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Doctoral Thesis
Property predictions for short fiber and platelet filled materialsby finite element calculations
Author(s): Lusti, Hans Rudolf
Publication Date: 2003
Permanent Link: https://doi.org/10.3929/ethz-a-004554826
Rights / License: In Copyright - Non-Commercial Use Permitted
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DISS. ETH NO. 15078
Property Predictions for
Short Fiber and Platelet Filled Materials by
Finite Element Calculations
A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH
for the degree of
Doctor of Sciences
presented by
HANS RUDOLF LUSTI
Dipl. Werkstoff-Ing. ETH
born September 7, 1973
citizen of Nesslau, SG
accepted on the recommendation of
PD Dr. A.A. Gusev, examiner
Prof. Dr. U.W. Suter, co-examiner
Prof. Dr. P. Smith, co-examiner
2003
Diese Arbeit widme ich
meinen Eltern
Rösli & Christian
sowie meiner Frau
Natacha
DANKSAGUNG
Ich möchte mich bei PD Dr. Andrei Gusev ganz herzlich für die grossartige
persönliche und fachliche Unterstützung bedanken. Er hat mich während meiner
Doktorarbeit mit Rat und Tat unterstützt und ich konnte durch zahlreiche
Diskussionen von seinem grossen Wissen und seiner langjährigen
Wissenschafts-Erfahrung profitieren.
Ein besonderer Dank geht auch an Prof. Dr. Ulrich W. Suter, der in seiner
Forschungsgruppe ein hervorragendes Arbeitsklima aufgebaut hat und
bereitwillig auf Wünsche und Probleme eingegangen ist.
Ich möchte mich ebenfalls bei Dr. Peter Hine von der Universität Leeds, UK,
für die gute Zusammenarbeit auf dem Gebiet von kurzfaserverstärkten
Kompositen bedanken, die schon viele Früchte getragen hat.
Ein Dankeschön geht auch an:
• Dr. Chantal Oberson, für ihre Hilfeleistungen, wenn ich mit meiner Mathematik
am Ende war und für die anregenden Diskussionen,
• Ilya Karmilov für die gute Zusammenarbeit und die feinen Spezialitäten, die er
mir regelmässig aus Moskau mitgebracht hat,
• Martin Heggli für die gute Zusammenarbeit und die hilfreichen Ratschläge
bezüglich Mathematica,
• Albrecht Külpmann für die interessanten und anregenden Diskussionen,
• Dr. Marc Petitmermet für seinen prompten und kompetenten Computersup-
port und seinen grossartigen Einsatz bei der Wiederinbetriebnahme der HP-
Workstation,
• Sylvia Turner und Christina Graf für ihre Hilfe bei administrativen Angelegen-
heiten,
• alle anderen Mitarbeiter der Forschungsgruppe für das angenehme Arbeits-
klima.
ZUSAMMENFASSUNG
Eine neue, mächtige Finite-Elemente (FE) Simulationsmethode wurde kürz-
lich von Gusev entwickelt, die es erlaubt, die linear-elastischen, elektrischen,
thermischen und Transport-Eigenschaften von mehrphasigen Werkstoffen, ba-
sierend auf realistischen 3D-Computermodellen, zu studieren. Im ersten Teil die-
ser Doktorarbeit wurde dieses neue Verfahren validiert, indem gemessene
thermoelastische Eigenschaften mit den numerischen Voraussagen von FE-Mo-
dellen verglichen wurden, die aufgrund von mikrostrukturellen Daten von spritz-
gegossenen, kurzfaserverstärkten Zugproben generiert wurden. Die
numerischen Voraussagen zeigten eine ausgezeichnete Übereinstimmung mit
allen gemessenen Eigenschaften. Die erfolgreiche Validierung erlaubte es dann,
die Genauigkeit sowohl von den in der Praxis am weitesten verbreiteten mikro-
mechanischen Modellen (Halpin-Tsai und Tandon-Weng) zur Voraussage der
elastischen Eigenschaften von unidirektional kurzfaserverstärkten Kompositen
als auch des Orientierungsmittelungs-Verfahrens zu beurteilen. Die Untersu-
chungen haben gezeigt, dass das Modell von Tandon-Weng wesentlich genauer
ist als dasjenige von Halpin-Tsai. Trotzdem sind die Abweichungen zu gross, als
dass es für Auslegungszwecke im Engineering taugen würde. Der Vergleich zwi-
schen den Voraussagen von numerischen Berechnungen und dem Orientie-
rungsmittelungs-Verfahren haben ergeben, dass die Orientierungsmittelung sehr
geeignet ist, um die thermoelastischen Eigenschaftstensoren von jeglichen Fa-
ser- und Plättchen-Orientierungszuständen zu bestimmen. Das unter der Bedin-
gung, dass die Orientierungsmittelung mit zuverlässigen Eigenschaftsdaten von
unidirektionalen Kompositen durchgeführt wird. Mit dem numerischen Verfahren,
das in dieser Arbeit verwendet wurde, können die Eigenschaften von unidirektio-
nalen Kompositen problemlos bestimmt werden.
Numerische Berechnungen zu den thermoelastischen und Barriere-Eigen-
schaften von Polymer-Schichtsilikat-Nanokompositen mit perfekt ausgerichteten
Silikatplättchen haben gezeigt, dass der Abfall, sowohl der Gaspermeabilität als
auch des thermischen Ausdehnungskoeffizienten, durch eine gestreckte Expo-
nentialfunktion beschrieben werden kann, die von x = af abhängt, wobei a das
Achsenverhältnis und f die Volumenfraktion der Plättchen ist. Diese Masterkur-
ven erlauben eine rationale Auslegung der Barriere- und der thermischen Aus-
dehnungs-Eigenschaften von Nanokompositen mit perfekt ausgerichteten
Plättchen. Es wurde ausserdem demonstriert, wie die thermische Ausdehnung
von Nanokompositen mit Auslegungsdiagrammen, die von der Masterkurve ab-
geleitet wurden, massgeschneidert werden kann. Der Minderungseffekt von
Fehlausrichtungen der Plättchen auf die Barriereeigenschaften wurde auch un-
tersucht. Die Voraussage von Fredrickson et. al. dass verdünnte Konzentrationen
von zufällig orientierten Plättchen hohen Achsenverhältnisses ein Drittel so effek-
tiv sind wie entsprechende Nanokomposite mit perfekt ausgerichteten Plättchen,
wurde durch numerische Berechnungen bestätigt. Es war allerdings nicht be-
kannt, dass dieser Minderungseffekt im halbverdünnten Konzentrationsregime
abnimmt, weil die fehlgerichteten Plättchen gemeinsam anfangen, die Diffusions-
wege der penetrierenden Moleküle zu vergrössern. Für typische Achsenverhält-
nisse und Volumenfraktionen der Plättchen in gegenwärtig existierenden
Nanokompositen bewegt sich der Minderungseffekt von zufällig orientierten Plätt-
chen im Rahmen von 40-50%.
ABSTRACT
Recently, a new powerful finite element (FE) based simulation technique
has been developed by Gusev, which allows to study the linear-elastic, electric,
thermal and transport properties of multi-phase materials based on realistic 3D
multi-inclusion computer models. In the first part of this thesis this new procedure
has been validated by comparing measured thermoelastic properties with
numerical predictions obtained with FE-models, which were generated based on
microstructural data of real injection molded short fiber reinforced dumbbells.
Numerical predictions showed excellent agreement with all the measured
properties. The successful validation then allowed to assess the accuracy of most
widely used in practice micromechanics-based models (Halpin-Tsai and Tandon-
Weng) which predict the elastic properties of unidirectional short fiber
composites, and also the accuracy of the orientation averaging scheme. It was
found that the Tandon-Weng model is considerably more accurate than the
Halpin-Tsai equations, but nonetheless deviations are too large to make this
model appropriate for engineering design purposes. Comparison of direct
numerical and orientation averaging predictions revealed that the orientation
averaging scheme is highly suitable to determine the thermoelastic property
tensors of any fiber and platelet orientation state. This under the condition that
orientation averaging is done based on reliable property data of unidirectional
composites. With the numerical approach employed in this work one can readily
determine the properties of unidirectional composites.
Numerical calculations of the barrier and thermoelastic properties of
polymer-layered silicate nanocomposites comprising perfectly aligned silicate
platelets elucidated that the decline both of the gas permeability and of the
thermal expansion coefficient can be described by a stretched exponential
function which depends on x = af, the product of the platelet aspect ratio a and
the platelet volume fraction f. These mastercurves allow to rationally design the
barrier and thermal expansion properties of nanocomposites with perfectly
aligned platelets. Furthermore, it has been demonstrated how the thermal
expansion coefficient of nanocomposites can be tailored by using design
diagrams adapted from the mastercurve. The degrading effect of platelet
misalignments on the barrier properties has also been investigated numerically.
