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

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

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

XX

Figure 2.12 Schematic representation of the two-step homogenization model. The

RVE is decomposed into a number of grains (sub-regions) followed by step 1:

homogenization of each grain , and step 2: second homogenization if

performed over all the grains. .................................................................. 44

Figure 2.13 Two-step homogenization procedure and implementation of damage

modelling proposed by Dermaux et al. [187]. .......................................... 53

Figure 3.1 Illustration of the drawing technique to produce steel fibers [217].66

Figure 3.2 Example of wavy fiber generated by the model for illustration. Black

dots represent ends of segments “control points”..................................... 72

Figure 3.3 Micrographs of short steel fiber reinforced polycarbonate sample

showing the fibers waviness (a) optical micrograph of the composite plate

(stainless steel 0.05VF%) and (b) scanning electron micrograph of the steel

fibers after a matrix burn-out procedure (stainless steel 2VF%), the figure

shows high entanglements of the fibers. .................................................. 74

Figure 3.4 Thresholding of steel fiber reinforced polycarbonate sample (a) 2D

gray-level 2D reconstructed images, (b) corresponding binary image and (c)

individual automatic global thresholds obtained from gray scale attenuation

histogram. The attenuation histogram consists of two overlapping bivariate

distributions. The peak corresponding to lower attenuation index is associated

with matrix material. Due to the low volume fraction (low probability) the peak

of steel fibers is not visible in the plot. The threshold value obtained from the

automatic global thresholding is shown with the red dashed line. ........... 77

Figure 3.5 Thresholded 3D model of a micro-CT scan of SSFRP built in Mimics

software package. The picture shows a green mask of rendered steel fibers and

the outline of the matrix mask in purple................................................... 78

Figure 3.6 Procedure for characterization of fiber length and orientation

distribution of SSFRP. (a) 3D reconstructed model in Mimics software, (b)

separation of single fibers and (c) fitting of centerline, automatic measurement

of fiber length and post-processing for measurement of fiber orientation.80

Figure 3.7 Length distribution of steel fiber reinforced polycarbonate composite

(a) probability density plots of achieved lengths of steel fibers fitted with

different statistical distribution functions i.e.: Normal, Lognormal and Weibull

distributions and (b) Weibull probability plot of the steel fiber length data.

................................................................................................................. 85

Figure 3.8 FOD of the short steel fibers (a) distribution of Φ angle and (b)

distribution of θ angle. ............................................................................. 86

Figure 3.9 Representative volume element of short wavy steel fiber composite

generated from micro-structural model with input parameters achieved from

micro-CT information. ............................................................................. 89

Figure 3.10 Micro-CT image of SSFRP and a comparison between real and

modeled waviness profiles using the developed micro-structural model. 90

Figure 3.11 Probability density of the straightness parameter Ps: comparison

between experimentally achieved (micro-CT) information and mathematical

XXI

model. Histograms are the probability distributions achieved from experiments

and model, fitting lines are normal probability fits of achieved histogram

showing a clear agreement between Ps calculated from model and experiments.

................................................................................................................. 91

Figure 4.1 Specimen preparation for single fiber test on the DMA machine.100

Figure 4.2 Length distributions of (a) GF-PA and (b) GF-PP and Lognormal

probability plots of (c) GF-PA and (d) GF-PP. ..................................... 103

Figure 4.3 Measured stress-strain curves and of the GF-PA and GF-PP materials.

............................................................................................................... 104

Figure 4.4 Stress-strain curve of the polyamide Akulon K222-D [273]. The tests

are stopped at the yield of the matrix. ................................................... 106

Figure 4.5 Stress-strain curve of the polypropylene matrix [274]. The tests are

stopped at the yield of the matrix. ......................................................... 107

Figure 4.6 Acoustic Emission (AE) diagrams during quasi-static loading of the (a)

GF-PA and (b) GF-PP materials. The figure shows plots of the stress, AE

events energy, and cumulative AE energy with the evolution of strains.109

Figure 4.7 Comparison of the cumulative AE energy registrations of the GF-PA

and the GF-PP materials. ....................................................................... 111

Figure 4.8 Distribution of AE amplitudes in (a) GF-PA and (c) GF-PP and AE

energies of (b) GF-PA and (d) GF-PP. .................................................. 113

