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Micro-Mechanics Based Fatigue Modelling of Composites Reinforced
With Straight and Wavy Short Fibers
Yasmine ABDIN
Supervisor: Prof. Stepan V. Lomov Prof. Ignaas Verpoest Members
of the Examination Committee: Prof. Albert Van Bael Prof. Andrea
Bernasconi Prof. Frederik Desplentere Dr. Larissa Gorbatikh Prof.
Patrick Wollants (Chairman) Prof. Willy Sansen (Chairman) Prof. Wim
Van Paepegem
Dissertation presented in partial fulfilment of the requirements
for the degree of PhD in Materials Engineering
September 2015
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© 2015 KU Leuven, Groep Wetenschap & Technologie
Uitgegeven in eigen beheer, Yasmine Abdin, Heverlee
Alle rechten voorbehouden. Niets uit deze uitgave mag worden
vermenigvuldigd en/of openbaar gemaakt worden door middel van druk,
fotokopie, microfilm, elektronisch of op welke andere wijze ook
zonder voorafgaandelijke schriftelijke toestemming van de
uitgever.
All rights reserved. No part of the publication may be
reproduced in any form by print, photoprint, microfilm, electronic
or any other means without written permission from the
publisher.
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V
Acknowledgements
First of all, I owe my deepest gratitude to my supervisors,
Professor Stepan
V. Lomov and Professor Ignaas Verpoest.
Professor Lomov has been more than a supervisor to me. This
thesis would
have not been possible without his mentorship, constant
guidance,
understanding and enormous support. He is a true mentor who
motivated
me to not only grow as a modeler and researcher, but most
importantly as
an independent and critical thinker, while always having an open
door for
me whenever I needed help.
I have also been very fortunate to have the guidance of
Professor Verpoest.
I learned a lot throughout the years from his deep
understanding, intuition
and passion for composites. He constantly provided me with
excellent
ideas for improvements of the various aspects of my research
work, both
experimental and modelling.
I wish to thank all the members of the jury: Professor Albert
Van Bael,
Professor Andrea Bernasconi, Professor Frederik Despelentere,
Doctor
Larissa Gorbatikh, Professor Wim Van Paepegem and the chairmen
of my
PhD committee, Professor Patrick Wollants and Professor Willy
Sansen
for their feedback, helpful comments and valuable time spent in
evaluating
this thesis.
It also gives me a great pleasure to acknowledge the support of
all the
members of the ModelSteelComp project. A heartfelt thanks goes
to
Christophe Liefooghe, Stefan Straesser, and Michael Hack from
the
Siemens Industry Software for all the help, feedback and
useful
discussions. I also thank Peter Persoone and Rik de Witte from
Bekaert for
their help and for providing me with the samples needed in this
PhD thesis.
And finally I thank Kris Bracke from Recticel and Vladimir
Volski from
ESAT, KU Leuven for valuable co-operations.
In the past years, I have also had the great privilege to be a
part of the
Composites Group in KU Leuven. I would like to thank all my
colleagues
and members of the CMG. Working within such a strong and
dynamic
group helped me to grow and shape my experience as a researcher.
It also
gave me the opportunity to gain knowledge about the different
fields of
composites.
I would like to thank Bart Pelgrims and Kris Van de Staey for
their help
and assistance in the experimental parts of this work.
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I am thankful to Atul Jain for being my colleague and research
companion
throughout the years. I am also really grateful for all the
friendships I have
made in Leuven. The list is too long to mention. For all of you,
your
friendships have made my stay in Leuven enjoyable and memorable
and I
am really grateful for the encouragement and emotional support
throughout
the years. A special thanks goes to: Farida, Lina, Yadian,
Valentin, Tatiana,
Eduardo, Baris, Marcin, MohamadAli, Aram, Oksana, Dieter,
Pencheng
and Manish.
Finally, and most importantly I would like to thank my family
and my
husband. I thank my parents for everything they have done and
for
allowing me to follow my goals and ambitions. Being in the
academic
career themselves, they have provided me with not only personal
but
professional guidance, in order to accomplish this important
phase of my
life. I would like to end this acknowledgment with deep
gratitude to my
husband Omar for his love, self-less support and continuous
encouragement. I especially thank him for the patience and
tolerance he
showed me to get through the stressful moments that were
necessary to
accomplish this work. The deep faith of my family is what got me
here,
and for that the least I can do is dedicate this work to them.
From all my
heart, THANK YOU!
Yasmine Abdin
Leuven, September 2015
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VII
Abstract
Short fiber composites, are extensively used in numerous
industrial fields,
and especially in the automotive industry, because of their
favorable
properties of high specific strength and stiffness. A
requirement for the use
of these materials in industrial applications is the ability to
evaluate the
behavior of the materials without the need for extensive, costly
and time
consuming testing campaigns. This can be achieved with the
development
of accurate predictive models.
In this PhD thesis, models are developed for the quasi-static
and fatigue
simulation of the short fiber composites. In addition to the
typical short
straight fiber composites, e.g. glass and carbon fiber
composites, the
models in this work are extended to the cases of complex short
wavy fiber
reinforced materials. The models are formulated in the framework
of the
mean-field homogenization techniques.
For simulating the behavior of wavy fiber composites, first, a
model is
developed for the generation of the representative volume
elements of the
complex random micro-structures of the wavy fiber composites
such as
short steel fiber composites. Second, a model is investigated
for the
extension of the mean-field techniques to wavy fiber composite.
A wavy
segment of the curved fiber is replaced with an equivalent
straight
inclusion whose elongation depends on the local curvature of the
original
segments.
Furthermore, models are developed for the prediction of the
quasi-static
stress-strain behavior of both the short straight and wavy fiber
reinforced
composites. The models take into account the plasticity of
the
thermoplastic matrices and the damage mechanisms of short
fiber
composites, mainly debonding. The matrix plasticity is modelled
using
secant formulations. In the damage model, a debonded inclusion
is
replaced with an equivalent bonded one with degraded properties
based on
a selective degradation scheme which takes into account the
local stress
states at the interface.
A novel model is developed for prediction of the fatigue S-N
behavior of
the short fiber composites. The model is based on the S-N curves
of the
constituents, and formulation of different failure criteria
which depends on
the local stress and damage states.
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Finally, in parallel with the developed modelling approach,
detailed
experimental characterizations were performed to achieve
better
understanding of the quasi-static and fatigue behavior and
damage
mechanisms of the short straight and wavy fiber reinforced
composites.
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Abstract
Korte vezelcomposieten worden vaak gebruikt in verschillende
industrieën, vooral in de automobielindustrie, omwille van hun
gunstige
eigenschappen zoals hoge specifieke sterkte en stijfheid. Een
vereiste voor
het gebruik van deze materialen in industriële toepassingen is
de
mogelijkheid om het materiaalgedrag te voorspellen zonder
uitgebreide,
kostelijke en tijdrovende testcampagnes. Dit kan bereikt worden
door het
ontwikkelen van nauwkeurige voorspellingsmodellen.
In deze doctoraatsthesis werden modellen ontwikkeld voor de
quasi-
statische en vermoeiingssimulatie van korte vezelcomposieten.
Naast de
klassieke korte vezelcomposieten met rechte vezels, zoals glas-
en
koolstofvezelcomposieten, werden de modellen ook uitgebreid naar
korte
vezelcomposieten met complexe, golvende vezels. De modellen
zijn
geformuleerd in het kader van de gemiddelde veld
homogenisatietechniek.
Voor het simuleren van het gedrag van golvende vezelcomposieten
werd
er eerst een model opgesteld om representatieve volume elementen
met een
complexe, willekeurige microstructuur van golvende korte
vezelcomposieten, zoals korte staalvezelcomposieten, te
genereren.
Daarna werd de gemiddelde veld homogenisatietechniek uitgebreid
naar
composieten met golvende vezels. Een golvende vezel werd
daarbij
vervangen door een equivalente rechte inclusie waarvan de lengte
afhangt
van de lokale kromming van het originele segment.
Bovendien werden modellen ontwikkeld voor het voorspellen van
de
quasi-statische spannings-rekgedrag van zowel rechte als
golvende korte
vezelcomposieten. De modellen houden rekening met de
plasticiteit van de
thermoplastische matrix en de schademechanismen van korte
vezelcomposieten, wat vooral ontbinding is. De
matrixplasticiteit werd
gemodelleerd met secant formulaties. In het schademodel werd
een
ontbonden inclusie vervangen door een equivalente, gebonden
inclusie met
gedegradeerde eigenschappen gebaseerd op een selectief
degradatieschema dat rekening houdt met de lokale spanningen aan
de
interfase.
