Substituting 22:0 into the functions for fiber domain, and
22:] into the functions for matrix domain, the equations for the periodic conditions can be
given. Then grouping the constant terms, sin(b*:1), and cos(b*:I ) together, respectively,
the linear system for solving unknowns can be achieved. For example. A [(17, A3”), A10"),
A20"), Blm, 8207, 8,0"). 83”") in Eqn. (3.2.41) are expressed as afI, af2, am], am2, be,
bf2, me, and me in Maple. By grouping the constant terms in zeroth order of a together,

the following equations are obtained.

eri I : =((-ef12kl+uf*afl +em 12kl-r*um *am I II.’

eri2: =((-ef22kl+af2 +e))122kI-r*m)12 I I:

(’(IOI3.'=(((’_I22I\I-(lf2-kk/Lffl I I I *(csz *t'I +I)j2-umZ *t'I-ImzZII):

eri-Iszl(efl2kl-zq/‘*aflI'm/cf] I I I *(afl *t'l +I)I'I-t1))zI “‘t'I-InnIIII.‘

qup/s=1(-¢{II2I<I+1(I*((II+em I2kI-I"5‘III)1‘*‘(1)))I I):

erpZ: =((-t{I22kI+ajZ+e)))22kI-r*a))12)I:

185
erp3 .° =( ( ( bf2-amZ -b)n2 ) ) I;
eq0p4:=(((be-am1-me))),°
bf] : =0;bf2: =0;
SOIVBI I erII . eriZ. eq0i3 . eq0i4. eq0p3. eq0p4 I, I afI ,af2,am I ,an12.bm I , bm2 I I;

bml .'= -I/(-uf*km *vI +uf‘"km+uf"ef1 I II *r*um+km *vI *r*um)*(-

km *vI *ef12k1+km *vl *emIZkI+uf"‘efI I I I *emIZkl);

am2:= (efI I I I *em22kI-kk*vl *ef22k1+kk*vl *en122kII/( efI I II *r+kk *vI *r-kk*i'l +kk);

aj’l : = (-km *t'l *ch 2kI+ejI2kI *km+efl2k1*efl I I I *r*um+km *1’] *6») I 2kI-em I 2kI “km I/(

1(I*km *t' I +uf*km+uf*cfl I I I *r*mn+km *t'I *r*mn).’

af2.’= (ej22kl*t{II I I I *r-kk*vl *ch22kI+ej22kI*kk+kk *t'I *emZ2kI-em22kI*kkI/

(efI I I I *I‘+I(I(*1‘I *r-kk*vI+kkI;

am If: I/I-III*I<))1“‘\'I+uf*km+uf*efl I I I *r*um+km*t'l *r*m))) “(-

km *t'l*t{f12kl+km *vl *em IZkI+uf*efI I I I *em I2kII.‘

- a-;_..§m._r.r

 

 

186

me:= -(efI I I I *em22kI-kk *vI *ef22k1+kk *t'I *en122kI)/(efII I I *r+kk*vI *r-kk *vl +kk);

Since the expressions for the other coefficients are very long, they are not listed here.
With all these solved coefficients, it is not difficult to obtain the effective moduli.

For example, given material constant

EI-lll = 55222 = 3006’)“, Eiizz = Eiziz = IOOGP“

Einin = 531222 = 300p“, Einizz ‘ 5TH) = IOGP“
two cases of effective moduli can be achieved. In case 1, if the composite considered is of
25% fiber volume fraction and with perfect bonding between fiber and matrix. then the
effective moduli depending on fiber waviness are:
EH 1 I I I :=-5 7. 04627106 *a"2+90. 967 74026:
EH22] I : = I 0.50] 08444 *a"2 + I 2. 90322079:
EH2222: = I . I 50095499*a"2+38. 70966243 .°

EHI2 I2=5.888983268 *u"2+ I2. 903224 I4

In case 2. if the composite considered is of 50% fiber volume fraction and 3% fiber

waviness. and the two bonding coefficients are equal. then the effective moduli are:

 

