Modelling and Predicting Textile Behaviour (Woodhead Publishing in Textiles)

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Modelling and predicting textile behaviour

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The Textile Institute and Woodhead Publishing The Textile Institute is a unique organisation in textiles, clothing and footwear. Incorporated in England by a Royal Charter granted in 1925, the Institute has individual and corporate members in over 90 countries. The aim of the Institute is to facilitate learning, recognise achievement, reward excellence and disseminate information within the global textiles, clothing and footwear industries. Historically, The Textile Institute has published books of interest to its members and the textile industry. To maintain this policy, the Institute has entered into partnership with Woodhead Publishing Limited to ensure that Institute members and the textile industry continue to have access to high calibre titles on textile science and technology. Most Woodhead titles on textiles are now published in collaboration with The Textile Institute. Through this arrangement, the Institute provides an Editorial Board which advises Woodhead on appropriate titles for future publication and suggests possible editors and authors for these books. Each book published under this arrangement carries the Institute’s logo. Woodhead books published in collaboration with The Textile Institute are offered to Textile Institute members at a substantial discount. These books, together with those published by The Textile Institute that are still in print, are offered on the Woodhead web site at: www.woodheadpublishing. com. Textile Institute books still in print are also available directly from the Institute’s website at: www.textileinstitutebooks.com. A list of Woodhead books on textile science and technology, most of which have been published in collaboration with The Textile Institute, can be found towards the end of the contents pages.

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Woodhead Publishing Series in Textiles: Number 94

Modelling and predicting textile behaviour Edited by

X. Chen

CRC Press Boca Raton Boston New York Washington, DC

Woodhead publishing limited

Oxford    Cambridge    New Delhi

iv Published by Woodhead Publishing Limited in association with The Textile Institute Woodhead Publishing Limited, Abington Hall, Granta Park, Great Abington Cambridge CB21 6AH, UK www.woodheadpublishing.com Woodhead Publishing India Private Limited, G-2, Vardaan House, 7/28 Ansari Road, Daryaganj, New Delhi – 110002, India www.woodheadpublishingindia.com Published in North America by CRC Press LLC, 6000 Broken Sound Parkway, NW, Suite 300, Boca Raton, FL 33487, USA First published 2010, Woodhead Publishing Limited and CRC Press LLC © Woodhead Publishing Limited, 2010 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 Woodhead Publishing Limited. The consent of Woodhead Publishing Limited 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 Limited 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 978-1-84569-416-6 (book) Woodhead Publishing ISBN 978-1-84569-721-1 (e-book) CRC Press ISBN 978-1-4398-0107-9 CRC Press order number N10008 The publishers’ policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp which is processed using acid-free and elemental chlorine-free practices. Furthermore, the publishers ensure that the text paper and cover board used have met acceptable environmental accreditation standards. Typeset by Replika Press Pvt Ltd, India Printed by TJ International Limited, Padstow, Cornwall, UK

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Contents

Contributor contact details

xi

Woodhead Publishing Series in Textiles

xv

Preface

xxi

Part I Modelling the structure and behaviour of textiles 1

Structural hierarchy in textile materials: an overview



X. Chen and J. W. S. Hearle, The University of Manchester, UK

3

1.1 1.2 1.3 1.4 1.5 1.6 1.7

The textile hierarchy Modelling of fibres from the molecular level Modelling fibre behaviour Modelling yarns and cords Modelling fabrics Sources of further information References

3 4 11 14 18 37 37

2

Fundamental modelling of textile fibrous structures

41



S. Grishanov, De Montfort University, UK

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

Introduction Fibre classification Fibre functions in textile materials and composites Modelling fibre structure Statistical models of fibre geometry Modelling mechanical behaviour of single fibres Viscoelastic properties of fibres Modelling fibre friction Modelling fibre assemblies Conclusions References

41 43 43 47 52 68 76 86 89 102 103

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Contents

3

Yarn modelling



R. Ognjanovic, Innoval Technology Limited, UK

3.1 3.2 3.3

Introduction Yarn construction Types of models to predict the structure and properties of yarns Applications and examples Future trends in yarn modelling Sources of further information and advice References

114 136 138 139 140

4

Modelling the structures and properties of woven fabrics

144



E. Vidal-Salle and P. Boisse, INSA Lyon, France

4.1

Introduction: The importance and objectives of modelling woven fabrics The mechanical behaviour of woven fabrics Different approaches for modelling the mechanical behaviour of woven fabrics at different scales Structure and geometry of the unit woven cell Specific experimental tests 3D simulation of the deformation of the unit woven cell at the mesoscopic level Image analyses: Full field digital image correlation measurements and X-ray tomography Conclusions and future trends References

3.4 3.5 3.6 3.7

4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5

Modelling of nonwoven materials



N. Mao and S. J. Russell, University of Leeds, UK

5.1 5.2 5.3

Introduction Constructing physical models of nonwoven structure Modelling of pore size and pore size distribution in nonwoven fabrics Tensile strength Modelling the bending rigidity of nonwoven fabrics Modelling the specific permeability of nonwovens Thermal resistance and thermal conductivity Acoustic impedance Particle filtration in nonwoven filters Future trends and sources of further information and advice References

5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11

112 112 113

144 145 148 153 155 161 172 174 175 180 180 182 185 189 192 193 204 207 212 219 220

Contents

6

Modelling and visualization of knitted fabrics



Y. Kyosev, Niederrhein University of Applied Sciences, Germany, W. Renkens, Renkens Consulting, Germany

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

Aim and objectives of modelling knitted structures Classification of knitted structures Scales in the structure Structural elements of knitted structures at the meso-scale Modelling steps Model building Post-processing Other types of model Application areas of the simulated fabrics and future trends Conclusions References

6.10 6.11

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225 225 226 228 231 236 237 256 257 257 258 259

Part II Case studies 7

Modelling of fluid flow and filtration through woven fabrics



M. A. Nazarboland, X. Chen and J. W. S. Hearle, University of Manchester, UK, and R. Lydon and M. Moss, Clear Edge Group, UK

7.1 7.2 7.3 7.4 7.5

Introduction Various techniques in modelling fluid flow and filtration Model design and analysis Influence of fabric parameters on flow performance Influence of fluid flow and fabric parameters on filtration performance Influence of particle properties on filtration performance Application of fluid flow and filtration modelling Future trends References

299 306 316 317 318

8

Modelling, simulation and control of textile dyeing

322



R. Shamey, North Carolina State University, USA

8.1 8.2 8.3 8.4

Introduction Sorption isotherms Dye diffusion models Models relating dyeing parameters to the quality of dyeing Numerical simulation of package dyeing Applications

7.6 7.7 7.8 7.9

8.5 8.6

265 265 267 271 285

322 323 326 328 347 353

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Contents

8.7 8.8 8.9

Future trends Acknowledgment References

356 356 356

9

Modelling colour properties for textiles

360



D. P. Oulton, The University of Manchester, UK

9.1 9.2 9.3 9.4 9.5 9.6

Introduction Types of model used Case study in colour communication Future trends in colour modelling Commercial vendors and their products References

360 364 376 381 384 385

10

3D modelling, simulation and visualisation techniques for drape textiles and garments

388



F. Han and G. K. Stylios, Heriot-Watt University, UK

10.1 10.2 10.3 10.4 10.5 10.6

Introduction Review of 3D textile models Automatic measurement of fabric mechanics Drape measurement and evaluation Key principles of 3D mass–spring models Clothing simulation: strengths, limitations and suggested improvements 10.7 Experimental results and discussions 10.8 Applications and examples 10.9 Conclusions and future trends 10.10 References 11

Recognition, differentiation and classification of regular repeating patterns in textiles



M. A. Hann and B. G. Thomas, University of Leeds, UK

11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9

Introduction Study of pattern: historical precedents Symmetry in pattern: fundamental concepts Classification of motifs The seven classes of border patterns The 17 classes of all-over patterns Colour symmetry Conclusions References

388 389 392 393 398 403 406 410 418 419 422 422 423 426 430 432 436 450 452 453

Contents

12

Mathematical and mechanical modelling of 3D cellular textile composites for protection against trauma impact



X. Chen, The University of Manchester, UK

12.1 12.2 12.3

Introduction Mathematical description of cellular textile structures Computer aided design/computer aided manufacturing (CAD/CAM) of 3D cellular woven fabrics Experimental study of properties of 3D cellular composites Theoretical characterisation of 3D cellular composites Discussions and conclusions Future trends References

12.4 12.5 12.6 12.7 12.8

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457 457 459 462 468 476 490 492 492

13

Development and application of expert systems in the textile industry



R. Shamey, W. S. Shim and J. A. Joines, North Carolina State University, USA

13.1 13.2 13.3 13.4 13.5 13.6 13.7

Introduction System principles Strengths and limitations of expert systems Applications of expert systems in the textile industry Future trends Sources of further information and advice References

494 499 507 509 514 514 514



Index

521

494

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xi

Contributor contact details

(*= main contact)

Chapter 1

Chapter 3

Dr Xiaogang Chen* and Professor John Hearle School of Materials The University of Manchester Manchester M60 1QD UK

Dr Rade Ognjanovic Innoval Technology Limited Beaumont Close Banbury OX16 1TQ UK

Email: xiaogang.chen@manchester. ac.uk

Chapter 2 Dr Sergei Grishanov TEAM Research Group De Montfort University The Gateway Leicester LE1 9BH UK Email: [email protected]

Email: rade.ognjanovic@innovaltec. com

Chapter 4 Dr Emmanuelle Vidal-Salle and Professor Philippe Boisse* INSA Lyon 20, rue Albert Einstein 69621 Villeurbanne Cedex France Email: [email protected] [email protected]

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Contributor contact details

Chapter 5 Dr Ningtao Mao,* S. J Russell Centre for Technical Textiles University of Leeds Leeds LS2 9JT UK Email: [email protected]

Chapter 6 Professor Dr Yordan Kyosev* Department of Textile and Clothing Technology Niederrhein University of Applied Sciences D-41065 Mönchengladbach Germany Email: [email protected] Dipl.-Ing. Wilfried Renkens Renkens Consulting Tittardsfeld 102 D-52072 Germany Email: [email protected]

Chapter 7 Dr Mohammad Ali Nazarboland,* Dr Xiaogang Chen and Professor John W. S. Hearle School of Materials The University of Manchester Manchester M60 1QD UK Email: [email protected] [email protected]

Professor Richard Lydon and Martin Moss Clear Edge Group Knowsley Rd Industrial Estate Haslingden BB4 4EJ UK

Chapter 8 Professor Renzo Shamey North Carolina State University Raleigh NC 27695-8301 USA Email: [email protected]

Chapter 9 David P. Oulton School of Materials Sackville St Building The University of Manchester Manchester M60 1QD UK Email: david.oulton@manchester. ac.uk

Contributor contact details

Chapter 10

Chapter 13

Dr Fan Han* Middlesex University London NW4 4BT UK

Professor Renzo Shamey,* Dr W Shim and Professor J.A. Joines North Carolina State University Raleigh NC 27695-8301 USA

Email: [email protected]

Professor George K. Stylios Heriot-Watt University Edinburgh EH14 4AS Scotland Email: [email protected]

Chapter 11 Professor Michael Hann* and Dr Briony G. Thomas School of Design University of Leeds Leeds LS2 9JT UK Email: [email protected] [email protected]

Chapter 12 Dr Xiaogang Chen School of Materials The University of Manchester PO Box 88 Manchester M60 1QD UK Email: xiaogang.chen@manchester. ac.uk

Email: [email protected]

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xv

Woodhead Publishing Series in Textiles

1 Watson’s textile design and colour Seventh edition Edited by Z. Grosicki 2 Watson’s advanced textile design Edited by Z. Grosicki 3 Weaving Second edition P. R. Lord and M. H. Mohamed 4 Handbook of textile fibres Vol 1: Natural fibres J. Gordon Cook 5 Handbook of textile fibres Vol 2: Man-made fibres J. Gordon Cook 6 Recycling textile and plastic waste Edited by A. R. Horrocks 7 New fibers Second edition T. Hongu and G. O. Phillips 8 Atlas of fibre fracture and damage to textiles Second edition J. W. S. Hearle, B. Lomas and W. D. Cooke 9 Ecotextile ’98 Edited by A. R. Horrocks 10 Physical testing of textiles B. P. Saville 11 Geometric symmetry in patterns and tilings C. E. Horne 12 Handbook of technical textiles Edited by A. R. Horrocks and S. C. Anand 13 Textiles in automotive engineering W. Fung and J. M. Hardcastle

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Woodhead Publishing Series in Textiles

14 Handbook of textile design J. Wilson 15 High-performance fibres Edited by J. W. S. Hearle 16 Knitting technology Third edition D. J. Spencer 17 Medical textiles Edited by S. C. Anand 18 Regenerated cellulose fibres Edited by C. Woodings 19 Silk, mohair, cashmere and other luxury fibres Edited by R. R. Franck 20 Smart fibres, fabrics and clothing Edited by X. M. Tao 21 Yarn texturing technology J. W. S. Hearle, L. Hollick and D. K. Wilson 22 Encyclopedia of textile finishing H.-K. Rouette 23 Coated and laminated textiles W. Fung 24 Fancy yarns R. H. Gong and R. M. Wright 25 Wool: Science and technology Edited by W. S. Simpson and G. Crawshaw 26 Dictionary of textile finishing H.-K. Rouette 27 Environmental impact of textiles K. Slater 28 Handbook of yarn production P. R. Lord 29 Textile processing with enzymes Edited by A. Cavaco-Paulo and G. Gübitz 30 The China and Hong Kong denim industry Y. Li, L. Yao and K. W. Yeung 31 The World Trade Organization and international denim trading Y. Li, Y. Shen, L. Yao and E. Newton

