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Stamping Analysis of Textile Composite Preforms

presented by
Ran Mohan Jit Singh Sidhu

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Master of Science degreein Dept, of Materials Science
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11m W.“

STAMPING ANALYSIS OF TEXTILE COMPOSITE PREFORMS

By

Ran Mohan Jit Singh Sidhu

A THESIS

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

MASTER OF SCIENCE

Department of Materials Science and Mechanics

2000

ABSTRACT

STAMPING ANALYSIS OF TEXTILE COMPOSITE PREFORMS

By

Ran Mohan Jit Singh Sidhu

A ﬁnite element model has been developed for the stamping analysis of
plain weave textile composite preforrns. This model is simple, efﬁcient and can
be used with existing ﬁnite element codes. The model represents the preform as
a mesh of 3-D truss elements and 3-D shell elements. The tows are modeled
explicitly through the truss elements. Inter tow sliding has been taken into
account in the model. The shell elements represent a ﬁctitious material, which we
called the transforming medium that models the effects of inter tow friction and
ﬁber angle jamming. The model takes into account large strains and large
deformations. ln-plane uni-axial tension tests were performed on plain weave
specimens for determining the constitutive law of the transforming medium. Use
of the model is demonstrated by simulating the stamping of a preform over a
sphere. The results from the simulation show good correlation with the results

from the experiments.

Copyright by
Ran Mohan Jit Singh Sidhu
2000

Dedicated to my Grandparents and Parents

ACKNOWLEDGMENTS

I would like to thank my advisor Dr. Averill for his incredible guidance,
expertise and foresight throughout the research. I would also like to thanks Dr.
Pourboghrat for his support and advice and Sakeb for his hard work and
assistance with the experimental work. I would also like to thank the MSM faculty
for the immense knowledge that I gained through the courses.

Furthermore I would like to thank my family for all their support and

guidance throughout the period of my research.

TABLE OF CONTENTS

ACKNOWLEDGMENTS ...................................................................................... V
LIST OF FIGURES ........................................................................................... VIII
CHAPTER 1 .......................................................................................................... 1
1 INTRODUCTION ........................................................................................... 1
1.1 Introduction ............................................................................................... 1
1.2 Literature review ....................................................................................... 3
1.2.1 Dry Preforms ................................................................................... 4
1.2.2 Soft Matrix Fiber Reinforced Composites ....................................... 6

1.3 Current model ............................................................................................ 7
CHAPTER 2.. ........................................................................................................ 8
2 THE PREFORM MODEL ............................................................................... 8
2.1 Objectives .................................................................................................. 8
2.2 Deformation Modes ......................... 8
2.3 Calculation of the Jamming Angle .......................................................... 16
2.4 Unit Cell .................................................................................................. 21
2.4.1 Rack Edge Pinned Model .............................................................. 21
2.4.2 Rack Edge Sliding Model .............................................................. 25
2.4.3 Tongue and Groove Model ............................................................ 29
2.4.4 Checkerboard Model ..................................................................... 32

2.5 Constitutive Model .................................. ............................................... 35
CHAPTER 3 ........................................................................................................ 41
3 EXPERIMENTATION AND NUMERICAL RESULTS .................................. 41
3.1 Large Scale Local Model ........................................................................ 41

vi

3 .2 Uniaxial Experiments .............................................................................. 42

3.3 Uniaxial Sliding Experiment ................................................................... 52
3.4 Uniaxial Tensile Simulation .................................................................... 58
3.4.1 Rack Edge Models ........................................................................ 59
3.4.2 Tongue and Groove Model ............................................................ 62
3.4.3 The Checker Board Model: ............................................................ 65

3.5 Draping Experiments .............................................................................. 77
3.6 Draping Simulation ................................................................................. 81
CHAPTER 4 ........................................................................................................ 86
4 CONCLUSIONS .......................................................................................... 86
APPENDIX A ...................................................................................................... 88
5 USER MATERIAL CODE ............................................................................. 88
6 BIBLIOGRAPHY .......................................................................................... 93

vii

List of Figures

Figure 1. A simple representation of a plain weave preform. The square drawn
on the preform represents one unit cell .......................................................... 9

Figure 2. Undulation of a tow as it goes over one tow and then under the next
tow.. .......................................................................................................... 10

Figure 3. Scissoring deformation of the unit cell with the tow changing ﬁber
angle and forming a bow with a different aspect ratio .............................. 11

Figure 4. Part of four tows which form a unit cell. The spacing between the tows
is magniﬁed in this ﬁgure. ............................................................................ 13

Figure 5. A part of the preform with the tows at ﬁber angle equal to the initial
jamming angle,9u ......................................................................................... 14

Figure 6. Sliding deformation in a unit cell. (a) The initial conﬁguration. (b) The
initial position and the ﬁnal position of the tows after deformation. The point
of intersection can be seen to have displaced. ............................................ 15

Figure 7. A part of 4 tows at a ﬁber angle equal to the initial ﬁber jamming
angle. W in the ﬁgure is the tow width at this ﬁber angle and is the
perpendicular distance between the boundary lines of the tow .................... 18

Figure 8. The exploded view of the triangle ABC from Figure 7. ...................... 18

Figure 9. Scissoring from initial ﬁber angle to the initial jamming angle with the
corresponding dimensions of the shell element. .......................................... 20

Figure 10. Unit cell for the rack edge pinned model. The ﬁgure shows the
elements at a reduced scale and separated to show the connectivity. Dark
circles indicate nodes ................................................................................... 22

Figure 11. The complete assembled preform for the rack edge pinned
model...... .................................................................................................... 23

viii

Figure 12. The shear stress-strain curve for the mooney-rivlin material. The
shear strain values are not absolute shear strain values but increment
numbers for the incremental shear strain application. ................................. 24

Figure 13. Shear deformation of the shell elements corresponding to the
uniaxial elongation of the preform. The smaller interior angles of the shell are
the inter ﬁber angle. ..................................................................................... 25

Figure 14. The unit cell of the rack edge sliding model. The plus marks in the
ﬁgure show the displaced nodes of the truss elements. The ﬁgure shows
elements at 80% of their actual size. ........................................................... 26

Figure 15. The complete modeled preform for the rack edge sliding model. ..27

Figure 16. The connectivity between the shell elements and the tmss
elements. The dark circles are the nodes of the shell element. The dark lines
represent the node at which the tows are connected to the shell and the
direction. ...................................................................................................... 27

Figure 17. The unit cell for tongue and groove model .................................... 30

Figure 18. The tow-to-shell element connectivity in tongue and groove
model.. ........................................................................................................ 31

Figure 19. The complete modeled preform for the tongue and groove
model.. ........................................................................................................ 31

Figure 20. The unit cell for the checkerboard model ....................................... 32

Figure 21. The two types of tow-to-shell element connectivity in the

checkerboard model. ................................................................................... 33
Figure 22. The complete modeled preform for the checkerboard model. ....... 34
Figure 23. The normal stress-strain curve for transition medium .................... 37
Figure 24. The shear stress-strain curve for the transition medium ................ 39

ix

Figure 25. The experimental large scale model of a part of a plain weave

preform. The ﬁgure shows the tows in a sheared position. .......................... 42
Figure 26. Deformed preform at 10% elongation. ........................................... 45
Figure 27. Deformed preform at 15% elongation. ................................... . ........ 46
Figure 28. Deformed preform at 20% elongation. ........................................... 47
Figure 29. Deformed preform at 25% elongation. ........................................... 48

Figure 30. Force versus % elongation of the preform from the uniaxial
experiment. .................................................................................................. 49

Figure 31. The various Zones formed in a Preform during the Uni-Axial
extension ..................................................................................................... 51

Figure 32. Sliding at the junction of Zone I and Zone II and some sliding in

Zone II... ...................................................................................................... 53
Figure 33. Sliding at the clamped edge at 25% elongation ............................. 54
Figure 34. Almost no sliding in Zone III except at the free edges. .................. 55
Figure 35. Sliding at the junction of the Zones and some in the Zone l| ......... 56
Figure 36. Sliding at the free edge of preform at 20% elongation ................... 57

Figure 37. Preform with boundary conditions for the uniaxial test. The dotted
rectangle indicates the nodes on the edges of the preform. ........................ 59

Figure 38. Deformed preform mesh at 40% elongation. ................................. 60

Figure 39. Comparison of center ﬁber angle from analysis and study by
Mauget etal. ................................................................................................. 61

Figure 40. The deformed mesh of preform with tongue and groove model. ...62

Figure 41. Deformed preform model with only truss elements present ........... 63

Figure 42. A part of the preform showing the varying strain in the shell
elements more clearly (tongue and groove model). ..................................... 64

Figure 43. The whole preform mesh with just shells. Shell element distortion
indicates sliding. .......................................................................................... 67

Figure 44. Sliding at the interface between Zones I and II mainly. ................. 68

Figure 45. The details of sliding at the interface between Zones II and III. ..... 69

Figure 46. Showing primarily Zone III. All the shell elements deform through
normal strain only except the interfaces and the free edges indicating no
sliding in the middle and signiﬁcant sliding at the interface and at the

edges ........................................................................................... 70
Figure 47. Preform at 10% elongation as predicted by the simulation. ........... 72
Figure 48. Preform at 20% elongation as predicted by the simulation. ........... 73
Figure 49. Preform at 25% elongation as predicted by the simulation. ........... 74

Figure 50. Center ﬁber angle comparison between experiment and
simulation ..................................................................................................... 75

Figure 51. Force versus % elongation plot for experiment and simulation. ....76

Figure 52. Preform contour at 1 in. penetration. ............................................. 78
Figure 53. Preform contour at 1.5 in. penetration. .......................................... 79
Figure 54. Preform contour at 2.25 in. penetration. ........................................ 80

xi

Figure 55. Preform contour at 0.25 in. penetration from simulation. ............... 83

Figure 56. Preform contour at 1.0 in. penetration from simulation. ................. 84

Figure 57. Preform contour at 2.25 in. penetration from simulation. ............... 85

xii

List of Tables

Table1. Properties of the perform specimen used for experiments. ............... 43

xiii

Chapter 1

1 INTRODUCTION

1. 1 Introduction

The importance of fabric reinforced composites hardly needs to be
emphasized. They have a variety of applications in various industries, especially
as structural components in the automobile, aerospace, marine, infrastructure
and recreation industries. The main advantage of using these materials is their
high speciﬁc strength. In the past these fabric reinforced composites were
manufactured by laying up the yarn bundles in the mold directly before injecting
the resin. However, this process of manufacturing was very labor intensive and
also very cost inefﬁcient. The manufacture of these composites has been made
more efﬁcient by weaving yarn bundles into a fabric sheet and then forming this
preform to the shape of the desired object prior to introducing a matrix material.

