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THESIS

’7
4.

2.0m

Date

0-7 639

This is to certify that the

thesis entitled

A Simplified Discrete-Tow Model for Modeling

Stiffness and Failure within Woven Composites

presented by

Craig Ryan Carrier

has been accepted towards fulﬁllment
of the requirements for

Master's Engineering Mechanics

 

 

degree in

Mam

Major professor

 

/' 0/ 3/ /w

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A SIMPLIFIED DISCRETE-TOW MODEL FOR MODELING STIFFNESS AND
FAILURE WITHIN WOVEN COMPOSITES

By

Craig Ryan Carrier

A Thesis

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

MASTER’S OF SCIENCE

Materials Science and Mechanics

2000

 

 

 

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ABSTRACT

A SIMPLIIFIED DISCRETE-TOW MODEL FOR MODELING STIFFNESS AND
FAILURE WITHIN WOVEN COMPOSITES

BY

Craig Ryan Carrier

Using a simpliﬁed discrete-tow model of a woven composite structure, it is
possible to accurately predict the stiffness of a woven composite. The simpliﬁed
discrete-tow model is based on the idea of discretely representing the ﬁber tows
and resin independently as beam and shell ﬁnite elements respectively. The use
of the discrete-tow model in progressive failure allows for the ability to better
predict the behavior and physical orientation of the tows throughout the analysis
in a manner that is more computationally efﬁcient than a fully three-dimensional
model. In the present study progressive failure is conducted using stiffness
reduction in both the beam and shell elements of the discrete-tow model to

represent ﬁber and matrix damage respectively.

Copyright by
Craig Ryan Carrier
2000

To my loving parents

ACKNOWLEDGMENTS

This work was sponsored by the Department of Energy and the Automotive
Composites Consortium under USAMP/DOE Co-operative agreement DE - FCO5
- 95OR22363. | wish to express my gratitude for the ﬁnancial support as

well as the many helpful comments that have molded this work. I would also like

to thank Dr. Averill for his thoughtful guidance and depth of knowledge that he
provided for me.

 

 

 

 

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USIC

CHAPT

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TABLE OF CONTENTS

LIST OF TABLES .............................................................................................. VIII

LIST OF FIGURES ............................................................................................. IX

CHAPTER 1 INTRODUCTION ............................................................................. 1

1.1 Introduction .......................................................................................... 1

1.2 Literature Review of Stiffness Prediction Models ................................. 2

1.3 Organization of the Thesis ................................................................... 8

CHAPTER 2 THREE-DIMENSIONAL ANALYSIS OF A WOVEN COMPOSITE 10

2.1 Introduction ........................................................................................ 10

2.2 Model Description .............................................................................. 11

2.2.1 Loading ................................................................................. 12

2.2.2 Geometry .............................................................................. 12

2.2.3 Material Properties ............................................................... 14

2.3 Finite Element Models ........................................................................ 14

2.3.1 Fully Three-Dimensional ....................................................... 15

2.3.2 Brick and Beam .................................................................... 15

2.3.3 Brick and Plate ..................................................................... 16

2.4 Numerical Results .............................................................................. 16

2.4.1 Load case 3: Axial Loading .................................................. 17

2.4.2 Load case b: Bending about the Y-axis ................................ 18

2.4.3 Load case c: Bending about the Z-axis ................................ 19

2.4.4 Load case d: Torsional Loading ............................................ 20

2.5 Conclusions ........................................................................................ 20

CHAPTER 3 THE SIMPLIFIED DISCRETE-TOW MODEL ................................ 42

3.1 Introduction ........................................................................................ 42

3.2 The Simpliﬁed Discrete-Tow Model .................................................... 42

3.2.1 Geometry of a Triaxial Braid ................................................. 43

3.2.2 Effective Resin Properties of a Triaxial Braid ....................... 44

3.2.3 Effective Resin Properties of a Plain Weave ........................ 49

3.2.4 Tow Curvature Modiﬁcation .................................................. 49

3.2.5 Determining Effective Properties .......................................... 52

3.3 Numerical Results .............................................................................. 53

3.3.1 Analysis of a Plain Weave .................................................... 53

3.3.2 Effect of Waviness Ratio on Effective Material Properties....54

3.3.3 Analysis of a Triaxial Braid ................................................... 55

3.4 Conclusions ........................................................................................ 55
CHAPTER 4 MODELING PROGRESSIVE FAILURE USING THE SIMPLIFIED

DISCRETE-TOW MODEL ........................................................................ 73

4.1 Introduction ........................................................................................ 73

vi

CHAP”.

 

 

 

BIBLIO

4.2 Progressive Failure Model .................................................................. 73

4.2.1 ﬁnite element mesh .............................................................. 74

4.2.2 Modelling progressive failure ................................................ 74

4.2.3 Progressive Failure Algorithm ............................................... 75

4.3 Numerical Results .............................................................................. 76

4.4 Conclusions ........................................................................................ 77
CHAPTER 5 CONCLUSIONS ............................................................................. 91
5.1 Conclusions ........................................................................................ 91

5.2 Future Work .................................. . ..................................................... 91
BIBLIOGRAPHY ................................................................................................. 93

vii

 

 

 

 

Table 2

 

 

Table 3

LIST OF TABLES

Table 2.1 Load case summary. .......................................................................... 22
Table 2.2 Tow geometrical properties. ............................................................... 22
Table 2.3 Material properties of the constituent materials of the woven
composite beam. ......................................................................................... 22
Table 2.4 Mid-span deﬂection data of points A and B for each load case ......... 23
Table 3.1 Material properties for comparison of predicted stiffness ................... 57
Table 3.2 Comparison of results for a plain weave- ........................................... 57
Table 3.3 Geometric data for the three waviness ratios studied ......................... 58
Table 3.4 Material properties for all waviness ratios studied ............................... 58
Table 3.5 Comparison of predicted effective properties for a plain weave
composite. ................................................................................................... 59
Table 3.6 Geometric and braid data for triaxial weaves. .................................... 60
Table 3.7 Properties for the constituent materials of the triaxial weaves. .......... 60
Table 3.8 Comparison of effective properties for a triaxial weave with numerical
and experimental methods ........................................................................... 61
Table 4.1 Failure criteria used in the resin and tow ............................................ 79
Table 4.2 Applied boundary for the linear and failure cases studied. ................. 79

viii

 

 

 

 

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Figure 2.1 Geometry of the cantilever beam with a single undulating ﬁber ........ 24

Figure 2.2 Reﬁned brick and beam model. ........................................................ 25
Figure 2.3 Reﬁned brick and plate model. ......................................................... 26
Figure 2.4 Fully three-dimensional ﬁnite element model. ................................... 27
Figure 2.5 Arrangement of the brick elements in the brick and beam and brick
and plate models. ........................................................................................ 28
Figure 2.6 Axial loading condition applied to each model. ................................. 29

Figure 2.7 Normal (X-Y bending) loading condition applied to each model. ...... 29
Figure 2.8 Transverse (X-Z bending) loading condition applied to each model. 30

Figure 2.9 Torsional load case applied to each model. ...................................... 30
Figure 2.10 Piece-wise linear approximation of the ﬁber tow path-line using
beam and shell elements (a) coarse model, (b) reﬁned model. ................... 31
Figure 2.11 Axial displacement (u) along the tow centerline of the coarse brick &
beam and brick & plate models due to an axial load .................................... 32
Figure 2.12 Axial displacement (u) along the tow centerline of the reﬁned brick &
beam and brick & plate models due to an axial load .................................... 32
Figure 2.13 Normal displacement (v) along the tow centerline of the coarse brick
& beam and brick & plate models due to an axial load. ............................... 33
Figure 2.14 Normal displacement (v) along the tow centerline of the reﬁned brick
& beam and brick & plate models due to an axial load. ............................... 33
Figure 2.15 Axial displacement (u) along the tow centerline of the coarse brick &
beam and brick & plate models due to a normal (y) load. ............................ 34
Figure 2.16 Axial displacement (u) along the tow centerline of the reﬁned brick &
beam and brick & plate models due to a normal (y) load. ............................ 34
Figure 2.17 Normal displacement (v) along the tow centerline of the coarse brick
& beam and brick & plate models due to a normal (y) load .......................... 35
Figure 2.18 Normal displacement (v) along the tow centerline of the reﬁned brick
8 beam and brick & plate models due to a normal (y) load .......................... 35
Figure 2.19 Axial displacement along the tow centerline of the coarse brick &
beam and brick & plate models due to a transverse (2) load. ...................... 36
Figure 2.20 Axial displacement along the tow centerline of the reﬁned brick &
beam and brick & plate models due to a transverse (z) load. ...................... 36
Figure 2.21 Normal displacement (v) along the tow centerline of the coarse
brick & beam and brick & plate models due to a transverse (2) load. .......... 37
Figure 2.22 Normal displacement (v) along the tow centerline of the reﬁned
brick & beam and brick & plate models due to a transverse (2) load. .......... 37
Figure 2.23 Transverse displacement (w) along the tow centerline of the coarse
brick & beam and brick & plate models due to a transverse (2) load. .......... 38
Figure 2.24 Transverse displacement (w) along the tow centerline of the reﬁned
brick & beam and brick & plate models due to a transverse (2) load. .......... 38
Figure 2.25 Axial displacement (u) along the tow centerline of the coarse brick
& beam and brick & plate models due to a torsional load. ........................... 39

 

 

 

 

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Figure 2.26 Axial displacement (u) along the tow centerline of the reﬁned brick

& beam and brick & plate models due to a torsional load. ........................... 39
Figure 2.27 Normal displacement (v) along the tow centerline of the coarse
brick & beam and brick & plate models due to a torsional load .................... 40
Figure 2.28 Normal displacement (v) along the tow centerline of the refined
brick & beam and brick & plate models due to a torsional load .................... 40
Figure 2.29 Transverse displacement (w) along the tow centerline of the coarse
brick & beam and brick & plate models due to a torsional load .................... 41
Figure 2.30 Transverse displacement (w) along the tow centerline of the reﬁned
brick 8. beam and brick & plate models due to a torsional load .................... 41
Figure 3.1. (a) Plain weave and (b) one possible repeating volume element of a
plain weave. ................................................................................................. 62

Figure 3.2 (a) Geometry of a triaxial braid (b) cross section of the triaxial braid
made perpendicular to the axial tows (c) cross section of the triaxial braid
made alone the direction of the braider tows. .............................................. 63

Figure 3.3 The area enclosed by A,B,C,D represents one possible RVE for a
[0&9] triaxial braided composite. ................................................................. 64

Figure 3.4 (a) Assembled ﬁnite element model of the unit cell and (b) exploded
view of the ﬁnite element model for a plain weave material system ............. 65

Figure 3.5 (a) Discretized representation of a RVE in a triaxial braid. (b)
Assembled and (c) exploded views of the ﬁnite element model for a triaxial

braid ............................................................................................................. 66
Figure 3.6 Division of the RVE of a triaxial braid into two regions for calculation of
the effective properties (a) plan view (b) side view. ..................................... 67
Figure 3.7 (a) Curved ﬁber tow. (b) Straight ﬁber tow. ...................................... 68
Figure 3.8 Curved beam on an elastic foundation (a) geometry of the undulating
region and (b) applied load. ......................................................................... 69
Figure 3.9 Geometry deﬁning the waviness ratio (A =Lub /LI) ............................. 70
Figure 3.10 Plot of the effective stiffness E11 as the waviness ratio (A) is
increased. .......................................................................... . .......................... 71
Figure 3.11 Plot of the effective Poisson’s ratio v12 as the waviness ratio (It) is
increased. .................................................................................................... 71
Figure 3.12 Plot of the effective shear modulus G12 as the waviness ratio (A) is
increased. .................................................................................................... 72
Figure 4.1 Implementation of the progressive failure algorithm. ........................ 80
Figure 4.2 Stress strain curve illustrating progressive damage (not immediate).
..................................................................................................................... 81
Figure 4.3 Stress strain curve illustrating immediate material failure. ................ 81
Figure 4.4 Illustration of applied boudary conditions to (a) a plain weave and (b)
a triaxial braid. ............................................................................................. 82
Figure 4.5 Force deﬂection curve for a plain weave under tensile loading
(without failure). ........................................................................................... 83
Figure 4.6 Force deﬂection curve for a plain weave under compressive loading
(without failure). ........................................................................................... 83

Figure 4.7 Linear validation of one RVE of a plain weave under a tensile load. 84

 

 

 

 

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Figure 4.8 Linear validation of one RVE of a plain weave under a compressive

load .............................................................................................................. 84
Figure 4.9 Force deﬂection curve for a plain weave under tensile loading (with
failure) .......................................................................................................... 85
Figure 4.10 Force deﬂection curve for a plain weave under compressive loading
(with failure). ................................................................................................ 85
Figure 4.11 Failure validation of one RVE of a plain weave under a tensile load.
..................................................................................................................... 86
Figure 4.13 Failure validation of one RVE of a plain weave under a compressive
load. ............................................................................................................. 86
Figure 4.14 Force deﬂection curve for a triaxial weave under tensile loading
(without failure). ........................................................................................... 87
Figure 4.14 Force deﬂection curve for a triaxial weave under compressive
loading (without failure) ................................................................................ 87
Figure 4.15 Linear validation of one RVE of a triaxial weave under a tensile load
(without failure). ........................................................................................... 88
Figure 4.16 Linear validation of one RVE of a triaxial weave under a
compressive load (without failure). .............................................................. 88
Figure 4.18 Force deﬂection curve for a triaxial weave under tensile loading (with
failure). ......................................................................................................... 89
Figure 4.19 Force deﬂection curve for a triaxial weave under compressive
loading (with failure) ..................................................................................... 89
Figure 4.19 Failure validation of one RVE of a triaxial weave under a tensile
load. ............................................................................................................. 90
Figure 4.21 Failure validation of one RVE of a triaxial weave under a
compressive load. ........................................................................................ 90

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Chapter 1
INTRODUCTION

1.1 Introduction

The popularity and interest in composites has grown considerably in
recent years. Today laminated composite structures are engineered lighter,
stronger and stiffer than their traditional counterparts. Typically, laminated
composites are ideal for situations where the majority of the load is applied in the
plane of the composite. The use of composites in structural applications has
been revolutionized by the introduction of woven textiles composites. While
traditional unidirectional composites have great strength and stiffness in the
direction of the ﬁbers, woven fabric composites (WFC’s) have high strength and
stiffness in both the warp and ﬁll directions. Additionally, the woven nature of
WFC’s makes them less likely to fail due to delamination while increasing a
structure’s ability to carry large transverse loads.

