Micro-Mechanics Based Fatigue Modelling of …...VII Abstract Short fiber composites, are...
Transcript of Micro-Mechanics Based Fatigue Modelling of …...VII Abstract Short fiber composites, are...
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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
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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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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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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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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
휀�̇� Matrix strain rate
휀𝑝∗ 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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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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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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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
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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 휀𝑦𝑦 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 flow of the melt, both in the screw area and
in the mold, results in variations of fiber orientations over the part, locally
characterized by a fiber orientation distribution function (FOD). The
orientation of a single fiber, and consequently the orientation distribution
of the assembly of fibers, can be described in a spherical coordinate system
by two angles: Φ and [34, 37, 38] as shown in Figure 2.2.
Figure 2.2 Fiber orientation described with a direction 𝒑 and corresponding
angles Φ and .
Where 𝒑 is the fiber orientation (or direction) vector, Φ and are the in-
plane and out-of-plane orientation angles respectively. The exact resulting
fiber orientation distribution of the final part depends on different factors
𝑝
Φ
3
1
2
State of the Art
19
such as the part geometry, mold design, viscoelastic behavior of the matrix,
melt and mold temperature and the processing parameters [39].
Instead of using a detailed statistical orientation distribution, Advani and
Tucker [32] developed a concise description of the orientation distribution
known as the “orientation tensor”. In their well-known paper, the authors
gave two variations of the orientation tensors namely the 2nd order and 4th
order orientation tensors 𝑎𝑖𝑗 and 𝑎𝑖𝑗𝑘𝑙 respectively. The orientation
distribution is represented more accurately with the higher order tensor.
The authors have shown though that the 2nd order tensor can represent the
fiber orientation well enough to predict the elastic properties of short fiber
composites. The details of the formulations of the component of the
direction vector 𝒑 and the Advani and Tucker’s orientation tensor will be
given in Chapter 3 of this thesis.
For the injection molded RFRCs, a preferential orientation of the fibers is
commonly found due to the shear stresses between the mold and the melt
[21, 40, 41], this results in the development of a layered-like micro-
structure. Detailed morphology of a typical random fiber reinforced
composite is shown in Figure 2.3 (a).
The different layers can be explained as follows:
The first layer, closest to the mold walls, is called the “skin layer”
and comprises fibers with random orientation. This can be
attributed to the temperature difference between the melt and the
mold walls, where as soon as the melt is in contact with the
relatively cold walls, the melt solidifies instantly with a random
orientation. The skin layer is often very thin.
The next adjacent layer is called the “shell” layer. This region is
governed by the shear forces which tend to cause alignment of the
fibers in the melt flow direction (MFD).
A “transition” layer then follows at a relative distance from the
mold walls where the effect of the shear forces, and hence the
alignment of the fibers decreases.
Finally, in the “core” at the middle of the sample the fibers tend to
orient perpendicular to the MFD due to the uniform velocity
profile in the layer and no shear induced displacements of the
polymer melt [42].
CHAPTER 2
20
Notion of the layered morphology of RFRCs is often simplified to the
terminology of “skin-core” morphology.
The morphology depicted in Figure 2.3 (a) can be described by the above
mentioned Advani and Tucker orientation tensors. Figure 2.3 (b) shows the
evolution of first component 𝑎11 of the 2nd order orientation tensor through
the thickness of the sample. Noting that due to the layered morphology the
value of 𝑎11 is different in each layer.
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].
skin layer
shell layer
shell-core
transition
layer
core layer
MFD
(a)
(b)
State of the Art
21
The orientation of RFRCs can be obtained in different ways. The first way
is by direct measurement. Different experimental methods for
characterization of the fiber orientations will be discussed in Chapter 3.
Another way is through numerical simulation of the manufacturing process
of the part. Different models exist in the software packages to simulate the
processing, flow of the melt and diffusion kinetics. The first commonly
used model was the Folgar-Tuker model. Recent improvements include the
reduced strain closure (RSC) model and the anisotropy rotary diffusion
(ARD) model. A review of the accuracy of the models can be found in
[44]. The exact geometry of the part is modelled and the processing
conditions such as temperature, pressure and injection speed are inputted
to the model. The result is a prediction of the orientation tensor at each
point of the component. While the experimental approach can be used for
simple lab-scale components such as simple plates or dog-bone samples,
which typically depict few variations of the orientation, this approach
cannot be used for real structural components. Typically, SFRP
components exhibit variable local statistical fiber orientation distribution
leading to different material properties at different points in the part. This
makes commercial simulation tools very attractive as a first step for
simulating the overall mechanical behavior, which necessitates the
knowledge of the local micro-structure. The most commonly used
commercial software are: MoldFlow, EXPRESS, Moldex3D and
SigmaSoft.
2.3.2 Factors affecting the quasi-static and fatigue behavior of
RFRCs
The mechanical response of RFRCs depends not only on the properties of
the individual constituents but also on the local micro-structure, as
explained above. The detailed list of morphological parameters which
affect the effective response of RFRCs include the fiber volume fraction,
fiber shape, fiber length and orientation distributions, the spatial dispersion
(arrangement) of the fibers in the matrix and finally the state of the
interface, i.e. strong or weak bonding of fibers and matrix at the interface.
The effect of micro-structure is even more significant on the damage and
failure properties of the composite, than the stiffness, as damage in
composites is a local phenomenon.
A large number of available literature is dedicated to the experimental
investigation of the effect of the above mentioned parameters on the quasi-
static behavior of short fiber composites, especially the fiber volume
fraction, fiber length (and consequently aspect ratios) and orientation
CHAPTER 2
22
distributions e.g. [29, 45-47]. In general, the trend reported by authors is
the increase of the composite stiffness and strength with increasing fiber
content as well as with increasing fiber length as shown in Figure 2.4. The
figure presents experimental strengths values of a number of different
random discontinuous fiber composites, namely random short glass fiber
polypropylene composites with fiber diameters of 14 and 19 µm (SF 14
and SF19 respectively), and a long random glass fiber reinforced
polypropylene composite with 19 µm (LF 19). For all presented
composites, a trend of the increase of the tensile strength with the
increasing fiber volume fraction is observed. The comparison between the
SF 19 and the LF 19 composites shows the effect of the increase of fiber
length on the increase of the tensile strength of discontinuous composites.
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].
Similarly, authors reported the significant effect of the fiber orientation on
the overall stress-strain behavior of the composite, e.g. Bernasconi et al.
[21] and De Monte el a. [22] . The results of Bernasconi et al. are shown
in Figure 2.5.
♦ LF19, ■ SF19, ▴ SF14
State of the Art
23
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.
The spatial arrangement in random fiber reinforced composites is a result
of the manufacturing process and is difficult to control. For this reason, the
effect of the varying spatial distributions is difficult to characterize
experimentally. The effect of interface cannot be easily characterized also,
although it can be estimated from studies concerned with improvement of
the fiber-matrix interface using e.g. treatment of the fiber surface with a
coupling agent. The general trend is the strong increase of the strength of
composites with interface improvement. This is a consequence of the
significant effect of the interface on the damage behavior of the composite,
where weak interfaces result in early and extensive debonding which
inhibit the load-transfer between the matrix and fiber and result in reduced
efficiency of the fiber.
All of the above factors influence both the quasi-static and fatigue behavior
of RFRCs. Moreover, the fatigue behavior of those composites is
influenced by other factors such as the mean applied stress, or the stress
ratio, and the frequency of testing. Also, attention is given to the effect of
environmental factors such as temperature and humidity.
In the following, focus will be given to the effect of different factors on the
fatigue behavior of RFRCs. A selection of some of the most relevant
(a) (b)
CHAPTER 2
24
effects will be discussed and some key figures will be shown. The objective
is to have an insight and understanding of the fatigue behavior of complex
RFRCs.
Very few studies have been conducted for investigation of the effect of the
fiber length on the fatigue properties of RFRCs. Grove and Kim [48]
remarked that in tension-tension regime composites with longer fibers
display superior fatigue strengths in the low cycle fatigue regime (LCF)
whereas in high cycle fatigue (HCF), i.e. at lower stresses, the fiber length
has no significant influence. This was attributed to the fact that in LCF, the
stresses are high enough to induce fiber breakage, whereas in HCF the
predominant mode is debonding. The authors showed however that in
flexural fatigue, the fiber length has a significant effect both in the high
cycle and low cycle regimes. The same observation regarding flexural
fatigue was confirmed by Lavengood and Gulbransen [49]. Nevertheless,
more investigations are needed to conclusively identify the effect of fiber
length on the S-N curves of RFRCs.
Different attempts for evaluation of the effect of the fiber orientation on
the fatigue behavior of RFRCs can be found in [21-23, 41, 43, 50-54]. The
effect of orientation was investigated by two means; first, studies such as
the ones conducted by Horst and Spoormaker [41] and Arif et al. [43]
explored the effect of the layered “skin-core” morphology. The authors
reported that specimens where the core layer was thinner showed higher
fatigue strengths (samples loaded in the melt flow direction). Thinner core
layers are obtained near the edges of an injection molded plates (compared
to the middle of the plate) or by reducing the thickness of the sample.
Another type of investigation is similar to the above described
experimental studies on the effect of fiber orientation on quasi-static
behavior. In this type of experiments, samples were machined from
injection molded plates at different orientation angles. The clearly
observed trend, confirmed in most studies, is that as the specimen angle
increases (relative to the MFD), the strength of the composite decreases.
An example is shown in Figure 2.6 for a glass fiber reinforced polyamide
6 composite (GF-PA 6), investigated, by Bernasconi et al. [21] where a
wide range of angles was studied. It is noticed that going from samples
from 0o to 90o samples, the strength significantly decreases. This decrease
can be as high as 50% as reported by Zhou and Mallick [55]. Guster et al.
[53] found that the slope of the S-N curve can also be steeper in the 90o
samples compared to the 0o samples.
State of the Art
25
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].
The effect of the interface on the fatigue behavior of RFRCs was studied
within the above discussed context of modification of the interface by a
surface treatment of the fibers. An example is the study of Takahara et al.
[56] who studied the fatigue behavior of 30 wt% short glass fiber
polybutylene terephthalate composites (GF-PBT). They applied a silane
treatment on glass fibers and observed 15% increase of the fatigue strength
of the surface modified composite compared to the unmodified composite.
The same conclusion was found by Yamashita et al. [57], who applied an
amino silane coupling agent for short GF-PA 6 composites. Mandell et al.
[58] showed that for composites with weak interface bonding (they studied
a GF-PEEK composite), the S-N curve in the high cycle fatigue regime
could converge to the behavior of unreinforced material. They explained
that the poor interface strength resulted in loss of the glass fiber
reinforcements due to the interfacial debonding.
The previous paragraphs discussed the effect of the microstructural
parameters on the fatigue of RFRCs. Next, the effect of load and testing
parameters are discussed. These include the effect of the minimum to
maximum stress ratio (the R ratio), effect of cyclic frequency, and effect of
the testing environment e.g. temperature and humidity. The main trends of
the effect of the testing parameters are shown in Figure 2.7.
Different authors investigated the effect of the mean stress and
consequently the R ratio on the fatigue behavior of short fiber composites
CHAPTER 2
26
e.g. [55, 59-61]. The generally agreed trend is a decrease of the fatigue
strength with increasing of the R ratios as shown in Figure 2.7 (a). The
authors generally attributed this decrease of strength to the effect of cyclic
creep.
(a) (b)
(c) (d)
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].
State of the Art
27
The frequency effects were explored in e.g. [51, 60, 64, 65]. It was shown
that frequency has a considerable role in changing the behavior of the
material. The effect is manifested in increased hysteresis self-heating of
the samples at elevated frequency values. For example the GF-PA 6.6
material studied by Bellenger et al. [64], an increase of specimen
temperature as high as 100oC was found at a frequency of 10 Hz. With such
excessive heating, the matrix undergoes a transition from the glass state to
the rubbery state, which is known as thermal softening. This results in a
decrease of fatigue strength, as shown in Figure 2.7 (b). At very high
temperature rises, the mode of testing changes from mechanical to thermal
fatigue. This observation should apply to all thermoplastic based
composites. The polyamide 6 and polyamide 6.6 (PA 6 and PA 6.6)
materials were especially found however to be sensitive to frequency
induced heating.
Finally, environmental effects such as temperature and humidity were also
shown in few papers. For the effect of temperature, authors agreed on the
general trend of significant decrease of fatigue strength with higher testing
temperatures as shown in Figure 2.7 (c). Nevertheless, for example for
similar GF-PA 6.6 material, two different patterns were reported e.g. in the
studies of Handa et al. [62] and Noda et al. [66] found that the slope of the
S-N curve changed with increase of temperature, while De Monte el al.
[22] found no dependency of the slope of S-N curves on temperature.
Studies on the moisture and humidity effects can be found in [63, 67-69].
Attention was given to polyamide based materials and natural fiber
composites due to the high sensitivity of those materials to water
absorption and hygrothermal effects. Humidity leads to swelling of the
polyamide matrices. This in turn has significant effects on the state of the
interface and increases the debonding rate. As a consequence the strength
of the composite decreases as shown in Figure 2.7 (d).The same effect of
swelling occurs in natural fibers and results in reduction of the reinforcing
efficiency.
2.3.3 Fatigue damage in RFRCs
Different damage mechanisms of short random fiber reinforced composites
are reported in literature. These include: fiber breakage, fiber matrix-
debonding, and matrix cracking. Damage can be observed by fractography
analysis using e.g. scanning electron microscopy (SEM) or recently by
high resolution micro-CT. An example of micro-CT observed damage
mechanisms, from the study of Arif et al. [43], is shown in Figure 2.8.
CHAPTER 2
28
Although fiber breakage can be found in short fiber composites, as shown
in Figure 2.8, the probability of breakage of the fiber is low. Horst and
Spoormaker [41] explained that the low possibility of fiber breakage is due
to the relatively short fiber length of RFRCs, where only a limited number
of fibers are longer than the critical length.
Using SEM fractographs on GF-PA materials, the authors showed that
fiber-matrix debonding is the main damage mechanism of RFRCs. The
same was concluded by Meneghetti et al. [70] who studied short and long
glass fiber polypropylene composites and confirmed using SEM
micrographs that debonding was the predominant fatigue damage
mechanism for both composites. This shows that even with long fibers,
fiber-matrix debonding is still the most significant cause of damage.
Matrix cracking was also observed by authors. However, a detailed study
by Arif et al. [43] has shown that matrix microcracks generally develop in
the core layer, in direction transverse to the applied load (samples loaded
in melt flow direction). The authors also found some voids located at fiber
ends (mechanism b) but explained that they may not be related to
interfacial debonding.
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].
State of the Art
29
2.4 Geometry Generation Models
The concept of “Representative volume element” first appeared in the early
work of Hill in [71]. The term referred to a sample that “ (a) is structurally
entirely typical of the whole material on average (b) contains a sufficient
number of inclusions (structural elements) for the apparent overall moduli
to be effectively independent of the surface values of traction and
displacements, as long as these values are macroscopically uniform. ”
Similar definitions can be found in [72, 73]. In such a way, an RVE
presents a bridge of length scales by correlating the macroscopic behavior
of the materials to the properties of microscopic constituents and micro-
structures. The definition of Hill imposes the statistical representation of
RVE to the real micro-structure of the heterogeneous medium (in this,
composite).
Drugan and Willis [72] added another definition regarding the
representation of the RVE to the macroscopic response. In this definition,
emphasis is given on the “mean response”, where accuracy of estimates is
defined by the average response obtained from of different simulated RVEs
of the same size regardless of the precision of individual values obtained
from the results of the response from each RVE. In this case RVEs do not
have to contain large numbers of inclusions as required in the description
of Hill [71].
In this section, a review will be given on the concept of the critical
(minimum) RVE size as described by different literature investigations. In
the second part, different algorithms available for micro-structure
generation of typical RFRCs will be given along with a brief description
of the commercially available software for generation of RVEs with
RFRCs.
2.4.1 Critical RVE size
From a numerical point a view, an important question is the minimum size
of the element that can be considered as representative. This is known as
“critical size of RVE”. The critical RVE size to determine a certain
material property with a prescribed accuracy depends on the material’s
micro-structure [74].While the size of an RVE is clearly defined for
periodic micro-structures as explained in [75], for random micro-structures
such as RFRCs it is not inherently described. The typical strategy is the
CHAPTER 2
30
computation of the desired property with increasing sizes of RVEs until
achieving convergence with a desired allowable accuracy.
Following this strategy, Drugan and Willis [72] and Gusev [76] analyzed
the elastic properties of RVEs of composites reinforced with randomly
dispersed, non-overlapping identical spheres. Gusev [76] performed FEA
on RVEs with different sizes and demonstrated that the critical RVE size
of this composite was unexpectedly small within the elastic regime.
Convergence of the overall elastic constants of the micro-structure was
obtained with RVEs of only 64 spheres. Nevertheless, scatter in the order
of few percent was achieved with even smaller RVEs (comprising about
27 inclusions).
For short fiber reinforced composites, a similar study was performed by
Hine et al. [77] on randomly dispersed aligned short fibers. The fibers had
an aspect ratio of 30 and a volume fraction of 0.15. Figure 2.9 shows the
results of the study of Hine et al. [78] for the estimated values of the
longitudinal elastic modulus 𝐸11 depending on the size of modelled RVE.
The authors reported that with an RVE size of only 30 fibers, the
predictions deviated with only a few percent with different random
realizations. Larger RVE sizes, i.e. comprising 125 fibers showed hardly
any scatter.
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.
State of the Art
31
Following the results of Hine [77], Pierard et al. [79] analyzed the
minimum size of RVE for prediction of the stress-strain curve of
composites reinforced with homogeneously distributed aligned ellipsoids.
They reported that for all modelled sizes of RVE the stress-strain curves
were superimposed in the elastic region. Significant differences were found
in the plastic region. Nevertheless deviations of stress-strain curves of
different realizations as low as 1% were achieved with RVEs of 30
ellipsoids.
By analyzing the reported literature observations, it is noticed that the
minimum size of RVE depends on the stiffness mismatch (phase contrast)
between constituents. By comparison of composite reinforced with
randomly dispersed spheres, RVEs with larger number of spheres were
needed to produce the same accuracy in case of elasto-plastic matrix [74]
compared to the case of the linear elastic matrix. The same was concluded
for an elasto-viscoplastic composite in [80]. The size of RVE is also
dependent on the randomness of the structure. For the homogeneously
distributed ellipsoids investigated by Pierard et al. [81] few percent scatter
was achieved with very small RVEs (with only about 15 inclusions) even
with an elasto-plastic matrix. This may suggest that the effect of
randomness of micro-structures on the size of RVE is more significant than
effect of phase contrast.
Another important conclusion is that the minimum RVE size strongly
depends on the concept of the “desired accuracy”. While most authors
concluded that RVEs of about 30 fibers is sufficient, the desired accuracy
obtained with this size of RVE was different in different papers as
discussed above. In some cases this RVE size led to a very low scatter of
the estimates obtained from different individual realizations. In other cases
this RVE size was intended as a size at which the “average” estimated
values corresponded to the exact solution (following the definition of
Drugan and Willis) although relatively large fluctuations between
individual estimates, obtained from the different realizations, existed. It
should be noted that in the above literature, mostly all investigations were
performed with FEA. This adds the constraint of computational efficiency
with respect to the choice of the size of RVE, i.e. favoring small as possible
RVEs. Such constraint is much less pronounced in analytical models e.g.
the mean-field homogenization models which will be considered in the
present study.
Finally, as mentioned, authors were looking at effective properties;
investigations did not include computation of the local stress states at the
constituents’ level, or imperfect composites (i.e. composites with weak
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32
interfaces, defects, etc.). When looking at the local states, which are
imperative in modelling phenomena such as damage, it might be that the
size of RVE should be significantly larger. This in fact was confirmed by
studies of e.g. Swaminathan et al. [82, 83] who found that for unidirectional
composites, when modelled without damage convergence of effective
properties was reached with RVEs of containing about 50 fibers while with
damage convergence was reached with RVEs of 150 fibers.
2.4.2 RVE generation algorithms
The complex and random micro-structure of RFRCs was discussed in
section 2.3.1. In the following an overview will be given on the different
methods for generation of RVEs describing the geometry of those
composites. In literature, random generation algorithms of random fiber
composites included: the Random Sequential Absorption method (RSA)
and Monte-Carlo simulations method.
Due to its simplicity, the RSA is more common. Examples can be found in
[79, 84].The method consists of creating a desired volume and adding one
fiber at a time. Constraints are added to the algorithm, which define the
minimum allowed distance between two fibers, to avoid overlapping of the
fiber. Other constraints can be added e.g. the minimum distance between
the fiber and the edges of the bounding volume of the RVE. If after the
creation of the fiber it doesn’t satisfy the constraining conditions, the fiber
is rejected and a new fiber is created. The procedure continues until the
desired number or volume fraction of fibers in the RVE is reached. Each
fiber can be assigned a random orientation following a certain orientation
distribution/tensor. Examples of attempts of generation of RFRCs using
this method can be found in [79, 85-87]. Figure 2.10 shows a generated
RVE of random fiber composites using the RSA method.
State of the Art
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Figure 2.10 Generated RVE of RFRCs using the RSA method (13.5% volume
fraction and aspect ratio of 10) [87].
In the Monte-Carlo simulation methods, the fibers are created all at once
and placed at initial positions in a large volume in the first step. At
subsequent steps, the bounding volume of the RVE is reduced and
intersections of the fibers are checked. The positions of the fibers are
changed in each step and translated into a new random position in case
intersections occur. Similar to the RSA, the steps are repeated until the
desired volume fraction is obtained. The method was used e.g. in [88, 89]
The main limitations of the RSA and the Monte-Carlo simulations method
is the limited achievable volume fraction which is also referred to as the
jamming limit restriction [90].
2.5 Mean-Field Homogenization Schemes
Mean-field homogenization methods are powerful tools that enable the
prediction of the mechanical behavior of heterogeneous materials, and
particularly composites, at very attractive computational cost [79]. Mean-
field models rely on the concepts of the strain and stress concentration
tensors, i.e. 𝐀 and 𝐁 respectively, first introduced by Hill [71]. The tensor
𝐀 relates the average strain in an inclusion α to the macroscopic strain and
the tensor 𝐁 relates the stress in the inclusion to the macroscopic stress as
presented in Equations (2.1) and (2.2) respectively.
ε𝛼 = 𝐀 ε̅ (2.1)
σ𝛼 = 𝐁 σ̅ (2.2)
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34
Using the strain concentration tensor 𝐀, the properties of the matrix and
reinforcement and the fiber volume fraction 𝑐𝛼, the effective composite
stiffness tensor 𝐂𝑒𝑓𝑓 can be expressed as follows (Equation (2.3)).
𝐂𝑒𝑓𝑓 = 𝐂𝑚 + 𝑐𝛼(𝐂𝛼 − 𝐂𝑚) 𝐀 (2.3)
Mean-field homogenization methods include simple models e.g. the Voigt
and Reuss algorithms and more advanced models which are based on
Eshelby’s solutions for ellipsoidal homogeneities [91]. The simple models
are typically used as rigorous upper and lower bounds for estimation of
composite moduli using assumptions of uniform strain (Voigt model) and
uniform stress (Reuss model) in the composite. These bounds however are
too far apart and do not take into account micro-structure details such as
the shape, aspect ratio and orientation of the inclusions [92]. They also do
not provide means for solving the localization problem, i.e. calculation of
the local stress and strain fields in the constituents. For these reasons the
second family of models, i.e. based on the Eshelby solutions are more
attractive for modelling the behavior of RFRCs. In the following sections,
an overview of the original Eshelby solution and the most common mean-
field models based on it will be given. These include: the Dilute Eshelby,
Mori-Tanaka, Self-Consistent models as well as the Hashin-Shritkamn
bounds and Lielens method.
2.5.1 Eshelby’s solution
In this section, the fundamental Eshelby solutions [91] will be briefly
recalled. Consider an infinite medium (matrix) in which is embedded an
inclusion. Both the inclusion and matrix have identical isotropic and elastic
properties denoted with the stiffness tensor 𝐂𝑚. Only the inclusion is
subjected to an eigenstrain (also called transformation strain) denoted by
ε∗.
If the inclusion was not embedded in the matrix material, it would undergo
a stress-free deformation corresponding to the strain ε∗. However, since it
is constrained by the matrix, the inclusion is in constrained strain state εc
and the reaction forces from the embedding matrix results in distortion
stress fields 𝜎𝑐. The main finding of Eshelby [91, 93] is that for an
ellipsoidal inclusion, when the eigenstrain ε∗ is constant, the constraint
strain εc is equally constant and the relationship between the eigenstrain
and the constraint strain associated with it is given by a constant tensor as
shown in Equation (2.4).
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εc = 𝐒 ε∗ (2.4)
Where 𝐒 is the constant tensor, later named the Eshelby tensor. The fourth
order Eshelby tensor depends only on the stiffness tensor of the
homogeneous medium (the matrix material) and the shape of the inclusion.
For isotropic material, the computation of the Eshelby tensor is simplified
and direct analytical formulae are available, e.g. in [94-96]. For anisotropic
materials, the tensor has to be numerically calculated by solving different
elliptical integrals as reported in [97].
2.5.2 Eshelby’s based homogenization models
2.5.2.1 Dilute Eshelby Model
The eigenstrain concept can be applied to a “heterogeneity” with domain
Ω𝛼 and properties 𝐂𝛼 which is different from the matrix using the so-called
Eshelby transformation principle. The principle is illustrated in
Figure 2.11. The composite is subjected to a far field strain ε∞. The
heterogeneity will undergo an equal (mechanical) strain ε𝛼. To apply
Eshelby’s transformation principle, the heterogeneity is replaced with an
equivalent inclusion with the stiffness 𝐂𝑚 (again the same as matrix). The
equivalence is in such a way that the stresses and strains σα and εα in the
heterogeneity and equivalent inclusions are the same.
Figure 2.11 Illustration of Eshelby's transformation principle.
Similar to above discussion, the constraining matrix leads to the
development of the fictitious constraint strain εc. The final strain in the
Ω𝛼
ε∞
Ω𝛼
ε𝛼 = ε∗ ε𝛼 = 0 ε𝛼 = ε∞ + ε𝑐
ε∞
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36
equivalent inclusion will then be a result of the externally applied strain
and the fictitious constraint strain as shown in Equation (2.5).
εα = ε∞ + εc = ε∞ + 𝐒 ε∗ (2.5)
The stress in the equivalent inclusion can be found using Equation (2.6)
[98]. The equation directly solves the localization problem for a single
heterogeneity embedded in an infinite matrix.
σα = 𝐂α(ε∞ + 𝐒 ε∗) = 𝐂𝑚 (ε∞ + 𝐒 ε∗ − ε∗) (2.6)
From Equation (2.5) and (2.6) a relationship between the strain in the
inclusion 𝑒𝛼 and the applied strain can be directly established as in
Equation (2.7).
ε𝛼 = [𝐈 + 𝐒 𝐂𝑚−1(𝐂α − 𝐂𝑚)]−1
ε∞ (2.7)
By comparison with Equation (2.1), the strain concentration tensor for the
dilute Eshelby model is then given by Equation (2.8).
𝐀dilα = [I + 𝐒 (𝐂𝑚)−1(𝐂𝛼 − 𝐂𝑚)]−1 (2.8)
The concentration tensor 𝐀dilα can then be used directly in Equation (2.3)
to calculate the effective properties of a composite with identical and
aligned fibers, as was done e.g. in [99, 100]. However, the model is limited
to dilute concentrations (i.e. very low fiber volume fractions). This is
because the model is based on the assumption that the interactions between
the individual inclusions are negligible [101, 102].
2.5.2.2 Mori-Tanaka model
Eshelby’s dilute model is valid for small inclusion concentrations (up to
𝑐𝛼 = 1% [103]). In actual applications of fiber reinforced composites, the
fiber volume fractions are much higher and can typically reach up to 30%
reinforcement volume fractions for RFRCs and even higher in continuous
fiber composites. Different models have evolved for extension of the dilute
Eshelby model to finite concentrations by involving the effect of inclusion
interactions. The most common among these methods are: the Mori-
Tanaka (M-T) and the Self-Consistent (S-C) models. In this section the
formulation of the M-T model will be given. An overview of the other
models will be presented below.
The original Mori-Tanaka method, originally published in the well-known
paper [104], was concerned with calculating the average internal stress in
the matrix of a heterogeneous material containing precipitates. Benveniste
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[105] presented a reformulation of the Mori-Tanaka method for direct
application of the model to reinforced composite materials.
The M-T method takes into account the inter-inclusion interactions with
the assumption that each inclusion feels the presence of the other inclusions
indirectly through the total strain in the matrix. In this context, in addition
to the far field strain and the transformation strain, each inclusion will be
further strained due to the presence of the other inclusions. The additional
strain is referred to as “image strain” denote εim [104].
The strains and stresses in each inclusion are then found from Equations
(2.6) and (2.7) by adding the additional image strain.
σα = 𝐂𝛼(ε∞ + ε𝑖𝑚 + 𝐒 ε∗α) = 𝐂𝑚(ε∞ + ε𝑖𝑚 + 𝐒 ε∗α − ε∗α) (2.9)
ε𝛼 = ε∞ + ε𝑖𝑚 + 𝐒 ε∗ (2.10)
The average strain in the matrix will be a result of the applied strain and
the average of the image strains of all inclusions, as in Equation (2.11).
ε̅ 𝑚 = ε∞ + ε̅𝑖𝑚 = ε∞ + ε𝑖𝑚 (2.11)
The last equality in Equation (2.11) denotes an important principle
explained in the following. A main question of the application of the Mori-
Tanaka formulation is the way the image strain is sampled between the
different inclusions. The formulation, as commonly used since its
foundation, implicitly assumes that the image strain in the matrix resulting
from the presence of the inclusions is equally distributed among the
different inclusions. From this assumption arose the concept of “mean-
field”, and consequently mean-field models, which assume that the image
strain is equally sampled by all constituents regardless of their shapes and
orientations [106]. By definition then the two quantities ε̅im and ε𝑖𝑚 become interchangeable.
From the equality in Equation (2.11) and from Equation (2.10), a so-called
dilute strain concentration tensor, this time relating the strain in the
inclusion to the strain in the matrix, is given by Equation (2.12).
𝐀𝑚(dil)𝛼 = [I + 𝐒 (𝐂𝑚)−1(𝐂𝛼 − 𝐂𝑚)]−1 (2.12)
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38
The strain concentration tensor 𝐀MTα associated with the M-T model and
relating the strain in each inclusion to the applied far-field strain is given
by Equation (2.13).
𝐀MTα = 𝐀𝑚(dil)
α [𝑐𝑚 𝐈 + ∑ 𝑐𝛼 𝐀𝑚𝛼
𝑀
𝛼=1
]
−1
(2.13)
where 𝑐𝑚 and 𝑐𝛼 are the matrix and inclusion volume fractions
respectively.
The composite effective stiffness 𝐂MT𝑒𝑓𝑓
is given by Equation (2.14).
𝐂MT𝑒𝑓𝑓
= 𝐂𝑚 + ∑ 𝑐𝛼 (𝐂𝛼 − 𝐂𝑚) 𝐀MTα
𝑀
𝛼=1
(2.14)
As can be seen from this section, the main advantage of the M-T scheme
is that its equations for prediction of the macroscopic moduli are of the
explicit nature and are rather simple. The method reformulated by
Benveniste [105] provides direct formulations for estimation of the local
fields in the reinforcement and average local fields in the matrix. Finally,
for short fiber composite applications, the model was found to give very
good agreement of the predicted moduli compared to experimental data for
a wide range of materials, including both aligned and random fiber
reinforced composites, e.g. in [107-111].
2.5.2.3 Self-Consistent model
The Self-Consistent scheme in its current formulation was introduced by
Hill [112, 113] for spherical reinforcements, although earlier formulations
of the model were proposed by Hershey [114]. The model was applied to
short fiber composites only in few studies, e.g. in the papers of Laws and
McLaughlin [115], Chou et al. [116] and recently Müller et al. [117].
The method assumes that each inclusion is isolated and embedded in a
fictitious matrix having a stiffness tensor 𝐂SC𝑒𝑓𝑓
corresponding to the
effective stiffness tensor of the homogenized composite. The problem then
becomes similar to the dilute Eshelby model. Hence, the strain
concentration tensor associated with the Self-Consistent method can be
directly obtained from Equation (2.8) by replacing the stiffness of the
reference Equation (2.15).
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𝐀SCα = [I + 𝐒 (𝐂𝑒𝑓𝑓)−1(𝐂𝛼 − 𝐂SC
𝑒𝑓𝑓)]
−1 (2.15)
The effective composite stiffness is given by Equation (2.16).
𝐂SC𝑒𝑓𝑓
= 𝐂SC𝑒𝑓𝑓
+ ∑ cα(𝐂𝛼 − 𝐂SC
𝑒𝑓𝑓) 𝐀SC
α
M
α=1
(2.16)
Since the composite stiffness is initially unknown, an iterative scheme is
needed to solve the model. This starts with an initial guess of 𝐂SC𝑒𝑓𝑓
, the
Eshelby tensor 𝐒 and strain concentration tensor 𝐀SC𝛼 are then calculated
and a new (improved) value of the effective stiffness is obtained from
Equation (2.16). A second iteration is computed using this new value, and
the iterations continue until convergence of the results of 𝐂SC𝑒𝑓𝑓
.
In general, the S-C model was found to give good predictions for
polycrystalline materials but is less accurate in case of two phase
composites [92]. For short fiber composites, it was found that the effective
elastic constants estimated by the S-C method are generally too stiff
compared to experimental values [117, 118].
2.5.2.4 Hashin-Shtrikman bounds
The Hashin-Shtrikman bounds [119, 120] are similar to the Self-Consistent
model, however, the material surrounding the inclusion is a reference
(comparison) material instead of the matrix material. Like the Voigt and
Reuss models, the single variational approach gives both bounds (upper
and lower) by making the appropriate choice of the reference material
[118]. The method has been further explored in different studies, e.g. [121,
122]. Weng [123] presented explicit formulae for composites reinforced
with aligned ellipsoids.
For the upper bound, the reference material is as stiff as or stiffer than any
phase in the composite and in the lower bound the stiffness of the reference
material is equal or lower than that of the composite phases. The resulting
bounds are tighter than the Voigt and Reuss and are often used to assess
the physical validity of the different predictive models.
For RFRCs, the fibers are generally the stiffer phase in the composite. The
lower bound can then be obtained by choosing the matrix as the reference
material and the upper bound can be obtained by choosing the fiber as the
reference material. The assumptions of the two bounds yield the following
CHAPTER 2
40
lower and upper concentration tensors in Equations (2.17) and (2.18),
respectively.
𝐀HS𝑙𝑜𝑤𝑒𝑟 = [𝐈 + 𝐒𝑚 (𝐂𝑚)−1(𝐂𝛼 − 𝐂𝑚)]−1 (2.17)
𝐀HS𝑢𝑝𝑝𝑒𝑟
= [𝐈 + 𝐒𝛼 (𝐂𝛼)−1(𝐂𝑚 − 𝐂𝛼)]−1 (2.18)
It is then concluded that the lower bound is identical to the Mori-Tanaka
model [124]. Tucker and Liang [118] explained that this lends theoretical
support to the M-T model as it always obeys the bounds. It should be noted
that the formulations given in this section are for aligned ellipsoids
following the above mentioned study of Weng [123].
2.5.2.5 Lielens’s model
Lielens et al. [125] suggested that with higher volume fraction of
inclusions, the effective stiffness should be closer to the upper Hashin-
Shtrikman bound and at lower volume fractions the estimates drifts closer
to the lower bound. They then proposed a model that interpolates between
the upper and lower Hashin-Shtrikman bounds, in which case the strain
concentration tensor is given by Equation (2.19).
𝐀Lielens = {(1 − 𝑓)[𝐀HS𝑙𝑜𝑤𝑒𝑟]
−1+ 𝑓[𝐀HS
𝑢𝑝𝑝𝑒𝑟]−1
}−1
(2.19)
Where 𝑓 is the interpolating factor which should depend on the inclusion
volume fraction. The authors suggested the following relation for the
interpolating factor (in Equation (2.20)).
𝑓 = 𝑐𝛼 + 𝑐𝛼
2
2
(2.20)
The expression of the interpolating factor was based on fitting of the model
to experimental results and not on clear physical background.
2.5.3 Criticism of Mori-Tanaka model
While the Mori-Tanaka method has been effectively used for modelling a
wide number of composites, the method was often subject to a number of
criticisms. In the following, an overview is given on the limitations and
criticisms of the Mori-Tanaka model. Insight will be given on the extent
and significance of the limitations on the accuracy of the model predictions
and the alternative solutions reported in the literature, if applicable. The
focus will be on the application of M-T model to RFRCs.
State of the Art
41
The first limitation is the mean-field (averaging assumption). It is clear
that this limitation is not restricted to the M-T approach but is applicable
to all the different mean-field based models. In these approaches
homogenization is performed using the average strains of the constituents
as an input into the constitutive laws, i.e. consideration of only average
fields. Certainly, full-field approaches using detailed Finite Element
methods provide a more realistic representation of the local stress and
strain states of heterogeneous materials. Nevertheless, FE approaches are
much more computationally expensive and the computational cost
increases with increasing complexity of the boundary value problem, e.g.
complex micro-structures (which also entail large RVE sizes), material
non-linearities or geometric non-linearities [126]. For this reason, despite
the approximations, using mean-field techniques for modelling complex
materials such as RFRCs remains attractive.
The second limitation is that the model is valid for ellipsoidal
inclusions. While as mentioned above, ellipsoids can be generalized to a
number of reinforcement shapes, fibers in most short fiber reinforced
composites are more accurately approximated with a discontinuous
cylindrical shape. Using accurate Finite Element modelling, Steif and
Hoysan [127] investigated the predictions of moduli of single short
cylindrical fibers and equivalent ellipsoidal inclusion models. They found
very reasonable agreement even for very low aspect ratios of the inclusions
(2.9% difference for aspect ratio of 4). For longer fibers with an aspect
ratio of e.g. 12 the difference was only 1.3%. Typical short fiber
composites have aspect ratios which are generally higher than 25 and hence
ellipsoidal inclusions are an acceptable assumption for these materials.
The third limitation is about the limited range of reinforcement
concentrations for which the model can be applied. The criticism was
motivated by observations of Ferrari [128] who noticed that according to
Equation (2.14), the effective composite stiffness will still depend on the
properties of the matrix even for fiber volume fractions equal to unity. The
authors reported significant inaccuracies by calculating the elastic moduli
with 𝑐𝛼 = 1 which is supposed to yield the properties of the inclusions. In
fact, this conclusion lead to the development of the Lielens’ model [125].
Berrymann and Berge [129] explored this limitation and stated that a
reasonable recommendation would be to limit the use of the M-T scheme
to situations where the host (matrix) have the dominant volume fraction.
They explained that matrix volume should at least be about 70 – 80% of
the overall volume. This is well in the range of the typical matrix volume
fractions for RFRCs.
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42
The fourth limitation, is about physical inadmissibilities of the effective
stiffness tensors estimated using the M-T method. This drawback is also
applicable to other mean-field schemes, e.g. the S-C model. As large
attention in literature is given to this issue, and it is often addressed as the
main drawback of the M-T model, therefore it will be discussed in more
details.
Two different inadmissibilities are reported by authors for the M-T
method. On one hand, Qiu and Weng [130] investigated the application of
the M-T model to different cases of transversely isotropic spheroidal
inclusions. The authors concluded that for multi-phase systems, which is
generally the term assigned to composites comprising inclusions with
different shapes or orientations, the M-T estimated moduli may violate the
Hashin-Shritkman bounds. However, the reported cases were needle (thin
cylinders with vanishing diameters and infinite aspect ratio) and disk
inclusions, which are not representative of the actual fiber shapes in
RFRCs.
On the other hand, it is well known that for energy considerations the
effective stiffness and compliance tensors of heterogeneous materials
should be diagonally symmetric [131, 132]. Authors reported that the
effective stiffness tensors predicted by the M-T model are generally
asymmetric and hence physically inconsistent [128, 129, 131, 133-136].
Huysmans [137] suggested that for the M-T model, the physical
inconsistencies can be attributed to the assumption of the uniform image
strain sampling. Note that for the S-C algorithm the same applies although
the method does not explicitly use the image strain concept, but the
inclusions interactions are reflected in the anisotropic Eshelby tensor
which leads to the same physical inconsistencies. The suggestion of
Huysmans [137] is reasonable in view of reported literature observations
where Benveniste et al. [134] stated that the dilute Eshelby solution (in
which inter-inclusion interactions are not taken into account) always leads
to symmetric effective tensors. Li [131] explained that the original Mori-
Tanaka model is intended for composites comprising inclusions with
similar shapes and that it is the extension of the model to multiple-phase
composites that cause these inconsistencies.
In what appears to be parallel work, Ferrari [128, 138] and Benveniste et
al. [134] explored different composite systems, on which they applied the
M-T model to have an assessment of the diagonal symmetry of the
estimated effective stiffness tensor. Benveniste et al. [134] provided
numerical values of the components of the resulting stiffness tensor for the
State of the Art
43
different cases. The authors concluded that diagonal symmetry is
guaranteed with M-T model only in case of: spherical inclusions, perfectly
aligned inclusions of the same shape, randomly dispersed inclusions of the
same shape. For those cases, it is clear that indeed that uniform image strain
approximation is reasonably accurate.
The above cases cannot be applied to short fiber composites. As explained
in previous sections, short fiber composites exhibit length distributions of
the inclusions and hence, inclusions have different shapes. Also, neglecting
the fiber lengths distributions, flow conditions lead to orientation
distributions of the fibers somewhere between the extreme conditions of
aligned and full random.
In literature, the opinions are conflicting between general acceptance of the
asymmetry problem and suggestions of limiting the applicability of the M-
T model to cases which yield admissible results. Different ways have been
reported to circumvent the physical inconsistency. Dvorak and Bahei-El-
Din [136] and Nemat-Nasser and Horri [102] remarked that the strain
concentration tensors are the source of asymmetry of predicted composite
tensors. Nemat-Nasser and Horri attributed this to violation of the
requirement of volume average of the strain concentration tensor of matrix
and inclusion phases to equal unity, except in the case of aligned
composites. To achieve symmetry of stiffness tensors, they then suggested
to normalize the strain concentration tensors. Ferrari [132] proposed to
replace the averaging term in Equation (2.13) of the Mori-Tanaka strain
concentration tensor by a concentration tensor of aligned fibers. These
solutions present a mathematical treatment of the inconsistency but are not
supported by solid physical background. Schjødt-Thomsen and Pyrz [139]
proposed to replace the averaging term in the Mori-Tanaka stiffness tensor
equation by the averaging of the stiffness of reinforcements over all
possible orientations of the orientation distribution function of the RVE to
obtain a weighted average stiffness, letting go of the concentration tensors.
This in fact is analogous to the orientation averaging scheme which is
known to be a simplified approximation.