It is known that bending, stretching, and shearing of textile preforrns
affects the orientation, the cross-sectional shape, the crimp angle, and the
density of tows in the preform. As a result, draping or forming of preforrns may
result in large changes in the microstructure of the preform. These changes in
turn strongly affect the permeability of the preform as well as the local stiffness
and strength of the ensuing composite. Deleterious microstructural distortion is

usually most severe near regions of high spatial curvature, making these already

critical stmctural regions even more prone to premature failure. Prediction of
failure in these regions cannot be based on nominal properties of ﬂat coupon
specimens. Rather, some estimate of the local stiffness and strength (both
dependent on the microstructure and the processing conditions) must be known
in order to provide a reasonable estimate of structural failure loads. Also, the
evolved microstructure after deformation is important for the permeability
calculations for modeling mold ﬁlling processes. The evolution of the orientation
of the tows of the preform during the deformation process thus needs to be
accurately determined in order to determine these properties of the deformed
preform.

Also it is necessary to know if the preform being considered can be
deformed onto the desired shape without wrinkling, and if the ﬁber orientation is
unique or is sensitive to the method of forming as the formability of the fabric is
greatly dependent on its architecture. The prediction of the microstructural
evolution of the preform undergoing draping will result in a large saving in terms
of time and money required in the designing of a die, without knowing if it will
yield the desired results. If the preform does not deform to the shape of the die
without wrinkling, the die is useless. This requires a simple ﬁnite element model,
which can be used to simulate the draping process and predict accurately the
deformed microstructure of the preform. A new ﬁnite element model has been
proposed for the stamping analysis of a plain weave textile composite preform.
This model is simple, efﬁcient and can be used with existing ﬁnite element codes.

The model represents the preform as a mesh of 3-D truss elements and 3-D shell

elements. The truss elements model the tows in the preform and are practically
inextensible during the deformations resulting from the draping or stretch forming
of the preforrns. The truss elements are not pinned at the points of intersection,
which allows for the modeling of sliding between the tows. The truss elements,
representing the tows, are connected to each other, indirectly, through the shell
elements. The shell elements represent a ﬁctitious material, called the transition
medium. This transition medium has a non-linear orthotropic material behavior
and represents the effects of inter tow friction and the ﬁber angle jamming in the
model. The model takes into account large strains and large deformations. ln-
plane uni-axial tension tests have been performed on plain weave preform
specimens. These tests were used to determine the constitutive law of the
transition medium and also to show that the inter ﬁber sliding in the preform is
signiﬁcant during even moderate strains on the overall preform. Use of the model
is demonstrated by simulating the stamping of a preform over a sphere. The
results from the simulation show good correlation with the results from the

experiments.

1.2 Literature review

Several different approaches have been used to model the deformation
behavior of the preform accurately but they fail to take into account all the
aspects of the deformation behavior of the preform. Some of these techniques

have been discussed below.

1.2.1 Dry Preforms

Robertson et al.1'6 use the ﬁsherman’s net analogy for the computation of
the evolution of the tow orientation. The tows are assumed to be inextensible and
pinned at the inter-sections. They assume that the only available mode of
deformation is the in-plane ﬁber shear. The position of each node (the point of
inter-section of the tows) is found by solving the equations of the intersection of
the weft and warp tow segments and the surface. 80 starting at the top the
fabric is laid on the hemisphere one node at a time until the bottom edge is
reached and then they can predict if wrinkling occurs. The ignoring of the ﬁber
angle jamming and the pinned ﬁbers lead to over prediction of ﬁber angles in the
areas of high deformation.

Advani et al.“ and Rudd et al.5 used the ﬁshnet technique to model the
preform behavior. The fabric is replaced by an in-extensible net that allows only
shearing of the ﬁbers. As in the above models the inter-tow slippage has been
ignored. However the jamming is implemented in their code. The information
from this code is then used to calculate the ﬁber volume fractions and ﬁnally the
permeability calculations.

Boisse et a|._7'8 represented the overall mechanical behavior of the fabric
by combining the tensile behavior of a single thread and the position of the tows
at that instant. The fabric is modeled using 3 and 4 noded membrane elements
due to the low bending stiffness. They have modeled the complete tensile
behavior of the tow including the undulation effect and the damage of the ﬁbers

obtained from the uni-axial and bi-axial tensile tests. Again in this model it has

been assumed that there is no sliding between the tows and the jamming has
also been ignored.

McBride et al.?“ consider the entire architecture of the tows in modeling
the mechanical behavior of the preforrns. They identiﬁed the unit cell and
described its conﬁguration by a set of four cosinusoids. The geometry of the unit
cell was formulated as a function of the fabric thickness, the yarn width, the yarn
spacing and the yarn angle. The fabric thickness and the yarn spacing have been
assumed to be independent of the yarn angle but the yarn width has been
deﬁned as a function of the yarn angle. So knowing the yarn angle the geometry
of the fabric can be found. They use this formulation in combination with the
ﬁshnet model used in [1] to give the orientation and the geometry of the fabric
with deformation.

Ascough11 et al. used a beam element large displacement model for the
simulation of fabric draping behavior. The fabric in this study is a cloth fabric with
a much ﬁner weave than that of the structural material preforrns being considered
here. In this model, the bending behavior of the cloth is governed by the stiffness
properties of the beam elements. The ﬁbers were not modeled explicitly. The
goal was to predict the overall shape of the draped cloth rather than investigate
the position or orientation of the ﬁbers in the cloth. The beam elements used
have mass and stiffness properties corresponding to a cloth width equivalent to
the element pitch. Large displacement behavior is included. A Newmark time
stepping approach is used for the analysis. The results compare well with the

experiments, but a reﬁned mesh requires high computational time.

1.2.2 Soft Matrix Fiber Reinforced Composites

All the above techniques have been used for the modeling of preforms.
There are some other techniques, which have been used for the modeling of

reinforced composite fabrics. Mauget et al.12

used the laminate analogy to predict
the ﬁber orientation of a soft composite structure. Since here the stiffness of the
resin is very small as compared to the ﬁbers it is very similar to a preform. They
again assume the ﬁbers to be pinned at the intersections and inextensible so the
deformation of the fabric is represented as a change in the aspect ratio of the unit
cell. The composite is modeled by thick shell elements or brick elements. The
laminate theory is used to calculate the properties of the elements. The Poisson’s
ratio of the elements is updated with the change in the aspect ratio of the
elements. The ﬁber angle reaches ﬁve degrees before jamming, which is less
than the actual jamming angle for typical preforrns.

Cherouat et al.”14 used a bi-component element to model reinforced
composite preforrns. They accounted for the non-linear behavior of the tows by
taking into account the undulation effect of the tows. This renders a very small
initial stiffness in the tows, as the initial force is used to straighten out the ﬁbers.

The resin is modeled as a visco-elastic material. Tow jamming and tow sliding

have been ignored in the formulation.

1.3 Current model

Nearly all the approaches to date ignore the inter tow sliding, and the inter
ﬁber jamming has not been modeled effectively in a few places where it has been
considered. Another aspect that is missing in all the above approaches is the
modeling of fabric imperfections. During the manufacturing process the tows are
not aligned perfectly and there may be a lot of waviness in the tows. This is
particularly the case in the tri-axially braided preforrns where the axial tow is not
straight. In our approach the tows are modeled explicitly and therefore the
imperfections in the specimen can be modeled exactly as they are.

In the present paper a new model is developed to study the mechanical
behavior of textile preforrns. The aim of this study is to represent accurately the
deformation behavior of a preform using a model that is simple to build. The unit
cell of the preform is identiﬁed and is discretized with 2 noded 3-D truss elements
and 4 noded 3-D shell elements. The truss elements are used to model the tows
and the shell elements have been used to model the ﬁctitious material (the air
gaps between the tows), which we call the transition medium. The constitutive
law of the ﬁctitious material is characterized through uni-axial tensile tests on the

specimen.