Een nieuw model werd ontwikkeld voor de voorspelling van het
S-N
vermoeiingsgedrag van de korte vezelcomposieten. Het model is
gebaseerd
op de S-N curves van de samenstellende fases, en de formulering
van
falingscriteria die afhangen van de lokale spanningen en
schadetoestanden.
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Uiteindelijk werden er, in parallel met de ontwikkelde
modelleeraanpak,
gedetailleerde experimenten uitgevoerd om een beter inzicht te
krijgen in
zowel het quasi-statische en vermoeiingsgedrag als de
schademechanismen van rechte en golvende korte
vezelcomposieten.
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Table of Contents
CHAPTER 1: INTRODUCTION
.................................................... 1
1.1 General Introduction
..................................................................
3
1.2 Scientific & Technological Context
............................................ 5
1.3 Objectives of the PhD research
.................................................. 7
1.4 Structure of the thesis
.................................................................
9
CHAPTER 2: STATE OF THE ART
............................................ 13
2.1 Introduction
...............................................................................
15
2.2 Injection Molding of RFRCs
.................................................... 16
2.3 Micro-structure and Mechanical Behavior of RFRCs ...........
18 2.3.1 Micro-structure of RFRCs
.................................................................
18 2.3.2 Factors affecting the quasi-static and fatigue behavior of
RFRCs ..... 21 2.3.3 Fatigue damage in RFRCs
.................................................................
27
2.4 Geometry Generation Models
.................................................. 29 2.4.1
Critical RVE size
...............................................................................
29 2.4.2 RVE generation algorithms
...............................................................
32
2.5 Mean-Field Homogenization Schemes
..................................... 33 2.5.1 Eshelby’s solution
.............................................................................
34 2.5.2 Eshelby’s based homogenization models
.......................................... 35 2.5.3 Criticism of
Mori-Tanaka model
....................................................... 40
2.6 Modeling the non-linear quasi-static behavior of RFRC
....... 45 2.6.1 Matrix non-linearity
...........................................................................
45 2.6.2 Composite damage and failure
.......................................................... 49
2.7 Modeling the fatigue behavior of RFRCs
................................ 56
2.8 Discussion of the state of the art and adopted approaches
.... 58
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CHAPTER 3: GEOMETRICAL CHARACTERIZATION AND MODELING OF SHORT
WAVY FIBER COMPOSITES............... 63
3.1 Introduction to Steel Fiber Composites
................................... 65
3.2 Challenges in characterization and modelling the geometry of
SFRP composites
...................................................................................
66
3.3 Description of the Geometrical Model
..................................... 69
3.4 Materials and Experiments
....................................................... 73 3.4.1
Steel fiber samples
............................................................................
73 3.4.2 X-ray micro-tomography
...................................................................
74
3.5 Analysis
.......................................................................................
75 3.5.1 Image
segmentation...........................................................................
75 3.5.2 Three-dimensional image analysis tool
............................................. 78
3.6 Results and Discussion
.............................................................. 83
3.6.1 Fiber length distribution
....................................................................
83 3.6.2 Fiber orientation distribution
............................................................. 86
3.6.3 RVE of steel fibers
............................................................................
87 3.6.4 Straightness parameter
......................................................................
90
3.7 Conclusions
................................................................................
92
CHAPTER 4: EXPERIMENTAL CHARACTERIZATION OF QUASI-STATIC
BEHAVIOR OF SHORT GLASS AND STEEL
FIBER COMPOSITES
.........................................................................
93
4.1 Introduction
...............................................................................
95
4.2 Materials and Methods
............................................................. 95
4.2.1 Materials
............................................................................................
95 4.2.2 Specimen preparation
........................................................................
96 4.2.3 Fiber length distribution measurement
.............................................. 97 4.2.4 Tensile
testing
...................................................................................
98 4.2.5 Micro-CT analysis
.............................................................................
99 4.2.6 Fractography analysis
........................................................................
99 4.2.7 Single steel fiber tensile tests
.......................................................... 100
4.3 Results and Discussion
............................................................ 101
4.3.1 Fiber lengths measurements
............................................................ 101
4.3.2 Tensile behavior of the short glass fiber composites
....................... 104
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4.3.3 Micro-CT observations of the morphology of the short glass
fiber composites
....................................................................................................
115 4.3.4 SEM fractography analysis of the short glass fiber
composites ...... 117 4.3.5 Tensile behavior of the short steel
fiber composites ........................ 120 4.3.6 Micro-CT
observations of the morphology of short steel fiber composites
....................................................................................................
132 4.3.7 SEM fractography analysis of the short steel fiber
composites ....... 136
4.4 Conclusions
..............................................................................
138
CHAPTER 5: EXPERIMENTAL CHARACTERIZATION OF THE FATIGUE BEHAVIOR
OF SHORT GLASS AND STEEL
FIBER COMPOSITES
.......................................................................
141
5.1 Introduction
.............................................................................
143
5.2 Materials and Methods
........................................................... 143
5.2.1 Materials
..........................................................................................
143 5.2.2 Fatigue testing
.................................................................................
143 5.2.3 Stiffness degradation
analysis..........................................................
145 5.2.4 Fatigue tests performed on the quasi-static tensile test
machine ..... 147 5.2.5 Fractography analysis
......................................................................
148
5.3 Results and Discussion
............................................................ 149
5.3.1 Fatigue S-N curves of the short glass fiber composites
................... 149 5.3.2 Fatigue damage of the short glass
fiber composites ........................ 151 5.3.3 Fatigue damage
of the short steel fiber composite........................... 157
5.3.4 Fatigue tests of the SF-PA on the tensile tester
............................... 161 5.3.5 Fatigue tests of the
GF-PA on the tensile tester ............................... 163
5.3.6 SEM fractography analysis of the short glass fiber samples
........... 164
5.4 Conclusions
..............................................................................
167
CHAPTER 6: LINEAR ELASTIC MODELING OF SHORT WAVY FIBER
COMPOSITES
......................................................... 169
6.1 Introduction
.............................................................................
171
6.2 The Poly-Inclusion (P-I) Model
.............................................. 173
6.3 Problem statement and methods
............................................ 174 6.3.1 Test cases
.........................................................................................
175 6.3.2 Implementation of Poly-Inclusion model
........................................ 177 6.3.3 Generation of
finite element models
................................................ 177
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6.4 Results and Discussion
............................................................ 178
6.4.1 VE containing a single half circular fiber with constant
curvature . 178 6.4.2 VE-Single sinusoidal fiber with varying
smooth local curvature .... 187 6.4.3 VE-Micro-CT reconstructed
assembly of short steel fibers with random local curvature
.................................................................................
192
6.5 Conclusions
..............................................................................
196
CHAPTER 7: NON-LINEAR PROGRESSIVE DAMAGE MODELLING OF SHORT
FIBER COMPOSITES........................ 199
7.1 Introduction
.............................................................................
201
7.2 Formulation of the Damage Model
........................................ 201 7.2.1 Matrix
non-linearity
........................................................................
201 7.2.2 Fiber-Matrix debonding
..................................................................
203 7.2.3 Fiber breakage
.................................................................................
208
7.3 Implementation of the Damage Model
.................................. 209
7.4 Description of Validation Test Cases
..................................... 213 7.4.1 Own experiments –
glass fiber reinforced composites .................... 214 7.4.2
Own experiments – steel fiber reinforced composites
..................... 219 7.4.3 Experiments of Jain – glass fiber
reinforced composites ................ 221
7.5 Results and Discussion
............................................................ 223
7.5.1 Own experiments – glass fiber reinforced composites
.................... 223 7.5.2 Own experiments – steel fiber
reinforced composites ..................... 225 7.5.3 Experiments
of Jain – glass fiber reinforced composites ................
230
7.6 Conclusions
..............................................................................
233
CHAPTER 8: FATIGUE MODELLING OF SHORT FIBER COMPOSITES 235
8.1 Introduction
.............................................................................
237
8.2 Objectives and Formulation of the Fatigue Model
............... 238
8.3 Implementation of the Fatigue Model
................................... 243
8.4 Description of Validation Test Cases and Model Input .......
245 8.4.1 Own Experiments
............................................................................
245
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8.4.2 Experiments of Jain
.........................................................................
249
8.5 Results and Discussion
............................................................ 250
8.5.1 Own-experiments
............................................................................
250 8.5.2 Experiments of Jain
.........................................................................
