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THESIS

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UNIT CELL ANALYSIS OF COMPOSITE
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UNIT CELL ANALYSIS OF COMPOSITE MICROSTRUCTURES
By

Shawn Christopher Klann

A THESIS
Submitted to
Michigan State University

in partial fulﬁllment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Mechanical Engineering

2008

ABSTRACT
UNIT CELL ANALYSIS OF COMPOSITE MICROSTRUCTURES
By
Shawn Christopher Klann

Composite materials are replacing traditional metals in many applications due to
their superior stiffness and strength to weight ratios. However, one problem with
composites is the tendency of successive layers in a structure to delaminate in the
thickness direction. This problem has been addressed in many ways such as z-pinning,
stitching, and woven fabrics that help to maintain the structural integrity of a composite.
Fiber weaving does not introduce the high stress concentration present in the other two
cases. A new type of composite, the Quasi-Three Dimensional (Q3D) weave, has shown
great promise in combating delamination. Unlike the popular 2D Plain Weave, this
weave holds successive layers together in order to increase the delamination resistance.
There are no ﬁbers aligned in the thickness direction but the tight weave creates a 3D
network. An analytical technique is needed to predict the properties of the weave. This
thesis presents an analytical method based on Classical Laminate Theory, using unit cell
analysis, to predict the mechanical properties of weaving patterns. It is derived on the 2D
Plain Weave and compared to other composite microstructures, such as laminates and the
Q3D Weave. It was found that the Q3D weave has very similar properties to the 2D
Plain Weave while having increased delamination resistance, by experimental tests.
Several case studies involving the material and geometric parameters of the composite
weave are also presented. The undulation zones in the weave are found to be the most

critical for the weave properties.

ACKNOWLEDGMENTS

I would like to thank my professor, Dr. Dahsin Liu, for both the ﬁnancial support
of my research and the directing of the research work. His insight allowed me to ﬁnd
suitable topics for research and complete this thesis in a timely manner. I would also like
to thank Dr. Patrick Kwon and Dr. Alfred Loos for agreeing to serve on my thesis
committee. Special thanks are required for Dr. Elias Shakour, Dr. Guojing Li, and Mr.
Kirit Rosario for their assistance in the research work. Lastly, I would especially like to

thank my parents, Dan and Vicki Klann, for their love and support through years.

iii

TABLE OF CONTENTS

LIST OF TABLES .................................................................................... v

LIST OF FIGURES ................................................................................. vi

NOMENCLATURE ................................................................................ vii
CHAPTER 1

INTRODUCTION ..................................................................................... 1

1.1 Background .......................................................................... l

1.2 Statement of Problem ............................................................... 6

1.3 Organization of Thesis ............................................................. 7
CHAPTER 2

ANALYTICAL METHOD. . ................................................................... 8

2.1 Model Derivation .................................................................... 8

2.1.1 Unit Cell Derivation ....................................................... 8

2.1.2 Undulation Model ........................................................ 17

2.1.3 Subregion Integration .................................................... 21

2.1.4 Computer Code ........................................................... 24

2.2 Comparison with Published Articles ........................................... 26

2.3 Summary ........................................................................... 27
CHAPTER 3

STUDY OF MICROSTRUCTURE GEOMETRY ............................................. 28

3.1 Yarn Parameters .................................................................... 28

3.1.1 Fiber Volume Fraction ................................................... 28

3.1.2 Yarn Thickness ........................................................... 30

3.2 Cell Parameters .................................................................... 33

3.2.1 Undulation Length ....................................................... 33

3.2.2 Weaving Angle ........................................................... 35

3.2.3 Cell Size ................................................................... 37

3.2.4 Undulation Angle Effects ............................................... 39

3 .3 Layer Parameters .................................................................. 42

3.3.1 Matrix Layer Thickness .................................................. 42

3.3.2 Number of Layers (2D) .................................................. 43

3.3.3 Number of Layers (Q3D) ................................................ 45

3.4 Summary ........................................................................... 47
CHAPTER 4

COMPARISONS OF WEAVE PATTERNS ................................................... 48

4.1 Weave Patterns .................................................................... 48

4.1.1 Laminate .................................................................. 48

iv

4.1.2 Quasi-3D Weave ......................................................... 49

4.2 Comparisons ....................................................................... 50
4.2.1 Moduli ..................................................................... 50
4.2.2 Bending Stiffness ......................................................... 55
4.3 Summary ........................................................................... 57
CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS ............................................... 58
5. 1 Conclusions ........................................................................ 58
5.2 Recommendations ................................................................. 59
APPENDICES ....................................................................................... 60
Appendix A — Poisson Ratios of Case Studies ......................................... 61
A.1 Fiber Volume Fraction ................................................... 61
A2 Yarn Thickness ........................................................... 62
A3 Undulation Length ....................................................... 62
AA Weaving Angle ........................................................... 63
A5 Cell Size ................................................................... 63
A6 Undulation Angle Effects ................................................ 64
A7 Matrix Layer Thickness ................................................. 64
A8 Number of Layers ........................................................ 65
A9 Q3D Number of Layers ................................................. 65
Appendix B - Mathematica Code ........................................................ 66
B.1 Laminate .................................................................. 66
B.2 2D Plain Weave .......................................................... 71
B.3 Q3D Weave ............................................................... 82
REFERENCES ...................................................................................... 94

Table 1.

Table 2.

Table 3.

Table 4.

Table 5.

Table 6.

Table 7.

Table 8.

Table 9.

Table 10.

Table 11.

Table 12.

LIST OF TABLES

Composite Yarn Properties ............................................................ 18
Matrix Material Properties ............................................................... 18
Unit Cell Geometric Properties ....................................................... 18
Comparison of Undulation Models for 2D Plain Weave ......................... 19
Comparison of the Subregion Alignments .......................................... 23
Comparison with Published Articles ................................................ 26
Comparison of 3D and 2D Models .................................................. 27
Undulation Angle Parameters ........................................................ 39
Composite Yarn Properties ............................................................ 51
Matrix Material Properties ............................................................... 51
Unit Cell Geometric Properties ....................................................... 52
Comparison of the Properties for Different Weaving Patterns .................. 52

vi

Figure 1.

Figure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.
Figure 12.
Figure 13.
Figure 14.
Figure 15.
Figure 16.
Figure 17.
Figure 18.
Figure 19.

Figure 20.

LIST OF FIGURES

Quasi3D Weave (a) Projected View, (b) Fill Side View, and (c) Warp Side

View ....................................................................................... 4
2D Plain Weave (a) Projected View and (b) Side View ............................ 9
2D Plain Weave Unit Cell (a) Projected View and (b) Side View .............. 10
Region 1 of 2D Plain Weave Unit Cell .............................................. 10
Weaving Angle for 2D Plain Weave ................................................ 15
Undulation Angle of 2D Plain Weave ............................................... 16
Side Views of (a) Sinusoidal Undulation, (b) Linear Undulation, (c) Vertical
Undulation, (d) Horizontal Model, and (e) Mosaic Model ........................ 18
Undulation Angle of Linear Model ................................................... 19
Subregion Alignment of (a) Parallel and (b) Series ................................ 21
Series-Parallel Undulation Alignments with (a) Horizontal Slices and (b)
Vertical Slices .......................................................................... 22
Flow Chart for Unit Cell Computer Code .......................................... 25
Effective Moduli for Varying Yarn Fiber Volume Fraction ..................... 29
Yarn Fiber Volume Fraction for Varying Yarn Fiber Volume Fraction ....... 30
2D Plain Weave Side View ............................................................ 31
Effective Moduli for Varying Yarn Thickness .................................... 32
Effective Moduli for Varying Undulation Length ................................. 34
Top View of the Weaving Angle .................................................... 35
Effective Moduli for Varying Weaving Angle .................................... 36
T0p View of Unit Cell Size .......................................................... 37
Effective Moduli for Varying Cell Size ............................................. 38

vii

Figure 21.
Figure 22.
Figure 23.
Figure 24.
Figure 25.
Figure 26.
Figure 27.
Figure 28.
Figure 29.
Figure 30.
Figure 31.

Figure 32.

Figure 33.
Figure 34.
Figure 35.
Figure 36.
Figure 37.
Figure 38.
Figure 39.
Figure 40.
Figure 41.
Figure 42.

Figure 43.

Parameters Affected by the Undulation Angle .................................... 39

Geometric Changes for Varying Undulation Angles .............................. 40
Effective Cell Moduli for Varying Undulation Angles ........................... 41
Side View of Matrix Layers in 2D Plain Weave Structure ....................... 42
Effective Moduli for Varying Matrix Layer Thickness ........................... 43
Projected View of Multiple 2D Unit Cells with Matrix Layers .................. 44
Effective Moduli for Varying Number of Layers ................................. 45
Projected View of Q3D Unit Cell with 5 Layers .................................. 46
Effective Moduli for Varying Number of Q3D Layers ........................... 46
0/90 Laminate Unit Cell (a) Projected View and (b) Side View ................ 48
Quasi3D Weave Unit Cell (a) Projected View and (b) Side View .............. 49
Edge Replication of the (a) Laminate, (b) 2D Plain Weave, and (c) Q3D

Weave .................................................................................... 51
Comparison of Weave Pattern Young’s Moduli for Varying Weave Angle...54
Comparison of Weave Pattern Shear Moduli for Varying Weave Angle ...... 54

Comparison of Weave Pattern Poisson’s Ratios for Varying Weave Angle...55

Bending Stiffness for Varying Weave Angle ...................................... 56
Poisson Ratios for Varying Yarn Fiber Volume Fraction ........................ 61

Poisson Ratios for Varying Yarn Thickness ........................................ 62
Poisson Ratios for Varying Undulation Length .................................... 62
Poisson Ratios for Varying Weaving Angle ........................................ 63
Poisson Ratios for Varying Cell Size ................................................ 63
Poisson Ratios for Varying Undulation Angles .................................... 64
Poisson Ratios for Varying Matrix Layer Thickness .............................. 64

viii

Figure 44. Poisson Ratios for Varying Number of Layers65

Figure 45. Poisson Ratios for Varying Number of Q3D Layers .............................. 65

ix

Aeﬂective
Amk

Amk

Q

23"”;‘96335‘1
G
‘9

ﬁle's 3
leg
D

a“

NOMENCLATURE

effective extensional stiffness matrix for unit cell
extensional stiffness matrix for an individual constituent

= extensional stiffness matrix for region integrated over its area

unit cell length

= undulation length

non-undulating length
length of specified square area

= coupling matrix

area of the unit cell

stress-strain relation of an individual constituent
bending stiffness matrix

Young’s modulus in the x direction
Young’s modulus in the y direction
Young’s modulus in the z direction
in-plane shear modulus
out-of-plane shear moduli
thickness of the composite yarn
number of subregions per region
directional cosine matrix

resultant moment

resultant force

number of plies

number of regions per unit cell
quasi3D weave

transformation matrix

thickness of matrix layer

ﬁber volume fraction

z coordinates of the composite constituents
strain

Lekhnitskii coefﬁcients

undulation angle

Chentsov coefﬁcients

ﬁber volume fraction

in-plane Poisson ratio

out-of-plane Poisson ratios

stress

weaving angle

CHAPTER 1 - INTRODUCTION

1.1 Background

Composite materials have become increasingly popular over the last several
decades. This is in large part due to their adaptability to meet the requirements of a
speciﬁc design. Composites have been proven to have extremely high strength and
stiffness to weight ratios, as compared to conventional metals. This, coupled with the
fact that they may be manufactured to any shape has contributed to their increased usage.

Composite materials are usually anisotropic materials consisting of two
constituents, reinforcing ﬁbers and a bonding matrix. The strength of the material comes
from the ﬁbers, while the addition of matrix allows the desired shape to be achieved and
transfers the mechanical loading among the fibers. The ﬁbers bundles, known as yarns,
may be stacked in unidirectional plies or woven together in any number of patterns. This
is known as the composite microstructure. The ﬁbers may be oriented in various
directions. This feature allows the designer to orient the yarn ﬁbers in the directions that
require stiff and strong properties. Consequently, this makes analysis of composite
materials much more complex.

The ability to align the ﬁbers in any direction gives composites excellent in-plane
properties. However, they are vulnerable to out-of-plane loading. Impact, bending, and
fatigue loadings have the potential to cause delamination of the composite. This is the
separating or debonding of successive layers. This occurs due to the weak matrix
bonding between layers. Delarnination causes a large reduction in the composites

strength, especially compressive. Many different methods have been used to remedy this

problem. The three most popular methods are stitching, z-pinning, and fiber weaving.
Stitching involves the tying of all the layers together with very high strength fibers.
Stitching the layers together can cause damage of the fibers and induce resin rich and
stress concentration areas that become weak points in the material. Z-pinning is the use
of rods that are inserted through the thickness of the composite structure that help to tie
the layers together for speciﬁc loadings, especially bending. However, Mouritz [1] has
shown through ﬁnite element analysis that the introduction of the pins has been shown to
moderately reduce the in-plane moduli of the composite. Also the creation of holes leads
to stress concentrations throughout the part.

Fiber weaving, where the yarns are interlaced with each other, has been effective
in keeping the layers together to maintain the integrity of the structure. Once again, these
weaves require more complex analysis in the determination of the composite properties.
Kim and Sham [2] studied impact and delamination properties in woven fabrics and
unidirectional plies. Crack propagation, damage area, fracture toughness, and max load
were analyzed by experimentally impacting specimen. They found the woven fabrics to
have superior out-of-plane loading properties. The woven fabrics were found to hold
their structural integrity after impact better than for laminated plies, as well as increase
delamination resistance. Atas and Liu [3] experimentally explored the effect of weaving
angles on two-dimensionally woven composite samples for out-of-plane loading. Special
attention was paid to peak force, contact duration, maximum deﬂection, and absorbed
energy of the impacted specimens. They found that small angles differences between
yarns improved the impact properties of the samples. This was attributed to a decreased

bending stiffness in the plies.

The most common weave pattern is the 2D Plain Weave. Two-dimensional
weaves interlace the yarns together for only one layer. However, the layers are only held
together by the weak bonding matrix, as with the unidirectional laminates. 2D weaves
have been used to derive most of the available analytical models. This is due to the
simplicity of the weave to model and the popularity of this weave in industry. Three-
dimensional fabrics are increasing in popularity. The interlocking of successive layers
creates a 3D structure and helps prevent delamination of the composite. Companies such
as 3TEX [4] offers 3D weave and braids for combating delamination.

A new class of composite weaves, the so-called Quasi-Three Dimensional (Q3D)
Weave [5], as seen in Figure 1(a) was introduced to combat delamination. The green
portions represent the fill yarns while the white areas are the warp yarns. The weaving of
the ﬁll and warp yarns together forms the basis of the composite structure. In the ﬁgure,
the yarns have been infused with resin to give it a rigid structure. The gray areas in the
ﬁgure are regions of pure matrix. There are actually no ﬁbers aligned in the thickness
direction in the Q3D Weave, but the tight weave creates a solid, 3D structure. It was
designed to avoid large undulation angles that weaken the in-plane stiffness properties
and created the high stress concentration. As seen in Figures 1(b) and (0), each layer is
woven into the layers above and below. Counting from the bottom, the ﬁrst ﬁll (green)
yarn is woven into the third ply. Each odd numbered ply follows the same pattern. The
third will be woven into the ﬁfth, and so forth. The warp (white) yarns follow the same
trend. The even numbered plies are woven into each other. The only difference is the
top and bottom layers that are unable to be woven into the layers above or below,

respectively, as there is none.

 

Figure l. Quasi3D Weave (a) Projected View, (b) Fill Side View, and

(c) Warp Side View

In a composite weave, the yarns are woven in a particular pattern. The
anisotropic behavior of these materials means that the properties are changing
periodically with the weave. Therefore, the stress-strain equation may not be used with
the assumption of constant elastic properties. This signiﬁcantly increases the complexity
of the problem. However, the periodicity of the weave allows for an assumption to be
made. Within the weave, there is a repeatable unit called the unit cell. Finding the
average values of the properties for the unit cell and using those values would allow the
equations for constant elastic properties to be used. The averaged values are known as
effective properties. Effective properties provide accurate averages so that classical
methods may be used. Analysis of the unit cell will give a good understanding of the
properties of the composite material upon which it is based. Optimization of cell
structure and hence the composite material may then follow.

There are many methods based on the unit cell that have been used to find the

effective elastic properties of weaves. Raju and Wang [6] derived a method for

determining the mechanical properties of a unit cell based on Classical Laminate Theory.
The constitutive matrix was integrated over the area of the unit cell and averaged. The
constitutive matrix was found by dividing the cell into regions and summing the
components. However, only 2D analysis was performed. Naik [7] divided the unit cell
into many slices. An arbitrary slice was analyzed based on volume fraction, orientation,
and the components of the slice. Formulas for finding the effective properties of a slice
were programmed in the commercial code TEXCAD. Inputting the speciﬁc weave
pattern allowed the effective properties of the unit cell to be calculated. A minimum of
12 slices would cause the value of the properties to converge. Naik and Ganesh [8] used
a small unit cell composed of a warp tow, a ﬁll tow, and resin pockets. The cell was then
idealized as a cell with the constituents composed of ﬂat layers one on top of the other.
Laminate Theory was then applied to ﬁnd the overall properties. Scida et a1. [9] divided
the unit cell into inﬁnitesimal slices. The averaged constitutive matrix for each slice was
calculated using Classical Laminate Theory based on the volume fractions of the
components. The slices were then added up to ﬁnd the overall matrix of the unit cell.
Hut and Kapania [10] divided the unit cell into many slices and integrated the strain
energy density function over the volume of the slices. The constitutive relation was
found from the results and summed over all the slices. Hsieh and Tuan [11] evaluated
particle reinforced composites on the basis of parallel and series alignment of the
particles with the matrix phase. They realized that the alignment at the interfaces of
constituents had a signiﬁcant impact on the properties. This deﬁned an upper and lower

bound for the range of the properties.

Computational methods have also been used to analyze the unit cell. Whitcomb
[12] used ﬁnite element analysis to find the effective properties of the unit cell. The cell
was modeled and had loads applied in the three primary directions. From the computer
generated displacements, strains were determined. The effective properties could then be
found through energy equations. Foye [13] used inhomogeneous elements to analyze
subregions of the unit cell. This required personal calculation of the subregions
properties for each subregion. Loads were then applied and the effective properties were

calculated, as a result of the displacements.