Chapter 2

2 THE PREFORM MODEL

2. 1 Objectives

The main objective of this study was to develop a model for representing a
textile preform undergoing an arbitrary shape change due to stretching or die
forming, with the ability to accurately represent the microstructural evolution. The
model should incorporate all the different modes of deformation during the
draping or stretch forming of the preform. Another feature desired in the model
was that it should lend itself for analysis using existing commercial ﬁnite element
codes and should be simple to build. With all these desired features laid down
the various modes of deformation taking place during the preform draping were
identiﬁed. Figure 1 shows the general scheme of the tow layout in a plain weave

preform.

2.2 Deformation Modes
The different modes of deformation of a woven or braided preform are
tow straightening due to the undulations, ﬁber angle change due to scissoring of
the tows, and inter tow sliding. These are discussed individually below.
The morphology of the plain weave composite preforrns is such that the
braider tows go over and under each other to form the entire preform. As a result,

the tows in the plain weave composite preform are not straight, but, have a

curvature associated with them. This undulating curvature of the tows can be
compared to a sine wave with a very small amplitude. This is shown in Figure 2.
There are a few existing models that represent the tows as a sinusoidal curve,

which is a function of the fabric geometry, and the ﬁber angle.

A Unit Cell

 

Figure 1. A simple representation of a plain weave preform. The square
drawn on the preform represents one unit cell.
Due to this waviness associated with the tows, when a tensile load is
applied to the preform the tows tend to straighten out the undulations ﬁrst. The
initial deformation, therefore, is not due to the elongation of the ﬁbers but just to

the ﬁber bundle straightening. The transverse shear stiffness of the tows in the

preform is very small as it is just a bundle of dry ﬁbers. The low value of the
transverse shear stiffness and a very small moment of inertia result in a very low
bending stiffness of the tows. Thus, the initial stiffness is very low until the tows
are nearly straightened or until they cannot straighten any further due to
interaction between the tows, after which the stiffness is almost equal to that of
the ﬁbers. This mode of deformation does not affect the orientation of the tows or
the shape of} the fabric [8]. This mode of deformation, however, is not very
signiﬁcant when compared to the deformation due to scissoring and sliding.
Because this mode of deformation is not very signiﬁcant in the draping process, it
is not considered in the current model.

Tow cross—sections (0° direction)
Tow (90° direction)

    

Figure 2. Undulation of a tow as it goes over one tow and then under the next
tow.

The second mode of deformation in the plain weave preform is due to the
scissoring of the tows. This is the primary mode of deformation in the prefonns.
This mode leads to the most signiﬁcant changes in the microstructure of the

preform. The scissoring refers to the change in the inter tow angle without any

10

change in the length of the tows. This can be described as the change in the
aspect ratio of a rectangle, which has the tows as its diagonals. The length of

these diagonals remains constant. This is depicted in Figure 3.

 

 

 

 

 

 

 

 

Figure 3. Scissoring deformation of the unit cell with the tow changing ﬁber
angle and forming a bow with a different aspect ratio.

The only change that occurs during this mode of deformation is the
change in the angle between the tows, i.e. the change in the ﬁber angle. During
this mode of deformation there is no relative motion between the tows and they
basically act as being pinned at the point of intersection between any two tows.
This mode accounts for most of the deformation of the preform. The friction
between the tows produces a resistance to the rotation of the tows about the
point of intersection thereby producing a resistance to the deformation of the
preform. The magnitude of this frictional resistance is very small and is assumed
here to remain constant. This fact makes the scissoring mode the most
predominant mode of deformation in the preform as the defamation will follow

the path of least resistance to minimize energy. The effect of friction between the

11

tows has been modeled in the current model, even though the friction has not
been modeled directly as friction between the tows. The details about the method
and approach used to model the effect of friction and the deformation behavior
have been dealt with in the sections to follow.

Unloading of the preform has not been considered in this model. The
current model is only valid for the loading of the preform and does not truly
capture the deformation behavior during the unloading of the preform. Energy is
lost in the preform due to work done against friction during scissoring and sliding
between the tows. Since this process is irreversible, this energy cannot be
recovered. As a result, the actual preform will not follow the same stress-strain
curve during unloading as it does during the loading and this has not been
accounted for in this model.

As mentioned earlier, during the scissoring deformation of the preform the
ﬁber angle changes. It has been shown experimentally that the ﬁber angle
between the tows does not reach zero degrees with progressively increasing
deformation of the fabric. There is a limit to the ﬁber angle to which the fabric can
be sheared. As the preform is subjected to an increasing tensile strain, the tows
scissor to a smaller ﬁber angle to accommodate the deformation of the fabric.
Due to this the ﬁber angle keeps decreasing progressively, with increasing
overall fabric tensile strain. As the ﬁber angle gets smaller it reaches a point
where the spacing between the tows vanishes, i.e. the adjacent tows are in
contact with each other at this point. This is shown in Figures 4 and 5. Figure 4

shows a part of four tows in the undeforrned position with the angle between

12

them equal to the initial ﬁber angle of the preform. Figure 5 then shows the tows
in the position where they just come in contact and the angle between the tows at
this instant is the angle at which the stiffening behavior starts. This ﬁber angle is
called the initial jamming angle, 9n and is deﬁned as the ﬁber angle at the point in
time when the adjacent parallel tows come in contact with each other.

After reaching this angle, any further scissoring defamation takes place
with the change in the width of the tows, as they are pressed closer together with

the increasing tensile force on the preform.

 

Figure 4. Part of four tows, which form a unit cell. The spacing between the
tows is magniﬁed in this ﬁgure.

13

GIJ

}

t\

//.
é
\ //

\V
)7

Figure 5. A part of the preform with the tows at ﬁber angle equal to the initial
jamming angle,9u.

The initial stiffness during this made of defamation depends upon the
transverse stiffness of the tows. Even though the transverse stiffness of the tows
is small, the resistance to change in width is more than the frictional resistance
between the tows. The transverse stiffness of the tows increases with decreasing
tow width due to the interaction with other tows. Moreover, since the tow width
cannot decrease to zero there is a limit to the ﬁnal ﬁber angle to which the fabric
can shear. After reaching this angle no more shear deformation can occur. As a
result, the stiffness to the shear defamation between the tows increases until it
reaches this limiting shear angle, which is called the ﬁnal ﬁber jamming angle.
The value of this angle depends upon the type of weave, tightness of the tows
(i.e., the tow spacing) and the original angle between the tows, which namally

ranges from 15° to 20°. These values are half ﬁber angles, where half ﬁber angle

is deﬁned as half of the ﬁber angle between the tows at that point. So after
reaching this angle the fabric cannot deform any further by changing the ﬁber
angle between the tows. In the current model the jamming has been incorporated
through the constitutive law of the transition medium, which has been modeled
by 3-D shell elements. The modeling and the material laws of the model are
explained in the next section.

After reaching the jamming angle, the only possible mode of defamation
that the tows can undergo to accommodate the tensile strain being applied to the
fabric is the inter tow sliding, as the ﬁber angle between the tows cannot change
any further. The sliding defamation of the tows is illustrated in Figure 6 below.

Tows

 

 

C:>

 

 

 

 

 

 

Point of Intersection Final point of intersection

(a) (b)
Figure 6. Sliding defamation in a unit cell. (a) The initial conﬁguration. (b)
The initial position and the ﬁnal position of the tows after defamation.
The point of intersection can be seen to have displaced.
This mode of defamation is signiﬁcant in stamping processes involving
large defamations. This aspect of defamation has been neglected in most other

‘ models proposed in this ﬁeld as being insigniﬁcant. But as mentioned earlier this

mode becomes signiﬁcant during large defomations and henceforth cannot be

15

neglected. The sliding defamation has been accommodated through the
connectivity of the elements and is controlled in the current model through the
transition medium. The next section explains the unit cell and how all these
modes of defamation have been incorporated through the constitutive law of the

transition medium.

2.3 Calculation of the Jamming Angle

As discussed earlier the stiffening behavior of the prefom starts when the
adjacent tows came in contact with each other. At this point the force required to
deform the prefom increases as defamation now involves the change in the
width of the tows in addition to the friction between the tows against scissoring
and sliding. This is modeled by stiffening the shell elements when the ﬁber angle
becomes equal to the initial jamming.

The jamming angle of the fabric is calculated from the geometry of the
fabric. The theory and the equations used in this calculation have been taken
from McBride and Chen’s geometric model of the unit cell for plain weave fabrics.
The geometry of the tows is modeled using a set of four sinusoidal curves. The
width of the tows can be expressed as a function of the initial fabric geometry
and the ﬁber angle. It has been shown by experiments that the fabric thickness is
independent of the ﬁber angle [9]. For the present purpose, the ﬁbers have been
assumed to be pinned at the intersection of the tows. These equations are used
in the current model for the calculation of the initial jamming angle, which marks

the start of the stiffening behavior of the prefom. Until this point the tows behave

16

basically as pinned thereby the assumption that the tows are pinned at the point
of intersection is valid for these particular calculations. This is shown later in the
experiments also validating the assumption as there is practically no sliding in the
center portion of the prefom specimen, where the ﬁber jamming occurs. The
equation relating the tow width to the trellis angle is given below.
M9) = Wo/(8in 9)"o’"o”"o (1)

where w = is the tow width at ﬁber angle 9

Wu = is the initial tow width at 9 = 90°

9 = is the ﬁber angle

so = is the tow spacing which remains constant

‘30 = half period of sinusoid describing yarn cross-section at 9 = 90°

[30: nwo/(Zarccos[sin2(nwol4so)])

The criterion that has been used for the calculation of the initial jamming
angle is that the stiffening behavior of the prefom starts as soon as the adjacent
tows come in contact with each other, i.e. when the when the gap between two
adjacent tows in the prefom is zero. This has been shown earlier in Figure 5.
Using this criterion the initial ﬁber jamming angle is calculated in tems of the
initial fabric geometry and the initial ﬁber angle.