254
8.6 Summary of the Overall Micro-Scale Solution
..................... 257
8.7 Component Level Simulation
................................................. 260 8.7.1 Current
framework of the component level simulation ................... 260
8.7.2 Description of the validation test case
............................................. 263 8.7.3
Experimental tests
...........................................................................
263 8.7.4 Description of the simulations
......................................................... 264 8.7.5
Results and discussion
.....................................................................
265
8.8 Conclusions
..............................................................................
270
CHAPTER 9: CONCLUSIONS AND FUTURE RECOMMENDATIONS
....................................................................
273
9.1 Global Summary of the Thesis
............................................... 275
9.2 General Conclusions
................................................................
275 9.2.1 Geometrical characterization and modelling
................................... 275 9.2.2 Quasi-static behavior
of short fiber composites............................... 276 9.2.3
Fatigue behavior of short fiber composites
...................................... 276 9.2.4 Linear elastic
modelling of wavy fiber composites ......................... 277
9.2.5 Quasi-static damage
modelling........................................................
277 9.2.6 Fatigue modelling
............................................................................
277
9.3 Future Outlook
........................................................................
278 9.3.1 Manufacturing of short steel fiber
composites................................. 278 9.3.2 Matrix
plasticity
...............................................................................
278 9.3.3 Component level solutions
.............................................................. 279
9.3.4 Multi-axial and variable amplitude fatigue
...................................... 279 9.3.5 Different modes of
the fatigue loading ............................................
279
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List of abbreviations (in alphabetical order)
AE Acoustic Emission
ARD Anisotropy Rotary Diffusion
BMC Bulk Molding Compound
CNT Carbon Nanotube
D.a.m Dry As Molded
DIC Digital Image Correlation
EAUI Equivalent Anisotropic Undamaged Inhomogeneity
EMI Electro-Magnetic Interference
FEA Finite Elements Analysis
FLD Fiber Length Distribution
FOD Fiber Orientation Distributions
FPGF First Pseudo-Grain Failure
HZ Higher Zone
IM Injection Molding
LFT Long fiber Thermoplastics
LZ Lower Zone
Micro-CT Micro-Computer Tomography
M-T Mori-Tanaka
P-I Poly-Inclusion
RFRC Random Fiber Reinforced Composites
ROM Rule of mixtures
RSA Random Sequential Absorption
RSC Reduced Strain Closure
RVE Representative Volume Element
S-C Self-Consistent
SEM Scanning Electron Microscopy
SFRP Short Fiber Reinforced Polymers
SMC Sheet Molding Compound
S-N Wohler Curve (applied fatigue stress against fatigue
life curve)
SSFRP Short Steel Fiber Reinforced Polymers
VE Volume Element
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XVII
List of symbols (some symbols are introduced
locally)
β Efficiency factor of the Poly-Inclusion model
γ Damage parameter: total amount of the debonded interface
area
which is loaded on traction.
δ Damage parameter: percentage of the frictional sliding
interface,
i.e. relative amount of the of the debonded interface area
which
loaded in compression.
ε𝛼 Inclusion strain
𝜀�̇� Matrix strain rate
𝜀𝑝∗ Effective matrix plastic strain
Out-of-plane orientation angle
𝜐𝑚 Poisson’s coefficient of the matrix
𝜎∗ Effective Von Mises stress in the matrix
𝜎𝐶 Critical interface strength
𝜎𝑓 Fatigue strength coefficient
𝜎𝑖𝑗′ Deviatoric component of the matrix stress tensor
�̇�𝑚 Matrix stress rate
𝜎𝑚𝑎𝑥 Maximum fatigue stress
𝜎𝑚𝑖𝑛 Minimum fatigue stress
𝜎𝑦 Initial yield stress
Φ In-plane orientation angle
𝜓1,2 Phase shifts
AMTα Strain concentration tensor according to Mori-Tanaka
method
Co𝑚 Elastic stiffness tensor of the matrix
C𝑒𝑓𝑓 Effective composite stiffness tensor
C𝑒𝑝 Continuum elasto-plastic tangent operator
C𝑚 Matrix stiffness tensor
C𝑠 Secant stiffness tensor
𝐸𝑑𝑦𝑛 Dynamic fatigue modulus
𝐸𝑚 Matrix elastic Young’s modulus
𝐸𝑚𝑠 Secant Young’s modulus of the matrix
𝑎𝑖𝑗 2nd order orientation tensor
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XVIII
𝑎𝑖𝑗𝑘𝑙 4th order orientation tensor
𝑎𝑟 Aspect ratio of the equivalent inclusion
𝑐𝛼 Fiber volume fraction
𝑛1,2 Waviness number
d Damage parameter: total percentage of the debonded
interface
area
ℎ Strength coefficient
S Eshelby tensor
𝐴 Amplitude of the wavy fiber
𝐿 Fiber length
𝑁 Number of cycles
𝑅 Radius of curvature
𝑅 Fatigue stress ratio
𝑈 Displacement vector
𝑏 Fatigue strength exponent
𝑑 Fiber diameter
𝑛 Work hardening exponent
𝑝 Fiber orientation vector
𝑟(𝑠) Radial position in relation to a certain axis of the wavy
fiber
𝑠 Coordinate along the curved fiber axis
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XIX
List of figures
Figure 1.1 Overview of the multi-scale predictive methods for
modelling the
fatigue behavior of RFRC parts.
................................................................
7
Figure 1.2 Outline of the PhD thesis.
............................................................ 10
Figure 2.1 Schematic illustration of the injection molding
process (adapted from
[25]).
........................................................................................................
17
Figure 2.2 Fiber orientation described with a direction 𝒑 and
corresponding angles Φ and .
...................................................................................................
18
Figure 2.3 Development of fiber orientation in injection molded
RFRCs (a)
morphology as analyzed using micro-CT scanning (b) associated
orientation
tensor component 𝑎11 through the thickness of the sample where
direction 1 is the MFD [43].
......................................................................................
20
Figure 2.4 The effect of fiber aspect ratio and volume fraction
on the strength of
RFRCs. SF 19, SF 14 refer to short discontinuous glass-fiber
reinforced
polypropylene (GF-PP) composites reinforced with fibers of
diameters 19 µm
and 14 µm respectively. LF 19 is a long discontinuous GF-PP
composite with
19 µm diameter [46].
...............................................................................
22
Figure 2.5 Effect of fiber orientation on the stress-strain
behavior of short fiber
composites (a) illustration of the general practice of producing
samples with
different orientation tensors where coupons are machined at a
certain
orientation angle from an injection molded plate [22] (b)
stress-strain plots of
an RFRC showing the effect of the different orientation on the
behavior of the
composite.
...............................................................................................
23
Figure 2.6 Effect of specimen orientation on the fatigue S-N
curves of RFRCs.
The graph shows plots of the S-N curves of GF-PA 6 material
[21]. ..... 25
Figure 2.7 Effects of various tests parameters on the fatigue
behavior of RFRCs
namely effect of (a) stress ratio [55], (b) cycling frequency
[62], (c)
temperature [22] and (d) water absorption (humidity), the blue
curve belongs
to GF-PA 6.6 samples containing 0.2wt% water content at 50%
humidity, the
red curves belongs to the same composite with 3.5wt% at 90%
humidity [63].
.................................................................................................................
26
Figure 2.8 Damage mechanisms observed in a fatigued sample up to
60% UTS.
(a) fiber/matrix debonding, (b) void at fiber ends, (c) fiber
breakage [43].28
Figure 2.9 Predictions of longitudinal elastic modulus E11as a
function of the
number of fibers in the RVE. [78]. The black dots represent
average of three
different random RVE realizations with the same size of RVE.
Error bars
represent 95% confindence intervals.
...................................................... 30
Figure 2.10 Generated RVE of RFRCs using the RSA method (13.5%
volume
fraction and aspect ratio of 10) [87].
....................................................... 33
Figure 2.11 Illustration of Eshelby's transformation principle.
..................... 35
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XX
Figure 2.12 Schematic representation of the two-step
homogenization model. The
RVE is decomposed into a number of grains (sub-regions) followed
by step 1:
homogenization of each grain , and step 2: second homogenization
if
performed over all the grains.
..................................................................
44
Figure 2.13 Two-step homogenization procedure and implementation
of damage
modelling proposed by Dermaux et al. [187].
.......................................... 53
Figure 3.1 Illustration of the drawing technique to produce
steel fibers [217].66
Figure 3.2 Example of wavy fiber generated by the model for
illustration. Black
dots represent ends of segments “control
points”..................................... 72
Figure 3.3 Micrographs of short steel fiber reinforced
polycarbonate sample
showing the fibers waviness (a) optical micrograph of the
composite plate
(stainless steel 0.05VF%) and (b) scanning electron micrograph
of the steel
fibers after a matrix burn-out procedure (stainless steel 2VF%),
the figure
shows high entanglements of the fibers.