Figure 4.9 Global micro-CT scan of the overall width of the GF-PP sample.116

Figure 4.10 Representative view of the skin-core morphology in the central part

of a GF-PP sample. ............................................................................... 117

Figure 4.11 SEM micrographs of the fracture surface of the GF-PA quasi-statically

failed sample. Green arrows denote the debonding damage mechanism, red

arrows denote fiber pull-out, and the blue arrows denote “hills” of matrix

around the fiber indicating strong fiber-matrix interface of the GF-PA. 118

Figure 4.12 SEM micrographs of the fracture surface of the GF-PP quasi-static

failed sample. Green arrows denote the debonding damage mechanism and red

arrows denote fiber pull-out .................................................................. 120

Figure 4.13 Tensile stress-strain curves of the neat Durethan B 38 PA 6 material

(matrix material in SF-PA composite samples) at a cross-head speed of 2

mm/min. Tests stopped at 150% strain. ................................................. 121

Figure 4.14 Measured stress-strain curves of single steel fibers (fiber diameter 𝑑 = 8 μm, gauge length 𝐿 = 25 μm). .......................................................... 122

Figure 4.15 Measured stress-strain curves of the SF-PA samples with the different

investigated volume fractions. ............................................................... 123

Figure 4.16 The obtained quasi-static mechanical properties of the SF-PA material

plotted against the fiber volume fractions of the samples...................... 125

Figure 4.17 Acoustic Emission (AE) diagram of SF-PA materials with the

different volume fractions considered in the present study. Plots of the tensile

stress of each AE events energy, and cumulative energy of the events against

XXII

the strain for (a) SF-PA 0.5VF%, (b) SF-PA 1VF%, (c) SF-PA 2VF%, (d) SF-

PA 4VF% and (e) SF-PA 5VF%. ........................................................... 129

Figure 4.18 Comparison of the cumulative AE energy registrations of the SF-PA

materials with the different fiber volume fractions. ............................... 130

Figure 4.19 Distribution of AE amplitudes in (a) SF-PA 2VF% (c) SF-PA 4VF%

and AE energies of (b) SF-PA 2VF% (d) SF-PA 4VF% ....................... 131

Figure 4.20 Micro-CT scanned volumes of the undeformed SF-PA samples with

different fiber volume fractions (a) 0.5VF%, (b) 2VF%, (c) 4VF% and (d)

5VF%. .................................................................................................... 132

Figure 4.21 Small volumes of the micro-CT scanned undeformed SF-PA samples

(a) 0.5VF% and (b) 2VF%. .................................................................... 134

Figure 4.22 View of voids formed in the undeformed 4VF% SF-PA samples.135

Figure 4.23 High magnification SEM images showing the irregular quasi-

hexagonal cross-section of the steel fibers embedded in the matrix. ..... 136

Figure 4.24 SEM micrographs of the fracture surface of the short steel fiber

composite samples with (a) 0.5VF%, (b) 1VF%,, (c) 2VF%, (d) 4VF%, and (e)

5VF%. .................................................................................................... 137

Figure 4.25 SEM micrographs of the voids observed at the fracture surface of the

SF-PA samples of (a) 4VF% and (b) 5VF%. .......................................... 138

Figure 5.1 Representative hysteresis loop (stress-strain deformation curve) and the

linear regression fitting analysis for calculation of the dynamic modulus of a

fatigue cycle. .......................................................................................... 146

Figure 5.2 Representative applied load diagram of the fatigue tests on the tensile

tester performed on the SF-PA 2VF% samples. ..................................... 147

Figure 5.3 Measured S-N curves of the GF-PA and GF-PP samples. Dashed lines

indicated 90% confidence level intervals. Arrows denote run-out samples.150

Figure 5.4 Evolution of the measured hysteresis loops at 𝜎𝑚𝑎𝑥 =70% 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ, for the (a) GF-PA and the (b) GF-PP

materials. N/Nfailure indicate the stage of the sample life with respect to the

failure cycle. ........................................................................................... 152

Figure 5.5 Evolution of the cyclic mean strain for the glass fiber reinforced

composites with the load cycles, tested at 𝜎𝑚𝑎𝑥 =70% 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ,. ............................................................ 153

