3D Textile Reinforcements in Composite Materials

3-D textile reinforcements in composite materials

© 1999,Woodhead Publishing Ltd

3-D textile reinforcements in composite materials Edited by Professor Antonio Miravete University of Zaragoza, Spain

Cambridge England

© 1999,Woodhead Publishing Ltd

Published by Woodhead Publishing Limited, Abington Hall, Abington Cambridge CB1 6AH, England Published in North and South America by CRC Press LLC, 2000 Corporate Blvd, NW, Boca Raton FL 33431, USA First published 1999, Woodhead Publishing Ltd and CRC Press LLC © 1999, Woodhead Publishing Ltd The authors have asserted their moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the authors and the publishers cannot assume responsibility for the validity of all materials. Neither the authors nor the publishers, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from the publishers. The consent of Woodhead Publishing and CRC Press does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing or CRC Press for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress. Woodhead Publishing ISBN 1 85573 376 5 CRC Press ISBN 0-8493-1795-9 CRC Press order number: WP1795 Cover design by The ColourStudio Typeset by Best-set Typesetter Ltd., Hong Kong Printed by TJ International, Cornwall, England

© 1999,Woodhead Publishing Ltd

Contents

Preface List of contributors

ix xiii

Introduction: Why are 3-D textile technologies applied to composite materials? A. Miravete

1

Manufacturing costs The problem in the thickness direction Examples of application Conclusions

1 2 6 8

1

3-D textile reinforcements in composite materials F.K. Ko

9

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Introduction Classification of textile preforms Structural geometry of 3-D textiles Tailoring fiber architecture for strong and tough composites Modeling of 3-D textile composites Application of the FGM Conclusions and future directions References

9 11 14 21 24 30 37 40

2

3-D textile reinforced composites for the transportation industry K. Drechsler

43

2.1 2.2 2.3

Introduction The mechanical performance of conventional and 3-D reinforced composites Manufacturing textile structural composites

43 44 49 v

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vi

Contents

2.4 2.5 2.6 2.7

3-D composites in aerospace structures Textile structural composites in automotive structures Conclusions References

56 58 63 65

3

Mechanical modelling of solid woven fabric composites P. Vandeurzen, J. Ivens and I. Verpoest

67

3.1 3.2 3.3 3.4 3.5 3.6 3.7

Introduction Review on solid woven fabric composites Elastic model: the complementary energy model Strength model Conclusions Acknowledgements References

67 67 78 88 95 97 97

4

Macromechanical analysis of 3-D textile reinforced composites A. Miravete, R. Clemente and L. Castejon

4.1 4.2 4.3 4.4 4.5

4.6 4.7 4.8

Introduction Determination of the stiffness and strength properties of 3-D textile reinforced composite materials Determination of the stiffness and strength properties of braided composite materials Determination of the stiffness and strength properties of knitted composite materials Application of macromechanical analysis to the design of a warp knitted fabric sandwich structure for energy absorption applications Application of macromechanical analysis to the design of an energy absorber type 3P bending Conclusions References

100 100 101 116 124

128 134 145 146

5

Manufacture and design of composite grids S.W. Tsai, K.S. Liu and P.M. Manne

151

5.1 5.2 5.3 5.4 5.5 5.6

Introduction Grid description Manufacturing processes Mechanical properties of grids Failure envelopes of grids Grids with skins

151 152 155 159 164 172

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Contents

vii

5.7 5.8 5.9 5.10 5.11

Flexural rigidity of isogrids Coefficients of thermal expansion Conclusions Acknowledgements References

175 177 179 179 179

6

Knitted fabric composites H. Hamada, S. Ramakrishna and Z.M. Huang

180

6.1 6.2 6.3 6.4 6.5 6.6 6.7

Introduction Description of knitted fabric Tensile behavior of knitted fabric composites Analysis of 3-D elastic properties Analysis of tensile strength properties Conclusions References

180 184 184 188 205 214 215

7

Braided structures T.D. Kostar and T.W. Chou

217

7.1 7.2 7.3 7.4 7.5

Introduction 2-D braiding 3-D braiding Summary References

217 219 220 237 238

8

3-D forming of continuous fibre reinforcements for composites O.K. Bergsma, F. van Keulen, A. Beukers, H. de Boer and A.A. Polynkine

241

8.1 8.2 8.3 8.4 8.5 8.6 8.7

Introduction Forming of continuous fibre reinforced polymers Simulation of the forming process Finite element simulation Optimization of CFRTP products Conclusions References

241 251 256 265 275 280 281

9

Resin impregnation and prediction of fabric properties B.J. Hill and R. McIlhagger

285

9.1 9.2 9.3

Introduction Hand impregnation Matched-die moulding

285 288 289

© 1999,Woodhead Publishing Ltd

viii

Contents

9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13

Degassing Preimpregnation Vacuum bagging Autoclave Liquid moulding with vacuum assistance Resin film infusion Pultrusion Conclusion Prediction of fabric properties References

© 1999,Woodhead Publishing Ltd

290 291 291 292 295 297 298 298 298 305

Preface

Laminated composite materials have been used since the 1960s for structural applications. This first generation of materials penetrated the majority of highly structural sectors because of the materials’ high stiffness and strength at low-density, high-specific energy absorption behaviour and excellent fatigue performance. High cost and the impossibility of having fibres in the laminate thickness direction, which greatly reduces damage tolerance and impact resistance, are two main limitations of these materials. However, the manufacturing costs are considerably reduced when using 3-D textile reinforced composite materials, the second generation of materials, which are obtained by applying highly productive textile technologies in the manufacture of fibre preforms. On the other hand, the damage tolerance and the impact resistance are increased since the trend to delamination is drastically diminished because of the existence of reinforcements in the thickness direction. However, methods for predicting mechanical properties of 3-D textile reinforced composite materials tend to be more complex than those for laminated composites because the yarns are not straight. Also, the existence of undulations or crimps in the yarns may reduce some mechanical properties such as tension or compression strengths. Even though this second generation of composite materials is clearly more advantageous than laminated composite materials in terms of cost, damage tolerance and impact resistance, some disadvantages have been identified. These demand research and development work in the following areas: • • • •

Textile preforms Micro- and macromechanical modelling Manufacturing processes Characterization

If researchers are able to go deeper into these four areas and overcome the problems related to the existence of crimps in the yarns, the 3-D textile reinix © 1999,Woodhead Publishing Ltd

x

Preface

forced composite material will emerge as a highly competitive family of materials for all those applications where the structural behaviour and weight are critical. The aims of this book are to describe the manufacturing processes, to highlight the advantages, to identify the main applications, to analyse the methods for prediction of mechanical properties and, finally, to focus on the key technical aspects of these promising candidates in order to know better how to exploit their main features and overcome their disadvantages in relation to the laminated composite materials. The present book focuses on the textile technologies, which use 3-D textile reinforcements for composite materials. Manufacturing techniques, design methodologies, key application fields and specific issues are studied. The first chapter is devoted to the general description of the 3-D fabrics for composite materials. The typologies available will be described and the manufacturing techniques are also explained. The needs of the transportation industry provide the most powerful reasons for the development of new materials since weight, stiffness, fatigue strength and energy absorption constitute the key design factors. The second chapter analyses the 3-D textile reinforcements for the transportation industry. The analysis of the 3-D textile reinforcements is more complex than that for unidirectional and 2-D fabrics composite materials. In our case, both micro- and macromechanical analyses must be developed. The micromechanical study is essential to determine the behaviour of the material in a fibre scale. The stiffness and strength properties must be determined as a function of type of fibre and matrix, fibre fraction, interface and fibre configuration. Chapters 3 and 4 are devoted to micro- and macromechanical analyses, respectively. The composite grid structure constitutes an amazingly efficient system to combine high stiffness and strength with lightness for high load-bearing composite applications. In the present system, the 3-D reinforcement is obtained by the inclusion of a system of ribs in the core of the material and the implementation of two outer skins. Both in-plane and out-of-plane stresses can be controlled by a proper sizing of the skins and an adequate design of the ribs. Chapter 5 discusses this type of reinforcement. Chapter 6 analyses knitted fabric composites. The knitting system is one of the most interesting manufacturing techniques for reinforcing the composite material in the thickness direction. By an appropriate design of the knitted fibres, both interlaminar normal and shear stresses can be easily controlled. The braiding technique is a very efficient method for reinforcing a large number of composite structures. Plates, beams, profiles and 3D structures can be braided nowadays, the result being a robust structure in terms of

© 1999,Woodhead Publishing Ltd

Preface

xi

stiffness, strength, energy absorption and impact loading. Hybrid schemes can be used and fibres of different materials can be oriented in the critical directions to optimize the desired criterion. This technology is discussed in Chapter 7. Chapter 8 studies the 3-D forming of continuous fibre reinforcements for composite materials. Finally, resin impregnation and xyz prediction of fabric properties are analysed in Chapter 9. I want to acknowledge the contributions from my colleagues Dr Frank Ko (Drexel University, USA), Dr Klaus Drechsler (Daimler Benz, Germany), Dr Ignaas Verpoest (KUL Leuven, Belgium), Dr Steve Tsai (Stanford University, USA), Dr H Hamada (Kyoto Institute of Technology, Japan), Dr Timothy D Kostar (Foster Miller, USA), Dr Adriaan Beukers (TU Delft, The Netherlands) and Dr Brian Hill (University of Ulster, Northern Ireland) and their teams. It has been a great pleasure for me to share with all of them the challenge to go deeper into this attractive subject. Antonio Miravete

© 1999,Woodhead Publishing Ltd

List of contributors

Bergsma, O.K., Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands Beukers, A., Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands Boer, H. de, Faculty of Mechanical Engineering and Marine Technology, Delft University of Technology, Delft, The Netherlands Castejon, L., Mechanical Engineering Department, University of Zaragoza, Spain Chou, Tsu-Wei, Center for Composite Materials, University of Delaware, Newark, DE, USA Clemente, R., Mechanical Engineering Department, University of Zaragoza, Spain Drechsler, K., Daimler-Benz AG, Ottobrunn, Germany Hamada, H., Faculty of Textile Science, Kyoto Institute of Technology, Kyoto, Japan Hill, B.J., University of Ulster, Belfast, UK Huang, Z.M., Department of Mechanical and Production Engineering, Institute of Materials Research and Engineering, National University of Singapore, Singapore Ivens, Jan, Katholieke University of Leuven, Belgium xiii © 1999,Woodhead Publishing Ltd

xiv

List of contributors

Keulen, F. van, Faculty of Mechanical Engineering and Marine Technology, Delft University of Technology, Delft, The Netherlands Ko, Frank K., Department of Materials Engineering, Drexel University, Philadelphia, PA, USA Kostar, Timothy D., Foster Miller, Waltham, MA, USA Liu, K.S., Applied Materials, Santa Clara, USA Manne, P.M., ESA/ESTEC, Noordwijk, The Netherlands McIlhagger, R., University of Ulster, Newtownabbey, Co. Antrim, UK Miravete, Antonio, University of Zaragoza, Spain Polykine, A.A., Faculty of Mechanical Engineering and Marine Technology, Delft University of Technology, Delft, The Netherlands Ramakrishna, S., Department of Mechanical and Production Engineering, Institute of Materials Research and Engineering, National University of Singapore, Singapore Tsai, S.W., Stanford University, Department of Aeronautics and Astronautics, California, USA Vandeurzen, Philippe, Katholieke University of Leuven, Belgium Verpoest, Ignaas, Katholieke University of Leuven, Belgium

© 1999,Woodhead Publishing Ltd

Introduction: Why are 3-D textile technologies applied to composite materials? ANTONIO MIRAVETE

Composite materials have been used for the past 30 years in many sectors such as aeronautics, space, sporting goods, marine, automotive, ground transportation and off-shore. These materials emerged in such areas because of their high stiffness and strength at low-density, high-specific energy absorption behaviour and excellent fatigue performance. Among the main limitations of these materials, however, are their high cost and the inability to have fibres in the laminate thickness direction, which greatly reduces damage tolerance and impact resistance.

Manufacturing costs 3-D reinforced composite materials are manufactured by impregnating a preform. This issue is critical in terms of manufacturing costs, since the process is reduced and simplified in comparison with traditional manufacturing technologies. Figure 1 represents the manufacturing steps of a standard sandwich composed of two skins and a core (left) and a warp knitted structure (right). The warp knitted sandwich structure has arisen as an efficient configuration for applications where interlaminar stresses (peeling or interlaminar shear) are critical. The warp knitted sandwich structure is characterized by having a series of fibres in the thickness directions or plies, which bridge the top and the bottom skins (Fig. 1, right). The standard sandwich requires three steps to be manufactured. First, both skins must be made. Hand lay-up or vacuum bag processes may be used for this purpose. Second, the core must be manufactured: honeycomb or foam materials are usually applied for the central part of the sandwich construction. Finally, the three sub-structures must be assembled by means of vacuum bag or press technologies. The foam may also be injected once the skins are positioned in the tooling. The manufacturing process of a warp knitted sandwich structure is much simpler and, therefore, much cheaper. The impregnated preform is pulled 1 © 1999,Woodhead Publishing Ltd

2

3-D textile reinforcements in composite materials

1 Manufacturing steps of standard sandwich (left) and warp knitted sandwich structure (right).

until the top skin reaches the design height. The resultant warp knitted sandwich is an extremely efficient structure in terms of peeling, shear, impact, damage tolerance and energy absorption behaviour, as we will show in this book.

The problem in the thickness direction The first generation of composite material consists of a number of plies composed of a matrix and unidirectional fibres oriented in a certain direction. This concept of laminate is very efficient since the fibres may be oriented in the optimal directions. However, there are a number of associated problems: the two directions perpendicular to the fibres showed very low stiffness and strength. This problem became a key issue for the composite material designers, since in-plane transverse strains and stresses appeared in many cases. In static conditions, when multidirectional loads were applied, the in-plane transverse stresses generated premature matrix cracking in the transverse

© 1999,Woodhead Publishing Ltd

Introduction

3

2 Stiffness and strength in the plane of the ply (left) and out of the plane (right).

directions. In fatigue analyses, matrix crackings were also reported for a low number of cycles. In crash problems, non-linear deformations occurred owing to the low strength and stiffness in the in-plane transverse direction and, finally, dynamic studies concluded that these transverse properties induced low natural frequencies and therefore low dynamic stiffness of a large number of composite structures. To overcome this problem, two solutions were proposed: • The implementation of multidirectional laminates. • The use of 2-D fabrics. 2-D reinforced composite materials were implemented because, by means of this typology, the two perpendicular directions were covered with fibres and, therefore, the weakness of the in-plane transverse direction vanishes. Not only were bidirectional fabrics [0/90] implemented, but multidirectional 2-D fabrics were also incorporated in order to increase stiffness and strength in a number of directions: [0/60/-60], [0/45/90/-45], etc. By using multidirectional plies or 2-D fabrics, it is possible to optimize the directions of the fibres in the plane of the ply. However, the perpendicular direction to the plane of the ply exhibits very low stiffness and strength (Fig. 2). Figure 3 shows the values of elastic moduli and strengths (MPa) in tension and compression of a carbon fibre/epoxy matrix unidirectional laminate in directions 1 (longitudinal) and 2 (thickness direction). As expected, both tension and compression elastic moduli and strengths are much higher in direction 1 than in direction 2. This problem is a major one in those cases where out-of-plane stresses are predominant over in-plane stresses. Generally speaking, when the laminate is thin and the case is static, the in-plane stresses are the most important components.

© 1999,Woodhead Publishing Ltd

4

3-D textile reinforcements in composite materials

3 Comparison of elastic moduli and strengths in MPa (tension and compression) in 1- and 2-directions.

4 Delamination due to an impact transverse load (left) and compression after impact (right).

However, the three out-of-plane stress components (peeling and the two interlaminar shear components) can become critical if the following conditions are applied: • • • • • •

Thick laminate Fatigue loading Dynamic effects Impact loads Crash problems Stress concentrations

Figure 4 shows a delamination failure due to an impact transverse load (left) and a compression after impact loading case (right). For those cases, the 3-D fabric reinforcement provides a solution to the problem detailed above. The manufacturing costs are considerably reduced when using 3-D textile reinforced composite materials, which are obtained by applying highly productive textile technologies in the manufacture of fibre preforms.

© 1999,Woodhead Publishing Ltd

Introduction

5

Table 1 Comparison of stiffness and strength properties of a standard sandwich structure and a warp knitted sandwich structure Standard sandwich Core

Height: Material:

Skins

Thickness: 2 mm Material: E-glass/ polyester (0°/90°) fabrics Vf = 30%a

Bending stiffness EI (N/mm2/mm length) Maximum bending moment (N mm/ mm length) Peeling strength (N/mm2) Interlaminar shear strength (N/mm2) a

Warp knitted sandwich 50 mm polyurethane foam r = 40 kg/m3

Core

Height: Material:

Skins

Thickness: 2 mm Material: E-glass/ polyester (0°/90°) fabrics Vf = 30%a

25 ¥ 106

25 ¥ 106

812

812

0.51

23

0.28

10

50 mm polyurethane foam r = 40 kg/m3 Plies each 5 mm (d = 0.6 mm)

Vf = fibre volume fraction.

The damage tolerance and the impact resistance are also increased since the trend to delamination is drastically diminished because of the existence of reinforcements in the thickness. Nowadays there are a number of manufacturing techniques available for composite materials: • • • • •

Braiding Stitching Warp knitting Weft knitting Weaving

These textile technologies have made possible a second generation of composite materials, specially designed for bearing high stresses in three directions, impact, crash, energy absorption and multiaxial fatigue. Table 1 represents the results of both configurations in terms of bending stiffness, peeling and interlaminar shear effects. The values of the bending

© 1999,Woodhead Publishing Ltd

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3-D textile reinforcements in composite materials

stiffness are similar, while the interlaminar strengths are two orders of magnitude higher for the warp knitted sandwich structure, owing to the existence of the plies oriented in the thickness direction.

Examples of application A comparison study between different material systems has been carried out in order to assess the weight saving in a 3P bending crash of a box beam. Braided carbon, glass fibre and a hybrid carbon–aramide system were compared with steel (Fig. 5). The maximum weight saving was obtained by the braided hybrid carbon–aramide system (67%). Also 61% and 59% of weight saving were also reported for carbon and glass fibre materials respectively. The floor of a vehicle was studied by using several material systems. The width of the floor was 1.2 m and the length was 1476 m. Both bending and torsion moment load cases were studied (Figs. 6 and 7, respectively). For the bending case, the maximum rigidity corresponded to the

5 Weight and weight saving of various material systems.

6 Bending moment load case.

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Introduction

7

warp knitted sandwich structure, followed by steel, aluminium and quasiisotropic glass fibre (Fig. 8). In terms of specific rigidity, the optimum material is the warp knitted sandwich structure, followed by the aluminium, quasi-isotropic glass fibre and steel (Fig. 8).

7 Torsion moment load case.

8 Rigidity and specific rigidity for the bending case.

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3-D textile reinforcements in composite materials

9 Rigidity and specific rigidity for the torsion case.

Finally, for the torsion case, the maximum rigidity was obtained by the warp knitted sandwich structure, followed by steel, quasi-isotropic glass fibre and aluminium (Fig. 9). In terms of specific rigidity, the optimum material is the warp knitted sandwich structure (FSS), followed by the quasiisotropic glass fibre (GF Q1), aluminium and steel (Fig. 9).

Conclusions This introductory chapter has been written in order to clarify why 3-D textile technologies are being used in conjunction with composite materials. It is obvious that the main disadvantages of standard laminated composite materials may be overcome by implementing the 3-D textile technologies available nowadays. However, methods for predicting mechanical properties of 3-D textile reinforced composite materials tend to be more complex than those for laminated composites because the yarns are not straight. Also, the existence of undulations or crimps in the yarns may reduce some mechanical properties such as tension or compression strengths. It is clear that further work must be done before the efficiency of this generation of materials for structural applications can be finally assessed. In the following nine chapters, a number of subjects related to this area are studied.

© 1999,Woodhead Publishing Ltd

1 3-D textile reinforcements in composite materials FRANK K. KO

1.1

Introduction

Textile structures are known for their unique combination of light weight and flexibility and their ability to offer a combination of strength and toughness. Textile structures have long been recognized as an attractive reinforcement form for applications ranging from aircraft wings produced by Boeing Aircraft Co. in the 1920s to carbon–carbon nose cones produced by General Electric in the 1950s. Textile preforms are fibrous assemblies with prearranged fiber orientation preshaped and often preimpregnated with matrix for composite formation. The microstructural organization of fibers within a preform, or fiber architecture, determines the pore geometry, pore distribution and tortuosity of the fiber paths within a composite. Textile preforms not only play a key role in translating fiber properties to composite performance but also influence the ease or difficulty in matrix infiltration and consolidation. Textile preforms are the structural backbone for the toughening and net shape manufacturing of composites. When combined with high-performance fibers, matrices and properly tailored fiber/matrix interfaces, the creative use of fiber architecture promises to expand the design options for strong and tough structural composites. Of the large family of textile structures, 3-D fabrics have attracted the most serious interest in the aerospace industry and served as a catalyst in stimulating the revival of interest in textile composites. 3-D fabrics for structural composites are fully integrated continuous fiber assemblies having multiaxial in-plane and out-of-plane fiber orientation. More specifically, a 3-D fabric is one that is fabricated by a textile process, resulting in three or more yarn diameters in the thickness direction with fibers oriented in three orthogonal planes. The engineering application of 3-D composite has its origin in aerospace carbon–carbon composites. 3-D fabrics for composites date back to the 1960s, responding to the needs in the emerging aerospace industry for parts and structures that were capable of 9 © 1999,Woodhead Publishing Ltd

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3-D textile reinforcements in composite materials

withstanding multidirectional mechanical stresses and thermal stresses. Since most of these early applications were for high-temperature and ablative environments, carbon–carbon composites were the principal materials. As indicated in a review article by McAllister and Lachman [1], the early carbon–carbon composites were reinforced by biaxial (2-D) fabrics. Beginning in the early 1960s, it took almost a whole decade and the trial of numerous reinforcement concepts, including needled felts, pile fabrics and stitched fabrics, to recognize the necessity of 3-D fabric reinforcements to address the problem of poor interlaminar strength in carbon–carbon composites [2–4]. Although the performance of a composite depends a great deal on the type of matrix and the nature of the fiber–matrix interface, it appears that much can be learned from the experience of the role of fiber architecture in the processing and performance of carbon–carbon composites. The expansion of global interest in recent years in 3-D fabrics for resin, metal and ceramic matrix composites is a direct result of the current trend in the expansion of the use of composites from secondary to primary load-bearing applications in automobiles, building infrastructures, surgical implants, aircraft and space structures. This requires a substantial improvement in the through-the-thickness strength, damage tolerance and reliability of composites. In addition, it is also desirable to reduce the cost and broaden the usage of composites from aerospace to automotive applications. This calls for the development of a capability for quantity production and the direct formation of structural shapes. In order to improve the damage tolerance of composites, a high level of through-thickness and interlaminar strength is required. The reliability of a composite depends on the uniform distribution of the materials and consistency of interfacial properties. The structural integrity and handleability of the reinforcing material for the composite is critical for large-scale, automated production. A method for the direct formation of the structural shapes would therefore greatly simplify the laborious hand lay-up composite formation process. With the experience gained in the 3-D carbon–carbon composites and the recent progress in fiber technology and computer-aided textile design and liquid molding technology, the class of 3-D fabric structures is increasingly being recognized as serious candidates for structural composites. The importance of 3-D fabric reinforced composites in the family of textile structural composites is reflected in several recent books on the subject [5,6]. This chapter is intended to provide an introduction to 3-D textile reinforcements for composites. The discussion will focus on the preforming process and structural geometry of the four basic classes of integrated fiber architecture: woven, knit and braid, and orthogonal non-woven 3-D structure.

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3-D textile reinforcements in composite materials

11

Table 1.1. Fiber architecture for composites Level Reinforcement system

Textile construction

Fiber length

Fiber orientation

Fiber entanglement

I II III IV

Chopped fiber Filament yarn Simple fabric Advanced fabric

Discontinuous Continuous Continuous Continuous

Uncontrolled Linear Planar 3-D

None None Planar 3-D

1.2

Discrete Linear Laminar Integrated

Classification of textile preforms

There is a large family of textile preforming methods suitable for composite manufacturing [7]. The key criteria for the selection of textile preforms for structural composites are (a) the capability for in-plane multiaxial reinforcement, (b) through-thickness reinforcement and (c) the capability for formed shape and/or net shape manufacturing. Depending on the processing and end use requirements some or all of these features are required. On the basis of structural integrity and fiber linearity and continuity, fiber architecture can be classified into four categories: discrete, continuous, planar interlaced (2-D) and fully integrated (3-D) structures. In Table 1.1 the nature of the various levels of fiber architecture is summarized [8]. A discrete fiber system such as a whisker or fiber mat has no material continuity; the orientation of the fibers is difficult to control precisely, although some aligned discrete fiber systems have recently been introduced.The structural integrity of the fibrous preform is derived mainly from interfiber friction. The strength translation efficiency, or the fraction of fiber strength translated to the non-aligned fibrous assembly of the reinforcement system, is quite low. The second category of fiber architecture is the continuous filament, or unidirectional (0°) system. This architecture has the highest level of fiber continuity and linearity, and consequently has the highest level of property translation efficiency and is very suitable for filament wound and angle ply tape lay-up structures. The drawback of this fiber architecture is its intraand interlaminar weakness owing to the lack of in-plane and out-of-plane yarn interlacings. A third category of fiber reinforcement is the planar interlaced and interlooped system. Although the intralaminar failure problem associated with the continuous filament system is addressed with this fiber architecture, the interlaminar strength is limited by the matrix strength owing to the lack of through-thickness fiber reinforcement. The fully integrated system forms the fourth category of fiber architecture wherein the fibers are oriented in various in-plane and out-of-plane

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3-D textile reinforcements in composite materials

1.1 The Noveltex® method.

directions. With the continuous filament yarn, a 3-D network of yarn bundles is formed in an integral manner. The most attractive feature of the integrated structure is the additional reinforcement in the through-thickness direction which makes the composite virtually delamination-free. Another interesting aspect of many of the fully integrated structures such as 3-D woven, knits, braids and non-wovens is their ability to assume complex structural shapes. Another way of classifying textile preforms is based on the fabric formation techniques. The conversion of fiber to preform can be accomplished via the ‘fiber to fabric’ (FTF) process, the ‘yarn to fabric’ (YTF) process and combinations of the two. An example of the FTF process is the Noveltex® method developed by P. Olry at SEP (Société Européenne de Propulsion, Bordeaux, France) [9]. As shown in Fig. 1.1, the Noveltex concept is based on the entanglement of fiber webs by needle punching. A similar process is being developed in Japan by Fukuta [10] using fluid jets in place of the needles to create through-thickness fiber entanglement. The YTF processes are popular means for preform fabrication wherein the linear fiber assemblies (continuous filament) or twisted short fiber (staple) assemblies are interlaced, interlooped or intertwined to form 2-D or 3-D fabrics. Examples of preforms created by the YTF processes are shown in Fig. 1.2. A comparison of the basic YTF processes is given in Table 1.2. In addition to the FTF and YTF processes, textile preforms can be fabricated by combining structure and process. For example, the FTF webs

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3-D textile reinforcements in composite materials

13

Biaxial woven

High modulus woven

Multilayer woven

Triaxial woven

Tubular braid

Tubular braid laid in warp

Weft knit

Weft knit laid in weft

Weft knit laid in warp

Weft knit laid in weft laid in warp

Square braid

Square braid laid in warp

Warp knit laid in warp

Weft inserted Weft inserted warp knit warp knit laid in warp

Warp knit

XYZ laid in system

Biaxial bonded

Flat braid

Flat braid laid in warp

XD

3-D braid

Stitchbonded laid in warp

3-D braid laid in warp

1.2 Examples of yarn-to-fabric preforms.

Table 1.2. A comparison of yarn-to-fabric formation techniques YTF processes

Basic direction of yarn introduction

Basic formation technique

Weaving

Two (0°/90°)

Braiding

One (machine direction) One (0° or 90°)

Interlacing (by selective warp and fill insertion of 90° yarns into 0° yarn system Intertwining (position displacement)

Knitting Nonwoven

Three or more (orthogonal)

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Interlooping (by drawing warp or fill loops of yarns over previous loops) Mutual fiber placement

14

3-D textile reinforcements in composite materials Eight 9-ply elements

Basic 9-ply subelement

Stiffener

AS4/3501 1/2-in. stitch spacing

0° +45° 0° +45° 0° +45° 0° –45° 90°

Eight 9-ply segments stitched with 200d Kevlar to form blade Stiffened panel Skin

Fold blade web ends open Stitching head Six 9-ply segments

Holding pin Web locating bar Glide bar Folding frame

Stringer flanges stitched to skin

Panel Table top

Stitched flap on skin

1.3 Combination of FTF and YTF processes.

can be incorporated into a YTF preform by needle or fluid jet entanglement to provide through-the-thickness reinforcement. Sewing is another example that can combine or strategically join FTF and/or YTF fabrics together to create a preform having multidirectional fiber reinforcement [11] (Fig. 1.3).

1.3

Structural geometry of 3-D textiles

The structural geometry of 3-D textiles can be characterized at both the macroscopic and the microscopic levels. At the macroscopic level, the external shape and the internal cellular structures are the result of a particular textile process and fabric construction employed in the creation of the structure. Similar shape and cellular geometry may be created by different textile processes. For example, a net shape I-beam can be produced by a weaving, braiding or knitting process. However, the microstructure or the fiber architecture produced by these three processes are quite different. This will lead to different levels of translation efficiency of the inherent fiber properties to the composite as well as different levels of damage-resistant characteristics. The efficient translation of fiber properties to the composite depends on the judicious selection of fiber architecture which is governed by the directional concentration of fibers. This directional fiber concentration can be quantified by fiber volume fraction Vf and fiber orientation, q. Depending upon the textile manufacturing process used and the type of fabric construction, families of Vf - q functions can be generated. These Vf - q functions can be developed by geometrical modeling as

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3-D textile reinforcements in composite materials

15

1.4 3-D woven fabrics.

detailed by Ko and Du [12]. Accordingly, the structure–property relationship of 3-D textile composites is a result of the dynamic interaction of microstructural and macrostructural geometries. In this section, the structural shapes, cellular structures and fiber architectures expressed in terms of the Vf - q functions are presented for the four basic classes of 3-D textile reinforcements.

1.3.1 3-D woven fabrics 3-D woven fabrics are produced principally by the multiple-warp weaving method which has long been used for the manufacturing of double and triple cloths for bags, webbings and carpets. By the weaving method, various fiber architectures can be produced including solid orthogonal panels (Fig. 1.4a), variable thickness solid panels (Fig. 1.4b, c), and core structures simulating a box beam (Fig. 1.4d) or a truss-like structure (Fig. 1.4e). Furthermore, by proper manipulation of the warp yarns, as exemplified by the angle interlock structure (Fig. 1.4f), the through-thickness yarns can be organized into a diagonal pattern. To address the inherent lack of inplane reinforcement in the bias direction, Dow [13] modified the triaxial weaving technology to produce multilayer triaxial fabrics as shown in Fig. 1.4(g). Through unit cell geometric modeling the Vf - q functions can be generated for various woven fabrics. Figure 1.5 plots total fiber volume fraction versus web interlock angle for an angle interlock 3-D woven fabric, with three levels of linear density ratio. For purposes of calculation, the fiber packing fraction is assumed to be 0.8, which provides the upper limit for possible fiber volume fraction. The fabric tightness factor (h) used is 0.2.

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3-D textile reinforcements in composite materials

1.5 Process window of fiber volume fraction for 3-D woven (l w/q is linear density of warp or web yarn, l f is linear density of filled yarn).

1.3.2 Orthogonal non-woven fabrics Pioneered by aerospace companies such as General Electric [14], the nonwoven 3-D fabric technology was developed further by Fiber Materials Incorporated [15]. Recent progress in automation of the non-woven 3-D fabric manufacturing process was made in France by Aérospatiale [16], SEP [9] and Brochier [17,18] and in Japan by Fukuta and Coworkers [19,20]. The structural geometries resulting from the various processing techniques are shown in Fig. 1.6. Figure 1.6(a) and (b) show the single bundle XYZ fabrics in a rectangular and cylindrical shape. In Fig. 1.6(b), the multidirectional reinforcement in the plane of the 3-D structure is shown. Although most of the orthogonal non-woven 3-D structures consist of linear yarn reinforcements in all of the directions, introduction of the planar yarns in a non-linear manner, as shown in Fig. 1.6(c), (d) and (e) can result in an open lattice or a flexible and conformable structure. Based on the unit cell geometry shown in Fig. 1.7, assuming an orthogonal placement of yarns in all three directions, the Vf - q function was constructed for an orthogonal woven fabric. Figure 1.8 plots the fiber volume fraction versus dy/dx (fiber diameter) ratios, assuming a fiber packing fraction of 0.8. For all three levels of dz/dx ratios, the fiber volume fraction first decreases with the increase in dy/dx ratio, reaches a minimum, and then increases. As can be seen in the figure, the maximum fiber volume fraction is about 0.63 at either high or low dy/dx ratios, whereas the minimum fiber

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1.6 Orthogonal woven fabrics.

1.7 Unit cell for orthogonal non-woven fabrics.

volume fraction of about 0.47 is achieved when both dy/dx and dy/dx ratios are equal to 1.

1.3.3 Knitted 3-D fabrics The knitted 3-D fabrics are produced by either the weft knitting or warp knitting process. An example of a weft knit is the near net shape structure

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3-D textile reinforcements in composite materials

1.8 Process window of fiber volume fraction for orthogonal nonwoven fabrics.

knitted by the Pressure Foot® process [21] (Fig. 1.9a). In a collapsed form this preform has been used for carbon–carbon aircraft brakes. The unique feature of the weft knit structures is their conformability [22]. By strategic introduction of linear reinforcement yarns, weft knitted structures can be used effectively for forming very complex shape structures. While the suitability of weft knit for structural applications is still being evaluated, much progress has been made in the multiaxial warp knit (MWK) technology in recent years [23,24]. From the structural geometry point of view, the MWK fabric systems consist of warp (0°), weft (90°) and bias (±q) yarns held together by a chain or tricot stitch through the thickness of the fabric, as illustrated in Fig. 1.10(b). The logical extension of the MWK technology is the formation of circular multiaxial structures by the warp knitting process. This technology (Fig. 1.9d) has been demonstrated in the Institute of Textiles of the University of Aachen [25]. An example of MWK is the LIBA system, as shown in Fig. 1.9(c) and (d). Six layers of linear yarns can be assembled in various stacking sequences along with a fiber mat and can be integrated together by knitting needles piercing through the yarn layers. The unit cell geometric analysis of a four-layer system is used as an example to generate the Vf - q functions for the MWK fabric [26]. This analysis can be generalized to include other MWK systems with six or more layers of insertion yarns. The fiber volume fraction relation in Fig. 1.10 shows that for the fixed parameters selected, only a limited window exists for the MWK fabric construction. The window is bounded by two factors: yarn jamming and the point of 90° bias yarn angle. Fabric constructions corresponding to the curve marked ‘jamming’ are at their tightest allowable point, and constructions at the q Æ 90° curve have the most open structure.

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3-D textile reinforcements in composite materials

0°

–45°

Chain

19

Tricot 45° a

Warp inlay yarns

Knitting yarns 0°

90°

–q, –45°

90°

+q, –45°

90°

45°45°0°90° b

Unit cell c Knitted welt yarn layers

Nonwoven material

1.9 3-D knitted fabrics.

1.10 Fiber volume fraction versus ratio of stitch-to-insertion yarn linear density (tricot stitch, k = 0.75, r = 2.5 kg/m3, fi = 5 and h = 0.5).

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3-D textile reinforcements in composite materials

1.11 3-D braided fabrics.

When q < 30°, jamming occurs in the whole range of yarn linear density ratio from zero to infinity. When q is in the range of 30° to 40°, the fiber volume fraction decreases with an increase in yarn linear density ratio until jamming occurs. When q = 45°, the fiber volume fraction decreases with an increase in yarn linear density ratio to a minimum at about ls/li = 1, and starts to increase until jamming occurs (ls = stitch yarn diameter, li = insert yarn diameter). When q ≥ 60°, the fiber volume fraction has the same trend as when q = 45°, but yarn jamming never occurs. The fiber packing in the yarns, taken as 0.75, limits the maximum fiber volume fraction in the fabric.

1.3.4 3-D braided fabrics 3-D braiding technology is an extension of the well-established 2-D braiding technology wherein the fabric is constructed by the intertwining of two or more yarn systems to form an integral structure. 3-D braiding is one of the textile processes wherein a wide variety of solid complex structural shapes (Fig. 1.11a) can be produced in an integral manner, resulting in a highly damage-resistant structure. Figure 1.11(b) shows two basic loom setups in circular and rectangular configurations [27]. The 3-D braids are produced by a number of processes including the track and column method [28] (Fig. 1.11c), the two-step method [29] (Fig. 1.11d) as well as a variety of displacement braiding techniques by discrete or continuous motions [30]. The basic braiding motion includes the alternate X and Y displacement of yarn carriers followed by a compacting motion. The formation of shapes is accomplished by the proper positioning of the carriers and the joining of various rectangular groups through selected carrier movements. Based on unit cell geometry analysis, Fig. 1.12 shows the Vf - q relationship prior to and at the jamming condition [31]. The fiber packing fraction,

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1.12 Relationship of fiber volume fraction to braiding angle for various tightness factors [31].

k, is assumed as 0.785. As can be seen, there are three regions of fiber volume fraction. The upper region cannot be achieved owing to the impossible fiber packing in a yarn bundle. Jamming occurs when the highest braiding angle is reached for a given fabric tightness factor h. The non-shaded region is the working window for a variety of Vf - q combinations. Clearly, for a given fabric tightness, the higher braiding angle gives a higher fiber volume fraction and, for a fixed braiding angle, the fiber volume fraction is greater at higher tightness factors.

1.4

Tailoring fiber architecture for strong and tough composites

Strength and toughness are usually considered to be concomitant mechanical properties for traditional engineering materials. By properly manipulating fiber architecture, the degree of freedom permitted in the engineering of strong and tough composite materials is greatly increased. The strengthening effect of fiber on polymer matrix composites is well established. The role of fiber in the toughening of ceramic matrix composites is now generally recognized. A major challenge in advanced composite materials is to achieve a balance of strengthening and toughening effects. While well-oriented linear unitape composites provide the maximum strengthening effect when loading is along the fiber direction, the interlaced non-linear integrated systems tend to maximize damage containment and

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3-D textile reinforcements in composite materials

enhance toughness. Expanding from these strengthening and toughening concepts, one may hypothesize that a combination of the strategically placed linear fibers in a 3-D integrated fiber network would provide the necessary ingredients for a strong and tough composite. The linear fibers and the fibers that make up the 3-D fiber network can be from the same material family but differ in diameter or bundle size. Alternatively, the two material systems can be quite different, each contributing unique properties such as strength, stiffness and thermal stability to the composite structure. To demonstrate this geometric/material hybrid concept, let us examine three examples, polymer matrix composites (PMC), metal matrix composites (MMC) and ceramic matrix composites (CMC), wherein different proportions of linear fibers were placed in a 3-D braided fiber network.

1.4.1 Polymer matrix composites The hybrid concept has been demonstrated in several previous studies for PMC. For example, it was found that a combination of linear fibers in a 3-D braided structure improves the compression-after-impact strength of PEEK/carbon composites [32]. It was further demonstrated that the linear fibers provide additional flexibility in modifying the failure modes of the composites whereas the 3-D fiber network greatly reduces the damage area caused by impact loading. In a controlled study [33], the compressive stress–strain behavior of carbon, glass and Kevlar fibers in an Epon 828/DETA matrix with linear fiber to 3-D braided fiber ratio ranging from 0/100 to 75/25 was examined. It was found that, in all three cases, both the compressive strength and the elongation to break increase as the proportion of linear fibers increases. This strengthening and toughening effect is illustrated in Fig. 1.13, which shows that the compressive strength of the carbon/epoxy composites increases from 415 MPa to 760 MPa as the proportion of linear fibers in the 3-D braided structure increases from 0/100 to 75/25. Likewise, elongation to breaking increases from 0.7% to 0.9%.

1.4.2 Metal matrix composites The concepts of geometric and material hybrids for MMC are demonstrated using a combination of SCS-6 SiC filaments in a 3-D braided Nicalon SiC reinforced Al 6061 composite [34]. The hybrid effect was studied by tensile, notched beam three-point bending and compact tension tests of the 3-D braided composites. The hybrid braided SiC/AL-6061 composites studied include a 0/100, 25/75, 50/50 and 75/25 combination of linear SCS with 3-D braided Nicalon. Addition of the strong and stiff SCS filaments to the Nicalon reinforced aluminum strengthens the composites and

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1.13 Compressive stress–strain behavior of carbon/epoxy composites.

modifies the composite failure mode from matrix-dominated failure to linear yarns controlled failure. Figure 1.14 shows the stress–strain curves of the MMC reflecting the nonlinear behavior of the composites. In comparison to the pure cast aluminum sample, the 100% braided composites show only a slight reinforcement effect whereas the hybrid composites show a remarkable improvement in strength (121 to 599 MPa) and toughness (0.15% to 0.81% failure strain) as the percentage of longitudinal SCS-6 lay-in yarns increases from 0 to 50%. The effect of geometric and material hybrids on the fracture behavior can be illustrated using the response of the MMC to 3-point bending tests. As shown in Fig. 1.15, the onset fracture load increases as the percentage of SCS-6 filaments increases. When the propagating crack reaches a bundle of SCS-6 filaments, the crack propagation rate is delayed and gradual failure occurs, as illustrated in the zig-zag pattern of the load–deflection curves.

1.4.3 Ceramic matrix composites Employing the same material–geometric hybrid system used in the MMC study, various proportions (ranging from 0/100, 25/75, 50/50 to 75/25) of

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1.14 Tensile stress–strain behavior of 3-D braided SiC/Al composites.

SCS-6/Nicalon 3-D braid were fabricated in a lithium aluminum silicate (LAS III) matrix. Tensile, flexural and fracture tests were carried out on the 3-D braided CMC [35]. Besides enhancing strength, the thermally stable SCS filament played a significant role in preserving the structural integrity of the composite during processing and under high-temperature oxidation end-use environments. By placing the strong and stiff SCS filaments in the axial (0°) direction in the 3-D braided Nicalon fiber network, a significant improvement in tensile strength as well as the first cracking strength were achieved in the SiC/SiC/LAS III structure (Fig. 1.16). It is remarkable to observe that the elongation to break of the hybrid composites also increases with the increase of the proportion of SCS filaments, resulting in a much strengthened and toughened CMC.

1.5

Modeling of 3-D textile composites

Given a large family of fiber architectures which can be generated by an impressive array of textile preforming techniques, it is quite evident that one can tailor composite properties to meet various end-use requirements.

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3-D textile reinforcements in composite materials

25

180 50.8 mm

160

6.35 mm

25 % Nicalon /75% Avco

140

Load (P)

6.35 mm

Load P (kg)

120 50% Nicalon/50% Avco 100

80

60 75 % Nicalon /25% Avco 40

20 100 % Nicalon 0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Deformation D (mm) 1.15 Fracture behavior of 3-D braided SiC/Al composites by threepoint bend test.

In order to facilitate the rational selection and to stimulate the creative design of fiber architecture by textile preforming, a science-based design framework must be established to bridge the communication gap between textile technologists, composite materials engineers and structural design engineers. This framework must be capable of relating preform manufacturing parameters to fiber architectural geometry as well as material properties. This design framework can be constructed through three levels of modeling including topological, geometrical and mechanical models. The topological model is a quantitative description of the preforming process. The geometrical model is the heart of the design framework which quantifies fiber orientation and fiber volume distribution in terms of fiber bundle

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3-D textile reinforcements in composite materials 1000 140 900

266 HYBRID (75% SCS-6)

120 800

GRIP FAILURE

261 HYBRID (50% SCS-6)

700

100 251 HYBRID (25% SCS-6)

LOOSING STRAIN GAGES

80 500

60 400

239 BRAID

STRESS (Kal)

STRESS (MPa)

600

300 40 200 119 BRAID [REF 5]

20 100

STRAIN (%)

0.2

0.4

0.6

0.8

1.0

1.2

1.16 Tensile stress–strain behavior of 3-D braided SiC/LAS III composites.

(yarn) and fabric structural geometry created by the preforming process. The mechanical model provides the link between the mechanical properties of the material system and the fiber architecture. The product of the mechanical model is the stress–strain response of the textile reinforced composite for a given boundary condition as well as a stiffness matrix reflecting the material and fiber architecture contribution to the properties of the composite system. With the stiffness matrix, the structural designer can perform finite element analysis with meaningful information, thus facilitating the integration of material design concepts, manufacturing processes and structural design to product engineering. The mechanical properties of textile reinforced composites can be predicted with a knowledge of the fiber properties, matrix properties and textile preform fiber architecture through a modified laminate theory

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3-D textile reinforcements in composite materials

27

approach. Geometric unit cells defining the fabric structure (or textile preform) can be identified and quantified to form a basis for the analysis. For 2-D woven fabric reinforced composites, Dow [35] and Chou and Yang [36] have developed models for the thermomechanical properties of plain, twill and satin reinforced composites. Examples of these include the mosaic, crimp and bridging models developed by Chou. In the mosaic model, fiber continuity is ignored and the composite is treated as an assembly of crossply elements. With the crimp model, the non-linear crimp geometry as well as yarn continuity is considered. Based on the geometric repeating unit cell, each yarn segment is treated as laminar. While the crimp model was found to be most suitable for plane weave composites, the bridging model was found to be best for satin weave composites, as it takes the relative stiffness contribution of the linear and non-linear yarn segments into consideration. Over the past decade, as reviewed by Cox and Flanagan [38], an impressive number of models have been developed for textile composites. The modeling of 3-D fiber reinforced composites begins with the establishment of geometric unit cells. Using 3-D braided composites as an example the following illustrates an approach which focuses on the integration of fiber architecture design for manufacturing. From a ‘preform processing science’ point of view, Pastore and Ko [39] developed a ‘fabric geometry model’ (FGM) based on the unit cell geometry shown in Fig. 1.12. The stiffness of a 3-D braided composite was considered to be the sum of stiffnesses of all its laminae. The unit cell for the 3-D braid can be represented by several yarns running parallel to the body diagonal of the cell. However, in some instances, yarns are placed in longitudinal (0°) and transverse (90°) directions of the fabric and are referred to as longitudinal and transverse reinforcements (or lay-ins) respectively. The preform processing parameters are specifically related to corresponding unit cell geometries. The geometric descriptions form the basis for an FGM which models a characteristic volume. Accordingly, the generation of the stiffness matrix through the FGM provides a link between microstructural design and macrostructural analysis. In order to establish a geometric model and method for analyzing the properties of the 3-D braid, it is necessary first to identify the orientation of the yarns in the 3-D fiber network. Figure 1.17 shows a typical 3-D braided structure with an enlarged view of the unit cell. Processing parameters U, V and W, representing the thickness, width and height of the unit cell, are related by the following equation: W=

U2 + V2 tanq

[1.1]

where q, the interior angle, defines the orientation of the yarn with respect to the longitudinal axis of the panel. Given this relationship, it is possible

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3-D textile reinforcements in composite materials

1.17 Unit cell geometry for the fabric geometry model.

to identify the angle q associated with a particular fabric system, or to determine the value of W necessary to manufacture a fabric with fiber orientation q. Once the interior angle q has been identified for a given system, the relation between the desirable fiber volume fraction and the total number of yarns needed in the composite can be established as follows: Vf =

N y Dy Cd Ac r cos q

where: Ny = total number of yarns in the fabric, Cd = 9 ¥ 105, is a constant, Ac = cross-sectional area of finished composite (cm2), r = density of fiber (g/cc),

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[1.2]

3-D textile reinforcements in composite materials

29

q = interior angle, defined in Equation 1, Dy = linear density of fiber (denier), Vf = fiber volume fraction. Once the fabric geometry has been quantified, the result can be used together with the fiber and matrix properties to predict the mechanical properties of the composite system through a modified lamination theory. From the geometry of a unit cell associated with a particular fiber architecture, different systems of yarn can be identified whose fiber orientations are defined by their respective interior angle q and azimuthal angle b, as previously shown in Fig. 1.17. Assuming each system of yarn can be represented by a comparable unidirectional lamina with an elastic stiffness matrix defined as follows: Ê c11 Á Á [C ] = ÁÁ Á Á Á Ë where: c11 c22 c12 c23 c44 c55 c66 K*

c12 c 22

c13 c 23 c33

0 0 0 c 44

0 0 0 0 c55

0 ˆ 0 ˜ ˜ 0 ˜ 0 ˜ ˜ 0 ˜ ˜ c 66 ¯

[1.3]

= (1 - n232)E11/K*, = c33 = (1 - n12n21)E22/K*, = c13 = (1 + n23)n21E11/K*, = (n23 + n12n21)E22/K*, = G23, = G13, = G12, = 1 - 2n12n21(1 + n23) - n232,

then the elastic stiffness matrix [C ¢] of this yarn system in the longitudinal direction of the panel can be expressed as

[C ¢] = [T ][C ][T ]

-1

[1.4]

in which the transformation matrix 2

Ê l1 Ál 2 Á 22 l [T ] = ÁÁ 3 Á l 2 l3 Á l1l3 Á Ë l 2 l1

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m12 m22 m32 m2 m3

n12 n22 n32 n2 n3

2 m1 n1 2 m2 n2 2 m3 n3 m2 n3 + m3 n2

2l1 n1 2l 2 n2 2l3 n3 l 2 n3 + l3 n2

m1 m3 m2 m1

n1 n3 n2 n1

m1 n3 + m3 n1 m1 n2 + m2 n1

l1 n3 + l3 n1 l1 n2 + l 2 n1

2l1 m1 ˆ ˜ 2l 2 m2 ˜ 2l3 m3 ˜ l 2 m3 + l3 m2 ˜˜ l1 m3 + l3 m1 ˜ ˜ l1 m2 + l 2 m1 ¯ [1.5]

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3-D textile reinforcements in composite materials

where l1 = cos q l2 = sin q cos b l3 = sin q sin b

m1 = 0 m2 = sin b m3 = -cos b

n1 = -sin q n2 = cos q cos b n3 = cos q sin b

It is noted that the material properties of a unidirectional lamina, E11, E22, G12, . . . , can be obtained easily using the well-established micromechanical relationships.

1.6

Application of the FGM

1.6.1 Prediction of stress–strain relationships In order to determine the stress–strain behavior of the fabric reinforced composites, it is necessary to utilize each of the yarn systems. A model for yarn system interaction has been chosen wherein the stiffness matrices for each system of yarns are superimposed proportionately according to contributing volume to determine the fabric reinforced composite system stiffness:

[C ] = Â ki[C ¢] j

[1.6]

i

where: [C] = total stiffness matrix, ki = fractional volume of the ith system of yarns. In order to account for the potentially non-linear behavior of the materials, the system stiffness matrix should be calculated anew at each strain level. Thus the stress–strain behavior of the composite can be expressed as Ds = [C ]s [De ]

[1.7]

where: Ds = incremental stress vector (6 ¥ 1), De = incremental strain vector (6 ¥ 1). From this, the stress vector can be determined as s = s + Ds

[1.8]

where: s = stress vector (6 ¥ 1). A failure point for the composite is determined for each system of yarns by a maximum strain energy criterion. If the strain energy on the fiber exceeds the maximum allowable, that system of yarns has failed. Mathematically, if the following expression is true, the system has failed: Uc,i ≥ Um

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[1.9]

3-D textile reinforcements in composite materials

31

e fu

where: Um = Vf (s fu (e fu ) 2) + (1 - Vf ) Ú s m (e ) e de , 0

sfu = fiber ultimate strength, efu = fiber ultimate strain, Uci = strain energy of the ith yarn system with strain ek, sci = stress on the ith composite system. Using these maximum energy criteria, a failure point for each system of yarns can be found. When a system of yarns fails, its contribution to the total system stiffness is removed. When all systems have failed, the composite is said to have failed. In this way, the entire stress–strain curve for the composite can be predicted up to the point of composite failure. The FGM has been employed with satisfactory results to predict the stress–strain properties of polymer, metal and ceramic matrix components. Shown in Figs. 1.18, 1.19 and 1.20 are the theoretical and experimental tensile stress strain relatively of carbon/epoxy, SiC/Al and SiC/LAS III composites respectively.

1.18 Theoretical and experimental compressive stress–strain relationship of 3-D braided carbon/epoxy composites.

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3-D textile reinforcements in composite materials

1.19 Theoretical and experimental tensile stress–strain relationship of 3-D braided SiC/Al composites.

1.20 Theoretical and experimental tensile stress–strain relationship of 3-D braided SiC/LAS III composite.

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1.21 An integrated design methodology for 3-D fabric composite structures.

1.6.2 Structural analysis of a 3-D braided composite turbine blade By using the FGM analysis, one can obtain the elasticity matrix [C] of a braided composite. In a displacement-based finite element formulation the stiffness matrix [K] can be obtained from the equation K = Ú BTC B dy

[1.10]

where [B] is the strain-displacement transformation matrix. Therefore, the FGM can be incorporated into general finite element analysis programs for solving braided composite structures with complex shapes. Data input to this program include fiber/matrix properties and braiding parameters, i.e. surface angle, inclined angle, fiber volume fraction and braiding percentage. For illustration purposes, the integrated design for manufacturing methodology has been applied to the design and analysis of various complex-shape engine components. A brief outline of the procedure for the design of a composite turbine blade is shown in Fig. 1.21. For a more detailed description and solution to the problem, readers are referred to reference [40]. In the design of the composite turbine blade, it has been established that the structure must be able to survive a certain

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3-D textile reinforcements in composite materials

level of centrifugal force due to a given level of rotational speed and the thermal loading due to heated gas flow from the combustion chamber. In order to assess the fatigue life of the turbine blade, the natural frequencies of the blade must also be known. On the other hand, the aerodynamic analysis of a turbine rotor may impose certain restrictions on the thickness of blades, and the fiber architecture used may have to provide the structure with enough fiber volume fraction in some critical areas and directions. The selection of material systems and fabrication techniques will greatly influence the mechanical properties of the composite system and the responses of structural components, and is therefore the most important step in this design framework. For each material system and fabrication technique selected as a possible candidate, the corresponding fiber architecture and material properties can be identified using the FGM. Structural analysis of the blade may then be conducted using the finite element method to verify whether the selected material system and fabrication technique meet the design criteria, and fabrication technique may be modified accordingly. This process may be iterated until a final design with an optimal fiber architecture and proven structural behavior is identified. The structural analysis of the composite blade is performed using a general-purpose finite element code [41]. Based on hypothetical blade specifications (Fig. 1.22), a 3-D model is first generated on a computer-aided design (CAD) system; the co-ordinates of the geometry for constructing a finite element mesh are then obtained from this model. Thirty-six 20-node 3-D isoparametric elements are used to model the blade, resulting in a total of 720 nodes. Figure 1.23 shows the finite element mesh established for the turbine blade. After consideration of several fabrication techniques with different fiber volume fractions, a final design was chosen. The mechanical properties of each fiber architecture are computed from the FGM. The example here is a 3-D braided Nicalon SiC structure in a lithium alumina silicate (LAS III) ceramic matrix. The input material properties and structural parameters for the computation of the system stiffness matrix are presented in Tables 1.3 and 1.4. The resulting elastic stiffness matrix and the coefficients of thermal expansion for the shank region are given by 0.4 1.2 1.2ˆ Ê 157.6 49.4 50.2 Á 49.4 156.1 52.3 -0.2 -0.4 -0.9˜ ˜ Á Á 50.1 52.3 156.5 -0.2 -0.9 -0.4˜ ( Á 0.4 -0.2 -0.2 46.7 -0.6 -0.6˜ GPa) ˜ Á Á 1.2 -0.4 -0.9 -0.6 43.9 0.4˜ ˜ Á Ë 1.2 -0.9 -0.4 -0.6 0.4 41.9¯

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3-D textile reinforcements in composite materials

1.22 Low-pressure turbine blade.

1.23 Hypothetical turbine blade showing finite element mesh for Fig. 1.22.

Table 1.3. Materials properties

E11 (GPa) E22 (GPa) E33 (GPa) G12 (GPa) r12 (g/cc) a (10-6/°C)

Nicalon SiC

LAS III

139 32 32 43 0.20 3.1

181 181 181 60 0.26 1.6

Table 1.4. Geometric parameters Braid construction Braiding angle q Vf

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35

1¥1 ±20 0.4

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3-D textile reinforcements in composite materials

with coefficients of thermal expansion a1 = 1.53 ¥ 10-6 a2 = 1.61 ¥ 10-6 a3 = 1.65 ¥ 10-6 and those of the dovetail region are given by 1.9 2.3 2.3ˆ Ê 154.3 51.0 51.0 Á 51.0 153.9 54.0 -1.0 -0.6 -2.0˜ ˜ Á Á 51.0 54.0 153.9 -1.0 -2.0 -0.6˜ ( Á 1.9 -1.0 -1.0 49.5 -1.1 -1.1˜ MPa) ˜ Á 1.7˜ Á 2.3 -0.6 -2.0 -1.1 43.5 ˜ Á Ë 2.3 -2.0 -0.6 -1.0 1.7 43.5¯ with coefficients of thermal expansion a1 = 1.53 ¥ 10-6 a2 = 1.61 ¥ 10-6 a3 = 1.65 ¥ 10-6 respectively. With the stiffness matrix generated according to the input material and geometric properties, one can examine the response of the 3-D composite system to a given thermomechanical condition. For a fixed boundary condition on the bottom surface of the blade, assuming a 1000 °C operating temperature and a rotational speed of 4500 rpm, we can evaluate the centrifugal force field as well as the displacement contour on the turbine blade. Figure 1.24 illustrates a typical stress distribution on the tension surface of the blade for axial (1) stresses. Using this preliminary analysis of this component, the axial stresses in the blade are relatively constant at a value of 50 MPa. In the radial direction, stresses on the shaft of the blade are in the range of 50–80 MPa, with an expected stress concentration at the dovetail region, reaching a predicted level of 170 MPa. Likewise, the shear stress and radial displacement as well as the critical natural frequency of the blade can be determined. To determine the critical natural frequency of the blade, eigen value analysis was carried out. Figure 1.25 shows the second natural mode of the blade. The original mesh is shown with dashed lines, the displaced mesh with solid lines. The frequency associated with this mode is 2.742 ¥ 106 rad/s for mode 2.