The prediction of Fredrickson et al. that dilute concentrations of randomly oriented
high-aspect-ratio platelets are 1/3 as effective compared to a corresponding
nanocomposite with perfectly aligned platelets was confirmed by numerical
calculations. It has, however, not been known that the degrading effect decreases
in the semidilute concentration regime due to the fact that the misaligned platelets
start to collectively increase the tortuosity of the penetrant’s diffusion path. For
platelet aspect ratios and volume fractions which are typical of currently existing
nanocomposites the expected degradation effect of randomly oriented platelets
is in the range of 40-50%.
PUBLICATIONS AND PRESENTATIONS IN CONNECTION WITH THIS THESIS
Articles
• A.A. Gusev, H.R. Lusti, Rational Design of Nanocomposites for Barrier Appli-
cations, Adv. Mater. 2001, 13, 1641-1643.
• H.R. Lusti, P.J. Hine, A.A. Gusev, Direct Numerical Predictions for the Elastic
and Thermoelastic Properties of Short Fibre Composites, Compos. Sci. Tech.
2002, 62, 1927-1934.
• P.J. Hine, H.R. Lusti, A.A. Gusev, Numerical Simulation of the Effects of
Volume Fraction, Aspect Ratio and Fibre Length Distribution on the Elastic
and Thermoelastic Properties of Short Fiber Composites, Compos. Sci. Tech.
2002, 62, 1445-1453.
• A.A. Gusev, H.R. Lusti, P.J. Hine, Stiffness and Thermal Expansion of Short
Fiber Composites with Fully Aligned Fibers, Adv. Eng. Mater. 2002, 4, 927-
931.
• A.A. Gusev, M. Heggli, H.R. Lusti, P.J. Hine, Orientation Averaging for Stiff-
ness and Thermal Expansion of Short Fiber Composites, Adv. Eng. Mater.
2002, 4, 931-933.
• H.R. Lusti, I.A. Karmilov, A.A. Gusev, Effect of Particle Agglomeration on the
Elastic Properties of Filled Polymers, Soft Materials 2003, 1, 115-120.
• M. Wissler, H.R. Lusti, C. Oberson, A.H. Widmann-Schupak, G. Zappini, A.A.
Gusev, Non-Additive Effects in the Elastic Behavior of Dental Composites,
Adv. Eng. Mater. 2003, 3, 113-116.
• P.J. Hine, H.R. Lusti, A.A. Gusev, The Numerical Prediction of the Elastic and
Thermoelastic Properties of Multiphase Materials, in preparation.
• H.R. Lusti, O. Guseva, A.A. Gusev, Matching the Thermal Expansion of Mica-
Polymer Nanocomposites and Metals, in preparation.
Poster presentations
• Materials Workshop, Crêt Berard, Switzerland, September 9-12, 2001:
H.R. Lusti, A.A. Gusev, “Rational Design of Nanocomposites for Barrier Appli-
cations”
• Top Nano 21 Third Annual Meeting, Bern, Switzerland, October 1, 2002:
H.R. Lusti, V. Mittal, A.A. Gusev, “Numerical Permeability Predictions for
Nanocomposites comprising Morphological Imperfections”
Oral presentations
• Materials Science Seminar, Department of Materials, ETH Zürich, November
14, 2001
• C4-Workshop, ETH Zürich, November 22, 2001
• Workshop in Analysis Techniques for Polymer Nanostructures, St. Anne’s Col-
lege, Oxford, UK, April 8-10, 2002
• CAD-FEM User's Meeting, Friedrichshafen, Germany, October 9-11, 2002
• 5. Werkstofftechnisches Kolloquium, Chemnitz, Germany, October 24-25,
2002
Table of Contents
TABLE OF CONTENTS Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 Importance of Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Short Fiber Reinforced Parts Made by Injection Molding . . . . . . . . . . 6
1.3 Polymer Nanocomposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2. Analytical and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Micromechanical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Orientation Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Gusev’s Finite-Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3. Short Fiber Reinforced Composites . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 Fiber Length Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Comparison between Micromechanical Models, Numerical Predictions
and Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.1 Micromechanical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.3 Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Stiffness and Thermal Expansion of Short Fiber Composites with Fully
Aligned Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.1 Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Table of Contents
3.3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Prediction of Stiffness and Thermal Expansion by the Orientation
Averaging Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.1 Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4. Polymer-Layered Silicate Nanocomposites . . . . . . . . . . . . . . . . . . . . 59
4.1 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.1 Morphologies with Perfectly Aligned Platelets . . . . . . . . . . . . . . 60
4.1.2 Morphologies with Misaligned Platelets . . . . . . . . . . . . . . . . . . . 64
4.2 Thermoelastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2.1 Morphologies with Perfectly Aligned Platelets . . . . . . . . . . . . . . 74
4.2.2 Morphologies with Misaligned Platelets . . . . . . . . . . . . . . . . . . . 83
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Notation & Acronyms
- 1 -
NOTATIONa Aspect ratio
aN Number average of a ARD
aS Skewed number average of a ARD
aRMS Root-Mean-Square average of a ARD
aW Weight average of a ARD
aij 2nd order orientation tensor
aijkl 4th order orientation tensor
Cijkl Stiffness tensor GPa
Cij Stiffness tensor in Voigt notation (6x6 Matrix) GPa
d Fiber diameter µm
Ef Young’s modulus of isotropic fibers GPa
Em Young’s modulus of isotropic matrix GPa
f Fiber or platelet volume fraction %
Gf Shear modulus of isotropic fibers GPa
Gm Shear modulus of isotropic matrix GPa
Km Bulk modulus of isotropic matrix GPa
L Fiber length µm
LN Number average of a FLD µm
LS Skewed number average of a FLD µm
LRMS Root-Mean-Square average of a FLD µm
LW Weight average of a FLD µm
N Number of inclusion in a computer model
Vf Fiber volume fraction
Vm Matrix volume fraction
P Permeability Barrer
p Unit vector pointing along the symmetry axis of
a fiber or platelet
Sijkl Compliance tensor GPa
αm Thermal expansion coefficient of isotropic matrix K-1
αL Thermal expansion coefficient lamellar composite K-1
Notation & Acronyms
- 2 -
αkl Thermal expansion tensor K-1
δij Unit tensor
εkl Effective mechanical strain
ζ Empirical Halpin-Tsai parameter
φ, θ Angles which define the orientation of fibers and platelets deg
µf , λf Lamé constants of isotropic fibers GPa
µm , λm Lamé constants of isotropic matrix GPa
νf Poisson’s ratio of isotropic fibers
νm Poisson’s ratio of isotropic matrix
σij Effective mechanical stress GPa
σTij Effective thermal stress GPa
ψ(p) Normalized probability density function of
a fiber orientation state
Notation & Acronyms
- 3 -
ACRONYMSARD Aspect Ratio Distribution
CTE Coefficient of Thermal Expansion
FE Finite Element
FEA Finite Element Analysis
FEM Finite Element Method
FLD Fiber Length Distribution
MC Monte Carlo
PBC Periodic Boundary Conditions
PDF Probability Density Function
RMS Root Mean Square
RVE Representative Volume Element
SCORIM Shear Controlled Orientation Injection Molding
Notation & Acronyms
- 4 -
Chapter 1 - Introduction
- 5 -
1. INTRODUCTION
1.1 IMPORTANCE OF COMPOSITE MATERIALS
Heterogeneous and composite materials, like hardened steel, bronze or
wood were valued since ancient times because they provide a better performance
compared to the individual phases or components which they consist of.
Nowadays the idea of combining eligible materials to form a composite material
with new and superior properties compared to its individual components is still
subject of ongoing research. For example, polymers, which by nature have a low
density, can be reinforced by highly stiff and strong carbon fibers, both continuous
and discontinuous. Such fiber reinforced composites excel in high specific
mechanical properties. For lightweight structures high specific stiffness and
strength are crucial requirements. The higher the specific mechanical properties
are the lighter a part or construction can be designed. This is of great importance
for moving components, especially in the automotive and airplane industry, where
reductions in weight result in greater efficiency and reduced energy consumption.
The expression “fiber reinforced composites” already says that the focus for these
materials is on the mechanical performance. There exist, however, a variety of
other composites where the functionality is more important. For example carbon/
polyethylene composites which suddenly increase electrical resistivity by several
order of magnitudes upon heating because the carbon particles get separated
due the thermal expansion of the surrounding polymer matrix.
In former times people only had an empirical knowledge about the property
changes taking place when combining different types of materials in a composite.