1.2 Statement of Problem

The complexity of composite analysis requires the use of an accurate model for
calculating the properties of various composite patterns. Confidence in the model
requires analytical studies that are comparable to accepted, published results. The 2D
Plain Weave unit cell has been used in the development of most unit cell models. This is
the most fundamental weave and will allow for comparison with other previously
constructed models. The model will need to be adaptable to analyze various weaving
patterns, including the Q3D Weave. Evaluation of property dependence and composite
response on geometric constraints is important for designing superior weaving patterns.
This includes studying weaving angle, undulation size, unit cell size, and other variable
geometric constraints. This will require a computer code that will allow material and
geometric properties to be input for quick calculation of the desired unit cell properties
and trends to be found. The primary goal of this thesis is to (1) create a model and code

for evaluating the properties of composite materials, (2) justify the code with other

published results, and (3) evaluate the properties of the Q3D weave. Complex geometry
requires computational investigation to sort out the effects due to individual parameters.

Throughout the study, perfect bonding between constituents is assumed.

1.3 Organization of Thesis

The approach taken in this paper is analytical. Chapter 1 is the introduction.
Chapter 2 will outline the analytical approach used in the model and the theory behind
the model. Justiﬁcation of the model by comparison with published data of similar
problems is also included. Multiple case studies of composite unit cells are examined in
Chapter 3. This involves property dependence on geometric and material constraints.
Chapter 4 compares the properties of various weaving patterns. Finally, Chapter 5

presents the conclusions of the study and recommendations for future research.

CHAPTER 2 - ANALYTICAL NIETHOD

2.1 Model Derivation
2.1.1 Unit Cell Derivation

The analytical approach used in the model is based on Classical Laminate Theory
[14]. It follows a similar approach to that of Raju and Wang [6]. However, there are
several modiﬁcations made to the model both for simpliﬁcation and greater accuracy.
These include three dimensional analysis, undulation modeling, and subregion integration
(parallel and series alignments). All of these modifications will be explained throughout
the chapter. Three dimensional analysis is used because the weaving patterns create a
three dimensional structure. It is important to know all of the properties of the structure.

One layer of the 2D (two-dimensional) Plain Weave is shown graphically in
Figure 2. This is the most fundamental composite weave. The unit cell of the 2D Plain
Weave is outlined in Figure 2(a). This is the subset of the weave that is repeated
throughout the whole structure. As before, the green and white areas represent the ﬁll
and warp composite regions, respectively. These regions are called yarns and are fiber
bundles that have been mixed with matrix. The gray areas represent pure matrix zones.

Figure 2(b) shows the side view of the weave.

 

 

 

 

 

(b) ' . .
Figure 2. 2D Plain Weave (a) Projected View and (b) Side View

Classical Laminate Theory is based on plies that are stacked on top of each other.
Only thickness is taken into account in the analysis. Therefore, regions must be
identiﬁed that allow this analysis to be conducted. In order to achieve this, the cell is
divided up into the four regions shown in Figure 3(a). The details of the three geometric
dimensions a, an, and h, representing cell length, undulation length, and yarn thickness,
respectively, are shown in Figure 3(b). These values are measured after the structure has
been manufactured. The length, a0, is defined as the difference between the cell length

(a) and the undulation length (an).

 

 

 

. - Mb

I I

l l
I a,/2 au ao/2
I

(b a
Figure 3. 2D Plain Weave Unit Cell (a) Projected View and (b) Side View

 

 

 

 

 

Each of the regions has a ﬁll component (green), a warp component (white),
matrix channels (gray), and a matrix pocket (gray). The regions can then be further

broken down into four subregions as Figure 4 displays.

 

 

Figure 4. Region 1 of 2D Plain Weave Unit Cell

Subregion 1 contains both ﬁll and warp plies laying ﬂat on top of each other.
Subregion 2 contains undulating warp yarns and matrix channels. Similarly, subregion 3
contains undulating fill yarns and matrix channels. Finally, subregion 4 is a matrix
pocket. These regions of pure resin are unavoidable results of the weave that must be
taken into account.

The purpose of the model is to find the effective mechanical properties of the
composite unit cell. This will allow the composite weaves to be analyzed as anisotropic
materials without using position-dependent functions. This is achieved through the
constitutive relation, the relation between forces (N) and bending moments (M) to strain

(a) and curvature (x). The constitutive relation is shown in the following equation.

N A B E

{MHB Dl-{x}
The A, extensional stiffness, matrix is of particular interest in this study as it relates in-
plane loading to in-plane straining. From this equation, the mechanical properties of the
unit cell may be calculated. The D, bending stiffness, matrix relates out-of-plane loading
to out-of-plane curving while the B, coupling stiffness, matrix couples the in-plane and

out—of-plane loadings and deformations together.
For three-dimensional analysis, all matrices are of size 6x6. They represent the
properties of the material in all three directions. Each of the constituents of the unit cell

has matrices. The constituents are the ﬁll and warp yarns, as well as the matrix. The

contributions from each of the subregions must be integrated over the volume of the cell.

All matrices are calculated from the stiffness matrix, C, and geometry. The C

matrix is associated with the material properties, such as Young’s modulus (E), shear

ll

modulus (G), and Poisson’s ratios (v). It also holds the relationship between stress and

strain as shown in Equation (2).
{o}=ICl{e} (2)

The stiffness matrix for orthotropic materials is seen in Equation (3)

 

 

 

 

 

 

 

 

 

 

 

E v,,v .—1 E, vyzvu +11“. Ex mil/n+1!er _
iﬂt.) _,(_ ,)_ <.. )0 O O
E ﬂ E ﬂ
x szVz. +11“, E, .. ,_ _1 x anu +110.
__ ( _ ,) ,(vkvd ) _ ( g ,) O 0 0
[3 ﬂ
C = E,.(v,_yvyz +11”) E, (1!”va +vyz) Ex(vxyvyx —1) 0 O 0 (3)
ﬂ ,3 ﬂ
0 0 0 G),z 0 o
0 0 o 0 0,, 0
0 O 0 0 O G”.
where
,8 = 140(va +Vszai+Vszzy +V.-Z(Vu +v),xv,y)-1 (4)

The properties substituted in Equation (3) are those of a unidirectional
composites, since the warp and fill yarns are prepared with unidirectional yarns.
However, the C matrix for the matrix charmels and pockets are based on pure matrix
properties. The properties of the yarns are determined by the ﬁber properties (Ef), matrix
properties (Em), and ﬁber volume fraction (Vf). Depending on which property is being
determined, one of the following equations is used, where EC represents the property of
the composite. Properties along the ﬁber direction use Equation (5). All other properties

use Equation (6).

EC=Efvf +E,,,(1—v,) (5)

12

 

1..
_1_=_‘:_,l__"_r_l (6)

f Em

The extensional stiffness matrix of each subregion, Amk, is simply the integration
of transformed stiffness matrices through the thickness. The differing material properties
between successive plies causes the integration to take the form of Equation (7), where n

is the number of plies.

Am, = Em, -A2 (7)

j=l
Once the A matrix for each subregion is found, it is integrated over its area. The four
subregions are then summed for the integrated regional matrices. The tilde represents
that the A matrix has been integrated over the region area, RA.

4

Km, = 2 [Am (IA (8)

i=1 RA
The summation integer, i, ranges from 1 to 4 due to the four subregions in each region.
The integration is dependent on the geometric parameters of the unit cell. These include
the undulation length (a), unit cell lenth (a), and yarn thickness (h). The effective
extensional stiffness matrix, Aeﬁ‘, is then the summation of the regional matrices divided

by the cell area, CA, of the unit cell.

4 ~
A...
A = ”=‘ 9
eff CA ( )

 

The summation integer, p, ranges from 1 to 4 due to the four regions in the unit cell. The
A matrix holds the relation between the force and strain for the unit cell. The effective

elastic constants can then be found through the inverse A matrix.

a: A-leff (10)

13

From the inverted A917 matrix, a, the properties are easily calculated. The elastic
constants are calculated as shown.

1
E,=— E,=—1- Ez=—1— (11)
a“! ' an! 6133!

The shear moduli are calculated similarly.

a, =; G =_1_ a =_1_ (12)

xy X: \z

The Poisson’s ratios are calculated as follows:

V :_& V :——— V =—— (13)

x)! xz yz

The Lekhnitskii coefficients, which relate shear stress to normal strain or shear strain to

normal stress, are calculated using Equation (14).
”my: = 041011 "max: = 6151011 "waxy = “start
77yy._vz = “42022 ”nun = 052022 nyy,xy = “626122 (14)
"wyz : “43033 ”12.x: = 053a” "awry = “63033

Chentsov’s coefﬁcients, which relate shear stress in one plane to shear strain in another

plane, are determined from Equation (15).
:uyz.xz = 054044 ﬂyzxy = a64a44 lunacy = 0656155 (15)
Now that the overall procedure has been layed out, there are several additional

calculations that need to be made. One possibility in the weave is that the ﬁll and warp

yarns are not orthogonal to each other. There is a weave angle, (,0, which is the angle

14

between the yarns. This is displayed in Figure 5, below. The lines represent the direction

of the fibers in the yarns.

 

 

 

 

Figure 5. Weaving Angle for 2D Plain Weave

This weave angle must be accounted for in the determination of the transformed, C’,
matrix for each subregion. If there is indeed a non-orthogonal weave angle, the ﬁbers in
the warp yarns are now aligned in a different direction and the tensorial transformation
must be used to find the transformed matrix, C’, for the new direction. This is shown in
Equation (16).

C’ = TCTT (16)
T is the tensorial transformation matrix and is composed of directional cosines. The
directional cosines are the cosines of the angles between the original coordinate system
and the new coordinate system that is aligned with the fibers in the warp yarns. The

directional cosine matrix is shown below.

cos(¢) cos(¢ + 90°) cos(90°)
l = cos(90° — ¢) cos(¢) cos(90°) (17)
cos(90° ) cos(90° ) cos(0° )

The transformation matrix is then composed from the directional cosines as shown in

Equation (1 8).

1121 1221 [321 2121131 2131111 2121111

[122 [222 [322 2122132 2112132 2112122

[123 [223 [323 2123133 2113 133 2113123
[12113 [22123 132133 122133 + [32123 112133 +132113 l12123 +122113
[“113 [21123 131133 121133 + [31123 111133 +1311” [11123 +121113
[11112 121122 131132 12l132 +131122 111132 +131112 l11122 +121112

(18)

The transformed C matrix, C’, is then used in place of the original C matrix for the warp
yarn portions.
The same type of transformation is used for the undulation portion of the yarns.

The undulation angle, 6, is displayed in Figure 6.

 

. , . _ , 1211*
l a./2 ’ a. ' a./2*

l a "
Figure 6. Undulation Angle of 2D Plain Weave

 

 

 

 

 

 

The curved nature of the yarn undulation causes the angle, 6, to vary with position. The
height of the undulation of a ﬁll yarn in the x direction is given by a piecewise function.

The following sinusoidal function is just an approximation of the height.

0 —a/2SxS—au/2
h(x)= [1+sinﬂj-g -a,/2Sx5a,/2 (19)
all
h au/ZSxSa/Z

The undulation angle is found by taking the derivative of Equation (19)

 

 

)= dh(x) = ”'h cosﬂ (20)

tan(19
dx 2 - a" a

ll

As the curved yarn is a function of position, the integration of the regions over the cell
area must be done numerically. The directional cosines to be used in the transformation

matrix for the undulating regions are shown below.

cos(l9) cos(90°) cos(6 — 90° )
I = cos(90°) cos(0°) cos(90°) (21)
cos(6 + 90° ) cos(90°) cos(6)

These directional cosines are then used in the transformation matrix for the undulating

subregions.

2.1.2 Undulation Model

The complexity of the sinusoidal undulating region motivated potential
simpliﬁcation of the model. Several other models, including linear and vertical
undulations, along with horizontal and mosaic simpliﬁcation models, were compared to
the sinusoidal undulation model. In order to achieve this, numerical values are useful.
The following tables provide the material and geometric properties used to calculate the
properties. These particular values [7] were chosen as they were commonly used in unit
cell models and will allow for comparison with other published results. These properties

will be used in all the calculations unless otherwise noted.

17

Table 1. Composite Yarn Properties

 

 

 

 

 

 

 

 

 

 

En (GPa) 144.8
E22 (GPa) 11.728
E33 (GPa) 11.728
G12 (GPa) 5.516
G13 (GPa) 5.516
623 (GPa) 5.516
V12 0.23
V13 0.23
V23 0.3
Vf 0.64

 

 

 

 

Table 2. Matrix Material Properties

 

 

 

| EM(GPa) 3.448 |
[ GM(GPa) 1.276 ]
vM 0.35 |

 

 

Table 3. Unit Cell Geometric Properties

 

 

 

I a(mm) 10 I
I au(mm) 1.467 |
| h(mm) 0.989 |

 

 

 

 

 

 

.I—JF—i. W 42:3
(d)if?i (”r

Figure 7. Side Views of (a) Sinusoidal Undulation, (b) Linear Undulation, (c) Vertical
Undulation, ((1) Horizontal Model, and (e) Mosaic Model

 

 

 

 

The graphical versions of the ﬁve models are presented in Figure 7. The models
(b)-(e) would all greatly simplify the calculation in the undulation region. The sinusoidal
model is traditionally believed to be most like the actual, physical unit cell. Obviously,

the horizontal model and the mosaic model could not physically occur. Only their use as

18

a mathematical model was analyzed. The thickness of the yarn in the vertical model is
kept constant at h.
For the linear model, the undulation angle remains constant through the

undulation. It is therefore determined from the following equation.

9 = tan "[1] (22)

a

ll

The equation is derived from inspection of Figure 8.

 

 

 

 

 

 

 

Figure 8. Undulation Angle of Linear Model
The vertical, horizontal, and mosaic models were calculated the same as the subregions
with stacked yarns and Classical Laminate Theory was used for the calculation.
The results of the different undulation models are shown in Table 4. Due to the
balance of the unit cell in the x and y directions, properties such as EJ, and Ey are
equivalent.

Table 4. Comparison of Undulation Models for 2D Plain Weave

 

 

 

 

 

 

 

 

 

 

 

 

 

Sinusoidal Iﬂear Vertical Horizontal Mosaic
Ex (GPa) 62.23 61.41 66.46 67.49 78.81
E,(GPa) 62.23 61.41 66.46 67.49 78.81
EZ(GPa) 12.83 12.25 35.11 11.31 12.62
ny (GPa) 4.894 4.89 5.49 4.89 5.52
Ga (GPa) 6.204 6.44 5.49 4.89 5.52
Gyz (GPa) 6.204 6.44 5.49 4.89 5.52
vxy 0.0313 0.0284 0.0524 0.0362 0.0345
vxz 0.3321 0.3613 0.1052 0.2942 0.2925
vyZ 0.3321 0.3613 0.1052 0.2942 0.2925
Angle (0) 19.31 33.99 90 O 0

 

 

 

 

 

 

19

As seen in Table 4, the values of the sinusoidal and linear models are very similar. The
vertical model has greater in-plane properties and much greater out-of—plane properties.
This is due to the yarns aligned vertically. The horizontal model has increased in-plane
properties and decreased out—of-plane properties, compared to the sinusoidal model. The
mosaic model has significantly larger in-plane properties as there are no pure matrix
zones and the yarns are ideally aligned hrizontally. The major reasons for the differences
in the models can be concluded to be the pure matrix content and the undulation angle.
Based on these differences, the linear model was chosen as an acceptable model to be
incorporated into the unit cell model. This greatly simpliﬁed the calculations.

The incorporation of the linear model means that the A matrices of all the
subregions remain constant for each constituent and no longer vary with position.
Therefore, the integration becomes algebra and simpliﬁes the equations. The subregion

calculations reduce significantly and are shown in the following equations.

Region 1 - Subregion l

 

 

Am,l = (5,, )0. -h + (5",, )¢. -h (23)

Region 1 - Subregion 2
Amkz = (Emit )¢°,undulared . h + (Emir )Mam'x . h (24)

Region 1 — Subregion 3
A""‘3 : (Emk )0'.undulared ' h + (Em): )Mam'x ' h (25)

20

Region 1 — Subregion 4

Amk4 = (attic )Marri.r . 2h (26)

These four subregions are summed to give the integrated A matrix of Region 1. The sine
term from the integration is a result of the change in volume associated with the weaving

angle.

Region 1

 

 

2 2
Km, = Am,l (go—simm] + 4.5“?" 5mm) + Am3 (“in sing») + Am [07.9mm] (27)

The other three regions have identical constituents that are arranged in different locations.

Their calculation is carried out in the same manner.

2.1.3 Subregion Integration

Thus far, in this thesis and in published models, during the summation of the
regions and subregions, the constituents have been assumed to be in a parallel or series
(Parallel/Series) arrangement. This is shown in Figure 9. That is true for all the ﬂat
regions, but the undulation portions are a special case. They are neither arranged in
parallel or series. For small undulation angles an assumption could be made that the
series or parallel arrangements would be a good approximation. For relatively large
undulation angles, this requires separate analysis involving both series and parallel

arrangements.

 

 

‘u-r —..— ,.» . T T . , 1*

Figure 9. Subregion Alignment of (a) Parallel and (b) Series

 

 

 

 

 

 

 

 

 

 

 

 

21

The undulation region was divided into many slices allowing both series and
parallel (Series-Parallel) arguments to be used. By analyzing the slices in series first, the
addition could be carried out as before. Therefore, the slices would be either arranged
vertically or horizontally depending on the direction of summation. Figure 10 shows
both of these alignments. The red portions represent the composite portions of the slices
while the gray portions are pure matrix. As the number of slices increases, the slices
cause the regions to become more like the linear undulation and greater accuracy of the
model is achieved. As the number of slices increases, the geometry will tend to match
that of the linear model. Enough slices were used so that the numbers would converge on
a value. The horizontal alignment required approximately one-thousand slices for the

number to converge, as seen in Table 5.

 

 

 

 

 

 

 

 

 

(a) (b) ,
Figure 10. Series-Parallel Undulation Alignments with (a) Horizontal Slices and (b)
Vertical Slices

 

The slices allow the undulation regions to have the same alignment as the ﬂat
regions. However, the size of the slices must be determined. For the horizontal slices,
the equation of the linear lines that bound them was used, as each slice does not have the
same length. From these equations, the length of the yarn (red) portion of each slice can
be found. For the vertical case, each of the slices has the same dimensions. Simple
trigonometry was used to determine their heights. Finding the A matrix of the slices

permits them to be treated as subregions with constant properties. They can then be

22

integrated through the thickness and then over the area, as done with the other
subregions. This allows the previous summation to be conducted.

