ADDRESSING VARIABILITY OF FIBER PREFORM PERMEABILITY

IN PROCESS DESIGN FOR LIQUID COMPOSITE MOLDING

by

Hatice Sinem Sas

A dissertation submitted to the Faculty of the University of Delaware in partial

fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical

Engineering

Summer 2015

© 2015 Hatice Sinem Sas

All Rights Reserved

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ADDRESSING VARIABILITY OF FIBER PREFORM PERMEABILITY

IN PROCESS DESIGN FOR LIQUID COMPOSITE MOLDING

by

Hatice Sinem Sas

Approved: __________________________________________________________

Suresh G. Advani, Ph.D.

Chair of the Department of Mechanical Engineering

Approved: __________________________________________________________

Babatunde A. Ogunnaike, Ph.D.

Dean of the College of Engineering

Approved: __________________________________________________________

James G. Richards, Ph.D.

Vice Provost for Graduate and Professional Education

I certify that I have read this dissertation and that in my opinion it meets

the academic and professional standard required by the University as a

dissertation for the degree of Doctor of Philosophy.

Signed: __________________________________________________________

Suresh G. Advani, Ph.D.

Professor in charge of dissertation

I certify that I have read this dissertation and that in my opinion it meets

the academic and professional standard required by the University as a

dissertation for the degree of Doctor of Philosophy.

Signed: __________________________________________________________

James L. Glancey, Ph.D, P.E.

Member of dissertation committee

I certify that I have read this dissertation and that in my opinion it meets

the academic and professional standard required by the University as a

dissertation for the degree of Doctor of Philosophy.

Signed: __________________________________________________________

Rakesh, Ph.D.

Member of dissertation committee

I certify that I have read this dissertation and that in my opinion it meets

the academic and professional standard required by the University as a

dissertation for the degree of Doctor of Philosophy.

Signed: __________________________________________________________

Pavel Simacek, Ph.D.

Member of dissertation committee

iv

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my advisor, Prof. Suresh G.

Advani, for the continuous support, patience, and enthusiasm he has provided during

my Ph.D. journey. I feel extremely lucky to have been mentored by someone with

deep knowledge, solid experience and infinite motivation to learn, teach, investigate

and invent.

I would like to thank Dr. Pavel Simacek for his support and advice on process

modeling and numerical simulation, and agreeing to serve on my dissertation

committee.

I acknowledge and give thanks to my other two committee members, Prof.

James Glancey and Prof. Rakesh for generously devoting time to judge my work and

providing insightful comments.

I would also like to thank many colleagues who worked with me during my

time in Delaware. I had the privilege to work with Jeffrey Lugo and Eric Wurtzel, and

with Louis Agostino and Minyoung Yun. I was lucky to have Richard Readdy for his

help in countless lab and LabVIEW problems. I want to express my gratitude to Dr.

Volkan Eskizeybek for his valuable advice. Also, I am grateful for the support and

friendship of the rest of my research group: Dr. John Gangloff, Thomas Cender, Jiayin

Wang, Dr. Hang Yu, Hong Yu and Michael Yeager.

I would also like to thank all of my office mates in Spencer007, CCM118 and

CCM123 (The Pit). Thank you all for the good times and friendship that we share.

v

I would also like to thank the administrative staff of the Mechanical

Engineering Department: Lisa Katzmire, Ann Connor and Letitia Toto and Center for

Composite Materials: Corinne Hamed, Robin Mack, Penny O’Donnell, Therese

Stratton and Megan Hancock. I am thankful for your hard work and smiles.

I would also like to thank all of my dear friends for making the bad times good,

and the good times even better. I want to give special thanks to Sumeyra Yildirim,

Deniz Ozdiktas, and Sezin Zengin for their friendship and support. Also, I am lucky to

have Ozan Erol and Sinan Boztepe both as friends and colleagues in CCM. I am

feeling lucky to have the chance to meet my dearest friend Sevil Buzcu. She made my

Ph.D journey colorful and cheerful. We shared our moments for five years and we will

continue to do so. I am also grateful to my friend Furkan Cayci for the perspectives he

brought into my life. I am also thankful to him for his contribution to my research

using his software skills that made this dissertation complete. I also want to thank Filiz

Yesilkoy for not only being a best friend, but also being an inspiration and motivation

for my studies and my life. She says, “Life is all about asking the right question.” and

I know we will keep looking for questions together.

Lastly, I would like to thank my family. Mom, Dad and my sister Senem,

thanks for your unconditional love and endless support. I am deeply thankful to all of

my family members for their support.

I want to dedicate this dissertation to my grandmother who foresees my future

in academia. We built this dream with her. I know she is watching me from heaven

and will continue to send her blessings.

Rumi says, “Be grateful for whoever comes, because each has been sent as a

guide from beyond.” and I am grateful to everyone who touched my life.

vi

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................ ix

LIST OF FIGURES ........................................................................................................ x ABSTRACT ................................................................................................................. xv

Chapter

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

1.1 Liquid Composite Molding ....................................................................... 1

1.1.1 Materials used in LCM .................................................................. 2

1.1.1.1 Reinforcements ............................................................... 2 1.1.1.2 Matrices .......................................................................... 4

1.1.2 The LCM Family of Processes ...................................................... 5

1.1.2.1 The Resin Transfer Molding .......................................... 5 1.1.2.2 Vacuum Assisted Resin Transfer Molding ..................... 8

1.1.2.3 Seemann’s Composite Resin Infusion Molding

Process ............................................................................ 8

1.2 Manufacturing Challenges in Vacuum Resin Transfer Molding............. 10

1.2.1 Permeability variation ................................................................. 10

1.2.2 Race-Tracking ............................................................................. 13

1.3 Modeling of LCM Processes ................................................................... 15 1.4 Objective and Dissertation Outline ......................................................... 19

2 PERMEABILITY MEASUREMENT TECHINIQUES .................................. 21

2.1 Historical Background ............................................................................. 21

2.2 Analytical and Predictive Methods ......................................................... 23 2.3 Numerical Methods ................................................................................. 25 2.4 Experimental Measurement Techniques ................................................. 26

2.4.1 Rectilinear Flow .......................................................................... 27

vii

2.4.2 Radial Flow ................................................................................. 29 2.4.3 Transverse and Three-Dimensional Flow ................................... 32

2.5 Skew terms .............................................................................................. 34

2.5.1 Introduction ................................................................................. 35 2.5.2 Methodology ................................................................................ 37 2.5.3 Results and Discussion ................................................................ 41 2.5.4 Summary ...................................................................................... 47

3 THROUGH THICKNESS PERMEABILITY ................................................. 48

3.1 Introduction ............................................................................................. 48

3.1.1 Effective Permeability of Preform Stacks ................................... 52

3.1.2 Unidirectional fabrics and their orientation ................................. 54

3.2 Through-thickness permeability characterization ................................... 54

3.2.1 Numerical Analysis ..................................................................... 54 3.2.2 Experimental Validation .............................................................. 57

3.3 Results and Discussion ............................................................................ 59

3.3.1 Experimental Study ..................................................................... 59

3.3.2 Parametric Study ......................................................................... 61

3.4 Summary .................................................................................................. 65

4 CHARACTERIZATION OF LOCAL VARIABILITY OF FABRICS ........... 67

4.1 Introduction ............................................................................................. 67 4.2 Mathematical Implementation ................................................................. 70

4.3 Experimentation ...................................................................................... 77 4.4 Results and Discussion ............................................................................ 80

4.4.1 Characterization of permeability variation .................................. 80 4.4.2 Characterization of the defects within a fabric ............................ 83

4.5 Summary .................................................................................................. 87

5 OPTIMIZED DISTRIBUTION MEDIA LAYOUT ........................................ 88

5.1 Introduction ............................................................................................. 88

5.2 Flow Control Mechanisms for Flow Through Fibrous Domain .............. 88

viii

5.3 Methodology and Implementation .......................................................... 90

5.3.1 Discrete Optimization .................................................................. 91

5.3.1.1 Tree Search Algorithms ................................................ 91

5.3.2 Pedagogical Example .................................................................. 93 5.3.3 Algorithm for Optimum DM lay-out ........................................... 97 5.3.4 Partition method .......................................................................... 98

5.4 Experimentation ...................................................................................... 99

5.5 Results and Discussion .......................................................................... 101

5.5.1 Experimental Validation ............................................................ 101 5.5.2 Complex Geometries ................................................................. 106

5.6 Summary ................................................................................................ 111

6 CONCLUSIONS, CONTRIBUTIONS AND FUTURE WORK .................. 112

6.1 Conclusions ........................................................................................... 112 6.2 Contributions of this work ..................................................................... 114

6.3 Future Work ........................................................................................... 116

REFERENCES ........................................................................................................... 118

Appendix

A MATLAB SCRIPTS FOR DISTRIBUTION MEDIA OPTIMIZATION ..... 132

A.1 Scissors.m: Main m-file ......................................................................... 133

A.2 Rock.m: Evaluation of all race-tracking possibilities ............................ 136

A.3 Paper.m: Finding the optimum region to place DM .............................. 138

B REPRINT PERMISSION LETTERS ............................................................. 142

B.1 “EFFECT OF RELATIVE PLY ORIENTATION ON THE

THROUGH-THICKNESS PERMEABILITY OF

UNIDIRECTIONAL FABRICS” .......................................................... 143

B.2 “FRACTAL CONCEPTS IN SURFACE GROWTH” ......................... 150

ix

LIST OF TABLES

Table 2.1. Parameters for virtual experiment ........................................................... 45

Table 2.2. Predicted permeability for the experiment .............................................. 46

Table 3.1. Experimental and numerical comparison of through-thickness

permeability. Case 1 and case 2 of 0o and 5o refers to all six

unidirectional layers being aligned along those angles respectively. In

case 3, case 4 and case 5, the successive layers were rotated by 5o,

45o and 90o degrees respectively. ........................................................... 60

Table 4.1. Characterization of the roughness exponent ........................................... 82

Table 5.1. Properties of E-glass fabric, DM and corn syrup .................................. 101

x

LIST OF FIGURES

Figure 1.1. Type of reinforcements ............................................................................. 2

Figure 1.2. Different fabric types: (a) E-glass-plain weave, (b) E-glass random

mat, (c) Aramid twill weave, (d) Carbon twill weave ............................... 3

Figure 1.3. 3D reinforcement architectures (generated via TEXGEN [3]) ................. 4

Figure 1.4. Schematic of RTM (left) and VARTM (right) steps (adapted from [1]) .. 7

Figure 1.5. Schematic of SCRIMP steps ..................................................................... 9

Figure 1.6. Examples of (a) macro-void and (b) micro-void [27] ............................. 10

Figure 1.7. Example of defects in the preform; (a) plain weave glass fabric, (b)

3D orthogonal glass fabric ...................................................................... 11

Figure 1.8. Thickness variation during vacuum infusion .......................................... 13

Figure 1.9. Race-tracking formation on the edges due to fray edges ........................ 14

Figure 1.10. Race-tracking examples: (a) Mid-layer of the preform with metal

insert spatially in the middle, and Flow front profiles at the bottom of

the preform at two different time steps with race-tracking along the

metal insert for two same experimental configurations: (b) experiment

1, (c) experiment 2 ................................................................................... 15

Figure 1.11. Liquid Injection Molding Simulation (LIMS) Structure ......................... 18

Figure 1.12. Permeability map approach ..................................................................... 18

Figure 2.1. Flow front profile with xyz mold coordinate, x’y’z’ principle direction

of the preform .......................................................................................... 22

Figure 2.2. One-dimensional permeability characterization experiment to find the

bulk permeability value in the direction of flow ..................................... 28

xi

Figure 2.3. Schematic of radial flow front profiles: (a) isotropic (R1=R2), (b)

anisotropic (R1≠R2), (c) anisotropic with non-zero in-plane skew term

(global coordinate frame doesn’t coincides with principle directions of

the preform) ............................................................................................. 31

Figure 2.4. 3D 25890 g/m3 E-glass fabric ................................................................. 36

Figure 2.5. Experimental set-up to monitor the resin flow at the top and bottom

surfaces of the preform (left: schematic, right: picture of the set-up) ..... 38

Figure 2.6. (Left) An image of isotropic flow from an experiment. (Middle) The

image after having the preceding flow image subtracted from it,

filtered, and converted to binary. (Right) An ellipse is fitted to the

edge of the resin flow front. .................................................................... 39

Figure 2.7. Algorithm for permeability prediction from experimental fill time of

top and bottom surfaces ........................................................................... 41

Figure 2.8. Flow front profiles at the top (solid lines) and bottom (dash-dot lines)

for different skew permeability at time equal to 700 seconds. The

jagged flow fronts are numerical artifacts because of fairly coarse

mesh. ........................................................................................................ 44

Figure 2.9. Flow front profiles comparisons with assigned and predicted

permeability values at the top and bottom surfaces ................................ 45

Figure 2.10. Flow front profiles at time 13.26 seconds at the top and bottom:

experimental, with predicted permeability and comparison ................... 46

Figure 3.1. Solid model of a unit cell and the corresponding cross-section of four

unidirectional plies stacked on top of each other (a) All plies aligned

along the y- axis (b) All plies are rotated by 10 degrees in the x-y

plane with respect to the y- axis (c) Each successive ply is rotated by

10 degrees resulting in a stacking sequence of 0/10/20/30 with respect

to the y- axis with the corresponding cross sections in the through-

thickness direction, respectively. ............................................................. 51

Figure 3.2. Front and back side of the unidirectional fabric ...................................... 52

Figure 3.3. Representation of the orientation of the plies .......................................... 54

Figure 3.4. (a) The Gambit model with each successive layer rotated by five

degrees. b) Gambit mesh of the model with 1,968,652 elements and

484,911 nodes. The cut-out shows the mesh density .............................. 56

xii

Figure 3.5. (a) Periodic boundary conditions to evaluate the permeability in z-

axis, (b) Evaluation of permeability in z-axis ......................................... 57

Figure 3.6. Experimental set-up: (a) Upper mold plate, (b) Lower mold plate, (c)

Mold assembly, (d) Resin flow through preform .................................... 58

Figure 3.7. Numerical through thickness permeability with different mesh

element sizes for incremental rotation angle 5o ....................................... 61

Figure 3.8. Effect on through-thickness permeability with increasing rotation

angle of the successive ply. The unit cell was created using the square

and hexagonal arrangement of the fiber tows in the unidirectional ply. . 63

Figure 3.9. Effect on through-thickness permeability with increasing rotation

angle of the successive ply. The unit cell was created using the square

and hexagonal arrangement of the fiber tows in the unidirectional ply. . 65

Figure 4.1. Radial injection and permeability tensor characterization: (a)

Schematic of flow front in an anisotropic fabric at a time step with the

principle direction 𝐱’𝐲′-axes, (b) Radial injection inlet gate and resin

propagation, (c) Permeability tensor. 𝐊𝐱𝐲 is non-zero as the principal

axis do not align with the selected coordinate axis ................................. 69

Figure 4.2. Flow front locations (height 𝐡(𝐫, 𝐭)) at various times with system size

L, and mean height (flow front position) 𝒉 ............................................. 70

Figure 4.3. (a) Change of the interface width with time (logarithmic scales for

both axes) for a fixed L value , (b) Growth of the interface width with

different system sizes (L). Reprinted with permission from [143] ......... 71

Figure 4.4. Top: LIMS mesh and random permeability assignment, Bottom: flow

front progression with time obtained via LIMS ...................................... 75

Figure 4.5. Assignment of the variation of the permeability of the defected zones:

left: 25% defective sample, right: variation of permeability within the

defective zone obtained from solution of Equation (4.5). Permeability

is higher in the center of the zone and reduces to the values prescribed

at the edges as described by the parameter Q in Equation (4.5). ............ 76

Figure 4.6. Flow through porous media experimental set-up with flow

visualization ............................................................................................. 78

xiii

Figure 4.7. Resin flowing into a fibrous preform with 25 cent coins placed inside

the fabric to simulate defective regions. On the left the defects were

evenly distributed on the right the defects are randomly distributed.

Measured experimental flow front profiles are also shown (flow front

contours at Δt = 25 seconds) ................................................................... 79

Figure 4.8. Characterization of the growth exponent: (a) Shape of flow front at a

time instant, (b) Bell curves with three different standard deviations

selected for the permeability values assigned in LIMS, (c) Change of

the variance of the interface with time from the simulated experiment

with permeability distributions shown in (b) .......................................... 81

Figure 4.9. Change in growth exponent, 𝛃 with increasing percentage of defective

zones (m) for different degree of defects, Q. A best fit functional

relationship is also plotted ....................................................................... 84

Figure 4.10. Change in roughness exponent, 𝛂 with increasing percentage of

defected zones, m, for different degree of defects, Q. A best fit

functional relationship is also plotted. ..................................................... 85

Figure 4.11. Defect tests via VARTM with 37.5% defect and flow front profiles

(Δt = 25 seconds), left: quarters right: tacky tape to represent the

defective zone. ......................................................................................... 87

Figure 5.1. Fill time contours for a VARTM and SCRIMP ...................................... 90

Figure 5.2. Tree search algorithms (a) example problem with two acceptable, H

and T, nodes, (b) Breadth-first search: finds node H, (c) Depth-first

search: finds node T ................................................................................ 92

Figure 5.3. Example to explain the methodology to determine the optimal DM

design using the DFS discretization method ........................................... 96

Figure 5.4. Flow chart of the algorithm to obtain optimal DM ................................. 98

Figure 5.5. Division of the domain with the built in k-means script in Matlab ......... 99

Figure 5.6. (a) 4th layer of the E-glass with metal insert placed in the center of the

fabric, (b) Experiment layup under vacuum .......................................... 100

xiv

Figure 5.7. DM layout design (a) geometry with inlet/vent locations with 4

race-tracking possibilities along the insert edges creating 24=16

different scenarios (b) 8 regions for placement of distribution media

when using discrete optimization, and (c) optimum DM design which

resulted in successful filling for all 16 scenarios. ................................. 102

Figure 5.8. Numerical Solution of flow front profiles of the top and bottom views

for 4 different race-tracking scenarios with time steps 10 seconds

apart, (a) with 95% of the top layer covered with DM, (b) with

optimized DM design ............................................................................ 104

Figure 5.9. Experimental flow fronts with the optimized DM design with flow

front locations in red 20 seconds apart. The background image of the

experiment at 60 seconds, (a) Top and (b) Bottom ............................... 106

Figure 5.10. Optimized DM design of trailer geometry with 1024 different possible

flow patterns .......................................................................................... 107

Figure 5.11. Void regions with full DM on top surface on the left hand side with

optimized DM design on the right hand side for three representative

scenarios from 1024 possible scenarios ................................................ 108

Figure 5.12. Time contours with full DM on top surface on the left hand side with

optimized DM design on the right hand side for three representative

scenarios from 1024 possible scenarios ................................................ 109

Figure 5.13. Pressure distribution at the instant resin reaches the vent with full DM

on the left and with optimized DM design on the right for the three

representative scenarios ......................................................................... 110

Figure 5.14. Change in CPU time with mesh size for optimized DM design ........... 111

xv

ABSTRACT

In Liquid Composite Molding (LCM) processes, reinforcing glass, carbon or

Kevlar fiber preforms are placed in a mold cavity and a liquid resin is introduced to

cover the remaining empty space to form a composite by curing the resin. The fiber

preform permeability plays a key role in the filling pattern of the mold, which dictates

if there will be any voids (empty spaces) in the composite. Permeability tensor

describes the resistance to fluid flow through the anisotropic fibrous porous media,

which may not be spatially uniform. The variability in the permeability due to the

variation in the preform or its placement in the mold can influence the filling pattern

and hence the quality of the part being manufactured. The permeability map of a

preform specifies the values of components of the permeability tensor at various

locations of the preform. The overall objective of this dissertation is to investigate

various approaches and tools to create a permeability map that will ensure filling to

achieve manufacturing success despite the variability of the filling pattern, a

requirement of robust process design.