It was in the 20th century that research on composite materials started and got
increasingly important also due to the need of new lightweight and high
performance materials in armor and astronautics. By the time composite
materials also found their way into civil applications e.g in passenger aircrafts,
cars, boats and sports equipment. But still the fraction of composite materials
employed in industrial goods is rather small because often traditionally used
lightweight materials are preferred to composite parts. The reasons for that are
manyfold. On the one hand there is a lack of experience in designing and
Chapter 1 - Introduction
- 6 -
constructing with this class of materials and on the other hand there are the higher
cost of composite parts caused by expensive raw materials, i.e. carbon fibers,
and costly production processes. To overcome these inhibitions one of the
measures is to reduce the production cost in order to promote a broader
application of these materials. However, it is not only about highly automated and
fast production processes but also about the development of new predictive tools
which can be used to reliably design composite parts. The design stage decides
if a composite part can show its merits in a specific application and if it will perform
successfully in operation.
1.2 SHORT FIBER REINFORCED PARTS MADE BY INJECTION MOLDING
In contrary to time-consuming, elaborate and expensive winding or
laminating techniques used to manufacture long fiber reinforced parts short fiber
composites can be fabricated into complex shapes using automated mass
production methods, such as injection molding, compression molding or
extrusion. One process, already widely used in the production of unreinforced
thermoplastic parts, which is also predestinated in order to manufacture short
fiber reinforced parts with complex shapes, is injection molding. This process is
efficient and offers due to the high degree of automation the possibility of making
complex shaped structural parts in large quantities at reasonable cost. Therefore
injection molded short fiber reinforced thermoplasts are increasingly finding their
way into industrial applications where high specific mechanical properties,
durability and corrosion resistance are required but where cost are a decisive
factor e.g. in car industry.
Figure 1: Injection molded glass fiber reinforced nylon 6 acceleration pedal which was developed for the Ford Focus. Picture taken from [1].
Chapter 1 - Introduction
- 7 -
Because the overall effective properties of a short fiber reinforced composite
can vary between isotropic (3D randomly oriented fibers) and highly anisotropic
(aligned fibers) it is of great importance to design the mold and to control the
process parameters, e.g in a way that everywhere locally across the finished part
the short fibers act along the axes of principal stresses. There exist commercial
software packages (e.g. Moldflow, Sigmasoft,...) which are used to simulate the
mold filling process and at the same time to determine the local fiber orientation
states in a finished injection molded part after cooling. It was shown that
simulation results agree remarkably well with measured microstructural data.[1]
Based on the local fiber orientation states one can calculate the local
thermoelastic properties which serve as input for structural FEM packages (e.g
Abaqus, Ansys,...). Structural FEA reveals if a particular part’s design is
appropriate to perform well under the expected loads during operation and if the
degree of shrinkage and warpage during cooling from the process temperature is
acceptable. At present, all mold filling simulation programs use one of the
micromechanical models (either Tandon-Weng or Halpin-Tsai model) to calculate
the thermoelastic properties of a unidirectional short fiber reinforced unit. The
elastic tensor for the unidirectional reinforced unit is subsequently used in
orientation averaging to determine the elastic tensor for all the actual fiber
orientation states which are present in the simulated injection molded part. The
procedure of using mold filling simulations followed by local property calculations
and structural FEA provides a very efficient way for product design with short fiber
composites. It is, however, unclear if this approach is accurate enough in order to
be used in practice for designing injection molded load bearing short fiber
reinforced structures. Therefore, one of the goals of this thesis was to investigate
if the combination of micromechanical models and orientation averaging is
accurate and reliable enough to successfully predict the local thermoelastic
properties based on the local fiber orientation states calculated by mold filling
simulations.
Chapter 1 - Introduction
- 8 -
1.3 POLYMER NANOCOMPOSITES
An interesting class of materials are polymer nanocomposites whose matrix
properties are promoted with the dispersion of high-aspect ratio, submicron-sized
particles, such as intercalated or exfoliated atomic-thickness sheets of layered
silicates, carbon nanotubes[2] and cellulose whiskers[3,4]. Especially polymer-
layered silicate nanocomposites have attracted a lot of attention in the last
decade.[5-15] An even dispersion of just a few weight percent of such 1nm thick,
high aspect ratio silicate platelets in a polymer can already significantly enhance
modulus, thermal stability, flame retardance, dimensional stability, heat distortion
temperature as well as the barrier properties and corrosion resistance.[16,17] For
example, a doubling of the tensile modulus and strength is achieved for a nylon 6
matrix comprising 2 vol% of montmorillonite.[18-20] In addition, the heat distortion
temperature increases by up to 100°C which opens new possibilities of
applications, for example for automotive under-the-hood parts.[18] In comparison
to conventional, highly filled microcomposites one can save up to 25% weight with
nanocomposites due to the low weight fraction of layered silicates, which have
about a 3 times larger density than polymers. At the same time one can benefit
from an improved and broadened property portfolio. Therefore such materials are
especially attractive for the automotive and the mobility sector in general. It is
crucial, however, that the cost/performance ratio for this class of materials
becomes attractive enough for this highly competitive industry sector. Since the
raw materials of nanocomposites are cheap it is mainly the processing cost which
are decisive for the final cost of these materials. There are already a few existing
applications of polymer nanocomposites e.g. automotive timing-belt covers[21]
and fascia. It is expected that this class of materials can provide property
portfolios which are not only interesting for the automotive[22] but also for the food
packaging industry where one can take advantage of the decreased gas
permeability while retaining flexibility and optical clarity of the pure polymer.
Research on polymer nanocomposites comprising layered silicates started
only 10 years ago and focused mostly on the synthesis of such materials by
elaborate and expensive processing routes revealing outstanding property
Chapter 1 - Introduction
- 9 -
enhancements. More recently researchers focused more on simpler and cost-
effective processing routes like melt compounding in order to promote this class
of materials for industrial applications. In parallel researchers also began to build
up a better theoretical understanding about the reasons which lead to the
outstanding property enhancements. An extensive and interesting review article
on the preparation, the properties and the use of polymer-layered silicate
nanocomposites has been published by Alexandre and Dubois.[23]
In this thesis the thermoelastic and barrier properties of polymer-layered
silicate nanocomposites were investigated by direct numerical FE-simulations.
The goal was to develop a more profound understanding of how the morphology
(aspect ratio, volume fraction and orientation of platelets) affects the overall
thermoelastic and barrier properties of nanocomposites and to predict the
possible property improvements both for ideal and imperfect morphologies.
Concretely, the geometry dependent enhancements of the thermoelastic and
barrier properties for nanocomposites comprising perfectly aligned mineral
platelets and the barrier losses due to platelet misalignments were investigated.
The findings of this thesis enable experimentalists to rationally choose suitable
morphologies for certain property requirements before any experiment is done in
the lab. This can obviously accelerate the development of nanocomposites for
barrier, load bearing and other applications.
Chapter 1 - Introduction
- 10 -
Chapter 2 - Analytical and Numerical Methods
- 11 -
2. ANALYTICAL AND NUMERICAL METHODS
2.1 MICROMECHANICAL MODELS
In addition to the experimental techniques of processing and measuring the
overall effective properties of composites their theoretical prediction starting from
the intrinsic properties of the constituents and the composite’s morphology has
been the subject of extensive studies.[24] The first theoretical considerations of
two phase systems go back to James Clerk Maxwell who derived an expression
for the specific resistance of a dilute suspension of spheres in an infinite isotropic
conductor.[25] Subsequently, accurate rational models of two-phase composites
with spherical or infinitely long cylindrical inclusions have been developed for
predicting elastic, thermal, transport and other properties.[24]
There exist numerous micromechanics-based models which were
developed to predict a complete set of elastic constants for aligned short-fiber
composites. One of the most popular ones is the Halpin-Tsai model which was
initially developed for continuous fiber composites and which was derived from
the self-consistent models of Hermans[26] and Hill[27]. The Halpin-Tsai
equations can be expressed in a short and easily usable form which might be one
of the reasons why they have found a broad usage:
(1)
Vf is the fiber volume fraction and stands for any of the moduli listed in
Table 1. E11 and E22 are the longitudinal and the transverse Young’s modulus,
G12 and G23 the in-plane and out-plane shear modulus, respectively. K23 is the
plane-strain bulk modulus and ν21 the longitudinal Poisson’s ratio of the
unidirectional transversely isotropic short fiber composite. The corresponding
values of the empirical parameter ζ are also listed in Table 1.
MMm--------
1 ζηVf+1 ηVf–
--------------------- with ηMr 1–Mr ζ+----------------==
MrMfMm--------=
Chapter 2 - Analytical and Numerical Methods
- 12 -
Table 1: Halpin-Tsai parameter ζ is listed for the different substitutions of Mf and Mmused in Eq. (1)
ζ is correlated with the geometry of the reinforcement and it was found empirically
that predictions for E11, the Young’s modulus in fiber direction, are best if ζ=2a,
where a is the fiber aspect ratio, defined as:
(2)
L is the fiber length and d the fiber diameter. It can be shown that for the
Halpin-Tsai equations become the rule of mixtures (Voigt bound) where fiber and
matrix experience the same, uniform strain:
(3)
The rule of mixtures is also applied to calculate the longitudinal Poisson’s ratio ν21
although predictions are not accurate when matrix and fibers have considerably
different Poisson’s ratios.