187

EH 1 I I I : =6.25 0000000 *( .234433464 7667 *km"4 +. 19094 973 I 6666 *km"5+.6494309855
637*km"21+. 7019783282646 *km"1 7+.54546 72209644 *km"18+.343 7425071642 *km"I9
+. I712 767461640*km"20+. 7455896323648*km"16+.6616375654650*km"15+.4947304
498652 *km"l4+.3l34471215654 *km"I3+.2513I54966629*km"24+. 1760777922635 *km
"22 +3 040668040632 *km"23 +. 6208 7661 49669*km+.2 I 8 9302 752668 *km"3 +. I 46 I 2450
81669*km"2+.I26I897I50670+.1229299582665*km"6+.64l6l91965663 *km"7+.27631
35686662 *km"8 +. 993 98 I 5 9] 0660 *km"9+. 7734 78 I 73665 7 *km"! I +. I 688022692656 *km
A12+.3012401823659"‘km"10)/
(. 2051441809626 *km"4+.66 18464 781624 *km"5 +.5 946529640629 *km +.450708884462 7
*km"3+.666l601491628 *km"2+.2424162693630+ 1552154248623 *km"6+.26664940746
2] *km"7+.33304574746 I 9*km"8+.295005902661 7*km"9+.6342180964612 *km" I I+ IO
45419866. *km"]2+. I759228701615*km"10)/
(.3641]98877620*k)n"4+.72340484I06I8*km"5+.3237590165624*km+.1217151295622
*km"3 +.2604805 944623 *km/‘2 +. I780] 84 I 09625 +. 95468820556] 6 *km"6+.8072 I 58 I 55
6 I 4 *km" 7 +.3 9688086006 I 2 *km"8+ 864687163. *km"9)/

(. [4283374336 I2 *km"2+ 9672 [2107. *k))t"3+.6910506445613 *km+.1089398644175):

EH22] I: =. I250000000 *I. [49552 I330697*km"4+.2224 732980696 *km"5+. 7823903 777
673 *km"2 I +.6 123916 I64680*km"l 7+. [3686103196 79*km"l8+.2 736384I85677*km"19
+.48940873266 75 *km"20+.2450l [5352682 *km" I6+.87542 I4 791683 *km"I5+.2787870
063685 *km" I4+. 789 I4 79222686 *km" I3+. I5998 75846668 *km"24+. I I I631 76646 72 *km

A22+. I4 I837] 9626 70 *k)))"23+. 628283 1034698 *km+. 764 7I37873697“*‘km"3+.2 7734 787

 

188
19698 *km"2 +.26307136336 95 *km"6+.2548153848694 *km"7+.206596 14 74693 *km"8+.

1424858790692 *km"9+.6552333073698+ 4371363618689*km"11+.1978283508688*km
A12+.8463351 798690 *km"10+. 1288828766642 *km"35+. 7292917776638*km"36+. 1105
543873645 *k))z"34+. 1072319128662 *km"2 7+. 71 1 666580065 9 *km"28 +.404582860065 7
*km"29+. 13994456 75664 *km"26+. 1595524503666 *km"25+. 1941895845655 *km"30+.6
1 3206 792464 7*km"33 +24 71 96 7264650 *km"32 +. 771 7725 868652 *km"31)/
(. 2051441809626 *km"4+. 6618464 781624 *km"5 +.24241 62693630+.5 94652 964062 9*km
+.4507088844627*km"3+.6661601491628 *km"2+. 1552154248623 *km"6+.2666404074
621 *km"7+. 33304574 746 I 9 *km"8+.2950059026617*km"9+.6342180964612 *km"1 1 +1
0454 1 9866. *km"12+. 1759228701615 *km"10)/

(.3641 198877620*km"4+. 7234048410618*km"5+.3237590165624*km+. 1217151295622
*km"3 +.2604805 944623 *km"2 +.1 780184109625+. 9546882055616 *km"6+.8072158155
614 *km"7+.3968808600612 *km"8+864687163. *km"9)/

(. 1428337433612 *km"2+ 967212107. *km"3+.6910506445613*km+.1089398644615)/

(. 1025720904626 *k))1"4+.3309232391624*km"5+.1212081346630+.2973264820629*km
+. 225354442362 7*km"3+. 3330800746628 *km"2 +. 7760771238622 *km"6+.1333202038
621*km"7+.1665228738619*km"8+.1475029514617*km"9+.3171090483612 *km"11+5

22 709933. *km"12+.8796143506614*km"10):

EH2222: =(. 9890186139694 *km"6+.8524885091697*km+. 793825918266 7*km"24+. 701
3047349669*km"23+.5497899569671 *km"22+. 132 759352 7677*km"19+.238781378267
5 *km"20+.38363085846 73 *k)))"2 1 +. I 162 993349682 *km"16+.65978592496 78 *km"18+.