The goal of the present research is to develop efﬁcient methods and tools
for the analysis of woven composite structures. First, an efﬁcient stiffness
prediction model is developed using traditional beam and shell ﬁnite elements for
use in analyzing two-dimensional woven fabrics. This stiffness prediction model
was then combined with a progressive failure algorithm to model the progressive
failure of large-scale woven composite structures.

A brief review of literature concerning the various stiffness prediction

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Due in large part to integrated multi-directional ﬁber reinforcement, textile
composite materials have many advantageous mechanical properties, including
excellent out-of-plane strength and impact resistance. The ability of textile
composites to conform to irregular shapes (i.e., to drape) and to integrate several
parts into one piece makes them especially appealing for many applications.
However, the same features that give rise to these sought-after mechanical
properties also make it very difﬁcult to analytically predict the mechanical
response of these materials, thus complicating the design process.

Textile composite materials contain a very complex microstructure, where
here the term microstructure is used to refer to the geometry and path of tows
that create the ﬁber network and the resin regions that lie within and between the
tows. The standard deﬁnition of the term microstructure does not always apply to
textile composite materials, since often the tow size and the size of a repeating
unit cell of the tow network are of the same order as the size of important
structural features. Therein lies the primary difﬁculty in analyzing many textile
composite materials: the microstructure is too complicated to be represented
explicitly throughout the structure yet extremely large tows may be too large to
homogenize.

Numerous analytical models have been developed to predict overall
stiffness and strength of textile composite materials. Excellent reviews of the
many classes of models and their assumptions are given by Cox and Flanagan

[1] and by Tan, et al. [2].

 

 

 

 

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Early research conducted by lshikawa and Chou [3-5] led to the
development of three homogenization techniques. The ﬁrst was the mosaic
model [3], which idealized a textile composite as an assembly of asymmetric
cross-ply laminates. The mosaic model divided the two—dimensional problem into
two one-dimensional models, the series (iso-stress) model and the parallel (iso-
strain) model. The mosaic model however did not account for the undulation or
continuity of the ﬁbers.

The Fiber Undulation Model (FUM) [4] was developed to consider the
continuity and undulation of the ﬁbers in one direction. The FUM is essentially a
series model, however, the extension, bending-extension, and bending stiffness
matrices of CLT are a function of the coordinate along the axis of the undulating
tow.

lshikawa and Chou [5] also developed the bridging model for predicting
- the behavior of satin weave composites. The bridging model idealizes the
hexagonal shape of the unit cell as a square to simplify calculations. The unit
cell is then divided into ﬁve regions, four of which are simply treated as cross-ply
laminates, while the remaining region contains the interlaced ﬁbers. The
continuity and undulation of the warp ﬁbers is ignored due to the fact that the
load is applied in the ﬁll direction and the effect of the warp ﬁbers is expected to
be small. The lower stiffness of the warp ﬁbers in the direction of the loading
causes the ﬁll ﬁbers to carry more of the applied load and act as a “bridge" to aid

in the load distribution throughout the structure.

 

 

 

 

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Pastore and Gowayed [6] proposed a self-consistent Fabric Geometry
Model (FGM). This model was a modiﬁcation to the original FGM [7] that treated
the ﬁbers and matrix as composite rods with its correct orientation and local
stiffness tensor. The local stiffness tensor of each rod is then transformed into
the global coordinate system and assembled to obtain the global composite
stiffness tensor based on each rod’s relative volume fraction (stiffness
averaging). The FGM was also used with an energy-based analysis by Ma, et al
[8] to predict the effective properties of three-dimensional braided textiles.

The Fiber Inclination Model (FIM) introduced by Yang, et al. [9] is another
method based on laminate theory. In the FIM the ﬁber tows are represented as
inclined unidirectional lamina. Because of this idealization the FIM does not
account for ﬁber interlock and bending.

Naik and Ganesh [10], Raju and Wang [11], and Aitharaju and Averill [12]
developed improved laminate theories that did not make the simplifying
assumptions associated with the mosaic model, the ﬁber undulation model, or
the bridging model. Without these simpliﬁcations these new theories could
account for undulation in both the warp and ﬁll directions along with ﬁber
continuity. Each method divided the unit cell of a textile into subregions.
Effective properties for each subregion were calculated and then reassembled
into the global model.

Two new modeling techniques that were introduced by Naik and Ganesh
[10] were the slice array model (SAM) and the element array model (EAM). The

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separately and then reassembled the slices using the isostrain formulation. The
second method, the EAM, also discretized the unit cell by slicing it, but the slices
could be made either parallel or transverse to the direction of loading. The slices
were again subdivided into elements, and the individual elements were
assembled in parallel or series to obtain the slice properties. The slices were
then assembled in series or parallel to obtain the effective properties of the unit
cell.

Raju and Wang [11] developed improvements to the ﬁber undulation and
bridging models, while retaining their simplicity without making any simplifying
assumptions. The unit cells of a textile composite were divided into subregions. ‘
Effective properties for these subregions were calculated through laminate theory
and reassembled using stress averaging.

Aitharaju and Averill [12] also subdivided the unit cell. Each subregion
was modeled as an eight noded brick element. The material properties of the
constituent material in each element were transformed into a local coordinate
system and combined through the use of an effective modulus. The ﬁnite
element method was then utilized to perform the analysis of a textile composite
mat.

Sankar and Marrey [13] used the selective averaging model (SAM) which
was similar to the slice array and element array models of Naik and Ganesh [10].
However, the SAM allowed the stiffness or compliance coefﬁcients to be

averaged selectively based on either an isostress or isostrain assumption.

 

 

 

 

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Finite element models of the explicit textile structure are tedious to
construct and computationally very expensive to solve. Some research has been
conducted on the fully three-dimensional geometry to study the effect of various
parameters on the effective moduli [14, 15]. Several other techniques exist that
are hybrids of homogenization and the ﬁnite element method. This is achieved
by homogenizing the material properties on the microscale level and
representing them with ﬁnite elements on the macro scale.

Dasgupta et al. [14] developed a two-scale homogenization method
that fully discretized the unit cell of a plain weave in the form of a ﬁnite element
model. The macroscale stress and strain ﬁelds were volume averaged over the
structure yielding the homogenized composite properties.

Whitcomb [15] applied the ﬁnite element method to the study of the effect
waviness ratio on the effective moduli. Predicted moduli for waviness ratios of
0.167, 0.25, and 0.5 were studied and quantitative comparisons were made with
a [0/90]3 tape laminate.

Whitcomb et al [16] also introduced the use of macro ﬁnite element
analysis to study woven composites. The macro element was subdivided into
subregions or subelements that modeled the ﬁber tows explicitly. A single
approximation for the displacement ﬁeld was used throughout the macro element
rather than a multiple ﬁeld approximation that would assume a different ﬁeld
within each subelement. In some instances the single ﬁeld approximation gave
very poor results compared to a multiple ﬁeld approximation. However, the

single ﬁeld method was computationally much cheaper than the multiple ﬁeld

 

 

 

 

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be used with good success on a wide range of problems.

Woo and Whitcomb [17] also used the macro element combined with a
local/global ﬁnite element analysis. The scale of textile composites contributes a
great deal to the difﬁculties that arise in their analysis. Often, the microstructure
of a textile composite is far too complex to model explicitly, while at the same
time it is too large to model as a single homogenized material. A local/global
ﬁnite element analysis can be used to obtain the overall response of the structure
with a coarse global mesh, then reﬁned meshes are used to capture local effects
of interest.

Finite element analysis was used by Marrey and Sankar [18] to determine
the laminate A, B, and D matrices along with the thermal expansion coefﬁcients
of a textile composite. The unit cell of a textile composite is modeled with three-
dimensional ﬁnite elements. The effective properties of the unit cell are
computed by assuming a uniform state of strain at the macroscale level.

Blackketter et al. [19] used the ﬁnite element method to model nonlinear
material behavior. This was combined with a progressive failure algorithm to
represent the effects of damage through the use of stiffness reduction. The
three-dimensional model of the unit cell represented the ﬁber tows and resin
individually for examining tensile and shear loading.

The Binary Model developed by Xu et al. [20] idealizes a three
dimensional weave using ﬁnite elements. Two-noded line elements are used to

represent the axial properties of the ﬁber tows while the transverse stiffness,

 

 

 

 

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SUCIl as
t0W crO:
basic to
demOns

a Varieti

1-3 Org

USe in i
modelin,

Needed 1

shear stiffness and Poisson’s ratio of the composite are model with three-
dimensional “effective medium” elements.

Lei et al. [21] combined computer aided geometric modeling (CAGM)
along with ﬁnite element procedures to represent three—dimensional braided
composites. The identiﬁed unit cells of a braided composite were modeled as
space structures and discretized using ﬁnite elements. A ﬁnite element analysis
was then performed to predict the elastic behavior of the composite stmcture.

In the present study, a model is developed for simulating the structural
behavior of textile composite materials throughout a wide range of scale sizes —
from tow level to structural level. The model attempts to capture the essential
features of the material behavior in the simplest possible form in order to allow
modeling of large sections of a textile composite structure. The main objective in
developing such a model is to allow the investigation of geometrical irregularities
such as uneven tow spreading or compacting, tow waviness, spatial variations in
tow cross-section due to tow pinching, spatially varying tow twist, etc. Here, the
basic formulation of the discrete tow model is discussed in detail and its utility is
demonstrated for the prediction of effective properties and progressive failure for

a variety of plain weave and triaxial braid materials.

1.3 Organization of the Thesis

The goal of the present research is to develop fast and efﬁcient tools for
use in analyzing woven fabric composites. Chapter 2 describes a simpliﬁed
modeling technique for a woven composite which is faster in terms of man hours

needed to build the model and computational solution time required. Rather than

 

 

 

 

explit

 

eleme

tow.

hexahe

repress

repress

 

 

discret
low we

Ofa cur

mOdeI

SthtL

from

DIESei

Each‘

explicitly model the ﬁber tow and resin matrix as fully three-dimensional
elements, traditional beam and shell elements are used to represent the ﬁber
tow. These tow elements are then ‘woven’ through a repeating array of
hexahedral brick elements that represent the resin matrix.

Chapter 3 extends the idea of using a simpliﬁed model by again
representing the ﬁber tow with beam elements. However, the resin matrix is
represented using traditional plane stress shell elements. While this simpliﬁed
discrete-tow model is two-dimensional, it captures the three-dimensional effect of
tow waviness through a stiffness reduction factor that is derived using the theory
of a curved beam on an elastic foundation.

Using the simpliﬁed discrete-tow model, Chapter 4 introduces the concept
of combining this modeling technique with a progressive failure algorithm to
model damage and failure propagation within a large-scale woven composite
structure.

The ﬁnal chapter, Chapter 6, presents the conclusions that can be drawn
from the present research. Recommendations for future study are also
presented in this chapter.

In an effort to facilitate ease of reading this thesis, tables and ﬁgures of

each chapter can be found immediately following the corresponding chapter.

 

 

 

 

 

 

2.1 lnt

scales.
and we
The 0Vl
How-eve

size of l

 

behavio
homogeI
mlcroles
fraction
Structure

C
Woven (
mOdels
these n.
Computz
USe SOr
iheSe IT

make si

“m“ thei

Chapter 2

Three-dimensional analysis of a woven composite

2.1 Introduction

The problem of analyzing woven composites is an issue of two conﬂicting
scales. The micro-level represents the scale of the weave. Tow shape, spacing
and waviness ratio are examples of dimensions on the order of the microlevel.
The overall shape and geometry of the global structure is considered macrolevel.
However, the size at the microlevel is such that it is small compared to the overall
size of the structure itself, but often it is not small enough that the microstructural
behavior and effects can simply be smeared together and represented as one
homogenized material. This means that small changes or variations at the
microlevel such as a change in tow spacing or a variation of the ﬁber volume
fraction can have a significant impact on the overall behavior of the global
structure.