(4)
The Halpin-Tsai model can deal both with isotropic and transversely
isotropic fibers e.g carbon fibers because the underlying self-consistent theories
of Hermans and Hill apply also to transversely isotropic fibers.
M Mf Mm ζ
E11 Ef Em 2a
E22 Ef Em 2
G12 Gf Gm 1
G23 Gf GmKmGm-------
KmGm------- 2+ ⁄
a Ld---=
ζ ∞→
M VfMf VmMm+=
ν21 Vfνf Vmνm+=
Chapter 2 - Analytical and Numerical Methods
- 13 -
Empirical and semi-empirical equations like the treatments of Halpin and
Tsai [28,29], which are widely used in industry can only be useful in reproducing
available experimental data.[30-32] They always reflect the existing technological
level and can therefore not be helpful in deciding if in principle the performance
of a certain composite can be improved further or not. To make this decision it is
necessary to quantify the degradation effects of imperfections like fiber or platelet
agglomerations and their poor adhesion to the polymer. Based on these
quantifications one could decide about the potential of further improvements of
the composite’s properties by controlling the degree of imperfections. Empirical
equations can not fulfill this task and therefore it is necessary to have another
method at hand which can predict the in principal achievable effective properties
of fiber- and platelet-reinforced composites.
A well established and theoretically well founded micromechanical model is
the one of Tandon and Weng[33] which is based on the Eshelby’s solution of an
ellipsoidal inclusion in an infinite matrix[34] and Mori-Tanaka’s average
stress[35]. This model is applicable to spherical, fiber- as well as to disk-shaped
particles, which are called “platelets” throughout this work. The Tandon-Weng
model predicts the five independent effective elastic constants of a transversely
isotropic composite for any fiber aspect ratio from zero to infinity by the following
analytical equations:
(5)
(6)
(7)
E11Em-------- 1
1 VfA1 2νmA2+( )
A6---------------------------------+
--------------------------------------------------=
E22Em-------- 1
1 Vf2νmA3– 1 νm–( )A4 1 νm+( )A5A6+ +[ ]
2A6---------------------------------------------------------------------------------------------------+
-------------------------------------------------------------------------------------------------------------------=
G12Gm--------- 1
Vfµm
µf µm–----------------- 2VmS1212+-----------------------------------------------+=
Chapter 2 - Analytical and Numerical Methods
- 14 -
(8)
(9)
(10)
Eq. (9) was derived by Tucker[32] and is not the original equation for ν21 which
was found by Tandon and Weng because the original equation is coupled with Eq.
(10) and could therefore only be solved iteratively. The parameters A1,...,A6,
B1,...,B5 and D1,...,D3 are defined as following:
(11)
(12)
G23Gm--------- 1
Vfµm
µf µm–----------------- 2VmS2323+-----------------------------------------------+=
ν21νmA6 Vf A3 νmA4–( )–A6 Vf A1 2νmA2+( )+------------------------------------------------------=
K23Km--------
1 νm+( ) 1 2νm–( )
1 νm 1 2ν21+( )– Vf2 ν21 νm–( )A3 1 νm 1 2ν21+( )–[ ]A4+{ }
A6----------------------------------------------------------------------------------------------------+
-----------------------------------------------------------------------------------------------------------------------------------------------------------=
A1 D1 B4 B5+( ) 2B2–=
A2 1 D1+( )B2 B4 B5+( )–=
A3 B1 D1B3–=
A4 1 D1+( )B1 2B3–=
A51 D1–( )B4 B5–( )
-----------------------=
A6 2B2B3 B1 B4 B5+( )–=
B1 VfD1 D2 Vm D1S1111 2S2211+( )+ +=
B2 Vf D3 Vm D1S1122 S2222 S2233+ +( )+ +=
B3 Vf D3 Vm S1111 1 D1+( )S2211+( )+ +=
B4 VfD1 D2 Vm S1122 D1S2222 S2233+ +( )+ +=
B5 Vf D3 Vm S1122 S2222 D1S2233+ +( )+ +=
Chapter 2 - Analytical and Numerical Methods
- 15 -
(13)
λm, µm and λf, µf are the Lamé constants of the matrix and the fibers,
respectively. Sijkl are the non-vanishing components of the Eshelby’s tensor
which depend themselves on the Poisson’s ratio of the matrix νm and the fiber
aspect ratio a. The expressions for the Eshelby’s tensor components Sijkl can be
found in [33]. The Tandon-Weng model was developed for isotropic inclusions but
Qiu and Weng extended it to transversely isotropic inclusions[36]. Therefore it is
also possible to predict the effective elastic constants of carbon fiber reinforced
composites. Although the Tandon-Weng model is widely perceived to give the
best predictions for fiber and platelet filled composites it has never been shown
by direct comparison with experimental results of unidirectional short fiber
composites that predictions are accurate. This due to the fact that it is almost
impossible to produce samples of short fiber composites with perfectly aligned
fibers.
Therefore, another concept, namely the one of the rigorous upper and lower
bounds was developed which is one of the most firmly established. Bounds are
clearly preferable to the use of uncertain micromechanical models because they
deliver rigorous upper and lower margins on the effective properties of a
composite. The most popular bounds are the Hashin-Shtrikman variational
bounds which were developed in order to predict both the elastic[37,38] and the
dielectric constants[39-41] if no morphological information apart from the volume
fractions of the phases is available. The bounds for the dielectric constant are
equally applicable in order to predict properties like the electric and thermal
conductivity as well as the diffusion coefficient. The main drawback of the
rigorous upper and lower bounds is that if the ratio of the constituent’s properties,
e.g , is rising, the bounds become increasingly widely separated and thus
D1 1 2µf µm–( )λ f λm–
----------------------+=
D2λm 2µm+( )λ f λm–( )
----------------------------=
D3λm
λ f λm–( )----------------------=
GfGm-------
Chapter 2 - Analytical and Numerical Methods
- 16 -
practically useless if one wants to predict the effective properties of a two-phase-
composite. Often one is interested in mixing two constituents with a preferably
large difference in their intrinsic properties because then the most attractive
property enhancements can be expected. In this case, however, neither
micromechanics-based models nor rigorous upper and lower bounds can make
firm predictions about the overall effective properties of the composite.
Furthermore both micromechanical models and rigorous upper and lower bounds
are only capable of dealing with two-phase composites. As soon as more than two
phases are present one is supposed to use a series of two-phase homogenization
steps as it is described in classical textbook guidelines. It has been found, though,
that using this additivity premise does not deliver reliable property predictions,
e.g. for composites which are highly filled with ceramic particles.[42]
Another approach to determine the effective properties of composites is FE-
modeling. The problem of this method is that the models are often rudimentary,
e.g. consisting of one or two aligned fibers with regular spatial symmetries, which
are hardly found in real composites. As a consequence numerical results are not
representative of real composites and therefore useless for practical design
purposes. At the end of the 1990’s, however, Gusev developed a FEM with which
it is possible to generate sophisticated multi-inclusion Monte-Carlo (MC) models
for a great variety of composite morphologies. By consistently using periodic
boundary conditions (PBC) throughout model and mesh generation as well during
the numerical solution for the overall, effective properties it has been shown that
this FEM delivers remarkably accurate predictions from surprisingly small
computer models. In chapter 2.3 the Gusev’s FEM is described in detail.
2.2 ORIENTATION AVERAGING
As soon as the inclusions have an anisotropic shape we observe anisotropic
overall properties of the composite. Maximal anisotropy is achieved when all
inclusions e.g. fibers or platelets are unidirectional aligned. In this case we
observe a maximum reinforcement for fibers in the longitudinal direction and for
platelets in the transverse directions. If the inclusions are randomly oriented
throughout the matrix, the composite shows macroscopically isotropic behavior.
Chapter 2 - Analytical and Numerical Methods
- 17 -
Between these two extremes the degree of anisotropy gradually decreases until
it disappears for randomly oriented inclusions. It was found that the anisotropy of
several material properties (elastic stiffness, thermal conductivity, viscosity) can
be directly related to the orientation state of the inclusions. As a consequence,
different methods have been developed which can be used to determine the
property tensors of anisotropic materials based on the orientation state of the
inclusions in a composite (equivalent to the treatment for the degree of crystalline
orientation in an unreinforced pure polymer). The orientation averaging scheme
is one of the methods to predict the overall properties of a known orientation state
of e.g. fibers1 by averaging the unidirectional property tensor T(p) over all
directions weighted by the orientation distribution function ψ(p).[43] The
orientation of a fiber is defined by a direction unit vector p with components p1,
p2, p3 in a cartesian coordinate system (see Figure 2).