.................................................. 74
Figure 3.4 Thresholding of steel fiber reinforced polycarbonate
sample (a) 2D
gray-level 2D reconstructed images, (b) corresponding binary
image and (c)
individual automatic global thresholds obtained from gray scale
attenuation
histogram. The attenuation histogram consists of two overlapping
bivariate
distributions. The peak corresponding to lower attenuation index
is associated
with matrix material. Due to the low volume fraction (low
probability) the peak
of steel fibers is not visible in the plot. The threshold value
obtained from the
automatic global thresholding is shown with the red dashed line.
........... 77
Figure 3.5 Thresholded 3D model of a micro-CT scan of SSFRP
built in Mimics
software package. The picture shows a green mask of rendered
steel fibers and
the outline of the matrix mask in
purple................................................... 78
Figure 3.6 Procedure for characterization of fiber length and
orientation
distribution of SSFRP. (a) 3D reconstructed model in Mimics
software, (b)
separation of single fibers and (c) fitting of centerline,
automatic measurement
of fiber length and post-processing for measurement of fiber
orientation.80
Figure 3.7 Length distribution of steel fiber reinforced
polycarbonate composite
(a) probability density plots of achieved lengths of steel
fibers fitted with
different statistical distribution functions i.e.: Normal,
Lognormal and Weibull
distributions and (b) Weibull probability plot of the steel
fiber length data.
.................................................................................................................
85
Figure 3.8 FOD of the short steel fibers (a) distribution of Φ
angle and (b)
distribution of θ angle.
.............................................................................
86
Figure 3.9 Representative volume element of short wavy steel
fiber composite
generated from micro-structural model with input parameters
achieved from
micro-CT information.
.............................................................................
89
Figure 3.10 Micro-CT image of SSFRP and a comparison between
real and
modeled waviness profiles using the developed micro-structural
model. 90
Figure 3.11 Probability density of the straightness parameter
Ps: comparison
between experimentally achieved (micro-CT) information and
mathematical
-
XXI
model. Histograms are the probability distributions achieved
from experiments
and model, fitting lines are normal probability fits of achieved
histogram
showing a clear agreement between Ps calculated from model and
experiments.
.................................................................................................................
91
Figure 4.1 Specimen preparation for single fiber test on the DMA
machine.100
Figure 4.2 Length distributions of (a) GF-PA and (b) GF-PP and
Lognormal
probability plots of (c) GF-PA and (d) GF-PP.
..................................... 103
Figure 4.3 Measured stress-strain curves and of the GF-PA and
GF-PP materials.
...............................................................................................................
104
Figure 4.4 Stress-strain curve of the polyamide Akulon K222-D
[273]. The tests
are stopped at the yield of the matrix.
................................................... 106
Figure 4.5 Stress-strain curve of the polypropylene matrix
[274]. The tests are
stopped at the yield of the matrix.
......................................................... 107
Figure 4.6 Acoustic Emission (AE) diagrams during quasi-static
loading of the (a)
GF-PA and (b) GF-PP materials. The figure shows plots of the
stress, AE
events energy, and cumulative AE energy with the evolution of
strains.109
Figure 4.7 Comparison of the cumulative AE energy registrations
of the GF-PA
and the GF-PP materials.
.......................................................................
111
Figure 4.8 Distribution of AE amplitudes in (a) GF-PA and (c)
GF-PP and AE
energies of (b) GF-PA and (d) GF-PP.
.................................................. 113
Figure 4.9 Global micro-CT scan of the overall width of the
GF-PP sample.116
Figure 4.10 Representative view of the skin-core morphology in
the central part
of a GF-PP sample.
...............................................................................
117
Figure 4.11 SEM micrographs of the fracture surface of the GF-PA
quasi-statically
failed sample. Green arrows denote the debonding damage
mechanism, red
arrows denote fiber pull-out, and the blue arrows denote “hills”
of matrix
around the fiber indicating strong fiber-matrix interface of the
GF-PA. 118
Figure 4.12 SEM micrographs of the fracture surface of the GF-PP
quasi-static
failed sample. Green arrows denote the debonding damage
mechanism and red
arrows denote fiber pull-out
..................................................................
120
Figure 4.13 Tensile stress-strain curves of the neat Durethan B
38 PA 6 material
(matrix material in SF-PA composite samples) at a cross-head
speed of 2
mm/min. Tests stopped at 150% strain.
................................................. 121
Figure 4.14 Measured stress-strain curves of single steel fibers
(fiber diameter 𝑑 = 8 μm, gauge length 𝐿 = 25 μm).
.......................................................... 122
Figure 4.15 Measured stress-strain curves of the SF-PA samples
with the different
investigated volume fractions.
...............................................................
123
Figure 4.16 The obtained quasi-static mechanical properties of
the SF-PA material
plotted against the fiber volume fractions of the
samples...................... 125
Figure 4.17 Acoustic Emission (AE) diagram of SF-PA materials
with the
different volume fractions considered in the present study.
Plots of the tensile
stress of each AE events energy, and cumulative energy of the
events against
-
XXII
the strain for (a) SF-PA 0.5VF%, (b) SF-PA 1VF%, (c) SF-PA 2VF%,
(d) SF-
PA 4VF% and (e) SF-PA 5VF%.
........................................................... 129
Figure 4.18 Comparison of the cumulative AE energy registrations
of the SF-PA
materials with the different fiber volume fractions.
............................... 130
Figure 4.19 Distribution of AE amplitudes in (a) SF-PA 2VF% (c)
SF-PA 4VF%
and AE energies of (b) SF-PA 2VF% (d) SF-PA 4VF%
....................... 131
Figure 4.20 Micro-CT scanned volumes of the undeformed SF-PA
samples with
different fiber volume fractions (a) 0.5VF%, (b) 2VF%, (c) 4VF%
and (d)
5VF%.
....................................................................................................
132
Figure 4.21 Small volumes of the micro-CT scanned undeformed
SF-PA samples
(a) 0.5VF% and (b) 2VF%.
....................................................................
134
Figure 4.22 View of voids formed in the undeformed 4VF% SF-PA
samples.135
Figure 4.23 High magnification SEM images showing the irregular
quasi-
hexagonal cross-section of the steel fibers embedded in the
matrix. ..... 136
Figure 4.24 SEM micrographs of the fracture surface of the short
steel fiber
composite samples with (a) 0.5VF%, (b) 1VF%,, (c) 2VF%, (d)
4VF%, and (e)
5VF%.
....................................................................................................
137
Figure 4.25 SEM micrographs of the voids observed at the
fracture surface of the
SF-PA samples of (a) 4VF% and (b) 5VF%.
.......................................... 138
Figure 5.1 Representative hysteresis loop (stress-strain
deformation curve) and the
linear regression fitting analysis for calculation of the
dynamic modulus of a
fatigue cycle.
..........................................................................................
146
Figure 5.2 Representative applied load diagram of the fatigue
tests on the tensile
tester performed on the SF-PA 2VF% samples.
..................................... 147
Figure 5.3 Measured S-N curves of the GF-PA and GF-PP samples.
Dashed lines
indicated 90% confidence level intervals. Arrows denote run-out
samples.150
Figure 5.4 Evolution of the measured hysteresis loops at 𝜎𝑚𝑎𝑥
=70% 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ, for the (a) GF-PA and the (b) GF-PP
materials. N/Nfailure indicate the stage of the sample life with
respect to the
failure cycle.
...........................................................................................
152
Figure 5.5 Evolution of the cyclic mean strain for the glass
fiber reinforced
composites with the load cycles, tested at 𝜎𝑚𝑎𝑥 =70% 𝑜𝑓 𝑡ℎ𝑒
𝑡𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ,.
............................................................
153
Figure 5.6 Evolution of the cyclic stiffness for the (a) GF-PA
and (b) GF-PP
materials.
................................................................................................
156
Figure 5.7 Evolution of the measured hysteresis loops of the
SF-PA material (at
55%UTS, 27.2 MPa). The legend indicates the cycle number of the
drawn
loops. The upper right graph shows more clearly the details of
the last
illustrated cycles.
....................................................................................
159
Figure 5.8 Evolution of the cyclic stiffness of the SF-PA
material at different stress
levels.
.....................................................................................................
160
-
XXIII
Figure 5.9 Representative evolution of the hysteresis loops of
the SF-PA in early
stages of the fatigue loading as observed in the short fatigue
tests performed
on a tensile tester.