29309542 12680 *km"1 7+.3623 745383686 *km"13+.4116136115683*km"15+. 129656024

189
5685 *km" I4+.4 7860648 72696 *km"4+. 7819666485695 *km/‘5 +.3 6 9845803 3 63 8 *km"36

+.69850435 72663 *km"26+.9775699321654 *km"30+. 7941484017665*km"25+.3569020
067659*km"28+.2033060997657*km"29+.5365669099661*km"27+.3891418185652 *km
"31 +.3100296 783647*km"33+. 1248198615650 *km"32+.5595860012644 *km"34+.6530
184336641 *km"35+.61 14936723697*km"2+.2127972553697*km"3+. 1007648427694 *k
m"7+.6046400380691 *km"9+.8496865313692 *km"8+.5574 7974 73689+.368395 126069
0*km"10+.1943183317689*km"11+.8950014171687*km"12)/
(. 1034769498623 *km"6 +.3 964353 094629*km+. I 36 762 78 72626 *k)))"4+.4412309853624
*km"5+.4441 06 7662628 *k))t"2+.3 004 7258 9662 7*km"3 +. I 777602 715621 *km" 7+. 19667
06017617*km"9+.16I6108462630+.2220304981619*km"8+.1172819134615*km"10+.4
228120642612 *km"1 1+6969465 7 7. *km"12)"2/
(.1591147009617*km"6+.5395983610624 *km+.6068664795620*km"4+. 1205674735619
*km"5+.4341343240623 *km"2+.2028585491622 *km"3+.1345359692615*km"7+14411
452 71. *km"9+.6614681000612*km"8+.2966973514625)/

(.6910506445613 *km+. 1428337433612 *km"2+967212107. *km"3+. 1089398644615);

EH1212:=.30000000006-
1 *(.2758902 724690 *k)))"5+.501822 7255680 *km"15+.24695220776 79 *km"16+. 1077759
921678*1<m"17+.4176982512676 *km"18+.4401 9090206 73 *k)))"20+.1438500505675*km
A19+. 1196009288672 *km"21 +. 1431914989683 *km"13+. 9024341948681 “*‘km"14+.6404
528138689 *km"6+. 182 7021339688 *km"8+.3263336945648*km"33+.8371062101650*I<
m"32+. 165244 7786653 *km/‘3 I +.2351550552687*km"9+.26I 1975 740655 “‘km"30+.3396

627439657*km"29+.3705068731659*km"28+.1191936924689 *k)))"7+.3439505327661*

 

190
km"2 7+. 1904881223665 *km"25+.2 74 754 7732663 *km"26+.1 1 53 9242 92667 *km"24+.28

81 I 094996 70*km"22+.6 139778242668 *km"23+. 1450013506640 *km"36+. 91871485066

45 *km"34+. 1661877937643 *km"35+.3579129538683+.2532492671691*km+.33945910

88691 *km"2+.19967951 76684 *km"1 2 +25 782 76468686 *km"10+.2434602816685 *kmAI
1+.9157309689690*km"4+.2197967675691*km"3)/

(. 7725237995622 *km"5+.3180583468621*km"6+.1988300910618*km"8+.29194492036

16 *km"9+. 9416082480619*km"7+. 1330130434627+.6189129040626 *km+. 1297939611

626 *km"2+418167949. *km"12+.2828998968614*km"10+. 1624505968612 *km"11+.134

0063913624 *km"4 +. 1620472509625 *km"3)/

(. I613420522618*km"5+.3846782 703616 *km"6+.4793300263612 *km"8+1729374333. *

km"9+.5714143005614*km"7+.5350832160622 *km+.8506507780621*km"2+.14651721

15623+.4379443473619*km"4+. 7709378273620*km"3)(. 1076623942612 *km"2+967212
109. *km"3+.3634989393613*km+.3631328825614I/

( .61 801 903 95623 *km"5 +.2544466 775622 *km"6+. 1590640728619 *km"8 +.233555 93626

17*km"9+.7532865985620*km"7+.1064104347628+.4951303232627*km+. 1038351689

627*km"2+3345343591. *km"12+.2263199174615*k))2"10+.1299604773613 *km"11+.10

7205 1 130625 *k)))"4 +. 12 963 7800 7626 *km"3 I:

