Ceramic and Polymer Matrix Composites: Properties, Performance and Applications (Polymer Science and Technology)

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POLYMER SCIENCE AND TECHNOLOGY

CERAMIC AND POLYMER MATRIX COMPOSITES: PROPERTIES, PERFORMANCE AND APPLICATIONS

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POLYMER SCIENCE AND TECHNOLOGY

CERAMIC AND POLYMER MATRIX COMPOSITES: PROPERTIES, PERFORMANCE AND APPLICATIONS

EROS DIMITRIOU AND

MARCO PETRALIA EDITORS

Nova Science Publishers, Inc. New York

Copyright © 2010 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Ceramic and polymer matrix composites : properties, performance, and applications / editors, Eros Dimitriou and Marco Petralia. p. cm. Includes index. ISBN 978-1-61668-467-9 (eBook) 1. Ceramic-matrix composites. 2. Polymeric composites. I. Dimitriou, Eros. II. Petralia, Marco. TA418.9.C6C385 2009 620.1'4--dc22 2009044321

Published by Nova Science Publishers, Inc.

New York

CONTENTS Preface

vii

Chapter 1

Surface Modification of Ultra High Modulus Polymeric Fibers: Effects on Interfacial Adhesion and Mechanical Properties of Epoxy Resin Composites Petroula A. Tarantili

1

Chapter 2

Manufacturing and Features of Acoustically Optimized Natural Fibre Reinforced Plastics N. Aisenbrey and L. Frormann

51

Chapter 3

Effects of Microstructure on Residual Stresses in DSE Al2O3/YAG Ceramic Composite by Experimental and Numerical Investigations J.J. Sha, S. Ochiai, H. Okuda, S. Iwamoto, K. Morishita, Y. Waku, N. Nakagawa, A. Mitani, T. Ishikawa and M. Sato

87

Chapter 4

Introduction of Particle Dispersion Reinforced Ceramic Matrix Composites Bin Li, Jianxin Deng and Hong Wang

119

Chapter 5

A Novel Strategy for Developing Polymer Nanocomposite with High Dielectric Constant Jing-Wen Wang and Shu-Qin Li

147

Chapter 6

Polymer Matrix Composites: Properties, Performance and Applications Milan Kracalik, Stephan Laske and Clemens Holzer

169

vi

Contents

Chapter 7

Contact Strength of Ceramic Materials Lucia Hegedűsová, Ladislav Ceniga and Ján Dusza

195

Chapter 8

New Analytical Model of Thermal Stresses and Analytical Fracture Mechanics in Two-Component Materials. Application to Two-Component Ceramics Ladislav Ceniga

247

Chapter 9

Finite Element Thermal Analysis of Ceramics Matrix Composites M.A. Sheikh

297

Chapter 10

Damage Reduction Methods in Drilling Polymeric Matrix Composites A. Di Ilio and A. Paoletti

341

Index

355

PREFACE In general, the strength of ceramic materials is influenced by the presence of flaws, originating during the material processing and representing fracture origins to be accordingly a reason of material fracture. In this book, fabrication processes of particle reinforced ceramic are introduced, which provide a useful set of variables for model experimental and theoretical studies. Toughening mechanisms of particle dispersion reinforced ceramic matrix composites are also discussed, and the practice of the composites containing the latest research results are introduced. In addition, the authors discuss the application of the contact and fourpoint bending tests on monolithic ceramic materials at room temperature to determine the strength of ceramic materials. Other chapters in this book examine fibre reinforced plastics, which are used worldwide in various applications. A novel nanofabrication strategy to develop high dielectric constant nanocomposite is also introduced, as well as a novel approach for systematic investigation of polymer nanocomposites. In Chapter 1, surface treatments of aramid fibers by methacryloyl chloride/carbon tetrachloride solution, hydrolysis with sodium hydroxide solution and coating with phenolic polymers, such as resole and novolac resins, were performed in an attempt to improve the adhesive bonding of those fibers with epoxy resin matrices. The modified aramid fibers were subsequently used for preparation of composite materials with epoxy matrix. The mechanical properties of these composites were investigated and the results explained taking into consideration the surface characteristics of modified fibers, as determined by pullout tests and contact angle measurements. Treatment of aramids with methacryloyl chloride was found an interesting modification for subsequent use of those fibers as reinforcement in epoxy matrices. Flexural properties of the above composites are significantly improved and the obvious effect on the fiber surface

viii

Eros Dimitriou and Marco Petralia

is a change in morphology. Coating of aramid fibers with phenolic resins seemed adequate to promote interfacial adhesive bonding to epoxy matrices, due to the changes in fiber surface profile, the affinity with the aramid substrate and the chemical reactivity to the epoxy resin. In the context of the above research, corona and chromate surface-treated UHMPE fibers, as well as calendered fibers were used as reinforcement for epoxy resins and the obtained composites were investigated for their mechanical properties as a function of the fiber morphology and characteristics. The treated fibers were tested by scanning electron microscopy (SEM), tensile and pull-out testing as well as contact angle measurements. Mechanical tests, such as flexural and interlaminar shear strength (ILSS), have also been performed with unidirectional specimens reinforced with the above fibers. The results showed that corona and chromate treated fibers give the highest adhesive bonding, whereas their tensile strength is reduced. The same is also observed with the flexural tests, where maximum strength corresponds to these two types of fibers for various filler volume fractions. Finally, the data derived from ILLS suggest a slight advantage for the corona treated fibers. Regarding the performance of treated UHMPE fibers it was shown that corona and chromate treatments modify their surface and give products with reduced strength, but lead to increased adhesive bonding. On the other hand, calendered fibers retain their strength with significant reduction of modulus due to some relaxation during hot calendaring. Properties dependent on the off-axis properties of the composite specimens, such as flexural and interlaminar shear strength, showed that corona treated fibers are advantageous as reinforcement for epoxies. As explained in Chapter 2, fibre reinforced plastics are used worldwide in various applications. Large quantities of natural fibre reinforced plastics are processed in the automobile industry due to their good mechanical and ecological characteristics, but particularly because of their low price. The application range is mainly in low loaded places, e.g., in the inside of automobiles like the side door panels or ceiling. Initially the market for natural fibres showed a rapid upswing that stagnated afterwards due to their limited range of applications. To force the application, supplementary functions must be integrated into the materials with no additional costs. Usually the focus of the research is on the increase of the mechanical characteristic values and the development of more economical and productive manufacturing methods. The high acoustic potential of the natural fibres is less investigated and only considered as a side effect. But this acoustic potential enables many application possibilities of the reinforced plastics. Particularly, the health endangerment by the constant increase of the noise

Preface

ix

pollution clarifies the necessity for the research. Acoustically effective structural elements can be manufactured as a sandwich. The building method of a sandwich element permits a large variation range. With simple changes in the structure, for example by the use of different layers with different characteristics, the material can be adapted to several applications. An effective sound absorption particularly in the middle and high frequency ranges is possible with layers of natural fibre nonwovens or open porous natural fibre reinforced plastics. Sound absorption values over 0.7 are reached at frequencies above 1 kHz. High weighted difference level Dw values of up to 30 dB can be achieved with very thin natural fibre reinforced composite plates, which exceed the mass law up to 4 dB. The sound absorption achievement of multilayered elements with firm surface layers and different core materials is very good with weighted difference level values of 34– 37 dB with low element weights between 13–16 kg/m². For these high strength lightweight construction units with positive acoustic characteristics in sound insulation and sound absorption, various applications exist in the architectural acoustics and in the automobile industry. The application of natural fibre reinforced elements as multi functional wall and ceiling panels permits a strong stimulation of the market for these materials. In Chapter 3, the effect of microstructure on the residual stresses in directionally solidified eutectic (DSE) Al2O3/Y3Al5O12 (YAG) ceramic composite were investigated by X-ray diffraction technique and finite element method (FEM). In the X-ray stress measurement, the YAG skeleton derived from the Al2O3/YAG composite by dioxidation of the Al2O3 phase was used as a reference specimen without thermally-induced stress, and the X-ray stress measurements with CuKα1 irradiation were performed on the two faces of a cubic specimen, namely, the faces parallel and perpendicular to the solidification direction, respectively. On the other hand, a numerical analysis using finite element method (FEM) which represents the actual microstructure features of the experimental specimen was carried out in different local regions with different morphologies to reveal the effect of microstructure on the distribution of residual stress in the composite. The distributions of residual stresses in both constituting phases were mapped by FEM calculation. Meanwhile, the mapping of residual stress indicated that the distribution of residual stress in the interior of each phase was not homogeneous being dependent on the solidification direction and local morphologies of constituting phases such as curvature of interfaces, array and volume fraction. The experimentally measured residual stresses were accounted for by the FEM analysis.

x


The study of brittle material mechanics is still a young and living science. Even now, the making of ceramic pieces for use in mechanical design, that are subjected to dimensional tolerances similar to those used for steel, or of complex geometry, gives rise to difficult problems, which hinder the large scale development of these materials, despite their extraordinary intrinsic properties. Particle dispersion reinforcing is an important way to improve the performance of ceramic matrix composites. In Chapter 4, fabrication processes of particle reinforced ceramic are introduced, which provide a useful set of variables for model experimental and theoretical studies. Toughening mechanisms of particle dispersion reinforced ceramic matrix composites are also discussed. When the second phase is added into the matrix, usually there existe the mismatch of structural, mechanical or physical properties, the mismatches of linear expansion coefficient and elastic modulus are related to the toughening and strengthening of ceramic. Meanwhile, the practice of the composites containing the latest research results is also introduced. In Chapter 5, a novel nanofabrication strategy to develop high dielectric constant nanocomposite was introduced using poly(vinylidene fluoride) (a piezoelectric polymer) as matrix and PCMS-g-CuPc [poly(p-chloromethyl styrene) (PCMS) grafted with copper phthalocyanine oligomer (CuPc, a planar multiring organic semiconductor with super high dielectric constant more than 105)] as filler. Improvement of the dispersibility of CuPc oligomer in polymer matrix was confirmed by TEM-observed morphologies. The PCMS-g-CuPc particles with a average size of about 80nm are dispersed in poly(vinylidene fluoride) matrix, while in PCMS-g-CuPc particles the PCMS acts as “matrix” which contains dispersed CuPc balls with a average size of ca. 25nm [about 1/20 of that of CuPc particles in the simple blend of poly(vinylidene fluoride) and CuPc]. The slution-cast nanocomposite film sample with only 15wt% CuPc can realize a dielectric constant of about 325 at 100Hz, more than 38-fold enhancement with respect to that of the pure PVDF. The enhanced dielectric response in the nanocomposite demonstrates the significance of the interface effect in raising the material responses far beyond that expected by simple mixing rules when there is a large dielectric contrast between the two components in the composite. Theoretic prediction using modified Cole-Cole equation indicates that, even at the high frequency limit, the dielectric constant of the nanocomposite can still be more than 70. Furthermore, the dielectric strength increase from 40 MV/m of the simple physical blend of poly(vinylidene fluoride) and CuPc to 51 MV/m of the nanocomposite sample. All these will lead to a nanocomposite with better controlled process conditions and are highly desirable for high dielectric constant composites.

Preface

xi

Preparation of polymer-clay nanocomposites by melt mixing has been intensively investigated in the last 20 years. However, only systems based on polyamide matrix have been commercialized. The problem consists mostly in question how to up-scale the results obtained using laboratory equipment. Most of research institutions dealing with nanocomposites compounding are equipped by "micro" twin screw extruders, which are able to process only small amount (in order of grams per hour) of material. In the industry, compounding throughput in order of kilograms up to tonnes per hour is usual. However, industrial compounders are normally not available for research purpose. Furthermore, the broad range of industrial compounders (differences in screw geometry, screw diameter, L/D relation etc.) makes any general conclusions impossible. In Chapter 6, approach for systematic investigation of polymer nanocomposites is presented. A way from "micro" to "advanced" compounding using laboratory as well as semi-industrial twin screw extruders is described. Chapter 7 deals with the determination of strength of ceramic materials by mechanical tests in bending and contact modes, and subsequently with nanostructural and microstructural analyses of strength-degrading defects. The bending and contact modes are simulated by a four-point bending test and a single-cycle contact tests using rollers or a single/multicontact test using spheres, respectively. Additionally, a combination of the multi-contact test using spheres and the four-point bending test as well as contact fatigue test, i.e. the multi-cycle contact test using spheres to be performed till material failure, are also considered. In general, the determination of strength of ceramic materials results from statistical methods, usually represented by the Weibull analysis to result in the determination of the characteristic strength σ 0 and the Weibull modulus m. Accordingly, the characteristic strength

σ 0,bend and σ 0,cont as well as the Weibull

moduli mbend and mcont, related to the four-point bending test and the single-cycle contact test using rollers, respectively, are determined along with the determination of an experimental relationships between σ 0,bend , σ 0,cont and an experimental relationship between mbend, mcont, where the experimental relationships consequently confirm the validity of the Fett theory. The nanostructural and microstructural analyses of strength-degrading defects are represented by the determination of fraction origins in the bending and contact modes to induce different types of cracks. Finally, in addition to the determination of the mechanical loading to cause material failure, a detailed analysis of parameters of the cracks related to different experimental conditions along with a detailed analysis of morphology of crack surfaces are also presented. The

xii


mechanical tests were applied to monolithic, composite and laminar ceramic materials, where the composite and laminar ceramic materials are additionally acted by thermal stresses originating due to different thermal expansion coefficients of material components. Chapter 8 deals with a new analytical model of thermal stresses in an isotropic continuum represented by periodically distributed isotropic spherical particles in an isotropic infinite matrix. The multi-particlematrix system to represent a model system regarding the analytical modelling is applicable to a two-component material of a precipitate-matrix type, consisted of isotropic components. The thermal stresses as functions of microstructural parameters (particle volume fraction, particle radius, inter-particle distance) originate during a cooling process as a consequence of the difference in thermal expansion coefficients. Additionally, integrals of elastic energy density along curves in the spherical particles and the matrix are derived to result in the determination of a critical particle radius as a consequence of the initiation of cracks in the components of any two-component material (ductile, brittle) along with the determination of functions describing the crack shapes in a plane perpendicular to a direction of the crack propagation in an ideal-brittle particle and an ideal-brittle matrix. With regard to the ideal-brittle particle and the idealbrittle matrix resulting in an application of the presented results to ceramic (brittle) components, the new analytical model, and consequently an analysis of the crack initiation and the crack propagation are applied to the SiC-Si3N4 and SiC-MoSi2 multi-particlematrix systems representing two-component ceramic materials of the precipitatematrix type. Modelling and analysis of a unique geometrically representative Unit Cell has been shown here as the key to predicting the macro thermal transport behaviour of composites, which otherwise requires the employment of a vast experimental infrastructure. Sophisticated materials, such as woven Ceramic Matrix Composites (CMCs), have very complex and expensive manufacturing routes, used by just a few research organizations. This broadens the scope of a modelling strategy to be adopted for the characterization of all possible material designs with various possible constituent volume fractions by using a commercial FE code such as ABAQUS. The variation of material constituents can be incorporated in the Unit Cell model geometry with subtle manipulation of key parameters dictated by quantitative SEM morphological data. Two CMC material systems have been modeled in the Chapter 9. The first material has been analysed with a focus on the homogenization of microscopic constituent material properties into the macroscopic thermal transport character. The actual set of property data used for the Unit Cell of this material is obtained from the cumulative property degradation

Preface

xiii

results extracted from the analyses of three sub-models based on the material’s unique porosity data. After validating the modeling methodology through a comparison with the experimental data, a geometrically more challenging CMC is modelled with a detailed incorporation of its morphological complexity in order to predict its macroscopic thermal transport behavior. Finally, it is shown how these models can be more efficiently analysed in a multi-processing parallel environment. Drilling of polymeric matrix composites may generate several kinds of damage, which can lead to unacceptable material degradation. The aim of Chapter 10 is to present a literature review on the principal methods adopted to enhance holes quality. The major damage is certainly the delamination that can occur both on the entry and exit sides of the workpiece. The level of delamination is related to a critical thrust force value which is dependent on the workpiece material and the uncut layer from the main body of the laminate.

In: Ceramic and Polymer Matrix Composites ISBN: 978-1-60741-896-2 Editors: E. Dimitriou et al, pp. 1-50 © 2010 Nova Science Publishers, Inc.

Chapter 1

SURFACE MODIFICATION OF ULTRA HIGH MODULUS POLYMERIC FIBERS: EFFECTS ON INTERFACIAL ADHESION AND MECHANICAL PROPERTIES OF EPOXY RESIN COMPOSITES Petroula A. Tarantili Polymer Technology Lab., School of Chemical Engineering, National Technical Univ. of Athens Heroon Polytechniou 9, Zographou, Athens, Greece, GR 15780

Abstract Surface treatments of aramid fibers by methacryloyl chloride/carbon tetrachloride solution, hydrolysis with sodium hydroxide solution and coating with phenolic polymers, such as resole and novolac resins, were performed in an attempt to improve the adhesive bonding of those fibers with epoxy resin matrices. The modified aramid fibers were subsequently used for preparation of composite materials with epoxy matrix. The mechanical properties of these composites were investigated and the results explained taking into consideration the surface characteristics of modified fibers, as determined by pull-out tests and contact angle measurements. Treatment of aramids with methacryloyl chloride was found an interesting modification for subsequent use of those fibers as reinforcement in epoxy matrices. Flexural properties of the above composites are significantly improved and the obvious effect on the fiber surface is a change in morphology. Coating of aramid fibers with phenolic resins seemed adequate to promote interfacial adhesive bonding to epoxy matrices, due to the changes in fiber

2

Petroula A. Tarantili surface profile, the affinity with the aramid substrate and the chemical reactivity to the epoxy resin. In the context of the above research, corona and chromate surface-treated UHMPE fibers, as well as calendered fibers were used as reinforcement for epoxy resins and the obtained composites were investigated for their mechanical properties as a function of the fiber morphology and characteristics. The treated fibers were tested by scanning electron microscopy (SEM), tensile and pull-out testing as well as contact angle measurements. Mechanical tests, such as flexural and interlaminar shear strength (ILSS), have also been performed with unidirectional specimens reinforced with the above fibers. The results showed that corona and chromate treated fibers give the highest adhesive bonding, whereas their tensile strength is reduced. The same is also observed with the flexural tests, where maximum strength corresponds to these two types of fibers for various filler volume fractions. Finally, the data derived from ILLS suggest a slight advantage for the corona treated fibers. Regarding the performance of treated UHMPE fibers it was shown that corona and chromate treatments modify their surface and give products with reduced strength, but lead to increased adhesive bonding. On the other hand, calendered fibers retain their strength with significant reduction of modulus due to some relaxation during hot calendaring. Properties dependent on the off-axis properties of the composite specimens, such as flexural and interlaminar shear strength, showed that corona treated fibers are advantageous as reinforcement for epoxies.

Introduction 1. Aramid Fibers 1.1. General Aspects It is well known that fully aligned, high performance polymer fibers display increased tensile strength and modulus and, therefore, many publications on fibers produced from rigid-rod polymers of high molecular weight have already focused on this fact.[1] The first type of aligned polymeric fibers commercialized for the composites industry were aramids. These fibers are produced from long-chain aromatic polyamides by the use of special wet spinning techniques. The most well known aramid fiber in current commercial production is Kevlar, which was introduced into the composites market by Du Pont, in the early 1970s. As mentioned above, aramid fibers are produced by extrusion and spinning process, which guarantees proper alignment of the stiff polymer molecules parallel to fiber axis. More specifically, their morphology includes rigid planar sheets with the chain-extended molecules hydrogen bonded together. The sheets are stacked to

Surface Modification of Ultra High Modulus Polymeric Fibers

3

produce a crystalline array in the form of a radial system of axially pleated lamellae. Each component of the pleat is about 500nm long and they are separated by short transitional bands. The angle between adjacent components is about 170o, as shown schematically in Figure 1.[2] Aramid fibers present a unique combination of stiffness, high strength, high toughness and thermal resistance. These excellent properties in combination with low density make them the recommended reinforcement for high performance composites structures used in production of spacecraft, aircraft, automobiles, military and sporting goods. However, a drawback restricting their use, is poor adhesion with most polymeric matrices due to their inert chemical structure and smooth surface, which prevents chemical and mechanical bonding to various substrates.[3] As a matter of fact, the mechanical properties of fiber-reinforced composites are highly dependent on the interactions between fiber and matrix, since the main role of the interface in composites is to transfer the load from the matrix to the fibers.

Figure 1. The radially arranged pleated structure of aramid fibers.

To take full advantage of the mechanical properties of the fiber and matrix, the interfacial shear strength between the fiber and matrix must be greater than the failure shear strength of the matrix or of the fiber. Several mechanisms that contribute to adhesion have been identified, namely mechanical, physical interaction and chemical bonding at the fiber/matrix interface.

4

Petroula A. Tarantili

Due to the poor adhesion of aramid fibers to most polymeric matrices, their composites are characterized by relatively poor off-axis properties. This limitation is further aggravated by the skin-core morphology and the weaker skin properties. In fact, it was observed that aramid-epoxy interfacial failure involves fibrillation at the fiber outer surface, which suggests the presence of a cohesive weak layer on the fiber exterior that can fail at low shear levels, resulting in low values of interfacial shear strength and consequently insufficient fiber-matrix load transfer.[4] It should be noted, however, that in the case of composites for impact performance, weak fiber–resin interactions and high sliding friction are desirable to enhance their energy absorption capabilities. Structural applications, on the other hand, demand strong interactions in order to ensure load transfer between fiber and matrix. Fiber–resin interactions, such as the degree of chemical bonding and physical interlocking are governed by the wetting behavior of the fiber. Intimate contact between solid and liquid is a necessary condition for good adhesion, which in turn, plays an important role for long-term durability of the composite.[5]

1.2. Chemical Treatments From the early stages of commercial application of aramids, a great variety of surface treatments and modification techniques, including chemical reaction, etching of the surface and coating with sizing agents, have been studied in the attempt to enhance adhesive bonding to polymeric matrices and efficient control of the fiber-matrix interfacial characteristics. Thus, the surface roughening of aramid fibers was investigated by water treatment.[6] The obtained product after this treatment exhibited lower adhesion properties than those of sized fibers, but superior bonding than that corresponding to untreated fibers. Many procedures of surfaces treatments have also been developed by Vaughan [7], who studied the effect of coupling agents on the fiber surface as a means of improving the performance characteristics of aramid containing composites. Various organic sizing agents were tested, but no evidence of improvement upon the baseline adhesion between bare fiber and matrix was detected.[8] A very interesting modification of the aramid surface was that of metalation, a process taking place in a solution of sodium methylsulfinylcarbanion, a chemical reaction claimed to be capable of introducing various functional groups to the fibers and tailor their interfacial properties according to the chemical nature of the matrix. Remarkable improvement in mechanical properties was reported in


5

the case of using matrices made of ionomeric polymers reinforced with fibers modified with carboxymethyl groups.[9] Similarly, Keller et al. [10] have studied a solution reaction scheme for Kevlar fibers, based on the hydrolysis of amide linkages in strong aqueous acidic and basic solutions. After hydrolysis, a bifunctional epoxy monomer was reacted with the free amine groups, that was expected to graft on the surface and yet retain reactive sites for further coupling with the epoxy matrix. Wu and Tesoro [11] incorporated amine functional groups on the fiber surface by bromination followed by amomonolysis, nitration and reduction. Immersion of Kevlar in bromine water followed by neutralization with ammonia solution was reported as a procedure capable of creating pores on the fiber surface. After the above treatment, the observed improvement in ILSS of composites was accompanied by some deterioration in tensile properties of the fiber.[12] Pen et al. [13] studied the adhesive properties of aramid fibers to epoxy resins, taking into consideration their surface energy after treatment. The performed modification consisted of surface-controlled nitration followed by nitration/reduction and was said to result in significant increase of interfacial bonding in these systems. The use of reactive chemicals, such as acid chlorides, has also been investigated and was found to improve adhesive properties of aramids with both, polar and non-polar matrices.[14,15] Particularly encouraging results have been reported for the systems epoxy/aramid.[16-18] Modification via the grafting of various organic compounds onto the aramid surface has been the topic of investigation of many researchers. Thus, Briscoe and Williams [19] reported the successful grafting of propane-diol, alkyl, epoxy and trimethyl silane groups on the surface of Kevlar fibers via an anhydrous reaction route. The changes in the contact angle and the works of adhesion for water and methylene iodide with the modified fibers confirmed the potential success of the grafting reactions. Furthermore, Chou and Penn [20] attached molecular chains with different chemical structures to the surface of aramids and studied the effect on the fiber adhesion to epoxy matrix. They came to the conclusion that inert chains did not alter the fiber-matrix adhesion. On the other hand, chains capable of forming chemical bonding with the matrix increased the adhesion significantly. It should be noted that Briscoe and Williams [21] studied the effect of acid-base interactions on the fracture toughness of aramid epoxy composites and found that the fracture process is not directly influenced by the interface quality but rather by

6


the structure of the system and, hence, the size and the dissipative character of the deformation zone associated with the crack. Another approach of improving the fiber/matrix interface is that of Hofsté et al. [22], who presented a procedure for preparation of ultra-high molecular weight polyethylene (UHMWPE) matrix reinforced with chopped aramid fibers. Surface oxidation of UHMWPE powder with chromic acid was found to improve the fiber-matrix interface and thus the physical, mechanical and tribological properties of the aramid-UHMWPE composite. Although only a relatively small improvement was observed in the modulus, yield and stress at break of 33, 17 and 9% respectively, a substantial enhancement in wear resistance of 117% was observed. Very interestingly, chemical treatment of aramid fibers with organic solvents improves the interfacial shear strength with epoxy resin to about 60%, as it was measured by multiple fiber pullout tests. This improvement could be explained by the presence of an oxygen rich fiber surface whose blistered, striated and undulated morphology adds to the mechanical interlocking to the matrix.[23] Regarding the use of coupling agents, Menon et al. [24] studied the surface modification of Kevlar fibers with titanate compounds for improved adhesion to treated Kevlar-phenolic composites. The specimens exhibited improved flexural properties and greater resistance to moisture penetration. The effect of surface treatment on the pull-out behaviour of aramid fibers from epoxy resins was studied by Young et al. [25], who made their investigation with untreated fibers (HM), and fibers containing a standard surface finish (HMF) as well as a special epoxy-based adhesion-activating finish (HMA). It was found that pull out of the HM fiber takes place by a debond propagating along the fiber/matrix interface at a low level of interfacial shear stresss. The HMF fiber showed better adhesion to the epoxy matrix with pull out occurring in a complex manner, through both separation of the fiber skin and failure at the fiber/finish interface. No evidence of debonding was found for the HMA fiber and failure occurred by fracture of the fiber at the point where it entered the resin block. A procedure similar to that of treatment with acid chloride, for the modification of aramid surface, was that of chlorosulfonation. Thus, Lin et al. [26] studied the surface modification of Kevlar 49 by chlorosulfonation and subsequent reaction with chemicals, such as allylamine, ethylenediamine and deionized water. By running microbond pull-out tests they found an increase of 7% - 62% in the interfacial shear strength of treated fibers compared to that of the original Kevlar. In another work, Shyu and his co-workers [27] found that treatment with chlorosulfonic acid greatly improves the interlaminar shear strength (ILSS) of Kevlar fiber/bismaleimide composites.


7

Metalation, bromination and grafting were studied by Lin [28], who also tried to establish an appropriate procedure for surface modification of Kevlar fibers. From the SEM micrographs it was observed a rougher surface in the case of bromine etched Kevlar fibers as compared with that of the untreated one. It was also observed that their tensile strength decreases with the increase of bromine treating time. The ILSS results showed an increase of 12% with samples of bromoacetic acid-grafted Kevlar whereas an increase of 8% was recorded for those of epichlorhydrin-grafted Kevlar. As already mentioned, a critical parameter for improving interfacial bonding of aramid fibers to polymeric matrices, is the controlled roughening of their surface, that can promote “anchorage” within the matrix. Treatments with less aggressive solutions, such as aqueous sodium chloride, were performed by Gu [29] and, very interestingly, extended attack was recorded. In fact, quartz, aramid and glass filaments were treated with NaCl solutions of various concentrations for different periods of time and the results were evaluated by investigating the appearance of the treated filaments. It revealed that for the quartz fiber the NaCl density and the treatment time have a minor effect as far as the decrease rate of the fiber strength is concerned. After three weeks of NaCl treatment, dramatic degradation of the aramid filaments was observed for all the NaCl densities used in the investigation. Kevlar fibers were functionalized with phosphoric acid (PA) of different concentrations by Li et al. [30], for the preparation of epoxy resin composites. Owing to the increase of polar functional groups on the fiber surface, as it was confirmed by XPS analysis, the compatibility between resin matrix and PAfunctionalized Kevlar fiber was greatly improved. Using optimally functionalized fibers, various combinations of epoxy resin and hardener were studied. In order to avoid strong attack of the aramid surface, which is likely to have adverse effects on mechanical properties of the fiber, many attempts were made to apply the suitable sizing on aramid fiber. Thus, Mai and Castino [31] measured the fracture toughness of epoxy composites containing Kevlar coated with a silicone vacuum fluid and a polyurethane varnish. The above authors reported significant improvement of both strength and toughness when treated fibers are used. Finally, techniques of fiber coating, already known from other types of fibers, such as glass fibers, have been used as a means of improving interfacial adhesion between aramids and polymeric matrices. Thus, Kim and Mai [32] reported that the transverse fracture toughness of unidirectional Kevlar-epoxy resin composites was substantially improved by coating the fibers with poly(vinyl alcohol) without any loss of flexural strength, but there was only a moderate improvement upon

8


coating with a carboxyl-terminated butadiene-acrylonitrile copolymer and poly(vinyl acetate) with some reduction in flexural strength. More recently, Varelidis et al. [33] studied the influence of fiber surface coating with polyamide on the mechanical properties of epoxy/Kevlar composites and found that the fracture toughness determined via double cantilever beam tests increases when aramid fabrics, coated via a solution process, are used as reinforcement. Additional research on the surface modification of aramids, by producing a coating that could act as the interface between fiber and matrix, includes the in situ polymerization of nylon-6,6 on the surface of plasma treated Kevlar fibers, studied by Salehi-Mobarakeh et al..[34]

1.3. Plasma Treatment It should be noted that fiber pre-treatments consisting of exposure to reactive environments, i.e. plasma and laser treatments, have become so far very popular approaches for the modification of chemical and morphological characteristics of the surface of high modulus fibers. This seemed a promising technique in order to overcome complicated and time-consuming chemical processes. Some of the advantages of plasma treatment are claimed to be simple processing, dry environment and short treatment time. In addition, plasma treatment shows the advantage of local action on the surface with minimum attack of the bulk characteristics of the substrate material and offers increased versatility by selecting various gases to create the appropriate atmosphere. Thus, Allred et al. [35] have employed gas phase oxygen plasmas for the oxidation of “Kevlar 49” fibers, whereas Wertheimer [36] used microwave plasma in atmospheres of O2, N2 and Ar to oxidize the surface of Kevlar incorporated into a triazine matrix polymer. They found that this treatment increased the strength of composite laminates. The mechanical performance of these materials seemed to be strongly dependent on the lapsed time between plasma reaction and sample preparation. The laminate strength was maximized when the time interval between plasma exposure and resin contact was minimized. Radio-frequency plasma treatment of Kevlar fibers was reported to increase by 45% their interfacial shear strength with an epoxy resin, as measured by using the microbond technique. [37] Furthermore, Schwartz et al. [38] explored the effect of oxygen and argon plasmas on the adhesion of epoxies to aramid fibers and found improvement in interfacial shear strength (IFSS) with both argon and oxygen treatments as well as with the combination of argon and oxygen.


9

Sheu and Shyu [39, 40] studied the effect of gas plasma on the adhesive properties of Kevlar 149 fibers to epoxy resins. The ILSS was essentially improved without any significant loss in fiber strength. The surface free energy and the work of adhesion of water on the fibers surface were also improved. Treatment of aramid fibers in a low-temperature microwave plasma resulted in a 12% increase in ILSS, which was attributed to surface cleaning accompanied by some slight surface etching.[41] Remarkable improvement in the ILSS of epoxy specimens reinforced with ammonia plasma treated aramid fibers was recorded which, however, was accompanied by a significant decrease in transverse ballistic impact properties.[42] Significant improvement in ILSS and flexural strength of resole and phenolic resin composites reinforced with ammonia and oxygen plasma treated aramid fibers was observed due to improved wetting by the phenolic resin.[43] More recent studies on the oxygen plasma treatments showed an improvement of the polar component of the surface free energy of aramid fibers in the order of 30%.[44, 45] Aramid fiber polar surface free energy improvements induced by oxygen plasma treatment have been shown by Park et al. [46] to be directly proportional to improvements in the measured interfacial shear strength of this fiber. In contrast to the mostly chemically based interfacial property improvements achieved by oxygen plasma treatment, laser ablation of aramid fibers induces surface micro-corrugations that lead to a 120% improvement of the IFSS.[47] Other techniques, including dielectric barrier discharge plasma and even γ-ray irradiation deriving from a Co60 source, were followed and studied.[48,49] Application of the above type of treatment to aramids revealed after characterization by SEM, XPS and wettability tests, that the surface roughness is improved, the O/C atomic ratio is increased from 15.99% to 27.15%, and the surface wettability is also enhanced significantly. It was also found that the improvements of physical and chemical properties increased with increasing power density and treatment time. Regarding the more powerful technique of γirradiation, it was shown the ILSS values of aramid/epoxy composites were improved to about 20%. Also, the surface elements of aramid fibers, determined by XPS analysis, indicated again that the oxygen/carbon ratio was increased. The surface of treated fibers was rougher than that of the untreated when examined by atomic force microscopy (AFM) and scanning electron microscopy (SEM). Fourier transform infrared (FT-IR) spectra confirmed that the epoxy groups were grafted onto the fibers. The wettability of the fibers' surface was also enhanced by the treatment. Nanoindentation technique analysis showed that the nanohardnesses of the various phases (the fiber, the interface and the matrix) in

10


the composite, whose fibers were treated, were correspondingly higher than those in the composite, containing untreated fibers as reinforcement. The authors claim that these results indicate that γ-ray irradiation grafting technique, which is a suitable batch process for industrialization, can modify the physicochemical properties of aramids and improve the interfacial adhesion of its composite. Studies were made on the penetration depth of atmospheric pressure plasma, in order to correlate this parameter with the mechanical properties of composites reinforced with aramid fabrics treated with this technique.[50] More specifically, three-dimensional aramid woven fabrics were treated with atmospheric pressure plasmas, on one side or both sides to determine the plasma penetration depth in the 3D fabrics and the influences on final composite mechanical properties. The properties of the fibers from different layers of the single side treated fabrics, including surface morphology, chemical composition, wettability and adhesion properties were investigated using scanning electron microscopy (SEM), X-ray photoelectron spectroscopy (XPS), contact angle measurement and microbond tests. Meanwhile, flexural properties of the composites reinforced with the fabrics untreated and treated on both sides were compared using three-point bending tests. The results showed that the fibers from the outermost surface layer of the fabric had a significant improvement in their surface roughness, chemical bonding, wettability and adhesion properties after plasma treatment; the treatment effect gradually diminished for the fibers in the inner layers. In the third layer, the fiber properties remained approximately the same to those of the control. In addition, three-point bending tests indicated that the 3D aramid composite had an increase of 11% in flexural strength and 12% in flexural modulus after the plasma treatment. According to the authors, these results indicate that composite mechanical properties can be improved by the direct fabric treatment instead of fiber treatment with plasmas if the fabric is less than four layers thick. Some researchers claim that the surface modification of aramid fibers by plasma treatment alone is not enough to improve adhesion. Combined exposure to plasma and subsequent treatment with a silicone adhesive as coupling agent was found effective in promoting adhesive bonding between aramid fibers and silicone rubber.[51] Wang et al. [52] applied the catalytic grafting technique to the oxygen plasma-treated aramid fibers-HDPE system. The reactive sites introduced by plasma treatment were used to chemically anchor a Ziegler-Natta catalyst followed by ethylene polymerization on the fiber surface. The grafted polyethylene acts as a transition layer between the polymer matrix and reinforcement and improves the interfacial adhesion. Improved mechanical properties were observed when a blend of polar polymers was used as matrix for the fabrication aramid fiber composites.[53]


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Exposure of Kevlar 49 fibers to radio- frequency plasma followed by coating of he fiber with a thermoplastic resin in the vacuum of the plasma reactor, appeared to improve the interfacial shear strength of a reinforced polycarbonate resin by 18%.[54, 55] Takata and Furukawa [56] managed to improve the adhesive bonding of aramid fibers to rubber, by applying low temperature plasma treatment and coating with a thin nylon film as well as with resorcinolformaldehyde latex. Finally, chemical modification of the aramid surface achieved by exposure to plasma has also been reported by Allred et al. [57], who presented a method to introduce amine groups to the filament surface by exposure to ammonia plasmas. Subsequent reaction of the amine groups with epoxides formed strong covalent bonds at the composite interface. It should be noted that a recent trend in the research and development of new, advanced technology composites, is the substitution of the low modulus thermosetting matrices with some thermoplastics of very high modulus of elasticity, such as ether ketones. It is, therefore, obvious that improvement of interfacial characteristics between aramid fibers and those thermoplastics must take into consideration some additional parameters. Chen at al. [58] studied the effect of oxygen plasma treatment method on the modification of aramid fiber surface for improving its interfacial adhesion to poly(phthalazinone ether sulfone ketone) (PPESK) matrices. Composite interfacial adhesion properties were determined by interlaminar shear strength (ILSS) using a short-beam bending test. The composite interfacial adhesion mechanism was explored by SEM, whereas the changes of chemical composition and wettability for plasma-treated fiber surfaces stored in air as long as 10 days were investigated by XPS and dynamic contact angle analysis (DCAA), respectively. The authors claim that the results obtained in the above work, indicated that oxygen plasma treatment was an effective method for improving interfacial adhesion.

2. Ultra High-Modulus Polyethylene Fibers 2.1. General Description The unique properties of Ultra-High Modulus Polyethylene (UHMPE) fibers are due to their fully extended and aligned chain configuration. Several methods are available for the production of these fibers, based on the deformation of a gel or a solid. Thus, the technique of ultradrawing polyethylene fibers crystallized from solution has been developed together with the “surface growth” process (i.e.

12


drawing of a wet gel of ultra high molecular weight polyethylene, with concurrent crystallization) and the melt extrusion technique. Commercially available UHMPE fibers, produced today by gel spinning/hot drawing technique, display high tensile strength (2.7 GPa) and ultra high modulus (87 GPa) combined with low density (0.97 g/ml). These features make UHMPE fibers a very promising material for advanced technology applications, including that of reinforcing polymeric matrices for high performance composites. However, those fibers present some weaknesses which restrict their application, namely low melting point, high creep, poor interfacial bonding to many matrices. In addition to their poor interfacial characteristics with most polymers, UHMPE fibers present very low strength in directions perpendicular to their axis, as compared to that corresponding to the axial direction. This is a serious weakness that further restricts their use as reinforcing agent, since the anisotropy in strength and modulus gives poor performance characteristics in cases of complex loading, enhancing the already discussed poor interfacial properties between fibers and polymeric matrices. The anisotropy in strength is due to the fiber morphology, which is a result of the production process. In fact, the morphology of UHMPE is microfibrillar, consisting of smooth or “shish-kebab” fibrils. More specifically, a shish kebab morphology is composed of core microfibrils with almost extended chains and a number of folded-chain platelets attached to the cores, as can be seen in Figure 2.[59] On the other hand, a smooth fiber contains microfibrils with almost extended molecules and a negligible amount of lamellar platelets. Many structural models have been proposed to explain the relationship between microstructure and mechanical properties of highly oriented linear polyethylene and two different approaches have generally been used. According to fiber reinforced model [60] the oriented polymer consists of a high strength needlike crystalline phase, embedded in a partially oriented amorphous phase, which is reasonable to act as reinforcement. A completely different model assumes that the reinforcement comes from taut-tie molecules [61] and intercrystalline bridges.[62] Finally, mechanical models such as Takayanagi’s model [63], recognize the two-phase nature of semicrystalline polymers and the mechanical behaviour of a fully oriented fibre, is explained in terms of two separate components representing crystalline and amorphous fractions placed in either parallel-series or series-parallel configurations.[64, 65] Real-time micro-Raman measurements on stressed UHMPE fibers have been carried out in order to investigate the macromechanical behaviour of this material and elucidate its structural status.[66] As a result of this research, a new model was established as schematically shown in Figure 3.


Figure 2. The “shish-kebab” fibrils structure of UHMPE fiber.

Figure 3. Proposed structural model of UHMPE fibers.

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14


In fact, Figure 3 presents a model similar to that of Takayanagi, but modified to allow shear displacement around the crystalline “bridges”. The fibrillar structure of UHMPE fibers requires that the crystalline blocks are fibrillar in nature, while some discontinuous interamorphous material may still be present. Due to their relatively low melting point UHMPE fibers are not recommended for use at temperatures above 100oC. In fact, upon heating UHMPE fibers show an orthorombic/hexagonal transition at 147oC accompanied by melting at 153oC. The fiber Tekmilon NA 310 (Mitsui Petrochemical Industries Ltd) retains 75-80% of its strength and modulus at 60oC, 50-55 at 80oC and only 30% at 100oC. Therefore, many attempts have been made aiming at the increase of heat stability by modifying the PE structure to a network. Furthermore, the formation of cross-links among the polymer chains is expected to reduce creep, since the network structure restricts the chain mobility and flow. The attempt of cross-linking ultra high strength PE fibers was based both, on chemical procedures and irradiation. Dicumyl peroxide (DCP), a widely applied crosslinking agent in PE technology, has been extensively used for this purpose. Furthermore, many kinds of high energy radiation, such as ultra-violet, X-ray, electron beam and γ-radiation, have been employed for cross-linking PE fibers. As above mentioned, the poor interfacial bonding to polymeric matrices is of increased significance when the related composite materials are intended to be used in constructions, where complex loading is rather expected. It is well known that the interface between reinforcement and matrix in fiber reinforced polymers plays an important role in determining the ultimate mechanical properties. The goal of many studies on the design and performance of composite materials is to tailor the interfacial properties in order to optimize their properties. In this context it should be emphasized the fact that UHMPE fibers have a very smooth surface and display low surface energy. Furthermore, the inert and nonpolar character of PE eliminates the possibilities of strong adhesive bonding between those fibers and the common polymeric matrices. Therefore, various surface treatments were proposed and investigated, including coating, chemical attack or exposure to reactive environments.

2.2. Surface Treatments More specifically, many studies have been published on the surface treatment of PE in order to improve its interfacial behaviour, based on chemical etching of the surface with oxidative chemical reagents. Also, some techniques were established including corona and plasma discharge treatments, which are common industrial processes for improving adhesive properties of polyolefins.


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2.2.1. Chemical Reagents Research work has been reported on the surface treatments of PE fibers, intended to increase the adhesive bonding to various polymeric matrices.[67-71] For this reason, chlorosulfonic acid was used to modify PE fiber surface and therefore promote the interfacial bonding.[72] It was found that the interfacial bond-strength between the polyethylene fiber and gypsum plaster can be improved at least 4.8 times by this procedure. After chlorosulfonation the tensile strength of the fibers decreases, whereas Young’s modulus can increase more than 50% of its initial value. Chlorosulfonic acid was also used by Ravichandran and Obendorf [73] who treated multifilaments and plain weave fabric of UHMPE fibers. Treatments were carried out at room temperature and 50oC for periods ranging from 10 to 60 min. Chemical tests revealed high concentrations of acid groups on the fibers and measurements of interfacial shear strength of treated fibers/epoxy system showed that the interfacial bonding can dramatically increase. The determination of changes of the contact angle led to the conclusion that better wetting can be achieved with treated fibers, whereas their mechanical properties in shear and tension displayed minimal decrease. The use of chromic acid as oxidative agent was studied by Rochette at al. [74] who investigated the influence of surface treatment of the fibers of a one-polymer composite made of PE reinforced with UHMPE fibers, on the properties of the composites. The fibers were first used with their initial sizing; then, they were used after either etching in chromic acid or plasma treatment. The performance of treated fibers was evaluated by stress–strain measurements in longitudinal and in transverse directions as well as by the results obtained from pull-out tests. The crystallization of the matrix on the fiber surface was followed by microscopy and by DSC analysis, whereas the structure of the interface was detected by wide angle X-ray diffraction. The use of silane coupling agents was also studied by applying these chemicals to untreated or plasma treated UHMPE fibers.[75,76] The oxygen plasma and γ-methylmethacryloxypropyltrimethoxysilane (γ-MPS)-treated UHMPE fiber/vinylester composites showed a slightly higher interlaminar shear strength than the oxygen plasma-treated UHMPE fiber/vinylester composites. The interfacial adhesion of the oxygen plasma-treated UHMPE fiber/vinylester composites in this study was mainly attributed to mechanical interlocking between the micropits formed by the oxygen plasma treatment and the vinylester resin. The γ-MPS molecules adsorbed onto the UHMPE fiber surface neither affected the morphology of the UHMPE fiber surface, nor reduced the extent of mechanical interlocking. The improved interfacial adhesion by the γ-MPS treatment was interpreted as the result of enhanced wettability and chemical

16


interaction through the chemically adsorbed γ-MPS molecules, as detected by FTIR spectroscopy. The γ-MPS molecules adsorbed onto the ultra-high molecular weight polyethylene (UHMWPE) plate surface also reduced the aging effect of the oxygen plasma-treated UHMWPE surface.[76] Graft polymerization of acrylamide (AAm) and glycidyl methacrylate (GMA) was performed onto the surface of UHMPE fibers pre-treated with Ar plasma [77], in an attempt to improve their wettability and adhesion to polymeric substrates. Following plasma treatment and the subsequent exposure to air for introducing peroxides onto the fiber surface, graft polymerization onto the UHMPE fiber was allowed to proceed from the polymer peroxides, either in deaerated monomer solution at an elevated temperature (degassing method), or in aerated monomer solution containing riboflavin at 30°C under UV irradiation (photoinduction method). Surface analysis of the grafted fibers was performed with ATR-FTIR and XPS, which revealed that PAAm and PGMA chains were grafted in the surface region of fibers. The grafting rate of PAAm by the photoinduction method was much higher than that by the degassing method when compared at the same concentration of the AAm solution. The amount of PGMA grafted was greatly affected by UV irradiation time, but depended on plasma treatment time to an insignificant extent if the treatment was carried out for longer than 30 s. Reaction of propylamine with the PGMA-grafted surface resulted in the appearance of a nitrogen peak in the XPS spectrum, suggesting the presence of epoxy groups on the surface of PGMA grafted fiber. Similar research has been conducted by Allmer et al. [78] who studied the surface grafting of glycidyl acrylate and glycidyl methacrylate on the PE surface. The reaction was started by photoinitiation and was said to introduce epoxy groups on the PE surface. The epoxies can further react with various chemical reagents and produce other kinds of groups, which enhance the possibilities of surface modification. Another modification of the surface of UHMPE fibers consists of grafting of gaseous butadiene to a range of morphological forms of gamma-irradiated polyethylene, including ultra-high-modulus fibers (UHMPE). The reaction rates of the above modifications have been measured in order to determine the availability of active free radicals over time at various temperatures. Blank experiments on unirradiated samples showed that monomer diffusion is not rate-controlling with film and natural draw ratio tapes, but is likely to be a major factor in the control of grafting rates in UHMPE fibers. Grafting rates from monomer loss versus time experiments with irradiated samples indicate that grafting is always in competition with free radical self-annihilation, the extent being influenced by temperature, dose and morphology, including prior sample annealing. At lower temperatures, graft-active radicals are produced over long periods of time, e.g. close to linear


17

grafting rates were monitored over 20 hours for PE tape at 0°C (50 kGy) and for gel-spun UHMPE at 40°C (100 kGy). At higher temperatures, grafting rates steadily decrease with time. Grafting rates are almost independent of irradiation dose in the early stages, however, the dose has an increasing positive influence as the reaction proceeds. At any given temperature and irradiation dose, the rates decrease in the series undrawn film; natural draw ratio tape; high draw ratio gelspun fiber; high draw ratio melt-spun fiber. An analogy is drawn between these results and the optimum conditions required for improving the creep properties of PE tape and UHMPE fibers by acetylene-sensitized irradiation crosslinking.[79] Gao and Mackley studied the swelling of the surface of ultra high molecular weight PE with decalin accompanied by cooling.[80] This technique seemed to have a significant effect on the surface properties of UHMPE fibers. As a matter of fact, the metal/polymer adhesion can be enhanced and, if graphite is added to the solvent, surface conductivity can be observed. The improved adhesive properties were attributed to an increased ductility of the pre-swollen polymer at elevated temperatures. Similarly, the effect of fine iron powder on the surface properties of UHMPE fibers have been studied by Drasnov et al., who used iron nanoparticles for this reason.[81] The effect of oxidative surface treatment of UHMPE fibers on the mechanical properties of their composites was studied by Taboudoucht et al..[82] They used fuming nitric acid at 85oC and immersed PE fabric for periods ranging from 2 to 30 min. From the obtained results the authors concluded that this technique leads to better adhesion of the fabric with epoxy resins. As revealed by SEM and FT-IR analysis, the mechanisms responsible for this improvement are better wetting of the fibers as well as chemical bonding across the interface. The authors found that the contribution of chemical reaction is smaller than expected, due to the low extent of the reaction of carbonyl groups with the epoxy. The Raman technique was also reported [83] as a means of analyzing the interface of model composites, after their deformation via micromechanical test methods. Various chemical treatments of UHMPE fibers have also been reported by Andreopoulos et al. [84, 85] who used oxidative agents, peroxides and seizing agents for this purpose. From the results collected in this research it can be concluded that treatments of the surface of UHMPE fibers can be divided into the following categories: 1. Chemical reaction with aggressive oxidative agents, such as chromic acid, chromosulphate solution, chlorosulphonic acid, permanganate solution, etc. These chemicals attack the surface and produce oxygen containing groups such as carbonyl, ketone, hydroxyl, that promote

18

Petroula A. Tarantili UHMPE’s affinity to most of the common polymer matrices. An additional effect is fiber etching, resulting in the formation of surface irregularities which contribute to a better interlocking between fiber and matrix. This latter mechanism is believed to proceed via degradation of the amorphous areas of polyethylene. 2. Coating of the fiber surface with sizing agents, provided that they strongly adhere to polyethylene and matrix. 3. Controlled swelling of the fiber with solvents such as toluene and decalin, containing various agents that can promote adhesion or even produce chemical bonding between fiber and matrix.

As already reported, treatments that attack the fiber surface are likely to reduce its strength while improving interfacial characteristics. On the other hand, methods that provide fiber coating seem to have small effects on the strength and interfacial properties. 2.2.2. Corona and Plasma Treatments It is well known that corona discharge treatment of polymer surfaces is a widely used industrial process for improving their adhesive properties [86]. Plasma treatments have been used for decades in modifying the surface of polymeric materials. It is regarded as an environmentally friendly process since no chemicals are involved and also as a effective way to modify the bondability and wettability of polymer surface by introducing polar groups or increasing surface roughness without affecting the bulk properties [87-89]. Most of plasma treatments have been performed under low pressure or high vacuum which would be expensive for many products with relatively low profit margin such as textiles. Therefore, much attention has been paid to plasmas operating at atmospheric pressure due to possible advantages of eliminating an expensive vacuum system, on-line processing capabilities, high efficiency, and the scalability to a larger area.[90-92] Typical atmospheric pressure plasma systems are arc plasma torches, corona discharges and dielectric barrier discharges (DBD). [93, 94] In recent years, an atmospheric pressure plasma jet (APPJ) is invented to produce the homogeneous plasmas at low temperature. It consists of an inner electrode, which is coupled to a 13.56 MHz radio frequency power source, and a grounded outer electrode. Typically a small fraction of reactive gases, e.g., oxygen or nitrogen is added to helium feed gas in order to generate chemically active species.[95]


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One of the problems that hinder the application of plasmas in surface modification of polymeric materials is the potential acceleration of aging of the treated material. In fact, after plasma treatment the polymer surfaces show a gradual hydrophobic recovery over the time and the surface free energy simultaneously decreases to the original value.[96-100] It has been suggested that the hydrophobic recovery is mainly due to two mechanisms: one is the reorientation of polar chemical groups on the treated surface towards the bulk of the material and the other is the diffusion of segments of non-modified macromolecules to the surface. [101-103] Limited number of papers have recently been published regarding the application of APPJ technology to polymer surface modification [91, 104, 105] and little has been reported about aging of the treatment effects of APPJ, as well as about the correlation of aging with treatment conditions. It has been suggested that aging behavior of a fiber can be influenced by many factors among which the composition of the gaseous environment during plasma treatment could play an important role.[106-109] UHMPE fibers were selected as model material to investigate the aging behavior of fiber surface treated by APPJ with pure helium, as well as helium mixed with oxygen at two levels. UHMPE fiber is a fiber with high tensile strength and modulus, low specific density but poor wettability and adhesive bonding to resins.[87, 88] The surface morphology, the adhesion between the fiber and epoxy, the surface chemical composition and the water contact angle were measured at 0, 7, 14, 21 and 40 days after the plasma treatment by atomic force microscopy, micro-bond tests, X-ray photoelectron spectroscopy and sessile drop method. Briggs and Kendall [110], in an attempt to study the mechanism of interactions taking place on the PE surface during corona treatment, used various derivatizing agents, such as pentafluorophenyl hydrazine, bromine, chloroacetylchloride, sodium hydroxide, etc. The exposure was carried out at 9 kV discharge and 50 Hz for 15 sec. Similar research was conducted by Gerenser et al. [111] who used ammonia, hydrogen chloride and sulfur dioxide as derivatizing gases. The techniques of gas-phase derivatization and ESCA were employed to identify the corona-introduced oxidative species in polyethylene. Five species were identified directly, namely: hydroperoxy, hydroxyl, isolated carbonyl, epoxy and carboxylic acid. The ESCA analysis showed the presence of ether and ester groups, suggesting homogeneous oxidation within the depth of 50Å of the PE surface. As far as a comparison of the treatments made via chemical reaction and corona discharge is concerned, George and Willis studied the interfacial bonding of hydroperoxidized and corona treated UHMPE fibers with an unsaturated

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polyester resin.[112] They found that hydroperoxidation minimizes the secondary oxidation of the fiber surface and provides an optimum concentration of graft sites for cross-linking of styrene present in the polyester mixture. This mechanism guarantees a maximum bond strength, as measured by the force to pull-out the treated yarn from the resin. Furthermore, SEM showed that the mode of failure for untreated fibers was a slippage at the fiber-resin interface, while for the hydroperoxidized surface the locus of failure moved to shear failure in the outer skin of the fiber or to fracture of the matrix. It was concluded that this was due to the strong covalent bonding between the fiber and resin becoming from the grafting of styrene during the cure. The yarns treated with corona discharge showed extensive secondary oxidation, much lower content of hydroperoxide and a pull-out force significantly lower than that hydroperoxidized material, but still higher than the pull-out load corresponding to untreated yarns. Analysis with SEM revealed that this may be attributed to slippage in a weak boundary layer on the oxidized surface. Another study of George et al. [113] was focused on the interfaces between reinforcing fibers and low temperature cured phenolics with X-ray photoelectron spectroscopy and static ion mass spectroscopy. The treatment of UHMPE consisted of exposure to oxygen and water plasmas. The improved adhesion of the as above treated fibers to phenolic matrices was assessed by fiber bundle pull-out and ILLS measurements. Furthermore, Gutowski et al. [114] developed a new surface engineering process, named SICOR, that is claimed to allow grafting of amine groups onto UHMPE and aramid fibers. The authors report significant improvement of interfacial adhesion based on the increase of interlaminar fracture energies and flexural moduli. The effectiveness of the process is comparable to the ammonia plasma treatment. The effect of plasma etching and that of treatment with chromic acid on the surface adhesion of ultra-high modulus PE fibers to an epoxy resin has also been studied.[115] The adhesion measured by pull-out tests showed a significant improvement for both treatments. The mechanism of failure with untreated or acid-treated monofilaments involves sliding along the fiber/resin interface, whereas plasma treatment seems to produce remarkable structure on the monofilament surface, that allows resin penetration and thus produces a mechanical keying effect. Failure in the pull-out test involved rupture within the fiber. The plasma treatment of UHMPE fibers in the presence of ammonia has been investigated by Holmes and Schwartz.[116] This study aimed at the investigation of the introduction of amine groups onto the surface of PE. The effect of treatment time and power of the equipment was studied. After treatment,


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the wettability of the fibers was found to increase dramatically, so that they became completely wettable at exposure times greater than 1 min. The primary amine concentration was maximum after a 5 min treatment with a power of 50 W. The surface texture seemed to remain unaffected, whereas tensile strength decreased approximately 10% at some treatment conditions. A comparative study was conducted by Choe and Jang [117], who investigated the promotion of adhesion of UHMPE fibers to epoxy substrates due to plasma etching and compared the results to those obtained from silane treated PE fibers. They found that plasma treatment increased the amount of oxygen on the PE fiber. On the other hand, measurements of ILSS of composite samples based on these reinforcements showed that the silane treated fibers display higher strength due to improved wettability and chemical bonding. Also, the effect of oxygen plasma treatment on UHMPE monofilaments has been investigated by Tissington and his co-workers [118] with particular attention to the adhesive properties of treated monofilaments to epoxy resins. The adhesion strength was monitored by pull-out tests, whereas the effect on the monofilament characteristics was followed by contact angle measurements, determination of gel content and SEM. The results of this investigation suggest that there are three parameters contributing to the improvement of adhesion achieved by plasma treatment. First, at relatively short treatment times, there is a uniform oxidation of the total surface. Second, at intermediate treatment times, cross-linking takes place on the surface, which increases the cohesive strength of the fiber surface. Finally, at long exposure times, a surface pitting is created, that could give rise to mechanical keying but may also reduce the fiber strength. The above authors measured the ILSS of unidirectional composite specimens made of epoxy resin and UHMPE fibers treated in oxygen plasma. The treatment conditions were: power 120 W, gas flow rate 17 ml/min, pressure 0.4 torr and duration 120 sec. They concluded that the oxygen plasma treatment greatly improves mechanical performance of the composites due to better adhesion between fiber and matrix. This was attributed not only to the formation of oxygen-containing groups on the fiber surface, but also to the modification of the weak boundary layer present in the outer surface of the PE fiber.[119] Modification of a commercial type of PE fiber, Spectra-900, was made by Rostami et al. [120] who used allylamine plasma for treatment. This technique seemed to allow grafting of the allylamine molecules onto the fiber surface. As a result, the interfacial bonding strength between the fibers and room-temperature-cured epoxy matrix was increased fivefold. Fiber covalently coated with allylamine plasma showed retention of their tensile strength, while pretreatment of the same fibers in an argon plasma caused up to 10% reduction in tensile strength depending on the energy and duration of

22


the treatment. Optimum treatment was obtained through a short argon plasma etching (15 sec) followed by allylamine polymerization and coating for 3 min. The coating process was expected to protect the fiber surface from etching by plasma bombardment. Two types of UHMPE fibers, Spectra 900 and 1000, were also treated by Hild and Schwartz [121] with various gas plasmas. The above researchers used nitrogen, argon and carbon dioxide plasmas with treatment times 1, 5 and 10 min. The significant improvement in fiber wettability was attributed to the formation of polar groups on the PE surface identified by ESCA. On the other hand, no change in surface topography could be observed at magnifications up to 2000x. Li et al. [122] studied the effect of ammonia plasma treatment on the mechanical and adhesive properties of PE fibers. They found that discharging conditions of 30 W, 5 torr for 1 min result in a more than four-fold increase of the IFSS. Gao and Zeng [123] studied the mechanism of surface modification of plasma treated UHMPE fibers. They concluded that the adhesion of modified fibers to epoxies depends on three factors: (a) chemical bonding effects after the plasma treatment through the formation of various oxygen containing groups, which greatly improves the surface energy of PE, (b) mechanical keying and (c) the non-polar dispersion force. These three factors can be considered as additive and optimum results are obtained when their respective contribution reaches about 60%, 30% and 10%. Biro et al. [124, 125] applied the micro-bond technique, a modification of the single fiber pull-out test for measuring the interfacial shear strength of plasma treated Spectra PE fibers. They found that the strength increased by 118% for PE fibers, whereas for aramid fibers the increase was only 45%. On the other hand, Chaoting and his co-workers [126] studied the influence of low-temperatureplasma on the adhesive properties of PE fibers. Their results showed that the surface energy and wettability of the treated fibers were increased greatly and the adhesive bonding to epoxy also increased by 10 times. Behnisch et al. [127] investigated the surface modification of PE in a remote nitrogen, oxygen and hydrogen DC discharge plasma and reported some differences on the effects of the above three techniques of gas plasma. The effect of plasma treatment of PE fibers on the properties of their composites with epoxies has also been studied by Woods et al. [128], who observed increased ILSS related to the improvement in the high speed flexural strength and impact energy absorption. UHMPE fibers were treated with oxygen plasma and a silane coupling agent, in an attempt to obtain improved interfacial adhesion between the fiber and vinylester resin.[75] The ILSS of the above composites showed a maximum for


23

plasma treatment time of 1 min. The totally absorbed energy of UHMPE fiber/vinylester composite during impact test decreased gradually with increasing plasma treatment time, due to the flaw-induced fiber breakage. 2.2.3. Morphological Modification of UHMPE Fibers A recognized drawback of UHMPE fibers is their low strength in directions perpendicular to their axis, i.e. to the drawing/orientation direction. This leads to highly anisotropic products when composites are produced since those materials too, present increased strength and modulus along their axis, provided that they are unidirectional composites. The anisotropy in strength is attributed to the packing of polyethylene chain during the production procedure of UHMPE fibers. As already mentioned, the combined action of spinning, ultra-drawing and orientation results in the formation of the so called “microfibrillar morphology” of fibers. This morphology further contains smooth or “shish-kebab” fibrils. Recently, attempts have been made to alter the characteristics of UHMPE fibers by treating them in a two-roll mill, at room and at elevated temperatures (Figure 4).[129-130] This treatment was expected to modify the interfibrillar areas and/or affect the lamellar morphology present in the “shish-kebab” fiber. Furthermore, these changes were anticipated to have an effect on the mechanical properties of composites containing the modified fibers. Finally, the transformation of the cylindrical shape into ribbon-like is an additional parameter which is expected to facilitate wetting of the fibers and provide better packing.