A more elaborate solution is the development of the now known as the
“two-step homogenization” method. This method was first proposed by
Camacho et al. [140] and developed into the current formulation by
Pierard et al. [92], and is now often referred to as the “pseudo-grain”
model. The basic idea of the model is illustrated in Figure 2.12. In this
model, the real RVE is decomposed into 𝑁 discrete grains or sub-regions
where the inclusions in each grain have the same aspect ratio and
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orientation. The two step homogenization procedure is then performed on
the new “numerical RVE” comprising the discretized grains. In the first
step, the Mori-Tanaka model is used for homogenization of each individual
grain. The new RVE at this point can be regarded as an aggregate material.
The second step homogenization is then performed using the Voigt model
explained above.
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.
The idea behind the pseudo-grain model is that when applied to each grain,
which is regarded as a two-phase composite with inclusions having same
shapes and orientation, the M-T model will satisfy the physical
requirements. For that reason, in the second step of the homogenization
albeit approximate, the Voigt model is used because the RVE comprising
homogenized pseudo-grains is in turn a multi-phase composite and the M-
T model will violate the consistency. While the model eliminated the
mathematical problem of the M-T formulation, it introduced additional
approximations, first by neglecting the interactions between the “isolated”
pseudo-grains and second with using the approximate Voigt model in the
second step.
Homogenized
RVE
Real RVE
Numerical
RVE
First step homogenization
Second step
homogenization
Homogenized
pseudo-grains
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A comparison between the original M-T approach and the pseudo-grain
approach can be found in [92, 141]. Generally, it is concluded that the
original M-T and pseudo-grain M-T do not give equivalent results but the
deviations of the effective properties estimated by both models are not
significant in most studied cases. However, the approach was investigated
in the recent work of Jain et al. [142] (co-authored by the author of the
present thesis), which focused on the evaluation of both methods for
prediction of the average local stresses in inclusions, which are significant
in damage modelling. The authors performed a validation using FE
benchmarks for random RFRCs and proved that the original M-T
formulation provides good correlation with FE calculations while the
pseudo-grain model failed to give good estimated of the local fields (both
models gave comparable predictions of the homogenized properties). The
study gives confidence in the use of the M-T model as a basis for modelling
RFRCs and provides means for concluding that the assumption of mean
image strain, which is also one of the concerns/limitations discussed above,
is a reasonable estimate even for the multi-phase RFRC materials.
To conclude, the M-T model provides an attractive way for predicting the
behavior of RFRCs. Despite the above mentioned limitations, the approach
has been extensively used in different contexts, i.e. linear elastic and non-
linear modelling and often results in accurate predictions. For the main
criticism of physical inadmissibility, the different ways proposed to
circumvent the mathematical problem leads to other approximations and/or
assumptions without adequate physical backgrounds. Moreover, numerical
values provided by Benveniste et al. [134] for the different test cases show
that the deviation from symmetry is usually not significant.
2.6 Modeling the non-linear quasi-static behavior of RFRC
2.6.1 Matrix non-linearity
While the Eshelby problem has an exact solution when dealing with linear
elasticity, it has no analytical solutions for systems with non-linear phases
(i.e. matrix or inclusions) [143]. To overcome this, approximate methods
have been reported in literature to describe the different non-linear material
deformation behavior. The general principle governing most of these
methods is replacing the non-linear properties of the individual
components with approximate representative linear properties, so called
“linear comparison materials”, for which the Eshelby solution can be
applied.
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46
SFRC materials typically consist of a non-linear elasto-plastic matrix and
brittle linear-elastic fibers. For this reason, the main focus of available
literature attempts is modelling the non-linearity of matrix materials. For
elasto-plastic materials, two different methods can be readily
distinguished, i.e. the secant modulus approach and the tangent modulus
approach.
2.6.1.1 The tangent (incremental) approach for rate-independent
plasticity
The tangent formulation was first proposed in the pioneer work of Hill
[112]. In this paper, Hill introduced an incremental approach for the
extension of the self-consistent method, in the context of flow theory by
making use of tangent modulus tensors [133]. Hill’s theory was further
explored in detail in [144] with the objectives of the assessment of model
accuracy and numerical implementation. The main advantage of the
tangent approach is the ability to model the complete load history in non-
monotonic loading cases.
In the incremental approach, the real matrix is replaced with a fictitious
matrix whose response is defined with Equation (2.21) relating the matrix
average stresses and strains in rate (or incremental) form [79]. In such a
way, homogenization models valid in linear elasticity can be applied in
each time interval.
σ̇ = 𝐂𝑒𝑝 ε̇ (2.21)
where �̇�𝑚 and 휀�̇� are the matrix stress and strain rates, 𝑡 is a time
parameter. 𝐂𝑒𝑝 stands for the uniform (continuum) elasto-plastic tangent
operator. The implementation of the tangent modulus approach is
numerically complex and requires a number of approximations.
A detailed study of the formulation of the incremental approach was
published by Simo and Taylor [145]. They explained that the integration
algorithm employed in the solution of the incremental problem, typically
based on Newton’s method, essentially requires the use of a consistent
(algorithmic) tangent operator 𝐂𝑎𝑙𝑔 , instead of 𝐂𝑒𝑝 in the linearization of
the discretized rate relations in order to preserve the quadratic rate of
asymptotic convergence.
In more recent investigations, Doghri [146, 147] and Doghri and Ouaar
[126] analyzed the concept of tangent operators. It was shown that the
consistent linearization of the time discretized constitutive equation (over
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47
a finite time interval [t𝑛, t𝑛+1]) using the tangent operator 𝐂𝑎𝑙𝑔 is generally
described as follows:
δσn+1 = 𝐂𝑎𝑙𝑔 δεn+1 (2.22)
Both tangent operators are anisotropic and the authors reported that 𝐂𝑎𝑙𝑔 →𝐂𝑒𝑝 only at vanishing plastic increments, otherwise the two operators can
be quite different.
Nevertheless, it was found that using the anisotropic tangent operators
within the framework of mean-field homogenization methods leads to too
stiff overall composite response [148-150]. In fact, this observation has led
to the development of the secant approach [126]. The origin of the error
was traced to the calculation of the Eshelby tensor with the anisotropic
tangent tensor of the reference (matrix) material [151]. An alternative
solution was the definition of an “isotropic” moduli 𝐂𝑖𝑠𝑜 using different
methods such as the spectral decomposition [133, 152] which will be only
used for computation of the Eshelby tensor while the anisotropic operators
are used for the other operations. No physical explanation was given for
this assumption.
When applied to different metal matrix composites (MMCs), the isotropic
operator leads to good predictions of the composite response [79, 126, 141,
153, 154]. However, when applied to a typical polymer matrix composite
such as polyamide 6 reinforced with glass fibers (GF-PA 6), the
incremental approach still leads to too stiff predictions of the modelled
stress-strain curves by comparison to experimental and full FE simulations
as can be found in [151, 155, 156]. In all variations of MMCs, both phases
(i.e. matrix and reinforcement) exhibited plastic behavior, aspect ratios of
the reinforcements were very low and no high stiffness mismatch between
matrix and reinforcements was present. The authors attributed the poor
predictions for the GF-PA 6 composite mainly to the later reason, i.e. the
high stiffness contrast of the glass fibers and PA 6 matrix. This has led to
further modification of the isotropization approach for the application of
the model on this material as explained in [155].
2.6.1.2 The secant approach for rate-independent plasticity
The secant formulation simulates the non-linear behavior of the composite
material and of each phase (here we focus only on the plasticity of the
matrix phase) within the framework of the deformation theory of plasticity.
The method was first proposed by Berveiller and Zaoui [157] and direct
formulations and implementation of the method using the Mori-Tanaka
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48
model were given by Tandon and Weng [158]. In the secant approach, the
stresses and strains of the elasto-plasic matrix are directly given as:
𝜎 = 𝐂𝑠 ε (2.23)
Where 𝐂𝑠 is the secant stiffness tensor. The secant stiffness tensor can be
directly determined from the plastic strain and equivalent (von Mises)
stress in the matrix. In the formulation of Tandon and Weng, the equivalent
stress is determined from the volume average of the stress in matrix (as
typically used within the M-T assumption). Suquet [159] developed a
“modified” secant approximation, where the equivalent stress in the matrix
is determined from the volumetric average of the second-order moment of
the stress tensor. The modified method was argued to present better
predictions. The modification was motivated by the fact that substituting
the complex stress state in the matrix, especially after the onset of
plasticity, by a volume average will lead to an overestimation of the flow
stress. Seguardo and Llorca [160] compared both the classical and
modified secant approaches and found that at the beginning of the plastic
regime, the modified formulation gives better agreement than the classical
formulation compared to FE estimations. However, it was shown that as
the plastic deformation became dominant, the hardening rate predicted by
the classical approach was very close to the numerical behavior, while the
modified approach predicted a softer response.
To conclude, since it was first proposed by Hill [112], the incremental
approach has been largely investigated. Different authors proposed
different modifications of the method to be validated for a certain category
of materials. Nevertheless, there are no physical grounds for the issues
associated with the method (discussed above) nor the proposed
modifications. In addition to the rather complex algorithms and numerical
implementations, no clear method is agreed upon in the literature. Also,
from our own point of view, the derivation of the different tangent
operators for the different method work-arounds is purely numerical and
cannot be directly related to the material behavior. A detailed critical study
of the different variants of the formulation can be found in [81], which
confirms the need for more analysis targeted towards the generalization of
the incremental formulation. The secant approach is simpler to implement,
and has adequate physical background. In this respect, it is attractive for
modelling the non-linear behavior of matrices. However, the drawback of
the method, is that it cannot simulate the full hysteresis behavior of the
material under fatigue loading.
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2.6.2 Composite damage and failure
In this section, an overview is presented for the available methods for
modelling damage of short fiber reinforced composites. As previously
explained, the damage mechanisms of short fiber composites are: fiber
breakage, fiber-matrix debonding and matrix cracking. The damage
(failure) models can be classified in three distinct categories, namely
strength models, phenomenological models and mean-field based models.
In the following details of the three types will be discussed.
2.6.2.1 Strength and Phenomenological models
In this work, we refer to strength models as the class of models which are
based on the simple rule of mixtures relationship (ROM). Cox [161] and
Kelly and Tyson [162] proposed extensions to the ROM by including the
aspect ratio of the fibers and introduced the dependency of the strength of
the composites on the interface shear strength. The models of Cox are now
well-known as Shear-Lag models and provide a strong basis for
understanding the mechanisms of stress transfer along the fiber length. A
length efficiency factor was introduced to account for the efficiency of
discontinuous fibers. Thomason [163] suggested the use of a single
orientation efficiency factor to account for the random fiber orientation of
RFRCs. Bowyer and Bader [164, 165] extended the Kelly-Tyson model
with the formulation of Thomason to simulate the complete stress-strain
curve.
Although the strength models are attractive, and have been widely used
e.g. in [163, 166-169], they have major simplifications e.g. elastic
constituent properties, the detailed length and orientation distributions are
not taken into account, instead a single factor is used for all fibers.
Moreover, debonding or physical damage mechanisms are not considered.
Some attempts have been done to account for fiber failure and fiber pull-
out [170]. Nevertheless the underlying assumptions are simplistic and the
models only predict the strength of the composite and were not extended
for modelling the complete stress-strain curve.
Phenomenological approaches involve the description of damage at the
macroscopic level without the need for information on the microscopic
states. In this way, the composite is viewed as a homogeneous material. In
the following, a brief overview is presented of the different attempts to
model the damage behavior of RFRCs with phenomenological based
models.
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50
Zhou and Mallick [171] used a combined phenomenological and statistical
approach to describe the stress-strain behavior of a short glass fiber
reinforced polyamide 6,6 composite. Damage was considered isotropic
(assumption of similar damage in all directions of the composite) and was
described using a statistical Weibull function. Although simplistic, an
advantage of their method is that it takes into consideration the strain rate
dependence of the polyamide based composite which is accounted for in
the Weibull scale parameter. Dano et al. [172] developed an anisotropic
damage model based on the work of Chow and Wang [173] using damage
mechanics theories. In this way, the model assumes that damage results in
anisotropy of the material which is initially considered isotropic. They
introduced a tensor accounting for damage effect, which describes the
degradation of the macroscopic effective stiffness. The damage evolution
was modelled in a thermodynamic framework by the associated
thermodynamic forces. Mir et al. [174] further improved the model by
Dano et al. by taking into account the residual strains using
thermodynamics potential although plasticity was not directly considered.
2.6.2.2 Micromechanics based damage approaches
Zhao and Weng [175] and Zheng et al. [176] investigated the effect of
interface debonding. In the former paper, two debonding configurations
were modelled; debonding on the top and bottom of an oblate inclusions
(inclusions with equatorial radius greater than the polar radius) and the
second configuration was debonding on the lateral surface of prolate
inclusions (inclusions with polar radius greater than the equatorial radius).
When significant debonding occurs, the debonded inclusion with original
isotropic properties is replaced with a transversely isotropic inclusion with
zero load-carrying capability in the debonded direction while in the bonded
direction the inclusion retains its load-carrying capacity. In the later paper,
the authors considered only oblate spheroids with single or double
debonding, and investigated the effect of the debonding angle, i.e. extent
of debonded surface on the effective moduli. The models were limited to
aligned inclusions and only effective properties were considered.
Micromechanics based modelling of random short fiber composites using
the Mori-Tanaka method was performed by Meraghni and Benzeggagh
[177-179]. The authors classified two distinct mechanisms of damage of
the short fiber composites, namely mechanisms relative to matrix
degradation (initiation and propagation of micro-cracks) and mechanisms
relative to interface failure (debonding, pull-out .. etc.). The authors used
an experimental approach based on acoustic emission (AE) analysis and
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51
SEM observations. They introduced parameters describing the two classes
of damage into the micromechanical models and successfully predicted the
stress-strain curves of the composite. The method however heavily relies
on rigorous experimental work.
Fitoussi et al. [180] modelled the stress-strain behavior of Sheet Molding
Compounds (SMCs). The authors considered damage of the SMCs solely
based on debonding and developed a model called “equivalent anisotropic
undamaged inhomogeneity” (EAUI). They introduced a local damage
criterion based on the Coulomb failure criteria and assessed the debonding
state of individual inclusions. The failure criterion was assessed on each
interface point along the inclusion’s equator. Different damage variables
are then assigned to the inclusion, taking into account how much of the
interface is debonded. The efficiency of stress transfer at the interface point
is based on whether the state of normal stress on the point is tensile or
compressive. The damage variables are used in calculating the degraded
components of the inclusion’s stiffness tensor. The model assumptions
were validated with accurate finite element modelling of the debonding of
the fiber.
Jao-Jules et al. [38] checked interface debonding at different interface
points along the fiber surface. They replaced the debonded fiber by the
same volume of matrix. Derrien et al. [181] explained that this assumption
results in a lower bound of the load carrying capability of the debonded
fiber. Hence, this assumption is expected to give an under-estimation of
the composite stress-strain curves which was indeed shown by Jao-Jules et
al.
A class of micromechanics based models use a statistical approach to
predict damage in random fiber reinforced composites. Zhao and Weng
[182] and Ju and Lee [183] proposed a micromechanics based model which
uses a Weibull probability function to describe the probability of
debonding of particulate composites. The model was extended to RFRCs
by Lee and Simunovic [184]. In their model, a two-parameter Weibull
statistical function is used to predict the probability of a partially debonded
inclusion. The governing parameter in the statistical function is the average
stresses inside of the inclusion. The initially isotropic partially debonded
inclusion is then replaced by an equivalent perfectly bonded transversely
isotropic inclusion with the assumptions of vanishing tensile and shear
stresses in the debonded directions while the stresses in the bonded
direction remain the same. A limitation of the approach is that all the fibers
are assumed to have the same aspect ratio and a random orientation is
considered with orientation averaging.
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52
Another approach was used by Fitoussi and Baptiste [185] using the Mori-
Tanaka model for a random sheet molding compound (SMC). They
assumed interface failure to be the principle failure mechanism of the
composite. The authors used the statistical formulations to describe the
failure probability of the interface strength. The model follows the same
formulation as the model proposed by Fitoussi et al. [180], however instead
of using a deterministic Coulomb failure criterion, the statistical interface
failure probability is applied to each interface point along the equator. The
same approach was used by Derrien et al. in [181]. The method was also
used by Zaïri [186], however instead of replacing the debonded fiber with
an equivalent anisotropic inclusion, it was replaced by a void.
Desrumaux et al. [187] used a statistical approach also in the Mori-Tanaka
scheme. However, the authors considered the problem of asymmetry of the
stiffness tensor discussed in section 2.5.3 for multi-phase composites. They
regarded their SMC as a three-phase composite (fibers, matrix and
microcracks) with the fibers having random orientations, and they
proposed the two-step homogenization procedure outlined in Figure 2.13.
They assigned different probabilistic damage function for fiber breakage,
matrix cracking and interfacial decohesion. When one fiber is broken it is
replaced by a void. The first step of the two-step homogenization scheme
involves homogenization of the micro-cracks with the matrix, this results
in the new homogenized damaged matrix. The source of the micro-cracks
can be one of three assumed damage phenomena namely, matrix cracking,
interfacial decohesion and fiber failure. For each of those failure
mechanisms a respective statistical failure probability equation is assigned.
In the next step, the intact fibers are introduced in the new homogenized
matrix and the second homogenization is performed.
The same model was used by Jendli et al. [188] and extended to include
effect the effect of strain rate on damage. Two inherent limitations are
associated with the approach. First, following the first homogenization the
new damaged matrix exhibits an anisotropic behavior. For this case, no
explicit formula exist for the computation of the Eshelby tensor and the
tensor should be solved numerically, e.g. using the method proposed in
[189]. The second limitation is that debonding is introduced as a micro-
crack and hence is treated as a void with null stiffness. This is a
simplification since a debonded fiber will continue to contribute to the load
carrying although with degraded properties. In this respect the scheme
proposed by Fitoussi et al. [185] seems more fitting.
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53
Figure 2.13 Two-step homogenization procedure and implementation of damage
modelling proposed by Dermaux et al. [187].
A final class of damage models, although generally categorized as
micromechanics based models, can actually be viewed as “meso-level”
modelling. The class of models uses the laminate analogy for modelling
the damage of RFRCs. A few key examples are given in the following.
The Halpen-Karoos model [190] views a random fiber reinforced
composite as a stacked laminate with different plies of aligned fibers. The
prescribed orientation distribution defines the thickness of each of the
composite plies. The homogenized properties of each ply can be done with
a simple model such as the Halpin-Tsai model. Laminates theory is then
applied to predict failure of the composite.
Van Hattum and Bernardo [191] used the Tsai-Hill failure criterion for
prediction of strength of a unidirectional short fiber composite. The
strength of the random composite is then obtained by orientation averaging
of the strength tensor. Lasplas et al. [167] used a similar approach, however
the Tandon and Weng model [192] was used instead of simple Kelly-
Tyson relationships used in Van Hattum and Bernardo’s paper. An
improvement was done by Nguyen et al. [193] who used the same model
of Van Hattum and considered the elasto-plastic behavior of the matrix
(which was assumed to follow the Ramberg-Osgood relation), using the
Hill-type incremental tangent approach discussed in section 2.6.1.1.
Fiber degradation
X2
X3
X1
Interface decohesion Matrix degradation
New damaged matrix Second homogenization step
First homogenization step
Homogenized material
CHAPTER 2
54
Finally, recent attempts were by Kammoun et al. [151, 194] for modelling
failure of the RFRCs using the pseudo-grain approach discussed in
section 2.5.3. In the first paper they proposed a First Pseudo-Grain Failure
(FPGF) model, by analogy with the laminate’s First Ply Failure model.
Progressive damage failure of the random fiber composite was modeled as
a succession of failure of the pseudo-grains. The Tsai-Hill failure criterion
is used on the ply level. Matrix non-linearity was also taken into account
using the incremental approach. The model has the same underlying
simplifications as the laminate’s ply discount assumption and, moreover,
lacks physical clarity of the First Ply Failure as “grains” are abstractions of
a set of fibers with the same orientation. Hence, the model showed
deviations compared to experimental stress-strain curves. An improvement
was performed in the second paper, where the authors introduced a First
Pseudo-Grain Damage model which results in more realistic damage
prediction by replacing the brittle Tsai-Hill failure criterion with the
continuum damage theory of Lemaitre and Desmorat [195]. It is worth
noting that within the context of failure the standard Mori-Tanaka/Voigt
two step homogenization could not be used due to the simplified iso-strain
assumption of the Voigt model. Instead a Mori-Tanaka/Self-Consistent
was used. The later is expected to still lead to the asymmetry of the
effective stiffness tensor.
To summarize, an overview was given of the different approaches and
assumptions used for modelling failure and damage of random fiber
composites. Strength models are not suited for modelling progressive
damage and for random fiber composites. They have major simplifications
which do not allow incorporating non-linear phases or real fiber length and
orientation distributions. Phenomenological models treat the composite on
a macroscopic level without consideration of the real micro-structure and
do not account for different physical modes of damage.
More sophisticated models based on the micromechanics approach were
described. They can be classified in three main categories, deterministic
models, statistical based models, and the final class of models combines
micromechanics with laminates analogy. The most commonly considered
damage mechanism of RFRC is fiber-matrix debonding as a result of
interface failure. In the first two categories of models, the general method
was replacing a debonded inclusion with an equivalent perfectly bonded
inclusion, on which Eshelby based models can be applied. Generally
different debonding criteria can be used, but the most common is the
Coulomb criterion, which will be discussed in more detail in Chapter 7.
Several assumptions were then used for the equivalent perfectly bonded
inclusions which are replacing the partially debonded inclusion with a
State of the Art
55
void, or with matrix with the same volume. Another way is replacing only
the debonded volume of the inclusion with matrix of the same volume, or
replacing the partially debonded inclusion (originally isotropic) by a
transversely isotropic inclusion retaining its same load carrying capability
in the non-debonded direction and with zero capability in the debonded
direction. And finally replacing it with a fully anisotropic inclusion with
selective degradation schemes for different directions. Out of these
assumptions, replacing the debonded fiber with a matrix is a lower bound
for the debonded fiber efficiency, and hence it is expected that replacing it
with a void is a strong underestimation. The selective degradation scheme
proposed by Fitoussi et al. [180] provides an attractive method for
modelling the realistic damage of RFRCs.
In general, the statistical approach should be regarded as an improvement
of the deterministic approach. This is due to the fact that it addresses the
stochastic nature of damage in real RFRCs. Nevertheless, while all
debonding models require parameters such as the interface strength which
are complex to obtain experimentally, the statistical approaches have in
addition, a number of statistical parameters describing damage probability
functions. These have no physical basis and cannot be obtained from
experiments, and are usually obtained by fitting to experimental results,
which adds a major drawback to the methods.
Laminate analogy models can be criticized on physical basis as the ply or
pseudo-grain is a fictitious entity that neglects the interaction of fibers of
different shapes and orientations. Though, they have an advantage of
computational efficiency, in comparison with the full micromechanics
based approach requiring exhaustive application of the damage criterion
on single inclusions. It was also shown, that incorporating damage defies
the advantage of the pseudo-grain approach which was motivated by the
objective of solving the problem of diagonal symmetry of the M-T method.
This is because integration of damage necessitates the use of a M-T/S-C
approach rather than the previously used M-T/Voigt approach.
Finally, other than debonding, very few attempts considered different
damage mechanisms such as fiber breakage and matrix cracking. This can
be due to the fact that fiber breakage is not a significant damage mode in
short fiber composites. The two assumptions used for modelling a broken
fiber is replacing it by a void or by two undamaged inclusions. One cannot
distinguish which assumption is better and both seem adequate. For matrix
cracking, own fractography analysis has shown that this mechanism cannot
be directly found in short fiber thermoplastic composites, nevertheless it
could be more important in sheet molding compounds typically made with
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56
thermoset matrices e.g. epoxy and polyester, which are generally more
brittle than thermoplastic matrices.
2.7 Modeling the fatigue behavior of RFRCs
Different approaches for modelling the non-linear quasi-static deformation
and stress-strain behavior of RFRCs were discussed in section 2.6. For
modelling the fatigue behavior, different works attempted at modelling the
hysteresis behavior from plasticity or energy point of view. Some examples
are given in the following.
Doghri and Ouaar [126] used the tangent plasticity model discussed in
section 2.6.1 to simulate the cyclic loading and unloading behavior of short
fiber composites. The same model was used in [153]. Launay et al. [196,
197] developed a nonlinear constitutive model for the fatigue behavior of
short fiber composites and mainly short glass fiber reinforced polyamide
6.6, based on the theory of standard generalized material. The theory is
described by parameters of elastic energy density and dissipation potential
by application in finite strain elastoplasticity. In this way, the theory takes
into account hysteresis development and cyclic softening of the material
but no notion of damage is taken into account. Nouri et al. [198] used a
coupled phenomenological and elastic strain energy model for prediction
of the progression of damage, where damage is viewed as crack growth.
The model was originally developed by Ladeveze and LeDantec [199] for
laminated composites and involves a set of 20 parameters for the full 3D
structure The model needs a considerable number of experimental tests.
In studying the fatigue behavior, as shown in section 2.3.2, focus is on
investigation of the S-N curves of the material. This is motivated by real
applications where a crucial step in the design and choice of the material is
the estimation of the lifetime of the part, under described loading
conditions.
Very few attempts can be found in literature for the prediction of the S-N
curves based on actual microstructural parameters. Some models are
available in literature to estimate the fatigue S-N curve of short fiber
composites using energy-based approaches. Examples include the model
of Meneghetti et al. [200] and Jegou et al. [201]. In these approaches, an
energy based fatigue failure criterion is assigned. Parameters of the failure
criterion relate the dissipated energy from the material during the fatigue
loading, which is evaluated experimentally from thermal measurements, to
State of the Art
57
the fatigue lifetime of the composite. Such models require difficult
experimental tests and extensive set-ups for the thermal measurements.
Wyzgoski et al. in [202] used an empirical relationship for the estimation
of the S-N curves of polyamide based composites from the single cycle or
quasi-static strength results. The empirical relationship indicates that the
normalized fatigue strength of short fiber reinforced polyamides decreased
by 10% per decade of frequency or lifetime. Nevertheless, this empirical
law was not found suitable for other materials, e.g. PBT reinforced
composites.
There exists as well a common type of methods which use the concept of
simple normalization (scaling) of the S-N curves of the composites by the
ultimate strength. These can describe the influence of parameters such as
the FOD, FLD and loading direction on the fatigue life. Failure criteria
such as the Tsai-Hill failure criterion can then be applied to predict the
fatigue life of the material. This class of models was applied in e.g. [21-23,
69, 203]. An extension of the Tsai-Hill failure criterion for multi-axial
fatigue can be found in [204].
An example of the methods of the normalization of the S-N curves by the
ultimate tensile strength is proposed by Bernasconi et al. [21] as shown in
Equation (2.24) with the purpose of modelling the fatigue life of specimen
with different orientations (as discussed in section 2.3.2).
(σ1,max
σ1,fat(N)
)
2
+ (σ2,max
σ2,fat(N)
)
2
−σ1,maxσ2,max
σ1,fat2 (N)
+ (τ12,max
τ12,fat(N))
2
= 1 (2.24)
Where the maximum values of the normal stresses σ1,max, σ2,max and
shear stresses τ12,max during one load cycle are related to the respective
experimental fatigue strength values σ1,fat(N), σ2,fat(N), and τ12,fat(N) at a
specimen life of N cycles. The relationship shown in Equation (2.24)
indicates that the fatigue strength corresponding to a particular fatigue life
can be obtained for any specimen direction by having the fatigue strength
at three principal directions (normally taken as the longitudinal, transverse
and bias directions). This requires experimental testing for the three
directions. The model has been shown to give good predictions of the S-N
curves compared to experimental results [21, 22].
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58
The limitation of these methods are the simplified assumptions, used for
extending the Tsai-Hill criterion to fatigue loading, which is that the fatigue
strength varies in the same way as tensile strength. An example of how the
behavior at the two types of loading is different is reported by Horst et al.
[41] who have clearly shown different damage behavior of tensile and
fatigue broken samples. The models also require a number of experimental
curves which reduce the efficiency of the modelling approach. Another
limitation is that the coefficients or strength parameters in the modified
Tsai-Hill equation are identified for one specific skin/core ratio and cannot
be easily adapted to industrial structures where both thickness and fiber
orientations are variable. Finally, the method does not take into account the
effect of variation of mean stress or loading conditions.
Zago and Springer [205, 206] suggested the use of a fatigue criterion based
on the stress tensor, which takes into account the influence of the local
orientation tensor. The principle governing this method is the use of the
generalized Miner’s rule for estimation of the fatigue life of the material.
The same approach was used by Gaier et al. [207] by adding the concept
of the critical plane for fatigue of orthotropic materials, by analogy to the
critical plane approach of metals. The authors used the assumption that
only the stress normal to the critical plane is responsible for fatigue
damage. The model parameters do not include the notion of particular
modes of damage.
Finally, within the mean-field homogenization technique, methods for
prediction the fatigue life of short fiber composites, to the knowledge of
the author, have only been implemented by Malo et al. [204] who used the
pseudo-grain approach discussed in section 2.5.3 and applied the Tsai-Hill
failure criterion on individual grains from which they can obtain the S-N
curve of the composite. Due to using the Tsai-Hill criterion, the same
limitations discussed earlier apply, i.e. treating damage on the macroscopic
level and no explicit formulation of individual damage mechanisms taken
into account as well as the need of the number of input S-N curves of the
composite as discussed above.
2.8 Discussion of the state of the art and adopted approaches
The review given in this chapter aimed at providing an understanding of
the general complex stochastic micro-structure and quasi-static and fatigue
behavior of random short fiber composites as well as the available methods
for modelling those behaviors.
State of the Art
59
First, details were given on the random micro-structure in terms of random
fiber length, orientations and spatial distributions of the short fiber
composites. The extent of the effect of these microstructural parameters on
the quasi-static and fatigue behavior has been shown. Also, different
testing parameters such as frequency, temperature and environmental
effects and stress ratio, which are inherent to fatigue testing, and their
effects on the fatigue behavior of the material were discussed.
From analysis of modelling attempts in literature, it can be seen that the
overall modelling approach starts with a model of the micro-structure
which is able to simulate the random geometry of RFRCs. The
representation of the actual stochastic micro-structure is a crucial aspect in
the final predictions of the desired mechanical properties. Similar models
are needed for simulation of the more complex wavy fiber materials.
This will also necessitate adequate geometry characterization methods for
wavy fiber composites. In the next chapter a methodology will be proposed
and validated.
In addition, an overview of mean-field homogenization techniques, as fast
methods with reasonable accuracy for homogenization of RFRCs, was
presented. From the different models, the Mori-Tanaka model is the
most accurate, and consequently the most commonly used model for
predicting the elastic properties of RFRCs. The model is often criticized
for a number of limitations. The main limitation is diagonal asymmetry of
the predicted stiffness tensor for multi-phase composites including RFRCs,
as they comprise inclusions with different shapes and orientations.
Different work-arounds proposed in the literature were described. The
most elaborate solution was the above described pseudo-grain approach.
While the method indeed solves the diagonal asymmetry problem, it adds
other non-physical assumptions by “isolating” families of similar
inclusions in each grain and neglecting the interaction between non-similar
inclusions. It was also shown that the method is not able to predict local
stress states. For this reason, despite the physical inconsistency, the direct
(original) Mori-Tanaka method will be used in this work. This is also
motivated by the fact that the maximum deviation from symmetry as
reported by authors was not significant.
As can also be concluded from the description of the mean-field models,
the models are applicable to straight fiber composites, for which each fiber
can be simulated with an ellipsoidal inclusion and this assumption does not
lead to significant inaccuracies. Applying these models to wavy fiber
composites, is not straightforward and hence, reliable methods are
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60
needed to extend the mean-field models and specifically the Mori-
Tanaka model used in this work to wavy fiber composites.
Next to choosing a suitable homogenization method, modelling approaches
should be adopted to simulate the non-linear behavior of thermoplastic
matrices and the different damage mechanisms of short fiber composites.
Different methods were shown in the context of mean-field
homogenization.
For matrix non-linearity, out of the two approaches, namely the tangent
and secant methods, the secant approach seems to be the most suitable.
Although this method cannot simulate full loading and unloading behavior
of the material in cyclic loading, the underlying parameters have more
physical grounds. Using the tangent approach requires the use of a number
of tangent operators that are handled in mathematical and numerical
contexts and cannot be directly related to actual material behavior
parameters.
For the damage behavior of random fiber composites, debonding was
reported to be the most dominant mechanism. Different approaches were
presented with analysis of advantages and limitations. The model
proposed by Fitoussi et al. [180] seems to give a good basis for
modelling the debonding behavior of random composites with detailed
damage parameters describing initiation and progression of
debonding. Similar models using statistical failure criteria which take into
account the stochastic nature of damage were discussed, but in this work
the deterministic approach is chosen as the statistical models involve a
number of statistical parameters that are usually obtained by fitting to
experimental results, which reduces the efficiency of the model. The model
of Fitoussi was only directly applied to SMC composites which consist of
long fibers that are assumed to be continuous. An interesting question will
be if this model will give good predictions for injection molded short fiber
composites with much smaller aspect ratios.
Finally, it was shown that mean-field based models for prediction of the
fatigue life of RFRCs are very limited and often based on a simple Tsai-
Hill based formulation. At present a fatigue model which describes the
actual progression of damage during cyclic loading is needed. Also, a
fatigue model which starts with the fatigue behavior of the constituents
as input, instead of an input of the fatigue behavior of the composite
at certain conditions (i.e. certain orientation, volume fraction, etc.)
taken as reference, does not exist for short fiber composites. Only a
State of the Art
61
limited number of similar models exist even for continuous fiber
composites. Such approach remains a fundamental and challenging
scientific problem which will be tackled in this work.
To date, there are few commercially available software for homogenization
and modelling of RFRCs. These softwares are Digimat from E-XTREAM
and Converse from PART engineering. Both softwares have the advantage
of linking manufacturing simulation to homogenization platforms in which
exact orientation maps predicted by process simulation software such as
Moldflow can be analyzed and homogenization of RVEs with different
orientations can be performed. The Converse software is limited to elastic
homogenization. An advantage of Digimat software is that it includes
elaborate models for modelling non-linear phases which includes models
for elasto-plasticity, visco-elasticity, elasto-viscoplasiticy, etc.).
Nevertheless, both software tools rely on the concept of the two-step
homogenization, or the pseudo-grain concept in which a certain orientation
tensor of an RVE is discretized in only few dozens of grains with the same
orientation. This results in the discussed inaccuracies regarding local fields
It was also discussed in this review that a much larger size of RVE is
needed for random composites especially in the case of damage modelling.
To date, the software also assume the same aspect ratio for all inclusions
and cannot take into account actual length distributions. Damage is only
included in Digimat software using the simplified pseudo-grain ply
damage model. Finally, fatigue models have been recently incorporated in
Digimat tools; however, the methods are also based on the simplistic Tsai-
Hill failure criterion for RFRCs.
For those reasons, commercially available homogenization tools will not
be used in this work and own toolkits need to be developed and
implemented for modelling the quasi-static and fatigue behavior of
RFRCs. In this respect, parallel to the scientific output of this PhD thesis,
numerical tools will be developed with a series of software toolkits
including geometry generators which are able to take into account
waviness of fibers and tools comprising algorithms for modelling the
quasi-static and fatigue behavior of RFRCs, starting from described micro-
structure and properties of constituents. As mentioned in the introduction
chapter, the micro-scale methods and tools developed in this work will then
be linked to the macro-scale fatigue solver (Virtual.Lab) for a complete
multi-scale approach of real RFRC components.
62
63
Chapter 3: Geometrical Characterization and Modeling of Short Wavy Fiber Composites
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
65
3.1 Introduction to Steel Fiber Composites
Short Steel fiber reinforced polymers (SSFRPs), composed of stainless
steel fibers embedded in a polymer matrix are a novel class of materials
with high strength and stiffness properties. The fibers can be tailored to
two different versions; ductile annealed fibers and brittle as-drawn fibers.
The inherent ductility of the annealed stainless steel fibers is an added
advantage in comparison with the brittle glass and carbon fibers [208, 209].
The as-drawn fibers have the advantage of very high strength with strength
values which are close to those of the high performance carbon fibers. The
advantage of steel fibers (and also possible disadvantage, due to high
transverse stiffness mismatch between the fiber and matrix) over carbon
fibers is the isotropy of the former, while the transverse properties of
carbon fibers are much lower.
Steel fibers and steel fiber reinforced polymer composites have been
widely used in strengthening of concrete structures and to improve their
durability and toughness. The addition of steel fibers results in conversion
of the failure behavior of concrete from brittle to more ductile [210-214].
The diameter of steel fibers used in the reinforcement of concrete is at least
10 times larger than the micron-sized steel fibers used in this study (namely
8 µm). It has been shown that in addition to their favorable mechanical
properties, micron-sized stainless steel fibers have intrinsic electrical
conductivity, heat and corrosion resistance [208]. Commercial steel fibers,
Bekaert Bekishield, used in this study are highly efficient in
electromagnetic (EMI) shielding applications: up to 60 dB EMI shielding
for 0.5% volume fraction (VF) of fiber concentration (15% weight fraction
- VF) [215].
In the following, the production process of the Bekishield steel fibers is
outlined. The fibers are produced using the bundle drawing technique as
illustrated in Figure 3.1. Prior to the drawing process, individual steel fibers
are separated from one another, and each wire is covered with a metal
matrix material such as copper. All the stainless steel wires covered with
the copper matrix are then enveloped in a metal envelope material and form
a “composite wire”. The composite wire is drawn into smaller diameters in
a number of subsequent drawing steps. Intermediate heat treatments are
performed between drawing steps to facilitate the deformability of the
wires. When the desired diameter is reached, the envelope and matrix
material are removed by electrochemical etching to obtain the individual
fibers [216]. Polymer coating can be applied on the individual fibers
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66
depending on the application. The wires are finally cut to obtain a single
fiber length of 4 to 6 mm.
Figure 3.1 Illustration of the drawing technique to produce steel fibers [217].
3.2 Challenges in characterization and modelling the geometry
of SFRP composites
Unlike continuous UD or textile fiber reinforced composites, short fiber
reinforced composites depict stochastic geometrical features that evolve
during processing [33]. As discussed in section 2.3.1 during the
compounding stage and further in the injection process, random fiber
breakage occurs resulting in a range of fiber lengths (l), which can be
described by fitting an appropriate length distribution function 𝜓𝐿(𝑙) [34-
36]. Similarly, the melt flow patterns in the mold during the injection
process result in random orientations of the short fibers which can be
described with a fiber orientation distribution function 𝜓(θ,Φ) [34, 37,
38]. Finally, while improved mechanical properties are achieved with a
homogeneous placement of the fibers in the matrix, some degree of random
spatial positioning of the fibers typically occurs. The later can be improved
for example by a compounding step prior to injection molding. Those
characteristics result in a more complex micro-structure compared to
continuous fiber composites.
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
67
Owing to the high aspect ratio of the stainless steel fibers and their low
bending rigidity, the fibers are plastically deformed during processing into
very curved shapes. Hence, an important characteristic of injection molded
short steel fiber composites is the high waviness of the fibers, which adds
to the complexity of the short fiber geometry.
Precise knowledge of the micro-structure, needed for accurate predictions
of the mechanical properties of a short fiber composite, imparts a particular
challenge for the three-dimensional wavy steel fiber thermoplastic
composites. In the past decades, different techniques have been
investigated for acquiring such information for composites with short
straight fibers. For measurement of the fiber orientation distribution,
microscopical observations on polished samples provide two-dimensional
sections of the fibers; simple geometrical calculations allow then to
generate the fiber orientation [31, 213, 218, 219]. Despite low equipment
cost associated with these methods, they are destructive and time
consuming, and hence only small volumes can be analyzed. More
importantly, they often are not capable of accurate extraction of three-
dimensional information [33, 218-221]. The fiber length distribution can
be determined with matrix burn-off techniques, which are again destructive
and are prone to significant errors due to degradation of the fibers and
altered geometries [222].
To overcome those problems, X-ray micro-computed tomography (micro-
CT) recently emerged as a powerful non-destructive tool for three-
dimensional fiber micro-structure analysis [220]. A number of studies thus
far aimed at the characterization of the geometrical parameters of short and
long fiber reinforced composites using X-ray micro-CT techniques [33,
222-225]. However, the primary focus of those investigations is straight
fibers or straight fiber segments. The quantification of the architecture of
wavy fibers reinforced composites remains yet a new topic of interest. The
use of X-ray micro-CT is especially suitable for the characterization of the
steel fiber reinforced polycarbonate samples considered in this study, due
to the large difference in X-ray absorption and hence high contrast imaging
between the metallic fibers and the polymer matrix.
The geometry of wavy fiber assemblies was studied before in the field of
non-woven textile materials. In a series of papers [226-229],
Pourdehyhimi et al. investigated different methods of evaluating fiber
orientation distribution functions (FODs) of non-woven fabrics using
image analysis, including direct fiber tracking, two-dimensional Fourier
analysis of images and a flow-field analysis to derive fiber orientation by
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68
analyzing local texture information. By applying these methods on
simulated structures with well described orientation and further, on real
non-woven webs, they concluded that direct tracking is the most accurate
technique for extracting fiber orientation distributions. Similar
investigations for characterizing FODs of wavy non-woven assemblies
using image processing include, among others, Gong et al. [230], Rawal et
al. [231], Masse et al. [232], Xu et al. [233] who explored Hough transform
image analysis algorithms. Nevertheless, all of the mentioned
investigations involve two-dimensional techniques based on early concepts
developed by Komori and Makishima [234], who considered that FODs of
curved fibers can be approximated by those of hypothetical straight
segments obtained by subdivision of fibers and replacement of divided
parts by straight segments. The main disadvantage of this concept is that
resulting FODs depend on fineness of subdivision of the fibers leading to
inaccuracies [226, 234]. Those can be especially more significant in the
case of three-dimensional wavy fibers. Thus, a technique allowing three-
dimensional analysis of complex wavy fibers and an accurate method
for the description of the FOD of three-dimensional wavy fibers is
needed.
Models for generation of RVEs of short random straight fiber composites
were discussed in section 2.4.2. These models do not allow generation of
curved fibers. Very few papers are available in literature, to the knowledge
of the author, which describe geometrical generation models for short
wavy fiber composites. Pan et al. [235] proposed a modified RSA
algorithm which allows modelling of only slightly curved fibers. Curvature
was only allowed at intersection sections of two fibers. The purpose of
introducing curvature was to avoid the “jamming limit” drawback and
increase the possible volume fractions that can be achieved using the RSA
model. A so-called “Random Walk Algorithm” was introduced by
Altendord and Jeulin [236] for modelling non-overlapping bent fibers
using a force-biased packing approach to model the fibers as chains of
balls. The model implementation is complex and do not take into account
the full geometrical parameters, e.g. length distributions. Also, the
maximum shown aspect ratio achieved by the model was 9 which is lower
than typical aspect ratios of RFRCs. Gaiselmann et al. [237] presented a
more elaborate model which was not applied to short fibers, but considered
the similarly curved non-woven fabrics (random mats of continuous curved
fibers). The model needs as input 2D SEM data images on which
geometrical parameters are fitted to produce the 3D model. However, the
3D morphology consists of independent layers of fibers stacked together.
The fibers are horizontally oriented and the model is not capable of taking
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
69
into account 3D orientations of the fibers. It also relies heavily on input
from experimental tests. Finally, a common limitation of these models is
that waviness of the fibers cannot be described with actual mathematical
formulations, which is an advantage of the proposed method in this work.