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3-D textile reinforcements in composite materials

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STRESS 1 I.O. VALUE 1 -4.00E+08 2 -3.50E+08 3 -3.00E+08 4 -2.50E+08 5 -2.00E+08 6 -1.50E+08 7 -1.00E+08 8 -5.00E+07 9 +0.00E-00 10 +5.00E+07 11 -1.00E+08

3

2

1

1.24 Typical stress distribution for axial stresses.

MAG. FACTOR – +3.4E-03 SOLID LINES – DISPLACED MESH DASHED LINES – ORIGINAL MESH

1.25 Natural mode of the blade: 2.742 ¥ 106 rad/s (Mode 2).

1.7

Conclusions and future directions

Textile preforms have much to offer to the toughening and to the economic manufacture of the next generation of high-performance structural composites. With a large family of high-performance fibers, linear fiber assemblies, and 2-D and 3-D fiber preforms, a wide range of composite structural performances may be tailored to meet specific requirements. By proper manipulation of the fiber architecture the engineering parameters of the

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3-D textile reinforcements in composite materials

3-D textile structures can be summarized in terms of their fiber volume fraction–fiber orientation relations. Table 1.5 provides a summary of the range of possible fiber orientation and fiber volume fraction as governed by their respective processing parameters. The increasing use of 3-D fiber reinforcements for structural toughening of composites poses important technical challenges. The first is the question of conversion of brittle fibrous materials to textile structures, especially for high-temperature and high-stiffness applications. As a rule, the higher the temperature capability of the fiber, the stiffer and more brittle it is. This processing difficulty with brittle fibrous structures calls for an innovative combination of materials systems such as the concept of material and geometric hybridization. The infiltration or placement of matrix material in a dense 3-D fiber network also creates new challenges. It demands an understanding of the dynamics of the process–structure interaction so that questions such as: ‘What is the optimum pore geometry for pore distribution, bundle size and matrix infiltration?’ can be answered. As the level of fiber integration increases, the chance of fiber to fiber contact will intensify at the crossover points. This also results in uneven distribution of fiber volume fraction. Guidance will be required to select the fiber preform and matrix placement method best suited to reduce the incidence of localized fiber-rich areas. Although 3-D fabric reinforcements have been proven to have superior damage resistance and they provide enormous design options in the tailoring of micro- and macro-structures of a composite, the adoption of 3-D reinforcements to commodity structural applications such as in automotive and building constructions has been slow.This can be attributed to the insufficient development of production economics and engineering design capability which must build on a solid engineering database and cost-effective manufacturing technology. In order to take advantage of the attractive features offered by textile structural composites, there is a need for the development of a sound database and design methodologies which are sensitive to manufacturing technology. An examination of the literature indicated that only a limited number of systematic experimental studies have been carried out on 3-D fabric reinforced composites.A well-established database is needed in order to broaden the usage of fabric reinforced composites for structural applications. The fabric geometry model developed thus far provides a useful framework to integrate fiber architecture design and processing parameters into structural analysis. The precision of strength prediction can be further improved with a better understanding of the failure behavior of 3-D fabric reinforced composites under various loading conditions. Future work in the modeling of fabric reinforced composites requires a better understanding

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Table 1.5. Engineering and processing parameters for 3-D textile reinforcements Preform

Fiber orientation, q (°)

Vf

Linear Assembly Roving yarn

q, yarn surface helix angle q=0 q = 5 ~ 10

0.6 ~ 0.8 0.7 ~ 0.9

Woven 3-D Woven

qf, yarn orientation in fabric plane qc, yarn crimp angle qf = 0/90, qc = 30 ~ 60

Non-woven 3-D Orthogonal

qx, fiber/yarn orientation along X axis qy, fiber/yarn orientation along Y axis qz, fiber/yarn orientation along Z axis qxy, fiber distribution on fabric plane qxy = uniform distribution, qz qx , qy , qz

Knit 3-D MWK

qs, stitch yarn orientation qi, insertion yarn orientation qs = 30 ~ 60, qi = 0/90/±30 ~ 60

Braid 3-D Braid

q, braiding angle q = 10 ~ 45

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Processing parameter Bundle tension, transverse compression, fiber diameter, number of fibers, twist level

~0.6

Fiber packing in yarn, fabric tightness factor, yarn linear density ratios, pitch count, weaving pattern

0.2 ~ 0.4 0.4 ~ 0.6

2-D non-woven: fiber packing in fabric, fiber distribution 3-D orthogonal: fiber packing in yarn, yarn cross-section, yarn linear density ratios

0.3 ~ 0.6

Fiber packing in yarn, fabric tightness factor, yarn linear density ratios, pitch count, stitch pattern

0.4 ~ 0.6

Fiber packing in yarn, fabric tightness factor, braid diameter, pitch length, braiding pattern, carrier number

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3-D textile reinforcements in composite materials

of the dynamic interaction among fiber, matrix and fiber architecture. Although the FGM has been used successfully in guiding the selection of materials and textile processing parameters, recent studies have also shown that the improvements in the prediction of shear properties will enhance the accuracy of the fabric geometry model [42].

1.8

References

1. McAllister, L.E. and Lachman, W.L., in Handbook of Composites, Vol. 4, Kelly, A. and Mileiko, S.T., eds, North-Holland, Amsterdam, 1983, p. 109. 2. Adsit, N.R., Carnahan, K.R. and Green, J.E., in Composite Materials: Testing and Design (Second Conference), Corten, H.T., ed., ASTM STP 497, Philadelphia, PA, 1972, pp. 107–120. 3. Laurie, R.M., Polyblends Composites, Appl. Polym. Symp., 15, 103–111, 1970. 4. Schmidt, D.L., SAMPE J., 8, May/June, p. 9, 1972. 5. Tarnopol’skii, Y.M., Zhigun, I.G. and Polykov, V.A., Spatially Reinforced Composite Materials, Moscow Mashinostroyeniye, 1987. 6. Ko, F.K., ‘Three-dimensional fabrics for structural composites’, in Textile Structural Composites, Chou, T.W. and Ko, F.K., eds, Elsevier, Amsterdam, 1989. 7. Ko, F.K., ‘Preform fiber architecture for composites’, Ceramic Bull., 68(2), 1989. 8. Scardino, F.L., ‘Introduction to textile structures’, in Textile Structural Composites, Chou, T.W. and Ko, F.K. eds, Elsevier, Covina, CA, 1989. 9. Geoghegan, P.J., ‘DuPont ceramics for structural applications – the SEP Noveltex Technology’, 3rd Textile Structural Composites Symposium, Philadelphia, PA, 1–2 June, 1988. 10. Fukuta, K., private communication, 1989. 11. Palmer, R., ‘Composite preforms by stitching’, paper presented at 4th Textile Structural Composites Symposium, Philadelphia, PA, 24–26 July, 1989. 12. Ko, F.K. and Du, G.W., ‘Processing of textile preforms’, in Advanced Composites Manufacturing, Gutowski, T.G., ed., Wiley InterScience, New York, 1998. 13. Dow, R.M., ‘New concept for multiple directional fabric formation’, in Proceedings, 21st International SAMPE Technical Conference, 25–28 September, 1989, pp. 558–569. 14. Stover, E.R., Mark, W.C., Marfowitz, I. and Mueller, W., ‘Preparation of an omniweave-reinforced carbon–carbon cylinder as a candidate for evaluation in the advanced heat shield screening program’, AFML TR-70283, Drexel University FMRC, March 1979. 15. Herrick, J.W., ‘Multidimensional advanced composites for improved impact resistance’, paper presented at 10th National SAMPE Technical Conference, 17–19 October, 1977. 16. Pastenbaugh, J., ‘Aerospatial technology’, paper presented at 3rd Textile Structural Composites Symposium, Philadelphia, PA, 1–2 June, 1988. 17. Bruno, P.S., Keith, D.O. and Vicario, A.A. Jr, ‘Automatically woven three dimensional composite structures’, SAMPE Quarterly, 17(4), 10–16, 1986. 18. O’Shea, J., ‘Autoweave: a unique automated 3-D weaving technology’, paper presented at 3rd Textile Structural Composites Symposium, Philadelphia, PA, 1–2 June, 1988.

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19. Fukuta, K. and Aoki, E., ‘3-D fabrics for structural composites’, paper presented at 15th Textile Research Symposium, Philadelphia, PA, September 1986. 20. Fukuta, K., Aoki, E., Onooka, R. and Magatsuka, Y., ‘Application of latticed structural composite materials with three dimensional fabrics to artificial bones’, Bull. Res. Inst. Polymers Textiles, 131, 159, 1982. 21. Williams, D.J., ‘New knitting methods offer continuous structures’, Adv. Composites Eng., Summer, 12–13, 1978. 22. Hickman, G.T. and Williams, D.J., ‘3-D knitted preforms for structural reactive injection molding (SRIM)’, in Proceedings of the Fourth Annual Conference on Advanced Composites, ASM International, 1988, pp. 367–370. 23. Ko, F.K. and Kutz, J., ‘Multiaxial warp knit for advanced composites’, in Proceedings of the Fourth Annual Conference on Advanced Composites, ASM International, 1988, pp. 377–384. 24. Ko, F.K., Pastore, C.M., Yang, J.M. and Chou, T.W., ‘Structure and properties of multidirectional warp knit fabric reinforced composites’, in Composites ’86: Recent Advances in Japan and the United States, Kawata, K., Umekawa, S. and Kobayashi, A., eds, Proceedings, Japan-US CCM-III, Tokyo, 1986, pp. 21–28. 25. ITA Research Information Bulletin, University of Aachen, 1998. 26. Du, G.W. and Ko, F.K., ‘Analysis of multiaxial warp knitted preforms for composite reinforcement’, paper presented at Proceedings of Textile Composites in Building Construction Second International Symposium, Lyon, France, 23–25 June, 1992. 27. Ko, F.K., ‘Three-dimensional fabrics for structural composites’, in Textile Structural Composites, Chou, T.W. and Ko, F.K., eds, Elsevier, Tokyo, 1989. 28. Brown, R.T. and Ashton, C.H., ‘Automation of 3-D braiding machines’, paper presented at 4th Textile Structural Composites Symposium, 24–26 July, 1989. 29. Popper, P. and McConnell, R., ‘A new 3-D braid for integrated parts manufacturing and improved delamination resistance – the 2-step method’, 32nd International SAMPE Symposium and Exhibition, 1987, pp. 92–102. 30. Du, G.W. and Ko, F.K. ‘Geometric modeling of 3-D braided preforms for composites’, paper presented at 5th Textile Structural Composites Symposium, Drexel University, Philadelphia, PA, 4–6 December, 1991. 31. Ko, F.K., Chu, J.N. and Hua, C.T., ‘Damage tolerance of composites: 3-D braided commingled PEEK/carbon’, J.Appl. Polymer Sci.:Appl. Polymer Symp., 47, John Wiley, New York, 1991. 32. Liao, D., ‘Elastic behavior of 3-D braided composites under compressive loading’, PhD thesis, Drexel University, June, 1990. 33. Lei, C.S.C. and Ko, F.K., ‘Mechanical behavior of 3-D braided hybrid MMC’, paper presented at ICCM-8, 12–16 July, 1991. 34. Ko et al., ‘Toughening of SiC/LASIII ceramic composites by hybrid 3-D fiber architecture’, paper presented at ICCM-7, 1989. 35. Dow, N.F. and Ramnath, V., ‘Analysis of woven fabrics for reinforced composite materials’, NASA Contract Report 178275, 1987. 36. Chou, T.W. and Yang, J.M., ‘Structure–performance maps of polymeric metal and ceramic matrix composites’, Metallurgical Transaction A, 17A, 1547–1559, 1986. 37. Ko, F.K., Pastore, C.M., Lei, C. and Whyte, D.W., ‘A fabric geometry model for 3-D braid reinforced FP/Al-Li composites’, paper presented at ‘Competitive

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38.

39.

40. 41. 42.

3-D textile reinforcements in composite materials Advancements in Metals/Metals Processing’, SAMPE Meeting, Cherry, NJ, 18–20 August, 1987. Cox, B. and Flanagan, G., Handbook of Analytical Methods for Textile Composites, Version 1.0, Rockwell Science Center, Thousand Oaks, CA, January, 1996. Pastore, C.M. and Ko, F.K., ‘Modelling of textile structural composites: part 1: a processing science model for three-dimensional braid’, J. Textile Inst., 81(4), 480–90, 1990. Tan, T.M., Pastore, C.M. and Ko, F.K., Engineering Design of Tough Ceramic Matrix Composites for Turbine Components, ASME, Toronto, 1989. ABAQUS Manual, Hibbit, Carlson and Sorensen, Inc, 1982. VanDeurzen, P., ‘Structure–performance modeling of two-dimensional woven fabric composites’, PhD Thesis, Katholieke Universiteit Leuven, Belgium, 1998.

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2 3-D textile reinforced composites for the transportation industry K. DRECHSLER

2.1

Introduction

Composites with directionally oriented long-fibre reinforcement have proven their potential for realizing high-performance, low-mass structural components in the aerospace industry over the past 40 years. Starting from the German glider ‘Phönix’, which was designed and manufactured using glass fibre reinforced resin, right to the Airbus carbon fibre fin, the material has helped to extend the limits of the performance and efficiency of planes, helicopters and space structures further and further. The benefits are reductions in fuel consumption and emission, improved payloads and extended service lives due to higher mass-specific stiffness, strength and energy absorption, as well as better fatigue and corrosion performance than metals. As a consequence, it is not very surprising that other fields of application outside the aerospace sector have an increasing interest in applying this kind of material, too. In the automotive industry, the need for cars with higher efficiency and no losses in terms of safety and comfort has become more and more important because of interest in improved environmental compatibility – low mass is one of the key factors in reaching this goal. Nevertheless, there are significant differences in the requirements for manufacturing methods and structural performance which prevent an easy transfer of know-how from aerospace to automotive applications. One of the most crucial differences is that of production rates. While aerospace components are usually manufactured at a rate of no more than a few hundred, the high-volume automotive market has a need for some hundred thousand components a year. Another difference is the costs allowed for weight reductions. While the space industry spends up to some US$10,000 just to save 1 kg of mass in a satellite, the automotive market currently accepts no more than some US$10–20. Thus, the big challenge for the next few years will be developing materials, processing methods and structural concepts which allow cost-effective, 43 © 1999,Woodhead Publishing Ltd

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3-D textile reinforcements in composite materials

high-volume manufacturing of low-mass composite components. A very promising approach to achieving this goal is the development and application of advanced textile technologies, such as 3-D weaving, 3-D braiding, knitting or stitching, offering the potential for automated manufacturing of near net shaped fibre preforms with optimized fibre reinforcement in 3-D space according to structural requirements. In combination with appropriate impregnation or consolidation techniques, a significant reduction of manual work can be realized compared with state-of-the-art aerospace technologies based on unidirectional fibre tapes or 2-D weavings. In this way, one of the most important requirements for cost reduction in aerospace and the introduction of composites in highvolume automotive applications can be fulfilled. Another very interesting feature is the possibility to produce a 3-D fibre reinforcement in the composite material. It has been shown that this results in significantly improved damage tolerance and structural integrity. The focal points in this chapter are the description of benefits and drawbacks involved in composite materials with conventional and textile reinforcement compared with metals, the requirements for the material with regard to aerospace and automotive applications, and discussion of first exemplary applications demonstrating the potential of textile structural composites for improving mechanical performance and reducing manufacturing costs.

2.2

The mechanical performance of conventional and 3-D reinforced composites

The mechanical performance of composites is mainly determined by the fibre type and the reinforcing fibre geometry. The most important fibre types are glass, carbon and aramide fibres. It has been shown that carbon fibres offer by far the best potential in terms of stiffness. Therefore, they represent the most important material for aerospace applications. They have so far not been considered as a structural material for high-volume automotive applications because prices are very high, ranging from $20 to 500/kg. In this field, glass fibres, which are priced at approximately $3/kg, represent the most important material. In the near future it can be expected that much cheaper carbon fibres will be launched on the market. Nevertheless, the mechanical performance and the textile processability of this new fibre class have to be proven, because it may be necessary to use much thicker fibre bundles to reach the cost reduction goal. The second factor of influence is the reinforcing fibre geometry, whereby composites can be broken down into two classes: material with non-directional (short) fibre reinforcement (mats, injection

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moulding) and with directionally oriented long fibres (unidirectional tapes, fabrics). For aerospace components, only directionally oriented long fibres are used, as this configuration alone allows full utilization of the fibre properties, and an optimal anisotropic design according to the structural requirements concerned. So far, use of this material in high-volume manufacturing has been limited to easily shaped components, because the manufacturing process requires a lot of manual work. In Fig. 2.1, the most important mechanical properties of the composites are compared with light metals and steel. It is shown that the most significant mass reductions can be achieved using carbon fibres and a nonisotropic fibre reinforcement, as required by the respective loads. When comparing quasi-isotropic composites with metals, one will find that mass savings of more than 30% compared with aluminium, and 60% compared with steel are feasible. Nevertheless, this comparison is based on ‘idealized’ laboratory-scale values determined under the following conditions: unidirectional reinforcement, high fibre volume fraction (60%), tensile load, no fibre undulation and no delaminations. In realistic applications several additional

2.1 Comparison between mechanical performance of metals and composite materials.

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3-D textile reinforcements in composite materials

2.2 Degradation of composite performance due to manufacturing and in-service effects.

factors of influence have to be taken into consideration which can be caused by the series manufacturing process, the structural service conditions, the component geometry and the fibre reinforcement. Figure 2.2 gives an idea of the magnitude of the various factors of influence. It is obvious that unidirectional tape-based composites offer the highest utilization of fibre properties and therefore the highest in-plane stiffness and strength, because the fibres are aligned without any curvature exactly in the loading direction and no resin-rich areas cause strain inhomogeneities within the material. All textile structures show a more or less high degree of fibre undulation. In 2-D weavings this effect is caused by the mutual crosslinking of weft and warp fibres, and weft knittings consist more or less of a mesh system with curved fibres. Additional degradations of in-plane properties are caused principally by a 3-D reinforcement, because the z-directional fibre fraction reduces the share of load-carrying fibres and generates resin-rich areas. Optimizing these effects is very important especially for aerospace applications: despite the growing need for cost savings, low weight is still the driving force for research and development in this field of application. In the past years, significant improvements have been realized for example in the field of 3-D weaving. In Fig. 2.3 2-D weavings, ‘conventional’ 3-D weavings and advanced 3-D weavings, manufactured by a process recently developed by the North Carolina State University, are compared. It is shown that, owing to reduced fibre undulation and fibre damage, the

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Tensile strength Compression strength

700

Compression strength after impact (6.7 J/mm)

600

MPa

500

400

300

200

100

0

2-D

3-D

3-D advanced

2.3 Mechanical performance of various 2-D and 3-D woven composites.

effect of the 3-D reinforcement can be compensated and the stiffness and strength of 2-D weavings can be achieved. Naturally, no problems with fibre curvature occur in multiaxial warp knittings, because the fibre layers are placed on top of each other without crosslinking. Nevertheless, the stitching fibres, holding the single layers together, can lead to a reduction of mechanical performance owing to fibre damage or to disturbance of the reinforcing fibre alignment. The mechanical performance of weft knittings can be improved by prestretching the meshes before curing or by an additional fibre system running straight through the mesh system. High stiffness and strength are just two criteria for the evaluation and selection of a structural material for automotive and aerospace applications. In particular, components that are susceptible to impact or crash loads have to be designed according to their mechanical performance after a first failure. This can lead to the necessity of high safety factors, reducing the weight reduction potential. Therefore, the ‘damage tolerance’ can be an important material property. Conventional 2-D reinforced composites based on tapes or weavings tend to delaminate owing to impact loads, because the bonding between the single layers is relatively poor, which leads to poor interlaminar performance. A significant improvement is possible by a 3-D through-thethickness fibre reinforcement, which can be realized by 3-D-weaving, 3-D-braiding or stitching.

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3-D textile reinforcements in composite materials

2.4 Damage tolerance of different composite materials.

In Fig. 2.4, the damage tolerances of various composite materials, characterized by the compression after impact test, are compared. In this test, composite plates are impacted and afterwards compression tested according to exactly defined specifications. The remaining strength and breaking elongation represents a value for the damage tolerance evaluation and the design of impact-susceptible structures. It is shown that performance after impact can be improved significantly by the 3-D fibre reinforcement. With a fibre share of below 5%, the design goal of 0.5% after impact that is required in aerospace can be reached even with brittle resin systems. The parameters that influence performance are type, thickness and distance of z-fibres as well as the reinforcing geometry. Figure 2.5 illustrates the reason for higher impregnation speed. Compared with 2-D composites, the z-fibres lead to a significant improvement in bonding of the single layers, as demonstrated by the peel strength. The structural integrity is of major importance, especially for automotive applications. After a crash, the structures have to maintain a minimum mechanical performance. Complete debonding of component parts has to be avoided. These criteria can be realized easily by metals owing to their plastic deformation characteristics. The more or less brittle crush behaviour of conventional, especially carbon fibre reinforced composites is much more critical in this respect. This performance can also be improved by a 3-D fibre reinforcement.

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2.5 Notch growth in 2-D and 3-D reinforced composites.

Figure 2.6 shows double T-shaped beams, typical structural components in automotive and aerospace design, after longitudinal and transversal crash tests. The preforms for the composite structures are integrally 3-D braided by a new ‘n-step’ braiding process in an optimum configuration according to the loads. The integral fibre reinforcement guarantees high structural integrity with locally restricted damage area and high after-crash performance. An additional feature is the high mass-specific energy absorption owing to the complex, exactly controllable failure modes in the 3-D fibre structure. More complex preforms for composites with high structural integrity which cannot be made by one textile technology can be realized by stitching several basic preforms together. A stiffened panel is discussed in Chapter 5 as an example. It has been made by stitching the warp-knitted skin to a 3-D braided profile.

2.3

Manufacturing textile structural composites

The diverse textile processes, such as advanced weaving, braiding, knitting or stitching, allow the production of more or less complex fibre preforms. While weavings and warp knittings are predestined for flat panels, braidings allow the manufacture of profiles. The most complex preforms can be realized where warp-knitting is used. Tables 2.1 and 2.2 summarize the most important features of textile process and composites as well as the

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3-D textile reinforcements in composite materials

2.6 Structural integrity of 3-D braided profiles after crash.

© 1999,Woodhead Publishing Ltd

Table 2.1. Textile processes for composites: an overview Textile process

Principle – design

Preform geometry

Fibre orientation

Productivity mounting

Development goals

3D weaving

Flat fabrics Integral stiffeners Integral sandwich-structure Simple profiles

Limited to weft and warp direction (0/90) Various z-fibre reinforcements

High productivity Very high mounting time

Multiaxial 3D weavings with integrated 45° fibres

3D braiding

Open and closed profiles (I, L, Z, O, U, . . .) Flat fabrics

Braiding fibres 10–80° Local integration of straight 0° fibres

Medium productivity High mounting time

Varying cross-sections Varying fibre orientation

Knitting (weft)

Very complex preforms (knot-elements, curved structures)

Fibres mainly in mesh structure

Medium productivity Short mounting time

Integration of straight fibres in the mesh structure

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52 Table 2.1. (cont.) Textile process

Principle – design

Preform geometry

Fibre orientation

Productivity mounting

Development goals

Knitting (warp)

Flat fabrics Integral sandwichstructures

Multiaxial in-plane orientation 0°/90°/±45° Up to 7 layers fixed by knitting fibre

High productivity High mounting time

Integration of more layers in one production step

Embroidery

Attaching additional fibres on basic fabrics

Very complex fibre orientation, for example in main stress direction

Slow process Short mounting time

Improvement of production speed Optimization of control program for 3-D structures

Stitching

Very complex preforms by combining several textile structures

Depending on basic preforms

Very quick process Short mounting time

Optimization of control program for 3-D structures Optimization of stitchhead (damage, size)

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Table 2.2. Textile structural composites: an overview

53

Textile structure

Mechanical properties

Composite manufacturing

Typical structural elements

Typical applications

3-D woven

High stiffness and strength only in 0° and 90° directions Very high damage tolerance

RTM Pressing (thermoplast and thermoset, film and prepreg) Limited drapeability

Slightly curved panels under biaxial load Sandwich-panels Simply shaped profiles

Cell-skin elements Fitting elements Body-in-white structures Chassis structures

3-D braided

High stiffness and strength above all in 0° direction High structural integrity of complex profiles

RTM Pressing (see weaving) Pultrusion (profiles) High deformability

Complex, open and closed profiles Flat panels with limited cross-section

Stiffening elements Chassis structures Spaceframe elements

Knitted (weft)

High stiffness and strength only in fill-fibre direction High energy absorption High structural integrity

RTM Pressing (see weavings) Very good drapeability Minimum waste

Knot-elements Complex, curved panels with limited stiffness and strength

Spaceframe elements Helmet shells Cladding element with high damage tolerance

Knitted (warp)

High stiffness and strength also under shear loading

RTM Pressing (see weaving) 45° fibres in roll direction

Flat and curved panels under biaxial load Curved sandwich-panels

Cell-skin elements Fitting elements Body-in-white structures Chassis elements

Embroidered

High stiffness and strength also under shear loads (for example, load introduction)

RTM Pressing (see weaving) No waste of embroidery fibre

Local reinforcement Load-introduction and loadtransmission elements

Rotor-blade joining element Inspection-hole cover

Stitched

High damage tolerance High structural integrity Reduction of in-plane properties possible due to damage

RTM Pressing (see weaving) Complex moulds for complex structure

Stiffened panels consisting of woven or knitted flat panels with braided profiles Complex 3-D structures

Highly integrated panels for cell structures Complex fittings

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development goals. A detailed description of the various textile technologies can be found in Chapter 1. Impregnating the complex shaped 3-D fibre structures applies for suitable infiltration and consolidation techniques in order to obtain high and constant fibre volume fractions with low void content during the composite manufacturing process. In general, the manufacturing methods used in both aerospace and nonaerospace applications are quite different. While autoclave prepreg technology is the most important technique for aerospace components, injection moulding and pressing techniques (SMC, GMT) are used for high-volume applications. The reason is that the autoclave prepreg technique results in large fibre volume fractions (typically 60%) and high performance. On the other hand, there is a penalty in the form of extremely high cycle times, typically lasting several hours. SMC, GMT and injection moulding techniques allow cycle times of less than one minute. On the other hand, fibre volume fractions are relatively poor (typically 30%). In combination with textile preforms, the RTM process (resin transfer moulding) is of special interest. In this process, a resin is pressed under vacuum into a closed mould where the fibre preform is fixed. The achievable fibre volume fractions amount to more than 50%, while cycle times of less than 10 minutes can be realized with appropriate resin systems. Although the density of the 3-D fibre structures can be very high, the impregnation speed is more or less higher compared with conventional 2-D structures. The reason for this effect is that the additional fibres in thickness direction form ‘flow channels’ which support resin transfer through the thickness. The continuous pultrusion process is of greatest interest for the impregnation of profile-shaped fibre preforms with a constant cross-section. Interesting developments are performed, especially in combination with 3-D braiding. The most important impregnation techniques are summarized in Fig. 2.7. In general, thermoplastic matrix composites offer a high potential for realizing short cycle times because no chemical reaction has to take place in the mould, and quick hot-forming techniques, comparable to the pressing of metal sheets, can be applied. On the other hand, thermoplastics generally require higher temperatures and pressure, and thus more expensive tooling and higher energy consumption. This is especially true of PEEK, the only thermoplastic matrix material for aerospace structural components with a melting point of 400 °C. The use of hybrid structures, consisting of reinforcing fibres and thermoplastic fibres, is of special interest in combination with textile technologies. According to the level of fibre mixture, the process is called commingling, or co-weaving (or co-braiding). In the commingling process, the com-

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2.7 Impregnation techniques for textile structural composites.

bination of reinforcing fibres and thermoplastic ‘matrix’ fibres occurs at fibre level. In the co-weaving or co-braiding process the two fibre types are mixed during the textile process. The advantage of the first approach is that the textile processing of the reinforcing fibres can be improved because the tough thermoplastics protect the brittle glass or carbon fibres. Additionally, the composite quality is better than co-weaving and co-braiding, owing to increased homogeneity. With a view to an overall cost evaluation, tooling may play an important role in low- and medium-volume manufacturing. In general, composite tools are much cheaper than steel tools. Therefore, part costs may be lower because of the use of composites for small series production, although material costs are much higher than those of steel. Nevertheless, the tooling technology for impregnating very complex textile preforms requires special developments to allow cost-effective component manufacturing. Promising techniques are, for example, the differential pressure RTM or the so-called ‘Scrimp process’, where only one tooling half is hard and the other one is formed by a vacuum foil. Of special interest for complex hollow structures is the wax core technique. These cores can be melted in very complex geometries and act, for example, as a braiding core during the textile process and as part of the impregnation tool during composite manufacturing. After curing they can be melted out completely.

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2.4

3-D composites in aerospace structures

Table 2.3 summarizes the most important requirements for a material to be used in aerospace applications. While mechanical performance, long-term behaviour and behaviour under special environmental conditions are of the greatest importance, costs for production and service have also gained in significance. The significance of the various requirements is, of course, different for military and civil aircraft. Compared to the automotive industry, composites have a long history in aerospace applications. In passenger planes, their share has reached 15%, while in modern fighter aircraft more than 50% of the structural material consists of composites. Helicopters are designed almost exclusively using composites. All major aerospace companies have launched technology programmes to apply composites in structural components of passenger planes other than just the fins. Fuselage structures and wing components are under investigation. The focal points are new design, material and manufacturing concepts, and textile structural composites play an important role in the research and development projects.

Table 2.3. General requirements for materials to be used in structural applications Product improvement

Required material properties

Aerospace

Payload Range Fuel economy Direct operational costs Safety

High stiffness and strength High damage tolerance (compression after impact e > 0.5%) High reproducibility High energy absorption (helicopter) Processing cost reduction Optimization of scrap

Ground transportation

Fuel economy Payload Low noise level Emission

Low cycle times (<<5 min) Low material price (<$10 /kg) Limited manual work Medium stiffness and strength (E > 20 GPa) High energy absorption High design freedom Potential for recycling Potential for low investments Potential for improvements due to 3-D textiles

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In order to demonstrate the advantage of textile structural composites, Daimler-Benz has conducted a research programme to manufacture and test a fin-leading edge of a fighter aircraft. The component was produced using 2-D weavings and 3-D weavings with a thermoplastic and thermoset matrix. Figure 2.8 summarizes manufacturing time estimates. It shows impressively that the use of the 3-D woven textile preform may reduce manual work significantly. Furthermore, the damage tolerance of the impact-susceptible component has been improved due to the 3-D reinforcement – no delaminations occurred after impact testing. On the other hand, 3-D weavings do not allow a structural optimization of the component, because the torsional stiffness of this material is relatively poor due to the lack of ±45° fibres. A promising approach would be to apply new multiaxial 3D weavings or multiaxial warp knittings. Another demonstrator component based on textile structural composites has been produced independently by McDonnell Douglas, Daimler-Benz and others. At Daimler-Benz stiffened panels have been manufactured with 3-D woven or warp-knitted skin and 3-D braided stiffeners stitched together to a complex preform (see Fig. 2.9). The fibre structure was subsequently impregnated in an RTM process by the DLR in Braunschweig. Table 2.4 shows a cost and weight estimate for the typical aerospace component, compared with aluminium and a tape-based structure. The textile-

2.8 Manufacturing times for fin-leading edge.

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2.9 Stiffened panel consisting of braided profiles and warp knitted skins.

Table 2.4. Cost and weight estimates for stiffened panels

Aluminium (milled) Hand-lay-up (unidirectional-prepreg) Automated tape lay-up Integral preform (3-D + stitched, RTM)

Material (US$)

Labour (US$)

Total (US$)

Weight (%)

21 45

11 120

32 166

100 70

29 20

16 10

46 30

70 75

based part seems to represent a good compromise between costs and mechanical performance. In Fig. 2.10 the goals, typical structural components and possible applications of textiles for aeroplanes and helicopters are summarized.

2.5

Textile structural composites in automotive structures

Table 2.3 summarizes the general requirements that materials for groundtransportation applications have to meet. The most important features are

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2.10 Textile structural composites in aerospace applications.

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costs, manufacturability, mechanical performance, long-term behaviour, repairability and recyclability. As discussed before, composites offer some important benefits over metals, such as mechanical performance, long-term behaviour (no corrosion), high damping and low-energy expense for raw material manufacturing. On the other hand, there are currently no adequate solutions for high-volume manufacturing techniques for long-fibre reinforced composites, costs, repair and recycling. Therefore, no automotive manufacturer has so far applied this class of material for structural components in real high-volume models (50 000 to 300 000 cars per year). Of special interest is the use of composites in chassis applications, as in this field the anisotropy of the material can be used to create all-new design concepts, allowing a significant reduction in component numbers and a very marked weight reduction compared with steel. It is, for example, possible to realize beam elements with a very high bending stiffness and low torsional stiffness, or vice versa. In order to evaluate the applicability of textile structural composites, Daimler-Benz has manufactured and tested a 3-D reinforced engine mount at laboratory level within the framework of a research project. One component has been produced using 20 conventionally woven glass fibre layers; the other one is based on a single 3-D weaving integrating these layers in a single preform. The result is a significant decrease in manufacturing time (see Fig. 2.11), which is due to a reduction in manual work (cutting and laminating). It is expected that cycle times of less than 10 minutes are possible when applying an optimized RTM process. Bending and fatigue tests have shown that the composite components

2.11 Manufacturing time for composite engine mounts.

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meet all requirements at a weight of less than 50% of the steel components. A special feature of the 3-D reinforced component has been demonstrated in crash tests. Energy absorption and structural integrity are higher and prevent the component from separating into two parts (see Fig. 2.12). Typical structural components in transportation engineering are knot elements for spaceframe-like structures. Because of the complex geometry and loading of these parts, cost-effective manufacturing based on conventional 2-D composites is not possible. Metal casting is state-of-the-art or, if the loads are low, injection moulding. High-performance structural parts with a high weight-saving potential can be realized using new braiding or knitting technologies. Figure 2.13 illustrates the principle. The braiding process, developed by Muratec in Japan, allows positioning of fibres around a complex mandrel consisting of foam or a replaceable material such as wax. Completely different is the preform manufacturing concept based on the knitting technology developed for example by Stoll in Germany. It allows direct transfer of a CAD file of a structure into a knitted net shaped fibre system. In highly loaded component areas additional straight fibres can be integrated in order to improve mechanical performance. Another interesting application field is that of exterior components. An important requirement for such parts is high stiffness. A sandwich design

2.12 Crash behaviour of 2-D (top) and 3-D reinforced engine mounts.

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2.13 Braided, knitted and stitched knot elements for spaceframe structures.

would therefore be very effective. Nevertheless, the manufacturing of conventional sandwich structures based on foam or honeycomb cores is time consuming and expensive. Additionally, the poor damage tolerance prevents these structures from being used intensively in transportation applications. An interesting new design and manufacturing concept is feasible by using integrally woven sandwich structures. The integral structure combining the two skins and the ‘core’ during the textile process offers a high potential for automated, cost-effective manufacturing of sandwich structures with a high damage tolerance because of the 3-D fibre reinforcement. An exemplary demonstrator component is shown in Fig. 2.14. A hood for a passenger car has been made by using a glass fibre weaving in one shot. After impregnation in an epoxy resin bath and pressing to adjust the resin content, the wet preform was placed in a negative mould and pressed in the area of the structure borders by a frame. Owing to the sandwich design, the 3-D fibre reinforcement and the ability to realize very effective load introductions, the weight-saving potential of this structure is very high. Even compared with an aluminium sheet hood a weight reduction of nearly 50% is possible. Nevertheless, an important development goal for exterior applications is to improve the surface quality of textile-based composite materials. The goals, typical structural components and possible applications of textiles are summarized in Fig. 2.15.

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2.14 Prototype of a sandwich hood based on an integrally woven sandwich.

2.6

Conclusions

High stiffness and strength at low density, high mass-specific energyabsorption behaviour and good fatigue performance are the main features that have led to the successful application of composite materials in the aerospace industry over the past decades. Two typical examples are weight savings of approximately 25% in the Airbus fin as compared with aluminium, and an extension of the service lives of helicopter rotor blades by a factor of more than 40. Textile structural composites with 3-D fibre reinforcement may facilitate utilizing the unique material performance in other structural components as well, e.g. the fuselage or wing components of new generations of passenger planes, such as the ultra-high-capacity aeroplanes planned by Boeing and Airbus.The main material improvements include enhanced damage tolerance due to 3-D reinforcement, and reduced manufacturing costs resulting from the application of highly productive textile technologies in the manufacture of fibre preforms. This fact may also be a key aspect when applying high-performance composites in high-volume automotive productions, trucks or trains. Nevertheless, the research and development work on textile structural composites involves several technological fields (see Fig. 2.16). There is still

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2.15 Textile structural composites in ground transportation.

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2.16 Research and development tasks for textile structural composites.

a lot of work to do, starting with improving the productivity of the textile processes and optimizing the stiffness and strength of 3-D composites, and ending with fast impregnation processes and suitable simulation tools describing the mechanical behaviour. But there is no doubt that 3-D textiles will be a key factor for future developments and for the successful application of composites in aerospace and ground transportation engineering.

2.7

References Composite materials in general

M. Krämer and P.-J. Winkler, Möglichkeiten und Grenzen einer Synergie zum Leichtbau zwischen Automobil- und Luftfahrtindustrie, VDI-Berichte, 1993. H. Kellerer, ‘Stand und Perspektiven moderner Verbundwerkstoffe’, paper presented at BMFT-Symposium ‘Materialforschung’, Dresden, 1991. H. Kellerer, J. Brandt, R. Meistring and R. Rauh, Limits to Today’s Composites: Changes for Tomorrow’s Developments. ECCM, Bordeaux, 1989. F.J. Arendts and K. Drechsler, ‘Composites with knitted fibre reinforcement’, paper presented at 3rd International Symposium on Advanced Composites, Patras, 1990.

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F.J. Arendts, J. Brandt and K. Drechsler, ‘Integrally woven sandwich structures’, paper presented at ECCM 3, Bordeaux, 1989. J. Brandt and K. Drechsler, ‘The potential of advanced textile structural composites for automotive and aerospace applications’, paper presented at 4th Japan International SAMPE Conference, Tokyo, 1996. J. Brandt and K. Drechsler, ‘Application of composites in the transportation industry’, paper presented at Texcomp 3, Aachen, 1996.

Automotive applications O. Altmann, Karosserie- und Automobilkonzepte mit Polymerwerkstoffen, VDIBerichte N°c.968, 1992. E. Roth, H. Sigolotto, H. Keller and P. Kümmerlen, Fahrerhaus in CFK-Technology, Chance oder Utopie?, VDI-Tagung, 1995. H.G. Haldenwanger, Einsatz von FVK im Automobilbau- Übersicht und neue Ansätze, VDI-Verlag, 1993. K. Johnson, ‘Resin transfer molding of complex automotive structures’, paper presented at 41st Annual Conference of the Society of Plastics Industries, 1986. K. Johnson, ‘Automotive applications for textile-based composite structures’, paper presented at Texcomp 3, Aachen, 1996.

Aerospace applications W. Hartmann and H. Kellerer, Criteria for the Material Selection for Aircraft Structures, MRS Europe, 1985. J. Brandt and K. Drechsler, ‘The application of 3-D fibre preforms for aerospace composite structures’, paper presented at ESA/ESTEC-Conference, Noordwijk, 1990. J. Koshorst, Verbundwerkstoffe in der Luftfahrttechnik, Entwicklung und Perspektiven, paper presented at Verbundwerk 88, Frankfurt, 1988. H. Rieckhof, Einsatz von Faserverbundwerkstoffen im Airbus-Programm, DGLRJahretagung 1986. G. Behrendt, Neue Technologien in der allgemeinen Luftfahrt, DGLR-Bericht 87-01, 1987.

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3 Mechanical modelling of solid woven fabric composites PHILIPPE VANDEURZEN, JAN IVENS AND IGNAAS VERPOEST

3.1

Introduction

Solid woven fabric composites represent a class of advanced composites which are reinforced by 2-D or 3-D woven preforms [1]. These materials offer new and exciting opportunities for tailoring the microstructure to specific thermomechanical applications in the fields of aerospace, marine, medicine and sports technology. The variables under control include fibre and matrix materials, yarn placement, yarn size and type. Together with this emerging ability to engineer composite materials comes the need to develop computationally efficient micromechanics models that can predict, with sufficient accuracy, the effect of the microstructural details on the internal and macroscopic behaviour of these new materials. Computational efficiency is indispensable because there are many parameters that must be varied in the course of engineering a composite material. This chapter addresses the issue of developing micromechanical models for solid woven fabric composites. In the future, it is probably inevitable that the optimization of the microstructure of a woven fabric composite will require the marriage of such micromechanical models and optimization algorithms.

3.2

Review on solid woven fabric composites

3.2.1 Introduction This section provides a survey of the literature. First, an overview of woven fabric composites is presented. Solid woven preforms vary considerably in terms of fibre orientation, entanglement and geometry. Second, in order to exploit the advantages of these composites fully, it is important to create a link between the microstructural geometry and the thermomechanical performance [2]. In the past decade, a variety of micromechanical models have been employed to study the overall thermo-elastic behaviour of orthogonal 2-D woven fabric composites based on the properties of the constituents 67 © 1999,Woodhead Publishing Ltd

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3.1 Classification of textile preforms for composite structures [3].

and the fabric architecture. Some of these models also provide the opportunity to address strength properties. A review will assist in defining possible modelling strategies for 3-D woven fabric composites.

3.2.2 Classification Fibre reinforcement constitutes the structural backbone of a composite.The classification by Cox and Flanagan [3] of various textile preforms is reproduced in Fig. 3.1. The left column classifies textile preforms according to the machines and processes used to produce them. The major textile-forming techniques for composite reinforcements are weaving, knitting and braiding. Further, it is possible to make a distinction between the dimensionality of the textile preform. Following the definition of Cox [3], the division into 2-D and 3-D textile structures is determined by whether the fibre preform can transport an important load (higher than the load carried by the matrix alone) in two or three linearly independent directions. In general, an orthogonal 2-D woven fabric is made by weaving yarns together. A yarn is a continuous strand of textile fibres. The fabric is produced on a loom that interlaces yarns at right angles to one another [2–8]. The lengthwise yarns are called warps, while the yarns that are shuttled across the loom are called fillings or wefts. The individual yarns in the warp and filling directions are also called an end and a pick, respectively. The interlacing of the yarns causes yarn undulation or yarn crimp. The weave type is determined by the method of interlacing both sets of yarns. Figure 3.2 shows three basic constructions: plain, twill and satin weave. Even in rather simple woven fabrics, there are important geometric differences between the warp and the weft direction. Those differences are the result

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3.2 Basic weave constructions: (a) plain, (b) twill and (c) 5HS satin weave. The black box represents the fabric unit cell.

of numerous constructional and process parameters such as weaving density, warp tension, weft tension and beating motion. The term ‘hybrid’ is used to describe fabrics containing more than one type of fibre material. Hybrid fabrics are attractive preforms for structural materials for two major reasons. First, these fabrics supply an even wider variety of material selection for designers. They offer the potential of improved composites’ mechanical properties, weight saving or excellent impact resistance. Second, a more cost-effective use of expensive fibres can be obtained by replacing them partially with less expensive fibres. Hybrid fabrics are woven from fibrous materials such as glass, aramid, carbon, boron, ceramics and natural fibres. Advances in textile manufacturing technology are rapidly expanding the number and complexity of 3-D woven preforms. By changing the traditional weaving technique to produce 2-D fabrics, it is now possible to achieve a much higher degree of integration in the thickness direction of the textile. The two major classes of solid 3-D weaving are through-thickness angle interlock weaving [10] and orthogonal interlock weaving [1–3]. Angle interlock 3-D woven fabrics can be produced on a dobby loom or a jacquard loom. The warp yarns can now enter more than one layer of weft yarns. Other textile structures with laid-in straight yarns are also possible. By changing the number of layers, the pattern of repeat and the position of the laid-in yarns, an almost infinite number of geometric variations becomes possible. In an orthogonal interlock 3-D weave, the yarns are placed in three mutually orthogonal directions. These fabrics are produced principally by the multiple warp weaving method. Matrix-rich regions are created in composites reinforced with a 3-D woven orthogonal preform. In general, solid woven fabrics offer the advantages of handleability, dimensional stability, improved impact and damage resistance. However, these advantages are obtained at the cost of reduced stiffness and strength properties owing to the undulation of the yarns. There is thus a significant need to model the mechanical behaviour of these composites.

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3.2.3 Micromechanical models Considering the actual importance of 2-D woven fabric composites in the family of structural composites, the mechanical analyses of these composites are now extensively reviewed and presented. Most of the published data are related to stiffness properties of plain weave laminae. There are few publications on the internal stress distribution and on the damage and strength analysis problem of general woven fabric composites. The possible extension of the different micromechanical models to analyse 3-D woven fabric composites will be discussed. It should also be stressed here that in this rapidly evolving field of study any review will soon be incomplete. New results are always being presented or printed. Models of Ishikawa and Chou In the 1980s, an extensive amount of work on the thermo-mechanical modelling of 2-D woven fabric composites was done by Ishikawa and Chou. They developed and presented three analytical 1-D elastic models [11–13]. These models are known as the mosaic model, the fibre crimp model and the bridging model. The classical lamination theory forms the basic analytical tool for these developments [14]. The models of Ishikawa and Chou are labelled 1-D models because they only consider the undulation of the yarns in the loading direction. Notice the total absence of any geometric analysis. That is, the actual yarn crosssectional shape or the presence of a gap between adjacent yarns is not considered. Therefore, no predictions are made for the out-of-plane yarn orientation and the fibre volume fraction. Moreover, these models consider balanced closed weaves only, whereas in practice the fabric can be unbalanced and open. Since the classical laminated plate theory is the basis of each model only the in-plane elastic properties are predicted. The elastic models were extended to analyse the thermal properties, hybrid fabrics and the knee behaviour under uniaxial tensile loading along the filling direction only. However, an extension to treat 3-D woven preforms is not useful because of the geometric simplifications and the limitation to predicting only in-plane properties. Models of N. Naik, Shembekar and Ganesh N. Naik and Shembekar have developed 2-D elastic models for a 2-D nonhybrid plain weave fabric composite [15]. These models are essentially an extension of the 1-D models of Ishikawa and Chou. However, these 2-D models take into account the undulation of both warp and weft yarns, the presence of a possible gap between adjacent yarns, the real cross-section of

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the yarn and the possible unbalanced nature of the plain fabric lamina. The representative unit cell is discretized into slices along or across the loading direction. These slices are further divided into different elements such as straight cross-ply or unidirectional regions, undulated cross-ply or unidirectional regions and pure matrix elements. In the analysis of Naik and Shembekar, two schemes for combining the in-plane stiffness matrices of the different elements are used: parallel–series and series–parallel. In the parallel–series (PS) model, the elements are first assembled in parallel across the loading direction with the isostrain assumption (adding the stiffness matrices, weighted by their volume fractions). Then, those multielements are assembled in series along the loading direction with the isostress assumption. In the second scheme, all the infinitesimal elements of a section along the loading direction are assembled with an isostress assumption (adding the compliance matrices, weighted by their volume fractions). Then, all the sections along the loading direction are assembled with an isostrain condition. Such a scheme is called a series– parallel (SP) model. Both schemes yield a full 2-D stiffness matrix for the plain woven fabric composite. A full mathematical treatment of the problem has been presented in reference [16]. Based on experimental work, the PS model is recommended for the prediction of all in-plane elastic constants. Out-of-plane properties cannot be predicted. Hence, the extension of the model to treat 3-D woven preforms is not useful. Recently, Naik and Ganesh have presented an extension of their thermoelastic models to include the prediction of failure in plain weave composites under on-axis static tensile loading [17,18]. The load is assumed along the filling direction. Different stages of failure such as warp yarn transverse failure, filling yarn shear failure, filling yarn transverse failure, pure matrix element failure and filling yarn longitudinal failure are considered. The newness of the model lies in the calculation procedure for the stresses in the matrix and yarn elements. However, this is exactly where the model is most confusing. A lot of effort has been spent on describing material nonlinearities, geometric non-linearities and geometric effects of matrix element failures, while the available information on the stress prediction procedure is inadequate. The failure analysis is then carried out by comparing the local element stresses or strains with the admissible values of stress or strain. The Tsai–Wu failure criterion [19] is used to predict the failure in the filling yarn elements. The maximum stress and strain criteria are used to predict the failure in the warp yarn and matrix elements. If an element fails, the stiffness of that element is reduced (degraded stiffness). The final failure of the unit cell laminate is assumed to have occurred if the fibres in the filling yarn are broken. In conclusion, some more practical drawbacks and disadvantages of the strength model of Naik are provided. First, the stress model lacks logic and

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simplicity (when and why is the PS model to be preferred over the SP model?). Second, only on-axis uniaxial tensile loads can be considered along the warp or weft direction. Third, the model does not account for thermal stresses which are known to be important in the stress and strength analysis of fibre composites. Finally, only a non-hybrid 2-D plain weave composite can be considered in the present analysis. Model of Hahn and Pandy The 3-D thermo-elastic model of Hahn and Pandy [20] for non-hybrid plain fabric composites is simple in concept and mathematical implementation. This model is essentially an extension of the 2-D models of Naik. The geometric model accounts for the undulation of warp and weft yarns, the actual yarn cross-section and the presence of a gap between adjacent yarns. The yarn undulations are sinusoidal and described with shape functions. The gap between two neighbouring yarns, however, is introduced by terminating the yarn at the start of the gap. Hence, for large gaps the yarn cross-section becomes quasi-rectangular, which is not realistic. In the thermo-elastic model, the strain is assumed to be uniform throughout the composite unit cell. Therefore the effective stiffness of the woven fabric composite is obtained as a volume average of the local stiffness properties of yarn and matrix elements. This is a so-called isostrain model. Closed-form expressions are provided for the 3-D effective elastic moduli and effective thermal expansion constants for a 2-D plain weave composite. The model has the advantage of being simple and easy to use. The isostrain model can very easily be applied to analyse complex 3-D woven fabric composites. However, some disadvantages are here provided. First, the accuracy of the isostrain model still remains to be verified through more experimental verification of all 3-D elastic constants. It will be further shown in this chapter that the isostrain technique is not capable of accurately predicting all 3-D elastic constants [21]. Second, the model can certainly not be extended to solve the stress analysis problem accurately, and hence cannot be used for strength predictions. Model of R. Naik Recently, a micromechanics analysis tool labelled TexCad was developed by R. Naik to calculate the thermo-elastic properties along with damage and strength estimates for woven fabric composites [22]. This tool can be used to analyse non-hybrid plain weave and satin weave composites. It discretely models the yarn centreline paths within the repeating unit cell by assuming a sinusoidal undulation of the yarns. The 3-D effective stiffness matrix is computed by a yarn discretization scheme (which subdivides each

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yarn into smaller, piecewise straight yarn slices) that assumed an isostrain state within the unit cell. Hence, as in the Hahn and Pandy model, the isostrain model is applied. In the calculation for the strength, TexCad uses a curved beam-on-elastic-foundation model for yarn crimp regions together with an incremental approach in which stiffness properties for the failed yarn slices are reduced, based on the predicted yarn slice failure mode. Only on-axis tensile loadings and in-plane shear loadings were modelled and reported. Certainly, the most questionable assumption in this strength model is the calculation of the local stress fields in yarn and matrix slices based on the isostrain assumption. Basically, TexCad is well documented and easy to use. It is a thorough implementation of the isostrain approach which could be extended easily to analyse complex 3-D woven fabric composites. It will perform stiffness and failure analyses as correctly as can be expected for an isostrain model. Model of Paumelle, Hassim and Léné Paumelle et al. [23,24] developed a finite element method for analysing plain weave fabric composite structures. The periodic medium homogenization method is implemented. Basically, by applying periodic boundary conditions on the surface of the unit cell and by solving six elementary loading conditions on the unit cell, the complete set of elastic moduli of the homogenized structure can be computed.At the same time, the method provides a good approximation of the local distribution of force and stress fields acting in the composite components and at their interface. These microscopic stress fields give a strong indication of the types of damage that will occur. To the best of our knowledge, Paumelle et al. have not yet reported an extension of this finite element model to predict damage propagation or to analyse 3-D woven preforms. Moreover, outlined below are some problems encountered in a practical finite element analysis of solid woven fabric composites. First, this approach requires large computer calculation power and memory because of the 3-D nature and the complexity (size) of the yarn architecture. Second, a correct finite element model includes the generation of the fabric geometry and the finite element mesh of nodes and elements. Most of the time spent is related to the creation and the verification of a correct geometric model [25]. Finally, there are major problems in analysing and interpreting the results in a 3-D domain of a rather complex geometry [26]. Model of Blackketter Here, we will discuss in some detail the research work of Blackketter [27]. In our opinion, this work is certainly one of the first and most important

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efforts to model damage propagation in 2-D woven composites. The approach could also be applied to 3-D woven fabric composites. Blackketter constructed a simplified 3-D unit cell of a single ply nonhybrid plain weave graphite/epoxy composite. From this description 3-D finite element models were generated. Twenty node isoparametric hexahedron elements were used in generating the finite element meshes. Limits on element refinement were imposed by the computational time required for solution. An incremental iterating finite element algorithm was developed to analyse loading response. Each iteration or load step required about 30 real-time minutes using a VAX8800 computer. Tension and shear loadings were modelled. The finite element model included capacities to model nonlinear constitutive material behaviour and a scheme to estimate the effects of damage propagation by stiffness reduction. Results from this analysis compared extremely well with experimental stress–strain data. It was concluded that the non-linear stress–strain behaviour of the woven fabric composite is principally caused by damage propagation rather than by plastic deformation of the matrix. Let us describe now the damage propagation model as developed by Blackketter et al. At each Gaussian integration point (27 Gaussian quadrature integration points over each volume element), damage or failure was detected by comparing the actual stress state with a failure criterion. To simulate damage at an integration point, the local stiffness properties were reduced. Therefore, each element in the model can contain both intact and failed Gaussian integration points. After the occurrence of damage, the volume considered was capable of sustaining only reduced loads and stresses had to be redistributed to surrounding volumes. It is important to select an appropriate failure criterion for the matrix and yarn materials. For the isotropic matrix material a maximum normal stress criterion was used to detect damage. If the principal stress exceeded the strength, the tensile modulus was reduced to 1% of its initial value. The shear modulus was reduced to 20% of its initial value. After failure, the matrix was no longer isotropic. For the transversely isotropic yarns, it is necessary to account both for the type of damage and the orientation of that damage. Blackketter compared the actual stresses in the local coordinate system (123) with the respective ultimate strengths. This is a maximum stress criterion. The 1-axis corresponds to the longitudinal yarn direction. Table 3.1 presents the different failure modes and the stiffness reduction factors used by Blackketter. Each Gaussian integration point was allowed to fail in one or all modes. Finally, catastrophic failure of the unit cell was characterized by large displacements compared with the previous values. The analysis by Blackketter of graphite/epoxy plain weave fabric composites has shown that by carefully modelling the fabric geometry, using

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Table 3.1. Stiffness reduction scheme for the UD yarn elements, according to Blackketter [27] Failure mode

1 2 3 4 5 6

Longitudinal tension s11 Transverse tension s22 Transverse tension s33 Transverse shear t23 Longitudinal shear t13 Longitudinal shear t12

Mechanical property and degradation factors E11

E22

E33

G23

G31

G12

0.01 1.0 1.0 1.0 1.0 1.0

0.01 0.01 1.0 0.01 1.0 0.01

0.01 1.0 0.01 0.01 0.01 1.0

0.01 1.0 1.0 0.01 1.0 1.0

0.01 0.2 0.2 0.01 0.01 1.0

0.01 0.2 0.2 0.01 1.0 0.01

correct constituent stiffness/strength data and by applying an appropriate stiffness reduction scheme, it is possible to predict the stress–strain behaviour of woven fabric composites. The same ideas could certainly be applied to analyse 3-D woven fabric composites. However, Blackketter does not discuss in detail the limitations of the finite element modelling technique (meshing or calculation time). Models of Whitcomb Whitcomb and coworkers [28–30] have studied the effect of the yarn architecture on the predicted elastic moduli and stresses in plain weave composites. The work is restricted to linear elastic analysis. Three-dimensional finite element models were used. Only simple plain weaves were studied because these offer sufficient complexity for the task. A refined model of the complete unit cell would require immense amounts of computer memory and calculation time. However, by exploiting the geometric and material symmetries in the simple unit cell, it was sufficient to analyse 1/32 of the size of the complete plain weave unit cell. Twenty node isoparametric hexahedral elements were used. Two different yarn architectures were investigated. The first was the ‘translated architecture’ where the complete yarn is created by keeping the cross-section vertical along the yarn path. The second was the ‘extruded architecture’ wherein the yarn cross-section is always placed perpendicular to the yarn path. The extruded yarn architecture requires a more complex mesh generation. Whitcomb and coworkers also analysed progressive failure of plain weave fabric composites under in-plane tensile loading using a 3-D finite element model. The mechanical loading was parallel to one of the yarn directions. Thermal loading or thermal residual stresses were not considered. The effects of various characteristics of the finite element model on predicted behaviour were examined. There is no ‘right’ way to model

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damage evolution that is also practical [30]. The most simple method to account for damage is to modify the constitutive matrix of the damaged finite element. Therefore, the failure analysis becomes a series of linear analyses. A maximum stress criterion was used to evaluate the damage of the matrix and yarn elements. Withcomb and coworkers have applied and compared three different techniques to modify the constitutive matrix after damage. First, the total constitutive matrix was reduced to essentially zero when any of the allowable strengths was exceeded. In the second technique, only specific rows and columns of the constitutive matrix were reduced according to the damage mode. Third, specific engineering moduli were reduced. This is the reduction scheme developed previously by Blackketter. Essentially, it was concluded that the predicted strength decreased considerably with increased waviness of the yarns. The modification technique of the constitutive matrix has a major effect on the predicted stress–strain curve. However, more numerical experiments are necessary to establish guidelines for an accurate failure analysis. No final conclusions have been given yet concerning the different reduction schemes. No extension is made to treat 3-D woven preforms.

