rights / license: research collection in copyright - non ...26498/eth... · property predictions...

Research Collection

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

This page was generated automatically upon download from the ETH Zurich Research Collection. For moreinformation please consult the Terms of use.

ETH Library

https://doi.org/10.3929/ethz-a-004554826http://rightsstatements.org/page/InC-NC/1.0/https://www.research-collection.ethz.chhttps://www.research-collection.ethz.ch/terms-of-use

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


3.3 Stiffness and Thermal Expansion of Short Fiber Composites with Fully

Aligned Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.1 Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Table of Contents


3.4 Prediction of Stiffness and Thermal Expansion by the Orientation

Averaging Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.1 Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


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


- 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].


- 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.


- 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


- 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.


- 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--------=


- 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+=


- 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+-----------------------------------------------+=


- 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+ +( )+ +=


- 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-------


- 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.


- 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


- 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∫°= =


- 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=–=


- 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〈 〉= =


- 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+ +( )+=


- 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


- 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= =


- 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∑

-----------------=


- 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∑

------------------=


- 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


- 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


- 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


- 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


- 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)


- 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


- 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


- 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


- 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( )+=


- 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


- 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


- 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


- 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


- 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


- 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

rights / license: research collection in copyright - non ...26498/eth... · property predictions...

Documents