Currently there are two dominant approaches used in the analysis of
woven composites. The ﬁrst method has been to create fully three-dimensional
models of the woven fabric and its underlying microstructure. However, building
these models is extremely time consuming and requires a prohibitive amount of
computational time and resources for everyday use. The second method is to
use some form of homogenization scheme to simplify the analysis. Although
these methods require signiﬁcantly less computing resources, they in general
make simplifying assumptions that either degrade the quality of the results or

limit their robustness.

10

 

 

 

 

 

CL
computai
fully thre
between
modeling
a woven
represen'
the resin
beam or
through
Cerlterline
through r
use to se
di"lensio
VisuaIIZat
has Occu

ComPOSIiI

2.2 Mode

Th.
Study COn

Currently there is a great need for a modeling technique that combines the
computational beneﬁts of homogenization approaches with the robustness of a
fully three-dimensional analysis. Clearly some form of tradeoff is necessary
between the two approaches. This chapter presents two novel methods for
modeling a woven composite structure. The ﬁrst method models the ﬁber tows of
a woven composite as one-dimensional beam elements while the second method
represents the ﬁber tows as two-dimensional shell elements. In both methods
the resin matrix is modeled using hexahedral brick elements. The appropriate
beam or shell element that is used to discretize the ﬁber tow is then woven
through the three-dimensional grid of resin elements along the ﬁber tow
centerline. The attachments of the tow and resin elements are maintained
through nodal connectivity. The discrete modeling of the tows and resin gives
rise to several advantages of both the traditional homogenization and fully three-
dimensional methods currently used. The discrete models allow for the easy
visualization of the structure and the locations of the ﬁber tows after deformation
has occurred. Additionally, these models can be easily expanded to model large

composite structures in a computationally efﬁcient manner.

2.2 Model Description

The general model that was the basis for assessing the models in this
study consisted of a rectangular composite cantilever beam, that is intended as
an idealized section from a woven composite. The composite cantilever beam

was assumed to be a strip of resin with a single ﬁber tow woven along the axis of

11

the beam (Figure 2.1). Three models were derived from the base model. The
ﬁrst of these was a brick and beam model (Figure 2.2), which represented the
ﬁber tows and resin as beam and hexahedral brick elements respectively. The
second model was the brick and plate model (Figure 2.3), which used the same
hexahedral brick elements to represent the resin as in the brick and beam model.
However, in this case, the ﬁber tows were modeled using two-dimensional shell
elements. The ﬁnal model was a fully three-dimensional model (Figure 2.4),
which used tetrahedral elements to discretize both the resin and the ﬁber tow as
fully three-dimensional objects. The fully three-dimensional model was to serve

as a means of validating the previously discussed models.

2.2.1 LOADING

Four load cases were used to study the proposed ﬁnite element models.
Each case was intended to analyze a speciﬁc characteristic of the proposed
model and its ability to predict the fully three-dimensional equivalent behavior.
The four load cases are summarized in Table 2.1 and graphical illustrations of

each respective load case can be seen in Figure 2.6 - Figure 2.9.

2.2.2 GEOMETRY

The geometrical description of the model is found in Figure 2.1. The same
exterior dimensions were used for each of the three models. Parameters speciﬁc

to the geometry of the ﬁber tow were dependent on the type of model used.

12

The undulation of the ﬁber tow was assumed to be sinusoidal along the axis of
the beam. The cross-sectional shape of the tow was assumed to be lenticular
and to have the same sinusoidal shape as the tow centerline. The equations for
the tow shape and other pertinent data are summarized in Table 2.2. In each of
the models the path line of the beam or shell elements followed the centerline of
the ﬁber tow.

In the case of the brick and beam model it was only necessary to assign a
cross-sectional area and moments of inertia to the beam elements.
Consequently, the tow cross sectional area and moments of inertia were directly
assigned to the beam elements.

Modeling the ﬁber tow with shell elements in the brick & plate model
involves more consideration. It is desirable for the shell elements to have both
the correct axial stiffness along with the bending stiffness. However, the cross-
sectional shape of the ﬁber tow is assumed to be lenticular, while the cross-
sectional shape of a shell element is rectangular. This mismatch in cross-
sectional shapes makes it impossible to capture the correct axial and bending
stiffness of a lenticular cross-section with a simple rectangle. Noting the previous
argument, it is possible to make a sacriﬁce in the model’s accuracy to help
improve its ease of use. Therefore, the width and cross-sectional area of the
shell element and the lenticular ﬁber tow were chosen to be identical. Knowing
the width and area of the rectangular cross-section, the element thickness could

then be easily found.

13

2.2.3 MATERIAL PROPERTIES

The goal of this study was not to compare with experiments, but rather to
investigate the viability of the proposed models. To simplify the analysis process
the material properties used in this analysis are approximate values that are
typical of a glass/epoxy woven composite. Additionally, to aid in the
simpliﬁcation of the problem, the materials were assumed to be linearly isotropic.

The properties used in the analysis can be found in Table 2.3.

2.3 Finite Element Models

In total, six ﬁnite element models were created for the purposes of this
study, and their properties are summarized in Table 2.3. For the fully three-
dimensional, brick and beam and brick and plate cases studied, two models were
created of each case with different mesh reﬁnements. Similar models with
different levels of mesh reﬁnement serve several purposes. At the most basic
level, multiple models of the same problem help to eliminate possible user errors,
which could be noticed if large discrepancies appeared between the solutions.
Secondly, it is possible to conduct a convergence study with models of varying
mesh reﬁnement. This allows for the ability to see what level of reﬁnement is
needed when solving similar problems and to make sure that the solution is

converging in the right manner.

14

 

 

 

 

2.3.1 FL

A .
using 10
of the ﬁb

tow/resin

 

 

2.3.2 BR

Th!
beam. TI
block of
CC’mpo'siti
Simple St;
each Othe
beam ele
tow centr
elements
nodes_ A
rare that
centemnE
mess of
approxim

the path

2.3.1 FULLY THREE-DIMENSIONAL

A complete three—dimensional representation of the geometry was created
using 10 noded tetrahedral elements (see Figure 2.4). The ﬁnite element mesh
of the ﬁber tow and resin were created as separate objects. However, at the

tow/resin interface mesh connectivity was maintained.

2.3.2 BRICK AND BEAM

The brick and beam concept is an idealized version of the composite
beam. The ﬁrst step in creating the model was to construct a three-dimensional
block of 8-noded hexahedral elements with the same dimensions as the
composite cantilever beam. The hexahedral brick elements were arranged in a
simple stacked conﬁguration that resembled boxes placed next to and on top of
each other (see Figure 2.5). The ﬁber tow was then added by ‘weaving’ 2-noded
beam elements through the block of hexahedral resin elements along the ﬁber
tow centerline. The connectivity between the beam elements with the brick
elements was accomplished by simply attaching the elements at coincident
nodes. As a result of the stacked conﬁguration of the brick elements, it was quite
rare that the nodes of the brick elements coincided with the path of the tow
centerIine. Therefore, it was necessary to attach the beam elements to the
nodes of the brick elements that were closest to the tow centerline. This
approximation, combined with the linear shape of the beam elements, resulted in

the path of the actual ﬁber tow being represented as piece-wise linear (see

15

 

 

 

 

Figure 2
path cor
were bu‘
accurate

composit

 

2.3.3 BR

Sir
same ba
matrix (5.
elements
model al:
and plate

DOint ICC

Figure 2.10). Therefore, as the model is reﬁned, the approximation of the tow
path converges to the actual path of the ﬁber tow. Coarse and reﬁned models
were built in order to examine the necessary levels of reﬁnement required for
accurate results. The combination of the brick and beam elements to form the

composite beam model is illustrated in Figure 2.2.

2.3.3 BRICK AND PLATE

Similar to the brick and beam model, the brick and plate model uses the
same background mesh of hexahedral brick elements to represent the resin
matrix (see Figure 2.5). However, the ﬁber tow is modeled with 4~noded shell
elements (see Figure 2.3). As with the brick and beam model, the brick and plate
model also represents the path of the ﬁber tow as piece-wise linear. The brick
and plate model was conceived in order to overcome certain shortcomings of the

brick and beam model. These shortcomings are discussed later in this chapter.

2.4 Numerical Results

The differences between the brick models and the fully three—dimensional
models, such as the ﬁber volume fraction and the fundamental difference
between the ﬁnite element models, made it difﬁcult to compare results. For this
reason two different forms of the results are presented. First, ﬁber tow path-line
deﬂection results are compared and followed by deﬂection results of a single

point located on the structure.

16

Path-line deﬂection results of the analyses for each of the models are
shown in Figures 2.11 through 2.30. Each of the ﬁgures depicts a particular
displacement component of the ﬁber tow centerline along the axis of the
composite beam for a speciﬁed loading condition.

Table 2.4 contains deﬂection data from the top of the mid-span of the
composite beam. This location was chosen because its distance from where the
boundary conditions and loads were applied was great enough that their effect
on the solution results would be minimized. It is also important to note that the
three models were fundamentally different from each other and large variations in
the analysis could occur. This was most evident where the loads were applied.
In the models where the resin was represented with brick elements, extremely
large displacements occurred because the resin is more compliant than the ﬁber
tow and there was a considerably larger amount of resin in the brick models due

to the resin volume fraction equaling one.

2.4.1 LOAD CASE A: AXIAL LOADING

Figures 2.11 through 2.14 illustrate the ﬁber tow path-line displacement
along the beam axis under an axial load of 5 N. Table 2.4 contains displacement
data of two ﬁxed points A and B located on the top of the composite beam at its
mid-span (see Figure 2.6)

As can be seen from the ﬁgures, both the brick and beam and the brick
and plate models follow the same trends as the fully three-dimensional model.

For this case the axial displacement, u(x), is predicted quite well by both the

17

 

 

 

coarse a
displacer
reﬁned r

dimensio

 

the norm;
ﬁnite elen

Re
mmems

that occu

2.4.2 LC

along t)-

displace

trend a:
the em
attrjbut
reﬁned

model

both E

coarse and the reﬁned models (Figure 2.11 and 2.12) However, the normal
displacement, v(x), is not captured as well in the coarse model as in the more
reﬁned model (Figures 2.14 and 2.14), which agrees with the fully three-
dimensional model quite well. Additionally, the trend in solution convergence of
the normal displacement in the coarse and reﬁned models demonstrates that the
ﬁnite element solution is indeed converging in the correct manner from below.
Results for the transverse displacement, w(x), are meaningless as a result
of the displacements being essentially zero in comparison to the displacements

that occur in the other directions.

2.4.2 LOAD CASE B: BENDING ABOUT THE Y-AXIS

Figures 2.15 through 2.18 illustrate the ﬁber tow path-line displacement
along the beam axis under a transverse (2) load of 5 N. Table 2.4 contains
displacement data of two ﬁxed points A and B.

The axial displacement of the brick and plate model follows the same
trend as the fully three-dimensional model and agrees quite well except close to
the end of the beam where the loads were applied. This most likely can be
attributed to the boundary conditions affecting the results. The coarse and
reﬁned brick and beam models do not agree well with the fully three—dimensional
model when x > 6 in.

The normal displacement (see Figure 2.17 and 2.18) is predicted well by

both brick and plate models except near the location where the load was applied.

18

 

 

The l

displa

theb

dime

2.4..

gen
2.2.
plat
Drer
one
an

dis;

The brick and beam models again do a poor job of predicting the correct
displacements when v is greater than 6in.

The transverse displacements for both the coarse and reﬁned versions of
the brick and beam and brick and plate models agree well with the fully three-

dimensional model.

2.4.3 LOAD CASE C: BENDING ABOUT THE Z-AXIS

For the case of bending about the z-axis, Figures 2.19 through 2.24
illustrate the ﬁber tow path-line displacement along the beam axis under a normal
(y) load of 5N. The displacement data for points A and B can be found in Table
2.4.

Both the brick and beam and the brick and plate models follow the same
general trends as the fully three-dimensional model (see Figures 2.19 through
2.24). The axial displacement, u(x), of both the brick and beam and brick and
plate models agree well with each other, however, the two models seriously over
predict the axial displacement. This could be attributed to the fact that in the
brick models the interaction of the tow thickness with the surrounding resin plays
an important role in the overall behavior of the structure. However, the normal

displacement, v(x), is captured well in both the coarse and reﬁned models.

19

 

 

 

2.4.4 LC

F
along in
displace

Ir
and bea
largest c
was app
plate mc
model dc
The tran
beam or
notably,
elel‘ltents

form of a

2.5 Con c

Th
low in tt
proceSS a
free‘dOm.

brick and

2.4.4 LOAD CASE D: TORSIONAL LOADING

Figures 2.25 through 2.30 depict the ﬁber tow path-line displacement
along the beam axis under a torsional load of 5 N*cm and Table 2.4 contains
displacement data of two ﬁxed points A and B.