3.2.4 Conclusions In the past 15 years, a variety of different micromechanical approaches has been developed to study the effective behaviour of 2-D woven fabric composites. Tables 3.2 and 3.3 summarize those micromechanical models. Basically, the literature review reveals that considerable work addressing the effects of fabric architecture on the effective elastic and thermal expansion properties was done. However, this work has not been systematic or exhaustive in general. Research has been too focused on material systems based on plain weave fabrics, limited ranges of fibre volume fractions and specific material thermo-elastic properties. Second, the stress and strength analyses are still in their infancy. Here, research has focused on specific loading directions, knee behaviour and damage mechanisms. There is certainly a need for reliable strength models. Finally, the extension of the models to consider 3-D preforms can only be achieved in a few cases (Tables 3.2 and 3.3). In the analytical methods we observe a predominant use of the isostrain assumption to predict the effective thermo-elastic and strength properties. No data are available to verify the accuracy of this approximation. Moreover, most researchers have concentrated only on the primary determinant of mechanical and physical properties, namely the geometric orientation of the yarns. The idea that other geometric effects or boundary conditions could have an influence on the prediction of effective properties of woven fabric composites was ignored. The well-established finite element method

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Table 3.2. Modelling review of woven fabric composites: analytical mechanical models. First four columns indicate whether the full stiffness matrix C, the thermal expansion coefficient a and the strength are predicted, and whether FEM is used. The last column indicates whether this model can be extended to 3-D-woven fabric composites Model

C

a

Strength

FEM

Limitations

To 3-D

Chou & Ishikawa: mosaic model, 1985

Y

Y

N

N

No yarn undulation In-plane properties Isostrain/isostress

N

Chou & Ishikawa: crimp model, 1985

Y

Y

Knee

N

Plain weave Undulation in one yarn system In-plane properties Isostress

N

Chou & Ishikawa: bridging model, 1985

Y

Y

Knee

N

Satin weave Undulation in one yarn system In-plane properties Isostress

N

N.K. Naik et al., 1992–1995

Y

Y

Y

N

Non-hybrid plain weave In-plane properties Mixed isostress/isostrain On-axis tensile load

N

Hahn & Pandey, 1993

Y

Y

N

N

Non-hybrid plain weave Isostrain

Y

R. Naik, 1995

Y

Y

Y

N

Non-hybrid plain and satin Isostrain On-axis tensile and shear loads

Y

Table 3.3. Modelling review of woven fabric composites: numerical mechanical models Model

C

a

Strength

FEM

Limitations

To 3-D

Paumelle et al., 1992

Y

N

Stress

Y

Computational time Non-hybrid plain weave No damage or strength model

N

Blackketter et al., 1993

Y

N

Y

Y

Computational time Non-hybrid plain weave On-axis in-plane load

Y/N

Whitcomb and coworker, 1993

Y

N

Y

Y

Computational time Non-hybrid plain weave On-axis in-plane load

N

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was employed by most researchers to compute the local stress fields accurately and to predict the strength of woven fabric composites. A comprehensive stiffness reduction scheme for damage modelling has been offered by Blackketter [27]. While it is possible to use mesh generation programs to alter the geometry and while changing constituent properties is very simple, the cost in computing time for a parametric study is significant. It is becoming increasingly clear that approaches other than the finite element method are needed to develop computationally efficient analysis tools for solid woven fabric composites. As mentioned previously, this is because of the large number of parameters that must be varied in identifying an optimal yarn architecture. In conclusion, the current models for 2-D woven fabric composites all have limited applicability, in that either they are not sufficiently accurate to predict the local stress fields in yarn and matrix phases, or they are not computationally efficient.The struggle between accuracy and computational efficiency is a continuous one. Nowadays, the ability to engineer not only the material composition but also the internal yarn geometry of 2-D and 3-D solid woven fabric composites gives the designer of a composite material unmatched control over the material. Exerting that control intelligently, however, requires a body of theory. The objective of micromechanical modelling should always be to develop both accurate and computationally efficient approaches that can predict the behaviour directly, given the material composition and the internal yarn geometry.

3.3

Elastic model: the complementary energy model

3.3.1 Introduction In this section, we focus on the recent development at the Katholieke Universiteit Leuven of the complementary energy model for 2-D woven fabric composites [32–34]. This model is included here because it is a pertinent example of the vigour that exists in textile composites research. The model captures both the ‘orientation effect’ and the ‘position effect’, important features of the actual heterogeneous composite material. Currently, the model is being extended to the mechanical analysis of 3-D woven fabric composites, and of braided fabric composites. It should be stressed that the yarn architecture of a solid woven fabric composite is rather precisely determined by the textile processing route. Woven fabric composites provide new opportunities for tailoring the yarn architecture to specific applications. This is in contrast to random or unidirectional composites, where the precise control of the fibre orientation and spatial distribution is difficult and only statistically macroscopic arrangements are possible. Hence, for the mechanical modelling of random

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or unidirectional composites the geometric characteristics are taken into consideration in a certain average sense (fibre level), whereas for solid woven fabric composites the yarn interlacings, curvatures and locations, should be taken into account, reflecting the actual geometry (yarn level).

3.3.2 Geometric model Since the mechanical properties of woven fabric composites have a very strong dependence upon the reinforcing yarn geometry, it is essential to create a geometric concept or scheme for describing the fibre architecture. The model deals with a perfect, regular, one-layer fabric composite. Hence, the presence of voids, the misorientation of yarns and the nesting of fabric layers are neglected, just as they are neglected in most other fabric composite models [35]. The woven fabric is treated as an assembly of unit cells (Fig. 3.2). By definition, the unit cell is the smallest repeating pattern in the structure. Figure 3.3 shows an example of a checkerboard pattern for a complex fabric, namely a hybrid carbon/Dyneema® twill weave. The rows of the board represent the warp yarns, while the columns are the filling or weft yarns. At an interlacing point, the square is coloured black if the warp yarn runs over the weft yarn. The main complexity arises from the fact that the fabrics considered here can contain two different warp and weft yarn types. First, the extension is necessary to describe hybrid weave styles. Of course, the extension is also needed when using special fibres in woven constructions such as optical glass fibres or shape-memory alloy fibres with particular sizes.

3.3 Extended checkerboard pattern of a hybrid 2 ¥ 2 twill weave unit cell. The w and f type yarns are Dyneema® fibres, and the w* and f* type yarns are carbon fibres.

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3.4 Lenticular yarn cross-section. W is width of yarn, t is yarn thickness, R is radius of curvature.

The most detailed geometric analysis would consider the path of each single fibre in the unit cell. The greatest practical problem, however, is caused by the fact that the complete set of input data necessary for such a detailed geometric description is very large and difficult to quantify. Therefore, the geometric analysis is carried out on the yarn level. We assume that all individual fibres in the yarn run in the same direction as the yarn. The intra-yarn fibre volume fraction or fibre packing density K, defined as the fibre to yarn area ratio, is assumed to be a constant for the woven fabric composite. Interlacing of the yarns and processing of the composite leads to thread or yarn flattening. On the basis of microscopic observations, a lenticular shape was selected to describe the cross-sectional shape of the yarn (Fig. 3.4). The geometric characteristics of a hybrid weave can be subdivided in three groups (Table 3.4). The first group, the know group, contains those parameters that are supplied by the weaving company. All the parameters that one has to measure on a real woven fabric composite are put together in the second group, the measure group. This fabric information can be obtained by microscopic observation of warp and weft sections of the fabric composite. The aspect ratio f of the yarn, defined as the width w over the thickness t of the lenticular yarn cross-section, is the most important one. The crimp parameter hf describes the undulation of the filling yarns. Of course, the undulation of the yarns in the warp direction is related to this parameter because the increase of undulation in one direction of the fabric reduces the undulation in the other direction [36]. Finally, the third group, the calculate group, contains all values that are calculated from the previous parameters, using formulas based on simple geometric considerations.

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Table 3.4. Classification of the geometric parameters Know group

Nf d p

Number of fibres in the yarn Diameter of the fibre Yarn spacing

Measure group

f hf D K

Aspect ratio of the yarns Crimp parameter for the filling yarn Thickness of the composite Fibre packing density

Calculate group

hw,hw* t w b Vf

Crimp parameter for the warp yarns Thickness of the yarn cross-section Width of the yarn cross-section Orientation of the yarn Fibre volume fraction

Certainly, the most important output of the calculation procedure is the fibre volume fraction, the orientation of the yarns and the fractional volume of each cell. These data are the basis for a modelling of mechanical properties. Moreover, the geometric model as such is most useful in determining some textile properties as fabric thickness, but also in determining the allowable microstructural states of fabrics. A custom Microsoft Excel® application, called TexComp, has been developed to perform all geometric calculations [34].

3.3.3 Multilevel decomposition scheme The geometric model treats a woven fabric composite unit cell, shown in Fig. 3.5, as a hierarchical system that can be decomposed. Two major motivations are here formulated. First, the calculation and bookkeeping of geometric data should become a simple task. It is easy to calculate the geometric parameters that fully describe the yarn architecture only based on the presented ‘know’ and ‘measure’ group. Second, a logic and simple geometric meshing of the unit cell is essential for the computation of the mechanical properties. Basically, the composite unit cell level (1) is split up into block cells or macro-cells (2), micro-cells (3), matrix and yarn layers (4) and matrix and fibres (5). This five-level decomposition scheme could be considered as an ‘intelligent mesh generator’ for 2-D woven fabric composites. A logic extension towards 3-D woven preforms and to braids is currently being carried out. The block partition of the unit cell consists of discretizing the unit cell in a number of rectangular block cells. At each crossover zone of a warp yarn and a weft yarn, one ‘building block’ is defined. The size of each block can easily be computed as a function of yarn spacings p, yarn widths w and com-

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3.5 Multilevel decomposition of woven fabric composites: top-down.

3.6 Schematic of a hybrid twill weave unit cell (level 1) and a block cell (level 2).

posite thickness D. As can be seen in Fig. 3.6, we need 16 block cells to create the unit cell of a 2 ¥ 2 hybrid twill fabric composite. Basically, each block cell is uniquely identified by four macro-cells (Fig. 3.7). That is, at each crossover point of a warp and a weft yarn, one needs two macro-cells in one layer to define the path for the warp yarn and two in the other layer for the weft yarn. However, it should be stressed that the two layers of macro-cells, which are always present in the unit cell, yield a correct description of the geometry (for example, the fibre volume fraction is correct). In order to describe a general 2-D weave geometry, a library of 108 macro-cells has been put together. Even the most complex 2-D woven

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3.7 Schematic of a block cell or macro-cell (level 2) and a micro-cell (level 3).

structures can be composed with this library of rectangular macro-cells or building blocks. It should be pointed out that only the weave construction pattern provided by the weaving company is sufficient to determine automatically the number of each type of macro-cell present in the unit cell. No extra fabric information, nor any geometric assumptions, nor operator interventions are needed. Therefore, the macro-partition is simple in concept and easy to apply. It provides a theoretical basis for the design of woven fabric composites. The decomposition of the block in micro-cells is called the micropartition. It is assumed that within each small micro-cell the yarn follows a straight yarn path. The 2-D micro-partition results in a fully 3-D division of the unit cell, so that the variation of properties within the unit cell can be properly analysed. This is essential for understanding the mechanical performance of woven fabric composites. For the local stress analysis, it will further be necessary to define the micro-cell also in the 123 local axis system, where 1 corresponds to the longitudinal yarn direction.This is called a combi-cell. Therefore the label micro-cell (xyz) or combi-cell (123) is selected depending on the reference axis system. Then, the multilevel decomposition approach is based on simplifying the geometry within each combi-cell by assuming one matrix layer and one

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3.8 Schematic of a micro-cell or combi-cell (level 3), matrix and yarn layers (level 4) and a unidirectional lamina with fibre and matrix phases (level 5).

impregnated yarn layer (Fig. 3.8). This is level 4. Finally, each impregnated yarn is treated as a unidirectional lamina with matrix and fibre phases. This is level 5.

3.3.4 The complementary energy elastic model Previously, by constructing a multilevel decomposition scheme, the composite unit cell was split automatically into matrix and yarn cells. Presently, by a multistep homogenization procedure, a link is established between the external loading and the internal stresses.The principal idea lies in the interpretation that the stress ‘concentration factors’ can be computed at each step by applying the complementary variational principle. This principle states that from all the admissible stress fields, the true field is that which minimizes the total complementary energy (hence the name: complementary energy model, CEM). We achieve a straightforward analytical stress model for woven fabric composites. The four-step homogenization model CEM is now developed. The Venn diagrams in Fig. 3.9 show the link between the fractional cell volumes k, the stress concentration factors A and the compliance matrices S. With this

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3.9 Multistep homogenization of woven fabric composites: CEM [S].

bottom-up homogenization scheme, from geometric level 5 to level 1, it is easy to compute the effective stiffness matrix of the woven fabric composite. In the first homogenization step, the effective lamina or impregnated yarn layer properties are predicted in terms of their constituent material properties, and the fibre packing density K in the yarn. The empirical Chamis expressions [37], describing the elastic properties of a unidirectional lamina composed of transversely anisotropic fibres in an isotropic matrix, are used in this first step. Consider now the micro-cell homogenization problem. This is step two. The complementary variational principle is used to compute the stress concentration factors for both layers. One average stress tensor is specified for each layer. These computed stress concentration factors are extremely useful because they link the stress tensor applied on the micro-cell with the average stress tensors on both layers and because they allow a straightforward computation of the micro-cell compliance matrix [SMC]. More information on this topic can be found in [33]. As a next step, homogenization step three, the effective properties for the block cell are also determined by applying the complementary variational principle, taking into account the position and the properties of the 200 micro-cell constituents. The calculation procedure is shown in the Venn diagram of Fig. 3.9. Obviously, the 3-D compliance matrix of the block-cell is related to the fractional volume k, the compliance matrix [S] and the concentration factors [A] for the micro-cell’s components as follows:

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3-D textile reinforcements in composite materials 10

10

2

T

[ S BC ] = Â Â Â kMCrst [ AMCrst ] [ S MCrst ][ AMCrst ]

[3.1]

r =1 s =1 t =1

where the subscripts r, s and t refer to the position of the micro-cell in the block-cell as shown in Fig. 3.7. Already in the geometric model, the fractional volume kMC for each of the micro-cells has been expressed and calculated as a function of the yarn spacings, the yarn width and the yarn thickness. The compliance matrix [SMC] for each of the micro-cells was calculated in the previous homogenization step. Applying the complementary variational principle [39] will yield the concentration factors [AMC] which link the average block stress and the micro-cell stresses. Now, the variational problem is solved in the global xyz coordinate system. Because one constant stress tensor is assigned to each of the 200 micro-cells, there are a total of 1200 unknown stress constants to be determined. The assumed and admissible stress fields are imposed by conditions 3.2 and 3.3: 10

10

2

10

ÂÂÂ k

rst

ÂÂÂ k

rst

ÂÂÂ k

rst

sxMCrst = sxBC

r =1 s =1 t =1 10 10 2

2 rst

t yzMCrst = t yzBC

ÂÂÂ k

rst

t zxMCrst = t zxBC

ÂÂÂ k

rst

t xyMCrst = t xyBC

r =1 s =1 t =1 10 10 2

syMCrst = syBC

r =1 s =1 t =1 10 10 2

[3.2]

r =1 s =1 t =1 10 10 2

r =1 s =1 t =1

(a) (b) (c) (d)

10

ÂÂÂ k

szMCrst = szBC

r =1 s =1 t =1

sxMC1 st = sxMC2 st = . . . = sxMC10 st syMCr 1t = syMCr 2t = . . . = syMCr 10t

s = 1 . . . 10; t = 1 . . . 2 r = 1 . . . 10; t = 1 . . . 2

szMCrs1 = szMCrs 2 r = 1 . . . 10; s = 1 . . . 10 t yzMCr 11 = t yzMCr 21 = . . . = t yzMCr 101 = t yzMCr 12 = t yzMCr 22 = . . . = t yzMCr 102

[3.3]

r = 1 . . . 10

(e) t zxMC1 s1 = t zxMC2 s1 = . . . = t zxMC10 s1 = t zxMC1 s 2 (f ) t xyMC11t

= t zxMC2 s 2 = . . . = t zxMC10 s 2 s = 1 . . . 10 = t xyMC 21t = t xyMC12t = . . . = t xyMC1010t t = 1 . . . 2

Mathematically, these stress constraints are considered by the method of the Lagrangian multipliers [38]. Therefore, the optimization problem is replaced by a set of equations that can be solved directly for the unknown stress constants. Proceeding from the geometric block cell level to the assembled composite unit cell level, the complementary variational principle is used for the last time. This is homogenization step four. The unknown block cell stresses and block cell concentration factors [ABC] are computed directly by the method of Lagrange [38]. Here, only the final expression for the compliance matrix of the unit cell is presented in Equation 3.4. The subscripts m and n refer to the position of the block in the unit cell (Fig. 3.6),

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the constants F and W refer to the number of weft and warp yarns in the unit cell: F

W

T

[ S UC ] = Â Â kBCmn [ ABCmn ] [ S BCmn ][ ABCmn ]

[3.4]

m=1 n=1

Basically, the four-step procedure defined in the foregoing will result directly in the computation of the overall symmetric 3-D compliance matrix of the woven fabric composite unit cell. Moreover, a direct link is established between the average unit cell stress and the cell stresses at each geometric level. This most important result will serve as a solid basis for further strength modelling in the next section. In conclusion, Fig. 3.10 and 3.11 present a benchmark parametric study for a glass/epoxy plain weave fabric composite. The analytical model yields elastic moduli predictions comparable to those obtained by 3-D finite element modelling. This fact is put forward as an indication of the appropriateness of the present multilevel, multistep technique.

3.3.5 Conclusion The ability to specify the woven fabric geometry gives the designer control over the composite material. Many of the properties that influence how a composite can be used are determined by the ‘averaged’ behaviours of the fibres and the matrix. The ‘averaged’ stiffness properties are shaped by the internal yarn distribution, i.e. yarn orientation and position. With CEM, we now have a fast and efficient tool to predict the effect of each geometric

3.10 Predicted Young’s moduli for the benchmark composite: comparing results from an FEM study [24] and our CEM calculations (material: glass-epoxy plain weave).
Ex = Ey (CEM);  Ez (CEM);  Ex = Ey (FEM);  Ez (FEM).

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3.11 Predicted shear moduli for the benchmark composite: comparing results from an FEM study [24] and our CEM calculations (material: glass-epoxy plain weave).
Gyz = Gzx (CEM);  Gxy (CEM);  Gyz = Gzx (FEM);  Gxy (FEM).

variable on the 3-D elastic performance of the 2-D woven fabric composite. The proposed model can easily be extended to calculate the so-called thermal concentration tensors for the computation of the effective thermal expansion constants [33]. Moreover, the model can be generalized to 3-D preforms by simply extending the set of macro-cells.

3.4

Strength model

3.4.1 Introduction Woven fabric composite components are subjected to a variety of loading conditions during their service life. Therefore, an understanding of the mechanical response of these materials to various loading conditions is necessary for the safe design of a component. The prediction of strength is certainly one of the outstanding problems in the analysis of fibre composites. This section presents a method to predict the micro-stress fields, the first cell failure and the ultimate strength of woven fabric composites.

3.4.2 The complementary energy stress model In order to predict strength accurately, a sufficiently detailed stress distribution must be available for composites subjected to arbitrary combinations of applied stresses. The need for computationally efficient predictive tools is clear when one considers the large range of fibres, matrices and

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3.12 Multistep stress analysis of woven fabric composites: CEM.

woven fabric types available. The Venn diagrams in Fig. 3.12 show the link between the external mechanical loading, the stress concentration factors and the internal stress tensors. With this top-down stress calculation scheme it is easy to compute the stress fields at each geometric level. The entities corresponding to one of the stress calculation steps are grouped together in one Venn diagram. As an example, the calculation of the yarn and matrix layer stresses for an arbitrary mechanical loading of the composite unit cell is explained below. The overall average stress tensor on the unit cell is denoted as: T

{s UC } = {s xUC , s yUC , s zUC , t yzUC , t zxUC , t xyUC }

[3.5]

By considering the computed concentration factors [A] at each step and the yarn orientation through the calculation of the stress transformation matrix [Ts], the matrix and yarn layer mechanical stresses are given by

{s M } = [ AM ][Ts ][ AMC ][ ABC ]{s UC }

[3.6]

{s Y } = [ AY ][Ts ][ AMC ][ ABC ]{s UC } Two important observations are made. First, the yarn orientation and yarn position effects are included in the stress model due to the calculation of the concentration factors using the multilevel, multistep CEM. For example, the matrix material is not characterized by one stress state but by multiple stress states depending on the position of the matrix cell. Second, it should be stressed that any type of simple or combined 3-D mechanical loading

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can be applied, irrespective of symmetry, without resorting to different boundary-condition application strategies, as in the case of the finite element procedure. This is particularly important in the analysis of realistic woven fabric composite components where different loading conditions exist throughout the structure.

3.4.3 Development of the failure model The computed micro-stress fields are very useful to predict the appearance of damage. That is, the model is capable of predicting the point of initial failure using only strength values of the constituent matrix and yarn cells. This point of initial failure is called the first cell failure. For the isotropic matrix material, the paraboloid failure locus applied on the principal stresses is used for its simplicity. It is a flexible failure criterion that yields a unique solution for each loading path. Moreover, it conforms with the basic physical laws and experimental evidence [40]. For the transversely isotropic yarn material, a maximum stress criterion is used. The current yarn stresses {sY} are computed in the local 123 coordinate system and compared with respective ultimate strengths. It is assumed here that five strength parameters of the impregnated yarn can be estimated from available unidirectional composite strengths. These are: the longitudinal tensile strength XT, the longitudinal compressive strength XC, the transverse tensile strength YT, the transverse compressive strength YC and the shear strength S. In the progressive failure analysis, the effects of matrix and yarn failure are taken into account in an average sense. It is based on the assumption that the damaged material could be replaced with an equivalent material of degraded properties. The properties of the damaged material are adjusted as the loading and progression of damage continue. However, it is not an easy task to determine the degraded properties of the degraded material with certainty [27]. In the present study, the stiffness reduction method as proposed by Blackketter [27] will be used. First, the method accounts for the damage mode when modelling degradation of yarn materials (Table 3.1). If failure is detected, appropriate moduli are reduced. Second, the matrix failure is introduced by reducing the Young’s modulus to 1% and the shear modulus to 20% of their original values. After failure, the matrix is no longer isotropic. After the implementation of the damaged elastic properties in the CEM, another global load increment is applied on the composite. The detailed stress state in the woven fabric composite is updated and compared with the strength properties. The load is increased until (1) a new material cell has failed, or (2) another failure mode is detected for a damaged cell, or (3) catastrophic failure of the total unit cell has occurred. Catastrophic failure

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is determined by a large displacement or stress fall compared with the previous values. Finally, to clarify the present approach, some important remarks are provided: • The woven fabric composite is assumed to be initially free of damage (cracks, voids, etc.). • The non-linearity of the matrix is not taken into account for two reasons. First, the non-linear behaviour of the different matrix zones in the composite is different from that of the bulk material. This is mainly caused by the presence of local multiaxial stress states and thermal stresses.This information is usually not available. Second, the non-linear stress–strain behaviour of woven fabric composites was shown to be mainly influenced by damage propagation [27] and not by the non-linearity of the matrix. • The model does not calculate the fabric geometric deformation at each load step. This is acceptable for on-axis loading because the strain-tofailure is low. However, it is expected that the model will predict less accurate results for off-axis tensile tests, where the strain-to-failure is much higher. • The proposed deterministic modelling approach will yield a typical ‘peaked’ stress–strain curve as shown in Fig. 3.13 because several cells fail at the same moment. There is no drop in stress in the experimen-

3.13 Predicted warp stress–strain curves as a function of the yarn aspect ratio f.

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3-D textile reinforcements in composite materials Table 3.5. Thermo-elastic properties of the matrix and the fibre material Material

E (GPa)

n

a (/K)

Epoxy matrix Glass fibre

3.13 73

0.34 0.2

6.60 ¥ 10-5 4.80 ¥ 10-6

Table 3.6. Strength parameters for the matrix and the impregnated yarn cells (MPa) SC

ST

XC

XT

YC

YT

S

83

56

610

1462

118

50

72

tally obtained stress–strain curve because the failure of matrix and yarn cells is spread out over a large strain range.

3.4.4 Parametric study: strength analysis A Fortran code, called WCUnix, has been developed for computing the micro-stress fields, the first cell failure and the ultimate strength of woven fabric composites. The most novel feature offered is the simulation of progressive failure by a stiffness degradation scheme. That is, the analysis of loading becomes a series of elastic analyses. The source code has been compiled on a SUN SparcStation 10. A single load step for a plain fabric composite required about 30 seconds’ calculation time. Blackketter reports a calculation time of 30 minutes for each iteration or load step using 3-D finite element modelling on a VAX 8800 computer [27]. Therefore, the WCUnix code is certainly not computationally intensive. Recently, by introducing more time efficient mathematical subroutines, the calculation time has been further reduced drastically. In this parametric study, a glass/epoxy plain weave fabric composite is considered. The elastic constants of the matrix and fibre constituents are presented in Table 3.5.The strength parameters for the matrix and the transversely isotropic impregnated yarn cells are listed in Table 3.6. Geometric characteristics of the warp and weft yarns are given in Table 3.7. The format of this study is to change the yarn aspect ratio f for both yarn systems. The ratio is set equal to 3, 6, 9 and 12, ranging from rather round to very flat yarn cross-sections. Then, the yarn spacings in warp and weft direction are computed as the width of the yarn plus 20%. The thickness of the composite is computed as the thickness of the plain weave fabric plus

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Table 3.7. Yarn characteristics of the plain weave composite Number of fibres, Nf

Fibre diameter, d (mm)

Fibre packing density, K

1000

0.01

0.70

3.14 Predicted bias stress–strain curves as a function of the yarn aspect ratio f.

an extra 10%. The predicted fibre volume fraction of all the resulting woven fabric composites equals 35%. Hence, a comparison of the results is possible. The tensile stress–strain curves are predicted with the WCUnix code. Figures 3.13 and 3.14 plot the computed stress–strain curves in warp and bias direction, respectively. The stiffness reduction scheme does not affect the stress level at which damage initiates; only the final shape of the stress–strain plot after the first cell failure occurs is influenced. In the warp direction load case, we observe a very strong influence of the yarn aspect ratio f on the failure behaviour. If the yarn aspect ratio f equals 3, the ultimate strength is only 60 MPa and the first cell failure is due to matrix cell failure. However, if the ratio equals 12 (a very flat yarn cross-

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section), the strength reaches 240 MPa and the first cell failure is due to transverse weft yarn failures at points of maximum yarn curvature. These differences should be attributed to the different geometric architectures. The maximum yarn orientation b (which is directly linked to the yarn crosssection) plays a key role. Basically, the predicted strength decreases considerably with increased yarn undulation. For all four woven fabric composites considered, the ultimate failure is due to warp fibre breakage. The non-linearity of the curves is a result of progressive damage development. In the bias direction load case, we observe only a minor influence of the yarn aspect ratio f on the failure behaviour. The first cell failure is always due to transverse yarn failure. The strength of the woven fabric composite is related to the failure of a undirectional lamina or yarn cell. The failure of a single yarn cell in the bias direction is only weakly dependent on the out-of-plane lamina orientation. Therefore, the predicted strength of the composite is constant. Figure 3.15 compares the experimental and theoretical stress–strain curves for the RE280 glass/epoxy composite. The RE280 basket weave fabric has been supplied by the Syncoglas company, Belgium. The elastic and strength properties of the constituent materials are readily available and listed in Tables 3.5 and 3.6. Three important observations are presented here. First, a very good agreement is observed between experiment and theory on glass/epoxy composites. Second, the theoretical and experimen-

3.15 Theoretical and experimental stress–strain curves in warp and bias directions.

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tal stress–strain curves in the warp direction are apparently straight lines. However, the curves are nonlinear due to the characteristic knee behaviour for woven fabric composites. The knee is the result of transverse weft yarn failures.The position of the knee is predicted very accurately with the CEM. Finally, the yarn reorientation in the loading direction plays an important role in the bias specimen. At large strains, the experimental stresses are higher than the predicted ones. This is because the model does not account for the fabric yarn reorientation, which becomes significant at strains higher than 10%. The yarns rotate towards a smaller angle with respect to the loading direction, resulting in a stiffness increase. Therefore, the ultimate bias strength is not predicted very accurately.

3.5

Conclusions

This chapter has addressed the important issue of developing micromechanical models for woven fabric composites. Besides an extensive literature review on the modelling of 2-D woven fabric composites, a fully 3-D geometric, elastic and strength analysis has been presented as an example of the vigour in textile composites research. A brief summary of the principal conclusions follows. • Many researchers have used the isostrain technique to model the 3-D elastic properties of 2-D woven fabric composites. The model is indeed a ‘quick’ method to calculate an upper and lower bound for the effective stiffness matrix because it only requires yarn orientation data. Most researchers agree that the upper bound yields much better results than the lower bound. The technique can easily be applied to other textile composites. We have experienced that for most woven fabric composites, the isostrain models predict correct in-plane elastic properties but incorrect out-of-plane properties. The relative position of the predicted shear moduli is always false [21]. • Development of a geometric decomposition scheme for woven fabric composites with an arbitrary 2-D architecture. It is new because of its clear geometric concept (a library of building blocks for woven fabric composites), its easy bookkeeping of geometric data and its ability to describe non-traditional fabrics. Moreover, by extending the library of building blocks, solid 3-D woven fabric composites and braided fabric composites are actually analysed, using the same approach. • For modelling woven fabric composites one can find inspiration in the extensive modelling efforts on unidirectional and short fibre random composites. However, analogies can be misleading. The ‘yarn distribution’ of a textile composite is determined by the textile processing route (yarn level analysis). For woven fabric composites the characteristic

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yarn interlacings, curvatures and locations should be taken into account, reflecting the actual geometry. One of the critical determinants of effective woven fabric composites properties is definitely the yarn orientation. However, the yarn position is also critical for the accurate prediction of both stiffness and strength properties. Because it is not always easy to account for the position effect, it is neglected, minimized or concealed by most researchers. • The development of mechanical models or the application of optimization models to the design of large woven fabric composite structures is difficult because the number of geometric design variables and constraints is large. A remedy is to break the problem into several smaller subproblems. Although this is not in itself a new discovery, it is important to be fully aware of that fact and draw the appropriate conclusions. We are able to report that a ‘multilevel decomposition, multistep homogenization’ approach has been developed to solve the stiffness, stress and strength analysis problem for 2-D woven fabric composites. Although we have adopted fully the complementary energy approach, it should be stressed that different modelling strategies can be used to solve the different subproblems. The method is only limited by the capability to come up with an appropriate decomposition in subproblems. • The new and successful complementary energy model is based on solving the stress analysis problem first. Second, the computed stress concentration factors yield a solution for the 3-D stiffness properties. To the best of our knowledge, this kind of model has not yet been presented in the literature for woven fabric composites. Finally, the accurate stress fields serve as a useful tool for the strength analysis. One major advantage is that any type of simple or combined multiaxial loading can be applied. Another advantage is that the model can be extended to take into account the presence of residual thermal stresses. Although a considerable body of knowledge has been generated in the past years, more research is required to develop design guidelines for optimizing material performance by manipulating the woven fabric architecture. Future research could address the following topics: • Design of structural components such as aircraft parts, automobile chassis elements or bicycle frames tends to be very complex and timeconsuming. The development of efficient analytical pre-processors for woven fabric composites can decrease cost and make finite element modelling an economic and easy-to-use solution. Using pre-processors, the production of a correct finite element model will only require the generation of the component geometry, but not any more the detailed and elaborate description of unit cells, textile style, yarn size and yarn undulation.

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• The design of woven fabric composite materials will grow dependent on computer models. The optimization of a microstructure will require the marriage of micromechanical and optimization models. Therefore, optimization models should be developed which can optimize at once certain geometric, mechanical, thermal, process and economic properties.

3.6

Acknowledgements

P. Vandeurzen would like to thank the IWT (Institute for Science and Technology) of the Flemish Government, for providing him with a PhD grant. This research is carried out in the framework of a GOA project on ‘Textile composites’. The department is further supported by the IUAP programme of the Belgian Government.

3.7

References

1. Ko, F.K. and Chou, T.W., Composite Materials Series 3 – Textile Structural Composites, Elsevier Science, Amsterdam, 1989. 2. Chou, T.W., Microstructural Design of Fibre Composites, Cambridge University Press, Cambridge, 1992. 3. Cox, B.N. and Flanagan, G., Handbook of Analytical Methods for Textile Composites, Rockwell Science Center, Thousand Oaks, CA, 1996. 4. Verpoest, I., Ivens, J. and Van Vuure, A.W., ‘Textiel voor composieten: een oude technologie voor een modern constructiemateriaal’, Het Ingenieursblad, 12, 20–32, 1992. 5. Hearle, J.W.S. and Du, G.W., ‘Forming rigid fibre assemblies: the interaction of textile technology and composites engineering’, J. Textile Inst., 81(4), 360–381, 1990. 6. Yurgartis, S.W. and Jortner, J., ‘Measurement of yarn shape and nesting in plainweave composites’, Composites Sci. Technol., 46, 39–50, 1993. 7. Bailie, J.A., ‘Woven fabric aerospace structures’, in Handbook of Composites – 2: Structure and Design, Elsevier Science, London, 1982. 8. Dictionary of Fibre and Textile Technology, Hoechst Celanese Corporation, Charlotte, NC, 1990. 9. Hoffman, R.M.,‘Some theoretical aspects of yarn and fabric density’,Textile Res. J., 1952, 170–171. 10. Byun, J.H. and Chou, T.W., ‘Elastic properties of three-dimensional angleinterlock fabric preforms’, J. Textile Inst., 81, 538–548, 1990. 11. Ishikawa, T. and Chou, T.W., ‘Stiffness and strength behaviour of woven fabric composites’, J. Mater. Sci., 17, 3211–3220, 1982. 12. Ishikawa, T. and Chou, T.W., In-plane thermal expansion and thermal bending coefficients of fabric composites’, J. Composite Mater., 17, 92–104, 1983. 13. Ishikawa, T., Matsushima, M. and Hayashi, Y., ‘Experimental confirmation of the theory of elastic moduli of fabric composites’, J. Composite Mater., 19, 443–458, 1985.

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14. Tsai, S.W. and Hahn, H.T., Introduction to Composite Materials, Technomic Publishing, Lancaster, PA, 1980. 15. Naik, N.K. and Shembekar, P.S., ‘Elastic behavior of woven fabric composites: I Lamina analysis’, J. Composite Mater., 26, 2196–2225, 1992. 16. Naik, N.K. and Ganesh, V.K., ‘Prediction of the thermal expansion coefficients of plain weave fabric composites’, Composite Structures, 26, 139–154, 1993. 17. Ganesh, V.K. and Naik, N.K., ‘Failure behavior of plain weave fabric laminates under on-axis uniaxial tensile loading: I Laminate geometry’, J. Composite Mater., 30(16), 1748–1778, 1996. 18. Naik, N.K. and Ganesh, V.K., ‘Failure behavior of plain weave fabric laminates under on-axis uniaxial tensile loading: II Analytical predictions’, J. Composite Mater., 30(16), 1779–1822 (1996). 19. Halpin, J.C., Primer on Composite Materials: Analysis, Technomic Publishing, Lancaster, PA, 1984. 20. Hahn, H.T. and Pandy, R., ‘A micromechanics model for thermoelastic properties of plain weave fabric composites’, J. Eng. Mater. Technol., 116, 517–523, 1994. 21. Vandeurzen, Ph., Ivens, J. and Verpoest, I., ‘A critical comparison of analytical and numerical (FEM) models for the prediction of the mechanical properties of woven fabric composites’, paper presented at New Textiles for Composites TEXCOMP-3, RWTH Aachen (G), 1996. 22. Naik, R.A., ‘Failure analysis of woven and braided fabric reinforced composites’, J. Composite Mater., 29, 2334–2363, 1995. 23. Paumelle, P., Hassim, A. and Léné, F., ‘Composites with woven reinforcements: calculation and parametric analysis of the properties of the homogeneous equivalent’, La recherche aérospatiale, 1, 1–12, 1990. 24. Paumelle, P., Hassim, A. and Léné, F., ‘Microstress analysis in woven composite structures’, La recherche aérospatiale, 6, 47–62, 1990. 25. Wood, J.,‘Finite element analysis of composite structures’, Composite Structures, 29, 219–230, 1994. 26. Hewitt, J.A., Brown, D. and Clarke, R.B., ‘Computer modelling of woven composite materials’, Composites, 26, 134–140, 1995. 27. Blackketter, D.M., Walrath, D.E. and Hansen, A.C., ‘Modeling damage in a plain weave fabric reinforced composite material’, J. Composites Technol. Res., 15(2), 136–142, 1993. 28. Whitcomb, J. and Kyeongsik, W., ‘Enhanced direct stiffness method for finite element analysis of textile composites’, Composite Structures, 28, 385–390, 1994. 29. Kyeongsik, W. and Whitcomb, J., ‘Macro finite element using subdomain integration’, Commun. Numerical Methods Eng., 9, 937–949, 1993. 30. Whitcomb, J. and Srirengan, K., ‘Effects of various approximations on predicted progressive failure in plain weave composites’, Private Communication. 31. Vandeurzen, Ph., Ivens, J. and Verpoest, I., ‘A three-dimensional micromechanical analysis of woven fabric composites I. Geometric analysis’, Composites Sci. Technol., 56(11), 1303–1315, 1996. 32. Vandeurzen, Ph., Ivens, J. and Verpoest, I., ‘A three-dimensional micromechanical analysis of woven fabric composites II. Elastic analysis’, Composites Sci. Technol., 56(11), 1317–1327, 1996. 33. Vandeurzen, Ph., Ivens, J. and Verpoest, I., ‘Micro-stress analysis of woven fabric composites’, J. Composite Mater. 32(7), 623–651, 1998.

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34. Vandeurzen, Ph., Ivens, J. and Verpoest, I., ‘TexComp: a 3D analysis tool for 2D woven fabric composites’, SAMPE J., March–April, 25–33, 1997. 35. Mirzadeh, F. and Reifsnider, K.L., ‘Micro-deformations in C300/PMR15 woven composite’, J. Composite Mater., 26(2), 185–204, 1992. 36. Peirce, F.T., ‘Geometrical principles applicable to the design of functional fabrics’, Textile Res. J., 17(3), 123–147, 1947. 37. Chamis, C.C., ‘Simplified composite micromechanics equations for hygral, thermal, and mechanical properties’, paper presented at 38th Conference of the Society of the Plastics Industry (SPI), Houston, Texas, 7–11 February, 1983. 38. Bertsekas, D.P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982. 39. Leech, C.M. and Kettlewell, J., ‘Formulative principles in mechanics’, Int. J. Mechanical Eng. Education, 9, 157–180, 1981. 40. Theocaris, P.S., ‘Failure criteria for isotropic bodies revisited’, Eng. Fracture Mechanics, 51(2), 239–264, 1995.

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4 Macromechanical analysis of 3-D textile reinforced composites A. MIRAVETE, R. CLEMENTE AND L. CASTEJON

4.1

Introduction

Both laminated composite materials [1] and 3-D textile reinforced composite materials [2] are characterized by being composed of biphasic materials: fibres and matrices [3]. The macromechanical analysis of 3-D textile reinforced composite materials is defined by the strain–stress relations, the failure modes and the degradation properties from first failure (FF) to last failure (LF) of the material system. Fibre properties and geometry of the fibres [4,5] inside the matrix are considered in the micromechanical analysis, the result being the following macromechanical parameters: • the strain–stress relations; • the failure modes; • degradation properties from first failure to last failure. The 3-D textile reinforced composite material [6] is no longer considered a biphasic material, but as a system with the properties listed above, as a result of the micromechanical study. To carry out the macromechanical analysis of a certain complex structure made of 3-D textile reinforced composite materials, the following information is required: 1 The definition of the material models, which will govern the behaviour of the material system in terms of stiffness and strength at a macromechanical level. 2 The introduction of the stiffness and strength properties necessary for the total implementation of the material models of each of the composite systems manufactured by the textile technologies available nowadays. 3 The definition of the geometry of the structure to be analysed, including the geometry of the borders between the different substructures, which can be made by different manufacturing textile technologies. 100 © 1999,Woodhead Publishing Ltd

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4 The definition of the boundary of loading conditions, which can be constant or variable with time, owing to the existence of contact or friction conditions from dynamic loads. 5 The design requirements of the structure, object of study. This information is processed by means of a mathematical model. In most cases, the finite element technique seems to be the most appropriate to solve the numerical problem of obtaining the response of the structure to the loading conditions. Usually, 1, 3, 4 and 5 are known and 2, which refers to the strength and stress properties, must be either obtained from testing or estimated by means of micromechanical studies. Testing is recommended, when possible, since the accuracy of the results is extremely high when a proper statistical analysis is made. However, in those cases when testing cannot be carried out owing to the high complexity of the characterization, as is the case with some through-thickness normal and transverse properties, or for other reasons, the estimation of properties by means of micromechanical analyses and the finite element technique or analytical procedures constitutes an alternative method, although it is much less accurate.

4.2

Determination of the stiffness and strength properties of 3-D textile reinforced composite materials

The object of the macromechanical analysis is the mechanical prediction of structures made of 3-D textile reinforced composite materials, under given working conditions. The implementation of appropriate material models, simulating the behaviour of the material system under given working conditions, is necessary to carry out this type of analysis. In order to define the material model properly, simulating the behaviour of the material system, the introduction of a number of stiffness and strength properties is required. However, the material model varies as a function of the following issues: • type of macromechanical analysis; • type of theory; • type of 3-D textile technology. Owing to the fact that the textile technologies present differences in terms of the type of construction [7], the stiffness and strength properties will also vary in terms of the textile technology used. The type of macromechanical analysis also affects the material model to be used, and therefore the stiffness and strength properties needed to carry

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out the analysis. These parameters will be a function of the type of analysis; linear/non-linear, static or dynamic [8,9], stress or displacement-based, hygrothermal, buckling, modal, crash analysis [10], etc. Finally, the elastic and strength properties needed also depend on the theory on which the analysis is based.

4.2.1 Theories used for the macromechanical analysis of 3-D textile reinforced composite materials In this section, several theories applicable to the macromechanical analysis of 3-D textile reinforced composite materials are described. The textile technologies studied correspond to those composite materials constituted by preforms generated from 3-D textiles and those joined by means of the stitching technology. The stiffness properties needed are also analysed. The most appropriate theories for analysing every textile technology will be selected according to the material typology and the desired degree of accuracy [11].

The classical beam theory The classical beam theory [11,12] is based on the fourth-order differential equations used in the Euler–Bernoulli bending theory, the torsion and the axial tension–compression theories. The Euler–Bernoulli theory assumes that the transverse section perpendicular to the beam axis remains plane and perpendicular to this axis after deformation. The transverse deflection w is governed by a fourth-order differential equation: d2 Ê d 2w ˆ ( ) E I x Á ˜ = f ( x) for 0 < x < L x dx 2 Ë dx 2 ¯

[4.1]

where f(x) is the transverse distributed load, Ex is the elasticity modules in the beam axis direction (x), and I(x) is the inertia momentum as a function of the x-direction. A scheme of these variables is represented in Figs. 4.1 and 4.2. The following stiffness properties are needed when using the classical beam theory: Ex, Gxy and nxy. The following parameters must also be implemented: I(x), IO(x), A(x), AC(x), k1(x, y, z ) and k2(x, y, z ), where: I(x) is inertia momentum, IO(x) is torsion inertia momentum, A(x) is crosssectional area, AC(x) is shear cross-sectional area, k1(x, y, z) is the function dependent on the cross-sectional shape in position x, used to determine the strain component gxy and k2(x, y, z) is the function dependent on the crosssectional shape in position x, used to determine the strain component gxz.

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4.1 Scheme of a beam subjected to bending, torsion and tension.

4.2 Scheme of a general cross-section.

Classical laminated plate theory Nonisotropic and lamination aspects of composite materials are introduced by means of the classical laminated plate theory [2,11,13,14]. This theory takes into account in-plane and bending stresses. Interlaminar stresses are not considered, and therefore the application field of this theory is limited to thin plates with small displacements subject to uniform loads. Those structures subject to impact, free edge effects, stress concentrations, point loads, mechanical and bonded joint, or thick structures are beyond the scope of the classical laminated plate theory. In terms of order of magnitude, a plate is considered thin when: plate thickness < 10 characteristic length

[4.2]

This theory is based on the following assumptions: • linear variations of strains; • the perpendicular line to the mid-surface remains perpendicular after deformation; i.e. the strains generated by the shear forces are neglected. The local axes (x, y, z), the mid-plane x–y and the displacements associated with these axes are represented in Fig. 4.3. The displacement fields are: ux ( x, y) = u( x, y) uy ( x, y) = v( x, y) uz ( x, y) = w( x, y)

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4.3 Definition of a local axis.

4.4 Definition of plate displacements.

When this theory is applied, the following stiffness properties are needed: EX, EY, GXY and nXY. Irons theory This theory is based on the following assumptions: • The perpendicular line to the mid-surface of the laminated plate remains straight after deformation. • The strain energy corresponding to the stresses perpendicular to the mid-surface is neglected. However, the assumption that the perpendicular line to the mid-surface remains perpendicular after deformation is not imposed. Therefore, interlaminar shear stresses are accounted for in this case. Irons theory considers in-plane and shear stresses for each ply of the laminate. The relationship between stresses and strains proceeds from a threedimensional approach. The local axis and the definition of the displacements of the plate are represented in Fig. 4.4. The displacement fields are:

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4.5 Displacement field in a thin plate according to YNS theory.

u1 ( x, y, z) = u( x, y, z) u2 ( x, y, z) = v( x, y, z)

[4.4]

u3 ( x, y, z) = w( x, y, z) The strain vector is represented in expression 4.5: È ∂u ∂v ∂u ∂v ∂u ∂w ∂v ∂w ˘ + , + , e = [e x , e y, g xy, g xz , g yz ] = Í , , + Î ∂ x ∂ x ∂ y ∂ x ∂ z ∂ x ∂ y ∂ y ˙˚

[4.5]

The following stiffness properties are needed for this theory: EX, EY, GXY, GXZ, GYZ and vXY. First-order shear theory This theory [11] is based on the work from Yang–Norris–Stavsky (YNS), which is a generalization of Mindlin theory to laminated non-isotropic materials. In-plane, bending and shear stresses are accounted for. This theory is applicable to both thin and thick laminated plates, by using an appropriate correction factor. Figure 4.5 represents a plate with constant thickness h and the parameters needed to define the displacement field. The following equations govern the displacement field by applying YNS theory: u( x, y, z) = uO ( x, y, z) + zYY ( x, y, z) v( x, y, z) = vO ( x, y, z) + zYX ( x, y, z)

[4.6]

w( x, y, z) = wO ( x, y, z) where: u, v, w = displacement components in the x,y,z directions, uO, vO, wO = mid-plane linear displacements, YX, YY = angular displacements around the x,y axes. The following stiffness properties are needed: EX, EY, GXY, GXZ, GYZ and vXY.

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3-D textile reinforcements in composite materials Higher-order shear theory

According to the first-order shear theories, shear strains are constant through the laminate thickness, and therefore they do not satisfy the equilibrium equation at the top and bottom surfaces, where shear strain must be zero if no external force is applied. For thick laminate plates, an accurate shear strain distribution through the laminate thickness is essential. To satisfy the equilibrium equation above mentioned, a higher-order shear theory must be applied [11,15,16]. In this section, a theory developed by Reddy will be described. In-plane, bending and shear stresses are taken into account, the number of variables being the same as in the first-order shear theories. A parabolic shear strain distribution through the laminate thickness is implemented, the shear strains being zero at both top and bottom surfaces. The displacement field according to Reddy theory is: u( x, y, z) = uo ( x, y) + zYy ( x, y) + z2 x x ( x, y) + z3r x ( x, y) v( x, y, z) = vo ( x, y) + zYx ( x, y) + z2 x y ( x, y) + z3r y ( x, y)

[4.7]

w( x, y, z) = wo ( x, y) where: uO, vO, wO = linear displacements of a point (x,y) at the laminate mid-plane, YX, YY = angular displacements around the x and y axes, xX, xY, rX, rY = functions to be determined by applying the condition that interlaminar shear stresses must be zero at top and bottom surfaces: s xz ( x, y, ± h 2) = 0 s yz ( x, y, ± h 2) = 0

[4.8]

The following stiffness properties are needed: EX, EY, GXY, GXZ, GYZ and vXY. Elasticity theory The elasticity theory [17] is applicable to both isotropic and non-isotropic materials, owing to the fact that all the effects related to the elasticity are taken into account. This theory is very efficient in those analyses where the whole stress tensor must be considered, including the interlaminar normal or peeling stress. The displacement field is shown in Fig. 4.6. The strain tensor is given by: e = [e x , e y , e z , g xy , g xz , g yz ] È ∂u ∂v ∂w ∂u ∂v ∂u ∂w ∂v ∂w ˘ , + , , =Í , , + + Î ∂ x ∂ y ∂ z ∂ y ∂ x ∂ z ∂ x ∂ y ∂ y ˙˚

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4.6 Displacement field in a thick plate.

Table 4.1. Stiffness properties as a function of the theory used Theory

Needed stiffness properties

Beams theory Laminated plates theory Irons’s theory First-order shear theory Higher-order shear theory Elasticity theory

EX, EX, EX, EX, EX, EX,

GXY and vXY EY, GXY and vXY EY, GXY, GXZ, GYZ and vXY EY, GXY, GXZ, GYZ and vXY EY, GXY, GXZ, GYZ and vXY EY, EZ, GXY, GXZ, GYZ, nXY, nXZ and vYZ

The following stiffness properties are needed: EX, EY, EZ, GXY, GXZ, GYZ, vXY, vXZ and vYZ. Table 4.1 represents the stiffness properties needed as a function of the theory applied.

4.2.2 Stiffness and strength properties as a function of the 3-D textile preform The 3-D textile preform used as a reinforcement for the composite material affects the needed stiffness and strength properties for two reasons [18–23]: • On the one hand, every 3-D textile technology is associated with one or more theories among the ones described in Section 4.2.1. Each of these theories requires a specific list of stiffness properties for an appropriate implementation. • On the other hand, every 3-D textile technology requires one or more specific strength criteria and, therefore, a number of strength properties.