Figure 4. Treatment of UHMPE fibers in the two-roll mill.

24


This will allow the preparation of composites with increased filler volume fraction and consequently, better mechanical performance. This research has shown, in fact, that calendering of UHMPE fibers is a process which significantly improves their performance as reinforcement in cases of complex loading. Although no conclusive evidence has been found so far about the changes in the fiber microstructure, it was clear that calendered UHMPE fibers at 130oC give improved performance in composites especially for low filler volume fraction. Moreover, this study defined the recommended area of application for each type of treated fiber. The response of these systems under long term loading seems to be of increased importance and is essential to become the scope of future research into this field.

3. Discussion The above literature survey was necessary in order to provide useful information about the modifications of high strength-high modulus polymeric fibers as a means of improving their interfacial characteristics with various matrices and making them more attractive as reinforcement for preparation of advanced technology composites. Thus, an up-to-date literature review can show us the state of the art in this field and help us organize more focused research in the area of fiber modification. In fact, the related papers showed that surface modification of both, aramid and UHMPE fibers is an interesting field of research, with significant contributions from many scientists. This suggests that application of these high tech fibers in the area of composites is still today an interesting approach with strong potential. It also should be noted that the basic techniques of treatment, i.e. chemical etching, grafting, coating and exposure to corona and plasmas, have remained in use for the last 2-3 decades. Therefore, it is reasonable that the scope of recent attempts includes these methods. Another conclusion coming from the above survey is that a combination of various treatments is often a very efficient approach. For example, exposure to plasma may produce the suitable reactive sites on the fiber surface, capable of reacting in a subsequent stage with various chemicals, which may result in a tailor made nature of the fiber surface. This combined action takes advantage of both, i.e. etching of fiber surface by the plasma and modification of its composition by the chemical reaction. Finally, much attention should be paid to the fact that every type of treatment is specified for improving a certain property. In other words, the etching of fiber


25

surface ensures enhanced adhesive bonding of the fiber to polymeric matrices but greatly reduces the fiber’s strength. Despite this effect, the overall mechanical response of a specimen reinforced with treated fibers may be significantly improved as compared with that of specimens reinforced with the as-received fibers. Moreover, a treatment that promotes strong interfacial bonding is expected to improve properties such as the interlaminar shear strength. On the other hand, mechanical properties such as impact resistance are found to become poor upon these treatment, because the energy absorption in an impact specimen is strongly related on the way followed by the crack as it propagates within the specimen. From the above discussion it is clear that a global consideration of the problem would require today a screening of conventional and new methods in order to map the whole area. This will allow a fast and easy selection of fiber treatment depending on the system composition and the intended final application. It also should be noted that an element of originality of the present work, is the attempt to interfere with the microfibrilar structure of ultra high modulus fibers, since this aspect is rather a morphological characteristic and therefore has not been included in the research of treatments so far.

Experimental 1. Materials and Methods The components of the composite specimens prepared and studied in this research, i.e. the high modulus fibers and polymeric matrices, were as follows: The aramid fiber Twaron 1000 supplied by AKZO (The Netherlands), was used in this study. For the needs of the work commercial UHMPE multifilament yarns were supplied by Mitsui Petrochemicals Industries LTD., Japan (Tekmilon NA310,). Corona treated (Tekmilon NC310, Mitsui Petrochemicals Industries LTD., Japan) was also tested for comparison. Regarding the matrix of the investigated composites, a low initial viscosity epoxy resin was used (Epikote 828, Shell Chemicals Hellas) appropriate for preparation of high-strength composites. Curing was carried out at 40oC for 1 h and 60oC for 3 h, by the addition of an oligomeric amide (Epilink 175) as hardener, at a weight ratio to epoxy 1/2. In a subsequent stage, post curing took place at 150oC for 1 h.

26


2. Treatment of Aramid Fibers 2.1. Treatment with Methacryloyl Chloride Methacryloyl chloride (Purum, Fluka AG) and carbon tetrachloride (CCl4, Chemically pure, Mallinckrodt Chemical Works) were used. The acid chloride was dissolved in CCl4 (10/90 v/v) in order to prevent fast hydrolysis of this reactive chemical taking place upon contact even with atmospheric humidity.

2.2. Coating with Resole Resin The Resol resin was prepared according to the following procedure: 94 g of distilled phenol, 123 g of aqueous formaldehyde, 37% by weight and 4 g of calcium hydroxide dihydrate were added to a spherical flask equipped with stirrer and reflux condenser. The mixture was stirred and heated in an oil bath at 70oC for 2 hours and then, 10% solution of sulfuric acid was added to bring the pH to 6-7. Distillation under reduced pressure was applied for 1 hour in a subsequent stage in order to remove water. The obtained resin is still thermoplastic and soluble to a 10% aqueous solution of sodium hydroxide. Aramid fibers were impregnated with the above solution and after drying in a vacuum oven at 70oC, a dry weight uptake of 1.5% was determined.

2.3. Coating with Novolac Resin Novolac resin was synthesized by direct polycondensation of phenol and formaldehyde. A molar ratio of 1:0.8 was used and oxalic acid was the acidic catalyst. The reagents were added in a spherical flask and the mixture was stirred and refluxed for 2 hours. After cooling the novolac resin formed a light yellow, viscous layer. After separation of the water layer vacuum distillation took place to expel residual water. The obtained resin is a brittle material at room temperature. Due to the low formaldehyde content this material is still thermoplastic and can be cross-linked by adding a source of additional formaldheyde (e.g. hexamethylene-tetramine) and further heating. Thus, 20 g of novolac were dissolved in 100 ml acetone and the suitable amount of aqueous hexamethylenetetramine (4 g in 10 ml distilled water) was added to the solution. Twaron fibers were impregnated with the above homogeneous mixture and after drying in an air oven, a weight uptake of 1.5% was found.


27

2.4. Treatments of UHMPE Fibers UHMPE fibers were calendered at 130oC. Calendering of UHMPE fibers was performed in a motor driven two-roll mill, with adjustable rotation speed, clearance and temperature of the cylinder (Scamia France). Finally Tekmillon NA310 fibers were treated with an oxidative agent, i.e. chromosulphate solution. The consistency of chromosulphate solution was: 7 g K2Cr2O7, 150g H2SO4 (98%) and 12 g H2O. The fibers were immersed in the solution at 90oC for 60 min.

3. Characterization of Treated Fibers 3.1. Scanning Electron Microscopy A JSM-300 microscope, capable of high resolution was employed for the scanning electron microscopy. The mechanical properties of the treated fibers were compared to those of untreated monofilaments using a 20kN Hounsfield screw-driven tensile tester machine. Individual monofilaments were mounted across a hole in a paper card. At each end, the fiber was twisted around a bar and stacked in a card using the HY 5161/CY 210 (Ciba Geigy) epoxy resin system. The curing was carried out at room temperature. The monofilaments had to be strongly fixed in the card, in order to avoid fiber’s gripping during the experiment. Samples of 250mm gauge length were tested in tension at room temperature at a rate of 0.5 mm/min.

3.2. Surface Characteristics The pull-out load of aramid monofilaments from an epoxy matrix was determined via microbond tests [131], using a Zwick 455 tensile machine. The contact angle of the as received and treated fibers was determined by measuring the critical dimensions of a liquid epoxy droplet placed on a monofilament and the subsequent use of the appropriate software.[132] Measurements were carried out in the optical microscope under magnification 40x.

28


4. Preparation of Composites The unidirectional fiber composites were prepared by the leaky mould technique. This technique involves the use of an open-ended two-part mould as shown in Figure 5, which presents the preparation of a single composite bar. The procedure adopted was as follows: The fibers were soaked/impregnated with the epoxy resin/curing agent system. After coating the mould with a silicone release agent (Frekote 1711, Dexter Corporation), the fibers were laid down in the groove and liquid resin was poured in until soaking was complete. The upper part of the mould was then placed in position and the mold was then placed in hydraulic press with the appropriate spacers to produce the appropriate thickness. The excess resin was expelled by squeezing the mould. Chopped fibres

Toluene

Epoxy resin/ curing agent Impregnated aramid yarns

Compression moulding/curing

Specimen fabrication

Open ended mould

Machining

Figure 5. Moulding procedure for reinforced specimens.


29

The desired fiber volume fraction (Vf) was achieved by changing the number of aramid yarns in the mould. The samples prepared were cut into specimens with a fine bandsaw. The fiber volume fraction obtained range from 10 to 50%.

5. Mechanical Measurements on Composites Mechanical testing of unidirectional composite specimens was carried out in an Instron tensometer 4466, equipped with a load cell of maximum capacity of 5 kN. The tensile specimens were prepared according to ASTM D3039/D3039M93. In order to avoid the problems of stress transfer between the composite sample and the Instron grips, tabs made from glass fiber/epoxy resin system were used. The flexural properties of the composites were measured by three-point bending tests, according to ASTM S D790-92. The selected support span-to-depth ratio was L/d= 32/1 with a specimen length of 100mm. The Interlaminar Shear Strength (ILSS) was determined using a short-beam bending rig, according to ASTM D2344-84. The specimen support span to thickness ratio was 4 and the specimen length to thickness ratio was 6. Finally, the impact strength was measured according to the specification ASTM D256.

Results and Discussion As already mentioned, the experimental results of this study consist a comparative work aiming at the evaluation of interfacial characteristics of treated aramid fibers (Twaron 1000) and epoxy resins (Epikote 828/Epilink 175 system). The various treatments described in the Experimental section include: chemical treatment with (i) solution of methacryloyl chloride in carbon tetrachloride and (ii) hydrolysis with sodium hydroxide solution, as well as coating with phenolic resins with the following two systems: (i) alkaline aqueous solution of resole resin and (ii) solution of novolac resin containing hexamethylene-tetramine as cross-linking agent. The SEM micrograph of the as received fibers reveals, in general, a very smooth surface, as shown in Figure 6(a). Figure 6(b) shows that after treatment with methacryloyl chloride small flaws and randomly distributed grooves were created on the fiber surface. The changes in the surface morphology suggest that mechanical interlocking of the treated fibers and the epoxy matrix may be a possible mechanism contributing to the enhancement of adhesive bonding between fiber and matrix.

30


The chloride treated fibers display a sharp decrease in tensile strength, whereas coating of the fiber with phenolic resins does not have any significant effect on tensile properties of aramid monofilaments. The results obtained by the measurements of IFSS and contact angle suggest that chloride treatment, controlled surface hydrolysis with sodium hydroxide solution and coatings with both types of phenolic resins give much better surface characteristics in comparison with those of untreated fibers. The increase of IFSS was a clear evidence that strong adhesive bonding to the epoxy matrix was achieved, due to mechanical and chemical interactions with the epoxy matrix.

a

b

c Figure 6. SEM micrograph of (a) untreated (b) chloride-treated and (c) novolac-coated, aramid fibers.

We could speculate various adhesion mechanisms generated by the chloride treatment of aramid fibers. These include: (a) an enhanced degree of mechanical keying between the fiber and matrix because of the increased fiber surface roughness, as already established by SEM


31

(b) possible removal of the weak boundary layer due to etching or chemical reaction, which provides better fiber/matrix contact and contributes to a stronger interface (c) an increased surface energy and, therefore, more efficient wetting of the fiber by the epoxy (d) chemical grafting of methacrylic groups on the surface of the fiber (Figure 7) or coating of the fiber by species deriving from the hydrolysis of methacryloyl chloride. Cl C O HN

N

CO

CO

+ CH2 C

H

HN

CH3

N C CH2

CO

CO

+ HCl

O

C CH 3

Figure 7. Aramid fibers treated with methacryloyl chloride.

On the other hand, hydrolysis may also take place on the aramid surface, thus leading to an increase of concentration of reactive functional groups, namely amine and carboxylate groups, according to the scheme presented in Figure 8. It also should be noted that the influence of the coating on the enhancement of adhesion may also be significant. The geometry of the specimen is expected to change by the presence of resole and novolac as it can be seen in Figure 6(c), and therefore, to further affect the results of the test. In that case, the observed changes in IFSS could be attributed to the characteristics of an “interphase region” created between the treated fiber and the epoxy matrix, rather than to the change in the mode of establishment of the adhesive bonding. Furthermore, differences in wetting capacity should be related to wettability of the aramid surface. Resole appears more efficient due to its higher content of polar hydroxyl groups and easier to produce uniform films on the aramid fiber surface. This is consistent with the results obtained from the contact angle

32


measurements. As shown in Table 1, some improvement of the wettability of aramid takes place upon treatment with novolac whereas the resole coating leads to significant reduction of contact angle.

Figure 8. Aramid fibers before and after hydrolysis.

Table 1. Tensile characteristics, contact angle and interfacial shear strength (IFSS) calculated from pull-out test, of aramid monofilaments Fiber type Original Twaron Chloride-treated NaOH-treated Novolac-coated Resole-coated

Tensile strength (MPa) 3103.5 1927.5 2966.8 3080.0 3097.4

Contact angle (o)

IFSS (MPa)

18.74 13.05 16.08 16.98 12.87

25.4 31.2 31.1 36.6 36.2

Chemical characterization of aramid fibers using DRIFT technique did not reveal any changes in the spectra of treated as compared with that of untreated aramid fibers, as presented in Figure 9. No detection of chemical changes due to surface hydrolysis can be conclusively established with the obtained data, taking into account the sensitivity of the employed IR spectroscopy technique. However, Amornsakchai et al. [133] identified a new peak around 880 cm-1 in aramid fibers slightly hydrolyzed with sodium hydroxide solution. The unidirectional fiber composites were prepared by the leaky mould technique, a process that involves the use of an open-ended two-part mould, as shown in Figure 5 which demonstrates a mould designed to give single composite bars. The unidirectional fiber composites reinforced with untreated, chemical treated and surface coated fibers at various volume fractions were prepared with the above mentioned procedure and the following mechanical characteristics were evaluated: tensile (ASTM D3039), flexural (ASTM D790), interlaminar shear strength [ILSS] (ASTM D2344) and impact strength (ASTM D256).

529 469

731 678

789

897 864 825 1020 942 980

1938 1903

2221

1.0

2416

2923

1.5

2646

3051

Abs

2.0

1112

1335 1304

3329

2.5

1657 1640 1614 1559 1527 1506 1412

3.0 Treated with NaOH solution aramid fibre

0.5 3.0 No-treated aramid fibre 2.5

4000

3500

2500 2000 Wavenumbers (cm-1)

1500

Figure 9. FTIR spectra of aramid fibers before after treatment with NaOH solution.

1000

531

731 670

789

897 864 826

1112

1020 979

1412

1333 1306

1663 1612 1563 1529 1500

3000

1938 1903

0.5

2413

1.0

2646

3052

1.5

2925

3327

Abs

2.0

34

Petroula A. Tarantili 1100 1000

Tensile strength (MPa)

900 800 700 600 500 400 300

Untreated aramid fibers composites

200

Chloride treated fibers composites Novolac coated fibers composites

100

Resole coated fibers composites

0 20

24

28

32

36

40

44

48

52

Fiber volume fraction, Vf (%)

Figure 10. The tensile strength of aramid fiber composites as a function of fiber content. 16

Modulus of elasticity (GPa)

14 12 10 8 6 4

Untreated aramid fiber composites Chloride treated aramid fiber composites

2

Novolac coated aramid fiber composites Resole coated aramid fiber composites

0 20

24

28

32

36

40

44

48

52

Fiber volumer fraction, Vf (%)

Figure 11. The tensile modulus of elasticity of aramid fiber composites as a function of fiber content.

From the curves of Figure 10 it is clear that there is a significant deterioration in the strength of specimens reinforced with chloride treated aramid fibers, which is consistent with the decrease in strength determined for the treated fiber. Also, the


35

strength obtained by the tensile tests is considerably lower than that expected according to the law of mixtures. This is most enhanced at high fiber volume fractions because this consistency does not allow enough volume of resin for the complete wetting and embedding of every single fiber. On the other hand, the tensile modulus for specimens with original and treated fibers remain almost unchanged as shown in Figure 11. This suggests that the fracture mechanism of those specimens involves crack propagation through the epoxy resin. In order to further explore the effect of the observed improvement on the adhesive bonding between treated aramid fibers and the epoxy matrix, the flexural properties were recorded by performing the three-point bending test. From the results of ultimate flexural strength versus Vf presented in Figure 12, it can be seen that the specimens reinforced with chloride treated aramids have improved properties especially at lower Vf. For Vf=0.45, the specimens reinforced with treated and original fibers have the same behaviour. The improvement of the flexural properties at low Vf is explained by the improved interfacial properties of the chloride treated fibers due to the surface roughening after treatment. This does not apply for higher Vf, probably due to the reduction of the volume balance epoxy resin/aramid fibers. The interactions between the fibers and the epoxy matrix result in changes of the failure mechanisms of the composite specimens. According to Brown et al. [42], the increased flexural strength of composites reinforced with plasma treated fibers can be attributed to a lower degree of compressive fiber buckling due to enhanced interfacial bonding. From the curves of Figure 12 it can also be observed that both phenolic coatings lead to improved fiber performance as far as the flexural properties of the prepared composites are concerned. This is obviously related with the interfacial bonding between the epoxy matrix and the treated fibers. Very interestingly, the resole treated fibers appear to produce reinforced specimens with higher strength in comparison with novolac and this effect is more enhanced at higher fiber concentrations. At that high filler volume fraction, it is reasonable to expect reduced wetting capacity of the liquid epoxy system because of the high total surface area of the aramid fibers that must be coated. It seems therefore, that the treatment of aramids with resole contributes to easier wetting, probably due to the favorable surface tension of the system resole/epoxy. It also should be noted that the presence of resole coating is likely to give different interface/interphase than the observed in other fiber types. This should influence both the properties of the bulk and single-fiber composite. The effect of the phenolic resins used for coating on the bulk composite samples would probably be analogous to the use of a stiffer resin matrix.

36


However, the low content of phenolics in the specimens tested (less than 0.6% w/w) does not appear capable of contributing to their flexural strength characteristics, despite the higher modulus of phenolic resins as compared to epoxies. On the other hand, novolac and resole are both reactive due to their hydroxy groups and therefore they would be potential hardeners for epoxy resin contributing to the setting chemistry, but this effect was considered negligible due to the very small amounts (less than 0.5%) of those resins in the mixture. 370 350 Flexural Strength (MPa)

330 310 290 270 250 230 210

Untreated aramid fiber composites

190

Chloride treated aramid fiber composites Novolac-coated fiber composites

170

Resole-coated fiber composites

150 20

22

24

26

28

30 32 34 36 38 Fiber Volume Fraction, Vf (%)

40

42

44

46

Figure 12. The flexural strength of aramid fiber composites as a function of fiber content.

Similarly, the effect of the type of treatment on the interlaminar shear strength (ILSS) of epoxy resin specimens reinforced with the above modified fibers, can be seen in the bargraph of Figure 13. It is clear that aramids treated with methacryloyl chloride seem to leave unaffected the ILLS of epoxies reinforced with untreated fibers. On the other hand, coating with phenolic resins shows a significant improvement of the ILSS of reinforced epoxy specimens, with the resole resin being far superior than novolac. These results undoubtedly suggested that improved interfacial characteristics can be achieved with resole coating probably because this resin is water soluble and ensures easier handling during coating procedures. Very interestingly, the impact strength of composites reinforced with coated fibers shows a serious decrease, as in this test the interfacial bonding is not the critical parameter which determines the materials performance. In fact, Fig. 14 shows that the impact strength of epoxy specimens reinforced with as-received


37

aramid fibers shows a linear increase with fiber content, whereas those contained coated fibers show a constant impact strength, with resole resin giving lower strength than the novolac. This is an additional evidence that better interfacial adhesion does not contribute to improved impact characteristics. This latest statement is associated with the energy absorption by the tested material as a function of the path of crack propagation. From the above results the following conclusions can be drawn: Treatment of the aramid fibers with methacryloyl chloride seems to be an interesting modification for the subsequent use of those fibers as reinforcement in epoxy matrices. The obvious effect on the fiber surface is a change in morphology leading to irregularities capable of producing a keying effect with the epoxy matrix. On the other hand, changes in chemical composition of the fiber surface have not been identified by the analytical technique employed in this study, despite the expectation of grafting of the very reactive acid chloride. Due to the etching of the fiber its tensile properties are declined and the same is observed for the tensile strength of composites made of treated aramids. However, the flexural properties of composite specimens were improved with treated aramids as reinforcement, which clearly suggests that the adhesive bonding between fiber and epoxy matrix becomes stronger.

Interlaminal Shear Strength (MPa)

48 44 40 36 32 28 24 20 Untreated aramid Chloride treated fiber composites fiber composites

Novolac coated fiber composites

Resole coated fiber composites

Figure 13. The Interlaminar shear strength (ILSS) of aramid fiber composites at Vf: 34%, as a function of type of treatment.

38

Petroula A. Tarantili 130

Impact strength (kJ/mm2)

120 110 100 90 80 70 Untreated aramid fiber composites Novolac coated fiber composites Resole coated fiber composites

60 50 25

27

29

31

33

35

37

39

Fibre Volume Fraction, Vf (%)

Figure 14. The impact strength of aramid fiber composites as a function of fiber content.

Similarly, treatment with aqueous sodium hydroxide seemed to hydrolyze the surface skin of the fibers and destroy their smoothness. Furthermore coating of aramid fibers with phenolic resins was found adequate to promote interfacial adhesive bonding to epoxy matrices, due to the changes in fiber surface profile, the affinity with the aramid substrate and the chemical reactivity to the epoxy resin. More specifically, fiber coating with resole would be a very interesting process since it is a simple and cheap technique that employs the as received aqueous solution of the resin, without the use of solvents or other additives. Furthermore, the higher hydroxyl content of resole, as compared with novolac, provides uniform surface coatings and ensures their better reactivity with the epoxy resin matrix. Experiments were also run with the multifilament yarn Tekmilon NA310, in order to enrich this research with results obtained from the application of modified UHMPE fibers as reinforcement for epoxy resins. Corona treated (Tekmilon NC310, Mitsui Petrochemicals Industries LTD., Japan) and calendered at 130oC fibers were used for comparison. Calendering of UHMPE fibers was performed in a motor driven two-roll mill, with adjustable rotation speed and temperature of the cylinder. Finally Tekmillon NA310 fibers treated with an oxidative agent, i.e. chromosulphate solution, were also used. The SEM micrographs of the as-received fibers show a very smooth surface, as demonstrated in Figure 15(a), whereas some small corrugation accompanied with extended groves can be found on the surface of the chromate treated fibers,


39

as demonstrated by Figure 15(b). This profile of surface may probably be the result of etching of the fiber via an oxidative attack that the chromosulphate solution causes to the polyolefin.

a

b

c Figure 15. Micrographs of UHMPE fibers: (a) untreated, (b) treated with chromosulphate solution and (c) calendered fibers.

40


Similarly, Figure 15(c) shows a calendered fiber, which after this treatment was rather transformed into a ribbon with some surface irregularities. These can be attributed to distortion of the fiber at some points because of the overlapping of the filaments within the yarn. Thus, two or more single filaments are sometimes forced to pass together through the clearance of the rolls, resulting in higher deformation. These changes in the surface geometry of the treated fibers are expected to cause mechanical interlocking between them and the epoxy matrix. The tensile characteristics determined by testing individual monofilaments are presented in Table 2. It can be seen that calendered fibers display lower modulus than the original, which is probably due to some relaxation taking place during the hot calendering at 130oC. However, fibers break at the same stress as the original, whereas the corona treated show almost the same modulus and lower tensile strength in comparison with the original UHMPE fibers. Furthermore, calendered fibers present higher elongation, which is again a result of the chain relaxation at the calendaring temperature, which destroys to some extent the fully aligned chain configuration, responsible for the ultra high modulus. Very interestingly, the chromate treated fibers display very poor mechanical properties, with the exception of the tensile modulus, which seems to remain high. As to the repeatability, it is evident that calendered fibers present the highest standard deviations, which suggests that their preparation procedure cannot give uniform products, for the reason mentioned above. However, this could be avoided in an industrial scale process, where each single filament would enter the two-roll mill immediately after the hot drawing process. Table 2. Tensile characteristics of treated UHMPE monofilaments Fiber type Untreated Calendered Corona treated Chromated treated


Tensile modulus (GPa)

2.20 ±0.09 2.17 ±0.72 2.00 ±0.21 1.08 ±0.37

67.0 ±1.24 30.6 ±1.82 62.7 ±3.92 67.5 ±1.54

Strain at breaking point (%) 8.12 ±1.38 9.55 ±3.47 6.33 ±0.75 2.39 ±0.72

After the investigation of the fiber properties in tension, unidirectional specimens reinforced with original and treated UHMPE fibers were tested for their tensile characteristics. The dependence of tensile strength and modulus on the filler volume fraction is shown in Figures 16 and 17.


41

500 450


400 350 300 250 200 Untreated UHMPE fiber composites

150

Calendered UHMPE fiber composites

100

Corona treated UHMPE fiber composites Chromate treated UHMPE fiber composites

50 5

10

15

20 25 30 Fiber volume fraction, Vf (%)

35

40

Figure 16. The tensile strength of UHMPE fiber composites as a function of fiber content. Unteated UHMPE fiber composites Calendered UHMPE fiber composites Corona treated UHMPE fiber composites Chromate treated UHMPE fiber composites

9.5

Tensile modulus (GPa)

8.5 7.5 6.5 5.5 4.5 3.5 2.5 1.5 5

7

9

11

13

15

17

19

21

23

25


Figure 17. The tensile modulus of elasticity of UHMPE fiber composites as a function of fiber content.

From the Figure 16 it is evident that a linear relationship is established between tensile strength and Vf in the range 7-25%. Clearly, it can be seen that UHMPE fibers are divided into two groups, according to the strength of specimens. Untreated and calendered fibers display higher strength in comparison with corona and chromate treated fibers. This suggests that increased interfacial adhesion does not control the tensile properties and, therefore, the corona and

42


chromate treated fibers, which are expected to show better interfacial properties with epoxy, give lower strength products because of the reduction of the fiber strength itself. Similarly, the tensile modulus demonstrated in Figure 17 shows a linear dependence on Vf, which means that the “law of mixtures” is applied in this case. Table 3 shows the stress required to pull-out the fibers embedded into the epoxy matrix, which can be considered a direct measure of the adhesive bonding developed at the interface. Table 3. Contact angle measurements and interfacial shear strength (IFSS) calculated from pull-out test of UHMPE monofilaments Contact angle (o) 40.18 38.20 25.59

Fiber type Untreated Calendered Corona treated Chromate treated 120

Interfacial shear strength (MPa) 0.36 0.60 0.85 0.70

Untreated UHMPE fiber composites Calendred UHMPE fibre composites

Flexural strength (MPa)

110


100

90

80

70

60 20

25

30

35

40

45

50

55


Figure 18. The flexural strength of UHMPE fiber composites as a function of fiber content.

Clearly, it can be seen that corona treated fibers show the maximum adhesion followed by chromate treated fibers. It should also be noted that calendered fibers need a load 65% higher than that of the original for the pull-out, which might be attributed to their surface irregularities and the subsequent interlocking into the epoxy matrix.

Surface Modification of Ultra High Modulus Polymeric Fibers 22.5

43

Untreated UHMPE fiber composites Calendered UHMPE fiber composites

Flexural modulus (GPa)

20.0


17.5 15.0 12.5 10.0 7.5 5.0 20

25

30

35

40

45

50


Figure 19. The flexural modulus of UHMPE fiber composites as a function of fiber content.

Interarlaminar Shear Strength (MPa)

22 20 18 16 14 12 10 8 6 4 2 0 Untreated UHMPE fiber composites

Calendered UHMPE fiber composites

Corona treated UHMPE fiber composites

Chromate treated UHMPE fiber composites

Figure 20. The Interlaminar shear strength (ILSS) of UHMPE fiber composites at Vf ~ 28%, as a function of type of treatment.

In order to further explore the improvement of the adhesive bonding between treated UHMPE fibers and the epoxy matrix based on the pull-out experiments, the flexural properties of composite specimens were tested, by performing the three-point bending test. Figure 18 presents the flexural strength of specimens with various types of UHMPE reinforcements, and shows again two groups of

44


fibers, but this time those who display strong adhesion seem to give the best flexural properties. In the case of modulus, corona treated fibers show the best performance and, clearly, can be distinguished from the other types of fibers, as Figure 19 shows. As expected, the ILSS measurements, presented in the bargraph of Fig. 20, show again that corona treated fibers are superior than the untreated, calendered and the chromate treated fiber composites. From the above discussion the following conclusions can be drawn: Corona and chromate treatments modify the surface of UHMPE fibers and give products with reduced tensile strength, but are capable of leading to increased adhesive bonding. The effect of the above two types of treatment is etching of the surface accompanied by a controlled oxidative attack, that introduces oxygen containing species on the fiber surface. These changes are adequate of promoting improved interfacial bonding, as mentioned above and established by the ILSS measurement. On the other hand, calendered fibers retain their strength with significant reduction of modulus due to some relaxation during hot calendaring. Taking into account that the modulus of UHMPE fibers is extremely high its reduction upon calendering would not be a serious problem provided that calendered fibers offer improved interfacial characteristics. It also should be noted that even the changes of the cross-sectional shape, i.e. from circular to ribbon-like, may contribute to better interfacial characteristics since the new shape ensures increased contact surface between PE fiber and epoxy resin. Properties dependent on the off-axis properties of the composite specimens, such as flexural and interlaminar shear strength, showed that corona treated fibers are advantageous as reinforcement for epoxies. This effect could probably be the reason why many authors are focusing today their research on corona and plasma processes instead of chemical treatment. Obviously, the increased versatility of the above methods combined with the advantages of dry method with short time treatment and continuous process make them attractive for use in industrial production.

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Chapter 2

MANUFACTURING AND FEATURES OF ACOUSTICALLY OPTIMIZED NATURAL FIBRE REINFORCED PLASTICS N. Aisenbrey and L. Frormann Institut für Produktionstechnik, Westsächsische Hochschule Zwickau, Germany

Abstract Fibre reinforced plastics are used worldwide in various applications. Large quantities of natural fibre reinforced plastics are processed in the automobile industry due to their good mechanical and ecological characteristics, but particularly because of their low price. The application range is mainly in low loaded places, e.g., in the inside of automobiles like the side door panels or ceiling. Initially the market for natural fibres showed a rapid upswing that stagnated afterwards due to their limited range of applications. To force the application, supplementary functions must be integrated into the materials with no additional costs. Usually the focus of the research is on the increase of the mechanical characteristic values and the development of more economical and productive manufacturing methods. The high acoustic potential of the natural fibres is less investigated and only considered as a side effect. But this acoustic potential enables many application possibilities of the reinforced plastics. Particularly, the health endangerment by the constant increase of the noise pollution clarifies the necessity for the research. Acoustically effective structural elements can be manufactured as a sandwich. The building method of a sandwich element permits a large variation range. With simple changes in the structure, for example by the use of different layers with different characteristics, the material

52

N. Aisenbrey and L. Frormann can be adapted to several applications. An effective sound absorption particularly in the middle and high frequency ranges is possible with layers of natural fibre nonwovens or open porous natural fibre reinforced plastics. Sound absorption values over 0.7 are reached at frequencies above 1 kHz. High weighted difference level Dw values of up to 30 dB can be achieved with very thin natural fibre reinforced composite plates, which exceed the mass law up to 4 dB. The sound absorption achievement of multilayered elements with firm surface layers and different core materials is very good with weighted difference level values of 34–37 dB with low element weights between 13–16 kg/m². For these high strength lightweight construction units with positive acoustic characteristics in sound insulation and sound absorption, various applications exist in the architectural acoustics and in the automobile industry. The application of natural fibre reinforced elements as multi functional wall and ceiling panels permits a strong stimulation of the market for these materials.

1. Introduction The development of fibre reinforced plastics is based on the analysis of natural structures. The search for economically and environmentally compatible reinforcement fibres led to the use of natural fibres. Natural fibres exhibit many advantages compared to synthetic fibres. When they break, the fibres don’t splinter and burn at thermal utilisation with CO2 neutrality. The specific mechanical characteristics of natural fibres with a low density approximately 1.5 g/cm³ are comparable with those of glass fibres. Glass fibres exhibit with a density of 2.6 g/cm³ a clearly higher density than natural fibres. Hereby natural fibres are interesting also for the application in lightweight constructions. Natural fibres are isolated from the bast fibre bundles of fibre plants. Thereby they are divided into different categories, about which only bast and hard fibres are interesting for technical applications. An important representative of hard fibres is the sisal fibre. Flax and hemp fibres are affiliated to the bast fibres [1]. The last mentioned are used most frequently due to their price, characteristics and supply possibility as technical fibres. The hollow bast fibres are developed by the plant as composite material of cellulose, pectin and lignin. They are laid out on high mechanical load by tensile and bending. Good acoustic behaviour of natural fibres is assumed by their special structure. The different layers of the cell wall in a bast fibre consists of cellulose fibrillates, which are aligned in longitudinal direction of the fibre under tensile loads, for example at the ramie fibre. Plants are also exposed additionally to bending loads and primarily a reduction of load peaks is essential. So the fibres are laid out to absorb mechanical energy in bending direction by so-called screw texture of fibrillates in the secondary wall, for example hemp and flax fibres [2, 3,

Manufacturing and Features of Acoustically Optimized Natural Fibre…

53

4]. The hollow lumen of the natural fibres can substantially strengthen the internal absorption properties of the manufactured reinforced composites. The internal absorption describes the transformation of oscillation energy in warmth within a material. Figure 1 shows different processed hemp, flax and jute fibres.

Figure 1. Mechanical fibre separated hemp; field retted flax fibre and yarn; jute nonwoven.

For the extraction of the natural fibres, the fibre plants are divided, for example, by retting or by mechanical means. The mechanical breakup is most economical, thereby less time intensive and supplies constant fibre quality. In order to keep the price advantage of the natural fibres compared to the synthetic fibres, expensive processing methods like those for yarn and canvas production are avoided; instead, natural fibres are usually converted by the nonwoven procedure to a semi-finished fabric. For the production of yarns and canvas, very high quality fibres are needed. Depending on the application of the finished nonwoven, rougher fibre bundles or less cleaned-up fibres can be used in the nonwoven technology. With the nonwoven technology, a lot of variations in the thickness, weight per unit area and air permeability of the natural fibre fabrics can be realised. Purposefully very firm and very weak voluminous nonwovens can be manufactured by variations in the number and puncture depth of the needling machine. Thereby the air permeability as well as the impregnation possibility of the nonwoven decreases with rising weight per unit area. Mixing of various kinds of fibres is easily realisable. In particular, the possibility of adding thermoplastic pile fibres at the same time to natural fibres at the needling machine during the mixing process offers the advantage of a very good mixing of the fibrous components and thus a good form and draping ability of the semi-finished fabric. The wettability problems arising with thermoplastic matrices in more firmly needled and voluminous nonwovens can be reduced by the support of so-called hybrid technology, since there are only very short flow paths. In nonwovens the natural fibres are laid almost disordered in 3-D assembly, in order to achieve a preferable anisotropic behaviour of the material. A certain orientation of the fibres would be possible by card slivers, yarns or canvas. It is also possible to interconnect layers with different characteristics by the nonwoven

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N. Aisenbrey and L. Frormann

process into a semi-finished layered material. Natural fibre needled nonwovens can be adjusted well to different requirements. Natural fibre reinforced plastics (NFRP) are particularly common in traffic engineering due to their ecological aspects and a high saving potential in cost and weight. The largest range of applications lies in the inner area of passenger cars, for example, the doors’ inside panels, hat rack or ceiling. The particular emphasis of the research here is on cost reduction by more efficient processing procedures and simultaneous improvement of the mechanical characteristic values of the natural fibre reinforced plastics. Thereby quality fluctuations of several batches of the natural fibres affect the mechanical characteristics [5, 6]. Many additional characteristics like high internal absorption of the natural fibres are used in our days only as side effects. Since there is an increased attention to the environmental compatibility by products in production, use and disposal, the need of materials from renewable components should actually increase. However, the market for natural fibre reinforced plastics stagnated after a first rapid rise in the last years. Despite the many positive characteristics natural fibre reinforced plastics have been generally accepted only in niche applications. The composites are used despite their good specific mechanical characteristic values mainly for applications with very low mechanical load and high requirements in the lightweight construction. With few exceptions the application i.e. is particularly limited to inner construction units of automobiles. The natural fibre reinforced plastics are not frequently used, because glass fibre reinforced plastics (GRP) ensure more constant construction unit qualities compared to NFRP. The characteristics of the natural fibres vary partially substantially depending upon cultivation area and the environmental conditions. The cost-neutral integration of additional functions is necessary to extend the scope of natural fibre reinforced plastics and for a stimulation of the market for these materials [7, 8]. Natural fibre reinforced plastics offer options due to the high internal absorption for innovative applications. The environmental pollution by noise is increasing continuously. Innovations in the noise control is equalising rapidly by the rising number of noise sources. Therefore it is very important to protect human habitat against noise sources. The insulation and absorption of sound by custom made construction units can reduce the health endangerment by the constantly growing noise pollution. Into the architectural and room acoustics usually voluminous and heavy materials are used for the subsequent sound insulation of rooms. By the positive characteristics of natural fibres, the material and weight can be reduced in sound proof systems. This exhibits clear advantages in planning and installation. The installation of


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light elements decreases the cost without affecting the statics of the existing buildings. In the acoustical characterisation of a material sound insulation and sound absorption are differentiated. Sound insulation describes the obstruction of sound propagation by a separative element, if the obstruction is caused mainly by reflection. Sound absorption designates in contrast the transformation of sound energy in warmth, for example by friction in porous media. Both mechanisms arise mostly at the same time. Thus the sound insulating effect of a construction unit is often substantially improved by the high internal absorption of the oscillation energy. The weighted difference level Dw contains both types [9]. Some few investigations of the sound absorption of natural fibre reinforced plastics were accomplished with the result that these materials possess a large potential in the area of the acoustics [10]. In the literature usually the sound absorption coefficient and thus the absorption characteristics of natural fibre reinforced plastics are determined. In previous investigations, sound absorbing sandwich structures made up of flax, hemp fibres and polypropylene matrix were used. The influence of the natural fibres, their amount in polypropylene matrix and the density of the materials on the mechanical and acoustic characteristics were examined. The acoustic measurements were done in an alpha cab and in the impedance pipe. The determination of the characteristics of the base materials led thereby to a better understanding of the absorption behaviour of the natural fibre reinforced plastics. Recommendations for the characteristics of an optimal structure of sandwich element can be made from this result, which provides a high absorption coefficient. The uncompressed interior layer from a natural fibre polypropylene hybrid nonwoven should exhibit a high amount of polypropylene, very fine natural fibres and low puncture quantity. For the surface layer, a strong needled nonwoven with a high amount of natural fibres should be processed by press technology [10, 11, 12]. In order to use the advantages of natural fibre reinforced plastics in further acoustic applications, an extended knowledge is mandatory for example regarding the sound insulation characteristics. The research presented in this chapter focuses on the area of the acoustic characteristics of natural fibre reinforced plastics and a multi functional separation construction unit with positive sound insulation and absorption characteristics has been developed. The better knowledge about the processing, construction, material and its insulating and damping possibilities as well as the relationship between structure and characteristics is needed regarding the application of construction units for the room and architectural acoustics, equipment and housing production for traffic engineering. With foresight the

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products have to be developed to support the tendency for noise control, health protection, environmental compatibility, reusability and resources conservation. In spite of the potential characteristics of the natural fibres, an integrated approach is needed for the fabric construction and the following compound production. In particular the high variability of natural fibre nonwovens and natural fibre canvas can be used. The acoustic and mechanical characteristics of the elements for different applications can be optimised by the variation of structure and the characteristics of different layers. In the development of a special construction, sandwich structure possesses the largest potential at material and weight reduction.

2. Experimental Process 2.1. Specification Analysis For the development of elements from new material combinations, it is compellingly necessary to analyse the market for a requirement profile after detailed search and consultation with possible sales partners. Acoustically effective elements as multilayered constructions from natural fibre reinforced plastics offer an extremely wide scope. It is not possible to determine a single sandwich element which cover all requirements for the diverse range of applications. For this reason the construction of a custom made element is necessary for different applications. To afford the set up of a requirement analysis two areas of application were selected. One of the application areas was a multi functional wall element in the room and architectural acoustics to absorb and insulate airborne sound. The regularisation of the room acoustics usually takes place by the characteristics of spatially delineating surfaces, which must exhibit appropriate sound absorbing characteristics. Different absorber types are used for a wide band effective sound absorption. For the absorption of middle and high frequencies usually porous absorbers are used. Porous absorbers exhibit very good absorption factors for frequencies, at which the absorber thickness is ≥ ¼ of the airborne sound wavelength. For the absorption of the deep frequencies the necessary thickness of porous absorbers becomes too large, so that here so-called resonance absorbers are used which are developed from plate elements and which are well adjustable by the choice of the construction parameters in the frequency range. In practice combinations of porous absorbers and resonance absorbers are used. The point of interest for applications in the architectural acoustics is the


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protection of the inhabitant from unreasonable noise from the adjacent areas. Central task is to provide a sufficient sound insulation. To obtain a single value in measurements of the resistance of a construction unit against the sound transmission the weighted difference level Dw is used. A high weighted difference level and thus a good sound insulation can be obtained by sufficiently thick and/or heavy separation construction units or multi layered constructions, which however must be acoustically impermeable. Usually architectural and room acoustic functions are fulfilled by different construction units. Therefore a high sound insulation is achieved by the application of thick, heavy partitions and/or multi layered constructions. On the other hand by the application of plates, perforated tiles or porous absorbers a good sound absorption is realised as well. An insulating and absorbing structure from natural fibre reinforced plastics as multi functional separation construction units with very good characteristics in the room and architectural acoustics primarily demands at upper surface with open porosity for the sound up take, and at opposite surface with simultaneous sound tightness. This functionality is given by a layered structure, with small technical and cost expenditure. The second area of application was the use of the sandwich elements in trains e.g. in the lining of the inner walls of a passenger railway car compartment. Here the airborne sound resulting from the movement of the wheels on the rails as well as the structure borne noise over the external wall must be excluded from the compartment. Another usage of sandwich elements in trains can be in the interior of the compartments as base plate. The usage locations e.g. walls or ceilings, are the areas where no high load peaks are expected. Difficult site conditions such as moisture in humid rooms affect the materials adversely. For such locations the natural fibres would have to be particularly impregnated. In all the locations e.g. buildings, train walls or ceilings, the most important demand is light weight. In addition especially at walls of the trains an impact resistant surface layer at the viewable side is necessary to prevent vandalism. At the application of sandwich elements as base plates in the trains the materials should be impregnated at least at the surface layer to protect against humidity, aggressive cleaning chemicals and salts. The surface layer must exhibit a certain resistance against abrasion due to the contamination of dirt particle. The application of the multi functional separation unit in buildings requires the installation of multi layered fibre reinforced components with weak and hard layers. The hard reinforced composites form the outside layers and exhibit a thickness of 1 - 6 mm. The core must be open-porous and between 10 and 40 mm thick. By the choice of the materials such as nonwovens and canvas from natural fibres a bending modulus >

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3 GPa should be reached. A good sound insulation occurs at a weighted difference level above 33dB is requested and the optimisation of the sound absorption should takes place at frequency of 1 kHz.

2.2. Development Concept of Construction Elements for airborne sound insulation can be developed from very simple to complex. The simplest possibility is the use of a porous material e.g. a thick nonwoven, which particularly dams high frequencies. A wall boarding dams particularly within the low frequency range. A combination of several kinds of sound absorption should result in an airborne sound insulation over the entire frequency band width. Thereby the constructional systems become more voluminous. A further possibility to use thin plates as possible insulation material is to increase their weight by the use of heavy weighted materials. For example in train applications where lightweight construction is demanded these methods are not suitable. Acoustically effective elements can be manufactured as sandwich structure. For these high strength lightweight structures various applications exist among others in the architectural acoustics and in the automobile industry. The building method of a sandwich element permits a large variation range. Sandwich constructional units are elements with a relatively thick, light core and relatively thin surface layers. The surface layers take up tensile and bending forces and protect the core against external influences. The core keeps the surface layers on a given distance and takes up shear as well as pressure forces. In addition the core protects the surface layers against distortion, breaks and crinkling [13]. Sandwich structures are used for very diverse applications, since they represent high-rigid structures with a very small mass. They possess technical meaning i.e. in the aerospace, shipping or transportation industry. These elements consist usually of a light, thrust-rigid core with high-strength and high-rigid surface layers, for example of aluminium honeycombs as core material and surface layers of aluminium or fibre reinforced plastics. The characteristics of the sandwich elements predestine it for the usage areas where a high thermal insulation, high breakthrough firmness and sound insulation are demanded with a small density. The core materials usually require high pressure stability whereas some weaknesses in mechanical stability are acceptable. As core materials above all foamed plastics, light wood e.g. balsawood and honeycomb structures e.g. from aramide fibre reinforced plastics are used. Here the materials are selected particularly after the best price considerations. The characteristics of a sandwich core are co-ordinated always exactly with the respective application. The surface


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layers consist frequently of fibre reinforced thermoplastics or thermoset polymers. With high-rigid materials a sandwich element can be build with thinner surface layers leading to light weight and low cost. An overview of the conventional composition of sandwich elements as well as the consideration of new sandwich elements from NFRP is presented in figure 2. The surface layers can be varied in thickness, weight per unit area and material application. The mechanical characteristics can be varied from flexible and impact resistant to resistant to bending and inflexibly. Furthermore the core layer can be varied, by being composed of several layers with different characteristics. The sandwich element can be symmetrically developed or each individual layer possesses its own characteristics. The acoustic and mechanical characteristics of the sandwiches for different applications can be adjusted by variation of the structure and the characteristics of the different layers, whereby generally a high rigidity and a large dissipation factor are to be aimed.

Figure 2. Composition of sandwich structures [14].

For the execution of mechanical and acoustic tests different fundamental sandwich constructions were defined and manufactured. To fulfil the needs of the requirement list the sandwich elements for the application in wall and ceiling components must be on the one hand self-supporting and as light as possible. The material is permitted to reversibly yield on continuous pressure, nevertheless it must be resistant to impact load. Sandwich elements for the ground application should not yield under pressure and contain therefore firmer and concomitantly

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heavier core materials. In order to determine the influence of the different variable parameters, only some few parameters in each case can be changed for the individual variants. Due to these build variations it is possible to conclude on the characteristics of a large range of further conceivable optional variations. In figure 3 A the essential structure of a simple sandwich element for the application as wall or ceiling component is schematically represented, which already covers a large range of optional variations. The essential structure consists of two press moulded natural fibre reinforced composite plates, bonded with a nonwoven between. On the basis of this simple structure the influence of different materials in the core layer on the acoustic characteristics can be examined. Various nonwovens with varying puncture quantity, synthetic and natural fibres as well as different thicknesses are used. The nonwoven material used without matrix reversibly yields under pressure and bending force. Therefore the material is classified weak under bending and pressure. As matrices different thermoset systems are used for the production of the surface layers. These surface layers were build in different thicknesses. Furthermore it was examined by variation of the surface layer thickness the affects of symmetry in sandwich elements on the acoustic behaviour. For manufacturing a sandwich element from natural fibres with solidified core material higher production costs appear.

Figure 3. Construction of sandwich elements with natural fibre reinforced plastics.

Taken into account the aspect of cost reduction, the core material used should be a natural fibre nonwoven with as small as possible matrix amount. Variation options are also possible here in the form of differently thick core layers with varying matrix amounts and low to high compressed natural fibre nonwovens. By


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immersion of the nonwoven the weight of the sandwich element rises; however a reduction of the surface layer thickness is possible due to the rigidity of the core layer. As the number of assigned layers increases, the possibilities of variation rise as well. In the case of a five layered sandwich element two press moulded natural fibre reinforced plastic plates are used as surface layers. On each surface layer another layer of a natural fibre needled nonwoven is glued. A middle core layer, an open porous natural fibre reinforced plastic plate with low density < 1 g/cm³ is bonded between the natural fibre nonwovens. Here again possibilities of variations in surface layers thickness and composition are available. The unsaturated nonwovens can be varied in compression, fibres and the thickness. The middle layer solidified with a thermoset matrix can reduce the majority of the sound energy by oscillation due to its higher weight between the weak natural fibre nonwovens. Depending upon weight of the intermediate layer and solidification of the nonwovens the oscillation produced by the sound energy at the surface layer can be more or less strongly absorbed. Disadvantages of the multi layered sandwich structure are the thickness and the rising weight of the element with each layer. The exchange of one of the surface layers of the multi layered sandwich element by an open porous composite plate leads to an acoustically absorbing construction unit represented in figure 3 B. In addition the sound permeability of the sound up taking surface layer can also be produced by drilling in varying diameter and allocation.

2.3. Materials After defining the particular application requirements, suitable materials were selected from a wide range of semi-finished fabrics and matrices. One of the demands in the selection of the semi-finished fabrics was limited price. As most cost effective, highly variable and easily handled materials, different natural fibre needled nonwovens made up of flax, hemp and jute were used. Some of the nonwovens taken out of running production for industrial applications were used as standards. By the nonwoven technology variations of natural fibre fabrics in the thickness, weight per unit area and the air permeability are possible very easily. Here particularly the influence of the air permeability on the acoustic characteristics and the firmness should be examined on the basis of different puncture quantities and weight per unit areas. The preliminary tests showed that light and thin plates with high resistance to bending exhibited sufficiently good acoustic characteristics. As thermoset systems

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the epoxy resin HT2 with the hardener HT2, as well as the epoxy resin L1100 with the hardener EPH 294 were used. As thermoplastic the matrix polypropylene, used in industry into various applications, was selected. Since the reinforcement fibres are a natural product, therefore a matrix produced from renewable materials is expected to be interesting as well. Keeping in mind a natural product, as thermoset system a linseed oil acrylate and as thermoplastic system polylactid were used. The thermoplastics are used as fibres mixed with the natural fibres into hybrid fabrics and as granulates. The synthetic thermoplastic fibres can function thereby through consolidation with heat and pressure as matrix. In addition it is possible to use the good internal absorption possibilities of the fibres if the hybrid nonwovens are processed against their normal processing with a thermoset matrix and low temperatures in press moulding. As core material the application took place predominantly from natural fibre nonwovens without matrix.

2.4. Manufacturing Technology The manufacturing of the sound insulating sandwich layers could take place by means of press mould technology. The natural fibre reinforced plastic plates were manufactured from a thermoplastic matrix with a processing temperature from 170 to 200 °C and a pressure of 100-300 bar. Press moulding with thermoset systems could take place at temperatures between 60 and 130 °C. With different conventional manufacturing methods closed porous and open porous foams can be manufactured. The injection moulding or the extrusions of physically and chemically foamed plastics are standard technologies. With these procedures natural and synthetic fibres can be inserted into the matrix as short and long fibres. However the use of nonwoven fabrics and canvas as reinforcement fabrics is possible only conditionally. These fabrics are processed mainly with the press mould technology and resin infusion procedure. The manufacturing of the open porous sound absorbing NFRP structures required a modification of the conventional approach. The amount and size of developing pores can be defined here by the applied pressure. To avoid difficulties in connection of the different layers the press moulding process for the core layer will be interrupted after complete soaking of the fibres with the thermoset matrix before curing. The core layer is mechanically reopened and inserted with the already prefabricated surface layers into the press mould. Thereafter this dry/wet layer structure is press moulded up to the end of the necessary press time under decreased pressure below 2 bar. The necessary press


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force is dependent on the desired thickness of the core material as well as demanded porosity. The expiration of the manufacturing of open porous natural fibre reinforced plastics is presented in figure 4.

Figure 4. Manufacturing of open porous NFRP structures.

The thickness of the press moulded elements was controlled according to the necessary geometry of specimen by a variable mould. The definition of the testing and specimen geometry took place according to the standard testing methods for the used materials following the appropriate DIN standards [15, 16, 17, 18]. For the production of the fibre reinforced specimens for the mechanical and acoustic analysis a plane geometry with the dimensions 300 x 400 mm and different thicknesses between 1 - 6 mm was specified. For the cutting of the specimen from the manufactured composite plates two different techniques could be examined for their suitability. Half amount of the specimen for the mechanical examination was sawed with a circular saw from the composite plates. Despite restraint devices the specimen were subjected by the saw with a vibration load. At the same time the teeth of the saw blade tore partial fibres from the cutting edges. This led to the cutting quality presented in figure 5. The side of the specimen had to be worked over again by grinding. For the production of a very exact edge during processing of glass and carbon fibre reinforced plastics the water jet cutting is used. The application of this method is not suitable for natural fibre reinforced plastics. Due to the capillary forces of the hollow natural fibres at the edges water can be absorbed. The consequence is swelling of the specimen and consequently damage of the material. In order to produce very sharp edges with dry method, cutting of a part of the specimen from the composite plates for the bending test took place with the help of laser.