In the case of the axial displacements (Figures 2.24 and 2.25), the brick
and beam and brick and plate models follow the same general trends, with the
largest deviations occurring near the end of the composite beam where the load
was applied. The normal displacements are closely predicted by the brick and
plate model and most accurately in the reﬁned model while the brick and beam
model does not predict the displacement as accurately near the end of the beam.
The transverse displacement, (z), is not captured very well by either the brick and
beam or the brick and plate models, however, the basic trends are present. Most
notably, the brick and beam model results are poor due to the fact that the beam
elements have no length in the z direction and therefore respond poorly to any

form of an applied torque.

2.5 Conclusions

The intent of this study was to show the validity of representing the ﬁber
tow in the simplest yet most accurate form to expedite the model building
process and to decrease solution time by decreasing the number of degrees of
freedom. However, in the majority of the cases studied, it was evident that the

brick and plate model outperformed the brick and beam model. Consequently,

20

 

 

 

 

the men
within th
lr
evident
suited.

techniqm

 

the more accurate brick and plate model contained more degrees of freedom
within the problem.

In the cases where the brick and plate model performed poorly it was
evident that a fully three-dimensional representation of the model was best
suited. This of course is the paradox encountered with simpliﬁed modeling

techniques. With each simpliﬁcation some information is lost in the process.

21

Table 2.1 Load case summary.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Load Case Description Force Applied
(N) Direction
A Axial Loading 5 +x
B Bending about y 5 -z
C Bending about 2 5 -y
. 2 -y
D Torsron 2 +y
Table 2.2 Tow geometrical properties.
Property Value Location
Centerline Equation: y= % sin £575 x + 0.5 z=0
Cross section top: = %sin 2?” z + 0.8 x=0
. 1 . 27r
Cross section bottom: y=z srn ? z + 0.7 x=0

 

 

 

 

 

 

 

 

Width: 21.7 mm
Volume: 5463 mm"
Area: 54.6 mm2
lxx: 46.5 mm“
lyy: 124.0 mm“
lxy: -4.014E-02 mm“

 

 

Table 2.3 Material properties of the constituent materials of the woven
composite beam.

 

 

 

 

 

 

Young’s Modulus Poisson’s Ratio
(GPa)
Resin 4.3 0.34
Fiber Tow 72 0.2

 

 

22

 

 

Table 2.4 Mid-span deﬂection data of points A and B for each load case.

 

Displacement

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Dercti'idpetion MOSh 1.22:: P‘ﬁm PoBint % Error
- A 9.35E-08 9.35E-08 N/A
Fw'lﬁjﬁig‘l’ff' 7858 10-noded B -3.96E-05 5955-05 N/A
ﬁber tetrahedrons C -4.84E-06 -4.84E-06 N/A
D -1 .88E-06 1 .88E-06 N/A
Fuuvsomodel 2304010 3 itiiiii .2133: Eli
w/lenticular noded C V , . .. . . N / A
ﬁber tetrahedrons k, 4W .,
D -1.65E-06 1 .65E-06 N/A
1500 8-noded A 9.75E-08 9.75E-08 -4.29%
Brick and Beam hexahedrons B -3.88E-05 -3.88E-05 2.02%
coarse 40 2-noded c -4.24E-06 -4.24E-06 12.53%
beams D -8.86E-07 8.86E-07 52.94%
12300 8-noded A 9.49E-08 9.49E-08 -1.47%
Brick and Beam hexahedrons B -3.90E-05 -3.90E-05 1.67%
reﬁned 80 2-noded C -4.28E-06 -4.28E-06 11.66%
beams 0 -1.03E-06 1.03E-06 45.13%
1600 34.0090 A 9.58E-08 9.58E-08 -2.44%
Brick and Plate hexahedrons B -3.91E-05 -3.91E-05 1.34%
coarse 160 4410de C -4.10E-06 -4.1OE-06 15.47%
quad 0 -1.75E-06 1.75E-06 6.99%
12300 3410090 A 9.35E-08 9.35E-08 -0.03%
Brick and Plate hexahedrons B -4.01E-05 -4.01E-05 -1.23%
reﬁned 80 4-noded c -3.76E-06 -3.76E-06 22.32%
quad 0 -1.77E-06 1.77E-06 6.00%

 

23

 

 

Figure 2.1 Geometry of the cantilever beam with a single undulating ﬁber.

24

 

Figure 2.2 Reﬁned brick and beam model

25

 

Figure 2.3 Reﬁned brick and plate model.

26

 

x .1 i. x
<\ ,‘4'

Figure 2.4 Fully three-dimensional ﬁnite element model.

27

 

 

Fl / /

 

 

 

 

 

[ / f7

 

 

 

/ / / /

 

 

/7//

 

 

 

 

/ / / /

 

Illl‘"
z x I...-

Figure 2.5 Arrangement of the brick elements in the brick and beam and brick
and plate models.

28

 

Figure 2.6 Axial loading condition applied to each model.

 

Figure 2.7 Normal (X-Y bending) loading condition applied to each model.

29

 

Figure 2.8 Transverse (X-Z bending) loading condition applied to each model.

 

Figure 2.9 Torsional load case applied to each model.

30

 

 

 

It

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

> x (b)

— —- — True ﬁber pathline

Approximated pathline

 

Figure 2.10 Piece-wise linear approximation of the ﬁber tow path-line using
beam and shell elements (a) coarse model, (b) reﬁned model.

31

 

1 .0E-06

—Fully 3D Model
8.0E-07 ~

 

 

 

 

 

 

 

 

 

 

 

 

 

Brick 8. Beam (Reﬁned)

3 6'05'07 “ ------ Brick& Plate (Reﬁned)/ ..... \ I

:: 4.0E'O7 / .l. ....I‘ K I

: ,... .

“5’ 2.0E-07 / ; '----_- \ l :

is? ODE-+00 _%ﬂ‘ 1 :

,_ ‘.-. . . J

g -2.0E-07 em ______ ' 1 \r,

N 405-07 .'
-6.0E-07 ----- 1 ......
-8.0E-07

0 2 4 6 8 10
x-coordinate

Figure 2.11 Axial displacement (u) along the tow centerline of the coarse brick &
beam and brick & plate models due to an axial load.

 

2.0E-07
—Fully 3D Model

Brick & Beam (Reﬁned)
1.5E-07 - ------ Brick & Plate (Reﬁned) - --

.........

 

   

 

1.0E-07 «

5.0E-08 — —- —
0.05+00 .4 . .

O 2 sit-coordinates 8 10

 

 

 

x-displacement (u)

 

 

 

 

 

Figure 2.12 Axial displacement (0) along the tow centerline of the reﬁned brick &
beam and brick & plate models due to an axial load.

32

 

 

 

 

 

 

 

 

6.0E-07
_Fully 3D model
Brick & Beam (Coarse)

S """ Brick & Plate (Coarse)
E 4.0E-07 ‘
O
E
8
t.“
8
5 2.0E-07 -' -
>

0.0E+00 * - 1. .

0 2 4 6 8 10

x-coordinate

Figure 2.13 Normal displacement (v) along the tow centerline of the coarse brick
& beam and brick & plate models due to an axial load.

 

  
   

 

 

 

 

 

 

 

 

 

6.0E-07
_Fully 3D Model ........
Brick & Beam (Reﬁned)

S ------ Brick & Plate (Reﬁned)
*5 4.0E-07 -
o
E
8
E ..........
B
5 2.0E-07
>~

0.0E+00 * r , .

0 2 4 6 a 10
x-coordinate

Figure 2.14 Normal displacement (v) along the tow centerline of the reﬁned brick
& beam and brick & plate models due to an axial load.

33

 

8.0E-06
_Fully 30 Model

Brick & Beam (Coarse)
...... Brick & Plate (Coarse)

 

 

4.0E-06

/\
ﬂ /

 

 

 

 

 

 

3
5
0
5
3 0.05+00- ,
9 L—-\__/ I
O.
.2
1?, -40E-06
' I \___f
-8.0E-06
o 2 4 6 a 10

X-coordinate

Figure 2.15 Axial displacement (u) along the tow centerline of the coarse brick &
beam and brick & plate models due to a normal (y) load.

 

 

 

 

 

 

 

8.0E-06
_Fully 3D model
Brick & Beam (Reﬁned)

3 4.0E-06 _ ------ Brick & Plate (Reﬁned) ._.-
E A

0.0E+00 ~ - ‘
v '4
.6
3 -4.0E-06 —

-8.0E-06

O 2 4 6 8 10
X-coordinate

Figure 2.16 Axial displacement (u) along the tow centerline of the reﬁned brick &
beam and brick & plate models due to a normal (y) load.

4.0E-05

 

0.0E+00

 

 

 

 

  
    

 

 

-4.0E-05 - —

-8.0E-05

 

 

y-displacement (v)

-1 .25-04 .- —a— Brick & Beam (Coarse) T”
. . -A - - Brick & Plate (Coarse)

 

 

 

-1 .6E-04

 

0 2 4 6 8 10
x—coordinate

Figure 2.17 Normal displacement (v) along the tow centerline of the coarse brick
8 beam and brick & plate models due to a normal (y) load.

 

 

 

 

 

 

 

   
  

 

 

 

 

4.0E-05
0.0E+00
S
5 405-05
E
i
a -8.0E-05
.22
1?
> _1 2504 -0 _Fully 3D Model __
° —8— Brick & Beam (Reﬁned)
- - a - - Brick & Plate (Reﬁned)
-1.6E-04
0 2 4 6 8 10
x-coordinate

Figure 2.18 Normal displacement (v) along the tow centerline of the reﬁned brick
& beam and brick & plate models due to a normal (y) load.

35

0.0E+00

 

 

 

 

 

 

 

 

 

 

 

 

 

x—I-H\5 _ T \
-205-09 ,__- L ‘ X‘
75' 405-09 ‘
5
i
g -6.0E-09
.2
—8.0E-09 -——-— Pliny 3D MOdeI
Bnck & Beam (Coarse)
...... Brick 8. Plate (Coarse)
-1.0E-08
X-coordinate

Figure 2.19 Axial displacement along the tow centerline of the coarse brick &
beam and brick & plate models due to a transverse (2) load.

0.0E+00

-2.0E-09

-4.0E-09

x—displacement (u)

-8.0E-09

-1.0E-08

-6.0E-09

 

 

 

 

 

 

 

 

 

 

 

 

 

_Fully 3D Model \
___ Brick & Beam (Reﬁned) ~* -
------ Brick & Plate (Reﬁned)
X-coordinate

Figure 2.20 Axial displacement along the tow centerline of the reﬁned brick &
beam and brick & plate models due to a transverse (2) load.

36

1 .2E-08

 

 

 

 

 

     

 

 

 

 

 

8.0E-09 ,

E /
7.5, 4.0509 -— ~ .. ‘
E .- '
i
Tm; 0.0E+00
5
s. _Fully 3D Model

-4.0E-09 *5 Brick & Beam (Coarse)

------ Brick 8. Plate (Coarse) V
-8.0E-09
0 2 4 6 8 10
x-coordinate

Figure 2.21 Normal displacement (v) along the tow centertine of the coarse
brick & beam and brick & plate models due to a transverse (2) load.

1 .0E-08

6.0E-09 . /

 

 

 

 

 

 

 

 

 

 

/
E 20509 .
5 a.—
i ” " """ '

3 20509 I
3 _Fully 30 Model \ /
-505-09 - — Brick & Beam (Reﬁned) ‘ V

------ Brick & Plate (Reﬁned)
-1.0E-08
x-coordinate

Figure 2.22 Normal displacement (v) along the tow centerline of the reﬁned
brick & beam and brick & plate models due to a transverse (2) load.

37

0.0E+00 ~

 

 

 

 

     
   

 

 

 

 

 

A -4.0E-06
5
E
o
g -8.0E-06
E
O _Fully 3D Model
N -1.2E—05 ._ Brick & Beam (Coarse)
------ Brick & Plate (Coarse)
-1.6E-05
0 2 4 6 8 10
x-coordinate

Figure 2.23 Transverse displacement (w) along the tow centerline of the coarse
brick & beam and brick & plate models due to a transverse (2) load.

 

 

 

 

 

 

 

 

 

0.0E+00 ~
3: -4.0E-06
‘E
0
5 -8.0E-06
E
O.
.22
9 _Fully 3D Model
N -1 25-05 — Brick & Beam (Reﬁned) —
------ Brick 8. Plate (Reﬁned)
-1.65-05
x-coordinate

Figure 2.24 Transverse displacement (w) along the tow centertine of the reﬁned
brick 8. beam and brick & plate models due to a transverse (2) load.

38

4.0E-10

 

 

 

 

 

 

 

 

 

 

0.05+00 -

a

:: 405-10

c

o

5

g -8.0E-10 «

e .