As-drawn Steel fiber reinforced polycarbonate samples with initial fiber
length (pre-injection) of 5 mm are considered for this investigation. In this
work, we will be referring to that material as a short random discontinuous
fiber reinforced composite system following the definition by Phelps et al.
[238] and Tatara [239], among others, who classified long fiber
thermoplastics (LFTs) as materials reinforced with fibers longer than 10
mm.
To summarize, the aim of this chapter is two-fold: (a) to develop a
geometrical model for generation of the random RVE of short wavy steel
fiber reinforced composites, (b) validation of the model through X-ray
tomography techniques. For the later purpose, a novel methodology is
established for accurate three-dimensional quantitative measurement and
analysis of the micro-structural parameters of short wavy steel fiber
reinforced thermoplastic samples, which will be used as input for the
mathematical model. The generated RVEs are compared qualitatively
against real tomography reconstructed volumes. A quantitative comparison
is done using a straightness parameter (𝑃𝑠) outputted from both the
simulated and real volumes.
The developed models for generation of random RVEs of short wavy fibers
provide a necessary starting point for further predictive methods of
modelling of the mechanical behavior of the composite, which take into
account its complex internal geometry. A key characteristic of short steel
fiber composites is its stochastic nature which presents itself in fiber length,
orientation, position, in addition to the stochasticity of waviness. For this
reason, an accurate statistical description of the “randomness” of generated
RVEs is crucial for reliable modelling of the overall composite behavior.
3.3 Description of the Geometrical Model
In the present work, the terms “geometry” or “architecture” refer to the
local orientation and fiber length distributions, fiber positions and fiber
waviness. In the RVE generation algorithm fibers are modeled as solid
cylinders with a wavy central line.
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70
The geometrical model is based on the following input parameters:
1. Fiber volume fraction VF.
2. Fiber diameter given as one average value for all fibers (this
constraint can be easily waived in further model development, for
example, statistics of fiber diameters can be introduced).
3. Fiber length distribution 𝜓𝐿(𝑙); the type of the distribution
function is not fixed, it can be even, normal, Weibull, etc. The type
and values of the parameters of the selected fiber length
distribution are input, e.g. for a normal distribution the mean and
variance parameters are input.
4. Fiber orientation distribution, given as the 2nd order orientation
tensor [32], which is used for reconstruction of the orientation
function 𝜓(θ,𝛷). This constraint also can be waived, with input
of an orientation tensor of the 4th order, or approximated
orientation function itself. Orientation distributions here are
considered as the end-to-end orientation of the wavy fibers.
5. Fiber waviness profile. Fiber waviness is represented by a
combination of random harmonic functions to define the profile of
one fiber as shown in Equation (3.1):
𝑟(𝑠) = 𝐴 (𝒓𝟏 sin (𝑛1
𝜋𝑠
𝐿+ 𝜓1) + 𝒓𝟐 sin (𝑛2
𝜋𝑠
𝐿+ 𝜓2)) (3.1)
where:
r(s) is the radial position in relation to a certain axis, 𝑠 the coordinate along
the wavy fiber axis,
𝐴 is average amplitude of fiber waves generated randomly as uniformly
distributed on the interval [0, 𝐴𝑚𝑎𝑥], 𝐴𝑚𝑎𝑥 is maximum amplitude
parameter given by the user.
𝒓𝟏 and 𝒓𝟐 are two randomly generated orthogonal unit vectors (|𝒓𝟏| =|𝒓𝟐| = 1) normal to the axis.
𝑛1,2 are waviness numbers generated randomly (on log2 scale) as
uniformly distributed on the interval [1, 𝑛𝑚𝑎𝑥],
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
71
𝑛𝑚𝑎𝑥 is the maximum waviness number parameter given by the user.
𝐿 is the fiber length randomly generated following the FLD given by the
user.
𝜓1,2 are phase shifts randomly generated as uniformly distributed on the
interval [0, 2𝜋].
The geometrical model creates a realization of a random RVE via a
hierarchy of modeled objects:
1. A fiber segment of a wavy fiber, characterized by fiber cross
section shape (elliptical, with two given axis), length and
orientation, and local fiber curvature.
2. A wavy fiber, modeled as a sequence of fiber segments (which can
be straight or wavy) assembled together, and characterized by fiber
diameter, total length, end-to-end orientation of the fiber and shape
of the fiber cross section (elliptical), generated using the given
waviness parameters. If the fiber is straight, then it contains only
one segment. The wavy fiber comprises a random number of
harmonic waves based on the generated 𝑛1,2 parameters. The
number of segments per wave (i.e. the number of segments
comprised in one wave) of the wavy fiber is a variable in the model
which can be modified by the user.
3. Random realization of an assembly of a given number of wavy
fibers. Positions of the fibers are defined by random deposition of
the fiber centers of gravity within the RVE boundaries. The RVE
dimensions are calculated based on the given fiber volume fraction
and the number of fibers inside the RVE. A given number of fibers
is randomly generated by the model according to the given
distributions of length, orientation and waviness of the fibers.
Overlapping of fibers is allowed.
4. Some of the fibers will protrude out of the RVE. The whole RVE
fiber assembly should be seen as an element, periodically
translated in three directions to fill all the space with the fibers.
The implementation of the model uses the following sequence. First, the
wavy fiber is created with described waviness using Equation (3.1). The
parameters 𝒓𝟏, 𝒓𝟐, 𝑛1,2, 𝜓1,2 and the total length of the fiber 𝐿 are randomly
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72
generated based on the corresponding input from the user. A number of
segments are then created along the fibers. The exact number of segments
for one fiber is a random value, is dependent on the number of waves per
fiber and on the number of segments per wave given by the user. The ends
of segments can then be used as “control points” for plotting the path
(centerline) of the fibers as shown in Figure 3.2 where an example of a
wavy fiber generated by the model is illustrated. The black dots denote the
ends of segments. In the figure, straight segments are used. These can be
replaced by wavy segments. In all cases, the number of segments per wave
(and consequently the number of control points) controls the smoothness
of the generated fiber centerline. These fiber centerlines will be used in
further analysis for calculation of the mechanical properties of the wavy
fiber composites as will be shown in Chapters 6 and 7.
Creation of the random fibers continues until the desired number of fibers
in the RVE is achieved. Each fiber is also assigned random orientation
angles Ф and θ (defining the end-to-end orientation vector) based on the
user inputted orientation tensor.
The fiber is initially created with horizontal orientation. After creation of
the fiber, the fiber axis is rotated based on its assigned Ф and θ angles. In
this respect, the corresponding segments of each fiber are also rotated and
hence the control points depict new coordinates reflecting the new fiber
orientation. In a similar way, the fibers are translated to random positions
in the RVE to reflect the random placements of the fibers in the matrix.
Finally, boundaries (volume) of the RVE are calculated for the desired VF.
Figure 3.2 Example of wavy fiber generated by the model for illustration. Black
dots represent ends of segments “control points”.
The model is implemented in a C++ with visualization based on OpenGL
algorithms. For ease of use, the software tool is designed with a graphical
user interface with user-friendly handling of the model input. The output
segment
wavy fiber
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
73
of the model is an array of control points for each fiber in the RVE and the
dimensions of the boundaries of the RVE which can then be used for
plotting the fibers for visualization or for further analysis for micro-
mechanical calculations, as will be shown in the subsequent chapters.
3.4 Materials and Experiments
3.4.1 Steel fiber samples
Beki-Shield BU annealed as-drawn stainless steel fibers (Bekaert,
Belgium) were used [215]. The continuous fibers are commercially
available in the form of rovings. Fiber diameters are in the range 8 – 11
µm. Sizing is applied to the fiber surface for better compatibility with
polymeric matrices (each polymer type with a distinct sizing). For
manufacturing of the short fiber composite, the bundles were chopped in 5
mm initial length. The fibers can be processed with a large range of binders
and polymeric matrices. Due to the high density of steel (ρ:7.8 g/cm3), the
fiber loading is varied in a range of volume fractions as low as 0.05 – 3%
(hence, weight fractions between 0.32 – 17%). In this study, the fibers were
mixed with a transparent polycarbonate (PC) matrix, using an injection
molding process.
Production of the samples was done in VKC-Centexbel (Kortrijk,
Belgium). Plates of dimensions W x L x H: 150 x 150 x 2.5 mm were
injection molded using a central injection point. All plates were produced
under identical processing conditions of 260 – 340oC barrel temperature
and 325 bar back pressure. In this study, the lowest available fiber volume
fraction, i.e. 0.05 VF = 0.05% (equal to a weight fraction VF = 0.32%) was
used for tomography analysis for more clear determination of the
geometrical parameters of the fibers in a less crowded matrix. Figure 3.3
shows optical and SEM micrographs of the investigated steel fiber
reinforced composite. The figure clearly illustrates the waviness of the
steel fibers embedded in the matrix following injection molding.
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74
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.
3.4.2 X-ray micro-tomography
X-ray microtomography (also called micro-computed tomography or
micro-CT) was performed on a nano-CT system (Phoenix Nanotom S GE
Measurement and Control Solution, Germany). The Nanotom device is
equipped with a high-power nanofocus X-ray tube with 4 power modes. A
tungsten target was chosen for the high X-ray absorbing steel fiber
reinforced polycarbonate samples. A high power mode (mode 0 of the
nanotom) was used to allow focal spot and voxel sizes in the desired
micrometer range.
A sample size of 6 mm edge length was used in this study (square samples).
This sample size was chosen in relation to the steel fiber length (larger than
the length of the fibers) in order to attain higher probabilities of catching
complete fibers within the sample. This prevents to a large extent bias in
characterization of the fiber length distributions induced by having a high
probability of a given fiber in the bulk material to be cut by the boundary
of the sample volume [222].
The sample was mounted on a sample holder and fixed on a high-accuracy
computer controlled rotation stage. Alignment of the sample axis with
respect to the rotation axis of the scanner table was checked with a laser
beam.
(a) (b)
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
75
Fast scans (overall scanning time of 20 min) were carried out due to the
high X-ray absorbance of the steel fibers, hence reducing the time needed
for achieving high intensity/contrast at the detector. Exposure time was 500
ms. For each scan, 2400 X-ray 2D projection images were acquired on a
flat panel CCD detector (field of view 2304 x 2304 square pixels) obtained
from incremental rotation of the scanned samples over 360̊ with a rotation
step of 0.15̊. The resulting radiographic projections are grayscale 16-bit tiff
images with gray histogram values in the range of 0 – 65 536. Acquisition
parameters were fixed for all samples as follows: voltage = 65 kV, current
= 210A, voxel size = 3.5µm, no filter was used during scanning.
Reconstruction of the acquired 2D projections into 3D volumes was
performed using the GE Phoenix datos|x REC software supplied by the
Nanotom manufacturer. A calibration of the acquired images was
performed for compensation of sample drift-effects using the software’s
scan|optimiser module. Automatic geometry calibration was done using
the agc|module. Beam hardening correction and automatic ring artefact
reduction were carried out using the bhc+|module and rar|module,
respectively. Owing to the high power applied to the source, and hence the
high intensities on the detector, ring artefacts were nearly negligible.
Reconstructed XY datasets (slices) were exported from the software in 16-
bit tiff format for further analysis and visualization.
3.5 Analysis
The 3D tomographic dataset reconstruction allows visualization and
acquisition of qualitative information of the scanned samples. In order to
get quantitative data on the fibrous micro-structures, i.e. determination of
the fiber length and fiber orientation distribution, additional image analysis
operations outlined below were performed. Difficulties arose in attempting
to get such information for the wavy fibers using common imaging
software packages due to the complexity of the micro-structure, hence the
need for specialized tools for 3D characterization.
3.5.1 Image segmentation
Tomographic reconstructed datasets, which are generally grayscale
images, are often segmented to extract quantitative information, especially
in case of presence of different phases [240]. Thresholding is the simplest
and most efficient segmentation technique, converting grayscale images
into binarized (black and white) images by turning all pixels above a
threshold to a foreground value and all the remaining pixels to a
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76
background value [240-242]. In this work, thresholding was performed
using CT-Analyser software (CTAn v.1.13, Brucker microCT). Automatic
global thresholding was chosen based on the Otsu method [243]. Global
thresholding refers to a process in which a global threshold value is used
for all pixels of an image as opposed to complex adaptive thresholding
methods in which threshold values change dynamically over the image.
To summarize, the 3D datasets were binarized using histogram global
thresholds, applied individually on each image in the dataset (individual
automatic global thresholds), above which all voxels were considered to
belong to the fibrous phase, and below which all voxels were considered
matrix phase and noise [244].
Thus, new binarized datasets are obtained where fibers are separated and
appear as white pixels in 2D slices. This allows the compaction of data in
reconstructed slices as further analysis is done only on the fibers. Figure 3.4
(a), (b) shows 2D reconstructed slices and their corresponding binarized
slice. Global thresholding values derived from image gray level histogram
are shown in Figure 3.4 (c).
The large attenuation contrast between the metal fibers and polymer resin
allowed straightforward thresholding of the images. Validation of
thresholding values was carried out by 3D morphometry analysis which is
a process from which the fiber volume fraction (calculated from ratio of
white voxels to ratio of all voxels) were calculated and found to be 0.051%
which was close to the specified value of 0.05%. This value was not found
to be sensitive to small variations of thresholds.
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
77
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.
(a) (b)
(c)
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78
3.5.2 Three-dimensional image analysis tool
As previously mentioned, extraction of the geometrical parameters of the
wavy fiber micro-structure is a complex undertaking which is difficult to
realize using common micro-CT analysis software. In the present work, a
3D image processing software (Mimics v.15.01, Materialise NV, Belgium)
was used to accomplish this aim. Mimics is a software primarily developed
for medical image processing [245]. The use of the software was granted
in collaboration with professor G.H. van Lenthe and Dr. Leen Lenaerts
(BMe section, KU Leuven).
Due to computer limitations and large amount of data, construction of the
3D model from original gray scale reconstructed images was not feasible.
Therefore, compaction of data through segmented images as described in
section 3.5.1 was necessary, and allowed considerably fast building of 3D
models on Mimics (around 15 mins/model). As a first step, all fiber pixels
were comprised in a so-called mask (virtual object containing the pixels).
A 3D model of the fibers mask was created as shown in Figure 3.5.
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.
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
79
3.5.2.1 Fiber length distribution (FLD)
Using the Mimics software package, a procedure was developed for the
determination of the “real” length distribution of the wavy steel fibers.
Figure 3.6 (a) shows the reconstructed steel fibers in a “green” mask. In
order to obtain the fiber length distribution as well as other required
geometrical parameters, it was necessary to separate individual fibers from
their neighbors to perform analysis on single fibers. Region-growing
operations, which are procedures for elimination of noise and separations
of structures that are not connected, were performed. Hence, individual
fibers were separated, creating a new separate mask for each of the fibers.
An example of single fiber separation is shown in Figure 3.6 (b). Figure 3.6
(c) illustrates 3D objects constructed from individual fiber masks. The next
step is the determination of the wavy fiber length. For this purpose,
centerlines were fitted to the fiber objects using the MedCad module in the
Mimics software. Centerline fitting is a skeletonization “thinning”
operation which reduces 3D objects to their medial axes. The result is a set
of automatically generated control points which traces the wavy fiber
profile. The wavy length of the individual fibers (Lf) is directly given as
the “length of centerline” which is the sum of the distances between control
points.
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80
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.
A similar analysis was performed in [225], where the authors used the
Mimics software for the determination of the length distribution of short
straight fibers. In the case of straight fibers, separation of single fibers was
not needed, and automatic centerline fitting for the whole model was
possible. However, with the waviness and entanglements of the steel fibers
in this study, the separation was necessary for accurate analysis.
(a) (b)
(c)
p
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
81
In the case of the studied steel fibers, resolution errors in measurement of
the fiber length through micro-CT information are negligible due to the
large scale difference between the length scale (in the order of mm) and
the scan resolution (in the order of microns) [246, 247]. Analysis was
performed on a total of 150 fibers. Fibers at the specimen edges were not
considered in the analysis. Statistical analysis of the FLD was performed
on Statistica v.6 software.
3.5.2.2 Fiber orientation distribution (FOD)
Orientation distributions of short random fiber reinforced composites
require a three-dimensional description [34]. To determine the orientation
of individual fibers in the matrix, a spherical coordinate system is typically
used. The orientation of a fiber can be described in spherical coordinates
by the two angles, Φ and θ [32, 213, 248]. The in-plane orientation angle
Φ is assumed to follow symmetry conditions i.e. the probability of the Φ is
equal to the probability of Φ+180o.
While the definition of orientation and distribution is well established and
generally accepted for short fiber composites in which the constituent
fibers are straight, there is no such clear definition in situations of fibers
with arbitrary curvatures or waviness. Komori and Makishima [234]
considered the orientation distribution of curved fibers as the distribution
of straight segments resulting from a subdivision of the wavy fibers in a
number of segments. However, such description is computationally
complex, and may lead to inaccuracies related to the dependency on
segment lengths. In the present work the orientation of an individual wavy
fiber is considered as the end-to-end direction of the fiber centerline as
defined in [213], with the variation of orientation along the fiber handled
by the random waviness model (Equation 3.1). A graphical representation
of the orientation of the wavy steel fiber, as considered in this work, is
shown in Figure 3.6 (c).
Orientation tensors, as introduced by Advani and Tucker [32], are compact
representations of the fiber orientation state, typically used in cases where
a rigorous description through fiber orientation distribution is
computationally expensive [5, 32, 249]. Advani and Tucker presented two
types of orientation tensors, i.e. the 2nd and 4th order orientation tensors 𝑎𝑖𝑗
and 𝑎𝑖𝑗𝑘𝑙. While the 4th order tensor presents more accurate description of
the orientation distribution, the lower order tensor is typically used in
manufacturing simulation and micro-structure generation models due to its
compact description. This was also motivated by the results of the authors
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82
who have shown that the 2nd order tensor provides nearly similar accuracy
as the 4th order tensor in most cases. In this respect, the normal trend of
using the 2nd order tensor, as defined in Equation (3.2), will be followed in
this work.
𝑎𝑖𝑗 = ∫𝑝𝑖𝑝𝑗𝜓(𝑝)𝑑𝑝 (3.2)
The components of 𝑝 are related to the angles θ and Φ as described in
Equation (3.3).
𝑝1 = sin θ cosΦ
𝑝2 = sin θ sinΦ
𝑝3 = cos θ
(3.3)
In this study, the second order orientation tensor is calculated from micro-
CT data and used as input in the geometrical model. Individual fiber
centerlines are exported from the Mimics software. A Matlab algorithm
was created for analyzing the centerlines and calculating the θ and Φ angles
(end-to-end orientation angles of the wavy fibers) and the direction vector
𝑝 of each fiber as shown in Figure 3.6 (c). The second order tensor 𝑎𝑖𝑗 is
then calculated according to Equation (3.2).
3.5.2.3 Fiber waviness
An analysis was performed to quantify the degree of waviness of the steel
fibers. In a recent study, Rezakhaniha et al. [250] investigated the
waviness of collagen fibers and introduced a so-called straightness
parameter (𝑃𝑠) which is defined according to Equation (3.4).
𝑃𝑠 =𝐿0
𝐿𝑓
(3.4)
𝐿0 being the distance of visible end-points of the wavy fiber (Figure 3.6
(b)) and 𝐿𝑓 is the real “wavy” length of the fibers. Consequently, 𝑃𝑠 is
bounded in the range between 0 and 1, where 𝑃𝑠 = 1 indicates a totally
straight fiber. The straightness parameter is analogous to the textile fibers
“crimp parameter” which is often expressed as the percentage of
unstretched length of the crimped yarn [251, 252].
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
83
In the present work, the straightness parameter was analyzed for the wavy
steel fibers. Values of 𝐿0 were calculated for each fiber using the Matlab
algorithm performed on exported centerlines from the Mimics software.
The straightness parameter could be practically useful as a means of
providing quantitative assessment of the mathematical model in
comparison with the experimental micro-CT data. For that purpose, the
same parameter was calculated from RVEs generated from the
mathematical model, using the same Matlab algorithm applied on
datapoints of fibers obtained from the model. The validation of the
straightness parameter is considered a minimum requirement for
evaluation of the accuracy of the model. For this reason, in the present
work, a full validation of the model is achieved quantitatively using 𝑃𝑠 in
addition to other aspects, i.e. qualitative comparison of the generated
RVE’s with micro-CT reconstructed samples and a comparison of the
waviness profiles generated from the model with the true waviness profiles
of the fibers observed through micro-CT.
To summarize, the process of analyzing micro-CT information involved:
thresholding of reconstructed datasets to remove matrix and noise voxels,
construction of a 3D model using the Mimics software, separation of single
fibers, idealization of the fibers into centerlines, extraction of (L, 𝐿0, θ, Φ)
values of each fiber, analysis of FLD, FOD, Ps and 𝑎𝑖𝑗 information.
3.6 Results and Discussion
3.6.1 Fiber length distribution
Figure 3.7 (a) shows the resulting length distribution data of the steel fibers,
measured from the micro-CT reconstructed model. While the original fiber
length was 5 mm as described above, the figure illustrates that, during
injection molding, the fibers were subjected to breakage resulting in a
length distribution with the maximum at about half of the pre-processing
length. Although these data indicate significant fiber breakage during
processing, in the case of the ductile steel fibers the fiber fragmentation is
much less severe than that reported for brittle glass fiber [33]. The lack of
fragmentation is compensated (replaced) by extensive fiber bending
leading to their waviness.
A number of typical statistical distributions were fitted to describe the
experimentally obtained fiber length histogram as shown in Figure 3.7.
Asymmetric functions such as lognormal or Weibull functions with a tail
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84
at the longer fiber lengths were used to describe the length distribution of
brittle straight glass fibers in [33, 34]. Nevertheless, the fitted probability
density functions plotted in Figure 3.7 (a) indicate that the lognormal
distribution does not accurately fit the steel fiber lengths data. Normal and
Weibull distributions provide better descriptions. However, normal
distribution has a left “tail” in negative values of the lengths, which is not
physical. Therefore, the Weibull distribution, with shape parameter, 𝑘 =2.24 and scale parameter 𝜆 = 2053, was used as input to the mathematical
model (Figure 3.7 (b)).
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
85
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.
(a)
(b)
[µm]
[µm]
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86
3.6.2 Fiber orientation distribution
Figure 3.8 (a) and (b) illustrates the resulting θ and Φ distributions (end-
to-end) between 0 o and 180o. The histogram of the Φ angle distribution,
shown in Figure 3.8 (a), indicates that the fibers are almost homogeneously
distributed, with no preferred orientation in the XY plane. In contrast, the
distribution of the θ angle shows a peak (preferred orientation) at 90o,
suggesting that the fibers are oriented quite parallel to the plane of the plate.
Figure 3.8 FOD of the short steel fibers (a) distribution of Φ angle and (b)
distribution of θ angle.
For input of the mathematical model, the orientation tensor was determined
as explained in section 3.5.2.2. The resulting second order orientation
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
87
tensor 𝑎𝑖𝑗 of the end-to-end fiber orientations corresponding to the case in
Figure 3.8 was as follows:
𝑎𝑖𝑗 = [0.444 0.061 0.0790.061 0.465 0.0060.079 0.006 0.089
]
The diagonal components of the orientation tensor provide an idea about
the preferential orientation. The sum of all diagonal components is equal
to unity. In the case of perfect random 2D orientation, the diagonal
components of the orientation tensor should be (0.5, 0.5, 0). In agreement
with the analysis of the angle distributions, the resulting orientation tensor
can be approximated as a quasi-planar orientation tensor (in XY plane).
This is because of the nature of the samples having very small thickness
compared to planar directions (Z being the thickness direction, Figure 3.5).
Komori et al. [234] reported that in sheet-like assemblies, all fibers may be
conceived to be oriented parallel to the plane of the sheet.
It should be noted that the observed planar orientations of the steel fibers
explained here refers to the end-to-end orientation as discussed earlier.
This end directional orientation was considered for efficient numerical
calculations. However, the straight segments explained in section 3.2, will
exhibit more complex three-dimensional orientations which can be seen in
Figure 3.4 (a) and (b), where 2D sections perpendicular to the plane of the
plate, reconstructed from micro-CT data, comprised only dots (point
projections) of the fibers indicating 3D orientations of fiber segments
resulting from its waviness.
3.6.3 RVE of steel fibers
Table 3.1 summarizes the main geometrical input parameters used for the
generation of RVEs of the short wavy steel fiber reinforced polycarbonate
composite, considered in this study, using the micro-structural model. All
input parameters are calculated from micro-CT information (VF, d are
assumed nominal). The values of 𝐴𝑚𝑎𝑥 and 𝑛𝑚𝑎𝑥 are similarly determined
from the micro-CT images. Based on the analysis of the data points of each
fiber, the amplitudes of the fiber waves were calculated using the
developed Matlab algorithm. 𝐴𝑚𝑎𝑥 was then taken as the maximum
obtained 𝐴. Using the same algorithm, the number of waves per fiber was
analyzed for each fiber in the RVE. 𝑛𝑚𝑎𝑥 was taken as the maximum 𝑛 per
fiber.
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88
Table 3.1 Main geometrical input parameters used for the mathermatical model.
Input parameters
Volume Fraction, VF% 0.05
Fiber diameter, d [mm] 0.008
Fiber length distribution, 𝜓𝐿 Weibull distribution
𝑘 = 2.24
𝜆 = 2053
Fiber orientation Orientation tensor
𝑎𝑖𝑗 = [0.444 0.061 0.0790.061 0.465 0.0060.079 0.006 0.089
]
Fiber waviness Maximum wave amplitude, 𝐴𝑚𝑎𝑥 = 0.05
mm
Waviness number, 𝑛𝑚𝑎𝑥 = 4
Figure 3.9 shows a realization of the wavy steel fiber composite RVE
generated using the micro-structural algorithm explained in section 3.3.
The figure shows a qualitative agreement of the mathematical model to the
real steel fiber reinforced architecture observed through tomography
analysis (Figure 3.5, and Figure 3.6 (a)). Using a random positioning
algorithm, the model was able to mimic the clustering of the steel fibers.
The figure illustrates very comparable regions of entanglements of the
simulated fibers (indicated by black arrows in Figure 3.9) and other regions
of low local volume fractions of fibers (indicated by red arrows in
Figure 3.9), as was observed in the real micro-structure shown in Figure 3.5
and Figure 3.6.
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
89
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.
Due to the large number of fibers generated in the RVE instance shown in
Figure 3.9, exact modeled waviness profiles of each fiber is not clearly
visible. Figure 3.10 shows a close-up micro-CT image (higher
magnification, smaller sample, voxel size = 2 µm) of the steel fiber
reinforced polycarbonate samples and a comparison of the real waviness
of the steel fibers with the one simulated by the geometrical model. The
generated RVE is the same as that illustrated in Figure 3.9 with input
parameters summarized in Table 3.1. Figure 3.10 is created by manually
selecting similar real and simulated fibers. The purpose is to validate the
waviness patterns obtained by the model, which is not clear from the global
picture in Figure 3.9. It can be seen that the model is able to generate
waviness profiles that are very comparable to real waviness of the steel
fibers embedded in the matrix.
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90
Figure 3.10 Micro-CT image of SSFRP and a comparison between real and
modeled waviness profiles using the developed micro-structural model.
3.6.4 Straightness parameter
Figure 3.11 plots the probability density histograms of the straightness
parameter 𝑃𝑠 calculated from experimental observations and geometrical
model. In addition, the figure includes normal fits of the density functions.
The 𝑃𝑠 parameter allows a certain quantification of the waviness of the
fibers, and is considered a primary requirement of the model validation.
The experimental histogram demonstrates that the waviness of the steel
fibers follows a normal distribution with most fibers having 𝑃𝑠 values in
the range between 0.4 – 0.7 (µ = 0.58, σ = 0. 21). The results imply the
high degree of waviness of the injection molded steel fibers. Rezakhaniha
et al. [250] explained that the straightness parameter is the inverse of the
straightening stretch, which refers to the amount of stretch to be applied
along the fiber in order to get it straightened. The 𝑃𝑠 distribution may be
particularly interesting for consideration in the development of micro-
structural models of short steel fiber composites, especially with the high
degree of waviness exhibited by the steel fibers under consideration. The
significance of the parameter is expected to be higher with increasing
aspect ratios of the fibers.
In addition to giving quantified information about the waviness of the steel
fibers, the straightness parameter 𝑃𝑠 provided means for validation of the
Geometrical Characterization and Modeling of Short Wavy Fiber Composites
91
micro-structural model against the micro-CT information. Figure 3.11
shows a comparison of the histogram and normal fits of the 𝑃𝑠 distributions
calculated from the simulated RVE against that calculated from
experimental micro-CT data. The figure shows a very good agreement of
both histograms and normal distribution fits. The close agreement reveals
that the mathematical model provided successful representation of the
waviness of the steel fiber composites, in addition to the satisfactory
simulation of the nature of the local entanglements and local variations of
the fiber volume fractions of the steel fibers as explained above. The
correct simulation of the short steel fiber composite architectures imparts
the basis for accurate predictions of the behavior of the material through
further structural and mechanical models.
Figure 3.11 Probability density of the straightness parameter Ps: comparison
between experimentally achieved (micro-CT) information and mathematical
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.
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92
3.7 Conclusions
The model concept proposed in this work, provides a means for generation
of short random fiber reinforced micro-structures. The novelty of the
model relies in the capability of modelling complex wavy reinforcements
based on mathematical formulations. The model gives foundation for
further accurate predictions of mechanical properties of the wavy fiber
composites taking into account random and stochastic characteristics of the
local micro-structure. In the present work, the model was applied for a
short wavy steel fiber reinforced composite. The developed model can be
applied to other composite systems e.g. natural fiber composites, non-
woven fiber reinforced composites where waviness typically exists but is
seldom considered in generation algorithms. The mathematical
formulation of harmonic functions of fiber waviness can be easily adapted
to envelop other types of random crimped short/long fiber reinforcements.
Another aspect of this work is the development of a methodology for the
characterization of the geometrical parameters of short steel fibers
composites using micro-CT testing. The developed methodology presents
solid means for efficient complex 3D analysis of random wavy structures.
In the next stage of the research, micro-mechanical and damage modelling,
based on the well-known mean-field homogenization techniques, will be
performed on RVEs of short steel fiber composites generated by the
algorithms described in this chapter. The obtained predictions will be
validated against experimental results of real samples. This will promote
further validation of the geometrical model presented in this work.
93
Chapter 4: Experimental Characterization of Quasi-Static Behavior of Short Glass and Steel Fiber Composites
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
95
4.1 Introduction
The focus of the present chapter is the experimental characterization of the
quasi-static tensile behavior of the materials considered in this thesis. Two
types of materials were investigated, namely, typical short glass fiber
reinforced composites and short wavy steel fiber composites. Several
variations of these materials were explored. The objective of the
investigations is twofold. First, to achieve understanding of the quasi-static
behavior and damage mechanisms of both types of materials, which can
then be used in the development of the model concepts. Second, the
experimental results will serve for final validation of the proposed models.
4.2 Materials and Methods
4.2.1 Materials
In the present work two different short glass fiber reinforced systems were
employed, namely: 30wt% (16VF%) E-glass fiber reinforced polyamide 6
(Akulon K224-G6) and 30wt% (13VF%) E-glass fiber reinforced
polypropylene (Schulatec PP 30H). The fiber diameter in both materials is
10 µm. Both the polyamide and polypropylene are semi-crystalline
polymers with glass transition temperatures 𝑇𝑔 = 50 − 60℃ and 𝑇𝑔 < 0℃
respectively. From this point on, the two materials will be referred to as
GF-PA and GF-PP respectively.
For short steel fiber composites, pellets of polyamide 6 (PA 6, Durethan
B38 F KA) with 5 mm initial length steel fibers (Beki-shield) were used.
The pellets were supplied by the company Bekaert (producer of steel
fibers). The steel fibers diameter was 8 µm. Specimens of different volume
fractions were produced. To achieve the desired volume fraction, the
pellets were “diluted” with pure PA 6 in a compounding step. Samples of
volume fractions of 0.5, 1, 2, 4 and 5% were manufactured. These volume
fractions correspond to weight fractions of 4.5, 8.7, 16.5, 29.9, and 35.6%
respectively. The samples will be designated SF-PA.
It should be noted that the PA 6 matrices in the GF-PA and the SF-PA
composites had different commercial grades. The PA 6 in the GF-PA
material was of the commercial grade Akulon K222-D, while the PA 6 in
the SF-PA material had the grade Durethan B 38. With reference to
available literature, in contrast to e.g. polypropylene materials, different
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96
grades of polyamide materials may significantly differ in properties and
behavior.
For the application and validation of the models developed in this PhD
thesis, the tensile properties and full stress-strain curves of the matrix in
each composite to be modelled is needed. This information was readily
available for the Akulon PA 6. Details of the stress-strain curves of the
Akulon material will be shown in the description of the models validation
test cases described in section 7.4.1. Nevertheless, the full stress-strain
curve of the Durethan PA 6 was not available. In this respect, in this PhD
thesis, tensile testing were performed on neat Durethan PA 6 samples
produced by the injection molding technique. The samples had the same
dimensions as the composite samples discussed above. Results of the
tensile test on the Durethan PA 6 material will be reported in
section 4.3.5.1.
4.2.2 Specimen preparation
For all the materials, dog-bone “standardized ISO specimens” were
produced by injection molding. The sample dimensions were in
accordance to the ISO 527-4:1997 standard, specimen type 1B. The main
sample dimensions (nominal) were as follows: width at the gauge section
= 10 mm, thickness = 4 mm, gauge length = 50 mm, total length of the
specimen = 170 mm and the width of the gripping ends = 20 mm.
Prior to injection molding, the steel fiber materials were compounded using
a co-rotating twin-screw extruder at a screw speed of 210 rpm and melt
temperature of 270-280oC. Compounding of the SF-PA samples was
performed at the Technology Campus Ostend. In a similar way to the glass
fibers, steel fibers are compatible with the injection molding technique due
to the very high melting point of the fibers. As explained in Chapter 3, the
fibers are treated with specific polymer coatings which are tailored for
compatibility with the different polymer matrices for composites
applications.
Injection molding of all samples was performed at Technology Campus
Diepenbeek. The main injection parameters used for the production of the
different samples are summarized in Table 4.1.
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
97
Table 4.1 Injection molding parameters of the glass fiber and steel fiber samples.
GF-PA GF-PP SF-PP
Injection pressure [Bar] 2580 2500 1089-1378
Holding pressure [Bar] 200 300 700
Packing pressure [% of max
injection pressure]
12 12 12
Injection speed [mm/s] 70 75 100
Injection flow rate [cm3/s] 28.5 28.5 28.5
Mold surface temperature [oC] 65 40 90
Melt temperature [oC] 240 220 250-270
Screw speed [mm/s] 150 100 400
Packing time [s] 4 4 4
Cooling time [s] 25 20 20
For the steel fiber samples, for volume fractions up to 2%, the melt
temperature was 270 oC which is recommended in the manufacturing data
sheet of the Durethan PA 6 polymer. However, due to leaking problems of
the melt at higher fiber volume fractions of the SF-PA material, the melt
temperature needed to be reduced to 250 oC. The volume fraction of 5%
was the highest possible achieved volume fraction due to manufacturing
constraints. Higher volume fractions of the steel fibers resulted in blockage
of the extruder die in the compounding process.
4.2.3 Fiber length distribution measurement
Fiber length measurements were performed on the GF-PA and GF-PP
specimens to obtain the fiber length distributions of each material. The
measurements were conducted using the standard matrix burn-off
technique [5, 253]. Samples of 3 cm length were cut from the middle of
the gauge segment of the dog bone samples. Samples were placed in
crucibles in an electric oven at 500 oC for 12 hours (melting temperatures
of PP and PA 6 matrices are 170 and 210 oC respectively). The polymeric
matrices were burned out and the remaining glass fibers were observed
under an optical microscope. Red background paper was used to achieve
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98
high contrast. The fibers were spread on the paper to visualize the
individual fibers. For each material, at least 200 fibers were measured
randomly. Fiber lengths were measured using magnified images obtained
from the microscope observations and fiber length distributions were thus
determined.
Detailed measurement of the length distributions of steel fiber composites
was more difficult to achieve. Using the matrix burn-off and optical
microscopy techniques explained above, obtaining the “real” wavy fiber
length requires manual measurements of small segments along the fiber.
The method developed in Chapter 3 for measurement of wavy fiber lengths
based on micro-CT scans of the composite was used instead. However,
with increasing volume fractions of the steel fibers, the very crowded
matrix and high entanglements of the fibers led to difficulties in tracking
and segmenting a large number of individual fibers for analysis of
complete statistical distributions. Therefore, micro-CT measurements were
done to have an estimate of the average fiber length.
4.2.4 Tensile testing
Tensile testing at room temperature of the short glass fiber and short steel
fiber composite dog-bone samples was performed according to the ISO
527-2:1996 standard [254]. An Instron 4467 tensile machine, equipped
with a 30 𝐾𝑁 load cell, was used. All tests were carried out at a cross-head
speed of 2 mm/min corresponding to about strain rate of 0.0007 s-1. A
minimum of 5 samples were tested for each condition. Strain was measured
using an optical extensometer (more details will be given below in
section 4.2.4.1). Stiffness was measured in the range 0.05-0.25% strain
according to the ISO 527-1:1997 [255] and ISO 527-2:1997 [254]
standards. The tensile test was coupled with Acoustic Emission (AE)
registration for damage monitoring (explained below in section 4.2.4.2).
4.2.4.1 Digital image correlation (DIC)
During the tensile tests, strains were measured using the optical
extensometer, i.e. the digital image correlation method. Prior to the test,
random speckle patterns were applied on the surface of the samples by
using appropriate black or white paints. Consecutive images of the sample
surface were registered using a digital image acquisition system (LIMESS,
Messtechnik und Software GmbH) during loading. Images were registered
every 500 μs corresponding to 0.011% strain increments. Full-field strain
maps were then obtained from the registered images using the digital image
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
99
correlation method (VIC 2D, Correlated Solution inc.). The area of interest
was chosen in the gauge segment of the dog-bone specimen.
4.2.4.2 Acoustic emission (AE) registration
Acoustic emission registration was used in the present work to evaluate
and have an insight on the damage inside the tested short fiber composites.
The concept of Acoustic Emission is based on the detection of sound waves
inside the material when strain energy is released during the formation and
propagation of microcracks, i.e. damage events [256]. Two sensors
(V5375-M, Vallen Systems GmbH) were mounted on the surface of the
sample at the boundaries of the gauge length. The distance between the
sensors is noted. Prior to the test, a calibration of the AE sensors is
performed using “Automatic Pulsing”. The velocity of the sound within
the specimen material is calculated by knowing the distance between the
two sensors and the difference in time in which the signal travelled from
one sensor to the other [257]. This allows the signals (events) that occur
outside of the sensors to be filtered out and to locate the exact position of
the registered events. The procedure was repeated for each tested
specimen. The signals recorded during the tests are filtered and amplified
(AMSYS-5 system, Vallen Systems GmbH). A threshold of 35 dB was
used to filter out the noise. The sensors were kept on the sample until final
failure.
4.2.5 Micro-CT analysis
The morphology of the different undamaged samples was observed using
high resolution micro-CT techniques. Micro-CT characterization was
previously discussed in Chapter 3. Two different scanners were used,
namely Phoenix Nanotom S (GE Measurement and Control Solution,
Germany) and SkyScan 1172(SkyScan NV, Kontich, Belgium). The scans
resolution (voxel size) was 3 µm. The visualization of the three-
dimensional reconstructed volumes was obtained using the software
VGStudio MAX (Volume Graphics Solutions).
4.2.6 Fractography analysis
Fracture surfaces of the broken specimens were observed using a scanning
electron microscopy (SEM). The SEM equipment used was (PHILIPS
XL30 FEG). The acceleration voltage was 10.0 kV using the secondary
electrons for detection. Prior to the SEM evaluation, samples were coated
with a thin layer of gold to achieve electrical conductivity. The coated
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100
samples were then placed in a vacuum chamber for a minimum of 12 hours
to remove humidity. Images with varying magnifications were captured to
gain insight on the different damage mechanisms in the samples.
4.2.7 Single steel fiber tensile tests
Tensile tests were performed on single Beki-shield steel fibers, used in this
study. Due to the very low diameters of the fibers (𝑑 = 8 μm), the forces
expected in the single fiber tests are in the order of μN, which could not be
achieved within the accuracy of normal tensile tester. A quasi-static testing
procedure was set-up in the Dynamic Mechanical Analyzer (DMA Q800,
TA Instruments). An 18 N load cell was used with a resolution of 1 μN and
a crosshead with a displacement resolution of 1 nm. The tests were
performed according to the ASTM D3379 standard test method for tensile
strength and Young’s modulus for high modulus single-filament materials
[258]. A schematic of the specimen preparation for the single fiber tensile
test is shown in Figure 4.1.
Figure 4.1 Specimen preparation for single fiber test on the DMA machine.
The single fiber is placed between two sheets of paper using adhesive glue.
The paper sheets have a rectangular gap in the middle (with rounded
corners). Carbon cement was applied to prevent slipping of the fiber during
the test. After mounting on the machine, the paper is cut at its middle as
shown in Figure 4.1. The gauge length used in the tests was 25 mm and
the total length of the mounting paper was 70 mm. The exact gauge length
after mounting is also measured automatically on the machine before each
test. The stiffness of the fibers was calculated from the derived stress-strain
curves between 0.1 and 0.3% strain [259].
Fiber
Sections to be cut after gripping
Carbon cement
Gauge length Adhesive Paper
Grip area
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
101
4.3 Results and Discussion
4.3.1 Fiber lengths measurements
As mentioned in section 4.2.3 measurements were performed in order to
obtain detailed fiber length distributions of the GF-PA and GF-PP samples.
Figure 4.2 (a) and (b) show the resulting length distribution histograms of
the GF-PA and the GF-PP materials respectively.
The results in Figure 4.2 (a) and (b) show that both materials exhibit
skewed, wide and asymmetric distributions with higher probabilities of the
shorter fibers (with a tail at the longer fibers end). The achieved
distributions are in agreement with the published data on the length
distributions of the short brittle (straight) glass and carbon fiber
composites, e.g. in [260-263]. The reason is attributed to the increased
probability of damage of the brittle fibers during processing, as discussed
in section 2.3.1.
Such kind of distributions can be represented using a Lognormal or a
Weibull distribution probability function, as discussed in section 3.6.1.