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Figure 5. Cutting with circular saw.

Figure 6. Cutting with laser.

This produced very linear and accurate edge is compared to the sawing method in figure 6. The laser produced very sharp and parallel edges; the fibres were cut very clean. The edges of the composite plate with a thermoplastic matrix were partly sealed by the thermal influence. The thermal load led thereby to no optically recognisable damage of the edges.

3. Results 3.1. Characterisation of Mechanical Properties The mechanical properties of the natural fibre reinforced plastics can be adjusted by the choice of a suitable matrix, the degree of compression of the assigned natural fibre and the kind of the semi-finished fabric. For the evaluation of a fibre reinforced plastic the characteristics of the semi-finished fabric or the reinforcement fibres are of crucial importance. Therefore first a characterisation of the constituted semi-finished fabric take place. The mass per unit areas of the


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natural fibre needled nonwovens and hybrid nonwovens were approximately 0.4 kg/m² to 1.6 kg/m². Since canvas materials were particularly used as surface layers, here semi-finished fabrics with clearly lower mass per unit area of approximately 0.2 – 0.4 kg/m² were used. Flax and hemp fibre mixtures were used for the determination of the influence of the puncture quantity with a range of 35 55 punctures/cm². The coherence of the fibres in the needled natural fibre nonwovens can be determined by measurement of elastic modulus. The values of most nonwovens were between 0.4 - 0.8 MPa within the expected low range with high standard deviations. The natural fibre bundles used in the nonwovens possessed a length of 2 - 13 cm. During processing of the nonwovens the length of the assigned fibres and the applied needle parameters were not sufficient in the production of natural fibre semi-finished fabrics with interconnected fibres with more tensile strength. Thereby the coherence of the fibres in the fabric was smallest in low needled nonwovens whereas the firmness rose in high needled nonwovens with more puncuteres/cm² especially in hybrid nonwovens. A fibre mixture nonwoven with a high amount of jute fibres (70%) achieved a comparatively small value of 0.15 MPa. A hybrid nonwoven with hemp, flax and polypropylene fibres reached contrary to expectations a very high elastic modulus of 2.5 MPa. The coherence of the individual fibres was increased by a high puncture quantity. This hybrid nonwoven is particularly suitable for the application as core material in a sandwich element due to its desired elastic modulus property. However the particularly high coherence of the fibre mixture of the semi-finished fabric is destroyed while processing. For optimal properties of the composite the complete wetting of the fibres is necessary. The characterisation of the flow, melting and networking behaviour of the polymer matrices took place by means of different thermoanalytical procedures. All used matrices can impregnate the semi-finished fabric completely with the appropriate processing temperature and applied pressure. In all examined matrices, a possibility of decreased viscosity with temperature exists. For thermoplastics an inclusion of air could be achieved by increasing the temperature. The air inclusions were largely distributed in the specimens in a coincidental manner. Such a distribution declines the characteristics of the composite. The determination of the mechanical characteristics (tensile, pressure, bending and impact strength) of the manufactured specimens can take place according to the standard methods of the plastic examination following the appropriate DIN standards. The mechanical characteristics of the composites can be adjusted over a large range by the choice of the matrix and the degree of compression. The connection between natural fibre nonwoven and the used matrix

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systems regarding the efficiency of the reinforced plastic was concretised by the tensile testing. The determination of the elastic modulus values took place in production direction (II) and 90° turned to the production direction (┴) of the nonwovens used for the manufacturing of the reinforced plastics. The materials exhibited the higher elastic modulus values in production direction of the nonwovens. This suggests a strong orientation of the fibres during the nonwoven production. The connection between the production direction and higher elastic modulus values was reduced with increasing puncture quantity. The application of a mixed fibre nonwoven with 30% flax and 70% jute led to relatively low elastic modulus of approximately 8.6 GPa of the composite plates. Opposite composite plates with pure hemp fibre needled nonwovens or the admixture of hemp in mixed fibre nonwovens resulted in higher values for the elastic modulus. The reached elastic modulus depended on the quantity of the needle punctures. A higher puncture quantity had a negative effect on elastic modulus by the damaging of the fibres. A composite plate with the high punctured hemp nonwoven achieved around 20% lower values with approximately 9.0 GPa than a composite plate with a medium punctured hemp nonwoven with 11.0 GPa. In comparison, the application of a flax and hemp mixture led to high elastic modulus values under puncture quantity within a moderate range from 35 to 55 punctures/cm². The variation of the composite plates in the puncture quantity with hemp/flax mixture nonwovens with the epoxy resin matrix HT2 within a middle range between 35 to 55 punctures/cm² did not cause significant change of the elastic modulus value, which was at all puncture quantities around 10.5 GPa. The puncture quantity must be changed in a clearly higher range to show clear effect to the mechanical characteristics. The influence of the used matrix systems was very high during the tensile testing. The composite plates with epoxy resin matrix, in the variants hemp nonwoven with epoxy HT2 and hemp nonwoven with epoxy L1100, achieved the highest elastic modulus values of up to 10.2 GPa with the matrix L1100 and 11.0 GPa with the matrix HT2. A material combination consisting of hemp fibres and a linseed oil acrylate system as matrix reached values of maximum 9.2 GPa. The values of the specimen with thermoplastic matrices possessed lower values in comparison to the thermoset matrix systems as expected. However the values for hemp and polylactid composite with approximately 7.0 GPa was approx. 20% higher than hemp and polypropylene with 5.8 GPa. The influence of the fibre content on the elastic modulus was clearly recognizable with composite plates within 20 to 40 vol% hemp fibres. The composite plates with the fibre amount 40 vol% exhibited an increase of the elastic modulus up to 35% with a value of 11.0


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GPa in comparison to the low amount of 20 vol% fibre reinforced plate with 7.2 GPa. The mechanical characteristic potential of natural fibre reinforced plastics is determined apart from the characteristics of the natural fibres also by the interaction between natural fibres and surrounding matrix. The fibre matrix adhesion for the examined natural fibre reinforced plastics could be derived from bending modulus values at three point bending test. The better the matrix material connects the reinforcement fibres the higher can become the bending modulus values. A low fibre matrix adhesion leads in opposite to clearly lower bending modulus values. The bending characteristics of plane elements are important for the acoustic characteristics. Impinge acoustic waves lead to a bending deformation of the plates. The more rigid a plate reacts to the deformations, the less it can become stimulated to bending vibrations. An increased rigidity of plates can be achieved, as the plate thickness is increased. However with the demanded lightweight building units this would lead to a clear increase of the weight. Therefore the rigidity must be achieved by the increase of the bending strength of the plates. Additional to the characterisation of the 3-point bending characteristics, an examination of the influence of the sample preparation via the mechanical treatment by means of a circular saw or the thermal treatment by means of laser cuttings took place. Natural fibres are relatively tough and difficult to cut. Frequently individual fibres are pulled out from the cut edge when sawing. This leads to the formation of micro cracks, which can lead to a degradation of the mechanical properties. At the use of a laser the natural fibres are cut without tensile load, however the thermal load must be reduced by a very short load time of the laser and suitable cooling methods. Natural fibres are converted by needling process to nonwovens. The puncture quantity affects thereby the compression and the coherence of the nonwoven material. Individual fibre bundles are diverted by the needling in z-direction. A certain needling is necessary to enable the handling of the nonwovens. Under high puncture quantities, there exists the danger of fibre damage. In the reinforced plastics the fibre damage by the needle process becomes apparent by smaller mechanical strength. Therefore in figure 7 the influence of the puncture quantity is represented on the bending characteristics of hemp and flax composites. The puncture quantity has a small influence on the bending modulus with a variation of the parameters within a middle range. The values were highest with 9.9 GPa with 35 punctures/cm² and lowest with the composite plates with 8.3 GPa with 55 punctures/cm². The rising puncture quantity slightly damaged the natural fibres by mechanical straining. Additionally the fibres in z-direction were turned back and formed small fibre bars at those a notch effect during tensile load arose.

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Slightly higher weight per unit areas of 0.93 kg/m² at the laser cut specimen lead with same puncture quantity of 40 punctures/cm² with 10.2 GPa to much higher bending modulus values compared with maximum 9.2 GPa at a material with 0.84 kg/m² weight per unit area. An influence could be recognized with the laser cut specimen with an increase of approximately 10% of the bending modulus values compared to the sawed samples.

Figure 7. Comparison of the weight per unit area and the puncture quantity.

In order to be able to use as light and thin plane elements for the sandwich surface layers as possible, their rigidity must be increased. Especially the highstrength hemp fibre offers particularly high flexural rigidities to the natural fibre reinforced plastic elements in combination with epoxy resin. The bending strength of the manufactured composite plates with different strong needled nonwovens correlates with the values of the bending module. However the differences between the individual materials and the cutting methods with bending strength values of 115 - 130 MPa are not very pronounced. The characteristics of the used matrices affect the transmission of the tensions among the reinforcement fibres in composite plates. Thermoset matrices can transfer higher tensions than thermoplastic matrices. A comparison between two cold-curing epoxy resin matrices, one heat curing linseed oil acrylate and one polylacid is presented in figure 8. The differences in the bending modulus between composite plates from the two different cold curing epoxy resin matrices


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was about 10% at a value of 10.0 GPa with a hemp nonwoven with HT2 and 8.8 GPa with L1100. The fibre matrix adhesion was equally good with both examined epoxy resin systems. The difference resulted from somewhat smaller mechanical basic characteristic values of the matrix L1100. Laser cut of the specimen led with all assigned thermoset matrices in each case to an increase of the bending modulus around approximately 10% compared with the sawed specimen. The linseed oil acrylate matrix consisted of 85% renewable materials and led to a smaller bending modulus value of 7.6 GPa.

Figure 8. Comparison of the influence of different matrices on the bending values.

The difference for the composite plates with completely synthetic epoxy resin matrices was found to be maximally 20%. With the polylactid matrix the bending modulus was reduced with the laser cut specimen to 5.7 GPa compared with the sawed specimen with 5.8 GPa. The bending strength followed the trend of the bending modulus values with parameters from 55 to 140 MPa for the laser cut samples. For sawed specimen in the case of the polylactid matrix a clear difference appeared. When sawing the thermoplastic matrix was melted by the development of frictional heat. Fibres were parallelised and welded with the matrix by the rotating motion. By the adjustment of the fibres clearly higher

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bending strength with approx. 90 MPa could be achieved than with laser cut where it was 55 MPa. The fibre content of a composite plate has a crucial influence on the bending modulus values. The more fibres a composite contains, the better the load in bending direction can be distributed on the individual fibres. The individual fibres in each case leave a smaller amount of the force on the composite plate. However this function is only optimal if the fibre matrix adhesion is high enough. If the composite contains more fibres in a given volume less matrix is available to transfer the forces on fibres. Also a material with lower fibre firmness can achieve better bending modulus values. For each needled natural fibre nonwoven there is an optimum fibre amount in the composite. The dependence of the bending values of the fibre content is represented in figure 9.

Figure 9. Dependence of the bending values of the fibre content.

With the laser cut specimen no significant difference to the sawed samples arose with a fibre percentage of 20 by volume. The fibre amount was relatively low and the saw produced by pulled off fibres only few micro cracks. An increase of the fibre percentage by volume of 20 vol% on a fibre amount of 40 vol% increase the bending modulus value by approximately 30% from 5.5 GPa to 8.2 GPa. A further addition of approximately 20% fibre material increased the bending modulus again by 20% to maximum of 10.0 GPa. Hereby the composite plate with the weak jute fibres achieved just as high bending modulus values as


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the mixed fibre material flax/hemp with a high amount of 70% of high-strength hemp fibres. Likewise the bending strength pointed a positive correlation to the fibre content with only minimum differences in comparison to the kinds of cut. It could be increased by the rising fibre content from 90 MPa to over 140 MPa. From acoustic point of view for a good internal absorption of a material it can be of advantage to weaken the fibre matrix adhesion, so that sound energy can be reduced by an increased mobility of the fibres. A good internal absorption behaviour and a high firmness of a composite plate cause a high charpy impact resistance value. In interaction with high-strength hemp fibres and the finer flax fibres a composite resulted with very good mechanical absorption characteristics and from there very high charpy impact resistance values exhibited. The values of the natural fibre reinforced plastics with thermoset matrix lie in the range of 11 kJ/m² to 19 kJ/m². From this a composite plate with jute/flax mixture nonwoven and epoxy HT2 slightly deviated with 20 kJ/m². This composite plate contained an increased fibre amount of 60 vol% compared to the other plates. Here the high internal absorption of the fibres has been recognized. The used thermoplastic matrices are relatively flexible at ambient temperature and can despite high fibre matrix adhesion reduce loaded energy by deformation. The combination of a matrix from polypropylene with high-strength hemp fibres and finer flax fibres achieved the highest impact strength value in a composite plate with 37 kJ/m². The composite plate is suitable for the application as surface layer in a sandwich element as wall or ceiling panel.

3.2. Verification of Porosity Silencing materials need an open porous surface and a porous core to absorb noise. For the acoustic characteristics of the material the distribution and width of the pores are crucial. Normally nonwoven materials from natural fibres are always open porous. Porosity can be adapted by adjusting the needle parameters to the desired characteristics. The weak nonwoven materials were impregnated for the increase of the mechanical characteristics with an epoxy resin matrix. The matrix can seal the pores. Therefore a characterisation of the surfaces of the specimens was accomplished regarding the open porosity by means of the microscopy and 2x enlargement. The press moulded natural and synthetic fibre reinforced absorber plates were analysed for the determination of a reduction of the open porosity with rising puncture quantity, rising weight per unit area and increased pressing force. In the figures 10 and 11 two examples of the open porosity of nonwovens are represented. It is to be recognized at both figures that all fibres and fibre

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bundles are completely soaked and respectively coated by the matrix. The fibres stuck together with one another at the points of contact, the areas between are matrix free and can take up the airborne sound very well. From the irregular structure of the pores which results in this way, the sound is reflected and diffused. The mechanism corresponds in this case to that of a classical porous absorber. Furthermore since the fibres exhibit certain flexibility despite soaking with the matrix, a part of the airborne sound energy can be converted into deformation energy. The fibres and fibre bundles become stimulated between the points of contact by oscillations. The oscillation is obstructed and transferred by the gluing at these points to further fibres. Since each fibre becomes stimulated at different oscillations, it can on one hand strengthen the oscillation and on the other hand the oscillation by interference can be absorbed. Depending upon the thickness of absorber substance the sound energy can be reduced effectively by deformation by these mechanisms.

Figure 10. Hemp/polylactid with epoxy resin.

Figure 11. Hemp/flax with epoxy resin.


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3.3. Characterisation of Acoustic Properties 3.3.1. Sound Insulation Apart from the sound absorption of porous plates the sound insulation is of special interest in the acoustic characteristics. However during the sound insulation measurements because of the small specimens of only 30 x 40 cm² dimensions special efforts were necessary to obtain reliable results [19]. Since the dimensions of the plane specimens are smaller than the openings of acoustic test stands which are normally used for the standard measurement of sound insulation a high sound insulating sandwich element mask from chipboards and mineral wool with a suitable opening had to be built. The installation of the mask in the window test stand is represented in figure 12. The smaller the format of the test specimens, the more problematic becomes the interpretation of the results of measurement, in particular with deep frequencies. However the accuracy of the measurement depends not only on the specimen size, but also on the kind of mask. This limits both the maximally measurable sound insulation and the usable frequency range. Measurements accomplished in order to characterize these restrictions were once with high sound insulating closed test opening and once with empty test opening.

Figure 12. View and design detail of high sound insulating mask in window test stand.

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The measured sound insulation of a test specimen should be appropriate if possible for 10 dB below the border insulation. Otherwise measured sound insulation is lower than the actual. For the sound insulation of the empty opening in the test mask there would be expected 0 dB. However due to diffraction effects sound insulation measures to nearly 7 dB receives and even negative values at frequency above 100 Hz. Due to the strong diffraction effects around 200 Hz reliable values can be expected only starting from approximately 315 Hz. In order to characterize the usable frequency range somewhat more exactly, for the test mask opening a 2 mm thick steel plate with 15.3 kg/m² weight per unit area was cut and measured. The sound absorption should essentially follow the mass law in the higher frequency test range in architectural acoustic. The result of the evaluation was that a restriction of the usable frequency to a range of frequencies between 300 Hz and 5 kHz occurred on measurements of the test specimens. The evaluation of the sound tightness of a material takes place usually on the basis of the weighted difference level Dw, which reduces the measured values to an easily comparable singular value. Since the sound insulation of a material depends strongly on its mass, with the production of particularly light acoustic elements an excess of the mass law is demanded. Simplified the mass law indicates that the sound absorption of a material increases with doubling of the weight around 6 dB [20]. The sound insulation measurement of most of the examined single layer natural fibre reinforced composite plates approximately follows the mass law in the middle and upper frequency range. The weighted difference level is often with up to 4 dB a little higher than to be expected by the mass law. During the overall evaluation with the weighted difference level Dw lies nearly at all plates in a range, which is marked out by the curve for the mass law and a curve shifted upwards 3 dB. In practical application usually multi layered elements are in demand. Mainly mass law behaviour also can be found here with some sandwich elements. With same layer structure of the sandwich elements no significant influence of the matrices and reinforcement fibres in the firmly press moulded surface layers on the sound absorption characteristics was determined. Most sandwich elements show a thickness resonance minimum with following steep rise in their sound insulating curves. Some sandwich elements exhibit a process, which is not explainable with simple theoretical models. Here connections with inhomogenities or anisotropies of the core layers are to be assumed. Also with the multi layered elements a bending wave coincidence can lead to a drastic degradation of sound insulation.


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Obviously this did not arise at the examined sandwich elements. Rather the thickness resonance was the problem. Thereby result weighted difference levels in the case of many sandwich elements, which lie partially clearly below the mass law value. On the other hand some sandwich elements exceed the mass law over up to 3 dB. From there it is valid to avoid the thickness resonance or weaken it at least. An increase of the absorption in the weak core layers generally affects sound insulation in the thickness resonance minimum positively. A variation of the core layers changes in the material application, thickness and position. The puncture quantity of natural fibre needled nonwovens determines the compression and absorption characteristics of these materials. The influence of the weight per unit area and the puncture quantity on the sound insulation of nonwoven materials with the employment as core material was examined on the basis of three layered sandwich elements. The surface layers for all sandwich elements were from hemp/flax fibre reinforced composite plates with 4 mm thickness. The sound insulation of natural fibre nonwovens with variation of the weight per unit area and the puncture quantity is represented in figure 13.

Figure 13. Influence of needled nonwovens as core material on the sound insulation.

All four sandwich elements possess 10.9 – 11.3 kg/m² for instance the same weight per unit area. The weight of the sandwich element results to a large content on the heavy press moulded surface layers. A light and low needled nonwoven

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with 35 punctures/cm² as core material led to a clear thickness resonance minimum at a frequency of approximately 400 Hz. Afterwards a steep rise of sound insulation took place which again flattens slightly at the reach of the bending coincidence of the surface layers. The increase of the puncture quantity to a middle value of 40 punctures/cm² prevents the break-down at the bending coincidence. Somewhat higher weight of this core layer i.e. 0.93 kg/cm², compared to 0.84 kg/cm², increases the values of the sound insulation curve slightly but has however negative effects on the thickness resonance minimum. The best influence on the thickness resonance minimum had a high puncture quantity of 55 punctures/cm². However the slope of sound insulation decreases here. The difference in the sound insulation curves don’t affect the weighted difference level which is for all elements approx. 30 dB. The second possibility to increase the insulation characteristics of the core layers is to increase their thickness. The core layers of the three layered sandwich elements consisted of 1 - 3 sub layers of hemp and flax needled nonwoven with a puncture quantity of 40 punctures/cm² and a weight per unit area of 0.93 kg/m². Thereby a somewhat increased weight with an increase of 11.0 on 13.1 kg/m² must be tolerated. The influence of the core thickness on the sound insulation is represented in figure 14.

Figure 14. Influence of thickness of core material on the sound insulation.


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With the increase of the core thickness, the resonance minimum shifts to lower frequencies. The sound insulation curve flattens thereby with higher frequencies. Positively, with the threefold thickness increase of nonwoven core, this flattening stagnates and has an improvement in the sound insulation at the lower frequencies. Thereby the weighted difference level rises from 30 dB up to 34 dB. With a natural fibre reinforced three layered sandwich element with a weak core layer consisting of three sub layers of natural fibre needled nonwoven has already exceeded the mass law. The sound insulation can be further improved, if the middle sub layer of nonwoven is exchanged by a middle layer of an open porous press moulded nonwoven with epoxy resin. This solidified middle layer by its higher weight compared with the nonwovens can oscillate between that relatively weak natural fibre nonwovens and reduces a majority of the oscillation energy by transformation in warmth. Depending upon weight of the intermediate layer and solidification of the nonwovens the oscillation produced in the surface layer by the sound energy can be absorbed. The influence of the solidification of the nonwovens and the thickness of the surface layers on the sound insulation is represented in figure 15.

Figure 15. Influence of the compression and thickness of the core layers.

The thickness of the surface layers was changed between 1, 2.5 and 4 mm. Thus the weight per unit area of the five layered sandwich elements ranged about 7.6, 10.5 and 14.3 kg/m². For all core layers the same quantity of material was

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used. The compression of the nonwoven layers in the core should lead to an increase of the mechanical strength and thereby making the use of clearly thinner surface layers possible. However the sound insulation curves of the sandwich with thinner surface layers and lower weights shift to the higher frequencies. For a good insulation effect at low frequencies a weight minimum is necessary. The compression of the nonwovens within the core layers can dampen the oscillation of the middle heavier open porous core layer. By the oscillation of the open porous core layer from NFRP the thickness resonance minimum flattens somewhat. The open porous middle layer possesses the highest influence on the sound insulation within the uncompressed and thus very weak nonwoven layers. The sound insulation curve is shifted at the high frequencies over the measuring maximum with approximately 55 dB of the high sound insulating mask. Therefore the real sound insulation effect of the sandwich will be better than measured at frequencies above 1250 Hz. The influence of the change of the layer thickness and compression becomes apparent at the comparison of the weighted difference level. This rises from 24 dB up to 37 dB. A very high sound insulation effect can be reached with sandwich elements build from several layers with an oscillating open porous core layer which is dampened by nonwoven layers. Thereby the nonwoven layers and the surface layers should be as thick as possible in the context of the maximum weight.

3.3.2. Sound Absorption The sound absorption was determined on the basis of the open porous core layers. The frequency influenced sound absorption coefficient α is predominantly determined by the thickness and flow resistance of an open porous material. The natural fibre needled nonwovens used in the core layers possess a high sound absorption potential due to their structure. On the basis of sound absorption coefficient the effects of the change of the thickness and the degree of compression can be determined by changed needle parameters of natural fibre nonwovens or open porous natural fibre reinforced plastics. In figure 16 the comparison between a weak voluminous and a heavier more strongly consolidated natural fibre needled nonwoven is represented. The heavy natural fibre nonwoven made up of hemp and flax fibres possessed a puncture quantity of 55 punctures/cm² with a weight per unit area of approx. 0.93 kg/m². The second hemp and flax nonwoven was more weakly needled with 40 punctures/cm² with 0.84 kg/m².


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Figure 16. Sound absorption coefficient of needled natural fibre nonwovens.

The sound absorption coefficient curves are shifted with higher puncture quantities of the natural fibre nonwoven toward the low frequencies. The doubling of the thickness has the same effect with both materials. Here the same sound absorption coefficient can be achieved, as with two layers of the heavier hemp flax nonwoven with one layer of the light and somewhat thinner weakly needled nonwoven. For the acoustic characteristics of the sandwich elements a small degree of compression in the core layer is of advantage, in order to ensure a high admission at airborne sounds. The optimum of the compression can be determined by the measurement of flow resistance. The compression can be adapted with the manufacturing by quantity of puncture of the nonwovens. However weakly needled natural fibre nonwovens possess the disadvantage of very low mechanical characteristic values and are complicated in handling. Impregnating the natural fibre needled nonwovens with epoxy resin can work against this disadvantage. The compression of the nonwovens can be varied by the pressing force. A further possibility to optimize the sound absorption of the natural fibre nonwovens is to increase their thickness. The doubling of the thickness of both natural fibre nonwovens led to a clear improvement of the sound absorption coefficient. However the increased thickness also doubles the mass. The sound absorbing natural fibre reinforced plastics were manufactured relatively thin, in order to avoid the thickness increase

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of the total element. Therefore the sound absorption achievement is very good particularly at the middle and high frequencies. For the increase of the sound absorption achievement at low frequencies the increase of the mass and thickness of materials would be necessary. Sound absorbing materials need to be laid out purposefully for the frequencies respective to the selected application to produce the elements as economically as possible. The use of a singular value e.g. evaluated sound absorption coefficient αw to compare the materials is not advisable due to the strong frequency response. On the basis of the determined acoustic characteristics an optimised layer structure for a multi functional separative element with good sound insulation and sound absorption characteristics can be defined. The principle of such an ideally five layered sandwich element is represented in figure 17. In this sandwich element the internal layer of the material will take up a part of the airborne sound energy depending upon the quantity of puncture of the used nonwoven and depending upon porosity of the surface layer. The sound energy is reduced within the sandwich core by friction of the air at the fibres and reflected clearly weakened back into the area. However a sandwich element open on the sound up taking upper side leads to a clear degradation of the sound insulation characteristics.

Figure 17. Principle of sound reduction in acoustically optimised sandwich elements.


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To conquer this problem the porosity of the core layer must be reduced. This was achieved via embedding a more strongly consolidated open porously manufactured natural fibre reinforced plastic with thermoset matrix between two nonwovens. The sound is very well absorbed by the first layer of natural fibre nonwoven due to the open structure but however hardly weakened. The open porous NFRP layer possesses a clearly higher flow resistance and becomes stimulated by the sound energy to oscillations. The airborne sound energy is converted here at a large amount into structure borne sound. By the large bearing surface the second nonwoven can become very strongly stimulated by the oscillations forced by sound. The natural fibres possess very high absorption characteristics in relation to mechanical deformations and can therefore convert a large part of the sound energy into warmth. The remaining sound energy is transferred to the very heavy and thicker surface layer on the sound withdrawal side. Since the layer exhibits a closed surface, the remaining airborne sound and the remaining oscillation energy can be transferred only over bending vibrations to the sound adverted side.

4. Conclusion The promoted research project on the acoustically effective elements in sandwich structure from natural fibre reinforced plastic characterised and optimized them systematically regarding their mechanical and acoustic characteristics. For a multi functional separation construction unit from NFRP, which is sufficient for high mechanical requirements, sound insulating values with a weighted difference level Dw over 33 dB and high sound absorption values demands the open porosity of the sound taking-up surface. At the same time the opposite surface must exhibit a high sound tightness. The determination of the characteristics of the base materials led to a better understanding of the acoustic behaviour of the natural fibre reinforced plastics. From these characteristics, an optimised structure of sandwich was derived and verified experimentally. The entire sandwich element should exhibit a low weight and maximum thicknesses of 10–30 mm. The core material from a natural fibre needled nonwoven had to fulfil, above all, the demand for a low density and a high sound damping. The production of semi-finished fabrics, which could be varied in the weight per unit area, needle characteristics, thicknesses and the air permeability, was possible by nonwoven technology. As thermoset matrices the epoxy resins HT2 and L1100 as

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well as a linseed oil acrylate system were used. The thermoplastics polypropylene and polylactid were used in hybrid nonwovens. It is not possible to develop a single sandwich element for use in all acoustic applications. Therefore, as an application range, the use as a wall and ceiling component as well as a base plate in trains was selected. The processing of the nonwovens and the matrices took place by means of press moulding technology. Natural fibre reinforced composite plates with various thicknesses and the fibre contents were manufactured. The multilayered sandwich elements were prepared from surface layers of press moulded composite plates with core layers of natural fibre nonwovens or open porous NFRP with a small density below 1g/cm3. A recapitulate comparison of the represented results for the bending modulus values with data from the literature showed that the measured values of the natural fibre reinforced composite plates lie in the middle to upper range. Usual literature data values are between 2.7 GPa to 5.3 GPa for natural fibre reinforced plastics with polypropylene matrix and between 3.8 GPa and 6.2 GPa with thermoset matrix [21, 22, 5]. The production of the specimen by laser cut improved the bending modulus value with the thermoset matrices around 10–35% as compared to the treatment with a saw. On the other hand, at the thermoplastic matrices, the laser cut caused a thermal damage on the edges and the bending modulus value sank approximately 20%. A small puncture quantity of 40 punctures/cm² combined with a weight per unit area of 0.93 kg/m² with composite plates of hemp and flax nonwovens with epoxy led to the highest bending modulus values of 10.2 GPa. Therefore these composite plates are best suited as surface layers in the optimised sandwich elements. Likewise an increase of the fibre amount up to 60 vol% with a jute and flax mixed nonwoven with epoxy led to high bending modulus values of 10.0 GPa. The thermoplastic matrices polylactid and polypropylene achieved a lower bending modulus up to 5.8 GPa in the composite. However, the composite becomes with up to 20 KJ/m² clearly tougher. A more exact knowledge of the material was obtained by the characterisation of the acoustic characteristics of natural fibre reinforced plastics under analysis. Most of the manufactured natural fibre reinforced composite plates reached an excess of the weighted difference level Dw up to 4 dB in relation to the mass law at absolute sound insulation values up to 30 dB. The thin single layer elements do not fulfil the demand of the excess of a weighted difference level Dw of 33 dB. Sandwich elements with three layers reached at the maximum a weighted difference level Dw of 34 dB. Five layered natural fibre reinforced elements with mass per unit area between 14 and 16 kg/m² could attain the maximum sound insulation of Dw 37 dB. Clearly improved sound insulation values can be achieved by the change of the layer structure. For the


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sound absorption characteristics of the materials a low degree of compression in the core layer is of advantage in order to ensure a high admission at airborne sound at a frequency of 1 kHz. The achievement of great acoustic properties ensures that the multilayer NFRP elements are promising in the soundproof systems. Sandwich elements from natural fibre reinforced plastic showed a combination of good mechanical and acoustic characteristics leading to a possibility to replace oil-based materials in the range of high acoustic requirements. On the basis of all the characterisation and the abundance of consolidated findings it is now possible to analyse and optimise a sandwich structure for concrete applications with the methods used in the projects.

Acknowledgment The authors would like to acknowledge the contribution of the following participants during the course of research projects −

−

Fachagentur Nachwachsende Rohstoffe e.V. (Bundesministerium für Verbraucherschutz, Ernährung und Landwirtschaft, Germany) for financial encouragement of the research project “Schallisolierende Sandwich-Strukturen aus naturfaserverstärktem Kunststoff” and Arbeitsgemeinschaft industrieller Forschungsvereinigungen “Otto von Guericke” e.V. (AiF, Germany) for financial encouragement of the research project “Entwicklung naturfaserverstärkter KunststoffComposite-Werkstoffe mit absorbierenden Oberflächen für raum- und bauakustische Anwendungen”.

The research took place in cooperation with − − −

S. Langer, Institut für angewandte Mechanik at TU Braunschweig, Germany; Schmitz, TAC Technische Akustik in Korschenbroich, Germany; W. Maysenhölder, Fraunhofer Institut für Bauphysik in Stuttgart, Germany.

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References [1] [2] [3]

[4] [5]

[6]

[7]

[8] [9] [10] [11]

[12]

[13] [14] [15]

N.N. DIN 60001-1 Textile Faserstoffe – Naturfasern und Kurzzeichen, Beuth Verlag: Berlin, DE, 2001. Haudek, H.W.; Viti, E. Textilfasern, Verlag Johann L. Bondi and Sohn: Wien-Perchtoldsdorf, A, 2002. Frormann, L. Prozessmodellierung zur Untersuchung der Imprägnierung textiler Halbzeuge mit Thermoplasten, Verlag Papierflieger, ClausthalZellerfeld, DE, 2002; pp 38-40. Nultsch, W. Allgemeine Botanik, Thieme Verlag, Stuttgart, DE, 1996; Vol. 10, pp 125-130. Lampke, T. (2001). Beitrag zur Charakterisierung naturfaserverstärkter Verbundwerkstoffe mit hochpolymerer Matrix, ISSN 1439-1597, http://archiv.tu-chemnitz.de/pub/2001/0089 Sedlacik, G. (2003). Beitrag zum Einsatz von unidirektional naturfaserverstärkten thermoplastischen Kunststoffen als Werkstoff für großflächige Strukturbauteile, http://archiv.tu-chemnitz.de/pub/2004/0113 Karus, M.; Ortmann, S.; Gahle, C.; Pendarovski, C. (2006). Einsatz von Naturfasern in Verbundwerkstoffen für die Automobilproduktion in Deutschland, nova-Institut, Hürth, http://www.nova-institut.de/pdf/0611NF-VerbundAutoD.pdf Schlott, S. Kunststoffe 2006, Vol. 3, 83-87. Maute, D. Technische Akustik und Lärmschutz, Carl Hanser Verlag, München, DE, 2006 Schachtschneider, H.; Müller, D.H. GAK Gummi Fasern Kunststoffe – Fachmagazin für die Polymerindustrie 2005, Vol. 58, 2, 92-96. Cescutti, G.; Schachtschneider, H.; Müller, D.; Müssig, J. Akustische und mechanische Eigenschaften von naturfaserverstärkten Werkstoffverbunden, 5. Internationale AVK-TV Tagung, Baden-Baden, DE, 2002 Schachtschneider, H.; Müller, D. Gezielte Einstellung der akustischen Eigenschaften von FVW aus Naturfasern durch Variation des Aufbaus und der Oberflächen, 6. Internationale AVK-TV Tagung, Baden-Baden, DE, 2003. N.N. DIN 53290 Prüfung von Kernverbunden Begriffe, Beuth Verlag, Berlin, DE, 1982. Frormann, L. Möglichkeiten zum Einsatz von Naturfasern in technischen Bauteilen, Fachsymposium zur BauKunstStoff, Pirmasens, DE, 2003. N.N. DIN EN ISO 527-4 Kunststoffe - Bestimmung der Zugeigenschaften, Beuth Verlag, Berlin, DE, 1997.


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[16] N.N. DIN EN ISO 14125 Faserverstärkte Kunststoffe - Bestimmung der Biegeeigenschaften, Beuth Verlag, Berlin, DE, 1998. [17] N.N. DIN EN ISO 179-1 Kunststoffe - Bestimmung der CharpySchlageigenschaften, Beuth Verlag, Berlin, DE, 2000. [18] N.N. DIN EN ISO 178 Kunststoffe - Bestimmung der Biegeeigenschaften, Beuth Verlag, Berlin, DE, 2006. [19] N.N. DIN EN 20140-10 Akustik Messung der Schalldämmung in Gebäuden und von Bauteilen Messung der Luftschalldämmung kleiner Bauteile in Prüfständen, Beuth Verlag, Berlin, DE, 1992. [20] Veit, I. Trockenbau Akustik 2006, Vol. 8, pp 40-41. [21] Endres, H.-J. Mechanische Eigenschaften jute-, baumwoll- und hanffaserverstärkter duromerer Verbundwerkstoffe, 2004 Kurzbericht zu einem internen Forschungsvorhaben an der Fachhochschule Hannover, http://www.fhhannover.de/imperia/md/content/zentral/forschung/berichte/56.pdf [22] Riedel, U. et al. Industrielle Fertigungsstrategien am Beispiel von automobilen Anwendungen, Deutsches Zentrum für Luft- und Raumfahrt e.V., Institut für Faserverbundleichtbau und Adaptronik, Braunschweig, 2005, p 8, http://www.riko.net/download/kwst2005_riedel.pdf.


Chapter 3

EFFECTS OF MICROSTRUCTURE ON RESIDUAL STRESSES IN DSE AL2O3/YAG CERAMIC COMPOSITE BY EXPERIMENTAL AND NUMERICAL INVESTIGATIONS J.J. Sha1,2,∗ , S. Ochiai3, H. Okuda3, S. Iwamoto3, K. Morishita3, Y. Waku4, N. Nakagawa4, A. Mitani4, T. Ishikawa4 and M. Sato4 1

State Key Lab. of Structural Analyses for Industrial Equipment, Dalian University of Technology, 116024, Dalian, China 2 Ceramic Materials Engineering, University of Bayreuth, D-95440 Bayreuth, Germany 3 Department of Materials Science and Engineering, Kyoto University, Kyoto 606-8501, Japan 4 Ube Industries Ltd., 1978-5 Kogushi, Ube, Yamaguchi, 755-8633, Japan

Abstract The effect of microstructure on the residual stresses in directionally solidified eutectic (DSE) Al2O3/Y3Al5O12 (YAG) ceramic composite were investigated by X-ray diffraction technique and finite element method (FEM). ∗

E-mail address: [email protected]

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J.J. Sha, S. Ochiai, H. Okuda et al. In the X-ray stress measurement, the YAG skeleton derived from the Al2O3/YAG composite by dioxidation of the Al2O3 phase was used as a reference specimen without thermally-induced stress, and the X-ray stress measurements with CuKα1 irradiation were performed on the two faces of a cubic specimen, namely, the faces parallel and perpendicular to the solidification direction, respectively. On the other hand, a numerical analysis using finite element method (FEM) which represents the actual microstructure features of the experimental specimen was carried out in different local regions with different morphologies to reveal the effect of microstructure on the distribution of residual stress in the composite. The distributions of residual stresses in both constituting phases were mapped by FEM calculation. Meanwhile, the mapping of residual stress indicated that the distribution of residual stress in the interior of each phase was not homogeneous being dependent on the solidification direction and local morphologies of constituting phases such as curvature of interfaces, array and volume fraction. The experimentally measured residual stresses were accounted for by the FEM analysis.

Keywords: Ceramic composite; Residual stress; Numerical analysis; X-ray diffraction.

1. Introduction Because of their attractiveness in superior oxidation resistance, high stability of the microstructure and excellent high temperature mechanical properties in oxidative environment [1-19], directionally solidified eutectic (DSE) oxide/oxide ceramic composites have received considerable attention and been regarded as one of the most potential materials for making the structural components of advanced energy generation systems and aerospace propulsion systems. Since DSE ceramic composites are generally fabricated by solidification from melt and expected to be applied in ultra high temperature environment, residual stresses can be induced due to the incompatibility of coefficient of thermal expansion (CTE) and thermal elastic properties between/among the constituting phases on cooling either during fabrication or during use under thermal cycling [20-22]. Such residual stresses (compressive in one phase and tensile in the other) will certainly affect the fracture properties and machinability of components at room temperature [23-25]. For the application of DSE ceramic composites in high temperature technologies, one of the problems have to be solved, therefore, is to understand the situation of residual stresses in composites including the state, magnitude and distribution. Among many DSE ceramic composites, the difference in CTE between two constituting phases in the DSE Al2O3/Y3Al5O12 (YAG) ceramic composite is very

Effects of Microstructure on Residual Stresses…

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small [19-22,26-27]. It is noted that this composite has a clean and strongly constrained interface between two constituting phases without glass phase [6-7]. These characteristics also make the Al2O3/YAG ceramic composite to be a competitive material used for electronic devices of high-power white-LED light sources [28]. Extensive efforts have been made to characterize the residual stresses in ceramic composites both experimentally and theoretically. Generally, all experimental methods could be divided into two categories, namely, destructive and non-destructive methods. In the former method, the drawback is the fact that it cannot be used for in situ measurement since it requires the specimen to be broken during testing, leading to a low accuracy of stress measurement. Non-destructive method, such as X-ray diffraction, neutron diffraction and Raman spectroscopy, has been demonstrated to be an effective way in the estimation of residual stress of crystal materials. Previous studies using X-ray diffraction technique have successfully demonstrated the presence of GPalevel residual stress in some DSE ceramic composites [21-22,29], but the residual stress in DSE Al2O3/YAG eutectic composite is rather small [22,27]. The existence of residual stress in DSE ceramic composites was mainly caused by the CTE mismatch during cooling due to the strong interface constraint between eutectic phases. For such interfacial constraint stresses, the magnitude of the stress is weighted by the area fraction of each phase and the sighs of the stresses along the interface in constrained phases should be opposite. The eutectic microstructure and crystallography are varied with the process conditions [1,8,30-31]. This process-dependent eutectic microstructure and crystallography may result in an influence on the distribution of residual stresses in each constituting phase. Therefore, the accurate estimation of the thermal residual stresses is a complex problem, because residual stresses and their distribution depend not only on the CTE mismatch but on the morphology of the eutectic microstructure, crystallographic orientation, the stress relaxation mechanism and the cooling rate. Additionally, the most of constituting phases in DSE ceramic composites exhibited an anisotropic elastic properties. The complexity of microstructure and anisotropic elastic properties give rise to difficulties in modeling work. In the past decades, in order to understand and develop composite materials by correlating properties with known microstructures, many researchers attempted to produce approximate models for materials behavior by periodic arrays of simple items such as spheres and cylinders [32-36]. However, these methods are not always predictive because in many cases the material properties strongly depend on the stereological distributions of microstructure features. In such numerical analyses, the analytical models represent two-phase or multi-phase materials only approximately, as they ignore specific features of the

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J.J. Sha, S. Ochiai, H. Okuda et al.

microstructure. Simulation methods which allow an accurate estimation of composite properties from microstructure and properties of constituting phases would result in an increased efficiency in the research and development of ceramic composites. [33-34]. The microstructures of materials, which controlled by chemical composition, processing parameters, crystal structure, crystallography etc [1,8,19], are often complex in ceramic composites, especially in directionally solidified eutectic (DSE) ceramic composites. The DSE ceramic composites have been reported possessing excellent high temperature mechanical properties and superior microstructure stability in ultrahigh temperature environments. However, the characterizations of high temperature properties for such materials are time- and cost-consumption works. To enhance the understanding of micromechanical mechanism of deformation and fracture, the finite element method (FEM) analysis can be an economic and effective way in the properties simulation of DSE ceramic composites [37]. In our former works [38-39], the residual stresses in YAG phase in the Al2O3/YAG ceramic composites have been characterized by indentation fracture method, X-ray diffraction and finite element calculation, but not attempted to investigate the microstructure effect on the distribution of residual stress and to correlate the microstructure features with the magnitude of residual stress. Additionally, in the work [39], the residual stresses were measured from {8,8,8} diffraction family. In this diffraction family, only the principal residual stress could be estimated because the number of diffraction peaks for the stress measurement is less than six. In this chapter, the residual stresses including principal and shear stresses were measured from the YAG {10,8,4} diffraction family which has more than six diffraction members and the diffraction intensity of each plane in this diffraction family is higher than that of {8,8,8} diffraction family. This would result in an increased understanding in the stress measurement. Furthermore, we described a method for creating an image-based FEM model. With this model, the residual stress analysis was performed in different local regions with different morphologies in the DSE Al2O3/YAG ceramic composite. The effects of microstructure on the state, magnitude and distribution of residual stresses were analyzed to correlate the residual stresses with the microstructure features of composite. Finally, the residual stresses calculated by FEM simulations were compared with those of measurements by X-ray diffraction.


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2. Experimental Procedure 2.1. Materials High purity α- Al2O3 and Y2O3 powders were milled in ethanol with a eutectic mole ratio of 82:18 to make slurry. And then the slurry was dried and premelt by arc-melting technique to obtain initial ingots. By crashing the ingots, precursor powders were obtained and re-melted in Mo crucible by high-frequency induction heating system in inert atmosphere. The directionally solidified melt growth was performed at about 2223 K for 0.5 h by descending the Mo crucible at a constant speed. By this way, the fabricated Al2O3/YAG ceramic composite has strongly bonded crystal phase and has no glass phase appeared on grain boundaries. The fabrication procedure has also been described in more detail elsewhere [6-7]. The surface morphology on the face perpendicular to the solidification direction of the fabricated specimen is shown in Figure 1, in which the Al2O3 phase can be distinguished from the YAG phase by its darker color. From the observation of Figure 1, it can be seen that the Al2O3 and the YAG phases are incorporated each other and distributed randomly. The volume fractions for both constituting phases are same (0.5).

200 μm

Figure 1. Surface morphology on the face perpendicular to the solidification direction of Al2O3/YAG ceramic composite, where the Al2O3 and YAG phases are incorporated each other and distributed randomly, the Al2O3 phase can be distinguished from YAG phase by its darker color.

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S3//[100]Al2O3

Al2O3 eB

Face A

Solidification direction

YAG

c Fa

S2//[110]Al2O3

S1//[001]Al2O3 Figure 2. Image of cubic specimen reconstructed by photographs, where faces A and B are parallel and perpendicular to the solidification direction, showing that the Al2O3 and YAG crystals on the face parallel to the solidification are elongated in comparison with those on the face perpendicular to it.

In order to easily understand the experimental procedure, using the photographs obtained from the different faces, an image of cubic specimen is reconstructed and shown in Figure 2, where faces A and B are parallel and perpendicular to the solidification direction. The mutually perpendicular directions in the cubic specimen are defined as S1, S2, and S3 (principal direction in a specimen system), respectively. It should be noted that the S3 direction is parallel to the solidification direction. The crystallographic orientation relationships between the two eutectic phases in the composite are: Al2O3//YAG; Al2O3//YAG. In addition, it has also reported that the Al2O3 and YAG crystals on the face parallel to the solidification are elongated in comparison with those on the face perpendicular to it [11]. The specimen for the X-ray stress measurement has a dimension of 10 mm. During the X-ray stress measurement, the X-ray beam irradiated the center area of the specimen.


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The X-ray diffraction analysis (diffraction vector is normal to the measured surface of the specimen) on each surface of specimen revealed that this DSE ceramic composite exhibited a strong texture, namely, showing preferred growth orientations. An example of diffraction pattern obtained with CuKα1 irradiation on the face B perpendicular to the solidification direction is presented in Figure 3. Evidently, {110} diffraction for Al2O3 phase and {210} diffractions for YAG phase are found, which indicates that {110} Al2O3 and {210} YAG crystals preferentially arrange themselves along the solidification direction as shown in Figure 2. According to the pole figure analysis carried in this work, it was found that the preferred growth orientation slightly tilts from the normal of the corresponding specimen surface by 3-4º. Because of the single crystal-like structure, the method for the measurement of residual stress in polycrystal materials is not applicable. Single crystal X-ray diffraction technique has to be applied in the measurement of residual stress in the present specimen. The measurement principle and experimental procedure would be described in section 2.2. The micrographs from different regions with different morphologies were used for FEM simulation. The residual stress was measured on different faces with different morphologies by X-ray diffraction technique.

Intensity in arbitrary unit

Al 2 O3 110

YAG 420

Face B

Al 2 O3 220

20 30 40 50 60 70 80 90 100 110 120 130 140

2θ(degree) 2 Theta/degree Figure 3. An example of diffraction pattern obtained with CuKα1 irradiation on the face B perpendicular to the solidification direction, only {110} diffraction for Al2O3 phase and {210} diffractions fro YAG phase are found, showing a preferred growth orientation in this composite.

94


Figure 4. Surface morphology of the YAG skeleton on face A prepared by deoxidation of the Al2O3 phase from the composite, showing a porous microstructure.

In order to obtain the specimen used as the reference without thermallyinduced stress for the X-ray stress measurement, three slices with a dimension of 101×10w×0.6tmm were cut from the corresponding surfaces of a bulk composite, and then the Al2O3 phase in the slices was removed by deoxidization in carbon container in vacuum at 1873 K for 7.2 ks. The depth of the Al2O3-removed region in the thickness direction of the specimen was around 250 μm, being around 8 to 25 times the thickness of the eutectic phases (around 10-30 μm). Surface morphologies in deoxidized specimen were examined by the observation of scanning electron microscope (SEM) as shown in Figure 4. Hereafter the slices without Al2O3 phase are referred as the YAG skeletons which will be used as the reference for the X-ray stress measurement. The lattice spacing of the YAG skeleton d0, which determines the sign and magnitude of stress, has a strong influence on the precise measurement of residual stress, especially for the specimen with a low CTE mismatch.

2.2. X-Ray Diffraction Technique The stress in polycrystalline materials which contain many crystal grains with random orientations in an X-ray irradiation area, can be measured by using the sin2ψ method [40-41]; however, this method is not applicable to the present material with an strong texture because the identical crystallographic planes can not be found by varying the ψ angle. Nevertheless, the fundamental principle for


95

X-ray stress measurement is still same, which originates from the measurement of interplanar spacing, d. The interplanar spacing, d, for a particular set of hkl planes of a given phase can be measured from the corresponding peak in the diffraction pattern with Bragg’s law:

d=

λ

(1)

2 sin θ B

where θB was the Bragg angle and obtained from the measurement of Kα1 peak position, λ wave length of CuKα1 irradiation. Again it should be noted that the Al2O3 and YAG crystals on the face parallel to the solidification direction are elongated in comparison with those on the face perpendicular to it, indicating an anisotropic morphologies [11]. For studying the effects of microstructure on the residual stress, the X-ray stress measurements with CuKα1 irradiation were performed on the two faces (face A and B): parallel and perpendicular to the solidification direction. In order to find the diffraction plane which is suitable for the X-ray stress measurement, firstly X-ray diffraction pattern was obtained from the YAG precursor powder as shown in Figure 5. YAG 10 8 4

Intensity in arbitrary unit

YAG precursor powder

115

120

125

2θ(degree) Figure 5. X-ray diffraction pattern obtained from high angular position for the YAG precursor powder.

96


In general, for improving the accuracy of stress measurement, the diffraction peak at high angular position is preferred. Combining the Bragg law with the plane-spacing, the diffraction family of YAG {10,8,4} (2θ=118.73°) were chosen for the stress measurement. Obviously, this diffraction locates at the angular position higher than 100º and the intensity is relatively higher than its neighboring peaks as indicated in Figure 5. XRD patterns were measured using a Rigaku X-ray diffractometer (RINT 2000 series, Model D/max-2200) with a four-circle goniometer (φ, ψ, ω, 2θ). The X-ray diffraction setup and specimen mounting are shown in Figure 6. The specimen can rotate in its own plane about an axis (A-A’) normal to its surface, and about a horizontal axis (B-B’). The horizontal axis lies in the specimen surface and it is initially adjusted by rotation about the diffractometer axis (C-C’), to realize the equal angles with the incidence and diffraction beams. After such an adjustment, the horizontal axis was fixed (no further rotation about the diffractometer axis). In order to know the principal residual stresses (normal to each surface) in the specimen, two coordinate systems were defined, namely, the laboratory coordinate system and specimen coordinate one. The relationship between the two coordinate systems is presented in Figure 7. Specimen holder X-ray source

Goniometer

A

Detector C

B

B’

A’

Ψrotation Specimen

C’ Diffractometer axis

Φrotation

Figure 6. X-ray diffraction setup shows the specimen mounting and the rotation system.

Figure 7 shows a laboratory coordinate system Li with respect to a specimen coordinate one Si, and the definition of ψ and φ angles. The direction of the incident beam with respect to the specimen’s coordinates was determined by these


97

two angles: ψ and φ (see Figures 6 and 7). The specimen was mounted onto a goniometer head that was used to position a specimen in the X-ray beam and was oriented with respect to the specimen coordinate system (Si) so that the φ axis coincide with the normal of the surface of the specimen throughout the experiment (Figure 7). In Figure 6, the ψ angle is zero when the surface of specimen is vertical and has a value of 90º when the surface of specimen is in the horizontal position. Both angles ψ and φ can be extracted from the pole figure analysis. The X-ray pole figure analysis was performed on the {10,8,4} family of diffraction using the Shulz reflection method [41] to determine the orientation of the diffraction planes which were used to measure the residual stresses in YAG phase. The stereographic projection of crystallographic orientations for the YAG {10,8,4} diffraction family on the face B perpendicular to the solidification direction is shown in Figure 8. In Figure 8, the axis S1 was chosen as the reference direction for angle φ. From this pole figure, the orientation of each diffraction plane can be read out from it’s ψ and φ values. During the X-ray stress measurement, a CuKα1 irradiation was used at 40 kV and 200 mA. The relevant diffraction condition is summarized in Table 1. To ensure that only reflections in the vertical plane of the diffractometer were measured, a 2.0 mm horizontal window was placed at the position after the incidence slit.

Incidence beam Incident beam S x33S

L

Lx33

Ψ

θB

Diffraction beam Diffracted beam x2LL2 xS22S

(hkl) Φ

xS11S

L x1 L

Figure 7. Relationship between the laboratory coordinate system and the specimen coordinate system, and the definition of φ and ψ angles in the present reference systems.

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J.J. Sha, S. Ochiai, H. Okuda et al. S2

S3

S1

Figure 8. Stereographic projection of crystallographic orientations for the YAG {10;8,4} diffraction family measured by pole figure analysis on the face B perpendicular to the solidification direction.

Table 1. X-ray diffraction condition for residual strain measurement Parameter

Condition

Equipment

Rigaku XRD: CuKαX-ray tube; four circle goniometer

Power

8 kW; 40 kV, 200 mA

Radiation

CuKα, λ=1.540562 Aº

Reflection

{10, 84}

2θrange Divergence slit Soller slit

118.5-118.9º 0.25º; 2.0 mm horizontal window 0.25º

Receiving slit

0.15 mm

Source-to-specimen distance

185 mm

Scans

Step: 0.004º; 2θ/step; 2.4s/step

Knowing the complete orientation of each diffraction plane on the pole figure as shown in Figure 8, and rotating the goniometer by a combination of φ and ψ degrees, a particular diffraction normal was brought into the diffraction plane. Then a 2θ/θ scan was collected from each accessible member of the YAG


99

{10,8,4} diffraction family. The peak profiles for both composite and skeleton were measured in the range of 118.5-119.0º. An example of peak profile is shown in Figure 9(a).

Figure 9. (a) Diffraction peak profiles measured from YAG {10,8,4} diffraction family in the composite (closed circle) and in the YAG skeleton (open circle), the solid line is a fitting one of the background-subtracted peak to the Gaussian function; the peak profile for YAG phase in the composite is broadened and shifted to larger 2θ in comparison with that of YAG phase in the skeleton. (b) A comparison of FWHM (full width at half maximum) for the YAG phases in the composite and skeleton.

The peak profile of the YAG phase in the composite (closed circle) is broadened and shifted to larger 2θ in comparison with that of the YAG skeleton (open circle). The 2θ positions were determined by a fitting of the backgroundsubtracted peak to the Gaussian function. The fitting profiles are shown with solid lines in Figure 9(a). The quantitative peak broadenings were expressed as the FWHM (full width at half maximum) as shown in Figure 9(b). The peak shift and broadening observed in the composite (Figure 9(a)) implied the existence of the residual stresses and their inhomogeneous distribution. Based on the Bragg’s law, it would be clear that lattice spacing of YAG phase in the composite is smaller than that of the YAG skeleton, indicating the YAG phase in composite experiences the compressive stresses.

2.3. Creation of FEM Model and Simulation The numerical analysis of residual stress is performed using finite element code MARC/MentatTM. The detailed procedure for the creation of models are explained in following sections.

100


2.3.1. Drawing of Phase Profiles Based on Real Image The characteristics of microstructure in DSE ceramic composites are very unique and quite different from those of particle- and fiber-reinforced ceramic composites. Following features of microstructure for the present specimen can be read from the micrograph showed in Figure 1: (i) the morphology is inhomogeneous (size, array and shape of the YAG and Al2O3 phases are varied with positions); (ii) the YAG and Al2O3 phases are incorporated together, but their interfaces can be identified clearly; (iii) the two phases are distributed randomly and each phase is connected by its phase bridge. From these features of microstructure, it is considered that the classical simulation method may not simulate the microstructure pertinent properties precisely [33-36]. To make a model which represents the microstructure features of DSE ceramic composites is essential for the precise simulation of properties. In this work, we used a commercial software, Adobe Acrobat illustrator, to read an image, and then the profile of each constituting phase was drawn based on the real image. This software can import different formats of an image into image gallery, and there are a number of tools that are useful in working on micrographs. It should be noted that the resolution of image should be high enough so that the detailed microstructure features of interface between Al2O3 and YAG phases can be identified exactly at a high magnification. During the drawing of the profiles of constituting phases, the interface lines should be as smooth as possible and have no intersected loops. After finishing the drawing of the profiles, the image was removed, saving only the phase profile as a new file in a format readable by MARC/MentatTM code as shown in Figures 10(a) and 11(a). From the phase profiles shown in Figures 10(a) and 11(a), it can be seen that these phase profiles precisely represented the microstructure features of composite. Because the main purpose of the present work is to reveal the effect of microstructure on the residual stresses and their distribution in both constituting phases, thus, two models were created using the actual micrographs taken from different regions with different morphologies. Hereafter the models in Figures 10 and 11 are referred as the M-A (representing the morphologies of face A) and MB (representing the morphologies of face B), respectively. The difference between two models can be distinguished by comparing the images of models in Figures 10 and 11.

Effects of Microstructure on Residual Stresses… a

b

101

c

Figure 10. Finite element model and mesh for Model A (M-A): (a) phase geometry; (b) model composite, dark phase: Al2O3, gray phase: YAG; (c) mesh of constituent phases, totally 27312 elements and 27643 nodes were given for calculation. a

b

c

Figure 11. Finite element model and mesh for Model B (M-B): (a) phase geometry; (b) model composite, dark phase: Al2O3, gray phase: YAG; (c) mesh of constituent phases, totally 27312 elements and 27643 nodes were given for calculation.