 

191

Appendix B

Stress Distributions for Casel in Section 5.1 by ABAQUS

The commercial software ABAQUS is used to calculate the stress distributions of a
composite beam with sinusoidal-fiber waviness under uniaixal tension in x, direction,
which is identical to the csael in Section 5.1. The following graphics are one for finite
element model and the other two for the stress distributions of 0'” (SI 1) and 012 (S12)
respectively. Since ABAQUS can not give the distributions of 0'22 clearly inside the beam
(almost constant), the graphics for 0'22 (S22) is not presented. The reasons for this are that
the range of 522 is so small first, and the loading at ends results in relatively great
numerical value at ends. It is concluded from these two graphics and Figs. (5.4), (5.6),
(5.7), and (5.9) that the stress distributions from those two approaches have the same
tendency. 0'“ and S11 have sinusoidal distributions along x, direction. 0'” and S11
change monotonously along x2 (:2) direction. (5'2 and 312 have cosine distributions along

x, direction and their both have parabolic-like distributions along x: (:2) direction.

192

 

52085-5. E 53:8 353:: 8.6::
Egon 832.58 hung—«28:56 a ..8 .358 .5820 2.5m - ..m BawE

 

N
influvi -. i - .1 11*
w

.
.~
—._
.2
..

193

5:00:36 5 53:8 3x25. 82:.
Ewen— Bmmomaoo hung—353:5 a .8» :m - Nd 059m

a HEINfiUZH .n muam
ouukund "um-Ha confilldfl "mafia nim.m "BOHmmm; mbogfl
oo+nooo.o "EHH DHHGADIDUnvt Adah OHIMONN.N muam mHuH IH nuauumxou HIHH

H

 

 

 

194

 

5:953». E 5:28 3.3:: 602:.
E63 86an8 Bog—«28:56. a 8.“ a; - md Bzwfi

.n HaulnmvzH H mnam
onuvmuna "Bah omlgidfl "uhda wind uIon—“mmu> mDOflmt
oo+noo°.o HIHH nua‘ablanv‘ A‘s“ odlnoaa.u muhm «Hun EH auhunmtoo HIHH

._..

 

 

 

195

Appendix C
Pure Tension in x2 direction

Consider a sinusoidal-fiber composite loaded in tension in the .rZ-direction as

shown in Fig. C.1a. The unit cell of the sinusoidal-fiber composite is given in Fig. C.1b
and the transformed local coordinate system zl-zz for depicting stress distributions given

in Fig. C.1c.

. . . . k] . . . H . .
The distribution function x2 and the effective moduli EU“ for various fiber

waviness ratios, interfacial conditions, and volume fractions have been obtained in

Chapter 3. The effective moduli E3“ for 50% fiber volume fraction are shown in Figs.

3.13 and 3.14. Since the homogenized material is of an orthotropic material, the global

solutions for this homogenization problem as obtained in Chapter 2 are:

0'” = O I (C2)

6?, = 0 (C3)

 

Figure C.1a - A curved fiber composite under pure tension.

 

Figure C.1b — A unit of the curved fiber composite given in Figure C.1a.

196

197

Matrix

 

Figure C. lc - A unit of the curved fiber composite described by another
local coordinates (:1, :2).

198

From the orthotropic constitutive equations

0 H 0

911: 51111811 +

0 H O

922 = E1122€11+

0

the strains can be expressed in terms of stresses

 

where D = EflHEin—(Efizz

0_ 0 .
”i ‘3ii-‘i

0 0
“1 = 822‘:

with the assumptions that ule when x120. and 113:0 when x3=0.

By using the Eqn. (2.2.19). the local solution are:

H 0
9i2 = 2512125

H 0
£1123

3‘)

H 0
E2222922

12

,
) . The global deformations are:

(C4)

(C5)

(C6)

 

(C7)

(C8)

(C9)

(C. 10)

(CH)

199

1 k1 0 ll 0 22 0
”g : —Xg Ekl : —Xg Eli‘Xg €33 (C.12)
I “ E E k' 0 CB
Oi} — ( ,‘jk/(J’I‘ ijg/,(J’1Xg,/,(.Y))Ek1 ( . )

The total solutions for the problem then become

HO

O l 0
ug -- ug-I-sug — Eggxg—Exg 8k, . (CM)