Horizontal slices are used for finding in-plane properties. The vertical slices are
used for ﬁnding out-of—plane properties. When summing in the x direction, the warp
yarns are already aligned in parallel and do not need to be divided into slices. The same
holds true for the fill yarns when summing in the y direction.

The division of the undulation region into slices was incorporated into the original
model and compared using the same properties as before. The comparison of the two
undulation models is shown below in Table 5, based on one-thousand slices.

Table 5. Comparison of the Subregion Alignments

 

 

 

 

 

 

 

 

 

 

 

 

 

Parallel/Series Series-Parallel
Alignment Aligment

Ex (GPa) 61.41 62.77
Ey(GPa) 61 .41 62.77
EZ(GPa) 12.25 12.33
ny (GPa) 4.89 4.89
On (GPa) 6.44 6.45
Gyz (GPa) 6.44 6.45

vxy 0.0284 0.0279

v),Z 0.3613 0.3595

vyz 0.3613 0.3595

 

 

The change in the model is found to have an impact on the Young’s moduli and
the Poisson’s ratios. The shear moduli remain largely unaffected by the modiﬁcation to
the model. Depending on the geometric properties, the series-parallel arrangement can
cause the properties to either increase or decrease. The differences in the effective
properties were found to be fairly small. This study helps to justify the use of the
Parallel/Series alignment for small undulation angles. For this study, the undulation

angle was 34°.

23

2.1.4 Computer Code

A computer code based on the commercial package Mathematica was written to
complete all the calculations. The input to the code is the material properties of the yarn
ﬁbers and matrix as well as the geometric properties of the unit cell. The output is the
mechanical properties of the corresponding unit cell. This allows the properties to be
quickly found for varying inputs. Also, it allowed trends to be found by varying any of
the inputs and plotting the changes of the output. The ﬂow chart that covers the process
of calculation in the code is shown in Figure 11. The computer codes for all weaving
patterns analyzed are shown in Appendix B.

The geometrical models were generated using Unigraphics. This allowed
visualization of the unit cells and corresponding dimensions. A clear visual comparison

of the weaving patterns could also be achieved.

24

 

 

I Material and Geometry Input

 

 

 

Transformation of Warp C Matrix (Weaving Angle)

 

 

 

 

 

I Integration of Flat Zones Areas over Volume

 

 

 

I Integration of Matrix Pocket over Volume

 

 

 

Transformation of Fill and Warp C Matrices (Undulation Angle)

 

 

 

 

I Integration of Fill Undulation Zones over Volume

 

 

 

I Integration of Warp Undulation Zones over Volume

 

 

 

I Integration of Matrix Channel Zones over Volume

 

 

 

I Summation and Averaging of all Regions (A Matrix) I

 

 

Inversion of A Matrix and Calculation of Cell Properties

 

 

 

Figure 11. Flow Chart for Unit Cell Computer Code

25‘

2.2 Comparison with Published Articles

Upon completion of the model, it is necessary to determine the validity by
comparing with other results. The model was compared with four other models that
analyzed 2D Plain Weave unit cells that have been published in journals. For this
purpose, the model was derived using the Plain Weave.

The current model was found to be in very good agreement with the other
published models. The comparison is shown in Table 6. This comparison gives

conﬁdence into the validity and accuracy of the current model.

Table 6. Comparison with Published Articles

 

 

 

 

 

 

 

 

 

 

 

Current Model Naik [7] l-lur et al [10] Foye [13] Raju [6]
Ex (G Pa) 62.77 64.38 64.41 63.78 60.61
E,(GPa) 62.77 64.38 64.41 63.78 60.61
Ez (G Pa) 12.33 1 1.49 11.73 - -
6., (G Pa) 4.89 4.87 5.05 4.827 4.894
G,z (GPa) 6.45 5.64 5.77 - -
Gyz (GPa) 6.45 5.64 5.77 - -
v,y 0.0279 0.027 0.0312 0.03 0.041
v,z 0.3595 0.396 0.4033 - -
vyz 0.3595 0.396 0.4033 - -

 

 

 

 

 

 

 

Table 7 displays the differences in using a 2D and 3D analysis in the current
model. Besides the fact that none of the out-of-plane properties are known, most of the
in-plane properties increase. This is a result of inverting a 3x3 matrix as opposed to the
6x6 matrix in the calculation of the constants. The difference in the properties is not very

signiﬁcant, but depending on the material and geometric constraints the difference could

increase.

26

Table 7. Comparison of 3D and 2D Models

 

 

 

 

 

 

 

 

 

 

 

3D Analysis 2D Analysis
Ex (GPa) 62.77 64.1 1
FW(GPa) 62.77 64.11
Ez (GPa) 12.33 —
ny (GPa) 4.89 4.89
ze (GPa) 6.45 -
Gyz (GPa) 6.45 -
vKy 0.0279 0.0549
sz 0.3595 -
vyz 0.3595 -

 

 

 

 

Completion of the analytical model will now allow an in-depth study into the
material and geometric parameters of the unit cell and how they affect the properties.

Other weaving patters, such as the Q3D Weave, may also be analyzed.

2.3 Summary

A model based on Classical Laminate Theory was derived to calculate the
effective properties of composite unit cells. The model used three dimensional analysis.
It was found that the undulation regions could accurately be modeled as linear to reduce
the complexity of the calculations. The subregion integration was found to affect the in-
plane Young’s moduli of the cell. The undulation regions were forced to have a series
and parallel alignment by the addition of multiple slices acting as subregions. This would
give higher accuracy in the model. The values from the example unit cell were compared
with published articles and found to have good consistency. The next step involves

multiple case studies on the unit cell parameters.

27

CHAPTER 3 - STUDY OF MICROSTUCTURE GEOMETRY

There are many variations of composite weaving that can be used. The fibers
may be placed in different orientations, constrained by different geometries, resulting in
many different material properties. In this chapter, several case studies of the 2D Plain
Weave pattern will be analyzed. This will involve different material and geometric
variations that can be placed on the weave. The geometric constraints will be changed
over a domain to see the effect on the overall properties and to ﬁnd trends. The Young’s
and shear moduli will be displayed on graphs for each case study. The Poisson’s ratios of
these trends will be shown in Appendix A. In all of the cases, the geometric and material
properties of Chapter 2 are used, unless otherwise noted. Also, for the case studies
involving the changing of a parameter, only that parameter is varied and the others are
considered constant. The unit cell parameters may be divided into those that deal with

the yarns, the cell, and layers.

3.1 Yarn Parameters
3.1.1 Fiber Volume Fraction

One very important parameter is the fiber volume fraction of the cell. This is the
percentage of the volume that is occupied by the fibers. This parameter is affected by the
manufacturing of the composite and will affect the properties of the composite. The ﬁbers
give the composite its stiffness and strength, so it is expected that an increase in the ﬁber
volume fraction would also increase the effective cell properties. By increasing the

volume fraction, the individual ﬁll and warp yarn properties increase. As a result, this

28

was found to have a signiﬁcant effect on the cell properties. The composite moduli for

this case study have the following dependency, where EC represents the cell properties.

EC =f(a(A.,. (4(Alv,,E..........))))) (28)

As seen in Figure 12, the cell moduli increase in a linear fashion with the volume
fraction. The Young’s modulus in the z direction is not as dependent on the ﬁber volume
fraction as the other moduli. The domain of the ﬁber volume fraction in the graph covers
most practical applications. The in-plane Young’s moduli and the shear moduli have a

difference of 50% over the domain. The Young’s modulus is the z-direction varies 33%.

 

 

 

 

80 1 1
-I- Ex and Ey
70 ‘— + £2 "A

 

 

 

 

—e— ze and Gyz B/A
— . 4

/' B/zr’

0'1

0
S
I

 

 

 

 

 

Young's Moduli (GPa)
Shear Moduli (0.1 GPa)
8 8

 

N
O
l

 

 

 

IF

 

 

 

 

 

 

 

 

 

 

 

O

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75
Fiber Volume Fraction

.0
co

 

 

Figure 12. Effective Moduli for Varying Yarn Fiber Volume Fraction
One form of composite manufacturing is the use of prepreg tape. These are sheets
of ﬁbers that have matrix already in the sheet. The sheets are cut and woven or stacked

to create the desired microstructure. Then the woven or stacked material must be cured

29

 

and shaped. During the curing process, the matrix may become liquid and leave the
mold, resulting in the decrease of the yarns. This will increase the ﬁber volume fraction
of the yarns. Based on the original ﬁber volume fraction and the thickness of the ﬁnal
composite, the fiber volume fraction as a function of the yarn thickness may be found.
This is displayed in Figure 13. The volume fraction increases significantly as the
thickness of the yarns are decreased. This graph is based on an initial fiber volume

fraction of 0.64.

 

0.8

 

0.7 ~ ~—

 

0.6

 

 

 

0.5

0.4 "7 \‘I—K ‘ m

0.3 XI, A

 

 

 

Yarn Fiber Volume Fraction

 

 

 

 

 

 

 

 

 

0.2
0.75 1 1.25 1.5 1.75 2 2.25

Yarn Thickness (mm)

Figure 13. Yarn Fiber Volume Fraction for Varying Yarn Thickness

3.1.2 Yarn Thickness
Besides ﬁber volume fraction, there are many other geometric parameters that can
affect the unit cell properties. These include undulation length (au), non-undulated length

(a0), cell size (a), yarn thickness (h), weaving angle ((p), matrix layers, and number of

30

 

layers. Figure 14 displays four of these parameters. Variation of these parameters will

be seen in the following case studies.

 

e
\

 

 

 

a,/2 I a. a./2
a
Figure 14. 2D Plain Weave Side View

 

 

 

 

 

 

The thickness of the yarns is one geometric parameter that affects the cell
properties. For isotropic materials, the thickness will have no affect on the properties.
However, increasing the height of the cell will increase the undulation angle if the
undulation length, au, is ﬁxed. This case study assumes that the volume fraction of the
yarns is held constant for the changing thickness. Therefore, the following parameters

affect this case study.
EC = f(a (71,1, (.11 (au,a,h))» (29)
Figure 15 shows the effect of the yarn thickness on the cell moduli. For ease of

visualization the height ratio, h/a, is introduced. The domain was chosen so that

convergence of the properties was achieved.

31

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

70
9 a:
6, E; __ ________ __
o - ‘
32’ 50 T—Z’ETA T713 #1:}— ‘ZS‘ ' 27‘ if: ‘1: T 2.3 ”L: a
0- <5
‘3 v-_ —I— Ex and Ey
= e, 40 we
3 = +Ez
g g —A—ny
2 E 30 ‘7‘ —e—ze and Gyz
‘3' a
- a
“‘ 6‘5 20 —
10 — — — —
0
0 0.2 0.4 0.6 0.8 1
Height Ratio (hla)

 

 

Figure 15. Effective Moduli for Varying Yarn Thickness

The in-plane Young’s moduli decreases slightly with increasing height ratio and
then converges. Only for very small height ratios does a change in the ratio affect those
properties. The Young’s modulus in the z-direction is signiﬁcantly affected. This is
because the undulation length remains the same and thickness is increased, resulting in a
large undulation angle. Therefore, the volume of the ﬁbers aligning in the vertical
direction increases as the undulation angle is increased. Ez increases over 100% over the
domain. The out-of-plane shear moduli decrease with the height ratio and then converge
toward the same value as the in-plane shear moduli, which are unaffected. The out-of

plane shear moduli decrease 30% over the domain.

32

3.2 Cell Parameters
3.2.1 Undulation Length

A major geometric variation is the undulation distance, a... This is the horizontal
distance of the fiber undulation due to the weaving pattern. It is also the undulation
distance of weaving between adjacent ﬁbers. If this distance is varied, it will cause the
undulation angle to change. It will also change the volume of the matrix channels and
pocket that ﬁll the voids left by the fibers. These two changes have a dramatic effect on
the properties of the unit cell.

In order to view the effect of the undulation distance on the unit cell, the
undulation ratio, au/a, is deﬁned. This is the fraction of the unit cell length that is

undulated. The case study dependency is shown in Equation (30).

EC =f(a(A.. (46.4.12)») (30>
Figure 16 shows the cell moduli plotted versus the change in the undulation ratio. The
domain varies from an = h (au/a ’2 0.05) to a“. = a, which are the lower and upper bounds
of the unit cell, respectively. The lower bound is such as the length of the undulation
may not be any shorter than the thickness of the yarns. The upper bound is such as the

length of the non-undulation may vanish. Linear undulation was used as described in

Chapter 2.

33

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

80 250
-l— Ex and Ey
70 l +Ez
e “ ,_ 200
60 _ “A; +ny f)
.3. E g \g —e— ze and Gyz
{'5 (D A A ,. \a —B—Undulation Angle A
v '— 3 50 7 g "A “ 1 N
g :5 A ‘ -e—Matrix Volume / ~ 50 E
: 8 l‘ A ‘ \n‘ g
g '5 u 40 T ‘ , ° 1»
2 'O V A , § ’1 E
a g g, X A _ e‘ . =
'01 r: ‘ \a A -- 100 3
g g < 30 K I, v ~2\.‘ >
O ’ A\3
> w ’ ‘13“
20 ,4 ‘ “
“5:" i“ Q ‘r 50
1O ’1 h = u - _ '3‘
M I. = = = = =:‘-“ «:I
0 4 4 4 4 4 4 ‘4 4 4 4 4 4 4 4 . 4 4 4 : 0
0.05 0.2 0.35 0.5 0.65 0.8 0.95
Undulation Ratio (aJa)

 

 

 

Figure 16. Effective Moduli for Varying Undulation Length
The graph shows the Young’s moduli in the in-plane directions to be the most

sensitive to the changes. There is a decrease of over 90% over the domain of that

property. As the undulation length gets longer, there are fewer ﬁbers aligned completely
in the x—y plane. Even though the undulation angle is increased as the undulation length
decreases, the in-plane properties will increase. The decrease in size of matrix regions is

the major reason for the increase in moduli. The Young’s modulus in the through-
thickness direction is not nearly as sensitive to the change. It increases about 20% and
then decreases after a particular ratio value. There is an optimal point of the ﬁber
alignment and volume of the matrix regions for this property.

The shear moduli tend to decrease with increasing undulation length, as well. The

in-plane shear modulus decreases linearly at a much smaller rate than the in-plane

Young’s moduli. The out-of-plane shear moduli increase initially and then decrease at a

34

larger rate than the in-plane moduli. There appears to be an optimal point for the out-of-
plane shear moduli that balances the undulation length and angle. All the shear moduli

decrease approximately 75% over the domain.

3.2.2 Weaving Angle

The geometric change that affects the largest number of cell properties is the
weaving angle. Figure 17 displays the weave angle between the ﬁll and warp yarns.
Changing the direction of the warp (white) yarns signiﬁcantly affects almost all of the

cell properties.

 

 

 

Figure 17. Top View of the Weaving Angle

The use of small differences of weaving angles between successive layers, such as 15°,
has been shown to improve delamination resistance [3]. It is therefore important to know
how this parameter affects the cell properties. The parameters that effect this case study

are shown in Equation (31).
EC =f(a(A.. (71(A(Ef..,. (¢)))))) (31)
The trends associated with the elastic and shear moduli are displayed in Figure 18. The

graph domain varies from 0° to 90°, as that is the orthogonal position of the ﬁbers. Due

to the lack of symmetry for this parameter, none of the properties will be identical.

35

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

140 1
-l— Ex
120 + Ey
+ Ez
/ -e— ny
A a
E {‘3 100 -El— ze
g .-_ —e— Gyz
E E, 80 ‘* g “‘7
'5
E, E 60
E g y—e——< >
11.1 m 40 __
20
0 . . .
0 10 20 30 40 50 60 70 80 90
Weaving Angie (4))

 

 

 

Figure 18. Effective Moduli for Varying Weaving Angle
As the angle is decreased from the orthogonal alignment (90°), EJr stays fairly steady until
45° and then increases rapidly, almost 100%, with the decreasing angle. Ey decreases
rapidly, 75%, and then after 45° stays constant. It approaches the value of E, which
changes very little with the angle.

The in-plane shear modulus has a parabolic shape. It increases 140% as the
weaving angle decreases. The maximum value is between 35° and 40°. This is due to
the lack of perfect symmetry between the warp and fill ﬁbers. Otherwise the peak value
would be at 45°. Both of the out-of-plane shear moduli decrease about 25% as the angle
is decreased. They take somewhat different paths, however, with the changing values.
All three of the shear moduli converge upon the same value when the weave angle

approaches 0°.

36

3.2.3 Cell Size

Another geometric change is the size of the cell. In this study, a square area, a* x
a*, was deﬁned and ﬁlled by a varying number of unit cells, n x n. For example, the area
could be filled by one square cell at length a*, four cells at length a*/2, 9 cells at length
a*/3, and so forth. This is shown in Figure 19. The height of the cells was kept constant
so that only the effect of the cell length was analyzed. The domain is defined in terms of

the ratio a/a *.

 

 

 

 

Figure 19. Top View of Unit Cell Size

As the number of cells increased, all the dimensions of the cell except the thickness were
scaled down. The lower bound of the domain is when the undulation length, a“, is equal
to the yarn thickness, h. This occurred at 5 cells per length. Therefore, the undulation
angle would increase slightly with the number of cells. Equation (32) shows the affected

parameters.
EC = f(a (qur (A (a,a*,au JD)» (32)
Figure 20 shows the changes of the cell moduli as the number of cells in the speciﬁed

area is changed.

37

 

0'!
O

-I- Ex and Ey
+ E2

.5
O

—A— ny
—9— 6x2 and Gyz

Elastic Moduli (GPa)
Shear Moduli (0.1 GPa)
0)
O

N
O

 

0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
Unit Size Ratio (ala*)

 

 

 

Figure 20. Effective Moduli for Varying Cell Size

It is seen in graph that only the out—of-plane shear moduli are significantly
affected by the change in cell size. They increase 30% over the domain as the cell size is
reduced. This is believed to be a result of the shortened distance between yarns aligned
in the vertical direction (undulation regions). The other moduli change very little or not
at all. The in-plane Young’s moduli decrease only slightly as the cell size is reduced.
There is much less change in comparison to the undulation length case study as the
change in the undulation ratio, a./a, is much less. This is due to the increase of the

undulation angle as all the dimensions, except height, as scaled accordingly.