..................................................................................
162
Figure 5.10 Evolution of the cyclic stiffness of the SF-PA
material with the
different stress level measured from the short fatigue tests
performed on the
tensile tester.
..........................................................................................
163
Figure 5.11 Representative evolution of the hysteresis loops of
the GF-PA in early
stages of the fatigue loading as observed in the short fatigue
tests performed
on a tensile tester.
..................................................................................
164
Figure 5.12 SEM micrographs of the fracture surface of fatigue
failed sampled of
the GF-PA material for the (a) 55 UTS%, (b) 65 UTS%, and (c) 70
UTS%
stress levels.
...........................................................................................
165
Figure 5.13 SEM micrographs of the fracture surface of fatigue
failed sampled of
the GF-PP material. (a) 55 UTS%, (b) 65 UTS%, and (c) 70 UTS%
stress
levels......................................................................................................
166
Figure 6.1 Equivalent ellipsoid replacing the original curved
fiber segment [294].
...............................................................................................................
174
Figure 6.2 Models used for validation of the P-I model: (a)
VE-Single half circular
fiber with constant curvature, (b) VE-Single sinusoidal fiber
with smooth
variable local curvature, (c) VE-Assembly of short steel fiber
with random
curvatures based on micro-CT images.
................................................. 176
Figure 6.3 Illustration of the P-I model concept and the ffect
of variation of the
efficiency factor 𝛃 on the dimensions of equivalent inclusions
(a) original fiber, (b) equivalent inclusions with 𝛃 = 𝛑𝟒, (c)
equivalent inclusions with 𝛃 = 𝛑𝟐.
................................................................................................
179
Figure 6.4 Comparison of the P-I model predictions for overall
elastic moduli of
the first test case with variations of efficiency factor β
against full FEA.180
Figure 6.5 Comparison of P-I model predictions of average local
stresses in
equivalent inclusions of the first test case (half circular
fiber) with variations
of efficiency factor β against full FEA (a) for axial segment
stresses 𝛔𝟑𝟑, (b) for transverse segment stresses 𝛔𝟐𝟐.
..................................................... 182
Figure 6.6 Comparison of P-I model predictions of average local
stresses in
equivalent inclusions of the first test case (half circular
fiber) with variations
of number of segments against full FEA (a) for axial segment
stresses 𝛔𝟑𝟑, (b) for transverse segment stresses 𝛔𝟐𝟐.
..................................................... 184
Figure 6.7 Comparison of P-I model predictions of average local
stresses in
equivalent inclusions of the first test case (half circular
fiber) with different
volume fractions against full FEA (a) axial segment stresses
𝛔𝟑𝟑, (b) transverse segment stresses 𝛔𝟐𝟐.
.......................................................... 185
Figure 6.8 Comparison of FE simulations on VE of original wavy
fiber (full FE)
and VEs of equivalent inclusions (a) for axial segment stresses
𝛔𝟑𝟑, (b) for transverse segment stresses 𝛔𝟐𝟐.
.......................................................... 187
-
XXIV
Figure 6.9 Comparison of the global maximum principal stress
predictions
𝝈𝒑𝒓𝒊𝒏𝒄𝒊𝒑𝒂𝒍 of P-I model of the second test case (sinusoidal
fiber) against full FE (a) transverse loading, (b) longitudinal
loading. P-I model generated with
20 segments.
...........................................................................................
188
Figure 6.10 Comparison of P-I model predictions of average local
stresses in
equivalent inclusions of the second test case (sinusoidal fiber)
with variations
of efficiency factor β against full FEA (a) for axial segment
stresses 𝛔𝟑𝟑, (b) for transverse segment stresses 𝛔𝟐𝟐.
..................................................... 190
Figure 6.11 Comparison of P-I model predictions of average local
stresses in
equivalent inclusions of the second test case (sinusoidal fiber)
with variations
of number of segments against full FEA (a) for axial segment
stresses 𝛔𝟑𝟑, (b) for transverse segment stresses 𝛔𝟐𝟐.
..................................................... 191
Figure 6.12 Comparison of P-I model predictions of average local
stresses in
equivalent inclusions of the third test case (VE of real fibers)
against full FEA.
The figure shows the comparison for an example of two selected
fibers from
the VE for (a) for axial segment stresses 𝛔𝟑𝟑 and (b) for
transverse segment stresses 𝛔𝟐𝟐 of 10 fibers in the modelled VE.
....................................... 194
Figure 7.1 Determination of the outward normal and the local
interfacial stress
vectors around the equator of the inclusion. 𝑛 (or 𝑛𝑖 in index
notation) is the outward normal vector, 𝜎𝑖𝑜𝑢𝑡 is the stress vector
(𝜎𝑁, normal component and 𝜏, shear component) at an interfacial
point 𝐴 with an in-plane angle θ. 204
Figure 7.2 Example of a partially debonded inclusion (a)
computation of the
damage parameters (d, γ, δ) and (b) demonstration of the higher
and lower
zones of an inclusion quadrant for calculation of 𝛾ℎ and 𝛾𝑙.
................. 206
Figure 7.3 Flowchart of a single load step of the developed
damage model.211
Figure 7.4 Manufacturing simulation of the dog-bone samples.The
figure shows
(a) a schematic of the typical geometry of a dog-bone sample
[54] and (b) an
example of the results of the manufacturing simulation (of the
GF-PP in this
plot) at different points across the width of the samples.
....................... 217
Figure 7.5 Results of the main component of the orientation
tensor 𝑎11in the central section for the (a) GF-PA and (b) GF-PP
samples. .................... 218
Figure 7.6 Manufacturing simulation of the SF-PA samples. The
figure shows the
results of the main component of the orientation tensor 𝑎11 of
the SF-PA 2VF% as an example of the SF-PA materials.
....................................... 220
Figure 7.7 Experimental stress-strain curves of the GF-PBT
material with the
different orientations of the specimens 𝜙 = 0, 45, 90° . Data
obtained from [308].
......................................................................................................
222
Figure 7.8 Stress-strain curve of the BASF Ultraduur B4500
[273]. The tests are
stopped at the yield of the matrix.
.......................................................... 222
Figure 7.9 Comparison of the experimental and predicted
stress-strain behavior of
the GF-PA composite.
............................................................................
224
Figure 7.10 Comparison of the experimental and predicted
stress-strain behavior
of the GF-PP composite.
........................................................................
225
-
XXV
Figure 7.11 Simulated stress-strain curves of the SF-PA 2VF%
composite with
different values of critical interface strength 𝜎𝑐 in the damage
model. . 227
Figure 7.12 Comparison of the experimental and predicted
stress-strain behavior
of the SF-PA 0.5VF% composite.
......................................................... 228
Figure 7.13 Comparison of the experimental and predicted
stress-strain behavior
of the SF-PA 2VF% composite.
............................................................
228
Figure 7.14 Comparison of the predicted and experimental Young’s
modulus of
the SF-PA materials with the different fiber volume fraction.
.............. 230
Figure 7.15 Comparison of the experimental and predicted
stress-strain behavior
of the GF-PBT 0 composite.
..................................................................
231
Figure 7.16 Comparison of the experimental and predicted
stress-strain behavior
of the GF-PBT 45 composite.
................................................................
231
Figure 7.17 Comparison of the experimental and predicted
stress-strain behavior
of the GF-PBT 90 composite.
................................................................
232
Figure 8.1 Schematic diagram representing the objective of the
fatigue model
developed in the present study.
.............................................................
239
Figure 8.2 Schematic representation of the fatigue failure
functions 𝑋𝑓,𝑋𝑖 and 𝑋𝑚 at a current load cycle 𝑁𝑐 during the
fatigue simulation. ...................... 242
Figure 8.3 Flowchart of a single load cycle 𝑁 of the developed
fatigue model.
...............................................................................................................
244
Figure 8.4 S-N curve of single glass fibers used as input for
the fatigue model
[318].
.....................................................................................................
246
Figure 8.5 S-N curve of the PA 6 matrix used as input for the
fatigue model [58].
...............................................................................................................
247
Figure 8.6 S-N curve of the PP matrix used as input for the
fatigue model [319].
...............................................................................................................
248
Figure 8.7 Experimental S-N curves of the GF-PBT material with
the different
orientations of the specimens 𝜙 = 0, 45, 90°. Data obtained from
[308].249
Figure 8.8 S-N curve of the PBT matrix used as input for the
fatigue model [320].
...............................................................................................................