When unidirectional fabrics are used to manufacture composites, they are

typically stacked on top of each other to build up the desired thickness. A slight

misalignment during the stacking can change the through-thickness permeability

dramatically and the flow pattern due to the creation of low-resistance pathways.

Experimental characterization of the out-of-plane or through-thickness permeability of

a series of unidirectional fabrics stacked in various orientations is investigated. Also,

numerical simulations are conducted to predict the effect of change in fiber orientation

xvi

on the through-thickness permeability for unidirectional fabrics. Results demonstrate

that the stacking sequence of the unidirectional fabrics influence the through thickness

flow and hence the transverse permeability.

Next, variation in the permeability value of the fibrous domain caused by the

non-uniformity in fiber architecture is investigated. The time evolution and geometry

of the rough interfaces of the fluid flow in porous medium are analyzed using the

concepts of dynamic scaling and self-affine fractal geometry and is shown to belong to

the Kardar-Parisi-Zhang (KPZ) universality class. Additionally, this characterization

can be used to quantify the percentage of abnormalities within the preform from flow

front profile analysis using KPZ formulation.

Finally, a methodology is introduced to create a permeability map for a given

mold geometry along with inlet and vent locations which will allow the mold to

completely fill despite the variations in the preform and the flow disturbances caused

due to its placement. The resin flow pattern can be manipulated with a tailored highly

permeable layer (Distribution Media (DM)) layout to be placed on top of the preform

as it does impact the flow patterns significantly. Thus, a predictive tool to design an

optimal shape of DM, which accounts for the flow variability introduced due to

race-tracking along the edges of the inserts is presented by adapting a discrete

optimization algorithm.

1

Chapter 1

INTRODUCTION

1.1 Liquid Composite Molding

Polymer composite materials combine polymeric resin with reinforcing fibers

to fabricate products that are lightweight of tailored stiffness and strength with

improved fatigue life and corrosion resistance compared to traditional materials. There

are various manufacturing methods to combine the reinforcement and matrix materials

together. Injection molding, hand lay-up, filament winding, pultrusion, compression

molding of sheet molding compound (SMC), prepreg vacuum bagging and autoclave

curing and liquid composite molding (LCM) to name a few commonly employed

processes. Selection of the manufacturing method is mainly based on the geometrical

and structural properties constraints in addition to total volume and cost requirements.

For example, one should use filament winding for making composite pressure tanks

and pultrusion for long profiles such as poles and window frames. Moreover, the

manufacturing method should result in desired design properties with low cost and

cycle time.

LCM is one of the most popular manufacturing method for its short cycle

times, low cost, high quality and it can handle complex geometries. LCM is a class of

processes in which the dry fiber preform in the shape of final product is placed in a

mold and impregnated by the desired resin system. The types of materials and the

common processes that belong to this family are described in the following sections.

2

1.1.1 Materials used in LCM

The process design of the LCM starts with the selection of the reinforcement

and the resin system that can satisfy the design requirements.

1.1.1.1 Reinforcements

High strength and high stiffness reinforcements are the load carriers of the

composite materials and usually manufactured as continuous fibers. These continuous

fibers can be put together in the form of rovings, yarns, strands and tows [1]. Different

forms of yarns or tows (bundles of fibers) can be used to create fiber preforms. Fabric

reinforcements are generated by weaving and stitching the tows as shown in Figure

1.1.(a) or from chopped fibers and strands (Figure 1.1.(b)). The architecture of the

fabric also affects the characteristics of the reinforcement. In Figure 1.1.(a) two

different weaving types are given: plain and twill.

Figure 1.1. Type of reinforcements

Glass, Aramid and Carbon are the most common type of reinforcement

materials (Figure 1.2). Glass fiber is preferred for its tradeoff between mechanical

properties and cost (Figure 1.2.(a) and Figure 1.2.(b)). Aramid fibers have excellent

Unidirectional

00

Plain

(a) (b)

Twill Chopped Mat

3

damage tolerance with low density and high toughness (Figure 1.2.(c)). Carbon fibers

have better mechanical properties but are more expensive than glass and aramid fibers.

Based on the design criteria of the composite material, the selection of the

reinforcement type is dictated by the tensile strength, tensile modulus, compression

strength, density, impact strength, environmental resistance and cost. For example; for

tensile strength and cost glass fiber; for tensile modulus and compression strength

carbon; and for density and impact strength aramid fibers are preferred.

Figure 1.2. Different fabric types: (a) E-glass-plain weave, (b) E-glass random

mat, (c) Aramid twill weave, (d) Carbon twill weave

Fiber tows can also be braided and/or woven to create a 3D reinforcement

structure by orienting the tows in all three directions. The yarn in the through

thickness direction improves the mechanical properties, delamination resistance and

impact damage. Different types of 3D weave architectures are shown in Figure 1.3.

Figure 1.3.(a) is an example of orthogonal weave where tows are placed vertically

between the in-plane layers and promote the tensile strength by increasing the stiffness

[2]. Figure 1.3.(b) is an example of an interlock weave in which the tows in the

vertical direction have a pattern.

(a) (b) (c) (d)

4

Figure 1.3. 3D reinforcement architectures (generated via TEXGEN [3])

1.1.1.2 Matrices

The matrix is the other component of the composite materials that holds and

protects the fabrics. The matrix materials are resins that protect the fibers from

abrasion, transfer the load between fibers and provide inter-laminar shear strength to

the composite material.

The resins can be classified as thermosets and thermoplastics. Thermoset resins

are low viscosity liquids at room temperature but when mixed with a curing agent

initiate a chemical reaction forming long molecular chains by cross-linking. Once they

start to cross-link, the resin viscosity increases and as the resin approaches a gelation

state, the viscosity becomes very large and resin cannot flow anymore. So the goal

with thermoset resins is to ensure that they have reached their destination before they

gel. Gelation time is the time that polymer chains start to form 3-dimensional network

and when the resin viscosity starts to increase exponentially and ceases to flow.

(a) Orthogonal

(b) Angle interlock

5

Thermoplastic resins on the other hand are solid at room temperature and need

to be heated to get them to flow. Their viscosity even in molten state is two to three

orders of magnitude higher than thermosets. Hence, use of thermoset resins such as

epoxy or vinylester is preferred for LCM processes.

1.1.2 The LCM Family of Processes

LCM processes can be divided into two main groups: Resin Transfer Molding

(RTM) and Vacuum Assisted Resin Transfer Molding (VARTM). RTM family of

processes require two-sided rigid mold and resin is impregnated into the dry preform

with positive pressure. VARTM needs one-sided mold surface and the other side is

covered with nylon vacuum bag and the vacuum is applied to infuse the resin into the

preform.

1.1.2.1 The Resin Transfer Molding

RTM is the most traditional LCM process. The steps of the RTM process are

shown on the left hand side of Figure 1.4. First the reinforcements are stacked to form

the desired preform shape. The preform is placed in the mold cavity and the mold is

closed. As the mold is closed and placed in a hydraulic press, the preform takes its

final thickness and it is impregnated with the resin. The resin system is pushed into the

dry preform by the resin injection unit that applies constant pressure or constant flow

rate. After the resin completely wets the dry preform, the injection is closed and the

resin is allowed to cure in the mold. Finally, the cured part in the final shape is

removed from the mold [4].

The main advantages of the RTM process are good surface finish because of

the two-sided rigid mold and high quality final products. The positive pressure from

6

the inlet enables the resin to fill the dry preform at higher speeds, which decreases the

cycle time. Since the two-sided mold is kept closed via hydraulic press, mold

deflection can be prevented under high-pressure fillings. By preventing the mold

deflection the dimensional uniformity of the product can be satisfied [5,6]. Also,

higher fiber volume composites can be manufactured with RTM due to higher inlet

forces and rigid and stable mold cavity. The fiber content is described by the fiber

volume fraction, vf; the ratio of the volume of the fibers/preform to the mold cavity.

The mechanical properties of the composite material can be enhanced by increasing

the fiber content [7,8].

The main disadvantage of the RTM is the high initial cost for the mold. This

makes RTM more feasible for small-sized parts with high production rates. For large

and complex part the mold cost might be a deterrent factor. The other disadvantage is

the lack of resin flow monitoring during the impregnation process [9]. If there is a

problem during the impregnation, it cannot be seen until the part is de-molded. This

issue might yield an expensive and inefficient trial and error procedure. However, use

of experimental and simulation tools to design the process can overcome the resin

impregnation issues. Danisman et al. [10] lists a variety of experimental tools

(sensors) to monitor the resin flow and cure cycle in closed RTM mold such as

SMARTweave [11], dielectric [12], ultrasonic [13,14], fiberoptic [15], thermocouple

[9], pressure transducer [16], and point- and lineal-voltage sensors [17,18]. The

simulation tools can also help overcome these issues. Adapting the Finite

Element/Control Volume approach in RTM simulations first implemented by Fracchia

et. al [19] and following simulation tools are developed: RTM-FLOT [20],

PAM-RTM [21], MyRTM [22] and LIMS [23].

7

Figure 1.4. Schematic of RTM (left) and VARTM (right) steps (adapted from

[1])

Preform manufacturing:

Vacuum

pump

Vacuum bag Inlet

Cured part

Resin Transfer Molding

(RTM)

Vacuum Assisted Resin

Transfer Molding

(VARTM)

Preform lay-up:

Mold Closure:

Resin Injection:

Curing and De-molding:

Resin impregnates

fibers and cures Resin injection

8

1.1.2.2 Vacuum Assisted Resin Transfer Molding

VARTM process is similar to the RTM process except VARTM has one-sided

mold and a nylon transparent layer, vacuum bag, is placed on the other side. As shown

schematically on the right-hand side of Figure 1.4, the process starts with preparing

the preform by placing the layers of reinforcement in the final shape of the product.

The preform is then placed on one-sided mold or a tool surface and the other side is

sealed with a vacuum bag. The sealing between the mold surface and vacuum bag is

achieved with a sealing tape. The vacuum pump, which is placed at the vent, is turned

on to extract the air and creates the pressure gradient to invoke resin flow. The inlet is

closed after the resin wets the preform and reaches the vent. The vacuum is maintained

until the resin cures. Once the resin cures, the part is de-molded.

VARTM process only needs a one-sided mold or a tool surface which reduces

the cost by orders of magnitude and makes it possible to manufacture large structures

such as wind blades. Thus, for large and complex parts VARTM becomes the

manufacturers’ choice [24]. However, VARTM has limitations due to vacuum

pressure. The maximum driving pressure is atmospheric pressure. This limit increases

the fill time and the risk of fill time reaching the gelation time of the thermoset resin

arises, especially for large parts. The fiber content that can be reached with VARTM is

limited compared to RTM as the compaction of the preform is being achieved with

atmospheric pressure [25]. Finally, the surface finish of the vacuum bag side is not as

good as the mold-side.

1.1.2.3 Seemann’s Composite Resin Infusion Molding Process

Seemann Composites Resin Infusion Molding Process (SCRIMP) is a widely

used patented version of Vacuum Assisted Resin Transfer Molding (VARTM) in

9

which a highly permeable layer (distribution media (DM)) is placed on top of the dry

preform to distribute the resin with very low flow resistance to reduce the filling and

hence the manufacturing cycle time [26]. The schematic in Figure 1.5 shows the steps

of the SCRIMP. As DM is not part of the composite, a peel ply is placed between the

preform and the DM and after the entire assembly cures, peel ply is used to separate

the DM from the composite and discard it.

Figure 1.5. Schematic of SCRIMP steps

Preform manufacturing

Preform lay-up

Mold Closure Resin Injection

Curing and De-molding

Vacuum bag Inlet

Vacuum

pump

Distribution

Media

Distribution

Media (DM)

Resin

impregnates

fibers and cures Resin

injection

Cured part

Peel Ply

Peel Ply

DM

10

1.2 Manufacturing Challenges in Vacuum Resin Transfer Molding

LCM enables tailoring of physical and mechanical properties and creating

complex composite parts. The success of the process depends on the preforming,

impregnation and curing, whereas impregnation is the most challenging part.

Unsuccessful impregnation results in formation of macro- or micro-scale voids.

Macro-scale voids are the large air pockets, dry spots that form due to problems in the

flow front profiles (Figure 1.6.(a)) and micro-scale voids forms around the fiber tows

due to trapped air (Figure 1.6.(b)). Any void that remain in the part after the part cures,

damages the quality of the final part.

Figure 1.6. Examples of (a) macro-void and (b) micro-void [27]

1.2.1 Permeability variation

Flow through porous media with Darcy’s law is used to describe the movement

of resin in the fibrous preform [28]. When using Darcy’s law volume-averaged values

are employed for flow variables such as resin velocity, pressure, and material

(b) Micro-voids

2 4 cm

(a) Macro-voids

0

Dry spot

Trapped air

11

properties such as resin viscosity and fiber preform permeability. These averaged

quantities are defined at any location by averaging them over a selected volume

surrounding that region within the domain. However, these fabrics (for example ones

in Figure 1.2) are rarely homogeneous and there can be statistically significant

variation from one region to the next. As shown in Figure 1.7, these variations could

also be due to defects in the fabric. This local non-homogenous architecture of the

fabric can have a noticeable effect on the dynamics of resin flow behavior [29–33].

Figure 1.7. Example of defects in the preform; (a) plain weave glass fabric, (b)

3D orthogonal glass fabric

Most fabrics do have local variations caused by local changes in fiber

orientation and due to change in fabric density [29]. During composite manufacturing

many layers of fabrics are stacked together to build up a certain layer of thickness.

When these layers nest, the nesting may not be uniform across the length of the fabric

may also contribute significantly to permeability variations [30]. Endruweit and

Ermanni [34] found that for a coarse 2x2 twill weave fabric made of thick fiber tows

the variance of local permeability values was higher than for a fine 8-harness satin

(a) (b)

0 1 2 cm 0 0.5 1 cm

12

weave fabric for the same fiber volume fraction, which could be explained by the

intrinsic in-homogeneity of the fabric and the relatively high local variations of the

fiber configuration. Quantitative evaluation of injection experiments, which is

normally based on flow front tracking, implies averaging of local variations in

material properties and measuring averaged global permeability values. While the

experimentally determined permeability values characterize quasi-uniform materials,

the accurate predictive description of global flow for non-uniform materials requires

knowledge of the distribution of local properties.

Another challenge during vacuum infusion is the change in the preform

thickness during injection. In RTM resin impregnation is performed through the

preform that is kept between two rigid molds. Assuming the mold design’s stiffness

stands the pressure of the resin and the preform, the preform will not be expanded or

compacted during the filling. However, in VARTM one side of the mold is sealed with

flexible nylon vacuum bag which will not be able to prevent the expansion of the

preform during impregnation. As seen in Figure 1.8, the thickness of the preform

changes as the resin propagates [35,36]. This variation arises because of the resin

pressure decreases the compaction. However, this variations does not significantly

affect the resin flow behavior during impregnation [37].

13

Figure 1.8. Thickness variation during vacuum infusion

1.2.2 Race-Tracking

Resin finds low resistance pathways when there are open channels: (i) between

the mold and preform edges (Figure 1.9), (ii) along sharp bends in reinforcement

and/or (iii) between preform and inserts in the mold (Figure 1.10.(a)). This fast

movement of the resin along edges and surfaces is called race-tracking. The filling

pattern can change significantly with the presence of race-tracking pathways in the

mold (Figure 1.10). This potentially could allow the resin to reach the vent line before

impregnating the entire preform which will result in a large dry spot or void within the

part resulting in the part that needs to be discarded or re-worked [38–43]. In RTM,

resin racing along the mold edges is more common than VARTM as the mold is a

two-sided closed cavity. In VARTM, race-tracking will be more prominent along the

boundaries of the inserts in the mold or around sharp bends. This can be seen in the

experimental filling of the preform with a metal inset placed in the 4th layer of an 8

layer preform. As seen in Figure 1.10.(a) the metal insert is placed in the middle of

0.5mx0.5m layer and from the flow front profiles at two different time steps for two

experiments (Figure 1.10.(b) and Figure 1.10. (c)) race-trackings took place along the

mold

surface

vacuum

bag flow front

14

edges of the metal insert. The flow front profiles in Figure 1.10.(b) and Figure 1.10.

(c) are obtained from two different set of experiments with same process parameters.

For the first experiment (Figure 1.10.(b)) race-tracking channel permeability is higher

than the ones in the second experiment (Figure 1.10.(c)). Thus, the same part with the

same process parameters may end up with significantly different filling patterns.

Another issue with the race-trackings is their location can be predicted but their

occurrence and the strength of the race tracking cannot be predicted.

Figure 1.9. Race-tracking formation on the edges due to fray edges

preform

mold wall

fray edges

gap

15

Figure 1.10. Race-tracking examples: (a) Mid-layer of the preform with metal

insert spatially in the middle, and Flow front profiles at the bottom

of the preform at two different time steps with race-tracking along

the metal insert for two same experimental configurations: (b)

experiment 1, (c) experiment 2

1.3 Modeling of LCM Processes

Modeling the LCM process will allow one to predict the filling pattern, fill

time and the distribution of fluid pressure in the preform. Good understanding of these

issues will allow one to improve the quality and reduce the cost of the composite [44].

Inlet/s and vent/s positions, permeability of the preform, injection rates are the factors

that influence the filling [45]. Numerical simulations have been developed to optimize

processing window and process design to improve manufacturing with LCM [46–51].

In LCM simulations, resin is modelled as a Newtonian fluid that flows in porous

metal insert

inlet line

race-tracking channels

(a) (b)

(c)

16

media with averaged pore size. The mathematical description of the resin flow through

porous media that models the physics is Darcy’s law (Equation (1.1)) coupled with

continuity equation (Equation (1.2)), as given below;

⟨𝐯⟩ = −𝐊

µ∇P (1.1)

∇ ∙ ⟨𝐯⟩ = 0 (1.2)

∇ ∙ (−𝐊

µ∇P) = 0 (1.3)

where ⟨𝐯⟩ is the volume averaged resin velocity and P is the pore-averaged resin

pressure, µ is the resin viscosity and 𝐊 is the permeability tensor. The components of

the symmetric, positively definite permeability tensor, K, as shown (in Cartesian

coordinates) in Equation (1.4), represent how easily resin can flow in the

corresponding direction;

𝐊 = [

Kxx Kxy Kxz

Kyx Kyy Kyz

Kzx Kzy Kzz

]. (1.4)

Solution of those equations with the initial pressure and/or flow rate specified

at the inlet gate for the fibrous domain enables the estimation of time to fill the mold,

identification of the optimal locations for placement of gates and vents, and to find

regions which may be susceptible to formation of dry-spots. Fracchia et al. [19],

Bruscheke and Advani [52] and Trochu et al. [53] presented successful 2D resin

impregnation models. Those models are practical for thin parts (for most of the

17

composite materials). There are other successful models for both isothermal and

non-isothermal mold filling [49,50,54–61]. As the modeling tools are developed and

improved, those tools are used to optimize the LCM process. If the objective is to have

minimum fill time and/or avoid dry spots, various methodologies have been developed

and reported [4,62,63].

In this dissertation, the numerical simulations of the flow through fibrous

preform are performed via Liquid Injection Molding Simulation (LIMS), which was

developed at the University of Delaware [23]. LIMS is a dedicated finite

element/control volume based simulation of flow through porous media that is capable

of analyzing both 2D and 3D flows. It has both a built in scripted language and a user-

friendly graphical user interface that user can set and perform the simulations. It can

also be called from Matlab® as a function to be used for optimization routines. As

represented in Figure 1.11, the program requires the mesh geometry, viscosity of the

resin, permeability tensor, porosity/fiber volume fraction with boundary conditions

and it provides the flow front locations with time, the last region to fill and the fill

time along with pressure distribution and the success of the filling (location of dry

spots, if any). The permeability values of the reinforcements are relatively low, so the

resin flow is slow (low Re, Re<10). Therefore LIMS adopts the quasi-steady state

assumption. At each step the pressure distribution is obtained from Darcy’s Law

(Equation 1.1) and the Continuity Equation (Equation 1.2). From the pressure

distribution the flow front is advanced using the Darcy’s Law (Equation 1.1) for resin

velocity. Hence one can design gate and vent locations if the preform permeability

map and the permeability tensor in the preform domain (Figure 1.12), are known with

certainty and do not change from one part to the next.