0’30

(g = 1,2 andthereisnosumforgine g)or

1 k1 0
o..=o.-= (E--,—E-. x )8.
(j I} 11111 ugh g,I) AI
2 0 (C.15)
2)€11

11 ii 0 22 2
= (Eijll “EijiiXi. 1‘51122X2.2)€ii +(E1jll _Eijllxl, 1‘51122X2

The deformed and undeformed cells for both perfect bonding and imperfect bonding

. . . . . . 0 .
(u=k=0.lmm/N) are shown in Figs. C2 and C3, respectively. The uniaixl tension 0' is

chosen as 1 GPa in this study. The undeformed cell is shown in Fig C.1b. The dimensions

of the unit cell are 2118 by 1iE where IiE is the height of the unit cell. The dotted lines in

diagrams represent for the undeformed cell while the solid lines the deformed cell. It is
concluded that from these two diagrams that both global and local deformations contribute
to the deformation of the unit cell. In fact. the global deformation introduces a uniform
stretching in the .rz-direction and a uniform shrinkage in the .rl-direction while the local
deformation gives a slightly nonuniform amplification in the sinusoidal amplitude.

Stress distributions along :l-axis for a unit cell with a perfect interfacial bonding

 

 

. h 000., interface on matrix
.‘2/ e
---- undeformed shape
1.2“ —- deformed shape

 

 

 

 

 

0.2“
M; 1
00 0.5 M _--
-02 - .Vl/hs

 

Figure C.2 - Deformation of a unit cell with perfect interface

 

 

 

.... interface on matrix
---- undeformed unit

— deformed unit

   

O
00
I

 

P
1‘?

0 0 0:5 1W2
-o.2~

 

I

.Vt/hc

 

Figure C.3 - Deformation of a unit cell with imperfect perfect interface (k:
p. =0. lmm/N).

 

202

condition between fiber and matrix are shown from Fig. C4 to Fig. C.6. Results reveal

that the normal stresses 0'” and on reach their extreme values, maximum or minimum, at

the peak and trough points of the sinusoidal curve, i.e., :l = 0.5 . and :l = 1.5. On the

contrary, the shear stress on reaches its maximum and minimum values at the middle point

and the ends. All the stresses show no big difference between the fiber and the matrix domains

for the pure tension case. The stress distributions along :z-axis are shown in Figs. C.7-C.9.
The results indicate that on and 0'12 are discontinuous across the interface. i.e. 2220.5.
However, 022 is continuous across the interface. This is because that on is equal to the
normal traction T" if the zeroth-order and the first-order terms of Eqn. (3.2.16) are retained.
However, since T, is the combination of the three stresses components. the continuity of T,
cannot be concluded from the stress distribution.

The stress distributions for a sinusoidal-fiber composite with both normaly and
tangentially imperfect interface (e.g. k=tt=0. lmm/N) are shown from Fig. C. 10 to Fig. C.15.
Figs. C.10-C.l2 depict the stress distributions along :l-axis. The shapes of the stress
distributions are no longer symmetric with respect to the midpoint when compared with the
cases of perfect interface. The extreme values of o" I decrease in fiber domain while increase
slightly in matrix domain. The shapes ofozz have a greaterchange at all the points considered.
The stress distributions alone :3-axis are depicted in Figs. C.13-C.15. These results further
indicate that the difference across the fiber-matrix interface decreases in 0'” but increases
1“ 012.

Figs. C.16 and C.17 show the distributions of normal traction T, along :,-axis at

fiber-matrix interface for various bonding coefficients k and 11. Results show that the

203

 

 

oo .. global stress
stresses in fiber domain

 

(511160

0.4“ ++++stresses in matrix domain 3:!
0.2-”- 0 00 .. .o .0 ooooooooo :2:O.5
,0 /05\. 15
0
37:0

I

-02 "‘

 

I

-0.4i

I

-0.6:

 

 

 

Figure C4 - 0” along the :l-axis in a unit cell with perfect fiber-matrix
interface.

 

204

 

 

(522100
1.008 ‘

I

1.006:
1.004:
1.002-
1.
0.998:
0.996"
0.994:

 

0.992:

0000

global stress

— stresses in fiber domain
+ +++ stresses in matrix domain

 

 

 

 

Figure C.5 - 033 along the :l-axis in a unit cell with perfect fiber-matrix

interface.