Q. .

% -1.25-09 -—— _Fully 3D Model :

" Brick & Beam (Coarse) ' N
'1'6E-09 +— ------ Brick & Plate (Coarse) , ..
-2.05-09

0 2 4 5 a 10
X-coordinate

Figure 2.25 Axial displacement (u) along the tow centerline of the coarse brick
& beam and brick & plate models due to a torsional load.

 

 

 

 

 

 

 

 

 

 

 

 

4.0E-10

0.05+00 . . _ﬁ
:3: 405-10 - 1'
c '.
g \
3 -8.0E-1O ” T
E
a \ '
f, -1.2E-09 ~— —Fully 3D Model
x Brick & Beam (Reﬁned) \

'1-65'09 ------ Brick & Plate (Reﬁned)

-2.0E-09

x-coordinate

Figure 2.26 Axial displacement (u) along the tow centerline of the reﬁned brick
& beam and brick & plate models due to a torsional load.

39

5.0E-09

 

 

 

 

 

 

 

 

 

 

 

 

 

4.05.09 _Fully 3D Model J.
3 Brick & Beam (Coarse) '
:5 305-09 ------ Brick & Plate (Coarse)
5
3 2.05-09 » - -—-—
a
3 105-09
I? .
>~

0.05+00 -

4.0509

0 2 4 6 a 10

x-coordinate

Figure 2.27 Normal displacement (v) along the tow centerline of the coarse
brick 8r beam and brick & plate models due to a torsional load.

 

 

 

 

 

 

 

 

 

 

 

 

 

5.0E-09
—F l M I

40309 - uly3D ode
9 Brick & Bream (Reﬁned)
:05; 3-0E-09 - ------ Brick & Plate (Reﬁned)
E
8 2.0E-09
_19
8
5 1.0E-09
>'.

0.0E+00 ~

-1.0E-09

0 2 4 6 8 10

x-coordinate

Figure 2.28 Normal displacement (v) along the tow centerline of the reﬁned
brick & beam and brick & plate models due to a torsional load.

40

 

1 .0E-06

   
  

 

 

 

 

 

 

 

 

 

 

 

 

 

8050., _ _Fully 30 Model
6 0E-07 J Brick 8 Beam (Coarse) 1|
5‘ ' ------ Brick & Plate (Coarse) \ I
:: 4.05-07 / r - - _ _‘ \ l
c : u
g 2.05-07 ., .1
g 0.0E+00 N-WV/‘r-i
3 205-07 ‘ ' “ " __ .2 j"
N -4.0E-07 +4 -—A.- ' ' .
-6.0E-07 ——————_ _, ‘ ,, r;
5.0557
0 2 4 6 8 10

x-coordinate

Figure 2.29 Transverse displacement (w) along the tow centerline of the coarse
brick & beam and brick 8 plate models due to a torsional load.

1.0E-06
8.0E-07 ~

 

_Fully 3D Model
Brick & Beam (Reﬁned)

6.0E-07 — ______ .
4.0507 Bnck& Plate (Reﬁned) / 5 ________ l. \ “H‘f
/:"" L...\ :

 

 

 

 

2.0E-07
0.0E+00 ~
-2.0E-07
-4.0E-07 . .
-6.0E-07 - - _ 't -------
-8.0E-07

 

Z-Displacment (w)

 

 

 

 

 

 

 

0 2 4 6 8 10

x-coordinate

Figure 2.30 Transverse displacement (w) along the tow centerline of the reﬁned
brick & beam and brick & plate models due to a torsional load.

41

Chapter 3

The simpliﬁed discrete-tow model

3.1 Introduction

In most ﬁber-reinforced polymer-matrix composites, the ﬁbers are much
stiffer than the resin material. Therefore, the majority of the load in the structure
is carried by the ﬁbers while the resin serves to transfer load among the ﬁbers. it
is evident that the deformations and relative motions of ﬁbers dictate the

deformation of a ﬁber-reinforced composite stnlcture.

3.2 The Simpliﬁed Discrete-Tow Model

Based on the above observations and experience with other modeling
approaches, the present simpliﬁed discrete-tow model was conceived. The
current model is constructed by representing the ﬁber tows using bar (or beam)
elements and the resin using plane stress (or shell) elements. Repeating unit
cells (RUC’s) of plain weave and triaxial braid are shown in Figure 3.1a and 3.2a
respectively. The RUC’s of both weaves can be further broken down into
repeating volume elements (RVE’s), which can be seen in Figure 3.1b and 3.4.
The two-dimensional discrete-tow models of the same unit cells are shown in
Figure 3.4 and 3.7. As a result of the periodicity of the triaxial braid (Figure 3.5a),
the triaxial braid model can be further simpliﬁed by modeling half of the RVE as
shown in Figure 3.50

In the discrete-tow model, the true material properties of the resin and

ﬁber tow cannot be assigned directly to the elements. This is a result of a beam

42

element’s inability to represent the transverse stiffness, shear stiffness, and
curvature of the tow. The resin in-plane and shear moduli must be modiﬁed to
account for the transverse and shear stiffness of the tows. Also, the axial tow
stiffness must be reduced to account for tow curvature. A general method has
been derived that can be used to calculate the effective resin moduli for plain
weave and triaxial braid materials. The effective resin properties are found
through the use of a laminate analogy while the axial tow stiffness reduction is
calculated using the theory of curved beams on elastic foundations in the same
manner that Naik [22] used. It is important to note that the calculation of the
effective resin properties for a plain weave is actually a sub-problem of the
triaxial braid calculations. For this reason the effective moduli calculations will be
derived for a triaxial braid starting with the geometrical deﬁnition of the triaxial

braid.

3.2.1 GEOMETRY OF A TRIAXIAL BRAID

A half-model of the RVE of a triaxial braid is shown in Figure 3.5b. The
geometry of this half-model RVE can be fully described through the use of the
iterative procedure described by Naik et al. [23]. Using the same method as
[23], it is only necessary to know do, w, t, 0b,, V), pd, no, and rib, to describe the full
geometry of the RVE of a triaxial braid. Where do is the axial tow spacing, w is
the width of the half-model of the RVE, t is the total thickness of the individual
lamina, and 6],, is the braid angle. Additionally, V) is the overall ﬁber volume
fraction, pd is the packing density of the ﬁbers, while no and tip, are the number of

ﬁber ﬁlaments in the axial and braider directions.

43

In order to simplify the number of unknowns in the geometrical description
of the triaxial RVE, Naik et al. [23] make several assumptions. The axial tows
are assumed to be straight with no undulation (see Figure 3.2 b 86), while the
braider tows are assumed to follow a sinusoidal path-line in the direction of the
braid. It is also assumed that the axial and braider tow thicknesses, to and t3,
respectively, must satisfy the relationship t = to + 203,. The assumption is also
made that the cross sectional area of the axial and braider tows (A0 and AB,
respectively) remains constant when sections are taken perpendicular to the tow
path-line. The cross sectional area of the axial tow is idealized as a combination
of straight and curved sections which correspond to the straight and undulating
regions of the woven braider tows. The curved portion of the axial tow’s cross
section is assumed to follow a path that is parallel to the sinusoidal path of the

braider yarn.

3.2.2 EFFECTIVE RESIN PROPERTIES OF A TRIAXIAL BRAID

Once the geometry of the RVE for the triaxial braid is fully described, it is
then possible to calculate the effective resin properties needed to account for the
transverse tow stiffness and shear stiffness. The RVE of a [Clio] triaxial braid is
divided into two regions. The ﬁrst region has a width equal to the width of the
axial tow (w) as can be seen in Figure 3.7. This region is assumed to contain
the entire volume of the axial tow (Va), along with a portion of the braider tows
and of the resin material. The second region contains the remainder of the
braider tows and resin volume that was not contained in the ﬁrst region. Each

region is then assumed to be an assemblage of lamina, with each ply

44

representing one of the region’s components (tow or resin). This laminate
analogy is represented graphically in Figure 3.6. From Figure 3.7 it can be seen
that the materials within each region are treated as a parallel model using
classical lamination theory (CLT) while the two regions are considered to be in
series with each other.

The material stiffness matrix [Q] for each of the constituent (or ply)

materials takes the following general form:

 

 

i E11 V12511 0 -
1- V1EV21 1-[\;_12V21
V
[Q]: 1 12 11 1 22 0 (1)
-V12V21 -V12V21
0 0 G12

 

 

where E11, E22, v12 and G12 represent the material properties of the constituent
(ply) material under consideration. Recall that the current model discretely
represents the axial stiffness of each tow within a beam element. Because the

axial tow stiffness is already accounted for explicitly in the model, it is necessary

to set E:1 = 0 in the tow plies used to calculate the effective resin properties.

This assumption also requires Viz = 0. The tow transverse and shear stiffness

are not modiﬁed. The stiffness matrix of the tow plies can now be written as:

 

 

'0 0 0‘
il‘I= 0 652 0 (2)
t
_0 0 G12_

where the superscript, t, denotes properties of the axial or braider tows.

45

The orientation of the tows is taken into account by transforming the

material stiffness matrices (Jones [24]):

I5] = ITI’1IQITI’T (3)
where,
_ cos29 sin29 —2sin9cosG -
IT] = sin2 9 cos2 9 28inecos6 (4)
- sin9cos0 sin9cos9 cos2 6 — sin2 6

 

 

and 6 is the orientation angle of the ply (tow) under consideration. For a [0/145]

triaxial braid, 6 is 145° for the braider tow plies, 0° for the axial tow ply, and 0° for
the resin ply. Using CLT, the effective stiffness properties for region one, [6]“),

are given by

_ _ v“) _
1 — 1
[(2111) = Vf(0) [QAXNL + —f28C—(IQBI' 1+9 1' IQBr I—O )+ Vfin) [out I (5)
where the tow volume fractions of the constituent materials are deﬁned as

v“)

 

v“) = 0
f0 1
vtbt)
v11)
(1) _ Br
VfBr T v(1) (6)

tot
(1) _ (1) (1)
vfm _1_vf0 —VtBr
and the following deﬁnitions are used to calculate the respective volumes of the

materials

1
vgl=Aor

46

 

v“) = A “’0 7
Br Br Sin(98r) ( )
(1) _
Vtot — [Wot

where I is the length of the RVE and t is the thickness. In a similar manner, the

effective properties for region two can be calculated with

_ viz) _ _
IQLZ) = quQBr I+9 'I' IQBr I—a)+ Vfiyzq)[Qm] (8)

where the tow volume fractions of the constituent materials in the second region
are deﬁned as

v(2)

2 B 2 _ 2
Vf(Br) = v(g) Vin) _1—v,(B) (9)
tot

 

and the volumes of the respective materials in region two are

W — W0
Sin(98r)

 

viz) = It(w—w0) (10)

(2) _
VBr ‘ ABr tot

The effective resin properties of regions 1 and 2 can now be combined in a

series model to compute the overall effective properties. Using the relation

[SI—145], the effective compliance matrices for each region are found and the

effective material properties are found as:

1 1 2 1
Eir)=_ 51(1):—

"‘ 1 _ 2

81(1) 31(1)
1 1

E“) = —— 5‘” = _—
22 1 22 2

852) 52(2)

1 1 — 1 2 2 "' 2
Viz) = ’Ei1)51(2) Viz) = “Ei1)31(2) (11)

47

1 1—1 2 2—2
Vgr) = {£2351} V21) = ‘E£2)32(1)

1 1

C“) = — em = _—
12 1 12 2
3(6) 56(6)

Using the equations of mechanics of materials, the combined effective resin

stiffness in the axial and transverse directions, E1’1and E52 respectively, are

 

given by
5(1)E(2)
r _ (1) (1) (2) (2) r _ 11 11
E11 va E11 +Vt E11 E22 ‘V(1)E(2)+V(2)E(1) (12)
f 11 f 11
where
1 VOI1 2 Vol
vft ) = vft )= 2
VOItOt VOItOt

In the same manner the combined effective Poisson’s ratios, v,"2 and v;,, are

found using
(1) (1) (2) (2) E22V1rz
I’ I‘
V12 =V12 V12 “’12 V12 V21 =—E‘,— (13)
11
Finally, the effective shear modulus, 61’2, is given by
6(1)G(2)

 

1 2 2 1
Vr()Gi2)+Vr( )Giz)

48

_“

3.2.3 EFFECTIVE RESIN PROPERTIES OF A PLAIN WEAVE

The effective resin properties of a plain weave are calculated in a manner
very similar to the method previously discussed for the analysis of triaxial braids.
Note that a plain weave can be completely represented by region two of a triaxial
braid that only contains i6 braider tows. The effective stiffness matrix can then
be deﬁned by:

via

2, (IGBrIw +[6Br L6)+VMI—6m] (15)

[6F

 

where the volume fractions of the constituent materials are deﬁned as:
VfBr =_ me =1‘VfBr (16)

and the volumes of the respective materials are:

V
VBr = _fvtot Vtot = ”W (17)

Pd

3.2.4 TOW CURVATURE MODIFICATION

In the ﬁnite element representation of the simpliﬁed unit cell, the curved
tows are modeled with straight beam elements. Consider both a curved and
straight tow of the same axial length as pictured in Figure 3.7. The beam
elements should be assigned a modulus such that the end shortening of the
straight beam is the same as the end shortening of a curved tow when subjected
to the same in-plane load. A stiffness reduction factor can be derived from the

theory of curved beams on an elastic foundation [22]. This theory makes it

49

possible to include some of the three-dimensional tow curvature effects as well
as the tow shape in a two-dimensional simpliﬁed model.