18

Figure 1.11. Liquid Injection Molding Simulation (LIMS) Structure

Figure 1.12. Permeability map approach

K1

K2

K3 K

4

K5

K6

K7

K8

K9

K10

K11

19

1.4 Objective and Dissertation Outline

Research objective of this dissertation is to develop a methodology to create a

spatial distribution of the permeability tensor, called the permeability map for a given

geometry (Figure 1.12), that will allow the mold cavity to fill from a gate and arrive

last at the vent (which implies no voids) despite variability in the preform and flow

disturbances around the mold walls, inserts and corners. The simulation output is the

void area where the input is the geometry of the part along with the permeability map,

location and strength of flow disturbances, location of gates and vents and the inlet

pressure or flow rate boundary condition at the gates. The goal is to find a

permeability map for a selected gate and vent location that will give at most a small

void region despite flow disturbances and variability in preform permeability.

In order to achieve this objective one needs; (i) to develop a characterization

method for permeability, (ii) to be able to quantify the variability in the fabric due to

manufacturing variability of textiles, (iii) to identify race-tracking and other issues of

the fabric and their effect on permeability, replaced by, (iv) to develop optimization

methods that create permeability maps despite variations to achieve successful filling.

In this dissertation, after the introduction to LCM processes in Chapter 1,

permeability measurement and characterization methods are presented in Chapter 2.

Also, a methodology is introduced to characterize the six components of the

permeability tensor with non-zero skew components.

In Chapter 3, the 3D permeability characterization technique introduced in

Chapter 2 is used to demonstrate that slight variation in orientation during lay-up can

influence through thickness permeability variability dramatically and a permeability

map should take this into consideration.

20

Chapter 4 introduces a technique to quantify variation in permeability of a

fabric which will be taken into account when assigning a permeability map

Chapter 5 presents the formulation of a methodology and development of

optimization technique that will use the forward simulation allowing for variability in

permeability to create a permeability map that will fill the mold without voids despite

the variations and flow disturbances.

The last chapter lists the conclusions and contributions with suggestions for

future work.

21

Chapter 2

PERMEABILITY MEASUREMENT TECHINIQUES

2.1 Historical Background

In 1856 Darcy conducted sets of water flow through sand beds experiments

which was the first attempt to model fluid flow through porous domain [64]. He

introduced the term permeability to quantify the ease of fluid flow in porous domain

and developed an empirical relationship to relate the flow rate of water to the pressure

drop across the sand column as follows

Q = −KA

µ∙

∂P

∂x (2.1)

where Q is the flow rate, µ viscosity of water, A is the cross-sectional area, ∂P

∂x is the

pressure gradient along the flow direction and K is the scalar that characterizes the

permeability of the sand in the flow direction.

Darcy developed the equation for homogeneous and isotropic porous domain

with water flow in one-direction. In 1961 Liakopoulos [65,66] expanded the Darcy’s

empirical equation by introducing the permeability as a tensor. The permeability

tensor (Equation 1.4) characterizes the ease of resin flow in porous domain in all

three-directions.

Symmetric, positively definite permeability tensor, K has orthogonal set of

axes – principle directions (which are the diagonal terms when the non-diagonal terms

are zero). Figure 2.1 shows the mold coordinate (xyz) and the principle direction of

22

the preform (x’y’z’). The permeability tensor in mold coordinate frame which is given

in Equation 1.4 can be rotated by θ degree such that the mold coordinate axis

coincides with the principle directions of the preform, in which case the diagonal

permeability tensor, K’ in x’y’z’ coordinate frame can be expressed as;

𝐊′ = [

Kxx′ 0 00 Kyy

′ 0

0 0 Kzz′

] (2.2)

Figure 2.1. Flow front profile with xyz mold coordinate, x’y’z’ principle

direction of the preform

As mention in Chapter 1, in LCM processes the dry preform is placed into

mold cavity and after the mold is closed and sealed, the resin is introduced into the

mold to impregnate the porous fibrous media. Darcy’s law is used to mathematically

preform

injection

point

z flow front

x’

y’

z’

θ x

y

23

describe the flow of resin into a closed mold containing fiber preform. However, how

closely the mathematical model mimics the actual flow behavior depends on the

fidelity of the material input data such as viscosity of the resin and the permeability

values of the fabric placed in the mold. Hence it is important to characterize the

permeability data accurately. Over the years, researchers have presented many

different methodologies to characterize the permeability of the fabric. The overall

approach to permeability characterization can be investigated under three broad

categories (i) analytical and predictive methods, (ii) numerical methods and (iii)

experimental methods.

2.2 Analytical and Predictive Methods

As a mathematical model, Darcy’s Law, relates the pore-averaged velocity

with the pore-averaged pressure gradient with the permeability of the porous domain,

K and the viscosity of the resin, µ (Equation (1.1)). Darcy’s Law is a macroscopic

model and the microscopic physical properties are averaged using continuum

approach. Thus, the effect of the fiber volume fraction (namely porosity), tortuosity

and capillary effects are lumped under permeability in Darcy’ Law [67].

Kozeny-Carman (KC) tried to establish a relationship between permeability

and porosity by modeling the flow within a porous media as a series of cylinder

capillary channels coupled with Carman’s introduction of hydraulic channel [68]. The

Kozeny-Carman equation can be expressed as;

K = 𝑅𝑓

2

4𝑘0

(1 − 𝑣𝑓)3

𝑣𝑓2 (2.3)

with 𝑅𝑓 is the fiber radius, 𝑘0 is the Kozeny constant that empirically accounts for the

tortuosity to be determined experimentally and 𝑣𝑓 is fiber volume fraction.

24

KC equation is a semi-empirical relation with 𝑘0 empirical constant which is

later proved not to be constant [69]. The KC model improvements are performed to

estimate the permeability [70]. Ahn et al. [71] showed good agreement in permeability

estimation for woven fabrics using KC, however, Gauvin et al [72] reported KC model

is not sufficient for random mats. Also, unsuccessful experimental implementations

are presented [73]. Thus, researchers suggest the introduction of the capillary model

for the resin flow to improve the estimations.

Gebart [74] developed a geometric model for permeability prediction. Set of

analytical expressions are presented for an idealized unidirectional reinforcement with

regular, parallel fibers. The expressions consists of Navier-Stokes equations both for

flow along and perpendicular to the fibers. Solution for the flow along the fibers has

the same form with KC formulation, however, for perpendicular flow includes the

physical limit in terms of fiber volume fraction. Another predictive model is

introduced by Bruschke [75]. The model consists of regular array of cylinders to

represent the fiber tows. Close form solutions are derived for the upper and lower fiber

volume fraction values for Newtonian fluids. Good agreement is obtained between

closed form solutions and numerical models for mid-range fiber volume fractions. The

limitation of Gebart and Bruschke models is that the physical model used to describe

the system does not capture the structural details of real preform materials. The fiber

preform usually used in LCM processes consist of woven or stitched fiber bundles

known as tows or yarns, rather than of individual fibers and their geometric

arrangements are usually more complex than the one assumed in analytic models.

Since, predictive tools cannot represent the realistic geometrical arrangement,

25

experimental and numerical methods are more useful for permeability

characterization.

2.3 Numerical Methods

Numerical methods, as a tool to characterize the permeability, generally

involves the solution of the Navier-Stokes equation for well-defined cell geometry for

the preform. The solution involves either use of periodic boundary conditions or

implementation of the Lattice-Boltzmann method. All methods impose a pressure

drop across the porous domain and calculate the average flow through the unit cell or

prescribe a flow rate along one face of the unit cell and calculate the pressure drop.

The permeability of the unit cell is derived by using the Darcy’s Law.

Averaging of permeability in a unit cell is an example of homogenization

method. Over the macro-scale, the equivalent homogeneous medium represents the

average behavior of the heterogeneous medium. Mathematical theory of the

homogenization method is established in several studies [76]. The numerical method

solves the Navier-Stokes equation for the homogeneous medium, representative unit

cell using the periodicity boundary condition [77].

The Lattice-Boltzmann Method (LBM) is based on microscopic models and

mesoscopic kinetic equations. The methods models the fluid as set of particles that are

moving and interacting on a lattice. From the discrete data of the particles, one can

define the space and time aspects of the fluid flow.

LBM has been used to investigate the porous media by several authors [78–

81]. Koponen et al. [78] employed the nineteen velocity LB model to calculate

permeability of three dimensional random fibrous structure generated by a growth

algorithm in discretized space. Nabovati and Sousa [82] investigated the permeability

26

of sphere packs. Also Nabovati and Sousa [82] reported their work on fluid flow in

three-dimensional random fibrous media simulated using the lattice Boltzmann

method.

The LBM overcomes the major limitation of the homogenization method. It is

capable of simulating flow in realistic situations of complex fabric geometries and

structure. However, Belov et al. [83]reported that the Lattice Boltzmann calculations

are computationally intensive. But it can also incorporate the surface tension effects of

the fluid very easily.

The numerical solutions providing permeability data for the unit cell, may not

accurately represent the permeability of the preform at the macro-scale. To perform an

entire simulation within a preform with thousands of unit cells solving for Navier-

Stokes equation may be a formidable computational challenge. Hence, experimental

methods are used to determine the permeability coupled with phenomenological and

numerical methods since the permeability changes with fiber orientation and fiber

volume fraction; otherwise one would have to conduct many experiments for the same

fabric to find the dependence on fiber orientation and volume fraction.

2.4 Experimental Measurement Techniques

Permeability characterization experiments are performed by controlling either

inlet pressure or injection flow rate and grouped according to the pattern of fluid flow

through the preform: rectilinear, radial, transverse and three-dimensional. Each

approach has its own advantages and disadvantages.

27

2.4.1 Rectilinear Flow

In-plane permeability measurements are the most commonly reported in

literature as they are straight forward. Rectilinear experimentation is an in-plane

permeability measurement technique to characterize the permeability by conducting

linear flow channel experiments [84–86]. The preform can be placed either in a RTM

mold with one transparent side or a VARTM set-up. As the resin flows through the

preform, linear flow front profiles are tracked with time, as shown in Figure 2.2. For

an ideal experimentation, the flow front profiles will be linear and can be easily

monitored. The experimental data is the flow front position with time. Then, time

integration of one-dimensional Darcy’s Law yields the solution of the flow front

position with time, as given in Figure 2.2. In this equation, xf(t) is the flow front

position at time t, K is the permeability of the preform in the flow direction, ΔP is the

resin pressure drop along the flow, μ is the resin viscosity and 𝜙 is the preform

porosity (defined as (1-vf)). From the slope of the best line fit of the plot of the square

of the experimental flow front, the average bulk permeability of the fabric in the flow

direction can be evaluated for that particular fiber volume fraction.

28

Figure 2.2. One-dimensional permeability characterization experiment to find

the bulk permeability value in the direction of flow

Rectilinear flow experiment is easy to conduct and the experimental data is

easy to process and have high reproducibility [84,87,88]. However, appropriate

equipment, such as visualization tools and sensors, might increase the initial cost

which can be listed as a disadvantage. Another disadvantage arises due to the

race-tracking issue (as introduced in Chapter 1) which invalidates the linear flow

assumption and generates error in permeability data [42,89]. As mentioned before,

this approach is used to determine the permeability component only in the flow

direction. Set of experiments are required for characterization of all the six

components of the permeability tensor.

Preform Filled region

𝑥𝑓(𝑡): flow front location at time 𝑡

Linear flow fronts

used to find bulk

permeability

Actual flow fronts 𝑡1

𝑡2

𝑡3

29

2.4.2 Radial Flow

Radial flow is another method for in-plane permeability characterization. The

test fluid is injected through a gate which is a hole in the center of the fiber preform.

The resin entering in the circular cutout in the middle of the preform spreads radially

impregnating the preform in a circular or elliptical shape. The radius of this boundary

is important during the data analysis. As seen in Figure 2.3, if the preform is isotropic

the flow front profiles are circular and as the anisotropy of the preform increases the

major minor axes ratios increases. The ellipses’ major and minor axes align with the

principle direction of the preform and the ellipse is at an angle with respect to global

coordinate frame if the principal axis of the preform do not coincide with the

coordinate axis (as shown in Figure 2.3.(c)).

Radial injection eliminates many of the disadvantages of the rectilinear flow

[85,90–97]. However, flow front tracking requires visual monitoring through the

transparent mold surfaces [98,99], fiber optic sensors [71], thermistors [100], pressure

transducers [101], and ultrasound and electrical residence [102]. Then, data reduction

schemes are required. Additionally data reduction schemes is more complex than the

linear flow experimental data. Though, for isotropic preforms the analytical solution of

the in-plane permeability can be calculated easily [103];

K = {𝑅𝑓2[2 ln(𝑅𝑓 𝑅0⁄ ) − 1] + 𝑅0

2} 1

𝑡

𝜇𝜙

4Δ𝑃 (2.4)

where 𝑅𝑓 is the flow front radius at time t, and 𝑅0 is the radius of the injection gate.

Thus, using equation 2.4, permeability of isotropic preforms can be determined by

measuring the pressure gradient and monitoring the circular flow fronts with time.

Then, for the anisotropic fabrics Chan and Hwang [91] proposed an approach to

determine the principle permeability components using the major and minor radius of

30

elliptical flow front profiles (Figure 2.3.(b)). This work is followed by Weitzenbock et

al. [103,104] with the methodology to obtain the principle permeability components

without the knowledge of the principle axes (Figure 2.3.(c)).

Radial flow experimentation can be used to characterize in-plane permeability

components. Moreover, the race-tracking issue doesn’t occur in radial injection and

doesn’t affect the permeability evaluation. However, radial flow experiments are

consistent with linear flow but result in different values for the same preform at the

same fiber volume fraction. However, for a reliable in-plane permeability data Wang

et al. [85] suggests conducting both linear and radial flow experiments.

31

Figure 2.3. Schematic of radial flow front profiles: (a) isotropic (R1=R2), (b)

anisotropic (R1≠R2), (c) anisotropic with non-zero in-plane skew

term (global coordinate frame doesn’t coincides with principle

directions of the preform)

(a) Isotropic flow front

(b) Anisotropic flow front

(c) Anisotropic flow front

with non-zero

in-plane skew term

Injection point, R0

R1 R

2

R1

R2

θ

R2

R1

Preform

x

y

32

2.4.3 Transverse and Three-Dimensional Flow

A variety of methods that have been used to experimentally characterize the

in-plane preform permeability components are presented in the previous section. For

thin parts only in-plane permeability is required which has three independent

components – either two principal values and the orientation of principal axes or two

normal and one “skew” component. For thick parts, one must characterize through the

thickness permeability as well [95,100,105–107].

There are two approaches for transverse permeability measurement;

simultaneous measurement of three principle permeability components and

independent measurements (separate experimentations for in-plane and transverse).

Several researchers conducted transverse permeability studies [100,108–110].

One-dimensional channel flow apparatus is utilized to characterize this component

with Darcy’s Law [111].

Trevino et al. [112] developed a tool to evaluate the transverse permeability

based on one-dimensional flow and discretized Darcy’s Law. Wu et al. [113] includes

the three-dimensional flow simulation to one-dimensional flow model using steady

state flow profiles. Ahn et al. [71] presents a device that simultaneously measures the

transverse permeability and capillary pressure.

In order to model the three-dimensional resin impregnation, three-dimensional

permeability characterization is required. Three-dimensional flow experiments are

proposed to fully characterize both isotropic and anisotropic permeability with a single

experiment. Traditionally, LCM parts are thin but there are practical resin flow

problems seeking for three-dimensional permeability tensor [100].

A general methodology is presented by Woerdeman [110] for

three-dimensional permeability tensor characterization from set of one-dimensional

33

flow experiments. The permeability data is derived from numerical solution of six

nonlinear equations [114]. Whereas, Weitzenbock et al. [100] tracked the flow fronts

using thermistors and mentioned the importance of the capillary pressure on the

three-dimensional permeability characterization. Using the same measurement

principle Ahn et al. [71]monitored the flow fronts using embedded fiber optic sensors

which are placed inside the preform. Following that, Ballata et al. developed Smart

Weave as another flow monitoring technique [115].

Gokce et al. [116] introduced a new experimental method, Permeability

Estimation Algorithm (PEA). PEA processes flow front information during the

experimentation and process with a numerical process model. Its limitation is being

applicable only for VARTM process. Whereas, Breard et al. [117] used X-ray

radiography to monitor the flow but the cost of the system requires expensive tooling.

Nedanov et al. [114] presents a method to evaluate principle values of the

three-dimensional permeability tensor. This method is based on visual monitoring of

the in-plane flow front profiles. The shape and size of the in-plane flow front through

the transparent membrane as well as the amount of fluid in the preform and elapsed

time are recorded and allow for characterization of principle permeability in all three

directions. Similar approach is used by Okonkwo et al. [118]. Instead of transparent

plates, electrostatic sensors are placed on the top and bottom surfaces of 3-D radial

injection mold is used and instead of an analytic solution, numerical simulation is

utilized to characterize the permeability.

Each of these experiment approaches has their disadvantages. The use of

embedded sensors affects the pattern of flow and renders the experimental data

unreliable. Weizenbock [100] observed that the flow front in the part of the mold

34

where the thermistors sensors had been placed was lagging behind compared with

other undisturbed parts of the mold. Also since the sensors are normally embedded in

the preform manually, this requires time and effort. Numerous experiments are usually

required for reliable characterization of preform permeability and as such using

embedded sensors will require extensive time and labor rendering the methods less

efficient. And the method [117] that involves the use of X-ray spectroscopy to

measure the flow front through the thickness is rather expensive. In case of Nedanov

experiments [114], the results for through thickness could be unreliable as they used

only one data point to find the transverse permeability –which was when the arrival of

the resin was recorded at the bottom. Also the size of the gate had an effect on the

permeability calculations. The method developed by Okonkwo et al. [118] is

applicable to non-conductive fabrics, e.g. cannot be implemented to carbon fibers.

From the review of the existing experimental methods for permeability

characterization of fibrous media shows that while traditional methods for in-plane

permeability measurement are well developed the methods for transverse permeability

measurement need further investigation. Hence the need for reliable and fast method

to determine the components of the three-dimensional permeability tensor in a single

experiment and use of simple equipment is desirable.

2.5 Skew terms

For thin parts, only in-plane permeability (Kxx, Kxy, Kyy) are necessary. That

requires three components – either two principal values and the orientation of

principal axes or two normal and one “skew” component (Kxy). Several methods to

obtain these values were devised. For thick parts, particularly when flow media is used

35

on part surface, through the thickness permeability is needed as well and can be

measured [119–122].

Three-dimensional tensor contains not only this (normal) transverse

permeability but also two additional skew components. These are, in practice,

neglected as it is assumed that fabric layering produces symmetry needed to eliminate

them. This assumption is somewhat questionable in the first place, but it becomes truly

invalid when thick, three-dimensionally woven or braided reinforcements are

concerned. The geometry of weave allows these terms to appear and acquire some

significance for flow. So far, no methods have been developed to measure these terms

for thick preforms and it remains uncertain how important they are for the

manufacturing process. In this section, permeability tensor is investigated for thick

3-D woven fabrics, including the skew components via a multi-objective optimization

algorithm coupled with Liquid Injection Molding Simulation (LIMS) tool. The effect

of the skew components on the resin impregnation and the limitations on the

importance of those terms are evaluated.

2.5.1 Introduction

Three-dimensional fabric permeability tensor requires in addition to the

transverse permeability, two additional out of plane skew components if the through

thickness principal direction is not orthogonal to the plane of the fabric. Most

researchers assume that the principal Kzz direction of the fabric coincides with the

z-axis. This assumption may not be true for thick, three-dimensionally woven or

braided textile fabrics as the ones shown in Figure 2.4. The geometry of the weave

could be such that the principal Kzz direction may not be aligned with the normal

direction of the preform plane. This will influence the resin flow pattern in a mold due

36

to the non-zero skew components (Kxz and Kyz not being equal to zero). So far, no

methods have been developed to measure these terms for thick, unbalanced and

braided preforms and it remains uncertain how important they are for the

manufacturing process.