Figure 7 shows the tows at the point when they just come in contact with
each other and the associated dimensional details. Figure 8 shows the exploded

view of the triangle ABC from the Figure 7.

17

 

Figure 7. A part of 4 tows at a ﬁber angle equal to the initial ﬁber jamming
angle. W in the ﬁgure is the tow width at this ﬁber angle and is the
perpendicular distance between the boundary lines of the tow.

B

So / W/2

 

 

I— v

A C
Figure 8. The exploded view of the triangle ABC from Figure 7.

 

18

Using the geometry of the tows as shown in the ﬁgures above and using
simple trigonometric laws, a relation between the tow width w, the initial ﬁber
jamming angle 9..., and the initial tow spacing so can be fomulated. The distance
EB in Figure 9 is equal to the initial spacing between the tows, so , because of
the assumption that the tows behave as pinned with no inter ﬁber sliding until this

point. From triangle ABC

 

W
S1716” = /2 '-'- 1 (2)
S 30

R

In Equation (2) above, w is given by the Equation (1). Substituting the

value for w from Equation (1) into Equation (2) and solving for Sin 9“ :

 

Finally, 9” is given by :

ﬂo
(9” = ArcSin[(w0 /s0 ) ﬂ°”°-W° ]

 

where 6... = is the initial ﬁber jamming angle.

The ﬁber angle calculated using the above equation is used to calculate
the local unit cell nomal strain at this point, which is then used in the material
model to mark the start of the stiffening behavior. This strain calculation is again
based on the assumption that the local tow defamation in the prefom is mainly
due to scissoring and that the tows behave as pin jointed members until that
point. This way the unit cell can be represented as a rectangular box with the

tows as the diagonals and the angle between the diagonals equal to the ﬁber

19

angle between the tows at that point. Since the length of the tows does not
change during the scissoring defamation the change in the unit cell can be
regarded as just a change in the aspect ratio of the rectangular box. This is

shown in Figure 9.

 

 

 

 

 

 

 

 

 

 

 

Transition Medium element

Figure 9. Scissoring from initial ﬁber angle to the initial jamming angle with
the corresponding dimensions of the shell element.

Using the above ﬁgure the axial and the transverse strains can be

calculated using the standard fomula.

Nomal strain at point of stiffening, 8x and 8y are given by

 

ELLE
L,
W—Wo
8),:
W0

The ﬁnal jamming angle is found from the experimental data.

20

 

2.4 Unit Cell

Various models were tried before ﬁnally settling on the current model. The
description of all these models with their pros and cans are discussed in detail in
the following section.

The main characteristics that need to be included in the prefom model
were to be able to model each tow explicitly, include the effects of friction
between the tows, ﬁber angle jamming and also, incorporate the modeling of
sliding to capture the true defamation behavior of the prefom. The approach
used to represent all these features into the model was to model them one by
one, i.e. get one of the features working in the model and then build onto this
model to incorporate the rest of the features successfully. This stepwise
procedure followed in the fomulation of the model is outlined in the following

section. Each model is explained with its features and its limitations.

2.4.1 Rack Edge Pinned Model

The friction in the tows was modeled as a ﬁrst step towards the complete
model building. The unit cell for this model is shown in Figure 10. The unit cell is
famed of 1 shell element and 4 tmss elements. The shell elements and the truss
elements both are modeled as isotropic materials. The shell element modeling
the transition medium is shaped like a rhombus with the tows going along the
sides and pinned at the comers. The tows in this model are connected at their
point of intersection, i.e. they have a common node at the intersection and as a

result they cannot slide with respect to each other.

21

The tows are modeled explicitly as they are in the prefom using 3-D truss
elements. The shell elements are required in this model due to the fact that there
is no stiffness to ﬂexural loading of a pure truss element. The inclusion of the
shell elements overcomes this problem and also allows for modeling the effect of
friction between the tows. In this way the shells provided the initial stiffness to the

tensile extension of the prefom.

\

..At

Truss elements
Shell element . .

Figure 10. Unit cell for the rack edge pinned model. The ﬁgure shows the
elements at a reduced scale and separated to show the connectivity.
Dark circles indicate nodes.
1 The effect of the friction between the tows is modeled through the stiffness

of the shell elements. The unit cell and the entire modeled prefom are shown

below in Figures 10 and 11 respectively. This model is called the Rack Edge

22

Pinned model as the edges of this model resemble the teeth of a rack in rack and

pinion and the tows are pinned.

 

630333333333:

33333333333:
930393029362.
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039393036303
660600606606
o°o°o°o°o°o°
009000

03 0°
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o
°o°
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°o°
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°o°
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96%
. 363°
0"?
99%
9°?
, ‘ 9 9 _ ,.

Es.

 

 

 

Figure 11. The complete assembled prefom for the rack edge pinned model.

The main drawbacks and limitations of this model are: -
1. There is no ﬁber angle jamming allowed in this model.
2. Also, it does not allow for the inter tow sliding as the tows are pinned
together at the intersection.
The next step was to include the ﬁber angle jamming into the model.

Inclusion of the feature of representation of the jamming was accomplished by

23

using a non-linear material with a stiffening behavior. The Mooney-Rivlin material
model was used for the shell elements. The main purpose of using this material
model for the shell elements is that this material stiffens up with increasing strain.
The shear stress strain curve is shown in Figure 12.

This was the kind of material model that was needed for the shell
elements to model the tow jamming in the model. The defamation mode for the

shell elements in this model is primarily shear as shown in the Figure 13.

 

Shear Stress

 

12 3 4 5 6 7 8 910111213141516171819
StrainMeasure

 

 

 

Figure 12. The shear stress-strain curve for the mooney-rivlin material. The
shear strain values are not absolute shear strain values but increment
numbers for the incremental shear strain application.

The material parameters used in deﬁning the Mooney-Rivlin material

model were varied until the defamation behavior and the ﬁber angle change with

24

percentage elongation matched the experiments. The results were compared
with those from the work of Mauget et al. [12]. The details of the analysis and the

results are discussed in the results’ section in Chapter III.

:(>

Shell element
Figure 13. Shear defamation of the shell elements corresponding to the
uniaxial elongation of the prefom. The smaller interior angles of the
shell are the inter ﬁber angle.

The main drawbacks of this model are: -

1. this model does not allow the modeling of the sliding behavior the tows
are pinned to each other.

2. the material model needs to be simple and easily understood so that it
is easier to recover the experimental data with varying very few

material parameters.

2.4.2 Rack Edge Sliding Model

The next step was to include the sliding behavior of the tows into the
model while still maintaining the explicit modeling of the tows in the prefom. This

was accomplished by meshing the prefom in a particular way such that no two

25

tows are connected together directly i.e. they do not have a common node. The
overall connectivity of the entire prefom is maintained through the shell
elements, which indirectly connect all the tows to each other. In the complete
model, individually, each tow runs independently of all the other tows i.e. at no
point on the tow is it directly connected to any other tow. No two tows have any
common node between them. This property of this new mesh provides it with the
ability to model sliding behavior of the tows during defamation of the prefom.
These changes were incorporated into the initial rack edge model. This model is
called the Sliding Rack Edge model. The unit cell of this model is shown in Figure
14, and Figure 15 shows the assembled prefom mesh. The unit cell consists of 4
truss elements and 4 shell elements. The connectivity of the truss elements to

the shell elements is shown in Figure 16.

  

/

  

Figure 14. The unit cell of the rack edge sliding model. The plus marks in the
ﬁgure show the displaced nodes of the truss elements. The ﬁgure
shows elements at 80% of their actual size.

26

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“999° '¢°¢°¢°9°¢°9°9°¢$0099999999099909999090990‘996990960943999:
‘99°09o°o°o°o°o°o°o°o°o°o°o° 90°o°o°¢°o°¢°o°¢°¢°o°o°¢°¢°o°o°o°i
‘°o°o°o°o°o°o°o°o°¢°o°o°o°o°o°o¢o°o°o¢o°o°¢°o°o°o°o°o°¢°o°o9o°’
“99999 .9'999999999999099999990909?
, oo o co co ,
°¢°oo°oooococoa-90000090999 oooeoo o

'99908499°o°o°o°o°o°o°¢°o°¢°¢°o°¢°o° °o°o°o°¢° ° 9 °¢°o°o°o°¢°‘
’°o°o°o°o°o°o°o°o°o°o°o°o°o- -o°o°¢°¢°o°o°o°c°o°o°o°o°o°¢°o°¢°o°‘
999999999999999999¢¢¢¢99999¢¢9¢f
“9 99999 9°999°°°°°°°°°999999.94?

. n u .. u .. .o u . .. u . ..

woo ‘

u

¢°¢

Figure 15. The complete modeled prefom for the rack edge sliding model.

Figure 16. The connectivity between the shell elements and the truss
elements. The dark circles are the nodes of the shell element. The
dark lines represent the node at which the tows are connected to the
shell and the direction.

27

1)

2)

3)

The main drawbacks of this model are: -

at the edges of the modeled prefom two adjacent tows have no shell element
between them and hence there is no support to these elements which results
in extreme defomations at the edges.

The jamming behavior can be modeled using the shear modulus of the shell
elements but it is difﬁcult to identify the local shell element strains related to
the overall sliding behavior in the prefom.