250
Figure 8.9 Comparison of the experimental and predicted S-N
curves of the GF-
PA composite. Dashed lines indicate the experimental 90%
confidence level
intervals. Arrows denote run-out samples A parametric study of
the effect of
the variation of the slope of the S-N curve of the interface 𝑏
is shown. 251
Figure 8.10 Illustration of the theoretical fatigue S-N curves
of the interface of the
GF-PA material with the different valies of the fatigue strength
exponent 𝑏.
...............................................................................................................
252
Figure 8.11 Comparison of the experimental and predicted S-N
curves of the GF-
PA composite. A parametric study of the effect of the variation
of the slope of
the S-N curve of the interface 𝑏 is shown.
............................................ 253
-
XXVI
Figure 8.12 Illustration of the theoretical fatigue S-N curves
of the interface of the
GF-PP material with the different values of the fatigue strength
exponent 𝑏.
...............................................................................................................
253
Figure 8.13 Comparison of the experimental and predicted S-N
curves of the GF-
PBT 𝜙 = 0 composite. A parametric study of the effect of the
variation of the slope of the S-N curve of the interface 𝑏 is
shown. ............................... 254
Figure 8.14 Illustration of the theoretical fatigue S-N curves
of the interface of the
GF-PA material with the different values of the fatigue strength
exponent 𝑏.
...............................................................................................................
255
Figure 8.15 Comparison of the experimental and predicted S-N
curves of the GF-
PBT 𝜙 = 45 composite. A parametric study of the effect of the
variation of the slope of the S-N curve of the interface 𝑏 is
shown. .......................... 256
Figure 8.16 Comparison of the experimental and predicted S-N
curves of the GF-
PBT 𝜙 = 90 composite. A parametric study of the effect of the
variation of the slope of the S-N curve of the interface 𝑏 is
shown. .......................... 256
Figure 8.17 Schematic representation of the micro-scale
modelling methodology
developed in the present thesis.
..............................................................
259
Figure 8.18 Flowchart describing the current component level
solution for the
fatigue simulation of SFRPs.
..................................................................
260
Figure 8.19 Illustration of the considered industrial component.
The component is
denote “Pinocchio”.
...............................................................................
263
Figure 8.20 Boundary conditions in the simulations of the
Pinocchio component.
(a) “fixing” constraints in XY direction are applied on the
holes indicated by
the arrows, (b) Load is applied in Z direction along the
highlighted line to
simulate bending stresses.
......................................................................
264
Figure 8.21 Quasi-stating 3 point bending load displacement
curves of the
performed tests on the Pinocchio component.
........................................ 265
Figure 8.22 Stress fields in the Pinocchio component as
predicted by the FE model.
...............................................................................................................
266
Figure 8.23 Full field strain mapping during the quasi-static
tests of the Pinocchio
component and the definition of the location of the extraction
of strain values
for comparison with the FE model.
........................................................ 266
Figure 8.24 Comparison of the DIC and FE extracted 𝜀𝑦𝑦 plotted
against the axial position in pixels on the registered suface. The
figure show the plots for a
displacement of 0.96 (load of 1.02KN) for (a) Line 1, (b) Line 2
and (c) Line
3.
............................................................................................................
268
Figure 8.25 Comparison of the experimental and predicted S-N
curve of the
Pinocchio component.
............................................................................
269
-
XXVII
List of tables
Table 3.1 Main geometrical input parameters used for the
mathematic model. .. 88
Table 4.1 Injection molding parameters of the glass fiber and
steel fiber
samples................................................................................................................
97
Table 4.2 Average fiber lengths of the SF-PA samples with
different fiber volume
fraction. ………………………………………………………………………. 102
Table 4.3 Tensile properties of the short glass fiber polyamide
(GF-PA) and
short glass fiber polypropyelene (GF-PP) composites.
.................................... 105
Table 4.4 Tensile properties of the neat Durethan B 38 PA 6
material. Comparison
between achieved results and manufacturer’s datasheet values.
……………… 122
Table 4.5 Tensile properties of single steel fibers.
……………………………. 123
Table 4.6 Summary of the tensile properties of the SF-PA
composites with the
different fiber volume fractions.
........................................................................
124
Table 5.1 Tested stress levels in the fatigue tests of the
investigated glass fiber
reinforced composites. ………………………………………..........................
145
Table 5.2 Tested stress levels in the fatigue tests of the
investigated steel fiber
reinforced composites. .....…………………………………………................
145
Table 5.3 Summary of the cycle at which 50% of the stiffness
degradation of the
SF-PA material occurred with the different applied stress
levels. …………….. 161
Table 7.1 Summary of the micro-structural parameters of the
GF-PA and the GF-
PP materials of the present work used as input for validation of
the developed
models. …………………………………………………..................................
219
Table 7.2 Summary of the micro-structural parameters of the
SF-PA materials of
the present work used as input for validation of the developed
models.
………………………………………………………………………………... 221
Table 7.3 Summary of the micro-structural parameters of the
GF-PBT materials
used as input for validation of the developed models. …………………………
223
-
1
Chapter 1: Introduction
-
Introduction
3
1.1 General Introduction
In the recent years, there has been an increasingly growing
interest in fiber-
reinforced composites as a replacement of metals and alloys in a
number
of engineering structures, owing to the favorable
characteristics of
composite materials. The major advantage of composite materials
over
metals is their superior specific properties e.g., specific
strength and
stiffness (strength-to-weight ratio and stiffness-to-weight
ratio,
respectively). Major industrial sectors have contributed to the
growth of
composite technologies. On one hand, the aeronautics industry
has largely
invested in the development of composites design and
manufacturing
technologies. At present, more than 50% of the “next-generation”
Airbus
aircraft A350 XWB is made of composites [1]. On the other
hand,
stipulated by the lawful regulations of CO2 reductions, the
automotive
industry has become today the largest consumer of the overall
types of
composite materials, accounting for over 20% of total
consumption [2].
Composites are a vast group of materials presenting itself in
large
variations of matrix materials, reinforcement types and
micro-structures.
On the industrial scale, polymer composites and especially those
based on
thermoplastic matrices are the most attractive types, offering
the needed
weight reductions, superior mechanical properties and high
durability.
Thermoplastic composites exhibit the added advantages of
recyclability
and lower energy processing, compared to their thermoset
counterparts.
From a structural viewpoint, these materials can be
distinguished in two
main categories which are continuous and discontinuous (or
short) fiber
reinforced composites.
Composites with the best mechanical performance are those
with
continuous fibers. However, these materials cannot be adopted
easily in
mass production and are confined to applications in which
property
benefits outweigh the cost penalty [3]. In this respect, the
aerospace
industry has pioneered the use of high performance continuous
fiber
composites in structural applications regardless of cost and
using cost-
intensive manufacturing methods such as autoclave manufacturing
and
hand lay-up. In contrast, the focus of the automotive industry
has been on
semi-structural components using short fiber composites [4,
5].
A number of processing techniques exist for the production of
short fiber
reinforced polymers (SFRPs). For thermosetting materials the
most
common processes are Sheet Molding Compound (SMC) and Bulk
-
CHAPTER 1
4
Molding Compound (BMC) processes. Extrusion compounding and
Injection Molding (IM) are the conventional techniques for
production of
thermoplastics composites [6].
Injection molding remains the most attractive manufacturing
method
allowing the production of components with intricate shapes at a
very high
production rate, with reasonable dimensional accuracy and fairly
low costs.
The versatility and low cost of the injection molding process
led to its
increased use, largely in the automotive industry, but also in
different
applications such electrical and electronic industries, sporting
goods,
defense sector and other consumer dominated products.
Despite of those advantages, injection molded short fiber
composites
depict a more complex morphology compared to other composite
types.
Increased fiber damage and complex melt flow behavior during
processing
give rise to random micro-structures characterized by
statistical fiber
length distributions (FLD) and fiber orientation distributions
(FOD). An
important and distinctive feature of SFRP parts is then the
variability of
the material properties throughout the part and hence, the
anisotropy of the
local properties. As a result, those materials are often
referred to as random
fiber reinforced composites (RFRCs).
Another complexity of the short random fiber composites is the
nature of
the fiber matrix interface which is dependent on the
compatibility of the
fibers and matrix materials and on the processing conditions.
The quality
of the fiber-matrix interface has significant impact on the
efficiency and
load-carrying capability of short fiber composites.
Fibers used in SFRPs are typically glass fibers and carbon
fibers. A number
of studies investigated the potential of natural fibers as a
replacement of
synthetic fibers SFRPs [7-9]. Metal fibers have been used to
provide
shielding and electrical conductivity [10-12]. Among the
different metallic
fibers materials are steel fibers, which are highly efficient
in
electromagnetic shielding at very low fiber volume fractions.