Woodhead Publishing Series in Textiles

xvii

32 Chemical finishing of textiles W. D. Schindler and P. J. Hauser 33 Clothing appearance and fit J. Fan, W. Yu and L. Hunter 34 Handbook of fibre rope technology H. A. McKenna, J. W. S. Hearle and N. O’Hear 35 Structure and mechanics of woven fabrics J. Hu 36 Synthetic fibres: nylon, polyester, acrylic, polyolefin Edited by J. E. McIntyre 37 Woollen and worsted woven fabric design E. G. Gilligan 38 Analytical electrochemistry in textiles P. Westbroek, G. Priniotakis and P. Kiekens 39 Bast and other plant fibres R. R. Franck 40 Chemical testing of textiles Edited by Q. Fan 41 Design and manufacture of textile composites Edited by A. C. Long 42 Effect of mechanical and physical properties on fabric hand Edited by Hassan M. Behery 43 New millennium fibers T. Hongu, M. Takigami and G. O. Phillips 44 Textiles for protection Edited by R. A. Scott 45 Textiles in sport Edited by R. Shishoo 46 Wearable electronics and photonics Edited by X. M. Tao 47 Biodegradable and sustainable fibres Edited by R. S. Blackburn 48 Medical textiles and biomaterials for healthcare Edited by S. C. Anand, M. Miraftab, S. Rajendran and J. F. Kennedy 49 Total colour management in textiles Edited by J. Xin

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Woodhead Publishing Series in Textiles

50 Recycling in textiles Edited by Y. Wang 51 Clothing biosensory engineering Y. Li and A. S. W. Wong 52 Biomechanical engineering of textiles and clothing Edited by Y. Li and D. X.-Q. Dai 53 Digital printing of textiles Edited by H. Ujiie 54 Intelligent textiles and clothing Edited by H. Mattila 55 Innovation and technology of women’s intimate apparel W. Yu, J. Fan, S. C. Harlock and S. P. Ng 56 Thermal and moisture transport in fibrous materials Edited by N. Pan and P. Gibson 57 Geosynthetics in civil engineering Edited by R. W. Sarsby 58 Handbook of nonwovens Edited by S. Russell 59 Cotton: Science and technology Edited by S. Gordon and Y.-L. Hsieh 60 Ecotextiles Edited by M. Miraftab and A. Horrocks 61 Composite forming technologies Edited by A. C. Long 62 Plasma technology for textiles Edited by R. Shishoo 63 Smart textiles for medicine and healthcare Edited by L. Van Langenhove 64 Sizing in clothing Edited by S. Ashdown 65 Shape memory polymers and textiles J. Hu 66 Environmental aspects of textile dyeing Edited by R. Christie 67 Nanofibers and nanotechnology in textiles Edited by P. Brown and K. Stevens

Woodhead Publishing Series in Textiles

xix

68 Physical properties of textile fibres Fourth edition W. E. Morton and J. W. S. Hearle 69 Advances in apparel production Edited by C. Fairhurst 70 Advances in fire retardant materials Edited by A. R. Horrocks and D. Price 71 Polyesters and polyamides Edited by B. L. Deopura, R. Alagirusamy, M. Joshi and B. S. Gupta 72 Advances in wool technology Edited by N. A. G. Johnson and I. Russell 73 Military textiles Edited by E. Wilusz 74 3-D fibrous assemblies: Properties, applications and modelling of three-dimensional textile structures J. Hu 75 Medical textiles 2007 Edited by J. Kennedy, A. Anand, M. Miraftab and S. Rajendran 76 Fabric testing Edited by J. Hu 77 Biologically inspired textiles Edited by A. Abbott and M. Ellison 78 Friction in textile materials Edited by B. S. Gupta 79 Textile advances in the automotive industry Edited by R. Shishoo 80 Structure and mechanics of textile fibre assemblies Edited by P. Schwartz 81 Engineering textiles: Integrating the design and manufacture of textile products Edited by Y. E. El-Mogahzy 82 Polyolefin fibres: Industrial and medical applications Edited by S. C. O. Ugbolue 83 Smart clothes and wearable technology Edited by J. McCann and D. Bryson 84 Identification of textile fibres Edited by M. Houck

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Woodhead Publishing Series in Textiles

85 Advanced textiles for wound care Edited by S. Rajendran 86 Fatigue failure of textile fibres Edited by M. Miraftab 87 Advances in carpet technology Edited by K. Goswami 88 Handbook of textile fibre structure Edited by S. Eichhorn, J. W. S Hearle, M. Jaffe and T. Kikutani 89 Advances in knitting technology Edited by T. Dias 90 Smart textile coatings and laminates Edited by W. C. Smith 91 Handbook of tensile properties of textile and technical fibres Edited by A. Bunsell 92 Interior textiles: Design and developments Edited by T. Rowe 93 Textiles for cold weather apparel Edited by J. Williams 94 Modelling and predicting textile behaviour Edited by X. Chen 95 Textiles for construction Edited by G. Pohl 96 Engineering apparel fabrics and garments J. Fan and L. Hunter 97 Surface modification of textiles Edited by Q. Wei 98 Sustainable textiles Edited by R. S. Blackburn

xxi

Preface

‘Textiles’ refer to fibres and fibre assemblies that are used principally as raw materials for different types of products. Under this definition, textiles will include fibres (be they natural or manufactured, short staples or continuous filaments), yarns (be they single strand or cabled) and fabrics (be they woven, knitted, braided or non-woven, two dimensional or three dimensional). For garments, beddings, curtains, floor coverings, as well as technical end-use (such as a type of textile composites), textile fabrics are the raw materials, providing not only the appearance, texture and decorative features but also the various properties that make the textile suitable for the intended applications. Textiles are popular types of material that have been widely used domestically and industrially. However, textiles as types of material are special when compared to materials such as metal. Textiles are far from homogenous and isotropic because they are assemblies of fibres. On the other hand they are soft materials compared to their metallic counterparts. Furthermore, fibres are made of wide range of different chemical compositions and when different fibres are used for making textiles, the physical and chemical properties can be vastly different. Because of all these special features, prediction of textile behaviour has been drawing much attention and effort over the years. This book is presented with the aim of introducing the methods and techniques that have been developed in modelling and predicting fabric properties and behaviour for most current end-use applications. The textile hierarchy begins with fibre as the basic element. Fibres are the construction units of yarns and some non-woven fabrics. Then yarns are used as components for making fabrics based on weaving, knitting and braiding technologies. It is essential to understand fibre behaviour which is largely determined by the chemical structure of the polymer and physical configuration of the molecular chain. Based on the fibres, it could be claimed that the behaviour of a textile assembly is a function of the properties of the building block and the way that these building blocks are constructed in the assembly. Following this logic, the behaviour of yarn depends on the fibre property and the yarn construction and the fabric behaviour is determined by

xxii

Preface

the properties of the composing yarn and construction of the fabric. A fabric contains a tremendous amount of fibres of the same or different types and there are endless ways that a fibre is configured individually or collectively in a fabric. Phenomena such as these make the modelling of textiles very challenging. Textiles are used in many different ways for different functions. Mechanical behaviour certainly is one of the most important aspects, because strength and durability are the most essential requirements of a textile product. Part I of this book addresses the fundamental issues in modelling and predicting textile behaviour. The first chapter gives an overview of the structural hierarchy and outlines the techniques and progress made in modelling fibres, yarns and fabrics. Techniques for detailed fibre and yarn modelling are given in two following chapters. Modelling of woven, knitted and nonwoven fabrics are described in three separate chapters, giving detailed insight into the modelling of these three very different types of fabric. Part II of this book lays emphasis on modelling of textiles for particular applications and case studies of individual problems in individual applications. When textile fabrics are used as the filtering media for air and water purification, the orifice of the fabrics becomes important. Chapter 7 reports on a study where the woven fabrics are used in a filtration application, analysing the influence of the fabric structural parameters on the filtration performance and behaviour. Modelling dyeing of textiles is explained in Chapter 8, where numerous models for predicting textile dyeing process are introduced. After Chapter 9 on the modelling of colour properties for textiles where basic models and some case studies are given, Chapter 10 discusses the modelling and simulation of the drape of textiles and garments. In this chapter, the authors explain the key principle of the three-dimensional (3D) mass–spring models that facilitate dynamic drape. Parallel to the modelling of the technical aspects of textiles, Chapter 11 discusses the modelling of patterns that are used for fabric printing. Three-dimensional textiles become more and more of interest to industry for their structural integrity and special properties. Chapter 12 is included to explain the structural and mechanical modelling of 3D cellular textile composites for impact energy absorption and for force attenuation. The progress made in modelling leads to engineering and manufacture of better textiles. The book ends with Chapter 13 on the development and application of expert systems in the textile industry. I would like to take this opportunity to thank all contributors for their valuable time devoted to writing the chapters for this exciting book. I also wish to extend my gratitude to my beloved family for their support and for permitting me the time taken away from them for all those weekends and late evenings. Xiaogang Chen

1

Structural hierarchy in textile materials: an overview

X. Chen and J. W. S. Hearle, The University of Manchester, UK

Abstract: This chapter gives an overview of the structural hierarchy in textile materials and the techniques adopted for modelling textiles at different levels. Molecular structures are fundamental in affecting the behaviour in fibres and fibre assemblies and the best known methods in modelling natural polymer fibres, manufactured polymer fibres and inorganic fibres are reviewed. After introducing the modelling of fibre behaviour, methods used in modelling textile assemblies are discussed. Structures of yarns, woven and knitted fabrics are presented and approaches in modelling the geometry and mechanical behaviour are reviewed in some detail. Key words: fabrics, fibres, geometric, hierarchy, mechanical, modelling, yarns.

1.1

The textile hierarchy

Most textile materials are two-dimensional (2D) sheets, but they differ from other sheet materials, such as metal plates or foils, plastic films, rubber membranes and even paper, in the complexity of their structure. Figure 1.1 shows the structural hierarchy from fundamental particles to consumer products. The central core in bold type is the dominant route, which is the main concern of this book, but some by-ways are also shown. The relation of fundamental particles to atoms is part of physics: the textile interest starts with molecules. Also outside the scope of the book is the conversion of fabrics into clothes, household and industrial textiles, but the influence of such fabric properties as drape, hand, bulk and softness on their performance in products is relevant. With the flowering of textile research from the 1920s onwards, modelling, or ‘theory’ as it was more commonly called, increased understanding of the science of fibres and textiles. However, it did not lead to quantitative design predictions as it did in many other industries. The new knowledge did help in the empirical development of new materials, machines and processes, since it added greater understanding to the practical experience of thousands of years and the intuition of those working with fibres and textiles. There are several reasons for the slowness to introduce an engineering design approach. Two factors are the complexity of the textile hierarchy and 3

4

Modelling and predicting textile behaviour Protons, neutrons and electrons

Atoms Other molecules

Polymer molecules

Inorganic fibres

Major textile fibres Yarns

Cordage

Weaves, knits and braids → 3D fabrics

Nonwovens

Consumer and industrial products

1.1 Textile structural hierarchy.

the achievements of the textile industry using limited design calculations. Another is that, unlike bridges or aircraft, where the manufacturer needs to be sure that they will not fall down, textile manufacturers can rely on trialand-error; if a shirt tears, it is nuisance but not a disaster. In the 21st century when textiles are employed for technical applications as well as domestic ones, this must change for the following reasons: making prototype samples is increasingly expensive; the lifetime experience in a particular branch of textiles is no longer a part of contemporary culture; engineers using textiles in new industrial uses require quantitative data; the young, referred to by the head teacher of a primary school as ‘digital natives’, expect to be able to use computers in their work and to be able to rely on science to make predictions; and the growth of computer power has made it possible to model the complex textile hierarchy more completely.

1.2

Modelling of fibres from the molecular level

1.2.1 Formation, structure and properties There are two aspects of fibre modelling to be considered: how processing conditions affect structure and how structure affects fibre performance. For natural fibres, there are some useful models relating structure to properties, but the formation mechanisms are a more difficult area of biochemistry and biophysics. For the commonest manufactured fibres, the structure/property relations are less well modelled. The processes, particularly for melt spinning, have been modelled in phenomenonological terms. Ziabicki (1976) pioneered

Structural hierarchy in textile materials: an overview

5

models of the dynamics of fibre spinning, which take account of rheology and heat and mass flow. This was followed up with more studies by various researchers (Ziabicki and Kawai, 1985). Monte Carlo methods can be used to model the behaviour of polymer molecules. The polymer chain is treated as following a path on a regular lattice. Its simplest form is based on a square two-dimensional (2D) lattice. At any point on the lattice, the path of the polymer chain is subject to a random choice as to which of the three available directions the next link follows. In this way an ensemble of polymer configurations can be built up and linked to statistical mechanics, through the relation between entropy and number of options. For example, the number of ways in which a given end-to-end length of the polymer molecule can be achieved. Monte Carlo methods can be used to model transport processes for mass and radiation.