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In the following sections, the most important 3-D textile technologies will be analysed. Special attention will be paid to the strength criterion for each case. Braiding Depending on the type of braiding technology considered (2-D or 3-D), there are several options in terms of type of finite element used and type of theory applied. This issue is analysed in Table 4.2. The most appropriate failure criterion for static analyses of braided preforms is the 3-D Tsai–Wu criterion [1]. For dynamic studies, the maximum strain criterion gives very interesting results until the final failure occurs. For those studies where out-of-plane stresses must be considered, the introduction of interaction factors between normal and shear stress components in the 3-D Tsai–Wu criterion generates more accurate results. In this case, the general quadratic criterion to be applied is governed by the following equations: Fijsij + Fisii = 0

i, j = 1 ∏ 6

(criterion 1)

where: F1 =

1 1 X X¢

F11 =

1 XX ¢

F44 =

1 S

2

xy

F2 = F22 =

F55 =

1 1 Y Y¢

1 YY ¢ 1 S

2

xz

F3 =

F33 = F66 =

1 1 Z Z¢

F4 = F5 = F6 = 0

1 ZZ ¢

[4.10]

1 S 2 yz

F45 = F46 = F56 = 0 Fij = -0.5 Fii Fjj

i, j = 1 ∏ 6 and i π j

When other theories on general elasticity are applied, the stress tensor is considerably reduced: • For beam theory, the stress tensor is composed of sx and txy, and therefore the criterion can be simplified to the following expression: F11sx2 + F44txy2 + F1sx + 2F14sxtxy = 1

(criterion 2) [4.11]

• For the classical laminated plate theory, the stress components are sx, sy and txy, and the failure criterion corresponds to: F11sx2 + F22sy2 + F44txy2 + F1sx + F2sy + 2F12sxsy + 2F14sxtxy + 2F24sytxy = 1

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(criterion 3) [4.12]

Table 4.2. Properties to be applied as a function of the type of braiding technology Type of braiding

Type of element

Theory implemented

Requested stiffness properties

Requested strength properties

Strength criterion

2-D 2-D 2-D 2-D 2-D 2-D

Beam Shell Shell Shell Shell Solid

Unidimensional Laminated plates theory Irons’s theory First-order shear theory Higher-order shear theory Elasticity theory

EX, GXY and vXY EX, EY, GXY and vXY EX, EY, GXY, GXZ, GYZ and vXY EX, EY, GXY, GXZ, GYZ and vXY EX, EY, GXY, GXZ, GYZ and vXY EX, EY, EZ, GXY, GXZ, GYZ, vXY, vXZ and vYZ

X, X, X, X, X, X,

Criterion Criterion Criterion Criterion Criterion Criterion

3-D 3-D

Beam Solid

Unidimensional Elasticity theory

EX, GXY and vXY EX, EY, EZ, GXY, GXZ, GYZ, vXY, vXZ and vYZ

X, X¢ and Sxy X, X¢, Y, Y ¢, Z, Z¢, Sxy Sxz and Syz

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X¢ and Sxy X¢, Y, Y ¢ and SXY X¢, Y, Y ¢, Sxy Sxz and Syz X¢, Y, Y ¢, Sxy Sxz and Syz X¢, Y, Y ¢, Sxy Sxz and Syz X¢, Y, Y ¢, Z, Z¢, Sxy Sxz and Syz

2 3 4 4 4 1

Criterion 2 Criterion 1

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• When Irons’s first- or higher-order shear theories are applied, the stress components are sx, sy, txy, txz and tyz and thus the failure criterion is represented by the following expression: F11sx2 + F22sy2 + F44txy2 + F55txz2 + F66tyz2 + F1sx + F2sy + 2F12sxsy + 2F14sxtxy + 2F15sxtxz + 2F16sxtyz + 2F24sytxy + 2F25sytxz + 2F26sytyz = 1

(criterion 4) [4.13]

Table 4.2 shows the type of finite element, theory, needed stiffness and strength properties and failure criterion to be applied depending on the type of braiding technology used. 3-D weaving The models to be carried out for simulations of 3-D weaving preforms can be constituted by shell or solid finite elements. The most appropriate failure criterion seems to be the 3-D Tsai–Wu criterion, implementing interaction factors between normal and shear stresses. This criterion can be simplified (criteria 2, 3 and 4) should theories different from the one based on elasticity be applied. Table 4.3 shows the type of finite element, theory, needed stiffness and strength properties and failure criterion to be applied depending on the type of braiding technology used. Weft knitting The analysis for the weft knitting technology coincides with the study for 3-D weaving. The models to be carried out for simulations of the weft knitting can be constituted by shell or solid finite elements. The most appropriate failure criterion seems to be the 3-D Tsai–Wu criterion, implementing interaction factors between normal and shear stresses. This criterion can be simplified (criteria 2, 3 and 4) should theories different from the one based on elasticity be applied. Table 4.4 shows the type of finite element, needed stiffness and strength properties and failure criterion to be applied as a function of the type of theory used. Warp knitting Several options exist in terms of type of finite element and theory to be applied, depending on the kind of warp knitted composite material studied (Tables 4.5 and 4.6). When using plain warp knitting [24–26] the treatment is similar to the one described for weft knitting. However, if a

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Table 4.3. Properties to be applied as a function of the type of braiding technology Type of Theory element

Requested stiffness properties

Requested strength properties

Strength criterion

Shell

EX, EY, GXY and vXY

X, X¢, Y, Y ¢ and Sxy

Criterion 3

EX, EY, GXY, GXZ, GYZ and vXY EX, EY, GXY, GXZ, GYZ and vXY

X, X¢, Y, Y ¢, Sxy Sxz and Syz X, X¢, Y, Y ¢, Sxy Sxz and Syz

Criterion 4

EX, EY, GXY, GXZ, GYZ and vXY

X, X¢, Y, Y ¢, Sxy Sxz and Syz

Criterion 4

Shell Shell

Shell

Solid

Laminated plates theory Irons’s theory First-order shear theory Higher-order shear theory Elasticity theory

Criterion 4

X, X¢, Y, Y ¢, Z, Criterion 1 EX, EY, EZ, GXY, GXZ, GYZ, vXY, vXZ and vYZ Z ¢, Sxy Sxz and Syz

Table 4.4. Properties to be applied as a function of the theory used Type of Theory element

Requested stiffness properties

Requested strength properties

Strength criterion

Shell

EX, EY, GXY and vXY

X, X¢, Y, Y ¢ and Sxy

Criterion 3

EX, EY, GXY, GXZ, GYZ and vXY EX, EY, GXY, GXZ, GYZ and vXY

X, X¢, Y, Y ¢, Sxy Sxz and Syz X, X¢, Y, Y ¢, Sxy Sxz and Syz

Criterion 4

EX, EY, GXY, GXZ, GYZ and vXY

X, X¢, Y, Y ¢, Sxy Sxz and Syz

Criterion 4

Shell Shell

Shell

Solid

Laminated plates theory Irons’s theory First-order shear theory Higher-order shear theory Elasticity theory

Criterion 4

X, X¢, Y, Y ¢, Z, Criterion 1 EX, EY, EZ, GXY, GXZ, GYZ, vXY, vXZ and vYZ Z ¢, Sxy Sxz and Syz

3-D warp knitted material is studied, there are a number of possibilities, since the sandwich skins can be modelled as a 2-D warp knitted material but the core can be analysed by means of several methods. The 3-D Tsai–Wu criterion can be applied for analysing 2-D warp knitted materials or the sandwich skins of a 2-D warp knitted system. The interaction factors between normal and shear stresses must be implemented to

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Table 4.5 Properties to be applied as a function of the type of theory used for the 2-D warp knitting technology Type of Theory element

Requested stiffness properties

Requested strength properties

Strength criterion

Shell

EX, EY, GXY and vXY

X, X¢, Y, Y ¢ and Sxy

Criterion 3

EX, EY, GXY, GXZ, GYZ and vXY EX, EY, GXY, GXZ, GYZ and vXY

X, X¢, Y, Y ¢, Sxy Sxz and Syz X, X¢, Y, Y ¢, Sxy Sxz and Syz

Criterion 4

EX, EY, GXY, GXZ, GYZ and vXY

X, X¢, Y, Y ¢, Sxy Sxz and Syz

Criterion 4

Shell Shell

Shell

Solid

Theory of laminated plates Irons’s theory First-order shear theory Higher-order shear theory Elasticity theory

Criterion 4

X, X¢, Y, Y ¢, Z, Criterion 1 EX, EY, EZ, GXY, GXZ, GYZ, vXY, vXZ and vYZ Z ¢, SXY Sxz and Syz

obtain more accurate results. If a theory different from the one based on the elasticity is applied, the number of stress components is drastically reduced, and therefore the expression of the failure criterion becomes more simple (criteria 2, 3 and 4). The sandwich core is constituted by a pile structure and optionally by a foam. The failure criterion number 5 is considered the most appropriate since shear and peeling stresses are accounted for. Since the pile structure may be different in the two principal directions, the core strength will also vary in both directions. The expressions of criterion 5 are represented in expression 4.14. There is no peeling failure if: Szcn £ sz £ Sztn There is no shear failure if: t xz

2

S xz n

2

where: Szcn = Sztn = Sxzn = Syzn =

+

t yz

2

S yz n

2

£1

(criterion 5) [4.14]

core compression strength in the z-direction, core tension strength in the z-direction, core shear strength in the x–z plane, core shear strength in the y–z plane.

Tables 4.5 and 4.6 represent the type of finite element, needed stiffness and strength properties and failure criterion to be applied as a function of

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Table 4.6. Properties to be applied as a function of the type of theory used for the 3-D warp knitting sandwich technology Type of element

Theory

Stiffness properties requested1

Strength properties requested1,2,3

Strength criterion

Shell including skins and core

Theory of laminated plates

Skins: EX, EY, GXY and vXY Core: EXn, EYn, GXYn and vXYn

Skins: X, X ¢, Y, Y ¢ and Sxy Core: Sxzn and Syzn

Skins: criterion 3 Core: criterion 5

Shell including skins and core

Irons’s theory

Skins: EX, EY, GXY, GXZ, GYZ and vXY Core: EXn, EYn, GXYn, GXZn, GYZn and vXYn

Skins: X, X ¢, Y, Y ¢ and Sxy Core: Sxzn and Syzn

Skins: criterion 4 Core: criterion 5

Shell including skins and core

First-order shear theory

Skins: EX, EY, GXY, GXZ, GYZ and vXY Core: EXn, EYn, GXYn, GXZn, GYZn and vXYn

Skins: X, X ¢, Y, Y ¢ and Sxy Core: Sxzn and Syzn

Skins: criterion 4 Core: criterion 5

Shell including skins and core

Higher-order shear theory

Skins: EX, EY, GXY, GXZ, GYZ and vXY Core: EXn, EYn, GXYn, GXZn, GYZn and vXYn

Skins: X, X ¢, Y, Y ¢ and Sxy Core: Sxzn and Syzn

Skins: criterion 4 Core: criterion 5

Skins: Shell Core: Solid

Skins: theory of laminated plates Core: elasticity theory

Skins: EX, EY, GXY and vXY Core: EXn, EYn, EZn, GXYn, GXZn, GYZn, vXYn, vXZn and vYZn

Skins: X, X ¢, Y, Y ¢ and Sxy Core: Sxzn and Syzn, SzCn and SzTn

Skins: criterion 3 Core: criterion 5

Skins: shell Core: solid

Skins: Irons’s theory Core: elasticity theory

Skins: EX, EY, GXY, GXZ, GYZ and vXY Core: EXn, EYn, EZn, GXYn, GXZn, GYZn, vXYn, vXZn and vYZn

Skins: X, X ¢, Y, Y ¢ and Sxy Core: Sxzn and Syzn, SzCn and SzTn

Skins: criterion 4 Core: criterion 5

Skins: shell Core: solid

Skins: First-order shear theory Core: Elasticity theory

Skins: EX, EY, GXY, GXZ, GYZ and vXY Core: EXn, EYn, EZn, GXYn, GXZn, GYZn, vXYn, vXZn and vYZn

Skins: X, X ¢, Y, Y ¢ and Sxy Core: Sxzn and Syzn, SzCn and SzTn

Skins: criterion 4 Core: criterion 5

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114 Table 4.6. (cont.) Type of element

Theory

Stiffness properties requested1

Strength properties requested1,2,3

Strength criterion

Skins: shell Core: solid

Skins: higher-order shear theory Core: elasticity theory

Skins: EX, EY, GXY, GXZ, GYZ and vXY Core: EXn, EYn, EZn, GXYn, GXZn, GYZn, vXYn, vXZn and vYZn

Skins: X, X¢, Y, Y ¢ and Sxy Core: Sxzn and Syzn, SzCn and SzTn

Skins: criterion 4 Core: criterion 5

Skins: solid Core: solid

Elasticity theory

Skins: EX, EY, EZ, GXY, GXZ, GYZ, vXY, vXZ and vYZ Core: EXn, EYn, EZn, GXYn, GXZn, GYZn, vXYn, vXZn and vYZn

Skins: X, X¢, Y, Y ¢, Z, Z¢, Sxy Sxz and Syz Core: Sxzn and Syzn, SzCn and SzTn

Skins: criterion 1 Core: criterion 5

Skins: solid

Skins and core: elasticity theory Piles: unidimensional

Skins: EX, EY, EZ, GXY, GXZ, GYZ, vXY, vXZ and vYZ Core: model of Krieg and Key Piles: EX p, GXY p and vXY p

Skins: X, X¢, Y, Y ¢, Z, Z¢, Sxy Sxz and Syz Core: ST, SC, tmax Piles: smaxp and tmaxp

Skins: criterion 1

Core: solid Piles: beams

Core: criterion of hydrostatic pressure failure Piles: criterion 2

1 The superscript(n) indicates that the stiffness and strength properties refer to the sandwich core, and the superscript(p) indicates that the properties refer to the piles. 2 ST, SC, tmax are the tension, compression and shear strengths of the core, respectively. 3 smaxp and tmaxp are the maximum tension and shear strengths that the pile fibres can bear.

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Table 4.7. Properties to be applied as a function of the type of theory used for the multilayer knitting 4/5-D technology Type of Theory element

Requested stiffness properties

Requested strength properties

Strength criterion

Shell

EX, EY, GXY and vXY

X, X¢, Y, Y ¢ and Sxy

Criterion 3

EX, EY, GXY, GXZ, GYZ and vXY EX, EY, GXY, GXZ, GYZ and vXY

X, X¢, Y, Y ¢, Sxy Sxz and Syz X, X¢, Y, Y ¢, Sxy Sxz and Syz

Criterion 4

EX, EY, GXY, GXZ, GYZ and vXY

X, X¢, Y, Y ¢, Sxy Sxz and Syz

Criterion 4

Shell Shell

Shell

Solid

Theory of laminated plates Irons’s theory First-order shear theory Higher-order shear theory Elasticity theory

X, X¢, Y, Y ¢, Z, Z¢, EX, EY, EZ, GXY, GXZ, GYZ, vXY, vXZ and vYZ Sxy Sxz and Syz

Criterion 4

Criterion 1

the type of theory used for the 2-D and 3-D warp knitting technologies respectively. Multilayer knitting 4/5-D This technology generates textile plane layers, as is the case for 3-D weaving and warp and weft knitted plain techniques. For all these cases, the numerical models to be implemented from multilayer knitting 4/5-D preforms may be constituted by shell or solid elements. There are a number of theories available, with various degrees of accuracy [27]. The 3-D Tsai–Wu criterion can be applied for analysing multilayer knitting 4/5-D technologies. The interaction factors between normal and shear stresses must be implemented to obtain more accurate results. If a theory other than the one based on elasticity is applied, the number of stress components is drastically reduced, and therefore the expression of the failure criterion becomes more simple (criteria 2, 3 and 4). Table 4.7 shows the type of finite element, needed stiffness and strength properties and failure criterion to be applied as a function of the type of theory used for the multilayer knitting 4/5-D technology. Stitching The stitching technology can be differentiated from the other techniques above described, since this is a joint technology and not a process to obtain

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textile preforms. However, this technique is essential for joining, from a highly structural point of view, sub-structures manufactured by different preform technologies. Therefore, the analysis of the stitching technique is critical when a macromechanical analysis of a complete structure composed of a number of preformed substructures is carried out [28]. Since the joint area between two preforms by means of a stitching technology is constituted by stitched fibres, this material system can be considered unidirectional. For this case, the Hashin criterion seems to be the most appropriate (criteria 6 and 7): • Fibre failure: the Hashin criterion takes into account the interaction between compression and shear. The failure of the fibre occurs when one of the following conditions is met: sx = sita for s1 > 0 2

2 2 2 Ê s1 ˆ Ê t 12 + t 13 ˆ +Á = 1 for s1 < 0 Ë s ica ¯ Ë t 2 ˜¯ 12 sa

(criterion 6) [4.15]

• Matrix failure: by means of this criterion, the matrix failure occurs when the stresses exceed an interactive combination of normal and maximum shear stresses (criterion 7): 2

2 2 2 Ê s n ˆ Ê t 23 + t 13 ˆ +Á ˜ = 1 for s n < 0 2 Ë s 2 ta ¯ Ë ¯ t sa 2

Ê t 23 2 + t 13 2 ˆ ˜ = 1 for s n < 0 Á 2 ¯ Ë t sa where: sica sita t12sa tsa

= = = =

(criterion 7) [4.16]

allowable compression stress in i-direction, allowable tension stress in i-direction, allowable shear stress in the 12-plane, allowable shear stress in the plane perpendicular to the fibres.

In order to assess the stress components in the joint area, it is not necessary to build a finite element mesh of this area, but the stress values in the joint surfaces must be obtained from the finite element mesh of the preformed substructures.

4.3

Determination of the stiffness and strength properties of braided composite materials

An analytical model for the prediction of stiffness and strength properties of 2-D triaxial braided composite materials [29] is presented in this section.

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The final properties of the braided material will be a function of the mechanical properties of fibre, matrix and fibre orientations [30]. In order to assess the accuracy of the model presented, a finite element micromechanical model [31], was built. Fibre and matrix were modelled separately [32]. This finite element model for 2-D braiding material can also be used to understand the material behaviour when subjected to different in-plane stresses. This model allows us to obtain information about the stiffness, strength and strain to failure, as well as about the degradation process in the material after the first failure occurs. The analysis focuses on 2-D braiding containing some percentage of 0° orientation fibres, the rest of the fibre having ±a orientations.

4.3.1 Analytical formulation The following parameters are used in the 2-D braiding analytical predictive expressions: a = braided fibres orientation V0 = 0° fibre volume percentage Va = a° fibre volume percentage Vf = V0 + Va = fibre volume percentage Vm = matrix volume percentage Ef = Young modulus of fibre Em = Young modulus of matrix emaxf = strain to failure of fibre emaxm = strain to failure of matrix Gf = shear modulus of fibre Gm = shear modulus of matrix gmaxm = angular strain to failure of matrix nm = Poisson coefficient of matrix Ea, Eb, ea, eb, dependent stress–strain graphs may be obtained from the analytical formulation, as is shown in Fig. 4.7. Direction 1 Ea = Ef cos4 aVa + Ef V0 + EmVm Eb = Ef cos4 aVa + EmVm ea = emaxf eb = emaxf/cos2 a

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4.7 Parameters obtained from analytical formulation.

Direction 2 Ea = Ef sin 4 aVa +

Ef Em (Vm + V0 ) E f V ¢m + Em V ¢0

V ¢m =

Vm Vm + V0

V ¢0 =

V0 Vm + V0

Eb =

Ef Em (Vm + V0 ) Ef Vm¢ + Em V0¢

e a = e maxf sin 2 a e b = e maxm Figures 4.8 and 4.9 show the parameters obtained from an analytical formulation on directions 1 and 2. In-plane 1–2 shear stress Ga = Ef sin 2 a cos 2 aVa + V ¢m =

Vm Vm + V0

V ¢0 =

V0 Vm + V0

Gb =

Gf Gm (Vm + V0 ) Gf V ¢m + Gm V ¢0

ga =

e maxf sin a cos a

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4.8 Parameters obtained from analytical formulation on direction 1.

4.9 Parameters obtained from analytical formulation on direction 2.

4.10 Parameters obtained from analytical formulation on plane 1–2, shear stress.

g b = g maxm =

e maxm (1 + um ) 3

2

Figure 4.10 shows the parameters obtained from an analytical formulation on plane 1–2, shear stress.

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4.11 Fibre model.

4.12 Finite element model of 60° fibre.

4.3.2 Finite element model for a braided composite (0°65%/60°35%)V =50% f

Model properties The finite element [33] model consists of 14 849 nodes and 24 592 elements classified as follows: • 8624 eight-node, linear brick, constant pressure, reduced integration, hourglass control elements; • 2656 six-node linear triangular prism, hybrid, constant pressure elements; • 13 312 four-node linear tetrahedron elements. The properties of the 2-D braided material modelled are the following: a = 60° Vf = 0.5 Vm = 0.5 V60 = 0.175 V0 = 0.325 Figures 4.11–4.14 show the finite element model.

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4.13 Finite element model of 0° fibre.

4.14 Finite element model of matrix.

Calculation procedure Explicit integration procedures were used with Abaqus Explicit 5.6 code [34]. The calculation time is divided into small steps. Some nodes are subjected to different velocities in order to simulate the different stress states. When failure occurs, the material properties are degraded. Materials Carbon fibre reinforced epoxy composite was used. The properties of the materials are as follows [35]: • Carbon fibre: Young modulus E = 250 GPa Strength X = 2700 MPa Strain to failure er = 1%

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4.15 s1–e1 curve for a carbon fibre (Type II) epoxy resin braided composite (065/6035)Vf=50% obtained by means of a micromechanical finite element model. Table 4.8. Predicted properties

Direction 1 Direction 2 Shear stress in plane 1–2

Ea (GPa)

Eb (GPa)

ea (%)

eb (%)

85.826 29.57 10.044

4.626 4.96 1.84

1 1.33 2.3

4 4.5 7.015

• Epoxy resin: Young modulus E = 3.684 GPa Strength X = 123 MPa Strain to failure er = 4.5% Poisson coefficient n = 0.35 Analytical predicted behaviour The predictive analysis yields the properties given in Table 4.8. Results The results of the finite element calculation for a carbon fibre epoxy resin braided composite (0°65%/60°35%)Vf =50% can be seen in Figs. 4.15–4.17.

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4.16 s2–e2 curve for a carbon fibre (Type II) epoxy resin braided composite (065 /6035)Vf=50% obtained by means of a micromechanical finite element model.

4.17 s6–e6 curve for a carbon fibre (Type II) epoxy resin braided composite (065 /6035)Vf=50% obtained by means of a micromechanical finite element model.

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Low-modulus, high-strength (Type II) carbon fibre reinforced epoxy composite was used, owing to its better ability to absorb deformation energy. The properties of the material used in the calculation are shown below: •

Carbon fibre: Young modulus E = 250 GPa Strength X = 2700 MPa Failure strain er = 1%

4.4

Determination of the stiffness and strength properties of knitted composite materials

The current developments related to weft knitting technology focus on the incorporation of straight fibres to the base fabric in several directions, in order to increase the mechanical properties of the material system. A unit cell of weft knitting is shown in Fig. 4.18.The same cell reinforced with fibres in longitudinal and transverse directions is represented in Fig. 4.19. In this section, the properties of a reinforced weft knitted material will be calculated by means of a finite element micromechanical model [36]. Also, the

4.18 Weft knitting.

4.19 Reinforced weft knitting.

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same weft knitted material without reinforcement will be analysed. The results will be compared with experimental data in order to assess the accuracy of the model used [37].

4.4.1 Model characteristics Interphase elements have been implemented in the finite element model in order to analyse the starting point of the material failure, where fibres are debonded from the matrix [38–49] or microbuckling phenomena under compression load [42,50–54]. The model used exhibits the following characteristics [37]: • Fibre fraction volume: Vf = 11% without reinforcement fibres Vf = 17.09% with longitudinal and transverse fibres • Fibre diameter: 0.238 mm • Interface thickness: 1.5% of fibre diameter • Unit cell dimensions are represented in Fig. 4.20.

4.4.2 Materials The material properties are given in Table 4.9.

4.4.3 Finite element model The finite element model is composed of 16 000 nodes and 17 464 elements: • 12 272 eight-node solid linear elements with reduced integration; • 1824 six-node triangular linear elements; • 3368 four-node linear tetrahedron elements.

4.20 Dimensions of the unit cell.

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Table 4.9. Material properties

E-glass fibre Epoxy resin Fibre–matrix interface

Elastic modulus, E (GPa)

Poisson coefficient, n

Tension strength, X (MPa)

Strain to failure, efailure (%)

73 3.684 3.684

0.24 0.35 0.35

2336 116 45

3.2 6.3 4.5

4.21 Resin.

4.22 Fibre.

The elements are divided into three groups according to the material to be modelled (Figs. 4.21–4.23). This model will be used as an example weftknitted nonreinforced material. In this case, a number of finite elements that modelled the fibre and the interface in the previous case will now pass to model the resin. Initially, each material is considered to behave in an elastic linear manner. The elastic properties progressively degrade as the deformation increases. When the final failure occurs in a given finite element, that element is removed. Contact has been defined between elements modelling fibre and nodes modelling resin, and between elements modelling fibres. Therefore,

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4.23 Fibre–matrix interface.

when an element modelling resin or interface is eliminated, the material is still consistent since the elements modelling the fibres are in contact with the rest of the elements modelling the existing matrix. The following failure modes can be analysed by means of the present model: • fibre failure; • matrix failure; • fibre–matrix interface.

4.4.4 Loading cases and boundary conditions Nine loading cases have been analysed in order to calculate all the elastic and strength properties of the material system. For each case, velocity has been applied to all the nodes that compose a side of a unit cell. The velocity is linearly increased until a final value of 20 mm/min, then remains constant. An explicit dynamic code has been used to perform this analysis (Abaqus Explicit 5.6) [33].

4.4.5 Results The stress–strain curve is obtained for each loading case.The average curves for the reinforced weft knitted materials are shown in Fig. 4.24. Directions X, Y and Z correspond to directions 1, 2 and 3. The tension and compression curves in directions X, Y and Z are shown as shear curves in the XY, XZ and YZ planes.

4.4.6 Experiment–theory correlation Table 4.10 represents the experimental data [37] and the theoretical results obtained by applying the present model for nonreinforced weft-knitted materials. The stiffness and strength properties in tension in X and Y directions are compared, the error being about 1% for the elastic modulus and about 10% for the ultimate tensile strength.

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4.24 Stress–strain curves corresponding to knitted composite systems.

Table 4.10. Theory–experiment correlation Elastic modulus, E (GPa)

X-tension Y-tension

4.5

Theory

Experiment

4.331 5.424

4.37 5.35

Strength, X (MPa)

Error (%)

Theory

Experiment

E

X

40.8 56.93

35.5 62.83

0.89 1.3

13 9.3

Application of macromechanical analysis to the design of a warp knitted fabric sandwich structure for energy absorption applications

A new concept of energy absorber is described in this section. This design can be applied to several air- and ground-transportation means for energy absorption purposes [55–57]. The constitutive materials of the energy absorber, the object of study, are preforms of 3-D warp knitted sandwich structures [58–62]. The macromechanical analysis carried out for the 3-D fabric sandwich structures under given working conditions was essential to design the energy

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4.25 Concept of energy absorber.

absorber device [58,62]. A numerical–experimental correlation is also shown in order to assess the accuracy of the macromechanical analysis.

4.5.1 Definition of the problem The concept studied for the energy absorption problem is represented in Fig. 4.25. In a first step, the wedge, which acts as an initial triggering device, attacks the middle part of the 3-D warp knitted sandwich structures (3-D WKSS) and the piles and the foam that compose the core of the sandwich are totally crushed, absorbing a large amount of energy. The failure mechanism of this initial energy absorption mechanism is shear failure. Once the central part of the 3-D WKSS is destroyed, the skins are pushed against the curved part of the triggering device. By following a bending failure mechanism, the skins are bent and crushed along both sides in a symmetrical way. The geometrical design of the curved part of the triggering device is critical in order to obtain a maximum level of energy absorbed. Several 3-D warp knitted sandwich panels can be assembled to build a grid showing high performance in terms of energy absorption (Fig. 4.26). This grid can be applied to: • the undersides of helicopters to absorb the crash energy in case of falling down from a low altitude; • bus front and lateral parts to absorb the energy generated in the case of a front and roll-over crash, respectively; • car front parts to absorb front crash energy, etc. X and T joints between 3-D warp knitted sandwich panels also absorb a large amount of energy and their design is considered essential to obtain an optimum level of energy.

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4.26 Absorber energy grid composed of 3-D WKSS.

4.5.2 Requirements for the energy absorption problem The behaviour of an energy absorption structure must meet the following requirements in terms of the force–displacement curve: • High energy absorption capability. In other words, the area below the force–displacement curve must be as high as possible. • The maintained load must be constant along the displacement. • The initial load peak must be low. • The maximum displacement must be limited. • The initial elastic behaviour can be obtained by using an appropriate base structure. • The specific energy absorption per weight and volume unit must also be high. An ideal force–displacement curve is represented in Fig. 4.27.

4.5.3 Macromechanical analysis of 3-D warp knitted sandwich panels A macromechanical analysis of 3-D warp knitted sandwich panels has been carried out.This analysis was dynamic, using an explicit code.The skins were modelled by means of shell finite elements and a higher-order shear theory. The foam was simulated by means of solid elements and elasticity theory and, finally, the piles were modelled by means of beam elements (see Table 4.6). The panels were subjected to a compression load and the force– displacement curve was obtained for the complete crash process. The following data were used:

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4.27 Ideal force–displacement curve in terms of energy absorption.

•

•

Materials – 3-D warp knitted material: PARABEAM Ref 89020-1 Fibre: E-glass Matrix: epoxy Core: polyurethane foam 40 kg/m3 – Reinforcing skins: Fibres: carbon fibre and aramide Matrix: epoxy Lay-up: Fabric [+45/-45] C/E (0.25 mm) Fabric [0/90] C/E (0.25 mm) Fabric [+45/-45] A/E (0.25 mm) Dimensions Crash length 75 mm Height 100 mm Sandwich thickness 17.5 mm Core thickness 15 mm 3-D warp knitted skin Thickness 0.75 mm Reinforcing skin thickness 0.5 mm

Two triggering devices were analysed: • A-type. A sharp wedge-type triggering device for provoking shear failures at an early stage in the material core with curved lateral parts for crushing the sandwich skins (failure mode).

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4.28 Series of deformations for the A-type triggering device.

• B-type. A progressive wedge-type triggering device for provoking moderate shear failures in the material core with curved lateral parts for crushing the sandwich skins (failure mode). A series of deformations along the crash process is represented in Figs. 4.28 and 4.29 for A and B-types, respectively. By comparison of deformations from A and B-type triggering devices, it is observed that type B is more efficient than type A in terms of energy absorption. This is due to the fact that the skins follow the linear device profile before deformation, increasing the energy absorption, the shear failure of the core is more progressive, and therefore the initial peak is lower and the maintained load more constant. The theory–experiment correlation analysis for the B-type triggering device is shown in Fig. 4.30. The results of the study are outstanding in terms of energy absorption since the initial peak does not exist owing to the fact that the triggering device has been optimized in terms of wedge geometry (shear mode) and curved parts (bending mode). The implementation of a linear profile above the curved part has been essential to eliminate the initial peak. Quite a constant force is also registered along the displacement, the final absorbed energy being excellent compared with other material systems and typologies. As the wedge penetrates the core with no failure, the value of the force increases up to a point. This phenomenon is represented in Fig. 4.30 by means of a number of increases of the force–displacement curve. These

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4.29 Series of deformations for the B-type triggering device.

4.30 Comparison between theory (left) and experimentation (right) obtained from the B-type triggering device.

increases generate a number of peaks, which are quite constant along the displacement. Once the shear failure of the core (piles) occurs, a decrease is observed in the force–displacement curve. A number of shear failures is registered in the curve mentioned. The minima values are also quite constant along the displacement. The progressive core failures are represented by a number of peaks along the displacement. However, these maxima values are quite constant. Finally, the theory–experiment correlation is excellent since the values of the forces are very similar from both studies and the number of increases and decreases registered in the experimental analysis due to the

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progressive failure of the piles is very accurately simulated by means of the macromechanical study.

4.5.4 Dynamic macromechanical analysis of X and T joints A dynamic analysis of the X and T joints has been carried out in order to evaluate the energy absorbed for the whole structure [58,62]. The following data were used: •

•

Materials 3-D warp knitted material: PARABEAM Ref 89020-1 Fibre: E-glass Matrix: epoxy Core: polyurethane foam 40 kg/m3 Dimensions Crash length 75 mm Height 50 mm Sandwich thickness 16.4 mm Core thickness 15 mm 3-D warp knitted skin Thickness 0.7 mm

The dimensions are represented in Fig. 4.31. The progressive deformation of X and T joints is represented in Figs. 4.32 and 4.33. These graphics are the result of a macromechanical study applied to both configurations. The load–displacement curves obtained in the macromechanical analysis are represented in Figs. 4.34 and 4.35.

4.5.5 Conclusions The main conclusion of this study is that 3-D warp knitted sandwich structures are more efficient than conventional sandwich structures in terms of bending, peeling and crash behaviour, owing to the fact that the connection between the two skins by means of the piles makes this configuration stronger for the cases mentioned above.

4.6

Application of macromechanical analysis to the design of an energy absorber type 3P bending

A material model obtained from Section 4.3 will be used in this section to study the 2-D triaxial braiding technology. A 3P bending dynamic analysis will be performed, considering the material behaviour from the initial step

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4.31 Dimensions of X and T joints.

4.32 Progressive deformation of the X joint.

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4.33 Progressive deformation of the T joint.

4.34 Force–displacement curve of a T joint.

which is linear and elastic until the final catastrophic failure [63–65]. The aim of this study is to obtain the energy absorbed by a square cross-section beam, subjected to a 3P bending case. Several 2-D braiding and steel configurations will be compared, in terms of energy absorbed, maximum reaction force and weight. This analysis can be considered as a starting point

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4.35 Force–displacement curve of an X joint.

4.36 Finite element model.

for the materials selection in the front crash design of an automotive vehicle [66,67].

4.6.1 Finite element model and boundary conditions The finite element model used in this study, composed of shell elements, is shown in Fig. 4.36. This model consists of a quarter of the beam, owing to the double symmetry of the problem. The problem, the object of the study, is shown in Fig. 4.37, in which the beam is supported on two rigid cylinders.The load is applied by means of a third rigid cylinder at the middle of the span. Velocity has been imposed in this cylinder, as can be seen in Fig. 4.36. This problem will be analysed by means of an explicit integration procedure. Contact between the surfaces of the cylinders and the beam has been defined. The material model for 2-D braiding has been introduced by means

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4.37 Three-point bending.

of a Fortran subroutine into the commercial finite element code (Abaqus explicit 5.6) [34].

4.6.2 Configurations analysed Four different material configurations have been analysed: 1 Steel configuration used as a reference. 2 Carbon fibre–epoxy resin braided configuration (0°50%/45°50%)Vf =50%. A braided composite material has been considered, constituted by 50% of the fibre oriented along the 1 direction and the other 50% oriented at 45°. Total fibre volume is 50%. 3 Glass fibre–epoxy resin braided composite (0°50%/45°50%)Vf =50%. In this case, the braided composite material considered is composed of 50% of the fibre oriented along the 1 direction and the other 50% oriented at 45°. Total fibre volume is 50%. 4 Hybrid carbon–aramide fibre–epoxy resin braided composite [65] (0°50%/45°50%)Vf =50%. Finally, the braided composite material is constituted by 50% of the total fibre in carbon oriented along the 1 direction and the other fibre in aramide oriented with 45°. Total fibre volume is 50%. The thickness distribution along the beam for every configuration is shown in Fig. 4.38 and Table 4.11.

4.6.3 Materials The stress–strain curves in the plane of the material are detailed for each material system in this section. These curves have been obtained by applying the analytical formulation described in Section 4.3. The properties of the constitutive materials [35] are the following: •

Epoxy resin: E = 3.6 GPa n = 0.38 emax = 4.5%

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Table 4.11. Thickness distribution for every configuration Zone

Thickness Thickness Thickness Thickness (mm) for (mm) for (mm) for (mm) for configuration 4, configuration 1, configuration 2, configuration 3, hybrid carbon– steel carbon fibre glass fibre aramide

A B C D

2 2 2 2

7 6 5 3

5 4 3 3

6 5 4 3

4.38 Zones of the beam.

•

Carbon fibre: PAN Type II (high strength) EII = 250 GPa eII max = 1.08% sII max = 2700 MPa

•

Glass fibre: Type E E = 76 GPa emax = 3.17% smax = 2400 MPa

•

Aramide fibre: E = 133 GPa emax = 2.1% smax = 2800 MPa

The behaviour of the configurations is shown in Figs. 4.39–4.42.

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4.39 Behaviour of the steel [68].

4.40 In-plane stress–strain relationships of carbon configuration.

4.41 In-plane stress–strain relationships of glass configuration.

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4.42 In-plane stress–strain relationships of hybrid carbon–aramide configuration.

4.43 Load–displacement curves for the configurations calculated.

4.6.4 Results The curves of the energy absorption versus displacement of the impact cylinder, and reaction force versus displacement of the cylinder are shown in Figs. 4.43 and 4.44. Another important result is the weight of each configuration, and the weight saving obtained (Fig. 4.45). The progressive deformation of the square cross-section beam is represented in Figs. 4.46–4.51.

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4.44 Deformation energy–displacement curves for the configurations calculated.

4.45 Weight and weight saving on every configuration.

4.6.5 Conclusions The energy absorption obtained by using 2-D braided composites subjected to bending is very similar to the energy absorption obtained with steel. The weight saving reported for the braided material is about 60%. This issue is critical for automotive vehicle design, where weight saving is becoming increasingly important, owing to contaminant emission reduction and fuel

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4.46 Deformation for 0 mm displacement at the middle of the span.

4.47 Deformation for 24.72 mm displacement at the middle of the span.

4.48 Deformation for 38.63 mm displacement at the middle of the span.

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4.49 Deformation for 52.53 mm displacement at the middle of the span.

4.50 Deformation for 63.65 mm displacement at the middle of the span.

4.51 Deformation for 65 mm displacement at the middle of the span.

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consumption saving, and the maximum energy absorbed occurs when the transverse elements break in a bending mode due to a crash impact [69–71].

4.7

Conclusions

Laminated composite materials present some deficiencies, owing to the low out-of-plane properties they exhibit and high manufacturing costs. 3-D textile-reinforced composites are characterized by having their fibres in the thickness direction and being made by means of preforms which can be made automatically. The result is a new generation of composite materials with superior out-of-plane properties, damage tolerance and reasonable manufacturing costs. Shell and plate finite elements have been used extensively to perform macromechanical analyses of laminated composite materials. Laminated plate and first-order shear theories are being used for thin and thick laminates, respectively. In order to carry out macromechanical analyses of 3-D textile reinforced composite structures, other theories must also be applied, such as higherorder shear theories, when the interlaminar normal stress component is negligible, or the elasticity theory, which takes into account the whole stress tensor. Generally speaking, to perform a macromechanical analysis of a given 3-D textile reinforced composite structure, the following information must be known: • Material models, governing its behaviour in terms of stiffness and strength at a macroscopic level. • Geometry of the structure. • Boundary conditions. • Loading conditions. • Design requirements. • Stiffness and strength properties needed, according to the material models defined previously. All these input data are usually known, except for the last, associated with the stiffness and strength properties. Therefore, the assessment or estimation of these parameters becomes critical for carrying out a macromechanical study of a certain 3-D textile reinforced composite structure. As has been explained in this chapter, mechanical testing is the best way to evaluate the stiffness and strength properties. However, in some cases, the experimental procedure is not feasible for the determination of certain properties (compression strength, out-of-plane stiffness and strength properties, etc.). In these cases, there are a number of procedures to estimate the properties of the object of study:

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• Nowadays, computers are powerful and fast enough to simulate the behaviour of a given 3-D textile reinforced composite material at a fibre level. The estimation of stiffness and strength properties is therefore possible with reasonable precision, by means of the same technology used for macromechanical studies – the finite element method. The interphase can be accurately simulated and the moment of initiation of debonding between fibre and matrix can also be predicted. Non-linear behaviour and contact problems can also be reproduced in order to predict the ultimate strength of the material system. This will let us build the complete curve of the material behaviour in the three directions. • Several simple analytical formulations exist, which are considered to be approximate, and whose application is usually limited to some inplane stiffness and strength properties. In any case, the accuracy is very variable. • Finally, there are also micromechanical studies where more complex analytical formulations are presented. The stiffness and strength properties needed to carry out a certain macromechanical study depend on the following factors: • Type of textile technology used. • Type of macromechanical analysis applied (linear/non-linear, static/ dynamic/fatigue/hygrothermal . . .). • Type of theory implemented. In this chapter, the factors above outlined have been extensively discussed. Two application examples have also been presented. The finite element method has been applied to two dynamic analyses of 3-D textile reinforced composite structures, taking into account contact problems. The material failure has also been analysed until the final failure occurred. A theory–experiment correlation study has been carried out, in order to analyse the accuracy of the numerical modelling of a 3-D warp knitted sandwich structure subjected to a crash loading case (dynamic loading, nonlinear material and geometry and contact problem).The results of this study have been excellent since both force-displacement curves present very close values.

4.8

References

1. Tsai, S.W., Theory of Composites Design,Think Composites, Dayton, Ohio, 1992. 2. Tsai, S.W., Miravete, A. and Diseño, Y., Análisis De Materiales Compuestos, Editorial Reverté, SA, Barcelona, 1988. 3. Philips, L.N., Design with Advanced Composite Materials, Springer-Verlag, 1989. 4. Vandeurzen, Ph., Ivens, J. and Verpoest, I., ‘A three-dimensional micro-

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25. Gommers, B. and Verpoest, I., ‘Tensile behavior of knitted fabric reinforced composites’, in Proceedings of ICCM-10, Whistler, B.C., Canada, August 1995. 26. Huysmans, G., Gommers, B. and Verpoest, I., ‘Mechanical properties (2D) warp knitted fabric composites, An experimental and numerical investigation’, paper presented at 17th Int. Sampe Europe Conference, Basel, Switzerland, May 1996. 27. Frederiksen, P.S., ‘Experimental procedure and results for the identification of elastic constants of thick orthotropic plates’, J. Composite Mater., 31(4), 360–382, 1997. 28. Moll, K. and Wulfhorst, B., ‘Determination of stitching as a new method to reinforce composites in the third dimension’, in The Design of Textile Reinforcements with Specific Engineering Properties, Hill, B.J., McIlhagger, R. and Abraham, D. eds, Tex Comp 3, New Textiles for Composites, Aachen, 9–11 December, 1996. 29. Masters, J.E., Foye, R.L., Pastore, C.M. and Gowayed, Y.A., ‘Mechanical properties of triaxial braided composites: experimental and analytical results’, J. Composites Technol. Res., 15(2), 112–122, 1993. 30. Tsai, J.S., Li, S.J. and James Lee, L., ‘Microstructural analysis of composite tubes made from braided preform and resin transfer molding’, J. Composite Mater., 32(19), 829–850, 1998. 31. Marrey, R.V. and Sankar, B.V., ‘A micromechanical model for textile composite plates’, J. Composite Mater., 31(12), 1187–1213, 1997. 32. Witcomb, J. and Srirengan, K., ‘Effect of various approximations on predicted progressive failure in plain weave composites’, Composite Structures, 34, 13–20, 1996. 33. Reddy, J.N., An Introduction to the Finite Element Method, McGraw-Hill, 1994. 34. Hibbitt, Karlsson and Sorensen, Abaqus Explicit Version 5.6, Abaqus Standard Version 5.6, User Manual, Inc. Pawtucket, RI. 35. Hull, D., An Introduction to Composite Materials, Cambridge University Press, Cambridge, 1981. 36. Zhu, H., Sankar, B.V. and Marrey, R.V., ‘Evaluation of failure criteria for fiber composites using finite element micromechanics’, J. Composite Mater., 32(8), 766–782, 1998. 37. Ramakrishna, S., ‘Characterization and modeling of the tensile properties of plain weft-knit fabric reinforced composites’, Composites Sci. Technol., 57, 1–22, 1997. 38. Rydin, R.W., Varelidis, P.C., Papaspyrides, C.D. and Karbari, V.M., ‘Glass fabric vinyl–ester composites: tailoring the fiber bundle/matrix interphase with nylon coatings to modify energy absorption behavior’, J. Composite Mater., 31(2), 182–209, 1997. 39. Frantziskonis, G.N., Karpur, P., Matikas, T.E., Krishnamurthy, S. and Jero, P.D., ‘Fiber–matrix interface – information from experiments via simulation’, Composite Structures, 29(3), 231–248, 1993. 40. Gao, Z. and Reifsnider, K.L., ‘Tensile failure of composites: influence of interface and matrix yielding’, J. Composites Technol. Res., 14(4), 201–210, 1992. 41. Drzal, L.T. and Madhukar, M., ‘Fibre–matrix adhesion and its relationship to composite mechanical properties’, J. Mater. Sci., 569–610, 1993. 42. De Moura, M.F.S.F., Gonçalves, J.P.M., Marques, A.T. and De Castro, P.M.S.T.,

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60. Miravete, A., Castejón, L. and Alba, J.J., ‘Nuevas Tipologías de Fibra de Vidrio en Transportes’, Ibérica Actualidad Tecnológica, 390, 492–494, 1996. 61. Verpoest, I., Wevers, M., De Meester, P. and Declerrg, P., ‘2.5D and 3D-fabrics for delamination resistant composite laminates and sandwich structures’, Sampe J., 25(3), 51–56, 1989. 62. Clemente, R., Miravete, A., Larrodé, E. and Castejón, L., 3-D Composite Sandwich Structures Applied to Car Manufacturing, SAE Technical Papers Series, Detroit, MI, 1998. 63. Nakai, A., Fujita, A., Yokohama, A. and Hamada, H., ‘Design methodology for a braided cylinder’, Composite Structures, 32, 501–509, 1995. 64. Chiu, C.H., Lu, C.K. and Wu, C.M.,‘Crushing characteristics of 3-D braided composite square tubes’, J. Composite Mater., 31(22), 2309–2327, 1997. 65. Karbhari, V.M., Falzon, P.J. and Herzerberg, I., ‘Energy absorption characteristics of hybrid braided composite tubes’, J. Composite Mater., 31(12), 1165–1185, 1997. 66. Cuartero, J., Larrodé, E., Castejón, L. and Clemente, R., New Three Dimensional Composite Preforms and its Application on Automotion, SAE Technical Papers Series, Detroit, MI, 1998. 67. Castejón, L., Cuartero, J., Clemente, R. and Larrodé, E., Energy Absorption Capability of Composite Materials Applied to Automotive Crash Absorbers Design, SAE Technical Papers Series, Detroit, MI, 1998. 68. Hill, R., The Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1950. 69. Thornton, P.H. and Jeryan, R.A., ‘Crash energy management in composite automotive structures’, Int. J. Impact Eng., 7(2), 167–180, 1988. 70. Farley, G.L. and Jones, R.M., ‘Crushing characteristics of continuous fiberreinforced composite tubes’, J. Composite Mater., 26(1), 1992. 71. Thornton, P.H., ‘The crush behavior of glass fiber reinforced plastic sections’, Composites Sci. Technol., 27, 199–224, 1986.

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5 Manufacture and design of composite grids S.W. TSAI, K.S. LIU AND P.M. MANNE

5.1

Introduction

Composite materials technology has emerged as the darling of many industries over the past 30 years. This class of materials is light, corrosion and fatigue resistant and can be manufactured in a variety of methods. Most successes can be found in sporting goods and satellites where graphite composites are the dominant materials. Here performance is the primary goal. Other notable achievements include components of aircraft, and many industrial applications where corrosion is critical. Composite materials have the potential to increase their market size significantly. As artificial fibers have all but replaced natural ones, we see composites as the structural materials of the future because they have unlimited supply and require less energy to process than metallic materials. There are many inhibitors to the growth of composites. They come from technological, economical and government regulatory sources. Maturing of any technology takes time, particularly if the technology involves public safety; however, innovation and favorable government regulation can hasten this process. Composite grids form the theme for this chapter. Grids are fundamentally different from stiffened and sandwich constructions in that the load transfer mechanisms are different. Grids can be made by the widely available filament winding and pultrusion. We believe that both high performance and low cost can be achieved. Current manufacturing processes of composite materials and structures are based on weaving, braiding, pultrusion and/or lamination. They require expensive facilities, and costly manufacturing equipment and processes. As a result, processing costs are many times the material cost. We intend to show that the cost of manufacturing composite grids can be reduced to the level of materials cost. Such composite structures can then compete against most traditional materials. Grids are like the skeleton of a human body or the frame of old airplanes made of wood and cloth cover. The grid is the primary load-carrying 151 © 1999,Woodhead Publishing Ltd

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member. Skins or covers are there for another function. Optimally grids are formed by a network of ribs made of unidirectional composites. These ribs are many times stronger and lighter than metallic materials. The key is to exploit the unidirectional properties.While concrete and metallic grids have been made, their performance is limited because the ribs are isotropic. Only when ribs are unidirectional can the true potential of grids be realized. We will show how to capitalize on this principle and combine it with low-cost manufacturing. Grid structures are not new; they have been used in civil engineering for many years. The aeronautical industry used metallic grids as early as in World War II, for example in the British Vickers Wellington bomber. The grid was metallic and offered exceptional battle damage tolerance. This extra assurance made it the favorite among the flight crew. Nowadays, jet engine covers and some hulls of the International Space Station feature integral grids machined from aluminum plates. Based on our understanding, these applications do not constitute a very effective use of grids. On the other hand, Airbus A330 and A340 have composite grid reinforced skins in their horizontal and vertical tails. Presumably they are cost effective. They are, however, hand-made. Our interest lies in developing new automatable manufacturing processes. It is hoped that with these processes, the outstanding performance of composite grids can be achieved at an affordable cost.

5.2

Grid description

We wish to describe the geometric and material characteristics of grids and show why composite grids are unique.

5.2.1 Rib orientation Since grids have directionally dependent properties, we chose to adopt terms analogous to those commonly used for laminated composite materials. In Fig. 5.1, grids are described based on the orientations of their ribs: square, angle and p/3 isogrids, respectively. In this figure all ribs are assumed to be in the same plane and to have the same height. But that restriction is not always followed: for example, ribs may run in different planes, like plies in a laminate. All grids shown here have identical rib intersections or joints. In particular the p/3 grid is isotropic and is often called an isogrid.

5.2.2 Rib construction There are at least two ways of making grids. The wrong way is to start with a slab of material and produce a grid by machining. As illustrated on the

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5.1 Designation of grids by rib orientation, analogous to laminated composites.

5.2 Two ways of making grids. Left: the wrong way, by machining a quasi-isotropic laminate. Right: the correct way, by forming unidirectional ribs.

left in Fig. 5.2, a quasi-isotropic laminate is taken as starting material and machined into an isogrid. We call the resulting grid an ‘iso’ isogrid, indicating that the starting material is isotropic. This class of grids is very costly and a very poor utilization of the material. The rib has the same stiffness as the starting material. The right way is to use directional materials such as composites. Instead of machining, unidirectional fibers are rearranged or regrouped to form unidirectional ribs as shown on the right of Fig. 5.2. We call this class of grids ‘uni’ isogrids. Here the superior stiffness of unidirectional composites is fully utilized. We will show later that the ‘uni’ isogrids are nearly three times stiffer than the ‘iso’ isogrids made from the same composite materials. This is indeed the right way. For the same reason, metallic grids are not effective. In fact, there is a close relation between composite laminates and composite grids. Grids can be viewed simply as a special case of laminates, and this will be used in deriving the stiffness and strength of grids.

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5.3 Rib volume fractions of sparse and dense grids.

5.2.3 Rib geometric parameters The principal geometric parameters of grids are the length L, width b and height h of the ribs. A useful dimensionless measure is the rib area fraction f within a unit cell. This fraction is related to the length and width of the ribs and their orientations in the grid. Two values of f are shown in Fig. 5.3: for a sparse grid on the left and for a dense grid on the right. A dense grid can also be called a waffle plate, characterized by the fact that its ribs would not buckle. The value of f is the same as the rib volume fraction as long as the grid pattern remains constant along the grid height. The rib fraction is analogous to the fiber volume fraction of a composite material. But fiber fraction in composite plies is not a common design variable because such a fraction is often predetermined by material suppliers. For grids, however, rib fraction is an important design variable and must be deliberately selected for a given design. We recommend f-values in the range shown in Fig. 5.3. Rib height h is also a critical design parameter, in determining flexural rigidity in particular. A low height-to-width ratio or h/b is a shallow grid; a high ratio, a tall grid. We assume in the present work that this ratio is higher than 1. Euler buckling of ribs occurs only in the lateral direction. It is then governed by the length-to-width ratio, L/b. Such a failure mode must be compared with failure by compressive strength. Whichever is lower will be the controlling failure mode. The relation defining the area fraction f of a grid is a function of the grid configuration. In Fig. 5.4, we show the definition of f for iso- and square grids. A visual presentation of an isogrid compared with square grids is featured. All grids have the same rib width. The smaller square grid on the left has the same area fraction f, whereas the larger square grid on the right has

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5.4 Definition of area fraction f of iso- and square grids. Slenderness ratio L /b is related to Euler buckling of ribs.

the same slenderness ratio L/b as the actual isogrid. For the smaller square grid, the length L is reduced by 3 ; for the larger square grid, the area fraction f is reduced by 3 . While there is a one-to-one relation between fraction f and L/b, each serves its own purpose in the design of composite grids. Area fraction f can be treated as a material property that governs both in-plane and flexural stiffnesses in a consistent manner. Slenderness ratio, L/b, is useful in its direct relation to Euler buckling of the ribs. We prefer the use of area fraction f because it reflects the weight and amount of material used in a grid. Another geometric parameter of grids is their height or height-to-width ratio, h/b. Grids have characteristics similar to those of solid and sandwich panels. The ribs of a grid should be as tall as possible, i.e. having a high height-to-width ratio. Like plates, flexural rigidity increases with the cube of the height. Short or shallow ribs are not effective. For sandwich panels, flexural rigidity depends on both the height of the core and the laminated face sheets. If a grid has one or two face sheets, its flexural rigidity is like that of a sandwich panel. The rigidity factors are more numerous than for a grid without facing.

5.3

Manufacturing processes

Composite grids have been explored in the former Soviet republic, South Africa, Germany as well as in the USA for over 20 years. In the USA, James Koury of the USAF Phillips Laboratory (now retired), Larry Rehfield of Georgia Institute of Technology (now with the University of California,

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Davis) and McDonnell Douglas Astronautics Company [1] have been pioneering the use of aluminum and later composite grids principally for the fairing and the interstage cone of missiles. W. Brandt Goldsworthy of Rolling Hills, California, pioneered not only pultruded but also filament wound grids. He first proposed this for the Beechcraft Star Ship in the 1970s. Recently, Burt Rutan of Scaled Composites in Mojave, California, built the fuselage of a corporate jet out of composite grids. The USAF continues to explore composite grids with new applications. The McDonnell Douglas Handbook [1] has been updated with the use of composite materials by Chen and Tsai [2] and by Huybrechts [3]. The modeling used in this work draws heavily from these earlier publications. The software developed by these authors is instrumental in the analysis and figures used throughout the current effort. It has been recognized by many people that filament winding would be an optimal method for manufacturing grids if the composite tows could be guided by some soft tooling. Grids are assembled by carving out slots or grooves in a rubber tool.

5.3.1 Assembly methods We believe that new approaches can improve performance and, at the same time, lower cost. A variation in the grid assembly is the configuration of the rib intersection or joint. Three possible joints are shown in Fig. 5.5.

5.5 Three types of joints in a grid. The slotted joint is not recommended. Stacked and TRIG joints can be produced more easily and have better properties.

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Slotted joint grids The traditional slotted joint grids are shown on the left in Fig. 5.5 and are most frequently used in carpentry. Slots are cut into ribs and assembled. The disadvantages of this design include: • • • •

cost of machining slots, difficult assembly of many ribs having multiple slots, low rib strength introduced by machined slots and notches, low grid stiffness and strength from imperfect fit at slotted joints.

We understand that Composite Optics Incorporated of San Diego, California, used [p/3] laminates as the rib in order to increase the rib strength. The use of laminates for ribs, however, degrades the grid stiffness by a factor of 3 from unidirectional ribs. It is therefore our opinion that slotted joint grids should remain as a popular technique for carpenters and cabinet makers. Stacked joint grids We believe that the stacked joint grids shown in the middle in Fig. 5.5 can be as effective as slotted joint grids and can be simpler to manufacture. An example of stacked joint grid is the bird cage, which has been in existence for centuries. To build a stacked grid, longitudinal and hoop or cross members are stacked. Members run on separate planes, similarly to the plies in laminate. There are at least two variations. The longitudinal members (longis) are pultruded, filament wound or made in a female mold by blow molding. The cross members (circs) can be skins applied by filament winding to form a circular or conical grid or shell. The longitudinal tubes may be fan-shaped, for example, and serve the same purpose as a sandwich core between the inner and outer filament wound skins. Although winding can also have a helical pattern if an increase in shear rigidity is desired, such a process increases the cost of manufacturing over pure hoop winding. The longitudinal and cross members may be fully or partially populated, i.e. the longis do not have to be placed adjacent to one another. The hoop wound plies can be continuous or discontinuous like bands or rings. An example of a ring reinforced cylinder is shown in Fig. 5.6. Other examples of a stacked grid include cross-members made by molding or vacuum infiltration. A multi-hole bar or ring through which longitudinal rods or tubes are threaded and bonded forms a bird cage-like structure. There are many possible configurations for different applications. Stacked grids, however, are currently limited to orthogrids. Isogrids, for example, are difficult to make because ribs in three levels must be stacked and joined.

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5.6 A stacked grid with round or fan-shaped longitudinal members sandwiched between inner and outer windings.

5.7 A filament wound cylinder made by the TRIG process. Left: tooling from contoured tubes. Right: wound interlacing fills the V-shaped grooves for grid strength.

Interlaced joint grids For the interlaced grid, the thin wall tubes, again, are the starting components. The filament wound tubes with all-hoop plies provide maximum stiffness for the final grid. The tubes are sliced to a contour that fits a mandrel. They are then positioned as tooling on the mandrel. This is shown on the left in Fig. 5.7. The V-shaped gaps between tooling are filled with interlacing tows, as shown on the right. The interlacing tows carry sufficient resin to bond the tooling and interlacing together to form a solid, continuous rib. The interlacing gives superior strength to the grid. The tooling becomes part of the finished grid and provides high stiffness to the grid. Although tooling contributes to the grid stiffness, it terminates at the rib joints. The discontinuity is small relative to the length of the rib. The effect on the grid stiffness is small.

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Significant cost savings can be obtained when the interlacing is filament wound with one helical winding angle. Having a V-shaped groove along a helical pattern would allow high-speed winding. That would further reduce the cost of assembly.

5.3.2 Features of grids We have discovered theoretically that grids are more efficient if the ribs are tall and thin. This process, identified as the tooling reinforced interlaced grid (TRIG), yields a high geometric definition for the ribs and also high grid stiffness. Several current interlacing and fiber placement processes use rubber or foam as guide and tooling. These processes do not produce the high definition and stiffness that the TRIG process does. The advantages of composite grids are derived from the availability of mass-producible rods and tubes, and from the final assembly by filament winding. This winding process is one of the most advanced and widely available processes. Curing is done at room or elevated temperature. Debulking, bagging and autoclaving are not required. With this process the cost of making a grid can be close to the cost of materials, not many times the cost. Assembly by adhesive bonding in the case of some stacked grids can also be cost effective. Although the stiffness of composite grids is nearly equal to that of laminates, the strength is many times higher. This is because unidirectional ribs do not fail by microcracking or delamination, but by loss of strength or buckling. Where foamed tubes are used, the grids will have superior damping and acoustic properties that cannot be matched by metallic structures. Composite grids are also more resilient. There is no permanent deformation upon unloading. Thus composite grids do not dent or crumple like sheet metals. While the advantages of composite grids are high strength and low cost, there are also disadvantages. As of now, grids can only be made in simple geometric shapes. Such a limitation is often imposed by filament winding. Circular and conical shells are the easiest. Spherical shells can be done using the TRIG process. But doubly curved or concave surfaces are not suitable for grids. Bolting is not recommended without local reinforcement. Finally we recommend that grids be designed to carry all the loads. Skins are present for functional reasons only: in sandwich panels the skins carry the load.