Figure 3.8a depicts the undulating portion of the ﬁber that is to be
considered, while Figure 3.8b describes the load applied to the ﬁber tow. The
ﬁber tow path-line is assumed to be sinusoidal. Taking the x-coordinate along

the tow axis, the tow path-line is deﬁned by the following equation
20 = A0 sin(px), where a = L (18)
Lab

where Lu), is the length of the undulating portion of the ﬁber tow and A0 is the

amplitude of the sine wave. As shown in [20], the axial strain of the curved ﬁber

 

 

 

is given by
ex = —:1-(,(32A1(A1+2A0)l1+cos(28x)]+e,-_S (19)
where
—F 2
A1 = t 43 A02 (20)
EHIYB +F[3 +k
a=3+-t2'- (21)
A0
Art2
I = Br
4E A a a
k = 22 ’7 [ 4] (23)
ttBrAO \IG2 —1
F
€i—s = t (24)
E11At‘t

50

E; is the effective through thickness modulus of the woven composite, 5:1 is the

axial modulus of the tow, and to, is the thickness of the undulating tow. The ﬁber

tow cross-sectional area A); can be estimated from the following

_ ﬂ— VO/ tot

Arr
Pd LT

(25)

where V) is the ﬁber volume fraction, pd is the tow packing density, Vol10, is the
total volume of the RVE and L7 is the total length of the undulating ﬁbers in the

RVE. The axial component of displacement in the curved tow (uc) is

 

au
uc= Iexdx+ [7'0 (26)
0 E11AT

where the total cross-sectional area A7 of the RVE can be calculated using Ar =
th. The axial component of displacement in the straight tow (us) due to an
applied load is given by

_ FL
" t
E11AT

 

Us , L = Lub + Lo (27)

The stiffness reduction factor (R.F.) necessary to model the effective curved tow
deformation correctly using a straight beam ﬁnite element is taken to be the ratio

of the displacements calculated above
R. F. = 5e- (28)

Uc

For many practical materials, R.F. takes a value above 0.9.

51

3.2.5 DETERMINING EFFECTIVE PROPERTIES

The effective properties of the RVEs for the weave geometries studies in
this work were determined through the use of three sets of boundary conditions
applied to a single RVE. For each boundary condition set a ﬁnite element
analysis was performed. The results of the analysis can then be used to
determine the effective properties of the various weave geometries under
consideration. The three boundary condition sets and the simplifying stress-
strain relations for each are summarized below
Boundary condition #1: (a1 = 1, 32 = 0, 712 = 0)

01 = 61151 (29)

0 2 = 61251 (30)
Boundary Condition #2: (a1 = 0, 82 = 1, 712 = 0)

c’1 = 52152 (31)

02 = cm (32)
Boundary Condition #3: (61 = 0, 82 = 0. 712 = 1)

T2 = 666Y12 (33)

where the stresses 01, 0’2, and 112 are found using

P F F
"—2 1 02 =——§ 2 T12 =——§ 1 (34)
2 2

Knowing both the state of stress and strain from the analysis and

boundary conditions respectively, the effective stiffness matrix of the RVE can be

52

calculated using Equations 29-33. After calculating the effective stiffness matrix,

the effective compliance matrix can be found through

— — 1
IS] =loi <35)
and the effective composite properties of the RVE can be determined using the

following relations that are similar to those found in Equation 11.

E11 i;-
11
1
5 =—
22 —
822
V12 = "51512 (36)
V21 = ‘522321
1
G =—
12 —
S66

3.3 Numerical Results

3.3.1 ANALYSIS OF A PLAIN WEAVE

The ﬁrst case considered is the prediction of the effective stiffness of two
plain weave composites. As an example of the input used for the properties in
the ﬁnite element analysis of this plain weave unit cell, the following effective
resin properties were found to be E11 = E22 = 7.27 GPa, v12= -.258, 612 = 2.69

GPa. These properties were then assigned to the shell elements in the model.

53

The tow stiffness reduction factor was found to be 0.968. Therefore, a value of
140.13 GPa was assigned to the axial stiffness of the beam elements.

Results from the current model are compared with results from Sankar
and Marrey [13], Naik [25], and Foye [26]. The material and geometric properties
are summarized in Table 3.1. The results for the two cases are presented in
Table 3.2. It can be seen that the effective stiffness, E", and the shear modulus,
612, predicted by the discrete-tow model agrees well with the other models. The
Poisson’s ratios predicted by the current model are lower than the other results

presented in cases 1 and 2, however all of the results are small compared to

unity.

3.3.2 EFFECT OF WAVINESS RATIO ON EFFECTIVE MATERIAL
PROPERTIES

The ability of the simpliﬁed discrete-tow model to capture tow curvature

effects makes it possible to investigate the effect of waviness ratio on the

predicted effective properties. The waviness ratio, 11., is deﬁned as

Lab Lab
= 37
L Luo + l-o ( )

where L, La, and Lu), are deﬁned in Figure 3.9 for a plain weave. Three waviness

ratios of 0.167, 0.25, and 0.5 were examined and compared to results presented
by Raju and Wang [11], Whitcomb [15], and Aitharaju and Averill [12]. The
waviness ratio for the three cases was determined by holding Lu), constant, while
varying L0. The geometric and material data for the three cases is found in Table

3.3 and Table 3.4 respectively. The predictions of effective modulus, Poisson’s

ratio, and shear modulus are shown graphically in Figures 3.12 through 3.14 and
in tabular form in Table 3.5. The current predictions of the effective stiffness, E11,
(Figure 3.10) and shear modulus, 612. (Figure 3.12) for the various waviness
ratios agree well with those in the literature. Predicted Poisson’s ratios are lower
than the other models (Figure 3.11), however, all of the results are quite small
compared to unity. All of the present results follow the expected trends as the
waviness ratio is increased. The effective stiffness and the shear modulus both

decrease with an increase in waviness ratio while Poisson’s ratio increases.

3.3.3 ANALYSIS OF A TRIAXIAL BRAID

Four different triaxial braids were examined using the current model, and
overall stiffness predictions were compared to the numerical and experimental
results of Naik et al. [23]. The study Investigates the effect of braid angle ((93,),
relative tow sizes and axial tow content on the predicted effective properties.
The material and geometrical data for each of the cases is summarized in Table
3.6 and Table 3.7. A comparison of the numerical results of the four cases can
be found in Table 3.8. The present results, in general, compare very well with

the numerical and experimental data found in Naik et al. [23].

3.4 Conclusions

A simpliﬁed discrete-tow model has been formulated and validated for
predicting overall mechanical stiffness properties of plain weave and triaxial braid

textile composite materials. In this model, the ﬁber tows and resin are discretely

55

represented using beam and shell ﬁnite elements, respectively, resulting in a
simple, accurate and computationally efﬁcient approach. Overall stiffness
predictions using the current discrete-tow model agree very well with other

analytical and experimental results found in the literature.

56

Table 3.1 Material properties for comparison of predicted stiffness

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Case1 Case2
(Glass/Epoxy) (Graphite/Epoxy)
Length (mm) 1.1879 3.9909
UnitCPll Width (mm) _ 1.1879 3.9909
Dlmenswns Height (mm) 0.228 0.2557
51? (GPa) 58.61 144.8
52 (GPa) 14.49 11.73
W2 0.25 0.230
TOW G12 (GPa) 5.38 5.52
pd 0.7836 0.75
vf 0.26 0.64
E (GPa) 3.45 3.45
Matrix v 0.37 0.35
G(GPa) 1.26 1.28
Table 3.2 Comparison of results for a plain weave-
E11(GPa) V12 612(GPa)
[Present Study 12.97 0.075 2.03
Case1 bankers Marrey (1997): s.A.M 12.46 0.162 1.67
(Glass/Epoxy) [SankaraMarrey (1995): F.E.A. 11.81 0.181 2.15
IDasgﬁupta et al (1990) 14.38 0.167 3.94
Present Study 65.358 0.003 4.898
Gas” ISankar& Marrey (1997): S.A.M 63.41 0.027 4.24
(Graphite/Epoxy)ISankar&Marrey(1995):F.E.A. 53.61 0.128 4.72
[vark (1994): TEXCAD 64.38 0.027 4.87
[Foye (1992): F.E.A. 63.78 0.031 4.82
[Foye (1992): Test 61.92 0.11 NA

 

 

 

 

 

57

 

Table 3.3 Geometric data for the three waviness ratios studied

 

 

 

 

 

 

 

 

i. 0.167 0.25 0.5
llh 24 16 8
Vi 0.580 0.569 0.520
pa 6 6 6

 

 

Table 3.4 Material properties for all waviness ratios studied

 

 

 

 

 

 

 

 

 

 

I511 (GPa) 134
Tow I522 (GPa) 10.2
IV12 (GPa) 0.3
I612 (GPa) 5.52
IE (GPa) 3.448
Matrix Iv (GPa) 0.35
lo (GPa) 1.28

 

58

Table 3.5 Comparison of predicted effective properties for a plain weave

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

composite.
t 0.167 0.25 0.5 |
IP 511 68.8 67.1 60.2]
resent Study v12 0.001 0.001 0.003]
or; 5.38 5.30 4.95]
5,, 67.7 65.1 56.2]
ltharaju & Averill (1999)) v12 0.043 0.044 0.044]
[A 612 5.36 5.29 5.08]
. 5,, 66.7 63.8 54.4]
ilvnitcomb (1991) v12 0.037 0.034 0.026]
612 5.30 5.19 4.80]
511 68.4 66.2 57.1]
item & Wang (1994) v12 0.045 0.046 0.052]
612 5.34 5.24 4.80]
511 72.5 72.5 72.5]
0,90). v12 0.043 0.043 0.043]
I on 5.52 5.5 5.52]

 

 

 

 

 

59

Table 3.6 Geometric and braid data for triaxial weaves.

 

 

 

 

 

Case Material SxFrIQTfnvilm) Ply 12$? ess Vr pa
1 [2339,1323] 6.05 0.7075 0.55 0.75
2 [36552523113] 11.04 0.935 0.59 0.75
3 [gagé‘fgigf] 5.52 0.94 0.58 0.75
4 Iggggigf 5.52 0.556 0.56 0.75

 

 

 

 

 

 

 

 

Table 3.7 Properties for the constituent materials of the triaxial weaves.
- 511 I522 G12
Meier'a' (GPa) (GPa) V12 (GPa)

AS4/1895 Yarn 144.8 11.73 0.23 5.52
1895 Matrix 3.45 3.45 0.35 1.28

 

 

 

 

 

 

 

 

 

 

60

Table 3.8 Comparison of effective properties for a triaxial weave with numerical

and experimental methods.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

61

 

 

EL ET GLT
(GPa) (GPa) V” V“ (GPa)
Present Study 54.18 46.77 0.12 0.10 9.39
[03001706,] Naik 55.67 49.30 0.15 0.13 9.68
46%-axial Naik Measured 60.25 47.50 0.20 0.09 11.65
Naik Measured] 59.15 43.60 ‘ ' '
Present Study] 58.67 51.13 0.11 0.10 10.18
[om/17015,] Naik | 59.63 53.38 0.15 0.13 10.38 |
46%-axia| Naik Measured] 62.46 48.12 0.16 0.05 11.03
INaik Measured] 59.77 45.77 na na...
Present Study] 61.89 19.84 0.61 0.20 17.54
[om/14515..) Naik ] 62.67 20.78 0.65 0.21 18.04 ]
46%-axial Naik Measured] 65.63 19.30 0.57 0.22 16.96
iNaik Measured] 61.70 19.23 na na na
Present Study] 27.35 19.45 0.72 0.51 25.83
[oak/s4515k] [Naik ] 27.92 20.13 0.73 0.53 26.00 |
12%-axia| [Naik Measured] 33.92 20.55 27.92
[Naik Measured] 29.92 27.92

 

-6

 

  

 

 

 

fig?
1‘

tBr 0

(b)

Figure 3.2 (a) Geometry of a triaxial braid (b) cross section of the triaxial braid
made perpendicular to the axial tows (0) cross section of the triaxial
braid made alone the direction of the braider tows.

63

 

   

K Axial Yarns j

Figure 3.3 The area enclosed by A,B,C,D represents one possible RVE for a
[0&0] triaxial braided composite.

Tow Element

Resin Element

qv O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

(a) (b)

Figure 3.4 (a) Assembled ﬁnite element model of the unit cell and (b) exploded
view of the ﬁnite element model for a plain weave material system.