Fitting of both distributions was performed on the two investigated
materials and it was found that for both materials, the Lognormal
distribution was fitting best. Figure 4.2 (c) and (d) show the Lognormal
probability plots of the FLDs of the GF-PA and GF-PP materials. The
figures show good agreement between the experimental data and the fitted
distributions. From the graphs, it is seen that the GF-PP material depicted
much higher fiber lengths compared to the GF-PA material. Although both
materials were reinforced with the same weight percentage of fibers, the
average fiber length in the GF-PP was about 1.1 mm, which is about three
times higher than the average length found in the GF-PA samples
(0.3 mm). The range of obtained fiber lengths in GF-PP was up to 5 mm,
while the fibers in the GF-PA samples were all shorter than 1 mm. This is
also reflected in the parameters of the fitted distributions. The difference
in length between the GF-PA and the GF-PP maybe related to the initial
length of the pellets before injection molding. It is expected that the ranges
of the fiber lengths distributions of the GF-PP pellets before injection were
higher than those of the GF-PA. This observation was reported in previous
studies, e.g. [264-266]. In these studies, the fiber lengths distributions
before and after the injection molding process were reported for a number
of short fiber composites. For all composites, the higher the fiber lengths
before injection molding, the higher the fiber lengths distributions in the
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102
final product. Nevertheless, the information of the pre-injection fiber
lengths distributions were not available in the present study.
As input to the models developed in Chapters 7 and 8, Lognormal
distributions of parameters μ = 5.6 , 𝜎 = 0.5 and μ = 6.8 , 𝜎 = 0.7 will
be used for the GF-PA and GF-PP respectively, where μ is the log mean
and 𝜎 is the log standard deviation of the Lognormal function (the data of
fiber lengths were defined in μm) .
As mentioned above, for the short steel fiber composites, measurements
were performed to obtain the mean length of the samples with the different
volume fractions. The results are summarized in Table 4.2.
Table 4.2 Average fiber lengths of the SF-PA samples with different fiber
volume fraction
Volume fraction
(%)
Average length
(𝐿𝑎𝑣𝑔) [µm]
0.5VF% 605
1VF% 527
2VF% 557
4VF% 385
5VF% 352
103
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.
Lognormal
distribution fit
(b)
(d)
Lognormal
distribution fit
(a)
(c)
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104
Table 4.2 indicates that the final lengths of the fibers in all the investigated
samples was less than 1 mm. The ranges of the lengths for all samples were
around 200 - 900 µm. A trend of the decrease of the mean fiber length with
increasing fiber volume fraction can be noticed. While between similar
volume fractions, e.g. 0.5, 1, and 2% only subtle differences can be found,
a clear difference can be found between 0.5 and 5%. This observation is in
agreement with literature data [46, 260, 262, 267] reported for the typical
glass fiber composites. The decrease of fiber length with increasing
concentration of the fibers is attributed to the higher melt viscosity and
increased tendency for more fiber-to-fiber contacts and hence fiber
damage.
4.3.2 Tensile behavior of the short glass fiber composites
4.3.2.1 Tensile properties
The tensile curves of the GF-PA and GF-PP are shown in Figure 4.3 and
their tensile properties are summarized in Table 4.3. The table also shows
a comparison between the actual achieved values of the tensile properties
and the values reported in the manufacturing datasheets of the commercial
materials. It can be noticed that for the GF-PP samples, the achieved values
are comparable to the manufacturer’s data.
Figure 4.3 Measured stress-strain curves and of the GF-PA and GF-PP materials.
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
105
For polyamide based materials, product datasheets often provide two
distinct values for the different properties, one corresponding to the dry as
molded behavior, and one for the conditioned material behavior. This is
due to the strong humidity dependence of the hygroscopic PA 6 material,
discussed in section 2.3.2. Dry as molded (d.a.m) refers to properties
obtained from a sample with equivalent moisture content as when it was
molded (usually less than <0.2%) while conditioned refers to properties of
the sample at 50% relative humidity [268, 269]. The difference in
mechanical properties between the d.a.m and the conditioned samples in
terms of decrease of stiffness and strength is attributed to the loss of
adhesion between the fiber and matrix [270]. It has also been reported that
water absorption results in an increase of the ductility and toughness of the
material, due to the plasticizing effects of the water molecules in
polyamides, as discussed in [270-272]. While the samples in the present
work were kept in sealed bags to avoid moisture effects as much as
possible, the obtained properties lied in between the d.a.m and conditioned
datasheet values. The probable variation of the exact moisture content of
the samples is also the reason for the higher standard deviations of the GF-
PA.
Table 4.3 Tensile properties of the short glass fiber polyamide (GF-PA) and
short glass fiber polypropyelene (GF-PP) composites. A comparison is given
between the actual measured properties in the present work and the data reported
in the manufacturer’s datasheets.
GF-PA GF-PP
Actual Data Sheet
dry/cond
Actual Data Sheet
𝐸 [GPa] 8.05 ± 0.38 9.5/6.0 6.75 ± 0.35 6.9
𝜎𝑢𝑙𝑡 [MPa] 112.3 ± 2.6 180/110 83.7 ± 1.29 116
𝜈 0.46 ± 0.06 - 0.36 ± 0.05 -
휀𝑢𝑙𝑡 [%] 4.39 ± 0.45 3.5/7 2.25 ± 0.08 2.6
𝐸 is the Tensile modulus, 𝜎𝑢𝑙𝑡 the ultimate tensile strength, 𝜈 Poisson’s
ratio, 휀𝑢𝑙𝑡 is the ultimate strain at break.
It is apparent from Figure 4.3 and Table 4.3 that, despite similar weight
fractions of the GF-PA and GF-PP, and the much smaller lengths of the
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106
fibers in the GF-PA samples, the GF-PA has higher stiffness and strength
compared to the GF-PP values. Moreover, due to the longer fiber length,
the GF-PP is expected to have higher fiber orientation (exact orientation of
the materials will be discussed in Chapter 7).
The observation of the superior mechanical properties of GF-PA compared
to GF-PP materials with similar weight fractions is in agreement with
literature findings, e.g. in [264]. There are several possible explanations
for this observation. The first explanation is the slightly higher volume
fractions of the GF-PA which is about 16% compared to 13% for GF-PP
(as a result of the lower density of the PP matrix 𝜌 = 0.9 compared to 1.14
for the PA polymer).
The second explanation can be the higher strength of the PA polymer over
the PP polymer. Details of the stress-strain curves of the PA and the PP
polymer are as follows. For the GF-PA material of the present work, the
matrix had the commercial grade Akulon K222-D. The stress-strain curve
of the matrix was available in the available manufacturing datasheet and
could be obtained from the CAMPUS plastics database [273] as shown in
Figure 4.4. The tensile modulus of the PA matrix was reported in the
database as 1200 MPa and the yield strength as 55 MPa at a 25% strain.
Figure 4.4 Stress-strain curve of the polyamide Akulon K222-D [273]. The tests
are stopped at the yield of the matrix.
For the GF-PP, the stress-strain curve of the exact neat polypropylene
material grade was not available in the manufacturer’s datasheet.
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
107
Nevertheless, the stress-strain curves of the (unmodified) polypropylene
polymer were found to be generally reproducible in the literature and the
materials databases. The stress-strain curve for the PP matrix in the present
work was obtained from the paper of Jao Jules et al. [274] as shown in
Figure 4.5. The tensile modulus of the PP matrix was 1450 MPa and the
yield strength of the PP matrix was about 31 MPa at a strain of 4.9%.
As discussed, the higher strength of the PA over the PP matrix can be a
reason for the improved strength of the GF-PA composite. However, the
two polymers have almost similar stiffness moduli (the PP had a slightly
higher stiffness than the PA), which does not explain the increased stiffness
of the GF-PA composite.
Figure 4.5 Stress-strain curve of the polypropylene matrix [274]. The tests are
stopped at the yield of the matrix.
A more significant reason for the superior mechanical properties of the GF-
PA over the GF-PP, can be the weaker interfacial adhesion of the glass
fibers and the PP matrix. This can inhibit effective load transfer from the
matrix to the fibers. A detailed study on the assessment of the interfacial
shear strength of glass fiber composites including PA and PP reinforced
composites was performed by Desaeger and Verpoest [275]. The authors
performed micro-indentation tests and have shown that the interface
adhesion of the polypropylene and the glass fibers is so weak that
debonding already occurred during specimen preparation. It should be
noted that in this study of Desaeger and Verpoest, a special type of
specimen preparation (cutting and polishing, etc.) was performed, that’s
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108
why debonding resulting from the weak interface could be observed at the
surface of the specimen, where the fibers stuck out of the matrix. The weak
interface of GF-PP composites is often improved and optimized in
commercial materials, by application of suitable fiber treatments and/or
additives to the PP matrix, to achieve full potential but are nevertheless not
comparable to that of GF-PA composites.
4.3.2.2 Damage development
Representative AE registrations recorded during the tensile tests on the
GF-PA and GF-PP materials are shown in Figure 4.6 (a) and (b)
respectively. For each of the materials, three distinct zones can be
distinguished. In the first zone A, no AE events can be detected. This
“silent zone” can be seen in both the GF-PA and GF-PP. For both materials
the onset of AE events is at approximately 휀 = 0.005.
It should be noted that the onset of damage or the registration of AE events
might be influenced by the threshold value for the signal amplitude, used
for AE noise filtering (35 dB). Hence, events with an amplitude lower than
the threshold might have not been registered. However, by comparison of
the stress-strain curves in the same plot, it can be observed that the onset
of AE events roughly corresponds to strain values at which non-linearity
of both the GF-PA and GF-PP becomes apparent (휀 ≈ 0.005). This leads
to the conclusion that the onset of damage predicted by AE is realistic and
that events with an amplitude below the threshold value are negligible.
The notation (behavior) of the second and third zones shown in Figure 4.6
is different for the GF-PA and GF-PP materials. For the GF-PA material,
as shown in Figure 4.6 (a), the AE events appear in zone B while
surprisingly in zone C no AE events were registered. The observation is
remarkable as it is expected that registration of AE events continue up to
failure, with the possibility of having high energy AE events close to
failure of the composite. The behavior was reproducible for all measured
samples. In contrast, for the GF-PP material zone B depicted a large
number of AE events and the last zone indeed included high energy events
very close to the failure strain.
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
109
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.
(a)
(b)
A B C
A B C
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110
The presence of the AE “silent zone” cannot be directly explained. A
suggested reason could be the ductility and failure mode of the PA 6
matrix. Mandell in [276] provided insightful observations on the tensile
and fatigue strength in different short fiber composites and on the
progression of damage. The author investigated the fatigue behavior of
short fiber composites, but some insight can be given on the general failure
aspect of different short fiber systems. By observing the failure surface of
fatigued glass fiber reinforced PA 6.6 composites (which is very similar in
nature of the glass fiber reinforced PA 6 composite considered in this
study), the author reported a dominance of ductile failure modes. Failure
of the GF-PA 6.6 composite was then assumed by the author to occur when
some local areas of the matrix ruptures quickly spread to produce failure.
Another reason can be that the during loading debonding occurs in fibers
with lower lengths, orientation, etc., afterwards further loading does not
lead to initiation of new damage in Zone C.
The much higher number of damage events registered for the GF-PP matrix
suggests a lower interface bonding compared to GF-PA systems as
described above. The same fact is also supported by the weaker stress-
strain curves, despite the much longer fibers in the GF-PP material, as
previously discussed. These conclusions led to reasonable assumptions that
there is less debonding in the GF-PA system and that local yielding or
microscopic rupture zones in the matrix contributed to a large extent to the
final failure of the composite. Such (matrix dominated) damage
mechanisms are expected to generate lower amplitude events that may
have not been registered with AE due to the filtering thresholds.
Apart from the stronger interface of the GF-PA composites compared to
the GF-PP composites [275], also the difference in fiber length between
both materials may influence the damage modes. Czigány and Karger-
Kocsis [277] investigated short and long (discontinuous) glass fibers
polypropylene composites. They have shown that for the same material
combination (same fibers and matrix), composites with short fiber lengths
resulted in ductile failure surfaces whereas those with longer fibers showed
much more brittle behavior.
Figure 4.7 shows a comparison of the cumulative AE energy plots of the
GF-PA and GF-PP composites. The patterns of AE cumulative energy
plots are different in both materials. The GF-PA materials showed the so-
called “knees” in the cumulative energy plots. While for unidirectional and
textile composites three different jumps have been repeatedly reported and
related to distinct failure events, e.g. in [256, 257, 259, 278], for the short
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
111
fiber composite systems in this study, such correlation between the jumps
in the cumulative AE energy curves and the damage mechanisms cannot
be directly confirmed.
Figure 4.7 Comparison of the cumulative AE energy registrations of the GF-PA
and the GF-PP materials.
Nevertheless, by analysis of the jumps in Figure 4.7 for the GF-PA and the
plots of the individual events shown in Figure 4.6 (a) it can be concluded
that the first jump corresponds to the threshold of damage (beginning of
acoustic emission events). The second jump corresponds to the start of a
region of more “intense” acoustic emission activity and hence damage
events. Since the energy levels of the registered events in both regions are
similar (mainly ≤ 103 a.u.), which are generally low level events, there is
no reason to associate the second jump to the development of a new
damage mechanism, but this jump should be rather associated to a strain
level corresponding to occurrence of a higher number of events.
Based on the literature information analyzed in section 2.3, it has been
shown that the main damage mechanism in short fiber composites is fiber-
matrix debonding. Hence, it can be assumed that the registered events are
related to debonding where at low strains, only few debonding events are
registered and at higher strains more intensive debonding occurs. This is
supported by the detailed study in [278] using the same AE system.
Although the scope of the study was textile composites, the authors
1
2
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112
reported that events with energy levels lower than 103 a.u are related to
microscopic debonding where higher energy levels can be related to meso-
scale damage mechanisms which are intrinsic to textile composites, such
as transverse matrix cracking and delamination.
In contrast, for the GF-PP material, no “knees” or significant jumps can be
distinguished. This suggests a smooth and progressive damage
development characterized with an increasing number of damage events
since the first onset of acoustic emission activity. The energy of AE events
were also mostly ≤ 103 a.u. similar to the energy levels in GF-PA. The
appearance of higher energy events can be attributed to the coalescence of
debonding voids into a crack until failure as suggested above. Another
possibility is the appearance of fiber fractures, which can occur because of
the high fiber length, resulting in built-up of higher stresses, possibly up to
the failure stress of the fibers. Although in general, the events are mostly
considered low energy events, a remarkable feature in the AE events
pattern (Figure 4.6 (b)) is the progressive increase of AE energy levels with
increasing strains in the range up the ≤ 103 a.u. This can be due to the
wide range of lengths distributions of the fibers, where longer fibers
debond at higher energies, as shown in the study of Czigány and Karger-
Kocsis [277].
Figure 4.8 shows the distributions of amplitudes and energies of AE events
in the GF-PA and GF-PP materials. For both materials asymmetric
distributions of amplitudes and energies can be noticed with higher
occurrences (probabilities) of lower amplitude and energy events. The
peak (mean) of the asymmetric distribution for the GF-PA and GF-PP were
about 40 dB and 45 dB respectively. For AE energies, the peaks of the
GF-PA and the GF-PP are at about 6 a.u. and 32 a.u. respectively which
essentially means that the peak of the distribution is at very low energy
events. As can be seen for GF-PA fewer events were found in the range
102 − 103 a.u. compared to the GF-PP and negligible events higher than
103 a.u. can be found for GF-PA while few such higher energy events
occurred in the GF-PP.
113
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.
(a) (b)
(d) (c)
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114
In the following, the observed amplitude distributions of the AE events are
compared with literature data. In their early work on short glass fiber
polypropylene composites, Barré and Benzeggagh [279] first assigned
distinct levels of AE amplitudes to different damage events. .According to
the authors ranges of 44-55 dB correspond to matrix cracking, 60-65 dB to
interface failure, 65-85 dB to fiber pull-out and 85-95 dB to fiber fracture.
Similar values were adopted later in literature for different materials, e.g.
in [177] for glass-epoxy SMC composites, [280] also for a short glass fiber
polypropylene composite, [47] and for a short glass fiber polyamide 6.6
composite.
Matrix cracking generally refers to the development of cracks which are
often perpendicular to the main loading direction, as discussed in
section 2.3. In this study, this damage mechanism was not observed. The
analysis of literature data reported in section 2.3 has shown that the
occurrence of this damage mechanism is very limited and is restricted to
the core layer where fibers are oriented perpendicular to the loading
direction. According to the classification of the authors above, the
distribution of events in the materials of the present study should be
dominated by matrix micro-cracks (with amplitudes between 44 and 55
dB), which is not the case since this damage mode was not observed.
Czigány and Karger-Kocsis [277] and Karger-Koscis et al. [281]
investigated discontinuous glass fiber polypropylene materials. The values
reported in the two papers are very similar, the different categories of AE
events amplitudes were: <30 dB for matrix deformation (crazing, i.e.
appearance of fine cracks on highly deformed regions of the material
during tensile loading, shear yielding, etc.), 30-60 dB for fiber-matrix
debonding, 60-90 dB for pull-out and >90 dB for fiber fracture. Czigány
and Karger-Kocsis [277] associated values of 80-85 dB to fiber fracture.
The AE amplitude categories in [277] and [281] are more in-line with the
damage modes found in the present studies (as will be discussed later,
using SEM micrographs). This categorization suggests then that the
dominant damage mechanism in both the GF-PA and GF-PP is the fiber-
matrix debonding. This also suggests that the GF-PP will have more pull-
out occurrences than the GF-PA as indeed will be shown in SEM
fractography analysis in section 4.3.4.
Moreover, all authors agreed that fiber fracture occurs at amplitude levels
at least higher than 80 dB. For both the GF-PA and GF-PA such high
amplitudes were not found. This suggest that fiber breakage, was to the
least negligible if at all existent.
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
115
The second AE characteristic is the energy content, expressed in arbitrary
units (a.u). It is difficult to relate the values of the distributions of AE
energies in Figure 4.8 to the energies of different damage mechanisms
reported in literature. This is because the unit [a.u.] is a virtual energy unit
used in the AMSYS-5 system to characterize the energy of AE events
[256]. However, as discussed above, both materials showed tendencies
towards low energy events which have been typically associated with
interface phenomena [278]. Since no events were found in the range of
amplitudes above 80 dB, it can be assumed that the higher energy events
≫ 103 a.u which appeared in Zone C in Figure 4.6 are not coming from
one single event, but from the brittle-like coalescence of debondings into
larger cracks, leading to final failure.
4.3.3 Micro-CT observations of the morphology of the short glass
fiber composites
The purpose of the micro-CT observations for the GF-PA and GF-PP
materials is to characterize the fiber orientations in the sample and the skin-
core morphology discussed in section 2.3.1. Figure 4.9 shows a
representative global micro-CT scan of the entire width of the GF-PP
sample used in this study.
In chapter 7, results of manufacturing simulation for the calculation of the
orientation tensor of the samples in this study will be shown. Micro-CT
analysis was performed as means of qualitative validation of the
manufacturing simulation results (to confirm the predicted orientations by
the manufacturing simulation software).
CHAPTER 4
116
Figure 4.9 Global micro-CT scan of the overall width of the GF-PP sample.
Figure 4.9 shows that the skin-core region found in the present samples
were located only in the center of the specimen: the “core” region,
highlighted red zone in the figure, is surrounded by the skin region, both
in thickness and in width direction, showing a preferential fiber orientation
in the mold flow direction. This indicated that the skin-core morphology is
localized centrally and is not developed homogeneously over the width of
the sample. A similar morphology was found for the GF-PA material. The
observation is in complete agreement with the detailed investigation of
Brunbauer et al. [54] who have clearly stated the difference in morphology
between injection molded plates and injection molded standardized testing
samples (dog bone samples). For the plates, the skin-core morphology
spread out over the entire width of the samples, whereas for the
standardized sample (injection molded in the final shape) the core layer
was only present in the center of the specimen.
Figure 4.10 shows a “close-up” view of the centralized skin-core structure.
t w
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
117
Figure 4.10 Representative view of the skin-core morphology in the central part
of a GF-PP sample.
The figure indicates that the region highlighted in red indeed shows lower
orientation (higher misorientations of the fibers is depicted in this central
region of the specimen, exact orientation values will be discussed in
Chapter 7) of the fibers and can be considered a core layer. The region
however is very narrow and therefore the core layer was limited in the
present samples and was surrounded by highly oriented shear layers in the
testing cross-section as indicated by the above mentioned authors [54].
4.3.4 SEM fractography analysis of the short glass fiber composites
Figure 4.11 shows SEM micrographs of the fracture surface of the GF-PA
material.
t
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118
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.
Figure 4.11 (a) shows an overview of a representative area of the fracture
surface of the GF-PA material. As can be seen in the figure, the pull-out
length of the fibers is short and in general the probability of the fiber pull-
out damage mechanism is low. Figure 4.11 (b), (c) and (d) show higher
magnification SEM micrographs of the fracture surface. The green arrows
denote the debonding damage mechanism. The figures also show relatively
low occurrence of debonding. The ductility of the fracture surface can be
observed through the high deformation of the matrix (micro-ductile
fracture). Another interesting aspect, denoted by the blue arrows is the
presence of “hills” of matrix around the pull-out zones of the fibers. This
observation, together with the relatively low number of pulled-out fibers,
confirm\ the strong interface bonding between the polyamide matrix and
the glass fibers. This helps to explain the improved mechanical
(a) (b)
(c) (d)
100 µm
50 µm 20 µm
200 µm
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
119
performance of the GF-PA material over the GF-PP material, as shown in
Figure 4.3, despite the shorter fiber lengths.
Figure 4.12 (a) shows an overview of a representative area of the fracture
surface of the GF-PP. Figure 4.12 (b) and (c) show higher magnification
SEM fractography images of the GF-PP. It can be seen that the main
damage mechanisms observed in the sample were also the debonding and
pull-out phenomena. In general, more debonded and pulled-out fibers can
be observed, in agreement to the higher number of AE events of the GF-
PP as shown in Figure 4.6 (b). In addition, generally longer pull-out length
of the GF-PP can be observed compared to the short pull-out length of the
GF-PA. Finally, in contrast to the “hills” of matrix material observed on
the fibers at the fracture surface of the GF-PA, the surface of the fibers in
the GF-PP appears to be “clean” with no traces of matrix on them. All those
observations support the assumption of the lower interface strength of the
glass fibers and polypropylene matrix compared to the interface strength
of the same fibers with the polyamide matrix.
For both the GF-PA and the GF-PP as shown in Figure 4.11 and
Figure 4.12, the ends of the fibers appear to be broken. The source of fiber
breakage cannot be distinguished from fracture surface analysis, namely
whether it is due to damage of the fibers during loading or due to the
damage of the fibers during processing.
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120
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
4.3.5 Tensile behavior of the short steel fiber composites
4.3.5.1 Tensile properties of the neat PA 6 matrix
A full stress-strain curve of the (Durethan B 38) polyamide 6 material is
not available in the manufacturing datasheet. For this reason, tensile tests
were performed on the neat PA 6 material (used for the steel fiber samples).
The tests were done at room temperature. Two series of tests were
performed. The first tests were done at a cross-head speed of 2 mm/min
(0.0007 s-1) to obtain the stress-strain curve of the material as well as the
tensile modulus and strength, which will be used as input to the models
developed in the next chapters (Chapter 7, 8). Due to the long duration of
the tests, the tests were stopped at 150% strain. The resulting stress-strain
(a) (b)
(d)
200 µm 100 µm
50 µm
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
121
curves of the neat PA 6 samples are shown in Figure 4.13. In order to have
an idea about the strain to failure of the PA 6 material and how it is
influenced with the addition of the steel fibers, a second series of tests were
performed at a higher cross-head speed i.e., 5 mm/min (0.002 s-1) till failure
to obtain the ultimate strain (strain at break). The results are summarized
in Table 4.4. It should be noted, that following the general practice of the
mechanical properties of plastics, the ultimate strength is defined as the
yield strength of the polymer instead of the stress at break as shown in
Figure 4.13.
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.
Table 4.4 indicates that the actual obtained properties of the PA 6 matrix
lie in between the dry as molded and the conditioned values reported in the
manufacturer’s datasheet. The strain to failure was however much larger
than the dry as molded reported value.
Test stopped
Ultimate strength
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122
Table 4.4 Tensile properties of the neat Durethan B 38 PA 6 material.
Comparison between achieved results and manufacturer’s datasheet values.
Neat PA 6 (Durethan B 38)
Actual Data Sheet
dry/cond
𝐸 [GPa] 1.6 ± 0.4 2.8/0.8
𝜎𝑢𝑙𝑡 [MPa] 59.6 ± 2.3 75/35
휀𝑢𝑙𝑡 [%] (5mm/min) 241.2 ± 0.2 20/>50
4.3.5.2 Single steel fiber tensile tests
Figure 4.14 shows the obtained stress-strain curves of the single 8 𝜇𝑚
diameter single steel fibers used in the SF-PA composites in the present
study. The graph shows very reproducible behavior of the different steel
fiber samples. The stress-strain curves show brittle behavior of the end-
drawn single steel fibers with quasi-linear curves up to final failure.
Figure 4.14 Measured stress-strain curves of single steel fibers (fiber diameter
𝑑 = 8 μm, gauge length 𝐿 = 25 μm).
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
123
The tensile properties of the steel fibers are summarized in Table 4.5. The
fiber depicted very high stiffness and strength properties compared to the
more typical glass fibers. The low strain to failure is in agreement with the
brittle nature of the as-drawn fibers.
Table 4.5 Tensile properties of single steel fibers.
SF
𝐸 [GPa] 184.5 ± 4.1
𝜎𝑢𝑙𝑡 [MPa] 1743.7 ± 140.5
휀𝑢𝑙𝑡 [%] 0.95 ± 0.07
4.3.5.3 Tensile properties of the short steel fiber reinforced materials
Figure 4.15 shows the obtained stress-strain curves of the short steel fiber
composites with the different fiber volume fractions.
Figure 4.15 Measured stress-strain curves of the SF-PA samples with the
different investigated volume fractions.
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124
For clarity the stress-strain curves of only 3 samples for each volume
fraction are shown. From the figure it can be seen that the 0.5VF% and the
1VF% samples show comparable behavior with the 0.5VF% having only
slightly higher strength. The 2VF% exhibited improved stiffness and
strength compared to the lower volume fractions as expected. Although it
is also expected that the further increase of fiber concentration will lead to
further improvement of the mechanical properties, the figure shows a
downwards shift of the stress-strain curves of the 4 and 5VF% compared
to the lower volume fractions.
The results of the overall quasi-static mechanical properties are
summarized in Table 4.6 and Figure 4.16. Figure 4.16 also shows a
comparison of the properties of the unreinforced PA material and the
different conditions of the investigated SF-PA samples.
Table 4.6 Summary of the tensile properties of the SF-PA composites with the
different fiber volume fractions
SF-PA
GF-PP 0.5VF% 1VF%
2VF% 4VF% 5VF%
𝐸 [GPa] 2.03 ±
0.24
1.8 ±
0.16
2.35 ±
0.24
2.04 ±
0.20
1.18 ±
0.39
𝜎𝑢𝑙𝑡 [MPa] 45.65 ±
0.78
45.30 ±
0.94
49.47 ±
0.91
45.98 ±
0.38
42.95 ±
1.08
𝜈 0.29 ±
0.05
0.31 ±
0.03
0.36 ±
0.05
0.34 ±
0.08
0.39 ±
0.02
휀𝑢𝑙𝑡 [%] 0.26 ±
0.05
0.32 ±
0.06
0.32 ±
0.07
0.49 ±
0.13
0.74 ±
0.12
It should be noted that the definition of the tensile strength is different for
the reinforced and unreinforced materials. For the reinforced steel fiber
composites, the tensile strength 𝜎𝑢𝑙𝑡 value in Table 4.6 refers to the real
strength of the composite, i.e. the highest value of the stress-strain curve.
For the unreinforced (neat PA 6), 𝜎𝑢𝑙𝑡 in Table 4.4 refers to the strength at
yield of the PA 6, following the common polymer terminology.
As can be seen from Figure 4.16, the addition of steel fibers to the PA
matrix resulted in an increase of the stiffness compared to the unreinforced
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
125
PA. A more significant increase of the stiffness is found at the 2VF%
(about 45% increase). As mentioned above, a decrease of the stiffness
values, instead of the expected increase was found for the higher volume
fractions, i.e. the 4VF% and the 5VF% samples. This behavior will be
discussed in details in sections 4.3.6 and 4.3.7.
Figure 4.16 The obtained quasi-static mechanical properties of the SF-PA
material plotted against the fiber volume fractions of the samples.
For the strength properties, as can be seen from the figure, for all
concentrations of steel fibers, the reinforced SF-PA samples exhibited
lower strength than the unreinforced PA 6 matrix. Noting that in the plot
the strength of the PA 6 matrix is considered the yield strength as
mentioned in section 4.3.5.1. Own tests showed that the actual strength at
break of the PA 6 matrix was even higher (about 74 MPa). If this value is
considered instead of the yield, it can be seen that even more strength
decrease occurs in the matrix as a result of the addition of steel fibers. The
2VF% samples gave the best strength of the reinforced materials.
Another interesting observation is the increase of the strain to failure with
increasing fiber content. This observation is counter intuitive as the steel
fibers have shown very brittle linear elastic behavior as discussed in
section 4.3.5.2. The 4 and 5VF% samples have also shown large variations
(standard deviation) of the failure strain. In sections 4.3.60 and 4.3.7, the
morphology of the steel fiber samples and fractography analysis will be
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126
discussed with a focus on gaining insight on the above observations and
the overall behavior of the SF-PA materials.
4.3.5.4 Damage development
Representative AE registrations recorded during the tensile tests of the SF-
PA samples with the different volume fractions are shown in Figure 4.17.
The results of the AE registration showed a logical dependence on the fiber
volume fraction. For the lower fiber volume fraction samples, a low
number of AE events were found.
In contrast, in the higher fiber volume fractions, a significant number of
AE damage events were found. Important observations can be deduced
from Figure 4.17. The first observation is that the onset of AE damage
starts very early in the stress-strain curves compared to the final strain to
failure of the composites. Since as mentioned earlier, the damage in short
fiber composites is dominated by fiber-matrix debonding and fiber pull-
out. The very early onset of damage as well as the significantly higher
number of AE events in the SF-PA material (compared e.g. to the GF-PA
in this work, with the same matrix material), despite the lower fiber volume
fractions, may lead to the conclusion that the fiber-matrix interface
between steel fibers and the PA matrix is weaker than that of the glass
fibers and polyamide. Another important reason is the high stiffness mis-
match between the steel fibers and the PA matrix which leads to high stress
concentrations and hence more significant damage.
Figure 4.19 shows the distributions of amplitudes and energies of AE
events in the SF-PA samples. Distributions of the 2VF% and the 4VF%
samples are shown for comparison of the low and high concentration
samples. For the amplitude distribution, by comparison to the distributions
of the glass fiber materials shown in Figure 4.8 (a) and (b), it can be seen
that the SF-PA materials similarly exhibit asymmetric amplitude
distributions. The peak (mean) of the distributions of the SF-PA samples
was very close to the threshold value, i.e. in the range of 35-36 dB. The
GF-PA materials had a peak of 40 dB. Nevertheless, the distributions of
the SF-PA materials were significantly narrower, with higher probability
of lower amplitude events. Only a few number of events were found with
amplitudes higher than 45 dB. Difficulties arise in directly applying the
same ranges of the amplitudes of the glass fiber materials and their
associations to corresponding damage mechanisms due to the difference in
materials, mainly fiber types. Direct application would result in the
assumption that no pull-out damage occurs in the SF-PA samples, which
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
127
is not in agreement with experimental observations of fracture surface that
will be discussed in section 4.3.7. Another explanation can be that the
debonding and pull-out in the SF-PA material, i.e. the damage mechanisms
found by present experimental observations, occur at lower amplitudes by
comparison to the typical GF-PA materials. In a similar way the energy
levels of the SF-PA (Figure 4.19 (c) and (d) events are shifted to lower
values compared to the GF-PA. The lower amplitudes and energies of the
damage events can be a result of a lower interface strength of the SF-PA
samples which will be confirmed in section 4.3.7.
128
(a) (b)
(c) (d)
129
(e)
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 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%.
CHAPTER 4
130
Figure 4.18 shows a comparison of the cumulative AE energy curves of
the SF-PA materials with the different volume fractions. The trends of the
0.5VF% are omitted due to the very low number of events. Owing to the
lower number of events, the 1VF% and the 2VF% samples exhibited lower
AE cumulative energy curves compared to the higher volume fraction
samples.
Figure 4.18 Comparison of the cumulative AE energy registrations of the
SF-PA materials with the different fiber volume fractions.
For the high volume fractions samples, although the 5VF% had a higher
number of events, the cumulative energy trends of the 5VF% samples were
generally relatively lower compared to the 4VF%. It can be concluded that
the damage in 5VF% occurred at generally lower energies supporting the
hypothesis of the lower interface properties. This can also be observed in
Figure 4.17 where most of the events in the 5VF% occurred at very low
energies, i.e. ≤ 10 a. u. As opposed to the cumulative energy curves of the
GF-PA material, as shown in Figure 4.7, the curves of the higher volume
fraction samples did not include any “jumps” after the first onset of
damage. Instead, a progressive increase of energy levels was observed until
reaching “plateau-like” behavior. This is due to the very low interfacial
strength, where debonds occur already at very low strain. From these strain
on, a continuous development of new debonds will occur resulting in a
continuous AE energy curve without jumps.
131
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%
(a) (b)
(d) (c)
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132
4.3.6 Micro-CT observations of the morphology of short steel fiber
composites
Figure 4.20 shows scanned samples of the SF-PA materials with the
different volume fractions.
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%.
The figure shows that for the low volume fraction samples, e.g. the 0.5VF%
sample in Figure 4.20 (a) no apparent defects can be observed. Voids start
to appear at the 1 and 2VF% samples, an example is shown in Figure 4.20
(a) (b)
(c) (d)
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
133
(b) for the 2VF% sample, but the extent of the volume of the voids was
found to be small. The voids were confirmed visually by inspection of the
samples and with optical microscopy. The voids also had different
greyscale values which were distinct from those of the matrix and were
further confirmed by thresholding attempts of 2D sections. Much larger
voids were found in the higher volume fraction samples, i.e. the 4VF% and
5VF% samples in Figure 4.20 (c) and (d) respectively. The volume and
extent of the voids increased with increasing volume fractions. The
observations were reproducible for different scanned samples of the same
condition.
The increase of the volume of the voids with increasing concentrations of
steel fibers is due to the difficulties in processing of the samples. The very
crowded matrix (observed in the 4 and 5VF% in Figure 4.20 (c) and (d))
and the high entanglements of the fibers resulted in problems with the
mixing and wetting of the steel fibers with the polymer matrix.
Additionally, leakage of the melt material from the injection nozzle
occurred. The later is a result of the trapping of the melt at the sprue
bushing which was observed at higher fiber volume fraction materials. To
overcome the leakage problem, the temperature of the melt was reduced as
mentioned in section 4.2.2. Those reasons caused the occurrence of
significant void defects in the high concentration samples as low melt
temperature results in “trapped air” in the material. Although it was
attempted to decrease the entrapped air problem by the increase of
decompression rates and increased screw rotation, the samples still showed
large void defects as discussed. Finally, shrinkage problems may occur in
the part during the cooling stage as a result of the large thickness of the
part where the outer surfaces close to the mold walls cool at a much faster
rate compared to the center. The solidified outer layers inhibit the
shrinkage of the part and lead to internal holes. This could contribute to the
voids found in the samples of the present study. Nevertheless, the
shrinkage problem is more common in components with variations of the
thickness profile. The dog-bone samples of this study had constant
thickness. Moreover, the voids appeared only more significantly in the
higher volume fraction samples, although the lower volume fraction
samples had the same geometry and thickness. This suggests that the main
cause of the voids is the lower melt temperature in higher volume fraction
samples.
It is believed that the increase of void content and larger voids in the higher
volume fraction samples is the reason for the increase of strain to failure of
the steel fiber composites with increased fiber content shown in
Figure 4.15 and Figure 4.16. In general, voids are considered defects which
CHAPTER 4
134
lead to a decrease of the failure strain of the composites. This is however
true in cases of small voids in the samples which initiate stress
concentrations and initiations of cracks. Nevertheless, with such large
voids (mm sized) as the ones observed by micro-CT in the high volume
fraction samples of the SF-PA, the size of the voids are significantly large
in relation to the sample sizes. It is expected than that the loads are
somehow damped (taken up) by these voids until large enough elongation
causes final failure. The presence of these voids may also contribute to the
increase of the strain to failure of the higher volume fraction samples by
the mechanism of cavitation and void growth with tensile loading. The
same mechanism can initiate from debonding and pull-out sites (which are
much more significant in the higher volume fraction samples as will be
shown in Figure 4.24) during deformation.
Another reason can be due to observations illustrated in Figure 4.21.
Figure 4.21 Small volumes of the micro-CT scanned undeformed SF-PA samples
(a) 0.5VF% and (b) 2VF%.
The figure shows a comparison of the micro-structure of very small
volumes of the 0.5VF% and the 2VF% digitally cut-off from the
reconstructed micro-CT volumes. These can be thought of as
representative volumes of the micro-structure. It can be seen that the lower
volume fraction samples depict less uniform dispersions of the fiber in the
matrix compared to the higher volume fraction one. In that way, the lower
volume fraction samples will have variations of local volume fractions
which has been discussed previously in Chapter 3. The less uniform
distributions of the fibers may result in lower overall strain to failure of the
lower volume fraction composites. The less uniform distribution may
result in very high local concentrations, leading to high local stresses and
hence failure at lower strains Nevertheless, the significant extent of the
(a) (b)
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
135
voids observed in Figure 4.20 lead to the conclusion that their effect on the
strain to failure of the composites should be more pronounced than the
effect of the fiber dispersions.
Figure 4.22 View of voids formed in the undeformed 4VF% SF-PA samples.
Similarly, it can be concluded that the presence of the voids contributed to
the decrease of the stiffness of the higher volume fraction samples
compared to the lower volume fraction illustrated in Figure 4.16. This can
be attributed to the lower “actual” fiber volume fractions compared to the
nominal values as a consequence of the voids. Another significant reason
for the low stiffness, is the decrease of the wetting and interface adhesion
between the steel fibers and the matrix due to the manufacturing limitations
during processing of higher volume fraction samples as previously
discussed. A final reason is illustrated in Figure 4.22. The figure shows a
“close-up” view of a void in the 4VF% sample shown in Figure 4.20 (b).
The figure shows that a number of fibers exist inside of the voids and the
voids are not “empty”. These fibers are isolated and do not contribute to
the load carrying capability of the composite. This further reduces the
actual effective fiber volume fraction of the higher concentration samples.
The high variations of the strain-to-failure values of the same conditions at
higher volume fractions samples of the steel fibers is then a result of the
variability of the extent and volumes of the voids in the samples.
CHAPTER 4
136
4.3.7 SEM fractography analysis of the short steel fiber composites
Figure 4.23 shows high magnification SEM micrographs of fracture
surfaces, showing the steel fibers embedded in the polyamide matrix. It can
be seen from the pull-out sites that (denoted with the arrows in Figure 4.23
(a)) that the fibers have irregular (quasi-hexagonal) cross-sections. A
focused image in Figure 4.23 (b) demonstrates the irregular cross-section
of the fibers. The irregular cross-section of the steel fibers will result in
high stress concentrations on the fiber-matrix interface. This was clearly
shown by modelling of UD steel fiber composites by Sabuncuoglu et al.
[282]. This characteristic, together with the previously mentioned low
interface properties and high stiffness mismatch between the steel fibers
and polymer matrices lead to significant damage, namely initiation and
progression of debonding compared to the typical glass fiber composites.
The increased debonding of the steel fiber composites is reflected in the
trends of acoustic emission analysis shown in Figure 4.17.
Figure 4.23 High magnification SEM images showing the irregular quasi-
hexagonal cross-section of the steel fibers embedded in the matrix.
Figure 4.24 shows SEM micrographs of the fracture surface of the SF-PA
samples with the different fiber volume fractions. For all volume fractions,
significant damage is observed as most of the fibers are pulled-out from
the matrix. In general, the SF-PA samples have shown higher amount of
pull-out compared to the GF-PA samples. A comparison can be done again
between the GF-PA samples and the 4VF% SF-PA samples having the
same weight fractions. The increased pull-out damage is in agreement with
the high number of events found by AE analysis in Figure 4.17. This
portrays the low interface strength of the steel fibers and the polyamide
matrix.
(a) (b)
20 µm 5 µm
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
137
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%.
Another evidence of the weak interface bonding between the steel fibers
and is the “clean” surface of the fibers at the fracture surface. This is in
contrast with the presence of the matrix residues on the fibers in the GF-
(e)
(a) (b)
(c) (d)
200 µm 200 µm
200 µm 200 µm
200 µm
CHAPTER 4
138
PA as observed in Figure 4.11. A representative micrograph of the 5VF%
SF-PA is shown in Figure 4.24 (e) where very long pull-out lengths of the
fibers is frequently observed at the higher volume fraction samples. This
further indicates the reduced interface strength. This observation at the
higher volume fraction samples can be a result of the above mentioned
manufacturing limitations with the difficulties in compounding samples
with the high concentrations of steel fibers. Finally, In agreement with the
micro-CT observations of the large void defects in the higher volume
fractions SF-PA samples, Figure 4.25 shows the defects which are also
observed on the fracture surface. It can also be seen that the number and
the size is significant and that in general the severity of the voids increase
with increasing volume fractions due to the manufacturing limitations.
Figure 4.25 SEM micrographs of the voids observed at the fracture surface of the
SF-PA samples of (a) 4VF% and (b) 5VF%.
4.4 Conclusions
In this chapter, detailed experimental characterizations of the quasi-static
behavior of typical short glass fiber composites were shown in addition to
the novel short wavy steel fiber composites considered in this thesis.
The experimental results shown in this chapter serve for gaining insight on
the behavior of the investigated materials which will be reflected in the
development and validation of the models.
An important aspect of the chapter is the combined use of different
experimental characterization methods, e.g. acoustic emission and
fractography analysis. This allows the determination of the different quasi-
static damage mechanisms of the short straight and wavy fiber composites
(a) (b)
Experimental Characterization of the Quasi-Static Behavior of Short Glass and
Steel Fiber Composites
139
and the qualitative analysis of the relative differences in behavior of the
different materials. This in turn can be especially important for having
insight on the physical characteristics of the short fiber composites that are
difficult to directly characterize, such as the nature of the bonding and the
interface strength between the different fibers and matrices.
Another main outlook of the chapter is the reflection on the behavior of the
novel steel fibers and its relation to processing and manufacturing
conditions. The results achieved in this work have shown different aspects
of the use of the steel fibers as a reinforcing material in short fiber
applications. On one hand, the manufacturing of the fibers which results in
the shown quasi-hexagonal cross-section, together with the high stiffness
mismatch (especially in transverse directions) of the isotropic steel fibers
result in high stress concentrations and hence increased damage in the
composite. On the other hand, current manufacturing limitations result in
constraints in the maximum achievable concentrations of the fibers where
only very low volume fractions of fibers can be compounded and
processed. Such low concentrations have shown to be sufficient in
providing satisfactory electromagnetic shielding in electrical applications.
However, for mechanical applications higher fiber volume fractions are
needed to achieve efficient increase of mechanical properties compared to
unreinforced materials. The steel fibers have also shown weaker interface
properties with the polymer matrix compared to the glass fiber
counterparts.
Finally even in the range of achievable volume fractions, i.e. up to 5VF%,
difficulties in processing have resulted in significant defects in the samples
(voids) that resulted in significant decrease of properties. From those
viewpoints and based on the results achieved in this work, it can be
concluded that future investigations and efforts need to be put forth mainly
towards the optimization of the manufacturing parameters as well as the
interface properties of the short steel fiber composites.