38

3.2.4 Undulation Angle Effects

From many of the previous case studies, it has been observed that several of the

parameters indirectly affect the cell properties by changing the undulation angle. Table 8

shows the parameters that affect the undulation angle and their bounds. The angle 5.65°

refers to a unit cell composed entirely of undulation.

Table 8. Undulation Angle Parameters

 

 

 

 

 

 

 

 

 

 

Lower Bound Upper Bound
Parameter Bound Angle Bound Angle
au h 90° a 5.65°

h Fiber Diameter 0° Infinity 90°
a* 3h 90° a 565°

 

 

It was determined that a detailed study was needed covering the effects of the

undulation angle on all important parameters. Through this study, the thickness of the

yarns, h, is assumed to be constant except in the undulation area. Therefore, any changes

in the undulation angle come from a change in the undulation length. This will change

the volume of the pure matrix regions in the unit cell. Figure 21 shows the parameters

involved. The distance d is thickness of the yarns in the undulation area, which decreases

as the undulation angle increases.

 

 

Tzh

 

 

 

a,/2 '

au

a,/2

 

 

a

 

 

Figure 21. Parameters Affected by the Undulation Angle

The overall ﬁber volume fraction of the cell is assumed to stay constant.

Therefore, the volume of ﬁber and matrix in the cell does not change.

Only the

positioning of the constituents is varied. Therefore, the average yarn fiber volume

39

fraction will change. All of the geometric changes as a result of the undulation are shown

in Figure 22.

 

 

1.2

\i
O

  

  

 

 

 

 

 

 

 

 

 

-I—vt - total " 60
c 4}: -yams

+
g 0.3 E .. ‘ +d —— 50 A

A ‘ +au A"

E E 3 - ~‘ ‘ﬁ—ao E E
o .5, ‘ = .. ,s -e—matn'x volume -- 40 v V
E *‘ 06 " -— =‘ :5 — _ J: 0
2 '5. “--'-' —-—"_—_ a E
g 7’ — ~‘ 30 c 2
g I \\ 3 g
E r 20

 

0.2

o
.h
A. /”

 

 

 

 

 

Undulation Angle (degrees)

 

 

 

Figure 22. Geometric Changes for Varying Undulation Angles

The minimum angle of the unit cell for this geometry is approximately 15°. This
is because angles lower than this value require the yarn ﬁber volume fraction to exceed
the maximum geometric allowance of 90.7% for the ﬁbers packed as close as possible.
From the graph, it is seen that the overall ﬁber volume fraction of the unit cell is kept
constant. The yarn ﬁber volume fraction decreases as the undulation angle is increased.
This is a direct result of the volume of the pure matrix regions. With a low undulation
angle, the regions of pure matrix have a large volume. However, they are decreased as
the undulation angle increases. The matrix is forced into the yarns, thus reducing the

yarn ﬁber volume fraction. As expected, when the matrix volume is zero at 90°, the

40

overall and yarn fiber volume fractions are equal. Also shown in the graph are the
corresponding lengths, au and do. All of these parameters are shown in the case study

parameter equation.

EC = f(a (Aeﬁ‘ (IZ (au’a’h’A(vf »))) (33)
For each undulation angle, the corresponding geometry from Figure 22 was

programmed into the computer code. This allowed the determination of the effective cell

moduli. The trends are displayed in Figure 23.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

80 1
70 - it -~ 0.9
. E 60 - c
‘5 o
a. ._
9‘3 50 ~ 3
§ 3 I;
g? 40 —l—Exand Ey 0.5 g
p g +Ez 0 4 E
g5 30 151—ny ' e
g s -e—ze and Gyz __ 0.3 g
m u.
20 —E}— vf yams
W412
10 1 __ 01
0 t t : i : i : i t t *r i : i : t : i : i : i t i : t : i : 4 “r 0
15 30 45 60 75 90
Undulation Angle (degrees)

 

 

 

Figure 23. Effective Cell Moduli for Varying Undulation Angles
As seen in the graph, the in-plane Young’s moduli increase with the undulation
angle. However, they do not increase nearly as rapidly as in the previous undulation
length case study. The decrease in the yarn ﬁber volume fraction causes the properties to

not be as sensitive. The out-of-plane Young’s modulus increases slightly to an optimal

41

point and then comes back toward the initial value. The change is not very large. The
geometric changes work against each other to prevent any significant change. The in-
plane shear moduli are unaffected by any of the changes that come from the increase in
the undulation angle. The out-of—plane shear moduli increase initially to an optimal point
and the decrease toward the in—plane value.

However, it is seen that the Young’s moduli and out-of—plane shear moduli can be
increased 10% or greater by simply varying the orientation of the constituents while
keeping their volume constant. This study also showed that the parameters are all linked

and that the changing of one parameter affects many other parameters.

3.3 Layer Parameters
3.3.1 Matrix Layer Thickness

During manufacture of composites, it is possible that a matrix layer would be
present in between the layers of ﬁber. This is shown graphically in Figure 24. The

thickness of the matrix layer is represented by t,,..

 

 

t...
h

Figure 24. Side View of Matrix Layers in 2D Plain Weave Structure

 

 

 

 

 

The introduction of matrix layers would have a tendency to lower the effective properties.
This is the case as it would decrease the overall ﬁber volume fraction of the composite.

The case study is a function of the parameters shown in Equation (34).

42

EC = f(a (Ad, (Z (A (tm,h,n))») (34)
Figure 25 shows the effects of the resin layer on the cell properties. The layer thickness
is presented as a ratio of the ply height, tm/h. The graph was generated using a three layer
composite with, therefore, two layers of matrix as seen in the above picture. The domain

of the matrix layer varies from no matrix layer to half the thickness of the yarns.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

70 0.56
A 50 \S\ 0.52
(.3 9, g: ’3 ’3 “Emir—~25 3
-5 r540 -I— Ex and Ey \S\ 0.5 g
g ‘3 + Ez \S\\B\ ""
.3 E30 1:- ny ~ — 0.48 §
g g —e— ze and Gyz \El :3
o -
>_ (020 —8— vi total 0.45
H+++++++++a
10 0.44
0 t : i .L J. t : t : t 41 0.42
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Matrlx Layer Ratio (tn/h)

 

 

 

Figure 25. Effective Moduli for Varying Matrix Layer Thickness
As shown in the graph, all of the properties decrease linearly by 15% over the domain.
The same trend holds true for the shear moduli. This is due to the decrease in overall

ﬁber volume fraction of the cell, as seen in the graph.

43

3.3.2 Number of Layers (2D)

In most composite structures, multiple layers will be used. It is important to know
how this will affect the properties. The addition of a matrix layer would cause the
properties to vary as the number of layers, n, changes. Figure 26 displays the unit cells

with matrix layers.

 

 

 

Figure 26. Projected View of Multiple 2D Unit Cells with Matrix Layers
The case study was performed with a matrix layer ratio, tn/h, of 0.1. The case study is
dependent on the following parameters.
E. = MA... (MA an») (as)
Figure 27 shows the trends in the moduli as the number of layers is increased. The solid
lines indicate the presence of the matrix layers. The dashed lines indicate the properties
without the presence of any matrix layers. These dashed lines are horizontal and do not

vary.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3‘75
{‘50
§é40 +ExandEy
g3 +Ez
so; 30 +ny
a
g‘af —e—zeandez
£5 20 ---'ExandEy
............ +.....----Ez
1o ""ny
---'zeandez
0
1 2 3 4 5 6 7 8

Number of Layers
Figure 27 . Effective Moduli for Varying Number of Layers

 

 

All of the moduli decrease slightly, 4%, and then converge upon a value after
approximately 5 layers. The dashed lines show that the decrease in the moduli is not very
signiﬁcant as the layer number is increased. Also, the properties do not change with the
number of layers without matrix layers present. It is therefore concluded that the 2D

Plain Weave has very similar properties regardless of the number of layers.

3.3.3 Number of Layers (Q3D)

The number of layers, n, case study was also performed on the Q3D Weave. The
details of analyzing the Q3D weave will be explained in detail in the next chapter. Figure
28 shows the Q3D Weave with 5 layers. With the Q3D weave, each layer is woven into

layers above and below.

45

 

 

 

 

Figure 28. Projected View of Q3D Unit Cell with 5 Layers

The lack of symmetry in the z direction will cause the properties to change as the number

of layers is changed. Equation (36) shows the varied parameters.

EC = f(a (Ad, (2 (A (n,h))))) (36)

Figure 29 shows the changes in the cell moduli for several different layer numbers. The

minimum number of layers for the Q3D Weave is three.

 

70

60

8 8

Young's Moduli (GPa)
0)
O

Shear Moduli (0.1 GPa)
N
O

A
o

 

 

3 4 5 6 7 8

—I—Exand Ey
+Ez

—A—- ny
-e— ze and

 

Number of Layers

 

Figure 29. Effective Moduli for Varying Number of Q3D Layers

46

As seen in the graph, the properties decrease slightly and then converge. The

properties only change about 1% as the number of layers is changed.

3.4 Summary

From the case studies conducted, it was found which unit cell parameters have the
largest effect on the cell properties. It was determined that the ﬁber volume fraction of
the yarns is very important. Small changes in this parameter have a strong effect on the
cell properties. It is also discovered that the location of the matrix in the cell can cause
deviation in the cell properties.

Many of the geometric parameters were found to affect the cell properties by
altering the undulation angle. The undulation regions are the most important geometric
parameters that change the cell properties. Variations in this area control the amount of
pure matrix regions and the orientations of many of the ﬁbers. These two changes cause
many of the variations in the cell properties.

The addition of multiple layers for the 2D and Q3D Weaves did not cause the cell
properties to change signiﬁcantly. Even with the addition of a small resin layer the
differences were very small. Therefore, it gives conﬁdence to the ability to use a single
layer unit cell to predict the properties of a structure.

All of the above case studies aid in the understanding of composite properties.
Optimization of the cell and composite properties may take place with this knowledge of

the cell parameters. The composite design may be altered to best ﬁt expected loadings.

47

CHAPTER 4 — COMPARISON OF WEAVING PATTERNS
4.1 Weave Patterns
The previous analysis on the 2D Plain Weave allowed for the derivation of the
analytical model and for various case studies to be conducted. Two other composites

structures to be analyzed are the 0/90 Laminate and the Quasi-3D (Q3D) Weave.

4.1.1 Laminate
The Laminate composite is two plies of unidirectional yarns stacked on each
other. One layer consists of two plies. Figure 29 shows the unit cell and side view of a

three layer 0/90 Laminate, L[0/90]3.

 

 

Win-w it v G‘tﬁ‘ﬁ’ﬁaisﬁr

J

        
  

 

 

 

 

(a (b)
Figure 30. 0/90 Laminate Unit Cell (a) Projected View and (b) Side View

 

The green plies have fibers running parallel to the page. The white plies have fibers
running into the page. This microstructure will give the highest possible in-plane
properties as the ﬁbers are perfectly aligned in the in-plane directions. There are no
undulation areas or pure matrix zones. For designs that require only in-plane loading,
this is a very good design. However, this microstructure is very susceptible to
delamination. The plies are only held together by the weak matrix material in the yarns.
Therefore, bending, impact, and fatigue loadings have a strong tendency to lead to

delamination. This will result in weakened properties or even failure of the structure.

48

The properties of this microstructure were calculated using Classical Laminate Theory
without any required modiﬁcations or averaging.

For loadings that can lead to delamination, this design must be modiﬁed. The
shortcomings of stitching and z-pinning were discussed in Chapter 1. The 2D Plain
Weave was one possible weave pattern that was introduced. It is effective in preventing
delamination in each individual layer. However, the layers are only held together by the
matrix material and have a tendency to delaminate. For structures requiring more than

one layer, delamination will be an issue. Therefore, the weave must be further modified.

4.1.2 Quasi-3D Weave

A weave introduced to combat delamination is the Quasi—3D (Q3D) Weave. The
weave does not align any ﬁbers directly in the thickness direction but the weave creates a
solid, 3D structure. Similar to the 2D Plain Weave, the ﬁll and warp ﬁbers are woven
together with a two harness weave. This is simply the frequency that the warp yarns
overlap the fill yarns. However, as explained in Chapter 1, each ply is woven into the
plies above and below. Therefore, there is not a stacking of layers, but an integration of
the layers. This does not allow the plies to easily peel apart when the matrix holding
them together fails. This design has been shown experimentally [15] to have higher
structural integrity and greater delamination resistance than the 2D Plain Weave. The

projected and side views of the Q3D unit cell are shown in Figure 31.

49

 

 

 

 

 

(a) (b
Figure 31. Quasi3D Weave Unit Cell (a) Projected View and (b) Side View

 

Analysis conducted on this weave was done using the same method as outlined in
Chapter 2. As before, there were four regions that all had four subregions. The number
of plies in the subregions was greater. There are two different undulation types in the
Q3D weave. As seen in Figure 31(b), some of the yarns at the top and bottom of the cell
do not have as large of an undulation as the other yarns. This is because these yarns at
the top and bottom extremes are unable to be woven with yarns in the layers above or
below as there are none.

The weaving pattern of the Q3D Weave is similar in style to that of the 2D Plain
Weave. The improvement to the delamination resistance makes this weave an attractive
design. However, it is important to know how much the additional weaving reduces

effective properties of the weave.

4.2 Comparisons
4.2.1 Moduli

As mentioned, the composite microstructures of the Laminate, 2D Plain Weave,
and Q3D Weave were all analyzed. Figure 32 shows the edge replication of the weaves

that have been created with prepreg tape.

50

 

 

 

 

 

  

Figure 32. Edge Replication of the (a) Laminate, (b) 2D Plain Weave, and (c) Q3D
Weave

The same material properties and geometric constraints were used in the calculation of all
the weaves. These are displayed in Tables 9-11. Obviously, the undulation length (au) is

not applicable to the Laminate.

Table 9. Yarn

    

144.8
11.728
11.728
5.516
5.516

5.516
0.23
0.23

0.3
0.64

Table 10. Matrix Material Properties

 

 

 

 

EM (GPa) 3.448
GM (GPa) 1.276
vM 0.35

 

 

 

51

Table 11. Unit Cell Geometric Prtmerties

 

 

 

 

| a<mm> 10 I
L au(mm) 1.467 |
| h(mm) 0.989]

 

Linear undulations with Series-Parallel alignments were used in the calculations.
Table 12 shows the properties of these weaves. Along with the Young’s moduli, shear
moduli, and Poisson’s ratios the Lekhnitskii and Chentsov coefﬁcients are tabulated, for

completeness. Therefore, all 21 independent coefficients are compared.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 12. Comparison of the Pro rties for Different Weaving Patterns
Laminate 2D Q3D
F.x (GPa) 78.81 62.77 60.33
Ey(GPa) 78.81 62.77 60.33
Ez (GPa) 12.62 12.33 14.91
ny (GPa) 5.516 4.89 4.894
ze (GPa) 5.516 6.45 6.41
Gyz (GPa) 5.516 6.45 6.41
vxy 0.0345 0.0279 0.0341
v,,z 0.2925 0.3595 0.2968
v),z 0.2925 0.3595 0.2968
1111,23 0 -5.33E-18 -8.07E-19
1111,31 0 -2.24E-17 -1.4lE-18
1111.12 0 1.53E-36 -1.67E—36
1122,23 0 2.24E-17 1.41E-18
1122,31 0 5.33E-18 8.07E-19
T122,12 0 1.53E—36 -1.67E-36
1133,23 0 2.67E—16 3.78E-17
1133.31 0 -2.67E-16 -3.78E-17
1133.12 0 4.78E-35 -2.11E—34
1423.31 0 -7.40E-17 -2.57E—17
1123,12 0 5.56E-34 -2.00E-33
1131.12 0 -5.56E-34 2.00E-33

 

 

As expected, there is a significant difference in the properties when switching

from laminate plies to woven fabrics. The in-plane Young’s moduli reduce by 20%

52

between the Laminate and the 2D Plain Weave. There is little difference in the out-of-
plane Young’s modulus as the increased undulation angle and the introduction of the
matrix regions offset each other. The in-plane shear moduli decrease and the out-of-
plane moduli increase more than 15%. This is also due to the undulation of the woven
yarns as opposed to the unidirectionality of the laminate. The Poisson Ratios follow the
same trends as the shear moduli. On the average, the in-plane properties decrease 20%
and the out-of—plane properties increase 20% when switching from laminates to woven
fabrics. The exception is the Young’s modulus in the z-direction as it does not change
significantly.

The table shows that the differences between the 2D Plain Weave and the Q3D
Weave are not very large. The differences in the in-plane Young’s moduli are less than
5%. The out-of-plane Young’s modulus increases by 20%. The shear moduli have no
signiﬁcant change between the weaving patterns. The Poisson’s ratio differences are
20% or less. The in-plane ratios increase while the out-of-plane ratios decrease. The
only signiﬁcant property change between the two patterns is the increase of E2.

The Lekhnitskii and Chentsov coefﬁcients are either zero or extremely small for
all the microstructures. These are the relations between normal stress/strain to shear
strain/stress and the relation between shear stress and strain in different plains,
respectively.

For better visual aid in the differences in the properties between weaves, Figures

33-35 show the comparison for varying weave angles.