Figure 2.4. 3D 25890 g/m3 E-glass fabric

In this chapter a methodology is presented to determine these skew terms using

radial flow experiments in an instrumented mold. Flow front profiles at the top and the

bottom of the mold are used to construct the flow front pattern at the top and bottom of

the preform respectively. Using a multi-objective optimization algorithm, the input

values for the permeability tensor in the flow simulation program LIMS are varied

until the flow front patterns at the top and the bottom of the mold in the simulation

match with the experimental results.

37

2.5.2 Methodology

To characterize the permeability, the fibrous preform is placed in the mold,

which consists of transparent acrylic top and bottom surfaces connected via aluminum

spacers and steel bolts. A resin injection hose is connected to the center of the bottom

surface, and the resin is contained in a pressurized vessel. Two video cameras are

placed and synchronized so as to capture images of the top and bottom surfaces of the

mold simultaneously. The recorded images contain a timestamp in their file name.

After recording the fiber volume fraction and viscosity, the resin is introduced into the

mold under constant pressure from the pressurized tank. The experiment is recorded

until the flow front reaches any edge of the mold. The experiment set-up is given in

Figure 2.5.

The fill times and flow front characteristics are determined experimentally via

image-processing, an example of which is shown in Figure 2.6. The images captured

during the experiment are input into a MATLAB® script, which processes and

analyzes them. The images are filtered and converted to binary, so that it only contains

a white elliptical ring, which represents the progress of resin flow from the preceding

image to the current one. The script iterates through each image to record the locations

of pixels in the flow front. These pixel locations are provided as input to a MATLAB

function that generates an ellipse for the flow front using a least squares fit method.

Given a set of ideal images (distinct resin flow edges with no noise) taken from a

virtual experiment so that the actual fill times were known, the approximated fill times

from image processing were shown to have an average relative error of less than 2.0%.

From the fitted ellipses, the script can also determine characteristics such as; the semi-

major and minor diameters, the angle of rotation, and centroid location.

38

Figure 2.5. Experimental set-up to monitor the resin flow at the top and bottom

surfaces of the preform (left: schematic, right: picture of the set-up)

The permeability tensor is evaluated by minimizing the difference between the

experimental flow-fronts and the flow-fronts numerically predicted with six

permeability components as variable parameters. The minimization uses Nelder-Mead

simplex method. This method is designed to solve unconstrained multi-objective

minimization/maximization problems. The method needs initial function values to

form the initial simplex but does not require any gradient input. The method is a

simplex-based method where a simplex, S, in n-space is defined as a convex hull of

n+1 vertices. For example, a simplex in a 2D space is a triangle and in a 3-D space is a

tetrahedron. The method begins with n+1 points (vertices of initial simplex) and

function values at those points for n-variable optimization. The method then performs

a sequence of transformations of the working simplex, S, aimed at decreasing the

function values at its vertices. At each step, the transformation is determined by

39

computing one or more test points, together with their function values, and by

comparison of these function values with those at the vertices. This process is

terminated when the working simplex, S, becomes sufficiently small that satisfies the

assigned tolerance. For this study the generation of the new simplex algorithm is

adopted from Mathews and Fink [17].

Figure 2.6. (Left) An image of isotropic flow from an experiment. (Middle) The

image after having the preceding flow image subtracted from it,

filtered, and converted to binary. (Right) An ellipse is fitted to the

edge of the resin flow front.

In this optimization routine the objective function is to minimize the residual

sum of square of the experimental and numerical fill times of the filled nodes located

at the top and bottom surface of the mold (Figure 2.7). The flow with initial

permeability values is simulated via a numerical tool called Liquid Injection Molding

Simulation, LIMS, which is a finite element/control volume based program that uses

Darcy’s law to describe the flow of resin inside a fibrous media and tracks the flow

front during the impregnation process [45] with input values provided for the

permeability and viscosity. LIMS then outputs the calculated node fill times, and the

-0.1 -0.05 0 0.05 0.1 0.15

-0.1

-0.05

0

0.05

0.1

0.15

x

y

a (x2)+b x y+c (y2)+d x+e y+f = 0

40

computation is made to converge using the Residual Sum of Squares (RSS) method

with the difference between the experimental fill times and the simulated fill times.

This process is iterated by updating the input values for permeability until the LIMS

fill times are sufficiently close to the experimental fill times. Those final input

permeability values in LIMS are the permeability values for the fabric

One of the main advantages of this method to characterize the permeability

tensor is that it is inexpensive and easy to implement. Okonkwo et al. [118] proposed

a similar algorithm, however the experiment involved a heavy mold with an expensive

set of electric resistance sensors. The use of electric resistance sensors excludes the

use of carbon fiber preforms, as carbon fiber is electrically conductive. The method

described in this paper requires no sensors, as it uses image-processing as its primary

means of calculating fill times. Also, the mold is relatively light and simple to set up

and clean, while a senor-based experiment would require a more tedious clean-up

process; if the resin is not completely cleaned off of the sensors, it could obstruct the

sensors and the subsequent experiment could yield unreliable data. Other proposed

methods have used embedded optical fiber sensors, which obstruct resin flow, leading

to inaccurate data. Another benefit of this method is that K can be characterized in a

single radial flow experiment, and the top and bottom surfaces of the mold are

accounted for so as to allow one to determine all the six independent components of

the three-dimensional permeability tensor. Weitzenböck et al. [93] proposed a method

for measuring the permeability components in one radial flow experiment, however

the resin was to be injected uniformly in the through-thickness direction. Therefore

only in-plane components of K could be determined.

41

Figure 2.7. Algorithm for permeability prediction from experimental fill time of

top and bottom surfaces

2.5.3 Results and Discussion

First, the methodology is tested with a parametric study of radial injection from

the bottom center of the mold cavity injected under constant pressure. Figure 2.8

demonstrated the effect of non-zero skew terms; Kxy, Kxz and Kyz on the flow pattern.

In Figure 6.a-c the magnitude of Kxy is increased while other two skew terms are zero

to investigate its influence on the flow front. As expected increase in-plane skewness,

Kxy, yields an increase in the in-plane rotation of the elliptical flow fronts. The center

of the ellipses observed along the top and the bottom surfaces coincide and do not shift

with the magnitude of Kxy. Nor do the ratios of the major and minor axes change,

only the angle of the in-planar rotation increases with increasing Kxy. However, as the

42

other two skew terms, Kxz and Kyz are assigned a non-zero value, the center of the top

ellipse in no longer coincident with the ellipse at the bottom surface. The shift between

the two centers is proportional to the magnitude of the skew terms. As the Kxz value is

increased the distance between the centers of the ellipse on the top surface and the

bottom surface increases along the x-direction (Figure 2.8.d-f), similarly as the Kyz

value is increased, the distance between the top and the bottom center increases in the

y-direction (Figure 2.8.g-i). The numerical approach presented in Figure 2.7 is applied

to the ellipses of Figure 2.8 in order to predict the permeability tensor just using the

top and bottom ellipse flow front information. The maximum error between the

assigned permeability and the predicted one is found to be 3.33 %

Furthermore, the methodology is validated by assigning six non-zero

permeability components in the LIMS simulation. The flow fronts on the top and

bottom surface of this virtual experiment are used in the approach presented in Figure

2.9, to find the six components of the permeability tensor based on just this

information as follows; the permeability tensor values on the left side in Table 2.1 are

assigned into LIMS and resin arrival time are obtained with the developed image

processing tool from the images of the top and bottom at different time steps. These

arrival times are used as the experimental fill time (Ti,exp in Figure 2.7). Also, from the

flow front profiles at the top and bottom the ratios of the in-plane permeability

components and the angle between the principal directions and the global coordinate

frame can be estimated. These estimations are assigned as the initial simplex. The

algorithm converged to the assigned permeability values listed on the right side of

Table 1 in about 6 hours of CPU time on a PC computer. In Figure 2.9, the flow front

profiles are compared with the assigned and predicted permeability values for the top

43

and bottom surfaces. At time equal to 90 seconds the resin reaches the top for the first

time and as it impregnates the top surface to form the ellipse, the center of the top

ellipse is not coincident with the center of the ellipse at the bottom because of the

non-zero Kxz and Kyz. This virtual experiment also shows the methodology can predict

the six components of the permeability tensor accurately.

Next, the methodology is applied to characterize the 3D glass fabric with 2627

g/m2 areal weight. Three layers of 3D fabric are placed in the mold cavity with

dimensions of 25.4 cm x 25.4 cm x 0.6 mm with 60% fiber volume fraction. Due to

Newtonian behavior, colored corn syrup diluted with water with viscosity of 107 cP is

used as simulated resin. The images with time stamps are analyzed via the image

processing routine and the fill times at the bottom and top nodes are derived as the

experimental data. Then the algorithm given in Figure 2.7 is implemented. Table 2.2

shows the results of predicted permeability components. From the predicted

permeabilities, the Kxz component is seen to be negligibly small, but Kyz values is

found to have effect on resin impregnation.

In Figure 2.10, experimental flow front profiles are compared with the profiles

obtained using the predicted permeability values provided in Table 2.2 at time equal to

13.26 seconds. As it can be seen in the figure, the match between experimental and

predicted profiles is quite good. The major and minor axes values are compared are

also compared. For the top, experimental major and minor radius are 0.055 m and

0.028 m and the prediction data are 0.050 m (9.09% error) and 0.029 m (3.57% error),

respectively. For the bottom, experimental major and minor radius are 0.086 m and

0.051 m and the prediction data are 0.085 m (1.16% error) and 0.052 m (1.96% error),

respectively.

44

Figure 2.8. Flow front profiles at the top (solid lines) and bottom (dash-dot lines)

for different skew permeability at time equal to 700 seconds. The

jagged flow fronts are numerical artifacts because of fairly coarse

mesh.

Kxz = Kyz = 0

Kxy

= Kyz

= 0

Kxy

= Kxz

= 0

(a) Kxy

= 1.0x10-11 (b) K

xy =3.0x10

-11 (c) K

xy = 7.0x10

-11

(d) Kxz

= 1.0x10-12

(e) Kxz

= 3.0x10-12

(f) Kxz

= 7.0x10-12

(g) Kyz

= 1.0x10-12

(h) Kyz

= 3.0x10-12

(i) Kyz

= 7.0x10-12

*For all simulations: Kxx

= 2.0x10-10

Kyy

= 1.0x10-10

Kzz

= 1.0x10-12

δx δx δ

x

δy δ

y δ

y

x

y

45

Table 2.1. Parameters for virtual experiment

Parameter: Numerical value:

Inlet pressure: 1.0 bar

Fiber volume fraction of E-glass: 50%

Viscosity of corn syrup 100 cP

Assigned

Permeability (m2)

Predicted

Permeability (m2)

Percentage

Error (%)

Kxx 2.0x10-10 1.979x10-10 1.05

Kxy 1.0x10-11 9.895x10-12 1.05

Kyy 1.0x10-10 9.884x10-11 1.16

Kxz 5.0x10-12 4.947x10-12 1.06

Kyz 2.5x10-12 3.875x10-12 3.12

Kzz 1.0x10-12 9.798x10-13 2.02

Figure 2.9. Flow front profiles comparisons with assigned and predicted

permeability values at the top and bottom surfaces

x

y

Top Bottom Assigned

Predicted

Level Time(sec)

1 300

2 400

3 500

4 600

5 700

δ

46

Table 2.2. Predicted permeability for the experiment

Kxx Kxy Kyy Kxz Kyz Kzz

2.405x10-11 7.451x10-12 1.736x10-10 5.777x10-16 7.365x10-13 1.911x10-12

Figure 2.10. Flow front profiles at time 13.26 seconds at the top and bottom:

experimental, with predicted permeability and comparison

-0.1 -0.05 0 0.05 0.1

-0.1

-0.05

0

0.05

0.1

x

y

a (x2)+b x y+c (y2)+d x+e y+f = 0

Experimental With predicted permeability Comparison

TOP

BOTTOM

Experimental With predicted permeability Comparison

47

2.5.4 Summary

This work presents a methodology to characterize all the six independent

components of a three dimensional second order permeability tensor. The approach

employs a multi-objective optimization algorithm coupled with Liquid Injection

Molding Simulation (LIMS) tool to calculate the permeability values. The effect of the

non-zero skew components on the flow front progression and flow patterns is

investigated through a virtual study to underline when the skew terms could change

the nature of filling and influence the manufacturing process.

48

Chapter 3

THROUGH THICKNESS PERMEABILITY

3.1 Introduction

When unidirectional stitched fabrics are used as reinforcement in composites,

plies are typically stacked on top of each other to build up the desired thickness.

Strength and stiffness requirements dictate the orientation of individual layers and the

accuracy of angular alignment is limited. A pressure differential across the thickness is

used to distribute the resin, either from a pre-impregnated fabric or injected from a

resin source, to occupy all of the empty spaces between the fibers. This process is

commonly modeled using Darcy’s law, which describes flow of resin through porous

media in which the flow rate is directly proportional to the applied pressure

differential by the through-thickness permeability of the fabric. A different orientation

between layers or even a slight misalignment during the stacking can change the

through-thickness permeability dramatically due the change of resin pathways. In this

chapter, characterization of the through-thickness permeability of a series of

unidirectional fabrics stacked in various orientations is studied to address both the

effect of stacking sequence and those of misalignment of the individual layers.

Numerical simulations are conducted to predict the effect of change in fiber

orientation on the through-thickness permeability. The results from the numerical

model are compared with experimental measurements. Results show that averaging

approach is not suitable to calculate the through-thickness permeability component

when using unidirectional fabrics and that the stacking sequence of the unidirectional

49

fabrics may significantly influence the through-thickness permeability. Also, it has

been shown that the effects of misalignments smaller than 5 degrees rotation between

individual layers do not significantly modify the transverse flow.

This chapter will analyze through-the thickness flow depending on relative

orientation of individual reinforcement layers. It will show that such an averaging

approach can result in large errors in calculation of the through-thickness permeability

component when the preform consists of layers of unidirectional fabrics stacked in

different desired fiber orientations. It will be shown both experimentally and by

modeling, that the stacking sequence can significantly influence the through-thickness

flow and hence the transverse permeability. This study will show that a limited

misalignment less than 5 degrees rotation between the neighboring layers – which can

be attributed to inaccuracy of the layup process – does not significantly modify the

through the thickness permeability.

A numerical study for a simplified model fabric is shown in Figure 3.1.

Through-the-thickness direction is aligned with the z-axis. It is assumed that this is the

principal direction of permeability and hence Kyz and Kxz are assumed to be zero. This

is a reasonably valid approximation, as these components are usually insignificant.

Figure 3.1 shows a solid model of three unit cells along with the corresponding cross-

sections. Each unit cell has four layers of unidirectional fabrics with different fiber

orientation sequence in the in plane direction. In Figure 3.1.(a), all plies are aligned in

the y-direction, while in Figure 3.1.(b), all plies are still aligned but rotated by 10

degrees with respect to the y axis in the x-y plane, and in Figure 3.1.(c) plies are

rotated by 10 degrees with respect to the previous ply in the x-y plane as they are

stacked on top of each other so the fiber orientation sequence with respect to the y-axis

50

will be 0/10/20/30 degrees. The change in orientation may arise from two sources:

First, the design commonly requires that the unidirectional reinforcement is oriented

with stacking sequence in pre-determined directions for desired strength and stiffness.

This change of orientation from one layer to the next is usually in increments of 15

degrees or more even though several subsequent layers might have the same

orientation. Second, the change in orientation may arise due to small unintentional

misalignments. We address both these cases.

Note that the cross-sectional area for the lay-up in Figure 3.1.(c) has a very

different profile for through-thickness flow of the resin compared to no rotation and

rotation of the plies by the same rotation degree (Figure 3.1.(a) and Figure 3.1.(b)

respectively). Thus, the in plane orientation of unidirectional fabrics and their stacking

sequence can create different pathways for resin flow in through-thickness direction

resulting in different through the thickness permeability values (Kzz component). In

this study the effect of the pathways formed by different in plane orientations of the

plies on the through-thickness component of the permeability tensor are investigated.

The simplified nature of the model is demonstrated by circular cross sections

of the fiber tows and by the absence of stitching which (Figure 3.2) may actually form

a very sparse “weft” layer. The only effect this stitching has in our model is that we do

not allow any interpenetration of subsequent layers.

51

Figure 3.1. Solid model of a unit cell and the corresponding cross-section of four

unidirectional plies stacked on top of each other (a) All plies aligned

along the y- axis (b) All plies are rotated by 10 degrees in the x-y

plane with respect to the y- axis (c) Each successive ply is rotated by

10 degrees resulting in a stacking sequence of 0/10/20/30 with respect

to the y- axis with the corresponding cross sections in the through-

thickness direction, respectively.

(a)

(b)

z

x

z

x

z

x

z

x y

z

x y

z

x y

(c)

52

3.1.1 Effective Permeability of Preform Stacks

Traditionally, the permeability tensor of a set of unidirectional plies stacked

together to form the thickness of the composite is calculated by using the laminate

analogy and the tensor transformation rules taking into account the orientation of the

plies with respect to a coordinate system [119]. This approach serves reasonably well

for two-dimensional (in-plane) permeability components, though some issues have

been noted [123]. However, for the three-dimensional permeability tensor – mainly the

through-the-thickness component(s), this method has two major shortcomings. First,

many models conclude that the permeability component (Kzz) in the thickness

direction will be the same irrespective of the lay-up and the stacking sequence

[124,125]. This is definitely not the case for unidirectional fabrics. It will be shown

that, for example, 6 plies of unidirectional fabric that are all in the zero direction, their

Kzz value will be very different from the same 6 plies if they have 0/90 sequence

repeated three times. Physically this is true because the permeability depends on easy

pathways for flow which will change in the thickness direction as one changes the

orientation lay-up.

Figure 3.2. Front and back side of the unidirectional fabric

Back

Front

0.0 1.0 2.0 3.0 mm

53

The effect of ply-angle misalignment has been studied in detail on in-plane

permeability of woven textiles components, but these studies have not addressed its

effect on the through-thickness permeability component [34,123,126,119]. The

relation between woven, random and stitched preforms and their effect on transverse

permeability has been studied both numerically and experimentally

[124,125,127,128,97,129,130]. Tahir M.A. et al. and Stylianopoulos, T. et al. stated

that transverse/ through-thickness permeability is independent of in-plane fiber

orientation [124,125]. Permeability studies for various preform configurations have

measured the through thickness component along with other components of the

permeability tensor [127,128,97,129]. Chen et al. [130] presented statistical analysis in

terms of inter-fiber spacing for through thickness permeability only for disordered

fiber arrays. For non-crimp fabric, Nordlund study [131] bears some similarity to our

approach but concentrated on in-plane permeability components, while Drapier [132]

did investigate the through-thickness permeability variations but dependent only on

the stitching density. The change in the transverse direction with ply-angle

misalignment is only investigated in terms of transverse matrix crack formation and

propagation [133,134]. The effect of ply angles on the through-thickness permeability

for unidirectional fabrics has not been addressed in the literature.

By accurately measuring this permeability component, a much more accurate

resin infusion prediction can be made in the through-thickness direction especially for

processes such as Vacuum Assisted Resin Transfer Molding Processes (VARTM) in

which a distribution media is placed on one side of the preform and in Out of

Autoclave processing as the resin flow is mainly through the thickness and plays a key

role in filling the empty spaces between the fibers [135–137].

54

3.1.2 Unidirectional fabrics and their orientation

Experimental characterization is conducted for the unidirectional glass fabric

with the areal density of 1397(+/-42) g/m2. As seen from Figure 3.2, the stitching of

the fabric (at the back) eliminates the nesting of the layers and because it is rather

sparse it does not affect the unidirectional characteristic of the fabric. The layers are

cut manually to stack up in desired orientations to build the required thickness. Figure

3.3 illustrates how each ply is rotated schematically to obtain the successive rotation

of the plies with a picture of the corresponding fabric layer.