140
141
Chapter 5: Experimental Characterization of the Fatigue Behavior of Short Glass and Steel Fiber Composites
142
Experimental Characterization of the Fatigue Behavior of Short Glass and Steel
Fiber Composites
143
5.1 Introduction
The focus of the present chapter is the characterization of the fatigue
behavior of the materials considered in this work, i.e. the typical straight
short glass fiber reinforced composites and wavy steel fiber composites.
Similar to the quasi-static characterization detailed in the previous chapter,
the fatigue characterization in this chapter provides an important insight on
the behavior and damage mechanisms of the composites under
investigation in cyclic loading. The obtained results will be used for the
validation of the developed model in Chapter 8.
5.2 Materials and Methods
5.2.1 Materials
The same glass fiber reinforced materials, i.e. the GF-PA and GF-PP
materials, tested in quasi-static loading conditions, as reported in
section 5.2.14.2.1, were characterized in fatigue loading. For the SF-PA
material, the 2VF% samples were used as they provided the best
mechanical properties in the quasi-static regime. Similar to the quasi-static
testing discussed in the previous chapter, the ISO (dog-bone) samples of
each material were used in the fatigue tests.
5.2.2 Fatigue testing
Fatigue tests in the tension-tension regime were performed using a
hydraulic fatigue machine (Schenck) for the GF-PA, GF-PP and SF-PA
materials investigated. The machine was equipped with a 10 kN load cell.
Tests were conducted in load-controlled mode in the range of cycles to
failure up to 106 cycles. A constant amplitude sinusoidal load function was
applied. For all tests, the load ratio R = min. load
max. load was 0.1. The failure
criterion for the performed fatigue tests was specimen separation (rupture).
The fatigue tests were interrupted and run-outs were assumed when the
number of cycles reached 106. The applied stresses were calculated by
dividing the load by the initial reference cross-section area of the samples.
All tests were conducted at a room temperature of 20℃.
The test frequency was selected in such a way as to reduce the temperature
rise in the specimen. Temperature monitoring was performed using a film
type NiCr-NI thermocouple clamped at the central part of the specimen
CHAPTER 5
144
surface, throughout the fatigue tests. For the glass fiber materials (both the
GF-PA and GF-PP) the frequency was fixed at 4 Hz. Such frequency
allowed obtaining reasonable test durations without excessive increase of
the specimen temperature due to hysteresis effects. The temperature rise
was below 6℃ for all materials and loading conditions in this work. This
is reasonably below the maximum allowable temperature rise as
recommended by the (EN ISO 13003:2003 standard). It should be noted
that with such temperature rises of the specimens, all tests of the GF-PA
material were conducted below the glass-transition temperature 𝑇𝑔 = 50 −
60℃ of the polyamide polymer. The glass transition temperature of the
polypropylene material is generally < 0℃ and hence all tests are
conducted anyways above 𝑇𝑔 regardless of the temperature rises.
For the steel fiber material (SF-PA 2VF%) a lower frequency of 2 Hz was
used. This is due to the low stiffness and significant hysteresis already at
low loads of the steel fiber composite which results in high increase of the
temperature measured at the surface of the specimen at higher frequencies.
The maximum temperature rise for the SF-PA was about 4℃ for the tested
conditions.
During the load-controlled fatigue tests, the axial strain was measured for
monitoring the cyclic stiffness degradation of the material. For the strain
measurements, a dynamic extensometer (Instron 2620-
824,+ −⁄ 40% strain) was mounted on the central part of the specimen. In
such a way, the stresses and strains of each cycle were registered in the
output file of the fatigue tester. A Matlab script was used for analyzing the
stiffness evolution with the fatigue cycles using the method explained
below in section 5.2.3.
In order to obtain the full S-N behavior, fatigue tests for each material are
performed at different stress levels. Tables 5.1 and 5.2 summarize the stress
levels applied to the glass fiber and steel fiber composites, respectively.
Due to the low frequency of the fatigue tests of the SF-PA material, and
hence long test durations, a lower number of stress levels were tested
compared to the glass fiber composite samples, as shown in Table 5.2. At
least 5 samples were measured for each condition for the glass fiber
composites and 3 samples for the steel fiber composites.
It should be noted that for the SF-PA material, one condition (65% UTS)
was performed on an (Instron 8801) axial fatigue machine in the
department of Materials Science and Engineering, Ghent University. The
Experimental Characterization of the Fatigue Behavior of Short Glass and Steel
Fiber Composites
145
author gratefully acknowledge the help of Prof. W. Van Paepegem and Dr.
I. De Baere.
Table 5.1 Tested stress levels in the fatigue tests of the investigated glass fiber
reinforced composites.
Maximum applied stress [MPa]
Stress level [% UTS] GF-PA GF-PP
45% - 37.7
55% 61.8 46.0
65% 72.9 54.4
70% 78.6 58.6
80% 89.8 66.9
Table 5.2 Tested stress levels in the fatigue tests of the investigated steel fiber
reinforced composites.
Maximum applied
stress [MPa]
Stress level [% UTS] SF-PA
40% 19.8
55% 27.2
65% 32.2
5.2.3 Stiffness degradation analysis
The dynamic modulus, i.e. cyclic stiffness, is defined as the slope of the
straight line between the points of the minimum and the maximum stress
and strain of the stress-strain or hysteresis loop as described in Equation
5.1.
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146
𝐸𝑑𝑦𝑛 = 𝜎𝑚𝑎𝑥 − 𝜎𝑚𝑖𝑛
휀𝑚𝑎𝑥 − 휀𝑚𝑖𝑛
(5.1)
where 𝐸𝑑𝑦𝑛 is the dynamic modulus, 𝜎𝑚𝑎𝑥 and 𝜎𝑚𝑖𝑛 are the maximum and
minimum stresses respectively of the fatigue cycle, and 휀𝑚𝑎𝑥 and 휀𝑚𝑖𝑛 are
the maximum and minimum strains respectively of the fatigue cycle.
By calculating the dynamic modulus for every cycle (or for cycles with
predefined intervals), the loss of stiffness of the material during the fatigue
tests can be analyzed. Due to the noise in obtained signal (an example is
shown in Figure 5.1), which makes it difficult to obtain maximum and
minimum values, the dynamic modulus is calculated for each fatigue cycle
by correlating the stress and strain values using linear regression fitting as
shown in Figure 5.1. In such a way, the modulus corresponds to the slope
of the linear fit. For all materials under investigation in this thesis, a high
value of the error of the regression fit, 𝑅2 was generally obtained (was
obtained > 0.98). This indicates the accuracy of the correlation of the data
points and hence of the stiffness values.
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.
Experimental Characterization of the Fatigue Behavior of Short Glass and Steel
Fiber Composites
147
5.2.4 Fatigue tests performed on the quasi-static tensile test machine
The steel fiber reinforced composite samples have shown a very high
stiffness degradation during the early cycles of the fatigue tests. This will
be discussed in details in section 5.3.3. Most fatigue testers require, at the
start of the test, a number of cycles, depending on the material tested and
on the test settings, before the imposed maximum and minimum load
values are accurately achieved. The early stiffness degradation, observed
for the steel fiber samples, required that the load values should be accurate
from the very first fatigue cycle. This was achieved by performing short
cyclic tests on a quasi-static tensile test machine.
Cyclic fatigue tests were performed on an Instron 4467 tensile machine
equipped with a 30 𝑘𝑁 load cell. Tests were performed at a cross-head
speed of 50 mm/min, corresponding to a test frequency of about 0.5 Hz.
Figure 5.2 shows a representative diagram of the applied load of the first
few cycles during the fatigue tests on the tensile machine.
Figure 5.2 Representative applied load diagram of the fatigue tests on the tensile
tester performed on the SF-PA 2VF% samples.
Due to the constraints of application of cyclic loading on a tensile tester for
long test durations, only 200 cycles were performed in the present tests.
The main advantage of the test is that the applied load is accurately
controlled from the first fatigue cycle as shown in Figure 5.2. This provides
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the opportunity for accurate monitoring of the stiffness degradation in early
cycles. A limitation is that only triangular load cycles instead of true
sinusoidal load cycles could be applied on the tensile tester. Another
limitation can be the difference in frequencies where in the actual fatigue
tests the frequencies were four times the test frequencies in the fatigue tests
on the tensile machine. Nevertheless, it is believed that the errors induced
by the triangular wave form and lower frequencies are less than those from
the inaccurate application of the maximum loads of the fatigue cycle. The
purpose is, in any case, only to gain an insight on the severity of stiffness
loss of the material and not to obtain exact values. Another objective is to
analyze the evolution of the fatigue hysteresis loops in the early cycles.
The tests were performed on the SF-PA with the different stress levels
described in Table 5.2.
The same tests were performed on the GF-PA and GF-PP materials, with
the load levels outlined in Table 5.1, similar to the SF-PA to analyze the
evolution of hysteresis loops. However, because of the same constraint on
cyclic loading on the tensile tester, only a few dozen cycles were applied
for each sample.
5.2.5 Fractography analysis
Fracture surfaces of the broken fatigue specimens were observed using a
scanning electron microscope (SEM). The same equipment and procedure
discussed in section 4.2.6 is used for the fatigue samples.
Experimental Characterization of the Fatigue Behavior of Short Glass and Steel
Fiber Composites
149
5.3 Results and Discussion
5.3.1 Fatigue S-N curves of the short glass fiber composites
Figure 5.3 shows the obtained fatigue lifetime (S-N) curves of the GF-PA
and the GF-PP materials. In the S-N curves the maximum fatigue stress
𝜎𝑚𝑎𝑥 is plotted against log (cycles to failure) or log (𝑁𝑓).
The relationship between the maximum applied stress 𝜎𝑚𝑎𝑥 and the cycles
to failure 𝑁𝑓 is often described using a power equation as follows [21, 55]:
𝜎𝑚𝑎𝑥 = 𝜎𝑓 (𝑁𝑓 )𝑏 (5.2)
where 𝜎𝑓 is the fatigue strength coefficient and 𝑏 is the fatigue strength
exponent. The fatigue strength exponent reflects the slope of the linear
form of the S-N curve (when considering log (𝜎𝑚𝑎𝑥) vs. log (𝑁𝑓)).
𝑇𝜎 denotes the data scatter index and is defined as the ratio between the
stress values at the lower and upper limits of the 90% confidence limit
(10% and 90% survival probability, respectively) [21, 22]. The index, as
suggested by the name, is an indicator of the scatter observed during the
fatigue test. Assuming a constant standard deviation of number of cycles
to failure for all stress levels, the scatter in the entire S-N diagram of a
material can be described with one 𝑇𝜎 value.
From Figure 5.3, it can be seen that the obtained strength coefficient
values, by fitting Equation (5.2) to the experimental data, of the GF-PA
(121 MPa) is higher than that of the GF-PP (95 MPa). This is consistent
with the higher quasi-static strength of the GF-PA material compared to
the GF-PP as discussed in section 4.3.2.
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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.
The lower strength exponent of the GF-PA (-0.047) compared to that of
the GF-PP (-0.079) indicates a higher fatigue resistance of the former
material. This observation is again despite the lower fiber length and
orientation of the GF-PA material compared to the GF-PP, which will be
discussed in details in Chapter 7. It will also be shown in Chapter 8 that
the slopes of the S-N curve of the unreinforced polyamide and
polypropylene matrices are comparable, with the polypropylene matrix
being slightly more fatigue resistant. The higher fatigue resistance of the
GF-PA materials then suggests that the fatigue behavior of the material is
dependent on the interface properties. This is in line with the explanation
given in Chapter 4, namely that quasi-static properties and the fractography
analysis suggested that the GF-PA material exhibited stronger interface
properties compared to the GF-PP material.
The higher value of 𝑇𝜎 = 1.14 of the GF-PA compared to 𝑇𝜎 = 1.11 of
the GF-PP indicates a higher scatter of the data of the former material. This
is also consistent with the higher standard deviation of the quasi-static
strength values of the GF-PA material as shown in Table 5.2. The generally
higher scatter of the strength values of the polyamide based composite can
be attributed to the hygroscopic nature of the polymer, as previously
discussed in section 4.3.2.1.
Experimental Characterization of the Fatigue Behavior of Short Glass and Steel
Fiber Composites
151
5.3.2 Fatigue damage of the short glass fiber composites
Figure 5.4 (a) and (b) shows the hysteresis loops at different cycles for the
GF-PA and GF-PP materials, for samples on which the maximum cyclic
stress was 70% of the tensile strength (79 MPa and 59 MPa for the GF-PA
and the GF-PP, respectively, as in Table 5.1). As can be noticed, for both
materials, the obtained cyclic hysteresis loops showed a shift of the loops
along the strain axis. Similar behavior was found for all the investigated
stress level conditions.
The observed behavior of the hysteresis loops in the present work is in
agreement with literature data, e.g. [21, 55, 283]. The observation was
attributed to the cyclic creep phenomenon in the samples. Cyclic creep is
defined as the phenomenon of strain accumulation or the shift of the stress-
strain cycle along the strain axis resulting in permanent deformation during
fatigue tests [284]. This phenomenon is despite the fact that with the
chosen frequency (4Hz), no significant increase of temperature of the
specimens was found as explained in section 5.2.2.
Klimkeit et al. [283] investigated the fatigue behavior of a short fiber
composite (namely a 30wt.% glass fiber reinforced PBT+PET) with
different 𝑅 ratios of the fatigue load. They explained that for loading
without a mean stress, e.g. their considered case of 𝑅 = −1 the obtained
hysteresis loops of the fatigue cycles were narrow and the change of the
minimum strain of the cycles with the fatigue life is negligible.
Nevertheless, for a loading condition with a positive mean stress as 𝑅 =0.1, the hysteresis loops showed a pronounced evolution where they
moved towards higher strains throughout the fatigue life. Similar findings
were reported by Mallick and Zhou [55] and De Monte et al. [22] and
Avanzini et al. [284].
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152
(a)
(b)
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.
Experimental Characterization of the Fatigue Behavior of Short Glass and Steel
Fiber Composites
153
The presence of cyclic creep effects, associated with the presence of mean
stress, is a typical phenomenon for composite materials with thermoplastic
materials [21]. Mallick and Zhou [55] clarified that at room temperatures
and with no noticeable temperature increase at the specimen surface, their
investigated GF-PA 6.6 material exhibited cyclic creep behavior.
The papers [21, 22, 55, 283] reported the cyclic creep effects in polyamide
based (PA 6 and PA 6.6) composites. As mentioned above, the same effect
was observed for the investigated GF-PP material in the present work.
The cyclic mean strain is plotted in Figure 5.5 against the number of cycles
for the GF-PP and the GF-PA materials. As can be seen from the figure,
for both materials, the mean strain continuously increases with increasing
number of cycles, reflecting the creep phenomenon.
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,.
Both materials showed a high rate of increase of the mean strain. A direct
comparison of the rates of increase of the mean strains at the same stress
level (% UTS) may not be straight forward because of the difference of the
actual applied stress values. The increase of mean strain of the GF-PA was
found to be around 93% and that of the GF-PP was found to be around
74%, at the same stress level 70%. Another observation is that the mean
strain values of the fatigue cycles of the GF-PP increased at an almost
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154
constant rate throughout the fatigue life (quasi-linear slope of the increase
of mean strain) as opposed to the GF-PA material, where the slope slowly
decreased with increasing number of cycles. The nature of the progression
of the slopes of the mean strains is generally dependent on the material,
applied stresses, loading frequency and other testing conditions, as
suggested by Bernasconi et al. [21].
By analyzing the hysteresis loops, shown above, the dynamic modulus can
be obtained. The evolution of the dynamic stiffness with the fatigue life is
the focus of the rest of the section. Dynamic moduli evolution or cyclic
stiffness degradation plots are considered to be an indicator of the material
degradation or damage [22, 43] in fatigue loading.
There is considerable uncertainty in literature concerning stiffness
degradation in short fiber composites. Also, very limited published data on
the loss of stiffness during fatigue of short fiber composites is available.
Most of the available published results are analyzed below.
In general, it is expected that the loss of stiffness is higher for higher fatigue
stresses (maximum fatigue stresses). This is reported e.g. in the work of
Arif et al. [43] and Avanzini et al. [285]. The results of Klimkeit et al. show
that this trend is not always respected, as in some cases stiffness loss under
higher loads were lower than that of the lower loads. The data reported by
De Monte et al. [22] show an opposite trend of higher stiffness loss for the
lower load levels. They attributed this to the fact that, when the hysteresis
curves are measured using extensometers placed in the central part of the
specimen, they are not fully characterizing the fatigue damage evolution in
the specimen. The authors also explained that the cyclic stiffness
degradation is not well understood and that two samples with the same
applied load and similar fatigue life can show significantly different
stiffness degradation curves.
There is also uncertainty with regards to the amount and trends of the
stiffness degradation. For a similar glass fiber reinforced polyamide 6.6,
De Monte et al. [22] reported stiffness losses of 10 - 15%, depending on
the stress level, while Arif et al. [43] reported negligible stiffness losses of
less than 5% for all their tested conditions (the two materials had similar
weight fractions of 35% in the former study and 30% of the later). Another
difference is that the trends of De Monte et al. showed a fast degradation
of the dynamic modulus in the first stage of the fatigue life, followed by a
second stage of lower degradation rates and finally a final region in which
an abrupt drop in stiffness occurred. Arif et al. showed no stiffness loss up
for a large part of the fatigue life followed by a gradual degradation up to
the 5% stiffness loss.
Experimental Characterization of the Fatigue Behavior of Short Glass and Steel
Fiber Composites
155
The dynamic stiffness as described in section 5.2.3, for the glass fiber
composites in the present work, normalized to the initial value and plotted
as a function of the fraction to the number of cycles to failure is shown in
Figure 5.6.
It can be seen from the figure that, in the present work, the general trend
of a higher stiffness loss with higher stress levels is observed.
The trend of the stiffness loss of the GF-PA in the present study was in
agreement with the trends of De Monte et al. [22] mentioned above, except
for the highest load level (80%, 90 MPa). Indeed, for all other cases (GF-
PA and GF-PP), a high decrease of stiffness is found at the beginning of
the fatigue loading followed by a decrease at much lower rates in the
second stage. In the GF-PP, however, the initial decrease is less significant
compared to the GF-PA composites, but it is followed by a steeper decrease
at constant rates in the second stage up to final failure.
A comparison of the overall loss of stiffness behavior between the GF-PP
and GF-PA of the same stress ratio conditions may not be straightforward
due to the difference in the actual stress values. For all tested samples for
both materials, the general total stiffness loss revolved around 15-20%.
The highest load level condition of the GF-PA showed a more pronounced
degradation. This can be due to the sensitivity of the PA 6 polymer to
fatigue at higher load levels, similar to the observations of Mandell et al.
[58].
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156
(a)
(b)
Figure 5.6 Evolution of the cyclic stiffness for the (a) GF-PA and (b) GF-PP
materials.
Experimental Characterization of the Fatigue Behavior of Short Glass and Steel
Fiber Composites
157
It can also be seen by comparison of the materials considered in this work
(the GF-PA and GF-PP) with the S-N curves in Figure 5.3 that those
stiffness loss values occurred at a generally higher number of cycles for the
GF-PA compared to the GF-PP (with the exception of the 80% UTS load
levels). This suggests a higher resistance to fatigue damage of the GF-PA
material.
5.3.3 Fatigue damage of the short steel fiber composite
Similar to the glass fiber composite samples analyzed above, this section
focuses on the fatigue behavior of the steel fiber composite samples.
The steel fiber composites showed some unusual behavior with regards to
the fatigue life and loss of stiffness. First, in all three stress levels (65%,
55% and 40% UTS) no fracture of the specimen was observed up to 106
cycles. One sample was loaded at the highest stress level (65% UTS, 32
MPa) to 2 x 106 cycles, also with no failure of the specimen. Higher stress
levels (70% or 80% UTS) were not possible to achieve because of high
temperature rises in the specimens, even at very low frequencies of 1 Hz,
and rapid thermal failures.
The observation that the samples did not fail at the tested stress levels, can
be explained by the low applied stresses. It was shown in Figure 4.16 that
the strength of the reinforced steel fiber polyamide composites was, for all
concentrations of steel fibers, lower than the strength of the neat polyamide
matrix. In this graph, the strength of the polyamide was actually taken as
the yield strength, following the terminology of the manufacturer’s
datasheet. However, the polyamide 6.6. polymer has an actual strength
defined by the stress at break of about 75 – 80 MPa, which is much higher
than the maximum applied stress of 32 MPa for the reinforced sample
tested in fatigue. Another main effect is the very low interface strength, as
suggested by the experimental results of the tensile tests and especially of
the acoustic emission and fractography analysis. It was confirmed that
significant debonding occurs in the steel fiber reinforced composites.
A similar phenomenon was analyzed in the early work of Dally and
Carrillo [286] who investigated several short glass fiber reinforced
composite systems. Among these systems was a polyethylene reinforced
composite, which exhibits weak interface properties with the glass fibers.
The authors have shown that massive debonding occurred in the specimens
during cyclic loading and that in effect, it reduced the glass fibers from
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reinforcement to unbonded inclusions (defects). As complete debonding
occurred, the load was accommodated by the matrix without failure.
In this respect, it can be concluded that with the very weak interface
properties of the short steel fiber composite, the fibers were completely
debonded after a small number of cycles, and the applied stresses, being
lower than the matrix failure strength, were carried by the polyamide
matrix. This behavior of the short steel fiber composites provides insight
on the general aspects of failure of short fiber composites. The behavior
suggests that even with the degradation of the composite properties as a
result of fiber-matrix debonding, final failure of the composite occurs when
matrix fails. In cases of the composites with stronger interface, e.g. the GF-
PA and the GF-PP materials, the stresses were high enough compared to
the matrix strengths. Although the stresses in these composites can be
much lower than the overall composite stresses as a result of the reinforcing
fibers, expected that with the progression of debonding, the fibers lose their
reinforcing efficiency leading to higher stresses in the matrix which can
ultimately cause final failure.
Another unexpected behavior of the SF-PA was the “stiffening” effect of
the material at a certain moment of the fatigue life. This can be observed
by the evolution of the hysteresis loops as shown in Figure 5.7 (the Figure
shows the example of the stress level 55% UTS, 27.2 MPa). As can be seen
from the figure, the slope of the hysteresis loops (defined as the line
connecting the extremes of the loop) decreased, whereas the area of the
hysteresis loops increased up to about 5800 cycles, resulting in the
expected stiffness degradation with cyclic loading. Afterwards, it was
found that the slope again increased and area of the loops decreased at the
higher fatigue cycles, leading to the stiffening of the material.
Experimental Characterization of the Fatigue Behavior of Short Glass and Steel
Fiber Composites
159
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.
This decrease in stiffness in the first 5800 cycles could be explained by the
gradual development of the debonding, leading to gradual damage and
hence decrease in stiffness and increase in the width of the hysteresis loops.
The stiffening at higher cycle numbers is not well understood. Two
explanations can be thought of.
The first explanation is related to the phenomenon of cyclic chain
orientation of the polymeric material. This phenomenon was mentioned in
the work of Dally and Carrillo [286]. This explanation is supported by
several necking sites observed on the surface of the samples. This suggests
that when all the fibers are debonded, the gradual cross-section reduction
due to necking leads to a higher orientation of the polymeric chains and
hence to an increasing stiffness of the material.
Similar stiffening effects can be observed in the results published by
Ramkumar and Gananamoorthy [287]. The authors reported dynamic
stiffness loss diagrams of a neat PA 6 material. However, no explanations
or comments were given on this observation. A stiffening of the PA 6 at
different reported stress levels showed clear trends of increase of stiffness
after about 103 cycles. Images of the deformed PA 6 samples published in
the paper also show neck formation with multiple necking sites along the
specimen gauge length as observed in the samples of the present study.
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A second possible explanation can be the reorientation of the steel fibers
leading to a decrease of the waviness of the fibers and increased stiffness.
The steel fibers can be completely debonded from the matrix, however due
to their curved nature, they can still be “locked” into the matrix and reorient
during loading. The reorientation of the fibers then contribute to the
increase of stiffness of the composite even with the loss of the load carrying
capability of the fibers due to debonding.
Figure 5.8 shows the evolution of the cyclic stiffness of the SF-PA material
at the different tested stress levels. In the figure, the cycles N are
normalized to N = 106 at which the tests were stopped (run-out).
Figure 5.8 Evolution of the cyclic stiffness of the SF-PA material at different
stress levels.
The figure clearly illustrates the above mentioned phenomenon. For the
55% and 65% UTS (27 and 32 MPa, respectively), a very steep and
significant decrease of the stiffness of the composite was found in the very
early stages of the fatigue loading. This is in agreement with the hypothesis
that extensive damage (debonding) occurs in early loading stages. The
material then starts to stiffen, and gradually regains about 20% of it is
stiffness. For the 40% UTS this phenomenon was not observed. At the
early loading cycles, a stiffness loss of about 15% occurred and the
stiffness value seemed to stabilize for the rest of the test. The applied
stresses were too low to activate one of the two above mechanisms
(polymer chain reorientation, or reorientation of the steel fibers).
Experimental Characterization of the Fatigue Behavior of Short Glass and Steel
Fiber Composites
161
Due to the fact that none of the samples showed failure for all the tested
stress levels, the S-N curve of the material could not be obtained. In order
to get quantitative data on how fast the damage of the SF-PA developed,
the cycle at which 50% of the modulus degradation (dynamic fatigue
modulus) is analyzed. The results are summarized in Table 5.3.
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.
Stress level [% UTS] Cycle at 50% 𝑬𝒅𝒚𝒏
degradation
40% -
55% 545 ± 140
65% 598 ± 27
The table indicates that at the 40% UTS the stiffness degradation did not
reach 50% for any of the samples. For the higher stress levels, this
degradation occurred at generally comparable fatigue lives in the range of
500 – 600 cycles, which means in very early stages of loading. The
progression of the fatigue degradation however, was more pronounced at
the highest stress level as shown in Figure 5.8.
5.3.4 Fatigue tests of the SF-PA on the tensile tester
As mentioned in section 5.2.4, fatigue tests on the tensile tester were
performed to have an insight on the early stiffness loss of the SF-PA
explained in the previous section. Figure 5.9 shows a representative
diagram of the hysteresis loops of the SF-PA material at a stress level of
55% UTS (27 MPa). As can be seen from the figure, the loads, and hence
stresses, were accurately controlled from the first cycles.
An interesting aspect shown in this graph is the presence of cyclic creep
effects, as the hysteresis loops moved along the strain axis from the first
few cycles. This confirms the statement of Mallick and Zhou who reported
that creep effects occur in very early stages of the fatigue life of short fiber
composites [55].
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162
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.
Figure 5.10 shows the normalized stiffness degradation of the SF-PA with
different stress levels, which was the main objective of the short fatigue
tests on the tensile tester. The figure shows that indeed for the SF-PA,
subjected to fatigue loads at the 55% and 65% UTS stress levels (27 and
32 MPa, respectively), fatigue damage occurs at very early cycles, because
a stiffness decrease of around 12-14% occurred already after the first cycle,
depending on the load level. For the lowest loaded sample (40% UTS), up
to the measure 200 cycles, the stiffness reduction stabilized around 15%,
whereas for the higher load levels, stiffness degradations at the end of the
test (200 cycles) were about 19% and 22% for the 55% and 65% UTS,
respectively.
Experimental Characterization of the Fatigue Behavior of Short Glass and Steel
Fiber Composites
163
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.
Using the fatigue stiffness degradation as a damage indicator, the high
stiffness loss at very low fatigue cycles of the SF-PA confirms the very
weak interface properties of the steel fibers and the polyamide matrix. An
indirect comparison can be performed between the SF-PA and GF-PA
materials. The hysteresis evolution of the same cyclic test on the tensile
tester for the glass fiber materials (though only for 30 cycles) will be shown
in the next section in Figure 5.11. A comparison between the lowest load
level (40% UTS, 20 MPa) of the SF-PA and the highest load level of the
GF-PP (80% UTS, 90 MPa) after 30 cycles yields stiffness loss values of
around 15% for the SF-PA samples and only 1.3% for the GF-PA sample.
A similar stiffness loss of 1.1% after 30 cycles was found for the GF-PP.
It is evident that there are differences between the materials, however, this
comparison shows the extent of pronounced damage in very early loading
stages of the SF-PA versus the stiffness losses at these stages in typical
short fiber composite systems.
5.3.5 Fatigue tests of the GF-PA on the tensile tester
Figure 5.11 shows a representative evolution of the hysteresis loops of the
GF-PA for the first 30 cycles of loading obtained from the short cyclic tests
on the tensile machine. The figure shows the example of the 80% UTS (90
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164
MPa). The stiffness loss of this condition was discussed in the previous
section for comparison with the SF-PA material. It was shown that, at this
very early stage of loading the stiffness loss was negligible. Another aspect
observed from those tests is that, similar to the SF-PA, the glass fiber
materials also have shown cyclic creep effects from the first cycles of
loading. The observation was found for all other loading conditions. The
same was also found for all the conditions of the GF-PP, although the creep
rates in the GF-PP were less pronounced, which is in agreement with the
results of the fatigue tests up to failure discussed in section 5.3.2.
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.
5.3.6 SEM fractography analysis of the short glass fiber samples
SEM micrographs of the fatigue fracture surfaces of specimens that failed
at different stress levels are shown in Figure 5.12 and Figure 5.13 for the
GF-PA and the GF-PP materials, respectively.
Experimental Characterization of the Fatigue Behavior of Short Glass and Steel
Fiber Composites
165
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.
A very important behavior can be observed from the SEM micrographs of
the fatigue failed samples, in both figures (Figure 5.12 and Figure 5.13),
by comparison to the quasi-static tested samples (Figure 4.11 Figure 4.12).
The comparison between the figures shows a much higher amount of pull-
out in the fatigue failed samples compared to the tensile failed samples.
This suggests a degradation of the interface properties in during
deformation in fatigue loaded samples.
(a) (b)
(c)
200 µm 200 µm
200 µm
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166
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.
A similar behavior is found in published data. Lang et al. [288] analyzed
the fracture surface of a glass fiber reinforced polyamide 6.6. The authors
observed that while the surface of the fibers sticking out from the fracture
surface was covered with matrix in the tensile failed samples, the fatigue
failed samples did not show any traces of matrix being still bonded to them.
The same observation was confirmed in the study of Horst and
Spoormarker [40], who published a detailed paper on the differences in the
damage mechanisms in short fiber composites, including the differences
between tensile and fatigue samples, and the study of .Mandell et al. [58].
The authors in both papers explained that the reinforcing efficiency of the
fibers in short fiber composites is progressively lost due to the progressive
failure of the interface in fatigue.
The above SEM micrographs, together with literature observations suggest
a degradation of the interface during fatigue loading or the so-called
(a) (b)
(c)
200 µm 200 µm
200 µm
Experimental Characterization of the Fatigue Behavior of Short Glass and Steel
Fiber Composites
167
“fatigue of the interface”. It is difficult from the SEM micrographs to
accurately assess the extent of the interface failure of one material in
relation to the different stress levels. However, it can be clearly seen that
for all stress levels, a much more pronounced damage (debonding) is
observed compared to the tensile failed samples. This behavior presents a
very significant aspect in the fatigue of the short fiber composites as it has
direct consequences on the degradation and failure of these composites,
which in turn needs to be reflected in the development of accurate
predictive models.
5.4 Conclusions
In this chapter, a detailed experimental characterization of the fatigue
behavior of the short glass fiber and steel fiber reinforced composites,
considered in the present work, is presented. The experimental
observations will serve for the development and validation of the fatigue
model as will be shown in Chapter 8.
The obtained S-N curve behavior of the short fiber composites indicated
that, as expected, the material with the better interface properties (GF-PA)
exhibit a higher fatigue resistance, reflected by a lower slope of the S-N
curves, compared to the composite with weaker interface (GF-PP).
Analysis of the trends of dynamic stiffness degradation of the present
materials, as well as the published literature data, leads to the conclusion
that a stiffness loss of about 15-20% of the short fiber composites occur
life up to failure. These values however can occur at different fatigue lives
where e.g., at a higher load level, this stiffness loss occurs at a lower life
of the composite compared to the lower load levels. For a material with
higher interface properties and higher fatigue resistance, the loss of
stiffness similarly occurs at a higher number of cycles.
The short steel fiber composites showed an unusual fatigue behavior where
at all load levels, no failure of the samples were observed till run-out. This
was due of the low applied stresses (as a result of the low tensile strength
of the samples). It is expected that failure of a short fiber composite occurs
by failure of the embedding matrix. In this respect, the low applied stresses
during the fatigue testing of the short steel fiber composites were lower
than the strength of the matrix, in such a way that no failure occurred in
the composite.
Analysis of the stiffness degradation showed a very steep loss of stiffness
in the early fatigue life, followed by an observed stiffening of the material.
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168
The initial fast loss of stiffness was linked to the very weak interface as
well as the high stiffness mismatch between the steel fibers and the PA 6
matrix. The stiffening effect was attributed to the possible chain re-
orientation of the polymer matrix after complete debonding of the fibers
and necking of the samples and/or the re-orientation of the wavy steel
fibers embedded in the matrix as a result of cyclic loading.
A final main conclusion found in this chapter is the behavior of the fiber-
matrix interface during fatigue loading. By SEM analysis of the fracture
surface, it was found that much more debonding and higher amounts of
pull-out were found in the fatigue failed specimens compared to the tensile
failed specimens (as was also observed on the glass fiber materials). This
was also confirmed with literature published observations and lead to the
conclusion that a progressive degradation of the interface strength occur
during fatigue loading. This behavior will be then taken into account in the
development of the fatigue model, as will be shown in Chapter 8.
169
Chapter 6: Linear Elastic Modeling of Short Wavy Fiber Composites
Linear Elastic Modeling of Short Wavy Fiber Composites
171
6.1 Introduction
Random fiber composites can contain wavy fibers, as well as straight ones.
For short glass fibers with length in the range 0.5 - 5 mm, corresponding
to an aspect ratio in the range 25 – 250, the later case is typical. If the fiber
aspect ratio is higher, the fibers in the composite can easily be wavy.
Examples are polymers reinforced with carbon nanotubes (typical aspect
ratio over 1000) and composites reinforced by discontinuous steel fibers.
As mentioned in the previous chapters, one of the objectives of this PhD
thesis is modelling the micro-structure and the mechanical properties of
random wavy fiber composites. The example material considered in this
thesis are the injection molded short steel fiber composites, which have
been shown to have (Chapter 3) highly curved geometries.
In chapter 3, the complex micro-structure of short steel fiber composites
was described. A methodology for the characterization of the geometry of
wavy short steel fiber composites using micro-computed tomography was
presented and a geometrical model was developed for the generation of
representative volume elements of the short random wavy steel fiber
composites.
The focus of the present work is the prediction of the mechanical behavior
of short wavy fiber composites, applicable to SSFC. The goal is to develop
models that can be used to assess the effect of the fiber waviness on the
behavior of SSFCs and to predict accurately the local composite response.
Mean-field homogenization approaches, most common among them is the
Mori-Tanaka (M-T) formulation, are analytical methods which provide a
very cost-effective way of predicting the effect of micro-structure, volume
fraction, aspect ratio and orientation of inclusions on the overall composite
properties [104]. These models are based on the dilute Eshelby’s solution
[289] for single ellipsoidal inclusions and hence have been typically
applied in literature on simple geometries of straight fibers.
Several approaches were used in previous studies for the application of
mean-field models to wavy fiber composites. The first was proposed by
Fisher et al. [290], who combined Finite Element Analysis (FEA) and the
Mori-Tanaka mean field homogenization method (M-T approach) for
modelling the effective properties of composites reinforced with wavy
carbon nanotubes (CNT). The authors performed FE analysis on single
nanotubes with a given waviness to calculate the reduced effective moduli
𝐸𝑤𝑎𝑣𝑦 (i.e. reduced by comparison of a straight CNT with the same aspect
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172
ratio) of each CNT with different magnitude of waviness; the effective
moduli are utilized in the multi-phase M-T model to calculate the elastic
properties of the composite with wavy nanotubes. In a similar way,
Bradshaw et al. [291] performed 3D FEA to compute the dilute strain
concentration tensor of wavy nanotubes to be used in M-T model.
Another approach was suggested by Gommers et al. [292] for knitted
fabric composites. Curved yarns in the knitted loops were subdivided in
small segments and each segment was replaced by an infinitely long
straight inclusion. The same method was used e.g. in [293] for wavy carbon
nanotubes composites. Huysmans et al. [294, 295] proposed a so-called
Poly-Inclusion (P-I) model in which they extended the methodology of
Gommers et al. by taking into account the effect of curvature of the yarn.
This is realized by decreasing the aspect ratio of the equivalent inclusion
depending on the local segment curvature. The method was later used and
showed good predictions of the elastic properties of different textile
composites [296-299].
The above studies were primarily focused on the prediction and validation
of the effective mechanical properties of wavy fiber composites. For
predictions of damage of composites using mean-field models reliable
estimates of the average local stress fields in the inclusions are needed. To
date, the accuracy of predictions of local stresses in inclusions has not been
validated.
It is clear that performing FEA, which requires meshing of the fibers and
the matrix, is computationally expensive, even if less heavy FE
formulations, e.g. embedded elements [300], are used. While it may be
reasonable for composites with smooth and prescribed wavy fiber
geometries, performing FEA calculations to back-calculate the effect
modulus of CNT with different waviness conditions, significantly reduces
the efficiency of the mean-field techniques for composites with random
waviness. In this respect, the methods developed by Gommers et al. and
Huysmans et al. are attractive alternatives for micro-mechanical modelling
of wavy fiber reinforced composites.
In this chapter, the P-I model of Huysmans et al. [294] is further developed
for short wavy fiber composites. The aim of the study is validation of the
predictions of the average local stress state in the equivalent inclusions by
comparison of the P-I predictions with the stress state in original wavy
segments obtained from full FE calculations. A number of models with
different short wavy fiber architectures are considered in order to
investigate the validity domains of the model.
Linear Elastic Modeling of Short Wavy Fiber Composites
173
6.2 The Poly-Inclusion (P-I) Model
The previously mentioned formulations (section 2.5) of the Eshelby
solution and the Mori-Tanaka mean-field approximation deals with
straight ellipsoidal inclusions for which the dilute Eshelby solution is
known either analytically or numerically. Nevertheless, in more complex
composite structures, such as short random steel fiber composites, the
fibers are curved and the evaluation of the Eshelby tensor in Equation
(2.12) is not possible. For this reason, a modelling approach is needed for
the transformation of the original wavy composite into an equivalent
system with ellipsoidal inclusions.
In early attempts, Gommers et al. [292] developed a methodology for
modelling the effective properties of knitted fabric composites, which
depict inherent curvatures of the knitting loops, based on the Mori-Tanaka
homogenization method. Each repeating knitted loop is subdivided into
straight fiber segments which are then replaced by ellipsoidal inclusions in
the homogenization model. The equivalent ellipsoids retain the same
orientation, cross-sectional shape and volume fraction as the corresponding
segment. Gommers et al. considered an aspect ratio of equivalent inclusion
equal to infinity.
Huysmans et al. [294] showed that this assumption leads to a strong
overestimation of the predicted equivalent elastic properties in case of
curved yarns. To overcome this shortcoming, they proposed taking into
account the reduction of the load carrying capability of the curved
segments by the so-called Poly-Inclusion (P-I) model. In the P-I model, a
curved fiber is divided into a sufficiently large number of smaller
segments. The effect of segment curvature is taken into account by
assuming a simple inversely-proportional relationship between the
equivalent inclusion’s length and the original segment’s curvature as
follows:
𝑎𝑟 = 𝛽 ∗𝑅
𝑑
(6.1)
where 𝑎𝑟 is the aspect ratio of the equivalent inclusion, β is the efficiency
(proportionality) factor, 𝑅 is the radius of curvature of original segment as
shown in Figure 6.1, and 𝑑 is the segment diameter . As indicated by
Equation (6.1), segments with higher local curvatures are modeled with
equivalent inclusions with lower aspect ratios, reflecting lower efficiency
of the curved segment.
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174
Figure 6.1 Equivalent ellipsoid replacing the original curved fiber segment [294].
Huysmans et al. [294] performed an evaluation of 𝛽 factor for the highly
curved yarns in knitted fabric composites and found that the best
correlation with the experimentally measured stiffness of the composite is
obtained with 𝛽 “lying around 3”, and the chosen value was 𝛽 = 𝜋. It
should be noted that in the range 𝛽 = 1.5…3 the influence of the choice
of 𝛽 value on the stiffness of the composite is weak (difference in the
stiffness values below 5%).
6.3 Problem statement and methods
The P-I model as described by Huysmans et al. has been only validated for
prediction of the overall macroscopic elastic constants by comparison to
experimental values. While the model is based on the assumption that the
average stress state in the curved fiber segments is correctly represented by
the stress state in the equivalent ellipsoids, this assumption has not yet been
validated. The predictions of average stresses in individual inclusions are
moreover essential for modelling damage events such as fiber matrix
debonding and fiber failure [142].
In the present chapter, the P-I model is applied to discontinuous wavy
fibers, with the main application to short steel fibers reinforcing a
thermoplastic polypropylene matrix. The average stresses in equivalent
z 3
y
x
2
1
θ
ϕ
R
3
Linear Elastic Modeling of Short Wavy Fiber Composites
175
inclusions, predicted by the P-I model, are compared with full-scale FE
results of the average stresses in the original wavy fiber segments.
6.3.1 Test cases
The considered test cases can be grouped into three categories (Figure 6.2).
In the first test case, a volume element (VE) containing a single half
circular fiber is considered (Figure 6.2 (a)). The fiber has a constant
curvature 𝑘 of 0.1. Three different models with fiber volume fraction 𝑉𝐹 =14.7%, 5.29% and 2.35% are considered; 𝑉𝐹 is varied by variation of the
fiber diameters.
The second set of test cases represents a VE containing a single sinusoidal
fiber exhibiting variable but smooth local curvatures over the fiber path
(Figure 6.2 (b)). The geometries of the fibers in the models are replicated
from the paper of Fisher et al. [290] for verification of the accuracy of
predictions of the P-I model by comparison to the combined FE-MT
method proposed by Fisher et al described in section 6.1. The fibers have
a sinusoidal shape: 𝑦 = 𝑎 cos(2𝜋𝑧
𝐿). 𝑦 is the wave path, 𝑎 is the amplitude
of the wave, 𝑧 is the axial fiber direction, 𝐿 is the length of the sinusoidal
fiber projection of the z axis (“wavelength”). Different models are
generated with variations of the waviness ratio (𝑊 = 𝑎/𝐿) in the range of
0-0.5, a fiber with 𝑊 = 0 being a straight fiber. Figure 6.2 (b) shows
examples of two considered cases with different waviness ratio. Similar to
Fisher et al. the wavelength of the sinusoidal fiber was set to 𝐿 = 100 with
the aspect ratio of the fibers 𝐿
𝑑= 40 where 𝑑 is the fiber diameter.
In the third test case, the VE contains wavy steel fibers extracted from
micro-CT scans of steel fiber reinforced composites (Figure 6.2 (c)).