53

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

160
140 EJHE -4i__ A,. -44 __—; ..- v -.- .. .....- g.__._.-- -
"um --E}--ExL -B—EyL -l—EzL
... 120 - “ --<>-ExZD —e—Ey2D +E22D
g El“ --A-- Ex 03D —A—Ey Q3D +Ez Q3D
9100 ~-- $47—13 __ A————--»————~~~
5 8:
g 807" i" Er-I'E‘r-LJ E
9
s 60 4 —
a
o
>.
40 W- __
20
0 I I r I I I I I I I
0 10 20 30 40 50 60 70 80 90
Weave Angle (Degrees)

 

 

 

Figure 33. Comparison of Weave Pattern Young’s Moduli for Varying Weaving Angle

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B--- ~13»ny L
12 '4 ,-' El'.‘ "0"ny20 __
[El " El... --A—- ny Q3D
0 “ﬁn, 'EJ —B—ze and GyzL
1 ., - __
a $ +3.2 030
; 3 E" , ' 94, '31-.. +Gyz 20 _
a L , A i . +6 30
3 03" .a u a -~ ’2 Q
2 . .-
5 6 i . --.
g .I L: LJ “.9".
4
2 _ _
0
0 10 20 30 4O 50 60 70 80 90
Weave Angle (Degrees)

 

 

Figure 34. Comparison of Weave Pattern Shear Moduli for Varying Weaving Angle

54

 

0.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.45 e -~~~~——— __E,... -— -—~ ----__-
0.4 -»—— - —-
0.35
.3
‘6 0.3 ’. - —-
I h' .
G ‘ '
g 0.25J ,. .
.. 0.2 '52-:
8 wit}.
0 15 ' [ f 1 '1:
01 ”El-'vxyL —B——vyzL -l—vsz 935$.
' --<>--vxy20 +vy220 +vx220 4::
0-05 --A—--vxyoso +vyzoso +vsz3D i “an?
0 l T 1. l 1 4
0 10 20 30 40 50 60 70 80 90
Weave Angle (Degrees)

 

 

Figure 35. Comparison of Weave Pattern Poisson’s Ratios for Varying Weaving Angle

The graphs show the gap between the properties of the laminated composites and woven
fabrics. However, the difference in properties between the 2D Plain Weave and the Q3D

Weave is very small.

4.2.2 Bending Stiffness

The different orientations of the warp and ﬁll ﬁbers cause a bending mismatch.
The difference in the resistance to bending in the yarns will cause them to delaminate if
the bending moment is great enough. If the warp ﬁbers have a different resistance to
bending than the ﬁll ﬁbers, there will be a stress concentration at the interface that will
lead to delamination. The resistance to bending is governed by the D matrix. The D

matrix is calculated by the following equation.

55

 

n

Am, = 5,, -Az’ (37)

1:1
Figure 36 shows the bending resistance coefficients of a woven fabric for different
weaving angles. The graph displays the difference in the bending resistance between the
fill and the warp yarns. The smaller the difference between yarns, the greater the

delamination resistance.

 

 

35 0.45

mi 0.4

 

 

0)
O
l

 

-- 0.35

J
V -_ 0.3

 

~I-D11
+016

 

N
01

 

k

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

”E
E
is
o.
9 a
3 0.20 +026 1 -- 0.25 d“:
C a
g d' -e—014 ﬂ 0 2 :‘z‘
.2 -* . a

E 5 ‘5 -B—D15 f
g 4 0.15
g. 10 EEK I
E ﬂ Bk 0 1

5 ‘ ﬁr Q ‘” 0.05

0 - n09.— 0

0 15 30 45 60 75 90
Weaving Angle (degrees)

 

 

 

Figure 36. Bending Stiffness for Varying Weave Angle
The coefﬁcients in the graph represent the relation between bending along the x-
axis and curvature in various planes. These represent delamination from the three
possible modes. As seen in the graph, the resistance for all of the coefﬁcients is very
small for small weaving angles. Therefore, the probability of delamination can be greatly

reduced by employing small weaving angles. This could be especially effective for the

56

bottom layers of a structure as this is a very common location of delamination from

impact and bending loadings.

4.3 Summary

This chapter shows the property differences between laminated composites and
woven fabrics. The in-plane moduli of the laminated composites are much higher than
that of the woven fabrics. However, the possibility of delamination makes the laminated
microstructures a poor choice in many applications. The 2D Plain Weave greatly
improves the delamination resistance for one layer designs. The major tradeoff in this
weave is the sacrificing of 20% of the maximum (laminate) in—plane Young’s moduli.
The other properties all have smaller changes that are very acceptable for the improved
delamination resistance. The increase in the structural integrity of a composite with only
one layer, while still achieving 80% of the maximum stiffness, is not a great tradeoff.

The Q3D weave protects against delamination for weaves with a higher number
of layers. As seen in the above tables and graphs, it has very similar properties to the 2D
Plain Weave. Therefore, it also achieves approximately 75% of the maximum stiffness
while allowing multiple layers that are integrated together. This is a much more

acceptable tradeoff.

57

5.1

CHAPTER 5 —- CONCLUSIONS AND RECOMIVIENDATIONS

Conclusions

. A model for calculating the properties of composite weaving patterns from unit

cell analysis, based on Classical Laminate Theory, was derived. It is a 3D model

that considers linear undulation and employs series-parallel subregion integration.

. The area found to have the most critical effect on the unit cell properties is the

undulation zone. Many of the unit cell parameters that were discussed changed
the properties because of their effect on the undulation angle. These include the
undulation length ((1.), yarn thickness (h), and cell size (a*). This zone controls
the orientation of a significant portion of the yarn ﬁbers, as well as the amount of
pure matrix regions in the unit cell. A 1% increase in the undulation length
caused a 1.2% decrease in the in-plane properties. This translates into a 0.5 mm

increase causing a 14 GPa decrease in the modulus for the given example.

. The addition of multiple layers beyond the minimum number of layers does not

signiﬁcantly affect the composite properties. Therefore, the minimum layer unit
cell will give a very good representation of the entire structure pr0perties. The
minimum layer unit cell is one layer for the 0/90 Laminate and the 2D Plain
Weave. For the Q3D Weave it is three layers because of the weave design. It
was seen in the case studies involving layer numbers that the properties changed

less than 1% as the number of layers increased for any of the weaves.

. There is a loss of in-plane stiffness of 20-25% when switching to 2D or Q3D

weaves from laminated composites. The woven fabrics are used to combat

58

 

5.2

delamination. Laminated composites are held together by only matrix bonding.
The 2D Plain Weave improves delamination resistance over the Laminate for only
one layer and sacrifices 20% of the Laminate’s in-plane stiffness. The
interlocking of the yarns in the 2D Plain Weave helps to prevent delamination in a
single layer. However, successive layers are held together by only matrix
bonding, as with the Laminate. However, the Q3D Weave improves delamination
resistance for multiple layers and sacriﬁces 25% of the Laminate’s in-plane
stiffness. This weave interlocks the yarns for all layers and thus giving
delamination resistance through the thickness. Therefore, the Q3D weave seems

to be a much more practical and useful design.

. Based on the geometric case studies of Chapter 3, the following recommendations

will cause the properties of woven fabrics to approach those of laminated
composites:

a) Thin Yarns

b) Longer Cell Length (Higher Harness)

0) Small Undulation Length

Recommendations

. Unit cell analysis of the Q3D Weave with the three and ﬁve harness tows would

be useful. The harness is the frequency of undulation. This would mean that the
length of the non-undulated portions of the ﬁbers would be longer for these
harnesses. Also, the increased complexity of this weave causes the much of the

pure matrix zones to be ﬁlled with yarns. It is therefore expected that the in-plane

59

properties of those weaves to increase compared to the basic two harness Q3D
weave.

The effective properties of the composite weave could be substituted into a finite
element code. This would allow for analysis of a complete composite structure
without concern for the weaving pattern. There would be a significant savings in
time and data if the composite structures were treated as an anisotropic material

with constant properties instead of modeling the entire weave.

60

APPENDICES

Appendix A - Poisson Ratios of Case Studies

A.l Fiber Volume Fraction

 

 

 

 

 

 

 

 

—l—vxy
+vxzandvyz

 

 

 

 

 

.0
-L
0'1
|
I

 

 

 

Poisson's Ratio
0
N

 

.0
i

 

 

 

 

0.05 - , A *

 

 

 

 

 

 

 

 

 

 

0 I I I I I l
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75
Fiber Volume Fraction

 

 

 

Figure 37 . Poisson’s Ratios for Varying Yarn Fiber Volume Fraction

61

A.2 Yarn Thickness

 

0.4

 

 

 

 

 

 

 

 

 

 

 

 

Poisson's Ratio
0
N

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.15 —-—
0.1
0.05
0 T . .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Height Ratio (Na)

 

 

 

Figure 38. Poisson’s Ratios for Varying Yarn Thickness

A.3 Undulation Length

 

0.35

0.3 - ~ . -~
-I-vxy

+ vxz and vyz

 

 

 

 

 

.0
N
0'!

 

 

 

 

 

P
N

 

 

Poisson's Ratio
0
a

.0
—L
l

 

0.05

 

 

 

 

 

 

 

 

 

 

 

 

 

0 .
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
Undulation Ratio (nu/a)

 

 

 

 

Figure 39. Poisson’s Ratios for Varying Undulation Length

62

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A.4 Weaving Angle
0.35
0.3 -» -- —~~ 4““ _ -—» 4 ~~ A a
0.25 «4 ---~ —— 4» * ~4- —
.2
E 0.2 __ _- ~4 r 4 A — —
_0
= ~I—vxy \
.3 0-15 ‘g +vxz
o
o. +vyz
0.1 ~ —— w -4 4 ~ — _.
0.05 - ~ ——*— ~—
0
0 15 30 45 60 75 90
Weaving Angie (0)
Figure 40. Poisson’s Ratios for Varying Weaving Angle
A.5 Cell Size
0.3 +\
0.25 \4 -——— ELL
\b-‘Kk‘rx—
\ﬂ
0 0.2 _
E
a:
p 0.15 +vxyandvyx
g +vxz and vyz
E
0.1 —
0.05
ﬁ—i—a— + 1
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Unit Size Ratio (818*)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 41. Poisson’s Ratios for Varying Cell Size

63

 

 

A.6 Undulation Angle Effects

 

0.4

Poisson's Ratio
0
N

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+ vxy
—I— vxz and vyz
0.15 - 4 —~— - - — -_ ~—
0.1 — we
0.05
0 ﬂ T l T
15 30 45 60 75 90
Undulation Angie (degrees)

 

Figure 42. Poisson’s Ratios for Varying Undulation Angles

A.7 Matrix Layer Thickness

 

 

0.25

0.3 #

 

L4 t + *i 4+

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Matrix Layer Ratio (t...lh)

 

% 0.2
a:
-: 015 NW
g +vxz and vyz
2
0.1
0.05
h I 1 I I
O .
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

 

Figure 43. Poisson’s Ratios for Varying Matrix Layer Thickness

64

 

 

 

A.8

Number of Layers

 

Poisson's Ratio

 

0.35

0.3 ~

0.25 ~

.0
m

0.15

.0
—L

0.05 -

 

 

 

 

 

-I—vxy

 

 

e vxzandvyz

 

 

 

 

 

___._ . _.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 4 5 6 7 8
Number of Layers

 

A.9

Figure 44. Poisson’s Ratios for Varying Number of Layers

Q3D Number of Layers

 

Poisson‘s Ratio

 

0.35

0.25

.0
to

0.15

.0
d

0.05

 

03*? —————— ~~-———-i- -— ﬂk-~—— — i— - —~—I— — —T

 

 

 

 

 

{vxy
{vxzandvyz

 

 

 

 

 

 

 

 

k a a I a

 

 

 

 

 

 

 

4 5 6 7 8
Number of Layers

 

Figure 45. Poisson’s Ratios for Varying Number of Q3D Layers

65

 

 

Appendix B - Mathematica Code

B.1 Laminate

 

 

 

 

 

 

 

ClearAlll]
I .1. “V!“ “A o o o ‘
E, E3, I5z
ﬂ- —1- ’sz o o o
E,‘ E}, Ez
ﬂ; “’3“ J— o o o

B = E, E], I:z ,

o o o i o o

o o o o 6:: o

o o o o 1

I. Gary ;

(*lnverting Stress-Straim)
Q = Simplify[lnverse[B]]; (*0 Matrix for Fibersx)
Ex =144.8 :r 109; (*Material Properties:«)
E, =11.728*109;
IEz =11.728*109;
0,, =5.516 . 109;
ze =5.516*109;
0,, =5.516:109;
EM =3.44s 4109;
vxy =0.23;
vxz =0.23;
It,z =0.3;
VM =0.35;
o... =1.276*109;
n =1; (*Number of Layerst)
a = (10 410'3); (*Geometric Propertiest)
h = 0.939 *10‘3;
ht = h;
30 = a - au:
r) = 90 :0: Jr! 180;
t=2thtn;

66

E
113,, = E—y 1: vxy; (*Polsson Definitiom)
X

z I
sz =— *VXZ!

X

z I
sz = E_ *Vyz,
Y

cosm Cos|¢+ g] o
= Cosh—g] 605”] o ;

o o 1
(*Fiber Direction Transformational)
L =
r Partll, 1, 1]2 Partll, 2, 112
Partll, 1, 2]2 Part[l, 2, 2]2
Partll, 1, 3]2 Partll, 2, 312

 

 

i
P = LQ.‘|'ranspose[L];
P:

FLAT = Tablelo, {1, 1, e}, {1, 1. 6}]:
(*Flat Fiber Reglom)

Partll, 3, 1 12
Partll, 3, 21‘
Partll, 3, 31‘

Part]l,1,2]xPart[l,1,3] Part[l,2, 21.1mm, 2, 3] Part[l,3, 214nm

Part[l, 1, 1].. Partll, 1, 3] 91111112 1]. Partli, 2, 3] Partll, 3, 1]. Part]
.Partu, 1, 11* Partil, 1, 2] Partll, 2, 11* Partll, 2, 2] Partll, 3, 11* Part]

PartlFLAT, 1,1] =Parth, 1, 1]*n*h+ Partla,1,1]xn*h;
Part[FLAT, 1,2] =Part[P, 1,2]xn 111+ Paqu,1,2]xnxh;
PartlFLAT,2, 1] =Part]P, 1,2]*n*h+ Partio,1,2]*nxh;
PartlFLAT, 2, 2] =Part[P, 2, 21m”: + Partial, 2, 2]: "11th;
Part[FLAT, 1,3] =Part[P, 1,3]*n*h+ PartlQ,1,3]*n*h;
Part]FLAT,3, 1] =Parth, 3, 1]*n*h+ Part[Q,3, 111nm;
PartlFLAT, 2, 3] =Part[P, 2, 31mm + PartlQ, 2, 3]: mh;
PaquLAT, 3,2] =Part[P, 3, 21mm + PartIQ, 3, 2]: nxh;
PartlFLAT, 3, 3] =Part[P, 3, 31m in + Paqu, 3, 3]: "1th;

67

PartlFLAT, 1, 4] =Parth, 1, 41m #11 + Partla, 1, 4]»: nxh;
PartlFLAT, 4,1] =Part]P,4,1]*n *h + Partla, 4, 11* mh;
PartlFLAT, 1, 5] =Part]P, 1, 5] *n *h + PartlQ, 1, 5] 1: mh;
PartlFLAT, 5,1] =Part]P, 5, 11m 111 + Part]0,5,1]* mh;
PartlFLAT, 1, 6] =Part[P, 1, 61m *h + PartlQ, 1, 61:: Ruth;
Part]FLAT,6, 1] =Part[P, 6, 11m *h + Part]Q,6,1]* fish;
PartlFLAT, 2, 4] = Parth, 2, 41* n :1: h + Part[Q, 2, 4] 1: n a: h;
PartlFLAT, 4, 2] = Parth, 4, 2] :1: n :1: h + Partlo, 4, 2] 1: n 4 h;
PartlFLAT, 2, 5] = Parth, 2, 5] 1: n :1: h + Partla, 2, 51* n 1: h;
PartlFLA'l', 5, 2] = Parth, 5, 2] 1: n :1: h + PartlQ, 5, 2] ... n 4 h;
PartlFLAT, 2, 6] = Parth, 2, 6] :1: n :1: h + Partlo, 2, 6] 1: n :1: h;
Part[FLAT, 6, 2] = Parth, 6, 2] :1: n :1: h + Part[Q, 6, 2] :1: n :1: h;
PartlFLAT, 3, 4] = Parth, 3, 4] :1: n :1: h + Partlo, 3, 4] :1: n at h;
PartlFLAT, 4, 3] = Parth, 4, 3] :1: n :1: h + Part“), 4, 3] 1: n 1: h;
PartlFLAT, 3, 5] = Parth, 3, 5] a: n lit h + Part“), 3, 5] :1: n x h;
PartlFLAT, 5, 3] = Parth, 5, 3] 1: n at h + PartIQ, 5, 3] 1: n 1: h;
PartlFLAT, 3, 6] = Parth, 3, 6] :1: n :1: h + Part[Q, 3, 6] 1: n :1: h;
PartlFLAT, 6, 3] = Part[P, 6, 3] :1: n :1: h + Part[Q, 6, 3] :1: n :1: h;
PartIFLAT, 4, 4] = Parth, 4, 4] 1: n a: h + Part“), 4, 4] :1: n :1: h;
PartlFLAT, 4, 5] = Parth, 4, 5] a: n :1: h + Part“), 4, 5] 1: n :1: h;
PartlFLAT, 5, 4] = Parth, 5, 4] :1: n 1: h + Part[Q, 5, 4] 1: n 1: h;
PartlFLAT, 4, 6] = PartIP, 4, 6] :1: n :1: h + Part[o, 4, 6] 1: n 1: h;
PartlFLAT, 6, 4] = Parth, 6, 4] 1: n :1: h + Partla, 6, 4] 1: n 1: h;
Part[FLAT, 5, 5] = Parth, 5, 5] :1: n :1: h + Pal’ﬂQ, 5, 5] :1: l1 :1: I1;
PartlFLAT, 5, 6] = Parth, 5, 6] :1: n :1: h + Partlo, 5, 6] :1: n :1: h;
PartlFLAT, 6, 5] = Parth, 6, 5] 1: n :1: h + 1'3de, 6, 5] :1: n :1: h;
Part[FLAT, 6, 6] = Parth, 6, 6] :1: n :1: h + Part“), 6, 6] :1: n :1: h;
FLAT;

FLATAREA = FLAT: (:02 .. 3mm;

FLA1TOTAL = F LATAREA;

68

TOTAL = FLATI'OTAL;
TOTAL
AVERAGE = ——.——:
(3)2 * 5mm
a: = lnverselAVERAGE];

Print["Ex="]
1

Partlor, 1,1141

 

Print]"E,,="
1
Partlax, 2, 2] *t

 

Print[”Ez="]
1

Partlor, 3, 31.1

 

Print["G,,y="]
1
Partlor, 3, 61:1:

 

rampage-1
1

Fifth, 5, 51.1

 

Print]"G,z="]
1
Partlor, 4, 4]*t

 

Printl"vxy="]
Partlor, 2, 1]
Partlar, 1, 1]

 

Print]"v,,,="]
Partlor, 1, 2]
Partlrz, 2, 2]

 

69

Printl"vxz="
Partloz, 1, 3]

 