The shell elements are not connected to any truss elements at one interior
node and this results in strange defamation modes in the elements leading to
element warpage. Also since the local defamation behavior is not identical in
all the elements it is difﬁcult to come up with a unique material constitutive law
for all the transition medium elements.

After trying all the above-mentioned models and ﬁnding the inadequacies of

these models the following three basic rules have been identiﬁed that need to be

followed while developing the prefom model.

1) Uniformity. The unit cell model for the prefom should be deﬁned such
that each shell element in the model is identical to the others in terms
of the number and type of tow (truss) element connections. The shell
elements may differ only by a simple rotation about the shell nomal.
This is necessary to ensure that one material law can be deﬁned for all
the shell elements.

2) Support at every node. The shell elements should be connected to a

tow at all its nodes. This is necessary in order for the shell elements to

28

have a realistic defamation mode and to avoid extreme defomations
that may result when there are unsupported nodes in the shell
elements.

3) Connectivity. The connectivity of the elements should be such that no
two tows have a common node. In other words, all the tows are
independent of one another, and they interact only through the shell

elements.

2.4.3 Tongue and Groove Model

Using the above-mentioned rules the Tongue and Groove model was
developed, which is free of all the shortcomings and errors of the models
discussed earlier. The complete conﬁguration of the unit cell is shown in Figure
17, and Figure 19 shows the model of the whole i45° plain weave prefom using
the tongue and groove model. In Figure 17 the nodes for the shell elements have
not been shown for clarity. As can be seen, the unit cell is made up of 4 two
noded line elements and 4 four noded quad elements. The unit cell in the ﬁgure
shows the elements at 90% of their actual size to give a clear view of the way the
elements are connected. The line elements represent the tows of the prefom
and the quad elements have been used to model the ﬁctitious material between

the tows, which we already have deﬁned as our transition medium.

29

 

 

Nodes

 

Shell elements

 

 

 

 

 

Tows

 

 

 

 

 

 

 

 

Figure 17. The unit cell for tongue and groove model

The connectivity at the nodes showing the sameness and the support at all
nodes is shown in Figure 18. The lines in this ﬁgure represent the tows in the
direction of the lines and they are connected at the node that they pass through.
The unifomity can be seen in the complete model Figure 19 as any shell
element in the model differs from this element by just a rotation. Also each shell

element has a tow connected to each of its nodes.

30

Figure 18. The tow-to-shell element connectivity in tongue and groove model

  
   
     
       
       
     
       
   
   
     
 
  

                    

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Figure 19. The complete modeled preform for the tongue and groove model.

The four line elements in the unit cell are not connected to each other, i.e.
they do not have any common node. The connectivity between the various tows

is famed through the shell elements, so the various tows interact only through

31

the transition medium. As a result of this when the whole of the prefom is
modeled with the unit cells, each line element is part of one continuous tow and
each tow is free from all the other tows. This property of the mesh gives us the
means to incorporate inter tow sliding into our model, as the tows are not pin
jointed. Although this model satisﬁes all previously mentioned criteria of
unifomity, support at all nodes and connectivity, there were some poor mesh
defamation issues with using this model due to lack of symmetry about the tows.
Due to this lack of symmetry, the shell elements deform in a cyclic non-unifom
mode. This is shown in the results section in the pictures of the stretched prefom
modeled using the tongue and groove model. Because of this, a new model,

called the checkerboard model was developed.

2.4.4 Checkerboard Model

The unit cell for the checkerboard model is shown in Figure 20 and the

complete model of the prefom is shown in Figure 22.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 20. The unit cell for the checkerboard model

32

In this case the model meets the support at all nodes and the connectivity
criterion completely and almost meets the unifomity criterion. Figure 21 shows
the two types of shell elements that exist in the complete model. These two
cases are not exactly the same but they are similar enough to give a unifom
defamation mode for the shells. This model satisﬁes the unifomity law as this
rule was selected so that the elements of the transition medium deform in a
similar fashion allowing the use of a simple and easy constitutive material model
for all the transition elements. Also the assembled model contains symmetry in
both the transverse and the axial directions. As a result of the symmetry in the
model, the non-unifom defamation modes that appeared in the defomed model
of the prefom with the Tongue and Groove model do not appear in the stretched

prefom modeled with the checkerboard model.

 

 

 

 

 

 

 

 

 

     

Figure 21. The two types of tow-to-shell element connectivity in the
checkerboard model.

33

 

Figure 22. The complete modeled prefom for the checkerboard model.

The straightening of the tows from their undulating initial fom has been
neglected. The tows have therefore been modeled as straight two noded truss
elements. The tows, which have been modeled as isotropic, have been assumed
to have a Young’s modulus equal to the Young’s modulus of the individual ﬁbers,
which has a very high value as compared to the stiffness of the Transition
Medium even after the jamming angle has been reached. Hence, the tows are
practically inextensible. As a result, even though the strains in the overall prefom
are very large, the strains in the tows themselves are insigniﬁcant during the
actual stamping or draping processes.

The Transition Medium in the prefom mesh is a critical part of the current

model. It is through this ﬁctitious material that the effects of all the tow

34

interactions, during the defamation, of a prefom have been modeled. The fabric
defamation can be divided into 3 stages: -

1) the pre-jamming stage;

2) the jamming stage; and

3) the post-jamming stage.

In the pre-jamming stage, defamation is primarily due to the scissoring
action. During this stage the only resistance to the defamation comes from the
friction between the tows as they rotate on the surface of other tows. The
magnitude of this frictional resistance is relatively small magnitude and
defamation within the stage can be produced without the application of high
loads. As the ﬁber angle gets smaller, it approaches the initial jamming angle at
which point the fabric becomes stiffer, requiring more force to deform the fabric.
This is the jamming stage where the stiffness of the fabric increases with the
defamation and the ﬁber angle change is small compared to that in the previous
stage. This occurs until the ﬁnal ﬁber jamming angle is reached. This marks the
beginning of the post-jamming stage. In this stage the defamation is primarily
due to inter ﬁber sliding and some scissoring in the regions where the ﬁber angle
is still less than the ﬁnal ﬁber jamming angle. This stage is appreciable only
during processes involving large defamations. which involve large ﬁber angle

changes.

2.5 Constitutive Model

This section discusses the constitutive material models used for the tmss

and the shell elements in the checkerboard model. The current model takes into

35

account the effects of friction, sliding and ﬁber angle jamming and these effects
are modeled into the checkerboard model through the material model of the shell
elements. The tows have been modeled as an isotropic material with the stiffness
properties equal to those of the ﬁbers. The material law for the transition medium
has been fomulated from the analysis of the above-mentioned stages to
represent the tow interactions of the prefom in a simple and accurate way. The
stiffness of the shell elements is much less than that of the truss elements, i.e.
the tows, and as a result the tows are practically rigid relative to the transition
medium. This is due to the fact that the shell elements represent only the effects
of friction, jamming and sliding which are very small as compared to the axial
stiffness of the tows. Nomal strain in the shell elements represent the tow
scissoring as has been shown In Figure 3 whereas sliding defamation requires
the shell elements to undergo shear defamation, which has been depicted in
Figure 5.

The material law for the transition medium is assumed to be non-linear
orthotropic. The criterion used for detemining the nomal stress strain curve is
that the stiffness of the transition medium is increased ﬁrst when the ﬁber angle
reaches the initial jamming angle and then, ﬁnally increased again when the ﬁber
angle reaches the ﬁnal jamming angle. The nomal stress-strain relationship for

the shell elements is shown in Figure 23.

36

 

Scissoring \Scissorin/g Primarily Sliding
& Sliding
Nomal
Stress

 

 

 

 

 

Su 8.:

Nomal Strain

Figure 23. The nomal stress-strain curve for transition medium

The relationship between E, G and v for isotropic materials is not
necessarily satisﬁed. The compressive behavior of the tows is not considered at
this time. The curve for the nomal stress-strain relationship is tri-linear. The ﬁrst
part of the nomal stress-strain curve represents the inter tow friction, which is
the resistance to scissoring. The second part marks the beginning of the phase in
which the adjacent parallel tows are in contact with each other, i.e. the
defamation between the initial ﬁber jamming angle and the ﬁnal jamming angle.
The ﬁrst transition point on the curve corresponds to the axial strain in the shell
element when the angle between the tows passing through this shell element is
equal to the initial ﬁber jamming angle. After reaching this point the stiffness of

the shell element increases. This also marks the increase in the force required to

37

deform the fabric through scissoring, which now requires the change in the tow
width together with the friction between the tows.

The third part of the nomal stress-strain curve marks the end of scissoring
in this local region, as the stiffness associated with further change in the width of
the tows is very high. As a result, any further defamation of the fabric is not due
to scissoring but rather due to the inter tow sliding. This is due to the fact that it
now takes more energy for the fabric to deform through scissoring of the tows,
which now requires a further change in the tow width, which is already reduced
from the initial width, than to deform through inter tow sliding. The Elastic moduli
for the ﬁrst, second, and the third part used for the shell elements are 10,20 and
190 psi respectively. The shear modulus of the transition medium is an indirect
measure of the resistance to inter tow sliding because in order to facilitate sliding
between the tows, the shell elements have to deform primarily in the shear mode.
The value of the shear stiffness is chosen such that the value is more than the
initial nomal stiffness but less than the ﬁnal nomal stiffness of the shell
elements. The shear modulus used for the shell elements is 60 psi. The shear
modulus of the transition medium is assumed to remain constant throughout the

defamation. The shear stress-strain relationship is shown in Figure 24.