5.4

Mechanical properties of grids

We wish to describe the stiffness and strength of grids and compare them with comparable properties of laminates.

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5.4.1 Stiffness of quasi-isotropic laminates It is useful to compare the stiffness of laminates and equivalent grids. The simplest comparison is that between isotropic laminates and isogrids. Laminates become quasi-isotropic with equally spaced ply orientations of [p/3], [p/4], [p/5] and so on. Similarly isotropy of the grid is assured when the three ribs are spaced 60° apart. There are closed-form solutions of the plane stress stiffness components [4]. The quasi-isotropic invariants are linear combinations or the ply stiffness components shown below: U1 =

3 1 1 (Q xx + Qyy ) + Q xy + Q SS 8 4 2

U4 =

1 3 1 (Q xx + Qyy ) + Q xy - Q SS 8 4 2

U5 =

3 1 1 (Q xx + Qyy ) + Q xy + Q SS 8 4 2

[5.1]

The quasi-isotropic Young modulus, Poisson ratio and shear modulus of the laminates are functions of the invariants: E[ iso ] =

D [ iso ] U 4 ,n = , G[ iso ] = U U1 U1

[5.2]

where D = U12 - U42. On the other hand, when the degree of anisotropy of a composite ply increases to the upper limit, the only dominant stiffness component is the longitudinal Young modulus Ex. The matrix-related components become vanishingly small. Then the invariants above approach: U1 =

3 1 1 E x , U 4 = E x , U5 = E x 8 8 8

[5.3]

The resulting engineering constants of this limiting quasi-isotropic laminate are: n[ iso ] =

1 1 1 2 1 , G[ iso ] = E x , D = E x , E[ iso ] = E x 3 8 8 3

[5.4]

The mathematical results in the last equation may be explained physically by viewing a laminate having three independent plies of equal thickness. The effective stiffness is equal to –13 of the unidirectional stiffness because each ply occupies –13 of the total laminate thickness. Having the same stiffness in 60° intervals, the laminate becomes isotropic. This can be shown by averaging the transformed stiffness components.

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Using the same explanation, the stiffness of a [0/90] laminate along the 1- and 2-axes is half the longitudinal stiffness. This square symmetric network is square symmetric, but not isotropic.

5.4.2 Stiffness of isogrids A composite isogrid is portrayed in Fig. 5.2 as a regrouped laminate where the matrix stiffness is approaching zero. The same –13 factor in Equation 5.4 can be applied for the contribution of rib stiffness to the isogrid stiffness. This global stiffness, however, must be weighted by the area fraction f. This factor is proportional to the ratio of width b and spacing L of each rib. The Poisson ratio can also be shown to have the same value of –13 as a laminate without matrix. A straightforward but rigorous derivation of the Poisson ratio of an isogrid as a truss will lead to this special value. The following global Young modulus and Poisson ratio of an isogrid are easy to use and to remember: E [ isogrid ] =

f 2 1 Ex = (L / b)Ex ; n[ isogrid ] = 3 3 3

[5.5]

where b = rib width and L = rib length as per the relations in Fig. 5.4. For square grids, the relation between the global stiffness and area fraction f or slenderness ratio L/b will be different (Section 4.3). The relations here apply to interlaced isogrids where all the ribs are in the same plane. For stacked joint grids, ribs of different orientations run in different planes and yield an effective stiffness lower than that of interlaced grids having the same overall rig geometry. The global thickness of the grid is the sum of the rib thicknesses. The effective stiffness of such grids would be lower than that of interlaced grids having the same rib geometry. If we use the in-plane stiffness matrix [A] where the unit is N/m there is no difference between interlaced and stacked joint grids. This is an ‘absolute’ stiffness rather than a normalized one, the unit of which is Pa (N/m2). The grid stiffness depends on the rib stiffness and geometry. It can be shown that the detail of the joint at rib intersections, which can be either pinned or fixed, does not affect the grid stiffness if the slenderness ratio is high. This is the case when the area fraction is small. This will not be the case of a grid that looks like a waffle plate. When ply anisotropy in a composite is moderate, like that of Eglass/epoxy composite, the matrix-related components are no longer negligible. The quasi-isotropic laminate stiffness will obey the relations in Equation 5.2. A comparison of laminate and grid stiffness for different materials is listed in Table 5.1. For comparison among different grid materials, the area fraction correction is the same. Its effect is included in this

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Table 5.1. Comparison of stiffness of isotropic laminates and isogrids for three materials on absolute and specific bases

A B C D E F G

Properties

Units

Source

T300/5208

Glass/epoxy

Aluminum

EX EY nx ES r[iso] r [isogd] E [iso] E [isogd] E [iso]/r[iso] E [isogd]/r[isogd] Efficiency[isogd]/[iso]

GPa GPa GPa GPa

Ply data [4] Ply data [4] Ply data [4] Ply data [4] Ply data [4] Fr[iso] Equation 5.2 fE X /3 C/A D/B F/E

181.00 10.30 0.28 7.17 1.60 1.60 f 69.70 60.30 f 43.56 37.68 0.87

38.60 8.27 0.26 4.14 1.80 1.80 f 19.00 12.90 f 10.55 7.17 0.68

70.00 70.00 0.30 26.90 2.60 2.60 f 70.00 23.33 f 26.92 8.97 0.33

GPa GPa GPa GPa

table; i.e. for a given fraction f, the global stiffness is this factor multiplied by the Young modulus shown in the table. Relative efficiencies between the grid and laminate are shown. Highly anisotropic material such as carbon fiber reinforced plastic (CFRP) is most efficient; the isotropic material is the least efficient. Thus comparison of the stiffness ratio between grids and laminates is a measure of the efficiency of grids. The most efficient rib stiffness is the longitudinal Young modulus Ex. Aluminum and other isotropic materials cannot compete in terms of efficiency. When the mass of the materials is factored in, as in the specific stiffness, composite grids can even be more effective than metallic ones. This is shown in Fig. 5.8. Note that glass grid is nearly as efficient as aluminum grid. We can also claim that if metals must be used, keep them in plate form and add composite grid as a hybrid construction. That would be preferable to adding a metallic grid to the construction. The manufacturing process for hybrid grid/plate combination must still be developed.

5.4.3 Stiffness of square grids and cross-ply laminates Stacked joint grids are most easily made with two orthogonal members. They are orthogrids. The stiffness comparison of orthogrids with cross-ply laminates is analogous to that of isogrids with quasi-isotropic laminates. We will compare the simplest case of an orthogrid and a cross-ply laminate by making them square constructions, i.e. the longitudinal and cross-members have the same stiffness contribution to the grid; the [0] and [90] plies of the cross-ply laminates are equal. Like Equation 5.5, we have the following relation for square grids:

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5.8 Specific stiffness of quasi-isotropic laminates and isogrids for CFRP, glass fiber reinforced plastic (GFRP) and aluminum.

Table 5.2. Comparison of stiffness of cross-ply laminates and square grids for three materials on absolute and specific bases Properties

Units

A B C

r[iso] r[isogd] E1[0/90]

D E F G

E1[sqgd] E1[0/90]/r[0/90] E1[sqgd]/r[sqgd] Efficiency[sqgd]/[0/90]

GPa

[ sqgrid ]

E1

= E2

GPa GPa GPa

[ sqgrid ]

=

Source

T300/5208

Glass/epoxy

Aluminum

Ply data [4] Fr[iso] Laminated plate theory [4] fE X /2 C/A D/B F/E

1.60 1.60 f 96.00

1.80 1.80 f 23.60

2.60 2.60 f 70.00

90.50 f 60.00 56.56 0.94

19.30 f 13.11 10.72 0.82

35.00 f 26.92 13.46 0.50

f b E x = E x ; n[ sqgrid ] = 0 L 2

[5.6]

The zero Poisson ratio is based on the two orthogonal ribs which are completely independent. Since there is no coupling, Poisson’s ratio vanishes.The same comparisons as those made in Table 5.1 are given in Table 5.2 for square grids versus cross-ply laminates for the same materials. Square grids and cross-ply laminates are not isotropic even though the properties in two orthogonal directions are equal. Relative efficiencies between the grid and laminate are shown. Again, the more anisotropic material like CFRP becomes more efficient; the isotropic grid is the least efficient. The

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Table 5.3. Comparison of stiffness of [±45] laminates and angle grids for three materials on absolute and specific bases Properties A B C D E F G

Units

r[iso] r[isogd] E6[±45] E6[angd] E6[±45]/r[±45] E6[angd]/r[angd] Efficiency[angd]/[±45]

GPa GPa GPa GPa

Source

T300/5208

Glass/epoxy

Aluminum

Ply data [4] Fr[iso] LPT [4] fE X /4 C/A D/B F/E

1.60 1.60 f 46.59 45.25 f 29.12 28.28 0.97

1.80 1.80 f 10.80 9.65 f 6.00 5.36 0.89

2.60 2.60 f 26.90 15.50 f 10.35 6.73 0.65

elastic constants and specific gravity of the materials can be found in Table 5.1.

5.4.4 Stiffness of angle grids and [±45] laminates A cross-ply laminate can be rotated by 45° to become an angle-ply laminate of [±45]. A square grid can also be rotated to form an angle grid with ±45° ribs. In fact, the International Space Station hull has this angle grid design. For square grids, the stiffness components in the 1- and 2-axes are equal and their value is E[sqgrid] = f/2 Ex as shown in Equation 5.6. If we apply the transformation relation of 45° from the 1- and 2-axes of the square grid, we can easily show that: E[angrid ] =

1 [ sqgrid ] f E = Ex 2 4

[5.7]

A comparison of the stiffness ratios of the three materials is given in Table 5.3. The same trend in efficiency prevails for this angle grid as well. The efficiency for aluminum reaches a value of 0.65 which is higher than 0.5 and 0.33 for square and isogrids, respectively. But it is still lower than T300/5208 composites by a wide margin in both absolute and specific values. It is still not competitive if aluminum [±45] ribs in a grid are used to increase the shear rigidity. The elastic constants and specific gravity of the materials can be found in Table 5.1.

5.5

Failure envelopes of grids

Failure envelopes for composite laminates can be based on a number of criteria; among the more common are the maximum strain and quadratic criteria. Since grids are modeled as ribs connected by hinges at the rib joints, the failure envelopes for grids are based on the lowest of three possible failure modes applied to the ribs: tensile strength, compressive strength or

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5.9 Combined normal strains or stresses for square and isogrids.

buckling. Since there is no biaxial effect on the failure of ribs, the maximum strain or maximum stress criterion seems most appropriate. Failure of the joints is not included in the present study. It is not expected to affect the compressive strength of the grid. For tensile and shear loading conditions, joints may have lower strengths. Whether joint strength controls the grid strength or not needs to be examined in the future. As is the case with failure envelopes for composite laminates, they can be shown in either strain or stress space. Strain space representation has the advantage of being invariant. The failure envelopes depend then only on ply orientation, not on the stacking sequence of the plies in the laminate. A [0] ply will retain its envelope in strain space whether or not there are other plies present with different orientations. Laminates having different ply orientations can be examined simply by superposing the envelopes of the desired orientations.

5.5.1 Failure envelopes in strain space Failure envelopes in normal strain or e1–e2 space can be constructed from a state of combined strains or stresses for square and isogrids, shown in Fig. 5.9. The simplest construction is the tensile failure strain along the 1-axis: e1 =

X =x Ex

for T300 5208 : e1 =

1500 = 8.3 181

for E-glass epoxy : e 2 =

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1062 = 27.5 38.6

[5.8]

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where x is the failure strain (¥ 10-3) along the rib as shown on the left in Fig. 5.9. Since the Poisson ratio of square grids, like cross-ply laminates, is practically zero, there is no interaction due to combined strains. Failure strains form a square envelope in strain space for square grids. For the [0] rib, the failure strain is controlled by the net strain along the 1-axis. Straining along this axis translates directly into the strain in the [0] rib. Straining along the 2-axis results in Poisson’s strain along the 1-axis. The sum of these combined strains yields the net strain. When the net strain reaches the value of x, failure in the [0] rib occurs. Thus, the failure strain is a vertical line for either the tensile or compressive strain along the 1-axis, for both square and isogrids, as given by Equation 5.8. For isogrids shown on the right of Fig. 5.10, we must determine the failure strain along the [±60] ribs in addition to that along the [0] rib. Failure strain along the 2-axis must be transformed into the normal strains along the rib axes. This strain transformation is straightforward. We can also rationalize the failure strain along the 2-axis by considering first the failure under hydrostatic strains, i.e. when the two normal strains are equal. The failure strain must be equal to x for hydrostatic tension and x¢ for hydrostatic compression. If we recall from Equation 5.5 that the Poisson ratio of the isogrid is –13 , it is a simple geometric problem to establish the failure strains along the 2-axis. The failure strain is –43 x and –43 x¢ for tension and compression, respectively. The geometric relations are shown on the right of Fig. 5.10. The [±60] rib fails at an applied tensile strain along the 2-axis equal to: 4 X 4 = x; e2 = 3 Ex 3 for T300 5208 : e 2 =

4 ¥ 8.3 = 11.0 3

[5.9]

5.10 Construction of failure envelopes in strain space for square and isogrids. Ultimate tensile and compressive strains are shown as x and x ¢, respectively.

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167

4 ¥ 27.5 = 36.7 3

where strain is ¥10-3.

5.5.2 Failure envelopes in stress space Failure envelopes in stress or stress-resultant space can be derived directly from those in strain space by applying stress–strain relations. For square grids where the Poisson ratio is zero, the rectangular envelope in one space translates directly into that in the other space. This is shown on the left of Fig. 5.10. The intercepts of the failure envelopes by the 1-axis are s1 =

N1cr ¢ f b N1cr f b = X = X ; s1 ¢ = = X¢ = X¢ h L h L 2 2

[5.10]

where N1cr, N1cr¢ = maximum tensile and compressive stress resultants; X, X¢ = tensile and compressive strengths of the unidirectional rib, respectively, h = total grid thickness. The intercepts by the 2-axis have the same corresponding numerical values as those in Equation 5.10. The envelope for isogrids is shown on the right of Fig. 5.11. The relations corresponding to Equation 5.10 are: s1 =

2b 2b N1cr f N1cr ¢ f = X= = X¢ = X ; s1 ¢ = X¢ 3 3 h h 3L 3L

[5.11]

The definitions are the same as those for Equation 5.10. For isogrids these intercepts apply to the 1-axis only. For the 2-axis or for a uniaxial stress applied in the [±60] ribs, the ultimate values are different. They can also be derived from the stress–strain relations of the isogrid. For example, the [0] rib is controlled by a constant strain x in the 1-axis for any value of strain

5.11 Construction of failure envelopes in stress space for square and isogrids. Ultimate tensile and compressive strengths are shown as X and X¢, respectively.

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in the 2-axis as shown on the right of Fig. 5.10. We can write the following stress–strain relation: e1 = a11N1 + a12N2 = X

[5.12]

We can differentiate this equation and find that ∂ N2 a11 1 == ∂ N1 a12 n

[5.13]

The same result can be obtained when the applied stress is compressive. In this case, the ultimate compressive strain x¢ must be used in Equation 5.12. We recognize that this [0] failure boundary intercepts the hydrostatic tensile and compressive points where ribs of any orientation must converge. Using a value of –13 for the Poisson ratio, the coordinates of these focal points are +–32 X and –32 X¢ for tensile and compressive pressure, respectively. We can rationalize the tensile or compressive failure of [±60] ribs by stress applied along the 2-axis. The failure by stress in these ribs is a function of the rib orientations. Any stress applied along the 1-axis changes these orientations only slightly. Thus the strength along the 2-axis remains constant, independent of the stress applied, along the 1-axis. The externally applied stresses on an isogrid can be seen in Fig. 5.9. The controlling envelope in stress space appears as horizontal lines on the right of Fig. 5.11. The values of the intercepts by the 2-axis are +–32 X and - –32 X¢.

5.5.3 Failure by rib buckling When the applied stress or strain is in compression, the rib can fail in either one of two modes: compressive strength or Euler buckling. We have just covered the former failure mode. The other rib failure mode can be most easily described by Euler buckling. The critical axial compressive load Pcr and ecr for the rib are: Pcr =

p 2 El p 2 Ehb3 = L2 12L2

Pcr Pcr p2 e cr = = = EA Ehb 12(L b)2

[5.14]

where E = rib axial stiffness, L = rib length, l = moment of inertia = hb3/12, b = rib width, h = rib height, A = rib cross-section = hb. It is assumed that h > b, thus buckling occurs by bending the width b of the rib. Each rib will buckle when a critical Euler strain is reached in it. The critical rib slenderness for buckling of grids is easily deduced from Equation 5.14:

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169

p

(12 x ¢)

for T300/5208:

Ê Lˆ = 10.0 Ë b ¯ cr

for E-glass/epoxy:

[5.15]

Ê Lˆ = 7.2 Ë b ¯ cr

In terms of area fraction f, Equation 5.14 is rewritten as follows. For isogrids, we obtain: 2

e cr =

Pcr Ê pf ˆ Ê Lˆ = , where f = 2 3 Ë b¯ EA Ë 12 ¯

[5.16]

For square grids, we obtain: 2

e cr =

(pf ) Pcr 2 = , where f = (L b) EA 48

[5.17]

Equations 5.16 and 5.17 can be rearranged to determine the critical area fraction f when the compressive failure strain and Euler buckling strain are equal. When fraction f is greater than this critical value, ribs fail under compressive strain. When fraction f is smaller than this critical value, ribs fail by buckling. Rewriting Equation 5.16 to determine the critical area fraction for isogrids, fcr =

12 x ¢ = 3.82 x ¢ p

for T300 5208: fcr = 0.35

[5.18]

for E-glass epoxy: fcr = 0.48 Rewriting Equation 5.17 for the critical area fraction for square grids, fcr =

48 p

x ¢ = 2.2 x ¢

for T300 5208: fcr = 0.20

[5.19]

for E-glass epoxy: fcr = 0.28 We can now define waffle plates as those where rib buckling does not occur. In fact the critical value for area fraction can also be viewed as the minimum value; i.e. fcr = fmin if rib buckling is to be prevented. For T300/5208 waffle grids, this area fraction f has to be larger than 0.35 for isogrids and 0.20 for square grids. The corresponding values for E-glass/epoxy are 0.48 and 0.28, respectively. Glass being less stiff than graphite, the higher area fraction or denser grid for rib buckling is expected.

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The buckling strain is thus proportional to the square of the area fraction. For example, if the area fraction f is reduced by a factor of 1.41 or 2 , the failure strain is reduced by one half. This is shown in Fig. 5.12 where the wavy lines represent rib buckling due to the reduced area fraction for square grids. The buckling controlled failure envelopes are designated by a superscript ‘b’ for rib buckling. The envelopes on the left are in strain space; on the right, in stress space. If the material is T300/5308 and the area fraction is f = fcr, the new area fraction f becomes f = 0.20/ 2 = 0.14. The envelopes in strain and stress space for isogrids are shown in Fig. 5.13. The buckling envelopes are again shown as wavy lines. The area fraction f that initiates rib buckling for T300/5208 is 0.35 as stated in Equation

5.12 Failure envelopes in strain- and stress-resultant space for a square grid. The solid lines represent failure by tensile or compressive strains/stresses; the wavy lines represent rib buckling.

5.13 Failure envelopes in strain- and stress-resultant space for an isogrid. The solid lines represent failure by tensile or compressive strains/stresses; the wavy lines represent rib buckling.

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5.18. The displayed inner buckling lines are defined by a reduction in the area fraction by a factor of 1.41 or 2 ; i.e. down to f = 0.35/ 2 = 0.25. The envelopes for compressive strains and stresses are again reduced by one half.

5.5.4 Examples of CFRP and GFRP failure envelopes In Fig. 5.13, we show the failure envelopes of isogrids in the normal strain space for T300/5208 graphite/epoxy and E-glass/epoxy composites. These envelopes are the same as the envelope on the left of Fig. 5.13 drawn to scale. For the T300/5208 envelope, shown on the left of Fig. 5.14, the critical area fraction fcr = 0.35. Compressive failure will be determined by strength when f ≥ 0.35. Alternatively we can set fcr = fmin so that rib buckling will not occur. Two lower area fractions of 0.28 and 0.23 are shown as successively contracting envelopes resulting from rib buckling. For E-glass/epoxy isogrids, the critical area fraction is fcr = 0.48, shown in Equation 5.17. When area fractions are less than this critical value, rib buckling will occur; i.e. f £ fmin. Two such lower area fractions of 0.35 and 0.28 are shown for an E-glass/epoxy grid on the right of Fig. 5.14. Comparing the two envelopes in Fig. 5.14, the E-glass strain envelope is much larger. Glass fibers have larger strain capability than graphite. The difference is about three times in favor of the glass composite grid. In terms of stress, glass grids are not as strong as graphite grids. The modes of failure, however, are similar. The tensile, compressive and buckling failures follow the same pattern as those for graphite grids.

5.14 Failure envelopes of T300/5208 and E-glass/epoxy isogrids in normal strain space. Rib buckling occurs when area fraction f becomes less than a critical value as shown by the two inner envelopes.

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Since glass is less stiff than graphite, the buckling occurs in shorter ribs or larger area fraction. The difference, for example, is (L/b)cr = 10.0 or fcr = 0.35 for T300/5208 and (L/b)cr = 7.2 or fcr = 0.48 for E-glass/epoxy.

5.6

Grids with skins

Grids can also have one or two skins. While we believe that grids are completely capable of carrying the applied external in-plane and flexural loads, skins can either share the load and/or perform functions like containing pressure or preventing penetration of an unwanted object. Interaction between skins and grids makes their combination not as simple as grids by themselves. The combination means the construction of two or more different material forms is no longer macroscopically homogeneous. The interaction of different material forms must be examined on a case by case basis. Buckling of ribs is constrained by the skins. Skin buckling is an added failure mode. Chen and Tsai [2] described various buckling failure modes. If we treat the grid and skins as two materials, the resulting in-plane stiffness follows the rule of mixtures. Because the density of the grid is much lower than that of the solid skins, the stiffness on a weight basis is different from that on a volumetric basis. This difference is analogous to the fiber fractions of a composite where the weight fraction is higher than the volume fraction. As a result, the relative performances of grids with and without skins are different on weight and volume bases. We have not been able to establish general rules. For simplicity, we examine only grids with two identical, symmetric skins. We purposely limit our illustration to an isotropic grid/skin construction with equal thicknesses; i.e. each quasi-isotropic skin is a quarter and the isogrid is half the total thickness. Failure envelopes are limited to in-plane loading only and are shown in absolute or normalized stress resultant. Each representation conveys different information on the behavior of the grid/skins construction.

5.6.1 CFRP grid with laminated skins In Fig. 5.15, we show on the left two envelopes in stress-resultant space. First, a T300/5208 isogrid without rib buckling ( f ≥ fcr = 0.35) is shown. For f = fcr, the weight of the grid equals 35% that of the skins, or approximately 25% the total weight. On the same side of this figure, the first failure (FF) envelope of an isogrid with quasi-isotropic laminated skins is shown. In the first, second and fourth quadrants, the grid has higher strength capability:

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5.15 Failure envelopes of T300/5208 isogrid with quasi-isotropic skins in absolute and specific stress-resultant space.

the laminate defines the first failure (FF) envelope. In the third quadrant, the laminate is stronger. The grid defines the FF envelope. While the grid envelope is diminutive as compared with the grid and skins, the substantial difference in weight must be considered. To compensate for this difference, the failure envelopes are plotted in specific stressresultant space shown on the right of Fig. 5.15. The normalization is done by dividing the stress resultant by the total weight of the structure. In this figure, the grid envelope is enlarged about four times (1/0.25) relative to the FF of grid with skins. This figure may help to visualize the respective capabilities of grid alone and grid/skin combination.

5.6.2 GFRP grid with laminated skins The same comparison between grid and grid/skin is made for E-glass/epoxy material. For the same absolute and specific stress-resultant representations, the failure envelopes of E-glass/epoxy grids and skins are shown in Fig. 5.16.The critical area fraction for this material is 0.48. So the grid weighs 48% the weight of the skins, or 32% of the total. In absolute space, shown on the left of this figure, the FF envelope of the grid and skins combination is again much larger than that of the grid alone. When the relative weight is taken into account as is the case in the specific stress-resultant representation on the right of Fig. 5.16, the grid shows much greater strength capability. Thus the first failure is determined by the weaker of the two components. In this case, it is the laminate that limits the entire FF envelope.

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5.16 Failure envelopes of E-glass/epoxy isogrid with quasi-isotropic laminated skins in absolute and specific stress-resultant space.

5.17 Failure envelopes of aluminum grid with skins in absolute and specific stress-resultant space.

5.6.3 Aluminum grid with skins For aluminum grids with skins, the FF envelope is larger than the grid envelope everywhere. The grid fails at significantly lower stresses. They are shown in absolute and specific stress-resultant spaces in Fig. 5.17. Thus metallic grids are not only less stiff than skins, they are weaker by a wide margin as well. Metallic grids are not recommended for applications where either stiffness or strength is critical. There must be other reasons to justify their use.

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5.18 Comparison of three grids in specific stress space. The ranking is as follows: CFRP, GFRP and aluminum.

To make a meaningful comparison of failure envelopes in stressresultant space, we use specific strength. The grids here do not have skins. The higher the specific strength, like specific stiffness, the more efficient the material in its load-carrying capability per unit weight. This is shown in Fig. 5.18. We have tried to show that grids have strong interaction in combination with skins. It must be assessed on a case-by-case basis. It is, however, clear that composite grids have superior strength and should be utilized to maximum extent. Metallic grids are not efficient and must only be used after careful consideration. Grids present a challenge for the designers. As we improve our understanding of the behavior of grids, with or without skins, we should now explore hybrid combinations such as composite grids with metallic face sheets. It may be feasible to design a structure with specific properties not possible otherwise with conventional constructions.

5.7

Flexural rigidity of isogrids

As stated earlier, the flexural rigidity of grids follows the same cubic relation with the grid height as the thickness of solid plates and laminates. In addition, rigidity [D] of isogrids is directly proportional to the area fraction f. Since the mass of the grid is also proportional to this area fraction, the

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specific flexural rigidity ([D*] = [D]/mass) will remain the same as the fraction changes. The poor conversion of rib to grid for the in-plane stiffness is carried over to the flexural rigidity. Again, we wish to emphasize that metallic grids are not effective. The examples in the following are intended to illustrate this point.

5.7.1 Flexural rigidity of a CFRP isogrid Find [D] and [D*] for a T300/5208 graphite/epoxy isogrid. Case 1 The grid dimensions are: rib height 10 mm, rib width 4 mm, rib length 50 mm. Area fraction: f = 0.28

Mass per area = 4.43 kg m 2

[5.20]

Flexural rigidity: D11 = D22 = 1577 N m, D66 = 532 N m

[5.21]

Specific flexural rigidity: D11* = D22* = 356 N m (kg m 2 )

[5.22]

D66* = 120 Case 2 If the rib width is reduced from 4 to 2 mm Mass per area = 2.21 kg m 2

[5.23]

Flexural rigidity: D11 = D22 = 785 N m, D66 = 263 N m

[5.24]

Specific flexural rigidity: D11* = D22 * = 354 N m (kg m 2 )

[5.25]

Area fraction: f = 0.14

D66* = 118 The slight difference in the specific [D*] is caused by the contribution from the twisting rigidity of the ribs which is not proportional to the rib width. Case 3 If a quasi-isotropic laminate [0/+60/-60] of the same height (10 mm) is added symmetrically to the grid, there are 80 plies. Area fraction: f = 0.28

Mass per area = 16 kg m 2

[5.26]

Flexural rigidity: D11 = D22 = 6364 N m, D66 = 2240 N m

[5.27]

Specific flexural rigidity: D11* = D22 * = 397 N m (kg m )

[5.28]

2

D66* = 140 There is a significant increase in [D] but only a modest increase in [D*].

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Case 4 If the height of the grid is increased to match the weight of the previous grid + laminate combination (case 3) Height = 36 mm Area fraction: f = 0.28

Mass per area = 16 kg m 2

[5.29]

Flexural rigidity: D11 = D22 = 73667 N m, D66 = 24585 N m

[5.30]

Specific flexural rigidity: D11* = D22 * = 4604 N m (kg m )

[5.31]

2

D66* = 1536 There is an order of magnitude increase in both [D] and [D*].

5.7.2 Flexural rigidity of an aluminum isogrid Find [D] and [D*] for aluminum grid with the same dimensions. Case 1 Grid dimensions: rib height 10 mm, rib width 4 mm, rib length 50 mm. Area fraction: f = 0.28

Mass per area = 7.21 kg m 2

[5.32]

Flexural rigidity: D11 = D22 = 633 N m, D66 = 236 N m

[5.33]

Specific flexural rigidity: D11* = D22 * = 87.9 N m (kg m ) 2

[5.34]

D66* = 32.7 Case 2 A solid aluminum plate of the same height. Height = 10 mm

Mass per area = 26 kg m 2

[5.35]

Flexural rigidity: D11= D22 = 6309 N m, D66 = 2208 N m

[5.36]

Specific flexural rigidity: D11 * = D22 * = 243 N m (kg m 2 )

[5.37]

D66* = 85 Note that the rigidity of the grid is much lower than that of the plate for both [D] and [D*].

5.8

Coefficients of thermal expansion

One unique feature of composite grids with graphite fibers is their low coefficients of thermal expansion (CTE). Five common graphite/epoxy

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5.19 CTE of CFRP quasi-isotropic laminates and isogrids, including fictitious composites having doubled longitudinal stiffness to simulate high-modulus fibers.

composites are shown in Fig. 5.19. The first of three columns for each composite material represents the CTE of quasi-isotropic laminates. The value is between 1.5 ¥ 10-6 and 2 ¥ 10-6 for all the composites shown. The next column shows the CTE for the same laminate if the longitudinal Young modulus is doubled. The CTE is decreased by a factor of 2 which is a direct result of having higher modulus fibers. The third column shows the CTE of an isogrid made of the same unidirectional composite. The value is not only significantly lower, it is near zero or even slightly negative. A negative CTE is possible for laminates in only one direction; for isogrids it is bidirectional in the plane of the grid. This unique feature of composite grids can be greatly utilized in structures for satellites. On the other hand, the CTE for E-glass/epoxy isogrids is 11.35 which is much higher than that of graphite/epoxy composites as shown above. This CTE is very close to that of steel (12 ¥ 10-6). Thus from the standpoint of matching CTE, a glass/epoxy isogrid will be perfect with steel face sheets. Aluminum is even higher at about 24 ¥ 10-6. But the CTE mismatch between GFRP and aluminum is less severe than that between CFRP and aluminum.

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5.9

179

Conclusions

We have tried to illustrate the unique properties and manufacturing processes offered by composite grids. If they can be produced in large quantities and sizes at low cost, such products will find markets in many fields. Most of the processes described herein can be automated. Low-cost and high-volume production are entirely feasible. Grids may emerge as a viable alternative to the conventional laminated, stiffened and/or sandwich constructions. We realize that composite grids are as simple as composite laminates. We enthusiastically embrace grids for many applications. For glass/epoxy grids, there are opportunities in the reinforcement systems of concrete structures and piling. We also envision vessels and piping for internal and external pressures, energy absorption devices and containment rings for rotating machinery. For graphite/epoxy grids, we see low-cost structures for fuselages, and several components of launch vehicles and satellites. Having a wide range of controllable coefficient of thermal expansion available, thermal matching of support structures is possible. Dynamic tuning and damping are also easily available with composite grids. Customized structures can be made in weeks instead of months or years. We hope to continue to gain design and manufacturing experience so composite grids can be recognized as something special.

5.10

Acknowledgements

The authors wish to thank their former and current employers for the support of this work. Financial support from US Air Force Office of Scientific Research, National Science Foundation, US Army Corps of Engineers, National Renewable Energy Laboratory, Stanford Integrated Manufacturing Association, and Industrial Technology Research Institute of Hsinchu are gratefully acknowledged.

5.11

References

1. Meyer, R.R., McDonnell Douglas Astronautics Company, Isogrid Design Handbook, NASA Contractor Report, CIR-124075, Revision A, 1973. 2. Hong-Ji Chen, H.-J. and Tsai, S.W., ‘Analysis and optimum design of composite grid structures’, J. Composite Mater., 30(4/6), 503–534, 1996. 3. Huybrechts, S.M., ‘Analysis and behavior of grid structures’, PhD thesis, Stanford University, Department of Aeronautics and Astronautics, 1995. 4. Tsai, S.W., Theory of Composites Design, Think Composites, Palo Alto, CA, 1992.

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6 Knitted fabric composites H. HAMADA, S. RAMAKRISHNA AND Z.M. HUANG

6.1

Introduction

In recent years, knitted fabric reinforcements have received great attention in the composites industry [1–10]. This is attributed to the unique properties of knitted fabrics compared with other reinforcement fabric structures such as woven and braid. Interlocking of loops of yarn makes knitted fabrics as shown in Fig. 6.1. Here, the term ‘yarns’ represents individual filaments, untwisted fiber bundles, twisted fiber bundles or roving. These loops can glide over each other and thus give a high degree of deformability to knitted fabrics. This deformability provides drapeability, which makes knitted fabric reinforcement formable into the desired complex preform shapes for liquid molding to produce the composite component. Moreover, the use of advanced knitting machines allows the production of near net shape fabrics such as domes, cones, T-pipe junctions, flanged pipes and sandwich fabrics. The use of near net shape preforms has the advantage of minimum material wastage. A combination of net shape fiber preforms and conventional liquid molding techniques has the potential to mass produce and to reduce the production time, and thus lower the cost of composite material. This is important especially when the applications for composite materials are changing from high-cost and high-performance products of aerospace industry to low-cost and mass-producible products of the general engineering industry. Knitted fabrics are basically categorized into two types, namely warp knit fabrics and weft knit fabrics, based on the knitting direction. Schematic diagrams of both the knitted fabrics are shown in Fig. 6.1. Warp knitted fabric is produced by knitting in the lengthwise direction (wale direction) of the fabric, as shown by a solid line in Fig. 6.1(a). Weft knitted fabric is produced by knitting in the widthwise direction (course direction) of the fabric (solid line in Fig. 6.1b). Several types of knitted fabrics are used in the garment industry for fashion purposes [11]. However, only a limited number of knit structures are being investigated for composites in engineering applications, 180 © 1999,Woodhead Publishing Ltd

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6.1 Schematic diagrams of (a) warp knitted and (b) weft knitted fabrics.

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Table 6.1. Classification of typical warp and weft knitted fabrics used in engineering applications Type

Fabric classification

Weft knitted fabric structure

Warp knitted fabric structure

I

2-D fabric

Plain, rib, Milano rib, inlaid fabrics

Dembigh, Atlas

II

2-D fabric – 3-D shape Plain, rib

Dembigh, Atlas

III

3-D solid fabric

Plain and rib fabrics with inlay fiber yarns

Multiaxial warp knitted fabrics or noncrimp fabrics

IV

3-D hollow fabric (sandwich fabric)

Single jersey face structure

Single Dembigh face structure

since: (a) most engineering applications require only simple knit structures and (b) unlike textile fibers (cotton and polyester), it is difficult to form stiff reinforcement fibers such as glass, carbon and aramid into complicated knit structures. Typical warp and weft knitted fabrics investigated for engineering applications are summarized in Table 6.1. Both the warp and weft knitted fabrics can be further classified into four types based on the dimensional (D) arrangement of yarns. Type I fabrics are simple 2-D flat knitted fabrics shown in Fig. 6.1. These fabrics can be cut to the required dimensions and laminated just as in conventional woven fabric composites. Using fully fashioned knitting machines it is possible to produce 2-D fabrics into the net shape of the components. Such 2-D fabrics with 3-D shapes may be categorized as Type II fabrics. As mentioned above, the combination of Type II fabrics with conventional composite molding techniques, makes it possible to cut down the fabrication costs. Type III fabrics are produced by stitching multiaxial layers of parallel yarn [12]. Because of minimum fiber crimp, they are also called non-crimp fabrics. A schematic diagram of a typical Type III fabric is shown in Fig. 6.2. Owing to their superior properties and better drapeability than the woven fabric composites, they are being considered for building buses, trucks, ships and aircraft wings. Type IV fabrics, also known as sandwich fabrics or 3-D hollow fabrics, are produced by binding 2-D-face fabrics together using pile yarns [13]. A schematic diagram of a typical Type IV fabric is shown in Fig. 6.3. These fabrics are sometimes referred to as 2.5-D fabrics, as the amount of fibers in the thickness direction is less than the fibers in the planar direction of the fabric. They are considered to achieve the optimum design of high-performance and damage-tolerant composite structures. The objective of this chapter is to model the mechanical behavior of

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6.2 Schematic representation of a multiaxial warp knitted fabric.

Face structure

n

tio

c ire

ale W

D

Course Direction 6.3 Schematic diagram of a warp knitted sandwich fabric.

Type I knitted fabric reinforced composites. However, the procedures described here can be easily generalized to the composites reinforced with other kinds of knitted fabrics. The presentations of this chapter begin with a geometric description of Type I plain weft knitted fabric, followed by a description of the tensile behavior of knitted fabric reinforced composites obtained from experimental studies. Analytical procedures for modeling the elastic and strength properties of knitted fabric composites are then presented. The analytical modeling work reported in this chapter is based upon references [11,14–21].

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6.2

Description of knitted fabric

Let us consider the plain weft knitted fabric shown in Fig. 6.1(b). The knit structure is formed by interlooping of one yarn system into continuously connecting vertical columns and horizontal rows of loops. This type of fabric can be produced using either flat bed or circular knitting machines. The vertical column of loops along the length of the fabric is called ‘wale’ and the horizontal row of loops along the width of the fabric is called ‘course’. The respective directions are called ‘wale direction’ and ‘course direction’. A single knit loop comprises a head loop, two side limbs and two sinker loops as shown in Fig. 6.4. Changing the structure of knit loops produces different knitted fabrics. Knitted fabrics are often specified using ‘wale density’ and ‘course density’. The wale density (W ) is defined as the number of wales per unit length in the course direction. Similarly, the course density (C) is the number of courses per unit length in the wale direction of the fabric. Both the wale and course densities are mainly determined by the gauge of the knitting machine, i.e. the number of needles per unit length of the machine bed. The product of C and W gives the stitch density, N, of the fabric. N is defined as the number of knit loops per unit planar area of the fabric.

6.3

Tensile behavior of knitted fabric composites

Composites are fabricated by impregnating knitted fabric of reinforcement fiber yarns with the matrix polymer. For a given knitted fabric structure, the mechanical behavior of composite material depends on the properties of

6.4 Schematic representation of various portions of a typical knit loop.

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185

the constituent fiber and matrix materials [22–26]. Typical tensile stress–strain curves of three different kinds of knitted fabric composites are shown in Fig. 6.5. These curves are obtained from tensile testing in the wale direction of the composite. The tensile stress–strain curve of composite made from knitted glass fiber fabric and epoxy matrix is grossly linear with

6.5 Typical tensile stress–strain curves of (a) knitted glass fiber fabric reinforced epoxy composite, (b) knitted glass fiber fabric reinforced polypropylene composite, and (c) knitted polyester fiber fabric reinforced polyurethane composite.

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6.5 (cont.)

a small ultimate failure strain, 1.3%. In the case of knitted glass fiber fabric reinforced polypropylene composite material, the stress–strain curve changes from an initial linearly elastic relationship to a significantly nonlinear relationship with an intermediate ultimate failure strain of 8.5%. The matrix polymer used in these composite materials mainly causes this difference. At the other end of the spectrum, a highly flexible stress–strain behavior could be achieved by reinforcing elastomeric material with a knitted fabric.A typical stress–strain curve of a knitted polyester fiber fabric reinforced polyurethane elastomer is shown in Fig. 6.5. The stress–strain behavior is characterized by a small initial linear elastic relationship, followed by nonlinear behavior with large ultimate failure strain of 60%. In other words, by selecting the type of matrix and reinforcement materials, the mechanical characteristics of a knitted fabric composite can be tailored from rigid to flexible. This chapter mainly concerns the mechanical behavior of the knitted glass fiber fabric reinforced epoxy composites, in which the stresses and strains are connected by fixed linear relationships. Hence, let us consider the tensile behavior of knitted glass fiber fabric reinforced epoxy composite in detail. The stress–strain curve is linear up to the knee point, which occurred at approximately 0.45% strain. Above the knee point, the material deformation and microfracture processes in the specimen cause the nonlinearity. A schematic representation of a typical fracture process in a knitted fabric composite is given in Fig. 6.6. At strain levels immediately

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6.6 Schematic representation of a typical fracture process in tensile tested knitted fabric composite.

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above the knee point, debonding of yarns oriented normal to the testing direction occurs. The cracks nucleated from the debonded sites propagate into resin-rich regions and coalesce into large transverse cracks. Unfractured yarns bridge the fracture plane. The ultimate fracture of the tensile specimen occurs upon the fracture of bridging yarns. In other words, the tensile strength of composite material is determined mainly by the fracture strength of yarns bridging the fracture plane.

6.4

Analysis of 3-D elastic properties

6.4.1 Methodology of analysis The plain weft knitted fabric reinforced composite material investigated in this study is assumed to have only reinforcement fiber yarns and polymer matrix. For analysis purposes, a unit cell representing the complete knitted fabric composite is identified. A geometric model is proposed to determine the orientation of yarn in the composite (Section 6.4.2). Section 6.4.3 outlines the procedure for estimating the fiber volume fraction of the composite. The unit cell is divided into four representative volumes, also called a ‘crossover model’. The crossover model is further divided into subvolumes, which are considered as transversely isotropic unidirectional fiber reinforced composites. A new micromechanical model is used to predict all the five independent elastic constants of the unidirectional fiber reinforced composites (Section 6.4.4). By considering the contributions of both the fibers and net matrix material, the compliance/stiffness matrix of each subvolume in the material co-ordinate system is calculated using the new formulae. This compliance/stiffness matrix of each sub-volume is then transformed to the global co-ordinate system (see Section 6.4.5). A volumeaveraging scheme has been applied to obtain the overall compliance/stiffness matrix of the knitted fabric composite (Section 6.4.6). The effects of fiber content and other parameters of knitted fabric on the elastic properties of the composite material are identified (Section 6.4.7).

6.4.2 Geometric model A schematic diagram of an idealized unit cell of the plain weft knitted fabric is given in Fig. 6.7. The basic assumption is that the projection of the central axis of the yarn loop on the fabric plane is composed of circular arcs. This assumption is reasonable as the knit loops are formed during knitting by bending the yarns round a series of equally spaced knitting needles and sinkers. The physical meanings of various symbols used below are also shown in the figure. The geometry of the unit cell can be described using

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Knitted fabric composites

6.7 Schematic diagram of an idealized unit cell of the plain weft knitted fabric.

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three parameters, i.e. wale density, W, course density, C, and the yarn diameter, d. In the fabric plane, we set the global rectangular axis Ox to be parallel to the wale direction and Oy to the course direction. Suppose that the OQ portion of the loop has a center at C with a total angle j, i.e. OCQ = j. ‘ad’ denotes the radius of projection of the loop, i.e. the length between O and C, where a is a constant. Q is the point at which the central axis of this loop joins the central axis of the loop with a center F. H and J are the points at which the yarns of adjacent loops (loops with centers at C and B) cross over. The angles OCB = y and HCB = f. Let P be any point on the central axis of the loop and the angle of the projection of the loop portion from O to P be q, OCP = q. The co-ordinates of P are given by x = ad(1 - cosq) y = ad sin q z=

[6.1]

hd È Ê qˆ˘ 1 - cos p ˙ Í Ë j¯˚ 2 Î

where h is a constant used for representing maximum height hd (at Q) of the central axis above the plane of the fabric. The parameters a, h and j in (6.1) are determined from the following formulae: a=

1 4Wd sin j

[6.2]

Ê C 2d j = p + sin Á 2 Á 2 Ë C + W 2 (1 - C 2 d 2 ) -1

[

È Ê yˆ Ê fˆ˘ h = Ísin p sin p Î Ë j ¯ Ë j ¯ ˙˚

ˆ C ˘ ˜ - tan -1 È 1 2 2 2 ˙ Í ˜ Î W (1 - C d ) ˚ ¯

]

[6.3]

-1

[6.4]

y = sin -1

Ê 2a ˆ sin j Ë 2a - 1 ¯

[6.5]

f = cos -1

Ê 2a - 1 ˆ Ë 2a ¯

[6.6]

The yarn diameter d can be expressed in terms of the linear density (Dy) of the yarn and packing fraction (K) of fibers in the yarn as d=

Dy 2 ¥ 10 -2 (cm) 3 10 pr f K

[6.7] 3

where rf is the density of fiber (g/cm ).

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From Fig. 6.7, it is clear that the orientation of the yarn in a knit loop (MNOQP) can be determined from knowing the orientation of the yarn in the portion OQ. We may assume that the OQ portion is an assemblage of a series of straight segments. Let (xn-1, yn-1, zn-1) and (xn,yn,zn) be the coordinates of start and end points of the (n - 1)th yarn segment (see Fig. 6.8). The orientation of the segment in 3-D co-ordinates can be specified using two angles, qx and qz, where qz is the angle between the z-axis and the yarn segment and qx the angle between the x-axis and the projected straight line of the segment on the x–y plane. These two angles are important in our geometric analysis. They are determined as q x = tg -1

Ê yn - yn -1 ˆ Ë xn - xn -1 ¯

È ( xn - xn -1 ) 2 + ( yn - yn -1 ) 2 ˘ q z = tg Í ˙ zn - zn -1 ÍÎ ˙˚ -1

6.8 Representation of a segment of yarn.

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[6.8]

[6.9]

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Equations 6.8 and 6.9 imply that only relative co-ordinates of the yarn are important. Therefore, we can replace the unit cell shown in Fig. 6.7 with the unit cell in Fig. 6.9(a). The unit cell in Fig. 6.9(a) can be further divided into four identical sub-cells. Each sub-cell consists of two impregnated yarns which cross over each other. This sub-cell is known as the crossover model [16] and is represented in Fig. 6.9(b). Using the crossover model, a unit cell can be constructed. Repeating the unit cell in the fabric plane will obviously reproduce the complete plain knitted fabric structure. We thus only need to investigate the crossover model which is taken as a representative volume. The co-ordinates of the first yarn in the model are given by Equation 6.1 with 0 £ q £ j. To determine the co-ordinates of the second yarn easily, we choose its starting point to be nearer to the end point of the first yarn. The co-ordinates of the points on the second yarn are thus given by x1 y1

2nd

2 nd

= 2ad =

1 2Wtg(y)

1 2W

z12nd = z1st 1 xn2nd = x12nd - xn1st yn2nd = y12nd - yn1st zn2nd = zn1st

n ≥ 2, 3, . . .

6.4.3 Estimation of fiber volume fraction Based on the above-mentioned geometric model, the fiber volume fraction of the knitted fabric composite is given by [11]: Vf =

nk Dy LsCW ¥ 10 -5 9r f At

[6.10]

where nk is the number of plies of the fabric in the composite, t is the thickness of the composite measured in centimeters, A is the planar area over which W and C are measured, and Ls is the length of yarn in one loop of the unit cell which can be represented approximately by Ls ª 4(ad)j

[6.11]

Let us apply Equation 6.10 to the knitted glass fiber fabric reinforced epoxy composites described in Section 6.3. Knitted fabrics with W = 2 loops/cm and C = 2.5 loops/cm, are made using 1600 denier (Dy) glass fiber yarns (fiber density rf = 2.54 g/cm3). Composites with single and four plies of

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6.9 Schematic diagrams of (a) unit cell and (b) crossover model.

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193

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6.10 Typical variation of volume fraction of fibers with the linear density of yarn.

knitted fabric contained 0.095 and 0.323, respectively, volume fraction of fibers. These fiber volume fractions are determined experimentally by the combustion method. The single-ply and four-ply composites have t = 0.06 and 0.07 cm, respectively. More details on the fabrication and testing of plain weft knitted glass fiber fabric reinforced epoxy composites can be found elsewhere [22,23]. Using Equation 6.10 the estimated fiber volume fractions are 0.0933 and 0.3198 for single- and four-ply composites, respectively. The predicted fiber volume fractions are found to be close to those determined from the experiments. Therefore, this equation will be used to study the variation of fiber volume fraction with the changes in the parameters of knitted fabric. By assuming that it is possible to use yarns of different sizes, the variation of Vf with Dy is computed theoretically. Figure 6.10 gives an estimate of Vf that can be expected when different sizes of yarn are used. For a given stitch density of knitted fabric (N = C ¥ W), the Vf increased linearly with increasing Dy. In other words, the fiber content of knitted fabric composites can be increased with increasing Dy. However, the maximum Vf that can be achieved is limited by the knitting needles used in the knitting machine. Increasing Dy means coarser yarns. In general, the coarser yarns are difficult to knit and the coarsest yarn that can be used is dependent on the yarn type, knitting needle size and other devices used on the knitting machine. For a constant Dy, the variation of Vf with N is shown in Fig. 6.11. The N

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6.11 Typical variation of volume fraction of fibers with the stitch density of knitted fabric.

of knitted fabric can be changed in two ways: (1) machine gauge and (2) stitch tightness control setting on the knitting machine. Machine gauge is defined as the number of needles per unit length of needle bed in the knitting machine. Most machines are equipped with a stitch tightness control button, so that it is possible to alter N in a range. Figure 6.11 suggests that Vf increases non-linearly with increasing N. This can be understood by examining Equation 6.10. Dy and other parameters in the denominator of Equation 6.10 are assumed to be constant. Hence, Vf is proportional to the product of Ls and N. An increase of N means smaller knit loops which implies that stitch length, Ls, decreases with increasing N. The inverse relationship between the N and Ls results in non-linear variation of Vf with increasing N. Nevertheless, it can be said that Vf can be increased with increasing N. The maximum Vf that can be achieved with increasing N is limited by the yarn diameter, d. With increasing stitch density the course spacing, 1/C, and wale spacing, 1/W, decrease. In other words, the side limbs of knit loop come closer with increasing stitch density or tightness of knitted fabric. The spacing between the limbs of a loop is approximately –12 W. The minimum spacing of side limbs is limited by the yarn diameter. To be able to stitch a knitted fabric, the condition ( –12 W ≥ d) needs to be satisfied. The plots in Fig. 6.12 give an approximate idea of the different Vf that can be achieved by changing N. The relation between Vf and number of plies of knitted fabric (nk) is shown in Fig. 6.12. For given Dy and N, the Vf can be increased with increas-

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6.12 Typical variation of volume fraction of fibers with the number of plies of knitted fabric.

ing nk. However, it is to be noted that the number of layers of knitted fabrics that can be used is limited by the thickness of the composite. Figures 6.10–6.12 give an approximate idea of variation of Vf with Dy, N and nk. The maximum Vf that can be achieved in knitted fabric composites is yet to be estimated, as it is dependent on many other parameters such as compressibility of knitted fabrics and composite fabrication conditions. Further efforts are needed to predict the theoretical maximum Vf that can be achieved in knitted fabric composites. Experimental research works reported in the literature suggest that a fiber volume fraction of 40% is realistically possible in knitted fabric composites.

6.4.4 Micromechanical model for unidirectional fiber reinforced composite The yarns in the crossover model shown in Fig. 6.9(b) can be divided into a number of small and straight segments. Each segment can be regarded as a transversely isotropic unidirectional composite. The conventional micromechanical models [27,28] give only four independent elastic constants (E11, E22, G12 and u12) for a transversely isotropic unidirectional composite if the constituent materials are both isotropic. We propose the following new micromechanical model that gives five independent elastic constants (E11, E22, G12, u12, and G23 or u23) of the unidirectional composite [20]. Let Ox1 be the direction parallel to the length of fiber in the sub-volume

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197

(see Fig. 6.8). The local rectangular co-ordinate system is Ox1x2x3, which is also called the material co-ordinate system. Let us assume that E, n and G are the Young modulus, Poisson ratio and the shear modulus, that V represents the fraction of volume of the material, that the subscripts ‘f’ and ‘m’ stand for the fiber and the matrix respectively, and that the subscripts 1, 2 and 3 denote the material co-ordinates x1, x2 and x3. Let us represent the macro-stress tensors of the fiber, the matrix and the composite in a selected sub-volume by [sijf], [sijm] and [sij] respectively. Correspondingly, the macro-strain tensors are [eijf], [eijm] and [eij]. The two sets of tensors satisfy the following micromechanical relationships

[s ij ] = Vf [s ij ] + Vm [s ij f

[e ij ] = Vf [e ij ] + Vm [e ij f

m

m

]

[6.12]

]

[6.13]

We use [Sijf], [Sijm] and [Sij] to denote the compliance matrices of the fiber, the matrix and the unidirectional composite. They have the forms

[S ] f

ij

È 1 Í Ef Í Í - vf Í Ef Í vf ÍEf =Í Í Í 0 Í Í Í 0 Í Í 0 ÍÎ

[S ] m

ij

È 1 Í Em Í Í- vm Í Em Í vm ÍEm =Í Í Í 0 Í Í Í 0 Í Í 0 ÍÎ

vf Ef 1 Ef vf Ef -

vf Ef vf Ef 1 Ef -

0

0

0

0

0

0 0

0

0

1 Gf

0

0

0

1 Gf

0

0

0

0

vm Em 1 Em vm Em -

vm Em vm Em 1 Em

-

˘ 0 ˙ ˙ 0 ˙ ˙ ˙ 0 ˙ f ˙ = È[Ss ] ˙ Í [0] 0 ˙ Î ˙ ˙ 0 ˙ ˙ 1 ˙ Gf ˙˚

0

0

0

0

0

0

0

0

1 Gm

0

0

0

1 Gm

0

0

0

0

0

[0] ˘

[Stf ]˙˚

[6.14]

˘ 0 ˙ ˙ 0 ˙ ˙ ˙ 0 ˙ ÈS m ˙=Í s ˙ 0 ˙ ÍÎ [0] ˙ ˙ 0 ˙ ˙ 1 ˙ Gm ˙˚

[ ]

[0] ˘ ˙

[S ]˙˚ m

t

[6.15]

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198

3-D textile reinforcements in composite materials È 1 Í E11 Í Í- v12 Í E11 Í v13 Í[Sij ] = ÍÍ E11 Í 0 Í Í Í 0 Í Í 0 ÍÎ

v12 E11 1 E22 v23 E22

v13 E11 v23 E22 1 E33

-

-

0

0

0

0

0

0 0

0

0

1 G23

0

0

0

1 G13

0

0

0

0

˘ 0 ˙ ˙ 0 ˙ ˙ ˙ 0 ˙ ˙ = È[Ss ] ˙ ÍÎ [0] 0 ˙ ˙ ˙ 0 ˙ ˙ 1 ˙ G12 ˙˚

[0] ˘ [St ]˙˚

[6.16] where [Ss] and [St] are 3 ¥ 3 sub-matrices relating normal stresses with elongation strains and shear stresses with shear strains respectively. With [Sijf], [Sijm] and [Sij], the macro-stresses and strains are connected by

{s } = [S ]{e } {s } = [S ]{e }

[6.17]

{s i } = [Sij ]{e j }

[6.19]

f

i

f

f

ij

j

m

i

m

m

ij

[6.18]

j

where {si} = {s11, s22, s33, s23, s13, s12}T and {ei} = {e11, e22, e33, 2e23, 2e13, 2e12}T. The critical step of the present model is to find a coefficient matrix [Aij] such that

{s } = [A ]{s } m

f

i

ij

[6.20]

j

Suppose [Aij] has been given. Combining 6.20 and 6.12, we get

{s } = (V [I ] + V [A ]) {s } {s } = [A ](V [I ] + V [A ]) {s } -1

f

i

f

m

ij

i

[6.21]

j

-1

m

ij

f

m

ij

[6.22]

j

where [I] is a unit matrix. By virtue of 6.12, 6.13, 6.17–6.19, 6.21 and 6.22, the compliance matrix [Sij] of the composite is derived as

[Sij ] = (Vf [Sij ] + Vm [Sij ][ Aij ])(Vf [I ] + Vm [ Aij ]) f

m

-1

[6.23]

The coefficient matrix [Aij] must be chosen so that the resulting compliance matrix [Sij] be symmetric. It is clear that [Aij] can be sub-divided into È[aij ] [0] ˘ [ Aij ] = Í ˙ Î [0] [bij ]˚ where [aij] and [bij] are 3 ¥ 3 sub-matrices such that

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Knitted fabric composites

[Ss ] = (Vf [Ss ] + Vm [Ss ][aij ])(Vf [I ] + Vm [aij ]) f

m

[St ] = (Vf [St ] + Vm [St ][bij ])(Vf [I ] + Vm [bij ]) f

m

199

-1

[6.24]

-1

[6.25]

Detailed discussions on the determinations of [aij] and [bij] are relatively lengthy and are out of the scope of this chapter. Here we only give a set of empirical formulae for aij and bij as below: a11 = Em Ef a22 = a33 = 0.5(1 + Em Ef ) a12 =

f

m

f

m

S12 - S12 S11 - S11

(a11 - a22 )

[6.26]

a13 = (c 22 d1 - c12 d2 ) (c11c 22 - c12 c 21 ) a23 = (c11d2 - c 21d1 ) (c11c 22 - c12 c 21 )

[6.27]

b22 = b33 = 0.5(1 + Gm Gf ) and all the other aij and bij but b11 being taken as zero. In 6.26, the parameters cij and di are given by c11 = S11m - S11f c12 = S12m - S12f

(

m

d1 = (a11 - a33 ) S13 - S13

( )(S

f

)

m

c 21 = (Vf + Vm a22 ) S11 - S11 c 22 = (Vf + Vm a11

m 22

f

- S22

(

f

) ) + V (S m

m

f

12 f

- S12

m

)a

12

)

(

m

f

)

d2 = (Vf + Vm a11 )(a22 - a33 ) S23 - S23 + (Vf + Vm a33 ) S13 - S13 a12 The expression for b11 is more complicated. However, we know that S44 is not an independent elastic constant but is determined by S44 =

1 2(1 + v23 ) = = 2(S22 - S23 ) G23 E22

[6.28]

Hence, b11 is actually immaterial. Combining 6.24–6.28 gives the compliance matrix. The stiffness matrix is simply obtained by the inversion of the compliance matrix, i.e.