Figure 3.3. Representation of the orientation of the plies

3.2 Through-thickness permeability characterization

3.2.1 Numerical Analysis

In this study the characterization of through-thickness permeability by

numerical analyses is performed using the commercial software ANSYS® Fluent Inc.

[138], implementing a mesh generated by the software, ANSYS® Gambit 2.4 [139].

The numerical study conducts a simulation of laminar viscous flow of resin through

55

the open regions of a unit cell of a preform created by stacking unidirectional fabric

layers in desired orientations. The unit cell preform, composed of open regions and

fiber tows, is represented in Figure 3.4. The fiber tows are modeled as a solid as the

permeability of the fiber tows is usually five orders of magnitude smaller than the bulk

permeability and can be neglected [140]. No slip boundary condition is applied on the

fiber tow surfaces. The boundary conditions of the unit cell model are defined for each

pair of parallel faces using the meshing software Gambit. An example of the

numerical model of the unit cell is shown in Figure 3.4.(a), in which the successive

unidirectional fiber layers are rotated by increments of 5 degrees. Thus the angular

difference between fiber orientations of tows in two successive fabric layers is five

degrees. The model was created with 6 layers for comparison with experimental

results as we used six layers in our experiments. For numerical parametric studies we

created a model with 10 layers of fabric in the through - thickness direction with 10

tows in the first layer of unidirectional fabric. In order to ensure equal spacing in

successive rotated plies, additional fiber tows were introduced in the unit cell. The

number of tows introduced will depend on the rotation angle as can be seen from

Figure 3.1. The mesh for the corresponding sample is presented in Figure 3.4.(b). In

order to ensure that the periodicity or periodic effect on all four faces (two x-z and two

y-z (as defined in the Figure 3.4) is satisfied, the unit cell dimension in x and y -

direction are incrementally increased until the through-thickness permeability values

(Kzz) converges. For numerical convergence, the cell dimension along in plane

direction (x and y direction) need to be increased as the rotation angle of the

successive unidirectional layer changes. The unit cell for the case with the largest

56

change in orientation (angle of 90 degrees from the adjacent ply) is studied for the

determination of the dimensions that satisfies numerical convergence criteria.

Figure 3.4. (a) The Gambit model with each successive layer rotated by five

degrees. b) Gambit mesh of the model with 1,968,652 elements and

484,911 nodes. The cut-out shows the mesh density

After the mesh with boundary conditions is generated using Gambit as shown

in Figure 3.4, it is exported to Fluent Inc. to be solved for the viscous flow within the

unit cell under a prescribed pressure gradient across the layers of the fabric as in

Figure 3.5.(a). The corresponding volumetric flow rates through the faces in the

through-thickness direction are obtained and one-dimensional Darcy’s law in the

through the thickness direction is used to find the through-thickness permeability, as

given in Figure 3.5.(b). The simulation results are presented and discussed in the

results and discussion section.

x

y

z

x

y

z

(a)

57

Figure 3.5. (a) Periodic boundary conditions to evaluate the permeability in z-

axis, (b) Evaluation of permeability in z-axis

3.2.2 Experimental Validation

For the experimental set-up, resin flow through fibrous preform is tracked

radially within a mold in all three directions simultaneously, yielding all three

permeability components from a single test [118]. During the experimentation of the

resin transfer molding process, stack of fabrics is placed between two horizontal plates

(top and bottom: 40 cm x 40 cm), as shown in Figure 3.6, that are separated by 6 mm

each with 96 electrical sensors in a radial configuration. Located in the center of the

bottom plate is an inlet hole, 6 mm in diameter, through which resin of known and

constant viscosity, µ ≈ 0.1 Pa.s, is injected applying a positive pressure of ∆P = 100

kPa.

x

y

z x

y

z

58

Figure 3.6. Experimental set-up: (a) Upper mold plate, (b) Lower mold plate, (c)

Mold assembly, (d) Resin flow through preform

Corn syrup is used as simulated resin because of its favorable characteristics; it

is a Newtonian fluid and nontoxic. As the fluid flows through the fabric in all three

directions, it wets the sensors, inducing a voltage drop that is subsequently recorded

by a LabVIEW™ data acquisition system. By having two sets of planar sensors, which

are separated by 6 mm in the z-direction, the flow can be tracked in the z-direction as

well, yielding experimental data for the through-thickness component of the

permeability tensor. A three dimensional numerical simulation of flow through

anisotropic porous fibrous media called Liquid Injection Molding Simulation (LIMS)

is used in which the permeability values are changed iteratively in a geometry, that is

(d) resin flow

(a) upper mold plate (b) bottom mold plate

(c) mold assembly

inlet

59

identical to the experimental mold, with numerical sensors placed at the same

locations as in the experiments until the residual sum of squares between the

numerical and the experimental arrival times converges to a minimum [118]. The

converged numerical values provide the three-dimensional permeability tensor for the

fabric lay-up which includes the through-thickness value, which is the main focus of

this study.

3.3 Results and Discussion

3.3.1 Experimental Study

Table 3.1 compares the numerical predictions with the experimental results for

five separate lay-ups of 6 unidirectional fabrics: (1) All plies are aligned along y axis

(zero degrees), (2) All plies are aligned and make an angle of five degrees with the y

axis (3) Successive plies are rotated in increments of five degrees, (4) successive plies

are rotated in increments of forty five degrees, and (5) successive plies are rotated in

increments of ninety degrees. For each case, the experiment was repeated three times,

from which the maximum standard deviation was observed to be 4.15 x10-12 m2 for

case 5 as shown in Table 3.1. As expected, the results for cases 1 and 2 are nearly

identical as they should be. This can be attributed to the fact that there is no relative

planar rotation of the plies with respect to each other. The planar permeability with

respect to the fixed coordinate system of the mold may change between cases 1 and 2

– and the skew permeability term (Kxy) should reflect the change - but the principal

through-thickness permeability should not be affected which, was confirmed with the

experiments. For case 3, the angle between successive layers was increased by five

degrees and this did not influence the through-thickness permeability in any

60

significant way. This answers one of the issues targeted by this study: Is a small

misalignment, such as caused by inaccurate lay-up, significant for the through-

thickness flow. Hence one does not notice any significant change in permeability due

to small misalignments. However, when we conducted the extreme case of 45-degree

and 90-degree increment between successive layers, there is a sharp increase in

transverse permeability, as one would expect due to lower resistance pathways straight

across the thickness that increases the permeability dramatically. Numerical

simulations at the unit cell level confirm this behavior.

Additionally, in Figure 3.7 numerical convergence study is conducted for

through thickness permeability values with increasing mesh density.

Table 3.1. Experimental and numerical comparison of through-thickness

permeability. Case 1 and case 2 of 0o and 5o refers to all six

unidirectional layers being aligned along those angles respectively. In

case 3, case 4 and case 5, the successive layers were rotated by 5o, 45o

and 90o degrees respectively.

Case 1 2 3 4 5

Experiment

Kzzx10-11 (m2)

Angle

0o

Angle

5

Increment

angle 5o

Increment

angle 45o

Increment

angle 90o

vf = 54 %

1 0.210 0.290 0.200 0.920 1.630

2 0.332 0.312 0.335 1.070 2.190

3 0.290 0.350 0.350 0.823 1.380

Average 0.277 0.317 0.295 0.938 1.733

Standard

Deviation 0.062 0.030 0.083 0.124 0.415

Numerical

Results 0.377 0.395 0.498 0.898 1.810

61

Figure 3.7. Numerical through thickness permeability with different mesh

element sizes for incremental rotation angle 5o

3.3.2 Parametric Study

After the numerical model is validated with experiments as shown in Table

3.1, the effect of ply angle is further investigated using the numerical approach only.

For this analysis ten layers of plies are stacked on top of each other to systematically

predict the effect of planar rotation between successive unidirectional plies. In order to

observe the effect of the cell configuration, for the numerical simulations the tows are

modeled using both square and hexagonal grid configurations

The permeability of a unit cell in which all ten plies were aligned along the

same angle was calculated for various angles and it was found that the through-

thickness permeability was not affected by changing the in-plane angle as long as the

62

unidirectional fibers in all ten layers were aligned in the same direction. This is

physically necessary as long as the through-the-thickness direction constitute the

principal direction of the permeability. In this case due to symmetry, this is true and

this result verifies the fact.

To explore the effect of the stacking sequence of ten layers on the through-

thickness permeability, a numerical unit cell was constructed in which one could vary

the fiber orientation of the unidirectional fabric in the successive layers by a fixed

number of degrees. Permeability in a total of 14 different unit cells in which the

angular in plane rotation of the successive plies was incremented by 0, 1, 2, 5, 10, 20,

30, 40, 45, 50, 60, 70, 80, and 90 degrees was predicted and is compared with zero

rotation (all layers aligned in the same direction) using square grids in Figure 3.8.

Figure 3.8 emphasizes the fact that larger the difference in the angular rotation of the

successive ply, higher is the through-thickness permeability of the fabric. Obviously, a

small change (such as one caused by lay-up inaccuracy) will only have a negligible

effect.

63

Figure 3.8. Effect on through-thickness permeability with increasing rotation

angle of the successive ply. The unit cell was created using the square

and hexagonal arrangement of the fiber tows in the unidirectional

ply.

It is also clear that the rotation effect is a non-linear one, as the probability of

higher permeability pathways increase with larger degree of rotations. Additionally

those 14 unit cell configurations are repeated for hexagonal unit cells and no

significant variations are observed with the square unit cells counterparts.

As the stacking sequence of ply orientation may vary depending on mechanical

property requirements, we explored if the order of the rotated plies would influence

the permeability value. Figure 3.9 represents the effect of the order of the ply rotation.

The first set of bars in Figure 3.9 shows the through-thickness permeability value for

12 plies in which each successive ply was rotated by 30 degrees. This configuration

was compared with 12 layers where layers with the same orientation were grouped

together (stacked as 0/0/30/30/60/60/90/90/120/120/150/150). These were also

64

compared with all plies aligned (no rotation) as the baseline case. This was repeated

for 45-degree and 90-degree increment of the successive plies, with number of plies

reduced to eight and six, respectively. By grouping the layers, the effective through

the thickness permeability drops significantly, though it is always higher than the

baseline case of the unidirectional sample with no rotation. This is shown in Figure

3.9. This result suggests the wider resin pathways are generated by successive rotation

of the plies. As seen from the results when layers of zero and ninety degree are

grouped together with a single angle change in the stack, the permeability of the

assembly is closer to the permeability of unidirectional layers without any rotation

(baseline case). The change in orientation between successive layers does influence

the transverse through-thickness permeability value even if the average orientation of

the stacked sequence is the same. Consequently, if a permeability value is to be

determined from component permeability, the number of crossovers at certain angular

difference (compared to total number of layers) should be taken into account. The

commonly used averaging scheme to find the transverse permeability component does

not account for this and will result in large errors in permeability predictions.

65

Figure 3.9. Effect on through-thickness permeability with increasing rotation

angle of the successive ply. The unit cell was created using the square

and hexagonal arrangement of the fiber tows in the unidirectional

ply.

3.4 Summary

The simplified numerical model that quantifies the effect of change of fiber

orientation direction in the successive ply of a laminate formed with unidirectional

fabrics on its through-thickness permeability has been created and compared with the

experimental results. The comparison was favorable and the model was used to

determine the dependence between the successive plies and through-thickness

permeability. Note, however, that we did not allow any nesting in our model between

the adjacent layers, as the stitching pattern essentially prevents it. For a different

material this may not be true.

The experimental and numerical results demonstrate that as the angle between

successive plies increases, the permeability in the through-thickness direction

66

increases in a non-linear fashion. Small deviations from alignment (such as five-

degree rotation between successive plies) did not noticeably affect the through-

thickness permeability, showing that lay-up inaccuracy is not significant; at least as far

as through-thickness flow properties are considered.

For higher angular rotation, experiment and model agree that the permeability

increases significantly in strongly non-linear fashion. This may be explained by the

architecture of the fiber layout. As the fiber alignment between plies decreases it

changes the layout of empty spaces, creating lower resistance pathways between

successive layers and increasing the bulk through-thickness permeability of the

laminate. Second important finding is that the grouping and order of the rotated plies

(stacking sequence) influences the through-thickness permeability value. Note that the

determination of this grouping is not for processing engineers to decide. It is dictated

by structural needs. The latter finding leads us to hypothesize that, in order to build a

successful through-thickness permeability model based on the through-thickness

permeability of unidirectional plies one must also account for the number and relative

angle of layer contacts into the model.

67

Chapter 4

CHARACTERIZATION OF LOCAL VARIABILITY OF FABRICS

4.1 Introduction

Local non-homogenous architecture of the fabric can have a noticeable effect

on the dynamics of resin flow behavior. There is no available standard characterization

method to characterize the fabric variation and defects from the observed variation in

the flow front motion. Thus, the objective of this chapter is to present a quantitative

way to characterize the local permeability variation of a fabric by monitoring the flow

front movement.

To study the effect of local permeability variations on the global permeability,

de Parseval et al. [141] simulated one-dimensional flow with stochastic and regular

local permeability variations. They observed that the global permeability is the spatial

harmonic mean of the local permeability values. Padmanabhan and Pitchumani [31]

performed stochastic analyses of non-isothermal injection processes based on

simulation of one-dimensional flow. For a rectangular mold with linear injection gate,

Sozer [32] simulated two-dimensional flow applying local random permeability

variations to observe the effect of preform non-uniformity on mold filling. Random

variations of the local permeability of up to +/- 35% were reported to have no

significant effect on the flow pattern, while variations in a specific pattern caused a

more significant effect on the mold filling results. Using a similar approach for

studying global permeability variations, Desplentere et al. [33] assigned local

permeability values for injection simulation not only randomly, but also imposed a

68

correlation between the properties of adjacent material zones along the principal flow

direction. It was found that for random assignment of local permeability values to

discrete material zones, the variation of global permeability values was influenced by

the size of the zones. For correlated local permeability values varying only along the

principal flow direction, the results for the global permeability were in agreement with

the observations of de Parseval et al. [141]. Lundstrom et al. [142] determined non-

uniform local permeability values from the dimensions of flow channels with variable

widths between the fiber tows. For a completely random distribution of the local

permeabilities, they found that the global permeability decreased with the maximum

variation at the unit cell level, while for a correlated distribution, the global

permeability could either increase or decrease.

The current state of the art to measure permeability of a fabric in a certain

direction assumes that the fabric is uniform and hence the permeability is spatially

uniform. Those assumptions allow one to conduct a one-dimensional experiment as

shown in Figure 2.2 and from the flow rate pressure drop relationship obtain an

averaged value of permeability in that direction. If one wants to find the principal

permeability values in the plane of the fabric, one would conduct a radial experiment

and from the elliptical spreading domain and the flow rate pressure drop relation one

can calculate the in-plane permeability tensor as shown in Figure 4.1 [118].

Quantitative evaluation of injection experiments, which is normally based on flow

front tracking, implies averaging of local variations in material properties and

measuring averaged global permeability values [29,30,34]. While the experimentally

determined permeability values characterize quasi-uniform materials, the accurate

69

predictive description of global flow for non-uniform materials requires knowledge of

the distribution of local properties.

Figure 4.1. Radial injection and permeability tensor characterization: (a)

Schematic of flow front in an anisotropic fabric at a time step with

the principle direction 𝐱’𝐲′-axes, (b) Radial injection inlet gate and

resin propagation, (c) Permeability tensor. 𝐊𝐱𝐲 is non-zero as the

principal axis do not align with the selected coordinate axis

In this chapter, emphasis will be on the characterization of the variation of the

permeability within the fibrous preform. The goal is to determine the variability of the

preform before it is used in the manufacturing process and measure the permeability

with its variations so it could be included in the process design. This characterization

will be based on the mathematical descriptions derived from surface growth equations

and interface/flow front analysis.

(a) (b) (c)

70

4.2 Mathematical Implementation

In order to generate a model to characterize the local variation of the fabric and

preform, the surface growth equations that provide mathematical description for

disorderly surface growth in random media is adopted [143].

Growth phenomena can be divided into two groups based on the driving

factors: local interactions and non-local interactions. Examples of local interactions

are spreading of fire and fluid flow in porous media, whereas examples for non-local

ones are formation of the snowflakes and metallic dendrites [144]. The local growth

can be formulated by what is known as the Kardar-Parisi-Zhang (KPZ) equation:

∂h(r⃗, t)

∂t= ν∇2h(r⃗, t) +

𝜆

2[∇h(r⃗, t)]2 + F + η (4.1)

where h(r⃗, t) is the height of the variable which depends on position and time as

shown in Figure 4.2 for fluid flow through porous media, ν and 𝜆 are constants, F is

the driving force and η is the white noise in the system.

Figure 4.2. Flow front locations (height 𝐡(�⃗�, 𝐭)) at various times with system size

L, and mean height (flow front position) �̅�

71

Thus, the flow through porous media can be defined as local interaction on the

macro scale in which certain universality can be obtained. Then, the surface growth in

random media forms self-affine shapes of interfaces. In order to define that self-

affinity the local height h(r⃗, t) is obtained as the flow front propagates, then interface

width, w(L, t) which is the variance of the h(r⃗, t) on a flat surface over the inlet gate

length, L, can be defined as

w(L, t) = √1

𝐿(ℎ(𝑖, 𝑡) − ℎ̅(𝑡))

2 (4.2)

where h(i, t) is the local height at a specific location and h̅(t) is the averaged value of

the h(i, t) (Figure 4.2).

Figure 4.3. (a) Change of the interface width with time (logarithmic scales for

both axes) for a fixed L value , (b) Growth of the interface width with

different system sizes (L). Reprinted with permission from [143]

(a) (b)

L increases

wsat

wsat(L1)

wsat(L2)

wsat(L3)

wsat(L4)

72

Typical plot of time evolution of the interface width has two regions (in Figure

4.3.(a)). First, interface width w(L, t) increases linearly with time in logarithmic scale

until the cross-over time, tx. The slope in the first region characterizes the time-

dependent dynamics of roughening, β and is called the growth exponent. Then at time

tx it reaches saturation point when the variance, w(L, t) reaches its saturation point,

wsat(L). As seen in Figure 4.3.(b), the saturation point, wsat(L) increases with system

size, L. Thus, self-affinity of the interfaces can be defined under the universality class

of KPZ using the following power law relationship [145].

w(L, t)~tβFw(Lt−1/γ)~ {tβ for t ≪ tx

Lα for t ≫ tx (4.3)

with Fw is a scaling function. β is the growth exponent and α is the roughness

exponent. The roughness exponent, α characterizes the roughness of the saturated

interfaces at different system sizes, L, (in Figure 4.3.(b)) with the relationship given in

Equation (4.3).

Moreover, cross-over time, tx is dependent on the system size, L as follows:

tx~Lα/β (4.4)

where α/β is called dynamic exponent, γ.

For phenomena such as paper burning, fluid flow through fibrous media and

two-phase viscous flows, the interface fluctuations can be measured experimentally or

numerically to determine the exponents α and β. For example- for flow through

fibrous porous media as the resin front advances the fluctuations of the flow front

73

shape can be visualized and recorded at different time steps as seen in Figure 2.2 and

using the flow front shapes with time data in Equation (4.2) and Equation (4.3) one

can determine the growth exponent, β and roughness exponent, α.

For different applications, the exponents may be different; however, there is

universality/consistency of those exponents despite the randomness of the medium

[146–148]. Thus, the above theory could be used for the characterization of the

morphology and the dynamics of the growth/propagation of the resin through the

fibrous preform. A conclusion can be formed from the investigation of the three

exponents listed above, α, β, and γ in terms of the variation of the permeability values

within the fabric via the monitoring of the interface/flow front position data with time.

These values can then serve as the indicator of the homogeneous nature of the fabric

from manufacturing viewpoint.