Details of the micro-CT scans and of the geometry of the VE can be found
in[301]. The considered VE includes 30 fibers, which has been previously
reported as a sufficient number of inclusions to form a VE of random fibers
[79]. The diameter of all fibers was set to 𝑑 = 8 𝜇𝑚 which is the nominal
diameter given in the fibers datasheet [215]. The fiber volume fraction of
the VE containing 30 fibers is the same as of the scanned composite
sample. The objective of this test case is the validation of the P-I model
on fibers with non-uniform random waviness and to verify the accuracy of
the P-I model in predicting local stress fields in case of fiber interactions.
It should be noted that in the first two test cases, the dimensions of the VE
are arbitrary. The third test case is generated based on the real dimensions
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176
of VE and fibers. In all cases, enough matrix material is considered to
assume dilute concentrations.
(a) (b) (c)
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.
In the models, a polypropylene matrix and steel fibers are used and are
considered isotropic linearly elastic with the following properties: 𝐸𝑚= 1.5
GPa, 𝜈𝑚 = 0.4, 𝐸𝑓= 193 GPa, 𝜈𝑓 = 0.25. 𝐸 and 𝜈 represent the Young’s
modulus and Poisson’s ratio, subscripts 𝑚, 𝑓 denoting the matrix and fiber,
respectively. A perfect fiber-matrix interface is assumed.
For all cases, a homogeneous uniaxial strain 휀̃ of 1% is applied in
transverse (Y-Y, see the coordinate notation in Figure 6.2) direction.
Transverse loading is considered due to its significant influence on
interface failure, and consequently fiber-matrix debonding, the major
damage mechanism in short fiber composites [180, 302].
Linear Elastic Modeling of Short Wavy Fiber Composites
177
6.3.2 Implementation of Poly-Inclusion model
The input to the micro-mechanical model contains information about
mechanical properties of the fibers together with geometrical data for each
individual fiber in the representative volume element. The fiber geometry
is described on a per segment basis, whereby each segment is characterized
by the following parameters:
- Segment centroid coordinates
- Segment local co-ordinate system [1-2-3] (Figure 6.1) described
in the model by direction cosines
- Segment average local curvature
- Segment volume fraction in relation to the total volume of the
model
The local curvature at each segment centroid is calculated as follows:
𝑘 = |𝑟′𝐱 𝑟′′|
|𝑟′|3
(6.2)
Where 𝑟(𝑠) is the radial position in relation to a certain axis, 𝑠 the
coordinate along the curved fiber axis. For fibers represented by a set of
straight segments finite difference schemes are used to compute the
derivatives in Equation (6.2).
The number of fiber segments 𝑁𝑠 and the efficiency factor β are
parameters chosen by the user. Based on the local curvature and the β
factor, the length of the equivalent ellipsoid is calculated according to
Equation (6.1). The dimensions of the equivalent ellipsoid are then re-
scaled so that the volume fraction of the equivalent ellipsoid is equal to the
volume fraction of the original segment. The overall composite elastic
properties can be computed, as well as the average micro-strains and
stresses in individual inclusions.
6.3.3 Generation of finite element models
The short wavy fibers were generated using SolidWorks software and
imported in Abaqus finite element software. In the third test case
(Figure 6.2 (c)), 30 fibers were picked randomly from a micro-CT scan of
steel fiber reinforced composites, and the fiber centerlines (paths) were
extracted using Mimics software and imported into SolidWorks. Details of
the method used for extraction of the fiber centerlines can be found in
[301]. In SolidWorks, the solid geometries of the fibers were created and
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178
imported to the Abaqus solver for creation of the VE. All fibers were
assigned a constant diameter 𝑑 = 8 𝜇𝑚 as explained in section 6.3.1.
For all models, the matrix box was meshed using quadratic 10 nodes
tetrahedron elements, type C3D10. Fibers were meshed using linear
hexahedron elements, type C3D8R. Periodic boundary conditions were
applied in all models. To ensure correct application of boundary
conditions, three boundary faces of the matrix box were meshed and
meshes were copied on the corresponding opposite faces. The
displacements fields under periodic boundary conditions are prescribed
using the following generalized system of equations:
𝑈(𝐴𝑋) − 𝑈(𝐴𝑋 + 𝛿𝑥) = 휀̃ . 𝛿𝑥,
𝑈(𝐴𝑌) − 𝑈(𝐴𝑌 + 𝛿𝑦) = 휀̃ . 𝛿𝑦,
𝑈(𝐴𝑍) − 𝑈(𝐴𝑍 + 𝛿𝑧) = 휀̃ . 𝛿𝑧 (6.3)
where 𝑈 is a displacement vector, 𝐴𝑋, 𝐴𝑌, and 𝐴𝑍 are arbitrary points on
boundary faces of the VE, 𝛿𝑥,𝑦,𝑧 are translation vectors and 휀̃ is the average
strain tensor. The discretization of the wavy fibers into the required number
of segments in FE was done automatically, with a Python script, using
Abaqus element sets, each set including mesh elements belonging to one
segment on the wavy fiber. All segments have the same length. The volume
averages of the stresses in each set (segment) are then computed for
comparison with the P-I model.
6.4 Results and Discussion
6.4.1 VE containing a single half circular fiber with constant
curvature
Figure 6.3 illustrates the concept of the P-I model and the effect of variation
of the β factor. Figure 6.3a shows the original fiber and the subdivision into
segments. Figure 6.3 (b) and (c) show the equivalent ellipsoidal systems
considered by the P-I model with β = 𝜋
4 and β =
𝜋
2 respectively. In both
cases the segment local orientation is the same. The equivalent system with
β = 𝜋
2 depicts longer length of ellipsoids. Due to the constraint of keeping
the volume fraction of equivalent ellipsoid the same as the original curved
segment, the diameter of ellipsoids generated with lower β factors are
Linear Elastic Modeling of Short Wavy Fiber Composites
179
larger, giving rise to lower aspect ratios, and hence lower efficiency of
equivalent ellipsoids.
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 = 𝛑
𝟒, (c) equivalent inclusions with 𝛃 =
𝛑
𝟐.
Figure 6.4 shows a comparison of the overall predicted elastic moduli of
the P-I model with variations of the efficiency factor β. The figure shows
very good agreement of predictions of the P-I model compared to FEA for
the first test case with β = 𝜋
2. Higher values of the β factor lead to an
overestimation of the overall composite properties. As mentioned before,
an infinite value of the β factor assumes that each segment behaves as if it
were taken from a continuous straight fiber.
For better visualization, the predictions of the P-I model with β = ∞ are
omitted, but have shown to lead to a very high overestimation of the
longitudinal elastic properties; for example, the predicted longitudinal
(c) (b)
(a)
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180
modulus was 𝐸𝑧𝑧 = 12140 MPa as opposed to 𝐸𝑧𝑧 = 2282 MPa obtained
from full FEA. The P-I model with β = ∞ leads as well to an
overestimation of the transverse elastic properties 𝐸𝑥𝑥 𝑎𝑛𝑑 𝐸𝑦𝑦 but to a
lesser extent than that of longitudinal properties; for instance, the predicted
transverse modulus was 𝐸𝑦𝑦 = 3342 MPa compared to 𝐸𝑦𝑦 = 2075 MPa
obtained from full FEA.
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.
Figure 6.5 shows the predictions of the local stresses of the equivalent
inclusions obtained from the P-I model with variations of the β factor. The
results are shown in the local segment coordinate system 1-2-3 both for
axial segment stresses 𝜎33 and transverse stresses 𝜎22 plotted as a function
of the normalized distance along the longitudinal axis (Z axis) of the VE.
The figure is based on the discretization of the half circular fiber into 5
segments (as illustrated in Figure 6.3). As expected, the axial stress (𝜎33 )
distribution reaches a maximum value for segments oriented near to the
direction of applied load (in the Y-direction), hence at the fiber extremities.
For the transverse stresses (𝜎22 ) in the inclusions, the maximum values are
reached for segments perpendicular to the loading direction, hence in the
middle of the fiber.
Similar to the results of the overall VE elastic properties, the P-I model
with β = 𝜋
2 shows a good agreement with full FEA results.
Linear Elastic Modeling of Short Wavy Fiber Composites
181
Figure 6.5 (a) shows a strong increase of the local axial stresses with
increasing β values.
The results indicate that even a factor β = 𝜋 significantly overestimates
the stresses in the inclusions. This confirms the assumption of the P-I
model that the presence of fiber curvature significantly reduces the load
carrying capability of the fiber. The dependency of the equivalent stresses
in inclusions on the β factor is less pronounced in the case of transverse
stress 𝜎22 as shown in Figure 6.5(b). The increase of the transverse stresses
with decreasing β is due to the larger relative diameters of equivalent
inclusions (in relation of the size of the VE), resulting from the decrease of
ellipsoid length with decrease of β, which requires larger diameters of
equivalent ellipsoid (as shown in Figure 6.3(b)) to maintain the constraint
of segment volumes.
The value of β = 𝜋
2, that resulted in the best correlation with the overall
elastic properties and the local stresses in inclusions for this test case, is
lower than the value chosen by Huysmans et al. [294], namely β = 𝜋 for
knitted fabric composites. However, it has to be emphasized that this
choice was based on the best correlation of the homogenized mechanical
properties. This can be attributed to a number of reasons, first of which, is
the high curvature of the geometry considered in this study compared to
that considered by Huysmans et al. While knitted fabrics composites are
more curved structures than typical woven composites, their curvature
values are expected to be much lower compared to the relatively high
curvature 𝑘 = 0.1 mm−1 of the investigated model in this test case. In
such case, the efficiency of the segments is more dependent on their
curvature than expressed by the linear efficiency factor (constant β).
Another reason can be the difference in stiffness mismatch between the
matrix and reinforcement. Huysmans et al. considered composites
reinforced with knitted glass fiber yarns, having a rather low transverse
stiffness (around 12 GPa). In the present study, the models are applied to
steel fiber composites that depict a very high stiffness mismatch between
the fiber and matrix (stiffness of the isotropic steel fibers is around 200
GPa). This can emphasize the decrease of the efficiency of the fiber with
increased waviness.
However, as noted above, the difference in the predictions of the
homogenized stiffness with β = 𝜋/2 and β = 𝜋 in [294] is within 5%
difference. This corresponds to our calculations (Figure 6.4). However, the
predictions of the local stresses in the segments, shown in Figure 6.5, is
much larger. This leads to the choice of β = 𝜋/2 as the most suitable
segment curvature efficiency parameter, as confirmed in other test cases.
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182
(a)
(b)
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.
Linear Elastic Modeling of Short Wavy Fiber Composites
183
Another interesting aspect is the effect of the variation of the number of
segments 𝑁𝑠 in the P-I model. Huysmans et al. proposed the subdivision of
the curved fiber into a sufficiently large number of segments. Since
𝑁𝑠𝑒𝑔𝑚𝑒𝑛𝑡𝑠 is a parameter controlled by the user, the stability of the
predictions of the P-I model needs to be investigated, especially for the
prediction of the local stress fields.
Figure 6.6 shows a comparison of the predictions of the P-I model and full
FEA with variations of 𝑁𝑠 for segment local longitudinal and transverse
stresses of the first test case. It can be seen that for both the longitudinal
and transverse stresses, the predictions of the P-I model converge with
increase of the number of segments starting from the value of 𝑁𝑠 = 5.
(a)
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184
(b)
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.
A final aspect of comparison is the effect of fiber volume fractions on the
accuracy of the P-I model. Figure 6.7 shows a comparison of the
predictions of the P-I model and full FEA with different fiber volume
fractions, for segment local longitudinal and transverse stresses of the first
test case. The increase in axial stresses with decreasing 𝑉𝐹 is attributed to
the increase of segment aspect ratio (decrease of fiber diameter) as can be
seen in Figure 6.2 (a). Figure 6.7 (a) demonstrates that the deviations of
the results of the P-I model in comparison with full FEA increases with
decreasing fiber volume fraction. This can be due to the more significant
effect of the loss of connectivity between the fiber segments in the P-I
model with increased matrix volume. For transverse segment stresses as
shown in Figure 6.7 (b) the difference of predictions of transverse stresses
in inclusions is not affected by fiber volume fractions.
Linear Elastic Modeling of Short Wavy Fiber Composites
185
(a)
(b)
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.
CHAPTER 6
186
In order to investigate the validity of the concept of the P-I model without
the influence of the intrinsic assumptions of the Mori-Tanaka model, FEA
simulations of VEs with the equivalent inclusions (with 𝛽 = 𝜋
2) were
performed. An illustration of the considered VE is shown in Figure 6.3 (c).
The VEs were generated with 5 different random positions (placements) of
the equivalent ellipsoids in the matrix volume. Figure 6.8 shows a
comparison of FE simulations on the VE containing the original wavy fiber
(full FE), the results of FE simulations on the equivalent inclusions, and
the predictions of the analytical P-I model. The figure shows that the trends
of analytical P-I model corresponds to the trends of the FE simulations of
the equivalent inclusions. This leads to the conclusion that the
discrepancies between predictions of analytical P-I model and real stress
fields in wavy fiber is due to loss of connectivity of equivalent segments,
a common characteristic of the P-I model and the FE-model of the
equivalent inclusions.
A limitation of all mean-field homogenization models is that the effect of
the positions of the inclusions is not taken into account. The effect of
inclusion interactions is only taken into account by the concept of image
strain.
(a)
Linear Elastic Modeling of Short Wavy Fiber Composites
187
(b)
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.
6.4.2 VE-Single sinusoidal fiber with varying smooth local curvature
As mentioned in section 6.1, Fisher et al. [290] used a combined FE-MT
approach where they performed FE simulations on the sinusoidal fibers
with different waviness ratios (𝑤 = 𝑎/𝐿) to be further used in M-T
simulations of VEs of nanotube reinforced composites. We use the
geometry, proposed in that work, but, following the method of the present
work. The trend of the average maximum principal global stresses
𝜎𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 in the wavy fibers, is considered, instead of deriving equivalent
wavy fiber moduli, as it is done by Fisher et al. [290] . Figure 6.9 shows
the predictions of P-I model compared to full FEA of the average
maximum principal stresses. An efficiency factor 𝛽 = 𝜋
2 is used.
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188
(a)
(b)
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.
Linear Elastic Modeling of Short Wavy Fiber Composites
189
The trends for longitudinal loading are shown for qualitative comparison
with the results of Fisher et al. [290]. The figure shows excellent agreement
of the predictions of P-I model and full FEA in both loading directions.
Figure 6.9 (b) indicates that the P-I model is able to successfully predict
high decrease of reinforcing efficiency of the fiber with increasing
waviness. This observation was reported by Fisher et al. and several other
investigations [291, 303-306]. This leads to the conclusion that the P-I
model can be used to calculate the reduced efficiency of wavy fibers
without the need for computationally expensive FEA.
Figure 6.10 shows the predictions of the local stresses of the equivalent
inclusions obtained from the P-I model with variations of the β factor for
the sinusoidal fiber. While the predictions of the P-I model in the semi-
circular fiber model were very dependent on variations of the β factor, the
dependency on β is less significant for the axial stresses in the sinusoidal
fiber and diminishes for transverse stresses as shown in Figure 6.10 (a) and
(b), respectively. For the considered sinusoidal fiber, simulations with
𝑊 = 0.3, an efficiency factor β = π
2 gives the best correlation with FEA
results similar to the first test case for axial stresses and indeed improves
the predictions of M-T model compared to the uncorrected model with β =∞ for average inclusion axial stresses.
It is observed that, for the corrected P-I model, the deviations from FEA
values are higher for segments with lower curvatures where the P-I model
over-estimates the average axial local stresses. The situation is reversed for
average transverse segment stresses where the P-I model over-estimates
the load carrying capability of less wavy segments. This can be a result of
the over-estimation of the equivalent inclusion aspect ratio in the P-I model
in case of straight segments.
Figure 6.11 shows a comparison of the P-I model and full FEA with
variations of 𝑁𝑠 for segment local longitudinal and transverse stresses of
the second test case. The trends converged at 𝑁𝑠 = 10 afterwards, no
dependencies on the 𝑁𝑠 can be found.
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190
(a)
(b)
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.
Linear Elastic Modeling of Short Wavy Fiber Composites
191
(a)
(b)
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.
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192
6.4.3 VE-Micro-CT reconstructed assembly of short steel fibers with
random local curvature
The purpose of this test case is the validation of the P-I model for VEs with
considerable number of fibers, depicting the actual state of fiber
interactions, as opposed to the simpler cases of single fiber VEs considered
above. Another objective is to investigate the validity of the P-I model for
wavy fibers with random variations of local curvatures along the fiber path.
Figure 6.12 shows the predictions of the P-I model compared to FEA. It
should be noted that for the whole VE, the same value of the efficiency
factor which gave the best matching in the previous test cases, i.e. β =𝜋
2 was used for all fibers and segments. For all fibers, 20 segments were
used. For clarity, an example of 10 fibers selected randomly from the
modelled VE, with exact matching of segments between P-I and FEA, is
shown. The figure shows that the P-I model gives very good agreement
with FEA for the segments axial stresses with the β =𝜋
2 efficiency factor.
Linear Elastic Modeling of Short Wavy Fiber Composites
193
CHAPTER 6
194
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.
Also, for the axial stress states in equivalent inclusions, a better agreement
of trends was found compared to the second test case of the sinusoidal
fiber. This can be due to the lower variations of local curvatures in the
random wavy fibers compared to the sinusoidal fiber with higher range of
variations of local curvatures between segments. This can lead to the
conclusion that the linear relationship (constant β) between the equivalent
(a) (b)
Linear Elastic Modeling of Short Wavy Fiber Composites
195
inclusion aspect ratio and local segment curvature (Equation 6.1) is a
reasonable assumption for wavy fibers with constant local curvatures or
for fibers with low variations of local curvatures between segments.
However, in case of high variations of local curvature, the assumption of
linear relationship may lead to overestimation of the axial stress fields in
the straight segments. This has been consistently found for all fibers as
shown in Figure 6.12 (a), where the P-I model leads to an overestimation of
the stress in segments with lower curvatures (lower 𝑘 values). This result
can be attributed to the actual stress transfer situation between connected
segments with high variability in curvatures, which is not accurately
depicted by the simple linear equation.
Instead, future improvements may be achieved with new derivation of
Equation 6.1 which can be modified to a non-linear relationship between
the equivalent inclusion aspect ratio 𝑎𝑟 and the radius of curvature 𝑅.
For the average local transverse stresses, as shown in Figure 6.12 (b), the P-
I model gives relatively good predictions compared to FEA. However, as
was found for the sinusoidal test case, trends of the transverse stresses
generally shows relatively less agreement with FEA compared to axial
stress state. Huysmans et al. [294] reported similar observations for
transverse stresses by comparison of predictions of P-I model with Mori-
Tanaka method and Self-Consistent method. The authors reported that the
transverse stress state predicted by both models were substantially
different, where the Self-Consistent method showed three times higher
stresses. It was not known which of the models gives more realistic results.
In the present work, although less agreement with FEA is reported for
transverse stresses compared to axial stresses, no clear trend of over-
estimation or under-estimation of transverse stresses was found.
To summarize, it has been shown that the P-I model generally gives good
agreement of the homogenized global elastic response of VEs of wavy
fibers as well as local stress fields in inclusions. Predictions of axial stress
states in equivalent inclusions depicted very good agreement with FE
results. Trends of normal stress states in equivalent inclusions showed less
agreement with FE results in cases of fibers with variable local curvatures.
Nevertheless, overall predicted values were still within reasonable
accuracy.
Parametric studies on the effect of P-I model parameters showed that the
number of segments per fiber considered in the P-I model did not seem to
significantly affect the predictions of the model.
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196
A limitation of the analytical model is the effect of the position of the
segments and edge effects which is captured by detailed FEA and is not
reflected in P-I model. This limitation however is applicable to all mean-
field models and is not a specific aspect of the investigated model.
In the present model, a linear relationship is assumed between the aspect
ratio of equivalent ellipsoid and the local curvature. This simplified
relationship has been found to give good predictions in the case of fibers
with constant local curvature. In case of fibers with high variations of local
curvatures this may lead to high over-estimation of the local stress fields
in straight segments by the analytical model which does not capture the
actual state of stress transfer between the connected curved and straight
segments. This can be improved by assigning a non-linear relation in
Equation (6.1), as it was proposed in Huysmans’ thesis [307].
Another aspect is the stiffness mismatch between fiber and matrix. In this
work, steel fiber reinforcements were considered. In such case a high
mismatch of stiffness between the fiber and matrix is present. It is expected
that the efficiency of the equivalent inclusion is affected by the load
transfer mechanism between the matrix and fiber, and hence will be
dependent on the relative difference of the fiber and matrix stiffnesses.
Therefore, in further studies it would be interesting to investigate as well
the possible material dependency of the efficiency factor, especially in
cases of high fiber curvatures.
6.5 Conclusions
Based upon full finite element benchmarks of different wavy structures and
of a real VE of random wavy fiber composites, it has been shown that the
Poly-Inclusion (P-I) model generally shows excellent predictions of global
elastic response of VEs of wavy fiber reinforced composites, as well as
local stresses in the inclusions. The results of this work show that the P-I
model provides relatively good predictions of the local stress states of the
equivalent inclusions. Predictions of axial stress state in equivalent
inclusions depicted very good agreement compared to FE results of
stresses in original segment. The predicted trends of transverse stresses in
the segments showed less agreement with FE results but resulted in overall
reasonable accuracy.
The results of this work showed that the P-I model can be used with
relatively good accuracy for predictions of local stresses in wavy fibers.
This validation is essential for performing further damage analysis on VEs
Linear Elastic Modeling of Short Wavy Fiber Composites
197
of wavy fiber composites, using the fast mean-field homogenization
technique, which primarily depends on local fields in inclusions.
Although in the present work, the model was applied to steel fiber
reinforced composites, the methodology and validation is applicable for a
wide range of reinforcements, e.g. long carbon fibers, carbon nano-tubes,
natural fibers, crimped textiles, all of which depict inherent local
curvatures.
Improvements of the model can be achieved by modification of the linear
relationship in Equation (6.1) to a non-linear relationship between
equivalent inclusion elongation and local curvature, to improve predictions
in extreme cases of straight segments connected to highly curved segments.
199
Chapter 7: Non-linear progressive damage modelling of short fiber composites
Non-linear Progressive Damage Modelling of Short Fiber Composites
201
7.1 Introduction
As discussed in the literature review, when subjected to loading, short fiber
reinforced composites exhibit a non-linear stress-strain behavior. This also
applies to all the materials investigated in the present work, as has been
shown in Chapter 7. The sources of non-linearities can be attributed to the
non-linear elasto-plastic behavior of the polymer matrix and the damage
development in the composite. It was shown in the present work (Chapter
7), in agreement to the published data discussed in Chapter 2, that the main
damage mechanism in short fiber composites is failure of the interface or
the fiber-matrix debonding mechanism.
This chapter focuses on modelling the quasi-static non-linear deformation
behavior of short fiber reinforced composites. First, a description of the
formulation of the damage model is provided. Next, the implementation of
the simulations in the present work, starting from the micro-structural
modelling to the prediction of the stress-strain behavior of the short fiber
composites, is discussed. An emphasis is given on the steps of the solution
of the non-linear damage model. Another section is further devoted to the
detailed description of the validation cases which include the typical short
straight fiber composites, in addition to the random wavy fiber composites.
Finally, the results of the performed validation are discussed.
7.2 Formulation of the Damage Model
7.2.1 Matrix non-linearity
In section 2.6.1 it was shown that there are two main methods for
modelling the non-linear behavior of the matrix, namely the tangent
(incremental) approach and the secant approach. The underlying
differences between the methods were discussed. The tangent approach has
the advantage of the ability to model the complete load history in non-
monotonic loading. Nevertheless, it was shown that the implementation of
the method, in the framework of mean-field homogenization, requires
operators that are handled in purely numerical contexts and cannot be
directly related to actual material behavior. For this reason, the secant
approach proposed by Tandon and Weng [158] is used in this PhD thesis.
As previously discussed in section 2.5, the Eshelby based mean-field
homogenization models, including the M-T method used in this work, can
only be used for composites with linear-elastic constituents. To overcome
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202
this, the concept of the “reference” material is introduced. In the secant
approach, the weakening constraint power of the matrix as a result of its
plastic deformation is represented by the evolution of its secant modulus.
The yielding matrix (with a prescribed elasto-plastic behavior) is then
replaced at each load step with a linear elastic reference material having
the same secant properties of the matrix during this load step. In the
following the formulation of the method is described.
The non-linear plastic behavior of the matrix is modelled using a von Mises
criterion together with an associated flow rule. An isotropic hardening flow
rule is adopted in the present work, represented by the modified Ludwik
equation (Equation (7.1)). This was found to accurately fit the non-linear
stress-strain behavior of all of the thermoplastic matrices considered in this
work.
𝜎∗ = 𝜎𝑦 + ℎ (휀𝑝∗)𝑛 (7.1)
where 𝜎∗ is the effective (von Mises) stress in the matrix, 휀𝑝∗ is the
effective matrix plastic strain, and 𝜎𝑦 , ℎ and 𝑛 are the initial yield stress,
strength coefficient and work hardening exponent respectively.
It should be noted that the above Ludwik hardening rule is considered in
the generalized form, i.e. using the effective (von Mises) stress and strain
states in the matrix as opposed to the uniaxial states. This is due to the
anisotropy in the short fiber composites, in which case the stress and strain
states in the matrix are usually triaxial. The von Mises effective stress and
plastic strain are given by Equation (7.2) and (7.3) respectively.
𝜎∗ = (3
2 𝝈𝒊𝒋
′ 𝝈𝒊𝒋′ )
1/2
(7.2)
휀𝑝∗ = (2
3 𝜺𝒊𝒋
𝒑 𝜺𝒊𝒋
𝒑)1/2
(7.3)
where 𝝈𝑖𝑗′ refers to the deviatoric component of the matrix stress tensor.
The secant Young’s modulus 𝐸𝑚𝑠 at a given effective plastic strain 휀𝑝∗ is
computed as follows:
Non-linear Progressive Damage Modelling of Short Fiber Composites
203
𝐸𝑚𝑠 =
1
1𝐸𝑚
+ 휀𝑝∗
𝜎∗
(7.4)
Using the assumption of plastic incompressibility, the matrix secant
Poisson’s coefficient 𝜐𝑚𝑠 can then be obtained from the secant Young’s
modulus and the elastic properties of the matrix.
𝜐𝑚𝑠 =
1
2− (
1
2− 𝜐𝑚)
𝐸𝑚𝑠
𝐸𝑚
(7.5)
where 𝐸𝑚 and 𝜐𝑚 are the matrix elastic Young’s modulus and Poisson’s
coefficient respectively.
The main assumption of the application of the non-linear plasticity model
is that the yield condition and further plastic deformation is based on the
mean (average) stress state in the matrix. This is a simplification which
essentially means that the plastic flow is homogenously distributed in the
embedding matrix and no localized plastic flow can take place. Although
actual yielding in the anisotropic short fiber composites can be a local
phenomenon, the simplification is an inherent assumption of the mean-
field models.
7.2.2 Fiber-Matrix debonding
Damage is another source of the non-linear deformation behavior of short
fiber composites. The main damage mechanism of short fiber composites
is failure of the interface which results in debonding of the fibers from the
embedding matrix and hence a degradation in the load carrying capability
of the fibers. Different micro-mechanics based approaches for modelling
damage of SFRCs were discussed in the literature review (section 2.6.2.2).
With the critical examination of the available methods, the approach of
Fitoussi et al. [180, 185] provided a good basis for modelling the
debonding behavior of the random composites by means of using detailed
damage parameters describing the initiation and progression of debonding.
The model however, was only applied in the published work for the
continuous sheet molding compounds. In the present study, the model is
investigated for the considered short random straight and wavy fiber
composites.
Similar to the incorporation of the matrix plasticity in the framework of
mean-field model, the main idea behind modelling fiber-matrix debonding
is replacing the debonded inclusion with a perfectly bonded one with
CHAPTER 7
204
degraded properties according to a suitable degradation scheme. This in
turn will reflect the reduced efficiency of load carrying capability of the
partially debonded inclusion.
7.2.2.1 Interfacial stress fields
The first step in modelling damage consists of assessing the local stress
states along the interface, illustrated in Figure 7.1.
Figure 7.1 Determination of the outward normal and the local interfacial stress
vectors around the equator of the inclusion. �⃗� (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.
At each point on the interface surface, the normal component of the
interfacial stress tensor 𝜎𝑁 is obtained by the projection of the interfacial
stress vector upon the outward normal 𝒏𝑖 to the interface at this point:
𝜎𝑁 = 𝝈𝑖𝑜𝑢𝑡𝑛𝑖 = 𝝈𝑖𝑗
𝑜𝑢𝑡𝒏𝑗𝒏𝑖 (7.5)
α
𝜎𝑁
�⃗�
𝟏 Inclusion
𝑨
𝟐 τ
𝝈𝒊𝒐𝒖𝒕
𝟏 𝟐
𝟑
Non-linear Progressive Damage Modelling of Short Fiber Composites
205
The interfacial stress vector outside an inclusion 𝝈𝑖𝑜𝑢𝑡 can be calculated
using the continuity conditions of the tractions and displacements across
the inclusion interface [178, 185, 295], from which we can obtain that:
𝝈𝑖𝑗𝑜𝑢𝑡𝒏𝑗 = 𝝈𝑖𝑗
𝑖𝑛𝒏𝑗 (7.6)
where the vector 𝒏𝑖 is the outward normal vector describing the equator of
the inclusion. For an ellipsoidal cross-section with an aspect ratio 𝑝 =𝑎1 𝑎2⁄ , 𝑎1 and 𝑎2 being the major and minor cross-section diameters, the
components of the normal vector at an interface point 𝐴 with an in-plane
angle α can be evaluated using Equation (7.7). As shown in Figure 7.1, the
outward normal vector 𝒏𝑖 is calculated in the local coordinate system (1-
2-3) of the inclusion.
(𝑛1, 𝑛2, 𝑛3) = (cosα
√1 + (𝑝4 − 1) sin2 α,
𝑝2 sinα
√1 + (𝑝4 − 1) sin2 α, 0)
(7.7)
For inclusions with circular cross-sections, which is a typical assumption
for modelling the reinforcing fibers in short fiber composites, Equation
(7.7) can be used by assigning 𝑝 = 1.
Finally, the tangential component 𝜏 of the interfacial stress vector can be
obtained from:
𝜏 = √‖𝝈𝑖𝑜𝑢𝑡‖
2− 𝜎𝑁
2 (7.8)
Thus, the normal and tangential components of the interfacial stress vector
at each point along the interface can simply be obtained from knowledge
of the local stresses inside of the corresponding inclusion.
7.2.2.2 Description of the debonding model
Once the interfacial stress states are evaluated, a suitable failure criterion
can be applied at each point along the interface surface. In the present
work, the well-known Coulomb criterion is used. The criterion takes into
account the combined effect of the normal debonding due to the normal
component and the shear slip due to the tangential component of the above
described interfacial stress vector.
206
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.
C C
C
T
T
T
HZ
LZ
Normal Stress
Vectors
Non Debonded point
Debonded point
Point on compression
Point on tension
C T
𝒅 = 𝟗
𝟐𝟎
𝜸 = 𝟕
𝟐𝟎
𝜹 =𝟐
𝟗
(a) (b)
Linear Elastic Modeling of Short Wavy Fiber Composites
207
The Coulomb criterion is hence given as follows:
𝜎𝑁 + 𝛽𝜏 ≤ 𝜎𝐶 (7.9)
where 𝛽 and 𝜎𝐶 are the shear coupling coefficient (contribution factor) and
the critical interface strength respectively. Both parameters describe the
interfacial resistance. The advantage of the linear Coulomb criterion over
other forms such as the quadratic criterion or the maximum stress criterion
is that the linear Coulomb formulation requires only the identification of
these two parameters (𝛽 and 𝜎𝐶).
In the model of Fitoussi et al. [180, 185], i.e. the “equivalent anisotropic
undamaged inhomogeneity (EAUI)”, as suggested by the name, the
partially debonded inclusion with originally isotropic stiffness tensor 𝐶𝑖𝑗𝑘𝑙𝑖𝑠𝑜
is replaced by an equivalent undamaged anisotropic inclusion with
degraded properties. The equivalent inclusion then has a new anisotropic
stiffness tensor 𝐶𝑖𝑗𝑘𝑙𝑒𝑞
reflecting its reduced efficiency. The motivation
behind the formulation of the model is that a debonded reinforcement still
contributes to the global stiffness of the composite and that local damage
induces a local anisotropy of the damaged fiber.
In this respect, the stiffness degradation is performed using a selective
scheme represented by three local damage parameters (d, γ, δ) defined as
follows:
d: total percentage of the debonded interface area.
γ: total amount of the debonded interface area which is loaded on
traction.
δ: percentage of the frictional sliding interface, i.e. relative amount
of the of the debonded interface area which loaded in compression.
These parameters can then be computed for each inclusion from the local
interfacial stress states (discussed in the previous section) along the equator
of the inclusion. Parameter d is calculated as the ratio between the number
of debonded interfacial points and the total number of interfacial points. γ
is calculated as the ratio between the number of debonded interfacial points
in traction and the total number of interfacial points. δ is calculated as the
ratio between the number of debonded points in compression and the total
number of debonded points. An example is shown in Figure 7.2.
Finally, the components of the anisotropic stiffness tensor of the
undamaged equivalent inclusion are calculated based on the local damage
CHAPTER 7
208
parameters according to Equations (7.10 a-d). Noting that the stiffness
tensor is given in the contracted notation.
𝐶11𝑒𝑞
= (1 − 𝛾𝑙) ∗ 𝐶11𝑖𝑠𝑜 (7.10 a)
𝐶22𝑒𝑞
= (1 − 𝛾ℎ) ∗ 𝐶22𝑖𝑠𝑜 (7.10 b)
𝐶33𝑒𝑞
= (1 − 𝛾) ∗ 𝐶33𝑖𝑠𝑜 (7.10 c)
𝐶44𝑒𝑞
= {𝛿 ∗ [𝑑 ∗ (𝜇 − 1) + 1] + (1 − 𝛿) ∗ (1 − 𝑑)} ∗ 𝐶44𝑖𝑠0 (7.10 d)
𝐶55𝑒𝑞
= 𝐶44𝑒𝑞
(7.10 e)
𝐶12𝑒𝑞
= 𝐶13𝑒𝑞
= 𝐶23𝑒𝑞
= {[𝛾 ∗ (1 − 𝑑)] + (1 − 𝛾)} ∗ 𝐶12𝑖𝑠0 (7.10 f)
𝐶66𝑒𝑞
= 𝐶66𝑖𝑠𝑜 (7.10 g)
It should be noted that the above formulations of the stiffness tensor of the
degraded equivalent inclusion were derived from analysis of the debonding
progression in an inclusion using full Finite Elements Analysis. With the
detailed calculations, the authors concluded that the components 𝐶11𝑒𝑞
and
𝐶22𝑒𝑞
(when axis 3 is the elongation axis of the ellipsoidal inclusion) of the
stiffness tensor of the equivalent inclusion actually depend on the position
of the “crack” initiated from the local debonds. This has led to the
definition of the damage parameters 𝛾ℎ and 𝛾𝑙 which correspond to the
percentages of the debonded interface in the higher zone (HZ) and to the
lower zone (LZ) respectively. As shown in Figure 7.2 (b). As shown in the
figure, the HZ and LZ are defined per quadrant of the fiber. The detailed
local damage parameters 𝛾ℎ and 𝛾𝑙 are taken into account in the present
modelling approach.
7.2.3 Fiber breakage
As discussed in the literature review, fiber breakage is rarely found in short
fiber composites as in these types of composites only a limited probability
of the fibers are longer than the critical length and/or depict weak interface
with the embedding, thus favoring the debonding damage mechanism.
Nevertheless, in the implementation of the damage models in the present
work, a criterion for fiber breakage is assigned as follows:
𝜎33𝛼 ≥ 𝜎𝑢𝑙𝑡
𝑓 (7.11)
Non-linear Progressive Damage Modelling of Short Fiber Composites
209
The criterion in Equation (7.11) assumes that breakage of the inclusion
occurs when the axial stress state (defined in the inclusion local coordinate
system) in the inclusion is higher than the ultimate strength of the
corresponding fiber material. In this work, the broken inclusion is replaced
by a void, i.e. with a null equivalent stiffness tensor 𝐶𝑖𝑗𝑒𝑞
= 0.
7.3 Implementation of the Damage Model
In this section, a description is provided for the details of the simulation of
the quasi-static non-linear behavior of the short fiber reinforced
composites. First, a brief idea about the general steps of the model solution
is provided. A detailed outline and overall solution flowchart will be given
in Chapter 8. In the second part of this section, an emphasis is given on the
implementation of the solution procedure of the damage model in the
framework of mean-field homogenization models.
The overall solution scheme starts with modelling the geometry of each
type of material. Based on the micro-structural parameters discussed above
for each distinct material, a representative volume element is generated
using the model described in Chapter 3. The model is able to generate the
geometries of straight or wavy short fiber composites. Using the micro-
structural model software toolkit, a parameter which describes the size of
the RVE i.e. the number of fibers in the RVE can be assigned. In the present
work, an RVE size of 1000 fibers is used for all simulations. This size was
chosen based on the discussion outlined in the literature review in
section 2.4.1 where it was shown that a large RVE size is needed for
modelling the non-linear behavior of the composites.
The next step in the solution is the translation of the generated fiber
information to an inclusions system. This step distinguishes between
straight and wavy fiber composites. In the straight fiber composites the
fibers and inclusions are interchangeable, where one fiber is directly
considered an ellipsoidal inclusion with the same aspect ratio as the
original generated fiber. In the case of the wavy fiber composites the model
described in Chapter 6 is used to discretize a wavy fiber into a number of
segments which are then replaced with equivalent ellipsoidal inclusions
with elongations (aspect ratios) obtained from the relationship in Equation
(7.1). An inclusion then represents a segment of the wavy fibers.
Finally, based on the properties of the constituents, and the suitable damage
parameters, the quasi-static non-linear simulation can be performed on the
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210
representative inclusions system. In the following, the details of the
implementation of the quasi-static simulation are discussed.
Figure 7.3 shows a flow chart of a single load step 𝑘 in the model. Similar
to the geometrical model described in Chapter 3, the model was
implemented in a C++ software toolkit.
Non-linear Progressive Damage Modelling of Short Fiber Composites
211
Figure 7.3 Flowchart of a single load step of the developed damage model.
Load step
𝑘 =1?
START
Plastic ? No
Yes Initialize
𝜺𝑘𝑚,𝑝
= 𝜺𝑘−1𝑚,𝑝
𝑪𝑘𝑚 = 𝑪𝑘−1
𝑚
Damage ?
Loop over inclusions (α)
Initialize
𝑑𝑘𝛼= 𝛾𝑘
𝛼 = 𝛿𝑘𝛼 = 0
Initialize
𝜺𝒌𝑚,𝑝
= 𝟎
𝑪𝑘𝑚 = 𝑪o
𝑚
Calculate
Composite Stiffness 𝑪𝑘𝑒𝑓𝑓
Increment Boundary conditions
Calculate Composite strains/stresses 𝝈𝑘
𝑐 , 𝜺𝑘𝑐
Calculate Matrix strains/stresses
𝜺𝑘𝑚, 𝜺𝑘
𝑚,𝑝, 𝝈𝑘
𝑚 Converged?
Solve Matrix plasticity
sub-model
Damage? Loop over inclusions (α)
Update inclusions
strains/stresses 𝜺𝑘𝛼 , 𝝈𝑘
𝛼
Inclusion
𝜎33𝛼 ≥ 𝜎𝑢𝑙𝑡
𝑓
Break
inclusion 𝑪𝛼 = 0
END
Inclusion
𝑅 ≥ 𝜎𝑐
Calculate
𝑑𝑘𝛼= 𝛾𝑘
𝛼 = 𝛿𝑘𝛼
Degrade inclusion
Yes No
No
No
No
Yes
Yes
Yes
No
Plastic?
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212
As shown in the figure, the specific steps of the solution depend on two
keywords: plasticity and damage. This denotes the options in the developed
methods where the quasi-static simulation can be performed with or
without damage consideration of the matrix plasticity and with or without
the application of the damage models.
In the beginning of each load step, if matrix plasticity is included, an
initialization of the matrix plasticity parameters, i.e. the stiffness tensor of
the matrix (taken as the secant stiffness) 𝐂𝑚 and the plastic strain in the
matrix 휀𝑝 are set to the values of the previous load step if 𝑘 > 1 or to initial
values 𝐂𝑚 = 𝐂o𝑚 and 𝜺𝑚,𝑝 = 0 where 𝐂o
𝑚 is the elastic stiffness tensor. If
damage is taken into account, the damage variables (𝑑, 𝛾, 𝛿) of each
inclusion α in the RVE are initialized to zero.
In the next step, homogenization is performed and the composite stiffness
tensor 𝐂𝑒𝑓𝑓 is calculated from the Mori-Tanaka formulation in Equation
(2.14). The boundary conditions are incremented to reflect the new far field
strain 𝛆∞ of the load step and the composite stress and strain states 𝛔𝑐 and
𝛆𝑐 are then updated.
If matrix plasticity is considered, once the yield criterion 𝜎∗ > 𝜎𝑦, an
iterative solution is needed to calculate the new matrix stiffness tensor and
average stress and strain states. The iterative solution is denoted the
plasticity sub-model. This is due to the formulation of the secant modulus
𝐸𝑚𝑠 in Equation (7.4), which depends on the effective plastic strain 휀𝑚,𝑝∗,
the later not being an initially known priori. A detailed description of the
iterative solution can be found in [158].
The final part of the solution is the assessment of the damage of the
inclusions, if the damage modelling is applied. For each inclusion, the
stresses and strains 𝝈𝛼 and 𝜺𝛼 are calculated, from which the interfacial
stresses are obtained according to Equations (7.5) to (7.8). The fiber
breakage criterion (Equation 7.11) is evaluated and if reached, the
inclusion is replaced with a void. If the inclusion is not broken, the
debonding failure criterion is then applied at each point along the interface
of each inclusion α and the associated damage parameters (𝑑, 𝛾, 𝛿) are
computed. Each damaged inclusion is then replaced with an equivalent
perfectly bonded inclusion with degraded stiffness components according
to Equations (7.10). A new RVE of equivalent inclusions is then used for
the next load step.
Non-linear Progressive Damage Modelling of Short Fiber Composites
213
7.4 Description of Validation Test Cases
In the present work, a number of materials are used for the validation of
the quasi-static behavior of short fiber composites. These include both the
short random straight fiber and wavy fiber reinforced composites
considered in the topic of this thesis.
For each validation material, a number of parameters are needed as input
for the developed micro-mechanical models. The first set of input
parameters is the mechanical properties of the constituents. This includes
the elastic properties of the fibers (Young’s modulus and Poisson’s
coefficient), and the overall stress-strain behavior of the matrix. The stress-
strain behavior of the matrix is used for modelling the elasto-plastic
behavior of the matrix using the above mentioned secant approach.
The second set is the micro-structural parameters of the composite. This
includes the volume fraction and the fiber length and orientation
distributions of the material. For the wavy fibers composites, parameters
describing the waviness of the fibers e.g. the maximum wave amplitude
and maximum waviness number, as described in Chapter 3 are added.