2.3.2. Generation of Mesh and Assignment of Material Properties The phase profiles drawn by Adobe Acrobat illustrator were imported to MARC/MentatTM, and then the interface lines were converted to closed curves. Applying an appropriate meshing method (advanced front method is used in the present work) to such closed curves, the mesh is generated with a desired curve division. However, in most finite element programs, when working with graphs, the geometry is not so well defined as the finite element program fits a mesh to mathematical boundaries, even in Figures 10(a) and 11(a), where the interface is smooth. In this case, an approach is to take every triangular or two triangular elements to form the quadrilateral elements by subdividing/combining triangles and removing/adding nodes. The generated mesh precisely represented the morphology as shown in Figures 10(b) and 11(b). In Figures 10(c) and 11(c), the mesh follows the overall interface quite well on the quadrilateral size scale. In order to assign material properties to this mesh, firstly the corresponding phases in the mesh were identified based on the phase features of the real image. Identifying

102


the phases means giving the names and storing them for later retrieval. On this basis, the material properties and different grayscale values were assigned to corresponding phases on the finite element mesh (Figures 10(b) and 11(b)). By this way, the features of generated meshes are very similar to those of real images. If the phases in Figures 10(b) and 11(b) have different coefficients of thermal expansion and thermal elastic properties, then the thermal residual stresses would be generated within two phases. Table 2. Material properties for FEM simulation and X-ray measurement Al2O3

YAG

Material model

Elastic

Elastic

Element type

4-node (plain strain)

4-node (plain strain)

Poisson ratio, ν

0.23

0.25

E (GPa)

423-0.0474T

299-0.0180T

CTE (×10-6/K)

6.50+0.00146T

6.09+0.00117T

As shown in Table 2, the main finite element implemented is a four-node plain strain element with linear elastic properties. In these models, as a first approximation, the temperature dependence of CTE and Young’s modulus were taken from our previous work [26]. The material properties including the thermoand elastic-properties of two constituting phases were listed in Table 2. Inputting such values into FEM model, the thermal residual stresses were calculated. The resulting finite element meshes and four-node plane strain elements are shown in Figures 10(c) and 11(c). The geometries for both constituting phases in each model are observed clearly in Figs. 10(a) and 11(a). The meshes for the portions picked up from Figures 10(b) and 11(b) are shown in Figures 10(c) and 11(c).

2.3.3. Finite Element Calculation The calculation was performed in two dimensions in plain strain. This method makes it relatively easy to add new materials, not only for binary systems (two eutectic phases coexisted), but also for ternary systems (more than two eutectic phases coexisted). The constituting phases are taken to be isotropic and elastic in properties during simulations. For the calculation of thermally-induced residual stress, a thermal load was imposed to each node of these meshes by a temperature gradient which is defined based on the temperature difference between the stress-


103

generated temperature and room temperature. Thus, the stress-generated temperature is a key factor for the calculation of thermally-induced residual stress. Considering the stress relaxation on cooling from process temperature to room temperature, the assumed temperatures for the stress generation should be lower than 2223 K (process temperature).

Finite element mesh

Figure 12. Finite element model with links and boundary conditions for the calculation of residual stress.

Figure 12 shows the boundary condition for these models which allow a free volumetric expansion of the composite. The bottom and left surfaces of model are fixed in displacement. Additionally, the boundary condition applied also provided an equal displacement of the surface nodes against the fixed surfaces to maintain the rectangular shape.

3. Result 3.1. Residual Stress Measured by X-Ray Diffraction Using the Eq. (1) and the measured peak positions, the total elastic strain, φψ

ehkl , is the strain measured for diffraction (hkl) located at φ and ψ, by using the φψ

lattice parameters ( d hkl ) of diffraction (hkl) in composite and the lattice parameter (d0) in YAG skeleton based on following equation:

104

J.J. Sha, S. Ochiai, H. Okuda et al. φψ

ehkl

φψ − d0 d hkl = d0

(2)

The direction of the measured strain with respect to the specimen is always along the bisector of the incident and diffracted beams, namely, along the L3 axis (Figure 7). Actually, the L3 is normal to the diffraction planes. Thus, by tilting the sample relative to the incident beam, this bisector can be made to make various angles, ψ, with the normal to the specimen surface. The direction of strain on the laboratory coordinate system (Li), is defined by the angle ψ and by the angle φ that the plane of the normal of the specimen’s surface (S3) and the normal of diffraction plane (L3) makes with a arbitrary direction in the specimen surface. Then, using the relationship between the two coordinate systems, the strain tensor εij in the specimen coordinate system (Si) can be determined by following equation: φψ ehkl = ai a j ε ij

(3)

where ai is the direction cosine between the (hkl) diffraction and axis i. It should be emphasized that the εij depends on the orientation of the laboratory coordinate system with respect to the specimen coordinate one if the sample is textured. Using at least six measured d-spacings and combining Eqs. (2) with (3), the strain tensor can be determined with generalized least-squares method [42] by: φψ ehkl = (ε 11 cos 2 φ + ε12 sin 2φ + ε 22 sin 2 φ ) sin 2 ψ + ε 33 cos 2 ψ + ε13 cos φ sin 2ψ + ε 23 sin φ sin 2ψ

(4) These strain components measured from the specific hkl planes are averaged over the X-ray beam irradiated volume. The precision can be improved by using data from more than six independent directions because it takes into account the effect of the nonhomogeneity in morphology and the uncertainties in the profile fitting due to counting statistics. The measured strain tensors from the YAG {10,8,4} diffraction families on both face A and face B are shown below, respectively. The strain tensor from the YAG {10,8,4} diffraction family on face A:


105

The stress tensor from the YAG {10,8,4} diffraction family on face B:

The standard deviations in the measured strain were obtained from the variances in the diffracted peak position. To facilitate a matrix formulation of the least-squares procedure the following definitions were made in Eq.(4): ε1=ε11, ε2=ε22, ε3=ε33, ε4=ε12, ε5=ε13, ε6=ε23, The variance in the strains may be calculated from the variance in each of the measured strain n

∂ε j

i =1

∂ei

var(ε j ) = ∑ (

) 2 var(ei )

(5)

The variance in e is computed from the variance in 2θ by following equation.

var(ei ) = (

1 2 π 2 λ cosθ i 2 var(2θ ) ) ( ) ( ) d 0 180 2 sin 2 θ i 2

(6)

where var(2θ) is given by the errors in 2θ which can be determined from nonlinear least-squares fits of peaks to analytical functions. The standard deviations were given by the square roots of the variances. For knowing the residual strain along the normal of each face of a cubic specimen, the measured principal residual strains in the specimen coordinate system are shown in Figure 13(a). Hooke’s law and the use of the material’s elasticity data allow one to relate the strain tensor to the stress tensor.


0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12

S2

S3

S1

0

Residual stress (MPa)

Residual strain (x104)

106

face B face A

S2

S3

S1

-100 -200 -300 -400 -500

face B face A

Figure 13. The principal residual strains and residual stresses measured from the YAG {10,8,4} diffraction family on both face A and B.

For simplifying the stress calculation, the YAG phase is treated as fully isotropic, and the triaxial stresses are calculated. This assumption is appropriate, because the Zener ratio (2C44/C11-C12) of YAG is 1.04 [22]. On the other hand, YAG is cubic so its thermal expansion tensor is fully isotropic. The residual stress tensor σij in YAG phase was calculated from total strains using the relation [41]:

ε ij =

1 +ν ν σ ij − δ ij σ kk E E

(7)

Here δij =1 for i=j, and δij =0 for i≠j, and ν and E are material’s poisson’s ratio and Young’s modulus as listed in Table 2, respectively. Substituting the Eq. (7) into Eq.(4) and making a rearrangement, then the total residual stress could be expressed as:

1 +ν (σ 11 cos 2 φ + σ 12 sin 2φ + σ 22 sin 2 φ − σ 33 ) sin 2 ψ E 1 +ν ν 1 +ν + σ 33 − (σ 11 + σ 22 + σ 33 ) + (σ 13 cos 2 φ + ε 23 sin φ ) sin 2ψ E E E

φψ ehkl =

(8)

Taking the Young’s modulus of the YAG phase to be 294 GPa with a Poisson’s ration, ν=0.25 as listed in Table 2, the residual stress tensors for the YAG {10,8,4} diffraction family on both face A and B were calculated and shown below, respectively: stress tensor from the YAG {10,8,4} diffraction family on face A:


107

stress tensor from the YAG {10,8,4} diffraction family on face B:

The error in the stress tensor is calculated from the error in the strain tensor. The principal residual stresses measured from the YAG {10,8,4} diffraction family on average are -321.3 MPa on face A and -173.7 MPa on face B, respectively.

3.2. Residual Stress Calculated by FEM Simulation In order to calculate the residual stress by FEM simulation, it is necessary to know the temperature, at which the elastic stresses begin to be generated. Since the elastic stresses were generated by the CTE mismatch and the anisotropic thermal elastic properties at temperature below the eutectic temperature, an approximate value of the stress-generated temperature can be estimated from the minimum temperature necessary to activate the slip systems in the eutectic crystal phases. Based on the stress-generated temperature, a temperature gradient applied to nodes in FEM model can be determined. However, the stress-generated temperature is unknown till now because it is quite dependent on the cooling rate and stress relaxation mechanism activated by creep/plastic deformation. Thus, it is difficult to determine the stress-generated temperature experimentally, but it is sure that residual stress would be generated below the eutectic point. Therefore, in order to understand the stress development, the average residual stresses were calculated in a wide range of temperature where the upper bound is 2023 K. The models used for finite element calculation are presented in Figures 10 and 11. Figure 14 shows the calculated residual stresses as a function of temperature for the Al2O3 and YAG phases by M-A and M-B models, respectively. The average values are given by analyzing the data read from the postprocessor.

108

J.J. Sha, S. Ochiai, H. Okuda et al. Temperature (K) 1200

1400

1600

Temperature (K)

1800

2000

2200

1200

-50

-150 -200 -250 M-A-YAG-x M-A-YAG-y M-B-YAG-x M-B-YAG-y

(b)

-180 -190 -200 -210 -220 -230

250

M-A-Al2O3-x M-A-Al2O3-y M-B-Al2O3-x M-B-Al2O3-y

(c) 230



2200

M-A-YAG-Avg. M-B-YAG-Avg.

-250

250 200 150 100 50 1400

2000

-170

-240

-350

300

1800

-160



(a)

350

1600

-150

-100

-300

1400

1600

1800

2000

Temperature (K)

2200

M-A-Al2O3-Avg. M-B-Al2O3-Avg.

(d)

210 190 170 150 130 1400

1600

1800

2000

2200

Temperature (K)

Figure 14. Showing the calculated residual stresses as a function of temperature by Model A and Model B: (a) residual stress in YAG phase in x and y direction; (b) average residual stress in YAG phase; (c) residual stress in Al2O3 phase in x and y direction; (d) average residual stress in Al2O3 phase.

It is clear that the values of residual stresses are dependent on the microstructure of composite. From these values, it could be seen that the Al2O3 and YAG phases experienced the tensile and compressive residual stresses respectively, and the absolute values of residual stresses increased with increasing the temperature linearly. In terms of CTE mismatch, the Al2O3 phase with larger value of CTE is subjected to a tensile stress and the YAG is under compression, which are consistent with those of FEM simulations. From the point of view of mechanical properties, the compressive residual stresses in YAG phase would increase the tolerance of composite under tensile loading condition because the YAG phase has a high stiffness in comparison with the Al2O3 phase.


109

The YAG phase is far creep resistant in comparison to that of the Al2O3 phase [3-4], indicating the accumulation of residual stresses in composite during cooling is mainly dependent on the deformation behavior of Al2O3 phase. Under this situation, we take the 1423 K as the stress-generated temperature which is the onset temperature of dislocation slip in prismatic and pyramidal planes in Al2O3 phase [5], to calculate the residual stress, and eventually we found that the calculated results are comparable with those of experiments by X-ray diffraction. The average residual stress calculated by M-A model is 161 MPa in Al2O3 phase, -170 MPa in YAG phase; the average residual stress calculated by M-B model is 147 MPa in Al2O3 phase, -162 MPa in YAG phase. In order to correlate the distribution of residual stresses with the microstructure, Figures 15 and 16 show the distribution of residual stress in DSE Al2O3/YAG ceramic composite accumulated during cooling from 1423 K to 298 K. From Figures 15 and 16, following information can be read: (i) the compressive and tensile residual stresses are coexisting in each constituting phase, but the most of YAG and Al2O3 phases are mainly in compressive and tensile state; (ii) the distribution of residual stresses in the interior of each phase are not homogeneous, being strongly dependent on the local morphologies, such as curvature of the interfaces, array and local volume fraction of constituting phases; (iii) the thermal residual stresses is much higher in the phases where their volume fractions are small and array are parallel to the axis of reference system, namely, the stress in the regularly aligned zone is greater than that in the randomly aligned zone; (iv) the stress concentration occurs at corners and sharp zones of phases. For understanding the distribution of residual stress quantitatively in the specific zones with different shapes and arrays, finite element analyses were performed to investigate the distribution of residual stress in the interior of straight zone and bridged zone in the YAG phase. Figs. 17 and 18 show qualitatively the stress distribution in the interior of YAG phases which were chosen from M-A model as shown in Figure 15. In Figs. 17 and 18, the stress at each node along the node path is almost same. The features in straight zone as shown in Figure 17 (a) were directly observed as follows: (i) the distribution of residual stresses is homogeneous; (ii) the residual stress in y direction is much higher than that of x direction, and the sighs of stresses in x and y direction are opposite. This would be quite possible, because the CTE of YAG is less than that of Al2O3, on cooling, the Al2O3 phase will contract more than that of YAG phase resulting in a compression stress applied to the YAG phase. In strongly constrained straight interface, the generated residual stress by CTE mismatch is along the constrained interface as shown in Figure 17(b).

110

J.J. Sha, S. Ochiai, H. Okuda et al. MPa

(a)

(b)

(c)

(d)

400 350 300 250 200 150 100 50 0 -50 -100 -150 -200 -250 -300 -350 -400

Figure 15. Mapping of the distribution of residual stress in both constituent phases calculated at 1423 K by Model A: (a) YAG phase in x direction; (b) YAG phase in y direction: (c) Al2O3 phase in x direction; (d) Al2O3 phase in y direction. MPa 400 350 300 250 200 150 100 50 0 -50 -100 -150 -200 -250 -300 -350 -400

(a)

(b)

(c)

(d)

Figure 16. Mapping of the distribution of residual stress in both constituent phases calculated at 1423 K by Model B: (a) YAG phase in x direction; (b) YAG phase in y direction: (c) Al2O3 phase in x direction; (d) Al2O3 phase in y direction.

Effects of Microstructure on Residual Stresses… MPa 400 350 300 250 200 150 100 50 0 -50 -100 -150 -200 -250 -300 -350 -400

(a)

(b)

A

C

B

D

Al2O3 Δα> 0

111

YAG Δα Matrix linear expansion coefficient

Radial compression stress Circumferential tension stress

(b) Reinforcement linear expansion coefficient < Matrix linear expansion coefficient

Figure 3.1. Residual stress caused by difference of linear expansion in composite [6].

130

Bin Li, Jianxin Deng and Hong Wang

The chemical compatibility is a precondition of fabricating; because we can get the better properties of the composite only there are no over chemical reaction between the two phases and a higher bonding strength at the interface of the ceramic. Composites of Al2O3/ZrB2/ZrO2 were fabricated by the hot-pressed sintering. Base on Table 3.1, the reaction or product doesn’t exit in reaction (1), (2) and (5) at 1700 ºC, with the exception of them, ΔG

θ T

>0 in the other reaction.

Table 3.1. The possible reaction in sintering Zr-O-B compounds reinforced Al2O3 ceramic materials at 1700ºC Code

ΔGθT

possible reaction

1

Al2O3+2ZrB2=2AlB2+ZrO+ZrO2

——

2

Al2O3+3ZrB2=2AlB2+B2O3+3Zr

——

3

2Al2O3+5N2=4AlN+6NO

8099090

4

2Al2O3+2N2=4AlN+3O2

1689020

5

2ZrB2+3N2=2ZrN+ 4BN

——

6

ZrO2+N2=ZrN+NO2

705173

7

2ZrO2 +N2=2ZrN+2O2

1093426

No occurrence

Al2 O3 ZrB2 m-ZrO2

1000 800

CPS/(S-1)

Result

t-ZrO2 600 400 200 0 20

30

40

50

60

70

80

2θ /(º) Figure 3.2. X-ray diffraction phase analysis of AZ20 specimen.

Introduction of Particle Dispersion…

131

The X-ray diffraction phase analysis of AZ20 (Al2O3 80%, ZrB2 18.44%, ZrO2 1.56%) specimen is shown in Figure 3.2. There exist the Al2O3, ZrB2, mZrO2, t-ZrO2 phases in AZ20. So, the standard reaction Gibbs energy can be used for chemical compatibility analysis. Elastic modulus E could generate the micro-stress redistribution effect only when the forces act upon materials and this effect the mechanical properties of the material relatively small. Residual stress field generated around the second phase particles by thermal expansion mismatch are the main source of toughening composite ceramics.

3.1. Toughening Mechanisms of Thermal Expansion Mismatch Thermal expansion mismatch is the most important mechanism of particle dispersion reinforcecment ceramic matrix composite. This is because it could generate residual stress field surrounding the second phase particles and matrix. Figure 3.3 shows the residual stress field caused by spherical particle in the infinite matrix.

σr Spherical particle

σt R

Infinite matrix p

Figure 3.3. Residual stress field caused by spherical particle in the infinite matrix [25].

When the second phase particle exists in a homogeneous infinite matrix, the forces act on the second phase particle can be expressed as Eq. (3.1).

p=

2ΔαΔTEm (1 + ν m ) + 2 β (1 − 2ν p )

(3.1)

132


Here, Δα = α p − α m ; ν the Possion’s ratio, E the elastic modulus. Subscript m and p represent the matrix and particle respectively; ΔT = Tp− Tτ, Tp is the temperature which can be neglected in plastic deformation, Tτ the room temperature; β = Em/Ep. The pressure will be formed radial normal stress σr and tangential normal stress σt in the matrix, as shown in Figure 3.3.

(3.2)

Here, r is the particle radius; R the distance from a point in stress field to the center of particle. So, σr and σt are proportional to (r/R)3. When the the expansion coefficient of second phase particles αp is bigger than the matrix αm, that is Δα> 0, according to Eq. (3.1) and Eq. (3.2), a static tensile stress p will generate in the second phase particle, and a radial tensile stress σr and tangential compressive stress σt will generate around the particle in the matrix, but the two forces decreas with (r/R)3. When the crack encounter the second phase particle (as shown in Figure 3.4), it does not towards the second phase particle to expand directly, but departure from the original direction firstly, expand to the direction that is parallel to the circumferential stress σt and vertical to radial stress direction σr. When the crack near the particle, it will deflect towards particle, because the increase of radial stress direction σr (σr is the largest in the interface). Crack can reach the interface of particle / matrix, and then extend along the original spreading direction. So, when the αp>αm, the common action of σr and σt can make the expansion path of crack longer in the matrix, toughening value is much bigger than without considering of residual stress field. As long as the radial tensile stress can maintain the level not leading the interface dissociation in the matrix, then the crack deflection mentioned above has no direct relationship with the bonding of interface. Therefore, the crack deflection caused by residual stress will have a toughening effect absolutely. In addition, when the second phase particle diameter is similar to the matrix grain size, the crack caused by deflection residual stress field is very small; when the second phase particle diameter is much bigger than the matrix grain size, the path of crack deflection longer, resistance of crack growth bigger, and fracture energy consumption much more, so a better toughening effect.


133

Tension σt Crack path

Compression σr Figure 3.4. Crack path for encounting the second phase particle [25].

When the the expansion coefficient of second phase particles αp is smaller than the matrix αm, that is Δα< 0, the crack will develop towards the direction of original crack (that is, vertical to σt, parallel to σr). When the cracks reach the interface of particle/matrix, the cracks will terminate if the applied stress no longer increases, referred as crack pinning. If the applied stress continues to increase, there are two possible ways for crack propagation: firstly, pass through the second phase particle and lead to crack, or fracture through the particle, also known as transgranular fracture. Secondly, expansion is along the interface of particle/matrix, also known as intergranular fracture, which can result in crack deflection. Which way the expansion will crack along depends on energy dissipation of crack propagation in the equilibrium stage. Considering the average surface energy of particle, surface energy required by new surface, the surface energy for unit interface area and other factors, we can see that if the size of second phase particle is larger, main crack is prone to change when extend to the grain boundary, once the interface generate the secondary crack, driving force of crack growth will be rapidly reduced to zero (expansion tends to infinity). Then, the main crack may fracture through the particle along the expansion path. So, if the size of second phase particle is larger, it is prone to transgranular fracture. It should be noted that the expansion of cracks in particle reinforced composite materials is a complex process, which contains a large number of random events. each micro-environment in which particles, particle shape, particle surface and crack the relative factors, particles near the the same system, all particles may provide different toughening mechanisms.

134


In addition, the effect of particle size for toughening, only descript qualitatively at present. With increase of the second phase particle size, tendency of micro-cracks generated in the material is greater. Most of the residual internal stress will be relax in the material, thus toughening mechanisms from residual stress field toughening to micro-crack zone toughening [25].

3.2. Toughening Mechanisms of Micro-Crack Zone Micro-cracks induced by stress will lead to shielding effect of crack tip, which is known as micro-crack toughening. Microcrack toughening mechanism had been proposed about 30 years ago, and much material toughening effects anastomose with the related model. However, the most convincing evidence is two phase ceramics, such as ZrB2 toughenes Al2O3. Basic concept of micro-crack toughening mechanism is shown as Figure 3.5. Micro-cracks usually exist in the local tension region which is the self-linear expansion coefficient mismatch or phase changed zone. Micro-cracks may also emerge along the path of smallest fracture energy, which could alleviate the local tension. The opens of Microcracks are along with volume expansion of the partial tension zone, which depend on the size and shape of process zone. Effect of crack shielding can be achieved by reducing the elastic modulus in the process region, because it can reduce the stress of crack tip. Different from expansion effect, modulus effect is only dependent on the shape of process zone.

Residual openings

σR

+σR Micro-crack toughening

σR

+σR

Figure 3.5. Basic concept of micro-crack toughening mechanism [6].


135

Expansion lags

σ

σ ij*

Modulus lags

E E

θ

ε ijT

Figure 3.6. Stress-strain curve of micro-crack toughening mechanism [6].

Basic stress - strain curves of micro-crack toughening is shown in Figure 3.6. Lag of curves shows the toughness changes caused by micro-crack. However, crack tip provide the opened surface for expansion of main crack, the contribution may be offset as the decrease of crack tip material performance. Sometimes, we can analyse toughening effect from a simple expansion perspective, the maximum toughness caused by microcrack toughening can be expressed as follow:

ΔTmax

CEe M h = 1− v

(3.3)

Here, eM is the volume strain for micro-cracking process region. Value of eM depends on the detail of micro-cracking process.

3.3. Toughening Mechanisms of Residual Stress Field When the external stress can not led to micro-crack toughening, crack deflection can be produced by interaction stress field between the crack tip and around particle, namely, the residual stress toughening mechanism, the model is shown as Figure 3.4.

136


Cyclical residual stress field exist in particles / ceramic composite materials. KIC value of composite is as Eq. (3.4):

K IC = K I 0 + 2q

2D

(3.4)

π

Here, KI0 is the critical fracture intensity factor of matrix; D for the length of compressive stress zone for uniform size, D = λ−d, λ is the distance between the center of adjacent particle,

λ = 1.085d / ν p ; q is the average stress field in the

matrix, and the expression as Eq. (3.5):

q= Here,

2 Emν p βα1*

(3.5)

A

α1* is the strain caused by the difference between linear expansion

coefficients in the particle. TR

α1* = ∫ (α p − α m )dT

(3.6)

Tp

Here, TR is the room temperature; TP the maximum temperature matrix not to be plastic deforming; A and β, respectively

A = (1 −ν p )( β + 2)(1 + ν m ) + 3βν p (1 −ν m )

β=

1 +ν m E p × 1 − 2ν p Em

(3.7)

(3.8)

Increment of the fracture toughness ΔKIC = (KIC− KIΔ) is expressed as:

ΔK IC = 2

2 Emν p βα1* A

×[

2d (1.085 −ν 1/p 2 )

ν 1/p 2

]

1 2

(3.9)


137

Here, when the νp is invariant, ΔKIC is directly proportional to the square root of the second-phase particle size d, which shows that, for residual stress field toughening, the second phase particle size the greater the better in a certain extent.

3.4. Toughening Mechanisms of Crack Bridging Some cracks can bypass the obstacles and maintain the original shape, so the obstacles may become the links of two splitting surface in the back of crack tip. The obstacles are similar to bridge role, so this phenomenon called the crack bridging. Usually, crack bridging can be divided into two types: ductility and fragility. Model of ductile particle crack bridging is shown in Figure 3.7. No. 1 and No. 2 ductile particles are transgranular damage; No. 3 ductile particle happens to be plastic deformation and crack bridging under the stress field; No. 4 ductile particle does not be ductile deformation but crack bridging. Closure force is produced by bridged ductile particle in the distance D from the crack tip to its rear as shown in Figure 3.7. When the linear expansion coefficient and elastic modulus of matrix are equal to ductile particles, the use of crack ductile bridging could achieve the best toughening effect.

1

3

2

4

5

C σ0 C'

D

Figure 3.7. Model of ductile particles crack bridging [25].

138


σ0 1 2

3

4

umax T(u) Figure 3.8. Model of fragile particles crack bridging [25].

If the ductile particles are metal particles, high temperature mechanical properties of ductile particle reinforced ceramic matrix material are not very good, because the high temperature properties of metals are lower than the ceramic matrix, but toughness can be improved in the low temperature significantly. Toughening mechanisms of ductile particle could be divided as crack bridging, grain deformation, grain pull out, crack deflection and crack termination in the particle, and the effect of crack bridging toughening mechanism is more significant than others. Ductile particle reinforced ceramic matrix composites can be used for wear resistant parts. Model of fragile particle crack bridging is shown in Figure 3-6. When the crack meets the brittle particles, transgranular damage may happens, such as No. 1 particle in Figure 3.8; in the first particle; crack extending along the grain boundary may also happens (crack deflection), such as No. 2 particle in Figure 3.8; No. 3 and No. 4 particles are crack bridging. If the bridge materials are fragile, the toughness of bridge is similar to of nonbridging material (matrix material). Bridging phenomenon is more subtle, and often depends on the residual stress of micro scale or weak interface. Residual stress caused by thermal expansion mismatch or relatively anisotropy can extend to partial cracks, so the bridge can exist in the rear of cracks. When the crack bridgeing is failure in the final wake zone, energy dissipates off in the form of sound waves, which playes a toughening role.

4. Practice of Particle Reinforced Ceramic Matrix Composites There are so many systems of particle dispersion reinforced ceramic composites, only several typical systems are introduced here. It can be divided to two categories: the first is the matrix for the oxide system, such as Al2O3-SiC,


139

Al2O3-TiC, Al2O3-ZrB2; the second is the matrix and reinforcement for the nonoxide system, such as Si3N4-SiC, . Toughening effect of the two types is not only with the enhanced content, but also with the particle size, shape and other related factors. Because the fabrication processes of particle reinforced ceramic is convenient and simple, the effect obviously, so many researchers and reports are in this area. Here, we only introduce several typical composite materials. Al2O3-SiC. SiC particle dispersion reinforced Al2O3 ceramic composites is a popular and widely concerned recently. Y.L. Dong [26] found that the fracture strength and fracture toughness of the nano-/micro-sized SiC/Al2O3 compositeswere significantly improved in comparison with the monolithic Al2O3, 7.6MPam1/2 was the highest fracture toughness andwas found in the compositewith 5% SiC, while the 20% SiC composite exhibited the highest flexural strength. During dry-pressing, orientation of the whiskers perpendicular to the pressing axis had been observed by Tiegs [27]. This resulted in anisotropic shrinkage of the whisker composites relative to the original uniaxial pressing direction during densification. For example, in an alumina 10vol.% SiC whisker compact, typical sintering shrinkages were 17~25% in the direction parallel and 10~15% in the direction perpendicular to the original uniaxial pressing direction. Nakahira [28] found that as the SiC content increased, maximum increase of composite ceramic fracture toughness was about 40%. Maximum of fracture toughness gradually decreased, when the hot pressing temperature from 1600°C to 1800°C. If the particle size of SiC is about 2μm, the volume fraction φ of greatest fracture toughness of SiC was about from 0.1 to 0.4. If the particle size of SiC is about 8μm, the increase of composite ceramic fracture toughness was less, the maximum could be got when φ was 0.1, and the increase was about 25%. Fracture toughness decreased as increasing hot pressing temperature, φ was about 0.03, when the temperature was 1800°C, decrease about 25%. Belmonte [29] found that the reduction in wear rate down to one order of magnitude by the addition of SiC particles (20% by volume) to Al2O3 is due to a change in the main wear mechanism: from intergranular fracture to plastic deformation. The effect of the SiC grain size becomes significant for long sliding distances. The wear rate decreases with the SiC grain size, the fracture toughness being the key parameter that governs the wear behaviour of the composites. A lineal fit was established for the dense composites. The sintering rate of Al2O3/SiC composites decreases with the SiC particle size due to a reduction in the interinclusion distance, which leads to lower densities and smaller matrix grain sizes for composite with SiC particles sizes below 1μm.

140


Al2O3-TiC. TiC particle dispersion reinforced Al2O3 ceramic composites is applyed in cutting tools or wear resistant parts for long time use. Jianxin Deng [30-31] had studied the microstructure and mechanical properties of hot-pressed Al2O3/TiC ceramic composites with the additions of solid lubricants. The additions of different solid lubricants such as MoS2, BN, and CaF2 were produced. Effect of the solid lubricants on the microstructure and mechanical properties of this ceramic composite has been studied. No trace of MoS2 was found in the sintered Al2O3/TiC/MoS2 composite owing to its low melting point and escaping during the hot-pressing process. The flexural strength, fracture toughness, and hardness of Al2O3/TiC/MoS2 composite continuously decreased with the increasing of MoS2 content. AlN phase resulted from the reaction of Al2O3 with BN was formed in Al2O3/TiC/BN composite after sintering. Significant microcracks resulted from the residual stress owing to the difference in the thermal expansion coefficient were found on the polished surface, and caused large mechanical properties degradation for Al2O3/TiC/BN composite. While Al2O3/TiC/CaF2 ceramic composite showed relative higher flexural strength, fracture toughness, and hardness compared with that of Al2O3/TiC/MoS2 and Al2O3/TiC/BN composites owing its absence porosity and finer microstructure. Jianghong Gong [32-33] studied the effect of TiC particle size on the toughness of Al2O3–30 wt.% TiC composites, which was examined by quantitatively analyzing the R-curves measured with direct indentation method for four samples having different TiC particle sizes. It was shown that the toughening effect in the composite is strongly dependent on the size of the TiC particles. N. Liu [34] studied the effect of starting powders size on the Al2O3–TiC composites. Al2O3–TiC composites with a content of 30wt% TiC with various size of starting powders were manufactured by hot pressing. The Vickers hardness, bending strength and fracture toughness were studied by N. Liu. The experiment results show that the starting powder size has a significant effect on the properties of the Al2O3–TiC composites. The maximum bending strength of the submicron Al2O3 with the fine TiC powders addition is 712MPa, while the maximum fracture toughness of the same Al2O3 matrix with the large TiC powders addition is 6.5MPam1/2. Al2O3-ZrB2. Author [35-36] had studied the performances of Zr-O-B compounds reinforced Al2O3 ceramic composites. Composites of Al2O3/ZrB2/ZrO2 are fabricated and the fundamental properties of the composites such as hardness, fracture toughness and bending strength are examined. Meanwhile, the effects of ZrB2 and ZrO2 on densification rate, mechanical properties, and microstructures of alumina matrix ceramic materials are analyzed. The addition of Zr-O-B compounds made the microstructures of the composites


141

finer and more homogeneous. The fracture mode turns to the combination of transgranular failure and intergranular failure as the addition amount of ZrB2 was 20vol.%, which may bring on the increase in bending strength and fracture toughness. One of the main causes was that the grain boundaries of the composites are strengthened and the other was the grains of Al2O3 in composites may be restrained from abnormal growing owing to the addition of Zr-O-B compounds. When the addition amounts of Zr-O-B was more than 20vol.%, the densification and mechanical properties of the composites decreases, as the densification of more ZrB2 was difficult. Si3N4-SiC. M. Balog [37] studied the nano/macro-hardness and fracture resistance of Si3N4/SiC composites with up to 13wt.% of SiC nano-particles. Relations between composition and mechanical properties of the Si3N4/SiC micro/nano-composites were studied by combination of nano-indentation and Vickers indentation techniques. The Si3N4/SiC composites were prepared from crystalline Si3N4/SiC powder doped with SiNC amorphous precursor and yttria as the sintering aid. During sintering the SiNC precursor crystallised to yield both SiC and Si3N4. The in situ formed SiC particles were located both inter- and intragranularly. The presence of SiC nano-particles enhanced the nano- and macrohardness, and the fracture toughness of the composites. The nano-hardness of Si3N4/SiC composites ranged between 20 and 24GPa, and depends on the volume fraction of SiC. The nano-hardness of individual Si3N4 grains exhibited large scatter as the consequence of the presence of intra-SiC inclusions, which directly influence the measured values as the harder phase, or by generating large thermal stresses within Si3N4 grains. Consequently the scatter of nano-hardness was much larger than in case of macro-hardness where the measured values are averaged over large area. The nano-indentation of grain boundaries indicates that the boundaries are much softer than the surrounding matrix phase. Koji Takahashi [38] studied the improvement of static fatigue strength of Si3N4/SiC crack-healed under cyclic stress. Si3N4/SiC composite ceramics were hot-pressed in order to investigate their crack-healing behaviour and the resultant static fatigue strength. Semi-elliptical surface cracks of 100 μm in surface length were made on each specimen. The pre-cracked specimens were crack-healed under a cyclic bending stress of 210MPa in air at 900, 1000, 1100, and 1200°C. The bending strength and static fatigue strength of the crack-healed specimens were systematically investigated at each healing temperature. The specimens which has been crack-healed and static fatigue-tested at 900 and 1000°C showed lower static fatigue strength than those tested at 1100 and 1200°C. Detailed investigation on the fracture surface of static fatigue-tested specimens showed that oxidation of the base material had strong effects on the static fatigue strength. It

142


was found that if the specimens were pre-oxidized in air at 1300°C, the surface was covered by a protective oxide layer, leading to a significant improvement of static fatigue strength at 900 and 1000°C. S.A. Baldacim [39] studied the development and characterization by HRTEM of hot-pressed Si3N4–SiC(w) composites. The activation of toughening mechanisms in Si3N4–SiC(w) composites depends essentially on interfacial characteristics. They were affected by the chemical compatibility and the difference of the thermal expansion coefficients between the SiC whiskers and the Si3N4 matrix. In this work, Si3N4–SiC(w) composites were produced by hotpressing, using AlN/Y2O3 as additives. The SiC whiskers–Si3N4 matrix interface had been investigated by high-resolution electron transmission microscopy (HRTEM). A thin amorphous phase had been detected at the whisker/matrix interface and at grain boundaries, triple junctions of matrix grains. This intergranular phase caused weak bonding between the whiskers and the matrix, enhancing crack deflection toughening mechanisms. As a result, and due to the high aspect ratio of the SiC whiskers, a significant improvement of the fracture toughness was achieved, when compared to Si3N4 monolithic ceramics. TiB2-SiC. Zhu Degui [40] had studied the in-situ HIP synthesis of TiB2/SiC ceramic composites. Dense TiB2/SiC ceramic composites were fabricated by means of in-situ HIP synthesis processing from TiH2, Si, C and B4C powder blends. The microstructure and mechanical properties of the ceramic matrix composites were investigated using X-ray diffraction, optical microscopy, scanning electron microscopy, transmission electron microscopy and microhardness. The grains of TiB2 and SiC in the ceramic matrix composites were fine, the TiB2 grains being regular-platelet, and the SiC grains being were irregular polygon. The mechanical properties of the composite were mainly affected by their densities. As the density increases, the properties of the composites improved. The microhardness, fracture toughness KIC and flexural strength were, respectively, 14.80GPa, 4.75MPa m1/2 and 408.62MPa in the 96% density TiB2/SiC composites. The crack propagation path is deflected in these composites. TiB2/SiC composites also had an excellent oxidation resistance. The average oxidation rate of TiB2/SiC is about 10−10m s−1 from 1073 to 1473 K. To sum up, there are two characteristics in particle dispersion reinforced ceramic composites. First of all, several types of fracture toughness depende on grain size. Fracture toughness generally can be enhanced with the increasing of particle size; at least there is an optimal particle size. Particle size so fine may be expanded effect of large particles, but sometimes there are other roles. Secondly, the relationship of the toughening effect and strain is more complex, which is caused by different linear expansion coefficient of particle and matrix. Linear


143

expansion coefficient of SiC and TiC is lower than that of Al2O3, the toughening effect of two particles is very obvious for Al2O3 matrix; linear expansion coefficient of SiC and TiC is higher than that of Si3N4, and the linear expansion coefficient of TiB2 is higher than that of SiC, the toughening effect is also obvious. Table 4.1. The mechanical properties of several common ceramic composite containing SiC Ingredient SiC-25% (wt.%) TiC SiC-50% (wt.%)TiB2 SiC-15% (vol.%) ZrB2 Al2O3-25% (vol.%) SiC SiC-85% (wt.%) AlN Y-TZP-20% (vol.%) SiC Al2O3-γ-AlON-20% (vol.%) SiC

σb/MPa 580 − 560 1000 476 1050 730

KIC/MPa·m1/2 6.5 7.5 6.5 7.6 3.7 8.0 4.1

Microcrack toughening may be the better toughening mechanism for some composites, such as TiB2-SiC. In view of the important role of SiC in the particle dispersion reinforced ceramic composites; Table 4.1 summarizes the mechanical properties of several common ceramic composite containing SiC.

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[24] R.J. Li, H.Q. Ru, X.D. Sun. (1995). Ceramic-metal composite materials. Beijing: Metallurgical Industry Press (in Chinese). [25] Y.J. Zhang, W.R. Zhang. (2004). Structural ceramic materials and applications. Beijing: Chemical Industry Press (in Chinese). [26] Y.L. Dong, F.M. Xu, X.L. Shi, C. Zhang, Z.J. Zhang, J.M. Yang, Y. Tan. (2009). Fabrication and mechanical properties of nano-/micro-sized Al2O3/SiC composites. Mater. Sci. Eng.:A, 504(1-2), 49-54. [27] T. N. Tiegs, P. F. Becher. (1987). Sintered Al2O3-SiC Composites, Am. Ceram. Soc. Bull., 66(2), 339–342. [28] A. Nakahira, K. Niihara, T. Hirai. (1987). Microstructure and mechanical properties of Al2O3-SiC composites. Int. J. High Tech. Ceram., 3(2), 174174. [29] M. Belmonte, M.I. Nieto, M.I. Osendi, P. Miranzo. (2006). Influence of the SiC grain size on the wear behaviour of Al2O3/SiC composites. J. Eur. Ceram. Soc., 26(7), 1273-1279. [30] J.X. Deng, T.K. Can, J.L. Sun. (2005). Microstructure and mechanical properties of hot-pressed Al2O3/TiC ceramic composites with the additions of solid lubricants. Ceram. Int., 31(2), 249-256. [31] J.X. Deng, T.K. Cao, Z.L Ding, J.H. Liu, J.L.Sun, J.L. Zhao. (2006). Tribological behaviors of hot-pressed Al2O3/TiC ceramic composites with the additions of CaF2 solid lubricants. J. Eur. Ceram. Soc., 26(8), 13171323. [32] J. Gong, H. Miao, Z. Zhao, Z. Guan. (2001). Effect of TiC particle size on the toughness characteristics of Al2O3-TiC composites. Mater. Lett., 49(34), 235-238. [33] J. Gong, Z. Zhao, H. Miao, Z. Guan. (2000). R-curve behavior of TiC particle reinforced Al2O3 composites. Scripta Mater., 43(1), 27-31. [34] N. Liu, M. Shi, Y.D. Xu, X.Q. You, P.P. Ren, J.P. Feng. (2004). Effect of starting powders size on the Al2O3-TiC composites. Int. J. Refract. Met. Hard Mater., 22(6), 265-269. [35] B. Li, J. Deng, S. Zhao. (2008). Fabrication and properties of Al2O3/ZrB2/ZrO2 composite ceramic materials. J. Chin. Ceram. Soc., 36(11), 1595-1600 (in Chinese). [36] B. Li, J.-X. Deng. (2008). Effect of grain refinement on mechanical properties of Al2O3/ZrB2/ZrO2 composite ceramic materials. J. Funct. Mater., 39(11), 1863-1866 (in Chinese). [37] M. Balog, J. Keck, T. Scherl, D. Galusek, F. Hofer, J. Krestan, Z. Lenc, J.L. Huang, P. Sajgal. (2007). Nano/macro-hardness and fracture resistance of

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Si3N4/SiC composites with up to 13wt.% of SiC nano-particles. J. Eur. Ceram. Soc., 27(5), 2145-2152. [38] K. Takahashi, H. Murase, S. Yoshida, K. Houjou, K. Ando, S. Saito. (2005). Improvement of static fatigue strength of Si3N4/SiC crack-healed under cyclic stress. J. Eur. Ceram. Soc., 25(11), 1953-1959. [39] S.A. Baldacim, C. Santos, K. Strecker, O.M.M. Silva, C.R.M. Silva. (2005). Development and characterization by HRTEM of hot-pressed Si3N4-SiC(w) composites. J. Mater. Process. Tech., 169(3), 445-451. [40] Z. Degui, L. Shikai, Y. Xiandong, Y. Liu, X. Chuanchun, Z. Haoming, Z. Jianyong. (1999). In-situ HIP synthesis of TiB2/SiC ceramic composites. J. Mater. Process. Tech., 89-90 (4), 457-461.


Chapter 5

A NOVEL STRATEGY FOR DEVELOPING POLYMER NANOCOMPOSITE WITH HIGH DIELECTRIC CONSTANT Jing-Wen Wang and Shu-Qin Li Department of Material Science and Engineering, College of Material Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China

Abstract A novel nanofabrication strategy to develop high dielectric constant nanocomposite was introduced using poly(vinylidene fluoride) (a piezoelectric polymer) as matrix and PCMS-g-CuPc [poly(p-chloromethyl styrene) (PCMS) grafted with copper phthalocyanine oligomer (CuPc, a planar multiring organic semiconductor with super high dielectric constant more than 105)] as filler. Improvement of the dispersibility of CuPc oligomer in polymer matrix was confirmed by TEM-observed morphologies. The PCMS-g-CuPc particles with a average size of about 80nm are dispersed in poly(vinylidene fluoride) matrix, while in PCMS-g-CuPc particles the PCMS acts as “matrix” which contains dispersed CuPc balls with a average size of ca. 25nm [about 1/20 of that of CuPc particles in the simple blend of poly(vinylidene fluoride) and CuPc]. The slutioncast nanocomposite film sample with only 15wt% CuPc can realize a dielectric constant of about 325 at 100Hz, more than 38-fold enhancement with respect to that of the pure PVDF. The enhanced dielectric response in the nanocomposite demonstrates the significance of the interface effect in raising the material

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Jing-Wen Wang and Shu-Qin Li responses far beyond that expected by simple mixing rules when there is a large dielectric contrast between the two components in the composite. Theoretic prediction using modified Cole-Cole equation indicates that, even at the high frequency limit, the dielectric constant of the nanocomposite can still be more than 70. Furthermore, the dielectric strength increase from 40 MV/m of the simple physical blend of poly(vinylidene fluoride) and CuPc to 51 MV/m of the nanocomposite sample. All these will lead to a nanocomposite with better controlled process conditions and are highly desirable for high dielectric constant composites.

Introduction Over the past decade or so, smart materials that can produce large mechanical actuation under external stimuli such as electric field, temperature, and stress have attracted a great deal of attention. [1-19] The development goals of these materials include achieving a large range of motion with high precision and speed, high strain energy density to generate large forces, and a low fatigue rate for a long lifetime and high reliability. [1,13] Conventional inorganic smart materials include, but not limited to, electroactive ceramics [EACs], shape memory alloys [SMAs], and McKibben actuators, etc. [1,12,13,19] Although EACs have low hysteresis and fast response speed, their strain levels are relatively low (~0.1%), and are typically fragile and expensive. SMAs can generate high strain and high force but are often associated with large hysteresis, very slow speed, short cycle life and unpredictable movement. McKibben actuators are sophisticated devices; however, they employ heavy electric motors and pneumatics. As a result, researchers have to investigate alternative technologies for various applications. [1,8-10,12,13] The fact that certain types of polymers can change shape in response to electrical stimulation has been known for decades. However, the induced strain was relatively small. Since the early 1990s, Electroactive polymers [EAPs] that respond to electrical stimulation with a significant size or shape change are emerging as a new class of actuation material. [8,12,20] The large induced displacement capability of these new EAP materials is attracting the attention of engineers and scientists from many different disciplines. The behavior similarity of these materials to biological muscles acquired them the moniker “artificial muscles”. [8,20] Since EAPs encompass a broad range of materials, based on their physical state or activation mechanism, EAPs can be routinely categorized into two major classes: ionic and electronic. The actuation mechanism for ionic EAPs involves the diffusion or mobility of ions, wherein such EAPs may require the presence of a liquid medium, while an electric field or Coulomb forces in general

A Novel Strategy for Developing Polymer Nanocomposite…

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drive electronic EAPs. [8,13,19,20] Electronic EAPs can exhibit fast response speeds, low hysteresis and high strain levels, with elastic energy densities even higher than those in the current available piezoelectric materials (elastic energy density >0.1J/cm3).[18] However, unlike their inorganic counterparts, the dielectric constant of most polymers are in a limited region ranging from 2 to 10, [1,9] whereas for a linear dielectric, the stored energy per unit volume is U=½Kε0E2, where K is the relative dielectric constant, ε0 is the vacuum dielectric permittivity (=8.85×10-12 Fm-1), and E is the applied electric field. As a result, for pure EAPs, to achieve high input electric energy density which is required to generate high elastic energy density measuring both the stress and strain generation capability of an electromechanical material, a high electric field (>100 MV/m) is required to make up for the low dielectric constant. Therefore, in order to reduce the applied electric field substantially in such type EAPs, one has to raise the dielectric constant of this class of polymers substantially. [1,9,12,14,21] In the past several decades, the polymer/ceramics 0-3 composite approach, in which high dielectric constant particulates (most of which are ferroelectric ceramics) were added to a polymer matrix, has been employed to raise the dielectric constant of polymer based materials. [1,9,12,14,21] High loading of the ceramic fillers needed in the composite, usually over 50 vol%, can increase dielectric constant by about ten times relative to the polymer matrix. [22] However, because ceramic materials have an elastic modulus much higher than that of polymers, this approach usually suffers from the adverse increase of the modulus of the polymer matrix, the loss of the flexibility, and the deterioration of processibility at the same time.[1,9,22] Furthermore, most ferroelectric fillers used in the composites are lead-based ceramics, which are not environmentally friendly.[11] Recently, two kinds of all organic composite approaches were exploited to fabricate high dielectric constant polymer composites.[14] One approach is to use the percolation phenomena observed in polymer/conductive polymer composites.[10,14,18,22,23] Dielectric constant enhancement of ca. 10~100 times that of polymer matrix has been observed in several such percolative composites. However, simultaneously, these composites also exhibit relatively high dielectric loss due to the insulator-conductor transition near the percolation threshold [11]. In addition, when the content of conductive fillers close to the percolation threshold, which is need to achieve a high dielectric constant, the composites will become very sensitive to the composition variation and the breakdown field will also become relatively low, all of which are not desirable for the electromechanical applications of the composites. The other approach is to increase the dielectric constant of the polymer matrix by dispersing some organic

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Jing-Wen Wang and Shu-Qin Li

semiconductors with super high dielectric constant. [1,6,9,14,16,] The most often used organic semiconductor is copper phthalocyanine oligomer (CuPc) (Scheme 1) with a very high dielectric constant more than 105. [9] The multiring CuPc represents a family of molecules which has proven to be of remarkable interest for both its chemical and physical properties. An important characteristic of CuPc molecule is that the nomadic polarization mechanism, [24,25] that is the pliant response of highly delocalized electrons under electric fields moving on the highly conjugated π-bonds within the entire sheet-like CuPc molecule, results in a high dielectric response. As an organic material, CuPc has a modulus comparable to that of the polymer matrix. Therefore, a high dielectric constant can be achieved in the polymer/CuPc composite without increasing the material modulus, thus the flexibility of the polymer matrix can be remained.[1,9,14] By exploiting an allorganic composite approach in which CuPc particulates (40wt%) were blended with poly(vinylidene fluoride-trifluoroethylene) [P(VDF-TrFE)], then treating the resulting composite with high-energy electron irradiation to turn the polymer into a relaxer ferroelectric thus enhance its dielectric response, a polymeric-like material can reach a dielectric constant higher than 400(at 1Hz and 13 V/μm-1), which results in a significant reduction of the field applied to generate high strain with high elastic energy density.[1] However, CuPc particles are susceptible to be agglomerated in the polymer matrix (the size of CuPc particles is ~1μm) due to incompatibility of CuPc with the polymer matrix, which will reduce the breakdown field and increase the dielectric loss. It is well known that in polymer composite the compatibility between the filler and the polymer matrix can be enhanced by addition of dispersant,[26] the formation of intermolecular hydrogen bonding, [27-29] cross-linking,[30] or grafting, [9,31] etc. In 2005, we developed a grafting approach to prepare a nanocomposite in which CuPc (25wt%) was partially grafted to chemically modified poly(vinylidene fluoride-trifluoroethylene-chlorofluoroethylene) [P(VDF-TrFECFE)], a polymeric relaxer ferroelectric with a dielectric constant of about 40.[9] Improvement of the dispersibility of CuPc in the terpolymer matrix was achieved (the CuPc inclusion size is about 60-100 nm), due to the compatible effect of the grafted product in the nanocomposite, and the dielectric constant of the resulting nanocomposite reaches nearly 175 at 100 Hz.

A Novel Strategy for Developing Polymer Nanocomposite… COOH

HOOC

HOOC

N N

HOOC

N

N Cu

N

N

N

N

HOOC

COOH

N

N N Cu

N

COOH

N

N

N

N

COOH

COOH HOOC

N

N N

HOOC

N

N

HOOC

Cu N

N

N

N

N N

N

N

COOH

HOOC

151

Cu N

16

COOH

N N

COOH

COOH

Scheme 1. Chemical structure of the CuPc. Shown on the right is a label to it.

We noticed that, according to a theoretical modeling proposed by Li [32] on such kind of composites, the interface exchange coupling effect between the dielectrically hard PVDF based ferroelectric and dielectrically soft CuPc can result in a significant change in the local polarization level. Since the exchange coupling exists only in the near interface region, if we further decrease the size of CuPc, dramatically improved dielectric properties can be achieved.[9,14] It was estimated that the grafted CuPc out of the CuPc totally filled in the above P(VDFTrFE-CFE) based nanocomposite is 12.8%. To further decrease the size of CuPc substantially, we need to increase the grafted ratio of CuPc by a wide margin. Furthermore, content of the filler should also be taken into consideration in view that low loadings of the filler will benefit the reduction of the amounts of voids/defects in the final composite and result in improvement of mechanical properties.[29,33] Here, we introduce a novel avenue to fabricate nanocomposite of poly(vinylidene fluoride) (PVDF) and poly(p-chloromethyl styrene) (PCMS) grafted with CuPc. PVDF is a commercially easy available piezoelectric polymer which possesses good performance and widely used electromechanical application.[5,12] To further decrease the size of CuPc particles, PCMS was selected for grafting. It imparts several advantages over the above mentioned grafting polymerization. [9] Anchoring of CuPc to PCMS backbone is much easy, thus the grafting ratio is very high compared with the above mentioned one, which will lead to substantially decreased CuPc inclusion size in PCMS “matrix”. A

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further benefit is that the dispersion of PCMS itself or PCMS-g-CuPc in PVDF is much better than that of CuPc in PVDF, which was found during our study. By grafting CuPc to PCMS, then blending with PVDF using the solution cast method, we developed a nanocomposite (with only 15wt% CuPc) in which the improvement of dispersion and decrease of size level of CuPc were achieved, consequently a large enhancement in dielectric response of PVDF based allorganic nanocomposite is realized, due to the strong interface effect between CuPc particles and polymer matrix.

Experimental Section Materials. The CuPc (Scheme 1) was synthesized following a procedure reported in referance 34. Copper sulphate pentahydrate, pyromellitic dianhydride, urea, ammonium chloride, and ammonium molybdate were ground together and then placed in a three-necked flask containing nitrobenzene. The temperature of the flask was maintained at 185 oC for 12 h. The as-synthesized solid materials was finely ground and washed with methanol to remove nitrobenzene completely. The powder was boiled with 2N hydrochloric acid saturated with sodium chloride and filtered after cooling to room temperature. The product was neutralized by 2N potassium hydroxide solution containing sodium chloride at 90oC. After centrifugation then washed with distilled water, the product was dried at room temperature under vacuum. PCMS (Mn=55,000) was purchased from Aldrich. Triethylamine (TEA) was dried with NaOH and distilled before use. Dimethylformamide (DMF) was dried with CaH2 followed by distillation in vacuo prior to use. The PVDF with a weight average molecular weight of 400,000 was purchased from Shanghai 3F New Materials Co., Ltd., China. Other reagents were of analytical grade and used without further purification. Synthesis of PCMS-g-CuPc. Scheme 2 shows the synthetic route to the PCMS grafted with CuPc. A 100mL three-neck round-bottom flask fitted with a magnetic stirrer, a thermometer and a condenser was used as the reactor. TEA (3.0mL) was added to a solution of PCMS (0.5g) and CuPc (0.5g) in DMF (40mL). The solution was stirred at 65oC for 12 hours under purified nitrogen atmosphere. After the TEA and DMF were removed by reduced pressure distillation, the mixture was washed with methylene dichloride to remove unreacted PCMS, if any, followed with distilled water to remove triethylamine hydrochloride. The final product was dried in vacuo at 50oC, and labeled PCMS-g-CuPc.


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CuPc TEA

Cl

DMF N2 65oC -HCl

O O HOOC

PCMS

15

PCMS-g-CuPc

Scheme 2. Schematic Drawing of the Synthesis of PCMS-g-CuPc.

Preparation of Films for Electric Measurement. Films were prepared using solution-cast method. For the composites of PVDF and PCMS-g-CuPc (PVDF/PCMS-g-CuPc) with the PCMS-g-CuPc of 15 wt%, 30 wt%, 40wt%, and 50 wt% (accordingly the contents of CuPc are 7.5 wt%, 15 wt%, 20wt%, and 25 wt%, respectively), PCMS-g-CuPc was added to the solution of PVDF in DMF, and then ultrasonically stirred for at least 2 hours. Afterward, the solution was poured onto a clean glass slide and dried in air at 70oC for 5 hours, then in vacuo at 50oC for 12 hours to remove DMF. Finally, the film was annealed at 120oC for 12 hours and then slowly cooled to room temperature. For comparison purpose, the film of the blend of PVDF and CuPc (PVDF/CuPc) was also prepared. The preparing procedure of the film of PVDF/CuPc containing 15wt% CuPc was the same as that of PVDF/PCMS-g-CuPc. The typical solution-cast film thickness was ~30μm. For the electric characterization, the films were cut into small pieces of about 10×10mm, and circular gold electrodes with 2.5mm radius were sputtered in the center of both surfaces. Characterization. FT-IR spectra were recorded with the sample/KBr pressed pellets using a Bruker Vector-22 FT-IR spectrometer. 1H NMR spectra were obtained in DMSO-d6 and collected on a Bruker DRX-500 spectrometer. Inductively coupled plasma atomic emission spectrometry (ICP-AES) was used to determine the graft ratio of CuPc in the synthesis of PCMS-g-CuPc. The unreacted CuPc in samples for test was removed by soaking PCMS-g-CuPc with 50mL 0.1mol/L NaOH aqueous solution, followed with distilled water to get rid of NaOH. The resulting product was dried in vacuo at 50oC, and then was soaked

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in a crucible which contained a mixture of 3 mL of 70% nitric acid and 0.6 mL of 70% perchloric acid for 12 hours. After all of the liquids were slowly evaporated off in air at 80 oC for about 2 hours, the organic components were burned up in the crucible. The residues were diluted by 5% nitric acid to a 10mL solution, and the metal contents were measured by a Jarrell-Ash J-A1100. For thermal analysis, a Perkin-Elmer DSC-2C calorimeter was used at a heating rate of 20oC/min. TEM was performed using An H-7650 Transmission Electron Microscope. The specimen was prepared by placing a drop of a solution with about 1.0 wt % of composite in DMF on carbon film coated copper grid and then dried in air at 75oC before observation. To elucidate the microstructure inside the PCMS-g-CuPc, an ultramicrotomed sample of PVDF/PCMS-g-CuPc was particularly observed. Xray study was carried out using a D8Advance X-ray generator with a copper target. The wavelength used was 1.5406×10-10m. For the characterization of frequency dependence of the dielectric properties, an Agilent 4194A Impedance Analyzer was used, and the dielectric constant K of the film was calculated by the formula of a parallel plate capacitor as: K=Ct/ε0A

(1)

where C is the capacitance of the metal-insulator-metal element, ε0 is the vacuum dielectric permittivity, A is the area of the electrode, and t is the thickness of the capacitor, respectively.