In
conjunction with electromagnetic properties, steel fibers depict
superior
mechanical properties (stiffness of about 200 GPa and strength
of about 2
GPa), which are comparable to high performance carbon fibers.
This
makes stainless steel fibers attractive for further
investigations in
mechanical applications.
One of the leading manufacturers of steel fibers is the Flemish
company
Bekaert. Since the 1990s the company has been performing
research on
-
Introduction
5
their steel fiber products available under the commercial name
Beki-
Shield. While the Beki-Shield fibers were initially targeted
only towards
Electromagnetic interference (EMI) shielding, recent research
efforts
include the investigation of steel fiber composites in
mechanical
applications.
An important characteristic of injection molded steel fiber
composites is
the waviness of the fibers embedded in the matrix. This
characteristic
waviness also exists in long carbon fibers, natural fibers,
crimped textiles
and non-woven composites. The inherent waviness of steel
fibers
embedded in the matrix, as a result of processing, further adds
to the
complexity of the RFRCs micro-structure.
Finally, automotive components, along with most other
engineering
applications, are often subjected to cyclic loading, resulting
in damage and
material property degradation in a progressive manner [13, 14].
The
penetration of short fiber composites in fatigue sensitive
applications
places focus on the durability aspects of those materials. This
leads to a
large interest in understanding the different durability and
fatigue behavior
aspects of this class of materials.
1.2 Scientific & Technological Context
Complete design of an SFRP component is a complex undertaking,
which
should simultaneously take into account different factors such
as loading,
weight reduction, part stiffness and durability. Exhaustive
testing and
trials-and-error are not effective ways due to the high
variability of material
and micro-structure parameters, part/mold geometries and
manufacturing
routes. In sectors where performance to cost ratios define
competitiveness,
like the automotive industry, a possibility to make design
decisions based
on accurate numerical models and virtual testing of the part is
a crucial
factor. Missing durability performance simulation tools are a
key
restricting factor for wider use of SFRP materials in cars.
To date, predictive models of fatigue behavior of composites are
largely
restricted to continuous fiber systems [13]. A large number of
the available
models for these composites are phenomenological models which
usually
require a large number of experiments and test data for each
kind of
material in question. Examples can be found in e.g. [15-18].
A challenging question remains if it is possible to model the
fatigue
behavior and lifetime of composites based on the behavior of
the
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CHAPTER 1
6
constituents (i.e. matrix, fibers, and interface) and actual
micro-scale
damage phenomena. The question is challenging, even for the
more
established continuous fiber composites where only a few
attempts can be
found in literature, e.g. in [19, 20].
The fatigue behavior of random fiber composites is much less
understood.
Similar to continuous fiber composites, a few phenomenological
based
models have emerged for modelling the fatigue behavior of
random
composites. Examples include e.g. [21-23]. Models linking the
fatigue
behavior of short random fiber composites to the behavior of
constituents,
do not exist, to the knowledge of the author. This results in
the need for
research efforts targeted towards the development and validation
of
efficient and robust models for prediction of the fatigue
behavior of RFRCs
based on the behavior of the underlying constituents, local
stress states and
actual damage mechanisms.
Additionally, modelling RFRC materials requires addressing the
multi-
scale behavior of the material. As mentioned above, a real
component of
random fiber composites produced with a manufacturing process
such as
the injection molding technique often has a complex geometry,
which
results in large variations of local micro-structure between
different points
along the part. In this respect, modelling the behavior of RFRC
materials
often requires multi-scale approaches.
Another challenge in the context of this work is understanding
and
modelling the behavior of short steel fiber composites. While
such material
is attractive due to the superior properties of steel fibers, it
exhibits several
differences from the generally used glass and carbon fiber
composites. On
one hand, the random waviness of the fibers adds to the
complexity of the
micro-structure. This also results in challenges in
incorporating the
waviness aspects of the fibers in geometrical and mechanical
models. On
the other hand, information about the mechanical behavior of the
steel fiber
composites as well as their distinct characteristics, such as
the nature of
fiber-matrix interface and the effects of the high stiffness
mismatch
between fibers and matrix, are not available due to novelty of
the material.
Finally, in the last decades, Finite Element (FE) based
simulation tools
have been commercially available. In the present technological
context,
one of the commercially available software packages is the
Siemens LMS
Virtual.Lab Durability software. Existing algorithms of the
software
include complete solutions for modelling metal fatigue under
variable
conditions of designs and complex loading states. A current
objective is
-
Introduction
7
the extension of the software solutions to the complex random
fiber
composites led by the increase of demand of the material in
automotive
applications.
1.3 Objectives of the PhD research
In view of the above mentioned scientific and technological
context, the
ultimate objective of the work is the formulation and validation
of
methodologies that enable the simulation of the fatigue behavior
of RFRC
components. As mentioned above, a complete fatigue simulation of
an
RFRC component requires a multi-scale modelling approach. Figure
1.1
illustrates an overview of the proposed solution used in this
PhD thesis.
Figure 1.1 Overview of the multi-scale predictive methods for
modelling the
fatigue behavior of RFRC parts.
The procedure starts with process (manufacturing) modelling
for
simulation of the injection molding of the component in
question. Such
simulations are available in different commercial packages such
as:
Moldflow, SigmaSoft, and Express, to name a few. Based on the
part
geometry and melt flow behavior of the material, the software
tools are
able to predict the local fiber orientation, which can be later
mapped to FE
meshes.
Virtual.Lab
Durability
Process model (MoldFlow,
SigmaSoft, etc.)
Fiber and matrix
data
Microscopic modelling
Material
parameters Pre-
Damage Feedback
loop
Local S-N
curves
FEA
Fatigue loading at elements
FE loading
Fatigue life of the
part
Local
stiffness
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CHAPTER 1
8
At the microscopic level, models need to be developed with the
end goal
of the accurate prediction of local lifetime, i.e. stress vs.
number of cycles
to failure (S-N) curves. This in turn can be achieved with a
series of
simultaneous micro-scale models. These include micro-structural
models
to generate statistically representative local geometries taking
into account
input of the preceding manufacturing simulation, quasi-static
mechanical
models for prediction of the local behavior and fatigue models
for
prediction of the local S-N curves.
At the macroscopic scale, Finite Element Analysis (FEA) is
performed on
the component level. Fatigue loading is applied and the
durability software
is able to solve the local multi-axial loading conditions at
each element.
The local stiffnesses and S-N curves are inputted to the
durability solver
by interaction with the micro-models. Based on the input of the
local
stiffnesses and S-N curves, the durability solver is able to
predict the
critical areas as well as the overall fatigue lifetime of the
component. The
solver includes so-called “feedback” algorithms.
While at the micro-scale full FEA modelling can be applied for
the
prediction of the local stress states, local damage and final
S-N curves at
each element, this approach leads to high computational
expensive
solutions which are inadmissible in consideration of the above
described
industrial requirements. The alternative route is the use of
suitable
analytical approaches which allow the estimation of the local
material
states with reasonable accuracy at efficient computational
speeds. Among
these approaches are the well-known mean-field homogenization
methods.
The position of this PhD work within the above described process
is the
micro-scale modelling (highlighted in Figure 1.1) of the
quasi-static and
fatigue behavior of RFRCs. For fatigue modelling, a novelty of
the work
is the ability to predict the S-N curves of the composite based
on the S-N
curves of the constituents (i.e. matrix, fibers and interface)
using detailed
micro-mechanics. As mentioned above, such methods are not
available in
literature. Another novelty of the work is that in addition to
the typical
short straight fiber reinforced materials, the thesis considers
the application
of micro-mechanical models to wavy fiber reinforced composites
e.g. the
steel fiber materials discussed above. The methodologies
developed in this
work can be applied to a number of other crimped fiber
systems.
-
Introduction
9
The main objectives of the thesis can then be summarized as
follows:
- Characterizing and modelling the complex micro-structure of
short wavy steel fiber composites and understanding the behavior of
this
novel class of materials.
- Assessment and validation of models for extension of the
mean-field homogenization techniques to short wavy fiber reinforced
composites.
- Development and validation of a modelling approach for the
prediction of the quasi-static behavior and progressive damage of
short fiber
composites, based on mean-field homogenization methods.
- Formulation and validation of a fatigue model in the context
of mean-field homogenization methods, for the prediction of the
fatigue
behavior based on the input of the fatigue properties of the
constituents.