[Cij ] = [Sij ]

-1

[6.29]

6.4.5 Elastic properties in global co-ordinates The compliance or stiffness matrix obtained above is in material coordinates. To obtain the overall mechanical properties of the composite, it

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3-D textile reinforcements in composite materials

is necessary to transform these matrices from a local co-ordinate system to the global co-ordinate system. Suppose that the direction cosines between the material co-ordinates Ox1, Ox2, Ox3 and the global co-ordinates Ox, Oy, Oz are denoted by (li, mi, ni) where li = cos( xi , x), mi = cos( xi , y), ni = cos( xi , z), i = 1, 2, 3

[6.30]

By means of Equations 6.8 and 6.9, Equation 6.30 can be represented as l1 = cos(q x ) sin(q z ), m1 = sin(q x ) sin(q z ), n1 = cos(q z ) l 2 = - sin(q x ), m2 = cos(q x ), n2 = 0 l3 = - cos(q x ) cos(q z ), m3 = - sin(q x ) cos(q z ), n3 = sin(q z ) With these coefficients, the two sets of co-ordinates are connected by Ï x1 ¸ Èl1 Ô Ô Í Ì x2 ˝ = Íl 2 Ô x Ô Íl Ó 3˛ Î 3

m1 m2 m3

n1 ˘Ï x¸ Ïx¸ Ô Ô Ô Ô ˙ n2 Ì y˝ = [eij ]Ì y˝ ˙ Ô zÔ n3 ˙˚ÔÓ zÔ˛ Ó ˛

[6.31]

We use [sijG] to denote the stress tensor in the global co-ordinate system. It has the form

[s ] G

ij

Ès xx = Ís yx Í ÎÍs zx

s xy s yy s zy

s xz ˘ s yz ˙ ˙ s zz ˚˙

The transformation between the global stress tensor [sijG] and the local stress tensor [sij] obeys the rule G

[6.32]

s kl = eik e jl s ij

where eij are defined in (20). By using Equation 6.32, the compliance matrix of the unidirectional fiber composite (one segment of yarn in the composite) is thus transformed into the matrix in the global co-ordinate system through the following formula: Y

[Sij ]n -1

T

= [Tij ] s [Sij ][Tij ]s

[6.33]

where the superscript Y stands for the yarn and the subscript n - 1 for the segment under consideration, [Tij]s is a transformation matrix given by 2

È l1 Í m2 Í 1 Í n2 [Tij ]s = Í 1 Í2 m1 n1 Í 2 n1l1 Í Î 2l1 m1

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2

2

l2 2 m2 2 n2 2 m2 n2

l3 2 m3 2 n3 2 m3 n3

l2 l3 m2 m3 n2 n3 m2 n3 + m3 n2

l3 l1 m3 m1 n3 n1 n3 m1 + n1 m3

2n n2 l2 2l2 m2

2 n3 l3 2l3 m3

l2 n3 + l3 n2 l2 m3 + l3 m2

n3 l1 + n1l3 l1 m3 + l3 m1

l1l2 ˘ ˙ m1 m2 ˙ ˙ n1 n2 ˙, m1 n2 + m2 n1 ˙ l1 n2 + l2 n1 ˙ ˙ l1 m2 + l2 m1 ˚

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Similarly, by using the rule transforming the strain tensor [eij] in the local co-ordinate system to the strain tensor [eijG] in the global co-ordinate system, we obtain the stiffness transformation formula as

[C ]

Y

ij n -1

T

= [Tij ]c [Cij ][Tij ]c

[6.34]

in which the transformation matrix [Tij]c is given by

[Tij ]c

È l1 2 Í 2 Í m1 Í n2 =Í 1 Ím1 n1 Ínl Í 11 ÍÎ l1 m1

2

2

l2 2 m2 2 n2 m2 n2

l3 2 m3 2 n3 m3 n3

2l 2 l 3 2 m2 m3 2 n2 n3 m2 n3 + m3 n2

2l3l1 2 m3 m1 2 n3 n1 n3 m1 + n1 m3

n2l 2 l 2 m2

n3l3 l3 m3

l 2 n3 + l3 n2 l 2 m3 + l3 m2

n3l1 + n1l3 l1 m3 + l3 m1

˘ 2l1l 2 ˙ 2 m1 m2 ˙ 2 n1 n2 ˙ ˙ m1 n2 + m2 n1 ˙ l1 n2 + l 2 n1 ˙ ˙ l1 m2 + l 2 m1 ˙˚

6.4.6 Assemblage in the crossover model Equations 6.33 and 6.34 give the compliance and stiffness matrices of only one segment of yarn. To obtain the overall compliance and stiffness matrices of the crossover model, it is necessary to consider the contributions of all the yarn segments. The contributions of all the yarn segments are assembled using the following volume-averaging method. To apply the volume-averaging method, the crossover model is divided into a number of sub-volumes. The material between two cross-sectional planes perpendicular to the wale direction represents one such sub-volume (Fig. 6.13). Certain sub-volumes contain one yarn segment and others contain two yarn segments. For easy analysis, each sub-volume containing two yarn segments is considered as two sub-volumes. Now each sub-volume with single yarn segment may be considered as a transversely isotropic unidirectional composite. Hence, the micromechanical formulae presented in Sections 6.4.4 and 6.4.5 can be used. In such a case, the Vf represents the overall volume fraction of fibers in the composite. The contributions of all the sub-volumes can be assembled using the following equations: M -1

[S ] = Â

1st

xn +1

ij

[C ] = Â

1st

[S ] ij

(2L)

n =1

M -1

- xn

1st

xn +1

ij

n =1

- xn

(2L)

1st

n

+Â

ij

n

xn +1

2 nd

1st

M -1

+Â n =1

- xn

2 nd

[S ] ij

(2L)

n =1

1st

[C ]

M -1

xn +1

2 nd

- xn

(2L)

2 nd

n

[6.35]

2 nd

[C ] ij

n

2 nd

[6.36]

where (M - 1) is the number of discretized yarn segments in the crossover model, the superscripts 1st and 2nd stand for the first and second yarns in the volume, and L is the projected length of one yarn on the x-axis (wale direction) of the crossover model, i.e.

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6.13 Schematic representation of a typical sub-volume of the crossover model.

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Table 6.2. Elastic properties of plain knitted fiber fabric composites Fabric (nkt, Vf)

Model

Exx (GPa)

Eyy (GPa)

1a 0.06b 0.095c

Exper.

5.38 (0.33)d 5.61 6.59

4.37 (0.07) 4.59 4.90

4.48 4.66

1.91 2.20

1.75 1.89

4 0.07 0.323

Exper.

10.28 (0.35) 9.47 13.55

8.49 (0.21) 7.21 8.53

7.00 7.65

3.13 4.43

2.78 3.38

6.35 6.36

6.35 6.36

Ezz (GPa)

Gxy (GPa)

Gxz (GPa)

nxy

nxz

nyz

1.63 1.67

0.48 (0.13) 0.369 0.382

0.354 0.353

0.367 0.375

2.53 2.70

0.371 0.408

0.351 0.342

0.368 0.378

Gyz (Gpa)

The parameters used are: Ef = 74 GPa, Em = 3.6 GPa, nf = 0.23, nm = 0.35, d = 0.0445 cm, Dy = 1600, K = 0.45, rf = 2.54 g/cm3, C = 2.5 cycles/cm and W = 2 cycles/cm. a nk (plies of the fabrics). b t (thickness of the composite). c Vf (fiber volume fraction). d Scatter deviation of the experiment.

L = xM

1st

1st

- x1

= xM

2 nd

- x1

2 nd

Equations 6.35 and 6.36 give the overall compliance and stiffness matrices of the crossover model respectively.

6.4.7 Elastic properties: results and discussion To validate the analytical procedures outlined in Sections 6.4.4–6.4.6, initially predictions were made for the knitted fabric composites whose experimentally determined elastic properties were known [22,23]. Both the experimental and theoretical elastic properties of single- and four-ply knitted glass fiber fabric reinforced epoxy composites are summarized in Table 6.2. The data clearly indicate that the present analysis procedure gives a good estimate of elastic properties of knitted fabric composites. It can also be noted that Equation 6.35 gives better prediction than Equation 6.36 and, hence, the following calculations are made using Equation 6.35. Let us investigate the role of Vf and parameters of knitted fabric on the elastic properties of the composite. Equation 6.10 suggests that Vf can be increased in three different ways: (1) by increasing the linear density of the yarn (Dy); (2) by increasing stitch density of knitted fabric (N); and (3) by increasing number of plies of knitted fabric (nk).The elastic constants versus various of these parameters are thus calculated and are shown in Figs. 6.14–6.17. Only the Young moduli and the shear moduli are reported since the Poisson ratio has shown little dependence on the Vf or knitted fabric

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6.14 Typical variation of elastic moduli of knitted glass fiber fabric reinforced epoxy composite with the fiber volume fraction.

6.15 Typical variation of elastic moduli of knitted glass fiber fabric reinforced epoxy composite with the linear density of yarn.

parameters as long as Vf is not large. The results clearly indicate that the elastic moduli are dependent almost linearly on either of the parameters, yarn linear density (denier), wale density (W) or course density (C), but slightly non-linearly on the Vf.

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6.16 Typical variation of elastic moduli of knitted glass fiber fabric reinforced epoxy composite with the course density of the fabric.

6.17 Typical variation of elastic moduli of knitted glass fiber fabric reinforced epoxy composite with the wale density of the fabric.

6.5

Analysis of tensile strength properties

6.5.1 Prediction of tensile strength As described in Section 6.3, the failure strength of knitted fabric reinforced epoxy composites mainly depends on the yarn bundles bridging the frac-

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ture plane (Fig. 6.6). The number of yarns bridging the fracture plane would depend on the testing direction with respect to the knitted fabric. The number of yarn bundles bridging the wale [nw]b and course [nc]b fracture planes are given by

[nw ]b = nk (2) [nc ]b = nk

W B 2

[6.37]

C B 2

where B is the width of tensile specimen in cm. The area fractions of yarn bundles bridging the wale [Aw]b and course [Ac]b fracture plane are given by

[ Aw ]b =

nkWpd 2 4t

[ Ac ]b =

nkCpd 2 8t

[6.38]

where t is the specimen thickness in cm, d is the yarn diameter given by Equation 6.7. The knitted fabric composite strengths in the wale (sw) and course (sc) directions are given by sw =

[ ]

nkWpd 2 s b

4t nkCpd 2 [s b ] sc = 8t

[6.39]

¯b is the mean strength of set of yarn bundles bridging the fracture where s ¯b can be estimated using the following procedure. plane. The s Assuming that all the bridging yarns possess the same tensile strength ¯b will be equal to and are aligned perfectly in the loading direction, the s the longitudinal tensile strength of unidirectional lamina (s1): s b = s1 = (s f )(Vyf ) + (s m )(1 - Vyf ) [6.40] where sf and sm are the tensile strengths of reinforcement fibers and matrix resin, respectively. Vyf is the volume fraction of fibers in the yarn bundle. However, owing to their looped architecture, it is reasonable to assume that the yarns in the fracture plane orient at an angle a with respect to the loading direction. For tensile testing in wale direction, an approximate estimate of a can be obtained using Equations 6.8 and 6.9: cosa = (cos q x )(cos q z ) The yarn bundle can be treated as off-axis loaded unidirectional lamina. Hence, the tensile strength of a yarn bundle is given by [28]:

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Table 6.3. Tensile properties of unidirectional glass fiber/epoxy lamina Fiber volume fraction (Vyf)

Longitudinal strength, s1 (MPa)

Transverse strength, s2 (MPa)

Shear strength, t12 (MPa)

0.45

885

45

35

6.18 Typical variation of sb with a.

È cos a 4 sin a 4 sin a 2 cos a 2 sin a 2 cos a 2 ˘ sb = Í + + ˙ 2 2 2 2 s2 t 12 s1 Î s1 ˚

-

1 2

[6.41]

where s1, s2 and t12 are respectively the longitudinal, transverse and shear strengths of unidirectional lamina given in Table 6.3. Typical variation of sb with a is shown in Fig. 6.18. sb decreased with increasing a. The decrease of sb was significant in the range 0° < a < 15°. Hence, the variation of sb with a in this range on the composite strength is analyzed. All the yarn bundles in the fracture plane may not have the same a value, since the fracture path is irregular and occurs at different positions of the knit loops. During tensile testing the yarn bundles are peeled (debonded) from the fracture surface and stretched before their failure. Owing to the peeling and stretching effect, the yarn bundles try to align in the testing direction. Determination of actual a just before the failure of yarn bundle is a difficult task. It may be the case that different yarn bundles

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orient at different a with respect to the loading direction. Because of different values of a, it can be expected that yarn bundles bridging the fracture plane possess different strength values. The yarn bundles may possess different strengths due to the statistical nature of fiber strength. Many researchers investigated the statistical nature of bundle strengths. The present study is mainly concerned with the variation of sb with a. From Fig. 6.18, an exponential relationship between sb and a is given by sb = Pe-Qa

[6.42]

where P and Q are parameters of exponential function and can be determined using Equations 6.43 and 6.44, respectively. When a = 0, P = s1

[6.43]

Assume that all the yarn bundles are oriented in the range 0 < a < ak. The maximum orientation, ak, can be determined from the fracture surfaces. sbk is the bundle strength corresponding to the maximum orientation ak. From Equations 6.42 and 6.43, a bk = s1e -Qa

k

Rearranging gives Q=

1 Ê s1 ˆ ln a k Ë s bk ¯

[6.44]

Equation 6.42 indicates the changes in sb with a. Equation 6.37 gives the number of yarn bundles bridging the fracture plane. It is necessary to know how many of these bundles orient at each value of a. The following exponential function f(a) was assumed for expressing the orientation distribution of yarn bundles in the fracture plane: f (a) = Re - Sa

[6.45]

where R and S are the parameters of the exponential function. This function suggests that more yarns orient close to the testing direction. This assumption is reasonable as the yarn bundles try to align in the loading direction due to the debonding and stretching mechanisms. Typical curves for the function f(a) are shown in Fig. 6.19. The area under a curve is unity, therefore

Ú

ak

0

ak

f (a)da = Ú Re - Sa da = -

R=

0

S (1 - e - Sa

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k

)

R - Sa (e - 1) = 1 S k

[6.46]

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6.19 Typical curves of function f(a).

R is dependent on the values of S and ak. Typical f(a) curves for different S and ak are shown in Fig. 6.19. These curves indicate that f(a) is more sensitive to the parameter S than ak. When S is small, the yarn orientation distribution is spread out. For large values of S, the distribution is skewed and more yarns are aligned close to the loading direction. Let g(sb) be the function of yarn bundle distribution with respect to the

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6.20 Typical curves of function g(sb).

bundle strength. Typical g(sb) curves are shown in Fig. 6.20. Using the variable transformation technique, g(s b )ds b = f (a)da Rearranging gives g(s b ) = f (a)

da da b

[6.47]

From Equations 6.42 and 6.47, g(s b ) =

R (Q - S ) a e PQ

[6.48]

From Equation 6.42, a=-

1 Ê sb ˆ ln Q Ë P¯

[6.49]

Combining Equations 6.48 and 6.49, g(s b ) =

R s ( S Q -1) QP S Q

[6.50]

Let G(sb) indicate the yarn bundles fractured due to the applied stress, sb. The surviving yarn bundles [1 - G(sb)] are given by s1

[1 - G(s b )] = Ús g(s b )ds b b

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or

[1 - G(s b )] =

R S Q S Q s1 - s b S Q SP

[

]

[6.51]

Let sbm be the value of bundle stress sb which gives sb[1 - G(sb)] its maximum value, namely d {s b [1 - G(s b )]}s ds b

b = s bm

=0

[6.52]

Equation 6.52 implies that the maximum yarn bundle stress, sbm, is found from the condition that at failure the load borne by the bundles is the maximum. Hence, 1 È ˘ s bm = P Í Î 1 + (S Q) ˙˚

Q S

[6.53]

¯b ) of surviving yarn bundles can be The maximum mean strength ( s obtained by substituting the value of sbm in sb[1 - G(sb)]: sb =

RP È 1 ˘ Í Q Î 1 + (S Q) ˙˚

Q S +1

[6.54]

For a given composite system, the parameter P is constant (Equation 6.43). Q is mainly dependent on the ak and sbk (Equation 6.44). Parameter ¯b mainly R is dependent on S and ak (Equation 6.46). In other words, s ¯ depends on S and ak. Typical variation of sb with S and ak is shown in Fig. ¯b is mainly influenced by the parameter 6.21, which clearly indicates that s ¯ S. The sb initially increased rapidly with increasing S from 0.2 to 2.5, above which it increased only marginally. This behavior is expected since large S means a greater number of yarns aligned close to the loading direction and hence higher mean bundle strength. Small values of S indicate that yarn orientation distribution is spread out and, hence, mean bundle strength is lower. Substituting Equation 6.54 in Equation 6.39, the knitted fabric composite tensile strengths in the wale (sW) and course (sC) directions are given by Q Ê nkWpd 2 ˆ Ï RP È 1 ˘ sW = Á ˜Ì Ë 4t ¯ Ó Q ÍÎ 1 + S Q ˙˚ Q Ê nkCpd 2 ˆ Ï RP È 1 ˘ sC = Á ˜Ì Ë 8t ¯ Ó Q ÍÎ 1 + S Q ˙˚

S +1

S +1

¸ ˝ ˛

¸ ˝ ˛

[6.55]

[6.56]

6.5.2 Tensile strength: results and discussion Tensile strengths of knitted fabric composites with different Vf are computed using Equation 6.55. The main assumptions are: (1) in the fiber

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6.21 Typical variation of ¯ sb with parameters S and ak.

content range investigated, the failure mechanisms of knitted fabric composites are similar and (2) the composite strength is determined mainly by the fracture strength of the yarns bridging the fracture plane. Again let us consider single- and four-ply knitted fabric composites with Vf = 0.0933 and 0.3198, respectively, described in Section 6.4.3. Figure 6.22 shows the variation of predicted tensile strength with the parameters S and ak. The composite strength is more sensitive to the parameter S than ak. This behavior ¯b with S and ak (Fig. is similar to the variation of mean bundle strength, s 6.21). The predicted strength increased rapidly with increasing S from 0.2 to 2, above which it increased marginally. Larger S means that more yarns aligned close to the loading direction and hence higher tensile strength. Smaller S indicates that yarn orientation distribution is spread out and hence lower tensile strength. Table 6.4 summarizes composite strengths for different S in the range from 0.2 to 10.0. For S = 0.2 and S = 10.0, the predicted values indicate lower and upper bounds of tensile strength of knitted fabric composites. The limit of lower bound would depend on the parameter S. It is necessary to determine S precisely for accurate estimation of composite tensile strength. For this purpose, the experimental tensile strengths are shown as dashed lines in Fig. 6.22. From Fig. 6.22, the critical value of parameter S corresponding to which predicted strength matches with the experimental result can be identified. In the case of single-ply composite, both the wale and course predicted tensile strengths match approximately with the respective experimental results when S = 1.0. In the case of four-ply composite, when S = 0.5 the wale and course predicted

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6.22 Variation of predicted tensile strength of knitted fabric composite with parameters S and ak: (a) wale specimen with Vf = 0.0933; (b) course specimen with Vf = 0.0933; (c) wale specimen with Vf = 0.3198; and (d) course specimen with Vf = 0.3198.

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Table 6.4. Tensile properties of plain knitted glass fiber fabric/epoxy composite Number of plies of knitted fabric, nk 1 4 1 4

Testing direction

Wale Wale Course Course

Experimental tensile strength (MPa) 62.83 (7.1) 152.7 (9.5) 35.5 (2.21) 75.4 (4.5)

Analytical tensile strength (MPa) S = 0.2

S = 1.0

S = 10.0

31.83 109.1 19.85 68.2

60.0 150.0 36.0 85.0

84.75 290.6 52.96 181.6

strengths match approximately with the respective experimental strengths. In other words, it appears that the critical S is dependent on the number of plies of knitted fabric used for reinforcing the composite material. This may be due to the mismatch between the adjacent plies of knitted fabrics. Further detailed experiments are necessary to establish clearly the dependence of critical value of the parameter S on the variables such as number of plies of knitted fabric, fabric stitch density and linear density of yarn. This will enable accurate prediction of tensile strengths of knitted fabric composites with different fiber volume fractions. In the present study, only the variation of orientation of bridging yarns is considered. The fracture process of a set of bridging yarns would depend on the yarn orientation distribution as well as the yarn strength distribution. The preliminary procedure outlined here may be further modified by considering the statistical nature of yarn strengths for accurate determination of composite strength. Both the experimental and predicted results (Tables 6.3 and 6.4) suggest that the plain weft knitted fabric composites possess superior tensile properties in the wale direction compared with the course direction. This is mainly due to the higher proportion of yarns oriented in the wale direction than in the course direction.Tensile properties increase with increasing fiber content.

6.6

Conclusions

Preliminary methodologies for predicting the tensile properties of plain knitted fabric reinforced composites are established. Elastic properties were predicted using the crossover model and volume-averaging method. Tensile strength properties were predicted by estimating the fracture strength of yarns bridging the fracture plane. The predicted tensile properties compare favorably with the experimental results. A more detailed

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analysis is necessary to assess fully the applicability and limitations of these analysis methods. Tensile properties of knitted fabric composites can be increased with increasing fiber content. It has been shown that the fiber content of the composite can be increased by increasing (a) the linear density of yarn, (b) the stitch density of knitted fabric and (c) the number of plies of knitted fabric.

6.7

References

1. Ramakrishna, S., Hamada, H., Kotaki, M., Wu, W.L., Inoda, M. and Maekawa, Z., ‘Future of knitted fabric reinforced polymer composites’, in Proc. of 3rd Japan International SAMPE Symposium, Tokyo, 1993, pp. 312–317. 2. Horsting, K., Wulhorst, B., Franzke, G. and Offermann, P., ‘New types of textile fabrics for fiber composites’. SAMPE J., 29, 7–12, 1993. 3. Dewalt, P.L. and Reichard, R.P., ‘Just how good are knitted fabrics?’, J. Reinf. Plast. Comp., 13, 908–917, 1994. 4. Mayer, J., Ruffieux, K., Tognini, R. and Wintermantel, E., ‘Knitted carbon fibers, a sophisticated textile reinforcement that offers new perspectives in thermoplastic composite processing’, in Proc. of ECCM6, Bordeaux, 1993, pp. 219–224. 5. Ramakrishna, S., Hamada, H., Rydin, R. and Chou, T.W., ‘Impact damage resistance of knitted glass fiber fabric reinforced polypropylene composite laminates’, Sci. Eng. Comp. Mater., 4(2), 61–72, 1995. 6. Ramakrishna, S. and Hull, D., ‘Energy absorption capability of epoxy composite tubes with knitted carbon fiber fabric reinforcement’, Comp. Sci. Technol., 49, 349–356, 1993. 7. Ramakrishna, S., ‘Energy absorption behaviors of knitted fabric reinforced epoxy composite tubes’, J. Reinf. Plast. Comp., 14, 1121–1141, 1995. 8. Ramakrishna, S., Hamada, H. and Hull, D., ‘The effect of knitted fabric structure on the crushing behavior of knitted glass/epoxy composite tubes’, in Impact and Dynamic Fracture of Polymers and Composites (ESIS19), Williams, J.G. and Pavan, A., eds, Mechanical Engineering Publications, London, 1995, pp. 453–464. 9. Rudd, C.D., Owen, M.J. and Middleton, V., ‘Mechanical properties of weft knit glass fiber/polyester laminates’, Comp. Sci. Technol., 39, 261–277, 1990. 10. Gommers, B., Verpoest, I. and Houtte, P.V., ‘Modelling the elastic properties of knitted fabric reinforced composites’, Comp. Sci. Technol., 56, 685–694, 1996. 11. Ramakrishna, S., ‘Characterization and modeling of tensile properties of plain knitted fabric reinforced composites’, Comp. Sci. Technol., 57, 1–22, 1997. 12. Ko, F.K., Pastore, C.M., Yang, J.M. and Chou, T.W., ‘Structure and properties of multilayer multidirectional warp knit fabric reinforced composites’, in Proc. of 3rd Japan–US Conference, Tokyo, 1986, pp. 21–28. 13. Ramakrishna, S., Hamada, H., Kanamaru, R. and Maekawa, Z., ‘Mechanical properties of 2.5 dimensional warp knitted fabric reinforced composites’, in Design and Manufacture of Composites, Hoa, S.V., ed., Corcordia University, Montreal, 1994, pp. 254–263. 14. Ramakrishna, S. and Hull, D., ‘Tensile behavior of knitted carbon-fiberfabric/epoxy laminates – part I: experimental’, Comp. Sci. Technol., 50, 237–247, 1994.

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15. Ramakrishna, S. and Hull, D., ‘Tensile behavior of knitted carbon-fiberfabric/epoxy laminates – part II: prediction of tensile properties’, Comp. Sci. Technol., 50, 249–258, 1994. 16. Ramakrishna, S., ‘Analysis and modeling of plain knitted fabric reinforced composites’, J. Composite Mater., 31, 52–70, 1997. 17. Ramakrishna, S., Hamada, H. and Cheng, K.B., ‘Analytical procedure for the prediction of elastic properties of plain knitted fabric reinforced composites’, Composites Part A, 28A, 25–37, 1997. 18. Tay, T.E., Ramakrishna, S. and Jin, W., ‘Three dimensional modeling of damage in plain weft knitted fabric composites’, Composites Sci. Technol. (in press). 19. Ramakrishna, S., ‘Analytical and finite element modeling of elastic behavior of plain-weft knitted fabric reinforced composites’, Key Eng. Mater., 137, 71–78, 1998. 20. Ramakrishna, S. and Huang, Z.M., ‘A micromechanical model for mechanical properties of two constituent composite materials’, Adv. Composite Lett., 6, 43–46, 1997. 21. Ramakrishna, S., Huang, Z.M., Teoh, S.H., Tay, A.O.O. and Chew, C.L., ‘Application of Leaf and Glaskin’s model for estimating the 3D elastic properties of knitted fabric composites’, J. Textile Inst. (in press). 22. Ramakrishna, S., Fujita, A., Cuong, N.K. and Hamada, H., ‘Tensile failure mechanisms of knitted glass fiber fabric reinforced epoxy composites’, in Proc. of 4th Japan International SAMPE Symposium & Exhibition,Tokyo, 1995, pp. 661–666. 23. Ramakrishna, S., Cuong, N.K., Fujita, A. and Hamada, H., ‘Tensile properties of plain weft knitted glass fiber fabric reinforced epoxy composites’, J. Reinf. Plast. Comp., 16, 946–966, 1997. 24. Ramakrishna, S., Hamada, H. and Cuong, N.K.,‘Fabrication of knitted glass fiber fabric reinforced thermoplastic composite laminates’, J. Adv. Comp. Lett., 3(6), 189–192, 1994. 25. Ramakrishna, S., Hamada, H., Cuong, N.K. and Maekawa, Z., ‘Mechanical properties of knitted fabric reinforced thermoplastic composites’, in Proc. of ICCM10, Vancouver, August, 1995, Vol. IV, pp. 245–252. 26. Ramakrishna, S., Tang, Z.G. and Teoh, S.H., ‘Development of a flexible composite material’, Adv. Composite Lett., 6(1), 5–8, 1997. 27. Chamis, C.C., ‘Mechanics of composite materials: past, present, and future’, J. Comp. Technol. Res., 11, 3–14, 1989. 28. Tsai, S.W. and Hahn, H.T., Introduction to Composite Materials, Technomic Publishing, Pennsylvania, PA 1980, chapter 9.

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7 Braided structures TIMOTHY D. KOSTAR AND TSU-WEI CHOU

7.1

Introduction

The design and fabrication of preforms for advanced composites has gained considerable attention in light of the recently developed textile preforming techniques. It is within this realm of preforming technology that the full advantage of the knowledge of process–structure–property relations may be realized. The fabrication history of these preforms directly determines composite microstructure and resulting mechanical properties. Textile preforms may be loosely classified into two-dimensional (2-D) and threedimensional (3-D) structures, depending on the degree of reinforcement between layers [1].

7.1.1 2-D fabrics 2-D fabrics woven on a loom generally contain two sets of yarns. These yarn groups are interlaced at right angles, with the longitudinal yarns being referred to as warp yarns and the cross yarns as weft. A basic loom consists of two harnesses that control warp yarn separation, a shuttle that passes the weft yarn through the separated warp yarns, and a beat-up mechanism that compacts the fabric. By controlling the separation sequence of the warp yarns, different fabrics may be formed. Two-dimensional woven fabrics offer a high degree of yarn packing, enhanced impact resistance and costeffective fabrication. However, some in-plane elastic properties, notably resistance to shear, and strength are sacrificed. Knitted (2-D) fabrics contain chains of interlaced loops. Depending on the orientation of the looping yarn, knits may be classified as either warp or weft. In warp knitting, the looping yarns run in the warp or longitudinal direction and in weft knitting the yarns travel in the weft or horizontal direction. Both fabrics are formed using similar fabrication schemes. The most common mechanism used is the latch needle. Many such needles are employed simultaneously in fabricating the knit. As the process is repeated, 217 © 1999,Woodhead Publishing Ltd

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the series of interlaced loops that are formed constitute the fabric. Knitted fabrics provide a high degree of formability and enhanced in-plane shear resistance. As a result, their application to high strain composites, such as inflatable skins, is readily apparent. Finally, increased directional stability can be obtained by adding laid-in yarns in the desired directions. Perhaps the most simple way of adding through-the-thickness reinforcement is the stitching process. An industrial size sewing machine is usually employed whereby a needle is used to penetrate the layers of fabric and pull the stitching yarn through the preform. Though cost-effective, considerable fiber damage occurs through needle penetration. The resulting reduction in composite strength can be appreciable, making stitching an unattractive option. In recent years, novel stitching techniques have been developed where the fibers are effectively spaced to reduce breakage greatly during needle penetration.

7.1.2 3-D fabrics 3-D knitted fabrics are akin to their 2-D brothers. They may be produced by either weft knitting or warp knitting process. Additional strengthening is accomplished by the use of laid-in yarns in the mutually orthogonal direction. The knitted preform which deserves the most attention is the multiaxial warp knit. The knit consists of longitudinal, latitudinal and bias (±q) yarns held together by a through-the-thickness tricot stitch. These 3-D knits possess the characteristics of unidirectional laminates while enjoying enhanced stiffness and strength in the thickness direction. 3-D weaving is achieved through a modification of the traditional 2-D weaving process. The two main types of 3-D woven fabrics are angleinterlock and orthogonal structure. Angle-interlock weaving is carried out by utilizing multiple harnesses on a conventional loom. The shifting sequence of the harnesses determines the undulation of the warp yarns. Many geometric variations are possible owing to the unlimited combinations of loom configuration and harness sequencing. These multilayer interlocked structures are ideal for thick-section composites. The reinforcement in the thickness direction may be tailor designed to enhance composite impact resistance. However, the low shear performance and limit to shape geometry make woven fabrics an undesirable option in many applications. Orthogonal woven fabrics possess three sets of mutually perpendicular yarns. Inherent in such a structure are matrix-rich regions between the intersections of the three sets of yarns. Fabrication of these preforms is accomplished by inserting alternating, in-plane yarns between the stationary thickness direction yarns. In this fashion, both Cartesian and cylindrical geometries are possible [2].

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In summary, the limitations of the weaving, knitting and stitching processes include poor shear resistance, limited strength in the primary loading direction, and the inability to produce complexly shaped parts. These shortcomings, as will be seen, are largely overcome with the adaptation of braiding.

7.2

2-D braiding

Braided fabrics (2-D) may be either circular or flat, where the flat braid is a special case of the more common circular braid. The similarity between the machines used to form these fabrics suggests a starting point to further explain their structure. Traditional circular braiders utilize a horngear arrangement as shown in Fig. 7.1(a). The gear train is covered by a track plate which has intertwining tracks used to guide the yarn carriers. The horngears ‘pass’ the yarn carriers to and from each other in an alternating fashion as shown in Fig. 7.1(b). For the case of flat braiders, the tracking system does not form a complete circle (Fig. 7.1c). In this configuration, the end horngears have an uneven number of slots which allow the yarn

7.1 Mechanisms and samples of 2-D braids after [1].

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7.2 A bank of flat braiders (compliments of Foster-Miller, Inc.).

carriers to reverse their paths and form a flat braid. Recently, flat braiding machines have been developed where a series of straight braider ‘banks’ are used to form thin-walled, structural shapes [3] (Fig. 7.2). Circular braids are usually formed over an axisymmetric mandrel which determines the final shape of preform. In addition, axial laid-in yarns may be used to increase longitudinal stiffness. Figure 7.1 also shows some common 2-D braids. By specifying the location of yarn carriers on the machine, different braiding patterns may be accomplished. The pattern of Fig. 7.1(d) may be loosely compared to a twill weave and that of Fig. 7.1(e) to a plain weave. Figure 7.1(f) shows a regular braid with axial in-laid yarns. Owing to the symmetric machine arrangement, braider yarns are oriented at equal and opposite angles about the longitudinal axis. This angle may be directly determined by machine operating conditions. A Wardwell 72 carrier circular braider is shown in Fig. 7.3. Finally, while 2-D braids offer cost-effective fabrication, the limitation in available braid geometry and their 2-D nature has restricted their use.

7.3

3-D braiding

3-D braids are formed on two basic types of machines. These are the horngear and Cartesian machines which differ only in their method of yarn carrier displacement. While the horngear type machines offer improved braid speed over the Cartesian machines, the Cartesian machines offer compact machine size, comparatively low development cost and braid architectural versatility.

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7.3 Wardwell 72 carrier circular braider (compliments of the Center for Composite Materials, University of Delaware).

Horngear machines with square or circular arrangement are employed in the fabrication of solid braids (Fig. 7.4a). Present-day machines are limited to 24 yarn carriers and therefore limit the size and shape of preform. The micro-geometry of braid is also restricted and is shown in Fig. 7.4(b). As can be seen, the braider yarns form intertwined helical paths throughout the structure. To allow for more flexibility in preform size, shape and microstructure, new braiding processes have been introduced. These include AYPEX [4], interlock twiner [5,6], 2-step [7], 3-D solid (Fig. 7.5) and Cartesian [8] which is more commonly referred to as four-step or track and column in the literature. An excellent recent review of textile preforming methods is supplied by Chou and Popper [9]. Of all the 3-D braiding processes, the 3-D solid and Cartesian methods represent the apex of braiding technology. Since they differ mainly in approach to yarn carrier displacement (horngear vs. track and column), we need only understand a single process and the structures that may be formed.

7.3.1 Cartesian braiding process The basic Cartesian braiding process involves four distinct Cartesian motions of groups of yarns termed rows and columns. For a given step, alternate rows (or columns) are shifted a prescribed distance relative to each

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7.4 Solid braid fabrication and geometry.

7.5 Method of advanced 3-D solid braiding (compliments of Toyoda Automatic Loom Works, Ltd).

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7.6 The Cartesian braiding process.

other. The next step involves the alternate shifting of the columns (or rows) a prescribed distance. The third and fourth steps are simply the reverse shifting sequence of the first and second steps, respectively. A complete set of four steps is called a machine cycle (Fig. 7.6). It should be noted that after one machine cycle the rows and columns have returned to their original positions. The braid pattern shown is of the 1 ¥ 1 variety, so-called because the relation between the shifting distance of rows and columns is one-to-one. Braid patterns involving multiple steps are possible but they require different machine bed configurations and specialized machines. This unique ‘multi-step’ braiding technique is what renders Cartesian braiding a versatile process. Track and column braiders of the type depicted in Fig. 7.6 may be used to fabricate preforms of rectangular cross-section such as Tbeam, I-beam and box beam if each column and row may be independently displaced. Cartesian braided composites offer excellent shear resistance and quasi-isotropic elastic behavior due to their symmetric, intertwined structure. However, the lack of unidirectional reinforcement results in low stiffness and strength, and high Poisson effect. To help eliminate this, some advanced machines allow for axial yarns to be fed into the structure during fabrication.

7.3.2 Braid architecture, yarn grouping and shapes If one allows for multiple steps in a machine cycle, independent displacement of tracks and columns, and non-braider yarn insertion, the Cartesian braiding process is capable of producing a variety of yarn architectures, hybrids and structures. Consider the eight-step braid cycle shown in Fig. 7.7, which also shows the phenomenon of yarn grouping. Yarn groups are sets of yarn tows that travel the same path. A multistep braiding process may have multiple yarn groups and a varying number of

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7.7 Sample multiple step cycle showing idealized architecture (repeat geometry) and yarn grouping.

yarns per group. It is possible to tailor the location of the yarn groups within the preform cross-section. That is to say, the braid cycle (i.e. shifting sequence of tracks and columns) that will yield the desired grouping of yarns may be determined and different fibrous material utilized for the tows that make up a given group. In this way, unique hybrid composite materials may be formed which benefit both from the 3-D integrated nature of the braid and from the hybrid effect and select yarn placement. The existence of yarn groups implies that sets of yarns trace the same path on the machine bed. After one complete machine cycle, each yarn in a group has moved to its leading yarn’s location. This in turn implies that the braid geometry produced during one machine cycle (repeat) is the repeating geometry for the entire structure. That is to say, a cross-sectional slab of preform with the length produced during one repeat may be ‘stacked-up’ on top of one another to reproduce the entire preform (Fig. 7.7). It is possible, within Cartesian braiding process limits, to specify this braid architecture and determine the braid cycle which will yield it. It may be seen that knowledge of this repeat braid geometry is essential for future prediction of braided composite properties. One way of producing a braided preform with a complex cross-sectional shape is through implementation of the universal method (UM) of braiding [10]. The basic concept behind the UM is to cut the complex crosssection of the preform into finite rectangular elements and then to braid these elements in groups. Since any shape may be estimated through a suitable number of rectangular elements, the UM provides a plausible means

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7.8 Five steps involved in implementing the universal method of braiding a complex shape.

to determine an appropriate braid plan. Additionally, yarns may be added to or removed from the braiding process in order to vary the cross-section along the length of the braid. The UM utilizes only one braiding pattern for a preform. It is essentially a series of four-step 1 ¥ 1 braid cycles which isolate the ‘rectangles’ of the complex cross-section and braid them in sequence. This method is demonstrated in Fig. 7.8 using an I-beam as an example. Since any shape may be estimated through a suitable number of rectangular elements, the UM is applicable to curved shapes as well. Additionally, the approach may be readily implemented through appropriate computer code and is piece-wise applicable to variations of the crosssection along the length of the braid.

7.3.3 Fabrication of braided structures The equipment used in the fabrication of 3-D Cartesian braided structures possesses five basic components. These are the machine bed, the actuating system, the take-up and braid compaction mechanism, the yarn carriers and the interface/control system. Inherent in the process of 3-D braiding is a limiting ratio of machine bed size to preform cross-sectional dimensions. The larger the spacing between yarn carriers on the machine bed (the spacing directly determines the

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amount of yarn a carrier can hold), the more difficult it becomes for the braid to be formed owing to the ‘pulling apart’ action of the yarns themselves. Some ingenious methods have been devised to overcome this limit to braidable cross-sectional size of preform [11]. However, as a rule, there is a trade-off between the length of preform and the cross-sectional size of preform which may be fabricated from a single machine set-up. With this aside, the number of tracks and columns and the resulting yarn carrier spacing on a Cartesian braider’s bed are important specifications. Figure 7.9 shows a 10 track by 24 column Cartesian braider that integrates stationary spacer tracks for the sole purpose of inserting axial (longitudinal) yarns. The transverse insertion (as seen in Fig. 7.9) is carried out manually. However, some advanced machines allow for this step of the process to be automated. The actuating system of choice for the Cartesian braiding machines is pneumatic. When one considers the required displacement forces, precision of displacement and number of actuators involved, a pneumatic drive system becomes an attractive option. Figure 7.10 shows a 20 track by 20 column Cartesian braider that is capable of displacing each track and column independently. To accomplish this, small pneumatic cylinders are utilized in series for each track and column. As previously mentioned, this results in the ability to fabricate complexly shaped or hybrid (yarn grouping) preforms for specialized applications. Figure 7.10 shows some samples of the types of braids that may be formed on a machine with this capability. Take-up and compaction of the braid is a critical part of the process. For a continuous fabrication process, the braid must be drawn or taken up. Take-up is carried out after a complete machine cycle and before compaction. As a result, the take-up distance directly determines the braid pitch length (i.e. the length of braid formed during one machine cycle) and resulting architecture. It is therefore essential to have precise control of the amount of take-up. This is most commonly accomplished by utilizing a motor in conjunction with a worm gear assembly. Without interyarn friction, the yarn orientation angle within the braid would be determined solely by the angle that the not-yet braided yarn makes with the braid axis. In reality, interyarn friction does exist and allows braider yarns to remain in place once compacted. As a result, a much greater orientation angle may be obtained. The idea behind the braid compaction is to pack the yarns up to the desired orientation and then allow interyarn friction and interlacing to hold the yarn in place. To the authors’ knowledge, this is commonly accomplished by manually inserting a rod in the braid convergence zone and gently compacting the braid after each complete machine cycle. It is suggested that the next generation of Cartesian braiding machines incorporate an automated version of this critical step. As

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7.9 A Cartesian (four-step) pneumatic braider with axial yarn insertion (compliments of the Center for Composite Materials, University of Delaware).

228

228

3-D textile reinforcements in composite materials 7.10 A Cartesian (multistep) pneumatic braider with independent track and column control (compliments of the Center for Composite Materials, University of Delaware and Atlantic Research Corporation).

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7.11 Schematics of conventional and adapted yarn carriers utilized in braiding 2-D and 3-D preforms, respectively (adapted yarn carrier schematics compliments of Atlantic Research Corporation).

mentioned earlier, however, larger bed arrangements cause the braid to be ‘pulled apart’ and even a compaction step may not be enough to form the braid. The design requirements of a yarn carrier include compact size, maintained yarn tension and yarn rewind. As a yarn carrier moves from the outside toward the center of the machine bed, the distance between carrier top and braided fabric shortens. The slack yarn so produced must be rewound by the yarn carrier or it will become entangled with other similar yarns. Figure 7.11 shows schematically the workings of both conventional and adapted yarn carriers used in braiding processes. For the adapted yarn carrier [11], rewind, tensioning and yarn feed are all accomplished through the yarn spool.

7.3.4 Braid consolidation The standard approach to preform consolidation, utilizing a thermosetting resin, is resin transfer molding (RTM). RTM is a straightforward method of injecting a resin system into a preshaped mold cavity that contains the 3-D braided preform. Although ideal for low to mid volume production runs, issues such as preform wet-out and residual stress warpage frequently arise. To address these issues, such adaptations as the utilization of preimpregnated yarn tows and room temperature cure resins are suggested. However, there is still much research to be done in this area. Additionally, the consolidation of complexly shaped parts presents a special challenge owing to the change in preform dimensions when it is removed from the braiding machine. Figure 7.12 shows a schematic of a typical RTM set-up used in the consolidation of 3-D braids.

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7.12 Schematic of typical RTM set-up for braid consolidation.

7.3.5 Braided composite characterization The characterization of braided composite microstructure may be investigated at two scales. The first is the yarn tow size and the second is the fiber (or filament) level. Braid packing during preforming and consolidation may be determined by a number of factors such as yarn tension, yarn twist, braid compaction and molding pressure, injection pressure and resin viscosity. The final arrangement of yarns and fibers directly determines the final composite elastic and strength properties and must first be quantified in order to be related to the processing history. The packing of yarns within a four-step 1 ¥ 1 (mono-fiber) braided composite is fairly well documented [12–15]. However, when one deals with a hybrid or complexly shaped braided composite (which is possible with multiple step track and column braid cycles), the variation in braid microstructure may be significant. As an example, consider a microstructural cross-section of a two-sided hybrid composite which has been consolidated through RTM (Fig. 7.13). The unique yarn paths that result from the yarn group producing braid cycle cause an unorthodox yarn packing. The goal is to quantify this effect so that a basic understanding of yarnto-yarn interaction and yarn cross-sectional deformation may be gained. In Figure 7.14, the microstructure of a four-step 1 ¥ 1 (Kevlar-49, 0.74 mm/0.029 in. diameter) braid with transversely inserted carbon tows (NG Corp., 0.30 mm/0.012 in. diameter) is shown. As expected, owing to the near uniform length-wise pressure from neighboring braider yarns, the transversely inserted carbon tows deform to a near rectangular cross-sectional shape. In return, they are also seen to cause a displacement and flattening of the braider yarn tows. The calculation or measurement of braided composite fiber volume

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a)

c)

b)

7.13 Cross-sectional microstructure of two-sided hybrid composite braids showing the yarn packing (a) and yarn interaction among the different tows in the hybrid composite (b and c).

7.14 Cross-sectional microstructure of a four-step braided composite with transverse carbon tow insertion.

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7.15 Measurement of fiber volume fractions for the two-sided braided hybrid composite sample through digital analysis.

fraction is more readily obtained through the identification of a unit cell of the structure [1]. However, when dealing with multistep, multiple fiber and filler material braided composites, the identification of a unit cell is tedious. Here, some possible representative cells for some sample braided composite microstructures are suggested. Focus is then on the measurement of the fiber volume fraction within the tows of the cell so that it may be quantitatively related to the aforementioned observed yarn packing. The measurement of the yarn tow fiber volume fraction may be carried out through use of digital image analysis. After a representative cell of the composite microstructure is chosen, a series of random image samples are picked from within the fiber bundles. These image samples are then thresholded. In other words, a gray-level value is chosen as a cut-off such that all image pixels above and below this value are made white and black, respectively. The pixels in the resulting binary (black and white) image may then be counted and a ratio of white pixels (fibers) to total pixels (fibers and matrix) computed. Ideally, this ratio should represent the fiber volume fraction within the yarn tow. It should be noted that some error is introduced by this method because of such factors as image resolution and improper thresholding. Figure 7.15 shows the chosen representative cell for the two-sided hybrid composite. The measured fiber volume fractions for carbon and Kevlar are 74% and 64%, respectively. This rather high fiber volume fraction within the tows (packing fraction in the literature [16]) is comparable to that found in a four-step 1 ¥ 1 braided composite [1,10,17–19]. It should be noted that the high fiber volume fraction measured in this sample is probably due to the high braid compaction during RTM of the preform. The slightly greater Vf of carbon over that of Kevlar may be attributed to the smaller fiber diameter (about 7 mm) compared with that for the Kevlar filaments (about

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7.16 Measurement of the fiber volume fractions for the four-step 1 ¥ 1 Kevlar braider, transversely inserted carbon fiber, hybrid composite sample.

35 mm). The transversely inserted carbon tows of the braided composite shown in Fig. 7.16 have the effect of pinching the braider yarns at additional contact areas along their length. The net result is a more highly packed tow which yields a measured fiber volume fraction of 73%.

7.3.6 Braided composite performance The prediction of the elastic and strength properties of 3-D braided composites presents an interesting challenge. Although much progress has been made in this area [1,14,15,20–23], there is still much to be done as it pertains to hybrid and complexly shaped braided composites. What is presented here is focused on the measured tensile response and hybridization effects of braided composites. The goal is to quantify some of the dominant parameters involved in determining the composite elastic constants so that 3-D braided hybrid composites of the future may be tailor designed to respond to the intended loading condition. Uniaxial tension tests were performed on a group of pure Kevlar and carbon/Kevlar hybrid composite samples (example shown in Fig. 7.13 with the braid cycle used for fabrication). From the literature [24], it is suggested that a strain gage size be selected such that its deformable length be greater than or equal to the unit

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cell size of the textile composite. This is to ensure that strain gage deformation corresponds to an average deformation across a representative unit of the braided composite microstructure. For the braided specimens in question, this size corresponds to the surface pitch length. The stress and strain data for each type of specimen (pure Kevlar and carbon/Kevlar hybrid) have been averaged and the results are presented below. For the pure Kevlar (PK) braided samples, the measured longitudinal strain is plotted with respect to the transverse strain (x - 1 for clarity) in Fig. 7.17. The measured Poisson ratio, which is taken from the initial linear portion of the graph shown in Fig. 7.17, is reported to be near unity. This strongly suggests a material elastic response that is dominated by the fiber architecture. The presence of the matrix, which is in the order of 40% by volume, is negligible. The braided composite appears to be behaving in a truss-like fashion. For increased loading, the slightly decreasing, non-linear nature of the Poisson ratio also suggests a fiber alignment or locking-out effect. Comparison of the measured Poisson ratios for the PK and the carbon/Kevlar (CK) hybrid composite samples is also shown in Fig. 7.17. For the CK samples, the initial slope of the curves yields a Poisson ratio of

7.17 Longitudinal vs. transverse strain in pure Kevlar and Kevlar/carbon hybrid tension samples.

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1.2 and 0.8 for measurements taken on the carbon and Kevlar sides, respectively. Since the carbon-side Poisson measurement is within 20% of the PK ratio and if we assume that an isostrain condition exists in the longitudinal direction, it may be concluded that the more stiff carbon fibers dominate the transverse contraction. The presence of the high modulus carbon fibers also produces a more pronounced nonlinearity of the Poisson contraction. It is believed that the lower shear resistance of the carbon fiber, compared with the Kevlar material, magnifies the fiber alignment effect. In Fig. 7.18, the nominal tensile stress is plotted versus the longitudinal strain. For both samples (PK and CK), a linear tensile stress–strain relation is seen to exist. The near equality of the slopes for the carbon and Kevlar sides of the CK sample determines that an isostrain condition exists in the longitudinal direction. The calculated tensile modulus for the PK sample is 41 GPa (6 ¥ 106 psi) while that for the CK sample (averaged) is 74 GPa (10.7 ¥ 106 psi). The fracture of all the specimens was catastrophic. Linear stress–strain behavior was observed until the ultimate strength was reached, at which time sudden and total fracture occurred. The average ultimate strengths of the PK and CK samples are 793 MPa (1.15 ¥ 105 psi) and 896 MPa (1.3 ¥ 105 psi), respectively. The average failure strain of the composites is found to be 1.9% and 1.1% for the PK and CK specimens, respectively. Figure 7.19

7.18 Nominal tensile stress vs. longitudinal strain for pure Kevlar and Kevlar/carbon hybrid tension samples.

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7.19 Typical fracture of braided composite specimens (left) and fracture across the shear face in the Kevlar/carbon hybrid sample (right) (compliments of the Center for Composite Materials, University of Delaware).

7.20 Micrographs showing crack initiation at a void within the intertow regions and propagation along the matrix/tow interface and crack arrest at the Kevlar interface.

shows typical fractured PK and CK specimens. For all samples, fracture occurred along a near 45° shear plane of the material (Fig. 7.19). Observation of the fracture surface near the carbon/Kevlar interface region reveals a dominant growth of cracks in the thickness direction of sample. It is believed that near the carbon/Kevlar interface region, the exaggerated difference in transverse strain is adding to a ‘pulling apart’ of the carbon tows. The ultimate result is the breaking away of carbon tows from the matrix, carbon tow failure and final tow pull-out. A likely source for the crack initiation, as with many polymer matrix composites, is voids. Figure 7.20 shows a series of cracks which initiate at an internal void. The cracks are seen to take the paths with the highest driving potential (minimizing energy of the system) and the least resistance. The cracks appear eager to cross the carbon tows but reluctant to negotiate the Kevlar tows. This

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crack initiation, propagation and arrest sequence may prove to be an exploitable quality of hybrid composites.

7.4

Summary

Textile preforms offer a wide selection of fabrication techniques. Ranging from simple 2-D weaves to the more complex 3-D braids, these fibrous arrangements have much to offer the composite industry. It is within this processing science that true control of yarn placement may be realized, resulting in the fabrication of unique structures. Although past work has added greatly to the existing science base, a comprehensive approach to the complete design of 3-D braided composites is continuously being developed. In general, the advantages of 3-D braiding as a method of preforming include the formation of a delamination resistant structure, the ability to fabricate thick and complex shapes, and single procedure, net shape preforming. Structural composites formed by this method which possess either a complex cross-section, a hybrid fiber arrangement or a desired microstructure are tailor designed to yield the required performance for the intended application. Innovative braid geometries were introduced to demonstrate the feasibility of fabricating a wide range of preform architectures given an advanced braiding machine. Additionally, interesting distributions of yarn groups have been shown, which suggest an application to hybrid composites. The development of prototype braiding equipment shows that a variety of structures may be automatically fabricated. Issues such as braid convergence, processing cost (time) and braid stability have also been addressed. The dominant limiting factors in braiding include: the entire supply of braiding yarns (packages or yarn carriers) must be moved, the machine size is large relative to the braidable cross-sectional size of preform, only limited lengths of braid may be formed, the range of fiber architecture is constrained by the process, and different machines are usually required to vary the braiding pattern. The development of advanced braiding processes and equipment is forever attempting to break free of these shackles. The consolidation of 3-D braided preforms is an issue in itself. While RTM offers a reliable method of preform infiltration, complexly shaped structural parts and open panel structures are but a few of the challenges that must be addressed. Braided composite microstructural characterization is the first step towards a study of elastic performance. The extent of yarn deformation (packing) resulting from preform consolidation was discussed through composite cross-sectional micrographs.Through digital image methods, the fiber volume fraction of select hybrid composites was measured and representa-

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tive cells of composite microstructure suggested. A comparison of the elastic performance of Kevlar/epoxy and carbon/Kevlar hybrid composites was presented. The tension test results show a linear stress–strain relationship for both specimen types within the range of the applied load. The calculated tensile moduli for the carbon/epoxy and hybrid composite were found to be 41 and 74 GPa, respectively. In addition, the Poisson ratio of near unity for both specimen types strongly suggests a fiber-dominated elastic material response. The difference in hybrid composite transverse strain due to the differing constituent fibrous materials is found to be appreciable. It is believed that this discrepancy in Poisson contraction causes the propagation of transverse cracks primarily within the carbon tows and ultimately leads to catastrophic composite failure. The initiation, growth and arrest of cracks due to the hybridization of the composite specimens were also observed to occur. Composite ultimate strength and strain to failure were found to be 793 MPa and 1.9% for the Kevlar/epoxy sample and 896 MPa and 1.1% for the carbon/Kevlar hybrid.

7.4.1 Future research In its present state, the braiding of 3-D articles, be it accomplished through use of a Cartesian (track and column) braider or a horngear type machine, has an inherent handicap. This shortcoming is the braidable size and length of preform. As it stands, 3-D braiding is only applicable, from a cost perspective, to the fabrication of high-performance, specialized structural composite parts. Inventive, novel methods of braiding need to be developed where more ‘braid for the buck’ is realized. It is suggested that the area of open structures be investigated so that the limited amount of braid which is formed is applied in an efficient manner.Additionally, fiber insertion techniques such as weaving and stitching may be coupled with the 3-D braiders of the future so that the maximum amount of fiber is introduced during the net shape braiding process. Special hybridization, use of piezo-ceramic materials and the imbedding of lineal sensors may also make the high cost of these high end-performance braided composites more attractive. It is strongly believed that the use of computer solid modeling for the simulation of braided composite microstructure will bear much fruit. Once any 3-D braid, be it hybrid, complexly shaped or voided in nature, is adequately represented through a simulation, the prediction of composite mechanical properties will be easy.