38

Shear Stress

 

 

Shear Strain

Figure 24. The shear stress-strain curve for the transition medium

The uni-axial tensile test on the specimen provided the overall behavior of
the whole prefom as well as the means to calibrate the constitutive law for the
transition medium. This was done by calculating the approximate of the initial
stiffness of the shell elements from the initial linear part of the force deﬂection
curve from the experiment, as during this part the defamation is primarily due to
scissoring. This was done by calculating the stress during the initial part using
the thickness of the tow and the width of the prefom as the dimensions of the

cross-section. Further it was assumed that the middle portion of the perform,

39

ll:

which deforms primarily through scissoring (third zone, the zones are described
later) takes almost all (about 90%) of the overall strain of the fabric, within this
linear initial part. So the strain in this region is calculated from the total strain in
the prefom. These stress and strain values are then used to calculate the
approximate initial stiffness of the transition medium. The stiffness values for the
second and the third part of the nomal stress-strain curve are assumed since the
defamation now involves more complex local behavior due to the fomation of
different zones, which is explained later. Performing the uni-axial extension
simulations using the approximate and the assumed values for the stiffness of
transition medium establish the ﬁnal stress-strain law for the transition medium.
The force deﬂection curve for the whole prefom is plotted for the results from
each simulation. Thus the ﬁnal stress-strain curve for the transition medium was
established by trial and error until the original force-deﬂection curve from the

experiment is recovered.

4o

Chapter 3

3 EXPERIMENTATION AND NUMERICAL RESULTS

Various sets of experiments were perfomed to ﬁrst get an understanding
of all the different local defamation behaviors in the tows to be able to accurately
model the overall behavior, to validate the model and then to show its use in

stamping and draping processes.

3. 1 Large Scale Local Model

A large-scale model of a part of the plain weave prefom was built to study
the behavior of the tows at a ﬁner level for a better understanding of the
defamation. 0.2-crn diameter plastic trimmer line was used to build tows that
were then braided to fom a i45° plain weave prefom. Each tow was built by
grouping together about 100 ﬁbers (where ﬁber refers to a strand of the trimmer
line) to get an effect similar to a real tow in a prefom. The weave was then ﬁxed
in a shear defomable frame made out of C-section steel bars. Special holders
were made out of plexi-glass to ﬁx the tows in the frame and also to keep the
ﬁbers together. The set up of the apparatus is shown in Figure 25. The apparatus
was then sheared to effect a change in the ﬁber angle, which was initially at 90°.
It was noticed that the change in ﬁber angle after the tows come in contact with
each other involves change in the width of the tows as has been mentioned

earlier. The jamming of the ﬁber angle was also studied and the increase in the

41

force required to shear the apparatus after reaching the initial jamming angle was
also studied to help understand the force deﬂection curve from the experiments
on the real preform. With good understanding gained into the mechanics involved
during the deformation of a perform, experiments were performed on an actual

prefom.

 

Figure 25. The experimental large scale model of a part of a plain weave
prefom. The ﬁgure shows the tows in a sheared position.

3.2 Uniaxial Experiments
Uni-axial tensile tests were performed on plain weave specimens to
formulate the constitutive behavior of the ﬁctitious material from the overall

response of the prefom specimen. A UTS machine with a load cell of 20 pounds

42

was used for the tensile tests, as the material S-2 ﬁberglass 145° plain weave
prefom was very compliant. The properties of the material of the prefom and the

geometric data of the perform specimen used is given in Table1.

 

 

 

 

 

 

 

 

Material S-2 glass

Type Plain weave 145°
Tow thickness 0.016 in.

Tow width 0.21 in.

Tow spacing 0.25 in.

Fiber count per tow 2000/ 0° tow
Young’s Modulus of ﬁbers 10.0E+06 psi
Ratio 0-90 53 : 47

 

 

 

 

Table1. Properties of the perform specimen used for experiments.

The prefom size used was 8 X 16 in. and the smaller edges were
clamped in the jaws of the UTS. The upper jaw of the UTS was then moved at a
unifom velocity of 0.2 in/min. A total of 1000 data points were recorded, for the
force deﬂection curve, from 0 to 30% total prefom elongation. At this percentage
elongation during the tensile experiment the tows start coming out of the prefom
at the comers of the clamped edges. Moreover the inter tow sliding becomes

extreme at this level of elongation. Figures 26 through 29 show the deformed

43

shapes of the prefom at elongations of 10, 15, 20, and 25 percent. The top free
edge of the prefom starts to fom a concave curve and the prefom keeps
deforming into the shape of a concave lens. This occurs until the ﬁber angle in
the Zone Ill gets closer to the ﬁnal jamming angle at which point the prefom
starts ﬂattening out at the middle part of the free edges since the ﬁber angle
cannot change any further. This starts at around 20% elongation and then the ﬂat
part increases in width as the elongation increases. The force versus %

elongation curve from the experiment is shown in Figure 30.

44

 

Figure 26.

'v1

.
7.
U
D
O
i
61‘
Ht
u
.

Deformed prefom at 10% elongation.

45

 

   

Figure 27. Deformed preform at 15% elongation.

46

 

-r.-qmuq¢mL

Figure 28.

 

Deformed preform at 20% elongation.

47

 

 

Figure 29. Deformed prefom at 25% elongation.

48

 

Force(lbs.)
O-wahUlmﬂmo

 

.3
O

 

+ Experiment

 

 

 

 

o 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Overall Longitudinal Strain

 

Figure 30. Force versus % elongation of the prefom from the uniaxial
experiment.

49

 

There are distinct zones that fom within the prefom when it undergoes a
uni-axial extension with the ends constrained from displacement in the
transverse direction. There are three basic defamation zones that can be
identiﬁed on the prefom and they are shown in the Figure 31 below. The various
modes of defamation taking place in these zones are explained below.

Zone I is famed of two isosceles triangles, on each end of the prefom,
with the base as the width of the prefom being held in the clamps and the other
two equal sides making an angle equal to one half the original ﬁber angle
between the tows of the undefomed prefom. This is the rigid zone. There is no
signiﬁcant defamation taking place in this zone. The ﬁber angle stays the same
as of the undefomed prefom and there is also hardly any sliding except on the
corners of the base and at the interface with the other zones. This is due to the
boundary conditions, as the ends of the prefom constrained from moving in the
Y and Z directions.

Zone ll consists of 4 identical triangular regions, neighboring the zone I.
These regions still see some effect of the boundary effects but not as much as
the Zone I. In this zone the defamation is a combination of both scissoring and
sliding defamation between the tows. There is not a lot of scissoring defamation

in this zone, and as a result the ﬁber angle does not change as much as it does

50

in the third zone. The sliding defamation is most appreciable at the edges, even

though there is some sliding throughout this zone

Zonell
ZonelH 5

ii
ii
ri
ii

Zonell

' I

 
 

o
I I
‘ ;

   

 

r I
I

 

v'

   

 

 
 

 

 

 

 

Zone |

Figure 31. The various Zones famed in a Prefom during the Uni-Axial
extension

The Zone III is the remainder of the prefom located between Zone I and
Zone II. It foms a shape that resembles a hexagon. This zone is free from the
boundary effects due to the constrained ends but exhibits some edge effects due
to the free boundary. Most of the defamation of the fabric occurs in this zone.

The main mode of defamation in this zone is scissoring. There is pure scissoring

51

in the middle of the prefom and here the tows behave as if they are pinned to
each other. However, there is a considerable amount of sliding that occurs
between the tows at the free edge of the prefom. These three zones were
identiﬁed in the uni-axial extension experiment and this zone fomation compares

well with the results from the numerical simulation of the uni-axial tensile test.

3.3 Uniaxial Sliding Experiment

Another type of experiment that was performed was to show the inter tow
sliding in the prefom. It is shown that the inter tow sliding defamation is not
insigniﬁcant in large defamation processes like draping and stamping of
prefoms. The experimental setup is the same as for the uni-axial tensile
experiment but here the prefom was coated with a very thin layer of red paint,
prior to application of any kind of load on it. The area under the tows remained
white as the paint did not get deposited on this region, and the white unpainted
part of the tows was revealed during inter-tow sliding. The amount of unpainted
part of the tows exposed represents the amount of sliding between them. The
experiment shows that the most severe sliding occurs at the interface of the 3
Zones and also on the free edges of the prefom and some sliding occurs in the
Zone ll. Zone Ill showed almost no sliding at all except for at the free edges of
the prefom. These results are in good correlation with the results from the
simulation as will be discussed below. The force required for the same
percentage elongation of the prefom increased when using a painted specimen,

implying that the resistance to sliding had increased since the paint should not

52

increase any resistance to tow scissoring. The results of the experiments are

shown in the Figures 32 through 36.

g i.

 

Figure 32. Sliding at the junction of Zone I and Zone II and some sliding in
Zone ll

53

 

Figure 33. Sliding at the clamped edge at 25% elongation

 

Figure 34. Almost no sliding in Zone lll except at the free edges.

55

  
 

Zone II
P.
re ..

Figure 35. Sliding at the junction of the Zones and some in the Zone ll

56

 

Figure 36. Sliding at the free edge of prefom at 20% elongation.

