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Micro-Mechanics Based Fatigue Modelling of Composites Reinforced With Straight and Wavy Short Fibers

Yasmine ABDIN

Supervisor: Prof. Stepan V. Lomov Prof. Ignaas Verpoest Members of the Examination Committee: Prof. Albert Van Bael Prof. Andrea Bernasconi Prof. Frederik Desplentere Dr. Larissa Gorbatikh Prof. Patrick Wollants (Chairman) Prof. Willy Sansen (Chairman) Prof. Wim Van Paepegem

Dissertation presented in partial fulfilment of the requirements for the degree of PhD in Materials Engineering

September 2015

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© 2015 KU Leuven, Groep Wetenschap & Technologie

Uitgegeven in eigen beheer, Yasmine Abdin, Heverlee

Alle rechten voorbehouden. Niets uit deze uitgave mag worden vermenigvuldigd en/of openbaar gemaakt worden door middel van druk, fotokopie, microfilm, elektronisch of op welke andere wijze ook zonder voorafgaandelijke schriftelijke toestemming van de uitgever.

All rights reserved. No part of the publication may be reproduced in any form by print, photoprint, microfilm, electronic or any other means without written permission from the publisher.

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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

𝜀�̇� Matrix strain rate

𝜀𝑝∗ Effective matrix plastic strain

Out-of-plane orientation angle

𝜐𝑚 Poisson’s coefficient of the matrix

𝜎∗ Effective Von Mises stress in the matrix

𝜎𝐶 Critical interface strength

𝜎𝑓 Fatigue strength coefficient

𝜎𝑖𝑗′ Deviatoric component of the matrix stress tensor

�̇�𝑚 Matrix stress rate

𝜎𝑚𝑎𝑥 Maximum fatigue stress

𝜎𝑚𝑖𝑛 Minimum fatigue stress

𝜎𝑦 Initial yield stress

Φ In-plane orientation angle

𝜓1,2 Phase shifts

AMTα Strain concentration tensor according to Mori-Tanaka method

Co𝑚 Elastic stiffness tensor of the matrix

C𝑒𝑓𝑓 Effective composite stiffness tensor

C𝑒𝑝 Continuum elasto-plastic tangent operator

C𝑚 Matrix stiffness tensor

C𝑠 Secant stiffness tensor

𝐸𝑑𝑦𝑛 Dynamic fatigue modulus

𝐸𝑚 Matrix elastic Young’s modulus

𝐸𝑚𝑠 Secant Young’s modulus of the matrix

𝑎𝑖𝑗 2nd order orientation tensor

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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 𝜀𝑦𝑦 plotted against the axial position in pixels on the registered suface. The figure show the plots for a

displacement of 0.96 (load of 1.02KN) for (a) Line 1, (b) Line 2 and (c) Line

3. ............................................................................................................ 268

Figure 8.25 Comparison of the experimental and predicted S-N curve of the

Pinocchio component. ............................................................................ 269

XXVII

List of tables

Table 3.1 Main geometrical input parameters used for the mathematic model. .. 88

Table 4.1 Injection molding parameters of the glass fiber and steel fiber

samples................................................................................................................ 97

Table 4.2 Average fiber lengths of the SF-PA samples with different fiber volume

fraction. ………………………………………………………………………. 102

Table 4.3 Tensile properties of the short glass fiber polyamide (GF-PA) and

short glass fiber polypropyelene (GF-PP) composites. .................................... 105

Table 4.4 Tensile properties of the neat Durethan B 38 PA 6 material. Comparison

between achieved results and manufacturer’s datasheet values. ……………… 122

Table 4.5 Tensile properties of single steel fibers. ……………………………. 123

Table 4.6 Summary of the tensile properties of the SF-PA composites with the

different fiber volume fractions. ........................................................................ 124

Table 5.1 Tested stress levels in the fatigue tests of the investigated glass fiber

reinforced composites. ……………………………………….......................... 145

Table 5.2 Tested stress levels in the fatigue tests of the investigated steel fiber

reinforced composites. .....…………………………………………................ 145

Table 5.3 Summary of the cycle at which 50% of the stiffness degradation of the

SF-PA material occurred with the different applied stress levels. …………….. 161

Table 7.1 Summary of the micro-structural parameters of the GF-PA and the GF-

PP materials of the present work used as input for validation of the developed

models. ………………………………………………….................................. 219

Table 7.2 Summary of the micro-structural parameters of the SF-PA materials of

the present work used as input for validation of the developed models.