65

 

 

 

 

 

 

 

 

 

ZN» C

(a)

 

Tow Element
Resin Element

(b) (C)

 

 

 

 

 

0

 

 

 

 

 

 

Figure 3.5 (a) Discretized representation of a RVE in a triaxial braid. (b)
Assembled and (c) exploded views of the ﬁnite element model for a
triaxial braid.

66

 

Region 2

 

:I:O° Ply
I 0° Ply
Resin

 

 

Figure 3.6 Division of the RVE of a triaxial braid into two regions for calculation of
the effective properties (a) plan view (b) side view.

(b)

67

 

Figure 3.7 (a) Curved ﬁber tow. (b) Straight ﬁber tow.

68

(b)

8x

   

 

Hg -._
Lo ’1

(a) (b)

Figure 3.8 Curved beam on an elastic foundation (a) geometry of the undulating
region and (b) applied load.

69

 
 
  
 
 
  
  
     

. C O C O I O C O C O C O C O V O I O C
"Iv “2v ’4" "W ”N ":0" ’4'! ”N "2" N N
”at ”I? “I" ’4? ’4': ”N "2°! "Iv "2v ’4? ’6"
v ”M "iv "2v “‘2" ’4': "iv "2v ’4? "2°: ’4" “24’
. "IV ”a ’4': ’4‘! "2" ”N "2v "2v ”Ev ”at ”at “ a
’43? ”iv ”2°: ’4'! ’4'! ’4'! "2" ”at "3" r" I ’4" ’4'!
t . Q$¢¢$¢¢$¢A4¢ .‘Z‘Zv ‘26". ’4”; (or. .’ 4v
843432-5424" ' ‘ ‘ i i ' ”
reset-9w ’

      
  

     

o
e

     

  

  
             

 

Figure 3.9 Geometry deﬁning the waviness ratio (it =Lub /LI)

70

 

— I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E11 vs Waviness Ratio 00)
l 80
70
r 60 f g . (:1 Aitharaju&Averill(1999)
i 7‘? 50 ___._. A Whitcomb(1991)
1 8.5 40 _r _ o Raju&Wang(1994)
E, 30 _ _ 9 Present Study
”J —(0,90)s Laminate
20 .
10 ~—
0 . .
0 0.2 0.4 0.6
Wavi ness Ratio (9,)

 

 

 

 

Figure 3.10 Plot of the effective stiffness E11 as the waviness ratio (A) is

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

increased.
v12 vs Waviness Ratio (A)
0.08 -
__ O
0'05 F, L Ci :1 Aitharaju 8. Aserill (1999)
i 0-04 “A‘ X A Whitcomb (1991)
' g 0.03 . —— o Raju & Wang (1994)
0 02 _ A 9 Present Study
' —(0. 90)s Laminate
0.01
0.00 +. c . 2
0 0.2 0.4 0.6
Wavi ness Ratio (1.)

 

 

 

 

 

(___’____ ##2—

Figure 3.11 Plot of the effective Poisson’s ratio v12 as the waviness ratio (A) is
increased.

71

 

G12 vs Waviness Ratio (2.)

 

 

 

 

 

(:1 Aitharaju & Averill (1999)
A Whitcomb (1991)

O

O

 

Raju 8. Wang (1994)
Present Study
—(0, 90)s Laminate

Grz (GPa)

 

 

0 0.2 0.4 0.6
Waviness Ratio (1)

 

 

 

 

 

Figure 3.12 Plot of the effective shear modulus 612 as the waviness ratio (A) is
increased.

72

Chapter 4

Modeling progressive failure using the simpliﬁed discrete-tow model

4.1 Introduction

Using the simpliﬁed discrete-tow model presented in Chapter 3, it is
possible to model progressive failure in woven composites. The simpliﬁed
discrete-tow model discretely represents the ﬁber tows and resin independently
as beam and shell ﬁnite elements respectively. The use of the discrete-tow
model in progressive failure allows for the ability to better predict the behavior
and physical orientation of the tows throughout the analysis in a manner that is
more computationally efﬁcient than a fully three-dimensional model.

In this Chapter, the utility of the simpliﬁed discrete-tow model developed
previously is expanded. Through the use of a stiffness reduction algorithm it is
possible to model the progressive failure that occurs in both the beam and shell
elements of the discrete-tow model to represent ﬁber and matrix damage

respectively.

4.2 Progressive Failure Model

When using the ﬁnite element method to predict progressive failure it is
necessary to accurately capture the energy release associated with the material
failure throughout the analysis. In a nonlinear analysis, if errors are made early,
they may propagate and result in erroneous results. The discrete-tow model has

the advantage of representing the energy release associated with damage in the

73

matrix or ﬁber tows independently because the matrix and ﬁber tows are

modeled discretely.

4.2.1 FINITE ELEMENT MESH

In order to effectively use the discrete tow model it is necessary to
carefully consider the ﬁnite element mesh that will be used to model the woven
composite structure. A ﬁnite element model of a single repeating volume
element for a plain weave and triaxial braid were developed previously and are
depicted in Figure 3.4 and 3.7. However, there are several competing
mechanisms occurring within the model that must be discussed. First, due to the
connectivity that exists between the shell and beam elements, singularities result
due to the application of point loads from the beam elements. Knowing this it is
desirable to have a ﬁnite element mesh that is coarse enough that the effects of
these singularities are effectively averaged out of the volume of the element.
Secondly, it is advantageous to use the discrete-tow model because of its ability
to explicitly model the ﬁber tows and their deformations within the woven
composite. Finally, if this model is used in a contact analysis, it is desirable to

have smaller element sizes to better capture the contact behavior that occurs.

4.2.2 MODELLING PROGRESSIVE FAILURE

Material damage within the tows and resin was modeled through the use

of stiffness reduction at each integration point within each element. Typically, in

74

 

a progressive failure analysis the integration point could be in one of three
stages, undamaged (a < err). damaged (air, < a < 8') , and completely failed (3' <
a). The stress vs. strain plot as shown in Figure 4.2 demonstrates the three
stages associated with progressive failure. However, for the purposes of this
study, the tows and resin were assumed to transition nearly instantly from
undamaged to completely failed (on, s 5') as illustrated in Figure 4.3.

Another advantage to the discrete-tow model is that it allows for different
failure criteria and values to be applied to the ﬁber tow and resin individually. In
the case of this study the ﬁber tow failure was based upon the maximum principle
strain failure criterion while the maximum principle stress failure criterion was
used in the resin. The actual values used in the current analysis are summarized
in Table 4.1. When a particular constituent material completely failed, its material
properties were reduced by several orders of magnitude (essentially zero, but not

exactly zero, to avoid numerical singularities in the analysis).

4.2.3 PROGRESSIVE FAILURE ALGORITHM

The progression of the failure in the ﬁnite element model is captured
through the use of an incremental iterative procedure associated with a standard
nonlinear analysis. This method of modeling progressive failure was ﬁrst
introduced by Dodds et al. [27] to model crack growth in concrete structures. In a
nonlinear analysis the applied load or displacement is applied in small
increments. For an implicit analysis, it may be necessary for the analysis to

iterate several times before an equilibrium position is found. Once equilibrium

75

has been satisﬁed within an acceptable tolerance, the next increment of load or
displacement is then applied. When a material point has failed, its stiffness is
reduced by calculating a new secant modulus and the analysis must iterate in
order to redistribute the load around the point of failure before the next increment
is started. The current progressive failure algorithm was implemented via a user
material subroutine in ABAQUS [28], a commercially available ﬁnite element
package. The subroutine implementation is summarized graphically in Figure

4.1.

4.3 Numerical Results

In order to validate the implementation of the progressive failure algorithm, RVEs
of both a plain weave and triaxial braid were constructed. Each RVE was then
subjected to four geometrically nonlinear analyses. Two analyses were
conducted without implementing the failure algorithm. The remaining two
analyses were completed with the failure algorithm turned on. In both cases,
with and without failure, one analysis was conducted using tensile loading while
the other used compressive loading. The geometry of the plain weave and
triaxial braid RVE’s are summarized in Table 3.1 and Table 3.6, while the
boundary conditions for each analysis can be found in Table 4.2 and illustrated
graphically in

Figure 4.4.

Results of the analyses of the plain weave are found in Figures 4.5

through 4.12, while results for the triaxial weave can be seen in Figures 4.14

76

 

through 4.21. When examining the results from the materially linear analysis of
the plain weave (Figures 4.5 through 4.8), the local and global behavior is nearly
linear as would be expected from a sample as small as one RVE. When failure
is included, and a geometrically nonlinear analysis is conducted of the plain
weave (Figures 4.10 through 4.12), the resulting nonlinear trends clearly
demonstrate that failure must be occurring within the constituent materials.

It is also important to note the signiﬁcant behavioral differences between
the plain weave and triaxial braid. Figure 4.9 and 4.12 illustrate how the majority
of the initial load is carried by the resin. Upon failure of the resin the load is
redistributed and carried by the tows until complete failure occurs. The load
carrying behavior of a triaxial braid (Figure 4.17 and 4.21) differs from the plain
weave due to the fact that the initial load carrying is accomplished by the axial
tow. This, of course, is dependent upon the loading conditions, however, in this
case the load was applied along the axis of the axial tow. Once the axial tow
fails in the triaxial braid, the braider tows assume the role of carrying the load

until complete failure occurs.

4.4 Conclusions

Using a standard nonlinear ﬁnite element analysis as a foundation, it is
possible to add the ability to predict progressive failure in woven composites
using the discrete-tow model. A progressive failure algorithm that represents
failure in the form of material stiffness reduction was combined with the discrete-

tow model to predict progressive failure of a plain weave and triaxial braid RVE.

77

 

Based upon the results of this study, it is quite reasonable to assume that this
simpliﬁed methodology of modeling damage in woven composites could be

extended to structures much larger than a single RVE.

78

 

Table 4.1 Failure criteria used in the resin and tow.

 

Case Resin Tow
Tensile Failure 81 = 0.138 GPa E1=0.02

Compressive Failure S3 = 0.138 GPa E3=0.02

 

 

 

 

 

 

 

Table 4.2 Applied boundary for the linear and failure cases studied.

 

 

 

 

Linear (without failure) With Failure
Tension Compression Tension Compression
8x 8y 8x 8y 8x 8y 8x 8y
0.126 0 0.126 0 0.126 0 0.126 0

 

 

 

 

 

 

 

 

 

 

79

Sit — — 1} Parse input ﬁle (once) \

. . Determine appropriate failure
Determrne material type —> criteria for each constituent

and test each for failure :

(Update property if failure I Calculate the new effective

has occurred properties

 

(Calculate the new material Calculate the new state 0f
stiffness matrix for the stress and return It to the
element calling subroutlne

I

O’ij

Figure 4.1 Implementation of the progressive failure algorithm.

80

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Linear Loading progressive Complete I
No Damage Damage Failure I
7 _ T7.
/ 2 h
/ \
b 1 I
/ \
/ I
74—-— — \
ﬁ _. —-I
’ t " ""' ' '
/
0 if .
0 8f 8*
8

 

 

 

Figure 4.2 Stress strain curve illustrating progressive damage (not immediate).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I _W
Linear Loading Complete
No Damage Failure
/
b /
I
/
/ l
__ i _ r
/ I -- 'l’ T I
.i . ; e.-
‘ 0
8
I -- -h

 

Figure 4.3 Stress strain curve illustrating immediate material failure.

81

 

(b)

Figure 4.4 Illustration of applied boudary conditions to (a) a plain weave and (b)
a triaxial braid.

82

 

 

0.03
0.025
0.02

0.015

RF1(KN)

0.005

 

 

0.01 .

 

+Beam & Shell Elements
____ —r:l— Beam Elements Only
-A— Shell Elements Only

 

 

 
 

 

 

-4

 

 

 

 

 

 

 

 

0 0.02

0.04 0.06 0.08 0.1
U1 (mm)

0.12

 

Figure 4.5 Force deﬂection curve for a plain weave under tensile loading

-0.12

 

 

l
I
I

(without failure).

 

 

-0.1 -0.08

U1 (mm)

-0.06 -0.04 -0.02 0
' 0

 

 

 

 

 

 

 

 

-o-Beam & Shell Elements —~
—l:l— Beam Elements Only

_. ———<

—A— Shell Elements Only

 

 

-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04

(Nx)lda

 

t

Figure 4.6 Force deﬂection curve for a plain weave under compressive loading

(without failure).

83

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10
3 _ +Shells (Gauss Point) -
/ I --l-- Beams (Gauss Point)
E 6 1 [i
9 /
g 4 - — /l/
/
I 2 /
/ ' 4* 4A
0 Rd:

 

 

 

811

 

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

._I

 

Figure 4.7 Linear validation of one RVE of a plain weave under a tensile load.