By monitoring the flow front progression as the resin impregnates the fibrous

media, we will describe the flow-front progression in “1D” flow, measuring the

(h(i, t)) values and calculating w(t, L) using Equation (4.2) at a series of time steps

and then determine the exponents α and β using Equation (4.3). These exponents are

directly related to the permeability variations/defects of the fibrous preform.

To correlate the fabric permeability variation with the two exponents’ α β we

need to simulate the effect of a series of known permeability distributions on flow

front progression variations which will allow us to calculate the exponents using

Equation (4.2) and Equation (4.3). The flow with known permeability

disturbances/distribution is simulated using Liquid Injection Molding Simulation,

LIMS [23,149]. In LIMS, every element in the finite element mesh can be assigned its

own permeability. Thus by populating the elements with a selected permeability

74

distribution, the effect of that distribution on flow front variation can be tracked to

measure h(x, t) and evaluate w(t, L) (Equation (4.2)) at each time step. Equation (4.3)

can then be used to find the three exponents. This could be repeated for various

selected permeability distributions to establish a correlation between the coefficients

and the variations in the permeability distributions.

As shown in Figure 4.4, in the LIMS mesh a distribution of permeability

values with a selected standard deviation with upper and lower limits are generated

and assigned randomly to each element in the mesh mimicking the variation one may

expect due to manufacturing of such fabrics. The one-dimensional flow is simulated

by introducing the resin along one edge of the mold. LIMS can capture the variations

in the flow front as the flow progresses from the inlet to the vent. Figure 4.4 shows an

example of LIMS simulation results in flow front interfaces at different time steps.

Thus, those flow front locations are used to obtain the interface width in Equation

(4.2), which will be then be used for the calculations of the exponents stated in

Equation (4.3).

75

Figure 4.4. Top: LIMS mesh and random permeability assignment, Bottom:

flow front progression with time obtained via LIMS

Additionally to characterize the physical defects of the fabric, the variation of

the permeability is assigned based on the solution of the Poisson’s equation with

Dirichlet boundary condition as stated in Equation (4.5).

∆K = Q in Ω , where K = Kbase on ∂ Ω (4.5)

Inlet

(constant

flow rate)

: random permeability values

i: numer of elements

t1

LIMS simulation

76

On the boundaries the permeability is assigned as the global (constant)

permeability and the defect/high permeability is placed in the middle and reduces

towards the edges as controlled by the parameter Q, which represents the size of the

defect. As Q increases, the variation of the permeability in the zone increases. Then,

Equation (4.5) is solved to obtain the variation of the permeability for the defected

zone (as shown in Figure 4.5), which represents the permeability data for the defected

zone. In Figure 4.5, the preform is divided into 4 x 4 equal to 16 zones of which four

zones are randomly selected as defective zones. In these defective zones, permeability

values are defined by the solution of Equation (4.5) and rest of the zones are assigned

constant permeability value equal to Kbase. This analysis can be used to determine the

extent of the defect within the fibrous domain.

Figure 4.5. Assignment of the variation of the permeability of the defected

zones: left: 25% defective sample, right: variation of permeability

within the defective zone obtained from solution of Equation (4.5).

Permeability is higher in the center of the zone and reduces to the

values prescribed at the edges as described by the parameter Q in

Equation (4.5).

77

4.3 Experimentation

Once the correlation between fabric permeability variation and flow front

variation is established with a series of LIMS simulations, one can experimentally

determine the three exponents for an actual fabric and using the correlation determine

the variability in permeability of the preform.

In order to visualize the movement of the flow front through the fibrous

medium with time, the one-dimensional test set up as shown in Figure 4.6, is created.

After the preform is placed on the acrylic table it is sealed with a vacuum bag and the

resin at atmospheric pressure is introduced from one end through a line gate while

drawing a vacuum at the other end. The resin impregnates into the preform due to the

pressure gradient of one atmosphere and the flow front movement is captured via the

flow visualization camera system along with the time stamp. Due to its Newtonian

characteristic, corn syrup with dark cloth dye and water is used as the simulated resin

to create a clear contrast between the dark resin front and the white fibrous porous

media. Using the set-up, the goal is to measure the variations at the flow front and use

the KPZ evaluation scheme to determine the growth exponents which have been

correlated to the permeability variation with the use of numerical simulations. This

allows us to directly relate the variations of the flow front to the variation in the

permeability of the porous media and identify the presence of defective zones in the

preform.

To validate identification of the defective zone and its quantification within the

fiber preform, 25 cent coins were placed within the fibrous media to simulate the

defective zones. The presence of coin within the layers will increase the permeability

in that zone due to imperfect fit. Eight layers of plain weave E-glass fabric (Figure

1.2.(a)) were used as a preform with in-plane permeability value, Kbulk equal to 8.43e-

78

11 m2. The resin was introduced from the line gate with the vacuum applied at the

other end. From Figure 4.7, the effect of randomly distributed and evenly distributed

defects can be observed on the progression and variation of the flow front profiles.

Figure 4.6. Flow through porous media experimental set-up with flow

visualization

79

Figure 4.7. Resin flowing into a fibrous preform with 25 cent coins placed inside

the fabric to simulate defective regions. On the left the defects were

evenly distributed on the right the defects are randomly distributed.

Measured experimental flow front profiles are also shown (flow front

contours at Δt = 25 seconds)

80

4.4 Results and Discussion

4.4.1 Characterization of permeability variation

To correlate the growth exponent to the permeability variation, the growth

exponent, β for three different permeability data sets as shown in Figure 4.9.(b) with

standard deviation for each distribution is shown in Figure 4.9.(c) were generated. The

values from each distribution set were assigned randomly to elements of the LIMS

mesh. Figure 4.8.(a) show flow front fluctuations for three different permeability

distributions shown in Figure 4.8.(b). Figure 4.8.(c) plots the change in the interface

width w(t) on a logarithmic scale with time (As given in Equation (4.3), the slope of

the log(w(t)) vs log (t) where t >> tx, will provide the exponent β. The LIMS time

and position data is transferred to a MATLAB script that calculates the w(t) and

generates the plots. As it can be seen from the plot in Figure 4.8.(c) the slope β is the

same for all the three different standard deviations. Its value is 0.77 ± 0.03 which

matches with the value reported in the fluid through fluid flow example analyzed in

random domain work [16]. These tests are repeated for different mesh sizes, different

permeability ranges, different inlet pressure values and β value was found to be

invariant.

81

Figure 4.8. Characterization of the growth exponent: (a) Shape of flow front at a

time instant, (b) Bell curves with three different standard deviations

selected for the permeability values assigned in LIMS, (c) Change of

the variance of the interface with time from the simulated

experiment with permeability distributions shown in (b)

In order to calculate surface roughness exponent, β, the simulated experiments

with LIMS are repeated for different inlet gate sizes, L (Figure 4.4) and the saturation

of the interface width is recorded. Using the relationship provided by Equation (4.3)

which states wsat(L)~Lα, α is the slope of the plot log (wsat) vs log (L) . The

calculations are performed for both anisotropic and isotropic fibrous porous media.

For anisotropic case, the permeability from the permeability distribution is only

assigned to the permeability component in the flow direction and the permeability

component in the perpendicular direction is held constant at 1.0e-12 m2. For the

isotropic case the perpendicular and parallel permeability components are equal for the

same element and randomly assigned to each element. The results for α are tabulated

(a)

(b) (c)

82

in Table 4.1 and for both anisotropic and isotropic cases change is observed with

different standard deviation values under the permeability range values of 1.0e-14 to

1.0e-8 m2. The α values are observed to decrease for both isotropic and anisotropic

cases as the standard deviation for the generated permeability distribution function

increases. Thus this decrease enables one to characterize the variation within the

preform in a quantitative manner. Higher the value of α, lower will be the permeability

variation within the preform.

Thus the Kardar-Paris-Zhang (KPZ) equation that models the surface growth

on random media can be adopted to characterize the variation of the permeability

within the preform.

Table 4.1. Characterization of the roughness exponent

Standard

Deviation

(m2)

𝛂 (𝐀𝐧𝐢𝐬𝐨𝐭𝐫𝐨𝐩𝐢𝐜 ) )

K⊥ = 1.0e − 12 m2

K∥ = 1. e − 11to 1e − 9 m2

𝛂 (𝐈𝐬𝐨𝐭𝐫𝐨𝐩𝐢𝐜 )

K⊥ = K∥ = 1.0e − 11 to 1e − 9 m2

2.0𝑒 − 11 0.72 0.57

5.0𝑒 − 11 0.67 0.48

2.0𝑒 − 10 0.52 0.42

Thus, the characterization of the local variation in the preform can be

determined by the change in the roughness exponent, α which is related to the

standard deviation in the fabric permeability variation but with no change in growth

exponent, β.

83

4.4.2 Characterization of the defects within a fabric

To characterize how the roughness exponent α and growth exponent β are

related to the size of the defect Q and the percent of the defect, m a numerical study

was performed via LIMS to obtain β and α at 4 different Q values (100, 75, 50 and 25)

and for 65 different percentages of defect, m, from 1% to 65%. For each Q and m

value the numerical analysis is repeated 5 times and the average value is used. The

domain was divided into 100 zones and for each Q and m value, defective zones are

randomly located and permeability variation in that zone is assigned using Equation

(4.5) with Kbase value of 1.00e-10 m2. The results are presented in Figure 4.9 and

Figure 4.10 for β and α, respectively, and an exponential fit with the equation is

obtained for each Q value. Thus through these set of simulations, strength of the

defect, Q and percentage of the defect, m are related to the growth, β and roughness, α

exponent. The analysis are not performed for m values larger than 65 % because a

region that has over 65% defects can be detected visually and does not need to be

analyzed. Also we found that the exponents calculated for regions over 65% were not

reliable as they showed very large degree of fluctuations.

84

Figure 4.9. Change in growth exponent, 𝛃 with increasing percentage of

defective zones (m) for different degree of defects, Q. A best fit

functional relationship is also plotted

85

Figure 4.10. Change in roughness exponent, 𝛂 with increasing percentage of

defected zones, m, for different degree of defects, Q. A best fit

functional relationship is also plotted.

Using the data presented in Figure 4.9 and Figure 4.10, which provide the

growth exponent and roughness exponent, respectively, a constitutive equation to

describe Q and m as functions of β and α, was formulated as shown below;

Q(β, α) = 125. ln(β. α) + 206 (4.6)

m(β, α) = 91.07 ln(β) − 51.79 ln(α) + 60.78 (4.7)

It was hypothesized that if there was a large variation in the permeability value

in a small region, one could capture that with change in the growth and roughness

exponents. To explore this, two types of experiments were conducted in which the

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local change in permeability was introduced by (i) placing 25 cent coins and (ii) tacky

tape within the preform respectively as shown in Figure 4.11. The goal was to relate

the change in exponent value with the percentage of the porous media in which the

permeability change was significant. The experiments are performed by placing

quarters and tacky tape at the center of the zone. In Figure 4.11, 48 quarters and 48

tacky tape pieces are randomly placed in total of 128 zones which represents a

preform with 37.5% defective zones for both cases. As shown in Figure 4.11, the flow

front profiles with time intervals of 25 seconds are obtained via VARTM set-up

(Figure 4.6). Those profiles are used to obtain β and α values. For defects introduced

with coins the β and α values were 0.5537 and 0.5621 respectively, and for defective

zones due to tacky tapes, the β and α values are 0.5271 and 0.5261, respectively.

When these values were substituted in Equation (4.7), the percentage of defective

zones calculated was 36.78% and 35.72% for the coins and tacky tape respectively

which was not far from the assigned 37.5%.

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Figure 4.11. Defect tests via VARTM with 37.5% defect and flow front profiles

(Δt = 25 seconds), left: quarters right: tacky tape to represent the

defective zone.

4.5 Summary

In this study the variation of the preform permeability is characterized via KPZ

formulations. Besides the growth exponent β belonging to the universality class of

KPZ equations, by using roughness exponent α one can determine the variation of the

permeability within the preform by recording the variations in the flow front profiles.

Additionally, this study enables the determination of the percentage of defects within a

preform, which is useful information to reduce variability due to material defects in

composites processing.

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Chapter 5

OPTIMIZED DISTRIBUTION MEDIA LAYOUT

5.1 Introduction

The flow patterns during filling may vary from part to part due to the

variability associated with the material, part geometry, and lay-up of the assembly,

which may result in race-tracking channels. The process is considered as reliable and

robust only if the resin completely saturates the preform despite changing filling

patterns caused by flow disturbances.

5.2 Flow Control Mechanisms for Flow Through Fibrous Domain

The resin flow pattern can be manipulated with a tailored DM layout as it does

impact the flow patterns significantly. The continuous DM layout over the entire part

surface works well for very simple geometries with no to little potential for

race-tracking along the edges. In this study we address complex cases, which require

placement of an insert within the assembly, which will introduce race-tracking along

its edges, and hence uniform placement of DM over the entire top surface will fail to

yield a void free part. We introduce a methodology using a predictive tool to design an

optimal shape of DM, which accounts for the flow variability introduced due to

race-tracking along the edges of the inserts. This iterative approach quickly converges

to provide the placement of DM on selective areas of the preform surface that ensures

complete filling of the preform despite the variability. This approach has been

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validated with an experimental example and will help mitigate risk involved in

manufacturing complex composites components with Liquid Molding.

In order the show the effect of the DM placement on top of the preform, a

numerical simulation via Liquid Injection Molding Simulation (LIMS) [23] (See

Section 1.3 for LIMS introduction) for VARTM and SCRIMP are performed and time

contours are compared (Figure 5.1). The simulation is performed for the cross-section

of the preform in the through-thickness direction. The fill time with the VARTM is

751 seconds and for the same configuration with the DM on 95% of the top layer

reduces the fill time to 378 seconds. The use of DM decreases the fill time because the

resin first flows through the DM layer and then resin is infused in the through

thickness direction [150,151]. Thus, SCRIMP eliminates the disadvantage of VARTM

in terms of fill time. However, SCRIMP might yield formation of voids (especially at

the leading edge of the flow front) and for complex geometries and parts containing

impermeable inserts [152].

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Figure 5.1. Fill time contours for a VARTM and SCRIMP

5.3 Methodology and Implementation

First, given the part geometry along with insert locations, one must identify all

possible race-tracking locations within the part and create possible scenarios, which

take into account all possible permutations of race-tracking that may occur during the

impregnation process. To determine a single optimal layout of DM for all these

possible scenarios, a discrete optimization method is adopted. Discrete zones are

generated by dividing the surface of the preform (where one places the DM) into a

finite number of regions. Optimal solution finds the regions where one should place

the DM such that for all scenarios the resin will arrive at the vent last after having

impregnated the entire preform. Mathematically, this is done by prescribing the cost

function that will minimize the region with no empty region or voids in the mold.

Evaluation of the cost function is performed with an existing numerical simulation

inlet vent

Distribution Media

SCRIMP

VARTM tfill = 751 sec

tfill

= 378 sec

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called Liquid Injection Molding Simulation (LIMS) [45] which simulates the flow in

any complex geometry in Liquid Molding. LIMS output provides the empty region

after each fill based on the inputs of preform and DM permeability, race-tracking

strengths along the edges and predefined inlet and vent locations.

5.3.1 Discrete Optimization

Gradient descent method is a first-order algorithm to find a local minimum of a

function. The method starts with an initial guess of the solution and the gradient of the

function at that point is evaluated. The solution is stepped in the negative direction of

the gradient and the process is repeated until the algorithm converges to a zero

gradient. This method works for the objective functions for which the gradient can be

evaluated. If the variables used in the objective function are only a finite or discrete set

of values, discrete optimization should be applied. The discrete optimization problem

can be defined as a set, S of finite possibilities that satisfies the objective function. The

objective function provides local minimum, xopt for all elements of the set S,

𝑓(𝑥𝑜𝑝𝑡) ≤ 𝑓(𝑥) for all 𝑥 ∈ 𝑆 (5.1)

Discrete optimization can be used for different problems such as VLSI layouts,

robot motion planning, test pattern generation, and facility location [153].

5.3.1.1 Tree Search Algorithms

The search of the optimal solution with discrete optimization consists of

computationally expensive problems. As Koft [154] states there are two parameters

that indicate the complexity of the searches: the branching factor of the problem space

and the depth of the solution to the problem. The branching factor represents the

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number of the new states that are generated and analyzed at each depth. The depth of

the search is the distance between the initial node and goal node/s. There are two basic

tree search algorithms; breadth-first search (BFS) and depth-first search (DFS). In

Figure 5.2.(a), a sample tree layout is given and there are two goal nodes satisfying the

objective function, H and T. BFS expands all the states one step away from the initial

state until a goal state is reached and converges to node H, before node T (Figure

5.2.(b)). DFS explores a path all the way to a leaf before backtracking and exploring

another path. Therefore, only path of nodes from the initial node to the current node

must be stored in order to execute the algorithm. DFS will find the node T, before the

node H (Figure 5.2.(c)) [155].

Figure 5.2. Tree search algorithms (a) example problem with two acceptable, H

and T, nodes, (b) Breadth-first search: finds node H, (c) Depth-first

search: finds node T

As the BFS converges, the solution at the minimum depth is found (Figure

5.2.(b)). When a DFS succeeds, the solution may not be at the minimum depth (Figure

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5.2.(c)). Thus, for a large tree, BFS may have large memory requirements and DFS

convergence may take a long time to reach the solution, goal node. For the solution for

DM layout design the minimum depth is not a concern but the memory is. So, DFS is

adopted by generating regions on the preform which represents the finite solution set

and the cost function is the unfilled area. Different race-tracking (RT) scenarios yield

different unfilled regions and our goal is a DM layout design that results in a

successful filling without voids for all possible race-tracking scenarios. Thus, the

approach adopted is to first investigate all possible RT scenarios and implement the

DFS algorithm to find a DM layout solution for the worst filling RT scenario defined

by largest unfilled region. Then, this updated DM layout solution is used for the

remaining cases of the RT scenarios and new worst filling RT scenario is identified. If

this new worst filling satisfies the acceptable tolerance, the algorithm stops. If not, the

DM layout is updated (superimposing on the DM layout from the previous iteration).

This procedure is repeated until the layout satisfies the filling of the mold within an

acceptable tolerance for all permutations of RT cases with the DFS algorithm. This

methodology is explained in more details in the next sub-section with an example

problem.

5.3.2 Pedagogical Example

In Figure 5.3, a pedagogical example is presented to explain the methodology.

For specified inlet and vent locations, three different race-tracking options are

identified, the permutations of which could result in 23=8 different race-tracking

scenarios (Figure 5.3.(a)). In the first step, all possible eight scenarios are simulated

without the use of DM with LIMS and the worst case of filling is identified by the

maximum percentage of unfilled region (dry spots or voids) by halting the simulation

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when the resin arrives at the selected vent. Unfilled region is calculated from the

percentage of number of empty nodes to total number of nodes. For this example

“Case 8” resulted in largest unfilled region (14.1%- worst case), as shown in Figure

5.3.(a). The methodology suggests finding the DM layout for the worst case “Case 8”

first. DM layout solution is found by dividing the domain into six areas and

conducting six separate simulations with DM placed in each of the six regions on top

of the preform successively and the percentage of unfilled region is recorded for each

placement as shown in Figure 5.3.(b). As seen from Figure 5.3.(b), by placing the DM

on the bottom right corner provides the best filling option (0.1% void) out of the six

configurations and this DM configuration will result in successful mold filling. Mold

filling is considered to be successful even when the dry region is typically below a

tolerance limit usually about 1 to 2%. This is because even after the resin reaches the

vent if one allows the resin to bleed for a short time before closing the vent it may

reduce the dry region to less than 1% and in some cases if the void is close to the vent,

it will be flushed out with the resin. Next, this DM layout that provides successful fill

for “Case 8” is applied to the remaining seven scenarios and it is found that the “Case

1” which has no race tracking along any edge has one of the largest unfilled region

(11.5%) as shown in Figure 5.3.(c) and is the worst case out of the remaining seven.