1.2.2 Polymeric form The great majority of textile fibres are partially oriented, partially crystalline assemblies of linear polymer molecules, although this statement covers a diversity of forms and there is a hierarchy of structure up to the whole fibre. The dominant atoms are carbon, hydrogen, oxygen and nitrogen with some fibres containing sulphur, chlorine or fluorine. Quantum theory provides data on the potential energies of interactions between the atoms. In principle, dynamic molecular modelling (DMM) could predict the formation of structure and the properties of the resulting fibres at the level of fine structure involving the interaction of a few hundred chain molecules. There are two problems. DMM treats the system as an assembly of particles obeying Newton’s Laws of Motion under the influence of the interatom potentials. There is a theoretical worry that the nanoscale systems do not follow the laws of classical physics, but allow for quantum superposition. More important, in practice, is that the limits of current computer power, coupled with the time needed for the systems to reach equilibrium or keep up with changes, mean that only small volumes can be effectively modelled. There have been some uses of DMM. In 1993, as illustrated in Fig. 1.2, BIOSYM Technologies produced images of the components of the fine structure of polyethylene terephthalate (polyester) fibres (Hearle, 1993). Crystal forms, with their regular structure of unit cells, are easily modelled. Amorphous regions are more difficult. Hints of what may be possible in future are found in the computational molecular modelling of the structure of keratin a-helices, for example by North et al. (1994), and transition temperatures by Knopp et al. (1996a, 1996b, 1997) though these do not include mechanical deformation. There is a continuous development of commercial software for DMM in association with academic groups such as the Harvard molecular modelling group who developed the CHARMM force field (MacKerell et al., 1998) for mechanical modelling.

6

Modelling and predicting textile behaviour

(a)

(c)

(b)

(d)

1.2 DMM of polyethylene terephthalate. (a) and (b) crystalline, (c) and (d) amorphous.

There are major differences in structure between natural fibres, which form slowly with the growth and lay-down of the polymer molecules under genetic control, and manufactured fibres which are formed rapidly under limited thermomechanical control following extrusion of melt or solution.

1.2.3 Natural polymer fibres In cotton, cellulose molecules grow from enzyme complexes into fibrils. A thin primary wall, which grows to the final size of the fibre, forms first and then the fibrils are laid down in helical layers at angles of 20–30o as a

Structural hierarchy in textile materials: an overview

7

secondary wall. At maturity, there remains a lumen at the centre of the fibre, which collapses to give the kidney shape and convolutions of the cotton fibre. These features are shown in Fig. 1.3. The mechanics of this structure with the contributions at the different levels were modelled by Hearle and Sparrow (1979b) and are summarised in Fig. 1.4. Other natural cellulose fibres, such as flax, hemp, sisal and jute, are multicellular with a helical structure within the cells; similar models apply, but the helix angles of 10o or less give a higher modulus. Lumen S3 S2 20°–30°

Reversal

S1

B

20°–35° Primary wall

C

Pectin Fats

Waxes

A

A (a)

N

(b)

(c)

1.3 (a) Structure of a cotton fibre, from Jeffries et al. (1969). (b) Cotton fibre cross-section from Kassenbeck (1970) with different tightness of packing of fibrils in zones A, B and C. (c) Convolutions of a cotton fibre, from Hearle and Sparrow (1979a); note reversal of twisting of convolutions.

8

Modelling and predicting textile behaviour Hollow tube

Collapse

Stress

Cellulose molecule

A

B B¢ C

D

D¢

C¢

Wet A: crystal lattice

Fibril

Strain

D: convoluted ribbon C: reversing assembly

B: helical

(a)

(b)

1.4 (a) Structural features of cotton, which determine the tensile properties. (b) Stress–strain plots. Line A is for extension of the crystal lattice, lowered to B owing to its helical structure, to C by untwisting of reversals and to D by pulling out convolutions. The dotted lines (letters with primes) are for wet cotton, from Hearle (1991).

The protein fibres have an even more complex physical and chemical structure. Chemically, proteins are chains of amino acid residues, –NH.C(R,H). CO–, with their great diversity coming from differing sequences of around 20 side-groups, R. Equally important for wool and hair is the pattern of lay-down, as genetic control switches on the formation of different proteins at different stages of growth. The resulting multilevel structure is shown in Fig. 1.5. Hearle (2003, 2005) has described a scheme for total mechanical modelling of the structural mechanics of wool and hair, as illustrated in Fig. 1.6. The first major influence is the fibril/matrix composite, which has been modelled by Chapman (1969). The crystalline fibrils undergo a phase transition from an a-helical form to an extended-chain b-lattice, controlled by a critical and an equilibrium stress. Behaviour above the yield point and the recovery are modulated by the elastomeric matrix. The second major influence is the difference between the axial arrangement of microfibrils in the macrofibrils of the para-cortex and the helical arrangement in the orthocortex. The resulting differential expansion on drying leads to the crimp of

Structural hierarchy in textile materials: an overview

9

Epicuticle High-S Low-S proteins proteins

Nuclear remnant

Exocuticle Endocuticle

High-tyr proteins

Righthanded a-helix I 1

Lefthanded coiledcoil rope I 2

Cuticle

Cell membrane complex

Matrix Microfibril

Macrofibril

I 7

I 200

Para cell Ortho cell I 2000

Cortex

I 20 000 nm

1.5 Wool fibre, as drawn by Robert C Marshall, CSIRO, Melbourne. Many wool fibres have a meso-cortex in addition to the ortho- and para-cortex, and some have a medulla.

[Fundamental particles – Quantum theory] Atoms {--potential function inputs to molecular modelling--} Keratin KAPS {-----molecular input-----} Computational sequence: IF--stress/strain--MATRIX [wet/dry, setting, time etc} Ø Ø Fibril/matrix composite {other directions} Ø Ø Ortho- [para/meso] macrofibrils [macrofibrils] Ø Ø [Macrofibril asembly] Ø Ø [Cell assembly (cmc)] {nuclear remnants etc} Ø Ø Cortical cells Cuticle [Medulla] Ø Ø Ø {variability} Whole fibre: Tensile; Bending + twisting {rheology} Ø wool and hair crimp Ø Ø Ø Lock of wool Wool garment Head of hair

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

1.6 The mechanical modelling sequence for modelling, from Hearle (2005).

10

Modelling and predicting textile behaviour

wool, as modelled by Munro and Carnaby (1999 and Munro (2001). Other features have a secondary effect. Silk has a simpler chemistry. It is a block copolymer with a limited range of amino-acid residues. The molecules are formed in solution in glands and then extruded. This results in a single-level structure, which is similar to that of solution-spun manufactured fibres. There is one notable difference. Because this is virgin polymerisation, the chains can line up in a fully extended form, so that chain folding in polymer crystallisation can be avoided.

1.2.4 Manufactured polymer fibres The polyester, polyethylene terephthalate (PET), is now the world’s dominant general purpose fibre. Other polyesters such as polybutylene terephthalate (PBT), with a different number of –CH2– groups, and polyethylene 2,6naphthalate (PEN), with naphthalate groups, are also produced. Polyester fibres are produced by melt spinning, as are nylon 6 and 66 fibres. There is a consensus that the fine structure of these fibres consists of crystalline micelles, with some chain folding at the crystallite ends, linked by tie molecules in amorphous regions. Typical pictures are shown in Fig. 1.7, but there is a dearth of quantitative information on the full three-dimensional (3D) structure. It is thought that the length-to-width ratio of the crystallites is greater in PET than in nylon. In contrast to modelling of cotton and wool, where there is quantitative agreement between structural features and



(a)

(b)

(c)

1.7 Views of fine structure of nylon fibres. (a) A common working model proposed by Hearle and Greer (1970). Angled ends are based on small angle X-ray diffraction pattern of nylon 66. (b) From Murthy et al. (1990), based on X-ray diffraction studies of nylon 6. (c) An alternative form, from Hearle (1978).

Structural hierarchy in textile materials: an overview

11

mechanical properties, models of structural mechanics of melt-spun fibres are more limited. Hearle et al. (1987) showed that simple mixture law models were unsatisfactory because the predictions were very different depending on the modelling procedure. A simplified network model of crystalline blocks linked by elastomeric tie molecules was developed in collaboration between DuPont and UMIST in the 1980s, but it contained many approximations and uncertainties. A brief account of the model has been given by Hearle (1991). The structural mechanics of this class of fibre is a prime target for modelling now that computer power has advanced, but there is less interest because PET and nylon are now regarded as commodity fibres with properties that are fit for purpose. The more recent high-performance fibres, para-aramids, Kevlar and Twaron, high-modulus polyethylene, Spectra and Dyneema and others, are more highly crystalline and highly oriented and have fully extended chains without folding back. These structures are easier to model. Northolt (1980) presented a theory of the effect of orientation, which was later extended to viscoelastic properties (Baltussen and Northolt, 2001). A model of time dependence of fracture is described by Termonia (2002). Manufactured cellulose fibres have more structural differences because the regeneration can involve chemical as well as physical changes. Standard viscose rayon has a micellar structure, but other forms have fibrillar forms. The consequences of this have been modelled by Hearle (1967).

1.2.5 Inorganic fibres Carbon fibres have a graphitic structure of interconnected, planar hexagonal rings. However, in contrast to graphite, which has poor cohesion between the planes, imperfections are needed to hold the fibres together. There may be curving of the planes to give geometrical connections or chemical bonds between planes. The literature contains a considerable diversity of models. Carbon fibres have a range of properties depending on how the manufacturing process influences the structure. Modelling could put this on a more predictable basis. Glass and ceramic fibres have three-dimensional network structures, which could be modelled as special cases of the bulk materials.

1.3

Modelling fibre behaviour

1.3.1 Mechanical responses The tension/extension properties of fibres are easily measured and a database of stress–strain values has been established. There are complications. The response is non-linear, inelastic and time dependent. Stress is not a single-

12

Modelling and predicting textile behaviour

value function of strain, but depends on the particular test conditions and on the prior mechanical history of the fibre. For modelling software that requires the input of a modulus, care is needed to choose an appropriate value. An example of this, although applied at the rope level, is in using modulus values to predict the response of moorings for oil-rigs in deep water (NDE/TTI, 1999). The minimum post-installation stiffness gives a limit for the displacement of the rig under given wind and current forces and the maximum storm stiffness gives a limit for the tensions in the rise and fall of the rig caused by wave action. A test sequence of cyclic loading between various percentages of break strength is proposed in order to give reasonable estimates of these two moduli. The viscoelastic properties of polymers have been extensively modelled, notably in the classic publication by Ferry (1970). The response can be represented by springs and dashpots. The two simplest forms are shown in Fig. 1.8 (a, b). Voigt’s parallel model shows primary creep at a reducing rate to a limiting value and creep recovery along the inverted form of the same curve. Maxwell’s series model shows instantaneous extension and secondary, irrecoverable creep at a constant rate. All the time dependent effects can be covered by three-elements, for example a Voigt model in series with a dashpot. To include all effects, a four element model of Voigt in series with Maxwell is needed. There are several limitations to this approach. Polymers respond differently to different frequencies of cyclic testing or rates of deformation in quasistatic tests. This can be accommodated by generalising the models to have spectra of Voigt elements in series or Maxwell elements in parallel. That

(a)

(b)

(c)

1.8 Spring and dashpot models. (a) Voigt parallel model. (b) Maxwell series model. (c) Eyring’s three-element model with non-linear dashpot.

Structural hierarchy in textile materials: an overview

13

is achieved with the highly developed theory of linear viscoelasticity, with ideal springs and dashpots, which follow the laws of Hooke and Newton. In reality, particularly for large deformations, the behaviour is non-linear. Eyring’s three element model, Figure 1.8(c), addresses this problem by having a non-linear dashpot, which follows a hyperbolic sine law of deformation. There is theoretical justification for this form in terms of the statistical mechanics of movement over energy barriers. Although there have been attempts to develop more advanced theories of non-linear viscoelasticity, these have limited value. A better approach to modelling is to adopt a linear four-element model, which is always valid for small increments of deformation, and allow the elastic moduli and coefficients of viscosity to change in successive increments according to an acceptable scheme. Sinusoidal cyclic deformation can be represented in various ways: by the magnitudes of stress and strain and the phase difference between them, by the elastic and viscous constants of the models and by a complex number notation, which separates the real (elastic) and imaginary (viscous) quantities. An important effect is the energy loss due to hysteresis, which results in the damping of vibrations and the generation of heat. The most commonly quoted quantity is tan d, where d is the phase difference. The energy loss increases with tan d, but it must be noted that in fibres the value of tan d may not be constant but can increase with the magnitude of the deformation. Other effects to be taken into account in modelling fibre responses are creep under load and stress relaxation for constant deformation. In its simplest form, strength is given by the break load in a tensile test, but this depends on rate of loading. For most fibres, there is a linear decrease with log(time-to-break), governed by a strength/time coefficient. Cyclic loading brings in fatigue effects. The tensile properties of fibres are the most studied, but other directions must also be considered. For small deformations, bending resistance follows standard beam-bending theory, but for large deformations account must be taken of the fact that most fibres (wool is an exception) yield more easily in compression than in tension, so that the neutral plane moves towards the inside of a bend. For small levels of twist, torque depends on the shear modulus; for large twists, the longer path on the outside of a fibre means that the tensile stiffness has a bigger effect. Fibre friction, which is relevant to many aspects of textile materials, can be represented by a coefficient of friction. Like so many other ‘physical constants’, this is not constant but varies with the nature of the contact, the contact pressure and the speed of separation. More detailed accounts of the mechanical and other properties of fibres and the way that they can be represented for modelling are given in the book by Morton and Hearle (2008).