-Part[o:,1,1]

Print["vzx="
Page, 1, 3]

 

_ Part[a, 3, 3]

Print["vyz="]
Paula, 2, 3]

 

Parﬂor, 2, 2]

Printrwz,="|
Partla, 2, 3]

 

Partlor, 3, 3]

70

8.2 2D Plain Weave

ClearAllu

 

1J— -..,“ Lu 0 o 0‘

 

 

 

 

 

—“”‘V L 'sz o o o
E, E, Ez
"V”- ‘”W L o o o
B = E, E, E2
0 o o A o o
Gyz
o o o o G1 o
it:
o o o 0 o ‘é‘L ,
I. W

(*lnverting Stress-Stralnx)
Q = Simplifyllnverse[B]]; (*9 Matrix for Fibersau)

Ex =144.s 4 109; (*Material Propertiesar)
E, =11.7281:109;
Ez =11.728*109;
0,, =5.516 1109;
c;xz =5.51e*109;
0,, =5.516 1:109;
EM =3.443 :1: 109;
v,,, =0.23;

vxz =0.23;

v, =0.3;
vM=03a

GM =1.276 4109;

n = 1; (*Number of Unit Celit‘nr)
m = 100000; (*Number of Elementsau)
b = m12 - 1;
(*Number of Elements in half undulatiom)
r = 2 :1: h! m; (*Thickness of element“)
area = au :1: 7; (*Area of elements”
au .
x = — ,
m

71

3,, =(1.4e7 .1 10‘3)/ n;
(*Geometric Properties*)
3: (10410'3)/n;

h = 0.989110'3:

ht =h;

ao = a — au;

3 = so am; 180;

9 = ArcTan[ 7:: :1: Cosl¢ — g";

t=2*h;
Ey _ _ .
Vyx = -E— * ny; (*POISSOH Deﬁnitions)
x
E: .
vzx = — * vxz,
x
E:
va = — * Vyz;

Y
Cos[¢] Cos[¢+ g] 0
I: °°5i¢‘§i Cos[¢] O 3
0 0 1

(*Fiber Direction Transformationau)
L =

 

 

i

p = L.Q.Transpose[L];
P;

r Part[l, 1, 1]2 Partﬂ, 2, 1? Par
Part“, 1, 2]2 Part[l, 2, 212 Par
Partll, 1, 3]2 Partll, 2, 312 Par

Part]l,1,2]azPart[|,1,3] Part[|,2,2]*Part[l,2,3] Part[l,3,
Partll, 1, 11* Partll, 1, 3] Part[l, 2, 11* Partll, 2, 3] Partll, 3,
tPartﬂ, 1, 1]. Partll, 1, 2] Partll, 2, 1].. Part[l, 2, 2| Partll, 3,

72

1““!115 1-vM2 ‘i—vm2 0 0 O
VEOEM EM VMOEM o o 0
1'VM2 1-Vu2 1-VM2
R = VMOEM VmoE" E" o o 0
1-Vu2 1-Vm2 1-VM2 ’
0 0 0 GM 0 0
0 0 0 0 GM 0
l 0 0 0 0 0 GM J

 

VMOEM

 

VMOEM

 

 

(1:0 Matrix for Resin.)

FLAT=TabIe[O, {1, 1, e], 1], 1, 5]];
(.FIat Fiber Region.)

Part[FLAT,1,1]=Part[P,1,1]1:h + Part[Q,1, 1].:11;
PartlFLAT, 1, 2] = Parth, 1, 2] .: 11 4 Part“), 1, 2] .: 11;
Part[FLAT, 2, 1] = Part[P, 1, 2]. 11 + Fargo, 1, 2]. 11;
PartlFLAT, 2, 2] = Parqr, 2, 2] . 11 + Part“), 2, 2].: h;
PartlFLAT, 1, 3] = Parqp, 1,3]. 11 + Page, 1, 3] . 11;
PamFLAT, 3, 1] = Part[P, 3. 1].: 11 + Parqo. 3, 1] .: h;
PaquLAT, 2, 3] = Parth, 2, 3].: 11 + Part“), 2, 3].: 11;
PartlFLAT, 3, 2] = Parth, 3, 2].: 11 + Fargo. 3, 2] .: 11;
PartlFLAT, 3, 3] = Parth, 3, 3] . 11 + Fargo, 3, 3].: 11;
Part[FLAT, 1, 4] =Part[P, 1, 4] .11 + Part“), 1, 4] .11;
Part]FLAT,4, 1] =Parth, 4, 1].:11 + Parqo, 4, 11.11;
Part[FLAT, 1, 5] = Part[P, 1, 5] .11 + Part“), 1, 5] .: 11;
Part[FLAT, 5, 1] = Parth, 5, 1].: 11 + Part“), 5, 1].: I1;
PartlFLAT, 1, e] = Parth, 1, s] .: 11 + Partio, 1, s] .: 11;
PartlFLAT, e, 1] = Parth, e, um + Parqo,e,1].: 11;
PaquLAT, 2, 4] = NEW, 2, 4] . 11 + 931110, 2, 4] . 11;
Part[FLAT, 4, 2] = Parth. 4, 2].: 11 + P311161. 4, 2] .: 11;
PartlFLAT, 2, 5] = Parth, 2, 5].: 11 + Fargo, 2, 5]. 11;
PartlFLAT, 5, 2] = Parth, 5, 2].: 11 + Part[Q, 5, 2] .: 11;
PartlFLAT, 2, e] = Parth, 2, e] .: 11 + 9mm, 2, e] .: 11;
PartlFLAT, e, 2] = Part[P, e, 2] .: 11 + Partlci, e, 2].: 11;
Part[FLAT, 3, 4] = Part[P, 3, 4].: 11 + Fargo, 3, 4] .: I1;
PartlFLAT, 4, 3] = Parth, 4, 3] .: 11 + P311113, 4, 3]. 11;
PartlFLAT, 3, 5] = Part[P, 3, 5] .: 11 + Part“), 3, 5]. I1;
PartlFLAT, 5, 3] = Parth. 5, 3] .: 11 + Partlo, 5, 3].: 11;

73

PartlFLAT, 3, s] = Parth, 3, e] 1: 11 + Part“), 3. 611: 11;
PartlFLAT, e, 3] = Parth, 5, 3] 1: 11 + PartlQ, e. 3] 1: h;
PartlFLAT, 4, 4] = Parth, 4, 4] 1: 11 + Partla, 4, 4] 1: 11;
PartlFLAT, 4, 5] = Parth, 4, 5] 1: 11 + Part[Q, 4, 5].: 11;
PartlFLAT, 5, 4] = Parth, 5, 4] 1: 11 + PartlQ, 5, 4] 1: 11;
PartlFLAT, 4, s] = Parth, 4, 611: 11 + Part“), 4, 3] 1: 11;
PartlFLAT, 3, 4] = Part[P, e, 4] 1: 11 + Partlo, e, 4].: 11;
PartlFLAT, 5, 5] = Parth, 5, 5] 1 11 + Part“), 5. 5].: 11;
PartlFLAT, 5, e] = Parth, 5, e] 1 11 + Partla, 5, e] 1 11;
PartlFLAT, 6,5] = Parth, 5, 5].: 11 1 Part“), 6, 5] 1: 11;
PartlFLAT, e, e] = Parth, 5, a] 1: 11 + Partla, e, e] .: 11;
FLAT;

30 2 .
FLATAREA = FLAT1: (7) .: 5mm];
FLATI'OTAL = FLATAREA .4;

MATRIX = Tablelo, (i, 1, 6}, (j, 1, 6]];
(11Matrix Pocket Regiom)
ParthATRlx, 1, 1] = Parth, 1, 1] 1:2 1: h;
ParthATRIX, 1, 2] = Part[R, 1, 2] 1:2 1: h;
Part[MATRIX, 2, 1] = Parth, 1, 2] 1:2 1: h;
Part[MATRIX, 2, 2] = Part[R, 2, 211:2 1: h;
ParthATRIX, 1, 3] = Parth, 1, 311:2 1: h;
ParthATRIX, 3, 1] = Parth, 1, 311:2 1: h;
ParthATRlx, 2, 3] = Parth, 2, 3] 1:2 1: h;
ParthATRIX, 3, 2] = Parth, 2, 3] 1:2 1: h;
Part[MATRIX, 3. 3] = Parth, 3, 3] 1:2 1: h;
Part[MATRIX, 1, 4] = PartIR, 1, 411:2 1: h;
ParthATRIX, 4, 1] = Part[R, 4, 111:2 1: h;
Part[MATRIX, 1. 5] = Part[R, 1, 5] 1:2 1: h;
ParthATRIX, 5, 1] = Parth, 5, 111:2 1: h;
Part[lVIATRIX, 1 . 6] = Parth, 1, 611:2 1: h;
ParthATRlx, 6, 1] = Parth, 6, 1].:2 1: h;
ParthATRIX, 2, 4] = Parth, 2, 4] 1:2 1: h;
Part[MATRlx, 4, 2] = Parth, 4, 2] 1:2 1: h;
ParthATRIX, 2, 5] = Parth, 2, 5] 1:2 1: h;
ParthATRIX, 5, 2] = Parth, 5, 2] 1:2 1: h;
Part[MATRIX, 2, 6] = Parth, 2, 6] 1:2 1: h;
ParthATRlx, 6,2] = Parth, 6, 211:2 1: h;

74

Part]MATR1x, 3, 4] = 1:3qu, 3, 4]. 2 . 11;
ParthATRiX, 4, 3] = Parth, 4, 3] .2 . I1;
ParthATRIX, 3, 5] = Part]R, 3, 5] .2 . 11;
ParthATRiX, 5, 3] = Parth, 5, 3] .2 . 11;
ParthATRIX, 3,6] = Part[R, 3, e] .2 . 11;
ParthATRlx, e, 3] = Part[R, 6, 3] .2 . 11;
ParthATRIX, 4, 4] = Parth, 4, 4] .2 . h;
PartIMATRIX, 4, 5] = Parth, 4, 5] .2 . 11;
Part[MATRIX, 5, 4] = Parth, 5, 4] .2 . 11;
Part]MATR1x, 4, e] = Parth, 4, e] .2 . 11;
Part]MATR1x, e, 4] = Part[R, e, 4] .2 . 11;
ParthATRIX, 5, 5] = Fart]R, 5, 5] .2 . 11;
Part]MATRI)(, 5, e] = Parth, 5, e] .2 . 11;
ParthATRIX, e, 5] = Parth, 6, 5] .2 . 11;
ParthATRIX, e, e] = Paqu, 3, e] .2 . 11;
MATRIX;

2
MATRIXAREA = MATRIX . (%) . 3mm;
MATRIXTOTAL = MATRIXAREA .4;

Cos[O] o coslo — g]
I = 0 1 o ;
Cos[0+ g] o Cos[0]

(.Undulation Transformation.)

M =
' Part[l, 1, 1]2 Partﬂ, 2, 1 12 P31
Partll, 1, 2]2 Part", 2, 2]2 Par
Partll, 1, 3]2 Perth, 2, 3]2 Par

Partll, 1, 2]. Partll, 1, 3] Partll, 2, 2]. Part", 2, 3] Partﬂ, 3,
Partﬂ, 1, 1]. Partll, 1, 3] Part[l, 2, 1]. Party, 2, 3] Part“, 3,
.Part]1,1,1].Fart]I,1,2] Part]l,2,1]1Part[I,2,2] Part]l,3,

 

 

J

Y = M.Q.Transpose[Ml:
X = L.Y.Transpose[L]:

75

FILL = Tablelo, (i, 1, 6], (j, 1, 6}];
(.Fill Undulation Region.)
Part]FlLL,1,1]=Part[Y,1,1]1:h;
Part[FILL, 1, 2] = PartIY, 1, 2] 1: h:
PartlFlLL, 2, 1] = PartlY, 2, 1]. h;
PartlFlLL, 2, 2] = PartlY, 2, 2] 1: h;
PartlFlLL, 1, 3] = PartlY, 1, 3] 1: h;
PartlFlLL, 3, 1] = Part[Y, 3.1]. h;
PartlFlLL, 2,3] = Part(Y, 2, 3] 1: h;
PartlFlLL, 3, 2] = PartlY, 3, 2] 1: h;
PartlFlLL, 3, 3] = PartlY, 3, 3] 1: h;
PartlFlLL, 1, 4] = PartlY, 1, 4] 1: h;
PartlFlLL, 4, 1] = PartlY, 4, 1]. h;
PartlFlLL, 1, 5] = PartlY, 1, 5] 1: h;
PartlFlLL, 5, 1] = PartlY, 5, 111: h;
PartlFlLL, 1, 6] = Part”, 1, 611: h;
PartlFlLL, 6, 1] = PartiY, 6, 1]. h;
PartlFlLL, 2, 4] = PartlY, 2, 4] 1: h;
PartlFlLL, 4, 2] = PartlY, 4, 2] 1: h;
Part[FILL. 2, 5] = PartlY, 2, 5] 1: h;
PartlFlLL, 5, 2] = Part[Y, 5, 2] 1: h;
PartlFlLL, 2, 6] = PartlY, 2, 6] 1: h;
PartlFILL, 6. 2] = PartiY, 6, 2] 1: h;
PartlFlLL, 3, 4] = PartlY, 3, 4] 1: h;
Part[FILL, 4, 3] = Part[Y, 4, 3] 1: h;
PartlFlLL. 3, 5] = PartlY, 3, 5] 1: h;
PartlFlLL, 5, 3] = PartlY, 5, 3] 1: h;
PartlFlLL, 3, 6] = Part”, 3, 6] 1: h;
PartlFlLL, 6, 3] = PartlY, 6, 3] 1: h:
PartlFlLL, 4, 4] = Part[Y, 4, 4] 1: h;
PartlFlLL, 4, 5] = Part[Y, 4, 5] 1: h;
PartlFlLL, 5, 4] = Part[Y, 5, 4] 1: h;
Part(FILL, 4, 6] = Part[Y, 4, 6] 1: h;
PartlFlLL, 6, 4] = PartlY, 6. 4] 1: h;
PartlFlLL, 5, 5] = PartlY, 5, 5] 1: h;
PartlFlLL, 5, 6] = Paqu, 5, 6] 1: h;
Part[FlLL, 6, 5] = PartlY, 6, 5] 1: h;
Part[FILL, 6, 6] = PartlY, 6, 6] 1: h;
FILL;

76

FILLAREA = FILL: 322- 1: 321 :1: Sinlﬂ;

AA = Table[0, (i, 1, bi]:
BB = Tablelo, {1, 1, 11)];
co = Tablelo, {i, 1, hi];
FF = Tablelo, (i, 1. hi];
MM = Table[0, (i, 1, 11]];
Q: = Tablelo, (i, 1, b}]:
A5 = Table[o, {1, 1, Di]:

DolPartlAA, i] = 1.1, {1, 1, 11]]
DolPart[BB, 1] = PartlAA, 1] . —:”— .1, {1, 1, 5}]

Do[Part[CC, i] = area —Part[BB, i], {1, 1, b]]
DolPartiFF, l] = PartlBB, iliarea, {1, 1, b}]
DolPartIMM, i] = Partlcc, illarea, {1, 1, b}]

1
Fart[FF,i] + PartIMMj] ’ 0’ 1’ ”I
Part[Y,1,1] Fart]R,1,1]
Do]Part[As, i] = Partlas. l]1:a._.1:'r1:ao, {1, 1, 11]]
T5 = 0;
Do": = Ts + PartlAs, i], {1, 1, b]]
PartlFlLLAREA, 1, 1] =Ts 1:2 1: Sin[¢];

 

DolPartle, i] =

 

 

 

— h K I
- Cos[01: 7:3] Tan|01$r
ar=21h1x;
fib0f=1111x§

matrix = (2 :1: i1 —1)4K:
ﬁbrat = ﬂberlar;
matrat = matrixiar;
1
QUS = 1':
1 + 1
PartlY,3,3] ParﬂR,3,3]
AUS = (GUS 4:21: hex»: ao)1:m;

Part[F|LLAREA, 3, 3] =AUS :1: SHIN];

 

F iLLTOTAL = FILLAREA 1: 4:

77

WARP =Tabiel0, {1, 1, 31,11, 1, e1];
PartIWARP, 1,1]=Part[X, 1, 11.11;
Part[WARP, 1, 2] = Partlx, 1, 21. 11;
PartIWARP, 2, 11 = Fart1x, 2, 1]. 11;
Part[WARP, 2, 21 = Partlx, 2, 2] . 11;
PartIWARP, 1, 3] =Partlx, 1.31.11;
ParthARP, 3, 1] =Fart]x, 3, 11.11;
Part[WARP, 2, 3] = Partlx, 2, 3]. 11;
Part[WARP, 3, 2] = Fart1x, 3, 2] . 11;
FartIWARF, 3, 3] = Part[X, 3, 3] . 11;
PartIWARP, 1, 4] = Partlx, 1, 4]. 11;
PartMARF, 4, 1] =Part[x, 4, 11.11;
PartIWARP, 1,5] = Partlx, 1, 51. 11;
Part]WARP,5,1] =Partlx, 5. 11.11;
PaTtIWARP, 1, s] = Partlx, 1, e1. 11;
PaTtIWARP, e, 1] = Partlx, e, 1]. 11;
PartIWARP, 2, 4] = Partlx, 2, 4] . 11;
ParthARP, 4. 2] = Partlx, 4, 2] . 11;
Part[WARP, 2, 51 = Partlx, 2, 5]. 11;
Part[WARP, 5, 2] = Partlx, 5. 2] . 11;
Part[WARP, 2, e] = Fart]x, 2, e1 . 11;
ParthARP, e, 21 = PartIX, e, 2] . 11;
Part[WARP, 3, 4] = Partlx, 3, 4]. 11;
Part[WARP, 4, 31 = Fart]x, 4, 3] . 11;
Part[WARP, 3, 5] = Partlx, 3, 5] . 11;
Part[WARP, 5, 31 = Parttx, 5, 3] . 11;
ParthARP, 3, 31 = Partlx, 3, e1. 11;
Part[WARP, e, 3] = Partlx, e, 3]. 11;
Fart1WARP. 4, 4] = Parqx, 4. 4] . 11;
Part[WARP, 4, 5] = Partlx, 4, 5] . 11;
PertIWARP, 5. 4] = Fart1x, 5, 4] . 11;
Pat‘tIWARP, 4. 3] = Fartpt, 4. 3] . 11;
Part[WARP, e, 41 = Partpt, 6,4]1: 11;
Part[WARP, 5, 5] = Partlx, 5, 5] . 11;
Part[WARP, 5, e] = Part[x, 5, 6]1: 11;
PartIWARP, s, 5] = Part1x, 6, 5] . 11;
Part[WARP, 3, e] = Part1x, 6,6]. 11;
WARP;