In order to find a successful filling solution for “Case 1”, the DM design from

the previous solution is retained and the approach to conduct six simulations with an

additional DM patch placed in each of the six areas successively is repeated and the

resulting unfilled area percentages are recorded. It should be noted that the 5th

configuration represents placement of 2 layers of DM on the bottom right corner. The

best filling solution for the “Case 1” results in 7.3% voids, which is greater than the

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tolerance (Figure 5.3.(d)). Therefore, the successive placement of DM in the six

regions is repeated for “Case 1” while retaining both the DM patches placed in the

earlier trials (Figure 5.3.(e)). This trial results in zero voids and successful filling for

the case in which the DM is placed in the middle of the top half while maintain the

two DM layers from the previous trials as shown in Figure 5.3.(e). Finally, the updated

design, which resulted in no voids for “Case1”, is used to perform mold filling

simulation for the remaining seven cases and the results are shown in Figure 5.3.(f),

which shows that the worst case scenario is “Case 8” with void region of 0.5% which

is within the tolerance limit. Thus, this represents the final DM layout design which

will provide successful resin impregnations for all 8 scenarios possibly expected

during manufacturing with the specified locations for the gate and vent. For this

example and for this gate and vent location we were successful in finding the DM

layout, which worked for all 8 cases. However, if this was not the case, one would

continue with this algorithm of adding a region and testing the cases until all six

regions were covered with DM. After covering all the regions if still one could not

find a solution, then the number of regions is increased from six to eight (or ten or

twelve) and the algorithm is repeated. If the gate and vent location are changed, one

would expect the DM design to change as well.

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Figure 5.3. Example to explain the methodology to determine the optimal DM

design using the DFS discretization method

2

1

4

3

6

5

Generate 6 regions

Case8

Worst case

2

1

4

3

6

5

2

1

4

3

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1

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3

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5

6.7% void 16.9% void 5.2% void 9.7% void 0.1% void 9.5% void

Bestfilling≤Tolerance

no void 7.5% void 4.4% void 0.1% void

23 RT cases 2

1

4

3

6

5

Case1 Case2 Case3 Case4

Case5 Case6 Case7 Case8

11.5% void 7.5% void 5.3% void 11.5% void

Worst case

Case1

Worst case

2

1

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10.5% void 8.3% void 28.0% void 9.1% void 10.2% void 7.3% void

Bestfilling≤Tolerance

23 RT cases 2

1

4

3

6

5

Case1 Case2 Case3 Case4

Case5 Case6 Case7 Case8

no void no void no void no void

no void %0.2 void no void %0.5 void

Bestfilling≤Tolerance

Case12

1

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6.3 void % 3.9% void 24.0% void no void 6.9% void 7.5% void

Worst case

2

inlet vent

1 323 RT cases

Case1 Case2 Case3 Case4

Case5 Case6 Case7 Case8

4.9% void 1.9% void 0.2% void 4.9% void

no void 1.9% void %2.3 void 14.1% void

(b)

(c)

(d)

(e)

(f)

(a)

Best filling ≤Tolerance

Best filling ≤Tolerance

Best filling ≥Tolerance

97

5.3.3 Algorithm for Optimum DM lay-out

In the previous section, the methodology uses six regions to seek the optimal

solution. However, one may not converge to a solution with the given regions. Thus,

the adapted algorithm is developed not only to find an optimized DM layout for all

possible RT scenarios for a given region but also to be able to update the number of

the regions, if necessary. In Figure 5.4, the flowchart of the algorithm is presented.

First, the domain is divided into 2n regions and solution is sought as explained in the

pedagogical example. At the end of the Discrete Optimization (DO) routine the filling

of the best case is selected and if that filling still has higher percentage of voids than

the prescribed tolerance limit; the number of DM regions is increased by 1 and DO is

repeated until the voids percentage is within the tolerance limit. If the number of DM

regions (m) is equal to the regions of the domain (2n) and still the voids percentage is

more than the tolerance limit, then the number of regions in increased by a factor of 2

and the entire cycle is repeated as shown in Figure 5.4.

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Figure 5.4. Flow chart of the algorithm to obtain optimal DM

5.3.4 Partition method

The presented methodology optimizes DM layout on a discrete domain. The

discrete domain is generated by dividing the domain, the top surface of the preform

where the DM is to be placed, into finite number of regions. This is accomplished by

Matlab® built in k-means++ algorithm [156]. The k-means clustering is a partitioning

99

method. The coordinates of the domain forms the data set and clustering divides the

data into k clusters and indexes the cluster. The objective is to have the points in the

same cluster being as close as possible to each other and as far as possible from points

in other clusters. It is an iterative method that minimizes the sum of the distances from

each object to its cluster centroid. Figure 5.5 presents an example region for the given

domain.

Figure 5.5. Division of the domain with the built in k-means script in Matlab

5.4 Experimentation

After the optimum DM layout is obtained via the proposed algorithm, one can

experimentally test the design. In order to visualize the movement of the flow front

through the fibrous medium with time and observe the filling of the preform, the same

test set up in Figure 2.2 except the additional camera system that monitors the bottom

layer is used. After the preform along with a steel insert is placed on the acrylic table,

it is sealed with a vacuum bag and resin at atmospheric pressure is introduced from

one end through a line gate while drawing a vacuum at the other end. The resin

propagates within the preform due to the pressure gradient of one atmosphere and the

flow front movement is captured via the flow visualization camera system along with

the time stamp.

2 regions

4 regions

6 regions

8 regions

100

In this experiment due to its Newtonian characteristic, corn syrup with dark

cloth dye and water is used as the simulated resin to create a clear contrast between the

dark resin front and the white glass fibrous porous media. The experiments are carried

out using 8 layers of 50cmx50cm Plain Weave E-glass and distribution media made of

polypropylene. The impermeable metal 1mm thick square metal insert (20 cm x 20

cm) is placed in the center of the 4th layer (Figure 5.6.(a)). After adding the remaining

4 layers of fabric, the DM layer is placed on the top and the preform is sealed and

vacuum is applied (Figure 5.6.(b)). Using the set-up, the flow front positions along the

top and the bottom are recorded with time stamps. Possible race tracking can occur

along the insert edges as the preform may not completely close the gap around the

insert. The material properties of the resin, fiber preform and the DM were measured

and are listed in Table 5.1 [157].

Figure 5.6. (a) 4th layer of the E-glass with metal insert placed in the center of

the fabric, (b) Experiment layup under vacuum

(a) (b)

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Table 5.1. Properties of E-glass fabric, DM and corn syrup

Parameter: Numerical value:

Density of E-glass: 2500 [kg/m3]

Fiber volume fraction of E-glass: 50%

Permeability of E-glass: Kxx=8.32e-11 [m2]

Kyy=5.88e-11 [m2]

Kzz=3.49e-12 [m2]

Density of DM: 946 [kg/m3]

Fiber volume fraction of DM: 15%

Permeability of DM: 3.46e-09[m2]

Viscosity of corn syrup 100 [cP]

5.5 Results and Discussion

5.5.1 Experimental Validation

The methodology presented in section 2 was applied to the mold geometry

presented in Figure 5.7 to find the DM layout to be placed on top of the preform so no

voids are created. The mesh representing the preform domain is shown in Figure

5.7.(a) with the line inlet at the left side and vent on the right side with 4 race-tracking

possibilities along the four edges (24 = 16 scenarios). Then the algorithm presented in

Figure 5.4 is executed for placement of DM. The algorithm was not successful in

finding a DM layout which would result in percentage of voids below the prescribed

limit with 2 regions, 4 regions and 6 regions (Figure 5.5), respectively, but the

algorithm converged to a successful filling solution with 8 regions as shown in Figure

5.7.(b). The algorithms finds the optimal DM layout design given in Figure 5.7.(c) as a

C-shape DM layer to be placed on the left side of the preform that provides successful

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filling for all 16 different possible scenarios. To arrive at an optimum layout for this

geometry, the algorithm executed LIMS simulations of 16 scenarios four times to

arrive at use of 8 regions and then had to execute 5 iterative LIMS simulations for the

placement of DM on 8 defined regions. Thus the total number of simulations executed

to arrive at the optimal DM layout were 104 (64+40) in 18 minutes on a PC computer

with the tolerance of 2% voids (unfilled volume).

Figure 5.7. DM layout design (a) geometry with inlet/vent locations with 4

race-tracking possibilities along the insert edges creating 24=16

different scenarios (b) 8 regions for placement of distribution media

when using discrete optimization, and (c) optimum DM design which

resulted in successful filling for all 16 scenarios.

Manufacturing using the convectional way by covering nearly the entire

preform (95%) with DM (leaving 5% gap at the end so the resin does not short circuit

the flow path and reach the vent through the DM which will result in large regions of

voids within the part. Figure 5.8 shows the resin flow front patterns along the top and

the bottom of the part in 4 of the possible 16 scenarios that can occur due to

permutations of race-tracking effects along the edges of the insert. These results are

inlet vent

insert

8 regions 1 layer DM

No DM

RT1

RT2

RT3

RT4

(a) (b) (c)

103

contrasted with the tailored DM design which do not result in any voids for all 16

scenarios. Use of DM on 95% of the top layer decreases fill time but as it can be seen

from the flow front profiles at the bottom (Figure 5.8.(a)) that large voids do form for

this DM layout. Simulations with the optimized DM design clearly show that

successful filling without entrapping any voids despite different flow front profiles

(Figure 5.8.(b)). Use of tailored DM design also saves DM material as one does not

need to cover the entire top of the mold with DM.

104

Figure 5.8. Numerical Solution of flow front profiles of the top and bottom views

for 4 different race-tracking scenarios with time steps 10 seconds

apart, (a) with 95% of the top layer covered with DM, (b) with

optimized DM design

Void

(b) With optimized Distribution Media

Void

Top view

Bottom view

Top view

Bottom view

Void

VoidVoid VoidRace-tracking channels

105

The design is also tested experimentally to validate it. Figure 5.9 shows the

flow front progression at the top and bottom of the preform at time intervals of 20

seconds. The experimental fill time is 221 seconds. The numerical fill times are in the

range of 183 seconds to 273 seconds. The experimental flow front profiles show

uniform flow front lines as the corn syrup reaches the vent and the filling is complete

without voids. However, the profiles do not exactly match with any of the 16

numerical scenarios, though they are “close”. This can be explained by several factors.

First and most importantly, the assigned of the “race-tracking strength” value for the

simulations is not exactly known value but is an estimate. The strength of the race-

tracking is the ratio of permeability along the race-tracking line to preform

permeability in the direction of the race-tracking line. For the simulations this value is

kept at a very high value, 1000, so the DM design will work for smaller race-tracking

strength values as well. Second, there are additional deviations between numerical

model and the experiment, related to inaccurately determined material properties

[48,158]. However, the flow pattern is reasonably captured despite these discrepancies

and the proposed solution is able to fill the mold for all scenarios even if the times do

not exactly match.

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Figure 5.9. Experimental flow fronts with the optimized DM design with flow

front locations in red 20 seconds apart. The background image of the

experiment at 60 seconds, (a) Top and (b) Bottom

5.5.2 Complex Geometries

The proposed methodology is tested with a complex geometry with corners

and edges. As seen in Figure 5.10, an optimized DM design is presented for a trailer of

a truck. For this geometry there are 10 different RT channel possible whose

permutations will yield 1024 different scenarios. The algorithm is used with defined

inlet and vent locations and DM layout solution is sought that considers all 1024

scenarios and converges to the DM design that provides successful filling for all 1024

cases. For this geometry the convergence is tested with 2, 4, 6, 8, 10, 12 and 14

regions, respectively, and the algorithm convergences to the design given in Figure

5.10 with 14 regions.

Void

Void

(a) Top view (b) Bottom view

107

Figure 5.10. Optimized DM design of trailer geometry with 1024 different

possible flow patterns

If the entire top surface is covered with the DM, in 886 scenarios out of 1024

the unfilled volume will be more than 1% with the maximum unfilled volume being

20%. If the optimized DM design is used, in all 1024 cases the void fraction is less

than 1%, the largest being 0.79%. Figure 5.11 shows void regions for the three

representative examples out of 1024 for the case of entire top surface being covered by

the DM on the left hand side and with the optimized DM design on the right hand side

that is free of voids. Figure 5.12 presents the time contours comparison for fully

covered DM situation and the optimized DM design for the same three scenarios. The

fill time in the optimized DM case is slightly higher but it is robust enough to provide

successful filling despite the variability introduced from possible race-tracking. The

lowest and the largest fill time for full DM case was 3710 and 3458 seconds where as

for the designed DM case, the fill time was 10459 and 8413 respectively from all 1024

DM

No DM

DM design algorithm

inlet

vent

inletvent

10 RT channels

Dividing into regions

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scenarios. Additionally, Figure 5.13 gives the pressure distribution data at the time

when the resin reaches the vent location for full DM and DM layout design. Besides

the improvement in the filling the DM design also provides uniform pressure along the

preform that yields uniform volume fraction and uniform material properties.

Figure 5.11. Void regions with full DM on top surface on the left hand side with

optimized DM design on the right hand side for three representative

scenarios from 1024 possible scenarios

void

filled

109

Figure 5.12. Time contours with full DM on top surface on the left hand side with

optimized DM design on the right hand side for three representative

scenarios from 1024 possible scenarios

Time (sec)

vent location void

110

Figure 5.13. Pressure distribution at the instant resin reaches the vent with full

DM on the left and with optimized DM design on the right for the

three representative scenarios

In order to examine the effect of mesh size on the DM design and CPU time

four mesh sizes starting with 0.05 m element size and halving it 3 times was

investigated. Except for the first coarse mesh, the rest of the meshes converged to the

same DM design but the CPU time increased exponentially with finer mesh size with

tolerance of 5% unfilled area. For all those 4 cases, the algorithm converged when the

domain is divided into 2 regions and 34 (16+2+16) LIMS simulations are performed

for each case. However, for the very coarse mesh (mesh size 0.05 m) required

tolerance is not satisfied with the C shape DM design as for the other cases and

convergence to the DM design is achieved when the domain is divided into 14 regions

which make the CPU time needed for the very coarse mesh the largest. Hence it is

Pressure (Pa)

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important to ensure that the mesh is fine enough to obtain DM design convergence

with fewer numbers of divisions.

Figure 5.14. Change in CPU time with mesh size for optimized DM design

5.6 Summary

In this study a methodology is introduced to design an optimum distribution

media layout that makes the process robust by successful filling for all possible

disturbances caused by different race-tracking scenarios around inserts. Depth First

Search, a tree search, algorithm is adopted via discretization of the domain into finite

regions to arrive at the DM design. The algorithm is demonstrated with an example

and an experimental validation is presented.

112

Chapter 6

CONCLUSIONS, CONTRIBUTIONS AND FUTURE WORK

6.1 Conclusions

In conclusion, this work has focused on various perspectives of permeability

characterizations which will bring the results of flow simulations closer to what is

observed in manufacturing practice

First, the dissertation study introduces a new methodology to characterize the

permeability tensor with non-zero skew terms via a single experiment. The available

permeability characterization techniques are discussed. Besides the lack of consistency

in most of the techniques, they mainly focus on the characterization of the in-plane

permeability components. However, the transverse permeability of the preform gains

importance especially for thick 3D fabrics. Additionally, the weaves in the through

thickness direction results in non-zero skew terms in the thickness direction. A new

methodology is introduced that records the flow fronts on the top and bottom surface

of the mold and with image processing uses these flow front profiles with a

multi-objective simplex optimization routine to find the six components of the

permeability tensor from one experiment. This work enables the understanding of the

effect of the non-zero skew components on the flow front progression and flow

patterns through a virtual study to underline when the skew terms could change the

nature of filling and influence the manufacturing process.

Then, the dissertation work investigates the through thickness permeability

characterization of uni-directional fabrics. During preparation of the preform using

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uni-directional fabrics, the fabrics are stacked on top of each other which might cause

misalignments. Also, design requirements might necessitate different degree of

rotation for each unidirectional fabric layer. Under this section of the work the effect

of those misalignments and degree of rotation of the layers with respect to each other

on the transverse permeability component of the preform is studied. At first a

numerical analysis is performed to model the change in the through thickness

permeability with fiber orientation using Gambit and Fluent. The numerical model

assumes the fiber tows are solid and impermeable and the laminar flow of the resin

occurs between the fiber tows. The work is followed by experimental characterization

of the permeability tensor using the electrostatic sensor embedded RTM mold plates to

record the resin arrival time. Numerical and experimental through thickness

permeability comparisons are reported to have good match. This study concludes that

the averaging approach to estimate the through thickness permeability of the

unidirectional fabrics’ when adjacent layers are rotated is not valid. The rotation in the

layup sequence does influence the transverse permeability. For example two 0 and two

90s (0/0/90/90) will have a different through thickness permeability than 0/90/0/90

layup. A significant change in through thickness permeability is observed for

misalignments larger than five degrees of rotation between individual layers.

The misalignment of the unidirectional fabrics between individual fabric layers

are showed to generate new pathways in through the thickness direction for resin flow

and these pathways are correlated with the change in the through thickness

permeability.

Another factor affecting the dynamics of the resin flow behavior is the non-

homogeneity of the permeability of the fabric. The characterization of the variation of

114

the permeability due to local non-homogeneous architecture of the preform lacks

standardization. A quantitative way to characterize the permeability variation within

the fabric by monitoring the flow front profiles of the resin impregnation with time is

introduced. The flow front profiles are processed using Kardar-Parisi-Zhang

formulation (KPZ). KPZ formulation requires the evaluation of two parameters;

growth exponent and surface roughness by evaluation the variance of the flow front

profiles. A major finding is the growth exponent falls into the universality class of

KPZ equations and roughness exponent can be used as parameter to quantify the

permeability variation in the preform. Moreover, the KPZ model is utilized to

determine the percentage of local defects in the preform, which can be a tool to

characterize the quality of the preform.

Finally, a methodology to optimize the distribution media layout is presented.

The optimized design for successful filling of the dry preform should not only work

for a single manufacturing scenario, it should also ensures successful filling for all

possible manufacturing scenarios caused by different disturbances within the domain

due to race-tracking. The algorithm adapts the Depth First Search, a tree search

algorithm, with domain discretization. The introduction of the methodology is

followed by experimental validation and application of the approach for a complex

part which should prove useful in manufacturing of large complex parts containing

inserts in VARTM. This methodology saves DM material and also provides more

uniform pressure reducing thickness variations in the part.

6.2 Contributions of this work

Some of the unique contributions of this work are summarized below.

115

First, the characterization of the permeability tensor with non-zero skew terms

with a single experiment is achieved. This approach enables the characterization of the

six components with a single radial injection experiment. The data reduction part

involves image processing to convert the set of flow front images with time stamps

into the fill time for predefined mesh geometry. Another improvement with this study

is the quick convergence to the optimized permeability tensor because of the

optimization algorithm adopted coupled with seamless interaction between LIMS and

Matlab®.

Second, the through thickness permeability of the unidirectional fabric is

investigated. The original contribution related to this work is the invalidation of the

averaging approach to determine the permeability in the through thickness direction

for plies rotated with respect to each other. Due to new pathways it has been showed

that the through thickness permeability tends to increase exponentially as the relation

rotation angle increases.

Third, the characterization of the variation of permeability within the preform

is studied for the first time. Using the growth exponent, the universality of the Kardar-

Parisi-Zhang formulation is validated for the resin flow thorough porous media.

Moreover, the roughness exponent is found to indicate the randomness of the

permeability value in the preform. Then, the formulation is utilized to develop a

correlation between growth and roughness exponents, and percentage of defects and

strength of the defects. This correlation suggests that for the growth exponent and

roughness exponent obtained from a single linear injection experiment for a fabric can

be used to characterize the randomness, namely quality, of the fabric.