The final set is the parameters of the damage model. This includes the
parameters of the failure criterion used for the assessment of damage
initiation and propagation. In this work, the Coulomb criterion is used as
discussed above and hence the damage parameters are: the critical interface
strength and the shear contribution coefficient. An approximate estimate
would be to assume the interface strength of the composite to be the same
as the yield strength in the matrix. This case is considered an upper bound
of the possible interface strength properties of the SFRCs. In general, it is
expected that most of the commercial grades SFRCs available in the
composite market depict optimized interface properties for achieving full
potential of the reinforced material. In principle, for composites with non-
optimized interfaces, the interface strength could be lower than the yield
strength. In this respect, in all simulations of the present thesis, the
interface strength 𝜎𝐶 in Equation 7.9 is taken as the yield strength of the
matrix. It will be shown later that this assumption leads to an over-
estimation of the composite properties only in the case of the SF-PA
materials, which confirms the conclusions of unoptimized interfaces of
these composites observed through experimental characterization.
The shear contribution coefficient is considered an empirical parameter
and is usually assumed in literature 𝛽 = 0.5. A detailed analysis was
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214
performed by Jain [308] in which a parametric study was performed for
determination of the optimal values of the 𝛽 parameter. The author has
shown that the value of 𝛽 = 0.5 was a suitable assumption by validation
on a number of different short fiber composites. This value will similarly
be used in all simulations of the present work.
Three distinct composite systems were characterized by own experiments
in the present work, i.e. two short glass fiber composites, as examples of
the typical random straight fiber composites, and a short steel fiber
composite, as an example of random wavy materials. A detailed
experimental analysis was performed on these materials as shown in
Chapter 4 and 5. In addition to own experiments, another set of data was
used for the validation of the developed quasi-static models and fatigue
models (as will be discussed in the next chapter), namely a glass fiber
reinforced polybutylene terephthalate. The experimental results of the GF-
PBT material is performed by colleague Atul Jain and reported in [308].
In the following sub-sections, a description of the main input parameters
of each material is provided.
7.4.1 Own experiments – glass fiber reinforced composites
Two different glass fiber reinforced composites were considered in the
present thesis, i.e. the GF-PA and GF-PP systems. Both materials were
produced in the form of standardized ISO specimens (dog-bone
specimens). A description of the materials and the manufacturing
parameters was shown in section 4.2. In the following, a description of the
input parameters for each material is given.
For all simulations in this work, the glass fibers are assumed to be linear
elastic materials with the following elastic properties: 𝐸 = 72 𝑀𝑃𝑎, 𝜈 =0.22. To assess the fiber breakage failure criterion, the ultimate strength
of the fiber is needed. The strength of the glass fibers in all simulations is
assumed 𝜎𝑢𝑙𝑡𝑓
= 2000 𝑀𝑃𝐴 as suggested in [5]. The properties and stress-
strain curves of the polyamide 6 and polypropylene matrices needed for
modelling the GF-PA and GF-PP materials, respectively, were previously
discussed in section 4.3.2.1.
The fiber volume fraction was calculated based on the weight fractions of
the composites and the density of the constituents. The GF-PA had a
slightly higher fiber volume fraction compared to the GF-PP. The length
distribution of both composites was obtained by experimental
characterization as described in section 4.3.1.
Non-linear Progressive Damage Modelling of Short Fiber Composites
215
To obtain the fiber orientation distribution, manufacturing simulation was
performed using the MoldFlow software. The simulations were performed
in Technology Campus Ostend. The author gratefully acknowledges the
help of prof. Frederik Desplentere and Dr. Bart Buffel.
Figure 7.4 shows plots of the results of the manufacturing simulation of the
dog-bone samples. In Figure 7.4 (a) a schematic of the typical orientation
and the skin-core layers in a dog-bone sample is presented. The figure is
taken from [54]. As shown in the figure, the core layer (characterized with
lower orientation of the fibers) is only present in a centralized part of the
injection molded standardized specimens. This is in contrast to specimens
milled from plates, where the core layer is pronounced over the entire
width of the specimen. This morphology of the dog-bone specimens was
confirmed by micro-CT analysis on the samples of the present work as
shown in Figure 4.9 and Figure 4.10. Figure 7.4 (b) shows an example of
the Moldflow model predictions of the dog-bone sample, the plot shows
the results of the GF-PP material at different points through the thickness,
for direct comparison of the micro-CT images in Figure 4.9. As can be seen
from the figure the model indeed predicts higher orientation of the fibers
at the lateral edges compared to the central points along the width.
Bernasconi et al. [21] discussed that in general, the core layer in the
standardized dog-bone specimens tend to be thinner than the core layer in
plates. Moreover, the lateral walls have the same orienting effect as the top
and bottom surfaces of the mold. This explains the larger skin (oriented)
layer in the standardized specimens as described in [54]. The thin core
layer was also confirmed in this work by micro-CT scanning. The
morphology of thin and centralized core layers and larger skin layers of the
standardized specimens is expected to result in high overall orientation of
the fiber with respect to the melt flow direction.
It should be noted that the manufacturing simulations were performed
using the ARD-RSC model. The model was chosen on the basis of
literature investigations where it was found to provide the best agreement
with experimental data, compared to the earlier models implemented in the
Moldflow software, e.g. the Folgar-Tucker model which was found to
over-predict the orientation tensor [44, 309].
The main inputs to the ARD-RSC model for the manufacturing simulation
of the GF-PA and the GF-PP materials are the processing (injection
molding) parameters described in Table 4.1. In addition to the parameters
in Table 4.1, the following settings were used in the model:
velocity/pressure switch over: automatic, fiber inlet conditions: fibers
aligned at skin/random at core. The fiber inlet conditions were applied at
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216
the gate. It should be noted that the automatic velocity/pressure switch over
is an option in the manufacturing simulation model which allows to
estimate an acceptable time to switch from velocity to pressure control to
avoid an end-of-fill pressure spike in the simulation. More information
about this setting and the different model parameter can be found in the
MoldFlow help file.
Figure 7.5 (a) and (b) show the resulting predictions of the main orientation
tensor component 𝑎11 plotted against the distance through the thickness in
the central gauge section of the GF-PA and the GF-PP samples
respectively. An interesting aspect of the manufacturing simulation is the
prediction of lower orientation of thin external layers at the surface of the
specimen [43, 44, 309-311]. This aspect was previously reported in
literature both through experimental measurements and manufacturing
simulation of the through-the thickness orientation of injection molded
specimens. It was attributed to the mold filling behavior in such a way that
as the melted material in the mold, the external skin layers are in direct
contact with the cold mold and solidify rapidly with lower orientation of
the fibers [312]. The more oriented skin layers then developed under the
lower oriented external layers.
From the plots of the main orientation tensor component of both materials
as shown in Figure 7.5 (a) and (b), it can be seen that the variation between
the 𝑎11 values in the skin and core layers are not significant. The difference
between the maximum difference in 𝑎11 (i.e. between the most oriented
and least oriented layer) was about 6% for the GF-PA and 14% for the GF-
PP. This is in contrast to the true skin-core morphology which may develop
in injection molded thick plates. A typical example was shown in
Figure 2.3 where it was shown that the reduction in the 𝑎11 component
between the skin and core layer can be in the range of 60-70%.
Non-linear Progressive Damage Modelling of Short Fiber Composites
217
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.
skin layer
core layer
(a)
(b)
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218
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.
Therefore for the numerical simulation of the present material, the entire
dog-bone coupon (specimen) can be treated as one single RVE, whose fiber
orientation distribution is described by the 2nd order orientation, tensor
which reflects an average of the components of the tensors of the though-
the thickness layers in the central gauge section. A summary of the micro-
structural parameters of the GF-PA and the GF-PP material used as input
in the model validation is presented in Table 7.1.
(a)
(b)
Non-linear Progressive Damage Modelling of Short Fiber Composites
219
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.
GF-PA GF-PP
Volume
Fraction, VF%
16 13
Fiber length
distribution,
𝝍𝑳
Lognormal
μ = 5.6 , 𝜎 = 0.5
Lognormal
μ = 6.8 , 𝜎 = 0.7
Fiber
orientation
Orientation tensor
𝑎𝑖𝑗 = [0.77 0 00 0.16 00 0 0.07
]
Orientation tensor
𝑎𝑖𝑗 = [0.86 0 00 0.12 00 0 0.02
]
7.4.2 Own experiments – steel fiber reinforced composites
The steel fiber reinforced polyamide 6 composites considered in the
present work were used for validation of the developed models as an
example of the short wavy fiber reinforced materials.
The mechanical properties of the steel fibers and the Durethan polyamide
matrix (neat PA 6 used in the SF-PA materials) were not available in
materials databases. Experimental characterization was performed in this
thesis for obtaining the constituents input parameters needed for the model,
as can be found in sections 4.3.5.1 and 4.3.5.2 for the fibers and matrix
respectively.
As mentioned in section 4.2.3, measurements were performed using the
developed micro-CT technique (discussed in Chapter 3) to obtain the mean
length of the SF-PA samples with the different fiber volume fractions.
Measurements of a large number of fibers in order to obtain the full fiber
length distributions was difficult as with increasing volume fractions of the
steel fibers, there were difficulties in tracking and segmenting a large
number of individual fibers. For the same reason, experimental
characterization of the full orientation tensor especially with the higher
volume fraction samples was not possible.
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220
In order to have an idea about the orientation of the samples, manufacturing
simulation was performed using the MoldFlow software, similar to the
glass fiber composites. In general, the models involved in the
manufacturing simulation software assume geometries of straight fibers in
the melt. Nevertheless, the results of the manufacturing simulation can be
used as an approximation of the orientation tensor of the wavy fibers SF-
PA material in this work. In such a case, the predicted orientation tensor
describe the end-to-end orientation of the wavy fibers as described
in 3.5.2.2. The manufacturing simulation was performed on all the SF-PA
materials with the different fiber volume fractions. Figure 7.6 shows the
results of the manufacturing simulation of the SF-PA 2VF% sample as an
example to the performed simulations on the SF-PA materials. Similar to
the glass fiber reinforced composites, the difference in the orientation
tensor components (reflected in Figure 7.6 by the main component 𝑎11 is
small (in the range of 10%). The same was found for all other volume
fractions. Therefore, the SF-PA coupons will similarly be treated in the
current micro-mechanical simulations as one RVE.
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.
Table 7.2 summarizes the micro-structural parameters of the lowest,
middle and highest volume fraction samples, i.e. the 0.5, 2 and 5VF%
conditions respectively. It was already seen in Chapter 4 that the mean
length of the fibers decreased with the increasing fiber volume fraction.
The table also shows that with the higher volume fraction an increase of
the fiber orientation is predicted. This is due to the fact that with the higher
Non-linear Progressive Damage Modelling of Short Fiber Composites
221
number of fibers, the fibers tend to orient themselves in the flow direction.
This was also previously found in [29].
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.
SF-PA
Volume
Fraction,
VF%
0.5 2 5
Fiber length
distribution,
𝝍𝑳
Constant
𝐿 = 605 μ𝑚
Constant
𝐿 = 557 μ𝑚
Constant
𝐿 = 352 μ𝑚
Fiber
orientation
Orientation tensor
𝑎𝑖𝑗
= [0.71 0 00 0.18 00 0 0.11
]
Orientation tensor
𝑎𝑖𝑗
= [0.85 0 00 0.13 00 0 0.02
]
Orientation tensor
𝑎𝑖𝑗
= [0.86 0 00 0.12 00 0 0.02
]
7.4.3 Experiments of Jain – glass fiber reinforced composites
Jain [308] performed tensile and fatigue tests on a 50wt% glass fiber
reinforced polybutylene terephthalate (PBT). The material was of the
commercial grade BASF Ultradur B4040 G10. Thin plates were produced
with the injection molding process. Coupons were machined from plates
in three directions in a way similar as the one described in section 2.3.2,
inclined at with angles of 𝜙 = 0, 45, 90° to assess the behavior of the
coupons with different orientation tensors. The material will be denoted
from this point forward as GF-PBT 𝜙.
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222
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].
The matrix material in the GF-PBT samples was of the commercial grade
BASF Ultraduur B 4500. Figure 7.8 shows the stress-strain curve of the
neat PBT material as obtained from the CAMPUS plastics database [273].
The yield strength of the matrix was 55 MPa at a strain of 3.2%.
Figure 7.8 Stress-strain curve of the BASF Ultraduur B4500 [273]. The tests are
stopped at the yield of the matrix.
Non-linear Progressive Damage Modelling of Short Fiber Composites
223
The fiber length distribution was not available, a mean fiber length of the
fibers was used instead as suggested by the author. Manufacturing
simulation was performed on the 0° coupons using the SIGMASOFT
software. Due to the low thickness of the plates, the fiber orientation was
found to be uniform throughout the specimen. Table 7.3 summarizes the
micro-structural parameters used as input of the developed for the
validation of the GF-PBT material.
Table 7.3 Summary of the micro-structural parameters of the GF-PBT materials
used as input for validation of the developed models. Data is taken from [308].
GF-PBT
Volume Fraction, VF% 35
Fiber length distribution,
𝝍𝑳
Constant
𝐿 = 200 μ𝑚
Fiber orientation Orientation tensor for 𝜙 = 0
𝑎𝑖𝑗 = [0.81 0.018 0.1370.018 0.11 0.0040.137 0.004 0.079
]
7.5 Results and Discussion
7.5.1 Own experiments – glass fiber reinforced composites
The quasi-static stress-strain behavior of the glass fiber reinforced
composites considered by own experiments, i.e. the GF-PA and the GF-PP
materials of the present work, was simulated using the above mentioned
procedure.
For each material, a comparison of the stress-strain curves predicted by the
modelling approach by taking into account only matrix plasticity, and the
combined matrix plasticity and inclusions damage models are shown and
compared to the experimental curves. This is to analyze the effect of the
different models on the predictions of the stress-strain behavior. A purely
linear elastic stress-strain curve (with elastic constituents and no damage)
is shown for reference.
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224
Figure 7.9 shows the simulated stress-strain behavior compared to the
experimental curve for the GF-PA material. As can be seen from the figure
both matrix plasticity and damage contribute to the non-linear stress-strain
behavior of the composite. The non-linearity due to fiber-matrix debonding
is however more pronounced, reflecting the significance of damage in short
fiber composites. The figure also shows that the overall modelling
approach with consideration of both plasticity and damage leads to
accurate predictions compared to the experimental curve.
Figure 7.9 Comparison of the experimental and predicted stress-strain behavior
of the GF-PA composite.
Figure 7.10 shows the predictions of the models for the GF-PP material of
the present work. Similar to the GF-PA material, the overall approach leads
to very good match of the predicted stress-strain behavior compared to the
experimentally achieved curve. It can also be noted that the deviation from
linearity due to only the consideration of the plasticity of the matrix is less
pronounced than that of the GF-PA. This is a result of the higher non-
linearity of the polyamide matrix compared to the polypropylene matrix as
shown in Figure 4.4 and Figure 4.5, especially given the difference in strain
to failure of both composites. The strain to failure of the GF-PA composite
is about twice higher than that of the GF-PP resulting in higher non-
linearity.
Non-linear Progressive Damage Modelling of Short Fiber Composites
225
Figure 7.10 Comparison of the experimental and predicted stress-strain behavior
of the GF-PP composite.
A significant difference between both materials, in addition to the
difference in matrix non-linearity, is the aspect ratios of the fibers. The GF-
PA material has a much lower mean of the fiber length distribution which
is about 250 compared to 950 for the GF-PP. As mentioned above, the
debonding model of Fitoussi et al. [180], used in the present work was
validated by the authors for sheet molding compounds (SMCs) with
random but continuous fibers. With the high aspect ratio of the GF-PP
composite, its morphology can be comparable to the SMCs. However, the
good predictions of the model for the GF-PA validate the use of model for
the class of composites with low aspect ratios.
For both materials, fiber breakage was not observed throughout the
simulations. This is in agreement with the experimental observations of the
fracture surface of the specimens where fiber-matrix debonding and pull-
out where the only damage mechanisms found in the specimens.
7.5.2 Own experiments – steel fiber reinforced composites
Validation of the developed models was similarly performed on the steel
fiber composites considered in the present work. As explained above in
section 7.3, the solution follows a slightly different route for the wavy fiber
composite, e.g. here the steel fibers composites and the typical straight
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226
fiber composites. For the wavy fiber composites, after generation of the
RVE using the geometrical model (discussed in Chapter 3), the P-I model,
described in the previous chapter (Chapter 6) is applied to different
segments of one wavy fiber. Each wavy segment is replaced with an
equivalent ellipsoidal inclusion. The length of this equivalent ellipsoid is
dependent on the radius of the curvature of the original segment, reflecting
the nature of its reinforcing efficiency. The model was validated for the
prediction of the overall elastic moduli previously in [294] for the
application of the model on textile composites and in this thesis for
discontinuous wavy fibers. But most importantly in this work, the model
was equally validated for the prediction of the local stress states in the
equivalent inclusions compared to the original segments, which gives the
basis for damage modelling. As discussed above in the framework of
mean-field homogenization, the damage models are significantly
dependent on the stresses in the inclusions.
An important aspect of the behavior of the steel fiber composites in this
PhD thesis is the weak interface of the fibers with the matrix. This behavior
was discussed in details in Chapter 4 and was observed by means of
detailed fractography and AE analysis. The behavior was a result of the
difficulties in processing the steel fibers which result in limitations in
compounding and hence weak interfaces of the composite.
It was discussed above that the assumption of the value of the critical
interface strength 𝜎𝑐 in the damage model, the same as the value of the
matrix yield strength is considered an upper bound reflecting optimized
interfaces, typically achieved in the commercially available composites.
Nevertheless this assumption was found to lead to over-prediction of the
stress-strain behavior of the steel fiber composites in the present work.
Figure 7.11 shows the simulated stress-strain behavior of the SF-PA 2VF%
as an example of the simulation of the steel fiber materials of the present
work. In the figure, the simulated stress-strain behavior with a critical
interface strength value 𝜎𝑐 = 55 𝑀𝑃𝑎 is shown. This value corresponds to
the average yield strength of PA 6 matrices, which was also used for
simulation of the GF-PA material above. The yield strength of the
particular PA 6 matrix (the Durethan B38) of the steel fiber composites
was found to have an even higher value of about 60 MPa as shown in
Chapter 4.
Non-linear Progressive Damage Modelling of Short Fiber Composites
227
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.
As shown in the figure, using the same value of the interface strength for
the steel fiber composite as that of the glass fiber composites, i.e. 𝜎𝑐 =55 𝑀𝑃𝑎 leads to an over-estimation of the stress-strain behavior. The value
of the interface strength which was found to give the best match with the
experimental results was around 35 𝑀𝑃𝑎. The lower value confirms the
unoptimized interfaces of the steel fiber composites. For the simulation the
value of 35 𝑀𝑃𝑎.
The resulting simulations of the SF-PA are shown in Figure 7.12 and
Figure 7.13 for the SF-PA 0.5VF% and the 2VF% respectively as examples
of the predictions of the models for the wavy steel fiber materials. The
simulations were stopped at 15% strain due to the difficulties of describing
the matrix stress-strain behavior using after up to the very high strain to
failures of the steel fiber composites.
Similar to the glass fiber materials, the overall approach was found to give
good match of the simulated stress-strain curves compared to the
experimental curves for the steel fiber composites. The good predictions
compared to the experimental stress-strain behavior validates both the
damage modelling approach as well the Poly-Inclusion models for dealing
with wavy fibers in Eshelby based models.
CHAPTER 7
228
Figure 7.12 Comparison of the experimental and predicted stress-strain behavior
of the SF-PA 0.5VF% composite.
Figure 7.13 Comparison of the experimental and predicted stress-strain behavior
of the SF-PA 2VF% composite.
Non-linear Progressive Damage Modelling of Short Fiber Composites
229
As can be seen from the figures, the plasticity of the matrix plays a
significant role in the non-linearity of the stress-strain behavior of the steel
fiber composites. The deviation from linearity due to consideration of only
the plasticity of the matrix was found to be much higher in the SF-PA
compared to the glass fiber composites due to the low concentration of the
fibers in the SF-PA composites as well as the much higher strains of the
stress-strain curves. This non-linearity due to plasticity was also found to
be higher for the lower volume fraction condition, i.e. the 0.5VF%
compared to the 2VF% samples. This is a direct result of the very low fiber
concentration in the 0.5VF% where the behavior of the composite is
significantly dependent on the matrix.
Nevertheless, despite the low concentration of the fibers in the steel fiber
samples, a high non-linearity due to damage was also found. This confirms
the assumption of high percentage of damage in the steel fiber composites
which was also observed through experimental characterization discussed
in Chapter 4.
Finally, the validation of the proposed model was not possible to achieve
on the higher volume fraction samples. Figure 7.14 shows a comparison of
the experimental and predictions of the longitudinal Young’s modulus of
the SF-PA materials with the different fiber volume fractions.
The figure shows that the current modelling approach provided very good
match of the stiffness of the steel fiber reinforced samples with the different
volume fractions up to the 2VF% volume fraction. A large difference can
be found between the predicted and experimental values for the higher
volume fraction samples.
The conclusions deduced from this comparison are two-fold. First, the
good match between the predicted and experimental stiffness shown in
Figure 7.14 and the full stress-strain behavior for the low volume fraction
as in Figure 7.12 and Figure 7.13 validates the overall modelling approach
for wavy fibers. This includes the P-I model for treatment of the wavy
fibers within the Mori-Tanaka model and the non-linear and damage
models discussed in this chapter.
For the higher volume fractions, e.g. for the plotted experimental and
predicted values of the stiffness of the 4VF% samples a large difference
can be found between the predicted and actual values. This is attributed to
the defects in the high volume fractions samples, discussed in details in
Chapter 4, which result from the difficulties of manufacturing the materials
with high concentrations of steel fibers.
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230
Figure 7.14 Comparison of the predicted and experimental Young’s modulus of
the SF-PA materials with the different fiber volume fraction.
Figure 7.14 gives an insight about the actual stiffness values expected with
the higher volume fraction samples in case of no defects. The expected
values were more than twice higher in case of the 4VF% samples and about
4 times higher in the case of the 5VF% (the model predictions of the 5VF%
are omitted from the figure for clarity).
The results in this figure indicate the potential of the steel fiber materials
as reinforcing materials where for a volume fraction as low as 4%, the
stiffness of the reinforced materials is expected to be 3 times that of the
neat PA 6 in case of no defects in the composite, even with the low
interface behavior of the composite discussed above. This in turn promotes
future research and development targeted towards the optimization of the
manufacturing and interface properties of the short steel fiber composites
to achieve full reinforcing potential.
7.5.3 Experiments of Jain – glass fiber reinforced composites
Likewise, the validation of the models was performed on the GF-PBT
material described above. Figure 7.15 to 7.17 show comparisons of the
predictions of the model with experimental stress-strain curves for the GF-
PBT materials with the different orientations 𝜙 = 0, 45, 90 respectively.
Non-linear Progressive Damage Modelling of Short Fiber Composites
231
Figure 7.15 Comparison of the experimental and predicted stress-strain behavior
of the GF-PBT 0 composite.
Figure 7.16 Comparison of the experimental and predicted stress-strain behavior
of the GF-PBT 45 composite.
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232
Figure 7.17 Comparison of the experimental and predicted stress-strain behavior
of the GF-PBT 90 composite.
As shown in Figure 7.15, the model gives good predictions of the stress-
strain behavior of the GF-PBT composite. It can be noted from the figure
that the deviation from linearity as a result of the plasticity of the matrix
was generally very limited for the GF-PBT composite. This is due to the
high linearity of the neat PBT matrix up to high strain (about 1.7%) which
can be seen in Figure 7.8.
As discussed above in section 7.4.3, the aspect ratio of the fibers in the GF-
PBT was about 200. Similar to the GF-PA material, the good predictions
of the model on GF-PBT gives further validation for the applicability of
the model on composites with short aspect ratios. The material also had a
higher volume fraction of the fibers than all the previous composites which
widens the scope of the validation.
Furthermore, an important aspect of the data of the GF-PBT of Jain [308]
is the different orientation angles of the tested coupons. As shown in
Figure 7.16 and Figure 7.17 the model give good predictions of the stress-
strain behavior of the coupons with 𝜙 = 45, 90 respectively. This results
in the validation of the proposed modelling approach for coupons with
distinctly different orientation tensors.
Non-linear Progressive Damage Modelling of Short Fiber Composites
233
It should be noted that, similar to the GF-PA and the GF-PP materials, no
fiber breakage was found during the simulation of the GF-PBT material
with all the different orientations.
7.6 Conclusions
A detailed methodology for modelling the quasi-static behavior of short
fiber composites was proposed in this chapter. The models were validated
for a wide range of materials. These include typical short straight fiber
composites with different volume fractions, lengths and orientations and
plastic behavior of embedding matrices. The model was also applied and
validated for wavy fiber reinforced composites such as the steel fiber
composites considered in this thesis.
The models also provided further insight on the damage mechanisms of
short fiber composites which supported the experimental observations. It
was confirmed for all materials that debonding was the main damage
mechanism and no fiber breakage was predicted for any of the materials
under investigation. This has also been shown by experimental
characterization in Chapter 4.
For steel fiber composites, as a specific structural material, the predictions
of the models led to important understanding of the potential of these
materials. It was shown that without the presence of defects, the predictions
show attractive potential of the steel fibers as reinforcing materials which
is possible at very low volume fractions of fibers. This then gives
perspective in future research focused on the optimization of the
manufacturing of these materials for semi-structural applications.
Two main aspects were also confirmed considering the modelling
approaches. First is the validity of the application of the debonding model,
which was initially developed and validated for random but continuous
sheet molding compounds, for discontinuous fiber composites. This is
despite the fact that damage in an inclusion is assessed based on the stress-
states and percentage of debonded areas along the equator of the inclusion.
The model hence assumes that the stress state along the length of the
inclusion is similar to that at the equator. This assumption is in agreement
with the overall mean-field concept which assumes mean stress and strain
states in constituents. Generally, this assumption can be considered a
simplification of the actual stress-states over the length of the inclusion
which can be different especially e.g. at the inclusion tips. Nevertheless,
although with stress concentrations debonding could initiate from the tips
CHAPTER 7
234
of an inclusion, the percentage of the debonded area as a result of this tip
debonding is negligible. Moreover, and most importantly, the minimum
aspect ratios in short fiber composites are around 20 which are considered
high. With these high aspect ratios the variations of the stress states along
the inclusion are expected to be very small. This is in agreement with the
statement of Meraghni et al. [178] who stated that inclusions with aspect
ratios of less than 2 (which is much lower than the typical aspect ratios of
short fiber composites) require considerations of the stress states along the
length of the inclusion.
The second main aspect is the validation of the overall modelling approach
for wavy fiber composites. This is a combination of the P-I model which
was discussed in the previous chapter for the treatment of wavy fibers in
the framework of Eshelby based models, which can only consider
ellipsoidal inclusions, and the damage models proposed in this chapter. In
the previous chapter, the validations of the P-I model were performed
solely in the linear elastic regime and with simplified RVEs. The good
match between predicted and experimental curves of the wavy fiber
composites (found in the conditions with no defects) shows the validity of
the P-I approach also in actual representative volume elements and in the
non-linear regime with presence of damage.
In a similar way, the results of predictions of stress-strain behavior of both
straight and wavy short fiber composites give further validation of the
geometrical models in such a way that the generated RVEs resulted in
accurate predictions of the overall stress-strain behavior. An advantage of
the present geometrical model compared to available commercial software
for RVE generation discussed in the literature review, is that the present
model is able to generate large RVEs which are needed for actual damage
modelling. This is in contrast to the commercial software which depend on
the representation of an RVE with a limited number of families of “grains”.
A more detailed insight on the limitation of consideration of small RVE
sizes will be discussed in Chapter 8 in relation to the overall micro-macro
scale, or component level simulation modelling.
Finally, the micro-mechanics models described and validated in this
chapter, as well as the insight gained from them, will be used to develop
the fatigue model in the next chapter (Chapter 8). A link will then be made
between the quasi-static and fatigue damage behavior of RFRCs.
235
Chapter 8: Fatigue Modelling of Short Fiber Composites
Fatigue Modelling of Short Fiber Composites
237
8.1 Introduction
For an overall component level (macro-level) simulation of a short fiber
composite part, each gaussian point in the FE model is considered an RVE
with a distinct micro-structure (local fiber VF, FLD, FOD), for which the
stiffness and fatigue behavior need to be estimated.
While a number of methods exist for the stiffness and overall elastic
problems of short fiber composites, among which is the M-T formulation
used in the present thesis, the fatigue behavior of these composites is much
less understood. To date, the predictive models for the fatigue behavior of
composites are mostly concerned with continuous fiber materials. Most of
these are phenomenological models which depend on empirical fitting and
require a large number of experimental tests for each considered material.
Only a few similar phenomenological models, could be found for the
fatigue of random short fiber composites e.g. in [21-23].
In this thesis, a novel approach is developed to predict the S-N curves of
short fiber composites. The objective is linking the fatigue behavior of the
short fiber composites to the behavior of the constituents and actual
damage mechanisms. The models are developed in the framework of the
computationally efficient mean-field homogenization theories as has been
seen throughout this thesis.
The proposed modelling approach is targeted towards the simplification of
the number of tests required for actual predictions of the fatigue of an SFRP
component. Moreover, the developed models are highly versatile. For this
reason, they can be used as design and optimization tools for assessment
of the influence of different material parameters and loading considerations
on the final behavior of an SFRP component, without the need for
exhaustive test based methods.
This chapter is devoted to the description and validation of the proposed
modelling approach for predicting the fatigue behavior of SFRPs. First, a
description of the objective and mathematical formulation of the fatigue
model is provided. Next, details will be given on the numerical algorithms
and the implementation of the developed model. The next sections will
then be devoted to the description of the test cases and the results of the
validations of the modelling approach. Finally a brief discussion will be
shown on the validation of the overall hybrid mutli-scale solution on an
actual short fiber composite using the solution discussed in section 1.3.
CHAPTER 8
238
8.2 Objectives and Formulation of the Fatigue Model
As mentioned above, the objective of the current fatigue modelling
approach is the prediction of the fatigue life of a short fiber composite from
the fatigue behavior at the constituents level and actual damage
mechanisms.
For short fiber composites, few models exist for prediction of the fatigue
life. Such models are approximate scaling methods targeted towards
solving the multi-scale component level problem described in the
introduction. These models rely on predicting the unknown fatigue
behavior of an RVE of a certain orientation tensor by scaling using suitable
proportionality criteria.
These include for example the approach discussed in the literature review
of scaling of the S-N curves of RVEs of different orientations to their
corresponding ultimate tensile strength (UTS). The approach hence require
direct measurement of the UTS of all composites in question. Moreover,
the method relies on the assumption of the direct proportionality between
the tensile and fatigue strength of a composite. Such assumption is a major
simplification and is not always found true due to the inherently different
failure behavior of tensile and fatigue specimens [41, 43, 206].
A more elaborate scaling method, was developed by Jain [308], in parallel
to the present work. The method is denoted the master S-N curve approach
and relies on scaling of the unknown S-N curve of a composite with a
certain orientation tensor to the known S-N curve of a reference composite
of the same material. Scaling is done of the basis of the difference in
damage between the composite in question and the reference composite. It
then requires testing of the fatigue behavior of the reference material. The
method relies on the assumption that the damage needed to cause failure
after a certain number of cycles in two composite with different FODs is
the same given that the constituents are the same. Damage is also
quantified as the loss of stiffness due to the first cycle loading and hence
doesn’t take into account the change of properties of the constituents
during the progression of the fatigue life. Albeit, approximate the model
was found to give good predictions of the S-N curves of the RVEs of
different orientation tensors compared to the experimental curves.
To date, models relating the fatigue behavior of random short fiber
composites to the fatigue of the constituents and actual micro-scale damage
phenomenon do not exist. The proposed modelling approach then
Fatigue Modelling of Short Fiber Composites
239
addresses this fundamental and scientifically challenging problem. These
constituents include the fiber, matrix and the interface. This can be
achieved with assignment of the suitable micro-scale failure criteria.
Figure 8.1 summarizes the objective of the proposed model.
Figure 8.1 Schematic diagram representing the objective of the fatigue model
developed in the present study.
The fundamental question of the prediction of the fatigue behavior of a
composite based on the fatigue of the constituents was considered in very
few attempts in published literature for continuous fiber composites. An
example is the model of Reifsnider and Gao [19] (also reported in [20])
for the fatigue life of a unidirectional composite laminates. The model is
based on the early theories of Hashin and Rotem [15] who adopted a
macromechanics based formulation to the fatigue of these composites.
According to Hashin and Rotem, failure of the composite is described by
macroscopic failure criteria in terms of the stress to which the composite
is subjected to. Reifsnider and Gao then proposed a modified approach by
considering failure criterion at the microscopic level relating the final
fatigue failure of the composite to that of the constituents.
In the model of Reifsnider and Gao [19] (and Subramanian et al. [20]) the
fatigue failure of a unidirectional laminated composite is assumed to occur
by two distinct mechanisms: fiber failure or matrix cracking. In this
respect, the authors suggested two failure criteria describing these two
mechanisms comparing the actual stress states in the fibers and matrix
with their fatigue strength at a certain loading cycle. Similar to the current
Fatigue model
Progressive
damage
Assignment of
suitable fatigue
failure criterion
S
N
Short fiber composite
timest
ress
max
min
S
N
matrix
interface
S
N
Fiber
CHAPTER 8
240
modelling context, the authors applied their model using the Mori-Tanaka
formulation.
The model of Reifsnider and Gao cannot be directly applied to short fiber
composites. This is due to the inherently different failure mechanisms of
the short random and the continuous fiber reinforced composites. In short
fiber composites, fibers exhibit different lengths and orientations. For this
reason, failure of a fiber results in degradation, but not overall failure of a
composite. Matrix failure in laminated composites, as described by the
authors, denote cracking of the matrix in the direction of fibers in matrix
dominated failure cases, e.g. in composites subjected to transverse loading
or off-axis loading with large deviations of loading angles from the fiber
direction. Also, the behavior of the interface was not taken into account.
The authors developed the models assuming perfect interfaces.
In the present work, different failure criteria are proposed, reflecting the
damage mechanisms observed by own experiments, as well as the analysis
of literature reported results of short fiber composites. The failure criteria
can be classified in two categories: primary and secondary criteria. A
description of the formulation of the criteria will be given below.
As mentioned before, failure of a fiber occurs when the average axial stress
state in the fiber reaches its ultimate strength. Although the later is a
constant value for quasi-static loading, the strength of the fiber decreases
progressively during fatigue loading which can be described using a
fatigue S-N curve of the individual fiber. Failure of a fiber in the short fiber
composite results in degradation of the stiffness and load carrying
capability of the short fiber composite but doesn’t lead to final failure,
hence can be considered a secondary failure criteria.
A novel aspect of this work is the consideration of the fatigue behavior of
the fiber-matrix interface. It has been shown by experimental observations
in the present work, that the interface depict progressive degradation
during the course of fatigue loading. Such behavior was also confirmed by
published data, experimentally e.g. in [40, 41, 288, 313, 314] and by direct
modelling of interface behavior under fatigue loading of single fiber
composites e.g. [315-317].
The next question is how to incorporate the fatigue of the fiber-matrix
interface within the proposed Mori-Tanaka based modelling approach. In
this work, the fatigue of the interface is considered using a modified
Coulomb criterion for fatigue loading. The constant criticial interface
strength 𝜎𝑐 at the right side of the quasi-static Coulomb criterion in
Equation (7.9) is replaced by the fatigue strength of the interface as a
Fatigue Modelling of Short Fiber Composites
241
function of the number of cycles of the fatigue loading. In this way, the
decrease of the fatigue resistance during cyclic loading is reflected in the
model. The same formulations described in section 7.2.2. are used to model
fiber-matrix debonding for each cycle (or block of cycles) during the
fatigue simulation. An assesment of the debonding for each point along the
surface of the inclusion will be done on the basis of the modified Coulomb
criterion for cyclic loading. The partially debonded inclusion, at each
fatigue cycle, is then replaced by an equivalent inclusion with degraded
properties according to the same degradation scheme described in section
7.2.2 (Equations 7.10).
Finally, the fatigue of a short fiber composite is assumed to occur when the
matrix fails. In that way, the fatigue failure criterion of the matrix can be
considered as the primary criterion governing the final failure of the
composite. In the context of the Mori-Tanaka model, this is obtained when
the average stress in the matrix reaches its fatigue strength at a certain
cycle. Since the stress state in the matrix in random short fiber composite
is typically triaxial, the criterion is then based on the equivalent von Mises
stresses in the matrix.
Using the above concepts, the proposed fatigue failure criteria for short
fiber composites are formulated as follows:
Fiber failure
𝜎33𝛼 ≥ 𝑋𝑓(𝜎, 𝑁, 𝑅) (8.1)
Interface failure
𝜎𝑁 + 𝛽𝜏 ≥ 𝑋𝐼(𝜎, 𝑁, 𝑅) (8.2)
Matrix failure
𝝈𝒎∗ ≥ 𝑿𝒎(𝜎, 𝑁, 𝑅) (8.3)
where 𝑋𝑓, 𝑋𝐼 and 𝑋𝑚 are called the fatigue failure functions of the
composite. They denote fatigue strength of the fibers, interface, and matrix
respectively under axial loading. These fatigue failure functions are
essentially the S-N curves of the constituents under axial fatigue loading
(Figure 8.2). They then depend on the applied stress 𝜎, the stress ratio 𝑅 ,
and the number of cycles 𝑁. In order to predict S-N curve of the composite
for a certain 𝑅 ratio, input S-N curves of the constituents of the same 𝑅
ratio should be used in the simulation.
CHAPTER 8
242
Figure 8.2 Schematic representation of the fatigue failure functions 𝑋𝑓 , 𝑋𝑖 and
𝑋𝑚 at a current load cycle 𝑁𝑐 during the fatigue simulation.
A brief description of the model process is given as follows: for a certain
applied load level, defined as the maximum stress of a fatigue cycle,
homogenization is performed. The failure criteria in Equations (8.1) to
(8.3) are applied in each block of loading cycles in the following way. At
a certain current cycle 𝑁𝑐, the equivalent stress in the matrix computed, if
the matrix failure criterion in Equation (8.3) is satisfied, failure of the
composite is assumed to occur at this cycle.
If not, the stress states for each inclusion are evaluated, if the fiber breakage
criterion (Equation 8.1) is reached for an inclusion, it is replaced with a
void (equivalent inclusion with null stiffness). Similarly, for each
inclusion, the interface fatigue failure criterion is applied at each point
along the surface of the interface. The damage model is then used to
degrade the partially debonded inclusions.
S
S
S
N
N
N
S
𝑋𝑓
𝑋𝑖
𝑋𝑚
𝑁𝑐
𝑁𝑐
𝑁𝑐
𝜎33𝛼
𝜎𝑚∗
𝜎𝑁 + 𝛽𝜏
Fatigue Modelling of Short Fiber Composites
243
8.3 Implementation of the Fatigue Model
Similar to the quasi-static damage models described in the previous
chapter, the proposed fatigue model of this PhD thesis is implemented in
C++ software toolkit.
Figure 8.3 shows a simplified flowchart describing the implementation of
a single load cycle 𝑁 of the fatigue model. A brief description of the
implementation is given in the following pragraphs.
The first step of the fatigue simulation for a certain applied stress, is
running a quasi-static virtual test (following the procedure in section 7.3)
up to the maximum applied stress of the fatigue loading. This occurs at the
first simulated load cycle in the fatigue solution. The exact solution route
then depends on the keyword: plasticity, as shown in Figure 8.3. If
plasticity is chosen, then for the initial virtual quasi-static test in the first
load cycle as well as for the subsequent modelled cycles, matrix plasticity
is taken into account. This was done to reflect the degradation of the matrix
during the course of the fatigue loading. It should be noted that in the
fatigue simulation, the matrix non-linearity is taken into account by means
of the same secant plasticity model described in the previous chapter. The
application of this model in fatigue loading can be a simplification as it
does not take into account the actual hysteresis fatigue degradation of the
thermoplastic matrix in fatigue.
The second step for a given load cycle is to find the fatigue failure functions
𝑋𝑚, 𝑋𝑓, 𝑋𝐼 corresponding to this cycle from the input experimental S-N
curves of the constituents. A homogenization step is then performed to
calculate the effective composite stiffness 𝑪𝑒𝑓𝑓. In fatigue loading, the
applied stress is constant, the strain states in the composite 𝜺𝑐 however,
need to be evaluated at each cycle, corresponding to the current 𝑪𝑒𝑓𝑓. The
matrix strains (including the plastic strain) and stress states 𝜺𝑁𝑚, 𝜺𝑁
𝑚,𝑝, 𝝈𝑁
𝑚 can
then evaluated from the composite strains. If plasticity is considered, the
new secant stiffness of the matrix need to be calculated using the iterative
scheme (the plasticity sub-model), mentioned in the previous chapter, until
convergence.
CHAPTER 8
244
Figure 8.3 Flowchart of a single load cycle 𝑁 of the developed fatigue model.
START
Cycle = 1?
Find
Fatigue failure functions 𝑋𝑚,
𝑋𝑓, 𝑋𝐼at cycle (N)
Plastic?
Run Quasi-static
L-Test
Run Quasi-static NL-Test
Calculate
Composite Stiffness 𝑪𝑁𝑒𝑓𝑓
Update
Matrix stress/strain
𝜺𝑁𝑚, 𝜺𝑁
𝑚,𝑝, 𝝈𝑁
𝑚
Calculate
Composite Strain 𝜺𝑁𝑐
Solve
Matrix plasticity sub-model
Plastic? Converged?
𝜎𝑁𝑚,∗ ≥ 𝑋𝑚
Loop over inclusions (i)
Update inclusions
Strains /stresses 𝜺𝑁𝛼 , 𝝈𝑁
𝛼
Interface
𝑅 ≥ 𝑋𝐼
Evaluate
𝑑𝑁𝛼 , 𝛾𝑁,
𝛼 𝛿𝑁𝛼
Degrade inclusion
Break inclusion
𝑪𝛼 = 0
END
Failure Cycle =
Current Cycle
Inclusion 𝜎33
𝛼 ≥ 𝑋𝑓
Yes
Yes
Yes
Yes
Yes Yes
No
No
No
No
No
Loop over interface points
Fatigue Modelling of Short Fiber Composites
245
Once the matrix stresses and strains are updated after convergence of the
plasticity sub-model, the primary failure criterion of matrix failure in
Equation (8.3) is evaluated. If the criterion is satisfied, failure of the
composite is assumed at this cycle.
If no failure of the matrix (and hence composite) occurred at this cycle, the
stress states 𝜎𝛼 in each inclusion 𝛼 are evaluated and the same damage
models (debonding and fiber breakage) are applied, only by using
secondary fatigue failure criteria in Equations (8.1) and (8.2) for fiber and
interface failure respectively instead of the static criteria of the previous
chapter. An assessment of the inclusion damage parameters 𝑑𝑁𝛼 , 𝛾𝑁
𝛼 , 𝛿𝑁𝛼 at
each cycle is then made for each inclusion based on the percentage of the
debonded interface, and the percentage surface subjected to tension or
compression stresses. The partially debonded inclusion is then replaced
with a perfectly bonded one with degraded stiffness tensor using the same
scheme presented in Equations 7.10 a – g. A new composite is then formed
with the equivalent inclusions and a new homogenization and assessment
of failure criteria is performed for the next block of cycles.