………………………………………………………………………………... 221

Table 7.3 Summary of the micro-structural parameters of the GF-PBT materials

used as input for validation of the developed models. ………………………… 223

1

Chapter 1: Introduction

Introduction

3

1.1 General Introduction

In the recent years, there has been an increasingly growing interest in fiber-

reinforced composites as a replacement of metals and alloys in a number

of engineering structures, owing to the favorable characteristics of

composite materials. The major advantage of composite materials over

metals is their superior specific properties e.g., specific strength and

stiffness (strength-to-weight ratio and stiffness-to-weight ratio,

respectively). Major industrial sectors have contributed to the growth of

composite technologies. On one hand, the aeronautics industry has largely

invested in the development of composites design and manufacturing

technologies. At present, more than 50% of the “next-generation” Airbus

aircraft A350 XWB is made of composites [1]. On the other hand,

stipulated by the lawful regulations of CO2 reductions, the automotive

industry has become today the largest consumer of the overall types of

composite materials, accounting for over 20% of total consumption [2].

Composites are a vast group of materials presenting itself in large

variations of matrix materials, reinforcement types and micro-structures.

On the industrial scale, polymer composites and especially those based on

thermoplastic matrices are the most attractive types, offering the needed

weight reductions, superior mechanical properties and high durability.

Thermoplastic composites exhibit the added advantages of recyclability

and lower energy processing, compared to their thermoset counterparts.

From a structural viewpoint, these materials can be distinguished in two

main categories which are continuous and discontinuous (or short) fiber

reinforced composites.

Composites with the best mechanical performance are those with

continuous fibers. However, these materials cannot be adopted easily in

mass production and are confined to applications in which property

benefits outweigh the cost penalty [3]. In this respect, the aerospace

industry has pioneered the use of high performance continuous fiber

composites in structural applications regardless of cost and using cost-

intensive manufacturing methods such as autoclave manufacturing and

hand lay-up. In contrast, the focus of the automotive industry has been on

semi-structural components using short fiber composites [4, 5].

A number of processing techniques exist for the production of short fiber

reinforced polymers (SFRPs). For thermosetting materials the most

common processes are Sheet Molding Compound (SMC) and Bulk

CHAPTER 1

4

Molding Compound (BMC) processes. Extrusion compounding and

Injection Molding (IM) are the conventional techniques for production of

thermoplastics composites [6].

Injection molding remains the most attractive manufacturing method

allowing the production of components with intricate shapes at a very high

production rate, with reasonable dimensional accuracy and fairly low costs.

The versatility and low cost of the injection molding process led to its

increased use, largely in the automotive industry, but also in different

applications such electrical and electronic industries, sporting goods,

defense sector and other consumer dominated products.

Despite of those advantages, injection molded short fiber composites

depict a more complex morphology compared to other composite types.

Increased fiber damage and complex melt flow behavior during processing

give rise to random micro-structures characterized by statistical fiber

length distributions (FLD) and fiber orientation distributions (FOD). An

important and distinctive feature of SFRP parts is then the variability of

the material properties throughout the part and hence, the anisotropy of the

local properties. As a result, those materials are often referred to as random

fiber reinforced composites (RFRCs).

Another complexity of the short random fiber composites is the nature of

the fiber matrix interface which is dependent on the compatibility of the

fibers and matrix materials and on the processing conditions. The quality

of the fiber-matrix interface has significant impact on the efficiency and

load-carrying capability of short fiber composites.

Fibers used in SFRPs are typically glass fibers and carbon fibers. A number

of studies investigated the potential of natural fibers as a replacement of

synthetic fibers SFRPs [7-9]. Metal fibers have been used to provide

shielding and electrical conductivity [10-12]. Among the different metallic

fibers materials are steel fibers, which are highly efficient in

electromagnetic shielding at very low fiber volume fractions. In

conjunction with electromagnetic properties, steel fibers depict superior

mechanical properties (stiffness of about 200 GPa and strength of about 2

GPa), which are comparable to high performance carbon fibers. This

makes stainless steel fibers attractive for further investigations in

mechanical applications.

