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I
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
II
© 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.
III
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.
VI
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
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.
VIII
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.
IX
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.
X
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.
XI
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
XIII
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
XIV
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
XV
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
XVI
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
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
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
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
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
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
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
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