 

 

 

i 811

 

 

 

 

 

 

 

 

 

 

 

 

 

 

£0.15 01 -0.05 0
i 0

.1 + ' M
i #____ -2

—0—Shells (Gauss Point) /
-—— --l-Beams (Gauss Point) f-.__-_ -4
/
r g _6
/

~-—-~~#-ef - ~ 1 « -8
-10

 

(eds) ”0

 

 

 

Figure 4.8 Linear validation of one RVE of a plain weave under a compressive

load

 

 

+ Beam 8 Shell Elements
—r:l— Beam Elements Only ’—
+ Shell Elements Only

 

 

 

 

 

 

 

 

 

 

 

 

 

0. 02 0.03 0.04 0.05 0.06

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I U1 (mm)
Figure 4.9 Force deﬂection curve for a plain weave under tensile loading (with
failure).
is __
U1 (mm)
-0.05 -0.04 -0.03 -0.02 -0.01 0
i i-ﬁ 'L ﬁf- f ‘ r / a 0
i/T/m
\ -0.001
0002 §
\\ a
- -0.003 3
+Beam & Shell Elements \\
/
.__.__ -El— Beams Elements Only ‘ ’ '0-004
«a— Shell Elements Only
0005

 

 

 

I

 

 

Figure 4.10 Force deﬂection curve for a plain weave under compressive loading
(with failure).

85

 

 

—o—Shells (Gauss Point)

 

 

 

2.5 7 ﬂ -—I——Beams (Gauss Point)
1A 2— /{ i
(U
69, 1.5 - r/ \\
v7; 1 _ / i

 

 

 

 

 

 

0.01

 

0.02 0.03

311

0.04

 

 

Figure 4.11 Failure validation of one RVE of a plain weave under a tensile load.

 

 

811

-0.05

 

 

 

 

 

 

 

 

 

 

-0.04 -0.03 -0.02 -0.01 0

I I O
, i ,1 -0.5

I F -1

—o—Shells (Gauss Point) I /

F “— —I-- Beams (Gauss Point) ] f '1'5

_-_ l‘ 7; -2
j e -2.5

-3

 

 

I

(eels) rrs

 

Figure 4.12 Failure validation of one RVE of a plain weave under a compressive

load.

86

 

140

+Beams & Shells
120 h ‘“ —u— Beams Only f
100 .- - -- —A— Shells Only _
.. f
60 f
40 f
20 ~
0 W—

0 0.1 0.2 0.3 0.4 0.5
U1 (mm)

 

 

 

RF1 (KN)

 

 

 

M‘—
, O

 

 

 

 

 

Figure 4.13 Force deﬂection curve for a triaxial weave under tensile loading
(without failure).

 

 

 

 

 

 

 

 

 

 

U1 (mm)
-O.5 -0.4 -O.3 -0.2 -0.1 0
w 0
» ~ -20
f -40
I)
f -60 j
1 _ﬁ 430 2
+ Beam & Shell Elements '100
.0. Beam ELements 120
’ _A_ Shell Elements —“ '
-14o

 

 

 

 

 

 

 

 

Figure 4.14 Force deﬂection curve for a triaxial weave under compressive
loading (without failure).

87

 

 

12
--O—- Braider Element (Gauss Point)

10 ~—— - I~ Axial Element (Gauss Point) ._I___

 

+Shell Element (Gauss Point) I I

 

 

 

 

 

s11 (GPa)

 

 

 

 

 

 

 

 

 

‘.
0 0.02 0.04 0.06 0.08

 

 

 

Figure 4.15 Linear validation of one RVE of a triaxial weave under a tensile load
(without failure).

 

e 11
—0.08 -0.06 -0.04 —0.02 0

 

 

 

 

 

 

 

 

 

 

 

 

 

it?
o.
_ e
.1’ . '8 m
.. I' I + Braider Element (Gauss Point)
__-_.,L - .- - Axial Element (Gauss Point) _4 -10
I’ _A—Shell Element (Gauss Point)

 

 

 

 

 

Figure 4.16 Linear validation of one RVE of a triaxial weave under a
compressive load (without failure).

88

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+ Beam & Shell Elements
30 _0_ Beam Elements -
1 _A_ Shell Elements
25
20
15 ..
10 y __
5 J M
0 WM—

0.2 0.3 0.4
U1 (mm)

 

0.5

 

 

Figure 4.17 Force deﬂection curve for a triaxial weave under tensile loading (with

failure).

 

U1 (mm)

 

 

 

 

 

t________.‘

 

+Beam & Shell Elements

—n— Beam Elements
—A— Shell Elements

 

 

 

 

 

 

(NM) L58

 

 

Figure 4.18 Force deﬂection curve for a tn'axial weave under compressive

loading (with failure).

89

“g..-

‘I -.-~\".'

 

 

 

 

 

 

 

 

 

 

 

 

-+— Braider Element (Gauss Point)

1 2.5 -— - l- Axial Element (Gauss Point)
I 2 " . —£:—Shell Element (Gauss Point)
i if .

0- .

9 1.5 — . .

m 1 __ l,

0.5 « .

i o
l O 0.02 0.04 0.06

 

0.08

811

 

lw EA.

Figure 4.19 Failure validation of one RVE of a triaxial weave under a tensile

load.

 

-0.03

A A A A

i -0.04

A

811

-0.02 -0.01 0

 

‘ f", A A 0

 

 

 

 

01
(ads) us

 

C,

 

 

 

. l2.
, . i ’ _-

 

 

—+—-Braider ELement (Gauss Point)
- I- Axial Elements (Gauss Point)
—A—Shell Element (Gauss Point)

 

 

 

 

Figure 4.20 Failure validation of one RVE of a triaxial weave under a

compressive load.

90

 

Chapter 5

Conclusions

5.1 Conclusions

A method for efﬁciently modeling two-dimensional woven composites was
successfully developed. This simple discrete-tow model has the ability to
accurately predict the stiffness of RVE’s for plain weave and triaxial braids. The
discrete-tow model also signiﬁcantly decreases the number of degrees of
freedom within a ﬁnite element model while greatly simplifying the model building
process.

The utility of the discrete-tow model was also demonstrated by applying it
to the problem of damage prediction within a woven composite. Although it is
necessary to validate this process with experimental results, it does prove to be a

promising tool for the analysis of woven composites.

5.2 Future Work

Although the discrete-tow model was shown to agree well with other
stiffness prediction models and with experimental results, the model has yet 0 be
applied to a large composite structure much larger than a single RVE. When the
discrete-tow model is used to model larger sections several addition items must
be addressed, including; variation of ﬁber volume fraction, tow cross-sectional
variation, and tow crimp. Each of these alone is not a trivial matter and should

be examined thoroughly.

91

Currently the application of the discrete-tow model to progressive failure
needs to be validated with experimental results. In practice, the failure algorithm
and subroutine derived from it perform correctly for single RVE models, however
a detailed analysis should be conducted to gain conﬁdence in its use for larger

structures.

92

Bibliography

1. Cox, B.N. and Flanagan, G., “Handbook of Analytical Methods for Textile
Composites,” Version 1.0, Rockwell Science Center Report, January, 1996.

2. Tan, P., Tong, L. and Steven, G.P., “Modeling for Predicting the Mechanical
Properties of Textile Composites - A Review,” Composites Part A, Vol. 28A,
1997, pp. 903-92

3. lshikawa, T. and Chou, T.-W., “Elastic Behavior of Woven Hybrid
Composites,” Joumal of Composite Materials, Vol. 16, January 1982, pp. 2-
19.

4. lshikawa, T. and Chou, T.-W., “One-Dimensional Micromechanical Analysis of
Woven Fabric Composites,” AIAA Journal, Vol. 21, NO. 12, 1983, pp. 1714-
1721.

5. lshikawa, T. and Chou, T.-W., “Stiffness and Strength Behaviour of Woven
Fabric Composites,” Journal Materials Science, Vol. 17, 1982, pp. 321 1-3220.

6. Pastore, CM. and Gowayed, Y.A., “A Self-Consistent Fabric Geometry
Model: Modiﬁcation and Application of a Fabric Geometry Model to Predict
the Elastic Properties of Textile Composites,” Journal of Composites
Technology & Research, JCTRER, Vol. 16, No. 1, January 1994, pp. 32-36.

7. K0, F.K., Pastore, C.M., Lei, C. and Whyte, D.W., “A Fabric Geometry Model
for 3-D Braid Reinforced FP/Al-Li Composites,” Proceedings, lntemational
SAMPE Metals Conference: Competitive Advances in Metals/Metal
Processing, Cherry Hill, NJ, Aug. 1987.

8. Ma, C.-L., Yang, J.-M., and Chou, T.-W., “Elastic Stiffness of Three
Dimensional Braided Textile Structural Composites,” Composite Materials:
Testing and Design (Seventh Conference), ASTM STP 893, J.M. Whitney,
Ed., American Society for Testing and Materials, Philadelphia, 1986, pp. 404-
421.

9. Yang, J.M., Ma, C.-L, and Chou, T.W., “Fiber Inclination Model of Three
Dimensional Stmctural Composites,” Journal of Composite Materials, Vol. 20,
September 1986, pp. 472-484.

10. Naik, MK. and Ganesh, V.K., “Prediction of elastic Properties of Plain Weave
Fabric Composites,” Composites Science and Technology, Vol. 45, 1992, pp.
135-152.

11.Raju, l. S. and Wang, J. T., “Classical Laminate Theory Models for Woven
Fabric Composites,” Journal of Composites Technology and Research,
JCTRER, Vol. 16, No. 4, October 1994, pp. 289-303.

12.Aitharaju, V. R. and Averill, R. C., “Three Dimensional Properties of Woven-
Fabric Composites” Composites Science and Technology, Vol. 59, pp. 1901-
1911, 1999.

93

13. Sankar, B.V. and Marrey, R. V., “Analytical Method for Micromechanics of
Textile Composites,” Composites Science and Technology, Vol. 57, 1997, pp.
703-713.

14. Dasgupta, A., S. Bhandarkar, M. Pecht and D. Barkar, “Thenno-elastic
Properties of Woven Fabric Composites Using Homogenization Techniques,”
Proceedings of the American Society for Composites, Fifth Technical
Conference, Technomic Publishing Co., 1990, pp. 1001-1010.

15.Whitcomb, J. D., “Three Dimensional Stress Analysis of Plain Weave
Composites, “ Composite Materials: Fatigue and Fracture (Third Volume),
ASTM STP 1110, T.K. O’Brien, Ed., American Society for Testing and
Materials, Philadelphia, 1991, pp. 417-438.

16.Whitcomb, J.D., Woo, K. and Gundapaneni, 8., “Macro Finite Element
Analysis of Textile Composites,” Journal of Composite Materials, Vol. 28, No
7, 1994, pp. 607-617.

17.Woo, K. and Whitcomb, J.D., “Global/Local Finite Element Analysis for Textile
Composites,” Journal of Composite Materials, Vol. 28, No. 14, 1994, pp.
1305-1320.

18. Marrey, RV. and Sankar, B.V., “A Micromechanical Model for Textile
Composite Plates,” Journal of Composite Materials, Vol. 31, No. 12, 1997,
pp.1187-1213.

19.Blackketter, D.M., Walrath, DE. and Hansen, A.C., “Modeling Damage in a
Plain Weave Fabric-Reinforced Composite Material”, Journal of Composites
Technology & Research, Vol. 15, No. 2, Summer 1993, pp. 136-142.

20.Xu, J., Cox, B.N., McGlockton, MA, and Carter, WC, “A Binary Model of
Textile Composites-ll. The Elastic Regime,” Acta Metallurgica et Materiallia,
Vol. 43, No. 9, 1995, pp. 3511-3524.

21 . Lei, 0., Cal, Y.J., and Ko, F., “Finite Element Analysis of 3-D Braided
Composites,” Advances in Engineering Software, Vol 14, No. 3, 1992, pp.
187-194.

22. Naik, Rajiv A., “Failure Analysis of Woven and Braided Fabric Reinforced
Composites,” Joumal of Composite Materials, Vol. 29, No. 17, 1995, pp.
2334-2363.

23. Naik, Rajiv A., lfju, Peter G., and Masters, John E., “Effect of Fiber
Architecture Parameters on Deformation Fields and Elastic Moduli of 2-D
Braided Composites,” Journal of Composite Materials, Vol. 28, No. 7, 1994,
pp. 656 — 681.

24. Jones, Robert M., Mechanics of Composite Materials, Second Edition, Taylor
8. Francis, Philadelphia, PA, 1999.

25. Naik, Rajiv A., “Analysis of Woven and Braided Fabric Reinforced
Composites,” Composite Materials: Testing and Design (Twelfth Volume),
ASTM STP 1274, RB. Dec and CR Saff, Eds., American Society for Testing
and Material, 1996, pp. 239-263.

94

26. Foye, R.L., “Approximating the Stress Field within the Unit Cell of a Fabric
Reinforced Composite using Replacement Elements,” NASA CR-191422,
National Aeronautics and Space Administration, Washington DC, 1993.

27.Dodds, R. H., Lopez, L. A., and Pecknold, D. A., “Numerical and Software
Requirements on General Nonlinear Finite Element Analysis,” Report No.
UILU-ENG-78-2020, Civil Engineering Studies, University of Illinois at
Urbana-Champaign, Sept. 1978.

28.“ABAQUS users manual, version 5.7,” Hibbit, Karlsson & Sorenson, Inc.
1996.

95