7.5

References

1. Chou, T.-W., Microstructural Design of Fiber Composites, Cambridge University Press, Cambridge, Chapters 6 and 7 (1991).

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2. Herrick, J., ‘Multidirectional orthogonal composite structures’, paper presented at 4th Textile Structural Composites Symposium, 24–26 July, 1989. 3. Thomson, D.T., ‘Braiding applications for civil infrastructure’, paper presented at Proceedings of the International Composites Expo, 19–21 January, 1998, Nashville, TN. 4. Brookstein, D., ‘Braiding of a three-dimensional article through select fiber placement’, US Patent 5 123 458, 23 October, 1994. 5. Spain, R. and Bailey, C., ‘Apparatus and method for braiding fiber strands and stuffer fiber strands’, US Patent 4 984 502, 15 January, 1991. 6. Ivsan, T.J., ‘Apparatus and method for braiding fiber strands’, US Patent 4 922 798, 8 May, 1990. 7. McConnell, R. and Popper, P., ‘Complex shaped braided structures’, US Patent 4 719 837, 19 January, 1988. 8. Florentine, R., ‘Apparatus for weaving a three-dimensional article’, US Patent 4 312 261, 26 January, 1982. 9. Chou, T.-W. and Popper, P., ‘Recent developments in the research of textile structural and functional composites’, paper presented at The Fifth International Conference on Composites Engineering, 5–11 June, 1998, Las Vegas, NV. 10. Li, Wei, ‘On the structural mechanics of 3-D braided preforms for composites’, PhD thesis, North Carolina State University, March 1990. 11. Brown, R.T., ‘Design and manufacture of 3-D braided preforms’, paper presented at 5th Textile Structural Composites Symposium, Philadelphia, PA, 5 December, 1991. 12. Pastore, C.M. and Ko, F.K., ‘Modeling of textile structural composites Part I: processing – science model for three-dimensional braiding’, J. Textile Inst., 81, 480–490, 1990. 13. Kishore, P. and Chou, T.-W., ‘Elastic property prediction from preform modeling for 3-D textile structural composites’, Composite Sci. Technol., 53(3), 213–219, 1995. 14. Ito, M., ‘Effects of yarn undulation on the stress and deformation of textile composites’, PhD Dissertation, University of Delaware, May, 1995. 15. Smith, L.V. and Swanson, S.R., ‘Micromechanics parameters controlling the strength of braided composites’, Composite Sci. Technol., 54(2), 177–184, 1995. 16. Hearle, J., Grosberg, P. and Backer, S., Structural Mechanics of Fibers, Yarns, and Fabrics, Vol. 1, Wiley-Interscience, New York, 1969, p. 80. 17. Ko, F.K., ‘Tensile strength and modulus of a three-dimensional braid composite’, in Composite Materials Testing and Design (Seventh Conference), ASTM STP 893, Philadelphia, PA, 1986, pp. 392–403. 18. Pastore, C.M. and Ko, F.K., ‘A processing science model for three-dimensional braiding’, SAMPE Quarterly, 19(4), 22–28, 1988. 19. Yang, J.M., Ma, C.L. and Chou, T.-W., ‘Fiber inclination model of threedimensional textile structural composites’, J. Composite Mater., 20(5), 472–483, 1986. 20. Byun, J.H., ‘Process–microstructure–performance relations of threedimensional textile composites’, PhD Dissertation, University of Delaware, May, 1992. 21. Mohajerjasbi, S., ‘Structural mechanical properties of 3-D braided composites’, PhD Dissertation, Drexel University, 1993. 22. Abusafieh, K. and Franco, E., ‘An experimental and numerical study of response

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of 3-D braided structural textile composites’, in Proceedings of American Society for Composites, 9th Technical Conference, Newark, Delaware, pp. 1118–1125, 1994. 23. Franco, E., ‘Finite element simulation of the micromechanical behavior of threedimensional braided composite materials’, MS Thesis, Drexel University, 1995. 24. Hartranf, D., Parvizi-Majidi, A., and Chou, T.-W., Tensile Testing of Textile Composite Materials, NASA Contractor Report 198285, February, 1996.

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8 3-D forming of continuous fibre reinforcements for composites O.K. BERGSMA, F. VAN KEULEN, A. BEUKERS, H. DE BOER AND A.A. POLYNKINE

8.1

Introduction

8.1.1 The earliest fibres, fabrics and composite structures The control of fire and the spinning of continuous strands of fibres are probably the most important discoveries humans ever made. Both inventions made it possible for a naked human to survive in non-tropical conditions. Yarns and derivatives, like robes and textile fabrics, provided humans with a portable and personal tropical microclimate by clothes and structures so that they could withstand most climatological conditions. It made humans able to migrate from the crowded and unhealthy tropical zones to the large, cool plains and mountainous areas, free of diseases, but rich in animals, vegetables, minerals and water. Compared with animal skin, flexible textile was a big step forward. The usage of light fabrics that were adjustable to local conditions made a big and relatively fast migration of hunters and gatherers possible over all continents, except Antarctica [1]. Humans could only use local natural growing fibres. They differ from modern artificial synthetic fibres in length. Instead of the continuous filaments, nature offers only short fibres, like animal hair. These protein-based fibres are provided by animals, such as sheep, goats, camels, llamas and rabbits. Various forms of vegetable cellulose-based fibres were available as well: in a hairy form taken from seeds (cotton) and fruits (coir) or as fibres extracted from basts and leaves, like jute, sisal, hemp, flax, yucca, palm, rice, grass, ramie and rattan. Several of these materials could instantly be used to make basket-like structures or to wattle hedges and walls. However, to handle and to make the staple fibres suitable for knitting and weaving, as shown in Fig. 8.1, the spinning and intertwining of yarns was essential. A distaff, a small portable wooden spinning wheel on a vertical axle (Fig. 8.2), had already been known in prehistoric times, far before the wooden wheel on the horizontal axle was invented to make wheeled transport possible. Depending on the local climate, people started to use ropes, felt (paper-like textile) and woven 241 © 1999,Woodhead Publishing Ltd

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8.1 Weaving of plain fabrics.

8.2 Spinning of yarns.

fabrics of different natural materials for several purposes. In many parts of the world the same ancient design of clothes, tapes, baskets and tents, all continuous fibre structures, are still in use and almost unchanged. Up to this day, 3-D textile structures still offer nomadic families the best protection against extreme temperatures. The peaked black tent, an example of a controlled draped fabric, is used in the hot dry deserts. In the cold snowy areas circular tents, yurts, are used. These circular trellis structures, limited in shear by a doorframe and a circumferential rope, are covered with wattle and felt. As soon as communities started to settle, the flexible and foldable textile structures were transformed, step by step, in more protective rigid wattle and daub or straw reinforced clay structures. In fact, it was the first creation of artificial composites, a combination of different materials to

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Table 8.1. Elastic properties of some natural composites compared with steel Material

Steel 0.2 carbon quenched Piano wires, springs

Density (kg/m3 ¥ 103)

Young’s modulus (N/m2 ¥ 109)

7.8

210

7.8

210

Yield stress (N/m2 ¥ 106)

Yield strain (%)

Elastic energy/ weight (J/kg)

773

0.2

99

3100

0.8

1590

Animal Sinew Buffalo horn Bovine bone Ivory

1.3 1.3 2.1 1.9

1.24 2.65 22.6 17.5

103 -124 -254 217

4.1 -3.2 -1.4 1.2

1620 1530 846 685

Hardwood Ash Birch Elm Wych elm Oak

0.69 0.65 0.46 0.55 0.69

13.4 16.5 7.0 10.9 13.0

165 137 68 105 97

1.0 1.0 1.0 1.0 1.0

1196 1050 740 950 703

Softwood Scots pine Taxus brevifolia

0.46 0.63

9.9 10.0

89 116

0.9 1.3

870 1100

Notes: 1 Northern hardwoods, sinew and horn were the basic structural materials for the laminated composite bows and chariots from Mesopotamia and Egypt. 2 Taxus baccata was used for medieval longbows. 3 Horn, a natural thermoplastic polymer was especially applied in the compression loaded areas. 4 Sinew, superior in tension, was employed for strings and bow-reinforcement; more in general it was used as a shrinking (smart) robe to encapsulate and to connect different components.

obtain improved or modified properties. The earliest laminated composite structures, like composite bows and chariots, were glued layered structures of natural composites such as wood, bone, sinew and horn [2]. They were all fibrous materials, based on cellulose, collagen and keratin, which had very specific capabilities, already discovered and understood by the prehistoric craftsman (Table 8.1). All applications mentioned in this part of the Introduction, from textile structures more than 50 to 8 millennia ago to the composite shelters, bows and chariots from 12 to 5 millennia ago were not developed overnight, the structures were sometimes very complex and took probably centuries

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of experimentation and evolution. The results are still striking for the craftsmanship, knowledge of materials and the sophisticated manufacturing processes [2]. For a successful introduction of modern artificial composites, equal understanding must be developed and supported by modern mathematical modelling and computers. Besides the application of new synthetic materials, the biggest break with the past will be the introduction of low-cost and fast manufacturing and simulation equipment to replace the time-consuming, high-priced and mystic skills of craftsmen.

8.1.2 The renaissance of fibre reinforced composites During the second half of this century, the last decade in particular, a true revival started of using light-weight composite structures for many technical applications. In the beginning, the introduction of fibre reinforced polymers was only driven by particular electromagnetic characteristics. More than a century ago, cotton reinforced rubbers and phenolics were used for insulators. Later, glass fibre reinforced polyesters were applied for radomes, minehunters and minesweepers. In the 1980s, high-technology composites based on carbon- and aramid-fibre reinforced epoxies became popular to improve the structure performance of spacecraft, military aircraft, helicopters and all kinds of sports and racing equipment. Initially, the sky was the limit as far as the price was concerned. Nowadays, cost reduction during manufacturing and operation is the technology driver, and examples are large structures in civil applications (carbon fibre reinforcement of bridges and buildings) or in corrosive chemical or marine environments (glass fibre reinforced bridges, piers, pipes, tanks, etc.). One of the latest developments is the application of continuous fibre reinforced polymers to protect people against impact and fire and a more general tendency to design means of transport which are less damaging to our environment. Some typical examples are shown in Fig. 8.3. Like in prehistoric times, the reinforcing fibrous materials are applied in different forms; short or continuous, as tapes, mats or plain weaves. Although the vegetable fibres mentioned earlier are gaining renewed interest, most structural applications are now reinforced with synthetic fibres with constant quality. Inorganic fibres are applied such as glass, metal and silica, organic fibres based on natural cellulose and protein polymers or synthetic fibres based on condensation or addition polymers. There are innumerable types of synthetic fibres, such as single filaments or tows, neat or post-treated, stretched or carbonized. Nowadays, the most popular reinforcing fibres with respect to price–performance are the low-cost (E) glass fibres and the high modulus (HM) aramid- and high-tenacity (HT) carbon fibres (Table 8.2).

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8.3 Some typical examples of continuous fibre reinforced products.

8.1.3 Industrial manufacturing of composite components A successful introduction of reinforced polymer materials and components depends on the availability of fast and reliable manufacturing techniques. In general, new materials are more expensive than the materials they have to compete with. Added value in mechanical, chemical or physical characteristics is only convincing when the price performance is competitive. No parameter is so determinant for the price–performance ratio of advanced structures as the cost to manufacture. Once the materials have been accepted and established, the performance per unit weight gains impor-

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Table 8.2. Fibre properties of some typical natural and synthetic fibres Density (kg/m3 ¥ 103)

Young’s modulus (N/m2 ¥ 109)

Tensile failure (N/m2 ¥ 106)

Strain failure (%)

Natural organic polymer base Jute Hemp Flax Sisal Coir Cotton

1.46 1.48 1.54 1.33 1.25 1.51

10–25 26–30 40–85 46 6 1–12

400–800 550–900 800–2000 700 221 400–900

1–2 1–6 3–2.4 2–3 15–40 3–10

Synthetic organic polymer base HT carbon (T300) HM carbon (M40) HM aramide

1.76 1.83 1.45

230 392 133

3530 2740 3500

1.5 0.7 2.7

Inorganic base E-glass S/R-glass

2.58 2.48

73 88

3450 4590

4.8 5.4

Note: Properties of natural materials are very variable, so the figures shown are averages and collected from a great variety of publications.

tance. Decreasing structural weight, often beneficial for performance improvement, not only reduces the quantity and cost of materials but also often reduces the production time, and consequently the cost of manufacturing. A powerful approach to reach this goal is the matrix reinforcement with proper fibres, to high possible volume fractions, continuous and with a complete control of fibre orientations, in other words to control anisotropy. The success of composite applications, by volume and by number, can be ranked by the success of the applied manufacturing techniques (Fig. 8.4). For all processes shown, suitable for short to continuous fibres, the introductory (pioneering) period was based on thermosetting polymers, from phenolics, polyesters, vinylesters to epoxies. In the case of injection moulding with short (<10 mm) and pressing with longer fibre reinforcements (<100 mm), thermoset polymers are being increasingly replaced by more expensive but technically equivalent or better thermoplastics. However, the main reason for this is the cost reduction by cycle time reduction. Most important among the technologies mentioned is injection moulding of generally small and complex parts. The reinforcement by fibres is limited with respect to length, volume percentage (<35%) and orientation control. Flow-moulding of larger thermoset and thermoplastic shellstructures (SMC and GMT) became important as well, especially for car

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8.4 Industrial manufacturing techniques.

parts. The length (<100 mm) and volume percentage (<45%) of the reinforcing fibres increase. The control of fibre orientations is similar to injection moulding, limited to keeping fibres as random and uniformly distributed as possible. In the case of modern advanced structures (high loads, low weight), where controlled fibre placement is essential, designers and manufacturers still rely on techniques that are labour or capital intensive (laminating by hand or the use of dedicated equipment). In the case of some advanced composite applications, human labour is only replaced by cost reducing and accurate machines in the stage of pre-impregnation and the cutting of patches. Industrial laminating by tape or fabric laying machines is still limited to a few (aircraft) shell structures. The most successful techniques in terms of volume usage are the filament winding of

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pressure vessels and the pultrusion of composite profiles. Typical for the advanced composite sector is the use of continuous fibres (glass, aramid and carbon) and the high fibre volume percentages (<70%). It is still the domain of thermosetting polymers. Although the application of advanced continuous fibre reinforced composites may result in highly satisfactory structural performances [3], the volume and number of applications are still limited. The success of the advanced composites depends completely on the availability of fast and reproducible industrial manufacturing processes. A development of such an industrial process based on 3-D deformation, i.e. draping, of (impregnated) textile fabrics, is the subject of this chapter. The major deformation mechanisms, the geometrical draping–simulation strategies, finite element simulation and the final product optimization, essential for designers and analysts, is outlined in the following sections. The draping process is part of a press-forming cycle, more specifically the press forming of textile fabrics which are impregnated to a certain extent with thermosetting or preferably thermoplastic polymers. Nowadays many industrial impregnation strategies for both thermosetting or thermoplastic polymers are available. Once the fabric has been impregnated and the polymer brought to a deformable state, e.g. by heating, the plain sheet can be formed into a shell structure in seconds by press forming and (re)consolidation in the last phase by application of matching dies. This technology can be used to produce high-quality preforms for the (thermosetting) resin injection or transfer moulding (RTM) of advanced aircraft and car components (Fig. 8.4). Major successes are, however, achieved in the press forming of continuous reinforced thermoplastic composite parts (Figs. 8.5 and 8.6). Similar to the already-mentioned technologies for advanced composites, high fibre volume contents and reproducible fibre orientations are typical for press forming. The speed is comparable with the ordinary injection moulding and flow-moulding of short fibre reinforced parts. The pressure levels are relatively low: for forming, 1 bar or less; for (re-) consolidation, 10–40 bar. When the heating and cooling times are considered as well (for thin-walled structures, a matter of seconds), it is clear that the press forming of advanced composites is not only attractive because of its manufacturing speed, but also because of the light-weight equipment and minimum energy required [4].

8.1.4 Outline of the simulation and optimization strategy The following sections deal with the simulation and optimization of 3-D formed continuous fibre reinforced components. In the scheme shown in Fig. 8.5 the role they play in an integral process of design and analysis is clarified. When automated structural optimization is applied, the scheme

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8.5 Design of advanced composite shell structures.

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8.6 Typical continuous fibre reinforced products, manufactured using a thermoforming process: (a) automotive chassis part; (b) bicycle wheel.

will alter somewhat, as the entire process must be controlled by the applied optimizer (see Fig. 8.19). A general description of thermoforming of continuous fibre reinforced thermoplastic (CFRTP) products is given in Section 8.2. Numerical simulation of the forming process is the topic of Section 8.3. In this section, the discussion is mainly restricted to geometrical approaches. This choice has

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been made because this type of simulation requires modest computer effort and is therefore especially suited for optimization processes. In addition to the simulation of the forming processes, the evaluation of design sensitivities is considered as well. Simulation of the mechanical behaviour of CFRTP products is discussed in Section 8.4, although the discussion is restricted to thin-walled structures. As descriptions of the finite element method can be found in many textbooks, this discussion is restricted. However, the evaluation of the required laminate stiffnesses is non-standard and is therefore included in the present chapter. Automated optimization is addressed in Section 8.5. Here, optimization is carried out on the basis of a so-called approximation concept. This approach replaces the actual optimization problem by a sequence of simpler approximate optimization problems. The main advantages of this approximation concept are: (1) it is applicable without information on design sensitivities and (2) noisy response evaluations can be dealt with.

8.2

Forming of continuous fibre reinforced polymers

8.2.1 Introduction With the development of high-performance continuous fibre reinforced polymers, the need for new production processes became clear. Hand layup, the most important production process for continuous fibre reinforced thermoset structures, is not suitable for most thermoplastic composites. This and the fact that large numbers produced with the hand lay-up process cannot lead to cost-competitive products, are the main reasons for the development of new industrial production techniques for composite materials. Some pressing processes have the potential of becoming cheap and fast, and are therefore suitable for mass production. In this section only the rubber forming process will be discussed, since this low-pressure form pressing process seems among the most promising of its kind.

8.2.2 Rubber forming The rubber forming process is a matched die press forming process. One of the dies, male or female, consists of rubber. Figure 8.7 gives a general outline of the rubber forming process. The important stages in the forming process are: • • • •

preconsolidation (depending on the prepreg type); heating stage (can be included in the preconsolidation stage); forming stage (draping); (re)consolidation stage.

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8.7 A sketch of the rubber forming process.

Heating stage The laminate can be heated by contact heat (conduction heating) between two hot plates. When this is done with sufficient pressure, pre-consolidation can become unnecessary [5]. A disadvantage of the conduction heating method is the fact that the laminate actually touches the heating equipment. Consequently, good release agents must be used to prevent sticking of the material to the plates. Convection heating in an oven is also possible, but will usually be time consuming, and the use of inert gas is sometimes necessary to prevent oxidation of the polymer at high temperatures. Inert gas environments are also recommended when using the very fast infrared heating methods. A disadvantage is that for thick sheets temperature gradients develop through the thickness and this will sometimes restrain formability during the forming stage. Radiation heating, however, is a clean and quick heating method in general. It provides a flexible and a well-manageable heating device. Forming stage When a fabric reinforced thermoplastic laminate is forced in a specific shape by the rubber forming process, the fibre reinforcement has to adjust to that same shape. The continuity of the fibres plays an essential role. In

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8.8 Schematic drawing of the intraply shear deformation mode.

contrast to metals or nonreinforced thermoplastics, continuous fibre reinforced plastics cannot undergo a deformation with only local adjustment of the material. Local bending or curving of the laminate material will influence the entire laminate. There are several ways in which the adjustments of the fibre reinforcement can take place. These adjustments are often called deformation modes. Although composite products generally are composed of laminates, that is to say of more than one layer, the deformation capacity of one individual layer plays a dominant role in the forming process. These deformations are called intraply deformations. In general, five different deformation modes of a single, flat layer of fabric can be identified, as was shown by Mack and Taylor [6], Robertson et al. [7], Heisey and Haller [8] and Robroek [4]: • • • • •

fibre stretching (elongation of the fibres); fibre straightening (undulation of the woven fibres); intraply shearing (trellis effect of the fibres); intraply slip (sliding of the fibres); bucking (in-plane and out-of-plane buckling).

As has already been shown by Robertson et al. [7] and confirmed by Potter [9] and Van West [10], the shear deformation mode (Fig. 8.8) is the most important mode for deforming fabrics into 3-D products. Simulations as discussed in the next section must therefore incorporate this deformation mode. Since the shearing dominates the deformation, it is important to know which forces are needed for the shearing. It appears [4] that the forming forces of these fabric reinforced plastics are small. Normally, a laminate consists of more than one layer. Such a laminate can be represented as a stack of fibre-rich layers alternated with thermoplastic resin-rich layers. In a thermoforming process (such as rubber forming), the resin-rich layers are softened by heating and will have a certain viscosity. They will allow the fibre-rich layers to slip with respect to each other when the laminate is bent

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8.9 Schematic drawing of the interply shear deformation mode.

by forming forces. This deformation mode is called interply slipping (Fig. 8.9). During this slipping the resin-rich layer must act as a lubricant, otherwise the generated tangential stresses become too high and can cause failure of the laminate bend [11]. Since the pressures needed for the forming are relatively low (<1 bar during forming, <40 bar in the final consolidation stage), tooling materials other than the ones used in a metal, matched die, forming process become feasible. The thickness of the fabric after deforming is mainly determined by the local amount of shear [5].When this thickness variation is obstructed, high pressures on the fabric will occur, which will obstruct the shearing of the fabric. Therefore, the gap between the male and female dies should not be constant in general, and a variation in the thickness must be allowed. A relatively simple way to accomplish this is to use a die made from a relatively soft material. This material should deform at places where the thickness varies. Another way of avoiding high local pressures is by adapting the shape of one of the moulds to the thickness variation. Since the increase in thickness is directly related to the deformation of the laminate, a detailed simulation of the fabric deformation is in that case essential for the design of the dies. Suitable materials for the relatively soft moulds are silicon rubbers, since they can withstand high temperatures for short periods (temperatures higher than 400 °C). The hardness of the silicon rubber is variable between 55 Shore A and 73 Shore A. A disadvantage of silicon rubber is that it is notch-sensitive. For lower temperature rubber forming processes (temperature lower than 250 °C) it is therefore preferable to use a tougher material, for instance PU (polyurethane) rubber [5]. The choice for a rubber male of female die depends on the product. The quality of the surface that is in contact with the metal die ranges from textured to super glossy. The surface that touches the rubber die is rough, in general. In most cases this determines which die should be made of rubber. At present most products are made with a metal female and a rubber male die. A disadvantage, related to the use of rubber as male die material, is that the shape of the die itself can change before the consolidation stage of the thermoplastic composite. This easily leads to folds and wrinkles. It can also lead to the unwanted effect of fibre bridging, as shown in Fig. 8.10. At places

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8.10 The yarns of the fabric deform the rubber mould.

where a large curvature of the surface is found, the fibres of the fabric will try to bridge the corner, resulting in an undefined shape of the product. To overcome this problem, extra rubber can be added locally. If preference is given to a metal male die and a rubber female die, the shape of the product cannot become undefined. Obtaining the proper pressure distribution for consolidation is often the main problem in this case. For mass production the only matching die candidate materials are metals. The function of the clamping device, also referred to as the buckling guide or blankholder, is different from similar metal forming processes. Although it is meant to prevent out-of-plane deformations of the laminate, the primary function of the guide is to make sure that the fibres are under tension during the forming process in order to avoid in-plane, rather than out-of-plane, buckling. Hence the buckling guide should act as a steering device. The forming of the product is governed by the pressure distribution on the laminate. Large local deformations, for instance near corners, can be stimulated by increasing the pressure locally. By using a clamping device the friction forces can be controlled. If the friction forces are not capable of deforming the fabric sufficiently, pins, locking the fabric at certain places, can also be used. Unfortunately, the forces exerted by these pins also result in a tearing of the fabric in the vicinity of the pins. It is therefore inevitable that the pins are placed at positions that remain outside the product, resulting in extra scrap material. Usually the fabric is placed on a supporting plate (Fig. 8.7). The fabric is held at the edges by the blankholder, i.e. the fabric is not supported in the middle. When the fabric is heated by conduction, the hole in the support-

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ing plate should be filled with a matching plate. This is to ensure a homogeneous temperature. (P)reconsolidation stage After the heating stage of the blank and the stage of product forming via intraply shear, bending and interply slip, the deformed shell laminae must be reconsolidated in a final solidification stage. The state of impregnation and adhesion is determinant for the final cycle of closed mould pressure and temperature. In the case of poorly impregnated fibres and fabrics, like the commingled, co-woven or powder impregnated polymer–fibre combinations, a preconsolidation of the laminate blanks can be inevitable. The fibres are properly impregnated with a viscous polymer and several layers of prepreg are bonded together by heat and pressure over a period of time, batchwise in a hot platen press or continuous in a double belt press. In that case of preimpregnation techniques based on solvent impregnation or (powder- or film-)melt impregnation a preconsolidation is no longer necessary. In that case, the heating, forming and final consolidation cycle of rubber forming takes less than a few minutes per part. In the case of mass production, where the forming cycle must be as short as possible, it is favourable to carry out a part of the impregnation and consolidation outside the real rubber forming cycle. Several prepreg materials are supplied in a state of partial or total consolidation [12].The cost saving by deleting the preconsolidation stage and the reduction of cycle time is in general balanced by a higher material price. A disadvantage of preconsolidated sheets is that they have to be supplied in the right configuration (lay-up and thickness) for each specific product.

8.3

Simulation of the forming process

8.3.1 Introduction Simulations of forming processes are required for several reasons. Firstly, questions regarding manufacturability should be answered in a costeffective manner. Often, the way of checking the manufacturability is by testing. In general, tests require long processing times, since tooling prototypes have to be manufactured and modified. Moreover, as testing is labour intensive, large costs may be involved. In such a setting, adaptations of the design are typically developed by trial-and-error. When these trial-anderror processes can be replaced by automated optimization processes, both the time to market and the amount of money spent in the design process can be reduced significantly. A second requirement for forming simulation techniques emerges from

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the point of view of structural analysis. If structural analysis is to be performed for continuous fibre reinforced products, information on local material properties is a necessity. A typical example is the local fibre orientations. The latter are required when laminate stiffnesses have to be calculated. Geometric predictions were carried out as early as 1956, when Mack and Taylor [6] showed how a continuous, differentiable surface of revolution can be covered with fabric. They concluded that the shear deformation mode is the most important in-plane deformation mode during the draping of fabrics. They also concluded that, as far as the shear deformation mode is concerned, the fabric behaviour is similar to the behaviour of a fisherman’s net, with the crossover points of warp and weft fibres as pivoting points. In general, geometric approaches assume that the thermoforming process is dominated by certain deformation modes of the reinforcing fabric. For this purpose the only deformation modes taken into account are intraply shear and bending. In the case of multiple layers, interply shear is allowed as well. Other deformation modes, such as fibre stretching, wrinkling and slip at crossover points, are not taken into account. These assumptions are quite commonly accepted [6–8,10,13]. It was shown that, generally speaking, a geometric simulation based on these assumptions can predict the local fibre orientations sufficiently accurately. By neglecting certain deformation modes, such as fibre stretching and wrinkling, the problem of finding the fibre orientations in the thermoformed configuration simplifies significantly. Clearly, each neglected deformation mode introduces additional constraints, which consequently reduce the number of unknowns of the forming problem at hand. Pioneering work on geometric approaches towards the simulation of thermoforming has been done by Bergsma and Huisman [14] and Van West et al. [15], among others. Both approaches start out with the definition of an initial warp and weft yarn. Subsequently, the remaining crossover points are found using the yarn’s inextensibility. A similar approach, but formulated in a more mathematical framework, is presented by Gutowski et al. [16]. The work of Van der Weeën [17] can be classified in the same group. The basic assumption is again that the only significant deformations are intraply shear and bending. In [17], the inextensibility constraints are, however, handled in three distinct ways. The first method is based on an energy approach. In this formulation the distance between crossover points is kept constant by trying to minimize the elastic energy in a single cell. This elastic energy is determined by the deformations in the yarn directions. The second approach tries to simulate the behaviour of an angler’s net. A cell is completed with line segments which run along geodetic lines. The integration along these geodetic lines makes the method expensive. Whereas

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the first two methods operate on curved surfaces, the third method described in [17] is intended for kinked surfaces. In the work of Trochu et al. [18] special attention is paid to an efficient method for describing the geometry at hand. The simulation of the draping process is basically the same as for other geometrically oriented methods. Noteworthy is the work by Aono et al. [19]. In [19] the effect of triangular darts has been incorporated as well. Later, the formulation was refined to distinguish between stitched and trimmed darts [20]. Most of the above geometrical approaches apply a choice for the locations of the first warp and weft yarn. Bergsma [21] attempted to avoid this initial guess. Bergsma’s algorithm applies a so-called ‘strategy’. This strategy provides a scheme that is used to add cells of the fabric during the simulation of the forming process. In [21] several strategies are reported and tested. In the context of design optimization, these strategies have the disadvantage that the evaluation of design sensitivities becomes somewhat more difficult. The major advantage of the geometric approaches is the fact that with little computational effort a good indication of the fibre orientations and manufacturability can be obtained. However, important effects, such as temperature effects, friction, interply shear, wrinkling and all effects related to the matrix material, are not incorporated. More accurate and detailed simulation results for forming processes can be obtained by using finite element models. An early application of finite element techniques to the modelling of cloth can be found in the work of Terzopoulos and Fleischer [22]. In there, the modelling was mainly used to achieve more realistic computer graphics. Among the early research devoted to the draping behaviour of fabrics was the work of Collier et al. [23]. Here, non-linear shell elements were applied in combination with orthotropic material behaviour. Only a single layer fabric without matrix material was used. Finite element models attempting more realistic simulations of thermoforming processes have been described in [24–26]. In Pickett et al. [24], stacking of shell elements was applied in an explicit finite element model. Both intra- and interply shear effects were taken into account. A uniform temperature distribution was applied. In De Luca et al. [25] the model of Pickett et al. [24] was further refined by the introduction of heat conduction. A similar approach was described by Johnson and Pickett [26]. With the detailed finite element models reported in [25,26], realistic results can be obtained. The required computer times are still relatively high, however. Particularly in the design stage or when automated optimization strategies are applied and many intermediate designs must be evaluated, this aspect may become prohibitive. For that reason, the present discussion will be restricted to geometrically based algorithms only.

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In general, structural optimization algorithms become much more efficient when design sensitivities [27] are available. Moreover, the availability of design sensitivities is also advantageous in the context of geometrical thermoforming simulations. When (design) sensitivities are available, it is possible to introduce an entirely new class of strategies. These can be based on the application of iterative processes that attempt to satisfy, iteratively, additional constraints and/or to minimize a global objective function. Such constraints can, for example, reflect a maximum shearing angle (locking angle) or constraints due to position pens.A typical objective function could be the estimated dissipated energy or the average shear angle. Hence, a fabric will be relocated in an iterative manner, until all constraints are satisfied and/or a characteristic function is minimized. The outline of the present section is as follows. Firstly, the geometric approach proposed by Bergsma [21] will be summarized. As this approach has some disadvantages in the context of structural optimization, a more classical geometric approach will then be described. In addition, the evaluation of design sensitivities will be described. Finally, some numerical examples will be presented.

8.3.2 Strategy approach Since interply interaction is neglected in a geometrical simulation, the present section will address single layer fabrics only. Multiple layers can be accounted for by stacking them on each other. The most commonly used method is based on an initial selection of the position of initial warp and weft yarns. This is depicted in Fig. 8.11. Typical approaches are to start with two orthogonal directions in a starting point and to define the initial warp and weft yarns as geodetic lines. All remain-

8.11 The fabric is uniquely defined when the positions of a single warp yarn and weft yarn are well known.

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ing grid points are now found using the inextensibility constraint. In contrast to this approach, Bergsma [21] used a so-called strategy for selecting a grid point where a new set of cells (incomplete row or column) must be added. The strategy also prescribes the shear angles of the newly introduced cells. This approach is depicted in Fig. 8.12. Note that without using a strategy, there are infinite possibilities to cover the product shape with the fabric. Only a few of them are physically acceptable. The strategy introduces additional constraints reducing the degree of freedom of the fabric. Hence, the uniqueness and the quality of the obtained solution are fully determined by the applied strategy. In order to illustrate the strategy approach, the strategy used for rubber forming of continuous fibre reinforced products will be discussed. It should be mentioned that in this case the strategy basically reflects the applied production process and is based on experimental observations. Proper formulation is therefore difficult and cannot be rigorously proven to be correct. Experimental observations show that during the rubber forming the fabric gradually contacts the mould surface. Pieces of the fabric that touch the die cannot move easily over it, since it is often cold, and the viscous matrix will cool and solidify. Moreover, the rubber die presses the fabric onto the mould surface, which leads to increased friction forces. Note that when using a heated mould, it is necessary to reformulate the strategy. Based on these observations, the present strategy is to find points that will be in contact first and will consequently define the corresponding fibre direction. This requires the following three subsequent steps: 1 Find the yarn segment of the fabric that is expected to be in contact with the product shape. This is done by comparing all possible pieces of yarn

8.12 The pieces of yarn that define the unique covering do not have to be part of a single warp or weft yarn.

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that are almost of the product shape and decide which one is closest to the product shape. 2 Find the expected direction of this piece of yarn. This is done by combining the direction of the part of the yarn that is already draped and the direction of the driving force of the forming. 3 Calculate the new crossover point as defined by this piece of yarn. This is done by using the inextensibility condition and the fact that the new crossover point must be located on the die. This way of calculating a new cell of the fabric can be repeated until the whole product shape is covered with fabric. The major advantage of this approach is that it gives a good reflection of the production process, especially in comparison with other geometrical approaches. However, the strategy approach has two disadvantages. Firstly, for every production process a strategy has to be defined, which can be very difficult for certain processes [21]. Secondly, the evaluation of design sensitivities is somewhat difficult.

8.3.3 Integration approach In a structural optimization setting, it is more convenient to start with positioning a single warp and weft yarn. As the placement of the first two yarns is computed by numerical integration, this approach will be denoted integration approach. The positions of these two yarns are considered as independent quantities, although, of course, constrained by the yarn inextensibility. Furthermore, the positions are controlled by an appropriate set of (design) variables. In this paragraph no distinction will be made between the first warp and weft yarn. The presented results are applicable to both. It is assumed that the surface consists of a collection of branches, each having its own geometric description, possibly on the basis of simple surfaces, e.g. Coons patches and cylindrical and spherical surfaces. The entire surface is finally determined by S ( x, d ) = 0, x ΠR3

[8.1]

where d is a vector of design variables. Moreover, it is assumed that for each branch a surface description with curvilinear coordinates is available. These coordinates may be discontinuous at interfaces between adjacent branches. For convenience we assume for each x, which satisfies 8.1, a set of unique surface coordinates aa, a = 1, 2, such that x = Rb (a a , d)

[8.2]

where the subscript b refers to a particular branch. A normal vector to the surface will be formulated as

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n=m

∂S ∂ (S ), where m = ∂x ∂x

-1

[8.3]

For each yarn a material coordinate y is introduced. The position of the yarn is assumed to be given by R(y, d). The corresponding tangent vector is defined as i=

∂R = Ry ∂y

[8.4]

A set of local basis vectors is completed by defining a vector j as follows: i ¥ j = n,

i·j=0

[8.5]

The material coordinate y is taken as the length measured along the yarn. Since the yarns cannot be extended, it consequently holds that i · i = j · j = 1. In order to control the position of the first yarn, it seems convenient to introduce a function P(y, d), which specifies the orientation by P( y, d ) = i y ◊ j

[8.6]

Hence P is a measure for the curvature of the yarn in the tangent plane. Note that an initial guess for P must be given in advance. The correct value of P can be computed iteratively by imposing additional constraints or by formulating the simulation as an optimization problem as discussed in the Introduction. Thus, the applied constraint functions and/or objective function basically reflect the forming process. This means that each forming process requires the formulation of a set of constraint functions and/or objective function, without affecting the actual implementation. The function P can be controlled by an additional set of parameters. Coming back to Equation 8.6, i y = Pj - (i ◊ n y )n

[8.7]

Integration of 8.4 and 8.7, using the initial conditions x(y = 0, d) = R0(d) and i(y = 0, d) = i0(d), yields the locations and orientations of the first yarn as a function of the co-ordinate y. Here, the initial conditions specify the location and orientation at y = 0, respectively. Once single warp and weft yarns have been specified, the locations of the remaining yarns can be found in a straightforward manner. At this point, it is necessary to distinguish between warp and weft yarns. All quantities related to the warp yarns will be denoted by. .¯ . . Consider the point of inter¯(y¯1, d) = R(y1, d). For both section of two yarns which is determined by R ¯ and D, yarns, increments of the co-ordinates y¯ and y are introduced as D respectively. ¯ and D, parts of the adjacent yarns can be conAfter the introduction of D structed, as depicted in Fig. 8.13. A new crossover point is constructed,

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8.13 Evaluation of new crossover points and adjacent yarn segments. The new yarn segments are plotted with dotted lines.

which is denoted as e, as shown in Fig. 8.13. When D¯ and D are sufficiently small, inextensibility of both yarns yields

[e - R (y

][

]

+ D , d ) ◊ e - R ( y1 + D , d ) = D2

[8.8]

[e - R( y1 + D, d )] ◊ [e - R( y1 + D, d )] = D 2

[8.9]

1

Moreover, the new crossover point e should be located on the die. Therefore, the condition S(e, d) = 0 has to be satisfied. The precise location of e can be determined iteratively using a Newton process. With a successive application of the above equations, the entire fabric can be constructed, as soon as initial warp and weft yarns are determined.

8.3.4 Design sensitivities In general, the derivative of a characteristic function with respect to a design variable is called a design sensitivity. In a structural optimization setting, it is useful to know how the draped fabric will re-orient if the shape of the product or the initial fabric orientation changes. That is, one wants to know the design sensitivity of, for example, the local fibre orientation with respect to a characteristic product dimension. As mentioned before, it is also useful to have sensitivities at hand within the context of thermoforming simulations. Suppose the function P, defined by 8.6, to be determined by a variable. This function has to be specified in advance, but obviously is not known a priori. The basic idea of the integration approach is to use an initial guess for P and to compute the corresponding fibre placement and sensitivities. Using the sensitivity information, a better guess for the function P can be formulated. This iterative process will be continued until all constraints are satisfied and/or a characteristic function is minimized. Design sensitivities on the basis of the integration approach can be obtained rather easily. To achieve a compact notation, partial derivatives

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with respect to a design variable are introduced as ∂ . . . /∂dk = . . .*. For the first warp and weft yarns, using 8.4, 8.7 and j = n ¥ i, follows i *y = R*yy = P * j + P(n * ¥ i + n ¥ i *) - (i * ◊ n y + i ◊ n*y )n - (i ◊ n y )n * [8.10] while corresponding initial conditions are formulated as x*(0, d) = R0* and i*(0, d) = i0*, respectively. Note that 8.10 provides a basis for determining R*, or equivalently x*, and i* by means of (numerical) integration. Design sensitivities for the entire fabric can be formulated on the basis of 8.8, 8.9 and S(e, d) = 0. Straightforward differentiation yields

[e - R (y

1

][

]

+ D , d ) ◊ e * - R* ( y1 + D , d ) = 0

[8.11]

[e - R( y1 + D, d )] ◊ [e * - R* ( y1 + D, d )] = 0

[8.12]

1 n(e , d ) ◊ e * + S * (e , d ) = 0 m

[8.13]

The design sensitivities e* can be solved from 8.11, 8.12 and 8.13. By a successive application of the above equations the design sensitivities can be evaluated for the entire fabric.

8.3.5 Examples A test example often considered is a thermoformed hemisphere. A typical simulation result is shown in Fig. 8.14(a). The initial warp and weft yarns are taken as geodetic lines. The starting point for these yarns is taken as the pole of the sphere. In Fig. 8.14(b) a thermoformed hemisphere is shown.

8.14 Draping simulation for a hemispherical surface: (a) grid representing the local yarn orientations; (b) experimental result using a hybrid reinforcement.

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As a simple example used to test the evaluation of design sensitivities, draping on an undulated surface is considered. The obvious orientations of the yarns as found by the simulation are depicted in Fig. 8.15(a). The design variable being considered is the blank orientation. In Fig. 8.15(b) the design sensitivities for this design variable are shown. It is stressed that these design sensitivities are obtained by direct application of 8.11, 8.12 and 8.13, rather than using finite differences.

8.4

Finite element simulation

8.4.1 Introduction For many practical applications it is necessary to satisfy certain mechanical behaviour constraints. Stress and strain criteria may be applied to guarantee that no failure occurs during normal operating conditions. Other constraints may reflect tolerable deflections and stability of the structure. Similar to the production process, numerical simulations may be used to investigate the mechanical behaviour of a CFRTP component or structure. Generally, such numerical simulations are carried out using finite element models. In many cases the CFRTP products at hand can be identified as being thin walled. Therefore, finite shell elements are the best candidates to use in the corresponding finite element models. Before a finite element analysis can be carried out, information on the material must be provided. For CFRTP products the laminate stiffnesses are determined by the local fibre orientations. As discussed in Section 8.3, these fibre orientations differ generally from place to place.Therefore, these orientations must be determined prior to a finite element analysis. Subsequently, the laminate stiffnesses must be evaluated using the properties of the constituents and the local fibre orientations. As little information on the evaluation of laminate stiffnesses for sheared CFRTP material can be found in the literature, we shall address this aspect in Section 8.4.2. In Section 8.4.3 some details on modelling thin-walled structures and in particular CFRTP products will be given. Since finite element analysis of thin-walled structures is described in many textbooks, little attention will be paid to this aspect. To a lesser extent the same holds true for design sensitivity analysis using finite element models.

8.4.2 Laminate stiffnesses for thermoformed CFRTP composites The correct determination of the mechanical material properties is a complicated problem. Often, the way to obtain strength and stiffness data is by

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8.15 (a) Draping simulation for an undulated surface. (b) Design sensitivities corresponding to the initial blank orientation.

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8.16 Transformation of a repeating cell into a four-layered laminate.

8.17 Assumed strand configuration in the undeformed (a) and deformed (b) states.

experiment [4]. This is because of the complex structure of CFRTP and the large influence of the manufacturing process on the local mechanical properties. The large number of parameters controlling the material properties makes it too expensive and impractical to characterize CFRTP through experiments only. Therefore, the necessity of theoretical models, which can predict the material properties of CFRTP accurately, becomes evident. The simplest woven fabric (WF) pattern is the plain weave (Figs. 8.16 and 8.17). A number of parameters determine the WF laminate structure, e.g. the fibre undulation, presence of a gap between adjacent strands, actual cross-sectional geometry of the strands and the strand fibre volume fractions. Various mathematical models have been proposed to describe the thermo-elastic properties of plain weave WF composites. Ishikawa and Chou [28] presented so-called mosaic and fibre undulation models, which apply classical laminate theory (CLT) for every infinitesimal piece of a repeating region of a WF lamina.The mosaic model idealizes the WF lamina

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as an assemblage of pieces of asymmetrical cross-ply laminates, which are stacked under iso-strain or iso-stress conditions. This model does not consider the strands’ undulation. It was noted by Rai [29] and Vu-Khanh and Liu [30] that the fibre undulation effect cannot be neglected.The fibre undulation model by Ishikawa and Chou [28] takes into account the undulation in one direction only. It is assumed that the strand geometry in the perpendicular direction does not vary. This model gives a satisfactory agreement between experiment and numerical prediction for the elastic moduli in the undulation direction, but the elastic properties in the transverse direction are not valid. Zhang and Harding [31] presented a plain weave fabric model based on the strain energy equivalence principle. A finite element model is used to evaluate the effective elastic properties. Again, the undulation is considered in one direction only. Another drawback is that it involves substantial computations. Kabelka [32] proposed a 2-D analytical model considering the undulation in both the warp and fill directions. A laminate is modelled as an assemblage of unidirectional (UD) warp and fill strands and a matrix layer. Considering the strand undulation, equivalent UD lamina properties are determined. Global elastic coefficients are defined on the basis of CLT. The actual strand cross-sectional geometry is ignored. The method gives a higher stiffness because the maximum strand thickness is used for calculations. Naik et al. published a series of papers [33–36] on refined plain fabric composite models. These models take into account the undulation of both the warp and fill strands and the actual strand cross-sectional geometry. The obtained results are in agreement with the experimental results. All above models are restricted to orthogonal weave structures. Therefore, these models are inadequate for thermoformed CFRTP products. In the present section, the analytical model of Polynkine and Van Keulen [37] for predicting the elastic properties of sheared plain weave composites is described. The approach is essentially based on the 2-D geometric representation of a plain single-layer WF composite proposed by Naik and Ganesh [34,35]. The model proposed in [37] requires four subsequent steps. Firstly, the equivalent properties, e.g. mean thicknesses and volume fractions, have to be calculated for each layer. It is emphasized that these properties depend on the actual weave geometry and the shear angle. Secondly, the equivalent elastic properties of the individual layers are computed using the composite cylinder assemblage (CCA) model given by Hashin [38]. Thirdly, the equivalent engineering constants according to the CCA model should be corrected for the effect of crimp, since the CCA model yields the elastic properties of straight strands. Finally, using the equivalent layer properties

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and the equivalent engineering constants, the effective elastic laminate stiffnesses can be computed using CLT. To account for multiple layers, it is convenient to subsequently solve the first three steps for each layer and then apply CLT to all sub-layers. Geometric consideration Before deriving expressions for the equivalent layer properties, the geometry of the repeating volume element will be described in more detail. The repeating cell is selected as ABCD, as shown in Fig. 8.17(a). For reasons of symmetry, only one quarter of the repeating cell will be considered. This part will be referred to as the unit cell. The unit cell will be transformed into a four-layered laminate as depicted in Fig. 8.16. In this way the coupling between membrane and bending behaviour cannot be described correctly. However, for a multilayered laminate, this coupling can be described, provided that a sufficient number of layers is available and/or the fabric reinforcements are sufficiently thin. In the initial undeformed configuration (Fig. 8.17a), mutually perpendicular strands (warp and fill) are assumed. In the deformed state, the intersecting angle is changed to p/2 - q as depicted in Fig. 8.17(b). The angle q will be referred to as the shear angle. Furthermore, it is assumed that the warp and fill strands in the fabric have a quasi-elliptical cross-sectional shape [34], i.e. their shape can be described by sinusoidal functions. The assumptions adopted for the material model are similar to those introduced for the geometrical simulation algorithms, as discussed in Section 8.3. With these assumptions, the geometry of the strands in the unit cell is characterized by the following parameters, which are also depicted in Figs. 8.17 and 8.18: S(k) = a(k) = g(k) = h(k) = hm = H = j(k) = j(k)max = k = k˜

the strand lengths in the unit cell, the width of the strands, the gap between two adjacent strands g(k) ≥ 0, the maximum thickness of the strands, the minimum thickness of the matrix, the total thickness, the local crimp angle, the maximum crimp angle, index which relates the above quantities to the warp (k = 1) and fill (k = 2) strands. = index, which refers to the opposite yarn, i.e. k˜ = 1 if k = 2 and k˜ = 2 if k = 1.

Now it is possible to derive expressions for the equivalent layer properties. A significant aspect of the shearing of a fabric reinforced laminate is

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8.18 Cross-section of the unit cell which is taken in the k-direction.

the change in thickness. Because of the constant material volume assumption, the thickness must increase during shearing deformation [4]. The actual thickness then becomes: H=

H0 sin(q)

[8.14]

where H0 is the total thickness in the undeformed configuration. Similar expressions can be derived for the mean thicknesses of the yarns as a function of the shear angle. From Fig. 8.17 immediately follows: k˜ g ( k ) = S ( ) sinq - a ( k )

[8.15]

If the gap is present, i.e. g(k) > 0, the width a(k) and the thickness h(k) of the strands are assumed to be constant: ( )

a ( k ) = a0 k , and h( k ) = h0

( k)

[8.16]

Assuming no strand penetration, a quasi-elliptical cross-sectional shape and a constant cross-sectional area, the following relations must hold after gap closure, i.e. g(k) = 0: a ( k ) = S ( k ) sin q = a0 h( k ) = h0

( k)

( k)

sin q sin a

sin a ( ) k˜ , with sin a = a0 k S ( ) sin q

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Here a is the shear angle where the gap becomes zero. Equations 8.17 and 8.18 describe the strand width and thickness as a function of the shear angle q. To derive an expression for the mean value of the strand thickness, it is necessary to know the shape of the strand cross-section. Experimental observations [4,29,33,34] have shown that sinusoidal functions yield good approximations of the actual geometry of plain weave fabric lamina crosssections. Therefore, the following shape functions for the yarn cross-section are used (see Fig. 8.18): ( k)

Ê py ˆ h( k ) ( y( k ) ) = h( k ) cos ( k ) Ë a ¯

[8.19]

Here, the mean thickness can be expressed as h ( k) =

2 h( k ) a ( k ) p[ a ( k ) + g ( k ) ]

[8.20]

Using 8.15–8.18, it can be shown that ( )

h ( k) =

h0 k 2 ( ) ( ) , where h0 k = h0 k sin a sin q p

[8.21]

After evaluating the effective thicknesses of the warp and fill strands according to 8.20, the thickness of the pure matrix layer is estimated by the remaining thickness, i.e. k˜ h m = H - h ( k) + h ( )

(

)

[8.22]

where H is the total thickness of the fabric unit cell. Calculation of effective elastic laminate stiffnesses The calculating procedure is performed by substituting the woven fabric reinforced composite by an equivalent four-layered laminate, i.e. an asymmetric angle-ply lamina between two pure matrix layers. In this method, the mean thicknesses of the strands obtained from equations 8.20–8.22 are taken as the thicknesses of the respective laminae. The equivalent properties of the individual layers ( k)

( k)

( k)

( k)

EL , ET , vLT , GLT , GTT

( k)

[8.23]

are evaluated by using the CCA model given by Hashin [38]. This model is probably the most commonly used for definition of the effective properties of fibre reinforced materials. Applying the CCA model [34,38], the values (8.23) are determined from the transversely isotropic fibre (subscript f)

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properties EfL(k), EfT(k), vfLT(k), GfLT(k), GfTT(k) and the isotropic matrix (subscript m) properties Em, vm, and the corresponding strand fibre volume fractions. Subscripts L and T denote unidirectional composite elastic properties along the fibre and transverse fibre directions. The experimentally determined fibre volume fraction of a CFRTP laminate is the overall fibre volume fraction vf0, i.e. the ratio of the volume of fibres with respect to the total volume of the laminate. In the present analysis, the strands are idealized as equivalent UD laminae. Therefore, the fibre volume fractions in such UD laminae with effective elastic properties (8.23) are referred to as the strand fibre volume fractions vfs. The latter is the ratio of the fibre volume with respect to the strand volume. Knowing the overall fibre volume fraction vf0 of a laminate and the strand volumes, the strand fibre volume fraction vfs can be obtained. The CCA model yields the elastic properties of straight strands with a specified strand fibre volume fraction. Thus, the undulation of the warp and fill strands will be brought into account using the technique described by Lekhnitskii [39], which is based on a transformation of the compliances. After this transformation, the locally reduced compliances S¢ij(j) are averaged over the length of the strands to determine the effective compliances of the strands. As an approximation, the mean value of the compliance can be defined in the interval (0, jmax) [32,34]: S ij =

j max

1 jmax

◊

Ú S ¢ (j)dj, ij

i, j = 1, 2, 6

[8.24]

0

The integration 8.24 is made under the assumption that in an actual lamina jmax is very small, and the functions sin j and cos j can be replaced by the first terms of their Taylor series. Finally, the effective elastic constants of the strands are given by [37]: EL

( k)

EL

= 1+

vTL

( k)

( k)

GLT

= vTL

j

2

( k) max

3 ( k)

( k)

( k)

( k)

È EL ( k) ˘ Í ˙ ( k ) - 2 vLT Î GLT ˚

j 2 max + 3

(

( k)

(v

( k) TT

, ET

( k)

= ET

)

- vTL

( k)

)

( k)

=

GLT ( k) ( k) ˘ j 2 max È GLT 1+ Í ( k ) - 1˙ 3 ˚ Î GTT

[8.25]

Notice that the straight fibre moduli are retrieved from 8.25 when jmax Æ 0. Now the stress–strain relations as used in the CLT become:

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3-D forming of continuous fibre reinforcements for composites Èsx ˘ Ís ˙ Í y˙ ÍÎs xy ˙˚

( k)

273

( k)

È ex ˘ ( k) Í ˙ = [Q] Í ey ˙ , ÍÎ g xy ˙˚ ( )

( k)

with [Q]

È EL k Í ( k) Í D Í ( k) ( k) = Í vLT EL Í D( k ) Í Í 0 Î

( k)

vLT EL D( k ) ( k)

ET D( k ) 0

( k)

˘ ˙ ˙ ˙ 0 ˙˙ ˙ ( k) GLT ˙˚ 0

[8.26]

with D(k) = 1 - v¯LT(k)v¯TL(k), k = 1, 2. The effective moduli for an isotropic material under plane stress conditions are taken for the matrix material. Knowing the thicknesses of all layers and the reduced effective moduli, the contribution to the equivalent laminate stiffnesses can be calculated using CLT and appropriate transformations to account for the orientations of the equivalent reinforcement layers. Note that at this stage the whole unit cell is modelled as an asymmetrical angle-ply laminate. Consequently, for a single reinforcement layer the coupling matrix will be set to zero. In case of multiple reinforcement layers, the present approach will be repeated for each layer.

8.4.3 Finite element analysis The thermoplastic products being considered in the present chapter are thin walled. Provided that the smallest local wavelength of the deformation pattern is sufficiently large compared with the wall thickness and moreover the smallest principal radius of curvature is large compared with the wall thickness, thin shell theory can be used as a starting point for mechanical analyses. In standard textbooks, several finite elements for thin shells can be found, see [40–42] among others. Examples presented in the present chapter were all obtained with the triangular shell element as described in [43] and the references given therein. When the smallest wavelength becomes too small it may be necessary to account for transverse shear deformations. The simplest theory accounting for transverse shear deformations is the Mindlin–Reissner theory. Many other higher-order theories with increasing complexity have been reported in literature, see for example [44]. Finite element analysis of CFRTP products is, as mentioned earlier, somewhat hindered by the fact that laminate stiffnesses must be specified that differ from place to place. As discussed in Section 8.3, the simulation of the forming process can be based on a geometrical algorithm, for which gen-

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erally a rectangular grid of crossover points is used. The requirements for this grid are entirely different from those for the finite element mesh as compared. The most striking differences are the following: • The grid size for the geometrical forming simulation is mainly determined by local details to be included in the forming simulation. For the finite element mesh, the required mesh density is in addition determined by the local stress and strain gradients. • Generally, when a geometrical forming simulation is applied there is no need for a more advanced selection of the local grid size, as the related numerical effort is relatively small. This is in contrast to finite element analysis, for which a graded mesh density is often required to achieve a requested accuracy against acceptable costs. • The domain being modelled for the forming process is often different from the corresponding finite element model. The latter covers only a sub-domain of the former. The reason for this is that after the forming process, the superfluous material will be removed and the topology may be adapted. The above differences imply that the fibre orientations cannot be transferred one-to-one from the grid that is being used for the forming simulation to the finite element model. One method is to search for each integration point in the finite element mesh, the closest crossover point available from the grid being applied in the forming simulation. As for the latter, sometimes relatively large grid dimensions can be applied, for which erroneous results may be found. Therefore, using additional checks is recommended. An efficient approach is to check the angle between the normal vectors to the surface at the crossover point and the integration point. If this angle differs too much from zero, then the crossover point found is to be rejected. In the context of structural optimization, the finite element model can be used to evaluate design sensitivities [27]. The simplest approach towards design sensitivities is by means of global finite difference techniques. The major disadvantage is that for each design variable an additional finite element solution must be determined. This makes the method inefficient. Analytical design sensitivities are accurate and efficient. Their implementation is, however, involved. A compromise between the above methods is the so-called semi-analytical method [27]. In the linear regime the starting point is the well-known equation K (d)u(d) = f (d)

[8.27]

where K is the system matrix, u is the vector of nodal degrees of freedom and f is the vector of nodal loads. For simplicity we have assumed a single design variable, which is denoted d. Differentiation of 8.27 gives

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3-D forming of continuous fibre reinforcements for composites ∂u È ∂ f ∂K ˘ = K -1 Í u ∂d Î ∂ d ∂ d ˙˚

275 [8.28]

The term ∂f/∂d - (∂K/∂d)u is often referred to as the pseudo-load vector. This pseudo-load vector is generally evaluated at element level, using central or forward finite differences for ∂K/∂d. The implementation of the semi-analytical method is relatively easy and the efficiency is good. A severe disadvantage is its sensitivity with respect to large rigid body motions for individual elements. Adequate remedies, which do not spoil its efficiency, have been proposed [45,46]. Once ∂u/∂d has been obtained, the design sensitivities for strains and stresses can be evaluated straightforwardly. Application of the semi-analytical method to CFRTP products requires additional information on the design sensitivities of the fibre orientations, as discussed in Section 8.3. Subsequently, corresponding design sensitivities for the laminate stiffnesses can be determined straightforwardly.