Figure 2: The orientation of a fiber can be defined by a unit vector p whose components p1, p2, p3 depend on the two angles θ and φ depicted in this figure.
1. From now on fibers are considered although the orientation averaging scheme is equally applica-ble to any other anisotropic shaped inclusions like platelets, spheroids, ellipsoids etc.
φ
θ
y
x
z
2
3
1
P
Chapter 2 - Analytical and Numerical Methods
- 18 -
The components can be expressed by the angles φ and θ as following:
(14)
Thus, the orientation averaging scheme can be expressed as:
(15)
The probability distribution function ψ(p) indicating the probabilities of finding
fibers with a certain orientation p in the composite is the most accurate form to
describe the fiber orientation state. It is, however, too cumbersome for numerical
calculations and therefore efforts have been made to find alternative ways of
describing orientation states.[45,46] One of the most general but nevertheless
most concise descriptions can be made by using orientation tensors. The
orientation state of a set of fibers, for example, can be defined by an infinite series
of even order orientation tensors. The 2nd order orientation tensor is determined
by forming dyadic products with all possible direction unit vectors p and
integrating the product of the resulting tensors with the distribution function ψ(p)
over all possible directions of p.[43]
(16)
(17)
The indices i, j, k, l run from 1 to 3. All orientation tensors are symmetric and the
2nd and 4th order tensors consist of 6 and 15 independent components,
respectively. If the laboratory frame coincides with the principal axes then all non-
diagonal components become zero and the number of non-zero components is
reduced to 3 and 6, respectively. In this case the components are defined as
follows:
p1 θcos=
p2 θ φcossin=
p3 θ φsinsin=
T〈 〉 T p( )ψ p( )dp∫°=
aij pipj〈 〉 pi pj ψ p( ) dp∫°= =
aijkl pipjpkpl〈 〉 pi pj pk pl ψ p( ) dp∫°= =
Chapter 2 - Analytical and Numerical Methods
- 19 -
(18)
(19)
Any tensor property T(p) of a unidirectional microstructure aligned in the
direction of p must be transversely isotropic, with p as its axis of symmetry. To be
transversely isotropic a 2nd order property tensor Tij(p) must have the form
(20)
where δij is the unit tensor.
Applying orientation averaging to Tij(p) gives:
(21)
Eq. (21) proves that the orientation average of a material property which can be
represented by a 2nd order tensor, e.g the permeability, is completely determined
by the 2nd order orientation tensor aij and by the underlying unidirectional
permeability tensor which determines the scalar constants A1 and A2 as following:
(22)
a11 θ2cos〈 〉=
a22 θ φ2cos
2sin〈 〉=
a33 θ φ2sin
2sin〈 〉=
a1111 θ4cos〈 〉=
a1122 θ2 θ φ2cos
2sincos〈 〉=
a1133 θ θ φ2sin
2sin
2cos〈 〉=
a2233 θ φ φ2sin
2cos
2sin〈 〉=
a2222 θ φ4cos
4sin〈 〉=
a3333 θ φ4sin
4sin〈 〉=
Tij p( ) A1pipj A2δij+=
T〈 〉 ij A1 pipj〈 〉 A2 δij〈 〉 A1aij A2δij+=+=
A1 P1 P2 and A2 P2=–=
Chapter 2 - Analytical and Numerical Methods
- 20 -
Therefore to calculate the permeability one only needs to know the 2nd order
orientation tensor aij of the actual composite morphology and the longitudinal and
transverse permeability coefficient P1 and P2 of the corresponding unidirectional
composite. The linear-elastic and the thermoelastic properties, however, require
knowledge of both the 2nd and the 4th order orientation tensor because the elastic
properties are characterized by a 4th order tensor. The orientation averaged
elastic tensor , is defined as:
(23)
are five scalar constants related to the elastic constants of a
transversely isotropic orientation state with fully aligned fibers[43, 44]
(24)
Although the thermal expansion is characterized by a 2nd order tensor the
orientation averaging of the thermal expansion tensor also requires the 4th
order orientation tensor. The reason is that the thermal expansion is directly
related to the elastic properties of a material. The orientation averaged thermal
expansion tensor is given by:
(25)
Cijkl〈 〉
Cijkl〈 〉 B1aijkl B2 aijδkl aklδij+( ) B3 aikδjl ailδjk ajkδil ajlδik+ + +( )+ + +=
B4 δijδkl( ) B+ 5 δikδjl δilδjk+( )
B1 … B5, , Cijkl
B1 C11 C22 2C12– 4C66–+=
B2 C12 C23–=
B3 C6612--- C23 C22–( )+=
B4 C23=
B512--- C22 C23–( )=
αkl
αkl〈 〉
αkl〈 〉 Cijklαkl〈 〉 Cijkl〈 〉1– D1aij D2δij+( ) Sijkl〈 〉= =
Chapter 2 - Analytical and Numerical Methods
- 21 -
where D1 and D2 are again two invariants which depend on the elastic and
thermal expansion tensor of the unidirectional composite.[44]
(26)
2.3 GUSEV’S FINITE-ELEMENT METHOD
A new FEM for predicting the properties of multi-phase materials based on
3D periodic multi-inclusion computer models has been developed by
Gusev.[47,48] This FEM excels that with remarkably small computer models one
can accurately determine the overall effective properties of ‘real’ composites with
complex morphologies comprising any desired number of anisotropic phases. In
the last few years the problem of obtaining accurate predictions from small
computer models has extensively been studied. It has been demonstrated that
PBC are most appropriate to predict the behavior and properties of multi-phase
materials from very small computer models. Numerical calculations have shown
that a unit cell comprising 25 spheres is already representative of a particle filled
composite with a random microstructure.[47] The same was done for short fibers
for which the minimal representative volume element (RVE) size is somewhat
larger. It was demonstrated that computer models comprising 100 parallel fibers
are appropriate to get accurate predictions for the longitudinal Young’s modulus
E11 (see chapter 3.1).
Often composite materials have a complex microstructure containing
inclusions of different size and shape featuring non-uniform orientation
distributions. Based on measured microstructural data a computer model
representative of a real composite morphology can be generated. Then the
computer model is meshed into an unstructured, morphology-adaptive FE-mesh
which is fully periodic.[48] In the first step of mesh construction a set of nodal
points is placed onto the inclusions’ surfaces. In the following step an additional
set of nodes is inserted on a regular grid inside the unit cell. A sequential Bowyer-
Watson algorithm[49] is used to uniquely connect both the surface and grid nodes
to a periodic, morphology-adaptive 3D-mesh consisting of tetrahedra following
D1 A1 B1 B2 4B3 2B5++ +( ) A2 B1 3B2 4B3+ +( )+=
D2 A1 B2 B4+( ) A2 B2 3B4 2B5+ +( )+=
Chapter 2 - Analytical and Numerical Methods
- 22 -
the Delaunay triangulation[50,51] scheme. The initial 3D-mesh normally contains
a large number of ill-shaped tetrahedra (sliver-, cap-, needle- and wedge-like
tetrahedra) which influence the speed of convergence and the accuracy of the
numerical results. The same problems occur for bridging elements which directly
connect two or even more inclusions. To get rid of these tetrahedra types the FE-
mesh is locally refined by inserting new nodes at the centers of the ill-shaped
tetrahedra circumspheres.[48]
When the FE-mesh is finished material properties are assigned to the
individual inclusions and the matrix. Like this each tetrahedron acquires certain
material properties depending on which phase it belongs to. One of the
outstanding possibilities of this FEM is that one can assign anisotropic properties
of crystalline materials belonging to any of the 7 crystal systems (triclinic,
monoclinic, orthorhombic, tetragonal, trigonal/rhombohedral, hexagonal and
cubic) both to matrix and inclusions.
To numerically calculate the overall, effective properties of the modelled
composites, a perturbation of certain type is applied to the computer model and
the material’s response on the perturbation is calculated numerically. For
example, to calculate the effective dielectric properties, one applies an external
electric field and solves the Laplace’s equation for the unknown local nodal
potentials by minimizing the total electric energy of the system. At the minimum
the nodal potentials can be determined and the local polarization fields inside
each tetrahedron are uniquely defined. The overall, effective dielectric tensor of
the multi-phase material is finally calculated based on the linear-response relation
between the effective induction and the external electric field. By successively
applying the external electric field in the 1-, 2- and 3-directions of the computer
model’s coordinate system one can calculate the complete dielectric tensor of an
anisotropic composite material. Analogous by numerically solving the Laplace’s
equation, also the overall, effective permeability as well as the electric and
thermal conductance can be determined.