14

Modelling and predicting textile behaviour

1.3.2 Other properties Depending on the needs of particular applications, other properties of fibres must be taken into account in modelling system responses. For example, heat and moisture transfer through fibres is influenced by the relation between moisture content and relative humidity and by the heat of sorption. This results in a coupling of moisture uptake and change of temperature. In principle, diffusion of water through fibres is also a factor, but in practice fibres are so fine that they can be treated as in transient equilibrium with the surrounding atmosphere. However, dye diffusion is technologically important. Chemical properties influence fibre degradation, which is also affected by heat and light. The optical properties of fibres and their shape determine light reflection and scattering, which influences fabric appearance. Thermal conductivity, specific heat and emissivity will influence the heating of fabrics subject to cyclic loading. Electrical conductivity plays a role in generation of static, with back-flow across the contact zone having been modelled.

1.4

Modelling yarns and cords

1.4.1 Types of yarn The simplest forms of yarn are the lightly interlaced, or previously lightly twisted, continuous filament (cf) yarns as they come from a fibre producer. If more lateral cohesion is required, these yarns are twisted. For bulk, stretch and appearance, cf yarns are textured. Many processes were tried, but the ones that became most important are false-twist texturing, either with a single heater to give stretch yarns or with two heaters to give set yarns, jet-screen or stuffer-box for bulked continuous filament (BCF) yarns, mainly used in carpets, and air-jet texturing (Hearle et al., 2001). These processes may be combined with fibre production or include fibre drawing. For short (staple) fibres, which are the only forms of cotton and wool and may be cut from tows of manufactured fibres, means must be found of holding the separate fibres axially in yarns. This can be achieved physically by twisting, wrapping or entangling or chemically by bonding. Ring twisting is the dominant process. There are some older methods, notably hand spinning and mule spinning. in the 1970s/80s, many other methods were developed. Rotor spinning and air-jet spinning are extensively used; some others have survived in specialist applications. For more information on staple fibre spinning, see Grosberg and Iype (1999), Lawrence (2003) and Lord (2003). The structure and mechanics of twisted cf and staple yarns is covered by Hearle et al. (1969). Singles yarns may be twisted together to give ply yarns. Ply yarns may then be twisted together to give cords and ropes. Usually twist direction alternates between levels.

Structural hierarchy in textile materials: an overview

15

1.4.2 Yarn geometry An idealised twisted yarn geometry is shown in Fig. 1.9(a). It consists of concentric helices with constant twist period, h and has yarn lengths in one turn of twist, l and L, at an intermediate radius r and the external radius R. By opening out the cylinders into flat rectangles, as shown in Fig. 1.9(b) and (c), the relations between helix angles q and a and the other quantities can be derived. Another feature of yarn geometry consists of the fibre packing, which influences yarn diameter and specific volume. The tightest form is hexagonal close packing of circular fibres; the packing factor is then 0.92, but many factors lead to lower values. In twisted cf yarns, values of 0.7–0.8 are common. In spun yarns of weakly crimped fibres, such as cotton and most manufactured staple fibres, values are nearer 0.5; in wool and in textured cf yarns, values are much lower. Packing becomes tighter as twist is increased. In real yarns, the fibres do not maintain a constant radial position, but migrate between inner and outer zones. The changing position is illustrated in Figure 1.10(a); the radially expanded view in Fig. 1.10(b) shows the change over longer lengths. In modelling cf yarns, migration can be neglected, because the structure is close to ideal over a short yarn element. In staple fibre yarns, migration has a more important role, as described in the next section. Ring-spun cotton and similar yarns have a helical structure, which is not too far from ideal, but there is an irregular partial migration with a period

r q

l h

l

h

q 2pr (b) L a

h L

a

R (a)

2pR (c)

1.9 (a) Idealised helical yarn geometry. Relations between helix angles q and a and other quantities are shown in (b) and (c).

h

16

Modelling and predicting textile behaviour

Yarn surface Yarn axis

(a)

(b)

1.10 Yarn migration: (a) changing position of fibres, (b) radially expanded view.

of around three turns of twist. The structure of crimped wool fibres is more open and less regular. In set textured cf yarns, the fibre paths are alternating left- and right- handed helices; in unset stretch yarns, the filaments can contract into pig-tail snarls. In rotor-spun yarns, there is less migration, but wrapper fibres have an important role.

1.4.3 Yarn mechanics The tensile stress–strain curve of twisted continuous filament yarns is the most successfully modelled example of the mechanics of a textile material, based on the idealised geometry of Fig. 1.9. In its simplest approximate form, modelling goes from the strain distribution in fibres, which reduces approximately as cos2q, to give a contribution to stress reduced by a further factor of cos2q through the axial component of tension and the oblique area on which fibre tension acts. The ratio of yarn to fibre modulus is thus the mean value of cos2q, which is cos2a for the ideal helical geometry. For a more exact model, an energy method is both simpler than the use of force methods and is easily adapted to cover large strains and lateral contraction. An important simplification in modelling is that most fibres extend at close to constant volume, in other words Poisson’s ratio is 0.5. This means that the deformation energy is the same when the fibre is extended at zero lateral pressure in a tensile test as it is when the fibre is subject to combined tension and pressure within a yarn. Consequently, the measured fibre stress–strain curve is a valid input. Except for a small deviation at low stresses where some buckling of central filaments reduces their contribution to tension, there is good agreement between the theory and experimental results.

Structural hierarchy in textile materials: an overview

17

Twisted yarns break when the central fibres, which are the most highly strained, reach their break extension. Consequently yarn and fibre break extensions are similar and the strength reduction is similar to the modulus reduction. The energy methodology can be extended to the multiple twist levels of ply yarns and ropes. Application of the principle of virtual work enables contact pressures to be determined (Leech et al., 1993). Since the main effect of twist is to increase the length of components at a distance from the centre, the method also gives torque–twist properties. For staple fibre yarns, the resistance to extension is reduced by slip at fibre ends, as illustrated in Fig. 1.11. The slippage factor SF equals (area OBBO/area OAAO). If the gripping pressure is too weak, the fibre is nowhere gripped. A low twist sliver or roving can thus be drafted with fibres sliding past one another. Above a critical gripping force, when the gripped zone BB is present, the yarn is self locking. The slippage factor, which depends on the fibre aspect ratio, the gripping force, and the friction between fibres, can be represented by the following basic relation:

SF = 1 –

a 2m JL

[1.1]

where a is the fibre radius, L is the fibre length, m is the coefficient of friction and J is an operator determining how fibre tensions are converted into transverse stresses. Tension in helically twisted fibres clearly leads to inward pressures. However twist alone is not enough. With the ideal structure of Fig. 1.9, a fibre on the outside of the yarn would be nowhere gripped and in turn could not grip fibres in the next layer. Slippage would be complete. The situation is saved by fibre migration, which occurs naturally in ring spinning. The structure is self locking, because fibres are gripped when they are near the centre of the yarn and provide gripping forces when they are near the outside. This was confirmed in an idealised model by Hearle (1965), which also yielded the following approximate relation for the ratio of yarn modulus to fibre modulus, g:

Tension A

O

B

B

Length along fibre

1.11 Slip at fibre ends.

A

O

18

Modelling and predicting textile behaviour



Ê aQ ˆ g = cos2 a Á 1 – 2cosec a 3 2m ˜¯ L Ë

[1.2]

where Q is the migration period. Worsted yarns, where a more open and low-twist structure is needed, are commonly produced as two-ply yarns. The pressure of the plies on each other generates the initial grip that can build up within the singles yarns. More recently, developments have led to increased migration, so that singles worsted yarns can be used in finer fabrics. Where there is less migration, the wrapping fibres, which occur naturally in rotor spun yarns, interact with twist to generate gripping forces. Wrapping is the sole grip mechanism in hollowspindle spinning in which a zero-twist strand of short fibres is wrapped by a continuous filament yarn. In woollen spun yarns, and to a greater extent in felted yarns, entanglement is the gripping mechanism. Contact pressures occur as fibres pass round one another in an irregular structure, which can be modelled as a set of overlapping helices. A similar model can be applied to air-jet textured yarns. The loops, which stick out from the yarns, are gripped by entanglement in the core of the yarns. For more bulky yarns, whether caused by the natural crimp of wool or the imposed helices or snarls of textured yarns, there is an initial stretch under low tension until the crimp is pulled out. The simplest model is to assume that there is no resistance to elongation until the fibres are taut at the shortest available path length. More detailed mechanics can be applied to the interaction of bending and twisting in the elongation of helices and snarls. Bending resistance of yarns is a more difficult problem because of a change from one mechanism to another. When inter-fibre slip is easy, the bending stiffness is given by summing the stiffness of the individual fibres. When the fibres are firmly gripped, yarns act as solid rods. The two extremes are easy to calculate, using beam bending models. The difficulty is the intermediate region, which is complicated by the fact that extension of fibre elements on the outside of a yarn can be relieved by slip along the helical path to the inside of the bend.

1.5

Modelling fabrics

Textile fabrics represent unique constructions of materials from individual fibres such as nonwoven or from a collective bundle of fibres by interlacing (weaving and braiding) or interloping (knitting) them together. Fabrics can be easily made to have the same or different materials as well as organising them according to certain structural parameters. Therefore, for given types

Structural hierarchy in textile materials: an overview

19

of materials the physical properties and the mechanical performance of textile fabrics are greatly dependent on how the fabrics are structured. Work in fabric modelling has been carried out to describe (i) the construction of fabrics (structural modelling), (ii) the physical limitations in constructing fabrics (geometrical modelling) and (iii) the influence of fabric structural parameters on fabric performance (property modelling), such as the mechanical and thermal properties.

1.5.1 Structure of woven and knitted fabrics Structure of woven fabrics Woven fabrics are produced by interlacing two systems of yarns perpendicular to each other; the one in the length direction of the fabric is the warp and the one that goes in the width direction of the fabric is the weft. There are many different ways of interlacing the warp and weft yarns into a fabric and a particular plan for constructing a fabric is known as a weave. Weaves can be classified into four different categories, elementary weaves, derivative weaves, combined weaves and complex weaves. In addition to these, woven fabrics can be made to have a considerable thickness from multiple sets of yarns in each of the two directions and these are sometimes termed threedimensional (3D) woven fabrics. The elementary weaves are defined as those having two floats in the weave repeat, with one, the float length, being 1. They typically include plain weave, simple twill weaves and satin/sateen weaves. The weave diagrams of some of these weaves are illustrated in Fig. 1.12. The elementary weaves can be manipulated to derive new weaves. The derivative weaves are created from the three types of elementary weaves. From the plain weave, two methods can be applied to create new weaves. The plain weave can be extended in the warp, weft or both directions in order to derive new weaves. Extending the plain weave in the warp direction will result in warp rib weaves and extending in the weft direction leads to weft rib weaves. When the plain weave is extended in both directions, hopsack weaves will be created. Figure 1.13(a), (b) and (c) shows a warp rib, a weft rib and a hopsack weave, respectively. A plain derivative can also be created

(a)

(b)

(c)

1.12 Elementary weaves: (a) plain weave, (b) 2/1 twill, (c) 5-end satin.

20

Modelling and predicting textile behaviour

(a)

(b)

(c)

(d)

1.13 Plain derivative weaves: (a) a warp rib, (b) a weft rib, (c) a hopsack and (d) a basket weave.

(a)

(b)

(c)

1.14 Twill derivatives achieved by mirroring (a) horizontal waved weave, (b) vertical waved weave and (c) diamond weave.

following the plain weave logic, where the derivative weave will have four quarters with the adjacent ones having opposite images, leading to a basket weave. This is illustrated in Fig. 1.13(d). Twill weaves are featured by the twill lines in either the Z or S direction. Therefore, ways of changing the twill line thickness and direction play an important role in creating twill derivative weaves. When the mirroring technique is applied, twill lines will be made to change their directions while keeping the continuity. This technique is used to create waved weaves and diamond weaves. Figure 1.14 shows a horizontal waved weave, a vertical  3 1   waved weave and a diamond weave derived from the S twill. When 2 2 changing twill line direction and breaking the continuity, that is, by applying the inverse mirroring technique, herringbone weaves and diaper weaves can be achieved. Figure 1.15 displays a horizontal herringbone weave, a vertical  3 1   herringbone weave and a diaper weave based on again the S twill. 2 2 More fanciful twill derivatives can also be created using different rules. Figure 1.16 demonstrates two other types of twill derivative weaves, an entwined weave and a saw tooth weave.

Structural hierarchy in textile materials: an overview

(a)

(b)

21

(c)

1.15 Twill derivatives achieved by inverse mirroring (a) horizontal herringbone weave, (b) vertical herringbone weave and (c) diaper weave.