116

Finally, with the methodology introduced to generate and optimize the

permeability map using distribution media of the preform for successful filling of the

preform is the first attempt to obtain a solution that is valid for all possible

disturbances and variations unlike previous studies. Also, the adaptation of the Depth

First Search algorithm is a novel idea to enhance the computational efficiency and find

optimal and automated solutions for optimum permeability map generation along with

location of inlets and vents for a specified geometry with variations in the permeability

characterized.

6.3 Future Work

Following the current dissertation study, the new permeability tensor

characterization can be improved with more experimental validation with various 3D

fabrics. The convergence of the methodology is compared with virtual

experimentation, but one can use the other permeability measurement techniques to

compare the permeability data. Also, the adapted simplex algorithm is a direct search

method, which doesn’t require the gradient calculation but this slows down the

convergence. The algorithm can be compared with other approaches, such as neural

network training, in terms of convergence performance and speed.

In order to improve the through thickness permeability study, the

experimentation can be performed on unidirectional fabrics with different tow size,

areal weight, material and/or fiber volume fractions. The numerical solution as a

validation tool can be also tested not only with circular tows but also at different

aspect ratios. The stitching can also be added as a parameter and its effect can be

investigated. In the numerical part the tows are modeled as solid walls with no

permeability. This assumption can be modified using multi-scale models or

117

homogenization to encounter the tow permeability. Furthermore, the numerical

approaches introduced for characterization of the permeability tensor with single

experiment and for the through thickness permeability can be adopted to different

material characterizations, such as thermal conductivity.

The characterization of the randomness work is the first attempt to quantify the

permeability variation in the preform using a mathematical model. The work on

randomness characterization proved that the growth exponent values for the flow

through porous media falls in the universality class of the KPZ formulation and it is

shown that the surface roughness can be used as a parameter to characterize the

randomness. This study can be further improved by the extension of the numerical and

the experimental work with wider variety in the dimensions and the standard

deviations of the preform domain. Additionally, the KPZ formulation is adapted for

the crystal structure radial domain. Thus, an investigation of the radial flow using the

same methods would be useful. Also, similar study can be used to investigate the

variation of the thermal conductivity.

The optimization of the LCM filling process is achieved using optimized

distribution media layout, which is experimentally validated for a single preform with

a metal insert and tested numerically for complex parts. This validation can be

extended with more experimental and numerical samples. The adapted algorithm

converges to a local minimum, so the uniqueness of the methodology can be tested.

The numerical solution with a complex geometry yields a more uniform pressure

distribution along the preform. This finding can be experimentally validated. Finally,

this methodology can be implemented for different problems, such as robot motion

planning and factory layout.

118

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MATLAB SCRIPTS FOR DISTRIBUTION MEDIA OPTIMIZATION

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A.1 Generation of the regions: regioning.m % generates the region#.txt files: % needs the nodes, elements, p (2p number of regions) function zones = regioning (nodes, elements, p) x_cord = nodes(:,2); %x coordinates y_cord = nodes(:,3); %y coordinates N = elements(:,3:6); %element connectivity % x_elem, y_elem x and y coordinates from element connectivity x_elem = [x_cord(N(:,1)) x_cord(N(:,2)) x_cord(N(:,3))

x_cord(N(:,4))]; x_elem = mean(x_elem,2); % x-cord center of the element y_elem = [y_cord(N(:,1)) y_cord(N(:,2)) y_cord(N(:,3))

y_cord(N(:,4))]; y_elem = mean(y_elem,2); % y-cord center of the element % generate regions zones = p*2; %number of zones X = [x_elem y_elem]; [IDX] = kmeans(X, zones); regions = [IDX elements(:,1)]; for i = 1:zones reg = regions(regions(:,1)==i,2); fName = sprintf('region%d.txt',i); fid = fopen(fName,'wt'); fprintf(fid, '%d\n',reg); end fclose all; % plot the regions Y = [IDX x_elem y_elem]; figure hold on for i = 1:zones sub_y = Y(Y(:,1)==i,:); plot(sub_y(:,2),sub_y(:,3),'o','color',rand(1,3),'marker', ... '.','MarkerSize',20) end end % Regioning ends

A.2 Scissors.m: Main m-file % Hatice Sinem Sas % Main % ROCK.m : Runs all possible race-tracking possibilities and return

the % worst case % PAPER.m: finds DM desing working for the worstcase tic % define variables load('nodes.mat'); % node matrix

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load('elements.mat'); % elements matrix - doesn't have DM on it. load('RTs.mat'); % RTs matrix-includes all possible racetrackings -

from allRTs.dmp load('channel.mat'); % to place 1D elements to the inlet and/orvent

location. remove for node vent cases n = length(nodes); % number of nodes m2d = length(elements); % number of 2D elements mchannel = length(channel); % ATTENTION m1d = length(RTs); opt = max(RTs(:,1)); Kxx = 8.32e-11; Kyy = 8.32e-11; h_preform = 5e-3; %thickness of the preform vf = 0.5000; %fiber volume fraction of the preform elements(:,9) = Kxx; elements(:,11) = Kyy; elements(:,7) = h_preform; elements(:,8) = vf; elementsmain = elements; KDM = 3.5e-9; % DM permeability h_DM = 1e-3; % thickness of 1-layer DM vfDM = 0.2000; % fiber volume fraction of KRT = Kxx.*1000; % ractraking permeability AreaRT = h_preform.*sqrt(12*KRT); vfRT = 0.010000; % vol. fraction for 1D elements KDM1 = (h_preform*Kxx+h_DM*KDM)/(h_preform+h_DM); KDM2 = (h_preform*Kxx+2*h_DM*KDM)/(h_preform+2*h_DM); vfDM1 = (h_preform*vf+h_DM*vfDM)/(h_preform+h_DM); vfDM2 = (h_preform*vf+2*h_DM*vfDM)/(h_preform+2*h_DM);

%% Generate the possibility matrix C = cell(opt ,1); [C{:}] = ndgrid([true, false]); %// Generate N grids of binary values p = cellfun(@(x){x(:)}, C); %// Convert grids to column vectors p = [p{:}];

putRT = cell(2^opt,opt); for j = 1: 2^opt for i = 1:opt putRT{j,i} = RTs(RTs(:,1) == i,:); RTput{j,i} = putRT{j,i} .* p(j,i); end end

pause(0.5);

for i = 1 : 2^opt RTcase(:,:,i) = cell2mat(RTput(i,:)'); end

pause(0.5);

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%% empty_percentALL = 100; %initial assignment r = 0; % first run , counts the RUN l = 0; % first loop, counts the LOOP %--------------------------------- % generates the region#.txt files: p =1; zones = regioning(nodes, elements, p); % Regioning ends %--------------------------------------------------------------------

----------------- while empty_percentALL>= 5.01 r = r+1; % function call to [index, NEN] = ROCK(r, elements, channel, nodes, m2d,

mchannel,RTcase, n, opt,AreaRT, vfRT, KRT); worstcase = index(1); empty_percentALL = NEN(worstcase,1)/n*100; % send the element and RT data to the PAPER.m fName = sprintf('run_%d_case_%d.dmp',r,worstcase); dummy = importdata(fName, ' ', n+7); elementDM = dummy.data(1:m2d,:); RTDM = dummy.data(m2d+mchannel+1:end,1:7); empty_percent = empty_percentALL; while (empty_percent >= 5.00) l = l+1; % index for the next loop if l > zones p = p+1; l = 1; elements = elementsmain; zones = regioning(nodes, elements, p); r =1; [index, NEN] = ROCK(r, elements, channel, nodes, p, RTs,

m2d, mchannel,RTcase, n, opt,AreaRT, vfRT, KRT); worstcase = index(1); empty_percentALL = NEN(worstcase,1)/n*100; % % send the element and RT data to the PAPER.m fName = sprintf('run_%d_case_%d.dmp',r,worstcase); dummy = importdata(fName, ' ', n+7); elementDM = dummy.data(1:m2d,:); RTDM = dummy.data(m2d+mchannel+1:end,1:7); empty_percent = empty_percentALL; end [indexDM, NENDM] = PAPER( l, worstcase, elementDM, channel,

nodes, RTDM, m2d, mchannel, n, zones); % [ indexDM, NENDM ] = PAPER( l, worstcase, elementDM, nodes,

RTDM, m2d, n, zones); empty_percent = NENDM(indexDM,1)/n*100; %update elementzDM fName = sprintf('loop_%d_%d_%d.dmp',l,worstcase,indexDM); dummy = importdata(fName, ' ', n+7); elementDM = dummy.data(1:m2d,:);

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DM_design = zeros(zones*2,1); DM_design(l) = indexDM; end elements = elementDM; end toc %prints the CPU time

%%

A.3 Rock.m: Evaluation of all race-tracking possibilities

% Hatice Sinem Sas, Feb. 2014 % Generate the .dmp files of all possible race-trackings function [index, NEN] = ROCK(r, elements, channel, nodes, m2d,

mchannel, RTcase, n, opt, AreaRT, vfRT, KRT) % Generate the .dmp files for drum = 1:2^opt % loop to write all the .dmp files fName = sprintf('run_%d_case_%d',r,drum); fileName = sprintf('%s.dmp',fName); o = fopen(fileName, 'w'); fprintf(o,'# \r\n'); fprintf(o,'Number of nodes : %5.0f \r\n',n); fprintf(o,' Index x y z\r\n');

fprintf(o,'===================================================\r\n'); for i = 1: n % loop to write node data fprintf(o,'%6.0f %12.6f %12.6f %12.6f \r\n',nodes(i,:)); end % loop to write DM data (if any) RTmatrix = RTcase(:,:,drum); RTmatrix(RTmatrix(:,1)==0,:)=[]; m1d = length(RTmatrix); % number of race-tracking elements : fprintf(o,'Number of elements : %5.0f \r\n',m2d+mchannel+m1d); fprintf(o,' Index NNOD N1 N2 (N3) (N4) (N5) (N6) (N7)

(N8) h(A) Vf Kxx Kxy

Kyy Kzz Kzx Kyz\r\n');

fprintf(o,'==========================================================

=====================================================================

===============================================\r\n'); for i = 1: m2d % loop to write 2D elements data fprintf(o,'%6.0f %4.0f %6.0f %6.0f %6.0f %6.0f %32.6f %15.6f

%15.4e %15.4e %15.4e \r\n',elements(i,:)); end

for i = 1:mchannel %loop to generate 1D elements for inlet and/or

vent fprintf(o,'%6.0f %4.0f %6.0f %6.0f %32.6e %15.6f %15.4e

\r\n',channel(i,:));

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end

if m1d ~=0 RTmatrix(RTmatrix(:,1)==0,:)=[]; m1d = length(RTmatrix); % number of race-tracking elements : RTelements = zeros(m1d,7); RTelements(:,1) = [1:m1d]+m2d+mchannel; RTelements(:,2) = 2; RTelements(:,3) = RTmatrix(:,2); % node 1 for 1D element

connectivity RTelements(:,4) = RTmatrix(:,3); %node 2 for 1D element

connectivity RTelements(:,5) = AreaRT; RTelements(:,6) = vfRT; RTelements(:,7) = KRT; for i = 1:m1d fprintf(o,'%6.0f %4.0f %6.0f %6.0f %32.6e %15.6f %15.4e

\r\n',RTelements(i,:)); end end fprintf(o,'Resin Viscosity model NEWTON\r\n'); fprintf(o,'Viscosity : 0.1\r\n'); fclose(o); fclose all; end % Generate lb file Pin = 1.000000e+005; %inlet pressure value for j = 1:2^opt fName = sprintf('run_%d_simulate_%d.lb',r,j); fid2 = fopen(fName,'w+'); fprintf(fid2,'PROC simu\r\n'); fprintf(fid2,'DO\r\n'); fprintf(fid2,'SOLVE\r\n'); fprintf(fid2,'EXITIF SOFILLFACTOR(1436) > 0.9\r\n'); fprintf(fid2,'LOOP WHILE ((SONUMBEREMPTY() > 0) AND

(SONUMBERFILLED() > 0))\r\n'); fprintf(fid2,'ENDPROC\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'READ "run_%d_case_%d.dmp"\r\n',r, j); for i = 1:41 % inlet node numbers fprintf(fid2,'SETGATE %d, 1, %d \r\n',i, Pin); end fprintf(fid2,'\r\n'); fprintf(fid2,'CALL simu\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'Print "%d # empty nodes =", sonumberempty\r\n', j); fprintf(fid2,'\r\n'); fprintf(fid2,'SETOUTTYPE "tplt"\r\n'); %fprintf(fid2,'SETOUTTYPE "dump"\r\n'); fprintf(fid2,'WRITE "run_%d_case_res_%d.tec"\r\n',r, j); %fprintf(fid2,'WRITE "case_res_%d.dmp"\r\n', j); fclose all;

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end % Run the lb files NEN = zeros(2^opt,2); % number of empty nodes for j= 1: 2^opt fName = sprintf('run_%d_simulate_%d.lb',r,j); fileName = sprintf('load run_%d_simulate_%d.lb',r,j); lims(3,6,2000); % set time-out to 2000 mS lims(3,3,50); lims(1,1); lims(5,1,'setoutputlevel 2'); lims(5,1,'setmessagelevel 0'); lims(5,1,fileName); lims(5,1,fName); output = 'ini'; pause(0.5); while ~isempty(output) output = lims(4,1,350); end lims(5,1,'print sonumberempty'); pause(5.0) lims(2,1) pause(5.0) fileName2 = sprintf('run_%d_case_res_%d.tec',r,j); result = importdata(fileName2, ' ', 3); emptynodes = result.data(1:n,6); % fill factors of the nodes NEN(j,1) = numel(emptynodes(emptynodes<0.9)); %number of empty

nodes filltime = result.data(1:n,5); % fill times of the nodes NEN(j,2) = max(filltime); fclose all; end % Worst case index = find(NEN(:,1) == max(NEN(:,1))); %index array can have more

than one elememts index = index(1); % index of one of the worst case f2name = sprintf('run_all_%d.mat',r); %stores all data save(f2name) fclose all; end

A.4 Paper.m: Finding the optimum region to place DM function [ indexDM, NENDM ] = PAPER(l, worstcase, elementDM, channel,

nodes, RTDM, m2d, mchannel,n, zones) if ~isempty(RTDM) m = RTDM(end,1); else m = length(elementDM)+mchannel; end

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m1d = length(RTDM); Kxx = 8.32e-11; Kyy = 8.32e-11; KDM = 3.5e-9; vf = 0.5000; vfDM = 0.2000; h_preform = 5e-3; h_DM = 1e-3; KRT = Kxx.*1000; % ractraking permeability AreaRT = h_preform.*sqrt(12*KRT); KDM1 = (h_preform*Kxx+h_DM*KDM)/(h_preform+h_DM); KDM2 = (h_preform*Kxx+2*h_DM*KDM)/(h_preform+2*h_DM); vfDM1 = (h_preform*vf+h_DM*vfDM)/(h_preform+h_DM); vfDM2 = (h_preform*vf+2*h_DM*vfDM)/(h_preform+2*h_DM); elements = elementDM; for drum = 1:zones % loop to write all the .dmp files fName2 = sprintf('region%d.txt',drum); data = importdata(fName2); data = data(:); data = data(~isnan(data)); data = sort(data); fName = sprintf('loop_%d_%d_%d',l,worstcase,drum); fileName = sprintf('%s.dmp',fName); o = fopen(fileName, 'w'); fprintf(o,'# \r\n'); fprintf(o,'Number of nodes : %5.0f \r\n',n); fprintf(o,' Index x y z\r\n');

fprintf(o,'===================================================\r\n'); for i = 1: n % loop to write node data fprintf(o,'%6.0f %12.6f %12.6f %12.6f \r\n',nodes(i,:)); end

if elementDM(data(1), 9) == KDM1 elementDM(data, 7 ) = (h_preform+2*h_DM); elementDM(data, 8 ) = vfDM1; elementDM(data, 9 ) = KDM2; elementDM(data, 11 ) = KDM2; else elementDM(data, 7 ) = (h_preform+h_DM); elementDM(data, 8 ) = vfDM2; elementDM(data, 9 ) = KDM1; elementDM(data, 11 ) = KDM1;

end fprintf(o,'Number of elements : %5.0f \r\n',m); fprintf(o,' Index NNOD N1 N2 (N3) (N4) (N5) (N6) (N7)

(N8) h(A) Vf Kxx Kxy

Kyy Kzz Kzx Kyz\r\n');

fprintf(o,'==========================================================

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=====================================================================

===============================================\r\n'); for i = 1: m2d % loop[ to write 2D elements data fprintf(o,'%6.0f %4.0f %6.0f %6.0f %6.0f %6.0f %32.6f %15.6f

%15.4e %15.4e %15.4e\r\n',elementDM(i,:)); end for i = 1:mchannel fprintf(o,'%6.0f %4.0f %6.0f %6.0f %32.6e %15.6f %15.4e

\r\n',channel(i,:)); end if ~isempty(RTDM) for i = 1:m1d fprintf(o,'%6.0f %4.0f %6.0f %6.0f %32.6e %15.6f %15.4e

\r\n',RTDM(i,:)); end end fprintf(o,'Resin Viscosity model NEWTON\r\n'); fprintf(o,'Viscosity : 0.1\r\n'); elementDM = elements; end

fclose(o); fclose all; % Generate lb file Pin = 1.000000e+005; for j = 1:zones fName = sprintf('loop_%d_simulate_%d.lb',l,j); fileName = sprintf('load loopd_%d_simulate_%d.lb',l,j); fid2 = fopen(fName,'w+'); fprintf(fid2,'PROC simu\r\n'); fprintf(fid2,'DO\r\n'); fprintf(fid2,'SOLVE\r\n'); fprintf(fid2,'EXITIF SOFILLFACTOR(1436) > 0.9\r\n'); fprintf(fid2,'LOOP WHILE ((SONUMBEREMPTY() > 0) AND

(SONUMBERFILLED() > 0))\r\n'); fprintf(fid2,'ENDPROC\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'READ "loop_%d_%d_%d.dmp"\r\n', l, worstcase,j); for i=1:41 fprintf(fid2,'SETGATE %d, 1, %d \r\n',i, Pin); end fprintf(fid2,'\r\n'); fprintf(fid2,'CALL simu\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'Print "%d # empty nodes =", sonumberempty\r\n', j); fprintf(fid2,'\r\n'); fprintf(fid2,'SETOUTTYPE "tplt"\r\n'); %fprintf(fid2,'SETOUTTYPE "dump"\r\n'); fprintf(fid2,'WRITE "loop_%d_%d_%d_res.tec"\r\n', l,worstcase,j); %fprintf(fid2,'WRITE "case_res_%d.dmp"\r\n', j); fclose all; end

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fclose all; NENDM = zeros(zones,2); for j= 1:zones fName = sprintf('loop_%d_simulate_%d.lb',l,j); fileName = sprintf('load loop_%d_simulate_%d.lb',l,j); lims(3,6,2000); % set time-out to 2000 mS lims(3,3,50); lims(1,1); lims(5,1,'setoutputlevel 2'); lims(5,1,'setmessagelevel 0'); lims(5,1,fileName); lims(5,1,fName); output = 'ini'; while ~isempty(output) output = lims(4,1,350); end lims(5,1,'print sonumberempty'); pause (2.0); lims(2,1) pause(2.0); fileName2 = sprintf('loop_%d_%d_%d_res.tec',l,worstcase,j); result = importdata(fileName2, ' ', 3); emptynodes = result.data(1:n,6); NENDM(j,1) = numel(emptynodes(emptynodes<0.9)); filltime = result.data(1:n,5); NENDM(j,2) = max(filltime); fclose all; end %Worst case indexDM = find(NENDM(:,1) == min(NENDM(:,1))); %index array can have

more than one elememts indexDM = indexDM(1); % index of one of the worst case f2name = sprintf('all_loop_%d.mat',l); save(f2name) end

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REPRINT PERMISSION LETTERS

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B.1 “EFFECT OF RELATIVE PLY ORIENTATION ON THE THROUGH-

THICKNESS PERMEABILITY OF UNIDIRECTIONAL FABRICS”

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B.2 “FRACTAL CONCEPTS IN SURFACE GROWTH”