Results and Discussion Synthesis and characterization of PCMS-g-CuPc. CuPc oligomer has 16 peripheral carboxyl groups (-COOHs) in itself and is hard to react with the PVDF backbone. To overcome this problem in this study, PCMS with hundreds of highly reactive chloromethyl functional groups was selected to modify CuPc, since CuPc can easily react to PCMS in the presence of TEA that acts as an acceptor to take in resulting hydrochloric acid in the solution, consequently the esterification is promoted. The evidence of the successful grafting procedure was provided by FTIR and 1H-NMR characterization. The FT-IR analysis of CuPc and PCMS-g-CuPc is schematically depicted in Figure 1. The absorption band at 1767cm-1 corresponds to carbonyl band of an ester linkage, which proves the successful esterification [9,31].


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Figure 1. FT-IR spectra of PCMS-g-CuPc (a) and CuPc (b).

The strong absorption band at 1717cm-1 is intrinsic to the stretching vibration of carbonyl group of –COOH of CuPc, indicating some unreacted -COOHs in PCMS-g-CuPc. As shown in the figure, the spectrum of PCMS-g-CuPc still keeps the characteristic peeks (such as 1100, 1130, 1450, 1510 and 3400 cm-1) of that of CuPc. Verification of the successful grafting of CuPc to PCMS can also be obtained from 1H NMR spectrum of PCMS-g-CuPc. The methylene of Cl─CH2─C6H4─ and CuPc─CH2─C6H4─ resonances occurred at 4.65 ppm and 5.32 respectively, which also confirms that the CuPc was grafted onto PCMS. The resonances at 6.8-7.4 ppm are assigned to aromatic hydrogen in ─C6H4─ of PCMS, and the peak lies at 8.14 ppm is attributed to hydrogen in >C6H2< of CuPc. To determine the grafting ratio of CuPc quantitatively, ICP-AES analysis was used, and it is estimated that the grafted CuPc in the synthesis procedure of PCMS-g-CuPc is as much as 96.8%. In effect, we have found that, in the preparing procedure of the sample for ICP-AES analysis, the color of the 0.1mol/L NaOH aqueous solution in which the PCMS-g-CuPc was soaked with to extract unreacted CuPc was almost not changed, whereas even 0.1 mg CuPc can turn 100ml 0.1mol/L colorless NaOH aqueous solution into visibly blue. Considering the inevitable measurement errors in the ICP-AES analysis, we can regard the grafting ratio as near 100%. Microstructure of the Nanocomposites. TEM micrographs of PVDF/CuPc (with 15wt% CuPc) and PVDF/PCMS-g-CuPc (with 30wt% PCMS-g-CuPc,

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accordingly the content of CuPc is also 15wt%) are shown in Figure 2. For the PVDF/CuPc, the CuPc particle size is about 500nm[Figure 2(a)].

Figure 2. (a) TEM photograph of the PVDF/CuPc. (b) Schematic illustration of the anticipated morphology of PVDF/PCMS-g-CuPc. (c) and (d) present the TEM photographs of the PVDF/PCMS-g-CuPc (the specimen was prepared by placing a drop of a solution of the nanocomposite in DMF on carbon film coated copper grid and then dried in air before observation) and the ultramicrotomed sample of PVDF/PCMS-g-CuPc, which confirm the above anticipation.

CuPc has a strong tendency to form stack assemblies and microaggregates due to its planar shape and aromatic nature.[35] The aggregation of CuPc in the


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PVDF/CuPc can hardly be prevented due to the incompatibility of CuPc with PVDF. For the PVDF/PCMS-g-CuPc the anticipated morphology as shown in Figure 2 (b) is based on the consideration that: (1) the PCMS is also incompatible with PVDF, in consequence the PCMS-g-CuPc is inclined to aggregate in PVDF matrix, and (2) in the PCMS-g-CuPc, part of CuPc oligomers attached onto PCMS can acts as nucleation centers, which further induced the growth of CuPc crystallite. Since the pendent CuPc groups were distributed separately along the PCMS backbone, the size of crystallite was restricted by the accessibility of adjacent CuPc molecules. Figure 2 (c) and (d) confirm our anticipation. From Figure 2 (c) we can learn that the particles of PCMS-g-CuPc with an average size of about 80nm are dispersed in PVDF matrix, and From Figure 2 (d), it is observed that the CuPc inclusion with an average diameter of ca. 25nm, about 20 times smaller than that of CuPc in PVDF/CuPc, was dispersed in PCMS “matrix”. In a word, in the PVDF/PCMS-g-CuPc nanocomposite, the grafted CuPc form nanophase crystallites (~25nm) in PCMS-g-CuPc particles, and the PCMS-gCuPc inclusion (~80nm) is dispersed in PVDF matrix. The DSC curves of PVDF and two composites are illustrated in Figure 3. The melting point of pure PVDF is 172.30oC, and for PVDF/CuPc and PVDF/PCMSg-CuPc the melting points are 169.25oC and 168.50oC, decreased by 3.05 and 3.80 °C respectively in comparison with that of the pure PVDF. The decrease of the melting point of the composites can be explained by the presence of heterogeneity of CuPc and PCMS-g-CuPc which hindered the crystal perfection and the reduced lamellar thickness of PVDF crystallites.[9,31,36-38] The degree of crystallinity (χ) can be calculated according to equation[39]:

χ(%) = (ΔH m / W ΔH m0 )

(2)

where ΔHm is the enthalpy of fusion of the melting transition, W is the PVDF content in the composites, ΔHm0 is the enthalpy of fusion of 100% crystalline PVDF which is 90.40J/g.[39] The ΔHm of PVDF/CuPc and PVDF/PCMS-g-CuPc are 44.68J/g and 35.06J/g, respectively. It is calculated using the Equation 2 that the crystallinity degree of PVDF/CuPc and PVDF/PCMS-g-CuPc is 58.2% and 55.4% separately which is much higher than that of the pure PVDF (35.9%). The possible origin of the increase of the crystallinity degree of the two composite can be attributed to the fact that both the two nanofillers can favor the nucleation of PVDF crystalline phase.[40]

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Figure 3. DSC curves of pure PVDF, PVDF/PCMS-g-CuPc (with 15wt% CuPc) and VDF/CuPc (with 15wt% CuPc).

Figure 4. X-ray data of PVDF (a), PVDF/PCMS-g-CuPc (b), PVDF/CuPc (c) and CuPc (d).


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Figure 4 presents the XRD results of the pristine PVDF, PVDF/PCMS-gCuPc, PVDF/CuPc and CuPc acquired at room temperature. For PVDF the diffraction peaks at 20.2° and 26.8° correspond to (110) and (021) reflections.[41] For the composites, the peak positions corresponding to (110) reflection almost do not change, and the wider diffraction peaks at ~27° are the (021) diffraction peak of PVDF overlapped with that of CuPc at ~27°. The Miller Index Lhkl can be estimated employing the Scherrer equation: [42]

Lhkl = 0.9λ /( B cos θ )

(3)

where λ is the X-ray wavelength (=1.5406×10-10m), B is the full width at halfmaximum of the diffraction peak in 2θ, and θ is the peak angular position. The Miller Index Lhkl in the direction perpendicular to the crystal planes can be deduced according to Equation 3. For the pure PVDF and the composites, L110 is about 4.8nm. These results lead to the conclusion that the CuPc oligomer is so bulky that it is totally excluded from the crystalline regions.[9,31] Dielectric Properties of the Composites. In a heterogeneous mixture of a relatively high conductive filler and an insulting polymer phase, it is generally expected that the breakdown field will be reduced. The breakdown field of the PVDF/PCMS-g-CuPc film was measured to be 51 V/μm, while for the PVDF/CuPc film, it is 40 V/μm. The higher breakdown field observed in the PVDF/PCMS-g-CuPc film was expected due to the notably reduced and more uniformly sized CuPc oligomer particulates in PVDF/PCMS-g-CuPc as compared with the PVDF/CuPc. In the latter, the excessive agglomeration of CuPc oligomer particulates can easily lead to conductive paths and lower breakdown field.[9,16,31] To determine the suitable weight fraction of CuPc in the nanocomposite, dielectric properties of PVDF/PCMS-g-CuPc with four selected content of CuPc (7.5 wt%, 15 wt%, 20 wt%, and 25 wt%) as a function of frequency were characterized at room temperature, and the experimental results are illustrated in the inset (a) of Figure 5. As generally expected, the dielectric constants (K) of composites are remarkably enhanced compared with that of the pure PVDF (8.4 at 100Hz), and, by and large, increased with weight percentage of the CuPc inclusion. However, in comparison to PVDF/PCMS-g-CuPc with 15 wt% CuPc, dielectric constant of the composites with more CuPc exhibits quite slow increase. Furthermore, at frequencies above 3250Hz, the dielectric constant of PVDF/PCMS-g-CuPc with 25 wt% CuPc is even lower than that of the sample with 15 wt% CuPc. Similar results were also observed in previous

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investigations.[16,43] This phenomenon is probably arising from the MaxwellWagner-Sillars (MWS) polarization mechanism[9,16,31,44,45] which is caused by the large difference in dielectric constant between the polymer matrix and the filler. V. Bobnar[46-48] has concluded that, the high dielectric response of the composites of CuPc and PVDF based copolymer/terpolymer is not due to the intrinsic high dielectric constant of CuPc oligomers but is rather governed by MWS interfacial effect.

Figure 5. Dielectric properties of PVDF/CuPc (triangles), PVDF/PCMS-g-CuPc (crosses) and the pure PVDF (squares) as a function of frequency at room temperature. The weight percentage of CuPc in both composites is 15wt%. Inset (a) illustrates the dielectric properties of PVDF/PCMS-g-CuPc with different weight fraction of CuPc (7.5wt%, 15wt%, 20wt%, and 25wt%) measured at room temperature as a function of frequency. Inset (b) shows the dielectric constant of P(VDF-TrFE-CFE) with partially grafted CuPc and the pristine P(VDF-TrFE-CFE) as a function of frequency at room temperature.


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For composites with high CuPc concentrations discussed here, the strong MWS relaxation results in large low frequency dielectric dispersion. We will come back to this aspect later. Due to its relatively high dielectric constant and lower dielectric dispersion compared with composites with higher loading of CuPc, sample with 15 wt% CuPc was selected for further investigation. The comparison of dielectric constant and dielectric loss (D) of PVDF/PCMS-g-CuPc, PVDF/CuPc , and the pure PVDF as a function of frequency from 100Hz to 1MHz at room temperature is presented in Figure 5. The dielectric constant of the two composites is substantially increased compared with the pure PVDF. For example, the PVDF/PCMS-g-CuPc film shows a dielectric constant of more than 325 at 100Hz, nearly 40 times higher than that of the pure PVDF, meanwhile, the dielectric constant of PVDF/CuPc is ~50 at the same frequency, nearly 6 times that of the pure PVDF. It should be pointed out that the dielectric constant of the composites is much higher than those derived from various models, especially for the PVDF/PCMS-g-CuPc nanocomposite. For instance, according to the mean field-type composite theory,[32,49] which is very reliable when the size effect is not important, the effective dielectric constant of the composite can be estimated as:

K=

2 K1 + K 2 − 2c2 ( K1 − K 2 ) K1 2 K1 + K 2 + c2 ( K1 − K 2 )

(4)

where K1 and K2 are the dielectric constant of polymer matrix and dielectric filler, separately, and c2 is the volume fraction of the filler. Since K2 (4.3×105 to CuPc and 6.6×103 to PCMS-g-CuPc) is much larger than K1 (8.4), after simplification to the Equation 4, we finally obtain:

K=

1 + 2c 2 K1 1 − c2

(5)

It is estimated that, the dielectric constants of PVDF/CuPc (with 15wt% CuPc) and PVDF/PCMS-g-CuPc (with 30wt% PCMS-g-CuPc) can only reach about 13 and 19, respectively. Obviously, this model cannot be used to explain the large enhancement of the dielectric constant as observed in both composites, especially for PVDF/PCMS-g-CuPc. The much higher dielectric constant of the composites could be arising from at least two characteristics of such composites. First, the MWS space charge phenomenon results in strong low frequency dielectric dispersion, especially for

162


the PVDF/PCMS-g-CuPc due to the large interface-to-volume ratios of CuPc particules, as observed in Figure 2. Second, the interface exchange coupling effect may play a much important role in the prominent enhancement of dielectric constant of composite. Li[32] has pointed out that, as the heterogeneity in the composite becomes smaller and smaller, the influence of the exchange layer, an interface layer in which the polarization is strongly affected by both phases, becomes more and more important, and eventually dominates when the heterogeneity size and the exchange length become comparable. Although the CuPc content in both composites is the same, the dielectric constant of PVDF/PCMS-g-CuPc with smaller CuPc nanoparticles is more than 7 times higher than that of PVDF/CuPc. Moreover, in one of our previous work, as shown in the inset (b) of Figure 5, P(VDF-TrFE-CFE) with 25 wt% partially grafted CuPc (the particle size is ca. 60-100nm) has a dielectric constant of only 175 at 100Hz. [9] Although the dielectric constant of PVDF (8.4) is much less than that of P(VDF-TrFE-CFE) (~40), the dielectric constant of PVDF/PCMS-g-CuPc is nearly one time higher than that of the P(VDF-TrFE-CFE) based composite. In consequence, the dramatic enhancement of dielectric response observed in PVDF/PCMS-g-CuPc is probably caused by the strong interface exchange coupling effect between the CuPc nanoparticles and the polymer matrix, as well as the MWS interface effect, due to the much smaller CuPc particle size as observed in TEM micrographs.[9,14,31,32] Figure 5 also demonstrates that the dielectric losses of composites are relatively low. CuPc suffers a high dielectric loss due to the long-range intermolecular hopping of electrons.[16,51] In composites, Compared with CuPc, the polymer matrix has very low dielectric loss, and as the matrix, polymer acts as insulation layers to significantly reduce the dielectric loss of CuPc. Over the frequency range observed, the loss of the PVDF/PCMS-g-CuPc (about 0.10 at 100Hz) is lower than that of the PVDF/CuPc which can be attributed to the reduced particle size and improved dispersibility of CuPc in PVDF/PCMS-gCuPc. The dielectric absorption of the composites with a maximum near 1 MHz is a simple relaxation process, as shown in Figure 6, which can be fitted quite well with the modified Cole–Cole equation:[52]

K * = K∞ +

ΔK 1 + (iωτ )1−α

= K '−iK " (6)

where K∞ is the dielectric constant at the high frequency limit, ΔK (=Ks-K∞) is the dielectric relaxation strength, Ks is the static dielectric constant, τ is the


163

characteristic relaxation time, and α is the parameter describing the distribution of relaxation time. Fitting the data in Figure 6 using the Equation 6 yields K∞=8.5, ΔK=31.3, α=0.27, τ=0.93μs for the PVDF/CuPc, and K∞=73.1, ΔK=191.9, α=0.30, τ=0.54μs for the PVDF/PCMS-g-CuPc, which reveals that the PVDF/PCMS-g-CuPc has a dielectric constant more than 8 times that of PVDF/CuPc at the high frequency limit. Moreover, the relaxation time obtained in the PVDF/PCMS-g-CuPc is much smaller than that obtained in the PVDF/CuPc. That is, the PVDF/PCMS-g-CuPc sample can exhibit a higher dielectric constant at a higher frequency than the PVDF/CuPc, which is consistent with the fitting results above.

Figure 6. Cole-Cole plot of the dielectric behaviors at room temperature for the PVDF/PCMS-g-CuPc and the PVDF/CuPc. The solid lines are the fitted results using the modified Cole–Cole equation.

Conclusion In summary, a novel strategy to fabricate high dielectric constant PVDF/PCMS-g-CuPc nanocomposite using PVDF as matrix and CuPc grafted onto PCMS backbone as filler was introduced. The size of the CuPc nanoparticles

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within the nanocomposite is ca. 25nm, representing a 20 times decrease in comparison with that of CuPc particles in the simple physical blend of PVDF and CuPc. The nanocomposite exhibits a high dielectric constant (325 at 100Hz, nearly 40 times that of the pure PVDF, and about sevenfold enhancement with respect to that of the physical blend of PVDF and CuPc), low loss (0.10 at 100Hz), all of which are highly desirable for high dielectric constant composites. Theoretic prediction using modified Cole-Cole equation indicates that, even at the high frequency limit, the dielectric constant of the nanocomposite can still be larger than 70, more than 8 times that of the PVDF/CuPc. Further property improvement can be expected through the amelioration in the nanocomposite fabrication process, especially by using high dielectric constant organic fillers with further reduced particle size as well as increasing the distribution uniformity of the filler particles in polymer matrix.

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Chapter 6

POLYMER MATRIX COMPOSITES: PROPERTIES, PERFORMANCE AND APPLICATIONS Milan Kracalik, Stephan Laske and Clemens Holzer Institute of Plastics Processing, University of Leoben, Leoben, Austria

Abstract Preparation of polymer-clay nanocomposites by melt mixing has been intensively investigated in the last 20 years. However, only systems based on polyamide matrix have been commercialized. The problem consists mostly in question how to up-scale the results obtained using laboratory equipment. Most of research institutions dealing with nanocomposites compounding are equipped by "micro" twin screw extruders, which are able to process only small amount (in order of grams per hour) of material. In the industry, compounding throughput in order of kilograms up to tonnes per hour is usual. However, industrial compounders are normally not available for research purpose. Furthermore, the broad range of industrial compounders (differences in screw geometry, screw diameter, L/D relation etc.) makes any general conclusions impossible. In this contribution, approach for systematic investigation of polymer nanocomposites is presented. A way from "micro" to "advanced" compounding using laboratory as well as semiindustrial twin screw extruders is described.

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1. Introduction Polymer-clay nanocomposites have become a new class of materials since Toyota patented first successful results based on polyamide and montmorillonite [1]. Since then, academic as well as industrial institutions worldwide have investigated this research field, in order to use the potential of layered silicates in further polymer matrices. Conventional polymer composites are based on reinforcement of the polymer matrix by micrometer scale particles. For example, isotactic polypropylene filled with talc, micro-ground calcium carbonate, wood powder, possibly with other suitable fillers, or epoxy and polyester resins filled with mineral particles offer an advantageous combination of mechanical properties and price. However, polymeric materials reinforced by nanoscale particles exhibit significantly higher performance, for example substantial increase in stiffness and decrease in gas and liquid penetration already for small amounts of the filler [2]. This advantage comes from immobilisation of polymer chains in close contact with layered inorganic filler possessing a large surface area. Moreover, uniaxial arrangement of inorganic platelets remarkably reduces gas and liquid permeability in perpendicular direction. Polymer nanotechnology involves not only incorporation of nanosized particles into the polymer but, more importantly, investigation of interactions between the polymer matrix and the enormously large nanofiller surface [3]. Especially for polymer/clay nanocomposites, the surface effects are responsible for improvement of barrier, mechanical and rheological properties, dimensional stability, heat-, flame- and oxidative-resistance. In comparison with traditional fillers (20−40 wt. % loading), 2−5 wt. % filling of layered clays is sufficient to achieve analogous material improvement [2, 4]. For preparation of polymer nanocomposites, layered silicates (especially montmorillonite, MMT) have been the most used nanofillers hitherto due to the opportunity to achieve aspect ratios ideally up to 1000 (by silicate platelet thickness of 1 nm). To achieve good adhesion with a hydrophobic polymer, chemical modification of MMT is required. Modification of natural Na+ type of MMT proceeds generally by organophilisation based on the ion-exchange method, where the natrium cation is replaced by quaternary ammonium salt. Such organically modified MMT is usually referred to as organoclay. According to the dispersion of MMT platelets in the polymer matrix, three composite structures can be formed: a) conventional composites, b) intercalated nanocomposites,

Polymer Matrix Composites

171

c) exfoliated nanocomposites. In the first case, the MMT particles (tactoids) are dispersed in the polymer matrix in micrometer scale with the tactoids acting as a micro-filler. On the other hand, intercalated (partially delaminated) systems show penetration of polymer chains into interlayer gallery of silicate platelets. Exfoliated (entirely delaminated) nanocomposite is characterized by homogeneous and uniform dispersion of silicate layers in the polymer. Polymer/clay nanocomposites can generally be prepared by three methods [5]: a) mixing during polymerisation („in situ“), b) mixing in solvent, c) melt mixing. Additionally, possibilities of nanoparticle dispersion by application of electric fields [6], by ultrasonic mixing [7] or direct chemical bonding of polymer chains onto the surface of silicate platelets [8–10] have been studied. Melt mixing (compounding) is the industrially most attractive method due to technological simplicity (usage of common polymer processing machines in contrast to special equipment and procedures in chemical laboratories), flexibility and environmental friendliness. Moreover, it is possible to use various polymers as a matrix (different molecular weight, branching degree, copolymers, etc.) including recycled polymers [11-13]. The production efficiency of this method is considerably higher than that of the others (“melt mixing” proceeds in the order of minutes, as compared to several hours characteristic for “in-situ” and “solution” methods) [14,15]. The principle of the melt mixing method consists in delamination of silicate platelets in the polymer melt by shear forces (in the extruder or kneader) and thermodynamical interactions between polymer chains and organoclay (the affinity between clay and polymer is usually increased by modification of silicate with organic compounds, in the case of polyolefine matrix also a compatibilizer is admixed). During compounding, penetration of polymer chains into the silicate gallery (intercalation) facilitates delamination of individual platelets, resulting in better dispersion of silicate layers in the polymer matrix [16]. The whole process has to be controlled in order to prevent degradation of the polymer or the organic part of organoclay (by high shear forces, temperature, etc.). Using the co-rotating twin-screw extruder or a continuous kneader (e.g. Buss kneader) as a continuous processing way is industrially preferred to melt mixing in a discontinous kneader. It is obvious that for the successful dispersion of silicate plates in polymer melt by continuous processing the following two

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requirements have to be fulfilled: sufficient shear energy and enough residence time. However, these two effects act against each other. With higher shear forces (e.g. usage of kneading blocks generating higher shear rate or increase in screw speed) the residence time is shortened. To our knowledge, we presented for the first time [17] that both, high shear rate as well as longer residence time can be matched by implementation of a melt pump in front of an open twin-screw extruder. The melt pump determines the throughput rate; the screw speed controls the shear intensity. In this contribution, selected results of nanocomposites preparation in labor and semi-industrial scale compounding are summarized. The systems investigated are based on layered silicates and hydrophilic (PET) as well as hydrophobic (PP) polymer matrix. It is shown that systematic investigation of the processing conditions (e.g. processing at the boarder of melt temperature) results in significant effect on material use properties (enhancement vs. deterioration). "Micro-compounder" has been successfully employed as a powerfull tool for determination of most suitable clay surface modification used for recycled PET enhancement. "Advanced-compounding" using a melt pump as a new opportunity for industrial production is discussed for PP nanocomposites. For development of new materials based on polymer nanocomposites it is advantageous to apply both technologies in order to optimize mixture composition (e.g. clay organophilization, polymer matrix) as well as processing conditions (e.g. screw geometry, screw speed) to reach defined material structure and properties.

2. Micro-Compounding of Recycled PET-Clay Nanocomposites 2.1. Materials Organo-modified clays were supplied by Southern Clay Products, Inc., Gonzales, TX, U.S.A. Recycled poly(ethylene terephthalate) from colour-selected beverage bottles (PET-R), with the intrinsic viscosity 0.73 dl/g was delivered by Polymer Institute Brno, Ltd., Czech Republic.

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Table 1. Characteristics of organoclay fillers a

Cloisite 6A

2M2HT

Modifier concentration (mequiv/100 g clay) 140

0 of the number n of cycles. 2. The functions α = α (n, Ri ), α = α (n, Rj ) for Ri 6= Rj , representing a relationship between the crack angle α and the variable n at the parameter R, are significantly different from each other within the same ceramic material, and accordingly a mutual relationship regarding Ri 6= Rj is not assumed to be defined.

Contact Strength of Ceramic Materials

a

237

b

Figure 28. SEM micrographs of a detail of cracks in MoSi2 , where chemical composition of the particles in the points A, B (a), C, D (b) and of the matrix is determined by the EDX analysis shown in Fig. 29. 3. A shape of cone cracks to exhibit a perpendicular course below a contact surface, consequently followed by a linear course with the angle ∠, corresponds to that presented in literature, where an increase of the contact loading results in an increase of a diameter of the cone cracks characterized by predominantly inter-granular propagation. 4. In contrast to the cone cracks in Si3 N4 (SN I, II) and SiC, the MoSi2 based ceramic materials exhibit median cracks perpendicular to a contact surface, formed from the contact surface to the centre of a specimen and vice versa. The initiation and the propagation of the median crack is assumed to result from the existence of higher density of pores. 5.2.4.

Multi-cycle Contact Test Using Spheres and Four-Point Bending Test

Fig. 32 shows surface optical micrographs of indentation damage sites in the Si3 N4 (SN I) material at the multi-cycle contact test using spheres along with an indentation damage site at the single-cycle contact test using spheres (see 32a). As also presented in [18], a transformation in the damage mode from a simple well-defined cone crack located outside a contact surface of the single-cycle contact test (see 32a) to the accumulated damage below the surface contact with incipient removal of a material in the multi-cycle contact test is observed. In

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Lucia Heged˝usová, Ladislav Ceniga and Ján Dusza

a

b

c

d

e Figure 29. The EDX analysis of the particles in the points A (a), B (b), C (c), D (d) and of the matrix (e) in Fig. 28.


239

Figure 30. SEM micrographs of a detail of a crack in MoSi2 +10%SiC, where chemical composition of the particle in the points A and of the matrix is determined by the EDX analysis shown in Fig. 31.

a

b

Figure 31. The EDX analysis of particles in the points A (a) and of the matrix (b) in Fig. 30.

contrast to the results in [18], attendant radial cracks are not present. Fig. 33 shows the strength σ bend as a function of the number n of cycles, representing a mean value of those (see Eq. (7)) determined by the four-point bending test applied after the multi-cycle contact test using, where σ bend for

240


n = 0 represents the characteristic strength σ0,bend determined by the fourpoint bending test without an application of the multi-cycle contact test using spheres, i.e. (σ bend )n=0 = σ0,bend . Solid curves are empirical fits through the data. Figs. 34 and 35 show fracture surfaces of Si3 N4 (SN I), SiC, and Al2 O3 , respectively, exhibiting the presence of processing flaws located at an edge of a specimen (see Fig. 34, 35), on a surface of a specimen (see Fig. 34), below a surface of a specimen (see Fig. 34, 35). With regard to lower values of n, i.e. n = 102 , 103 , the specimen failure caused by the four-point bending test applied after the multi-cycle contact test using spheres might be considered to be a consequence of the presence of the processing flaws, in contrast to the specimen failure at higher values of n, i.e. n = 104 , 105 , assumed to be a combination of an influence of cone cracks and the processing flaws. Similar to the analysis in Section 5.2.2. concerning the quasi-plastic character of a strain-stress state [13], the MoSi2 +10%SiC ceramic material without the presence of cone cracks exhibits median cracks perpendicular to a contact surface, and accordingly the specimen failure is assumed to be a consequence of higher density of the processing flaws in a form of pores and clusters of pores located on and below a surface of a specimen as well as in volume of a specimen (see Fig. 36). Conclusions. Results of the multi-cycle contact test using spheres followed by the four-point bending test are as follows: 1. The transformation in a damage mode from a simple well-defined cone crack located outside a contact surface of the single-cycle contact test (see Fig. 32a) to accumulated damage below the surface contact with incipient removal of a material in the multi-cycle contact test is observed. In contrast to the results in [18], attendant radial cracks are not present. 2. Fracture surfaces of Si3 N4 (SN I), SiC, and Al2 O3 exhibit the presence of processing flaws located at at an edge of a specimen, on a surface of a specimen, and below a surface of a specimen. With regard to lower values of n, i.e. n = 102 , 103 , the specimen failure might be considered to be a consequence of the presence of the processing flaws. With regard to lower values of n, i.e. n = 104 , 105 , the specimen failure is assumed to be a combination of an influence of cone cracks and the processing flaws.


241

a

b

c

d

e Figure 32. Optical micrographs of indentation damage sites for the single-cycle contact test using WC spheres (a) with the radius R = 2.5 mm, at P = 3 kN, and for the multi-cycle contact test using WC spheres with the radius R = 2.5 mm, at Pmin = 60 N, P = 3 kN, n = 102 (b), n = 103 (c), n = 104 (d), n = 105 (e), f = 10 Hz.

242


a

b

c

d

Figure 33. The strength σ bend of Si3 N4 (SN I) (a), SiC (b), Al2 O3 (c) and MoSi2 +10%SiC (d) as a function of the number of cycles n, representing a mean value of those determined by the four-point bending test applied after the multi-cycle contact test using WC spheres with the radius R = 2.5 mm at Pmin = 60 N, P = 3 kN (a), P = 2 kN (b,c), P = 1 kN (d), f = 10 Hz, where the dependence σ bend = σ bend (n) in Fig. 33c is related to three different values of the radius R. 3. Fracture surfaces of MoSi2 +10%SiC exhibit the processing flaws in a form of pores and clusters of pores located on and bellow a surface of a specimen as well as in volume of a specimen. Finally, the specimen failure is assumed to be caused by median cracks perpendicular to a contact surface without the presence of cone cracks.


a

243

b

Figure 34. Fracture surfaces of Si3 N4 (SN I) (a) and SiC (b) after the multi-cycle contact test using WC spheres with the radius R = 2.5 mm at Pmin = 60 N, P = 3 kN, n = 105 , f = 10 Hz (a), and Pmin = 60 N, P = 2 kN, n = 104 , f = 10 Hz (b), followed by the four-point bending test, where both tests were performed in air.

a

b

Figure 35. Fracture surface of Al2 O3 after the multi-cycle contact test using WC spheres with the radius R = 2.5 mm, at Pmin = 60 N, P = 2 kN, n = 104 (a), n = 105 (b), f = 10 Hz, followed by the four-point bending test, where both tests were performed in air. 5.2.5.

Contact Fatigue Test

Fig. 37a shows the critical load Pc to cause the specimen failure at the number n of cycles, and each of the values Pc = 2, 3, 4 kN is related to a relatively

244


a

b

Figure 36. Fracture surface of MoSi2 +10%SiC after the multi-cycle contact test using WC spheres with the radius R = 2.5 mm at Pmin = 60 N, P = 1 kN, n = 104 (a), n = 105 (b), f = 10 Hz, followed by the four-point bending test, where both tests were performed in air.

a

b

Figure 37. The relationship between the critical load Pc (a) to cause the specimen failure of Al2 O3 at the number n of cycles for the radius R = 3.5 mm of WC spheres, and for Pmin = 60 N, f = 10 Hz. The dependence Pc − n (a) is transformed the dependence lg Pc − lg n (b), where the slope d (lg Pc ) /d (lg n) = −3.12 was determined by fitting the data, considering a mean value of the number n of cycles related to each of the values P = 2, 3, 4 kN.


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narrow range of n as also presented in [14], where the relatively narrow range is in strong contrast to lifetime results obtained from cyclic bending test applied to Al2 O3 [14]. Additionally, the dependence Pc − n in Fig. 37a is transformed the dependence lg Pc − lg n in to Fig. 37b, where the slope d (lg Pc ) /d (lg n) = −3.12 in Fig. 37b is also in contrast to results obtained from a cyclic bending test for Al2 O3 [14]. The slope d (lg Pc ) /d (lg n) = −3.12 was determined by fitting the data, considering a mean value of the number n of cycles related to each of the values P = 2, 3, 4 kN. Finally, as mentioned in Section 4.2., fracture surfaces due to the contact fatigue test using spheres to be applied to Al2 O3 could not be identified due to the destruction of the fracture surface to pieces.

Acknowledgments This work was supported by the Slovak Research and Development Agency under the contracts No. APVV-0034-07, No. COST-0042-06, No. COST-002206, No. APVV-51-061505, No. APVV-0171-06; by the 6th FP EU NESPA; by the Slovak Grant Agency VEGA (2/0156/10, 2/0120/10); by NANOSMART Centre of Excellence (01/2007–12/2010) Slovak Academy of Sciences; by HANCOC: MNT.ERA-NET 01/2009-12/2011.

References [1] Skoˇcovský, P.; Bok˚uvka, O.; Palˇcek, P. Materials Science; EDIS Technical ˇ University: Zilina, SK, 1996. [2] Rektorys, K. Review of Applied Mathematics; SNTL: Prague, CZ 1973. [3] Fett, T.; Ernst, E.; Munz, D.; Badenheim, D.; Oberacker, R. J. Eur. Ceram. Soc. 2003, 23, 2031–2037. [4] Fett, T.; Munz, D. Eng. Fract. Mech. 2002, 69, 1353–1361. [5] Fett, T.; Munz, D.; Thun, G. Strength and Toughness Test Devices with Opposite Roller Loading. Report FZKA 6378; Forschungszentrum: Karlsruhe, Germany, 2000. [6] http://rsbweb.nih.gov/ij

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[7] Zimovˇca´ k, P.; Köves, T.; Dusza, J., Peˇsek, L.; Chalvet, F.; De Portu, G. Key Eng. Mater. 2005, 290, 316–319. [8] Alfredsson, B.; Olsson, M.. Fatigue Fract. Eng. Mater. Struct. 1999, 22, 225–237. [9] Fett, T.; Ernst, E.; Munz, D. J. Mater. Sci. Lett. 2002, 21, 1955–1957. [10] Ceniga, L. Analytical Models of Thermal Stresses in Composite Materials I; Nova Science Publishers: New York, USA, 2008. [11] Ceniga, L. In Ceramics and Composite Materials: New Research; Caruta, B.M.; Ed.; Nova Science Publishers: New York, USA, 2005; pp. 147-195. [12] Lawn, B. R.; Lee, S. K.; Peterson, I. M.; Wuttiphan, S. J. Am. Ceram. Soc. 1998, 81, 1509–1520. [13] Lee, S. K.; Lawn, B. R. J. Am. Ceram. Soc. 1998, 81, 997–1003. [14] Fett, T.; Ernst, E.; Rizzi, G.; Munz, D.; Badenheim, D.; Oberacker, R. Fatigue Fract. Eng. Mater. Struct. 2006, 29, 876–886. [15] Zeng, K.; Breeder, K.; Rowcliffe, D. J. Acta Metall. Mater. 1992, 40, 2595–2600. [16] Zeng, K.; Breeder, K.; Rowcliffe, D. J. Acta Metall. Mater. 1992, 40, 2601–2605. [17] Hu, S.; Chen, Z.; Mecholsky, J. J. Int. J. Fract. 1996, 79, 295–307. [18] Akimune, Y.; Katano, Y.; Matoba, K. J. Am. Ceram. Soc. 1989, 72, 1422– 1428.

In: Ceramic and Polymer Matrix Composites ISBN 978-1-60741-896-2 c 2010 Nova Science Publishers, Inc. Editors: E. Dimitriou et al, pp. 247-296

Chapter 8

N EW A NALYTICAL M ODEL OF T HERMAL S TRESSES AND A NALYTICAL F RACTURE M ECHANICS IN T WO -C OMPONENT M ATERIALS . A PPLICATION TO T WO -C OMPONENT C ERAMICS Ladislav Ceniga∗ Institute of Materials Research, Slovak Academy of Sciences, Watsonova 47, 040 01 Koˇsice, Slovak Republic

Abstract The chapter deals with a new analytical model of thermal stresses in an isotropic continuum represented by periodically distributed isotropic spherical particles in an isotropic infinite matrix. The multi-particlematrix system to represent a model system regarding the analytical modelling is applicable to a two-component material of a precipitate-matrix type, consisted of isotropic components. The thermal stresses as functions of microstructural parameters (particle volume fraction, particle radius, inter-particle distance) originate during a cooling process as a consequence of the difference in thermal expansion coefficients. Additionally, integrals of elastic energy density along curves in the spherical particles and the matrix are derived to result in the determination of a critical particle radius as a consequence of the initiation of cracks in the components ∗

E-mail addresses: [email protected], [email protected]

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Ladislav Ceniga of any two-component material (ductile, brittle) along with the determination of functions describing the crack shapes in a plane perpendicular to a direction of the crack propagation in an ideal-brittle particle and an ideal-brittle matrix. With regard to the ideal-brittle particle and the idealbrittle matrix resulting in an application of the presented results to ceramic (brittle) components, the new analytical model, and consequently an analysis of the crack initiation and the crack propagation are applied to the SiC-Si3 N4 and SiC-MoSi2 multi-particle-matrix systems representing two-component ceramic materials of the precipitate-matrix type.

1.

Introduction

Investigated usually by approximate computational [1]– [3] and experimental methods [4]– [8], thermal stresses represent an important phenomenon observed in materials. With regard to material science, thermal stresses originate in a composite material as a consequence of the difference in thermal expansion coefficients between individual material components, consequently influencing mechanical properties, diffusion processes in materials. Consisted of isotropic components, a real two-component material of a precipitate-matrix type with finite dimensions defined in Section 1.2. is replaced by an isotropic multi-particle-matrix system consisted of periodically distributed spherical particles embedded in an infinite matrix, respectively. Consequently, the multiparticle-matrix system, as a model system regarding analytical modelling, is imaginarily divided into cells with such shape to correspond to particle distribution, and the thermal stresses are investigated in the cell containing a central spherical particle.

1.1.

“Integration” and “Differential” Approaches

Analytical models of stresses, including the thermal stresses, in an inclusion (=precipitate)-matrix system are usually determined using Eshelby’s and MoriTanaka concepts [9]– [12] resulting from mathematical techniques belonging more or less to Theoretical Physics, represented by an “integration” approach to use the Green’s functions, ordinary Newtonian potential and biharmonic potential [9], applied to a two-component system consisting of an isotropic particle and an isotropic matrix, where the particle is assumed to be acted by a homogeneous (average) stress inducing accordingly a homogeneous strain, and an average value of a fluctuating stress in the matrix around the particle is equal

Thermal Stresses and Analytical Fracture Mechanics

249

to zero [12]. A similar “integration” approach is presented in the paper [13] to result in a partially numerical solution, using the boundary integral equation approach known as the boundary element method (BEM), as well as in [14] to consider the Green’s functions. Analytical models and consequently their applications presented in [9]– [12] describes the disturbance of an existing applied-stress field to be uniform in the infinity, in contrast to the Mizutani’s and Ceniga’s analytical cell models [15,16] to describe a thermal-stress field non-existing in the absence of the inclusion. The analytical cell models in [15, 16] to result from a “differential” approach consider isotropic multi-particle-matrix system as a model system represented by an isotropic infinite matrix with periodically distributed isotropic spherical particles with the radius R. The multi-particle(=precipitate)-matrix system of infinite dimensions replaces, with regard to analytical modelling of a state of deformation and stress, a real two-component material of a precipitate-matrix type with finite dimensions and aperiodically distributed precipitates, where the matrix consists of grains, neglecting an influence of grain boundaries as usual in Continuum Mechanics [17]. The thermal stresses in the multi-particle-matrix system are investigated within the spherical particle and a cell matrix. The cell matrix, representing the part of the infinite matrix to be related to one spherical particle, results from imaginary division of the infinite matrix into identical cells, and derived formulae are accordingly valid for any cell of the infinite matrix. Consequently, the cell matrix containing a central spherical particle both determine the particle volume fraction v ∈ (0, vmax ). The “differential” approach presented in the Ceniga’s analytical cell models [15, 18]– [21] and in this chapter considers suitable substitutions of the Cauchy’s equations, equilibrium equations, and the Hooke’s law, where the Cauchy’s and equilibrium equations are related to an infinitesimal spherical cap. With regard to cite15,18– [21] and this chapter, the substitutions result in a differential equation for the radial stress and displacement, σ11 and u1 solved by well-known mathematical techniques, and consequently, the thermal stresses are functions of σ11 and u1 , respectively. The Mizutani’s analytical model [16] considers spherical cells and a condition of zero radial stress on the cell surface. However, the matrix between three neighbouring spherical cells is replaced by the vacuum what is physically unacceptable. As an improvement of the Mizutani’s analytical model, the Ceniga’s works [15, 18] considers derived formula pc = pc (p) for the radial stress pc

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Ladislav Ceniga

on the cell surface and the condition ∂W/∂pc = 0 for the determination of the radial stress p acting on the particle-matrix boundary, corresponding to the extreme of the function W = Wp (p) + Wm (p) with the variable p, where W , Wp (p) and Wm (p) represent thermal-stress induced elastic energy in the cell, the spherical particle and the cell matrix, respectively. In addition to the achieved improvement, the mentioned work, dealing with such cells to fill the infinite matrix perfectly, presents an analytical model of the thermal stresses in a multi-particle-matrix system divided into cubic cells, considering the extremal condition and neglecting an influence of shear stresses and strains. In contrast to the Mizutani’s analytical model, formulae for the thermal stresses of the analytical model related to the multi-particle-matrix system [15, 18] are transformed for v = 0 to those related to the one-particle-matrix system [22]– [24], and accordingly the transformation condition results in the neglect. Finally, mathematical techniques to derive the thermal stresses in the isotropic multi-particle-matrix system with the perfect-filled infinite matrix lead to two mutually incompatible analytical models. On the one hand, formulae for the thermal stresses related to the isotropic multi-particle-matrix system as presented in [15, 18] are transformable on the condition v = 0 to those related to the isotropic one-particle-matrix system as presented in [22]– [24], where v is particle volume fraction. On the other hand, the condition v = 0 results in the neglect of an influence of shear thermal stresses, and additionally a geometrical condition, represented by zero radial displacement of a matrix on the cell surface, is not considered within the first analytical model. Finally, this chapter deals with the analytical model to consider the geometrical condition as well as an influence of the shear thermal stresses. Consequently, the mathematical techniques resulting from the “differential” approach for the determination of the thermal stresses in the isotropic multiparticle-matrix system with the perfect-filled infinite matrix lead to two mutually incompatible analytical models. On the one hand, formulae for the thermal stresses related to the isotropic multi-particle-matrix system as presented in [15, 18] are transformable on the condition v = 0 to those related to the isotropic one-particle-matrix system as presented in [22]– [24], where v is particle volume fraction. On the other hand, the condition v = 0 results in the neglect of an influence of shear thermal stresses, and additionally a geometrical condition, represented by zero radial displacement of a matrix on the cell surface, is not considered within the first analytical model. Finally, this chapter deals with an analytical model to consider the geometrical condition as well as


251

an influence of the shear thermal stresses.

1.2. Real Two-Component Material The analytical model of thermal stresses presented in this chapter is applicable to a real two-component material of a precipitate-matrix type represented by grains and precipitates, both exhibiting mutually different properties. With regard to the analytical models of the thermal stresses, the real twocomponent material with finite dimensions is replaced by a multi-particlematrix system with infinite dimensions, consisted of components represented by periodically distributed spherical particles embedded in an infinite matrix. To derive the thermal stresses acting in the multi-particle-matrix system, the infinite matrix is imaginarily divided into identical cells with the shape to correspond to particles distribution, and consequently the thermal stresses are investigated within the cell representing a part of the solid continuum related to one spherical particle. Considering the matrix infinity, formulae for thermal stresses related to a certain cell are consequently valid for any cell in the infinite matrix (see Fig. 1).

Figure 1. The infinite matrix divided into the cubic cells with the central spherical particles with the radius R in the point O of the Cartesian system (Ox1 x2 x3 ), and the inter-particle distance d along the axis xi (i = 1,2,3).

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Ladislav Ceniga

As presented in Section 1.2., the precipitates and the grains embedding the precipitates are considered to represent the periodically distributed spherical particles and the infinite matrix, respectively. Finally, boundaries of the grains representing the infinite matrix, and consequently the cell matrix, are considered to represent a coherent solid continuum. The thermal stresses investigated in this chapter are thus consequence of the difference αm − αp 6= 0 between the thermal expansion coefficients of the matrix and the particle, αm and αp , respectively, both included in the coefficient βq (see Eq. (60)).

1.3.

Mathematical Techniques

In contrast to the “integration” approach, the “differential” approach presented in [15, 18]– [21] and in this chapter considers an infinitesimal part in an arbitrary point within which a state of deformation and stress is investigated using a system of suitable coordinates, as usual in Continuum Mechanics [17]. Strictly speaking, an infinitesimal spherical cap in the point P with a position determined by the spatial polar coordinates [r, ϕ, ν] (see Fig. 1) are considered, where r = OP . The determination of a state of deformation and stress results from the Cauchy’s, and equilibrium equations, and from the Hooke’s laws, representing fundamental equations of Continuum Mechanics (see Eqs. (23)–(34)) [17]. Accordingly, the Cauchy’s equations (see Eqs. (23)– (26)), representing relationships between the strain εijq (i, j = 1,2,3) and the radial displacement u1q along with the derivations ∂u1q /∂ϕ, ∂u1q /∂ν, are substituted to the Hooke’s laws (see Eqs. (30)–(34)), and the stress σijq is consequently derived as a function of u1q , ∂u1q /∂ϕ, ∂u1q /∂ν (see Eq. (27)). The dependence σijq = σijq (u1q , ∂u1q /∂ϕ, ∂u1q /∂ν) is substituted to the equilibrium equations (see Eqs. (27)–(29)), representing relationships between the radial and tangential and shear stresses, σ11q and σ22q , σ33q and σ12q , σ13q , respectively. Finally, the equilibrium equations are accordingly transformed to a system of differential equations (see Eqs. (50)–(52)), representing a relationship between the derivations ∂ 3 u1q /∂r∂ϕ2 , ∂ 3 u1q /∂r∂ν 2 , ∂ 2 u1q /∂ϕ2 , ∂ 2 u1q /∂ν 2 (see Eq. (50)), and a relationship between u1q , ∂u1q /∂r, ∂ 2 u1q /∂r2 , ∂ 2 u1q /∂ϕ2 , ∂ 2 u1q /∂ν 2 (see Eq. (51)). Consequently, suitable mathematical techniques applied to the differential equations (50), (51) lead to a linear differential equations with a non-zero right side (see Eq. (57)) solved by the Wronskian’s method [25] to result in a solution


253

for u1q (see Eq. (58)) suitable for the determination of the thermal stresses in the spherical particle (q = p) and the cell matrix (q = m), considering the relationship σijq = σijq (u1q , ∂u1q /∂ϕ, ∂u1q /∂ν) and the boundary conditions defined in Section 4.. With regard to a mathematical point of view, the “differential” approach presented in this chapter can be assumed to be relatively simple in comparison to the “integration” approach belonging more or less to Theoretical Physics, and accordingly, derived formulae, as functions of microstructural parameters represented by the particle volume fraction v and the particle radius R, can be easily used by material scientists for the determination of a state of stress and deformation. Additionally, the analytical models of the thermal stresses presented in this chapter are transformable to analytical models of stresses originating in an isotropic crystalline lattice with a central substitutive atom (see Section 6.). Finally, the “differential” approach presented in this chapter can be assumed to compete with the “integration” approach developed by Eshelby [9]. Additionally, although the latest papers [26, 27] to result from [9] are devoted to a cell model, theoretical formulation and applications to homogenization both consider spherical cells (domain). Consequently, a state of deformation and stress in a matrix between three spherical cells is not considered what is physically unacceptable as similar to the Mizutani’s analytical model [16].

1.4.

Analytical Fracture Mechanics

Analytical fracture mechanics to result from the thermal-stress induced ’curve’ elastic energy density Wcq (see Section 2.1.) as an integral of the thermal-stress induced elastic energy density wq over a suitable curve in the spherical particle (q = p) and matrix (q = m) with the thermal expansion coefficients αp and αm , respectively, includes 1. the determination of the critical particle radius Rqc as a consequence of the crack initiation in the spherical particle (q = p) for αm < αp or in the cell matrix (q = m) for αm > αp , where the crack initiation is related to different directions of the crack propagation (see Figs. 8, 9), and formulae for Rqc = Rqc (v) representing functions of v ∈ (0, π/6i are valid for any two-component material (ductile, brittle) of a precipitate-matrix type, 2. an analysis concerning the different directions of the crack initiation (see Section 6.3.), where the analysis is valid for any two-component material

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Ladislav Ceniga (ductile, brittle) of a precipitate-matrix type,

3. the determination of the function fp or fm to consider the analysis in Item 2, describing a shape of the crack propagated in the spherical particle or in the cell matrix for the particle radius R > Rqc regarding αm < αp or αm > αp (see Eqs. (130), (131), (133)–(135)), respectively, where formulae for fp , fm are applicable to ceramic two-component material of a precipitate-matrix type to consist of ceramic particles and a ceramic matrix both generally considered to be ideal-brittle. Neglecting a redistribution of the thermal stresses during high-speed crack propagation, the crack propagation analysed in this chapter is accordingly related to an ideal-brittle spherical particle and an ideal-brittle matrix. Additionally, the analytical modelling of the crack formation to include the crack initiation and propagation in a two-component material of a precipitatematrix type, the latter related to ideal-brittle spherical particle and matrix, exhibits general applicability as explained in the following items: 4. With regard to the crack initiation, on the one hand, the determination of the critical particle radius Rqc results from the conditions (128), (136) to consider Wcq as a function of wq (q = p,m) (see Eq. (97)) derived by the analytical model related to elastic strains induced by thermal stresses presented in Section 5.. On the other hand, wq and consequently Wcq can be also derived by various analytical models related to elastic, elasticplastic, or plastic strains. 5. With regard to the high-speed crack propagation, the analytical modelling in Section 6.4. can also result from wq and consequently Wcq derived by various analytical models related to elastic, elastic-plastic, or plastic strains of elastic or elasto-plastic stresses. Finally, as mentioned above, the redistribution of stresses, derived by the various analytical models, during the high-speed crack propagation ideal-brittle spherical particle and matrix is required to be neglected. The engineering background and the practical importance of the scientific results presented in this chapter are as follows:


255

6. With regard to Item 1, the determination of the critical particle radius Rqc = Rqc (v) to define accordingly a limit state regarding the crack initiation in any two-component material (ductile, brittle) of a precipitationmatrix type is helpful for material scientists interested in material processing. Substituting numerical values of parameters of a real two-component material of a precipitate-matrix type, i.e. the particle volume fraction v ∈ (0, π/6i, the relaxation temperature Tr and the final temperature Tf of a cooling process, the Young’s modulus Eq , the Poisson’s ratio µq , the thermal expansion coefficients αq for the spherical particle (q = p) and the matrix (q = m), the material scientists are able to determine a numerical value of Rqc = Rqc (v) dependent on the microstructural parameter v ∈ (0, π/6i (see Eq. (1)), and consequently suggest such parameters of a heat treatment process (htp), e.g. a protective atmosphere, the temperature T , the holding time τ , resulting in the microstructural parameters Rhtp , vhtp to be required to fulfil the condition Rhtp < Rqc (v) for v = vhtp . 7. With regard to Item 2, materials scientists interested in analytical fracture mechanics of any two-component materials (ductile, brittle) of a precipitate-matrix type are able to analytically predict the different directions of the crack propagation in the spherical particle or in the cell matrix for αm < αp or αm > αp , respectively, and consequently verify the analytical prediction of the different directions with those of the crack propagation to arise in the real two-component materials. 8. Similarly, with regard to Item 3, materials scientists interested in the analytical fracture mechanics of ceramic two-component materials of a precipitate-matrix type are able to analytically determine the function fp or fm to consider the analysis in Item 2, and consequently verify the function fp or fm determined analytically with the function fpia or fmia describing a shape of the crack propagated in the ceramic spherical particle or in the ceramic cell matrix regarding αm < αp or αm > αp , respectively, where fpia , fmia can be determined by an image analysis within microscopical techniques applied to the real ceramic two-component materials. 9. In general, the crack initiation and the crack propagation both represent an usual topic of the computational modelling to include the finite element analysis (FEA) resulting in numerical values of the quantities defined in

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Ladislav Ceniga Items 1–3. Consequently, scientists interested in the computational modelling are able to compare the numerical values of FEA with those of the analytical modelling presented in this chapter (see Sections 6.3., 6.4.).

10. Finally, representing an analytical-computational approach to the modelling of the crack initiation and the crack propagation, a combination of FEA results with analytical ones can be considered, i.e. numerical values of the elastic energy density wq (q = p,m) derived by FEA can be substituted to the formulae in Sections 6.3., 6.4..

2.

Cell Model. Geometric Boundary Condition for Cell Matrix

With regard to the periodical distribution of the spherical particles with the radius R shown in Fig. 1, the infinite matrix is imaginarily divided into cubic cells defined by the inter-particle distance d along the axis xi (i = 1,2,3). Moreover, due to symmetry of the multi-particle-matrix system resulting from the matrix infinity and from the periodical distribution, the thermal stresses are sufficient to be investigated in the interval ϕ, ν ∈ h0, π/4i regarding the spatial polar coordinates [r, ϕ, ν] (see Figs. 1, 3). Considering zero displacement of the points O, the centres 1, 2, 4 of the abscissae O1O, O2O, O4O, respectively, exhibit zero displacement along these abscissae, as also valid for the arbitrary point C on the cell surface along the axis x′1 representing a radial direction (see Figs. 2, 3). Assuming non-zero radial displacement on the surface of a certain cell, the same condition is also valid on the surface of any cell, resulting from the matrix infinity, as well as on the surface of neighbouring cells. Accordingly, the space between neighbouring cells is replaced by the vacuum what is physically unacceptable. The following conclusions, including a geometric boundary condition for the cell matrix defined in Item 1 presented below, are considered: 1. zero radial displacement of an arbitrary point in the cell matrix on the cell surface, (u1m )r=rc = 0, accordingly represents a geometric boundary condition for the cell matrix, simultaneously valid for a cell of such shape to fill the infinite matrix perfectly,


257

Figure 2. The points C3 and C1 as points of intersection of the axis x′1 , representing a radial direction, with the surfaces 3657 and 1456 of one eighth of the cubic cell (see Fig. 1) for ν ∈ h0, ν ∗ ) and ν ∈ hν ∗ , π/2i, respectively. 2. conversely, such imaginary dividing of the infinite matrix is required regarding particles distribution that cells can fulfil the infinite matrix perfectly. Considering volume of the spherical particle and cubic cell, Vp and Vc , respectively, the particle volume fraction v has the form Vp 4π R 3 π ∈ 0, , (1) = v= Vc 3 d 6 and consequently, the inter-particle distance d is derived as 1/3 4π d=R ∈ (0, 2Ri , 3v

(2)

where vmax = π/6 in Eq. (1) results from d = 2R. The two-component material defined in Item 1 (see Section 2.) consists of the grains A and B with the volume fractions vA and vB = 1 − vA , respectively. Provided that vA ∈ (0, vmax i, the grains A and B represent the spherical particle and the infinite matrix, respectively.

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The analytical model of the thermal stresses acting in the multi-particlematrix system with a finite matrix is required to consider a position of each cell in the finite matrix. Additionally, boundary conditions for radial displacement u1m on surfaces of each cell are required to be defined for each cell separately. Accordingly, even though the boundary conditions are defined, an application of the analytical model to a real two-component material, resulting in numerical dependencies of the thermal stresses on d, R and v, would be probably timeconsuming.

2.1.

Determination of Distance rc

With regard to the interval ϕ, ν ∈ h0, π/4i, the parameter rc is distance from the particle centre O to the points C1 or C3 on the cell surfaces 1456 or 3627 with the normals x1 or x3 (see Fig. 2), where C1 and C3 are points of intersection of the cell surfaces 1456 and 3627 with the axis x′1 representing a radial direction (see Fig. 3) for ν = ∠ OC3 , x3 ∈ h0, ν ∗ ) and ν = ∠ OC1 , x3 ∈ hν ∗ , π/2i (see Eq. (22)), respectively. The angle ν ∗ has the form ! 39 1 ∗ ν = arctan = arctan , (3) O3 cos ϕ where, regarding Eq. (2), the length O3 and 39 of the abscissae O3 and 39, respectively, are derived as d O3 = = R 2 2

4π 3v

1/3

, 39 =

d R = 2 cos ϕ 2 cos ϕ

4π 3v

1/3

.

(4)

2.1.1. Intervals ϕ ∈ h0, π/4i, ν ∈ h0, ν ∗ ) The distance rc = OC3 between the points O and C3 , the latter on the surface 3627, i.e. for ϕ ∈ h0, π/4i and ν = ∠ OC3 , x3 ∈ h0, ν ∗ ), are derived by the following mathematical procedure. Considering the sine rule (see Fig. 2) [25] C3 9 O9 = (5) sin ∠ OC3 , O9 sin ∠ OC3 , C3 9 and the condition O9 = rc cos ∠ OC3 , O9 + C3 9 cos ∠ C3 9, O9 ,

(6)


259

we get rc =

O9

cos ∠ OC3 , O9 ( ) sin ∠ OC3 , O9 cos ∠ C3 9, O9 . × 1− sin ∠ OC3 , C3 9

(7)

where the length O9 of the abscissa O9 and the angles ∠ OC3 , O9 , ∠ C3 9, O9 , ∠ OC3 , C3 9 , regarding Eqs. (3), (4), are derived as q 2 2 O9 = O3 + 39 = R

4π 3v

1/3 p 1 + cos2 ϕ , 2 cos ϕ

∠ OC3 , O9 = ν ′ = ν ∗ − ν, ν ′ ∈ (0, ν ∗ i , π ∠ C3 9, O9 = − ν ∗ , 2 π ∠ OC3 , C3 9 = + ν ∗ − ν ′ 2 Considering the following relationships [25] π sin ± α = cos α, 2 π ± α = ∓ sin α, cos 2

(8) (9) (10) (11)

(12) (13)

sin (α ± β) = sin α cos β ± cos α sin β,

(14)

cos (α ± β) = cos α cos β ∓ sin α sin β,

(15)

sin ν ∗ =

1 |39| =p , |O9| 1 + cos2 ϕ

(16)

cos ϕ |O3| =p , (17) |O9| 1 + cos2 ϕ and with regard to Eqs. (8)–(11), the distance rc = OC3 (see Fig. 2) has the form (18) rc = OC3 = Rfc , cos ν ∗ =

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where the function fc , regarding Eq. (2), is derived as p 1/3 D πE 4π 1 + cos2 ϕ , , ν ′ ∈ (0, ν ∗ i . (19) ϕ ∈ 0, fc = 2 (cos ϕ cos ν ′ + sin ν ′ ) 3v 4 On the one hand, the distance rc = OC3 (see Fig. 2) can be also given by the relationship d , ν ∈ h0, ν ∗ ) . (20) rc = 2 cos ν On the other hand, in contrast to the relationship rc = rc (ϕ, ν) for ϕ ∈ h0, π/4i, ν ∈ hν ∗ , π/2i given by Eqs. (21), (22), the thermal stresses in the isotropic particle and the isotropic matrix related to the relationship rc 6= rc (ϕ) for ϕ ∈ h0, π/4i, ν ∈ h0, ν ∗ ) given by Eq. (20) are not functions of the variable ϕ ∈ h0, π/4i what is physically unacceptable, and accordingly, formula (20) is not considered. 2.1.2.