One of the leading manufacturers of steel fibers is the Flemish company

Bekaert. Since the 1990s the company has been performing research on

Introduction

5

their steel fiber products available under the commercial name Beki-

Shield. While the Beki-Shield fibers were initially targeted only towards

Electromagnetic interference (EMI) shielding, recent research efforts

include the investigation of steel fiber composites in mechanical

applications.

An important characteristic of injection molded steel fiber composites is

the waviness of the fibers embedded in the matrix. This characteristic

waviness also exists in long carbon fibers, natural fibers, crimped textiles

and non-woven composites. The inherent waviness of steel fibers

embedded in the matrix, as a result of processing, further adds to the

complexity of the RFRCs micro-structure.

Finally, automotive components, along with most other engineering

applications, are often subjected to cyclic loading, resulting in damage and

material property degradation in a progressive manner [13, 14]. The

penetration of short fiber composites in fatigue sensitive applications

places focus on the durability aspects of those materials. This leads to a

large interest in understanding the different durability and fatigue behavior

aspects of this class of materials.

1.2 Scientific & Technological Context

Complete design of an SFRP component is a complex undertaking, which

should simultaneously take into account different factors such as loading,

weight reduction, part stiffness and durability. Exhaustive testing and

trials-and-error are not effective ways due to the high variability of material

and micro-structure parameters, part/mold geometries and manufacturing

routes. In sectors where performance to cost ratios define competitiveness,

like the automotive industry, a possibility to make design decisions based

on accurate numerical models and virtual testing of the part is a crucial

factor. Missing durability performance simulation tools are a key

restricting factor for wider use of SFRP materials in cars.

To date, predictive models of fatigue behavior of composites are largely

restricted to continuous fiber systems [13]. A large number of the available

models for these composites are phenomenological models which usually

require a large number of experiments and test data for each kind of

material in question. Examples can be found in e.g. [15-18].

A challenging question remains if it is possible to model the fatigue

behavior and lifetime of composites based on the behavior of the

CHAPTER 1

6

constituents (i.e. matrix, fibers, and interface) and actual micro-scale

damage phenomena. The question is challenging, even for the more

established continuous fiber composites where only a few attempts can be

found in literature, e.g. in [19, 20].

The fatigue behavior of random fiber composites is much less understood.

Similar to continuous fiber composites, a few phenomenological based

models have emerged for modelling the fatigue behavior of random

composites. Examples include e.g. [21-23]. Models linking the fatigue

behavior of short random fiber composites to the behavior of constituents,

do not exist, to the knowledge of the author. This results in the need for

research efforts targeted towards the development and validation of

efficient and robust models for prediction of the fatigue behavior of RFRCs

based on the behavior of the underlying constituents, local stress states and

actual damage mechanisms.

Additionally, modelling RFRC materials requires addressing the multi-

scale behavior of the material. As mentioned above, a real component of

random fiber composites produced with a manufacturing process such as

the injection molding technique often has a complex geometry, which

results in large variations of local micro-structure between different points

along the part. In this respect, modelling the behavior of RFRC materials

often requires multi-scale approaches.

Another challenge in the context of this work is understanding and

modelling the behavior of short steel fiber composites. While such material

is attractive due to the superior properties of steel fibers, it exhibits several

differences from the generally used glass and carbon fiber composites. On

one hand, the random waviness of the fibers adds to the complexity of the

micro-structure. This also results in challenges in incorporating the

waviness aspects of the fibers in geometrical and mechanical models. On

the other hand, information about the mechanical behavior of the steel fiber

composites as well as their distinct characteristics, such as the nature of

fiber-matrix interface and the effects of the high stiffness mismatch

between fibers and matrix, are not available due to novelty of the material.