205

 

 

 

0' /0’0 o M. global stress
0 1 § 12 —— stresses in fiber domain
. H H stresses in matrix domain
0.1:-
22:0.25
0 .05 6
22:05
0 0 ...... v ..... 22:0.75’
-0.05 ..
-0.1 -
22:0. 0.5
-0.15 ..

 

 

Figure C.6 - 0'3 along the :I-axis in a unit cell with perfect fiber-matrix
interface.

 

206

 

 

 

 

(511/60
0.4:-
...... 31:15
....:EE:.E-°;;°“Innuull‘.°.:E:. ....... 21:0. I
0.2- ------ zl=0.5
\ /Zl=0.5
00 0,2 0,4, ' 0,6 0,8 .1
Zz/I’tE
-0.2 r
// 31:0, I
-0.4-- / \
--- stresses in fiber domain
-0.6 ~
++++ stresses in matrix domain
-0.8* 51:15

 

 

Figure C.7 - 0'“ along the :3-axis in a unit cell with perfect fiber-matrix
interface.

 

 

 

 

 

 

 

 

022/00 — stresses in fiber domain

I .+++ stresses in matrix domain
1.02“

.. 21:0.5
1.01:

10 ............................................... 31:0, 1
0.99- "
0.98:k

0 0f2 0f4 . 0:6 ofs 7i

ZZ/he

 

Figure C.8 - 6:: along the Cz-‘XmS in a unit cell with perfect fiber-matrix
interface.

 

208

 

 

—— stresses in fiber domain

0 .151” . . + + stresses in matrix domain

 

 

 

 

Figure C9 - on along the :z-axis in a unit cell with perfect fiber-matrix
interface.

 

 

 

 

 

 

O o . . . global stress
61 l/6 stresses in fiber domain
0.4+ ++++ stresses in matrix domain
............ 2’2:l
: i31::::............:::EE¢2=O.75
.1.- . °
0.2 7230.5
01
22:0
-0.2“-
22:0.25
04)-
22:0.)—
-0.6~- \
-0.8*

 

 

Figure C. 10 - 0'” along the :l-axis in a unit cell with imperfect fiber-matrix
interface (k: It =0. lmm/N).

 

210

 

022/00
1.0081

1.006:
1.004-

1.002 ‘

0.998:
0.996“

0.994:

 

I

0.992:

 

0°00

global stress

ois

I
ZI/h8

— stresses in fiber domain

+ + + + stresses in matrix domain

 

1T5 2

 

Figure C. l l - 0'33 along the :I-axis in a unit cell with imperfect fiber-matrix
interface (k: 1.1 =0. lmm/N).

 

 

 

G 12100 o . . . global stress
0,151 —— stresses in fiber domain
HM stresses in matrix domain
0.1"
0 .05" . olziggs
:2- .
60.22 ..... 2' 1 ~. '...Z,_Zz=l
. . . Zlyhg ' . . ° 22:0.75
-0.05:-
' =O.5
~2
-0017- ~7=O
-0.15 r

 

 

Figure C. 12 - 0'23 along the :l-axis in a unit cell with imperfect fiber-matrix
interface (k: p. =0. lmm/N).

 

 

 

 

 

0'1 1100
0.41“
................. ...o 31:15
....................:::EEEEE§§:cooHEESEEEE:::°..Zl=O,l
0.2,, ................. 3120.5
\ 21:0.5
00 , 0,2 0,4 /' 0,6 0,8 .1
-0 2I \
\i/ :1:0 I
' \ . .
-02,» — stresses in fiber domain
++++ stresses in matrix domain
'0o67- Zl=l.5

 

 

Figure C.13 - on along the :z-axis in a unit cell with imperfect fiber-
matrix interface (k: u =0. lmm/N).

 

 

 

 

 

 

 

022/00 . .
1 02+ —— stresses in fiber domain

- 0 ++++ stresses in matrix domain
1.014 :.= .5

1 ..u...................................EE;.[EEEE: 31:0,]
099 ~i-'-5
l.
0.98“ . , + *
0.4 Z2”,E 0.6 0.8 1

 

Figure C. 14 - 033 along the :3-axis in a unit cell with imperfect fiber-matrix
interface (A: p. =0.]mm/N).