A displacement-based, linear-elastostatic solver is used to numerically
compute the elastic constants and thermal expansion coefficients of multi-phase
materials. The effective elastic properties can again be calculated from the
Chapter 2 - Analytical and Numerical Methods
- 23 -
response to an applied perturbation in the form of a constant effective mechanical
strain εkl. The solver finds iteratively a set of nodal degrees of freedom thatminimize the total strain energy of the system. Conjugate gradient
minimization[52] is used to find this unique energy minimum in the space of
system’s degrees of freedom which is defined by a certain set of nodal
displacements. The knowledge of the nodal displacements allows to determine
the local strains in each tetrahedron and consequently the effective stress σij ofthe system. The effective elastic constants Cijkl can then be calculated from the
linear response equation:
(27)
Six independent strain energy minimizations conducted under 6 different
effective mechanical strains (tensile strains in each of the 3 directions and shear
strains in each of the 3 planes of the coordinate system) are necessary to
determine all the 21 independent components of the stiffness matrix. To obtain
the effective thermal expansion coefficient local non-mechanical strains
corresponding to a temperature change of one Kelvin are applied, assuming a
zero effective mechanical strain εkl. One last strain energy minimization isnecessary in order to calculate the effective thermal stress σΤij at the energyminimum. Using the previously calculated effective stiffness matrix Cijkl of the
composite it is possible to determine the 6 independent components of the
effective thermal expansion tensor αij:
(28)
In a similar way, one can evaluate the effective swelling coefficients and the
effective shrinkage caused by chemical reactions or the relaxation of residual
stresses.
It has already been shown that the FEM of Gusev delivers remarkably
accurate predictions for the overall, effective properties of multi-phase
materials.[53-55] Therefore this FEM has also been applied to identify the
technological potential of sphere and platelet filled polymers.[42,56-58]
σij Cijklεkl=
α ij Cijkl1– σkl
T Sijkl σklT= =
Chapter 2 - Analytical and Numerical Methods
- 24 -
Chapter 3 - Short Fiber Reinforced Composites
- 25 -
3. SHORT FIBER REINFORCED COMPOSITES
In injection molded short fiber composites microstructural variations like
polydispersed fiber lengths and arbitrary fiber orientation states are unavoidable,
and influence the overall elastic properties of the composite. For structural design
of short fiber reinforced parts one would like to be in the position to reliably predict
the thermoelastic properties either by micromechanical or numerical models.
Many micromechanical have been developed[32] but they are often based on
idealized composite morphologies, e.g. a matrix comprising aligned fibers of
equal size[28,33], or a single ellipsoid in an infinite matrix[34]. FEMs often deal
with rudimentary models comprising one or two fibers with regular spatial
symmetries which are rarely if ever found in real composites. In order to predict
the properties of realistic composite morphologies it is, however, necessary to
use models which take into account the ‘real’ composite morphology with all its
imperfections. In this chapter it is shown that with Gusev’s FEM one can make
accurate and precise predictions of the thermoelastic properties of short fiber
composites with complex morphologies which excellently agree with
experimental data. Furthermore it is demonstrated that the property prediction for
composites comprising morphological imperfections, like polydispersed fiber
lengths or arbitrary fiber orientation states can be simplified by eligible averaging
methods.
3.1 FIBER LENGTH DISTRIBUTIONS
One aspect of short fiber composites which can be difficult to address
analytically is the distribution of fiber lengths that are normally present in a real
material. The most popular approach is to replace the fiber length distribution
(FLD) with a single length, normally the number average length LN.
(29)LNNiLi∑Ni∑
-----------------=
Chapter 3 - Short Fiber Reinforced Composites
- 26 -
A number of proposals for this “mean length” have been published for special fiber
orientation states. Takao and Taya [59] and Halpin et al.[60] concluded that the
number average length LN of a distribution was an appropriate value. Eduljee and
McCullough [61] suggested a different average LS to take into account the
skewed nature of real FLDs, in particular to give a heavier weighting to shorter
fibers.
(30)
The Root-Mean-Square (RMS) average LRMS has also been suggested as a
possible descriptor of the FLD.
(31)
For completeness the weight average LW was also taken into account in this
study.
(32)
For fibers of constant diameter one can express Eq. (32) again by using Ni, the
frequency of fibers in a certain length interval.
(33)
It would seem, therefore, that there is merit in being able to model an assembly
of fibers with a ‘real’ FLD, in order to establish whether the distribution can be
replaced by one of the above mean values in order to establish what
LSNi∑NiLi-----∑
------------=
LRMSNiLi
2∑
Ni∑------------------=
LWWiLi∑Wi∑
------------------=
LWNiLi
2∑NiLi∑
------------------=
Chapter 3 - Short Fiber Reinforced Composites
- 27 -
McCullough[61] describes as ‘the appropriate statistical parameters to represent
the microstructural features of the composite’. The FEM of Gusev offers the
chance to establish which type of mean length is appropriate in order to replace
a length distribution. For this purpose a fiber length distribution measured by
image analysis of a typical injection molded plate was sampled in a MC-run,
producing computer models with an equivalent FLDs. Results from models with
polydispersed fibers were compared to models comprising assemblies of
monodispersed fibers to assess whether the length distribution could be replaced
by a single length.
3.1.1 NUMERICAL
Direct FE-calculations with 3D multi-inclusion computer models were done
under periodic boundary conditions in an elongated unit cell of orthorhombic
shape. All computer models comprised fibers perfectly aligned along the x-axis of
the unit cell and placed on random positions using a MC-algorithm[47]. A typical
example is shown in Figure 5A. For monodispersed fibers the length-to-width
ratio was set to 7.5. The computer models comprising fibers with a distribution of
lengths, however, were generated in a more elongated unit cell with a length-to-
width ratio of 25 (see Figure 5A). This due to the fact that the fibers must not be
longer than the box itself because this would imply self-overlaps under periodic
boundary conditions. Previously, it was checked with monodispersed fibers that
numerical predictions are not influenced by increasing the box aspect ratio from
7.5 to 25.
In order to assure that the computer models are representative of large
laboratory samples the minimal RVE size was investigated. Five computer
models were built with unit cells of different size comprising random dispersions
of 1, 8, 27, 64 and 125 aligned fibers of aspect ratio 30 at volume fraction 15%.
For each unit cell size three MC-runs were performed which delivered three
different fiber arrangements. The elastic properties of each set of three computer
models with a particular size were calculated numerically. From the results the
arithmetic mean and the 95% confidence interval of the longitudinal Young’s
Chapter 3 - Short Fiber Reinforced Composites
- 28 -
modulus E11 were determined. Figure 3 shows that with 27 fibers one can already
get predictions for E11 deviating only a few percent from the true value provided
that one averages the results of three individual calculations. In case of larger
computer models comprising 125 fibers the individual predictions for the different
MC-configurations show hardly any scatter.
Figure 3: Predictions for the longitudinal Young’s modulus E11 depending on the size ofthe computer models (number of fibers N). The filled circles indicate the arithmetic meanof three individual estimates and the error bars depict the 95% confidence interval.
Consequently, the effective elastic properties for composites with a
monodispersed fiber length were obtained from one single MC-configuration of a
computer model comprising 100 fibers. The short fibers were assigned the
isotropic elastic properties of glass fibers and the matrix the ones of a typical
thermoplast (see Table 2).
Table 2: Isotropic phase properties for glass fibers and a model matrix.
Glass fibres Model matrix
E (GPa) 72.5 2.28
ν 0.2 0.335
α (x 10-6/°C) 4.9 117
Chapter 3 - Short Fiber Reinforced Composites
- 29 -
Based on these isotropic phase properties the Young’s modulus E11 in fiber
direction was calculated for 15 models each comprising monodispersed fibers
with an aspect ratio between 5 and 50 at a volume fraction of 15%.
The next stage was to generate computer models using a distribution of fiber
lengths. In order to be representative, a real data set, measured with image
analysis facilities developed at Leeds, was used as the basis for the computer
model generation. The measured data for 27,500 fibers, collected by Bubb from
an injection molded short glass fiber filled plate[62], is shown in Figure 4. For this
non-symmetrical distribution the number average length was determined as
388µm and the weight average length as 454µm. Assuming a common glass fiber
diameter of 10µm gives aspect ratios of 38.8 and 45.4 for the length and weight
averages, respectively.
Figure 4: Experimentally measured fiber length distribution (FLD) collected by Bubb froman injection molded short glass fiber filled plate.[62]
As described earlier, the measured FLD was used to bias the MC-runs. For
this purpose the measured frequency distribution of the fiber lengths was
transformed into the cumulative PDF. The cumulative PDF was then sampled by
generating 100 random numbers in the interval [0,1], whence each random
number corresponds to a particular fiber length. The 100 fibers with the previously
0
1000
2000
3000
4000
5000
0 400 800 1200
L (µm)
frequ
ency
Chapter 3 - Short Fiber Reinforced Composites
- 30 -
sampled fiber lengths were randomly placed in parallel in the unit cell without
overlaps at a volume fraction of 15% (see Figure 5A). The fiber length distribution
was sampled in 3 different MC-runs in order to better approximate the measured
distribution. In each of the 3 MC-runs a different seed was used for the random
number generator, which consequently delivered computer models with three
different FLDs. Averaging these three fiber length distributions excellently
approximated the experimentally measured FLD (see Figure 5B). The computer
models with the polydispersed fibers, were meshed and solved numerically in
order to determine the longitudinal modulus, E11.