In the implementation of the present work, the fatigue simulation is
performed over fixed loading steps. Small increments are chosen to ensure
accuracy of the prediction of the cycles to failure. The steps are defined as
follows: for the first 5000 cycles, the failure criteria are evaluated each 10
cycles, afterwards, the criteria are evaluated each 50 cycles up to 50000
cycles and each 100 cycles up to 1 million cycles. The smaller increments
at the beginning of the simulations (up to 5000 cycles) are considered
because of the higher damage rates in the first loading cycles. The
simulations are stopped at 1 million cycles. If no failure is predicted at this
limit, run-out predictions are assumed.
The procedure above is performed for one stress level (one point on the S-
N curve), to obtain the full S-N curve, the simulations are repeated for the
desired number of points over the S-N curve by variations of the maximum
applied stresses and predicting their corresponding failure cycles.
8.4 Description of Validation Test Cases and Model Input
8.4.1 Own Experiments
The validation of the developed fatigue model was performed on the glass
fiber reinforced materials of the present work, namely the GF-PA and the
GF-PP. The properties of the constituents, and micro-structure of the
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composites were described in details in section 7.4.1 in the previous
chapter.
In order to predict the fatigue behavior using the proposed model, three
additional inputs need to be obtained namely: the S-N curves of the fibers,
matrices and interface.
The fatigue behavior of single glass fibers was investigated in only a few
previous studies. A detailed study was performed by Mandell et al. [318]
investigating the fatigue behavior of single glass fibers and impregnated
and unimpregnated glass fiber bundles in tension-tension loading [318].
Figure 8.4 shows the S-N curve of the single glass fibers, as reported by
Mandell et al. This S-N curve was used as input for all fatigue simulations
of the present work.
Figure 8.4 S-N curve of single glass fibers used as input for the fatigue model
[318].
It can be seen from the figure that the single glass fibers are relatively
fatigue resistant where the strength drops only by about 20% at 1 million
cycles. Most importantly, the range of strengths of the fibers from the first
cycle (about 2200 MPa) to 1 million cycles (about 1750 MPa) are much
higher than the typical stresses achieved in short fibers composites. In this
respect, the fatigue of the fibers generally play a minor role in the overall
predictions of the fatigue behavior of the short fiber systems.
An important input to the models however, is the S-N curves of the pure
matrix. This is due to the above mentioned primary failure criterion where
Fatigue Modelling of Short Fiber Composites
247
the failure of the short fiber composite is assumed to occur when the matrix
fails.
Nevertheless, unlike the use of the stress-strain curves of the specific
matrix grades as shown in the previous chapter, the S-N curve of an exact
matrix is difficult to obtain. Only a very limited number of published data
can be found for the S-N curves of the neat thermoplastics. This can be due
to difficulties in testing the pure polymers and especially in relation to very
long fatigue tests, as a result of the low testing frequencies. The low
frequencies required to avoid the rapid excessive self-heating of these
materials. Therefore, the S-N curves of a polymer of the same material, but
not necessarily the same grade was collected from different literature for
the validation of the models.
Figure 8.5 and Figure 8.6 show the collected S-N curves of the polyamide
6 matrix [58] and the polypropylene matrix [319]. The S-N curves are used
as input for validation of the models on the GF-PA and GF-PP materials
respectively.
Figure 8.5 S-N curve of the PA 6 matrix used as input for the fatigue model
[58].
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Figure 8.6 S-N curve of the PP matrix used as input for the fatigue model [319].
A more challenging model input is the fatigue S-N curves of the interface.
In the last chapter it was discussed that the values of the interface strengths
of the different materials needed for the assessment of the Coulomb criteria
are difficult to obtain experimentally. The difficulties in experimentally
characterizing these values in fatigue loading are even higher. In the quasi-
static models discussed in the previous chapter, it was discussed that the
values of the interface strengths can be approximated to be the same as that
of the yield strength of the matrix. Such approximation was considered an
upper bound which is expected to be a suitable estimate for the optimized
commercial grades composites such as those used in the present study.
The assumption of the interface strength the same as the matrix strength
means that both have the same strength coefficient 𝜎𝑓 in Equation (5.2),
i.e. the power function fitting relationship of the S-N curve.
In a similar way, the S-N curves of the interfaces can be assumed to be the
same as the S-N curves of the matrix. This basically means assuming the
same slope of the S-N curves 𝑏 (strength exponent in Equation 5.2) of the
interface as that of the matrix. This assumption is investigated in the
present work for all materials under investigation. A parametric study is
performed for each material for investigating the validity of this
assumption and the effect of variation of the slopes of the S-N curves of
the interfaces (i.e. the interface strength exponent).
Fatigue Modelling of Short Fiber Composites
249
8.4.2 Experiments of Jain
As mentioned in the previous chapter, Jain [308] performed tensile and
fatigue testing on the GF-PBT material with different orientation of the test
coupons. Figure 8.7 shows the experimental S-N curves of the GF-PBT
composite as reported by the author. As expected, the fatigue strengths of
the coupons decreased with increased misalignment of the tested coupons.
Similar to the GF-PP and the GF-PA materials, the properties of the
constituents and the composite micro-structure were described in the
previous chapter, in section 7.4.3.
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].
Figure 8.8 shows the collected S-N curve of the PBT matrix from [320].
Although the PBT matrix exhibited similar strength as that of the PA, as
can be seen its strength exponent was much higher indicating a decreased
fatigue resistance compared to the PA (also the PP) matrices.
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Figure 8.8 S-N curve of the PBT matrix used as input for the fatigue model
[320].
8.5 Results and Discussion
8.5.1 Own-experiments
The fatigue S-N behavior of the glass fiber reinforced composites
considered by own experiments, i.e. the GF-PA and the GF-PP materials
of the present work, was simulated using the above mentioned procedure.
For all materials (own experiments and data of Jain) used in the validations,
a parametric study is performed showing the effect of the different slopes
𝑏 of the S-N curve of the interfaces. The purpose is to show the significance
of the effect of the fatigue of interface on the predictions of the composite
fatigue behavior. The study also gives an insight on the possible
estimations for the S-N curves of the interfaces which as discussed above
are very difficult to characterize experimentally. For all the next
simulations, red square symbols indicate the predictions of the model with
taking the slope of the S-N curve of the interface the same as that of the
matrix. Blue symbols indicate the predictions of the models with the
assumption of constant interface strength. Green triangles indicate the
model predictions assuming the slope of the S-N curve of the interface
twice that the matrix (twice faster degradation of the interface).
Fatigue Modelling of Short Fiber Composites
251
Figure 8.9 shows the simulated S-N curves of the GF-PA material
compared to the experimental curve (discussed in Chapter 5).
As shown in the figure, the proposed model (with the assumption of the
slope of the interface similar to that of the matrix 𝑏 = 0.0508) leads to
very good agreement of the predicted S-N curves of the composite. All
predicted points lied between the 90% experimental confidence intervals.
The parametric study shown in the figure, shows that the assumption of the
same slope of the interface strength as that of the matrix is a reasonable
assumption for this material. The study also indicates the effects of the
fatigue of the interface.
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.
Figure 8.9 shows that for the GF-PA material, if no fatigue of the interface
is considered, i.e. the fatigue strength is constant during fatigue loading
(𝑏 = 0). The model predicts run-outs even at the highest applied stresses
in the S-N curve. By taking the slope of the interface to be twice as high as
that of the matrix (𝑏 = 0.116), i.e. the interface is assumed to degrade with
twice higher rates compared to the matrix, the model predicts very fast
failure of the composite, i.e. a high underestimation of the fatigue life for
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the different applied stresses, compared to the experimental curves.
Figure 8.10 illustrates the considered S-N curves of the interface, in the
parametric study, with the different 𝑏 values.
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.
The parameteric study confirms the discussed phenomenon of the fatigue
failure of the interface.The study also clearly shows the significance of the
effects of the interface degradation on the overall fatigue behavior of the
composite where interfaces with high fatigue degradation rates can lead to
a high decrease in the composite fatigue strengths.
Although performed in a simplified manner (with secant approach), the
incorporation of the plasticity of the matrix during the fatigue simulation
contributed to the accuracy of the predictions of the S-N curves of the
composite, this was especially significant in the low cycle regime.
Figure 8.11 shows the simulated S-N curves of the GF-PP material
compared to the experimental curve. In a similar way, the parametric study
is performed with the different slopes of the interface S-N curves
(illustrated in Figure 8.12).Likewise, the proposed model with the
assumption of the same slope of the S-N curves of the matrix and interface
leads to good agreement to the experimental curve. All predicted points
lied between the 90% confidence intervals. Although the assumption of
constant interface fatigue strength did not lead to predictions of run-out
already at the highest load levels as the GF-PA, this assumption clearly
leads to overestimation of the fatigue behavior of the composite. The
Fatigue Modelling of Short Fiber Composites
253
validations of the proposed model on the two materials then further
confirms that the fatigue of interface needs to be taken into account in the
fatigue models
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.
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.
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8.5.2 Experiments of Jain
The fatigue simulations were also performed on the GF-PBT composites
and validated with the experimental data of Jain [308]. The validation of
the model on the 0 degree coupons is shown in Figure 8.13.
As shown in the figure, the model successfully predicted the S-N curve of
the composite and all predicted points lied in the range of 90% confidence
intervals. The same above conclusions concerning the fatigue of the
interface can also be observed for the GF-PBT.
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.
Fatigue Modelling of Short Fiber Composites
255
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.
An important aspect of the experiments of Jain is the validation of the
developed models for the quasi-static behavior (as shown in Chapter 7) and
the developed fatigue models described in this chapter, of coupons with
different orientations.
Figure 8.15 and Figure 8.16 show the validation of the proposed fatigue
modelling approach on the GF-PBT with orientation angles 𝜙 = 45, 90
respectively.
As shown in the figures, the model predictions correspond well to the
experimental results. The assumption of the same strength of the interface
as that of the matrix leads to good predictions also for the coupons with the
different orientations.
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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.
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.
Fatigue Modelling of Short Fiber Composites
257
To summarize, based upon the performed simulations, the developed
fatigue model is validated for a range of materials with different variations
of morphology and different plastic and S-N behavior of matrices. The
assumption of the fatigue strength and slope of the S-N of the interface as
that of the matrix was found to be suitable for all composites investigated.
As mentioned before, this assumption is considered an upper bound
corresponding to optimized materials.
It can finally be noted that the above validations focused on the short
straight glass fiber composites. As has been shown in detail in Chapter 5,
the steel fiber composites of the present study showed the unusual behavior
of no failure up to 1 million cycles, for all considered stress levels. In this
respect, no S-N curve of the materials were obtained. Fatigue simulation
trials were performed for the steel fiber composites. The model led to
prediction of failure of the short steel fiber composites in fatigue, which is
not in agreement with the experimental results. The main reason for that is
the stiffening behavior of the steel fiber composites which was discussed
in Chapter 5. Such behavior is not depicted by the analytical model.
8.6 Summary of the Overall Micro-Scale Solution
A final summary of the micro-scale modelling approach and the relation
between the different models described throughout this PhD thesis is given
in this section.
Figure 8.17 shows a schematic representation of the developed models.
Each of the models can be used independently or as a part of a desired
simulation.
The solution starts with the geometrical model described in Chapter 3. The
focus of the model development was the accurate simulations of the local
micro-structure of the complex short wavy fibers. As shown in the
literature review, previous models for the generation of wavy fiber
composites are very limited. Available commercial software also do not
take into account fiber waviness. This has led to the motivation of the
development of the algorithm for generation of such complex micro-
structures as those of the short steel fiber composites of this thesis. The
versatility of the model allowed using it as well for generation of the
straight glass fiber composites in this work.
The Mori-Tanaka formulation was chosen as the suitable homogenization
model in the present work. Given that it is based on the Eshelby solution
for straight ellipsoidal inclusions, an extension of the model to incorporate
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wavy fiber geometries was necessary. Therefore, the P-I model, described
in Chapter 6, was adopted and validated for the predictions of the local
stress fields. Hence the present solutions exhibit two alternative routes
based on the type of material. For straight fiber composites, are directly
considered in the M-T model in which case a fiber and an inclusion are
interchangeable. For wavy fiber composites, the P-I model need to be
applied. Each curved segment in the wavy fiber is replaced by an
equivalent inclusion as shown in Chapter 6. The M-T model can be then
be applied on the new RVE of equivalent straight inclusions.
Finally, the developed quasi-static damage models discussed in Chapter 7
and fatigue models discussed in the present chapter can be applied on the
resulting RVEs, regardless of whether the inclusions in the RVE represent
straight fibers or are equivalent inclusions resulting from the discretization
of wavy fibers into a number of equivalent ellipsoidal inclusions.
259
Figure 8.17 Schematic representation of the micro-scale modelling methodology developed in the present thesis.
Geometrical model
Modeling of RVEs of
wavy fiber composites
Poly-Inclusion model
Replacing wavy fiber
segments with equivalent
straight inclusions
Mori-Tanaka model
Chosen homogenization
model
Damage model
Matrix plasticity
debonding
fiber breakage
Fatigue model
Progressive damage
Fatigue failure criteria
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8.7 Component Level Simulation
In the previous section, final summary of the micro-mechanics based
models developed in this PhD thesis was given. It was previously
mentioned in the introduction of this work that the models were developed
in the context of an overall multi-scale approach for the overall prediction
of the fatigue life of SFRP components. The rest of this chapter is devoted
to the discussion of the steps of the multi-scale approach and the attempts
of the validation of the methods for prediction of the fatigue S-N curves of
actual components.
A brief discussion of the key results and aspects of the performed
component level validation is given to provide insight on the current
position of the fatigue solution. An emphasis is given on the position of the
current work in the overall multi-scale solution. Details will also will be
given on the current limitations of the process, and outline for future
improvements. More detailed descriptions of the simulations can be found
in the thesis of Jain [308].
8.7.1 Current framework of the component level simulation
Figure 8.18 shows a schematic representation of the framework of the
fatigue simulation of the SFRPs as currently applied. A brief description
of the process is given below.
Figure 8.18 Flowchart describing the current component level solution for the
fatigue simulation of SFRPs.
Converse
Fatigue Micro-
Analyzer
Scaling
Manufacturing
simulation
FE solver
LMS Durability
Virtual.Lab
Fatigue Modelling of Short Fiber Composites
261
The first step of the overall simulation process is to perform a
manufacturing simulation to predict the FOD at each point of the
component.The information of the predicted local orientation tensors in the
component are received by a “multi-scale” platform. The main role of the
multi-scale platform is mapping of the local orientation tensor to FE
meshes which can then be transferred to an FE solver. As shown in the
figure, the Converse software is the used multi-scale platform software.
In the current solution, the Converse software is also used for generation
of RVEs representing the different local FODs over the component
model.The software also performs a homogenization step to obtain the
local stiffnesses, i.e. the stiffness of each RVE in the model. The generation
of the geometries and homogenization of the RVEs in the Converse
software are performed on the basis of the pseudo-grain method previously
discussed in the literature review. In such a case, an RVE consists of only
23 inclusions whose actual orientations are represented by weights of pre-
defined orientation vectors. It has been discussed in the literature review
that such small RVE size can be sufficient for prediction of the average
effective properties, however this size of RVE can be too small for
modelling of actual stress states and damage. This method however was
chosen for reasons of computational efficiency.
Next, a link exists between the Converse software and a software depicting
scaling algorithms for prediction of the local S-N curves. The scaling
algorithms were described above. The method rely on predicting the local
S-N curves (with different orientation tensors) by approximate scaling
(based on damage criteria) from a known reference S-N of the same
material and a known orientation.
The model developed in the present thesis (denoted the Fatigue Micro-
Analyzer), in the solution as applied today, is used for accurate prediction
of the “reference” S-N curve. This eliminates the steps of experimental
testing for obtaining reference S-N curves and hence the overall fatigue
simulation can be considered test-free (assuming the input for the present
models and the scaling algorithm are available in the databases).
In addition to the generation of the reference S-N curve, the model is
intended to be used as a fundamental design and optimization tool for
assessment of the effect of different constituent behavior (including the
interface), composite micro-structures and loading cases.
The reference S-N curve is usually the S-N curve of 0 degrees RVEs of the
same material, i.e. RVEs with fibers oriented in the main loading direction
(although it has been shown in the validations of the scaling approach in
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[308] that RVEs of any arbitrary orientation tensor can be used as
reference).
Nevertheless, it has been shown in section 8.5, that the developed fatigue
model is able to predict the S-N curves of RVEs of any orientation and not
necessarily restricted to 0 degrees materials. Hence the model can be used
for prediction of the local S-N curves in the component model independent
of the scaling algorithm. To date, this cannot be achieved due to the current
computational capabilities.
It is expected that the computational times for the developed models are
higher than that of the scaling algorithms because of the nature of the
solutions where the stress-states and failure criteria are evaluated on cycle
basis. In the scaling approach only the first cycle of the fatigue loading is
modelled and scaling is based on the assessment of the one cycle damage.
It should be still noted that the calculations of the present work are
performed in the framework of the fast Mori-Tanaka methods. The
calculations are then still much less computationally expensive than e.g
FE calculations. The solution times for one RVE, e.g. for generating the
first S-N curve are not significant. However, for an actual component with
thousands of RVEs, the computational cost can be considerable.
Once the reference S-N curves are generated and the scaling algorithm
predicts the local S-N curves of the different RVEs, the data is transferred
to the fatigue solver. As mentioned in the introduction, in the framework
of the current work, the LMS Virtual.Lab software is the employed fatigue
solver. The stress-states in the elements (homogenized RVEs) are
estimated from an FE solver (e.g Abaqus or SAMCEF). Based on these
stress states and the local S-N curves, a fatigue simulation is performed in
Virtual.Lab and the final failure of the component is predicted.
In the next sections, a brief discussion of validation attempts of the above
mentioned approach is given.
Fatigue Modelling of Short Fiber Composites
263
8.7.2 Description of the validation test case
The validation of overall multi-scale approach was performed on the
component shown in Figure 8.19.
Figure 8.19 Illustration of the considered industrial component. The component
is denote “Pinocchio”.
The component is a demonstrator for the design freedom of the injection
molding process and hence has no specific application. Nevertheless, the
geometry can represent a typical housing shell found in consumer
electronic devices or other products with housing bodies. The part is
denoted “Pinocchio” (for its resemblance to the cartoon character). The
Pinocchio component is produced with the GF-PBT material of Jain [308]
which was used for validation of the present models.
8.7.3 Experimental tests
Experimental characterization of the quasi-static and fatigue behavior of
the Pinocchio part were performed for the validation of the multi-scale
simulations. Three-point bending tests were performed.
The quasi-static three-point bending experiments were performed by the
author of this thesis. A special rig was manufactured for clamping of the
component for both the quasi-static and fatigue tests. The quasi-static tests
were performed on an Instron 4467 tensile machine. A 5 KN load cell was
used. Test speed was 1 mm/min. Full field image analysis was performed
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during the quasi-static tests. Image registration was done using the Limess
– Vic 2D system. A 12 bit 1392 x 1040 pix camera was used. Digital image
correlation was done using the software Vic 2D – 2009. Fatigue tests were performed by A. Jain. The tests were performed on a
horizontal Schenck machine equipped with a 10 KN load cell. The applied
loads had a stress ratio 𝑅 = 0.1. The testing frequence was 6 Hz.
8.7.4 Description of the simulations
The local FODs of the Pinocchio component were obtained by a
manufacturing simulation using the Moldex3D software. The chosen FE
solver was LMS Samtech SAMCEF. The fatigue simulations are
performed using LMS Virtual.Lab software as mentioned above. The
simulations were performed by the Siemens LMS durability team
(Kaiserslautern, Germany). Figure 8.20 shows the boundary conditions
applied in the simulations.
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.
(a)
(b)
Fatigue Modelling of Short Fiber Composites
265
It should be noted that in the current implementation of the solution, the
performed simulations, including the scaling approach, and FE and fatigue
simulations assume linear elastic constituents. Non-linearities of the matrix
are not taken into account.
8.7.5 Results and discussion
Figure 8.21 shows the obtained quasi-static load displacement curves from
the three-point bending tests performed on the Pinocchio component. As
shown in the figure, the load displacement curves exhibit clear and
significant non-linearities. This leads to the conclusion that the performed
linear simulations will result in inaccuracies of the overall predictions.
Figure 8.21 Quasi-stating 3 point bending load displacement curves of the
performed tests on the Pinocchio component.
Figure 8.22 shows the stress plots predicted by the FE solver for the quasi-
static loading of the Pinocchio component. The FE model was able to
successfully predict the critical area, which are highlighted in the figure.
0
0.5
1
1.5
2
2.5
3
3.5
0 1 2 3 4
Load
, KN
Displacement, mm
Component 1
Component 2
Component 3
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Figure 8.22 Stress fields in the Pinocchio component as predicted by the FE
model.
In order to get a quantitative idea of the predictions of the model compared
to the actual behavior observed in experiments, strain contours in the FE
models are compared with the results of the strain mapping using the
performed DIC. The strains were calculated on three lines on the surface
of the specimen as shown in Figure 8.23.
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.
The plots of the 휀𝑦𝑦 strain contours along the 3 lines obtained by the DIC
and the FE simulations are shown in Figure 8.24. As shown in the Figure
the experimental and simulated trends showed reasonable match, however
the actual values were highly overestimated in the FE model. A detailed
quantification of the errors can be found in [308].
Line 1
Line 2
Line 3
Fatigue Modelling of Short Fiber Composites
267
Finally, the S-N curves predicted by the overall multi-scale solution are
shown in Figure 8.25. In this simulation, the micro-mechanics based
modelling approach developed in the present thesis are used to generate
the reference S-N curve.
As shown in the figure, the simulations led to reasonable prediction of the
fatigue life of the component in the high cycle fatigue regime. In the low
cycle fatigue (high applied stresses), more significant deviations of the
predicted and experimental values can be observed.
The deviations were attributed to two main sources of inaccuracies. The
first is the low RVE sizes considered by the converse software. The low
RVE sizes lead to high overestimations of the stress-states in the RVEs.
This is especially the case since the scaling approach is based on the
damage in the RVE as the scaling parameter. The second main source of
errors is the assumption of the linearity of the matrix. As mentioned above
matrix plasticity is not taken into account in the simulations, starting from
the scaling algorithm.
Based on the above discussion, it can be concluded that the models
developed in the present PhD thesis can contribute to an improvement of
the predictions of the overall solutions as follows:
The geometrical model developed in the present work can be used for
generation of the RVEs. Using this model large RVEs (the size of RVE is
a parameter controlled by the user) can be generated. This can reduce the
errors induced by the small RVE sizes of the Converse software.
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Figure 8.24 Comparison of the DIC and FE extracted 휀𝑦𝑦 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.
(a)
(b)
(c)
Fatigue Modelling of Short Fiber Composites
269
Figure 8.25 Comparison of the experimental and predicted S-N curve of the
Pinocchio component.
In future implementations, the local S-N curves of the different RVEs can
be directly obtained using the model developed in this work. The model is
able to take into account the plasticity of the matrix and the change of
matrix stiffness (secant) during the fatigue simulation, hence is expected
to reduce the errors from the linear elastic assumption.
In the current formulation of the model, the (micro-level) fatigue
simulation is performed with very small cycle increments for accurate
prediction of the progressive degradation of the constituents and hence
accurate predicition of the S-N curves. Hence, it is not suitable to be
applied directly for obtaining the local S-N curves in large component
simulations in the framework of a commercial software. A future
recommendation can then be combining the model with efficient “cycle-
jump” algorithms and extrapolation of the damage in between blocks of
cycles hence reducing the computational cost.
A final consideration is the versatility of the proposed fatigue model.
Scaling algorithms depend on certain requirements e.g. the same material
of the reference RVEs and the scaled RVEs which essentially means the
same constituents and volume fractions. The scaling method has not been
validated for predicting the S-N curve of an RVE from a reference RVE
with e.g. different fiber volume fraction. Such requirements are not needed
in the present approach. It has been shown above that the model was able
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to succesfully predict the S-N curves of RVEs of different materials and
volume fractions. This can be useful in situations where there are
differences in the local volume fractions in the simulated component. In
such a case the assumption of a constant local volume fraction may lead to
errors.
It should be however mentioned that the inputs needed for the present
model can be a limitation in some cases. This is especially the case for the
S-N curves of pure matrix where the exact S-N curves of a certain grade of
polymer may be difficult to obtain. It was however, shown in the validation
of the present fatigue model that even though the S-N curve of the exact
commercial grade of the matrix (e.g. the Akulon PA 6) could not be found,
using an S-N curve of the same type of thermoplastic (PA 6) lead to good
predictions of the model.
8.8 Conclusions
A fatigue model was developed for the prediction of the S-N curves of
short fiber composites from the S-N curves of the constituents and actual
local stress states and damage phenomenon. The model relied on
description of different failure criteria describing damage of the short fiber
composites.
The model was applied on a number of validation test cases. And
simulations were performed in combination of the overall modelling
approach described in this PhD thesis. The validation cases depicted
variations in the constituents properties and morphology. For all the test
cases the model showed accurate predicitons of compared to the
experimental curves.
A novel aspect in the developed model is the incorporation of the fatigue
of the interface. The parametric studies performed in this work showed the
significance of the interface fatigue behavior on the overall fatigue
behavior of the composite. This phenomenon of the progressive failure of
the interfaces in fatigue loading was described in a number of previous
studies by means of experimental observations. The present study puts
forward the significance of the behavior by means of modelling evidence.
Finally the last part of the chapter was devoted to the description of the
component level simulation of the short fiber composites and the relation
of the present models with respect to the multi-scale solution. It was
shown that the solution results in promising predictions of the overall S-N
Fatigue Modelling of Short Fiber Composites
271
behavior of SFRP components, however a number of limitations result in
inaccuracies of the predictions in low cycle fatigue. This includes the
generation of small size RVEs and assuming linear elastic constituents. A
discussion is given on the possible future improvements of the micro-
macro solution using different algorithms of the present work.
273
Chapter 9: Conclusions and Future Recommendations
Conclusions and Future Recommendations
275
9.1 Global Summary of the Thesis
The main objective of this PhD thesis is the simulation of the quasi-static
and fatigue behavior of short fiber composites. A novel aspect of this work
is the consideration of complex wavy fiber micro-structures.
This objective is achieved with the development of a series of models
described in the different parts of the thesis. The proposed approach is in
the framework of the analytical mean-field homogenization techniques.
The developed models begin with algorithms for generation of
representative volumes of the short fiber composites. Models for
extensions of the mean-field techniques to wavy fiber composites are
explored. These are then followed by damage models for predictions of the
quasi-static stress-strain behavior of the short straight and wavy fiber
composites. Finally a fatigue model is proposed for the prediction of the
fatigue life of short fiber composites based on the fatigue of the
constituents and local damage.
In parallel to the developed models, detailed experimental
characterizations were performed with the aim of achieving more
understanding of the quasi-static and fatigue behavior of short fiber
composites. The experimental analysis also provided useful insight on the
behavior of the novel short steel fibers considered in this work (as an
example of short wavy fiber composites).
9.2 General Conclusions
9.2.1 Geometrical characterization and modelling
A model was developed for the generation of the micro-structures of short
random fiber composites. The model is able to generate both straight and
wavy reinforcement geometries. Particular focus was given on modelling
of the complex wavy reinforcements based on described mathematical
formulations, taking into account the random and stochastic nature of the
local micro-structure. In addition, an experimental methodology based on
micro-CT techniques was developed for characterization of the complex
3D wavy fiber morphology.
The developed model was used for generation of all the representative
volume elements (short straight and wavy composites) used in the
validation of the present models. The advantage of the ability of generating
CHAPTER 9
276
wavy fibers of the present model is that it can be used for generation of a
number of different other composites where waviness typically exists e.g.
crimped fabrics and non-woven composites and natural short or continuous
fiber reinforced composites.
9.2.2 Quasi-static behavior of short fiber composites
Experimental characterization of the quasi-static behavior of several short
fiber composites was performed. These included two short straight glass
fiber composites and different conditions (concentrations) of short wavy
steel fiber composites.
The obtained experimental curves were used for validation of the
developed models. Detailed experimental investigations, using different
techniques e.g. acoustic emission, micro-CT and fractography analysis,
were also performed with the aim of gaining better understanding of the
general deformation and damage behavior of random short fiber
composites.
The performed experimental analysis also provided an insight on the
behavior of the novel steel fibers and its relation to the current
manufacturing processes and constraints. The achieved insight can be used
in the future for optimizing the manufacturing and properties of these
composites.
9.2.3 Fatigue behavior of short fiber composites
In a similar way, experimental characterization of the fatigue behavior of
the short straight glass and wavy steel fiber reinforced composites were
performed. Different aspects of the fatigue behavior of short fiber
composites were discussed e.g. the S-N behavior, dynamic stiffness
degradation and creep effects during fatigue loading. A main conclusion
from the experimental characterization was the characteristics of the fiber-
matrix interface during fatigue loading. The observed behavior in the
present study, also reported in published literature, was the progressive
degradation of the interface strength during fatigue loading. This behavior
was taken into account in the development of the fatigue models in this
thesis.
Conclusions and Future Recommendations
277
9.2.4 Linear elastic modelling of wavy fiber composites
A so-called Poly-Inclusion (P-I) model was previously developed with the
aim of the extension of the Eshebly based mean-field homogenization
model to the case of crimped fiber composites. The model was only
validated for composite effective properties.
Based on Finite Elements benchmarks, the validity of the model for
application on short wavy fiber composites was investigated. The focus
was the validation of the accuracy of the predictions of the model for the
local stresses in inclusions. The model was applied on different test cases,
including an RVE generated from real micro-CT data of wavy steel fiber
composites. The results of the benchmarks showed that the P-I model can
be used with relatively good accuracy for predictions of local stresses in
wavy fibers. The validation provided means for further damage analysis on
VEs of wavy fiber composites, using the mean-field homogenization
technique, which depends on local fields in inclusions.
9.2.5 Quasi-static damage modelling
A quasi-static modelling approach was proposed in the present work. The
model takes into account the plasticity of the thermoplastic matrices in
short fiber composites and the damage mechanisms of the short fiber
composites, namely, debonding and fiber breakage. The model was applied
on a number of short straight and wavy fiber reinforced composites. For all
investigated materials, the model predictions were in good agreement with
the experimental curves. For the case of wavy fiber composites, the
damage models were applied in combination with the P-I model. The good
predictions of the stress-strain behavior then lead to further validation of
the P-I model in terms of the correct representation of the local stress fields,
based on which damage modelling is applied.
9.2.6 Fatigue modelling
A modelling approach was developed for the prediction of the fatigue S-N
behavior of short fiber composites. The method is based on the prediction
of the S-N curve of the composite based on the S-N curves of the
constituents and the local stress states. By combination of modelling of
progressive degradation and suitable failure criteria, the fatigue life of the
short fiber composite is obtained. Accurate predictions were obtained
using the model for a number of composites with different constituents and
micro-structures.
CHAPTER 9
278
A discussion was also given on the position of the present work as a part
of a multi-scale solution aimed at the simulation of the fatigue behavior of
SFRP components. It was shown that the present solution leads to
promising predictions. Deviations from experimental curves were found in
the high cycle regime. The possible reasons for the deviations were
identified. The different ways in which the developed models in this thesis
can contribute to more accurate predictions were suggested.
9.3 Future Outlook
Future work can be performed on a number of directions explored in this
PhD thesis. These are discussed in the following sub-sections.
9.3.1 Manufacturing of short steel fiber composites
As has been shown in the PhD thesis, the manufacturing of the short steel
fiber composites is not yet optimized. Difficulties were encountered in the
compounding of the material and injection molding processes. High
concentration of the steel fibers could not be achieved. The current
manufacturing processes also resulted in defects of the composites.
Modelling work has shown the good potential of the steel fibers as
reinforcements in short fiber materials. This encourages future research
and development efforts targeted towards the optimization of the
manufacturing parameters of these composites to achieve full potential.
In a similar way, it was found in the present thesis that the steel fibers
exhibit weak interfaces with the matrices. A significant improvement of
the properties of the short steel fiber composites can be achieved by
enhancement of the fiber-matrix interface by suitable methods such as the
applications of fiber treatments.
9.3.2 Matrix plasticity
In the present thesis, matrix plasticity was considered by means of the
secant approach. This approach is sufficient for the quasi-static models.
For fatigue modelling, the secant models cannot describe the full fatigue
loading histories (loading-unloading). It also does not take into account the
hysteresis properties of thermoplastic matrix. More sophisticated matrix
plasticity models can be investigated.
Conclusions and Future Recommendations
279
9.3.3 Component level solutions
In chapter 8, it was mentioned that the proposed models are used in the
present multi-scale solution to predict the “reference” S-N curve. A
discussion was given on the different ways the present models can
contribute to lower errors of the multi-scale predictions. This can be
investigated in future work. “Cycle-jump” based algorithms may be
implemented in the present fatigue models to be suitable for direct
application on complex components, with low computational costs.
9.3.4 Multi-axial and variable amplitude fatigue
The proposed fatigue models in the present thesis were only validated for
the cases of uniaxial, contant amplitude fatigue. In practice, actual
components are generally subjected to complex loads. The present model
formulation is based on a progressive cycle-by-cycle calculations and
hence fatigue cycles with variable amplitudes can be taken into account.
The implementation of the model also allows multi-axial loads to be
defined. The validation of the model on multi-axial and variable amplitude
loading cases can be explored in future work.
9.3.5 Different modes of the fatigue loading
In a similar way, all the validation cases shown in the thesis were on fatigue
loading in tension-tension regime. The model is able to take into account
different stress ratios and mean stress effects, by introducing the
corresponding input S-N curves of the constituents. However different
stress ratios and fatigue modes were not explored in this work. Alternating
tension-compression fatigue loads can be of particular interest for future
developments as such loading cases can underly different damage
mechanisms than the ones found in this thesis e.g. fiber matrix debonding
may not be the main damage mechanism.
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Curriculum vitae
307
Curriculum vitae
Yasmine ABDIN
Date of birth: 10 March 1987
Place of birth: Moncton, Canada
Nationality: Egyptian
Email: [email protected]
Address: Abdel Hamid Badawy st.
11351 Cairo,
Egypt
Educational Background:
2009-2011, Department of Mechanical Engineering, Ain Shams
University, Egypt
Master of science in Mechanical Engineering (specialization:
materials engineering)
Thesis title: Draping behavior of woven fabric for polymer
composites application
2004-2009, Department of Mechanical Engineering, Ain Shams
University, Egypt
Bachelor of Science in Mechanical Engineering
Bachelor thesis title: Modeling of the mechanical behavior of
short randomly oriented glass fiber polypropylene composites.
Work Experience:
10.2011 – present, Department of Materials Engineering, KU Leuven,
Belgium
PhD researcher
09.2009 – 10.2011, Department of Mechanical Engineering, The British
University in Egypt, Egypt
Teaching assistant (Demonstrator)
Curriculum vitae
308
Awards and Honors
Paper titled “Geometrical characterization & micro-structural
modelling of short steel fiber composites” was featured in the
science direct top 25 most downloaded articles in the period July
– September 2014 ranked 25th for journal Composites Part A.
Paper titled “Pseudo-grain discretization and full Mori-Tanaka
formulation for random heterogeneous media: Predictive
abilities for stresses in individual inclusion and matrix” was
featured in the science direct top 25 most downloaded articles in
the period Oct – Dec 2013 ranked 11th for journal Composite
Science and Technology.
List of publications
309
List of publications
International peer-reviewed journal publications
S.Kirchberg, Y. Abdin, G. Ziegmann, “Influence of Particle
Shape and Size on the Wetting Behavior of Soft Magnetic
Powders”, Powder Technology 207 (2011) 311–317.
I. Taha, Y. Abdin, “Modeling of Strength and Stiffness of Short
Randomly Oriented Glass-Fiber Polypropylene
Composites”, Journal of Composite Materials 45 (2011) 1805-
1821.
I. Taha, Y. Abdin, S. Ebeid, “Analysis of the Draping Behaviour
of Multi-Layer Textiles using Digital Image
Processing”, Polymers & Polymer Composites 20 (2012) 837-
845.
I. Taha, Y. Abdin, S. Ebeid, “Prediction of Draping Behavior of
Woven Fabrics over Double-curvature Moulds Using Finite
Element Tehniques”, International Journal of Material and
Mechanical Engineering 1 (2012) 25-31.
Y. Abdin, I. Taha, A. El-Sabbagh, S. Ebeid, “Description of
Draping Behaviour of Woven Fabrics over Single Curvatures by
Image Processing and Simulation Techniques”, Composites:
Part B 45 (2013) 792-799.
I. Taha, Y. Abdin, “Comparison of picture frame and bias-
extension tests for the characterization of shear behaviour in
natural fibre woven fabrics”, Journal of Fibers and Polymers 14
(2013) 338-344.
A. Jain, S.V. Lomov, Y. Abdin, I. Verpoest, W. Van Paepegem,
"Pseudo-grain discretization and full Mori-Tanaka formulation
for random heterogeneous media: Predictive abilities for stresses
in individual inclusion and matrix”. Composites Science and
Technology 87 (2013): p. 86-93.
Y. Abdin, S. V. Lomov, A. Jain, H. van Lenthe, I. Verpoest,
“Geometrical characterization & micro-structural modelling of
List of publications
310
short steel fiber composite”. Composites: Part A 67 (2014) 171-
180.
A. Jain, Y. Abdin, W. Van Paepegem, I. Verpoest, S. V. Lomov,
“Effective anisotropic properties of inclusions with imperfect
interface for Eshelby-based models”. Composites Structures
131(2015): p. 692-706.
Y. Abdin, A. Jain, I. Verpoest, S. V. Lomov, “A mean-field based
approach for micro-mechanical modelling of short wavy steel
fiber reinforced composites”. In preparation.
Y. Abdin, I. Verpoest, S. V. Lomov, “Micro-mechanics based
modelling and validation of the damage behavior of short wavy
fiber composites”. In preparation.
Y. Abdin, I. Verpoest, S. V. Lomov, “Micro-mechanics based
modelling of the fatigue behavior of short fiber composites”. In
preparation.
Book chapter
Y. Abdin, A. Jain, V. Carvelli, S. V. Lomov. “Fatigue analysis of
carbon and glass fibers”. Book title: fatigue of textile composites,
publisher: Woodhead publications.
Contributions to international conferences
Y. Abdin, S. V. Lomov, Atul Jain, G.H. van Lente, I.
Verpoest, “Geometric characterization & micro-structural
modelling of short steel fiber reinforced composites”, in
Tomographic Imaging of Displacement and Strain Fields at
Loughborough, UK , April 10, 2013.
Y. Abdin, S.V. Lomov, A. Jain, I. Verpoest, “Micro-mechanical
modelling of short wavy steel fiber reinforced composites”, in
COMPOSITES 2013 IV ECCOMAS Thematic Conference,
Azores, Portugal, September 25-27, 2013.
A. Jain, S.V. Lomov , Y. Abdin, S. Sträßer, W. Van Paepegem, I.
Verpoest, “Scaling of SN curves of short fiber composites -
hybrid multiscale approach”, in COMPOSITES 2013 IV
List of publications
311
ECCOMAS Thematic Conference, Azores, Portugal, September
25-27, 2013.
Y. Abdin, S.V. Lomov, A. Jain, I. Verpoest, “A mean-field based
approach for micro-mechanical modelling of short wavy
reinforced composites”, in TEXCOMP 11 (composite
week@leuven), Leuven, Belgium, September 16-21, 2013.
A. Jain, S.V. Lomov, Y. Abdin, S. Sträßer, W. Van Paepegem, I.
Verpoest. “Model for partially debonded inclusions in the
framework of mean-field homogenization”, in TEXCOMP 11
(composite week@leuven), Leuven, Belgium, September 16-21,
2013.
Y. Abdin, S. V. Lomov, A. Jain, G.H. van Lente, I.
Verpoest, “Geometric characterization & micro-structural
modelling of short steel fiber reinforced composites”, in
COMPTEST 2013, Aalborg, Denmark, April 22-24, 2013.
A. Jain, S. V. Lomov, Y. Abdin, Verpoest I., W. Van Paepegem,
“Pseudo-grain discretization and full Mori-Tanaka formulation
for random heterogeneous media: predictive abilities for stresses
in individual inclusions and the matrix”, in COMPTEST 2013,
Aalborg, Denmark, April 22-24, 2013.
A. Jain, S.V. Lomov , Y. Abdin, I. Verpoest , W. Van Paepegem,
M. Hack, “ Micromechanics and fatigue life simulation of
random fiber reinforced composites”, in NAFEMS world
congress, Salzburg Austria, June 9-12, 2013.
A. Jain, S.V. Lomov , Y. Abdin, S. Sträßer, W. Van Paepegem, I.
Verpoest, “Micromechanics and fatigue life simulation of
random fiber reinforced composites”, in 12th SAMPE BeNeLux
students meeting, Almere, Netherlands, Dec 17, 2013.
Y. Abdin, S. V. Lomov, A. Jain, I. Verpoest, “Micro-mechanical
modelling and validation of progressive elasto-plastic damage of
short wavy steel fiber composites”, in ECCM 16, Seville, Spain
June 22-26, 2014.
A. Jain, J. M. Veas, S. V. Lomov, Y. Abdin, S. Sträßer, W. Van
Paepegem, I. Verpoest, “Master SN curve method for short fiber
List of publications
312
composites- theory and experimental validation”, in ECCM 16,
Seville, Spain June 22-26, 2014.
Y. Abdin, S. V. Lomov, A. Jain, I. Verpoest, “Micro-mechanical
and progressive damage modelling of short steel fiber reinforced
composites with insight on cyclic behavior”, in MECHCOMP
2014, New York, June 8-12, 2014.
Y. Abdin, S. V. Lomov, A. Jain, I. Verpoest, “Micro-mechanics
based modelling and validation of the damage behavior of short
wavy fiber composites”, in COMPTEST 2015, Madrid, Spain,
April 8-10, 2015.
A. Jain, Y. Abdin, S. Sträßer, W. Van Paepegem, I. Verpoest,
S.V. Lomov, “Validation of the master SN curve approach for
short fiber reinforced composites”, in COMPTEST 2015, Madrid,
Spain, April 8-10, 2015.
Y. Abdin, A. Jain, I . Verpoest, S. V. Lomov. “A micro-
mechanics approach for modelling the fatigue behaviour of
short straight and wavy fiber reinforced composites”, in ESMC
2015, Madrid, Spain, July 6-10, 2015.
Y. Abdin, S. V. Lomov, A. Jain, I. Verpoest. “Mean-field based
fatigue damage modelling of composites reinforced with short
straight and wavy fibers”, in ICCM 2015, Copenhagen, July 19-
24, 2015.
A. Jain, Y. Abdin, S. Sträßer, W. Van Paepegem, I. Verpoest,
S.V. Lomov, “Master S-N curve approach – A hybrid multiscale
approach to fatigue simulation of short fiber composites”, in
ICCM 2015, Copenhagen, July 19-24, 2015.