Finally, in the last decades, Finite Element (FE) based simulation tools

have been commercially available. In the present technological context,

one of the commercially available software packages is the Siemens LMS

Virtual.Lab Durability software. Existing algorithms of the software

include complete solutions for modelling metal fatigue under variable

conditions of designs and complex loading states. A current objective is

Introduction

7

the extension of the software solutions to the complex random fiber

composites led by the increase of demand of the material in automotive

applications.

1.3 Objectives of the PhD research

In view of the above mentioned scientific and technological context, the

ultimate objective of the work is the formulation and validation of

methodologies that enable the simulation of the fatigue behavior of RFRC

components. As mentioned above, a complete fatigue simulation of an

RFRC component requires a multi-scale modelling approach. Figure 1.1

illustrates an overview of the proposed solution used in this PhD thesis.

Figure 1.1 Overview of the multi-scale predictive methods for modelling the

fatigue behavior of RFRC parts.

The procedure starts with process (manufacturing) modelling for

simulation of the injection molding of the component in question. Such

simulations are available in different commercial packages such as:

Moldflow, SigmaSoft, and Express, to name a few. Based on the part

geometry and melt flow behavior of the material, the software tools are

able to predict the local fiber orientation, which can be later mapped to FE

meshes.

Virtual.Lab

Durability

Process model (MoldFlow,

SigmaSoft, etc.)

Fiber and matrix

data

Microscopic modelling

Material

parameters Pre-

Damage Feedback

loop

Local S-N

curves

FEA

Fatigue loading at elements

FE loading

Fatigue life of the

part

Local

stiffness

CHAPTER 1

8

At the microscopic level, models need to be developed with the end goal

of the accurate prediction of local lifetime, i.e. stress vs. number of cycles

to failure (S-N) curves. This in turn can be achieved with a series of

simultaneous micro-scale models. These include micro-structural models

to generate statistically representative local geometries taking into account

input of the preceding manufacturing simulation, quasi-static mechanical

models for prediction of the local behavior and fatigue models for

prediction of the local S-N curves.

At the macroscopic scale, Finite Element Analysis (FEA) is performed on

the component level. Fatigue loading is applied and the durability software

is able to solve the local multi-axial loading conditions at each element.

The local stiffnesses and S-N curves are inputted to the durability solver

by interaction with the micro-models. Based on the input of the local

stiffnesses and S-N curves, the durability solver is able to predict the

critical areas as well as the overall fatigue lifetime of the component. The

solver includes so-called “feedback” algorithms.

While at the micro-scale full FEA modelling can be applied for the

prediction of the local stress states, local damage and final S-N curves at

each element, this approach leads to high computational expensive

solutions which are inadmissible in consideration of the above described

industrial requirements. The alternative route is the use of suitable

analytical approaches which allow the estimation of the local material

states with reasonable accuracy at efficient computational speeds. Among

these approaches are the well-known mean-field homogenization methods.

The position of this PhD work within the above described process is the

micro-scale modelling (highlighted in Figure 1.1) of the quasi-static and

fatigue behavior of RFRCs. For fatigue modelling, a novelty of the work

is the ability to predict the S-N curves of the composite based on the S-N

curves of the constituents (i.e. matrix, fibers and interface) using detailed

micro-mechanics. As mentioned above, such methods are not available in

literature. Another novelty of the work is that in addition to the typical

short straight fiber reinforced materials, the thesis considers the application

of micro-mechanical models to wavy fiber reinforced composites e.g. the

steel fiber materials discussed above. The methodologies developed in this

work can be applied to a number of other crimped fiber systems.

Introduction

9

The main objectives of the thesis can then be summarized as follows:

- Characterizing and modelling the complex micro-structure of short wavy steel fiber composites and understanding the behavior of this

novel class of materials.

- Assessment and validation of models for extension of the mean-field homogenization techniques to short wavy fiber reinforced composites.

- Development and validation of a modelling approach for the prediction of the quasi-static behavior and progressive damage of short fiber

composites, based on mean-field homogenization methods.

- Formulation and validation of a fatigue model in the context of mean-field homogenization methods, for the prediction of the fatigue

behavior based on the input of the fatigue properties of the

constituents.