(a)

(b)

1.16 Twill derivatives achieved using other principles (a) an entwined weave and (b) a saw tooth weave.

The traditional weaving technology is also capable of weave fabrics with thickness or 3D fabrics. Three-dimensional fabrics can be made as broad solid panels (3D solid), with porous cross-sections (3D hollow), or 3D shapes. 3D solid woven fabrics can be manufactured based different principles such the multilayer, orthogonal and angle interlock. Figure 1.17 shows the 3D model as well as the weave for an orthogonal woven fabric with four layers of warp yarn.

22

Modelling and predicting textile behaviour

(a)

(b)

1.17 (a) Model and (b) weave for an orthogonal woven fabric with four warp layers.

Structural hierarchy in textile materials: an overview

23

Modelling the structure of the woven fabrics is regarded as an important step towards the computerised generation of weaves. Chen and his colleagues started the structural modelling of woven fabrics by defining weaves into regular and irregular. A regular weave is one whose float arrangement and the step number do not change in a repeat of the weave. All other weaves are defined as irregular weaves. Many commonly used weaves are regular themselves or further developed from regular weaves. Whilst each type of the irregular weave would need a distinct mathematical model to describe its construction, all regular weaves will need the same mathematical model for their construction. A model describing the regular weave construction was reported by Chen et al. (1996), which yields the two-dimensional (2D) binary weave matrix, W, upon the specification of the float arrangement, Fi, and the step number, S. Suppose that Wx, y is the element of this matrix at a coordinate (x, y), where 1 £ x £ Re and 1 £ y £ Rp with Re and Rp being the warp and weft repeat, respectively, then the first column of the weave matrix can be generated using the equation [1.3]:



Ï 1 if i is an odd integer W1, y = Ì Ó 0 if i is an even integer

[1.3]

i Ê i ˆ where y = Á S Fj – Fi + 1 ˜ to S Fj ; and 1 £ i £ Nf. Nf is the number of j =1 Ë j =1 ¯

floats in the float arrangement. Then, the rest of the matrix will be assigned values as follows:

Wx, z = W1, y

[1.4]

where



Ï y + [S ¥ Ô z = Ì y + [S ¥ Ô y + [S ¥ Ó 2 £ x £ Re; and

(x – 1)] + Rp (x – 1)] (x – 1)] – Rp

if {y + [S ¥ (x – 1)]} < 1; if 1 £ {y + [S ¥ (x – 1)]} £ Rp ; if {y + [S ¥ (x – 1)]} > Rp ;

1 £ y £ R p.

Chen and colleagues also worked on weaves for other 2D fabrics and 3D fabrics (Chen and Potiyaraj, 1999; Chen and Wang, 2006). Structure of knitted fabrics Knitted structures differ from woven fabrics in that they are formed by inserting yarn loops into one other, or interloping. If the loops are developed in the length (or wale) direction of the fabric, the fabric created is known as a warp knitted fabric. On the other hand, if the loops are created in the

24

Modelling and predicting textile behaviour

width (or course) direction, a weft knitted fabric is formed. Weft knitted fabrics are more commonly used for domestic application but warp knitted fabrics have found applications for technical end uses as well as being used in everyday domestic life. The structural element of knitted fabric is the stitch. The plain stitch is the mostly used stitch type in forming fabrics and it has two sub-types, the L-loop stitch, which is front-to-back, and the R-loop stitch, which is back-to-front. The reverse of L-loop stitch is an R-loop stitch. The two types of stitches are illustrated in Fig. 1.18. Single layer knitted fabrics can be generated by arranging different types of stitches in the fabric. Knitted fabric structures can be represented using an integer matrix. The R-loop and L-loop stitches of the same type are symmetrical. The use of R-loop or L-loop plain stitch leads to a plain knitted fabric, and the use of R-loop and L-loop plain stitches alternately in the course and wale directions will result in purl and rib knitted structures respectively. An interlock fabric is formed by combining two rib fabrics. These form the four basic structures used for constructing knitted fabrics. Figure 1.19 shows models of knitted fabrics with these different knitted structures.

1.5.2 Geometrical modelling The performance of a textile fabric is basically a function of the property of the constituent fibre/yarn and the geometrical construction of the fibres/yarn in the fabric. For this reason, the study on the geometry of fabrics has been continuing for almost a century. Geometrical models of fabrics have led to the estimation of some structural and physical properties of fabrics, such

(a)

(b)

1.18 Plain stitches: (a) R-loop stitch and (b) L-loop plain stitch.

(a)

(b)

(c)

(d)

1.19 Basic knitted structures: (a) plain, (b) rib, (c) purl, and (d) interlock.

Structural hierarchy in textile materials: an overview

25

as the areal mass and the porosity, and they have also provided guidance in fabric manufacture by giving the maximum areal density of the fabric. Geometrical modelling of textile assemblies has become more important nowadays as geometrical models are the sole solution to providing geometrical information about textile assemblies for finite element (FE) models for performance simulation. Geometrical models of woven fabrics Peirce’s work (1937) is regarded as the beginning of modelling woven fabric geometries. Under certain assumptions, including circular yarn cross-section, complete flexibility of yarns, incompressible yarns and arc-line-arc yarn path, he derived the following equations describing the geometry of plain woven fabrics. The cross-section of plain woven fabric based on Peirce’s assumption is shown in Fig. 1.20.

D = d e + d p

[1.5]



he + hp = D

[1.6]



ce =

le –1 pp

[1.7]





lp –1 pe pp = (le – Dqe)cos qe + Dsin qe

[1.9]



pe = (lp – Dqp)cos qp + Dsin qp

[1.10]



he = (le – Dqe)sin qe + D(1 – cos qe)

[1.11]



hp = (lp – Dqp)sin qp + D(1 – cos qp)

[1.12]

cp =

[1.8]

pe

lp /2 hp/2 qp

qp

he/2

1.20 Peirce’s model for a plain woven fabric.

D

26

Modelling and predicting textile behaviour

where he, hp are the modular heights of the warp and weft yarns, respectively, normal to the neutral plane of the fabric; ce, cp are the crimps of the warp and weft yarns, respectively; D is the sum of the diameters of the warp and weft yarns; de, dp are the diameters of the warp and weft yarns, respectively; pe, pp are the thread spacing between adjacent warp and weft yarns, respectively; le, lp are the modular lengths of the warp and weft yarns, respectively, in one repeat; qe, qp are the weaving angles of warp and weft yarns, respectively. Subscripts ‘e’ and ‘p’ in the variables above refer to warp (ends) and weft (picks) respectively. There are thirteen variables in these eight equations. Therefore, with five variables known, such as the two spacings (pe, pp ), the two yarn diameters (de, dp) and one crimp (either ce or cp), these simultaneous equations can be solved. Ai (2003) presented an algorithm to calculate the geometry assuming that five variables, pe, pp, de, dp and one of ce and cp are specified. If pe, pp, de, dp and ce are known, the other fabric parameters can be worked out as follows:

le = pp(1 + ce)

[1.7¢]



f(qe) = (le – Dqe)cos qe + D sin qe – pp = 0

[1.9¢]



he = (le – Dqe)sin qe + D(1 – cos qe)



h p = D – h e



lp =

pe – D tan q p + Dq p cos q p

f(qp) = pesin qp – hpcos qp – D(1 – cos qp) = 0 cp =

[1.6¢] [1.10¢] [1.13]

lp –1 pe

Equations [1.9¢] and [1.13] are transcendental equations and have no analytical solution, but can be solved using a numerical approach. If de, dp, pe, pp and cp are specified, other parameters can be found in the similar way. The yarn cross-section in a real fabric is barely circular because of the pressure between the warp and weft yarns during the weaving process. Peirce himself proposed an alternative model for the plain woven fabric assuming the yarn cross-section to be elliptical. It proved to be mathematically too complicated to describe the relationship between the structural parameters. Peirce’s model of plain woven fabrics was extended by others, notably Kemp (1958), who assumed that the yarn cross-section is racetrack shaped and Shanahan and Hearle (1978) who proposed a lenticular yarn cross-section. These extended models retained all the assumptions Peirce used except for the yarn cross-section and are regarded as Peirce derivative models.

Structural hierarchy in textile materials: an overview

27

Figure 1.21 is the cross-sectional shape of a plain woven fabric where the yarn cross-section is racetrack shaped. The race track is formed by tangentially connecting two parallel lines with two half circles facing each other. The part between the plane S1 and S2 is identical to the diagram of Peirce’s circular cross-section model, so the racetrack model can be converted into Peirce’s model and equations solved in a similar way. In Figure 1.21, Ae and Be are the width and height of cross-section of warps. The variables he, hp, ce, cp, pe, pp, le, lp, qe, and qp have the same definition as for Peirce’s model. The variable p¢e is the distance between S1 and S2 and l¢p the length of the path between S1 and S2. The variable c¢p is the crimp of the weft between the S1 and S2. From this definition, we have:

p¢e = pe – (Ae – Be)

[1.14]



l¢p = lp – (Ae – Be)

[1.15]

cp¢ =

lp¢ – pe¢ cp pe = pe¢ pe – (Ae – Be )

[1.16]



p¢p = pp – (Ap – Bp)

[1.17]



l¢e = le – (Ap – Bp)

[1.18]



ce¢ =

le¢ – pp¢ ce pp = pp¢ pp – (Ap – Bp )

[1.19]



Shanahan and Hearle’s plain woven fabric model with lenticular yarn crosssection is illustrated in Fig. 1.22. A lenticular shape is formed by joining two identical arcs facing each other.

Ae(Ap)

Be(Bp)

S1

S2

l ¢p(l ¢e)

hp Êhe ˆ 2 ÁË 2 ˜¯

l p(l e)

he Êhp ˆ Á ˜ 2 Ë 2¯

p¢e(p¢p) pp(pe)

1.21 Kemp’s racetrack model for a plain woven fabric.

28

Modelling and predicting textile behaviour

1 2 Bp

hp

1 2 Be

qp

he

De/2

pe

1.22 Shanahan and Hearle’s lenticular model for plain woven fabrics.

All other assumptions used in the Peirce model also apply to this model apart from the fact that the yarn cross-section in this case is lenticular. Similar to Peirce’s model, the following equations have been derived to describe the relationships of the fabric parameters:

h e + h p = B e + B p

[1.20]

ce =

le –1 pp

[1.21]

cp =

lp –1 pe



[1.22]



pp = (le – Deqe)cos qe + De sin qe

[1.23]



pe = (lp – Dpqp)cos qp + Dp sin qp

[1.24]



he = (le – Deqe)sin qe + De(1 – cos qe)

[1.25]



hp = (lp – Dpqp)sin qp + Dp(1 – cos qp)

[1.26]



De = 2Re + Bp

[1.27]



Dp = 2Rp + Be

[1.28]

where he, hp, pe, pp, le, lp, ce, cp, qe, and qp have the same definition as those used for Peirce’s model. In addition, Be, Bp are the crimp heights of warp and weft yarns and Re, Rp are the radii of the lenticular arcs representing the warp and weft yarns. Also because of the similarities to the Peirce model, the fabric geometry can be definitely defined when seven of the parameters are specified.

Structural hierarchy in textile materials: an overview

29

Figure 1.23 is a snapshot of the geometric model created based on the lenticular assumption, where the warp and weft linear densities (related to the yarn dimensions) are 100 tex and 70 tex, respectively, and the warp and weft densities (related to thread spacings) are 13 ends/cm and 15 picks/cm, respectively. The warp crimp was specified as 7.2%. Under this specification, all fabric parameters that define definite fabric geometry and some structural attributes are shown as follows: ∑ ∑ ∑ ∑ ∑ ∑

warp density or warp spacing weft density or weft spacing warp crimp weft crimp

13.0000 cm–1 0.7692 mm 15.0000 cm–1 0.6667 mm 7.2000% 8.2918%

Yarn cross-section shape lenticular: ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

warp yarn width warp yarn height weft yarn width weft yarn height height of warp crimp weaving angle of warp yarn length of warp yarn height of weft crimp

0.6776 0.2710 0.5669 0.2268 0.2228 0.5767 0.7147 0.2750

mm mm mm mm mm rad mm (per unit) mm

1.23 Computer-generated geometrical model of a plain woven fabric.

30

∑ ∑ ∑ ∑ ∑

Modelling and predicting textile behaviour

weaving angle of weft yarn length of weft yarn fabric thickness fabric porosity fabric weight

0.6503 rad 0.8330 mm (per unit) 0.5018 mm 1.783 ¥ 10–2 2.600 ¥ 102 g m–2

Geometrical models of weft knitted fabrics Loop is the key element of knitted fabric geometry. Based on the yarn property and the knitting condition, the spatial shape of the loops in a knitted fabric assumes different geometric features. In the case of weft knitted fabrics, there are three well defined geometries, which are the Peirce’s model (1947), Leaf and Glaskin’s model (1955) and Leaf’s model (1960). Peirce’s model In Peirce’s initial compact stitch model for weft knitted fabric, the wale spacing w is four times the yarn diameter and the course spacing c is 2 3 times the yarn diameter, which are the minimum possible values. A loose plain stitch model was then created by increasing wale and course spacings, based on Peirce’s plain stitch model. The configuration of yarn paths in the loose model is shown in Fig. 1.24, where (a) is the top view and (b) is the side view. In this model, the projection of a loop is formed by an arc tangentially connected to two line segments. To determine the relative positions of yarn segments 1 and 2, it is assumed that Lc, the distance between yarns 1 and 2 where they cross the plane of the fabric at M (the plane M is shown in the side projection of yarn path), equals the diameter of yarn, i.e. Lc = d. The radius r of the loop head can be defined from the wale spacing w and Lc by

r=w+d 4 2

[1.29]

Other variables defining the geometry of the fabric can also be deduced from Fig. 1.24 as follows:



( )

q = arctan 2c w Ê Á a = arccosÁ Á ÁË

ˆ ˜ 2r ˜ 2 w + c2 ˜ ˜¯ 2

( ) b = a + (p – q ) 2

[1.30]

[1.31]

[1.32]

31

M

Structural hierarchy in textile materials: an overview

Ê wˆ y = tan q ¢ Á x – ˜ + c 4¯ 2 Ë R q¢O ¢r

O ¢c a b

g

X

Ls

Lc Oc

O1

q

r

1

M

d

2

Or S2

Td–

S1

X

2S

Td+

S = S1 + S2 (a)

(b)

1.24 Peirce loose plain stitch model for weft knitted fabrics. (a) Top view, (b) side view.



q¢ = p + a – q 2

[1.33]

The equation for the connecting line L between the two arcs is

(

)

y = tan q ¢ x – w + c 4 2



[1.34]

Calculations of other detailed geometrical quantities are described by Ai (2003). To avoid the two loops jamming together, the values of wale spacing w, course spacing c and diameter d follow the relationships:

w ≥ 4d

[1.35]



c ≥ 2 3d

[1.36]

Another geometrical constraint is that the distance of the part of diagonal

32

Modelling and predicting textile behaviour

line O c O¢r between the two arcs, Ls, must be larger than or equal to d. From Fig. 1.24(a):



( ) +c 2

w 2

Ls =

2

– 2r

i.e.