57

Two different types of analyses, a uni-axial and a draping analysis, were
run using the commercial ﬁnite element packages MARC and LS-DYNA. The uni-
axial simulations were run on MARC and LS-DYNA both to compare the results
from explicit and the implicit analysis. The purpose of these simulations was to
ﬁrst check the validity of the FE model and then to simulate the draping analysis
of the specimen over a hemisphere using LS-DYNA. The details of the analyses

are described in the following sections.

3.4 Uniaxial Tensile Simulation

These simulations were performed for all the models discussed in the
model section in Chapter II. The main purpose of these simulations is to check
the validity of the model against the experiments performed.

The following conditions are true for all the uni-axial extension simulations.
All the nodes were constrained from moving in the out of plane direction. The
ends of the modeled prefom were constrained in the transverse direction i.e. all
the nodes at the two ends had zero displacement in the transverse direction, to
simulate the effect of the rigid clamps in the jaws of the tensile testing machine. A
ﬁxed, equal and opposite, tensile, incremental displacement was applied to the
two ends in the axial direction. These boundary conditions are shown in Figure
37 below. The ﬁber angles at the center of the prefom at different values for the

percentage elongation were recorded and compared to the values from the

58

experiments. The details of these simulations for individual models and the

results are discussed in the sections that follow.

 

 

 

 

 

A Y
Ux - 'U UX "' U
+
uz =0 for all nodes on the preform
X

 

 

f

/

Prefom

 

 

 

 

 

 

 

Figure 37. Prefom with boundary conditions for the uniaxial test. The dotted
rectangle indicates the nodes on the edges of the prefom.

3.4.1 Rack Edge Models

The pinned model itself has been discussed in Chapter II under the model
fomulation. A 8 X 4 in. mesh of a 350° plain weave composite prefom was
constructed using this model. Isotropic material model was used for the tows with
the stiffness equal to the stiffness of the ﬁbers. These properties of the ﬁbers

were taken from [8]. The material model used for the shell elements was Mooney

59

Rivlin. The material constants were chosen to get the ﬁber angle to jam at half
ﬁber angle of 18°. The initial part of the curve represents the inter tow friction and
the ﬁnal part represents post jamming behavior. The intemediate part between
the two straight lines is the increasing stiffness due to the change in the tow
thickness. Figure 38 shows the deformed shape of the prefom at 40%

elongation.

 

Inc : 25 M ‘ \
Tim. : 2 .500¢+01 WWI/U \C.

 

 

10.991

 

 

 

Figure 38. Defomed prefom mesh at 40% elongation.

The change in the center ﬁber angle with % elongation is shown in Figure

39 where it is compared with the data from [8]. The initial part of the curve shows

60

good correlation with the results in [8] but since their model does not incorporate
jamming behavior the ﬁber angle keeps decreasing to 5° with increasing
extension. This is contrary to the experiments, which have shown that the ﬁber

angle jamms between 15-20 degrees half ﬁber angles.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
I 60
50

2 40

3’ +Pamr

<

a 30 \ +Arialysis

.D

O T T
0 20 _ 40 60
% Elongation
Figure 39. Comparison of center ﬁber angle from analysis and study by
Mauget etal.

The data from the simulation in this study shows the ﬁber angle jamming
which starts at around 30% overall elongation. These results were just qualitative
to show that the prefom behavior can be modeled accurately using this
technique and there was no experimental data to support the results other than
the data from [8].

The serrated model with sliding incorporated in it was modeled on the

same lines. The results, however, were not accurate due to the drawbacks in this

61

model, which have already been discussed in Chapter 2. The main reason for not
using this model was the asymmetric shell elements, which makes it impossible
to represent all the shell elements with a unique constitutive law. Also, the shell

elements did notohave support at all nodes.

3.4.2 Tongue and Groove Model

An 8 X 16 in. prefom, the same as used in the experiments, was modeled
using 1223 truss elements and 1153 shell elements. The meshed prefom has
been shown in Figure 19. The boundary conditions used are the same as in the
previous model. As discussed in the previous chapter this model was not used
because of the asymmetric connectivity of the shell elements to the tows. This is

shown in Figure 40, which shows the deformed prefom.

 

.‘lifg
a
l
A.
1)

 

 

 

1"

XXI.1)K

 

 

 

 

 

 

 

 

 

 

Figure 40. The defomed mesh of preform with tongue and groove model.

62

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As can be seen the shell elements see varying nomal strain so along the

Y direction. The ﬁgure 42 shows the varying strain at a larger scale isolated from
the complete model. As can be seen in the ﬁrst column the elements experience
increasing compressive strain along the negative Y direction, whereas in the next
column of elements it is the other way around. This goes on as we move towards
the middle of the prefom where the pattern reverses and goes on towards the
other end of the prefom. This strange defamation mode, which as has already

been explained, is due to the connectivity of the shell elements with the tows,

63

results in an inaccurate prediction of the behavior of the tows during the

extension of the prefom.

  

Column 1

     
  

Column 2

t...

  

Figure 42. A part of the prefom showing the varying strain in the shell
elements more clearly (tongue and groove model).

Figure 41 shows the prefom with the tows without the shell elements to
give a better picture of the tow defamation behavior predicted by this model. No
results or comparisons are shown in this model, as the defamation behavior that
the model predicted is erroneous. Another drawback in this model is that in order
to have a unifom mesh the truss elements at the edges need to be trimmed.
This is due to the model characteristics as the length of the truss elements is

twice the length of the diagonals of the shell elements. As a result of this the

model of the prefom could not represent sliding at the edges since the tows had
to be pinned at the point of intersection at the edges. Moreover, the defamation
shown in Figure 41 is the maximum defamation that the prefom underwent
before the analysis died because the element size became very small resulting in
an extremely small time step. As a result, further use of this model was
discontinued and the new and ﬁnal model, the Checker Board model, was

famed.

3.4.3 The Checker Board Model:

This is the ﬁnal model that was built and validated against the
experimental data. The same type of prefom was modeled using this approach
as was modeled with the tongue and groove model. The dimensions of the
prefom are 8 X 16 in. The actual preform specimen used for the experiments is
S-2 ﬁberglass i45°. The plain weave fabric was modeled using 4608 truss
elements and 4608 shell elements. The meshed prefom has been shown in
Figure 22. The boundary conditions again are the same as described eariier. A
maximum overall elongation of 25% was achieved with this model before the
simulation died out. This is due to the reason that the sliding between the tows at
the free edges becomes large, which is in agreement with the experiments. But
as a result of this the shell elements at the edges become warped to
accommodate this defamation as they are connected to the tow elements.
However, this value of overall elongation is more than sufﬁcient for most of the
foming or shaping processes in the industrial applications as reinforcement for a

composite part. So this inability of the current model to go beyond 25% overall

65

elongation is not a limitation of its capabilities as it can be seen that at this
elongation there is a signiﬁcant amount of sliding in the fabric.

Similar patterns of defamation modes were observed in the simulation
and experiments. The model predicts the sliding behavior at the same places as
shown by the experiments. The predictions are shown in Figure 43 through 46,
which are in very good correlation with the results from the experiments showing
the inter tow sliding (Figures 32 through 36). Figures 43 through 46 show the
mesh of the shell elements with and without the tows since the ﬁber sliding in the
model is predicted at the places where the shell elementsundergo considerable
shear defamation. The sliding defamation, as predicted by the simulation, is
signiﬁcant along the interface of Zone I with Zone II, the interfaces of Zone II and
Zone Ill and also on the free edges of the prefom. The model also predicts some
sliding in Zone II, which agrees well with the experiments. Also the interior shell
elements in the Zone lll undergo only nomal defamation indicating pure

scissoring behavior.

66

 

Figure 43. The whole prefom mesh with just shells. Shell element distortion
indicates sliding.

67

L.

Figure 44.

 

Sliding at the interface between Zones l and Il mainly.

68

 

 

tthe interface between Zones II and III.

I Inga

The details of sl

Figure 45.

69

 

Figure 46. Showing primarily Zone "I. All the shell elements deform through
nomal strain only except the interfaces and the free edges indicating
no sliding in the middle and signiﬁcant sliding at the interface and at

the edges.

70

The overall shape of the fabric at different percentage elongations
compared well with the shape of the prefom from experiments. Figure 47, 48
and 49 shows the prefom at 10, 20, and 25 percent elongation respectively as
predicted by the simulation. Figure 50 shows the comparison between the center
ﬁber» angles at different percentage elongations from experiments and the
predicted angles from the simulation. The ﬁgures show that there is good
correlation between the experiments and the simulation.

The force deﬂection curve for the simulation has also been compared with
the same curve from the experiments. This is shown in Figure 51, which shows
the force deﬂection curves for both the experiment and the analysis on the same
graph. As can be seen the curves are in very good agreement with each other,

thus validating the model.

71

     
    

  
   
  
 

  
    
 

 
 
 
 
 

   
     

       
   
  

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Figure 49. Prefom at 25% elongation as predicted by the simulation.

74

 

1 00
90
80
70
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+ Experiment
+Analysis

 

 

 

Center Fiber Angle

 

0 5 10 15 20 25 30

 

% Elongation

 

 

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Figure 50. Center ﬁber angle comparison between experiment and simulation.

75

 

 

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I 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Overall Longitudlnal Strain

 

 

Figure 51. Force versus % elongation plot for experiment and simulation.