Intervals ϕ ∈ h0, π/4i, ν ∈ hν ∗ , π/2i

Similarly, the ϕ, ν-dependent distance rc = OC1 between the points O and C1 , the latter on the surface 1456, i.e. for ϕ ∈ h0, π/4i and ν = ∠ OC1 , x3 ∈ hν ∗ , π/4i, is derived as D πE d = Rfc , ϕ ∈ 0, , rc = OC1 = 2 cos ϕ sin ν 4 (21) D E π ν = ∠ OC1 , x3 ∈ ν ∗ , , 2 ∗ where the function fc and the angle ν , regarding 39 = d/ (2 cos ϕ) and Eq. (2), have the form 1/3 D πE 1 4π fc = , ϕ ∈ 0, , 2 cos ϕ sin ν 3v 4 (22) D E ∗ π ν = ∠ OC1 , x3 ∈ ν , . 2

3. Thermal Stresses in Solid Continuum 3.1.

Coordinate System and Radial Displacement

Thermal stresses are investigated in the arbitrary point P of the solid continuum along the axes x′1 , x′2 , x′3 of the Cartesian system (P x′1 x′2 x′3 ) (see Fig. 3).


261

With regard to the spatial polar coordinates [r, ϕ, ν], the axes x′1 = xr and x′2 = xϕ k x1 x2 , x′3 = xν represent radial and tangential directions related to the spherical surface in the point P , respectively, where r = |OP | is radius of the spherical surface, and P = [r, ϕ, ν] for r ∈ h0, rc i, ϕ, ν ∈ h0, π/2i (see Section 2.). Finally, the infinitesimal part of the solid continuum in the point P is represented by an infinitesimal spherical cap described by the parameters dr, dϕ, dν.

Figure 3. The radial and tangential axes x′1 = xr and x′2 = xϕ k x1 x2 , x′3 = xν , respectively, and the point P with a position determined by the spatial polar coordinates [r, ϕ, ν] regarding the Cartesian system (Ox1 x2 x3 ), where O is the centre of the spherical particle, and x′2 , x′3 are tangents to the spherical surface with the radius r = |OP |.

The thermal stresses originate as a consequence of the condition βm − βe 6= 0, where the coefficient βq (q = p,m) (see Eq. (60)) is a function of the thermal expansion coefficient αq , of the relaxation temperature Tr below that the stress relaxation as a consequence of thermal-activated processes does not occur in a two-component material [28]– [31], and of the final temperature Tf of a cooling process. Considering βm − βp > 0 or βm − βp < 0, the cell matrix in an arbitrary point on the particle-matrix boundary is pushed or pulled by the spherical particle along the normal to the particle-matrix boundary in the arbitrary point, respectively, and vice versa. Accordingly, the pulling or pushing of any point of

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the normal is also realized along the normal direction only. Consequently, the normal in any point of the particle-matrix boundary is identical with the axis x′1 to represent a radial direction regarding the Cartesian system (Ox1 x2 x3 ). Finally, an infinitesimal spherical cap with the normal identical to the axis x′1 exhibit a radial displacement only.

3.2.

Equations of Solid Continuum Mechanics

Equations of solid continuum mechanics considering in this chapter are represented by the Cauchy’s equations (see Eqs. (23)–(26)), the equilibrium equations (see Eqs. (27)–(29)), both related to the infinitesimal spherical cap, and the Hooke’s laws for an isotropic elastic solid continuum (see Eqs. (30)–(34)), derived as [17] ∂u1q ε11q = , (23) ∂r u1q , (24) ε22q = ε33q = r ε12q = ε21q =

1 ∂u1q , r ∂ϕ

(25)

ε13q = ε31q =

1 ∂u1q , r ∂ν

(26)

2σ11q − σ22q − σ33q + r

∂σ11q ∂σ12q ∂σ13q + + = 0, ∂r ∂ϕ ∂ν

(27)

∂σ12q ∂σ22q + 3σ12q + r = 0, ∂ϕ ∂r

(28)

∂σ33q ∂σ13q + 3σ13q + r = 0, ∂ν ∂r

(29)

ε11q = s11q σ11q + s12q (σ22q + σ33q ) ,

(30)

ε22q = s12q (σ11q + σ33q ) + s11q σ22q ,

(31)

ε33q = s12q (σ11q + σ22q ) + s11q σ33q ,

(32)

ε12q = s44q σ12q ,

(33)

ε13q = s44q σ13q ,

(34)


263

and the elastic moduli s11q , s12q , s44q for an isotropic elastic solid continuum have the forms 1 s11q = , (35) Eq s12q = − s44q =

µ , Eq

(36)

2 (1 + µq ) , Eq

(37)

where Eq and µq are the Young’s modulus of elasticity and the Poisson’s number, respectively. The subscripts q = p and q = m are related to the spherical particle and the matrix, respectively. Finally, ε11q = εrq (σ11q = σrq ); ε22q = εϕq , ε33q = ενq (σ22q = σϕq , σ33q = σνq ); and ε12q = ε21q = εrϕ , ε13q = ε31q = εrν (σ12q = σ21q = σrϕq = σϕrq , σ13q = σ31q = σrνq = σνrq ) represent radial; tangential; and shear strains (stresses), respectively, where the subscripts p and m are related to the spherical particle and cell matrix, respectively. With regard to the analysis in Section 3.1., and accordingly as a consequence of radial loading along the normal to the particle-matrix boundary, the strain ε23q and the stress σ23q are equal to zero.

3.3.

Solution of Differential Equations for Radial Displacement

Considering Eqs. (23)–(26), the Hooke’s laws for an isotropic elastic solid continuum are transformed to the forms s11q σ11q + s12q (σ22q + σ33q ) =

∂u1q , ∂r

u1q , r u1q s11q σ33q + s12q (σ11q + σ22q ) = , r 1 ∂u1q , s44q σ12q = r ∂ϕ s11q σ22q + s12q (σ11q + σ33q ) =

s44q σ13q =

1 ∂u1q , r ∂ν

(38) (39) (40) (41) (42)

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and consequently, we get σ11q = (c1q + c2q )

∂u1q u1q − 2c2q , ∂r r

u1q ∂u1q + c1q , ∂r r 1 ∂u1q = , s44q r ∂ϕ

(43)

σ22q = σ33q = −c2q

(44)

σ12q = σ21q

(45)

σ13q = σ31q =

∂u1q , s44q r ∂ν 1

(46)

where the condition σ22q = σ33q results from ε22q = ε33q (see Eq. (24)), and the coefficients c1q , c2q , c3q to consider s11q , s12q , s44q (see Eq. (35)) have the forms c1q = c2q =

Eq s11q = , q = p, m, 2 (1 + µq ) (1 − 2µq ) s11q (s11q + s12q ) − 2s12q

(47)

s12q µq Eq =− , q = p, m, (48) 2 (1 + µq ) (1 − 2µq ) s11q (s11q + s12q ) − 2s12q c3q =

s44q c1q + 2 = −4 (1 − µq ) , q = p, m. s44q c2q − 1

(49)

With regard to the Poisson’s number of an elastic solid continuum and real materials, µq = 0.25 [17] and µq < 0.5 [28], respectively, the coefficient c3q < 0, and additionally, an investigation concerning the condition µq = 0.5 to result in c1q → ∞, c2q → −∞ and consequently in possible discontinuities is irrelevant. Substituting Eqs (43)–(46) to [∂Eq.(28)/∂ϕ] + [∂Eq.(29)/∂ν] and Eq. (27), the equilibrium equations (see Eqs. (27)–(29)) are transformed to the forms ∂U1q r = c3q U1q , (50) ∂r r2

∂u1q U1q ∂ 2 u1q + 2r − 2u1q + = 0. 2 ∂r ∂r s44q (c1q + c2q )

(51)

where the function U1q is derived as U1q =

∂ 2 u1q ∂ 2 u1q + . ∂ϕ2 ∂ν 2

(52)


265

The substitution of Eqs. (38)–(42) to the Saint-Venant’s equations [17] as relationships between strains represents tautology not leading to differential equations. Performing r [∂Eq.(50)/∂r], the differential equation (50) is transformed to the form ∂ 2 U1q ∂U1q r2 = 0, (53) + (1 − c3q ) r 2 ∂r ∂r and substituting Eq. (50) to Eq. (53), we get r2

∂ 2 U1q + c3q (1 − c3q ) U1q = 0. ∂r2

(54)

Assuming U1q = rλq (see Eq. (52)), we get U1q =

1 X ∂ 2 u1q ∂ 2 u1q C1+iq rλ1+iq , + = ∂ϕ2 ∂ν 2

(55)

i=0

where C1q , C2q are integration constants and the exponent λ1+iq (i = 0,1) is derived as √ i 1h λ1+iq = 1 + (δ0i − δ1i ) D , i = 0, 1, q = p, m, 2 Dq = 1 − 4c3q (1 − c3q ) = 1 + 16 (1 − µq ) [1 + 4 (1 − µq )] , (56) where c3q < 0 (see Eq. (47)) and the discriminant Dq > 0 both due to µq < 0.5 [17, 28], and accordingly the real exponents λ1q > 3, λ2q < −2. Considering Eq. (55), the differential equation (51) is transformed to the form 1

X ∂ 2 u1q ∂u1q r C1+iq rλ1+iq , + 2r − 2u = −3a 1q 1q ∂r2 ∂r 2

i=0

a1q =

1

3s44q (c1q + c2q )

(57)

,

representing a linear differential equation of the second rank regarding the variable r with a non-zero right side. Using the Wronskian’s method [25], the solution of the differential equation (57) for the radial displacement u1q has the form 1 X C1+iq Λ1+iq rλ1+iq , u1q = a1q i=0 (58) 1 1 Λ1+iq = − , i = 0, 1. λ1+iq + 2 λ1+iq − 1

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The integration constants C1q , C2q are derived from boundary conditions (see Section 4.), considering one or two of the terms u1rq and u2rq in the forms u1rq = rλ1q , u2rq = rλ2q ,

(59)

representing increasing and decreasing functions of the variable r, resulting from the exponents λ1q > 1 and λ2q < 0, respectively.

3.4.

Reason of Thermal Stresses

The thermal stresses are a consequence of the differences βm − βp 6= 0, where the coefficient βq (q = p,m) is derived as βq =

ZTr

αq dT, q = p, m,

(60)

Tf

where Tf and Tr = (0.35 − 0.4) × Tm [28] is final and relaxation temperature of a cooling process, respectively. The melting temperature Tm represents the minimum of the set {Tmp , Tmm }, where Tmp and Tmm are melting temperature of the particle and the matrix, respectively. Cooling down the multi-particle-matrix system from the relaxation temperature Tr to the temperature T ∈ hTf , Tr i, the spherical particle and the spherical hole in the cell matrix, both with the radius R at the temperature Tr , tend to exhibit the radii Rp and Rm , respectively, derived as Rp = R (1 − βp ) , Rm = R (1 − βm ) .

(61)

The thermal stresses in the multi-particle-matrix system originate as a consequence of the condition (∆ u1mp )r=R 6= 0 resulting from βp 6= βm , and accordingly the radial stress p is induced on the particle-matrix boundary, where the parameter (∆ u1mp )r=R , regarding Eq. (61), has the form (∆ u1mp )r=R = Rp − Rm = R (βm − βp ) .

(62)

Consequently, the distance RT in a radial direction at the temperature T ∈ hTf , Tr i from the particle centre O to a point on the particle-matrix boundary is derived as RT = Rp + (u1p )r=R = Rm + (u1m )r=R , (63)


267

where (u1p )r=R and (u1m )r=R , induced by the radial stress p acting on the particle-matrix boundary, are radial displacements in the spherical particle and cell matrix on the particle-matrix boundary, thus for r = R, respectively. Considering Eqs. (24), (61), (63), the compressive or tensile radial stress p > 0 or p < 0, respectively, can be determined from the condition (ε22m )r=R − (ε22p )r=R = βm − βp .

4.

(64)

Boundary Conditions

4.1.

Spherical Particle

With regard to the spherical particle with the radius R and with the centre O in the beginning of the Cartesian system (Ox1 x2 x3 ), the thermal stresses σiip , σ12+jp (i = 1,2,3; j = 0,1), and the radial displacement u1p are required to fulfil the boundary conditions determined as (u1p )r=0 = 0, (σiip )r→0 → / ± ∞, (σ12+jp )r→0 → / ± ∞, i = 1, 2, 3; j = 0, 1, (σ11p )r=R = −p,

(65) (66) (67)

where (65) and (66), (67) represent geometric and stress boundary conditions, respectively. The absolute value |u1p | is required to represent an increasing function of r ∈ h0, Ri, exhibiting a maximal value on the particle-matrix boundary, thus for r = R, and the integration constants C1p 6= 0, C2p = 0 (see Eqs. (58), (59)).

4.2.

Cell Matrix

With regard to the cell matrix, the thermal radial stress σ11m and the radial displacement u1m are required to fulfil the boundary conditions determined as (σ11m )r=R = −p,

(68)

(u1m )r=rc = 0,

(69)

where (68) and (69) represent stress and geometric boundary conditions for the cell matrix, respectively, and the integration constants C1m 6= 0, C2m 6= 0 (see

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Ladislav Ceniga

Eqs. (58), (59)); the ϕ, ν-dependent distance rc is given by Eq. (21). Finally, the absolute value |u1m | is required to represent a decreasing function of r ∈ hR, rc i, exhibiting a maximal value on the particle-matrix boundary, then for r = R. The decreasing course of the dependence |u1m | − r is ensured by the integration constants C1m , C2m .

5. 5.1.

Thermal Stresses in Isotropic Multi-particle-matrix System Spherical Particle

Considering Eqs. (24), (43)–(46), (64), (58) and the analysis in Section 4., we get p r r λ1p −1 u1p = − , (70) c4p R r λ1p −1 , (71) σ11p = −p R p c6p r λ1p −1 , (72) σ22p = σ33p = − c4p R ∂p r λ1p −1 1 , (73) σ12p = − s44p c4p ∂ϕ R 1 ∂p r λ1p −1 σ13p = − , (74) s44p c4p ∂ν R (ε22p )r=R = −

p , c4p

(75)

where the radial stress p is given by Eq. (90) and the coefficients c4p , c6p are derived as c4+iq = λ1+iq (c1q + c2q ) − 2c2q , i = 0, 1; q = p, m,

(76)

c6+iq = c1q − λ1+iq c2q , i = 0, 1; q = p, m,

(77)

and the coefficients c1q , c2q , c3q along with the exponent λ1+iq (i = 0,1; q = p) are given by Eqs. (47)–(49), (56).


5.2.

269

Cell Matrix

Considering Eqs. (24), (43)–(46), (64), (58) and the analysis in Section 4., we get 1 r λ1+im −1 X c8+im u1m = −p r , (78) R i=0

σ11m = −p

1 X

c4+im c8+im

i=0

σ22m = σ33m = −p

1 X

r λ1+im −1 R

c6+im c8+im

i=0

σ12m = −

σ13m = −

1 s44m 1 s44m

1 X

,

(79)

r λ1+im −1 R

,

(80)

∂c8+im ∂p r λ1+im −1 ∂ p + c8+im , ∂ϕ ∂ϕ ∂ϕ R

(81)

1 X ∂c8+im ∂p r λ1+im −1 ∂ p + c8+im , ∂ν ∂ν ∂ϕ R

(82)

i=0

i=0

(ε22m )r=R = −p (c8m + c9m ) ,

(83)

where c6+im (i = 0,1) is given by Eqs. (77). The coefficient c8+im along with the derivation ∂c8+im /∂η (i = 0,1; η = ϕ, ν), regarding Eqs. (81), (82), are derived as 1 , i = 0, 1, (84) c8+im = λ −λ c4+im − c5−im fc 1+im 2−im λ

−λ

∂c8+im c5−im (λ1+im − λ2−im ) fc 1+im 2−im = 2 ∂η λ −λ c4+im − c5−im fc 1+im 2−im

−1

∂fc , i = 0, 1; η = ϕ, ν, ∂η

(85) where the coefficient c4+im (i = 0,1) and the function fc are given by Eq. (76) and Eqs. (19) or (22) for ν ∈ h0, ν ∗ ) or ν ∈ hν ∗ , π/4i (see Eq. (3)), respectively. Consequently, the derivation ∂fc /∂η, regarding Eq. (9), have the forms sin ϕ (cos ν ′ − cos ϕ sin ν ′ ) ∂fc p = ∂ϕ 2 (cos ϕ cos ν ′ + sin ν ′ )2 1 + cos2 ϕ D πE , ν ′ ∈ (0, ν ∗ i . ϕ ∈ 0, 4

4π 3v

1/3

, (86)

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Ladislav Ceniga

p ∂fc ∂fc ∂ν ′ ∂fc (cos ν ′ − cos ϕ sin ν ′ ) 1 + cos2 ϕ 4π 1/3 , = =− = ∂ν ∂ν ′ ∂ν ∂ν ′ 3v 2 (cos ϕ cos ν ′ + sin ν ′ )2 D πE , ν ′ ∈ (0, ν ∗ i . (87) ϕ ∈ 0, 4 1/3 D πE D 4π tan ϕ πE ∂fc , ϕ ∈ 0, , ν ∈ ν ∗, , (88) = ∂ϕ 2 cos ϕ sin ν 3v 4 2 1/3 D πE D ∂fc cot ν 4π πE , ϕ ∈ 0, =− , ν ∈ ν ∗, . (89) ∂ν 2 cos ϕ sin ν 3v 4 2

5.3. Radial Stress p Finally, with regard to Eqs. (64), (75), (83), the radial stress p considered in Sections 5.1., 5.2. along with the derivation ∂p/∂η (η = ϕ, ν), regarding Eqs. (81), (82), has the form p=

c4p (βm − βp ) , 1 − c4p (c8m + c9m )

c24p (βm − βp ) ∂p = ∂η [1 − c4p (c8m + c9m )]2

∂c8m ∂c9m + ∂η ∂η

(90)

,

(91)

where the coefficients βq (q = p), c4p , c8+im and the derivation ∂c8+im /∂η (i = 0,1; η = ϕ, ν) are given by Eqs. (60), (76), (84) and (85)–(89), respectively.

6. 6.1.

Analytical Fracture Mechanics General Analysis

The analytical fracture mechanics presented in this chapter to include the crack initiation and propagation results from the following analysis concerning a solid continuum with the general shape as shown in Fig. 4a. The shaded area in Fig. 3a represents cuts of the solid continuum in the planes x1 x2 , x1 x3 , x12 x3 . The curves 1, 2, 3, 4 are outlines of the cuts in the planes x1 x3 , x1 x2 , x2 x3 , x12 x3 , respectively. The x12 is an axis in the plane x1 x2 with a general position determined by the angle ϕ = ∠ (x1 , x12 ) ∈ h0, π/2i, and then x12 ≡ x1 , x12 ≡ x2 for ϕ = 0, ϕ = π/2, respectively. The points P and P ′′ are arbitrary points on the axis x12 and the curve 4, respectively, the latter on the solid continuum


271

surface. The coordinate x12 related to the axis x12 determines a position of the points P , P ′′ , then |OP | = |OP ′′ | = x12 . Finally, x12 is replaced by x1 or x2 for x12 ≡ x1 or x12 ≡ x2 , then for ϕ = 0, π or ϕ = π/2, 3π/2, respectively. The general analysis presented in Section 6.1. is considered for each of octants determined by the axes ±x1 , ±x2 , ±x3 and for any plane x12 x3 perpendicular to x1 x2 with a general position given by the angle ϕ = ∠ (x1 , x12 ) ∈ h0, π/2i. With regard to briefness of this chapter, the analysis concerning the crack initiation and propagation is related to the plane x1 x2 only. As functions of a position given by the coordinate x, the thermal stresses induce the x-dependent energy density w = w (x) accumulated in R an arbitrary point of a two-component material. Accordingly the energy W = w dV accuV

mulated in the two-component material with the volume V tends to be released by the crack initiation and consequently by the crack propagation. With regard to Fig. 4b, the energy W is assumed to be released by a crack propagated in the plane x1 x2 , where the curves 5 and 6 represent crack shapes in the planes x1 x3 and x12 x3 , respectively. Consequently the curve 7 determines a position of the crack tip in the plane x1 x2 . The function f = f (x12 ) describes the crack shape in the plane x12 x3 with the crack tip in the point P0 determined by the coordinate x0 , then |OP0 | = x0 . Finally the analytical fracture mechanics considers the energy dW = dW (x12 ) accumulated in the infinitesimal volume dV of an infinitesimal prism with the height P P ′′ (see Fig. 4a,b) and with the surface area dS12 = x12 dϕ dx12 of a basis in the plane x1 x2 (see Fig. 4c). Consequently we get Z w (x12 ) dx3 = Wc (x12 ) x1 dϕ dx12 , (92) dW (x12 ) = x12 dϕ dx1 P P ′′

and the ’curve’ energy density Wc (x12 ) dependent on the solid continuum shape has the form Z w (x12 ) dx3 . (93) Wc (x12 ) = P P ′′

Resulting in the crack initiation and propagation, the energy dW (x12 ) is in the equilibrium state with the energy dWcs = γ dS for the creation of the new infinitesimal surface dS (i.e. the crack surface) with the surface area dS = 2 is crack surface energy ds x12 dϕ (see Fig. 4c). The parameter γ = s11 KIC per unit surface area [17, 28], where s11 is an elastic modulus given by Eq. (35)

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Ladislav Ceniga

a

b

c

Figure 4. Solid continuum with the general shape with the volume V (a,b). The axis x12 (a,b,c) is an axis in the plane x1 x2 with a general position determined by the angle ϕ = ∠ (x1 , x12 ) ∈ h0, π/2i, where x12 ≡ x1 , x12 ≡ x2 for ϕ = 0, π, ϕ = π/2, 3π/2, respectively. The shaded area represents cuts of the solid continuum in the planes x1 x2 , x1 x3 , x12 x3 (a,b). The curves 1, 2, 3, 4 represent outlines of the cuts of the solid continuum in the planes x1 x3 , x1 x2 , x2 x3 , x12 x3 (a,b), respectively. The points P and P ′′ are arbitrary points on the axis x12 and the curve 4, the latter on the solid continuum surface, respectively, where the coordinate x12 related to the axis x12 determines a position of the points P , P ′′ and then |OP | = |OP ′′ | = x12 . Finally, x12 is replaced by x1 or x2 for x12 ≡ x1 or x12 ≡ x2 , and then for ϕ = 0, π or ϕ = π/2, 3π/2, respectively. The curves 5 and 6 (b) represent crack shape in the planes x1 x3 and x12 x3 , respectively. Accordingly the crack is propagated in the plane x1 x2 . The curve 7 (b) determines a position of the crack tip in the plane x1 x2 . The function f = f (x1 ) (b,c) describes the crack shape in the plane x12 x3 with the crack tip in the point P0 determined by the coordinate x0 and then |OP0 | = x0 . The term ds (c) related to the plane x12 x3 is length of an infinitesimal part of the crack in the arbitrary point P ′ . The term dS (c) related to the plane x12 x3 is surface area of the new infinitesimal surface dS. Finally dS represents the crack surface related to the infinitesimal length ds in the arbitrary point P ′ on the curve 4 on the crack surface.

Thermal Stresses and Analytical Fracture Mechanics 273 q and KIC is fracture toughness. The term ds = dx12 1 + (∂f /∂x12 )2 [25] related to the plane x12 x3 is length of an infinitesimal part of the crack in the arbitrary point P ′ on the curve 4 on the crack surface. Accordingly the function f = f (x12 ), strictly speaking ∂f /∂x12 , describing a shape of the crack in the plane x12 x3 propagated in the plane x1 x2 has the form r 2 ∂fq 1 2 2 =± [W (x )] − s K . (94) c 12 11q ICq 2 ∂x12 s11q KICq The energy condition for the crack initiation and propagation is derived as 2 Wc (x12 ) − s11q KICq ≥0

(95)

2 2 = resulting from the condition [Wc (x12 )]2 − s11q KICq h i h i 2 2 Wc (x12 ) + s11 KICq × Wc (x12 ) − s11 KICq ≥ 0. The term

2 Wc (x12 ) − s11 KICq in Eq. (95) is considered only, because 2 Wc (x12 ) + s11 KICq > 0 due to wq (x12 ) ∝ [σq (x12 )]2 > 0 (see Eq. (96)), where σq = σq (x12 ) as a function of the variable x12 is a thermal stress. Provided that fq is either a decreasing or increasing function of the variable x1 , the sign either “−” or “+” in Eq. (94) is considered, respectively. With regard to the analytical modelling, as explained in Section 2., a real two-component material of a precipitate-matrix type (see Section 1.2.) is replaced by the multi-particle-matrix system (see Fig. 1). The beginning O of the Cartesian system (Ox1 x2 x3 ) is identical to a centre of the spherical particle with radius R (see Fig. 1). Accordingly, the subscript either (q = p) or (q = m) in Eqs. (94), (95) as well as in the following terms is related to the crack initiation and propagation either in the spherical particle or in the cell matrix for either αm < αp or αm > αp , respectively. As analysed in Sections 6.4.2., 6.4.3., the crack shapes shown in Figs. 8, 9 represent schematic drawing only and are expected to be quasi-linear with the minimum on crack tips [32, 33]. Finally, real two-component materials may be assumed to exhibit the crack propagation shown in Figs. 8b, 9b only. Resulting from a mathematical viewpoint, the crack propagation in Figs. 8a,c, 9a,c is presented with regard to completeness of the analysis to consider mutual combinations of the conditions [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =0 R [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =R and αm − αp R 0.

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6.2.

Ladislav Ceniga

Determination of Curve Integral Wc

Induced by elastic stresses acting in an arbitrary point of elastic solid continuum, the elastic energy density wq accumulated in the arbitrary point is derived as [17]   3 3 X X 1 σijq εijq  , q = p, m, σiiq εiiq + (96) wq =  2 i=1

i,j=1; i6=j

and considering Eqs. (24)–(26), (44)–(46), we get wq =

σ11q ε11q + σ22q ε22q + σ12q ε12q + σ13q ε13q , q = p, m. 2

(97)

The solid continuum of the general shape shown in Fig. 4a,b is replaced by one eighth of the cubic cell with the central spherical particle (see Fig. 5), and the curve integral Wc (x12 ) has the forms Wc (x12 ) = Wcp (x12 , R) + Wcm1 (x12 , R) , x12 ∈ h0, Ri , Z wp (x12 , R) dx3 , x12 ∈ h0, Ri , Wcp (x12 , R) =

(98) (99)

P1 P2

Wcm1 (x12 , R) =

Z

wm (x12 , R) dx3 , x12 ∈ h0, Ri ,

(100)

P2 P3

Wc (x12 ) = Wcm2 (x12 , R) =

Z

wm (x12 , R) dx3 , x12 ∈ hR, d/ (2 cos ϕ)i ,

P4 P5

(101) where P1 P3 = P4 P5 = d/2; P2 is a point of intersection of the particle surface with the abscissa P1 P3 k x3 for x12 ∈ h0, Ri (see Fig. 5). Accordingly, the analysis to result in Eqs. (94), (95) is related to the plane O893 ⊥ x1 x2 , where a position of the axis x12 = O8 is determined by the angle ϕ ∈ h0, π/4i. With regard to the length P1 P2 representing a decreasing function of the variable x12 ∈ h0, Ri, the ’curve’ energy density Wcp (x12 , R) as an increasing function of R is also a decreasing function of x12 ∈ h0, Ri. Similarly, with regard to the length P2 P3 representing an increasing function of the variable x12 ∈ h0, Ri, the ’curve’ energy density Wcm1 (x12 , R) dependent on R can


275

be an increasing function of x12 ∈ h0, Ri, but is not decreasing function of x12 ∈ h0, Ri. Accordingly, the sum Wc (x12 ) = Wcp (x12 , R) + Wcm1 (x12 , R) is a decreasing-increasing function of x12 ∈ h0, Ri, exhibiting minimum for x12 = xmin (R), where xmin (R) dependent on R is determined from the condition ∂ [Wcp (x12 , R) + Wcm1 (x12 , R)] /∂x12 = 0.

Figure 5. One eighth of the cubic cell with the arbitrary points P1 , P4 and P3 , P5 on the axis x12 ⊂ x1 x2 with a general position determined by the angle ϕ = ∠ (x1 , x12 ) ∈ h0, π/4i and on the surface C3 C6 C5 C7 , respectively; a position of P1 , P3 and P4 , P5 is given by the coordinate x12 ∈ h0, Ri and x12 ∈ hR, d/ (2 cos ϕ)i related to the axis x12 , then OP1 = |x3 , P3 | = x12 and OP4 = |x3 , P5 | = x12 , respectively; P2 is a point of intersection of the particle surface with the abscissa P1 P3 k x3 for x12 ∈ h0, Ri; O is the spherical particle centre; d is the cubic cell dimension. The absolute value |σm | of the thermal stress in the matrix along with 2 in the matrix [17] represent decreasing functhe energy density wm ∝ σm tions of the variable x12 ∈ hR, d/ (2 cos ϕ)i. Accordingly the ’curve’ energy density Wc (x12 ) = Wcm2 (x12 , R) is also a decreasing function of x12 ∈ hR, d/ (2 cos ϕ)i.

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The curve integrals Wcp (x12 , R), Wcm1 (x12 , R), Wcm2 (x12 , R) are derived by the following mathematical procedure to consider the determination of the thermal stresses as function of the variables ϕ, x12 ∈ h0, d/ (cos ϕ)i and x3 ∈ h0, d/2i (see Eq. (2)). Strictly speaking, the thermal stresses determined by the spatial polar coordinates [r, ϕ, ν] (see Fig. 1) are required to be derived as functions of the variables ϕ ∈ h0, π/4i, x12 ∈ h0, d/ (cos ϕ)i, x3 ∈ h0, d/2i (see Eq. (2)) related to the plane x12 x3 (see Fig. 5). Consequently, the determination of Wcp (x12 , R), Wcm1 (x12 , R), Wcm2 (x12 , R) results from the determination of the angles ν ∈ hν ∗ , π/2i and ν ′ ∈ (0, ν ∗ i (see Eq. (9)) to define a position of the arbitrary points Pν and Pν ′ with the coordinate [x12 , x3 ] for x3 ∈ h0, x∗3 i and x3 ∈ (x∗3 , d/2i (see Figs. 2, 6), respectively. The line Pν Pν ′ perpendicular to the axis x12 (see Fig. 6) represents a curve along that the curve integrals Wcp (x12 , R) and Wcm1 (x12 , R), Wcm2 (x12 , R) of wp and wm are determined (see Eqs. (97), (99)–(101)), respectively, where P ∗ = [x12 , x∗3 ] is a point of intersection of the line Pν Pν ′ with the abscissa O9 (see Fig. 5). Finally, the coordinate r = |OPν | and r = |OPν ′ | for x3 ∈ h0, x∗3 i and x3 ∈ (x∗3 ,d/2i, respectively, along with x∗3 as a function of the angle ν ∗ = ∠ Pν Pν ′ , O9 (see Eq. (3)) have the forms r=

x∗3

q x212 + x23 ,

πE = x12 cot ν = x12 cos ϕ, ϕ ∈ 0, , x12 ∈ 4 ∗

D

(102)

0,

d 2 cos ϕ

.

(103)

Finally, the determination of Wcp (x12 , R), Wcm1 (x12 , R), Wcm2 (x12 , R) required requires fc (see Eqs. (19), (22)) to be derived as a function of the variables ϕ, x12 ∈ h0, d/ (2 cos ϕ)i, x3 ∈ h0, d/2i (see Eq. (2)), where the function fc along with the derivations ∂fc /∂ϕ, ∂fc /∂ν are included in formulae for the thermal stresses (see Sections 5.1.–5.3.). 6.2.1.

Determination of Function fc

A. Intervals ν ∈ hν ∗ , π/2i, x12 ∈ (0, d/ (2 cos ϕ)i, x3 ∈ h0, x∗3 i. With regard to the point Pν = [x12 , x3 ] (see Fig. 6), the angle ν ∈ hν ∗ , π/2i (see Eq. (3)) is derived as x12 d sin ν = p 2 , x3 ∈ h0, x∗3 i , (104) , x12 ∈ 0, 2 2 cos ϕ x12 + x3


277

Figure 6. The coordinate r = |OPν | and r = |OPν ′ | (see Eq. (101)) along with the angles ν ∈ hν ∗ , π/2i and ν ′ ∈ (0, ν ∗ i (see Eq. (9)) defining a position of the arbitrary points Pν = [x12 , x3 ] and Pν ′ = [x12 , x3 ] for x3 ∈ h0, x∗3 i and x3 ∈ (x∗3 , d/2i (see Figs. 2), respectively. The line Pν Pν ′ ⊥ x12 represents a curve along that the curve integrals Wcp (x12 , R) and Wcm1 (x12 , R), Wcm2 (x12 , R) of wp and wm are determined (see Eqs. (97), (99)–(101)), respectively, where P ∗ = [x12 , x∗3 ] is a point of intersection of Pν Pν ′ with the abscissa O9 (see Fig. 5), and x∗3 as a function of the angle ν ∗ = ∠ Pν Pν ′ , O9 (see Eq. (3)) is given by Eq. (103). x3 cos ν = p 2 , x12 ∈ x12 + x23

0,

d 2 cos ϕ

, x3 ∈ h0, x∗3 i .

(105)

Substituting Eqs. (104), (105) to Eqs. (22), (88), (89), the function fc along with the derivations ∂fc /∂ϕ, ∂fc /∂ν for ϕ ∈ h0, π/4i, x12 ∈ (0, d/ (2 cos ϕ)i, x3 ∈ h0, x∗3 i are derived as p x212 + x23 4π 1/3 fc = , 2x12 cos ϕ 3v D πE d ϕ ∈ 0, , x12 ∈ 0, , x3 ∈ h0, x∗3 i , 4 2 cos ϕ

(106)

278

Ladislav Ceniga p ∂fc x212 + x23 tan ϕ 4π 1/3 , = ∂ϕ 2x12 cos ϕ 3v D πE d ϕ ∈ 0, , x12 ∈ 0, , x3 ∈ h0, x∗3 i , 4 2 cos ϕ

(107)

p x3 x212 + x23 4π 1/3 ∂fc , =− ∂ν 3v 2x212 cos ϕ D πE d , x3 ∈ h0, x∗3 i . , x12 ∈ 0, ϕ ∈ 0, 4 2 cos ϕ B. Intervals ν ′ ∈ (0, ν ∗ i, x12 ∈ (0, d/ (2 cos ϕ)i, x3 ∈ (x∗3 , d/2i. ing the sine rule (see Fig. 6) [25]

(108)

Consider-

OPν ′

P ∗ Pν ′ = , sin ν ′ sin ∠ OP ∗ , P ∗ Pν ′

(109)

and the following relationships [25]

sin (π − α) = sin α, cos α =

p

1 − sin2 α,

(110) (111)

the angle ν ′ ∈ (0, ν ∗ i, regarding Eq. (16), is derived as x3 − x12 cos ϕ , x212 + x23 (1 + cos2 ϕ) D πE d d ϕ ∈ 0, , x12 ∈ 0, , x3 ∈ x∗3 , , 4 2 cos ϕ 2 sin ν ′ = q

x12 + x3 cos ϕ , x212 + x23 (1 + cos2 ϕ) D πE d ∗ d ϕ ∈ 0, , x12 ∈ 0, , x3 ∈ x3 , , 4 2 cos ϕ 2

cos ν ′ = q

(112)

(113)

Thermal Stresses and Analytical Fracture Mechanics 279 ∗ where the length P ∗ P ν ′ of the abscissa P Pν ′ , regarding Eq. (103), and the ∗ ∗ angle ∠ OP , P Pν ′ have the forms P ∗ Pν ′ = x3 − x∗3 , x3 ∈

∗ d x3 , , 2

∠ OP ∗ , P ∗ Pν ′ = π − ν ∗ .

(114) (115)

Substituting Eqs. (112), (113) to Eqs. (19), (86), (5.2.), the function fc along with the derivations ∂fc /∂ϕ, ∂fc /∂ν for ϕ ∈ h0, π/4i, x12 ∈ (0, d/ (2 cos ϕ)i, x3 ∈ (x∗3 , d/2i are derived as p x212 + x23 4π 1/3 d ∗ d , x12 ∈ 0, , x3 ∈ x3 , , (116) fc = 2x3 3v 2 cos ϕ 2 p ∂fc x12 x212 + x23 sin ϕ 4π 1/3 , = ∂ϕ 3v 2x23 (1 + cos2 ϕ) D πE d ∗ d , x3 ∈ x3 , , , x12 ∈ 0, ϕ ∈ 0, 4 2 cos ϕ 2

(117)

p x12 x212 + x23 4π 1/3 d ∂fc ∗ d , x12 ∈ 0, , x3 ∈ x3 , . = ∂ν 3v 2 cos ϕ 2 2x23 (118) C. Intervals ν ′ = ν ∗ , x12 = 0, x3 ∈ h0, d/2i. The condition x12 = 0 corresponds to the curve integrals Wcp (x12 , R), Wcm1 (x12 , R) of wp and wm (see Eqs. (99), (100)) for x3 ∈ h0, Ri and x3 ∈ hR, d/2i along the axis x3 (see Fig. 6) i.e. for the angles ϕ = 0, ν ′ = ν ∗ (see Eqs. (3)), respectively. Considering Eq. (19) and Eqs. (117), (118) due to ν ′ = ν ∗ , the function fc along with the derivations ∂fc /∂ϕ, ∂fc /∂ν for x12 = 0 and x3 ∈ hx∗3 , d/2i = h0, d/2i (see Eq. (103)) have the forms

1/3

d , x12 = 0, x3 ∈ 0, , 2 ∂fc ∂fc d . = = 0, x12 = 0, x3 ∈ 0, ∂ϕ ∂ν 2

x3 fc = 2

4π 3v

(119)

(120)

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Ladislav Ceniga

The formulae (119), (120) can be also derived by Eqs. (86), (5.2.) for ϕ = 0, ν ′ = ν ∗ , considering Eqs. (16), (17). The condition ∂fc /∂η (η = ϕ, ν) (see Eq. (120)) results in [∂c8+im /∂η] = 0 (i = 0,1) (see Eq. (85)), and accordingly, (∂p/∂η) = 0 (see Eq. (91)). Finally, with regard to [∂c8+im /∂η] = 0 (i = 0,1), (∂p/∂η) = 0, considering Eqs. (73), (74), (81), (82), we get d σ12p = σ13p = σ12m = σ13m = 0, x12 = 0, x3 ∈ 0, . (121) 2 6.2.2. Determination of Curve Integrals Wcp (x12 , R), Wcm1 (x12 , R), Wcm2 (x12 , R) With regard to Fig. 7, the curve integrals Wcp (x12 , R), Wcm1 (x12 , R), Wcm2 (x12 , R) (see Eqs. (99)–(101)) are derived as Zx3

∗

Wcp (x12 , R) =

0

Zx32 wp dx3 + wp dx3 , x∗3

Zd/2 Wcm1 (x12 , R) = wm dx3 , x12 ∈ h0, x∗12 ) ,

(122)

x32

Zx3

∗

Wcp (x12 , R) =

0

Zd/2 wp dx3 , Wcm1 (x12 , R) = wm dx3 , x12 = x∗12 , x∗3

(123)

Zx32 Wcp (x12 , R) = wp dx3 , 0

Zx3

∗

Wcm1 (x12 , R) =

x32

Zd/2 wm dx3 + wm dx3 , x12 ∈ (x∗12 , R) ,

(124)

x∗3

Zd/2 Wcm2 (x12 , R) = wm dx3 , x12 ∈ hR, d/ (2 cos ϕ)i , 0

(125)


281

where x∗12 , x32 have the forms R , x∗12 = R sin ν ∗ = p 1 + cos2 ϕ x32

(126)

q = R2 − x212 .

(127)

a

b

c

d

Figure 7. The abscissae P1 P2 and P2 P3 , P4 P5 along which the curve integrals Wcp and Wcm1 , Wcm2 are determined, respectively, considering x12 ∈ h0, x∗12 ) (a), x12 = x∗12 (b), x12 ∈ (x∗12 , R) (c), x12 ∈ hR, d/ (2 cos ϕ)i (d).

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Ladislav Ceniga

The determination of the curve integrals Wcp (x12 , R), Wcm1 (x12 , R), Wcm2 (x12 , R) (see Eqs. (122)–(125)) considers the following conditions: 1. The integration of wq (q = p,m) (see Eq. (97)) along the abscissae P1 P3 and P4 P5 for x12 ∈ (0, Ri and x12 ∈ hR, d/ (2 cos ϕ)i (see Figs. 5, 7), respectively, considers the functions fc , ∂fc /∂ϕ, ∂fc /∂ν (a) given by Eqs. (106)–(108) for x3 ∈ h0, x∗3 i, (b) given by Eqs. (116)–(118) for x3 ∈ (x∗3 , d/2i, (c) given by Eqs. (119), (120) regarding Eq. (121). 2. The inter-particle distance d, the distance r, and the coordinate x∗3 , included in wq (q = p,m) (see Eq. (97)), are given by Eqs. (2), (101), and (103), respectively. Accordingly, with regard to d = d (R, v), the elastic energy density wq = wq (x12 , x3 , R, v) is thus transformed to a function of the particle radius R and the particle volume fraction v ∈ (0, π/6i (see Eq. (1)). 3. Substituting numerical values of parameters of a real two-component material, i.e. the material parameters Eq , µq , αq , KICq (q = p,m), Tf , Tr , and theR microstructural parameters v ∈ (0, π/6i, P R, to Eq. (97), the integral wq dx3 is replaced by the summation wq ∆ x3 , where ∆ x3 is taken as small as possible, e.g. the step ∆ x3 = (d/2) × 10−4 − 10−3 can be considered to be sufficient.

6.3.

Crack Initiation

The crack initiation is a consequence of the condition R = Rqc (≡ Rqc1 , Rqc2 ), where Rqc is critical particle radius to result in the formation of an infinitesimal crack with the length dx12 along the axis x12 , and the subscript either q = p for αm < αp or q = m for αm > αp is related to the crack initiation either in the spherical particle or in the cell matrix, respectively. Consequently, the condition 2 [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =a − s11q KICq = 0,

(128)

resulting from (∂fq /∂x12 )x12 =a = 0 for Wc (x12 ) = Wcp (x12 , R) + Wcm1 (x12 , R), and representing an equation with the term R considered to be a variable, is used for the determination of the critical particle radii Rqc1 and


283

Rqc2 to result in the crack initiation in the positions x12 = 0 and x12 = R, then for a = 0 and a = R, respectively. Provided that either Rqc1 < Rqc2 or Rqc1 > Rqc2 resulting from either [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =0 > [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =R or [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =0 < [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =R , one crack is initiated in the position either x12 = 0 or x12 = R, respectively. Provided that Rqc1 = Rqc2 resulting from [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =0 = [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =R , two crack are simultaneously initiated, i.e. in the position x12 = 0 as well as in the position x12 = R. Finally, with regard to the crack initiation, the condition (128) used for the determination of the critical particle radii Rqc1 and Rqc2 is considered for any two-component material (brittle, ductile) of a precipitation-matrix type. Substituting numerical values of parameters of a real two-component material, i.e. the material parameters Eq , µq , αq , KICq (q = p,m), Tf , Tr , and the microstructural parameters v ∈ (0, π/6i, R > Rqc , to Eq. (97), and considering the analysis in Section 6.2.2. (see Items 1–3), the critical particle radii Rqc1 and Rqc2 can be numerically determined by a suitable numerical method, e.g. the Newton’s method of tangents [25], applied to Eq. (128) representing a transcendental equation related to the variable R with the roots Rqc1 and Rqc2 for a = 0 and a = R, respectively.

6.4.

Crack Propagation

The crack propagation either in the spherical particle or in the cell matrix, i.e. either for αm < αp and q = p, or for αm > αp and q = m, respectively, is a consequence of the condition R > Rqc (≡ Rqc1 , Rqc2 ). Consequently, the condition 2 Wcp (x12 , R) + Wcm1 (x12 , R) − s11q KICq = 0, (129) resulting from ∂fiq /∂x12 = 0 for x12 ∈ h0, Ri and Wc (x12 ) = Wcp (x12 , R) + Wcm1 (x12 , R), is used for the determination of the position x0iq of the point P0i (i = 1, 2) on the crack tip in the spherical particle (q = p) or the cell matrix (q = m) (see Fig. 4c), where the subscript i = 1, 2 is mentioned below. Accordingly, the condition [∂fiq /∂x12 ]x12 =x0iq = 0, resulting in Eq. (129) and assuming the function fiq = fiq (x12 ) to be extremal on the crack tip (i.e. for x12 = x0iq ), may be considered for an ideal-brittle solid continuum [17,18].

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Ladislav Ceniga

With regard to Eq. (94), the decreasing and increasing functions f1q and f2q , respectively, in the interval x12 ∈ h0, x0iq i ⊂ h0, Ri has the form fiq =

1 2 s11q KICq

) ! Z (r 2 2 2 dx12 , [Wcp (x12 , R)+Wcm1 (x12 , R)] − s11q KICq × βiq −(δ1i −δ2i ) x12 ∈ h0, x0iq i ⊂ h0, Ri , (fiq )x12 =x0iq = 0, i = 1, 2; q = p, m,

(130)

where δ ij is the Kronecker’s symbol, and δ ij = 0 for i 6= j and δ ij = 1 for i = j. Similarly, with regard to the decreasing function Wc (x12 ) = Wcm2 (x12 , R) of the variable x12 ∈ hR, d/ (2 cos ϕ)i, the decreasing function f3m (x12 ) in the interval x12 ∈ hR, x03m i ⊂ hR, d/ (2 cos ϕ)i is derived as f3m (x12 ) =

1 2 s11m KICm Z q 2 2 2 dx12 , [Wcm2 (x12 , R)] − s11m KICm × β3m − d x12 ∈ hR, x03m i ⊂ R, , (f3m )x12 =x03m = 0, (131) 2 cos ϕ

where the position x03m of the crack tip in the point P03 is determined from the condition 2 Wcm2 (x12 , R) − s11m KICm = 0. (132) As mentioned in Section 6.1., formulae in Sections 6.4.2., 6.4.3. are valid for an ideal-brittle two-component material, neglecting a redistribution of the thermal stresses during the high-speed crack propagation. 6.4.1.

Determination of Crack Tip Position

Substituting numerical values of parameters of a real two-component material, i.e. the material parameters Eq , µq , αq , KICq (q = p,m), Tf , Tr , and the microstructural parameters v ∈ (0, π/6i, R > Rqc , to Eq. (97), (71)–(77), (79)–(91), the crack tip positions


285

1. x01p or x01m for αm < αp and R > Rpc1 , or αm > αp and R > Rmc1 respectively, provided that [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =0 > [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =R , 2. x02p or x02m for αm < αp and R > Rpc2 , or αm > αp and R > Rmc2 respectively, provided that [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =0 < [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =R , 3. x03m for αm > αp and R > Rmc2 can be numerically determined by a suitable numerical method, e.g. the Newton’s method of tangents [25], applied to Eqs. (129) and (132) representing transcendental equations related to the variable x12 with the roots x12 = x0iq (i = 1,2) and x12 = x03m , respectively. 6.4.2.

Crack Propagation in Spherical Particle

Considering the crack propagation in the spherical particle, i.e. for αm < αp and x12 ∈ h0, Ri, the terms fiq (i = 1, 2), s11q , KICq in Eq. (128) are replaced by fip , s11p , KICp , respectively. Provided that either R ∈ (Rpc1 , Rpc2 ) or R ∈ (Rpc2 , Rpc1 ), one crack described by either f1p or f2p (see Eq. (128)) is propagated from the position either x12 = 0 (see Fig. 8a) or x12 = R (see Fig. 8b) within the interval either x12 ∈ h0, x01p i ⊂ h0, Ri or x12 ∈ hx02p , Ri with the crack tip in the point either P01 or P02 , respectively. The integration constants β1p and β2p for R ∈ (Rpc1 , Rpc2 ) and R ∈ (Rpc2 , Rpc1 ), determined from the conditions [f1p (x12 )]x12 =x01p = 0 and [f2p (x12 )]x12 =x02p = 0 concerning the crack tip points P01 and P02 , respectively, are derived as βip = (δ 1i − δ 2i ) ) ! Z (r 2 2 2 × dx12 [Wcp (x12 , R) + Wcm1 (x12 , R)] − s11p KICp

X0 = δ 1i x01p + δ 2i x02p i = 1, 2.

, x12 =X0

(133)

Provided that either R ∈ (Rpc1 , Rpmin ) for Rpc1 < Rpc2 , or R ∈ (Rpc2 , Rpmin ) for Rpc1 > Rpc2 , two unconnected cracks described by f1p and f2p (see Eqs. (130), (133)) are propagated from the positions x12 = 0 and x12 = R within the intervals x12 ∈ h0, x01p i ⊂ h0, Ri and x12 ∈

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Ladislav Ceniga

hx02p , Ri (see Fig. 8c) with the crack tip points P01 and P02 , respectively, and additionally, provided that Rpc1 = Rpc2 , the two unconnected cracks are propagated simultaneously. Substituting numerical values of parameters of a real two-component material of a precipitate-matrix type, the particle radius Rpmin to result in a connection of the two non-simultaneously or simultaneously propagated cracks, i.e. for Rpc1 6= Rpc2 or Rpc1 = Rpc2 , respectively, is numerically determined from Eq. (128) for a = xmin (R) and q = p, where Rpmin > Rpc1 as well as Rpmin > Rpc2 . The positions x0pmin = [xmin (R)]R=Rpmin and x∗0pmin = [xmin (R)]R>Rpmin of the points ∗ P0min and P0min of the connection (see Fig. 8c) result from the substitutions R = Rpmin and R > Rpmin to xmin = xmin (R) representing a function of the variable R. Included in the functions f1p , f2p connected in the crack tip points ∗ P0min and P0min for R = Rpmin and R > Rpmin , the integration constants β1p , β2p , determined from the conditions [f1p (x12 )]x12 =x0pmin ; R=Rpmin = 0, [f2p (x12 )]x12 =x0pmin ; R=Rpmin = 0, are given by Eq. (129), where the terms δ 1i x01p + δ 2i x02p , R in Eq. (133) are replaced by x0pmin , Rpmin , respectively. 6.4.3.

Crack Propagation in Cell Matrix

Considering the crack propagation in the cell matrix, i.e. for αm > αm , the terms fiq for x12 ∈ h0, Ri (i = 1, 2), s11q , KICq in Eq. (130) are replaced by fim , s11m , KICm , respectively. Provided that R ∈ (Rmc1 , Rmc2 ), one crack described by f1m (see Eq. (130)) is propagated from the position x12 = 0 within the interval x12 ∈ h0, x01m i ⊂ h0, Ri with the crack tip in the point P01 (see Fig. 9a). Provided that R ∈ (Rmc2 , Rmc1 ), two cracks described by f2m and f3m (see Eq. (130), (131)), mutually connected in the point C (see Fig. 9b), are propagated from the position x12 = R within the intervals x12 ∈ hx02m , Ri and x12 ∈ hR, x03m i ⊂ hR, d/ (2 cos ϕ)i with the crack tip points P02 and P03 , respectively. The integration constants βim (i = 1, 2) and β3m in Eqs. (130) and 1/2 (131) determined from the conditions [fim (x12 )]x12 =x0im = R2 − x20im and [f3m (x12 )]x12 =x03m = 0 have the forms q 2 βim = s11m KICm R2 − x20im + (δ 1i − δ 2i ) Z q 2 2 2 × [Wcp (x12 , R) + Wcm1 (x12 , R)] − (s11m KICm ) dx12 i = 1, 2,

, x12 =x0im

(134)


a

287

b

c Figure 8. Detailed description of Figs. 8a,b,c concerning the crack propagation in the spherical particle is presented in Section 6.4.2.

β3m =

Z q 2 2 dx12 [Wcm2 (x12 , R)]2 − s11m KICm

. (135)

x12 =x03m

Provided that either R ∈ (Rmc1 , Rm min ) for Rmc1 < Rmc2 , or R ∈ (Rmc2 , Rm min ) for Rmc1 > Rmc2 , two unconnected cracks described by f1m and f2m (see Eqs. (130), (134)) are propagated from the positions x12 = 0 and x12 = R within the intervals x12 ∈ h0, x01m i ⊂ h0, Ri and x12 ∈ hx02m , Ri (see Fig. 9c) with the crack tip points P01 and P02 , respectively, along with the

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crack described by f3m (see Eqs. (131), (135)) and propagated in the interval x12 ∈ hR, x03m i ⊂ hR, d/ (2 cos ϕ)i with the crack tip point P03 , where f3m is connected with f2m in the point C. Additionally, provided that Rmc1 = Rmc2 , the two unconnected cracks are propagated simultaneously. Similar to the analysis concerning Rpmin within the crack propagation in the spherical particle, substituting numerical values of parameters of a real twocomponent material of a precipitate-matrix type, the particle radius Rm min to result in a connection of the two non-simultaneously or simultaneously propagated cracks, i.e. for Rmc1 6= Rmc2 or Rmc1 = Rmc2 , respectively, described by f1m , f2m , is determined from Eq. (128) for a = xmin (R) and q = m, where Rm min > Rmc1 as well as Rm min > Rmc2 . The positions x0m min = [xmin (R)]R=Rm min and x∗0m min = [xmin (R)]R>Rm min of the points P0min ∗ and P0min of the connection (see Fig. 8c) result from the substitutions R = Rm min and R > Rm min to xmin = xmin (R) representing a function of the variable R. Included in the functions f1m , f2m connected in the crack tip points ∗ P0min and P0min for R = Rm min and R > Rm min , the integration constants β1m , β2m , determined from the conditions [f1m (x12 )]x12 =x0m min ; R=Rm min = 1/2 1/2 , , [f2m (x12 )]x12 =x0m min ; R=Rm min = R2 − x20m min R2 − x20m min are given by Eq. (134), where the terms x0im (i = 1, 2), R in Eq. (134) are replaced by x0m min , Rm min , respectively. Finally, the condition 2 [Wcm2 (x12 , R)]x12 =d/(2 cos ϕ) − s11m KICm = 0,

(136)

similar to Eq. (128), representing an equation with the term R considered to be a variable, is used for the determination of the particle radius Rmc3 to result in a position of the matrix crack tip on the cell surface. Accordingly, provided that R ≥ Rmc3 , matrix cracks in neighbouring cubic ∗ , as shown in Fig. 9c for R > R cells are mutually connected in the point P03 mc3 . ∗ Provided that R = Rmc3 , P03 represents a point of intersection of the axis x12 with the cell surface. 6.4.4.

Determination of Integrals

Considering the analysis in Section 6.2.2. (see Items 1–3), the integrated functions with the variable x12 in the integrals in Section 6.4. are required to be derived by the Taylor series to consider the term (x12 − x120 )n , where the exponent n represents a natural number, and


a

289

b

c Figure 9. Detailed description of Figs. 9a,b,c concerning the crack propagation in the cell matrix is presented in Section 6.4.3. 1. x120 = 0 related to f1q , β1q (q = p,m), 2. x120 = R > Rc related to f2q , f3m , β2q (q = p,m), β3m . Finally, substituting numerical values of parameters of a real two-component material, i.e. the material parameters Eq , µq , αq , KICq (q = p,m), Tf , Tr , and the microstructural parameters v ∈ (0, π/6i, R > Rqc , to Eq. (97), and

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considering the analysis in Section 6.2.2. (see Items 1–3), the Taylor series can be determined by a suitable software (e.g. Mathematica) regarding such value of the exponent n to be suitable with regard to sufficient accuracy.

7.

Application to Two-Component Ceramics

Table 1 presents material parameters of the SiC particle along with the Si3 N4 and MoSi2 matrixes of the SiC-Si3 N4 and SiC-MoSi2 two-component systems of a precipitate-matrix type with the relaxation temperature Tr = 665◦ C and Tr = 710◦ C, respectively, determined by the formula Tr = 0.35 Tm [28], where Tm is the minimum of the set {Tmp , Tmm }, and the melting temperature Tmp , Tmm of components of the SiC-Si3 N4 and SiC-MoSi2 two-component systems are presented in Tab. 1. Table 1. Material parameters of the SiC particle and the Si3 N4 and MoSi2 matrixes of the SiC-Si3 N4 and SiC-MoSi2 two-component systems of a precipitate-matrix, where Eq and µq is the Young’s modulus and the Poisson’s ratio, respectively, αq is a thermal expansion coefficient, KICq is fracture toughness, and Tmq is melting temperature.