( ) + c – ( w2 + d ) ≥ d w 2

2

2

The above leads to the following two geometrical constraints: 4d 2 + 2wd



c≥



2 w £ c – 2d 2d

[1.37] [1.38]

Leaf and Glaskin’s plain stitch model In 1955, Leaf and Glaskin (1955) produced their loose plain stitch model assuming that the projection of the loop is comprised of tangentially connected arcs. Peirce’s non-penetration assumption was adopted in this model, shown in Fig. 1.25. The parametric expression of a loop is expressed as follows:

x = r(1 – cos q)



y = r sin q



[1.39]

Ê ˆ z = h Á1 – cos pq ˜ j¯ 2Ë

According to Fig. 1.25, the following can be obtained:



2 Ê ˆ a1 = 1 Á c – d 1 + 2 w 2 ˜ 2Ë c –d ¯

[1.40]



2 Ê ˆ a2 = 1 Á c + d 1 + 2 w 2 ˜ 2Ë c –d ¯

[1.41]

The sum of the above is the course spacing, i.e. c = a1 + a2. The radius of the arc forming the loop can be expressed as either



( )

r = d + 1 a12 + w 2 2 2

2

[1.42]

r

a1 S2

2 Lc j

Td+

R

C

P 1

q

M

B

d

Td–

A

Q

a2

Ls

E

33

S1

Structural hierarchy in textile materials: an overview

F

C¢ (a)

(b)

1.25 Leaf and Glaskin’s plain stitch model for weft knitted fabrics. (a) Top view, (b) side view.

or



( )

r = 1 a22 + w 2 2

2

[1.43]

To avoid jamming, the geometrical constraints are:

c≥

wd + d 2

[1.44]

and

w ≥ d + 2r

[1.45]

Leaf’s elastica plain stitch model In Leaf’s elastica model (1960), yarns are assumed to be elastic. In forming the loops, the bending of yarn is governed by the elastic property of the yarn. This model is illustrated in Fig. 1.26. Equation [1.46] describes the yarn path in the stitch:

34

Modelling and predicting textile behaviour B T1 1

2 I1

b

a

O

H2

c

S2

I2

S1

H1

A w /2

C (a)

(b)

1.26 Leaf’s elastica plain stitch model for weft knitted fabrics. (a) Top view, (b) side view.



Ï x = b[2E (e , j ) – F (e , j )] Ô Ì y = R sin (2be cos (j /R)) Ô z = R cos (2be cos (j /R)) Ó

[1.46]

where E(e, j) and F(e, j) are the elliptic integrals of the first and second kinds, respectively. Leaf parameterised the model with wale spacing w, course spacing c and yarn diameter d. Figure 1.27 compares the three geometric models.

1.5.3 Mechanical modelling Models have been created to analyse the mechanical properties of different types of fabric, based on the specification of the properties of the constituent material (yarns in this case) and the construction of these materials. In the

Structural hierarchy in textile materials: an overview

(a)

(b)

35

(c)

1.27 Comparison of different models. (a) Peirce’s model (b) Leaf and Glaskin’s model and (c) Leaf’s elastica model.

case of woven fabrics, the biaxial deformation of the fabrics depends on the two different features of crimp interchange, which depend on bending resistance and yarn extension. For many years, the dominant approach to modelling the mechanics was through force and moment equilibrium. Most of the work was limited to plain weaves and the curvilinear crimped yarn paths were usually replaced by an intersection of planar zig-zags. Based on such seemingly over-simplified models, some useful equations have been obtained. Equations [1.47] to [1.49] are the tensile, shear and bending moduli of plain woven fabrics reported by Leaf and his colleagues. Tensile moduli (mN mm–1): (Leaf and Kandil, 1980) E1 =



p1 p22 (1

È b p 3 (1 + c1 )3 cos2 q1 ˘ 12b1 1 + 2 32 Í 3 2 + c1) sin q1 Î b1 p1 (1 + c2 )3 cos2 q 2 ˙˚

È b1 p13 (1 + c2 )3 cos2 q 2 ˘ 12b 2 1 + E2 = b 2 p23 (1 + c1)3 cos2 q1 ˙˚ p2 p12 (1 + c2 )3 sin 2 q 2 ÍÎ

[1.47]

Shear modulus (mN mm–1): (Leaf and Sheta, 1984) Ï p [ p (1 + c1 ) – 0.8Dq1 ]3 p2 [ p1 (1 + c2 ) – 0.8Dq 2 ]3 ¸ G = 12 Ì 1 2 + ˝ b1 p2 b 2 p2 Ó ˛



–1

[1.48]

Bending moduli (mN mm2 mm–1): (Leaf et al., 1993)



B1 =

B1 p2 p1[ p2 (1 + c1) – 0.8758Dq1]

B2 =

B2 p1 p2 [ p1 (1 + c2 ) – 1.0778Dq 2 ]

[1.49]

In the above equations, subscripts 1 and 2 refers to quantities for the warp and

36

Modelling and predicting textile behaviour

weft respectively, where d is the yarn diameter in mm; D is the the sum of the warp and weft diameters, i.e. D = d1 + d2; b is the yarn flexural rigidity; q is the the weave angle in radians; p is the thread spacing, in mm, between two adjacent yarns in the fabric; and c is the yarn crimp in fabric. More recently, it has been recognised that an energy method is simpler and can be based on actual yarn paths. The principle is the minimisation of the sum of the energies of yarn extension, yarn bending and yarn flattening. This determines the internal geometry for given external dimensions. The difference in energies for an increment of deformation enables forces to be determined. For a simple plain weave, there is just one type of crossover to consider; for more complicated weaves, the different crossover elements need to be identified and modelled. Tensile energy is an easily determined yarn property. Bending behaviour is understood, but is more complicated and will be different depending on whether the yarns are under pressure in contacts at crossovers or they are free between crossovers. Flattening has been studied less. For bulky yarns, reduction in specific volume will occur, but a more general effect is a change of shape from an ideal circle to a flattened form. There is both a direct effect of the change and an interaction with the change in the bending of the yarn path. Although there have been many academic papers on woven yarn mechanics, practical application has been extremely limited. The benefit of increased computer power needs to be combined with clever modelling in order to address this problem. The academic approach has attempted to develop universal models. Practical advances are more likely to come from a concentration on the needs of specific applications, which enable appropriate simplifications to be adopted. For example, in fabrics of monofilament or hard-twisted yarns, flattening can be neglected. In contrast to this, very soft yarns will spread out to eliminate any free length between crossovers. In knitted fabrics, which are more open structures, yarn bending plays a more important role as the shape of loops changes under applied forces. Force and moment methods have been developed to model the behaviour of idealised, plain weft-knit fabrics (Hepworth, 1980; Konopasek, 1980), but the development of energy methods should be more useful. Other workers have taken a different approach to modelling fabric mechanics. They have adapted commercial finite element programs, such as ABACUS. This approach is particularly suitable for fabrics used in composites. ‘Nonwoven’ fabrics, which are better positively described by their German terminology, fliesstoffe, consist of a web of fibres that are bonded together. The extreme form, not regarded as a textile, is paper wet-laid from a dispersion of short fibres. If this is tightly bonded, it lacks the handle and drapabilty of woven or knitted fabrics; if it is weakly bonded, it is soft but weak. Nonwovens occupy an intermediate position.

Structural hierarchy in textile materials: an overview

37

Both web formation and bonding take several forms. Card webs can be parallel-laid in layers to give a high fibre orientation in the length direction; alternatively a card web can be cross-laid to give a more uniform orientation, but still showing peaks in the diagonal directions corresponding to the length of the card web. More random distributions are obtained when a bank of spinnerets extrudes filaments on to a moving belt, when staple fibres are laid from an air stream, or sometimes wet-laid, or when extruded polymer is melt-blown and collected. Fibre-to-fibre bonding is achieved by addition of adhesive in liquid form or as meltable powder or by direct thermal bonding under pressure without extraneous adhesive. Entanglement results from needling or impact of fluid jets. Stitch-bonded fabrics are held together by loops of yarn introduced by feeding a web through a warp knitting action. The mechanical modelling of nonwovens follows methods used for yarn modelling. For bonded fabrics, orientation distribution determines the relation between the external deformation and the change of length of fibre elements between bond points. Account must be taken of the degree of curl, which offers minimal resistance to elongation. When the curl is taken out, the fibre extension provides the input for an energy model of the mechanics. Entangled and stitch-bonded fabrics of short fibres depend on friction for cohesion, which leads to adaptation of the methods used in modelling staple fibre yarns.

1.6

Sources of further information

More information about modelling of textiles can be found in the following titles (see References for full details): Bogdanovich and Pastore (1996) Chou and Ko (1989) Hearle et al. (1969 and1980) Hu (2004) Long (2005) Miravete (1999) Morton and Hearle (2008) Postle et al. (1988)

1.7

References

Ai X (2003), Geometrical Modelling of Woven and Knitted Fabrics for Technical Applications, MPhil Thesis, UMIST. Baltussen J J M and Northolt M G (2001), ‘The viscoelastic extension of polymer fibres: creep behaviour’, Polymer, 42, 3835. Bogdanovich A E and Pastore C M (1996), Mechanics of textile and laminated composites, Chapman & Hall, London, England.

38

Modelling and predicting textile behaviour

Chapman B M (1969), ‘A mechanical model for wool and other keratin fibers’, Textile Res J, 39, 1102. Chen X and Potiyaraj P (1999), ‘CAD/CAM of the orthogonal and angle-interlock woven structures for industrial applications’, Textile Res J, 69(9), 648. Chen X and Wang H (2006), ‘Modelling and computer aided design of 3D hollow woven fabrics’, J Textile Inst, 97(No.1), 79. Chen X, Knox R T, McKenna D F and Mather R R (1996), ‘Automatic generation of weaves for the CAM of 2D and 3D woven textile structures, J Textile Inst, 87(Part 1, no. 2), 356. Chou T-W and Ko F K (editors) (1989), Textile structural composites, Elsevier, Amsterdam, Netherlands. Ferry J D (1970), Visco-elastic properties of polymers, John Wiley, New York, USA. Grosberg P and Iype C (1999), Yarn production; theoretical aspects, Woodhead Publishing, Cambridge, England. Hearle J W S (1965), ‘Theoretical analysis of the mechanics of staple fibre yarns’, Textile Res J, 35, 1060. Hearle J W S (1967), ‘The structural mechanics of fibers’, J Polym Sci, Part C, Polym Symp, 20, 215. Hearle J W S (1978), ‘On structure and thermo-mechanical properties of fibres and the concept of a dynamic crystalline gel as a separate thermodynamic state’, J Appl Polym Sci: Appl Polym Symp, 31, 137. Hearle J W S (1991), ‘Understanding and control of textile fibre structure’, J Appl Polym Sci: Appl Polym Symp, 47, 1. Hearle J W S (1993), Polyester: 50 years of achievement, Brunnschweiler D and Hearle J W S (editors), The Textile Institute, Manchester, UK, 6. Hearle J W S (2003), ‘A total model for the structural mechanics of wool’, Wool Technol Sheep Breeding, 51, 95. Hearle J W S (2005), ‘A total model for stress-strain of wool and hair’, Proceedings 11th International Wool Research Conference, Leeds, 2005. Hearle J W S and Greer R (1970), ‘On the form of lamellar crystals in nylon’, J Textile Inst, 61, 240. Hearle J W S and Sparrow J T (1979a), ‘Mechanics of the extension of cotton fibers. I. Experimental studies of the effect of convolutions’, J Appl Polym Sci, 24, 1465. Hearle J W S and Sparrow J T (1979b), ‘Mechanics of the extension of cotton fibers. II. Theoretical modelling’, J Appl Polym Sci, 24, 1857. Hearle J W S, Grosberg P and Backer S (1969), Structural mechanics of fibers, yarns and fabrics, Wiley-Interscience, New York. Hearle J W S, Thwaites J J and Amirbayat J (editors) (1980), Mechanics of flexible fibre assemblies, Sijthoff and Noordhoff, Alphen aan der Rijn, The Netherlands. Hearle J W S, Prakash R and Wilding M A (1987) ‘Prediction of mechanical properties of nylon and polyester fibres as composites’, Polymer, 28, 441. Hearle J W S, Hollick L and Wilson D K (2001), Yarn texturing technology, Woodhead Publishing, Cambridge, England. Hepworth R N (1980), ‘The mechanics of a model of plain weft-knitting’, in Mechanics of Flexible Fibre Assemblies, Hearle J W S, Thwaites J J and Amirbayat J (editors), Sijthoff and Noordhoff, Alphen aan der Rijn, The Netherlands, 175. Hu J (2004), Structure and mechanics of woven fabrics, Woodhead Publishing, Cambridge, England. Jeffries R, Jones D M, Roberts J G, Selby K, Simmens S C and Warwicker J O (1969), ‘Current ideas on the structure of cotton’, Cellulose Chem Technol, 3, 255.