Figure 5.6 Evolution of the cyclic stiffness for the (a) GF-PA and (b) GF-PP

materials. ................................................................................................ 156

Figure 5.7 Evolution of the measured hysteresis loops of the SF-PA material (at

55%UTS, 27.2 MPa). The legend indicates the cycle number of the drawn

loops. The upper right graph shows more clearly the details of the last

illustrated cycles. .................................................................................... 159

Figure 5.8 Evolution of the cyclic stiffness of the SF-PA material at different stress

levels. ..................................................................................................... 160

XXIII

Figure 5.9 Representative evolution of the hysteresis loops of the SF-PA in early

stages of the fatigue loading as observed in the short fatigue tests performed

on a tensile tester. .................................................................................. 162

Figure 5.10 Evolution of the cyclic stiffness of the SF-PA material with the

different stress level measured from the short fatigue tests performed on the

tensile tester. .......................................................................................... 163

Figure 5.11 Representative evolution of the hysteresis loops of the GF-PA in early

stages of the fatigue loading as observed in the short fatigue tests performed

on a tensile tester. .................................................................................. 164

Figure 5.12 SEM micrographs of the fracture surface of fatigue failed sampled of

the GF-PA material for the (a) 55 UTS%, (b) 65 UTS%, and (c) 70 UTS%

stress levels. ........................................................................................... 165


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


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)



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


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)



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)



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


of the SF-PA 0.5VF% composite. ......................................................... 228


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


of the GF-PBT 0 composite. .................................................................. 231


of the GF-PBT 45 composite. ................................................................ 231


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


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


GF-PP material with the different values of the fatigue strength exponent 𝑏.

............................................................................................................... 253


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


GF-PA material with the different values of the fatigue strength exponent 𝑏.

............................................................................................................... 255


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


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

CHAPTER 2

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.

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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∞

CHAPTER 2

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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44

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.

CHAPTER 2

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

State of the Art

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

CHAPTER 2

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.

State of the Art

49

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

State of the Art

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.

State of the Art

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

CHAPTER 2

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.

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.


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


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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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],


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


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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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)


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.


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.


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


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].


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)).


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


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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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.


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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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


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.



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



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



101


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.



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.



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.



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



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.



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).

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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



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



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



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

CHAPTER 4

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).



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



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

CHAPTER 4

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



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%.

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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)



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)



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



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)



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.

141

Chapter 5: Experimental Characterization of the Fatigue Behavior of Short Glass and Steel Fiber Composites

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


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


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


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.

CHAPTER 5

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.


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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.


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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.


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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.


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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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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.


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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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(a)

(b)

Figure 5.6 Evolution of the cyclic stiffness for the (a) GF-PA and (b) GF-PP

materials.


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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.


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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).


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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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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.


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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.


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165


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


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


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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.


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


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].


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.


6.4.1 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


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.


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)


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.


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)


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


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.


185

(a)

(b)


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.

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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)


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


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.


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.

CHAPTER 6

190

(a)

(b)


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



191

(a)

(b)


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


CHAPTER 6

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.


193

CHAPTER 6

194


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)


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


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

CHAPTER 7

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:


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


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)


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)


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

CHAPTER 7

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.


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?

CHAPTER 7

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.


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.


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%.


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)


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


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.


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

]


7.5.1 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.


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.


225


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.


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.

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228


of the SF-PA 0.5VF% composite.


of the SF-PA 2VF% composite.


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.


231


of the GF-PBT 0 composite.



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232



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.


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.

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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


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


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.

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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


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.

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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


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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246

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


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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248

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).


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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250

Figure 8.8 S-N curve of the PBT matrix used as input for the fatigue model

[320].


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).


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.


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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252

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


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


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.


the GF-PP material with the different values of the fatigue strength exponent 𝑏.

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254

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.



slope of the S-N curve of the interface 𝑏 is shown.


255


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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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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258

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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260

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


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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262

[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.


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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264

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)


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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266

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


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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268

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)


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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270

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


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

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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.


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.

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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.


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)

mailto:[email protected]

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


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


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


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


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.

micro-mechanics based fatigue modelling of …...vii abstract short fiber composites, are...

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