Eq [GPa] µq αq [10−6 K−1 ] KICq [MPam1/2 ] Tmq [◦ C]

Particle (q = p) SiC 360 0.19 4.15 3.25 2730

Matrix (q = m) Si3 N4 MoSi2 310 271 0.235 0.165 2.35 7.1 5.25 3 1900 2030

Fig 10 shows a dependence of the radial stress p (see Eq. (90)) on the particle volume fraction v ∈ (0, π/6i (see Eq. (1)), where the radial stress p < 0 (see Fig 10a) and p > 0 (see Fig 10b), acting on the SiC-Si3 N4 and SiC-MoSi2 particle-matrix boundaries, is tensile and compressive due to αm < αp and αm > αp , respectively. Substituting numerical values of material parameters of the SiC-Si3 N4 two-component system to Eqs. (99)–(101), we get [Wcp (x12 , R) +


a

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b

Figure 10. The tensile and compressive radial stress p < 0 (a) and p > 0 (b) (see Eq. (90)) as a function of the particle volume fraction v ∈ (0, π/6i (see Eq. (1)), acting on the the SiC-Si3 N4 and SiC-MoSi2 particle-matrix boundaries, for the relaxation temperature Tr = 665◦ C and Tr = 710◦ C, respectively, for the final temperature Tf = 20◦ C of a cooling process, and regarding the material parameters in Tab. 1.

Wcm1 (x12 , R)]x12 =0 < [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =R = [Wcm2 (x12 , R)]x12 =R , and with regard to αm < αp , a crack is accordingly initiated in the spherical particle for the radius R = Rpc2 and consequently propagated from the particle surface to the particle centre for R > Rpc2 (see Fig. 8b), where the critical particle radius Rpc2 = Rpc2 (v) as a function of v ∈ (0, π/6i (see Eq. (1)), shown in Fig. 11a, is determined by Eq. (129) for a = R and q = p, using a suitable numerical method as analysed in Section 6.3.. On the one hand, the condition (129) includes the curve integrals Wcp (x12 , R) and Wcm1 (x12 , R) both dependent on the variable ϕ ∈ h0, π/4i, but on the other hand, the critical particle radius Rpc2 exhibits an insignificant dependence on ϕ ∈ h0, π/4i. Accordingly, the insignificant dependence on ϕ ∈ h0, π/4i is also observed in case of the function f2p = f2p (x12 ) for x12 ∈ hx02p , Ri (see Fig. 8b), describing a shape of the crack in the spherical particle propagated from the particle surface to the particle centre for R > Rpc2 . Finally, with regard to the interval v ∈ h0.1, 0.2i of the SiC particle in the SiC-Si3 N4 two-component system, considering Rpc2 = 11 µm for v = 0.15, Fig. 12a shows the crack in the SiC spherical particle for v = 0.15, R = 15 µm,

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a

b

Figure 11. The critical particle radii Rpc2 (a) and Rmc2 (b) as functions of the particle volume fraction v ∈ (0, π/6i (see Eq. (1)), as well as a reason of the crack initiation in the SiC particle and the MoSi2 matrix of the SiCSi3 N4 and SiC-MoSi2 two-component systems, where the crack initiation is consequently followed by the crack propagation from the SiC particle surface to the SiC particle centre and to the MoSi2 cell surface for R > Rpc2 and R > Rmc2 , for the relaxation temperature Tr = 665◦ C and Tr = 710◦ C, respectively, respectively, for the final temperature Tf = 20◦ C of a cooling process, and regarding the material parameters in Tab. 1. ϕ = 0 (see Fig. 5) described by the function f2p = f2p (x12 ) in the form f2p = 18.02 − 5.18 x12 + 0.43 x212 − 0.01 x312 , x12 ∈ hx02p , Ri , v = 0.15, R = 15 > (Rpc2 )v=0.15 = 11, x02p = 12.2, ϕ = 0, [x12 , x02p , f2p , R, Rpc2 ] = [µm] .

(137)

Similarly, substituting numerical values of material parameters of the SiC-MoSi2 two-component system to Eqs. (99)–(101), we get [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =0 < [Wcp (x12 , R) + Wcm1 (x12 , R)]x12 =R = [Wcm2 (x12 , R)]x12 =R , and with regard to αm > αp , a crack is accordingly initiated in the cell matrix for the radius R = Rmc2 and consequently propagated from the particle surface to a surface of the cell matrix for R > Rmc2 (see Fig. 9b), where the critical particle radius Rmc2 = Rmc2 (v) as a function of v ∈ (0, π/6i (see Eq. (1)), shown in Fig. 11b, is determined by Eq. (129) for a = R and q = m, using a suitable numerical method as


a

293

b

Figure 12. The cracks in the SiC particle (a) and the MoSi2 matrix (b) of the SiC-Si3 N4 and SiC-MoSi2 two-component systems, described by the functions f2p = f2p (x12 ) and f2m = f2m (x12 ), f3m = f3m (x12 ) (see Figs. 8b, 9b) given by Eqs. (137) and (138), respectively.

analysed in Section 6.3.. The critical particle radius Rmc2 along with the functions f2m = f2m (x12 ) and f3m = f3m (x12 ) for x12 ∈ hx02m , Ri and x12 ∈ hR, x03m i, respectively, connected in the point C (see Fig. 9a), also exhibit an insignificant dependence on ϕ ∈ h0, π/4i. Finally, with regard to the interval v ∈ h0.1, 0.2i of the SiC particle in the SiC-MoSi2 two-component system, considering Rmc2 = 7.1 µm for v = 0.15, Fig. 12b shows the crack in the MoSi2 cell matrix for v = 0.15, R = 9 µm, ϕ = 0 (see Fig. 5) described by the functions f2m = f2p (x12 ), f3m = f2p (x12 ) in the forms

f1m = 89796.8 − 29974.42x12 + 3334.97x212 − 123.67x312 , x12 ∈ hx02m , Ri f2m = −519.56 + 168.49x12 − 18.04x212 + 0.639x312 , x12 ∈ hR, x03m i v = 0.15, R = 9 > (Rmc2 )v=0.15 = 7.1, x02m = 8.93, x02m = 10.135, ϕ = 0, [x12 , x02m , x03m , f2m , f3m , R, Rmc2 ] = [µm] .

(138)

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Acknowledgments This work was realized within the frame of the project ’Centre of Excellence of Advanced Materials with Nano- and Submicron- Structure’, which is supported by the Operational Program ’Research and Development’ financed through European Regional Development Fund.

References [1] Ivanˇco, V.; Kub´ın, K.; Kostolný, K. Finite element method I; Elfa: Koˇsice, SK, 1994; Vol. 1, pp 225–243 (in Slovak). [2] Ivanˇco, V.; Kub´ın, K.; Kostolný, K. Program COSMOS/M; Elfa: Koˇsice, SK, 2000; pp 36–45 (in Slovak). [3] G. Eggenberger, V. Ivanˇco, K. Kostolný, Comput. Mater. Sci. 2006, 37, 599-602. [4] Balázsi, C., Sedlácˇ ková, K., Czigány, Z. Composites Sci. Technol. 2008, 68, 1596–1599. [5] Balázsi, C., Sedlácˇ ková, K., Llobet, E., Ionescu, R. Sens. Actuators B: Chem. 2008, doi:10.1016/j.snb.2008.02.006. [6] Balázsi, C., Wang, L., Zayim, E.O., Szilágyi, I.M., Sedlácˇ ková, K., Pfeifer, J., Tóth, A.L., Gouma, P.I. J. Europ. Ceram. Soc. 2008, 28, 913– 917. [7] Balázsi, C., Wéber, F., Kövér, Z., Horváth, E., Németh, C. J. Europ. Ceram. Soc. 2007, 27, 1601–1606. [8] Balázsi, C., Wéber, F., Kövér, Z., Shen, Z., Kónya, Z., Kasztovszky, Z., Vértesy, Z., Biró, L.P., Kiricsi, I., Arató, P. Current Appl. Phys. 2006, 6, 124–130. [9] Eshelby, J.D. Proc. Royal Soc. London A 1957, 241, 376–396. [10] Fischer, F.D.; Böhm, H.J. Acta Mater. 2005, 53, 367–374. [11] Fischer, F.D.; Böhm, H.J.; Oberaigner, E.R.; Waitz, T. Acta Mater. 2006, 54, 151–156.


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[12] Mori, T.; Tanaka, K. Acta Metall. 1973, 21, 571–574. [13] Dong, C.Y.; Lo, S.H.; Cheung, Y.K. Eng. Analysis with Boundary Elements 2004, 28, 123–130. [14] Ben-Menahem, A.; Gibson, R.L. Geophys. J. Int. 1995, 122, 249–265. [15] Ceniga, L. J. Thermal Stresses 2004, 27, 425–432. [16] Mizutani, T. J. Mater. Sci. 1996, 11, 483–494. [17] Brdiˇcka, M.; Samek, L.; Sopko, B. Mechanics of Continuum; Academia: Prague, CZ, 2000. [18] Ceniga, L. In Ceramics and Composite Materials: New Research; Caruta, B.M.; Ed.; Nova Science Publishers: New York, USA, 2005; pp. 147-195. [19] Ceniga, L. In New Developments in Materials Science Research; Caruta, B.M.; Ed.; Nova Science Publishers: New York, USA, 2007; pp. 139-180. [20] Ceniga, L. Analytical Models of Thermal Stresses in Composite Materials I; Nova Science Publishers: New York, USA, 2008. [21] Ceniga, L. Analytical Models of Thermal Stresses in Composite Materials II; Nova Science Publishers: New York, USA, 2007. [22] Davidge, R.W.; Green, T.J. J. Mater. Sci. 1968, 3, 629–634. [23] Diko, P. Supercond. Sci. Technol. 1998, 11, 68–72. [24] Diko, P.; Krabbes, G. Supercond. Sci. Technol. 2003, 16, 90–93. [25] Rektorys, K. Review of Applied Mathematics; SNTL: Prague, CZ 1973. [26] Li, S.; Sauer, R.A.; Wang, G. J. Appl. Mech. 2007, 74, 770–783. [27] Li, S.; Sauer, R.A.; Wang, G. J. Appl. Mech. 2007, 74, 784–797. [28] Skoˇcovský, P.; Bok˚uvka, O.; Palˇcek, P. Materials Science; EDIS Technical ˇ University: Zilina, SK, 1996.

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ˇ [29] Ocel´ık, V.; Csach, K.; Kasardová, A.; Miˇskuf, J.; Svec, P.; Kriˇstiaková, K.; Mat’ko, I. Scrip. Mater. 1996, 35, 1301–1306. [30] Diko, P.; Kasardová, A.; Miˇskuf, J.; Csach, K. J. Alloys and Compounds 1992, 186, 361–367. [31] Jur´ıková, A.; Csach, K.; Miˇskuf, J.; Ocel´ık, V. Czech. J. Phys. 2004, 54, 129–132. [32] Liebowitz, H. Fracture I; Academic Press: New York, London, 1968. [33] Liebowitz, H. Fracture II; Academic Press: New York, London, 1968.


Chapter 9

FINITE ELEMENT THERMAL ANALYSIS OF CERAMICS MATRIX COMPOSITES M.A. Sheikh∗ School of Mechanical, Aerospace and Civil Engineering The University of Manchester, Sackville Street Building Manchester M60 1QD, United Kingdom

Abstract Modelling and analysis of a unique geometrically representative Unit Cell has been shown here as the key to predicting the macro thermal transport behaviour of composites, which otherwise requires the employment of a vast experimental infrastructure. Sophisticated materials, such as woven Ceramic Matrix Composites (CMCs), have very complex and expensive manufacturing routes, used by just a few research organizations. This broadens the scope of a modelling strategy to be adopted for the characterization of all possible material designs with various possible constituent volume fractions by using a commercial FE code such as ABAQUS. The variation of material constituents can be incorporated in the Unit Cell model geometry with subtle manipulation of key parameters dictated by quantitative SEM morphological data. Two CMC material systems have been modeled in the present study. The first material has been analysed with a focus on the homogenization of microscopic constituent material properties into the macroscopic thermal transport character. The actual set of property data used for the Unit Cell of this material is obtained from the ∗

E-mail address: [email protected]. Tel: +44 (0) 161 306 3802; Fax: +44 (0) 161 306 3803.

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M.A. Sheikh cumulative property degradation results extracted from the analyses of three submodels based on the material’s unique porosity data. After validating the modeling methodology through a comparison with the experimental data, a geometrically more challenging CMC is modelled with a detailed incorporation of its morphological complexity in order to predict its macroscopic thermal transport behavior. Finally, it is shown how these models can be more efficiently analysed in a multi-processing parallel environment.

1. Introduction Ceramic Matrix Composites (CMCs) are considered to be possible replacement for metallic super-alloys in high-temperature parts of aero-engines and heavy-duty gas turbines, and for applications such as nozzles in rocket engines. Increased operating temperatures from 900-1200°C for coated superalloys to above 1300°C for CMCs have the potential to achieve higher thermal efficiencies and lower emissions [1]. Such applications require good thermal properties for heat transfer in CMC engine components. Compared to metallic alloys, a deficiency of CMCs is the degradation of thermal transport properties due to internal damage. An ability to predict thermal transport in CMCs is a primary requirement at the design stage. The presence of damage and cracks can be introduced in either manufacturing or in service. Firstly, damage during manufacturing is introduced as a result of the different thermo-mechanical properties of the constituent materials, which during cooling introduce thermal gradients, thermal stresses, localized failure and hence damage. This manifests itself after cooling as micro-porosity. Secondly, damage is created in service [2], [3] by mechanical overloads, fatigue, time-dependent and environmental effects. As damage evolves a limiting condition is reached when an engineering component becomes mechanically unserviceable. It then requires either repair or replacement. The dominant effect on material serviceability is that the thermal transport properties are dramatically reduced due to the evolution of damage, which can be highly directional. The overall impact is to render the component thermally unserviceable due to impaired thermal efficiency. There is a very strong coupling between mechanical behavior and thermal properties which is not currently well understood; and is not capable of being accurately predicted at the present time. Hence one of the drivers for current research on CMCs is the need to describe and predict these effects at the design stage. This is to be achieved through the establishment of effective modeling procedures. The modelling of CMCs requires a number of factors to be taken into account, which influences their thermal transport, such as the architecture of the

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composite, the properties of constituent material, and the influence of defects [4]. Early approaches to the thermal finite element modelling were two dimensional; for example unidirectional long fibre composites have been studied by Lu and Hutchinson [5] for longitudinal thermal transport, and by Klett et al.[6] for longitudinal and transverse thermal transport. The main limitation of such models is their simplicity, which does not reflect the complexities of real composites. A complex weave model of a plain weave CMC was presented by Sheikh et al. [7]. This model was 3-dimensional, and represented a relevant development towards the modelling of complex composites architectures. They introduced the effect of directionality in thermal transport by introducing the individual properties of fibre and matrix. However, the model was deficient since the influence of initial porosity was not taken into account. A developmental modelling strategy is set, which took account of the following factors: • • •

categorise manufacturing or initial porosity; create an ability to numerically model it; and subsequently, further develop this to a point where growth and coalescence of initial porosity due to applied loadings and thermal strains can be accurately predicted; and, in addition, couple this with the degradation of mechanical and thermal properties.

But first, for each porosity classification, finite element analysis techniques are used to quantify the effect of each class of porosity on the spatial heat transport properties assessed at the level of a micro Unit Cell. In the analysis care is taken to accurately model porosity volume fractions and characteristic defect lengths. However, thermal properties determined using one model is included in subsequent models. It is in this way that the synergy between different classes of porosity is assessed. The two CMC materials used for modeling here are: (1) DLR-XT, a [0/90] plain weave laminate, and (2) HITCO, a complex 8-Satin Weave CMC. Geometric models for both are built from SEM Micrographs. For DLR-XT a Unit Cell model is built using 4 quarter parts which assemble together to form the Unit Cell with fibre tows and matrix together. In contrast, the Hitco Unit Cell is created as a single part since this Representative Volume Element (RVE) is the smallest and unique geometric entity that cannot be further simplified. For the thermal characterization of the two chosen materials through extensive finite element modelling, two thermal properties are evaluated using steady-state and transient analysis respectively. First is Thermal conductivity

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which is one of the driving forces in designing materials for thermal applications. The other is thermal diffusivity, which is an important parameter for controlling thermal transport, defined as the ratio of a material’s capacity to conduct heat versus its capacity to store it. Experimental work is conducted for validation purposes by making thermal diffusivity measurement by the laser flash method [8]. The materials are thermally analysed with and without the external mechanical load in order to establish a correlation between the mechanical damage on a CMC causing degradation of its thermal diffusivity. With the increase in the geometric complexity of the CMCs, the requirement of the computing resources rises considerably. This is reiterated by the current modeling effort with HITCO 8-harness satin weave geometry. It is shown that with the multiplication of the Unit Cell (each containing around a million elements) across the lamina and then the laminate, demand for computational power increases drastically. Special arrangement is therefore required for conducting analysis of such large FE models. The solution is shown here to be the use parallel processors in a High Performance Computing (HPC) environment.

2. Ceramic Matrix Composites (CMCs) Generally, ceramics cover a wide variety of non-metallic, inorganic compounds that are frequently processed at high temperatures. Starting compounds of ceramics can be borides, carbides, nitrides, oxides, silicides, phosphides, chalcogenides and their complex compounds, both natural and synthetic [9]. Compared to metals these compounds have higher melting temperatures, elastic moduli and hardness, and lower densities, and electrical and thermal conductivities. Ceramics can be broadly grouped into two classes, namely, conventional ceramics and high performance ceramics. Technical ceramics, which are employed for the fabrication of CMCs, exhibit extraordinary resistance to heat, chemicals and wear [10]. Fibres used in ceramic matrix composites fall into three general categories based on their diameter: monofilaments, textile fibres and whiskers. In addition, reinforcements in the form of particulates and platelets are also being utilised in ceramic composite designs. In ceramic composites the reinforcements usually increase the strength indirectly by increasing the toughness of the matrix. The selection of matrix materials for ceramic composites is strongly influenced by thermal stability and processing considerations. Common matrix materials include oxides, carbides, nitrides, borides and silicides.


301

Engineered ceramics are used in thermal and structural applications requiring high temperature resistance, high hardness and chemical inertness. Applications that exploit the thermal structural properties of ceramics commonly include cutting tool inserts, wear resistant components, ballistic armour, heat exchangers, burner tubes, prosthetics, dental implants, heat engine components and thermal barrier coatings [11].

2.1. Manufacturing of CMCs CMCs can be processed either by conventional powder processing techniques used for making polycrystalline ceramics or through novel techniques specifically developed for such purposes [11]. Since this chapter includes the modelling of a composite manufacturing porosity, it is instructive to outline some of the important processing techniques [2], [12], utilised in the fabrication of CMCs. Technical details of the manufacturing processes as well as the relative merits and disadvantages of the different fabrication methods are also discussed.

2.1.1. Infiltration Infiltration of a preform constructed of reinforcement can be executed with a matrix material in solid, liquid or gaseous form. Liquid infiltration demands proper control of the fluidity of liquid matrix. The process results in a high density matrix without pores in the matrix. Besides, only a single processing step is required to attain a homogeneous matrix. However, high processing temperatures can cause unfavourable chemical reactions between the reinforcement and the matrix. Ceramics have relatively high melt viscosities, which render the infiltration of the preform a difficult operation. In addition, thermal expansion mismatch between the reinforcement and the matrix, a large temperature interval between the processing and room temperature as well as the low strain to failure of ceramics are factors that impede the production of a CMC devoid of cracks. Similarly, the matrix is prone to cracks due to the differential shrinkage between the matrix and the reinforcement on solidification. Figure 1 schematically represents the liquid infiltration process.

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Figure 1. Schematic diagram of the infiltration technique.

2.1.2. Polymer Infiltration and Pyrolysis Ceramic matrix in a composite can be accomplished from the usage of polymeric precursors. Polymer infiltration and pyrolysis (PIP) constitutes an attractive processing route because of the relatively low cost as compared to the generally high costs of CMC fabrication. Apart from the cost factor, the process is able to maintain small amounts of residual porosity and minimal degradation of fibres. Furthermore, this approach allows near net-shape moulding and fabrication of composites near their full densities. In PIP, fibres are infiltrated with an organic polymer, which is heated with raised temperatures and pyrolysed to form a ceramic matrix. Due to the relatively low yield during the conversion from polymer to ceramic, multiple infiltrations are necessary to obtain an acceptable density of the composite. Polymeric precursors for ceramic matrices permits the usage of conventional polymer composite fabrication technology that is readily available and capitalise on processes used to make polymer matrix composites. Complex shape forming and fabrication are possible and the fairly low processing and pyrolysing temperatures prevent fibre degradation and the formation of unsolicited reaction products at the fibre/matrix interface.


303

2.2. Thermal Behaviour of CMCs The behaviour of composite materials is often sensitive to changes in temperature. Such trends are displayed due to the temperature-dependent response of the matrix to an applied load. Also, alterations in temperature can induce internal stresses because of a differential thermal contraction and expansion. These stresses ultimately affect the thermal expansion of the composite. Furthermore, thermal gradients are especially deleterious to CMCs as they possess relatively lower thermal diffusivities and are inherently heterogeneous. A thermal gradient is inversely related to thermal diffusivity (α) of a material, which is in turn related to thermal conductivity (k), specific heat (Cp) and density (ρ) of the material [2] as:

(1) During the cooling from a typically high processing temperature of CMCs, very high thermal stresses ensue as a result of the thermal mismatch between the reinforcement and the matrix. In the instance of a fibrous composite, an interfacial pressure develops during the cooling phase. Thermal stresses generated are dependent on the reinforcement volume fraction ‘V’, reinforcement geometry, thermal mismatch (αF < αM) temperature interval ‘ΔT’ and also the modulus ratio (EF/EM). Generally, during cooling, the matrix tends to contract more than the reinforcement putting the reinforcement in compression (αF < αM). This mismatch is difficult to eliminate, but it can be manipulated to extract desirable characteristics of the composite. Through a prudent choice of components, it is feasible to acquire a favourable residual stress pattern rather than adverse final attributes at the end of the processing. In CMCs, crack deflection at the fibre/matrix interface is beneficial for the augmentation of toughness in the composite. Frequently, interfacial coatings, that are primarily mechanical in nature, are introduced in CMCs to optimise stress distribution and bonding at the fibre/matrix interface. The longitudinal expansion coefficient of a composite is given by:

(2)

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M.A. Sheikh Similarly, the transverse expansion coefficient of a composite is expressed

by: (3) where: (4) and ‘V ’ is the volume fraction, ‘E ’ is the Young’s Modulus, ‘ν ’ is the Poisson’s ratio and the subscripts ‘C ’, ‘F ’ and ‘M ’ refer to composite, fibre and matrix, respectively [13]. For a unidirectional fibre reinforced composite, a prediction of thermal conductivity in the longitudinal (l) and the transverse (t) directions is proposed by Behrens [14] as:

k1 = k cl = k FlV F + k M V M

(5)

k Ft k M k FtV f + k M VM

(6)

k 2 = kct =

where, subscripts ‘1’ and ‘2’ denote the principal directions of the unidirectional composite. In case of woven fabric-reinforced composites, such formulation is not applicable since non-homogenous material phases are intermixed and intertwined in one volume. One direct solution to this problem is finite element simulations which entail the geometric details, property data and detailed boundary conditions to set up the model. Because of the inherent heterogeneity of a composite and thermal property mismatch in constituents, even a uniform temperature change will result in thermal stresses. In absence of any variable temperature gradient the thermal strain is [2]:

(7)


305

2.3. Experimental Measurements of Thermal Diffusivity The theory and experimental procedure for the thermal diffusivity measurement by the flash method requires the surface of a small sample being irradiated with a laser pulse, and the temperature response at the opposite surface recorded. The recorded data would be the temperature- time profile of the rear face. The plot for temperature rise ΔT ΔTmax measurement against time would highlight the peak value of 1 at a certain time. Considering the initial temperature condition of 20oC, the time for half temperature rise value 0.5 would be noted as half-rise time t1 2 . For each test these are obtained to calculate the thermal diffusivity α using Equation 8 for a thin disc specimen, which has one face uniformly irradiated in one-dimensional (1D) heat flow:

⎛

⎞

2 α = 0.139 ⎜ L t ⎟ ⎜ 1 ⎟

⎝

2

(8)

⎠

where L is the sample thickness and t1 2 is the experimentally obtained half-rise time.

Figure 2. Schematic diagram of apparatus in test rig 1.

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Figure 3. Schematic of apparatus in test rig 2.

Two experimental test rigs have been used in the current analysis for the measurement of thermal diffusivity. These will be referred to as Test Rig 1 and Test Rig 2. A schematic diagram of apparatus in Test Rig 1 is shown in Figure 2, and is fully described in [8]. This rig has been used to test materials for their thermal diffusivity over a wide temperature range (300 - 3000 K). It requires thin specimens, which are normally subjected to one-dimensional heat flow - but also allows both radial and axial heat flow by irradiating only the central region of larger specimens – Figure 2. Test Rig# 2 which is shown in Figure 3, is similar to Test Rig 1 in terms of pulse source, detection and amplification system, and recording and analysis facility. It is designed, however, to measure thermal diffusivity whilst the specimen is mechanically loaded. The rectangular plate specimens are held in the grips of a mechanical testing machine which allows continuous monitoring of the externally applied mechanical load and induced strains. Information about any possible bending of the specimen is also obtained to ensure that the measurements of thermal transport properties correspond to a case of purely tensile loading only.

3. Modelling of a Plain Weave CMC 3.1. Manufacturing Process One of the materials under consideration [15] is DLR-XT, a 10 laminate CMC material developed by the German Aerospace research Establishment,


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Stuttgart, Germany for aerospace applications involving high-temperature gas turbine components. T300 carbon fibres manufactured from carbon are arranged in tows, which have been used to form a plain weave (Figures 4 and 5) to generate a single laminate. Ten such laminates are assembled to form a sheet of the DLRXT composite.

Figure 4. DLR-XT - schematic view of a composite macro unit cell of a single laminate.

Figure 5. Diagrammatic plan view of a plane weave configuration.

The laminate bundle is infiltrated with a polymer, which is thermally decomposed to leave a carbon char. This in turn is infiltrated with liquid silicon, which reacts with the carbon char to give silicon carbide, which forms the matrix around the carbon fibre bundles. Ten sheets of the woven carbon fibre fabric are assembled between forming platens; which are then housed within a chemical vapour infiltration (CVI) chamber. The assemblage is then CVI processed, and the effect of temperature and pressure results in the fibres within each tow being embedded within a carbon matrix, created by diffusion of carbon from the fibres. The tows are, in turn, embedded within the infiltrated silicon carbide matrix. The process involves timedependent temperature and pressure distributions within the structure of the laminate. Figure 5 schematically represents a plain weave arrangement and

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Figures 4 and 6 illustrate the geometric configuration and coordinate global system of this material.

Figure 6. Schematic drawing of (a) a material laminate; (b) a lamina element and global coordinate system.

3.2. Classes of Porosities Internal damage in the form of cracks and voids can be introduced either during manufacturing or in service. During manufacturing, damage is caused by a mismatch of thermo-mechanical properties of the constituent materials, thus inducing thermal gradients, thermal stresses, localised failure and hence damage when cooled. This manifests itself after cooling as micro-porosity. Additionally, damage is created in service by mechanical overloads, fatigue, time-dependent and environmental effects. As damage evolves, a limiting condition is reached when an engineering component becomes mechanically unserviceable. It then requires either repair or replacement. The dominant effect on material serviceability is that the thermal transport properties are dramatically reduced due to the evolution of damage, which can be highly directional, ultimately rendering the component thermally unserviceable due to impaired thermal efficiency. Despite the existence of a very strong coupling between mechanical behaviour and thermal properties, these relations are currently ill comprehended and are not capable of being accurately predicted at the present time. Paolo et al [16] have achieved the quantification as well as classification of initial porosity through the usage of optical electron microscopy and SEM. These techniques have been selected for their ease of use and availability; and, for their


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potential to cross check observations and measurements. 4 classes of porosities have been identified and schematically shown in Figure 7.

3.2.1. Inter-Fibre Micro-Porosity (Class A) This type of porosity occurs between adjacent fibres contained within a tow. It comprises either of a series of voids or large cracks between the fibres. These can be seen within as black dots A1 or cracks parallel to fibre tows A2 in Figures 8(a)-(c) and schematically represented in Figure 7.

Figure 7. Schematic drawing of a general plane orthogonal to either the X or the Ydirection illustrating the four classes of porosity. [Circled letters denote the porosity type].

These two kinds of porosity are mainly generated by the manufacturing process. The spheroidal pores, shown in Figure 8(b), are the result of incomplete infiltration due to shrinkage from the processing temperature which is around 1550°C for molten silicon impregnation. The shrinkage de-bonding is due to different thermal contractions of the fibre and the matrix during cooling. This generates thermal stresses across the interface, which in turn causes interface debonding.

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a

b

c Figure 8(a)-(c). Three different images of orthogonal planes obtained using optical microscopy. (a) Plane normal to laminate and the Z axis, (b) plane normal to X axis, and (c) plane normal to Y axis. The locations of four different classes of porosity (A, B, C, D) are indicated.

3.2.2. Trans-Tow Cracks (Class B Porosity) This class of porosity may be observed as cracks B in Figure 8(a)-(c). The porosity is comprised of cracks, which run through the tows in planes parallel to the fibres. They occur in planes which are orthogonal to X, Y and Z directions. Porosity classes A and B are also shown schematically in Figure 7. Cracks of this type are formed during manufacture as a consequence of differential cooling, coupled with the different mechanical/thermal properties in adjacent orthogonal tows.

3.2.3. Matrix Cracks (Class C Porosity) This class of porosity is composed of cracks, embedded in the matrix, which surround the tow in planes perpendicular to the tow axes. C matrix cracks run


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circumferentially around the tows. The generation of the matrix cracks occurs during cooling of the composite due to tensile stresses generated in the matrix arising from the difference in the values of the linear coefficients of thermal expansion for the matrix and the fibre parallel to its axis. Class C porosity is schematically given in Figure 7. Figure 8(b) is a section taken in the Y-Z plane which cuts through the tow centres running in the Y direction. This is shown in Figure 10(a). Likewise, Figure 8(c) is a section taken in the X-Z plane which cuts through the silicon carbide matrix encapsulating the tows running in the Xdirection. Hence it can be seen that the Class C matrix cracks run circumferentially around the tows as shown schematically in Figure 10(b). In addition, Figure 9 shows a SEM primary electron image of Class C defects in a plane normal to the X direction.

Figure 9. Class C porosity (matrix cracks): SEM primary electron image of a plane normal to the X direction. Region (a) has fibres running normal to the plane; region (b) is silicon carbide matrix; and region (c) has fibres running parallel to the plane. The matrix region (b) surrounds the fibre tow (c). Two Class C periodic matrix cracks are shown in region (b) which are circumferential to the in-plane tow (c).

3.2.4. Denuded Matrix Regions (Class D Porosity) This type of porosity is due to the large voids, which occur at the intersection of four orthogonal tows. This class of porosity occurs during manufacture due to difficulties in the CVI process, these are largely due to the difficulty and time required to fill the large void which occurs at the intersection of four tows. Porosity of such type is illustrated in Figure 8(a)-(c), Figures 7 and 11.

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Figure 10(a). Two orthogonal tows sectioned normal to the X-axis in a Y-Z plane cutting through the centre of the lower tow.

Figure 10(b). Two orthogonal tows sectioned normal to the Y axis in a X-Z plane cutting through the silicon carbide matrix encapsulating the lower tow.

Figure 11. A schematic view of a section through a unit cell showing one unit of Class D porosity (spotted region).

3.3. Quantification of Porosity Various classes of porosities for DLR-XT are quantified by utilising a porosimeter as well as detailed observations of optical micrographs. The


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porosimeter received its input data in the format of a digital two dimensional surface image. A software was then used to edit black and white images by the selection of a group of varying levels of greyness.Discernment between levels of greyness and porosity area fractions was calibrated with standard reference test pieces. Eventually, the result generated was an area fraction expressed as a percentage. Pertinent area fractions in connection with porosity classes A and D are quantitatively determined with the porosimeter. Optical micrographs were used in the case of porosity classes B and C.For these two classes of porosities, the length scale is that associated with the height of the tow measured in the Zdirection. This is equal to the diameter of the circle inscribed in the tow, which corresponds to the tow diameter. In Class B porosity, cracks are characterised by their frequency (number of cracks per tow), their length in the tows’ crosssectional plane and their depth parallel to the tow axis. A Class C porosity crack is characterised by its length, depth and periodicity. As these cracks surround the entire tow, the length is equivalent to the tow circumference, hence related to the average tow diameter. All these measurements are summarised in Tables 1 and 2. Table 1. Area fraction for Class A and D porosity

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M.A. Sheikh Table 2. Crack geometry and periodicity for Class B and C porosities

3.4. Thermal Transport Modelling of DLR-XT This first 3-dimensional model for DLR-XT (C/C – SiC CMC) is developed from the SEM micrographs (Figure 12) and represents a relevant development towards the modelling of complex composites architectures. In this model the effect of directionality in thermal transport is taken into account by introducing the individual properties of fibre and matrix. However, the model, as shown in Figure 13, is deficient since the influence of initial porosity is not taken into account. Hence, for each porosity classification, finite element analysis techniques is used to quantify the effect of each class of porosity, discussed in section 3.2, on the spatial heat transport properties assessed at the level of a micro Unit Cell. In the analysis care is taken to accurately model porosity volume fractions and characteristic defect lengths, discussed in section 3.3 and given in Tables 1 and 2.


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Figure 12. Generation of a Geometric Model using SEM Micrographs.

Figure 13. Unit cell geometry assembled from 4 quarter parts having a fibre volume fraction of 65%.

The thermal properties determined using one model is included in subsequent models. It is in this way that the synergy between different classes of porosity is assessed. Figure 15 shows the flow chart for thermal transport modelling of this DLR-XT CMC material using sub-models shown in Figure 14 and incorporating porosity type D, shown in Figure 9.

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Figure 14. Porosity sub-models along with their respective FE meshes.

Figure 15. Flow chart for thermal transport modelling of DLR-XT CMC Material.


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3.4.1. Thermal Properties Thermal conductivity is one of the driving forces in designing materials for thermal applications. In a material, heat flow is proportional to the temperature gradient with the constant of proportionality being the thermal conductivity. Its general form is

q i = − k ij

dT dxj

(9)

where qi is the heat flux and k ij is the thermal conductivity. This is a secondorder tensor and in most cases a symmetric one. Another important parameter for controlling thermal transport is thermal diffusivity α . It is defined as the ratio of a material’s capacity to conduct heat versus its capacity to store it. It is related to thermal conductivity k , specific heat C p and density ρ as:

k = α ρ Cp

(10)

Relevant thermal properties of CMC constituents, shown in Table 3, along with air for the pores or cracks are employed for the thermal transport models. Table 3. Standard thermal property values of constituent materials Material Carbon Fibre Transverse Carbon Fibre Longitudinal

k (W m-1 K-1)

ρ

(kg m-3)

C p (x10-6 )(J kg-1 K-1)

4

1928

921

40

1928

921

Carbon Matrix

10

1800

717

SiC Matrix

70

3200

1422

0.001

1

1

Air

Finite-element methods for determining the thermal transport properties of solids are based on the two thermal analyses; Steady State and Transient. The

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steady state thermal analysis using FEM involves applying a temperature gradient ΔT Δx across the composite section in a 1D heat flow simulation. Using Fourier Law, k x , is given as:

kx = qx

Δx ΔT

(11)

where q x is the overall heat flux in the x -direction calculated by integrating the nodal flux values across rear face. In the current analysis, q x is obtained using the nodal flux values (in x -direction) given by the FE solution on one of the faces where the temperature boundary condition is applied as in Figure 16. However, due to a high degree of mesh non- uniformity and a significant difference in the thermal properties of the matrix and fibre, the nodal flux values vary quite considerably across that surface, and the summation employed to calculate the overall flux as: N

qx = ∑ qi Ai i =1

N

∑A i =1

i

(12)

In the case of transient thermal analysis, a heat flux is applied to one face (Figure 17) of the composite section for a short time and the temperature history is recorded on the opposite face to simulate the experimental conditions.

Figure 16. Boundary conditions for steady-state FE analysis.


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Figure 17. Boundary conditions for transient analysis.

The temperature profile is obtained by averaging temperature across the complete rear face. This is obtained from an expression similar to Equation 12, as here qi is replaced by the nodal temperatures Ti as: N

Tav = ∑ Ti Ai i =1

N

∑A i =1

i

(13)

Tav , the average temperature, is calculated for each time step through the transient analysis and a temperature history is recorded. Assuming 1D uniaxial heat flow, the half rise time related to this average temperature value is then used in Equation 8 to calculate thermal diffusivity α and thermal conductivity k is found from Equation 10. The remaining scalar properties, density ρ and specific heat C p for the composite are determined using the rule of mixtures in Equation 14 shown in Table 4. It is important to emphasize that two different volume fractions are involved here. One is 75% Carbon fibre with in the Carbon matrix, forming the fibre tow. The other is the 65% fibre tow in the composite with remaining being SiC matrix surrounding it. Using rule of mixtures, specific heat C p and density ρ as calculated as:

ρ C = ρ f V f + (1 − V f ) ρ m ⎫

⎬ CpC = Cp f V f + (1 − V f ) Cpm ⎭

(14)

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Table 4. C p and

ρ for CMC from constituent materials’ property values

Material Property Density

ρ (kg m-3)

Specific Heat C p (x 10-6 J kg-1 K-1)

Carbon Carbon SiC Matrix Fibre Matrix 1928 1800 1832 (fibre tow) 3200 2310.000 921 717.48 870.12 (fibre tow) 1422 1063.278

Composit e

3.4.2. Thermal Analysis of DLR-XT – Strategy and Results The morphology and the geometric structure of the material has been translated into a three-dimensional macro unit cell model, as shown in Figure 13. Following that, microscopic sub-models of porosity types A, B, C, as shown in Figure 14, have been created for steady-state finite element analyses similar to that performed for the macro unit cell model. Class D porosity is then introduced in the main unit cell model, where a denuded portion of the central section, as shown in Figure 11, is created. The results of these analyses in the form of the degraded fibre tow and matrix thermal properties are listed in Table 5. Table 5. Degradation of thermal conductivity with sequential introduction of each class of porosity


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Material properties for the fibre tow and the matrix in the macro unit cell are the prescribed values tabulated in Table 5. Assignment of degraded thermal properties hence addresses the influence of initial porosities on the thermal behaviour of the material. From Table 6 an obvious degradation of the overall composite thermal conductivity is noted during the sequential introduction of each type of porosity. In addition, it is also evident that Class B porosity (trans-tow cracks) has a dominant effect on the transverse thermal conductivity, ‘k’ value for the composite whereas Class C porosity (matrix cracks) mainly affects the longitudinal ‘k’ value. To a certain extent, the latter value is also influenced by shrinkage de-bonding or Class A2 porosity. These results conclude that Class C porosity is the primary form of defect that causes the most appreciable reduction of overall spatial thermal conductivity profile of the CMC when these values are used at the macro unit cell level. Table 6. Tabulation of degraded thermal conductivity values of throughthickness and in-plane flow direction along with its overall sequential degradation in the unit cell. Values in parenthesis show the percentage reduction with respect to virgin material C Tow Thermal Conductivity (W m-1 K-1)

SiC Matrix

Macro Unit Cell

||

⊥

||

⊥

In Plane

Through Thickness

Values for Virgin Material

32.407

5.086

70

70

34.89

16.29

Values for Material with Class A Porosity

32.344

5.036

70

70

34.85

16.23

Values for Material with Class A and B Porosity

32.24

4.441

70

70

34.55

15.52

Values for Material with Class A, B and C Porosity

29.68

4.441

29.683

70

26.94

15.09

Values for Material with Class A, B, and C Porosity used in Class D Porosity affected unit cell

29.683

4.441

29.683

70

25.06

13.64

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Table 7. Comparison of 1D experimental results vs. FE analyses results for thermal conductivity, k (W m-1 K-1) Direction

Experimental

FE Modelling Steady-State Transient

Through Thickness

14.39

13.639

11.55

In-Plane

22.45

25.06

25.16

Finally, the analysis of 3D thermal transport carried at the composite macro Unit Cell level, in combination with the fourth class of porosity, the Denuded Matrix Regions, produced the final k values are presented in Table 7. A further correlation can be established between the mechanical damage on a CMC causing degradation of its thermal diffusivity through the same modelling strategy. A range of property values are given to the class-C sub-model and the eventual overall effects plotted in Figure 18. It represents the resulting degradation in thermal diffusivity from a successive increase in matrix crack density. Although this does not account for the total reduction of the thermal property derived from the experimental data, given in Figure 19, a 50% reduction is shown to have occurred due to matrix cracking.

Figure 18. Degradation of through-thickness thermal diffusivity with increasing crack density due to external mechanical loading. ‘Rf’ represents the tow diameter and ‘p’ denotes the periodicity of matrix cracks.


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Figure 19. Experimental results for the degradation of thermal diffusivity of the DLR-XT CMC material with external mechanical loading.

4. Modelling of a Complex 8-Satin Weave CMC A recognition of the generic architecture of 8 Harness Satin weave is required that bears close resemblance with the actual morphology of the material such as shown in Figure 20. The architecture shown in Figure 21(a) has been used here for modelling as it follows the sequence seen in the micrograph in Figure 20(a) which shows the top side of the sample used for the thermal diffusivity measurement in the experimental rig described earlier. It is also important to specify the stacking sequence and orientation of each lamina in the composite laminate tested since the effects of the off-set lay up of multiple laminae in a composite laminate does have effects on the analyses. Woo and Goo [17] have effectively proven this parameter as quite dominating for, at least, through-thickness measurement. But in order to establish the basic sequence of modelling and analysis methodology, a single lamina is used in this simulation. A RVE Unit Cell is chosen from this single lamina and not of the stack in a laminate which will form the next phase of this study in regards to computational challenges involved.

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Figure 20. (a) Top view (b) Side view of a 8-Satin Weave Composite laminate with arrow showing fill fibre direction.

Figure 21. Generic architecture of the 8-harness satin weave (a) Warp side (b) Fill side, arrows showing fibre tows direction.

HITCO composite architecture comprises of a laminate with 10-12 laminae of 8-harness satin weave of Carbon-Carbon fabric stacked up to form the final composite laminate with Graphite matrix. The schematic with a single lamina considered as the Unit Cell in Figure 22 has been the starting point for modelling, with various other feature details that have been closely examined being mentioned during the course of RVE Unit Cell modelling. This RVE Unit Cell model has been formed with due correlation with the two micrographs showing the weave formation from the top and the side edges captured, as shown sketched in Figure 22. The sketch also shows schematically how simultaneously these features are to be captured in modelling. It is evident from the edge micrograph in Figure 20(b)


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that a single lamina can not be isolated from the laminate stack due to the present nesting of the weave, overlapping and encroachment of the top and bottom laminae on to the middle laminae, resulting in a very compact intertwined fibre tow structure. As a start the nesting factor, more geometrically random than repeated, has been ignored in modelling in order to avoid double complexity with the weave. Figure 22 shows the schematic view of the Unit Cell that is created with the help of micrographs of the HITCO CMC composite sections through the XY and XZ planes shown in Figure 20(a) and 20(b) respectively, coupled with the generic architecture details of the 8 harness satin weave. Bright areas in Figure 20(a) denote Carbon fibre tow of the warp and dark areas denote segments of the Carbon fill fibre bundles in the composite.

Figure 22. Schematic drawing of the 8 harness satin weave material single lamina.

From within these features a 3D Unit Cell is identified, which on replicating itself across in three spatial directions produces the macro-structure of the CMC. A brief but specific outline of the modelling undertaken with ABAQUS/CAE is given here. Standard modelling procedure in ABAQUS/CAE has been sequentially detailed earlier for the DLR-XT Unit Cell. It was built using 4 quarter parts that assemble together to form the Unit Cell with fibre tows and matrix together. In contrast, the HITCO Unit Cell has been created as a single part since this RVE is the smallest unique geometric entity that cannot be further simplified or divided. For HITCO, first the fibre tows, warp and fill, are separately created and then the

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matrix region is ‘filled’ around it in the cuboidal envelope of the 3D Unit Cell outline. The key to capturing the crimp form in the 8 harness satin weave has been the ‘loft’ part modelling tool in ABAQUS/CAE which allows the accurate generation of the changing cross-section, especially at the interlacement points where the warp tows pinch the fill carbon tows and vice versa. A minimal gap is left between these two crossing tows to model the Carbon matrix by ‘inserting cell’ as seen present there in Figure 20(b). Ideally the fibre bundles are considering having a lenticular cross-section shape shown in Figure 23(a). On one hand these bundles are seen compressed at the tow crossover locations and then these same tows join their neighboring tows in the same lamina and take up a clearly rectangular shape after compaction. This transformation from a rectangular area to a lenticular one and back can only be modeled with ‘lofting’ as explained in Figure 24. ‘Loft’ function in ABAQUS/CAE allows the creation of a 3D object from a pair of 2D sketches and also with the help of the tracer path, where needed. As an example a lenticular and a rectangular area are shown in Figure 24(a) and then ‘lofted’ together in Figure 24(b).

Figure 23. (a) The lenticular area, used for the ‘sweep’ and ‘loft’ function for creating volumes, (b) same sketch used to form the crossover in the weave between warp and fill tows.

a)

b)

Figure 24. Using Loft function (a) between two surfaces, creating (b) final volume.


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Figure 25. Warp and Fill Fibre Tow geometry with the HITCO composite Unit Cell developed from the optical micrograph with fibre-volume fraction Vf =50 %.

‘Extrusion’ function used earlier in the DLR-XT Unit Cell geometry is similarly used here for generating constant cross-section volumes as seen in Figure 25 model forming straight sections between the curved sections. The ‘sweep’ function has been used to create the crossover sections shown as an example in Figure 23. The Unit Cell is shown in Figure 25 without the Carbon matrix enveloping the 8-Harness Satin cloth. In designating an RVE Unit Cell with fibre tows that exhibit orthotropic thermal properties, the regions created with ‘Loft’ and ‘Sweep’ function have been assigned special polar coordinate systems. These have been necessary in order to dictate the fibre orientation and their directional property bias as shown superimposed upon a single fill fibre tow in Figure 26. The assignment of the local polar coordinate system requires selection of a discrete volume or region. This region is then assigned a unique local coordinate system, rectangular or polar, which ever is applicable. To obtain distinct regions, it was imperative to divide bigger regions of the fibre tow into smaller ones from places where the curvature sign was changing. This can be seen in Figure 25 where partitioning sections with lines are visible in both warp and fill fibre tows that run parallel to the lenticular area outline, used for the fibre tow generation with the ‘sweep’ function. After separating these unique regions, different local coordinate systems are created with relevant selection of coordinates for centers of curvature and assigned to each curved section. The concern of the various regions thermally interacting with each other that had arisen in the DLR-XT Unit Cell does not arise here as in the present case of HITCO material, the RVE Unit Cell has been generated as a continuous volume. Therefore, there is no need for any interaction property definition within itself. The need for these interactions may arise when multiple

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RVE Unit Cells are brought together to form a larger section of the composite lamina or during the stacking to form the prototype laminate. Moreover, the HITCO composite laminate has around 8 to 12 laminae stacked upon each other in various positions with respect to specific features of the weave pattern such as the location of fibre tow crossover points. For through-thickness interaction between the laminae, again, care is needed while assigning thermal properties at the common ‘tied’ surfaces i.e., between master and slave surfaces defined with proper thermal interaction properties. An important aspect of an FE analysis has always been the selection and discretisation density of elements for bringing in the governing equation within the mesh for solution. Having the luxury to selectively ‘mesh’ individual regions, with various element types available, makes the meshing process very thorough and interesting, but at the same time, certain constraints always restrict selection from all available options. The 3D tetrahedral, linear diffusion, 4-noded DC3D4 element has been used in the complete RVE Unit Cell model.

Figure 26. Material Orientation setup with local polar coordinate system, insets showing curved ends in close-up.

In spite of the big advantage of computational economy of the structured ‘brick’ elements in ABAQUS/CAE over the tetrahedral ones, the complexity of the weave and presence of sharp intricate curves and edge slivers between the


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fibre tow and matrix regions forced the only option as the tetrahedral elements. In order to reduce the problem size, some local mesh coarsening has been suggested at places where possible. An example shown in Figure 27 does highlight this possibility but the reduction in the computational load on the hardware platform has been minimal and therefore it is not considered worth the effort. A properly meshed Unit Cell is shown with the matrix region in Figure 28 and completely without them in Figure 29 where it shows 1,300,955 elements and 240,066 nodes. An inherent complexity faced during the meshing process was the internal mesh generation around the sharp edge slivers shown in Figure 30. A local seeding bias of further refined element edge size had been determined after many trials for an optimum value in order to manage a complete element formation at these odd geometric regions.

Figure 27. Suggested local coarsening in the mesh of HITCO RVE Unit Cell.

Figure 28. Meshed Unit Cell using a total of 1,300,955 DC3D4 elements.

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Figure 29. Meshed Unit Cell shown fibre tows but without matrix region.

Figure 30. Edge slivers meshed with refined local seed adjustment.

4.1. Thermal Analysis The boundary conditions set have been shown in Figure 31 for 1D Steady State heat flow in x-direction. A temperature gradient is applied across the section of the Unit Cell to allow heat flow in the direction for which the average thermal


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transport property is required. For the determination of kx, as in this case, a temperature gradient ‘ ΔT Δx ’ is applied in the x-direction and the average heat flux, qx, is calculated for use in Eq. (12). All intermediate outer surfaces are assumed perfectly insulated ( δ T δ n = 0 ). The adiabatic condition is set by default on all surfaces by ABAQUS/CAE when the temperatures on two faces are set in the initial step. The temperature is given a value of To = 100 degrees on front face, where as for the rear face, no change is made and it remains at the default zero temperature state, creating a difference of Δ T = 100 across. In spite of the minimal availability of precise constituent material data, a selection has been made from within reasonable bounds for material thermal property values. This exercise has also established a preliminary benchmark for assessing various computational aspects in the analyses. For true comparisons, first the basic execution of a single RVE Unit Cell analysis had to be perfected. More importantly, the choice of this RVE geometry had also been geared towards the fact that the complete CMC laminate could be modeled by simple replication of its Unit Cell. This has been achieved by sequentially joining multiple Unit Cells together, as shown in Figure 27, through X- and Z-directions to form a single lamina and then stacking these in Y-direction to form the full laminate prototype. Using the chosen set of constituent material properties, a steady-state analysis has been conducted. The local coordinate system assignment has insured that the heat flux must conform to the vectors dictated by fibre tow’s orthotropic orientation system, as seen in a vector plot in Figure 30.

Figure 31. Boundary Conditions for steady-state FE thermal analysis for HITCO RVE Unit Cell.

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Figure 32. Formation of the Composite test specimen sample using multiple HITCO RVE Unit Cells.

Figure 33. Vector Plot of a single warp tow for an in-plane steady-state heat flow, the inset showing a close-up view of the interlacement point showing flow vector conforming to the curvature.

Arrows in the enlarged section of Figure 31 clearly show the desired heat flow direction following the curves of the fibre tows. The flow has been enforced by the material property orientation enforced during assignment of local coordinate


333

system for fibre tow sections bending along the tow interlacement points. A profile of in-plane heat flux for the HITCO RVE Unit Cell has been shown in Figure 32 ‘without the matrix region’ so that heat flux in the warp fibre tows is clearly visible. Amongst the expected flux contours, there are certain flux hot spots present in the plot that can also be explained. These are typically present at the edges of the rectangular tow structure that changes its cross-sectional shape from rectangular to lenticular and back i.e. places where fibre tows have been pinched due to the interlacement and so these are bent abruptly, hence the heat flow is subsequently diverted. This results in the observed flux concentration.

Figure 34. ‘In-plane’ steady-state heat flux in X-direction (arrow) for the HITCO RVE Unit Cell showing only the ‘Warp’ fibre tows.

These are also signs of less sudden changes in flow direction at the crossover locations which forces flux concentration there as well although to a less degree than the shape change phenomenon mentioned above. This behavior has been expected within such a crimp weave composite cloth and shows a thermal behavior that could be expected of a woven composite. Although this behavior may be slightly reduced with the help of certain effort in smoothing the remaining sharp cuts on the lofted regions during the modelling process but its positive impact is expected to be minimal and therefore not a worthwhile exercise.

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a)

b)

c) d)

e)

Figure 35. ‘In-plane’ Steady-State heat flux in X-direction (arrow) for the complete HITCO RVE Unit Cell with flow contours seen (a) in an in-plane slice across XZ-plane, (b) across thickness in YZ-plane, heat flowing normal to the page, (c and d) across thickness in XY-plane, (e) in fibre tows only (without matrix).

a)

b)

c)

d)

e)

Figure 36. ‘Through-thickness’ Steady-State heat flux in Y-direction (arrow) for the complete HITCO RVE Unit Cell with flow contours seen (a) with in-plane slice across XZ-plane, (b) across thickness in YZ-plane, heat flowing normal to the page, (c and d) across thickness in XY-plane, (e) in fibre tows only (without matrix).

The heat flux concentration is 25% greater than the in-plane heat flux (light green) indicating that sudden change in flow direction has been dictated by fibre tow undulations and this is how the heat is expected to flow through the composite fabric across the dominant heat flow x-direction. An overall flow


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contour summary is given in Figure 32 and Figure 33 for in-plane and throughthickness heat flow scenarios respectively for the full composite RVE Unit Cell (except in Figures 32(e) and 33(e) where matrix is removed). The arrows depict the prevailing heat flux direction in each case. It is imperative that the final results obtained bare similarity to the experimental data and this has been shown in Table 8 with limited input information for the FE analysis as far as constituent material properties are concerned. The input values taken for the modified matrix region was calculated from Equation 14 using rule of mixtures as the matrix and air in voids and cracks, comprising the bulk porosity, do have isotropic thermal conductivity. Hence an input value of kmp = 18.72 has been used based on the values of k m and k p given in Table 9. Table 8. Values of Thermal Conductivity of the Constituent Materials Material Carbon Fibre Transverse Carbon Fibre Longitudinal Carbon Matrix Air (porosity)

k (W m-1 K-1) 7 70 28 0.001

Volume Fraction (%) 50 39 11

Table 9. FE analyses compared with Manufacturer’s Thermal Conductivity Data (W.m-1K-1)

In-plane Through-Thickness

Experimental Data 28 8

Numerical Analysis (SteadyState) 25.656 (+ 8%) 9.79 (- 22%)

5. Computational Aspects It has been highlighted earlier that with the increase in the geometric complexity of the CMCs, the requirement of the computing resources has risen. This has been reiterated by the current modelling effort with 8-harness satin weave geometry, apart from the modelling challenge of the HITCO materials’ fibre tow weave pattern which alone has been massive enough. It can be reckoned that with the multiplication of the Unit Cell (each containing around a Million elements) across the lamina and then the laminate, demand for computational

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power increases drastically. Special arrangement is therefore required for conducting analysis of such large FE models. Earlier DLR-XT Unit Cell had been managed within a single-CPU desktop machine operating on Windows XP with 80 Gigabytes storage space and 512 Megabytes physical memory. It has also been observed how the machine had been swapping its hard disk for need of fast storage; as much as 1.8 Gigabytes at times, almost 4 times the physical memory. In that case, the element count had not exceeded 200,000 whereas in the present case this figure exceeds 1 Million in the HITCO Unit Cell model. Moreover, the real simulation for HITCO samples may definitely involve the analysis of a full laminate such as the one formed at the end in Figure 27. This would mean the multiplication of element count for a single RVE Unit Cell with the total number of Unit Cells used to form the laminate, e.g., 18 from the example shown in Figure 27. It is emphasized here that even in the presence of a much more powerful PC - 3 GHz, single processor, 1GB physical memory - a thermal analysis with just two Unit Cells joined together has not been successful, clearly exhibiting the limitation of a single processor PC when ABAQUS/CAE [18] is conducting a simple steady-state analysis on such a model. Table 10. Speed-up observed for, a comparison (percentage) between 1-D steady-state heat transfer analyses for thermal conductivity measurement and 1-D monotonic tensile loading for determining composite stiffness employing parallel processing Processors

Thermal Analysis (%)

Mechanical Analysis (%)

2 4 8

7 9 10

12 20 24

The next option used is an SGI Onyx 300 machine with 32 SGI R14000 MIPS processors running at 600 Megahertz. Each such processor having a peak speed of 1.2 Gigaflops makes the total peak speed of the machine of approx. 38 Gigaflops. But the system is a shared memory machine with 16 Gigabytes of physical memory, permitting both shared memory and distributed memory programming models to all users simultaneously. Although the analyses conducted on this machine have already yielded better memory management scenarios compared to single processor Windows PC, but still certain ABAQUS code limitations regarding thermal analysis mentioned in the software reference documentation has made the scope for speedup gain rather narrow. Proof of this


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behavior has been seen through initial runs conducted using increasing number of parallel processors. Some trends for ABAQUS and its solver performance are clearly seen in Table 10. Here the comparison is made with tests that were run on the same Unit Cell with a monotonic tensile loading. Increased speed up hints at the advantage gained by the use of multiprocessor platform for the mechanical analysis of HITCO Unit Cell model and larger models are considered to be solved faster here but only in the regime of mechanical loadings. For thermal application simulation, this study suggests that in order to benefit from parallelization which is inevitable therefore crucial for larger models, further investigation are necessary in improving solver performance. This has also led the present study towards another domain of parallelization specially coded for finite element modelling and it is being pursued simultaneously now.