Structural hierarchy in textile materials: an overview

39

Kassenbeck P (1970), ‘Bilateral structure of cotton fibers as revealed by enzymatic degradation’, Textile Res J, 40, 30. Kemp A (1958), ‘An extension of Peirce cloth geometry to the treatment of noncircular threads’, J Textile Inst, 49, T44. Knopp B, Jung B and Wortmann F-J (1996a), Investigation of the alpha-keratin intermediate filament structure by molecular dynamic simulation’, Macromol Chem Macromol Symp, 102, 175–81. Knopp B, Jung B and Wortmann F-J (1996b), Comparison of two force fields in MDsimulation of alpha-helical structures in keratins’, Macromol Theory Simul, 5, 947–56. Knopp B, Jung B and Wortmann F-J (1997), Modeling of the transition temperature for the helical denaturation of alpha-keratin intermediate filaments’, Macromol Theory Simul, 6, 1–12. Konopasek M (1980), ‘Textile applications of slender body mechanics’, in Mechanics of Flexible Fibre Assemblies, Hearle J W S, Thwaites J J and Amirbayat J (editors), Sijthoff and Noordhoff, Alphen aan der Rijn, The Netherlands, 293. Lawrence C A (2003), Fundamentals of spun yarn technology, Woodhead Publishing, Cambridge, England. Leaf G A V (1960), ‘Models of the plain knitted loop’, J Textile Inst, 51, T49. Leaf G A V and Glaskin A (1955), ‘The geometry of plain knitted loop’, J Textile Inst, 46, T587. Leaf G A V, Kandil K H (1980), ‘The initial load-extension behaviour of plain-woven fabrics’, J Textile Inst, 71, 1. Leaf G A V and Sheta A M F (1984), ‘The initial shear modulus of plain-woven fabrics’, J Textile Inst, 75, 157. Leaf G A V, Chen Y and Chen X (1993), ‘The initial bending behaviour of plain woven fabrics’, J Textile Inst, 84, 419. Leech C M, Hearle J W S, Overington M s and Banfield S J (1993), ‘Modelling tension and torque properties of fibre ropes and splices’, 3rd ISOPE Conference, Singapore, II, 370. Long A C (editor) (2005), Design and manufacture of textile composites, Woodhead Publishing, Cambridge, England. Lord P R (2003), Handbook of yarn production, Woodhead Publishing, Cambridge, England. MacKerell A D, Bashford D, Bellotti M, Dubrack R L, Evanseck J D, Field M J, Fischer S, Guo H, Gao S, Ha S et al. (1998), All-Atom Empirical Potential for Molecular Modeling and Dynamics Studies of Proteins, J. Phys. Chem. B, 102, 3586. Miravete A (editor) (1999), 3-D textile reinforcements in composite materials, Woodhead Publishing, Cambridge, England. Morton W E and Hearle J W S (2008), Physical properties of textile fibres, 4th edition, Woodhead Publishing, Cambridge, England. Munro W A (2001), ‘Wool-fibre crimp. Part II: fibre-space curves’, J Textile Inst, 92(Part 1), 213. Munro W A and Carnaby G A (1999), ‘Wool-fibre crimp. Part I: the effects of microfibrillar geometry’, J Textile Inst, 90(Part 1), 123. Murthy N S, Reimschussel A C and Kramer V J (1990), ‘Changes in void content and free volume in fibers during heat setting and their influence on dye diffusion and mechanical properties’, J appl polym Sci, 40, 249. NDE/TTI (1999), Deepwater fibre moorings: an Engineers’ design guide, Oilfield Publications Ltd, Ledbury, England, Section EC-5, 5.

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Modelling and predicting textile behaviour

North A C T, Steinert P M and Parry D A D (1994), ‘Proteins: structure, functions’, Genetics, 20, 174. Northolt M G (1980), ‘Tensile deformation of poly(p-phenylene terephthalate) fibres; an experimental and theoretical analysis’ Polymer, 21, 1199. Peirce, F T (1947), ‘Geometrical principles applicable to the design of functional fabrics’, Textile Res J, 17, 123. Peirce, F T (1937), ‘The geometry of cloth structure’, J Textile Inst, 28, T45. Postle R, Carnaby G A and de jong S (1988), The mechanics of wool structures, Ellis Horwood, Chichester, England. Shanahan W J and Hearle J W S (1978), ‘An energy method for calculations in fabric mechanics, part ii: examples of application of the method to woven fabrics’, J Textile Inst, 69(No. 4), 92. Termonia Y (2002), ‘Fracture of synthetic polymer fibers’, in Fiber Fracture, Elices M and Llorca J (editors), Elsevier, Amsterdam, 287. Ziabicki A (1976), Fundamentals of fibre formation, Wiley, London. Ziabicki A and Kawai H (editors) (1985), High-speed fiber spinning, Wiley, New York.

2

Fundamental modelling of textile fibrous structures S. Grishanov, De Montfort University, UK

Abstract: This chapter presents an overview of mathematical models generated over the past 60 years for the description of the geometry, structure and properties of fibres and fibre assemblies. General fibre functions in various fibre-based products are briefly introduced and linked to the relevant fibre properties. This chapter then proceeds to discuss models that are currently employed in investigations of the structure–property relationship of natural and synthetic fibres at the nano-level including methods of molecular modelling based on quantum mechanics and molecular mechanics. Models of fibre geometry, including the distribution of fibre length and diameter (fineness), fibre cross-section and fibre spatial shape due to crimp are discussed. The linear elastic, linear viscoelastic and non-linear viscoelastic models that characterise the mechanical behaviour of fibres are considered in detail. Factors affecting friction in textiles and models of fibre friction are presented. Structural and micromechanical models of fibre assemblies based on a fibrous unit cell are discussed. Key words: fibre assembly, fibre diameter, length and cross-section, fibre crimp, linear elastic model, linear viscoelastic model, non-linear viscoelastic model, fibre friction, fibre blends, unevenness, tensile strength, compression, structure–properties relationship.

2.1

Introduction

Fibres are the main building blocks in the production of textile materials. The physical and structural properties of fibres to a large extent define the properties of the final product which includes yarns, threads and fabrics. The past two decades have seen a rapid growth in the application of computerbased methods to the design of new textile materials. Modern methods of computer-aided design of textiles use a number of mathematical models describing the relationships between the fibre properties and the properties of the end-use product for property/behaviour prediction and three dimensional (3D) visualisation. Detailed knowledge of these relationships is particularly important for achieving specific performance characteristics in medical and technical applications. The relationships can be established by carefully planned experimental trials, theoretical studies based on fundamental principles, or by a combination of these two approaches. The advantage of the experimental approach is its apparent simplicity, which sometimes may be deceptive because not all fibre properties can be 41

42

Modelling and predicting textile behaviour

estimated by direct measurement and some of them are very difficult to test (colour of an individual fibre and shear modulus of individual fibres are good examples). Testing equipment may be expensive and requires specific skills and experience in order to achieve good results. It is necessary to remember that any test is an interaction between the sample and the tester which creates the testing conditions and these may be different from the actual conditions in which the material is used. In many tests the interpretation of the results is based on certain assumptions about the behaviour of the sample in the test and consequently the test results are only as good (or as bad) as the assumptions made. If the relationship between the parameters being tested is known, then the determination of one of them through the knowledge of the others can be achieved. The experimenter must be aware of the variability of the parameters, limitations of the experimental method and the effects on the validity of the experimentally identified values. If the relationships are not known and there is no underlining theory that can give insight into possible mathematical formulations, then the aim of the experiment is to establish such relationships. The usual method employed is factorial design of the experiment (Montgomery, 1991) followed by statistical data processing (Leaf, 1984, 1987). This enables a regression equation to be obtained, which to some extent reflects the true relationship between the parameters being studied. This equation then can be used to find an optimal combination of parameters that provide the best performance characteristics of the product or the process in question. For example, Barella et al. (1976a, 1976b) successfully used the method of factorial design for the optimisation of the parameters of the open-end spinning process. The results of experimental investigations as a rule have a limited range of applications and their results cannot be easily extended to a wider range of conditions or scales. Very often in the case of new materials or structures the experimental investigation must be repeated, which may be a time consuming exercise. The theoretical approach to the problem of understanding the relationship between the fibre properties and the properties of the final product is much more complex, requiring as it does a specific knowledge of mathematics, physics, chemistry and any other discipline relevant to the area of investigation. The advantage of the theoretical approach is that a good theory, once developed and verified, can be used universally for designing new materials, predicting properties, explaining phenomena and carrying out virtual experiments. These virtual experiments can be undertaken in conditions that are difficult to create, at a minimum cost, and within a short period of time. The disadvantage of a theoretically based investigation is that the quality of the solution is affected by the assumptions that had to be made in order to find that solution. The solution from the mathematical model requires application of an appropriate numerical method, which in itself may affect the solution

Fundamental modelling of textile fibrous structures

43

by the approximations of the method. Further simplifications of the model may thus be needed to eliminate this source of error. The analysis of mechanical, surface and transfer properties of textiles shows that owing to the non-linearity and statistical nature of all properties of textile materials, only a limited number of relatively simple problems can be resolved by analytical means. The solution to the majority of real life situations requires a combination of mathematical models to be based on carefully formulated assumptions about the structure and properties of textiles, together with the application of sophisticated computational methods. A good account of the issues that should be considered in textile mechanics modelling can be found in a paper by Hearle et al. (1972). These methods of experimental and theoretical modelling are not independent. No theory can be developed without the knowledge of experimentally defined basic constants and a comprehensive verification against existing experimental data. Theoretical and experimental investigations must always go hand-in-hand to provide important mutual support and a well balanced approach based on combination of theoretical and experimental investigation gives the best results. This chapter gives an overview of modern fundamental models applied to fibrous structures which start with models of single fibre structure and properties and continues with the analysis being extended to fibre assemblies.

2.2

Fibre classification

According to Denton and Daniels (2002), textile fibre is defined as textile raw material, generally characterised by flexibility, fineness and high ratio of length to thickness. Further classification of textile fibres (as defined by Denton and Daniels, 2002) is given in Fig. 2.1. A different approach to the fibre classification has been taken in a research report by Burdett and Bard (2006) where fibres, in addition to their origin and chemical composition, were classified with regards to potential hazards to health. This classification is however subject to a continuous update following advances in fibre science research.

2.3

Fibre functions in textile materials and composites

The functionality required of textile materials can be very different depending on their end-use. Fibre functions in garments are extremely diverse and depend on the environmental conditions of wear and specific applications for which they have been designed:

Textile fibres

Vegetable Seed (cotton, kapok, coir)

Silk

Wool (sheep)

Man-made

Bast (flax, hemp, jute, kenaf, ramie, etc.)

Mineral Synthetic polymer (asbestos) Leaf (abaca or manila, henequen, phormium tenax, sisal, etc.)

Polyvinyl derivatives

Polyethylene Polypropylene

Lastrile (FTC) Anidex (FTC) Novoloid (FTC)

Other (carbon, glass, metal, ceramic, etc.)

Alginate Rubber (FTC) Regenerated Regenerated Cellulose (elastodiene) protein cellulose ester (azlon (FTC)) (rayon (FTC))

Hair (alpaca, camel, cow, goat, (mohair, cashmere), horse, rabbit (angora), vicuna, etc.)