- Detailed experimental investigations of the quasi-static and
the fatigue properties of random straight and wavy fiber reinforced
composites for
better understanding of the underlying damage phenomena and
for
validation of the developed models.
1.4 Structure of the thesis
The structure of the thesis follows the objectives described in
the previous
section. A schematic overview of the thesis is presented in
Figure 1.2.
Chapter 2 of the thesis is devoted to the study of the
literature and
introduces general knowledge of the available methods for RFRCs.
The
chapter gives an overview of the micro-structure of RFRCs and
the factors
affecting the mechanical behavior of RFRCs. A review is given on
the
different methods and concepts of simulation of the geometry of
RFRCs.
The chapter also gives a brief description of the different
mean-field
homogenization techniques as well as the available models for
the quasi-
static and progressive damage models of RFRCs. Finally,
different
attempts for micro-mechanical fatigue modelling of RFRCs are
discussed.
-
CHAPTER 1
10
Figure 1.2 Outline of the PhD thesis.
Motivation
Novelty
Chapter 2.
State of the art
Chapter 3.
Geometrical
characterization
and modelling
Chapter 1.
Introduction
Chapter 4.
Experimental
characterization
quasi-static
behavior
Chapter 5.
Experimental
characterization
fatigue
behavior
Chapter 6.
Linear elastic
modelling of
wavy RFRCs
Chapter 7.
Quasi-static
modelling of
RFRCs
Chapter 8.
Fatigue
modelling of
RFRCs
Chapter 9.
Conclusions and future
perspectives
-
Introduction
11
Chapter 3 describes the developed geometrical model for the
generation of
volume elements (VEs) of RFRCs. The model is able to generate
VEs of
both straight and wavy fiber composites. As in the published
literature,
different models are available for generation of random straight
fiber
composites, the chapter is focused on the aspects of the model
concerned
with the description of wavy fibers. In parallel to the
modelling attempts,
a novel experimental methodology for characterization of the
micro-
structure of complex wavy fiber samples, based on
micro-computer
tomography (micro-CT) techniques, is discussed.
Chapters 4 and 5 cover the performed experimental investigations
for
quasi-static and fatigue behavior respectively of short glass
fiber and short
steel fiber reinforced composites. The different
characterization techniques
e.g. mechanical testing, fractography analysis, full-field
strain mapping
and acoustic emission techniques are discussed. The achieved
experimental results provide a better understanding of the
behavior of
random fiber reinforced composites, which will be reflected in
the
development of the models. The results of those chapters also
serve as
validation for the models developed in the subsequent
chapters.
Chapter 6 deals with the extension of the existing
mean-field
homogenization methods for wavy fiber reinforced composites. A
model
for the transformation of wavy fibers into equivalent straight
fiber systems
that are able to be modelled using mean-field techniques is
presented and
validated with full FEA.
Chapter 7 presents the developed methods for the quasi-static
damage
modelling of RFRCs. This includes models reflecting the
damage
phenomena of short fiber composites i.e. fiber matrix debonding,
and fiber
breakage and models for the non-linear plastic deformation of
the matrix.
The models are applied on the VEs generated by the geometrical
model
explained in chapter 3. For wavy fiber composites, the
additional model
developed in chapter 6 is applied prior to the quasi-static
modelling. The
implementation of the model in a numerical tool is briefly
presented.
Validation of the models with experimental results is reported
in the
chapter.
Chapter 8 is devoted to the fatigue model. This in turn is
dependent on the
quasi-static models in Chapter 7. Similar to the quasi-static
models,
numerical implementation of the models is discussed. A detailed
validation
with the experimental results is presented. The chapter also
gives a brief
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CHAPTER 1
12
overview of attempts for component level simulation and
validation, with
the connection with the micro-scale models developed in this PhD
thesis.
Chapter 9 concludes the thesis and provides perspective for
future research
work.
-
13
Chapter 2: State of the Art
-
State of the Art
15
2.1 Introduction
In this chapter, a detailed overview of the available methods
for modelling
the geometry and the quasi-static and the fatigue behavior of
random short
fiber reinforced composites will be presented. In order to model
the
material behavior, an understanding of the unique
micro-structure of short
fiber composites and the different factors affecting its
mechanical behavior
is needed. This in turn can be achieved using a synopsis of
available
experimental observations.
The structure of the chapter will be explained in the following.
As
discussed in the introduction, the injection molding process is
the most
attractive and commonly used manufacturing technique for short
fiber
composites. In the first section of this chapter, this
manufacturing process
will be briefly discussed in order to understand the different
processing
factors affecting the final random fiber composite parts. Next,
details of
experimental observations in literature of the evolution of the
micro-
structure of short fiber composites will be given, followed by
an overview
of the factors affecting both the quasi-static and the fatigue
behavior of
RFRCs supported by key literature results. Injection molded
components
are considered in this thesis as the most common RFRCs as well
as the
ones with relatively more complex micro-structures. The
developed
concepts and models can also be applied to other types of
RFRCs.
The following parts of the review will be dedicated to modelling
the
behavior of RFRCs. This starts with an overview of the available
methods
for generation of representative volume elements which are able
to
simulate the complex micro-structure of RFRCs, and of important
factors
to be taken into consideration such as the size of those
representative
volumes. In the subsequent section, mean-field homogenization
methods
will be introduced and examination of the variations of the
different mean-
field models will be given. Focus will be given on the original
concepts of
the models, namely the Eshelby solution. The Mori-Tanaka model
which
is the most commonly used out of the different mean-field
methods for
modelling RFRCs will be discussed in more detail. Moreover,
an
important aspect considered in this review is outlining the
different
limitations of the Mori-Tanaka model and how these were
addressed in
literature.
Mean-field homogenization models, as will be shown in section
2.5, were
first intended for modelling the elastic behavior of composites.
In the next
section, the different methods for extending the mean-field
models to
-
CHAPTER 2
16
describe the non-linear behavior of short fiber composites will
be given.
The sources of non-linearity are typically the elasto-plastic
behavior of the
thermoplastic matrix and the different damage mechanisms of
the
composite. Finally, an outline will be given on the few attempts
conducted
in previous research for modelling the fatigue life of short
fiber composites.
It should be noted that this literature review discusses general
concepts of
short random fiber composites. An important part of this thesis
aims at
understanding and formulation of methods for modelling the
micro-
structure and mechanical behavior of wavy fiber composites. The
example
considered in this work is short steel fiber composites. The
next chapter of
this thesis is devoted to modelling the micro-structure of
complex wavy
short steel fiber reinforced composites. The chapter will also
include
details of the motivation for investigating this novel class of
materials, the
production process of micron-sized steel fibers and efforts
for
characterizing and modelling similar wavy micro-structures.
2.2 Injection Molding of RFRCs
As mentioned in section 1.1, injection molding provides a very
attractive
and cost effective way of manufacturing short fiber reinforced
composites
[24]. Figure 2.1 shows a schematic diagram illustrating the
injection
molding process.
-
State of the Art
17
Figure 2.1 Schematic illustration of the injection molding
process (adapted from
[25]).
The raw material used for the injection molding process are
compounded
pellets of the desired thermoplastic/fiber materials combination
and
volume fractions. Prior compounding can be performed using
methods
such as extrusion or high shear mixing. Compounding already
results in
damage of the fiber with stochastic nature and consequently
development
of a length distribution of the fibers in the pellets.
During injection molding, the pellets are fed to the hopper and
the injection
molding cycle begins. The material is heated and its viscosity
is reduced.
This enables flow of the polymer compound with the driving force
of the
injection unit, during which stage, shear forces are exerted by
the screw.
This adds a significant amount of friction on the material prior
to injection.
In the next stage, a desired amount of molten material is stored
in front of
the tip of the screw and is then pushed into the closed mold. A
cooling
cycle begins, and after the material is cooled down and
solidified in the
mold the part is ejected.
-
CHAPTER 2
18
2.3 Micro-structure and Mechanical Behavior of RFRCs
2.3.1 Micro-structure of RFRCs
The performance of short fiber composites is governed by the
complex
geometry of the fibers and their distribution in the part
[26-32]. Unlike
continuous UD or textile fiber reinforced composites, short
fiber reinforced
composites depict stochastic geometrical features that evolve
during
processing [33]. During the injection molding process, as
briefly discussed
in section 2.2, high shear stresses exerted in the melt by the
screw rotation,
in addition to fiber-fiber interactions, lead to further fiber
breakage (to the
already damaged fibers from the compounding process), resulting
finally
in a range of fiber lengths, characterized by a length
distribution function
(FLD) [34-36]. The complex