5.1. Parallel Processing Results are now presented of thermal analysis of a benchmark heat flow problem, as defined in Figure 34, in a multi-processor environment. Figure 35 shows the analysis speed-up against the number of processors with an increasing mesh size.

Figure 37. A benchmark steady-state thermal problem for parallel processing.

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Figure 38. Results of scalability studies.

Conclusions Unit Cell Modelling technique is useful for thermal transport analysis of composites. Based on the DLR-XT Unit Cell modelling with sub-modelling of manufacturing porosities for capturing sequential material degradation and also on just bulk porosity introduction in HITCO, this approach has been shown successfully for even complex weaves like 8-harness satin. A bulk porosity of 11% has been accommodated within the matrix with reduction in the matrix thermal conductivity, managed by using the ‘rule of mixtures’. Comparison of FE analyses results from ABAQUS with the experimental results have shown good agreement, with emphasis to the extent of individual role of different porosities in DLR-XT. In material by HITCO, the variability has been seen due to the variety of stacking orientation and phase shift between the laminas. The increased degrees-of-freedom per RVE Unit Cell has also challenged the computational prowess of the present facilities, prompting the use of parallel computing wisely.


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References [1] [2] [3] [4]

[5]

[6]

[7]

[8]

[9] [10] [11] [12] [13] [14]

A.G. Evans and R. Naslain. High-temperautre Ceramic-Matrix Composites, Vol. I and II, Ceramic Transaction, 57, 1995. K. K. Chawla, Ceramic Matrix Composites, Second Edition, Chapman and Hall, 2003. D. A. G. Bruggemann. Ann. Phy., 24, 1935, p. 636. P. Del Puglia, M. A. Sheikh, An overview of thermal transport models for composites, Proceedings of XXX AIAS (Associazione Italiana per l'Analisi delle Sollecitazioni) Conf., Alghero (Italy), University of Cagliari, Dep. of Ingegneria Meccanica, 2001, pp.687-696. T.J. Lu and J. W. Hutchinson, Effect of matrix cracking on the overall thermal conductivity of fibre-reinforced composites, High-temperature Structural Materials, Editors: R. W. Cahn, A. G. Evans and M. McLean London: Chapman and Hall, London, 1996, pp.177-192. J. W. Klett, V. J. Ervin, D. D. Edie, Finite-element modelling of heat transfer in carbon/carbon composites, Composit. Sci. Technol., 59, 1999, pp. 593-607. M. A. Sheikh, S. Taylor, D. R. Hayhurst, R. Taylor, Microstructural Finite Element Modelling of a Ceramic Matrix Composite to predict Experimental Measurements of its Macro Thermal Properties, Modelling Sim Mater Sci Eng, 9, 2001, pp. 7-23. M. A. Sheikh, S. Taylor, D. R. Hayhurst and R. Taylor, Measurement of Thermal Diffusivity of Isotropic Materials using the Laser Flash Method and its validation by Finite Element Analysis, J. Phys. D: App. Phys. 33, 2000, pp. 1536-1550. V. I. Trefilov. 1995. Ceramic And Carbon Matrix Composites. London: Chapman and Hall. K. K Chawla. 1998. Composite Materials: Science and Engineering (2nd Edition). Springer-Verlag, New York. S. T. Peters. 1998. Handbook of Composites (2nd Edition).Springer Verlag. F. L. Matthews, R. D. Rawlings. 1994. Composite Materials: Engineering and Science. London: Chapman and Hall. D. Hull, T. W. Clyne. 1996. An Introduction to Composite Materials (2nd Edition), Cambridge University Press. E. Behrens, Thermal Conductivities of Composite Materials, Journal of Composite Materials 2 (1968) 2-17.

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[15] J. K. Farooqi and M. A. Sheikh, Finite Element Modelling of Thermal Transport In Ceramic Matrix Composites, Computational Materials Science, 37, pp. 361-373, 2006. [16] P. Del Puglia, M.A. Sheikh, D.R. Hayhurst, Classification And Quantification Of Initial Porosity In A CMC Laminate, Composites Part A: Applied Science and Manufacturing 35 (2004) 223-230. [17] K. Woo and N. S. Goo, Thermal conductivity of carbon-phenolic 8-harness satin weave composites, Composite Structures, vol. 66, no. 1-4, pp. 521526, 2004. [18] ABAQUS Version 6.5 Documentation, Hibbit, Carlsson and Sorenson Inc, Providence, RI, 2005.


Chapter 10

DAMAGE REDUCTION METHODS IN DRILLING POLYMERIC MATRIX COMPOSITES A. Di Ilio∗ and A. Paoletti DIMEG, University of L’Aquila, 67040 Monteluco di Roio, L’Aquila, Italy

Abstract Drilling of polymeric matrix composites may generate several kinds of damage, which can lead to unacceptable material degradation. The aim of this chapter is to present a literature review on the principal methods adopted to enhance holes quality. The major damage is certainly the delamination that can occur both on the entry and exit sides of the workpiece. The level of delamination is related to a critical thrust force value which is dependent on the workpiece material and the uncut layer from the main body of the laminate.

Introduction Hole quality can be assessed by various morphological aspects, such as waviness/roughness of the wall surface, axial straightness and roundness of the hole cross-section, and hole damage. In drilling polymeric matrix composites (PMCs), a finish comparable to metals and tight dimensional and geometric tolerances are not easily achieved due to the intrinsic nature of constituents. The ∗

E-mail address: [email protected] (Corresponding author)

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hole damage involves cracks, fiber-matrix debonding, fiber pull-out, matrix cratering, thermal damage and delamination [1]. The major damage is certainly the delamination that can occur both on the entry and exit sides of the workpiece. Since PMCs do not exhibit visible plastic behaviour, the high axial contact force between the drill and the workpiece may cause separation of plies at the exit of the drill (push-out delamination). At the entrance of the hole, delamination is caused by a peeling action and may occur if the cutting angle is too small and the cutting edge is wedged between two plies (peel-up delamination). The level of delamination is related to the thrust force; in particular, there is a thrust force threshold below which delamination is negligible. This critical thrust force at the onset of delamination is dependent on the workpiece material and the uncut layer from the main body of the laminate. Damage in terms of delamination can be measured directly using parameters such as damage width, delaminated area or delamination factor or, indirectly, through thrust force monitoring. Damage reduction can be achieved by selecting proper cutting parameters (feed rate and cutting speed), drilling conditions and tool geometry and material [2]. A high cutting speed (vr) and a relatively low feed rate (vt) can greatly improve the quality of the holes, either from the point of view of delamination or the dimensional accuracy. Low feeds in some cases improve the surface roughness due to the reduction of thrust force. In other cases drilling at lower feeds and high speed leads to increased temperature, assisted by a low coefficient of thermal conduction and a low transition temperature of plastics. The accumulated heat around the tool edge destroys the matrix stability and produces fuzzy and rough cuts. Drill diameter together with feed rate strongly affect the delamination factor as greater diameter values yield thrust force increase due to the growth in shear area. The choice of drilling conditions concerns the application of particular cutting methods, such as the use of a pre-heated tool, the pre-drilling technique, the vibration-assisted drilling, the presence of a support under the workpiece and the employment of a damper for unsupported drilling. Conventional high speed steel (HSS) twist drills are used as much as cemented tungsten carbide (WC) drills, while polycrystalline diamond (PCD) tools are seldom employed. Particular attention is paid on the necessity of developing tools with special geometry in order to achieve best performances.

Damage Reduction Methods in Drilling Polymeric Matrix Composites 343

Special Drills Tool geometry is a relevant aspect to be considered in drilling PMCs. As far as the tool geometry is concerned, drills with special geometry, such as multifaceted drill, candle stick drill, core drill together with drills with modified geometry, such as various chisel lengths and rake, clearance, point and helix angles, are preferred when drilling with WC tools. On the other hand, when using HSS drills the use of standard twist drills and drills with special geometry is quite similar [3]. A different geometry of the tool from the traditional one can reduce push-out at exit by lowering the thrust force. Delamination at the exit can be reduced if a drill with a sharp tip angle is used. In fact, the sharp tip makes the drill punch through the last plies over a smaller area compared with a blunt tip. Among the drill point geometry parameters, the chisel edge and point angle have a major influence on the magnitude of the thrust force; increasing the chisel edge length results in an increase in the thrust force, which can cause separation of plies at the exit as the interlaminar bonding yields. In addition, the cutting action at the chisel edge region is extremely poor due to large negative rake angles and low cutting speeds leading to indentation process very close to the drill axis where the dynamic clearance angles are negative. With the use of tripod and web thinned point geometry drills, reduction of delamination factor can be achieved [4]. In the case of tripod point geometry, there is no cutting edge at the central portion and it is almost like a point thus ensuring positioning from the start of the hole and the peripheral cutting edges shear the composite materials leaving a clean hole. With the web-thinned drill, the thinning of the web alters the inclination angle which increases effective rake angle thus leading to reduced thrust level. Spiral point drills perform much better than the conventional ones, while multifaceted drills (MFD) allow to reduce the thrust force and to better distribute the cutting temperature. Delamination at the exit side can be also improved if the drilling-induced thrust with distributed circular load is used. The traditional twist drill provides a relatively low threshold of the thrust force in delamination compared to special drills such as saw drill, candle stick drill, core drill and step drill. The advantage of these special drills is illustrated theoretically as well as experimentally; their thrust force is distributed toward the drill periphery instead of being concentrated at the centre, allowing to increase feed rate without causing delamination [5]. The saw drill utilises the peripheral distribution of the thrust for drilling the composite laminates. The delamination of size less than the drill is not of much concern because being inside the hole radius, it is drilled out. When the

344


delamination grows beyond the drill radius, the saw drill, which applies circular force to material, can maintain a much larger thrust force without causing delamination than the twist drill, which applies concentrated force. The thrust force of the candle stick drill can be considered as a concentrated central load plus the peripheral circular load. Since the total thrust force is distributed towards periphery, the drill is expected to be advantageous in allowing a larger critical thrust force at the onset of delamination. The candle stick drill is physically an intermediate between the twist drill and the saw drill. The candle stick drill has a much smaller chisel edge than twist drill and, consequently, punches through the last plies over a small area when compared with that of twist drill. Thus a smaller extent of the last laminate is subjected to a bending force from the center, leading to a better quality of the hole. The thrust force of the core drill is uniformly distributed in an annular area rather than concentrated at the center. This circumstance allows the obtaining of a higher critical thrust force responsible for the onset of delamination. Among core drill parameters, the grit size of diamond is the most significant factor in reducing the thrust force, while the drill thickness shows limited influence [6]. The step drill can be considered as composed of a primary stage and a secondary stage. The primary stage of the tool reduces the central material removal and, particularly cancels the contribution of chisel edge to the thrust force. In Figure 1 the correlation between thrust force and feed rate for various drills is shown. Peel effect of plies along the edge of the major drill cutting edges takes place when the tool tip encounters the first ply. This defect increases with the rake angle and tends towards ply detachment [7]. To reduce or eliminate the entrance defect, a small rake angle prevents the first lamina from lifting up and tearing off. A rake angle lower than 6° is usually recommended. The principal cutting edge significantly affects the hole quality; in fact, less delamination can be achieved by making smaller contact length between drill and hole. A drill with special geometry, having three cutting edges, twist and rake angles of 0°, clearance angle of 6° and a minor cutting edge angle varying from 59° to 0°, allows to enhance the drilling quality of thin carbon/epoxy plates without a backing plate [8]. Additional problems arise whenever Aramid Fibres Reinforced Composites (AFRPs) are machined, owing to the high toughness and the low compressive resistance of the fibres, which buckle under bending stress instead of being sheared off.

Damage Reduction Methods in Drilling Polymeric Matrix Composites 345

Figure 1. Correlation between thrust force and feed rate for some special drills [5].

Figure 2. Geometry of twist drill with a C-shape cutting edge [9].

The resultant cut surfaces appear rough and fuzzy with a poor dimensional precision [9]. The above facts imply the need for the design of suitable tools different from the ones used in cutting Glass Fibre Reinforced Plastics (GFRPs) and Carbon Fibre Reinforced Plastics (CFRPs). Among the various geometries

346


studied, the optimised tools exhibit sharp edges with high values of axial rake angle (γa) and lip relief angle (α), as shown in Figure 2. Twist drills with a radial rake angle (γr) positive at the outermost periphery

are obtained by means of a C-shaped cutting edge and are currently produced by employing solid sintered carbide [10].

Drilling Conditions A proper choice of drilling conditions is difficult due to the coexistence of hard abrasive fibres and a soft matrix. Matrix materials, reinforcing shapes and, particularly, reinforcing volume content affect delamination. In drilling chopped GFRP composites, thrust force increases with the increase of the fibre volume fraction (Vf). Increasing the cutting speed in drilling cross-winding, woven and chopped composites, reduces the push-out delamination as a result of decreasing the thrust force. The thrust forces in drilling continuous-winding composite are more than three orders of magnitude higher than those in the cross-winding composites. Chopped composites have lower push-out delamination than those made from woven fibres. For the same fibre shape, peel-up and push-out delaminations of woven/epoxy composite are lower than that for woven/polyester composites [11]. Tool conditions have significant effect on hole damage, as thrust force increases when tool wear proceeds and drill run-out occurs. The effect of drill wear on the delamination factor becomes significant at higher cutting speeds. The variation of thrust for drilling carbon fiber reinforced composites with a carbide drill is different from HSS drill, which results from the higher resistance to wear of a WC drill than a HSS tool. Therefore, delamination can be improved when a carbide drill is used rather than a HSS one when drilling CFRPs [12]. Particular attention must be paid to tool regrinding as the critical thrust force decreases with increasing point eccentricity [13]. The use of a pre-heated tool drastically reduces the thrust force, yielding a narrower push-out delamination and a lower energy required to drill the hole. On the other hand, an increase of the twist drill temperature leads to a deep retraction and de-bonding effects on the matrix, due to its thermal vitrification, and negatively affects the structural properties of the composite. Therefore, the heating of the tool before or during the cut considerably reduces the resistance of material [14]. On the ground of this remark, it is even possible to hypothesize a cooling thermal pre-treating of the drill site through the application of liquid nitrogen. In drilling AFRPs, although there is an increase in drill thrust under

Damage Reduction Methods in Drilling Polymeric Matrix Composites 347 cryogenic conditions enhancing the probability of delamination, this shortcoming is more than offset the improved surface finish, hole quality and far superior tool life [15]. The effect of the chisel edge on the thrust force is linear and, besides adopting special drills, can be neutralised with a pre-drilled pilot hole [16]. In this process a hole is pre-drilled in order to eliminate the thrust caused by the chisel edge, thus the threat for delamination is significantly reduced. The diameter of the pre-drilled hole is set equal to the length of chisel edge. The smaller diameter of the pilot hole cannot fully cover the chisel edge, while a larger one tends to cause undesired delamination during pre-drilling. An optimal range of chisel edge length with respect to pilot hole diameter can be derived defining the process window of chisel length [17]. By controlling the ratio of chisel edge length, one can conduct medium to large hole of composite laminates drilling at higher feed rate without delamination damage. The window is influenced by the feed rate in such a way that, the smaller the feed rate is, the wider the window opens. Better quality holes and high efficiency can be obtained by vibration-assisted cutting technique [18]. The conventional drilling is a continuous cutting process, while the vibration-assisted drilling is a pulsed intermittent cutting process by piezoelectric crystal oscillator. Figure 3 shows the schematics of the set-up [12, 19]. Thrust force and delamination can be reduced by using a low frequency, high amplitude vibratory drilling which induces an axial vibration in the feed direction. This can be explained by the features of vibratory drilling. The impact experienced by the composite in the axial direction makes it possible for the fiber in the composite to be easily and sharply cut by a pulse/intermittent cutting mode, which changes the mechanics of chip formation, makes the cutting energy concentrate, reducing the chip deformation, and decreasing the thrust as well. In this process the workpiece is suitably clamped by a pre-drilled backup plate. The presence of the support beneath the workpiece, in any case, significantly influences the level of the push-out delamination because the threshold thrust force at the onset of delamination is increased, but mildly affects the internal damage. Consequently, a commonly followed practice in the industry is represented by the use of a support on the back to prevent deformations leading to exit delamination. On the contrary, the peel-up delamination is not affected by neither the presence of a support nor the pre-heating.

348


Figure 3. Schematics of vibration-assisted drilling [12].

However, the use of a backing plate makes the drilling operation longer and dearer; consequently, developing devices or procedures to allow the reduction of damaged area without the presence of a support, represents an important task. A system proposed is based on the constraining of the workpiece dynamics which can be obtained by using a viscous damper to constrain the relative workpiecedrill point speed during machining [20]. In the case of unsupported drilling, selecting high level of vr/vt ratio, the push-out delamination is sensibly wider with respect to supported drilling. The common thesis that delamination depends on the thrust force exerted by the drill point is valid only for the supported condition; in unsupported drilling the delamination mechanism is more complex. The main causes of delamination in unsupported drilling are hypothesised to be the presence of a peak in the actual feed rate, that changes the action of the drill point from cutting to punching, and the overload on the peripheral part of the cutting lips. Both these phenomena are a consequence of the elastic inflection of the workpiece under the drill point load and of the subsequent fast release movement that occurs as the chisel edge exits the workpiece. The effect of the support in reducing delamination is generally held to be the mechanical strengthening of the lower laminae. The experimental results comparing supported and unsupported drilling, indicate that the main effect of the support is to impede inflection, not to provide mechanical strengthening. Therefore, a significant reduction in delamination can be achieved

Damage Reduction Methods in Drilling Polymeric Matrix Composites 349 by limiting the workpiece dynamics [20]. The idea is to avoid the fast release movement of the workpiece by using a viscous damper to constrain the relative workpiece-drill point speed, as shown in Figure 4.

Figure 4. Schematics of damper for unsupported drilling [20].

A viscous damper is required because it exerts a force directly proportional to the velocity, acting consequently only during the fast release movement and not during the drilling operation, when the relative workpiece-drill point velocity is very low. Besides its direct effect on the speed of the workpiece, the damper may also alleviate the overload on the cutting lips, by greatly reducing the imbalance of forces produced with the exit of the chisel edge. The effect obtained on the release movement of the workpiece using a damper is similar to that produced with a support, but it is achieved in a more simple and practical way. The workpiece begins to be inflected before the engagement of the drill point, because for practical reasons the damper must touch the workpiece before the drill point does. During the first phase of drilling, when the chisel edge is engaged in the workpiece, the damper exerts a small force on the workpiece. The actual feed rate in this phase is very low, less than the one set, because the workpiece itself moves in the same direction as the drill point, reducing the feed rate.

Process Parameters The entity of the damage is deeply influenced by the ratio between cutting speed and feed rate. For a high level of vr/vt ratio, the internal surface of the hole is smooth and regular with small delamination, both in the case of supported or

350


unsupported drilling; in the latter case, the push-out delamination is sensibly wider. For a low level of vr/vt ratio, the hole is significantly damaged: the pushout delamination is wider and involves more layers, internal cracks and debonding of the fibres occur. Figure 5 shows the size of delamination zone (D) as a function of the ratio between cutting speed (vr) and feed rate (vt) [2].

Figure 5. Size of delamination zone as a function of the ratio between cutting speed and feed rate [2].

The influence of the vr/vt ratio on peel-up delamination can be explained taking into account the tool geometry. Since the axial forces acting on the side of the twist drill tend to peel-up the first laminae of the composite, the entity of these forces are connected to the dynamic rake angle of the twist drill, which is higher as the vr/vt ratio decreases. Consequently, higher values of the dynamic rake angle increase the peel-up component with wider delamination [21]. Increasing cutting speed results in lower thrust force and torque due to the higher temperatures produced by heat generation associated with the low coefficient of thermal conduction together with the low transition temperature of plastics. Higher twist drill temperatures negatively influence the internal damage of the hole, so that between the cutting parameters, the feed rate seems to be the most critical one and should be selected carefully in order to reduce all kinds of damages.

Damage Reduction Methods in Drilling Polymeric Matrix Composites 351 When feed rate is high, the failure modes show the features typical of impact damage, with step-like delaminations, intralaminar cracks and high density microfailure zones. If feed rate sufficiently low values are adopted, the failures consist essentially of delamination mainly originating near to the intersection between the conical surface generated by the main cutting edges and the cylindrical surface of the hole. At low feed rates most of the delaminations are induced near the tool exit edge and are generated when the chisel edge and the inner portion of the lips have already left the work material [21]. When drilling PMCs using HSS and WC drills, cutting speeds from 20 to 150 m/min are usually employed, whereas feed rate values lower than 0.3 mm/rev are frequent [3, 22]. Cutting speed is not a limiting factor when drilling polymeric composites, particularly with hard metals, therefore, the use of cutting speeds below 150 m/min may be explained by the maximum rotational speed of conventional machining tools, since drill diameters above 10 mm are rarely reported. Another reason for keeping cutting speeds below 150 m/min may reside in the fact that higher cutting speed values lead to higher cutting temperature, which in turn may cause the softening of the matrix. The use of feed rates below 0.3 mm/rev may be associated to the delamination damage caused when this parameter is increased. Generally, tungsten carbide tools are preferred when drilling at higher cutting speeds and, in contrast to metals, also at higher feed rates [10]. A decreasing feed rate improves drilling quality, but leads to increased wear and machining time. Consequently, variable feed rate strategies, based on adaptive control or neural network controller, can be useful [23]. With adaptive control with constraint (ACC), it is possible to regulate feed rate by a feed-back control system, in order to maintain the measured thrust force below the threshold critical value, optimising the machining time [24]. Experimental observations have evidenced that the delamination at entry side is much smaller with respect to that at the exit side. Delamination can be assessed by measuring the damaged area of the ring around the hole at entrance and exit side by means of a CCD camera. The extension of delamination is represented by the white damaged area of the ring around the hole, as shown in Figure 6. In Figure 7 a comparison between the damaged area (A) at the exit side of the hole under constant feed rate and with adaptive control, during drilling of AFRPs laminate with 33 fabric laminas, fibre volume fraction equal to 0.5 and thickness equal to 3.6 mm, is reported. As can be seen the holes obtained under feed control are characterised by a lower damaged area with respect to those produced under constant feed.

352


(a)

(b)

Figure 6. Typical extension of damaged area produced by delamination at hole exit side adopting constant feed rate (a) and adaptive control (b). standard deviation

Damaged area, A (mm2)

14

12

v r = 84.8 m/min

Constant feed v t = 0.05 mm/rev

10

ACC

8

6

0

20 40 Hole number

60

Figure 7. Damaged area at the exit side as a function of hole number with constant feed and with ACC.

The neural network controller allows to drill at desired thrust force profiles within an acceptable error margin, minimising delamination [25]. The effect of cutting parameters on surface roughness is influenced by composite characteristics and, in particular, by fiber volume fraction. For glass fiber reinforced/epoxy composites the hall wall finish can be improved by increasing cutting speed and fiber volume fraction. The drilled holes with lower vf

Damage Reduction Methods in Drilling Polymeric Matrix Composites 353 values at lower feed have greater roughness than that drilled at higher feed. Composites with high fiber volume fraction exhibit an opposite behaviour [26].

Conclusion A literature review of the principal techniques used to improve hole quality in drilling PMCs has been presented. Damage reduction can be achieved by selecting proper cutting parameters, drilling conditions and tool geometry and material. The major damage is certainly the delamination that can occur both on the entry and exit sides of the workpiece. The level of delamination is related to the thrust force; in particular, there is a thrust force threshold below which the delamination is negligible. Consequently, if thrust force is kept below a critical value by adopting, during the drilling process, different strategies, such use an adaptive control or a neural network controller, delamination can be reduced.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

W. Konig et al., Ann. CIRP, 34, 537 (1985). G. Caprino et al., Int. J. Mach. T. Man., 30, 77 (1990). A.M. Abrao et al., J. Mat. Proc. Tech., 186, 1 (2007). A.Velayudham and R. Krishnamurthy, J. Mat. Proc. Tech., 185, 204 (2007). H. Hocheng and C.C. Tsao, Int. J. Mach. T. Man, 46, 1403 (2006). C.C. Tsao and H. Hocheng, J. Mat. Proc. Tech., 192, 37 (2007). H. Hocheng and C.H.K. Dharan, J. Eng. Ind., 112, 236 (1990). R. Piquet et al., Comp.: Part A, 31, 1107 (2000). A.Di Ilio et al., Int. J. Mach. T. Man., 31, 155 (1991). W. Konig W, et al., in Fabrication of Composite Materials, edited by M. M. Schwartz, ASM, (Metal Park, Ohio, 1985). U.A. Khashaba, Comp. Str., 63, 313 (2004). X. Wang et al., J. Mat. Proc. Tech., 148, 239 (2004). C.C. Tsao and H. Hocheng, Int. J. Mach. T. Man, 45, 125 (2005). E. Capello and V. Tagliaferri, J. Comp. Tech. Res., 23, 131 (2001). D. Bhattacharyya and D.P.W. Horrigan, Comp. Sci. Tech., 58, 267 (1998). S. Jain and D.C.H. Yang, J. Eng. Ind., 115, 398 (1993). C.C. Tsao and H. Hocheng, Int. J. Mach. T. Man, 43, 1087 (2003). S. Arul et al., Int. J. Mach. T. Man., 46, 252 (2006). L.B. Zhang LB et al., Int. J. Mach. T. Man., 41, 641 (2001).

354 [20] [21] [22] [23] [24] [25] [26]

A. Di Ilio and A. Paoletti E. Capello, J. Mat. Proc. Tech., 148, 186 (2004). G. Caprino and V. Tagliaferri, Int. J. Mach. T. Man, 35, 817 (1995). W.C. Chen, Int. J. Mach. T. Man., 37, 1097(1997). A.Di Ilio et al., Proc. Mach. Comp. Mat. Symp., 199 (Chicago, 1992). A.Di Ilio and A. Paoletti, Proc. 3th Wor. Conf. IDPT, 245 (Berlin, 1998). R. Stone and K. Krishnamurthy, Int. J. Mach. T. Man, 36, 985 (1996). El-Sonbaty et al., Comp. Str., 63, 329 (2004).

INDEX A

B

acetone, 26 acid, 5, 6, 7, 15, 17, 19, 20, 26, 37, 152, 154, 179, 181 acrylate, 16, 62, 66, 68, 69, 82 actuators, 148, 165 additives, 38, 142, 201, 202 adhesion, 3, 4, 5, 6, 8, 9, 10, 16, 17, 18, 19, 20, 21, 22, 30, 31, 37, 41, 42, 44, 67, 69, 70, 71, 170 adhesion properties, 10 adhesive properties, 5, 14, 18, 21, 22 adjustment, 69, 96, 182, 184, 185, 189, 190, 330 aerospace, 58, 88, 307 AFM, 9 aggregation, 156 alcohol, 7, 124 alloys, 144, 148, 298 allylamine, 6, 21, 22 alters, 343 aluminum, 58, 143 ammonia, 5, 9, 11, 19, 20, 22 ammonium, 152, 174, 175, 181 amplitude, 347 anisotropy, 12, 23, 113, 138, 215, 217 annealing, 16 applied research, 191 argon, 8, 21, 22 asbestos, 128 atmospheric pressure, 10, 18 atomic emission spectrometry, 153 atomic force, 9, 19 automobiles, viii, 3, 51, 54

beams, 96, 104 behavior, xiii, 4, 19, 89, 109, 121, 145, 148, 164, 165, 298, 333, 337 bias, 327, 329 blends, 142, 166, 189, 193 blocks, 14, 172, 183, 191, 192 bonding, vii, viii, 1, 2, 3, 4, 5, 10, 11, 14, 15, 17, 18, 19, 21, 22, 25, 29, 30, 31, 35, 37, 38, 42, 43, 44, 112, 130, 132, 142, 150, 171, 303, 309, 343, 350 bounds, 331 branching, 171 breakdown, 149, 150, 159 brittleness, 120 Brno, 172 bromination, 5, 7 bromine, 5, 7, 19 Bruker DRX, 153 burn, 52 butadiene, 16

C calcium, 26, 170 calcium carbonate, 170 calibration, 128 capillary, 63, 189 carbides, 300 carbon, vii, 1, 9, 22, 26, 29, 63, 94, 154, 156, 166, 202, 307, 326, 339, 344, 346 carbon dioxide, 22, 52 carbon film, 154, 156

356

Index

carbon tetrachloride, vii, 1, 26 carbonyl groups, 17 cast, 152 casting, 202 catalysis, 179 catalyst, 10, 26 cation, 170, 179 cell, 29, 52, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 261, 263, 266, 267, 273, 274, 275, 282, 283, 286, 288, 289, 292, 293, 307, 312, 315, 320, 321, 326 cell surface, 249, 250, 256, 258, 288, 292 cellulose, 52 chain mobility, 14 chain scission, 179, 192 chalcogenides, 300 chemical etching, 14, 24 chemical interaction, 30 chemical properties, 9 chemical reactions, 178, 301 chemical reactivity, viii, 2, 38 chemical structures, 5 chemisorption, 166 classes, 148, 299, 300, 309, 310, 312, 313, 315 classification, 299, 308, 314 cleaning, 9, 57 clusters, 207, 213, 217, 220, 240, 242 CMC, xii, xiii, 297, 298, 299, 300, 301, 302, 306, 314, 315, 316, 317, 320, 321, 322, 323, 325, 331, 340 coatings, 30, 35, 38, 301, 303 coherence, 65, 67 cohesion, 122 compatibility, 7, 54, 56, 129, 130, 131, 142, 150 competition, 16 complexity, xiii, 89, 121, 298, 300, 325, 328, 329, 335 components, x, xii, 3, 12, 25, 53, 54, 57, 59, 88, 104, 120, 148, 154, 181, 192, 196, 207, 209, 230, 247, 248, 251, 290, 298, 301, 303, 307 composite mechanical properties, 10 composition, 10, 11, 19, 24, 25, 37, 59, 61, 90, 141, 149, 172, 192, 230, 231, 237, 239 compounds, 6, 124, 125, 126, 130, 140, 141, 179, 182, 300 compression, 61, 64, 65, 67, 75, 77, 78, 79, 83, 108, 109, 112, 129, 197, 303 computer technology, 122 concentrates, 120

concentration, 16, 20, 21, 31, 109, 173, 202, 221, 333, 334 conduction, 167, 342, 350 conductivity, 17, 299, 303, 304, 317, 319, 320, 321, 322, 335, 336, 338, 339, 340 configuration, 11, 40, 183, 191, 192, 307, 308 conservation, 56 consolidation, 62 constituent materials, 298, 308, 317, 320 construction, ix, 52, 54, 55, 56, 57, 58, 61, 81, 189, 190, 192 contamination, 57 contour, 335 control, 4, 10, 16, 41, 54, 56, 121, 129, 301, 351, 352, 353 conventional composite, 170 conversion, 302 cooling, xii, 17, 26, 67, 88, 89, 103, 107, 109, 112, 114, 128, 152, 213, 247, 255, 261, 266, 291, 292, 298, 303, 308, 309, 310, 311, 346 cooling process, xii, 213, 247, 255, 266, 291 copolymers, 165, 171 copper, x, 144, 147, 150, 154, 156, 166, 167 corona discharge, 18, 19, 20 correlation, 19, 300, 322, 324, 344 coupling, 4, 5, 6, 10, 15, 22, 151, 162, 166, 298, 308 covalent bond, 11, 20 covalent bonding, 20 creep, 12, 14, 17, 107, 109, 113, 114 critical value, 204, 205, 351, 353 crystal structure, 90 crystalline, 3, 12, 14, 141, 144, 157, 159, 167, 253 crystalline solids, 144 crystallinity, 157 crystallites, 157 crystallization, 12, 15, 122 crystals, 92, 93, 95, 112, 166 cultivation, 54 curing, 25, 27, 28, 62, 68 cycles, 205, 228, 229, 230, 231, 232, 236, 239, 242, 243, 244, 245 cycling, 88

D damping, 55, 81 danger, 67

Index decomposition, 179, 192 defects, xi, 151, 195, 196, 207, 212, 213, 216, 217, 218, 299, 311 deficiency, 298 definition, 63, 96, 97, 122, 327 deformation, 6, 11, 17, 40, 67, 71, 72, 90, 109, 113, 114, 137, 138, 139, 223, 249, 252, 253, 347 degradation, xii, 7, 18, 67, 74, 80, 140, 171, 177, 178, 179, 181, 192, 216, 298, 299, 300, 302, 321, 322, 323 degree of crystallinity, 157 density, 3, 7, 9, 12, 19, 52, 55, 58, 61, 81, 82, 122, 124, 142, 149, 231, 237, 240, 301, 302, 303, 317, 319, 322, 328, 351 dental implants, 301 destruction, 120, 203, 225, 245 detachment, 344 detection, 32, 306 deviation, 211, 215, 217 dielectric constant, vii, x, 147, 148, 149, 150, 154, 159, 160, 161, 162, 163, 164, 165, 166 dielectric permittivity, 154 dielectric strength, x, 148 dielectrics, 167 differential equations, 252, 265 diffraction, 15, 74, 88, 89, 90, 93, 95, 96, 97, 98, 99, 103, 104, 105, 106, 107, 112, 113, 115, 159 diffusion, 16, 19, 121, 122, 144, 148, 189, 191, 248, 307, 328 diffusion process, 189, 248 diffusivities, 303 diffusivity, 300, 303, 305, 306, 317, 319, 322, 323 direct measure, 42 directionality, 299, 314 discharges, 18 discs, 183, 191 dislocation, 109, 144 dispersion, vii, x, 22, 119, 120, 129, 131, 138, 139, 140, 142, 143, 152, 161, 170, 171, 177, 181, 191 displacement, 14, 103, 148, 188, 249, 250, 252, 256, 258, 262, 265, 267 dissociation, 132 distillation, 26, 152 distilled water, 26, 152, 153 distributed memory, 336

357

distribution, ix, 65, 71, 88, 89, 90, 99, 100, 109, 110, 111, 112, 113, 115, 163, 164, 166, 197, 207, 211, 215, 218, 219, 248, 251, 256, 257, 303, 343 division, 101, 249 DMF, 152, 153, 154, 156 doors, 54 dosage, 181 drawing, 12, 23, 40, 100, 273, 308, 309, 325 drying, 26, 125 DSC, 15, 157, 158 DSM, 173 ductility, 17, 137 durability, 4 duration, 21

E elasticity, 11, 34, 41, 105, 263 elastomers, 165 electric field, 148, 149, 150 electrical properties, 165 electrodes, 153 electron, 14, 94, 142, 150, 165, 173, 203, 205, 206, 213, 214, 308, 311 electron microscopy, 203, 205, 308 electrons, 150, 162 elongation, 40, 188, 191 elucidation, 114 employment, xii, 75, 191, 297, 342 encapsulation, 166 encouragement, 83 energy, xii, 4, 14, 21, 22, 23, 25, 37, 52, 53, 55, 61, 71, 72, 77, 80, 81, 88, 122, 126, 132, 133, 134, 138, 148, 149, 150, 172, 189, 191, 193, 235, 236, 247, 250, 253, 256, 271, 273, 274, 275, 282, 346, 347 energy consumption, 132 energy density, xii, 148, 149, 150, 247, 253, 256, 271, 274, 275, 282 enlargement, 71 environment, xiii, 8, 19, 88, 173, 298, 300, 337 environmental effects, 298, 308 epoxy groups, 9, 16 epoxy resins, 2, 6, 9, 17, 21, 29, 38, 81 equilibrium, 133, 249, 252, 262, 264, 271 ester, 19, 154 etching, 4, 9, 15, 18, 20, 21, 22, 24, 31, 37, 39, 44, 203, 209

358 ethanol, 91 ethylene, 10 European Regional Development Fund, 294 evaporation, 122 evolution, 121, 298, 308 experimental condition, xi, 196, 318 exposure, 8, 10, 11, 14, 16, 19, 20, 21, 24 external influences, 58 extraction, 53 extrusion, 2, 12, 188

Index

G

gases, 8, 18, 19 gel, 11, 12, 17, 21 generation, 88, 103, 111, 149, 311, 326, 327, 329, 350 Gibbs energy, 131 goals, 148 gold, 153 grain boundaries, 112, 141, 142, 232, 249 grain refinement, 145 grains, 94, 122, 141, 142, 203, 206, 207, 208, F 209, 213, 215, 217, 220, 232, 249, 251, 252, 257 fabric, 10, 15, 17, 53, 56, 64, 65, 307, 324, 334, graphite, 17, 125, 164 351 gravity, 126 fabrication, vii, x, 28, 88, 91, 113, 114, 119, groups, 4, 5, 7, 11, 15, 16, 17, 19, 20, 21, 22, 31, 125, 126, 139, 164, 300, 301, 302 36, 41, 43, 154, 157, 174, 178, 179 fatigue, xi, 141, 142, 146, 148, 195, 204, 206, growth, 11, 91, 93, 112, 122, 132, 133, 157, 216, 245, 298, 308 197, 218, 228, 299, 342 fiber content, 34, 36, 37, 38, 41, 42 fibers, vii, viii, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, growth rate, 122 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, H 38, 39, 40, 41, 42, 43, 44, 50, 126 fibrillation, 4 habitat, 54 filament, 11, 40 hardener, 7, 25, 62 filler particles, 164, 178 hardness, 140, 141, 300, 301 fillers, 149, 164, 170, 173, 175, 176 healing, 141 film thickness, 153 health, viii, 51, 54, 56 films, 31, 153, 166 heat, 14, 62, 68, 69, 170, 255, 298, 299, 300, finite element method, ix, 87, 88, 90, 93, 99, 301, 305, 306, 314, 317, 318, 319, 330, 331, 102, 107, 108, 111, 112, 113, 114, 115, 213, 332, 333, 334, 335, 336, 337, 339, 342, 350 318 heat transfer, 298, 336 flexibility, 72, 149, 150, 171 heating, 14, 26, 91, 154, 346 flow curves, 177 heating rate, 154 fluctuations, 54 height, 271, 313 fluid, 7 helium, 18, 19 foams, 62 hemp, 52, 53, 55, 61, 65, 66, 67, 68, 69, 71, 75, formaldehyde, 11, 26 76, 78, 79, 82 fragility, 137 heterogeneity, 157, 162, 230, 304 fragments, 200 hexafluoropropylene, 164 free energy, 9, 19 homogeneity, 177 free radicals, 16 hot spots, 333 friction, 4, 55, 80 housing, 55 FTIR, 16, 33, 154 HPC, 300 fuel, 120 HRTEM, 142, 146 functional separation, 55, 57, 81 humidity, 26, 57 fusion, 157 Hunter, 194

Index hybrid, 53, 55, 62, 65, 82 hydrazine, 19 hydrogen, 2, 19, 22, 150, 155, 179 hydrogen chloride, 19 hydrolysis, vii, 1, 5, 26, 29, 30, 31, 32 hydrophobicity, 177 hydroxide, 26, 152 hydroxyl, 17, 19, 31, 38, 178, 179 hydroxyl groups, 31, 178, 179 hypothesis, 348 hysteresis, 148, 149

I image, 92, 100, 101, 255, 311, 313 image analysis, 255 images, 100, 102, 310, 313 immersion, 61 impact strength, 29, 32, 36, 37, 38, 65, 71 implementation, 172, 189 impregnation, 53, 309 incidence, 96, 97 inclusion, 65, 150, 151, 157, 159, 248, 249 incompatibility, 88, 150, 157 indentation, 90, 140, 141, 237, 241, 343 induction, 91 industry, viii, ix, xi, 2, 51, 52, 58, 62, 169, 347 infinite, xii, 131, 247, 248, 249, 250, 251, 252, 256, 257 infrastructure, xii, 297 initiation, xii, 215, 218, 227, 237, 247, 248, 253, 254, 255, 256, 270, 271, 273, 282, 283, 292 insertion, 183, 191 institutions, xi, 169, 170 Instron, 29, 205, 206 insulation, ix, 52, 54, 55, 57, 58, 73, 74, 75, 76, 77, 78, 80, 82, 128, 162 integration, 54, 248, 249, 252, 253, 265, 266, 267, 268, 282, 285, 286, 288 interaction, 3, 16, 67, 71, 135, 166, 327, 328 interactions, 3, 4, 5, 19, 35, 170, 171, 174, 176, 327 interface, x, 3, 5, 6, 8, 9, 11, 14, 15, 17, 20, 31, 35, 42, 89, 100, 101, 109, 112, 129, 130, 132, 133, 138, 142, 147, 151, 152, 162, 209, 302, 303, 309 interfacial adhesion, 7, 10, 11, 15, 20, 22

359

interfacial bonding, 7, 12, 14, 15, 19, 21, 25, 35, 36, 44, 166 interference, 72 interphase, 31, 35 interval, 8, 224, 256, 258, 284, 285, 286, 288, 291, 293, 301, 303 Intervals, 258, 260, 276, 278, 279 intrinsic viscosity, 172 ion-exchange, 170 ions, 148 IR spectroscopy, 32 iron, 17 irradiation, ix, 10, 14, 17, 88, 93, 94, 95, 97, 150 isolation, 124 isotactic polypropylene, 170, 181

K KBr, 153 ketones, 11 kinetic parameters, 122

L laminar, xii, 196, 202, 213 laser ablation, 9 lattice parameters, 103 laws, 252, 262, 263 lignin, 52 limitation, 4, 190, 299, 336 line, 99, 196, 198, 276, 277 linear dependence, 42 linkage, 154 links, 103, 137 liquid phase, 123 liquids, 154 locus, 20 low temperatures, 62 LTD, 25, 38 lubricants, 140, 145 lumen, 53

M macromolecules, 19 manufacturer, 173, 174

360

Index

manufacturing, viii, xii, 51, 60, 62, 63, 66, 79, 297, 298, 299, 301, 308, 309, 338 mapping, ix, 88 market, viii, ix, 2, 51, 52, 54, 56 material degradation, xiii, 338, 341 measurement, ix, 10, 44, 65, 73, 74, 79, 88, 89, 90, 92, 93, 94, 95, 96, 97, 98, 102, 112, 113, 155, 188, 189, 198, 300, 305, 306, 323, 336 measures, 74 mechanical loadings, 337 mechanical properties, vii, 1, 3, 4, 7, 8, 10, 12, 14, 15, 23, 25, 27, 40, 64, 67, 88, 90, 108, 120, 131, 140, 141, 142, 143, 145, 248 mechanical testing, 306 median, 197, 215, 216, 217, 223, 227, 230, 231, 237, 240, 242 melt, xi, 12, 88, 91, 169, 171, 172, 173, 174, 177, 178, 179, 182, 183, 186, 187, 188, 189, 190, 191, 192, 193, 301 melting, 12, 14, 65, 140, 157, 266, 290, 300 melting temperature, 266, 290 melts, 177 membranes, 166 memory, 148, 336 metals, 138, 300, 341, 351 methanol, 152 micrometer, 170, 171 microscope, 27, 94, 173 microscopy, 9, 15, 19, 71, 142, 206, 207, 310 microstructure, ix, 12, 24, 87, 88, 89, 90, 94, 95, 100, 108, 109, 112, 115, 121, 140, 142, 144, 154, 165, 207, 208, 209, 230 microstructure features, ix, 88, 89, 90, 100, 115 microstructures, 89, 90, 121, 140 military, 3 mixing, x, xi, 53, 125, 148, 169, 171, 173, 178, 179, 183, 189, 191, 192 mobility, 71, 148 model system, xii, 247, 248, 249 modeling, xiii, 89, 151, 166, 298, 299, 300 models, xiii, 12, 74, 89, 99, 100, 102, 103, 107, 122, 144, 161, 248, 249, 250, 251, 253, 254, 298, 299, 300, 315, 317, 336, 337, 339 moisture, 6, 57 mold, 28 mole, 91 molecular weight, 2, 6, 12, 152, 171, 178 molecules, 2, 12, 15, 16, 21, 150, 157

morphology, viii, xi, 1, 2, 4, 6, 10, 12, 15, 16, 19, 23, 29, 37, 89, 91, 94, 100, 101, 104, 113, 156, 157, 196, 320, 323 motion, 69, 127, 148 moulding, 28, 62, 82, 302 movement, 57, 148, 348, 349 multiplication, 300, 335, 336 muscles, 148, 165

N NaCl, 7 nanocomposites, xi, 167, 169, 170, 171, 172, 174, 175, 176, 177, 178, 179, 180, 181, 182, 189, 190, 191, 192, 193 nanofabrication, vii, x, 147 nanofibers, 165, 166 nanoparticles, 17, 162, 163 nanotechnology, 170 National Research Council, 202 neglect, 250 neural network, 351, 352, 353 nitrides, 300 nitrobenzene, 152 nitrogen, 16, 18, 22, 125, 144, 152, 173, 174, 192, 346 NMR, 153, 155 nodes, 101, 103, 107, 329 noise, viii, 51, 54, 56, 57, 71 nucleation, 122, 157 numerical analysis, ix, 88, 99

O observations, 309, 312, 351 obstruction, 55 oil, 26, 62, 66, 68, 69, 82 oligomers, 157, 160, 166, 167 optical micrographs, 222, 223, 237, 312 optical microscopy, 142 organ, 170, 171, 173, 174, 176, 177, 178, 181, 183, 192 organic compounds, 5, 171, 178 organic solvents, 6 orientation, 23, 53, 66, 89, 92, 93, 97, 98, 104, 112, 113, 139, 191, 221, 323, 327, 331, 332, 338 oscillation, 53, 55, 61, 72, 77, 78, 81

Index overload, 348, 349 oxidation, 6, 8, 19, 20, 21, 88, 141, 142 oxidation rate, 142 oxidative agents, 17 oxides, 300 oxygen, 6, 8, 9, 10, 11, 15, 16, 17, 18, 19, 20, 21, 22, 44 oxygen plasma, 8, 9, 11, 15, 16, 21, 22

361

polycondensation, 26 polymer chains, 14, 171, 174, 176, 178, 191 polymer composites, 149, 165, 170, 189 polymer matrix, 10, 147, 149, 150, 152, 160, 161, 162, 164, 167, 170, 171, 172, 174, 178, 181, 189, 190, 191, 302 polymer molecule, 2 polymer nanocomposites, vii, xi, 169, 170, 172, 188, 191 polymeric materials, 18, 19, 167, 170 P polymeric matrices, 3, 4, 7, 12, 14, 15, 25 polymerization, 8, 10, 16, 22, 151 parallel processing, 336, 337 polymers, vii, 1, 2, 5, 10, 12, 14, 59, 148, 149, parallelization, 337 164, 165, 166, 167, 171 particles, x, xii, 120, 122, 124, 126, 129, 131, polyolefins, 14 132, 133, 136, 137, 138, 139, 140, 141, 142, polypropylene, 55, 62, 65, 66, 71, 82, 181, 190, 143, 147, 150, 151, 152, 157, 164, 170, 171, 193 179, 181, 209, 237, 238, 239, 247, 248, 249, polyurethane, 7 251, 252, 254, 256, 257 poor performance, 12 percolation, 149 porosity, xiii, 57, 63, 71, 80, 81, 140, 207, 298, periodicity, 313, 314, 322 299, 301, 302, 308, 309, 310, 311, 312, 313, permeability, 53, 61, 81, 170 314, 315, 320, 321, 322, 335, 338 permittivity, 149 porous media, 55 peroxide, 14 positive correlation, 71 PET, 166, 172, 173, 174, 175, 176, 178, 179, potassium, 152 180, 181, 192, 193 power, 9, 18, 20, 21, 128, 300, 336 phenol, 26 precipitation, 255 phenolic resins, viii, 1, 29, 30, 35, 36, 38 prediction, x, 148, 164, 255, 304 photoelectron spectroscopy, 10, 19 pressure, 18, 21, 26, 58, 59, 60, 62, 65, 121, photoinitiation, 16 125, 126, 132, 152, 182, 183, 189, 190, 191, photomicrographs, 176 192, 193, 303, 307 physical interaction, 190 probability, 198, 347 physical properties, x, 120, 129, 150 processing variables, 121 plasma, 8, 9, 10, 11, 14, 15, 16, 18, 19, 20, 21, production, 2, 3, 11, 12, 23, 44, 53, 54, 55, 56, 22, 23, 24, 35, 44, 153, 208, 209, 210 60, 61, 63, 65, 66, 74, 81, 82, 121, 171, 172, plastic deformation, 107, 114, 132, 137 189, 301 plastics, vii, viii, ix, 51, 52, 54, 55, 56, 57, 58, production costs, 60 60, 62, 63, 64, 66, 67, 71, 78, 79, 81, 82, 342, propagation, xii, 35, 37, 55, 133, 142, 215, 218, 350 220, 221, 222, 227, 232, 236, 237, 248, 253, platelets, 12, 170, 171, 177, 181, 189, 190, 191, 254, 255, 256, 270, 271, 273, 283, 284, 285, 300 286, 287, 288, 289, 292 PMMA, 126 proportionality, 317 Poisson ratio, 102 prototype, 328, 331 polar groups, 18, 22, 179 pulse, 305, 306, 347 polarity, 174, 175 purification, 152 polarization, 150, 151, 160, 162 purity, 91 pollution, ix, 51, 54 pyrolysis, 302 poly(ethylene terephthalate), 166, 172 pyromellitic dianhydride, 152 polyamides, 2 polycarbonate, 11

Index

362

Q quartz, 7 quaternary ammonium, 170, 173, 179

R radiation, 14, 98, 112, 173 radio, 11, 18 radioactive tracer, 144 Raman spectroscopy, 89 range, viii, ix, xi, 16, 29, 41, 51, 54, 56, 58, 60, 61, 65, 66, 67, 71, 73, 74, 82, 83, 98, 99, 107, 120, 148, 162, 169, 177, 181, 192, 203, 245, 306, 322, 347 reaction rate, 16 reactive sites, 5, 10, 24 reactivity, 38, 178 reagents, 14, 16, 26, 152 recognition, 323 recovery, 19 redistribution, 131, 254, 284 reference system, 97, 109 reflection, 55, 97, 159 region, 31, 94, 113, 115, 134, 135, 149, 151, 164, 306, 311, 312, 326, 327, 329, 330, 333, 335, 343 reinforcement, vii, viii, 1, 2, 3, 8, 10, 12, 14, 24, 37, 38, 44, 52, 62, 64, 67, 68, 74, 121, 129, 139, 170, 188, 190, 191, 192, 301, 303 reinforcing fibers, 20 relationship, xi, 12, 41, 55, 96, 104, 132, 142, 181, 195, 199, 230, 236, 244, 252, 253, 260 relaxation, viii, 2, 40, 44, 89, 103, 107, 114, 161, 162, 163, 165, 255, 261, 266, 290, 291, 292 relaxation process, 162 repair, 298, 308 replication, 331 residues, 154 resins, vii, viii, 1, 5, 19, 36, 170 resistance, 6, 25, 57, 61, 71, 78, 79, 81, 88, 132, 141, 142, 145, 213, 215, 221, 300, 301, 344, 346 resolution, 27, 100 resorcinol, 11 resources, 56, 300, 335 retention, 21

reusability, 56 rheology, 190, 192 rheometry, 190 riboflavin, 16 room temperature, vii, 15, 26, 27, 88, 103, 112, 136, 152, 153, 159, 160, 161, 163, 204, 205, 206, 301 roughness, 9, 10, 18, 30, 341, 342, 352, 353 rubber, 10, 11 rutile, 122

S salt, 57, 170 scanning electron microscopy, viii, 2, 9, 10, 27, 142, 203 scatter, 141 scattering, 113, 173 schema, 347 seeding, 329 selecting, 8, 342, 348, 353 SEM micrographs, 7, 38, 208, 209, 230, 233, 234, 235, 236, 237, 239, 314 semiconductor, x, 147, 150, 167 semiconductors, 150 semicrystalline polymers, 12 sensitivity, 32 separation, 6, 26, 57, 342, 343 shape, 23, 44, 100, 103, 121, 122, 133, 134, 137, 139, 148, 156, 178, 210, 215, 237, 248, 251, 254, 255, 256, 270, 271, 272, 273, 274, 291, 302, 326, 333, 346 shear, viii, 2, 3, 4, 6, 8, 9, 11, 14, 15, 20, 22, 25, 32, 36, 37, 42, 43, 44, 58, 90, 112, 171, 172, 173, 177, 183, 189, 191, 192, 193, 250, 251, 252, 263, 342, 343 shear rates, 177 shear strength, viii, 2, 3, 4, 6, 8, 9, 11, 15, 22, 25, 32, 36, 37, 42, 43, 44 Si3N4, 141, 142, 143, 146, 201, 202, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 216, 217, 218, 219, 221, 222, 223, 225, 226, 227, 233, 236, 237, 240, 242, 243, 290 side effects, 54 signs, 217, 333 silane, 5, 15, 21, 22 silica, 143 silicon, 125, 126, 143, 307, 309, 311, 312 silver, 144

Index simulation, 90, 93, 100, 102, 107, 122, 318, 323, 336, 337 sintering, 121, 122, 123, 124, 126, 128, 130, 139, 140, 141, 144, 202 SiO2, 202, 207 skeleton, ix, 88, 94, 99, 103, 115 skin, 4, 6, 20, 38 smart materials, 148 smoothing, 333 smoothness, 38 sodium, vii, 1, 4, 7, 19, 26, 29, 30, 32, 38, 152 sodium hydroxide, vii, 1, 19, 26, 29, 30, 32, 38 software, 27, 100, 203, 290, 313, 336 solidification, ix, 61, 77, 88, 91, 92, 93, 95, 97, 98, 113, 115, 301 solvents, 18, 38 space, 161, 256, 336 species, 18, 19, 31, 44 specific heat, 303, 317, 319 spectroscopy, 16, 20 spectrum, 16, 155 speed, 22, 27, 38, 91, 126, 148, 172, 173, 178, 181, 182, 188, 189, 191, 192, 336, 337, 342, 346, 348, 349, 350, 351, 352 stability, 14, 58, 88, 90, 170, 342 standard deviation, 65, 105, 352 standards, 61, 63, 65 starch, 166 steel, x, 74, 119, 204, 205, 216, 226, 229, 231, 232, 233, 234, 235, 342 STM, 32 storage, 179, 180, 181, 336 strain, 98, 102, 103, 104, 105, 106, 107, 112, 135, 136, 142, 148, 149, 150, 174, 248, 252, 263, 301, 304 stretching, 155, 191 strong interaction, 4 structural changes, 192 styrene, x, 20, 147, 151 substitution, 11, 179, 265 substrates, 3, 16, 21 sulfur, 19 sulfur dioxide, 19 sulfuric acid, 26 supply, 52 surface area, 35, 271, 272 surface energy, 5, 14, 22, 31, 133, 271 surface layer, ix, 10, 52, 55, 57, 58, 59, 60, 61, 62, 68, 71, 74, 75, 76, 77, 78, 80, 81, 82

363

surface modification, 6, 7, 8, 10, 16, 19, 22, 24, 172 surface properties, 17 surface region, 16 surface tension, 35, 144 surface treatment, vii, 4, 14, 15, 17 surfactant, 179 suspensions, 203 swelling, 17, 18, 63 symmetry, 60, 256 synthesis, 142, 146, 153, 155, 166

T Takeshima (Islets), 116 teeth, 63 temperature dependence, 102 tensile strength, viii, 2, 7, 12, 19, 21, 30, 34, 37, 40, 41, 44, 65 tension, 15, 27, 40, 112, 129, 134 textiles, 18 thermal analysis, 154, 318, 331, 336, 337 thermal degradation, 173, 179 thermal expansion, xii, 106, 112, 131, 138, 142, 196, 213, 220, 221, 247, 248, 252, 253, 255, 290, 301, 303 thermal properties, 298, 299, 308, 310, 315, 317, 318, 320, 327, 328 thermal resistance, 3 thermal stability, 300 thermal treatment, 67 thermoplastics, 11, 59, 62, 65, 82 thin films, 167 titanate, 6 toluene, 18 trajectory, 126 transducer, 188 transformation, 23, 53, 55, 77, 237, 240, 250, 326 transistor, 128 transition, 10, 14, 149, 157, 342, 350 transition temperature, 342, 350 transmission, 57, 68, 142, 173 transmission electron microscopy, 142, 154, 155, 156, 162, 173, 176, 177, 192, 210 transport, xii, xiii, 122, 167, 297, 298, 299, 300, 306, 308, 314, 315, 316, 317, 322, 331, 338, 339 tungsten carbide, 342, 351

Index

364

U

W

UHMPE, viii, 2, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 38, 39, 40, 41, 42, 43, 44 UV irradiation, 16

water absorption, 179 WAXS, 173, 174, 175, 176, 192 weak interaction, 175 wear, 6, 138, 139, 140, 145, 300, 301, 346, 351 weight ratio, 25 wettability, 9, 10, 11, 15, 16, 18, 19, 21, 22, 31, 32, 53 wetting, 4, 9, 15, 17, 23, 31, 35, 65 wood, 58, 170 wool, 73

V vacuum, 7, 11, 18, 26, 94, 125, 149, 152, 154, 249, 256 validation, 300, 339 variability, 56, 338 variables, vii, x, 119, 121, 276 variance, 105 VDF, 151, 158 vector, 93, 331, 332 velocity, 349 versatility, 8, 44 vibration, 63, 155, 347 vinylester, 15, 22, 23 vinylidene fluoride, x, 147, 148, 150, 151, 164, 165, 167 viscoelastic properties, 180 viscosity, 25, 65, 177, 178, 179, 181

X X-axis, 312 XPS, 7, 9, 10, 11, 16 X-ray analysis, 209 X-ray diffraction, ix, 87, 89, 90, 93, 95, 96, 98, 109, 112, 118, 130, 131, 142, 159, 167, 174

Y yarn, 20, 38, 40, 53

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