3-D Textile Reinforcements In Composite Materials (1999)

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3-D textile reinforcements in composite materials

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

Cambridge England

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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

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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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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

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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

290 291 291 292 295 297 298 298 298 305

Index

307

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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

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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

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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

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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

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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

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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

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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

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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.

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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.

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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

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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.

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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

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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

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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)

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

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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

Stringer flanges stitched to skin

Folding frame 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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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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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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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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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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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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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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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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26

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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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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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),

[1.2]

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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

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

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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

[1.9]

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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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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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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 Ê 157.6 49.4 50.2 Á 49.4 156.1 52.3 Á Á 50.1 52.3 156.5 Á 0.4 -0.2 -0.2 Á Á 1.2 -0.4 -0.9 Á Ë 1.2 -0.9 -0.4

0.4 1.2 1.2ˆ -0.2 -0.4 -0.9˜ ˜ -0.2 -0.9 -0.4˜ (GPa) 46.7 -0.6 -0.6˜ ˜ -0.6 43.9 0.4˜ ˜ -0.6 0.4 41.9¯

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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

35

1¥1 ±20 0.4

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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˜ (MPa) Á 1.9 -1.0 -1.0 49.5 -1.1 -1.1˜ 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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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 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

0.6 ~ 0.8 0.7 ~ 0.9

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

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

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

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

q, braiding angle q = 10 ~ 45

Linear Assembly Roving yarn

Woven 3-D Woven

Non-woven 3-D Orthogonal

Knit 3-D MWK

Braid 3-D Braid 0.4 ~ 0.6

0.3 ~ 0.6

0.2 ~ 0.4 0.4 ~ 0.6

~0.6

Vf

Fiber orientation, q (°)

Preform

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

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

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

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

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

Processing parameter

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

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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

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

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

Very complex preforms (knot-elements, curved structures)

3D braiding

Knitting (weft)

Preform geometry Flat fabrics Integral stiffeners Integral sandwich-structure Simple profiles

Principle – design

3D weaving

Textile process

Table 2.1. Textile processes for composites: an overview

Fibres mainly in mesh structure

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

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

Fibre orientation

Medium productivity Short mounting time

Medium productivity High mounting time

High productivity Very high mounting time

Productivity mounting

Integration of straight fibres in the mesh structure

Varying cross-sections Varying fibre orientation

Multiaxial 3D weavings with integrated 45° fibres

Development goals

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52 Attaching additional fibres on basic fabrics

Very complex preforms by combining several textile structures

Embroidery

Stitching

Preform geometry Flat fabrics Integral sandwichstructures

Principle – design

Knitting (warp)

Textile process

Table 2.1. (cont.)

Depending on basic preforms

Very complex fibre orientation, for example in main stress direction

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

Fibre orientation

Very quick process Short mounting time

Slow process Short mounting time

High productivity High mounting time

Productivity mounting

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

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

Integration of more layers in one production step

Development goals

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Mechanical properties

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

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

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

High stiffness and strength also under shear loading

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

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

Textile structure

3-D woven

3-D braided

Knitted (weft)

Knitted (warp)

Embroidered

Stitched

RTM Pressing (see weaving) Complex moulds for complex structure

RTM Pressing (see weaving) No waste of embroidery fibre

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

RTM Pressing (see weavings) Very good drapeability Minimum waste

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

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

Composite manufacturing

Table 2.2. Textile structural composites: an overview

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

Local reinforcement Load-introduction and loadtransmission elements

Flat and curved panels under biaxial load Curved sandwich-panels

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

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

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

Typical structural elements

Highly integrated panels for cell structures Complex fittings

Rotor-blade joining element Inspection-hole cover

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

Spaceframe elements Helmet shells Cladding element with high damage tolerance

Stiffening elements Chassis structures Spaceframe elements

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

Typical applications

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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

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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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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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•

•

3-D textile reinforcements in composite materials 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

2

3

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. 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. 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

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5

101

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. 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)

[4.3]

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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 ˙˚

[4.9]

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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

(criterion 3) [4.12]

Type of element

Beam Shell Shell Shell Shell Solid

Beam Solid

Type of braiding

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

3-D 3-D

Unidimensional Elasticity theory

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

Theory implemented

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

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

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

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

Requested strength properties

Requested stiffness properties

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

2 3 4 4 4 1 Criterion 2 Criterion 1

Criterion Criterion Criterion Criterion Criterion Criterion

Strength criterion

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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

Theory

Theory of laminated plates

Irons’s theory

First-order shear theory

Higher-order shear theory

Skins: theory of laminated plates Core: elasticity theory

Skins: Irons’s theory Core: elasticity theory

Skins: First-order shear theory Core: Elasticity theory

Type of element

Shell including skins and core

Shell including skins and core

Shell including skins and core

Shell including skins and core

Skins: Shell Core: Solid

Skins: shell Core: solid

Skins: shell Core: solid

Skins: criterion 3 Core: criterion 5 Skins: criterion 4 Core: criterion 5 Skins: criterion 4 Core: criterion 5 Skins: criterion 4 Core: criterion 5 Skins: criterion 3 Core: criterion 5 Skins: criterion 4 Core: criterion 5 Skins: criterion 4 Core: criterion 5

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

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

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

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

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

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

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

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

Strength criterion

Strength properties requested1,2,3

Stiffness properties requested1

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

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114

Skins: higher-order shear theory Core: elasticity theory

Elasticity theory

Skins and core: elasticity theory Piles: unidimensional

Skins: shell Core: solid

Skins: solid Core: solid

Skins: solid

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

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

Stiffness properties requested1

Skins: criterion 4 Core: criterion 5 Skins: criterion 1 Core: criterion 5

Skins: criterion 1

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

Core: criterion of hydrostatic pressure failure Piles: criterion 2

Strength criterion

Strength properties requested1,2,3

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.

Core: solid Piles: beams

Theory

Type of element

Table 4.6. (cont.)

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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

Gf Gm (Vm + V0 ) Gf V ¢m + Gm V ¢0

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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.

123

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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.

135

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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 2

Steel configuration used as a reference. 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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•

•

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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

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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

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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 2 L

[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 =

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 2 L h 2 L

[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 =

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

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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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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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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.

[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.

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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(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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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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[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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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

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

l3 2 m3

2

2

2l 2 l 3 2 m2 m3

2l3l1 2 m3 m1

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

n2 m2 n2

n3 m3 n3

2 n2 n3 m2 n3 + m3 n2

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

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 ij

n =1

M -1

[C ] = Â ij

n =1

1st

xn +1

- xn

1st

[S ] ij

(2L) 1st

xn +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

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

Knitted fabric composites

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

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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

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

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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)

b)

c) 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 Te