Figure 5: A: Orthorhombic unit cell containing 100 randomly situated, perfectly alignedfibers of different length at volume fraction 15%. The fiber lengths were determined bysampling the measured fiber length distribution (see Figure 4) during a MC-run. All fiberswere assumed to have a diameter of 10µm. B: Measured fiber length distribution (solidline) and the average fiber length distribution of 3 different MC-runs (bars).
3.1.2 RESULTS AND DISCUSSION
The numerical results of E11 are shown in Figure 6. The diamond symbols
represent the results for different aspect ratios of monodispersed fibers, and the
solid line the best fit through all the data. The random nature of the generated
microstructures is reflected by the scatter of the points around the best fit line. It
is typical of short fiber reinforced composites that E11 is levelling off towards
larger aspect ratios. Above a certain critical fiber aspect ratio no substantial gains
in E11 can be achieved.
A
12
3
B
Chapter 3 - Short Fiber Reinforced Composites
- 31 -
Figure 6: Numerical results for E11 are depicted as filled symbols for compositescomprising monodispersed fibers. The solid line fits the simulation data best.
The question to be answered in this chapter is: “What is the length of a
monodispersed distribution, which would have the same longitudinal modulus as
the ‘real’ distribution?” Figure 7 shows a comparison of the numerical results from
simulations with monodispersed and polydispersed fiber lengths.
Figure 7: A comparison of numerical results for E11 calculated with computer modelswhich comprised either of monodispersed or of polydispersed fibers. The solid, horizontalline shows the average E11 calculated from three different MC-configurations ofpolydispersed fibers. The triangles symbolize E11 which was predicted from severalcomputer models with monodispersed fibers of different aspect ratio.
4
6
8
10
12
0 10 20 30 40 50a
E11
(GP
a)
7
9
11
13
20 30 40 50
a
E11
(GP
a)
Chapter 3 - Short Fiber Reinforced Composites
- 32 -
The horizontal lines show the band of predictions made with the three
computer models comprising polydispersed fibers, in this case 10.9 ± 0.12 GPa.
The triangles represent the predictions from the simulations with monodispersed
fibers for six particular aspect ratios. The crossing point between the best line fit
through the diamonds and the horizontal lines, determines the monodispersed
aspect ratio which matches E11 of the composite with the real distribution. For this
set of data the equivalent monodispersed aspect ratio was 36.6 ± 2.5.
To explore different regions of the modulus versus aspect ratio curve shown
in Figure 6, fiber aspect ratio distributions (ARD) were generated by using the
FLD of Figure 5B assuming different fiber diameters of 15, 20 and 25µm. Figure
8 shows the ARD for fiber diameters of 10µm (as used so far) 15µm and 20µm.
One can see that as the fiber diameter is increased the distribution is pushed to
lower aspect ratios. As above, the monodispersed length needed to match the
modulus of the ‘real’ distribution was determined for each distribution.
Figure 8: ARD for fibers with diameter d of 10, 15 and 20 microns generated by usingthe measured FLD of Figure 5B.
Results are shown in Figure 9 and Table 3. Although the monodispersed
fiber aspect ratio that matches the properties of composites with polydispersed
fibers does not fit exactly with one of the four considered averages, the number
average LN appears to be the best choice to cover the whole range of likely aspect
0
1000
2000
3000
4000
5000
0 20 40 60 80 100 120
a
frequ
ency
10 microns 15 microns 20 microns
Chapter 3 - Short Fiber Reinforced Composites
- 33 -
ratios. This result explains why the number average LN has proved so successful
in substituting FLDs, although until this point there has been little justification for
its use.
Figure 9: For different fiber diameters d the monodispersed fiber aspect ratios a (filledcircles) are depicted which match the E11 predictions of computer models comprisingpolydispersed fibers. The four lines represent the different averages that wereconsidered.
Table 3: For different fiber diameters the monodispersed fiber aspect ratio a is listedwhich matches the E11 predictions from computer models comprising polydispersedfibers. Also the numerical values of the four considered average types are listed.
3.2 COMPARISON BETWEEN MICROMECHANICAL MODELS, NUMERICAL PREDICTIONS AND MEASUREMENTS
In this subchapter the focus is on another type of morphological
imperfection, namely the one of fiber misalignments. The goal was to reproduce
10
20
30
40
50
5 10 15 20 25 30d (µm)
a
Monodispersed fibers
Number average
Weight average
RMS average
Skewed average
d (µm) 10 15 20 25 a 36.6 ± 2.5 24.3 ± 1.4 20.7 ± 0.5 15.8 ± 0.5
aN 38.8 25.9 19.4 15.6 aW 45.4 30.2 22.7 18.1
aRMS 41.9 28.0 21.0 16.8 aS 33.2 22.1 16.6 13.3
Chapter 3 - Short Fiber Reinforced Composites
- 34 -
measured fiber orientation distributions in 3D-multi-inclusion computer models, to
numerically calculate their thermoelastic properties and to compare the results
with experimental measurements and micromechanical models. For this purpose,
the fiber orientation distributions of two differently processed short glass fiber
reinforced composites were determined experimentally and subsequently
reproduced in 3D multi-inclusion computer models. In analogy to the previous
subchapter, the two measured fiber orientation distributions were sampled during
a MC-run.
3.2.1 MICROMECHANICAL MODELS
Micromechanical models combined with the orientation averaging scheme
can be used to predict the elastic properties of composites with misaligned fibers.
For this purpose the composite is considered as an aggregate of elastic units
comprising perfectly aligned fibers, whose properties can be calculated by a
micromechanical model. The properties of the aggregate are predicted by
orientation averaging the unit properties according to the measured orientation
distribution via the tensor averaging scheme described in chapter 2.2. Crucially,
the averaging can be done either assuming constant strain between the units
(averaging the stiffness constants of the units) which leads to an upper bound
prediction, or by assuming constant stress between the units (averaging the
compliance constants of the units) which leads to a lower bound prediction. The
advantage of the numerical approach of Gusev employed here is that only a
single estimate is produced, with no assumptions of constant strain or stress
being imposed.
In terms of the unit predictions, the micromechanical models chosen here
were those accepted as the most appropriate in literature[32,53]. For isotropic
fibers (i.e. glass) the approach of Tandon and Weng [33] is widely accepted as
giving the best unit predictions. The Halpin-Tsai model was chosen because it is
the most widely used micromechanical model in industry.
With respect to the thermal expansion, the overall CTEs αi of two phase
composites with arbitrarily shaped phases are uniquely related to the overall
Chapter 3 - Short Fiber Reinforced Composites
- 35 -
elastic compliances , and one can use the explicit formula of
Levin:[24,71]
(34)
The superscripts 1 and 2 stand for the fiber and the matrix phases,
respectively, and the general summation convention is used for the indices
occurring twice in a product. Thus, for any composite with a single type, fully
aligned but not necessarily equal length fibers, the overall thermal expansion
coefficients αi are not truly independent entities and one can always use Eq. (34)
to obtain the αi in a simple calculation from the accurate in principle numerical Cik.If both fibers and matrix are isotropic the Levin formula can also be applied to
calculate the CTE of composites with misaligned fibers because still it can be
viewed as a two phase composite. However, for anisotropic fibers e.g carbon
fibers Eq. (34) is not valid any more because differently oriented fibers have
generally different laboratory-frame elastic constants. Since in this chapter we
deal with composites where both matrix and misaligned fibers are isotropic the
Levin formula was used to compute the CTEs of composites with misaligned
fibers.
3.2.2 EXPERIMENTAL
Circular dumbbells (see Figure 10) were injection molded by conventional
and shear controlled orientation injection molding (SCORIM) using a mold gated
at both ends. During processing, the flow from the larger to the smaller dumbbell
cross section produces preferred fiber alignment in the smaller central section
due to elongational flow.[64] The first set of samples was produced by
conventional injection molding where the polymer/glass-fiber melt was injected
into the mold through one gate of the mold before the sample was cooled down.
For the second set of samples the SCORIM process developed at the University
of Brunel was used. Again the polymer/glass-fiber melt was injected through one
gate but during cooling of the sample, the polymer melt containing the glass fibers
was forced back and forth through the mold cavity using both gates of the mold.