 

 

 

— stresses in fiber domain

+ + + + stresses in matrix domain

  

-
21-1
.0 ..oo°°° °"'°Ooo

O

"'32: ‘ 'o. . 21:0.5

0:4 z2/h8.,0.6 "'-~~~0;8._,..............1z.=i.s

4

21-0

 

015i

 

Figure C. 15 - on along the :z-axis in a unit cell with imperfect fiber-matrix
interface (k: It =0. lmm/N).

 

 

 

 

 

 

 

0
T,,/o k=u=0
1 g
0.8-
0.6-
k=u= 104
0.4-
0.2-
k=u=108
00 0:5 1 1i5 2

 

I ZI/hg

 

Figure - C. l 6 Normal traction T" on the fiber-matrix interface and along the

:l-axis at different bonding coefficients 1)., k.

 

 

 

 

 

 

 

Figure - C. 17 Tangential traction T, on fiber-matrix interface and along the
:l-axis at different bonding coefficients I1,1(.

 

 

217

extreme values of both T,, and T, decrease as the bonding coefficients 11 increases (i.e., the
bonding condition becomes poor). In addition, it should be noted that both Tn and T, are

continuous across the interface of both perfect and imperfect conditions.

 

218

Appendix D

Natural Vibration Frequency

The effects of sinusoidal-fiber waviness on natural vibration frequencies are
studied in Section 6.3. The formulae used to calculate the natural vibratibn frequencies by

classical plate theory (in Fig. 6.5) are from Lekhnitskin’s book [34]. It can be written as

The results from both elasticity analysis and classical plate theory, which are
presented by Fig. 6.5, are listed in Table D1 and Table D2. The normalized frequencies
are defined as the frequency 00 divided by 030 which is the frequency of a composite beam
(I/Iz=10) at 0% fiber waviness. (00 equals tol.225/./5 Hz.

Table D.1 Normalized Natural Vibration Frequencies (1/11210)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Waviness all 0 0.1 0.2 0.3 0.4 0.5
Elasticity l. .7265 .5837 .5441 .5384 .5380
Classical theory 1.0694 .7918 .6449 .6016 .5869 .5780

Waviness all 0.6 0.7 0.8 0.9 1.0 1.5
Elasticity .5375 .5371 .5367 .5362 .5358 .5307

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Waviness all 0.6 0.7 0.8 0.9 1.0 1.5

Classical theory .5726 .5682 .5651 .5625 .5600 .5486
Table D2 N orrnalized Natural Vibration Frequencies (I/h=2)

Waviness all 0 0.1 0.2 0.3 0.4 0.5
Elasticity 2.6367 2.4653 2.3755 2.2331 2.2196 2.2049
Classical theory 5.3469 3.9592 3.2245 3.0082 2.9347 2.8898

Waviness all 0.6 0.7 0.8 0.9 1.0 1.5
Elasticity 2.1918 2.1755 2.1589 2.1420 2.1282 2.0604
Classical theory 2.8629 2.8408 2.8253 2.8122 2.8000 2.7429

 

 

 

 

 

 

 

 

 

 

 

220

Appendix E

Critical Buckling Load

The effects of sinusoidal-fiber waviness on critical buckling load are studied in
Section 6.4. The results from elasticity analysis, which are presented by Fig. 6.6, are listed
in the following Table El and Table E2. The normalized critical buckling loadings are
defined as critical buckling loadings divided by P0 which is the one of a composite beam

(I/h=10) at 0% fiber waviness. P0 equals to 1.2424 GPa.

Table E.1 Normalized Critical Buckling Load (l/h=10)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Waviness all 0 0.1 0.2 0.3 0.4 0.5
Elasticity 1.0000 .7244 .5819 .5441 .5367 .5368
Waviness all 0.6 0.7 0.8 0.9 1.0 1.5
Elasticity .5361 .5365 .5363 .5361 .5358 .5308

 

 

Table E.2 Normalized Critical Buckling Load (l/h=2)

 

 

 

 

 

 

 

 

Waviness all 0 0.1 0.2 0.3 0.4 0.5
Elasticity 2.6368 2.4348 2.2939 2.2328 2.2255 2.2054

Waviness all 0.6 0.7 0.8 0.9 1.0 1.5
Elasticity 2.1901 2.1756 2.1563 2.1410 2.1249 2.0605

 

 

 

 

 

 

 

 

221

 

“11111111111 111111111111“