78

WARPAREA = WARP 1: 322 1: ii 1: Sin[¢];

ParthARPAREA, 2, 2] = 2 4T5 1: Sin[¢];
ParthARPAREA, 3, 3] =AUS :1: Sin[¢];
WARPTOTAL = WARPAREA :1: 4;

RESIN =Table[0, {1, 1, s1, {1, 1. s1];
ParthESiN, 1,1] =Part[R, 1.11.11;
ParthESIN, 1, 2] =Parth, 1.21.11;
ParthESlN, 2, 1] =Parth, 2, 11.11;
Part[RESIN, 2, 21 = 113qu, 2, 2] . 11;
Part[RE5|N, 1, 3] = Parth, 1, 3]. 11;
Part1REs1N, 3, 11 = Parth, 3, 1]. 11;
ParﬂRESIN, 2, 31 = Parth, 2, 3] . 11;
ParthESlN, 3, 2] = Parth, 3, 2] . 11;
ParthESIN, 3, 31 = Parth, 3, 3]. 11;
ParthESlN, 1, 4] = ram, 1, 4] . 11;
Part[RESIN, 4, 1] = Paqu, 4, 11. 11;
ParthESiN, 1, 5] =Fart1R, 1.51.11;
ParthESIN, 5, 1] = Parth, 5, 1]. 11;
Part[RES|N, 1, e] =Part]R, 1, 31.11;
ParthESiN, s, 1] = Parth, 3, 1]. 11;
ParthESIN, 2, 41 = Parth, 2, 4] . 11;
ParthESIN, 4, 2] = ram, 4, 2] . 11;
Part[RESiN, 2, 5] = Fart1R, 2, 5] . 11;
ParthESIN, 5, 21 = Parth, 5, 2] . 11;
Part1REs1N, 2, e] = Parth, 2, e] . 11;
Part[RE5IN, e, 2] = Faqu, s, 2] . 11;
PartIREsIN, 3, 4] = Fart]R, 3, 4] . 11;
Part[RESIN, 4, 3] = rum, 4, 3] . 11;
Part[RESIN, 3, 5] = Parth, 3, 5]. 11;
Part[RESIN, 5, 3] = Parth, 5, 3] 1: 11;
ParthESIN, 3, e] = Parth, 3,6]1: 11;
ParthESIN, e, 3] = Paqu, 5, 3] . 11;
ParthESIN, 4, 4] = Paqu, 4, 4] . 11;
Fan1ResIN, 4, 5] = Faqu, 4, 5] . 11;
Part1RE51N, 5, 41 = Paqu, 5, 4] . 11;
Part1REsm, 4, e] = Parth, 4, 611: 11;
Part1RE51N, e, 4] = Paqu, 6, 4] . 11;

79

ParthESIN, 5, 5] = 11311111, 5, 5] . 11;
FaquEsm, 5, s1 = Parth, 5, e1. 11;
ParthESIN, s, 51 = Parth, 6,511: 11;
Part[RESlN, e, e] = Parth, 5, e1. 11;
RESIN;

_ au ‘0 - .
RESINAREA _ RESIN . (7) . 1;). 31111.11],
RESINTOTAL = RESINAREA .3;

TOTAL = FLA1TOTAL + MATRIXTOTAL + F iLLTOTAL +
WARPTOTAL + RESINTOTAL;

:12 . TOTAL

AVERAGE = . :
(n 1: a)2 1: Slnlqi]

 

a: = lnverse[AVERAGEl;

Print["E,,="]
1
Part]a:,1,1]1:t

 

Printl'E,="
1
Fan]... 2, 21.1

 

Printl'Ez="]
1
Partla, 3, 31.1

 

Printl'Gx,="]
1

Part]... 6, 31.1

 

anq'ox,="]
1
Part]... 5, 51.1

 

Frintroﬁa]
1
Pert[o:, 4, 4] .t

 

Printl'vx,="]
Partlar, 2, 1]

Part]a:,1,1]

 

80

Prim [11yyx=11]
Part [a, 1, 2]
' P1111111, 2, 21

 

Print ["vxz="
Part[a, 1, 3]

-Part[a,1, 1]

 

Print [llvzx=ﬂ]
Part[a, 1, 3]

' Part [a, 3, 31

 

Print ["VR="]
Part [a, 2, 3]

 

— Part [a, 2, 21

Print ["VzFul
Part [a, 2, 3]

 

’ Part [a, 3, 31

81

B.3 Q3D Weave

 

 

 

ClearAlII]
r L ‘VW -sz
"V 1111 L ‘sz
Ex Ey Ez
_;_'VI 'VYZ L
0 0 O
O 0
0 O 0
i

 

 

 

 

O 0 O ‘
O O O
0 0 O
L o o ’
Gyz
1

O 6.2 O
o o 1

13,3, 1

(.Inverting Stress—Strain.)
Q = Simplify[lnverse[B]]; (1:0 Matrix for Fibers.)

Ex =144.3 .109;
E, =11.123.109;
Ez =11.72e.109;
6,, =5.516 .109;
3,2 =5.51e .109;
0,, =5.516 .109;
EM =3.443 . 109;
v,,, =0.23;

vxz =0.23;

11,, =0.3;

v... =0.35;

GM =1.27e.109;

n=1;

3,, =(1.437.10'3)/n;
a=(101:10'3)/n;
11=o.eas.10'3;

11: =11:

ao =a—a..:
¢=9043lf1803

3
p—2;

p
q:

(.Material Properties.)

(.Number of Unit Cells.)

(.Geometric Properties.)

(.Number of Layers.)

82

0 = ArcTanlg 1: Coslgi - g";

 

2 :1: h a
ﬁ = ArcTanl 1: Cosle - — l];
3,1 2
t = 2 :1: I1;
E!
v,,, = E— 1: vx,; (.Polsson Definition.)
X
E
sz = E—: 1‘ vxz:
sz -— E 1: 11,2,

6051111] Cos|¢+ g] 0
|= Cosh—g] Cosloi] o ;

o o 1
(.Fiber Direction Transformation.)
L .—.
' Partll, 1, 1]2 Partll, 2, 1]2 Partll, 3, 112
Partll, 1, 212 Partll, 2, 212 Partli, 3, 2]2
Partll, 1. 312 Part", 2, 3]2 Partﬂ, 3, 3]2

Partll,1,2]1:Part[I, 1,31 Part“, 2, 21111111111, 2, 31 Part", 3, 21.111111"
Partll, 1, 11.111111", 1.3] Part", 2, 11.1mm, 2, 3] Part[l,3,1]1:Part[l
,Partll,1, 11.111111", 1,2] Fait11,2, 1]1:Part[l,2,2] 951111.31, 11.Fart]1

 

 

1

P = LQ.Transpose[L]:
P;

1 Eu VuoE. VMeEu 1
1‘11“! ‘1-11f-2 1-Vuz o o o
VIOEE El VleE!

1-v" 1-v. 1-vu 0 0 O

E V 0E E
R: VIP ll I1 11 I o o o
1-Vu2 1-v1‘2 1-v“2

 

 

 

 

o o o 0],. o o
o o o 0 GM 0
, o o o o 0 GM 1
(.0 Matrix for Resin.)

83

FLAT = Tablolo, {1, 1, 6], {j, 1, 6}];

(1Flat Fiber Rogiom)

Part]FLAT,1, 1] =Part[P, 1,1]1:h + PartlQ,1,1]1:i1;
PartlFLAT, 1, 2] = Parth, 1, 2]::: h + Part“), 1, 2] 1: h;
Part[FLAT, 2, 1] = Parth, 1, 2] 1: h + Partla, 1, 2] 1: i1;
PartlFLAT, 2, 2] = Part[P, 2, 2] 1: n + Parqo, 2, 2]1: h;
PartlFLAT, 1, 3] = Part[P, 1, 3] 1: h + Part“), 1, 3] 1: h;
PartlFLAT, 3, 1] = Parth, 3, 1]1: h + Part[Q, 3, 1]1: h;
PartlFLA‘l', 2, 3] = Parth, 2, 3] 1: h + Part“), 2, 3] 1: h;
PartlFLA‘l', 3, 2] = Parth, 3, 2] 1: n + Partlo, 3, 2] 1: h;
PartlFLAT, 3, 3] = Parth, 3, 311: h + Part“), 3, 3] 1: h;
PartlFLAT, 1, 4] = Part[P, 1, 4] 1: n + Part“), 1, 4] 1: h;
PartlFLAT, 4, 1] = Partﬂ’, 4, 1]1: h + Part“), 4, 1]1: h;
Part[FLAT, 1, 5] = Parth, 1, 5] 1: n + Partlo. 1, 511: n;
PaquLAT, 5, 1] = Partﬂ’, 5, 1]1: h + ParﬂQ, 5, 1]1: h;
PartlFLAT, 1, 6] = Part[P, 1, 6] 1:11 + Part“), 1, 6] 1: h;
PartlFLAT, 6, 1] = Parth, 6, 1]1: h + Paqu, 6, 1]1: h;
PaquLAT, 2, 4] = Parth, 2, 4] 1: 11 + Part“), 2, 4]: n;
PartlFLAT, 4, 2] = Parth, 4, 2] 1: n + Partla, 4, 2] 1: i1;
PaquLAT, 2, 5] = Part[P, 2, 5] 1: h + Part“). 2, 5] 1: h;
Part[FLAT, 5, 2] = Parth, 5, 2] 1: n + Partm, 5, 2] 1: h;
PartlFLAT, 2, 6] = Part]P, 2, 61111 + Partlo, 2, 6] 1: h;
PaquLAT, 6, 2] = Parth, 6, 2] 1: h + Part“), 6, 2] 1: h;
PartlFLA'l', 3, 4] = Parth, 3, 4] 1: h + Partla, 3, 4] 1: n;
Part[FLAT, 4. 3] = Parth, 4, 3] 1: n + Part“), 4, 3] 1: h;
Part[FLAT, 3, 5] = Parth. 3, 5] 1: h + Part“). 3, 5] 1: n;
PartIFLA‘I', 5, 3] = Parth, 5, 3] 1: n + Partlo. 5, 3] 1: h;
PaquLAT, 3, 6] = Parth, 3, 6] 1: h + PartlQ, 3, 6] 1: h;
ParttFLAT, 6, 3] = Parth, 6, 3] 1:11 + Part“), 6, 3] 1: n;
Part[FLAT, 4, 4] = Part[P, 4, 4] 1: h + Part“), 4, 4] 1: i1;
Part[FLAT, 4, 5] = Parth, 4, 5] 1: h + Partla. 4, 5] 1: h;
PartlFLAT, 5, 4] = Part]P, 5, 4] 1: h + Part[Q, 5, 4] 1: I1;
PartlFLAT, 4, 6] = Parth, 4, 6] 11:11 + Part“), 4, 6] 1: h;
Part]l-'LAT, 6, 4] = Parth, 6, 4] 1: h + Part[Q, 6, 4] 1: n;
PartlFLAT, 5, 5] = Parth, 5, 5] 1: n + Paqu, 5, 5] 1: h;
Part[FLAT, 5, 6] = Parth, 5, 6] 1: n + Partla, 5, 6] 1: h;
Part[FLAT, 6, 5] = Parth, 6, 5] 1: h + Part[0, 6, 5] 1: h;
PartlFLAT, 6, 6] = Part[P, 6, 6] 1: h + Part“), 6, 6] 1: h;
FLAT;

30 2
FLATAREA = FLAT1: {—5) 1: 5mm;
FLA1TOTAL = 4 1: (2 + q) 1: FLATAREA;

84

MATRIX = Tablo[0, {i, 1, 6}, {j, 1, 6}];
(1Matrix Pocket Rogiom)
Part[MATRIX,1,1]=Part[R, 1,1]12 1:p1:h;
ParthATRlx, 1, 2] = Part[R, 1, 2] 1:2 1: p 1: h;
Part]MATRIX,2, 1] = Parth, 1, 2] 1:2 1: p 1: h;
Part[MATRIX, 2, 2] = Parth, 2, 2] 1: 2 1: p 1: n;
ParthATRix, 1, 3] = Parth, 1, 3] 1: 2 1: p 1: h;
ParthATRix, 3, 1] = Parth, 1, 311:2 1: p 1: h;
ParthATRIx, 2, 3] = Parth, 2, 311:2 1: p 1: h;
Part]MATRIX, 3, 2] = Parth, 2, 311:2 1: p 1: n;
Part[MATRlx, 3, 3] = Parth, 3, 311:2 1: p 1: h;
Part[MATRIx, 1, 4] = Part[R, 1, 4] 1: 2 1: p 1: h;
ParthATRlx, 4, 1] = Parth, 4, 1] 1: 2 1: p 1: h;
ParthATRIX, 1, 5] = Parth, 1, 5] 1:2 1: p 1: I1;
PartIMATRIx, 5, 1] = Parth, 5, 1] 1: 2 1: p 1: I1;
ParthATRIx, 1, 6] = Part[R, 1, 611: 2 1: p 1: n;
ParthATRlx, 6,1] = Parth, 6,1]12 1:p 1: h;
ParthATRlx, 2, 4] = Parth, 2, 4] 1: 2 1: p 1: h;
ParthATRlx, 4, 2] = Parth, 4, 2] 1:2 1: p 1: h;
ParthATRlx, 2, 5] = Paqu, 2, 5] 1:2 1: p 1: n;
ParthATRIX, 5, 2] = Parth, 5, 2] 1:2 1: p 1: n;
ParthATRlx, 2, 6] = Parth, 2, 6] 1:2 1: p 1: n;
ParthATRlx, 6, 2] = Part[R, 6, 2] 1:2 1: p 1: h;
Part[MATRlx, 3, 4] = Parth, 3, 4] 1:2 1: p 1: h;
ParthATRIX, 4, 3] = Parth, 4, 3] 1:2 1: p 1: h:
ParthATRlx, 3, 5] = ParﬂR, 3, 5] 1:2 1: p 1: h;
ParthATRIX, 5, 3] = Parth, 5, 311:2 1: p 1: h;
Part[MATRix, 3, 6] = Parth, 3, 611:2 1: p 1: I1;
ParthATRIX, 6, 3] = Parth, 6, 3] 1:2 1: p 1: h;
Part[MATRIx, 4, 4] = Part]R, 4, 4] 1: 2 1: p 1: h;
Part[MATRIX, 4, 5] = Parth, 4, 5] 1:2 1: p 1: h;
Part[MATRIX, 5, 4] = Parth, 5, 4] 1: 2 1: p 1: h;
Part[MATRIX, 4, 6] = Parth, 4, 6] 1: 2 1: p 1: h;
ParthATRlx, 6, 4] = Parth, 6, 411:2 1: p 1: i1;
Part[MATRIX, 5, 5] = Part[R, 5, 5] 1: 2 1: p 1: h;
ParthATRIX, 5, 6] = Parth, 5, 6] 1: 2 1: p 1: h;
ParthATRIX, 6, 5] = Parth, 6, 5] 1: 2 1: p 1: h;
Part[MATRIX, 6, 6] = Parth, 6, 6] 1: 2 1: p 1: h;
MATRIX;

2
MATRIXAREA = MATRIX 1: (4:3) 1: smm;
MATRIXTOTAL = MATRIXAREA 1: 4;

85

Cos[0] o eoslo— g]
1: o 1 a
«140+ g] o Cos[0]

(1Undulation Transformatiom)

M =
r Partll, 1,112 Partll, 2, 112 Partll, 3,1]2
P111", 1. 212 Partll, 2, 2]2 Partll, 3, 2]2
Part", 1, 3]2 Partll, 2, 312 Partll, 3, 312

Partli,1,2]1Part[l,1.3] Part]l,2,2]1:Part[i,2,3] Part||,3, 2119:1111
Partll,1,1]1:Part[I,1,3] Part]l,2,1]1:Part]l,2,3] 911111, 3.1119111"
:Part[1,1,1]1Part[l,1,2] Part[1,2,1]1:Part[l,2,2] Partll,3,1|1:Part[I

 

 

J

Y = M.O.Transposo[M]; (161nall Undulatiom)
x = LY.Transpose[L];

eosm 0 Cost —§
1: o 1 o
Cos[ﬂ+ g] o oosm

(1Un11ulatlon Transformatiom)

M =
* Partll, 1, 112 Partll, 2, 1]2 Pan", 3, 112
91:11:11, 1, 212 Part", 2, 212 Partll, 3, 212
Partll, 1, 312 Part“, 2, 312 Partll, 3, 312

Part[I,1,2]1Part[l,1,3] Pan[1,2, 2]1:Part[l. 2, 3] Part||,3,2]1:Part[|
Part]l,1,1]1Part]l,1,3] Part[I,2,1]1:Part]l,2,3] Part]l,3,1]1:Part[l
,Partn, 1, 111: Partﬂ, 1, 2] Partll, 2, 111: Part[l, 2, 2] Partll, 3, 111: Part[l

 

 

J

V = M.0.Transposo[M]; (1Largo Undulatiom)
U = LV.Transposo[L];