Sik Cik1–=
α i αk1( ) αk
2( )–( ) Skl1( ) Skl
2( )–( )1–Sli Sli
2( )–( ) α i2( )+=
Chapter 3 - Short Fiber Reinforced Composites
- 36 -
Due to the additional shear forces experienced by the melt during the SCORIM
process, the fibers are more aligned along the dumbbell axis than in
conventionally injection molded samples.[65]
Figure 10: Picture of a circular dumbbell manufactured by injection molding from a glass-fiber-polypropylene granulate.
The material used was a glass-fiber-polypropylene granulate from Hoechst,
Grade G2U02, containing 20 wt% of short fibers. The polypropylene was an easy
flowing injection molding grade with a melt flow index of 55. Specifications of the
thermoelastic properties of the polypropylene matrix and the glass fibers are
listed in Table 4.
Table 4: Thermoelastic properties of polypropylene and glass fibers that were used tocalculate the overall properties of short glass fiber reinforced composites bothnumerically and by the use of micromechanical models.
The degree of fiber orientation in each type of injection molded samples was
measured on a two-dimensional longitudinal cut through the axis of the central
gauge length section, using an image analysis system[66] developed in-house at
the University of Leeds. This image-analysis system, whose reliability and
accuracy has already been validated,[67] was used to measure the angular
deviations θ (see Figure 2) of the glass fibers from the ideal orientation along the
80mm
5mm 8.5mm
25mm
Polypropylene Glass fibres E [GPa] 1.57 72.5 ν 0.335 0.2 α [x 10−6 Κ−1] 108.3 4.9
Chapter 3 - Short Fiber Reinforced Composites
- 37 -
dumbbell axis. The orientation in both samples was found to be non-uniform with
a well aligned shell region around a central, less well aligned, core. This pattern
of fiber orientation was found to be symmetric about the centre line of the section
and consistent along the gauge length.
Typical image frames (700µm x 530µm) taken from the shell region of eachsample type are shown in Figure 11 with the injection axis in horizontal direction.
It is clear that in the SCORIM sample the fibers are more highly aligned along the
1-axis compared to the conventionally molded sample, which itself has a high
preferential alignment. To compare with mechanical measurements, the fiber
orientation distributions for each gauge length cross section was required. To
produce this distribution, the 2D image analysis data was divided into 10 strips
across the sample diameter and then normalized in terms of the appropriate
angular area. As the distributions were found to be transversely isotropic, for
orientation averaging purposes they can be described by only two orientation
averages, and . The measured values of these two averageswere 0.872 and 0.769 for the conventionally and 0.967 and 0.936 for the SCORIM
molded samples, for the second and fourth orders respectively.
Figure 11: Figure 11A and Figure 11B show longitudinal cuts through the gauge sectionof a conventionally and a SCORIM injection molded dumbbell, respectively. Typicalimage frames (700 x 530 µm) from the shell region of the samples’ gauge section aredepicted.
In order to measure the fiber length distribution of the two samples the
polypropylene matrix was first burnt off at a temperature of 450°C in a furnace.
The remaining glass fibers were spread onto a glass dish and their length
BA
Chapter 3 - Short Fiber Reinforced Composites
- 38 -
distribution was determined by image analysis. The burn off technique was also
used to confirm that the weight fraction of the glass fibers was 20%, which is
equivalent to a volume fraction of 8%.
The Young’s modulus E11, of the glass fiber reinforced samples was
measured in a tensile test at a constant strain rate of 10-3 s-1. The sample strain
was measured using a Messphysik video extensometer and 10 samples were
measured for both conventionally and SCORIM processed dumbbells. To
determine the properties of the matrix phase, compression molded plates were
made from pellets of the unreinforced polymer. The matrix Young’s modulus was
measured using the same technique as above, while the Poisson’s ratio was
determined using an ultrasonic immersion method.
CTEs were determined for both short fiber reinforced samples and
unreinforced polypropylene using a dilatometer by measuring the length change
of the samples for a temperature change from +10 to +30 °C both in the
longitudinal and the transverse direction of the dumbbells.
3.2.3 NUMERICAL
Computer models comprised 150 misaligned fibers of equal aspect ratio
randomly positioned in a cubic unit cell at a volume fraction of 8%. For both a
conventionally and a SCORIM molded sample the length distribution of the fibers
was measured. In the previous subchapter it was found that the number average
LN is the best choice to substitute a fiber length distribution by a single fiber
length. As a consequence in the computer models for the conventionally molded
composite the number average LN = 448µm was assigned to all fibers whereasfor the SCORIM molded composites a slightly smaller number average
LN = 427µm was employed. The diameter of the glass fibers was measured aswell and found to be 12µm in both samples. The specification of number N,length L and diameter d of the fibers determines the total fiber volume. Since we
know that the fiber volume fraction is 8% the length of the cubic unit cell is
determined. In a MC-run the cumulative PDF for each of the two measured θ -
distributions was sampled with 150 random numbers in the interval [0,1]. Each
Chapter 3 - Short Fiber Reinforced Composites
- 39 -
random number assigns an angle θ to one of the 150 fibers in the computer
model. Measurements elucidated that the angle φ is homogeneously distributed
in the interval [0°, 360°] which means that the gauge section of the dumbbell was
transversely isotropic. Therefore another 150 random numbers were used to
randomly determine the second angle φ in the interval [0°, 360°].
Figure 12: The average θ-distribution (grey bars) of 450 fibers in three different MC-snapshots which were generated by sampling the PDF (black curve) in three MC-runs isshown for the conventionally (left) and for the SCORIM molded dumbbell (right). Theangle θ characterizes the fibers’ misalignments in a transversely isotropic composite.
After having specified length, diameter and orientation of all 150 fibers they
were successively placed in the unit cell on random positions while a subroutine
checked for overlaps with already positioned fibers. If overlaps occurred the
position was rejected and the MC-algorithm repeated the procedure until the fiber
could be placed without overlaps and until all fibers were inserted into the unit cell.
Because the fibers were misoriented it was impossible to randomly place the
fibers without overlaps even at the relatively small volume fraction of 8%. This
problem was solved by increasing the box size and inserting the fibers at a dilute
volume fraction of 0.1%. The box was then compressed during a variable-box-
size MC-run to the desired volume fraction of 8% keeping the fiber orientations
constant while repeatedly displacing each fiber in the unit cell. In Figure 14 a
computer model for both the conventional and the SCORIM morphology is shown
together with a cut through the FE-mesh. In order to obtain information about the
Chapter 3 - Short Fiber Reinforced Composites
- 40 -
scatter of the numerical predictions three different MC-snapshots were generated
for both composite morphologies by sampling the cumulative PDFs with three
different seeds for the random number generator. By averaging the individual
orientation distributions of the three computer models the measured distribution
was better approximated (see Figure 12).
Figure 13: On the left side 3D multi-fiber computer models are shown for both theconventional (top) and the SCORIM (bottom) morphology. On the right side longitudinalcuts through the FE-mesh of both computer models are depicted.
The 6 computer models (3 for the conventional and 3 for the SCORIM
morphology) were meshed into unstructured, morphology-adaptive FE-meshes
and numerically solved for the overall, effective thermoelastic properties. The FE-
meshes of all 6 computer models consisted of about 2.4 x 106 nodes and 15 x 106
tetrahedra. Calculations were done on a HP Visualize J6700 Workstation with two
PA-RISC 8700 processors and took about 25 hours for 7 strain energy
1
2
3
Chapter 3 - Short Fiber Reinforced Composites
- 41 -
minimizations (6 minimizations to determine the elastic properties and 1
minimization to determine the CTEs) on a single processor.
3.2.4 RESULTS AND DISCUSSION
In this section we present both experimental and numerical results for glass
fiber reinforced polypropylene composites and compare them with values which
were computed by two micromechanical models, namely the ones of Tandon-
Weng[33-36,69] and Halpin-Tsai[28,70], together with the orientation averaging
scheme. Experimental, numerical and micromechanical results of the Young’s
modulus E11 in the longitudinal direction of the glass-fiber/polypropylene
dumbbells are listed in Table 5.
Table 5: The Young’s modulus E11 in the longitudinal direction of both conventionallyand SCORIM injection molded glass-fiber/polypropylene dumbbells. Measured andnumerical results are listed together with micromechanical model predictions.
For the longitudinal Young’s modulus E11 there is an excellent agreement
between the numerically calculated and measured values. The numerically
calculated E11 is nominally higher than the measured value for both the
conventional and the SCORIM sample but the difference is less than 1% and is
well inside the error range of the measurements. The Tandon-Weng model
combined with orientation averaging for determining the aggregate properties
Young’s Modulus E11 [GPa]
Conventional SCORIM
Measured 5.09 ± 0.25 5.99 ± 0.31
Numerical 5.14 ± 0.1 6.04 ± 0.02
Tandon-Weng + Orientation Averaging
Upper bound 5.13 5.91 Lower bound 3.94 5.51 Halpin-Tsai + Orientation Averaging
Upper bound 4.42 5.02 Lo