76

3.5 Draping Experiments

The next set of tests was perfomed using a hydraulic press machine for
foming the prefom onto a sphere. A spherical punch, with a radius of 2 inches,
was used to fom the prefom. The setup of the machine is such that the working
table has a circular hole about 6 inches in diameter. The punch comes from
under the table and goes up through the hole. A square plate 10 in. X 10 in. in
dimensions, with a cavity of 6 in. X 6 in. cut out of it in the center, was cut out of a
hard board to be placed over the prefom for support at the edges. The
dimensions of the prefom sample being famed onto the sphere were 8 in. X 8
in. The prefom was placed on the working bench of the machine over the
circular hole and the square plate was placed on the prefom. Very light weights
were placed on the ends of the plate, where there is no prefom under it to keep
it from lifting under the force from the punch motion into the prefom. The
boundary conditions were such that the edges of the prefom were free to move
in all directions except in the direction of movement of punch i.e. vertical
direction. The punch was then moved into the prefom at incremental
displacements of 0.25 in. and pictures of the defomed fabric were taken at
different angles. Qualitative results obtained from these pictures were taken and
compared with the results obtained from the simulation performed. Figures 52,53
and 54 show the prefom contour at punch penetration corresponding to 1.0 in.,

1.5in., and 2.25 in. respectively.

77

 

Figure 52. Prefom contour at 1 in. penetration.

78

 

Figure 53. Prefom contour at 1.5 in. penetration.

79

 

Figure 54. Preform contour at 2.25 in. penetration.

80

3. 6 Draping Simulation
The draping problem was simulated using LSDYNA-3D using the

checkerboard model for meshing the prefom. The specimen dimensions were 8
in. X 8 in. and it was modeled using 2304 truss elements and 2304 shell
elements. The setup for the analysis is the same as for the experiment. The
prefom is held between two square plates with a 6 in. X 6 in. square hole cut out
of the plated to allow the spherical punch to move up into the prefom. The
prefom is free to slide between the plates as the punch moves into it. The radius
of the punch is 2 in. The punch is moved into the prefom at a unifom speed of
0.3 in./min. The experiment was performed for a total punch penetration of 2.5 in.
into the prefom. Figures 55,56 and 57 show the prefom shapes with the punch
penetrations corresponding to 0.25, 1.0, and 2.25 in. as predicted by the
simulation. The contour of the prefom as predicted by the draping simulation
bears close resemblance to the contour shape from the experiments for the
corresponding punch penetrations. The ﬁgures from the experiments do not
show the part of the prefom under the square plate placed over the prefom. As
a result just 6in. X 6in. of the prefom, which comes out of the square hole in the
square plate is shown. However, for the pictures from the simulation the entire
preform is shown i.e. the part under the square plate is also shown. The overall
shape of the prefom is in good agreement with the experiments.

The fabric defamation is symmetric about the centerline, i.e. the axis of
the sphere along its direction of motion. The prefom foms a pyramid shape after

the punch motion into it with the top rounded, ﬁtting on the punch, and the edges

81

are rounded. The edges correspond to the prefom coming out at the corners of
the square hollow in the plate. The simulation also predicts this shape. There is
no wrinkling in the prefom during the shaping over the spherical punch in both
the experiments and the simulation. Almost unifom ﬁber scissoring occurs along
the slanted faces of the so famed pyramid shape of the prefom. The ﬁber angle
at the top of the punch does not change and the tows remain essentially
perpendicular to each other, both in the experiment and the simulation. The
results look very similar in both the experiments and the simulation. This study
performed only a qualitative comparison between the analysis and the
experiments for the shaping simulation as a good criterion for the quantitative
comparison could not be established. It is very difﬁcult to measure the ﬁber
angles on the deformed prefom shape in draping process as the faces are
slanted. So the comparison of the ﬁber angles could not be made. However, as
said earlier the qualitative results show a good correlation with the experimental

results.

82

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83

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85

Chapter 4

4 CONCLUSIONS

A ﬁnite element model has been presented for modeling deformation of a
plain weave textile composite preform during the stamping process. The preform
is modeled explicitly using this approach. This model takes into account the inter
ﬁber friction, tow jamming and ﬁber sliding. It was shown that inter tow sliding is
not insigniﬁcant during large defamation processes involving large shear angle
changes. The current model has been shown to predict accurately the overall

deformation behavior of the preform.

86

APPENDICES

87

Appendix A

5 USER MATERIAL CODE

subroutine umat41 (cm,eps,sig,hisv,dt1,capa,etype,time)

implicit real*8 (a-h,o-z)

isotropic elastic material (sample user subroutine)
variables

cm(1)= initial young's modulus
cm(2)=poisson's ratio

eps(1)=loca| x strain
eps(2)=local y strain
eps(3)=local 2 strain
eps(4)=local xy strain
eps(5)=local yz strain
eps(6)=local zx strain

sig(1)=local x stress
sig(2)=local y stress
sig(3)=local 2 stress
sig(4)=local xy stress
sig(5)=local yz stress
sig(6)=local zx stress

hisv(1 )=1 st history variable
hisv(2)=2nd history variable

hisv(n)=nth history variable

dt1=current time step size
capa=reduction factor for transverse shear
etype:

eq."brick" for solid elements

eq.”shell" for all shell elements

00000000000000000000000000000000000

88

000000000000000000

0

eq."beam" for all beam elements

time=current problem time.

all transformations into the element local system are
performed prior to entering this subroutine. transformations
back to the global system are performed after exiting this
routine.

all history variables are initialized to zero in the input
phase. initialization of history variables to nonzero values
may be done during the ﬁrst call to this subroutine for each
element.

energy calculations for the dyna3d energy balance are done
outside this subroutine.

character*(*) etype
dimension cm(*).eps(*).sig(*).hisv(*)

compute shear modulus, g

92 =cm(1)/(1.+cm(2))
g = 60.0

if (etype.eq.'brick') then
davg=(-eps(1 )-eps(2)-eps(3))/3.
p=-davg*cm(1 )/((1 .-2.*cm(2)))
sig(1 )=sig(1 )+p+92*(eps(1 )+davg)
sigl2)=si9(2)+p+92*(eps(2)+davg)
si9(3)=sig(3)+p+92*(eps(3)+davg)
si9(4)=si9(4)+g*eps(4)
si9(5)=sig(5)+9*eps(5)
si9(6)=si9(6)+9*ep6(6)

elseif (etype.eq.'shell') then

hisv(1) = hisv(1) + eps(1)

hisv(2) = hisv(2) + eps(2)

if (hisv(2).gt.-O.45) then
gc =capa*g
q1 =cm(1)*cm(2)/((1 .0+cm(2))*(1 .0-2.0*cm(2)))
q3 =1./(q1+92)
eps(3)=-q1*(eps(1)+eps(2))*q3
davg =(-eps(1)-eps(2)-eps(3))/3.

89

p =-davg*cm(1)/((1 .-2.*cm(2)))
sig(1)=sig(1)+p+92*(eps(1)+davg)
sig(2)=sigl2)+p+92*(eps(2)+davg)
sig(3)=0.0
sig(4)=sig(4)+gc*eps(4)
sig(5)=si9(5)+gc*eps(5)
sig(6)=sig(6)+gc*eps(6)
else
if (hisv(2).lt.-0.80) then
. e = 190.0
enu=0.3
gZ=e/(1 .0+enu)
gc =capa*g
q1 =e*enul((1 .0+enu)*(1.0-2.0*enu))
q3 =1./(q1+gZ)
eps(3)=-q1*(eps(1 )+ePS(2))*q3
davg =(-eps(1)-eps(2)-eps(3))/3.
p =-davg*e/((1.-2.*enu))
sig(1 )=si9(1 )+p+92*(epS(1)+daV9)
sig(2)=Si9(2)+9+92*(ePS(2)+daV9)
sig(3)=0.0
sig(4)=sig(4)+gc*eps(4)
sig(5)=sig(5)+gc*eps(5)
sig(6)=sig(6)+gc*eps(6)
else
e = 20.0
enu=0.1
92=e/(1 .0+enu)
gc =capa*g
q1 =e*enul((1 .0+enu)*(1.0-2.0*enu))
q3 =1./(q1+92)
eps(3)=-q1*(eps(1)+eps(2))*q3
davg =(-eps(1)—eps(2)-eps(3))/3.
p =-davg*e/((1.-2.*enu))
sig(1 )=si9(1 )+p+92*(eps(1 )+davg)
si9(2)=si9(2)+p+92*(eps(2)+davg)
sig(3)=0.0
sig(4)=si9(4)+gc*eps(4)
sig(5)=sig(5)+gc*eps(5)
sig(6)=sig(6)+gc*eps(6)
end if
end if

C

c
elseif (etype.eq.'beam' ) then

90

q1 =cm(1)*cm(2)/((1.0+cm(2))*(1 .0-2.0*cm(2)))
g=10000.0

q3 =q1+2.0*g

gc =capa*g

deti =1./(q3*q3-q1*q1)

c22i = q3*deti

023i =-q1*deti

fac =(c22i+c23i)*q1

eps(2)=-eps(1 )*fac-sig(2)*c22i-sig(3)*c23i

eps(3)=-eps(1 )*fac-sig(2)*c23i-sig(3)*022i

davg =(-eps(1)-eps(2)~eps(3))/3.

p =-davg*cm(1)/((1 .-2.*cm(2)))

sig(1)=sig(1 )+p+92*(eps(1)+davg)

sig(2)=0.0

sig(3)=0.0

sig(4)=si9(4)+90*698(4)

sig(5)=0.0

sig(6)=sig(6)+gc*eps(6)

endﬁ

return
end

91

BIBLIOGRAPHY

92

10.

11.

6 BIBLIOGRAPHY

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95

 

 

 

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