- Detailed experimental investigations of the quasi-static and the fatigue properties of random straight and wavy fiber reinforced composites for

better understanding of the underlying damage phenomena and for

validation of the developed models.

1.4 Structure of the thesis

The structure of the thesis follows the objectives described in the previous

section. A schematic overview of the thesis is presented in Figure 1.2.

Chapter 2 of the thesis is devoted to the study of the literature and

introduces general knowledge of the available methods for RFRCs. The

chapter gives an overview of the micro-structure of RFRCs and the factors

affecting the mechanical behavior of RFRCs. A review is given on the

different methods and concepts of simulation of the geometry of RFRCs.

The chapter also gives a brief description of the different mean-field

homogenization techniques as well as the available models for the quasi-

static and progressive damage models of RFRCs. Finally, different

attempts for micro-mechanical fatigue modelling of RFRCs are discussed.

CHAPTER 1

10

Figure 1.2 Outline of the PhD thesis.

Motivation

Novelty

Chapter 2.

State of the art

Chapter 3.

Geometrical

characterization

and modelling

Chapter 1.

Introduction

Chapter 4.

Experimental

characterization

quasi-static

behavior

Chapter 5.

Experimental

characterization

fatigue

behavior

Chapter 6.

Linear elastic

modelling of

wavy RFRCs

Chapter 7.

Quasi-static

modelling of

RFRCs

Chapter 8.

Fatigue

modelling of

RFRCs

Chapter 9.

Conclusions and future

perspectives

Introduction

11

Chapter 3 describes the developed geometrical model for the generation of

volume elements (VEs) of RFRCs. The model is able to generate VEs of

both straight and wavy fiber composites. As in the published literature,

different models are available for generation of random straight fiber

composites, the chapter is focused on the aspects of the model concerned

with the description of wavy fibers. In parallel to the modelling attempts,

a novel experimental methodology for characterization of the micro-

structure of complex wavy fiber samples, based on micro-computer

tomography (micro-CT) techniques, is discussed.

Chapters 4 and 5 cover the performed experimental investigations for

quasi-static and fatigue behavior respectively of short glass fiber and short

steel fiber reinforced composites. The different characterization techniques

e.g. mechanical testing, fractography analysis, full-field strain mapping

and acoustic emission techniques are discussed. The achieved

experimental results provide a better understanding of the behavior of

random fiber reinforced composites, which will be reflected in the

development of the models. The results of those chapters also serve as

validation for the models developed in the subsequent chapters.

Chapter 6 deals with the extension of the existing mean-field

homogenization methods for wavy fiber reinforced composites. A model

for the transformation of wavy fibers into equivalent straight fiber systems

that are able to be modelled using mean-field techniques is presented and

validated with full FEA.

Chapter 7 presents the developed methods for the quasi-static damage

modelling of RFRCs. This includes models reflecting the damage

phenomena of short fiber composites i.e. fiber matrix debonding, and fiber

breakage and models for the non-linear plastic deformation of the matrix.

The models are applied on the VEs generated by the geometrical model

explained in chapter 3. For wavy fiber composites, the additional model

developed in chapter 6 is applied prior to the quasi-static modelling. The

implementation of the model in a numerical tool is briefly presented.

Validation of the models with experimental results is reported in the

chapter.

Chapter 8 is devoted to the fatigue model. This in turn is dependent on the

quasi-static models in Chapter 7. Similar to the quasi-static models,

numerical implementation of the models is discussed. A detailed validation

with the experimental results is presented. The chapter also gives a brief

CHAPTER 1

12

overview of attempts for component level simulation and validation, with

the connection with the micro-scale models developed in this PhD thesis.

Chapter 9 concludes the thesis and provides perspective for future research

work.

13

Chapter 2: State of the Art

State of the Art

15

2.1 Introduction

In this chapter, a detailed overview of the available methods for modelling

the geometry and the quasi-static and the fatigue behavior of random short

fiber reinforced composites will be presented. In order to model the

material behavior, an understanding of the unique micro-structure of short

fiber composites and the different factors affecting its mechanical behavior

is needed. This in turn can be achieved using a synopsis of available

experimental observations.