86

FILL = Tablolo, {1, 1, 5}, 1], 1, 5]]; (15111 Undulation Regiom)
Part]FiLL,1,1] =Part[Y, 1,1]111;
PartlFlLL, 1, 2] =PartlY, 1. 2] 1:11;
PartlFlLL, 2, 1] = PartlY, 2, 1] 1: 11;
Part[FlLL, 2, 2] = PartlY, 2, 2] 1: 11;
PartlFlLL, 1, 3] =Part[Y, 1, 31111;
Part[FILL, 3,1] = PartlY, 3, 1]1: 11;
Part[FILL, 2, 3] = PartlY, 2, 3] 1: 11;
PartlFlLL, 3, 2] = Part]Y, 3, 2] 1: 11;
PartlFiLL, 3, 3] = PartlY, 3, 3] 1: n;
PartlFiLL, 1, 4] =Part[Y, 1, 4] 1:11;
Part[FlLL, 4, 1] = PartlY, 4, 1] 1: 11;
PartlFlLL. 1, 5] =PartlY, 1, 51111;
PartlFlLL. 5,1] =Part[Y, 5, 11111;
PartlFlLL, 1,5] =PartlY, 1, 51111;
PartlFlLL, 5.1] = PartlY, 5, 1]1: n;
Part[FILL, 2. 4] = PartlY, 2, 4] 1: 11;
PartlFiLL, 4, 2] = PartlY, 4, 2] 1: 11;
PartlFlLL. 2, 5] = PartlY, 2, 5] 1: h;
PaquILL. 5, 2] = Part]Y, 5, 2] 1: h;
PartlFlLL. 2, 5] = PartlY, 2, 5] 1: h;
PartlFlLL, 5, 2] = PartlY, 5, 211: h:
Partli-‘ILL, 3, 4] = PartlY, 3, 4] 1: i1;
Part[FlLL, 4, 3] = PartlY, 4, 3] 1: 11;
ParttFlLL. 3, 5] = PartlY, 3, 5] 1: 11;
Part]FiLL. 5, 3] = PartlY, 5, 3] 1: 11;
Part[FILL, 3, 5] = PartlY, 3, 5] 1: h;
Part[FlLL. 5. 3] = PartlY, 5, 3] 1: n;
PaquILL, 4, 4] = PartlY, 4, 4] 1: 11;
Part[FiLL, 4, 5] = PartlY, 4, 5] 1: n;
PartlFiLL, 5, 4] = PartlY, 5, 4] 1: 11;
PartlFiLL. 4, 5] = PartlY, 4, 5] 1: 11;
PartlFlLL, 5, 4] = PartIY, 5, 4] 1: h;
Part[FlLL, 5, 5] = 11mm, 5, 5] 1: n;
PartlFiLl. 5, 5] = PartlY, 5, 5] 1: 11;
PartlFiLL, 5, 5] = PartlY, 5, 5] 1: n;
PartlFlLL, 5, 5] = PartlY, 5, 5] 1: h;
FILL;

FILLAREA = FILL: % 1: g:— 1: Sin[¢];
FILLTOTAL = FILLAREA 1:4;

87

FILLER = Tablolo, {i, 1, 6], {j, 1, 6}];
(:::Fill Largo Undulation Regiom)
Part]FlLLER,1,1]=Part]V,1,1]111;
PartlFlLLER, 1, 2] = Part[v, 1, 2] 1: n;
Part[FlLLER, 2, 1] = Part]v, 2, 1] 1: h;
Part[FILLER, 2, 2] = Partlv, 2, 2] 1: n;
PartlFlLLER, 1, 3] = Parth, 1, 3] 1: 11;
PartlFlLLER, 3, 1] = Part[V, 3, 1]1: 11;
Part[FILLER, 2, 3] = Partlv, 2, 3] 1: n;
PartlFlLLER, 3, 2] = Partlv, 3, 2] 1: i1;
Part[FILLER, 3, 3] = Partlv, 3, 3] 1: n;
PartlFlLLER, 1, 4] = Partlv, 1, 4] 1: h;
Part[FiLLER, 4, 1] = Partlv, 4, 1] 1: 11;
Part[FILLER, 1, 5] = Part[v, 1, 5] 1: 11;
ParﬂFlLLER, 5, 1] = Partlv, 5, 1] 1: h;
Part[FILLER, 1, 6] = Part[v, 1, 6] 1: n;
PartlFlLLER, 6. 1] = Part[v, 6, 1]1: h;
PartlFILLER, 2, 4] = Partlv, 2, 4] 1: n;
PartlFILLER, 4, 2] = Partlv, 4, 2] 1: n;
PartlFlLLER, 2, 5] = Partlv, 2, 5] 1: 11;
PartlFlLLER, 5, 2] = Partlv, 5, 2] 1: h;
PartlFILLER, 2, 6] = Partlv, 2, 6] 1: n;
PartlFlLLER, 6, 2] = Partlv, 6, 2] 1: n;
Part[FlLLER, 3, 4] = Parqv, 3, 4] 1: n;
Part[FILLER, 4, 3] = Part[v, 4, 311: h;
PartlFlLLER, 3, 5] = Partlv, 3, 5] 1: n;
PaquILLER, 5, 3] = Partlv, 5, 3] 1: h;
PaqulLLER, 3, 6] = Partlv, 3, 6] 1: h;
PartlFlLLER, 6, 3] = Partlv, 6, 3] 1: h;
PartlFlLLER, 4, 4] = Partlv, 4, 4] 1: h;
Part[FlLLER, 4, 5] = Partlv, 4, 5] 1: h;
PartlFlLLER, 5, 4] = Partlv, 5, 4] 1: h;
PaquILLER, 4, 6] = Partlv, 4, 6] 1: n;
PartlFlLLER, 6, 4] = Partlv, 6. 4] 1: n;
ParttFlLLER, 5, 5] = Partlv, 5, 5] 1: h;
PartlFiLLER, 5, 6] = Partlv, 5, 6] 1: 11;
PartlFlLLER, 6, 5] = Partlv, 6, 5] 1: 11;
PartlFlLLER, 6, 6] = Parth, 6, 6] 1: n;
FILLER;

FILLERAREA = FILLER1: % 1: 3‘21 1: 5mm;
FILLERTOTAL = FILLERAREA1: (1 + q) 1:4;

88

WARP = Tablo|0, {1, 1, 6}, {j, 1, 6}];
Part]WARP, 1,1] = Partlx, 1,1]1: h;
Part]WARP, 1, 2] = Partlx, 1, 2] 1: h;
PartMARP, 2, 1] = Partlx, 2, 1]1: h;
Part[WARP, 2, 2] = Partlx, 2, 2] 1: h;
PartIWARP, 1, 3] = Part[x, 1, 3] 1: I1;
Part[WARP, 3, 1] =Part[x, 3, 1]1: h;
PartIWARP, 2, 3] = Part[x, 2, 3] 1: h:
Part[WARP, 3, 2] = Part[x, 3, 2] 1: n;
Part]WARP, 3, 3] = Part[x, 3, 3] 1: I1;
Part[WARP, 1, 4] = Part[x, 1, 4] 1: n;
Part[WARP, 4, 1] = Part[x, 4, 1]1: n;
Part[WARP, 1, 5] = Partlx, 1, 5] 1: h;
Part[WARP, 5, 1] = Partlx, 5, 1]1: h;
Part[WARP, 1, 6] = Partlx, 1, 6] 1: n;
Part[WARP, 6, 1] = Part[x, 6, 1]1: h;
Part[WARP, 2, 4] = Part[x, 2, 4] 1: n;
Part[WARP, 4, 2] = Partlx, 4, 211: n;
Part[WARP, 2, 5] = Partlx, 2, 5] 1: n;
PartnNARP, 5, 2] = Partlx, 5, 2] 1: n;
Part[WARP, 2, 6] = Parﬂx, 2, 6] 1: h;
Part[WARP, 6, 2] = Partlx, 6, 2] 1: h;
Part[WARP, 3, 4] = Part[x, 3, 411: n;
Part]WARP, 4, 3] = Parth, 4, 3] 1: h;
PartMARP, 3, 5] = Partlx, 3, 5] 1: h:
PartMARP, 5, 3] = Partlx, 5, 3] 1: i1;
Part[WARP, 3, 6] = Partlx, 3, 6] 1: h;
Part[WARP, 6, 3] = Part[x, 6, 3] 1: n;
Part[WARP, 4, 4] = Partlx, 4, 4] 1: n;
Part[WARP, 4, 5] = Partlx, 4, 5] 1: n;
Part[WARP, 5, 4] = Partlx, 5, 4] 1: n;
Part[WARP, 4, 6] = Partlx, 4, 6] 1: n;
Part[WARP, 6, 4] = Part[x, 6, 4] 1: n;
Part]WARP, 5, 5] = Part[x, 5, 5] 1: n;
Part[WARP, 5, 6] = Partlx, 5, 6] 1: h;
PartMARP, 6, 5] = Part[x, 6, 511: h;
ParﬂWARP, 6, 6] = Partlx, 6, 6] 1: i1;
WARP;
ao

WARPAREA = WARP 1: —2— 1: 1;: 1: 5mm;
WARPTOTAL = WARPAREA 1: 4;

89

WARPER = Tablolo, {1, 1, 6], [j, 1, 6}];

PartliNARPER, 1, 1] = Partlu, 1, 1] 1: h:
PartIWARPER, 1, 2] = Partlu, 1, 2] 1: h:
PartIWARPER, 2, 1] = Partlu, 2, 1]1: n;
Part[WARPER, 2, 2] = Partlu, 2, 2] 1: 11;
Part[WARPER, 1, 3] = Partlu, 1, 311: h;
Part[WARPER, 3, 1] = PartlU, 3, 1]1: i1;
PartIWARPER, 2, 3] = Partlu, 2, 3] 1: h;
ParthARPER, 3, 2] = Partlu, 3, 2] 1: h;
PartIWARPER, 3, 3] = Part[U, 3, 3] 1: h;
Part[WARPER, 1, 4] = Part[U, 1, 4] 1: l1;
ParthARPER, 4, 1] = Part[U, 4, 1]1: h;
Part[WARPER, 1, 5] = Partlu, 1, 5] 1: h;
PartIWARPER, 5, 1] = Partlu, 5, 1]1: h;
PartIWARPER, 1, 6] = Partlu, 1, 6] 1: h;
PartMARPER, 6, 1] = Partlu, 6, 1]1: h;
PaItIWARPER, 2, 4] = Partlu, 2, 4] 1: l1;
PartIWARPER, 4, 2] = Partlu, 4, 2] 1: h;
Part]WARPER, 2, 5] = Partlu, 2, 5] 1: n;
PartMARPER, 5. 2] = Partlu, 5, 2] 1: n;
Part]WARPER, 2, 6] = Partlu, 2, 6] 1: h;
PartIWARPER, 6, 2] = Partlu, 6, 2] 1: i1;
Part[WARPER, 3, 4] = Partlu, 3, 4] 1: n;
Part[WARPER, 4, 3] = Parqu, 4, 3] 1: h;
Part[WARPER, 3, 5] = Part[U, 3, 5] 1: i1;
ParthARPER, 5, 3] = Partlu, 5, 3] 1: n;
Part[WARPER, 3, 6] = Partlu, 3, 6] 1: n;
ParthARPER, 6, 3] = Parqu, 6, 3] 1: h;
Part]WARPER, 4, 4] = Partlu, 4, 4] 1: h;
PartMARPER, 4, 5] = Part]U, 4, 5] 1: h;
ParthARPER, 5, 4] = Partlu, 5, 4] 1: n;
Part[WARPER, 4, 6] = Part[U, 4, 6] 1: l1;
PartMARPER, 6, 4] = Part“), 6, 4] 1: h;
ParthARPER, 5, 5] = Partlu, 5, 5] 1: n;
PartIWARPER, 5, 6] = Paqu, 5, 6] 1: 11;
PartIWARPER, 6, 5] = Partlu, 6, 5] 1: n;
ParthARPER, 6, 6] = Partlu, 6, 6] 1: h;
WARPER;

30

WARPERAREA = WARPER: ? 1: :02— 1: Sln[¢];
WARPERTOTAL = WARPERAREA 1: (1 + q) 4;

9O

RESIN = ramp, {1, 1, 5}, {1, 1, 5]]:
Part]RESlN,1,1]=Part]R, 1,1]1:p 1:11;
ParthESIN, 1, 2] = Parth, 1, 2]1: p 1: h;
Part[RESlN,2, 1] =Parth, 2, 1]1: p m;
ParthESlN, 2, 2] = Parth, 2, 2] 1: p 1: h;
ParthESlN, 1, 3] = Parth, 1, 3] 1:p 1: n;
Part]RESlN,3, 1] =Parth, 3, 1]1:p 1:11;
ParthESiN, 2, 3] = Parth, 2, 3] 1: p 1: 11;
ParthESIN, 3, 2] = Parth, 3, 2] 1: p 1: 11;
ParthESlN, 3, 3] = Parth, 3, 3] 1: p 1: h;
Part[RESiN, 1, 4] = Parth, 1, 4] 1: p 1: h;
111151153111, 4, 1] = Parth, 4, 1]1: p 1: 11;
ParthESlN, 1,5] =Part[R, 1, 5] 1: p 1: 11;
Part]RE6|N,5, 1] =Part[R, 5, 1]1:p 1:11;
ParthESlN, 1, 5] =Parth, 1, 511p 1: 11;
ParthESIN, 5, 1] =Part[R, 5, 1]1: p 1:11;
ParthESIN, 2, 4] = PartIR, 2, 4] 1: p 1: h;
ParthESIN, 4, 2] = Parth, 4, 2] 1: p 1: h;
ParthESiN, 2, 5] = Parth, 2, 5] 1: p 1: h;
Part[RESIN, 5, 2] = Parth, 5, 2] 1: p 1: h;
Part[RESIN, 2, 5] = Parth, 2, 5] 1: p 1: h;
911-1111551111, 5, 2] = Parth, 5, 2] 1: p 1: h;
ParthESIN, 3, 4] = Parth, 3, 4] 1: p 1: h;
ParthESIN, 4, 3] = Parth, 4, 3] 1: p 1: h;
ParthESIN, 3, 5] = PartIR, 3, 5] 1: p 1: h;
ParthESiN, 5, 3] = Partlﬂ, 5, 3] 1: p 1: h;
ParthESlN, 3, 5] = Parth, 3, 5] 1: p 1: h;
ParthESIN. 5, 3] = Partlﬂ, 5, 3] 1: p 1: h;
ParthESIN, 4, 4] = Parth, 4, 4] 1: p 1: h;
ParthESiN, 4, 5] = 9mm, 4, 5] 1: p 1: h;
ParthESIN, 5, 4] = Parth, 5, 4] 1: p 1: h;
ParthESlN, 4, 5] = Parth, 4, 5] 1: p 1: 11;
ParthESIN, 5, 4] = Parth, 5, 4] 1: p 1: h;
ParthESIN, 5, 5] = Part[R, 5, 5] 1: p 1: h;
ParthESlN, 5, 5] = Parth, 5, 5] 1: p 1: h;
ParthESIN, 5, 5] = Parth, 5, 5] 1: p 1: 11;
ParthESIN, 5, 5] = Parth, 5, 5] 1: p 1: h;
RESIN:

_ 'u ‘0 .
RESINAREA _ RESIN 1]?)1: {—2—} 1: 51m],
RESINTOTAL = RESINAREA15;

91

TOTAL = FLA1TOTAL + MATRIXTOTAL + F ILLTOTAL + F ILLERTOTAL +
WARPTOTAL + WARPERTOTAL + RESINTOTAL;

n21TOTAL

AVERAGE = . :
(n 1: a)2 1: sIn[¢]

 

or = lnvorsolAVERAGE]:

Print[”Ex="
1

Partlor,1,1]1:p1:t

 

Print]"Ey="
1
Partla, 2, 2] 111‘! :1: t

 

Print]*5,="]
1
Partla', 3, 3]1:p1:t

 

Print["G,,y="]
1

Partla, 6, leptt

 

Print]"0xz="]
1

Pafﬂa', 5, Sltptt

 

Printl'Gyz="]
1

PartIar, 4, 4I¥P¥t

 

Print]~v,,,,="
Partlar, 2, 1]

Farm, 1, 1]

 

Print]"1,,,="]
Partla, 1, 2]
Paula, 2, 2]

 

92

Prin1:["wxz ="
Partloz, 1, 3]

 

—Part[o:, 1, 1]

Print["vzx="]
Partloz, 1, 3]

 

_ Paula, 3, 3]

Print["1:,,="]
Partlor, 2, 3]

 

_ Partlar, 2, 2]

Print["wzy=“]
P3111022, 3]

 

Part[o:, 3, 3]

93

[ll

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

REFERENCES

Mouritz, A.P. “Review of Z-Pinned Composite Laminates,” Composites Part A, 38,
2383-2397, 2007.

Kim, J. and M. Sham, “Impact and Delamination Failure of Woven-Fabric
Composites,” Composites Science and Technology, 60, 745-761, 2000.

Atas C. and D. Liu. “Impact Response of Woven Composites with Small Weaving
Angles,” International Journal of Impact Engineering, 35, 80-97, 2007.

<http://www.3tex.com>, 2007.

G. Coppen. “Effects of Three-Dimensional Geometry on Penetration and
Perforation Resistance,” Masters Thesis, Mechanical Engineering Dept., Michigan
State Univ., East Lansing, MI, 2004.

Raju, I. S. and J .T. Wang. “Classical Laminate Theory Models for Woven Fabric
Composites,” Journal of Composites Technology & Research, 16(4), 289-303, Oct
1994.

R.A. Naik, “Analysis of Woven and Braided Fabric Reinforced Composites,” NASA
Contract Report CR-I94930, 1994.

Naik, N.K. and V.K. Ganesh, “An Analytical Method for Plaine Weave Fabric
Composites”, Composites, 26, 281-289, 1995.

Scida, D., Aboura, Z., Benzeggagh, M.L., and E. Bocherens, “A Micromechanics
Model for 3D Elasticity and Failure of Woven-Fibre Composite Materials,”
Composites Science and Technology, 59, 505-517, 1999.

Hur, H., Johnson, E. R., and R. K. Kapania, “A Simple Homogenization of
Degraded Micro—Cracked Plain Woven and Braided Textile Composites,” The 47"I
AIAA Structural Dynamics and Materials Conference, Mayl-May4, 2006, Rhode
Island, Newport, AIAA-2006-1692.

Hsieh, C. and W. Hsing. “Elastic and Thermal Expansion Behavior of Two-Phase
Composites,” Materials Science and Engineering: A, 425(1-2), 349-360, 2006.

JD. Whitcomb, “Three-Dimensional Stress Analysis of Plain Weave Composites,”
Composite Materials: Fatigue and Fracture, 3, 417-438, 1991.

R.L. Foye, “Finite Element Analysis of the Stiffness of Fabric Reinforced
Composites,” NASA Contract Report CR-189597, 1992.

94

[14] Baker, A., Dutton, S., and D. Kelly. “Structural Analysis,” Composite Materials for
Aircraft Structures, 2“(1 edition, Reston: AIAA, 2004. 171-211.

[15] K. Rosario. “Harness and Quasi-3D Weaving Effects on Composite Material
Impact and Compressive Properties,” Masters Thesis, Mechanical Engineering
Dept., Michigan State Univ., East Lansing, MI, 2008.

95

 

 

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