8.5

Optimization of CFRTP products

8.5.1 Introduction As indicated in the previous sections, the design of thermoplastic products with a continuous fibre reinforcement is a difficult task. The designer not only has to meet the (mechanical) behaviour constraints, such as stability and upper limits on stresses and strains, but in addition restrictions are imposed by the applied forming process. This makes the effects due to changes of shape and topology and changes in the processing parameters difficult to predict intuitively. Consequently, controlling a design process towards an optimal solution manually becomes a difficult or even impossible task. Therefore, application of automated optimization techniques is required. In general, an optimization problem can be formulated as min[ F0 ( x)]

[8.29]

subject to F j ( x) £ 1, ( j = 1, . . . , M ), Ai £ x i £ Bi , (i = 1, . . . , N )

[8.30]

Here x is a vector of design variables; F0(x) is the objective function; Fj(x), (j = 1, . . . , M) are the normalized constraint functions; Ai and Bi are the lower and upper limits on the design variables, respectively. The objective function typically reflects the costs of a structure or weight. The constraint functions could reflect limits on deflections, strains, stresses, etc. In the present setting they also reflect the manufacturability. When

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8.19 Combination of optimization, pre- and post-processing, simulation of forming process and finite element analysis for continuous fibre reinforced thermoplastic products.

geometrical forming simulations are applied, the latter constraint is determined by the ratio between the absolute value of the shearing angle and the locking angle. The design variables could reflect parameters controlling the shape of the product, but also the initial position and orientation of the blank. Application of an automated optimization technique to 8.29 and 8.30 leads, for CFRTP materials, to a combination of the simulation and optimization tools as shown in Fig. 8.19. From this scheme it is seen that the mesh generator, which operates on a parametric model description, is used for both the triangulations of the die and the actual finite element model. The optimizer interacts with the simulation tools directly and through the mesh generator. Owing to the fact that a sequence of simulation is applied, accumulation of errors can occur. This causes the response function to become noisy. This will be illustrated on the basis of a simple optimization example, which is formulated as maximization of the strain energy of the conical shell with a constant volume requirement. Design variables are the height of the cone (x1) and the base radius (x2). At the top of the cone the radius must be kept unchanged and specified by a value of 10 mm. The top surface of the structure is loaded by a pressure load of p = 0.1 N/mm2. Owing to symmetry only

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8.20 (a) Conical shell made from CFRTP material. (b) Objective function for the CFRTP conical shell problem.

one quarter of the structure is analysed. A typical configuration and corresponding finite element mesh are depicted in Fig. 8.20(a). The behaviour of the response functions was analysed by a series of response evaluations using 30 increments of both design variables. The results for the objective function are shown in Fig. 8.20(b). The present objective function clearly shows a noisy behaviour which, obviously, can have a significant influence on the convergence characteristics of the optimization process. The above complication indicates that an optimization algorithm has to be applied which is relatively insensitive with respect to noisy response evaluations. In Section 8.5.2 the so-called multipoint approximation method will be outlined, which satisfies this requirement.

8.5.2 Multipoint approximation method The basic idea behind the multipoint approximation method [47–50] is to replace the initial optimization problem 8.29–8.30 by a sequence of approximate optimization problems. For the latter, the implicit response functions ¯j(x). Typically linear and Fj are replaced by approximate explicit functions F multiplicative approximation functions will be used, but virtually any struc-

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ture of approximate functions is possible. The initial optimization problem 8.29–8.30 is now replaced by:

[

( k)

]

min F0 ( x)

[8.31]

subject to Fj

( k)

( k)

( x) £ 1, ( j = 1, . . . , M ), Ai ( k)

( k)

Ai £ Ai , Bi

( k)

£ x i £ Bi

£ Bi , (i = 1, . . . , N )

[8.32]

where k is the current iteration number. The current move limits Ai(k) and Bi(k) define a search sub-region of the design variable space. The (k + 1)th iteration is started from x(k) which is the solution of 8.31 and 8.32. The size and location of the next search sub-region depend on the quality of the approximations, the location of the point x(k) in the current search subregion and the optimization history.A detailed description of the move limit strategy can be found in [51–53]. The approximation functions F¯j(k)(x) are determined using a weighted least-squares method [54], for which only function evaluations can be taken into account and information on the design sensitivities [47–50]. The selection of the weight factor must be done carefully. Details on the selection of the weight factors are given in [51–53]. Because of the fact that a weighted least-squares method is applied, the multipoint approximation method becomes relatively insensitive to noisy response functions. A further advantage of the method is that it can still be used when no information on design sensitivities is available.

8.5.3 Implementation The starting point for design optimization of a continuous fibre reinforced product is a parametric description of both the actual product and the die surface. The former is generally a subset of the latter. These parametric descriptions typically consist of a collection of interconnected primitives. The parametric descriptions together with a set of design parameters will be used to generate a non-parametric description, this being the input for the preprocessor. The preprocessor is then invoked to generate (a) a representation of the die surface and (b) an appropriate finite element mesh. As mentioned before, the requirements for these meshes may be totally different. After a representation of the die is generated, a simulation of the draping process is carried out. The local fibre orientations are used to evaluate the local laminate stiffnesses. Once laminate stiffnesses are calculated, the finite element simulation can take place. As shown in Fig. 8.19, objective and constraints functions are determined by the material model and the finite

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element model. With a post-processor, the relevant information is retrieved from the finite element solution and passed on to the optimizer. When design sensitivity information is available, the scheme in Fig. 8.19 becomes slightly more complicated. In that case, design sensitivities are passed from the forming simulation to the material model and subsequently to the finite element analysis. Afterwards, the design sensitivities are used by the optimizer.

8.5.4 Example A panel with a hole and flanged edges is redesigned with respect to its shape. Figure 8.21 shows a symmetric quarter segment of the panel analysed, the corresponding boundary and loading conditions, and the finite element mesh. As depicted, the upper plate is connected with the flanges by means of a cylindrical surface, and the edge of the hole is reinforced by a part of a torus. The objective is to minimize the volume of the structure under displacement, strain and linear buckling constraints. Design variables represent the radius of the hole x1, the width of the upper plate x2 and the flange width x3, respectively. The structure is assumed to be thermoformed of E-glass/ epoxy composite material. The warp and fill strands of the fabric were prescribed to be parallel to the X and Y co-ordinate directions, respectively, as depicted in Fig. 8.22(b). More details on the precise formulation of the optimization problem can be found in [55]. For the optimal solution, which was obtained after five iterations, the volume of the structure is reduced from 766.2 mm3 to 486.5 mm3.

8.21 A panel with a hole and flanged edges.

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8.22 The optimal configuration of the panel with a hole and flanged edges: (a) the finite element mesh; (b) the orientation of the yarns.

8.6

Conclusions

Composites combine the attractive properties of different materials, e.g. high mechanical and physical performance of fibres and the appearance, bonding and physical properties of polymers. Through this marriage, the poor capacities and drawbacks of the individual components often disappear, like the poor mechanical properties of polymers in general and fibres in compression. To justify the higher cost of this class of materials compared with other engineering materials, such as metals, the price-performance of a design must be competitive. Developments with respect to (minimum energy) structure design and manufacturing of composite components are therefore important. Tools for analysis, design and optimization which are under development must be user-friendly (labour cost), fast (time to market) and able to show designers and analysts in an easy way the effects of variations of the design variables on process parameters, shape, mass, stiffness, strength, durability, etc., to cost. In this chapter, on the 3-D forming of continuous fibre reinforcements for composites, an outline has been given of the ongoing research at Delft University of Technology. This research is mainly focused on the application of composites in advanced structures. That means the application of high-strength and high-modulus fibres, continuous in length, impregnated up to high volume percentages and with complete control of fibre orientations. The manufacturing processes based on the use of fabrics, such as RTM and press and diaphragm forming, are studied and optimized in such a way that the process control and the desired fibre architecture become part of the entire design process. The straightforward geometrical drape simulation

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is a fast and powerful (interactive) tool for the conceptual design phase of composite shell structures, varying from viability and approximate analysis to tool design. In addition, it can be used as a preprocessor to determine the permeability distribution for parts manufactured by resin transfer or resin injection moulding. The program can also be used as a post-processor in several existing CAD systems, and as a drapeability checker and material modeller for the determination of the local engineering constants of the part to be designed. For the embodiment phase of the design process, numerical finite element and design sensitivity calculations are under development as presented. Interactive manipulation, high speed and proper failure criteria are still missing links and therefore important subjects for further developments. Once the interface problems with the process simulation programs are solved and the specific behaviour of rubber dies is formulated in effective algorithms, the design loop can be completed. When this is fast and interactive, realistic integral and concurrent engineering of future advanced composite structure designs becomes possible.

8.7

References

1. McNeill, W.H., ‘Human migration in historical perspective’, Population Development Rev. 10, 1–18, 1984. 2. Keegan, J., A History of Warfare, Pimlico, London, 1993, pp. 126–136, 162–163. 3. Beukers, A., ‘Polymer composites versus metals, the structure efficiencies compared’, in SAMPE Proceedings, Tokyo, 1991, pp. 1113–1120. 4. Robroek, L.M.J., The Development of Rubber Forming as a Rapid Thermoforming Technique for Continuous Fibre Reinforced Thermoplastic Composites, Delft University Press, Delft, 1994. 5. Robroek, L.M.J., ‘Material response of advanced thermoplastic composites to the thermoforming manufacturing process’, paper presented at Proceedings of the Third International Conference on Automated Composites, 15–17 October, 1991, The Hague, the Netherlands. 6. Mack, C. and Taylor, H.M., ‘The fitting of woven cloth to surfaces’, J. Textile Inst., 47, T477, 1956. 7. Robertson, R.E., Hsiue, E.S. and Yeh, G.S.Y., ‘Fibre rearrangements during the moulding of continuous fibre composites II’, Polymer Composites, 5, 191, 1984. 8. Heisley, F.L. and Haller, K.D., ‘Fitting woven fabric to surfaces in three dimensions’, J. Textile Inst., 2, 250–263, 1988. 9. Potter, K.D., ‘The influence of accurate stretch data for reinforcements on the production of complex structural mouldings’, Composites, July, 1979. 10. Van West, B.P., ‘A simulation of the draping and a model of the consolidation of commingled fabrics’, PhD Thesis, University of Delaware, 1990. 11. Tam, A.S. and Gutowski, T.G.,‘Ply-slip during the forming of thermoplastic composite parts’. J. Composite Mater., 23, 587–605, 1989. 12. Cogswell, F.N., ‘The processing science of thermoplastic structural composites’, Int. Polymer Processing, 1(4), 157–165, 1987. 13. Wulfhorst, B. and Horsting, K., Rechnergestutzte Simulation der Drapierbarkeit

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14.

15.

16.

17. 18.

19.

20.

21.

22. 23. 24.

25.

26.

27. 28.

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von Geweben aus HL-Fasern für Faserverbundwerkstoffe (in German), Institut für Textiltechnik der RWTH Aachen, 1991. Bergsma, O.K. and Huisman, J., ‘Deep drawing of fabric reinforced thermoplastics’, in Computer Aided Design in Composite Material Technology; Proceedings of the International Conference, Southampton, Brebbia, C.A., de Wilde, W.P. and Blain, W.R., eds, Springer-Verlag, Berlin, 1988, pp. 323–334. Van West, B.P., Keefe, M. and Pipes, R.B., ‘A simulation of the draping of bidirectional fabric over three-dimensional surfaces’, J. Textile Institute, 81, 448–460, 1990. Gutowski, T., Hoult, D., Dillon, G. and Gonzalez-Zugasti, J., ‘Differential geometry and the forming of aligned fibre composites’, Composites Manufacturing, 2(3/4), 147–152, 1991. Van der Weeën, F., ‘Algorithms for draping fabrics on doubly-curved surfaces’, Int. J. Num. Meth. Eng., 31, 1415–1426, 1991. Trochu, F., Hammami, A. and Benoit, Y., ‘Prediction of fibre orientation and net shape definition of complex composite parts’, Composites: Part A, 27A, 319–328, 1996. Aono, M., Breen, D.E. and Wozny, M.J., ‘A computer-aided broadcloth composite layout design system’, in Geometric Modeling for Product Realization, Selected and Expanded Papers from the IFIP TC5/WG5.2 Working Conference on Geometric Modeling, Rensselaerville, NY, USA, 27 September–1 October 1992, Wilson, P.R., Wozny, M.J. and Pratt, M.J., eds, North-Holland, 1992, pp. 223–250. Aono, M., Denti, P., Breen, D.E. and Wozyn, M.J., ‘Fitting a woven cloth model to a curved surface: dart insertion’, IEEE Computer Graphics Appl., 16, 60–69, 1996. Bergsma, O.K., ‘Three dimensional simulation of fabric draping’, PhD Thesis, Faculty of Aerospace Engineering, Delft University of Technology, November 1995. Terzopoulos, D. and Fleischer, K., ‘Deformable models’, Visual Computer, 4, 306–331, 1988. Collier, J.R., Collier, B.J., O’Toole, G. and Sargand, S.M., ‘Drape prediction by means of finite element analysis’, J. Text. Inst., 82(1), 96–107, 1991. Pickett, A.K., Queckbörner, T., De Luca, P. and Haug, E., ‘An explicit finite element solution for the forming prediction of continuous fibre-reinforced thermoplastic sheets’, Composites Manufacturing, 6, 237–243, 1995. De Luca, P., Lefébre, P. and Pickett, A.K., ‘Numerical and experimental investigation of some press forming parameters of two fibre reinforced thermoplastics: APC2-AS4 and PEI-CETEX’, paper presented at Fourth Int. Conf. on Flow Processes in Composites Materials FPCM ’96, 7–9 September, Aberystwyth, UK, pp. 1–15, 1996. Johnson, A.F. and Pickett, A.K., ‘Numerical simulation of the forming process in long fibre reinforced thermoplastics’, in Computer Aided Design in Composite Material Technology V; CADCOMP 96, Blain, W.R. and De Wilde, W.P., eds, Computational Mechanics Publications, Southampton, pp. 233–242, 1996. Haftka, R.T. and Gürdal, Z., Elements of Structural Optimization, 2nd edn, Kluwer, Dordrecht, 1990. Ishikawa, T. and Chou, T.W., ‘One-dimensional micro-mechanical analysis of woven fabric composites’, AIAA, 21, 1714–1721, 1983.

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29. Rai, H.G., Rogers, C.W. and Crane, D.A., ‘Mechanics of curved fiber composites’, J. Reinf. Plast. Comp., 11, 552–566, 1992. 30. Vu-Khanh, T. and Liu, B., ‘Prediction of fibre rearrangement and thermal expansion behaviour of deformed woven-fabric laminates’, Composites Sci. Technol., 53, 183–191, 1995. 31. Zhang, Y.C. and Harding, J., ‘A numerical micromechanics analysis of the mechanical properties of a plain weave composite’, Computer Structures, 36, 839–844, 1990. 32. Kabelka, J., ‘Prediction of the thermal properties of fibre–resin composites’, in Developments in Reinforced Plastics-3, Pritchard, G., ed., Elsevier Applied Science, London, 1984, pp. 167–202. 33. Naik, N.K. and Shembekar, P.S., ‘Elastic behaviour of woven fabric composites: I–lamina analysis’, J. Composite Mater., 26, 2196–2225, 1992. 34. Naik, N.K. and Ganesh, V.K., ‘Prediction of on-axes elastic properties of plain weave fabric composites’, Composites Sci. Technol., 45, 135–152, 1992. 35. Naik, N.K. and Ganesh, V.K., ‘Prediction of thermal expansion coefficients of plain weave fabric composites’, Composite Structures, 26, 139–154, 1993. 36. Naik, N.K. and Ganesh, V.K., ‘An analytical method for plain weave fabric composites’, Composites, 26, 281–289, 1995. 37. Polynkine,A.A. and van Keulen, F., Calculation of Laminate Stiffnesses for Thermoformed Continuous Fibre Reinforced Thermoplastic Composites, Delft University of Technology, Report LTM 1079, July 1995. 38. Hashin, Z., ‘Analysis of composite materials – a survey’, J. Appl. Mech., 50, 481–505, 1983. 39. Lekhnitskii, S.G., Theory of Elasticity of an Anisotropic Elastic Body, HoldenDay, San Francisco, 1963. 40. Argyris, J. and Mlejnek, H.P., Die Methode der Finiten Elementen in der elementaren Strukturmechanik, Band I, Verschiebungsmethode in der Statik, Friedr. Vieweg, Braunschweig/Wiesbaden, 1986. 41. Bathe, K.J., Finite Element Procedures in Engineering Analysis, Prentice-Hall, New Jersey, 1982. 42. Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method, vols 1 and 2, 4th edn, McGraw-Hill. 43. van Keulen, F. and Booij, J., ‘Refined consistent formulation of a curved triangular finite rotation shell element’, Int. J. Num. Meth. Eng., 39, 2803–2830, 1996. 44. Noor,A.K. and Burton,W.S.,‘Assessment of shear deformation theories for multilayered composite plates’, Appl. Mech. Rev., 42(1), 1, 1989. 45. Olhoff, N., Rasmussen, J. and Lund, E., ‘A method of “exact” numerical differentiation for error elimination in finite-element-based semi-analytical shape sensitivity analyses’. Mech. Struct. Mach., 21, 1–66, 1993. 46. van Keulen, F. and de Boer, H., ‘Rigorous improvement of semi-analytical design sensitivities by exact differentiation of rigid body motions’, I. J. Num. Meth. Eng., 42, 71–91, 1998. 47. Toropov, V.V., ‘Simulation approach to structural optimization’, Structural Optimization, 1, 37–46, 1989. 48. Toropov, V.V., Filatov, A.A. and Polynkine, A.A., ‘Multiparameter structural optimization using FEM and multipoint explicit approximations’, Structural Optimization, 6, 7–14, 1993. 49. Toropov,V.V.,‘Multipoint approximation method in optimization problems with

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expensive function values’, in Computational Systems Analysis, Sydow, A., ed., pp. 207–212, Elsevier, 1992. Toropov, V.V., Filatov, A.A. and Polynkine, A.A., ‘Multiparameter structural optimization using FEM and multipoint explicit approximations’, Structural Optimization, 6, 7–14, 1993. van Keulen, F., Toropov, V. and Markine, V., ‘Recent refinements in the multipoint approximation method in conjunction with adaptive mesh refinement’, in Proceedings of the 1996 ASME Design Engineering Technical Conferences and Computers in Engineering Conference, 18–22 August, Irvine, CA, McCarthy, J.M., ed., 1996. Toropov, V.V., van Keulen, F., Markine, V. and de Boer, H., ‘Refinements in the multi-point approximation method to reduce the effects of noisy structural responses’, in 6th AIAA/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue WA, 4–6 September, Part 2,A Collection of Technical Papers, pp. 941–951, 1996. van Keulen, F. and Toropov, V.V., ‘New developments in structural optimization using adaptive mesh refinement and multi-point approximations’, Eng. Optimization, 29, 217–234, 1997. Draper, N.R. and Smith, H., Applied Regression Analysis, 2nd edn, Wiley, New York, 1981. Polynkine, A.A., van Keulen, F., de Boer, H., Bergsma, O.K. and Beukers, A., ‘Shape optimization of thermoformed continuous fibre reinforced thermoplastic products’, Structural Optimization, 11, 228–234, 1996.

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9 Resin impregnation and prediction of fabric properties B.J. HILL AND R. M c ILHAGGER

9.1

Introduction

It is apparent from the previous chapter that there is a very wide range of textile products that can be used as reinforcements for composite materials and components. Such a wide choice provides the designer with great difficulty since an appropriate reinforcement must be selected for a specific application. There is no hard and fast rule for this selection and, in many instances, factors such as ease of manufacture become dominant and reinforcements are often selected on the basis of this rather than for performance enhancement. In general, textile reinforcements for composites show good tensile strength but have poor performance in terms of compression or stiffness. This necessitates the use of a matrix to encapsulate the fibres, thus protecting them from damage but also enhancing the performance of the composite, in particular overcoming some of the weaknesses of textiles. Structural composites can be defined as products that use fibre reinforcements (50–70% by weight) of very high strength and stiffness in combination with polymeric, metal or other matrices. This class of composite has extremely unusual properties in which the matrix binds the reinforcing fibres together, forming a cohesive structure, providing a medium to transfer applied stresses from one filament through the matrix to the adjacent filaments. When polymeric matrices are used, composite structures with relatively low densities are produced which have very high specific properties, i.e. high strength/weight and high stiffness/weight ratios. Thus it is necessary to use a means of impregnating the reinforcements with a matrix system which can be polymeric or metallic, although the emphasis as far as this book is concerned is directed towards polymer matrix composites (PMC). The distribution of the matrix throughout the reinforcement is critical to the overall performance of the composite. Small variations in fibre volume fraction throughout the composite give rise to significant variations in properties. The simple rule of mixtures (9.1) for uni285 © 1999,Woodhead Publishing Ltd

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directional tape demonstrates the influence of fibre volume fraction (vf) on stiffness (E): E11 = vf ◊ Ef + (1 - vf ) ◊ Em

[9.1]

where the subscripts f and m refer to fibre and matrix respectively. If there is more than one type of fibre then this relationship can be modified: E11 = (1 - vf ) ◊ Em + vf1 ◊ Ef1 + vf 2 ◊ Ef 2 + vf 3 ◊ Ef 3 . . .

[9.2]

where vf is the overall fibre volume fraction and vf1, vf2 and vf3 the fibre volume fraction of the different fibre types. In order to achieve uniformity of properties, the resin must completely fill the interstices within the fabric and also, significantly, the spaces between the filaments making up the tows. When optimum packing is achieved, the spaces between the fibres account for 9.9% but in reality this is more likely to be in the order of 20–25% since ideal packing of the fibre filament bundle is unlikely to occur under normal circumstances. Further, when more complex textile structures are employed as reinforcements, the efficiency of fibre packing will decrease even further, making fibre volume fractions in excess of 60% very difficult to realize. The filaments have to be completely encapsulated in the matrix in order to ensure effective and efficient load transfer between fibres and matrix and also to protect the filaments from damage. To achieve this effective load transfer it is important that complete wetout of the fibrous mass is achieved. This implies that low-viscosity resins must be used which, in turn, suggests that thermosetting resins are employed and that the performance is developed and enhanced through the crosslinking of the resin system. In general, thermoplastic resins are of higher molecular weight, and hence of higher viscosity during processing, making complete wet-out, in particular in the interfilament spaces, very difficult to achieve satisfactorily. While the fibres dominate the tensile and stiffness properties, the matrix material influences high-temperature performance, transverse strength and moisture resistance of the composite. The resin is also a key factor in toughness, shear strength and in particular interlaminar shear stress (ILSS) resistance and oxidation and radiation resistance. Figure 9.1 demonstrates the significance of small deviations in fibre orientation on tensile modulus; these deviations also significantly reduce the tensile strength. Hence inplane misalignment or indeed reinforcement crimp can result in significant losses in mechanical performance. The matrix system has a significant influence on the fabrication process and associated parameters for forming the composite materials into intermediate and final components. Most carbon fibre composites are based on

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9.1 Typical variation of modulus with fibre orientation.

thermosetting epoxy matrices which offer low shrinkage during processing, excellent adhesion to the fibres, good property balance, particularly mechanical to electrical performance, and ease of fabrication. They also have a good heat resistance and stability over a wide range of environmental conditions. Typical fibre loading in high-performance composite materials is 60–65% by volume (65–70% by weight). Carbon fibres have a coefficient of thermal expansion which is a slightly negative sequence. Production of composites from fibres with a fairly broad range of coefficients of thermal expansion values permits the manufacture of components with an almost zero coefficient of thermal expansion. This feature can be exploited, particularly in aircraft, to hold critical instrumentation in a precise position as the composite properties of the supporting component can be tailored specifically at the design stage. For particular components this demonstrates the potential to design or engineer specific properties into materials to meet the performance requirements and hence optimize the structural design. In comparison with steel and aluminium, carbon fibre composites are lighter, have lower thermal conductivity, are stiffer and stronger and have superior fatigue resistance. A summary of the typical properties of high strength and high modulus carbon fibre composite materials in an epoxy resin is shown in Table 9.1. The marked differences in properties between uni-directional (0°), transverse (90°) and the quasi-isotropic (0°, ±45°, 90°) fibre orientations should be noted. The high-modulus fibre composite data

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Table 9.1. Typical properties of carbon fibre composite materials Property Unidirectional laminate Longitudinal (0°) Tensile strength (MPa) Tensile modulus (GPa) Ultimate strain (%) Compressive strength (MPa) Compressive modulus (GPa) Ultimate strain (%) Flexural strength (4 pt) (MPa) Flexural modulus (GPa) Interlaminar SS (short beam) (MPa) Transverse (90°) Tensile strength (MPa) Tensile modulus (GPa) Ultimate strain (%) Additional properties Density (kg/m3) Shear strength (in plane) (MPa) Shear modulus (in plane) (GPa) Poisson ratio (0 coupon) Coeff. thermal expansion ¥ 10-6/°C 0 °C 90 °C Quasi-isotropic laminate (0°, ±45°, 90°) Tensile strength (MPa) Tensile modulus (GPa) Ultimate strain (%)

High strength

High modulus

1785 145 1.2 120 140 1.1 1995 135 95

1165 215 0.55 840 190 0.45 1335 190 80

49 9.5 0.52

36 7.0 0.49

1550 72 4.8 0.30

1610 59 4.1 0.24

0.31 35.8 537 50 1.2

— — 305 73 0.42

reflect the lower-strength, higher-modulus properties and lower shear strengths inherent in high modulus composites associated with the much higher thermal treatments that these fibres undergo during their manufacture.

9.2

Hand impregnation

There are a number of ways in which fibres or reinforcements can be impregnated with resin. Initially, when composite materials were used for leisure goods such as sports canoes, the resin system was hand mixed and then applied by brush to each layer and consolidated using pressure applied through a hand-held roller. Chemical reaction proceeded in the presence of air to produce a crosslinked matrix. Such systems are highly labour intensive with long cure cycles and also there are significant hazards owing to the volatile products of reaction released into the atmosphere during the

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cure. Properties of materials produced in this way tend to be variable because of the lack of process control, in particular local variations in amount of resin applied. In addition, it is extremely difficult to occlude all the air entrapped between the plies since compaction of the layers is by hand only. With no direct escape route for this air, stress concentrations are set up during the exothermic cure reaction, creating large voids within the structure. In these applications, glass fibre, often in chopped strand mat form, and polyester resins were used and under these conditions, it is extremely difficult to achieve high fibre volume fractions and hence high performance. Hand lay-up techniques are used with open moulds to produce components with good surface finish characteristics. This is only possible on one surface. A gel coat is applied to the tool surface and allowed to cure. Plies of textile reinforcement are laid in on top of this hard gel coat finish, each being coated with resin and compacted. In this way the composite component is assembled and allowed to cure at room temperature. This labourintensive hand lay-up operation in open tools is used to produce large components.

9.3

Matched-die moulding

To achieve a more uniform distribution of resin throughout the reinforcement, more automated systems came into use [1]. Pre-mixed resin and hardener are injected, under pressure from a pressure pot, into the reinforcement placed in closed matched cavity tools. The resin spreads out radially from the point of injection, permeating through the reinforcement until the cavity is completely filled with resin. Under such conditions the flow paths must be fully understood and predictable, otherwise resinstarved areas are created even in very simple geometric configurations in which the resin front impinges on the cavity boundary wall when the flow front can no longer expand in the radial direction. Two such fronts on adjacent walls will result in the flow converging on a point within the reinforcement. Unless high pressures are used, this region will remain dry, i.e. not impregnated with resin. If high pressure is used, then compression of the enclosed air will occur, which will cause an increase in the air temperature. At best this temperature rise will accelerate the crosslinking reaction prematurely and at worst the temperature will rise to such a degree that thermal degradation of the resin will occur. The outcome of this will be burn marks on the component and significant loss of mechanical properties. Hence air vents must be accurately positioned in these areas to assist the removal of entrapped air and provide a quality composite. Such problems have led to a considerable amount of effort being made to model and predict the precise position of the molten resin front with

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respect to time [2,3]. These approaches, applied at the design stage, have permitted fill procedures to be developed by which resin-starved areas are eliminated through the use of accurately positioned vents and/or different resin injection points. These predictive approaches require greater knowledge of the properties of the reinforcements, particularly their permeability, which, depending upon the nature of the reinforcement constriction, may be different in the longitudinal and transverse directions. Modelling of isothermal flow of resin of constant viscosity through textile reinforcements with isotropic permeability is based on D’Arcy’s equation (9.3): Q=-

KA dp ◊ m dx

[9.3]

where K is the permeability, m the resin viscosity and dp/dx the pressure drop per unit length. For random mat non-woven reinforcements permeability is isotropic inplane while for other textile structures the permeability will be different in different directions depending upon the nature of the textile structure. This differential permeability will result in complex flow patterns in the tool, making flow prediction even more important, although the use of D’Arcy’s equation then becomes an over-simplification. The vast majority of the tows employed in woven, braided or knitted reinforcements comprise low twist or untwisted continuous filament yarns. The pressure flow of the low viscosity resins can be assisted by capillary flow in the parallel channels between the filaments and control of the filling operation must be exercised to ensure resin ‘racing’ or ‘tracking’ does not occur. If this is not controlled the resin flow front will race ahead (or fall behind) before rejoining the pressure flow front, leading to unimpregnated enclosed dry regions. Hence variations in fibre volume fraction will result. A variation of this process is to vacuum assist the resin into the tool. The cavity, with the reinforcement in situ, is evacuated and the resin is forced under pressure into the tool, thus wetting out the fibre. This approach is known as vacuum assisted resin injection (VARI) [4].

9.4

Degassing

One of the major difficulties associated with composite manufacture is that of void formation during impregnation and cure [5]. When these become entrapped within the matrix, stress concentrations can be established within the matrix. These may originate: • during mixing of the resin formulation; • during the complex chemical reactions that take place during the cure of thermosetting resins, when volatile gases are released and become encapsulated in the crosslinked resin;

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• during filling of the cavity as described above; • owing to the complex nature of the textile reinforcement, since air can become entrapped in the interstices of the fabric structure. This can be particularly evident when coarse yarns (or tows) are used or in complex 3-D braided or woven structures and may be most prevalent at the solid tool/composite interface. During the formulation stage of the resin system, mixing is necessary to ensure that the hardeners, the crosslinking agents or any other additives are uniformly distributed and dispersed. The agitation during this formulation draws air into the uncured polymer along with the air already absorbed within the low viscosity fluid. As indicated above, these ‘volatiles’ are potential problem areas and must be eliminated in high-performance composites. After rigorous mixing, the resin mixture is degassed, under full vacuum, giving a deaerated fluid ready for application to the reinforcement. This deaeration can be assisted by heating the resin, to reduce its viscosity, although great care must be exercised to ensure that crosslinking is not initiated.

9.5

Preimpregnation

One of the limitations of producing high-performance composite materials lies in the difficulty of achieving uniformity of fibre/resin distribution with low void content. Instead of relying on the pressure flow to force the resin throughout the reinforcement, dip coating and lick roll technology are used to apply a controlled and uniform amount of uncured resin to the reinforcement. The resin bath contains both the base matrix resin and the hardeners in a partially cured resin system. The rolls of ‘prepreg’ are wrapped in release film and can be stored under refrigerated conditions for a period of time before the shelf-life of the product expires (normally 90 days at -18 °C for aerospace quality materials). Adoption of this route ensures uniformity of resin distribution in the reinforcement and eliminates the need for the processor to handle resin systems but does require that lowtemperature storage facilities are available on the production site.

9.6

Vacuum bagging

The vacuum bagging system is used for producing non-critical components. Plies of thawed out and conditioned prepreg are cut into the appropriate shape either by hand or by an automated process such as a Gerber® cutter system. Plies are placed in a precise order and orientation on a tool surface. The lay-up sequence and orientation of the plies is critical to the performance of the composite. A layer of release film is laid on top of the ply lay-

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9.2 Vacuum bagging process for the production of composites.

up to prevent the resinous stack of plies from adhering to the fibrous breather cloth. This cloth is used to absorb any excess resin and distributes the applied pressure evenly over the lay-up. The complete assembly is enclosed in a sealed bag or the bagging layer is sealed to the surface of the tool surround beyond the boundaries of the component as shown in Fig. 9.2. A vacuum connector is inserted into this bagging film so that the ply stack can be consolidated under approximately one atmosphere of vacuum. This complete assembly, while still under vacuum, is placed in an oven at an elevated temperature to cure the resin system. While this route uses prepreg material, which should ensure an even distribution of resin throughout the reinforcement, it is only operated at a maximum pressure of approximately 1 bar to consolidate the plies into a ‘homogeneous’ layer. This low pressure is insufficient to compact the layers adequately to produce a high performance component with high fibre volume fraction and low void content.

9.7

Autoclave

For high-performance composites, high fibre volume and low void contents are essential. It is also important that distribution of both fibre and resin is uniform throughout the component. This is achieved by taking the vacuum bagging process one stage further. As previously described, prepregs in the form of unidirectional tows or woven fabrics impregnated with a partially cured resin system are used. The process follows the stages outlined in Fig. 9.3. A number of the steps in this process are similar to those used in the vacuum bagging process. In these steps care must be exercised both from the point of view of health and safety and to ensure that the lay-up is contamination free. Such contamination can seriously impair the performance of the composite component. A clean room is required and protective cloth-

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9.3 Route for composites production using the autoclave process.

ing should be worn at all times for the production of both defect-free components and health and safety reasons. The various steps in the manufacturing route are as follows. 1 Prepreg material is stored under refrigerated conditions at -18 °C. Prior to processing, rolls are removed and allowed to thaw and condition. After reaching room temperature, the fabric is cut into shaped plies, taking fibre orientation into account. Computer-based nesting is used to optimize fabric utilization. These plies are labelled. 2 The plies are hand laid into the thoroughly degreased and clean moulding tool in the correct sequence and orientation. Constant inspection and signing-off of the lay-up at each stage is necessary to ensure performance and quality. Where the component comprises a large number of plies, frequent debulking is required, i.e. the lay-up is compressed under vacuum, after which a further series of plies are laid-in. A balanced lay-up, i.e. symmetry of lay-up about the neutral axis, minimizes the extent of springback. Once the lay-up is completed, a layer of release film is placed on top of the plies, the breather cloth placed on top of the release film and the whole assembly is bagged and sealed. A vacuum nozzle is attached to the complete assembly. These steps are identical to those shown in Fig. 9.2 for vacuum bagging. Vacuum is then applied to the assembly. 3 After confirming the integrity of the seal, the bagged assembly, while still under vacuum, is placed in a computer-controlled autoclave which is programmed to follow a particular processing cycle of both temperature and pressure. A typical cycle is as shown in Fig. 9.4. The cycle is designed so that the maximum flow is achieved up to and including the hold period so that the fabric can be completely wettedout and the interstices and the interfilament regions in the fabric structure completely filled with resin. The application of pressure, through inert nitrogen gas at a very early stage in the cycle, consolidates the composite structure and the nitrogen minimizes the risk of fire and explosion. On ramping up the temperature to the maximum, the resin commences to crosslink through an exothermic chemical reaction. Heat-

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9.4 Autoclave cure cycle.

up rates, typically at 2–5 °C/min, are slow, to ensure that the exothermic reactions are kept under control. 4 Once the cure cycle has been completed and the component cooled, also at a slow rate, the excess resin around the periphery is trimmed off and holes drilled, etc., where necessary. 5 Finally, the various components are put together to form the final assembly. This route is used to manufacture high-quality composites mainly for aerospace applications. The fibre volume fraction for carbon fibre composites should be in the region of 60% and the void content <1%. Non-destructive quality control is performed using ultrasound scans or X-ray micrographs to confirm this. The autoclave can often form a bottleneck for composites manufacturing since commercial autoclaves are large and to be viable, full loads have to be assembled. Components of similar thickness are processed under the same cure cycle and hence production scheduling is crucial to the success of this operation. Since a single tool surface is utilized, a good finish is only secured on one surface of the component, the other surface being in contact with the release film. However, caul plates can be used to produce a good surface finish on both sides of the component, even for reasonably complex shapes. The example in Fig. 9.5 shows how T-pieces can be manufactured. These caul plates solve the additional problem of consolidating both the web and the flanges simultaneously once the pressure is applied during the cycle. While autoclave processing provides high-performance composites, the operation of the autoclave has associated high running costs. This route can

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9.5 Tooling for production of T-pieces.

9.6 Route for composites production using preforms.

also take advantage of preforming of reinforcements. Three-dimensional technical textiles, produced by weaving [6], knitting [7], braiding [8] or as non-crimp fabrics (NCF) [9] as dry near net shape fabrics, can be placed directly in the tool, thus reducing considerably the labour intensity of the operation. Resin, applied either by brush or in film form (see later), is heated to make it less viscous, and wets-out the fabric under pressure. This is followed by cure in a heated tool. The basic process is shown in Fig. 9.6.

9.8

Liquid moulding with vacuum assistance

Alternative routes have been investigated to provide more flexible processes [10], both to eliminate the bottleneck in the autoclave process and to address the problems of mass production of quality components particularly for automotive applications. The major drawback to the autoclave route, apart from the large capital outlay required, is the high cost associated with the operation of the process. Low-temperature storage space is costly to operate, the ply cut-out is time consuming and wasteful. The manual ply lay-up procedure is labour intensive in terms of both implementation and inspection to guarantee the quality of the component.

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9.7 Liquid moulding of composites using preforms.

As discussed above, resin impregnation under pressure also has its performance limitations but by using vacuum assistance, first of all to evacuate the cavity and then to draw the degassed resin into the reinforcement, high-performance composites can be produced. Adoption of such a route, as shown in Fig. 9.7, removes many of the disadvantages of the autoclave process, namely: • Low capital investment, although mould costs may be higher if matched tooling is used. • Cold storage areas are not necessary and hence high value added products are not held in stock. • Shelf-life constraints are eliminated. • When preforms are used, expensive cutting out and wastage are minimized and the labour-intensive hand lay-up is dispensed with. The removal of these offers a much more viable operation, although there may be some restrictions on the level of pressure that can be applied and hence the degree of consolidation of the composite. Investigations [7] into different combinations of gating and pressure/ vacuum injection have shown that peripheral gating and vacuum injection provide the most effective route to achieve high-performance composites. Under this arrangement, the flow surrounds the reinforcement and then the flow front converges towards the vacuum exit point. The position of this is not critical since initial vacuum, removing all the air from the sealed tool, encourages the in-flow of resin throughout the reinforcement. The progression of the fill of a rectangular plaque with the exit point deliberately off-set is shown in Fig. 9.8 [11]. Hence under these circumstances, venting, if at all necessary, is much less critical. The liquid moulding route with vacuum assistance is used to produce structural composites with high fibre volume fraction and low void content using either ply lay-up or more complex 3-D reinforcements. Clearly, the latter is a much less costly process particularly if near net shape textile reinforcements are employed. One of the objectives of adopting the resin transfer moulding route as a

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9.8 Sequence of mould filling using peripheral gating and off-centre vacuum exit.

manufacturing process is to reduce costs. However, utilization of matched tooling increases mould costs and indeed great care must be exercised to ensure that both sides of the tool are sufficiently stiff to resist plate deflection, thus introducing thickness variations in the component. Such variations will result in fibre volume fraction and performance variations throughout the composite structure. Stiffer tooling also has a large thermal mass which will, in turn, necessitate longer cycle times to complete the cure of the component. The use of ‘soft-top’ tooling, i.e. using a base metal tool and bagging the complete assembly as previously described, and applying vacuum, overcomes the problem of thickness variation. For good surface finish on both sides of the component, a caul plate with a release film can be placed on top of the preform. Under these conditions, the pressure applied between the caul plate and tool surface holds the reinforcement in position. This allows the resin to flow in the space around the outer edge of the reinforcement before being drawn into the reinforcement, completely wetting it out. Thickness variations, using this technique, are much less pronounced than for matched tooling under pressure unless very stiff tooling is used.

9.9

Resin film infusion

To overcome the time-consuming process of deaerating the resin system to ensure void-free composite manufacture, layers of resin, in film form, are laid into the ply assembly. When heat, pressure and/or vacuum are applied, the resin becomes less viscous and flows to fill the interstices in the reinforcement and the spaces between the filaments. This is a much more rapid

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method of manufacture and can be used for mass production of components. When complex 3-D reinforcements are used, resin film layers can be placed on the top and bottom surfaces, allowing resin, when heated, to be drawn throughout the reinforcement by vacuum before further consolidation.

9.10

Pultrusion

For constant cross-section composite components, the pultrusion method of impregnation offers a relatively low-cost operation for composite manufacture. A preformed 3-D reinforcement is drawn through a heated die, not dissimilar to a conventional extrusion die. Heated polymer is forced into the reinforcement as it is hauled at slow speed through the shaped die. The impregnated textile is passed through either a second die or an oven to cure the resin system. This process produces, for example, I-beams or hollow rectangular tubes of uniform section which can be used in construction applications.

9.11

Conclusion

With this large number of permutations and combinations of reinforcements and resins systems it is necessary to provide the designer with some means of assessing the potential of some of these alternatives. This must be by a means that predicts the volume of fibre in different directions within the composite material. Hence a modelling system has been devised that calculates the amount of fibre making up the reinforcement within a known composite volume. Using a modified Rule of Mixtures, which takes into account contributions from the transverse fibres, the composite modulus can be predicted. This will provide the designer with a means of selecting an appropriate reinforcement for a particular application without a costly trial and error experimental assessment.

9.12

Prediction of fabric properties

The thickness, and thus the fibre volume fraction, of the composite, together with the engineering properties of the reinforcement, which are a function of the proportions of yarn in each of the three mutually perpendicular directions, can be derived from the mass (areal density) of the textile reinforcement. Therefore the prediction of the areal density is important in the determination of composite properties. The areal density, expressed in g/m2, of the textile reinforcement can be determined by the summation of the masses of all the yarns within a measured area of the complete 3-D reinforcement which is large enough to

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contain sufficient weave pattern repeats to be representative of the structure. Consider the different yarn paths that can be located in a 3-D woven textile reinforcement (Figs. 9.9 and 9.10). In each of the warp and weft directions a yarn can play one of three roles and in Fig. 9.9 the role of each of the warp yarns is shown. For the purposes of calculation it is assumed that each yarn follows a rectilinear path, as shown in Fig. 9.9 and, that the role of the yarn is consistent during the weaving process. Therefore, the ‘straight’ length of each yarn in the warp or weft direction is equal to the length of the reinforcement in that direction and the vertical elements contribute either to the throughthe-thickness portion and/or to the ‘normal’ crimp content. To calculate the total areal density of the reinforcement, the mass of each of the constituent yarn elements in 1 m2 of the reinforcement is determined. d

dt

t weave pattern repeat

layer 1 layer 2

depth of penetration

na

layer 3

t

layer 4 layer 5 layer 6

one layer

9.9 Geometric representation of possible warp yarn paths.

d12 weave

weave

pattern repeat

layer 1

pattern repeat

layer 2

depth of penetration

layer 3

t

layer 4 layer 5 layer 6

warp yarn

(a) weft yarn

(b) weft integrating yarn

(c) weft stuffer yarn

9.10 Geometric representation of possible weft yarn paths.

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9.12.1 Mass of warp interlinking yarn (W1) in 1 m2 of reinforcement [12] The warp interlinking yarns are relatively crimp free and lie on the top and bottom surfaces of the reinforcement. They bind the structure together by linking through the entire thickness of the reinforcement. The path of a typical warp interlinking yarn is shown in Fig. 9.9(a). The mass of the warp interlinking yarn (W1) is calculated using Equation 9.4. W1 = mass of the ‘straight’ section of yarn (W2) + mass of the interlinking section (W3) tex1 Ê ˆ W1 = e1 ¥ 100 ¥ ¥ n1 Ë ¯ 1000 tex1 ˆ Ê t ¥ 2 e1 ¥ 100 ¥ f1 ¥ 100 + ¥ ¥ n1 ¥ Ë 100 d1 100 ¯ where: e1 tex1 n1 t f1 d1

[9.4]

= ends/cm/layer of interlinking yarn, = yarn count of the interlinking yarn, = number of layers of the interlinking yarn (usually 1 or 2), = thickness of the reinforcement in (cm), = picks/cm/layer, = number of picks between consecutive interlinks.

9.12.2 Mass of warp integrating yarn (W4) in 1 m2 of reinforcement [13] The warp integrating yarns are woven in the structure in a specific weave pattern and all of the warp yarns are engaged in binding the structure together. At predetermined and regular intervals the linking is accomplished by a yarn transferring from its original layer to another layer before returning to its original layer. Within a reinforcement, the depth of penetration of the interlinking yarns into the structure is the same, in order to give symmetry about the mid-plane and to reduce the possibility of spring back. In Fig. 9.9(b) the integrating yarn follows a plain weave pattern with the yarns interlinking every six picks and transferring from their original layer to one three distant from it. Thus layer 1 links to layer 4 (as illustrated), layer 2 to layer 5, layer 3 to layer 6, layer 4 to layer 1, layer 5 to layer 2 and layer 6 to layer 3. The mass of the warp integrating yarn (W4) is determined using Equation 9.5.

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W4 = mass of the ‘straight’ section of yarn (W5) + mass of the integrating section (W6) + mass of the crimp element (W7) tex 4 Ê ˆ Ê j 4 ¥ t ¥ e 4 ¥ f4 ¥ tex 4 ¥ n4 ˆ W4 = e 4 ¥ 100 ¥ ¥ n4 + Ë ¯ Ë ¯ 1000 5 ¥ d4 È t ¥ e 4 ¥ f4 ¥ tex 4 ¥ (d4 - w 4 ) ˘ +Í ˙˚ 5 ¥ d4 ¥ w 4 Î where: e4 tex4 n4 t f4 d4 j4

w4

[9.5]

= = = = = = =

ends/cm/layer of integrating yarn, yarn count of the integrating yarn, number of layers of the integrating yarn, thickness of the reinforcement (cm), picks/cm/layer, number of picks between consecutive interlinks, is a factor related to the depth of penetration of the interlinking yarn and is equal to the number of yarns the integrating yarn passes vertically when forming the link plus one divided by the total number of yarns in one column in the cross-section, = number of picks in one weave pattern repeat (for a plain weave w equals 2, for a 2 ¥ 1 twill w = 3 and 8-end satin w = 8).

It should be noted that in the equation for W4, the term relating to the mass of the crimp element does not contain a factor for the number of layers. This is because the crimp depth within a layer is defined as the thickness (t) divided by the number of layers (n) and therefore the total crimp element for n layers is equal to t ¥ n/n, i.e. t.

9.12.3 Mass of warp stuffer yarn (W8) in 1 m2 of reinforcement [12] These yarns are crimp free and are in-laid into the structure, as illustrated in Fig. 9.9(c) to give improved engineering properties to the composite component. The mass of the warp stuffer yarn (W8) is calculated using Equation 9.6: W8 = e8 ¥ 100 ¥

tex 8 ¥ n8 100

where: e8 = ends/cm/layer of stuffer warp yarn, tex8 = yarn count of the stuffer warp yarn, n8 = number of layers of the stuffer warp yarn.

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9.12.4 Mass of weft yarn (W9) in 1 m2 of reinforcement [14] These weft yarns take no part in binding the structure together and are part of the basic weave structure within each layer. In Fig. 9.10(a) the weft yarn follows a plain weave pattern. The mass of the weft yarn (W9) is determined using Equation 9.7: W9 = mass of the ‘straight’ section of weft yarn (W10) + mass of the crimp element (W11) tex 9 Ê ˆ Ê t ¥ e 9 ¥ f9 ¥ tex 9 ˆ W9 = f9 ¥ 100 ¥ ¥ n9 + Ë ¯ Ë ¯ 1000 5 ¥ w9 where: f9 e9 tex9 n9 t w9

= = = = = =

[9.7]

picks/cm/layer, ends/cm/layer of warp yarn, yarn count of the weft yarn, number of layers, thickness of the reinforcement (cm), number of picks in one weave pattern repeat.

9.12.5 Mass of weft integrating yarn (W12) in 1 m2 of reinforcement [12] These yarns, illustrated in Fig. 9.10(b), have an identical role to the warp integrating yarns and their mass is determined using Equation 9.8: W12 = mass of the ‘straight’ section of yarn (W13) + mass of the integrating section (W14) + mass of the crimp element (W15) tex12 Ê ˆ Ê j12 ¥ t ¥ e12 ¥ f12 ¥ tex12 ¥ n12 ˆ W12 = f12 ¥ 100 ¥ ¥ n12 + Ë ¯ Ë ¯ 1000 5 ¥ d12 È t ¥ e12 ¥ f12 ¥ tex12 ¥ (d12 - w12 ) ˘ [9.8] +Í ˙˚ 5 ¥ d12 ¥ w12 Î where: f12 tex12 n12 t el2 d12 j12

= picks/cm/layer of integrating yarn, = yarn count of the integrating yarn, = number of layers of the integrating yarn, = thickness of the reinforcement (cm), = ends/cm/layer, = number of ends between consecutive interlinks, = a factor related to the depth of penetration of the interlinking yarn and is equal to the number of yarns the integrating yarn passes vertically when forming the link plus one divided

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w12

303

by the total number of yarns in one column in the crosssection, = number of picks in one weave pattern repeat.

9.12.6 Mass of weft stuffer yarn (W16) in 1 m2 of reinforcement [1] These are the non-crimp yarns incorporated in the reinforcement to improve the composite performance and are illustrated in Fig. 9.10(c). They perform the same function as the warp stuffer yarns and their mass is determined using Equation 9.9: W16 = f16 ¥ 100 ¥

tex16 ¥ n16 1000

[9.9]

where: f16 = picks/cm/layer of stuffer weft yarn, tex16 = yarn count of the stuffer weft yarn, n16 = number of layers of the stuffer weft yarn. The total mass (areal density) (W) of 1 m2 of reinforcement is determined by the summation of the above values, using Equation 9.10: W = W1 + W4 + W8 + W9 + W12 + W16

[9.10]

Most woven reinforcement constructions will contain only some of these elements and therefore only those that are relevant would be included in the calculations to determine the total areal density and subsequently the fibre proportions.

9.12.7 Determination of the percentage of yarn in the X, Y and Z directions Once the overall mass has been determined then the proportions of yarn in each of the three mutually perpendicular directions can be determined. At this stage the ‘normal’ crimp is not considered as contributing to the X, Y or Z proportions even though it has been included in the determination of the overall weight. This is because the crimp elements do not lie in the principal stress directions and therefore do not contribute to either X or Y. In the Z direction, the crimp elements do not link the layers together; however, if nesting of the layers within the structure does occur then, under compaction during composite manufacture, the crimp undulations will fit into one another and provide resistance to interlaminar shear, thus aiding the Z component performance, and this may have to be taken into account later. The X proportion has three elements W2 (from Equation 9.4), W5 (from

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Equation 9.5) and W8 (Equation 9.6). These are summed together (Equation 9.11) to give the total mass of yarn contributing to the X direction. This value is expressed as a percentage of the total areal density (W) (Equation 9.12). Therefore WX = W2 + W5 + W8

[9.11]

and X% =

WX ¥ 100 W

[9.12]

The Y proportion is also made up of three elements W10 (from Equation 9.7), W13 (from Equation 9.8) and W16 (Equation 9.9). These too are summed together (9.13) to give the total mass of yarn contributing to the Y direction. This value is expressed as a percentage of the total areal density (W) (Equation 9.14). Therefore WY = W10 + W13 + W16

[9.13]

WY ¥ 100 W

[9.14]

and Y% =

The Z proportion is also made up of three elements W3 (from Equation 9.4), W6 (from Equation 9.5) and W14 (from Equation 9.8). These equations are summed together (9.15) to give the total mass of yarn contributing to the Z direction. This value is expressed as a percentage (9.16) of the total areal density (W ). Therefore WZ = W3 + W6 + W14

[9.15]

and Z% =

WZ ¥ 100 W

[9.16]

Thus, from the design concept of a woven reinforcement structure it is possible to predict the areal density, thickness and proportions of fibre in each of the three mutually perpendicular directions. From the total areal density (W ), the fibre volume fraction (vf) can be determined if the thickness (t) of the proposed composite is known using Equation 9.17: vf =

W 10 000 ¥ t ¥ r

where r = fibre density (g/cm3).

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9.12.8 Prediction of the composite modulus Known values of Ef and Em and calculated values of vfx, vfy and vfz can be substituted into a modified rule of mixtures, Equations 9.18, 9.19, 9.20, and the composite moduli in the three mutually perpendicular directions determined thus: E x = Ef vfx + (1 - vf )Em +

Ef Em vfy Ef Em vfz + Em vfz + (1 - vf )Ef Em vfz + (1 - vf )Ef [9.18]

where Ef is the modulus of the fibre and Em is the modulus of the resin matrix. Similarly, Ey = Ef vfy + (1 - vf )Em +

Ef Em vfx Ef Em vfz + Em vfx + (1 - vf )Ef Em vfz + (1 - vf )Ef [9.19]

Ez = Ef vfz + (1 - vf )Em +

Ef Em vfx Ef Em vfy + Em vfx + (1 - vf )Ef Em vfy + (1 - vf )Ef [9.20]

The directional fibre volumes (vfx, vfy and vfz) are fractional values of vf based on the percentage of fibre in the specified direction and, for example, vfx can be determined using Equation 9.21. Therefore vfx =

vf xX % 100

[9.21]

and similarly for vfy and vfz are determined using the values of Y% and Z%. In the particular case when the mass of the yarn making up the crimp element is not taken into account in determining the overall areal density, then vf can be calculated from the summation of the directional fibre volumes (9.22): vf = vfx + vfy + vfz

9.13

[9.22]

References

1. Kendall, K.N., Rudd, C.D., Owen, M.J. and Middleton, V., ‘Characterisation of the resin transfer moulding process’, Composites Manufacturing 3(2), 235–249, 1992. 2. Lui, B., Bickerton, S. and Advani, G., ‘Modelling and simulation of resin transfer moulding-gate control, venting and dry spot prediction’, Composites Part A, 27a(2), 135–141, 1996. 3. Ferland, P., Guittard, D. and Trochu, F., ‘Concurrent methods for permeability

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4. 5.

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measurement in resin transfer moulding. Modelling and simulation of resin transfer moulding’, Polymer Composites, 17(1), 149–153, 1996. Matthews, F.L. and Rawlings, R.D., Composite Materials: Engineering and Science, Chapman and Hall, London, 1995, p. 184. Lowe, J.R., Owen, M.J. and Rudd, C.D., ‘Void formation. Resin transfer moulding’, in Proc. of the 4th International Conference on Automated Composites, ICAC95, Nottingham, Institute of Materials, pp. 227–234, 1995. Hill, B.J., McIlhagger R., Harper C.M. and Wenger, W., ‘Woven integrated multilayered structures for engineering preforms’, Composites Manufacturing, 5(1), 25–30, 1994. Gommers, B. and Verpoest, I., ‘Tensile behaviour of knitted fabric reinforced composites’, in Proc. 10th International Conference on Composite Materials (ICCM10), Whistler, Canada, 1995, Vol. IV, pp. 309–316. Li, W., Hammad, M. and El-Shiek, A., ‘Structural analysis of 3D braided preforms for composites. Part 1 The 4 step process’, J. Text. Inst., 81(4), 491–514, 1990. Chou, T.W. and Ko, F.K., Textile Structural Composites, Vol. 3, Composite Materials Series, Elsevier, Oxford, 1989, pp. 140–142. Abraham, D. and McIlhagger, R., ‘A review of liquid injection techniques for the manufacture of aerospace composite structures’, Polymers Polymer Composites, 4(6), 437–443, 1996. Abraham, D. and McIlhagger, R., ‘Vacuum assisted resin transfer moulding for high performance carbon fibre composites’, in Proc. of the 4th International Conference on Automated Composites, ICAC95, Nottingham, 1995, pp. 299–306. Hill, B.J. and McIlhagger, R., ‘The development and appraisal of 3D interlinked woven structures for textile reinforced composites’, Polymers Polymer Composites, 4(8), 535–539, 1996. Hill, B.J. and McIlhagger, R., ‘The development and appraisal of 3D fully integrated woven structures for textile reinforced composites’, Polymers Polymer Composites, 5(1), 49–56, 1997. Hill, B.J., McIlhagger, R., Soden, J.A., Hanna, J.R.P. and Gillespie, E.S., ‘The influence of crimp measurements on the development and appraisal of 3D fully integrated woven structures’, Polymers Polymer Composites, 5(2), 1–10, 1997.

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