The structure of the chapter will be explained in the following. As

discussed in the introduction, the injection molding process is the most

attractive and commonly used manufacturing technique for short fiber

composites. In the first section of this chapter, this manufacturing process

will be briefly discussed in order to understand the different processing

factors affecting the final random fiber composite parts. Next, details of

experimental observations in literature of the evolution of the micro-

structure of short fiber composites will be given, followed by an overview

of the factors affecting both the quasi-static and the fatigue behavior of

RFRCs supported by key literature results. Injection molded components

are considered in this thesis as the most common RFRCs as well as the

ones with relatively more complex micro-structures. The developed

concepts and models can also be applied to other types of RFRCs.

The following parts of the review will be dedicated to modelling the

behavior of RFRCs. This starts with an overview of the available methods

for generation of representative volume elements which are able to

simulate the complex micro-structure of RFRCs, and of important factors

to be taken into consideration such as the size of those representative

volumes. In the subsequent section, mean-field homogenization methods

will be introduced and examination of the variations of the different mean-

field models will be given. Focus will be given on the original concepts of

the models, namely the Eshelby solution. The Mori-Tanaka model which

is the most commonly used out of the different mean-field methods for

modelling RFRCs will be discussed in more detail. Moreover, an

important aspect considered in this review is outlining the different

limitations of the Mori-Tanaka model and how these were addressed in

literature.

Mean-field homogenization models, as will be shown in section 2.5, were

first intended for modelling the elastic behavior of composites. In the next

section, the different methods for extending the mean-field models to

CHAPTER 2

16

describe the non-linear behavior of short fiber composites will be given.

The sources of non-linearity are typically the elasto-plastic behavior of the

thermoplastic matrix and the different damage mechanisms of the

composite. Finally, an outline will be given on the few attempts conducted

in previous research for modelling the fatigue life of short fiber composites.

It should be noted that this literature review discusses general concepts of

short random fiber composites. An important part of this thesis aims at

understanding and formulation of methods for modelling the micro-

structure and mechanical behavior of wavy fiber composites. The example

considered in this work is short steel fiber composites. The next chapter of

this thesis is devoted to modelling the micro-structure of complex wavy

short steel fiber reinforced composites. The chapter will also include

details of the motivation for investigating this novel class of materials, the

production process of micron-sized steel fibers and efforts for

characterizing and modelling similar wavy micro-structures.

2.2 Injection Molding of RFRCs

As mentioned in section 1.1, injection molding provides a very attractive

and cost effective way of manufacturing short fiber reinforced composites

[24]. Figure 2.1 shows a schematic diagram illustrating the injection

molding process.

State of the Art

17

Figure 2.1 Schematic illustration of the injection molding process (adapted from

[25]).

The raw material used for the injection molding process are compounded

pellets of the desired thermoplastic/fiber materials combination and

volume fractions. Prior compounding can be performed using methods

such as extrusion or high shear mixing. Compounding already results in

damage of the fiber with stochastic nature and consequently development

of a length distribution of the fibers in the pellets.

During injection molding, the pellets are fed to the hopper and the injection

molding cycle begins. The material is heated and its viscosity is reduced.

This enables flow of the polymer compound with the driving force of the

injection unit, during which stage, shear forces are exerted by the screw.

This adds a significant amount of friction on the material prior to injection.

In the next stage, a desired amount of molten material is stored in front of

the tip of the screw and is then pushed into the closed mold. A cooling

cycle begins, and after the material is cooled down and solidified in the

mold the part is ejected.

CHAPTER 2

18

2.3 Micro-structure and Mechanical Behavior of RFRCs

2.3.1 Micro-structure of RFRCs

The performance of short fiber composites is governed by the complex

geometry of the fibers and their distribution in the part [26-32]. Unlike

continuous UD or textile fiber reinforced composites, short fiber reinforced

composites depict stochastic geometrical features that evolve during

processing [33]. During the injection molding process, as briefly discussed

in section 2.2, high shear stresses exerted in the melt by the screw rotation,

in addition to fiber-fiber interactions, lead to further fiber breakage (to the

already damaged fibers from the compounding process), resulting finally

in a range of fiber lengths, characterized by a length distribution function

(FLD) [34-36]. The complex