Polymethylene urea Polyolefin (polycarbamide) (olefin (FTC))

Natural polymer

Polyurethane

Animal (casein)

Vegetable (arachin zein)

Viscose

Cupro (cupra (FTC))

Polyamide or Nylon

Aramid

Modal

Acetate

Triacetate

Deacetylated acetate

Lyocell

Polyester

Non-segmented Segmented polyurethane (elastane, spandex polyurethane (FTC)) Acrylic

Modacrylic Nytril (FTC) Chlorofibre

Synthetic polyisoprene (elastodiene)

Vinydal Fluorofibre Trivinyl (acetalised (PFTE) poly(vinyl alcohol), vinal (FTC))

Poly(vinyl chloride) (vinyon (FTC))

2.1 Classification table of textile fibres according to Denton and Daniels (2002).

Poly(vinylidene chloride) (saran (FTC))

Polystyrene

Modelling and predicting textile behaviour

Animal

44

Natural

Fundamental modelling of textile fibrous structures

45

∑

Everyday seasonal clothing is required to provide durability and comfort by body coverage in a range of ‘normal’ wearing and weather conditions. This includes heat insulation, mechanical strength, abrasion resistance, ultra violet (UV) protection, breathability, heat and vapour transport from inside the clothing, and wind- and waterproof characteristics. ∑ Clothing for protection against external actions in harsh environmental conditions such as those in arctic regions, deserts, high altitudes or underwater require similar properties to those above but with higher levels of performance. ∑ Sportswear requires a high level of breathability and moisture/vapour transfer combined with windproof, waterproof, heat insulation and UV protection. ∑ Industrial protective clothing needs to provide protection against extreme hazards and environments. These include impacts, heat, spillage of metals, acids and alkalis, X-rays and electromagnetic fields. In some specific cases, for example in the production of medical textiles, pharmaceutical products and electronics, clothing must work in reverse, i.e. by protecting the clean room environment from contamination by human factors. ∑ Typical functions for textiles in medical applications include absorbency (wound dressings, swabs, incontinence pads, nappies), strength (surgical sutures, artificial ligaments, bandages), air permeability (surgeon’s gowns, staff uniforms), durability (hospital bedding, pressure garments) and ability to serve as a membrane or to biodegrade (surgical sutures). In addition to their immediate functions, medical textiles must meet a number of requirements related to their biocompatibility with the human tissues and blood. ∑ Technical applications include textiles used in automotive, aerospace, construction, agriculture, geoengineering and maritime industries. Fibrebased products for technical applications are required to meet performance characteristics that combine high strength, elasticity, durability and light weight. In many applications, such as clothing, furniture and medical textiles, this functionality must be considered in conjunction with the interaction between the human body, the textile material and the environment. In this respect, it is necessary to consider a dynamic balance of energy and mass (vapour) transfer of an environment–garment–body system (EGB system). It is usual in these applications to see low demands on mechanical performance of textiles, since the external forces are relatively small and the main emphasis is always on the qualities of comfort and aesthetics. Common to all applications are demands for energy efficiency and a low environmental impact of the production process. This functionality is provided by geometrical, mechanical, physical, chemical and structural properties of

46

Modelling and predicting textile behaviour

fibres used to construct the material in combination with the structure of the material itself. In application to fibres and fibrous structures, the term structure can be understood in different ways depending on the scale at which it is considered. Such structures and levels can be described as follows: ∑ ∑

∑

∑

Nano-level – considers internal structure of individual fibres and filaments in terms of the arrangements of molecules constituting the fibre; Microscopic level – deals with internal structure of individual fibres and filaments as consisting of uniform regions with known properties; typical examples are the structure of bicomponent fibres generated by phase separation (Zhang and Hsieh, 2008) and the mechanics of bicomponent fibres (El-Sheikh et al., 1971; Batra, 1974a and 1974b); Mesoscopic level – considers the arrangement of individual fibres in yarns, threads and fabrics (in the case when such fabrics are produced directly from fibres such as nonwovens) without reference to the internal structure of the fibres. There is a wealth of publications on structure of yarns and nonwovens available; Macroscopic level – refers to binding patterns of interlacing threads in woven, knitted and nonwoven materials where the fibrous structure of threads themselves is not taken into consideration.

Usually at each level up the scale the structural details of the previous level are not considered owing to the large number of parameters and degrees of freedom involved and the properties of structurally inhomogeneous media are substituted with asymptotically equivalent properties of a uniform material. This process of simplification, known as homogenisation, has been widely used in application to composite materials mechanics (Bogdanovich and Pastore, 1996; Ye et al., 2004). Functionality of textiles and composites is achieved by using fibres with appropriate properties that can be classified as follows: ∑ ∑ ∑

∑

General mechanical properties (strength and elasticity in single/multiple cycles of tensile, bending, compression and torsion deformation, abrasion resistance and friction). Heat/mass transfer (heat insulation, thermal conductivity, water and vapour absorption, water and vapour transport and flammability). Optical/electromagnetic properties (absorption, reflection and transmission of visible light, UV and infrared (IR) radiation, formation of static electrical charge, piezoelectric effects, electroconductivity, effects of static magnetic field and electromagnetic fields). Chemical and biochemical properties (reaction to acids and alkalis, organic and inorganic solvents, salts, enzymes, vapours and gases).

Recent advances in the development of textile technology and materials for automotive and medical textiles, smart materials, sensory textiles,

Fundamental modelling of textile fibrous structures

47

wearable electronics, pervasive computing and other modern areas generate requirements for new fibre functions such as sensing, reacting and adapting. The problems associated with the application of textiles in these areas can be successfully resolved on the basis of an engineering approach towards the design of materials for the specified end-use and performance characteristics. Thus, for the successful implementation of this approach it is necessary to model structural, geometrical, mechanical and physical properties of individual fibres and fibre assemblies. These properties will be considered in the sections that follow.

2.4

Modelling fibre structure

The majority of fibres used in the production of textiles are natural or synthesised polymers with a complex internal structure which is partly crystalline and partly amorphous. The behaviour of amorphous parts of polymers changes with the change of temperature. The temperature known as the glass transition temperature, Tg, marks the point below which the solid polymer consists of rigid crystals and a rigid amorphous part. Above this temperature the molecular segments of the amorphous fraction are in motion thus making it viscous or rubbery and the whole polymer is flexible. The structural details and properties of natural and synthetic fibres have been thoroughly investigated and reported in many publications; see for example Hearle and Peters (1963), Mark et al. (1967), Hearle (1982), Postle et al. (1988), Hearle (2001), Feughelman (1996), Simpson and Crawshaw (2002), and Wallenberger and Weston (2004).

2.4.1 Modelling structure of natural fibres Cotton and wool are two main natural fibres that are distinctively different in their origins and chemical composition, i.e. cotton is cellulose based and wool is a protein (keratin)-based fibre. The structure and properties of cotton have been investigated for a long time by many researchers owing to the importance of this fibre in the textile industry. A comprehensive overview of structural, chemical, physical and mechanical properties of cotton and other cellulose-based fibres was summarised in a series by Ott et al. (1954, 1955) and Hearle and Peters (1963), many conference proceedings and research papers. Theory of cotton fibre structure progressed from the notion of micelles embedded into an intermicellar substance, to fringed micelles and then to fringed-fibril structures containing crystalline and non-crystalline regions and is still under development owing to the discovery of new experimental facts. Cotton fibre has a distinctive twisted structure which is explained by the presence of a helical assembly of fibrils in the outer winding and in the

48

Modelling and predicting textile behaviour

laminated secondary wall, (Kolpak and Blackwell, 1976; Atalla and van der Hart, 1984). The direction of helices in the secondary wall changes some 50 times along the fibre length causing numerous convolutions to appear in every fibre. The microfibrils themselves consist of crystalline regions of relatively high lateral order alternated with low order irregular parts, Hearle (1958). The micromechanics of single fibre deformation for both natural and regenerated cellulose fibres was investigated by Eichhorn et al. (2001a, 2001b) and Eichhorn and Davies (2006). One recent attempt to investigate the structure–property relationships of cotton fibres was undertaken by Abhishek et al. (2005) where microcrystalline parameters of cotton cellulose were examined using a wide angle X-ray scattering (WAXS) technique. A direct relation between the size and shape of the crystallites with the number of hydrogen bonds was found and it was suggested that the crystallites have the shape of an ellipsoid. There have been many experimental investigations of structural and mechanical properties of cotton fibre but no attempts have been made to generate a comprehensive micromechanical model based on fundamental principles. The models of wool fibre structure reviewed by Postle et al. (1988) and Feughelman (1996) were based on the assumption that the fibre consists of two or possibly three structurally different components: ∑

a matrix which can be considered as an amorphous viscoelastic glassy polymer that is readily swollen by water, ∑ viscoelastic water impenetrable microfibrils aligned parallel to the fibre axis and embedded in the matrix, and ∑ a water penetrable mechanical phase associated with microfibrils. Later Munro and Carnaby (1999) suggested an improved micromechanical model of wool fibre in which the fibre cross-section was composed of paracortical regions where microfibrils are parallel to the fibre axis and orthocortical regions within which microfibrils are arranged in twisted configurations with an angular deviation of up to 40o between them. Three models of fibre cross-sections were considered where paracortical microfibrils occupied a different proportion of the fibre cross-section: ∑ ∑ ∑

a half of the cross-section and arranged in annular layers with a varying number of microfibrils; a segment of fibre cross-section; a circular core positioned at a distance from the centre of the crosssection.

The application of finite-element method for the prediction of 3D fibre shape yielded realistic fibre curvatures which supported the validity of the proposed model. The models of wool fibre surface have been investigated in many papers and

Fundamental modelling of textile fibrous structures

49

in particular in publications considering the differential friction effect which is discussed in Section 2.8. The most effective method of characterising the surface topography of wool fibre is using 3D scanning electron microscopy (Bahi et al., 2007) which enables the width and the height of the scales to be measured.

2.4.2 Structure of synthetic fibres Synthesised polymers are long molecules consisting of many monomers arranged in long chains of various configurations. There are many monomers that can be polymerised resulting in addition or condensation polymers. Addition polymers are formed by the addition of unsaturated monomers to the end of the growing molecular chain. In this way, polymers such as, for example, polyethylene, polypropylene, polystyrene, poly(vinyl)chloride and poly(methyl methacrylate) are synthesised. The formation of condensation polymers (for example polyamide and polyester) is accompanied by splitting off a number of small molecules, usually water. These two methods produce polymers with linear, linear with side branches, or cross-linked structures. Polymers used for the production of fibres contain both crystalline and amorphous parts. Mechanical properties of polymers such as elastic modulus, strength and toughness depend on the proportion and orientation of crystals in the solid. The ability of polymer molecules to form crystals is determined by the forms of their symmetry (or stereo-regularity) which are termed isotactic, syndiotactic and atactic. This depends on the way in which the four bonds of the carbon atom are directed to the neighbouring atoms in the molecule making the carbon asymmetrical in either the right- (R) or lefthand (L) form. Isotactic polymers are formed by long chains of purely R or L units. Regular alternation of R and L units leads to syndiotactic polymers whereas their random alternation results in atactic polymers. Of these three, crystalline parts of the polymer can be formed from isotactic and sometimes syndiotactic polymers but not from atactic polymers. The length of the molecular chain is considerably greater than the length of the crystal; thus, the long molecules provide integrity to the solid by joining together crystal and amorphous regions of the polymer. The molecular chain can assume different random shapes which are termed conformations. The usual approach is to characterise the conformations through the probability that the end of the molecular chain will be in a volume dV = dxdydz, a distance r apart from the beginning. It can be shown that this probability is: Ê ˆ –Á r ˜ Ë r¯



P(x, y, z ) dxdydz = e r p

(

2

)

3

dxdydz,

[2.1]

50

Modelling and predicting textile behaviour

where

r = l 2n 3

and l is the length of one molecular segment (monomer) and n is the number of monomers in the polymer or the degree of polymerization. The average distance between the ends of molecular chain is r = l n . The main advantage of synthesized fibres is that their properties, such as elastic modulus, toughness, melt viscosity and thermal stability can be predicted depending on polymerisation conditions and composition of the (copolymer) mixture (McCrum et al., 2001). In recent years methods of molecular modelling based on quantum mechanics and molecular mechanics have been developed and used for predicting molecular structure and properties of polymers (Hinchliffe, 1996; Leach, 1996; Atkins, 1997). It can be noticed that the methods used in molecular modelling change according to the level of detail of molecular structure under consideration. Methods of quantum mechanics consider kinetic energy of motion of orbital electrons, potential energy of attraction between electrons and nucleus and repulsion between the electrons due to Coulomb forces. These methods are based on fundamental theories developed by Planck, Born, Shrodinger, Hartree, Fock, Dirac, Oppenheimer and other prominent physicists of the early 20th century. The motion of a single particle, such as an electron of mass m, in space under the action of an external potential field U, if it is assumed to be independent of time, is described by the time-independent form of the Shrodinger equation:



È ˘ h 2 Ê ∂2 ∂2 ∂2 ˆ Í – 8p 2 m Á ∂x 2 + ∂y 2 + ∂z 2 ˜ + U ˙Y(r ) = E Y (r ) Ë ¯ Î ˚

[2.2]

where h = 6.626 ¥ 10–34 J s is Planck’s constant; x, y and z are the particle’s coordinates in 3D Cartesian coordinate system XYZ, E is the energy of the particle and Y is the wavefunction which characterises the particle’s motion. In the simple case of a single electron in an isolated atom with N protons the external potential is represented by the electrostatic interaction which, according to the Coulomb equation, depends on the distance between the electron and the nucleus: 2 U = – Ne [2.3] 4 pe 0r where e is the absolute charge of the electron, e0 is dielectric constant in vacuum and r is the distance between the electron and the nucleus.

Fundamental modelling of textile fibrous structures

51

The exact solution of the Shrodinger equation can be found only for a limited number of simple cases such as the hydrogen atom. For polyelectronic atoms or molecules any solution will be an approximation of the real life scenario. Quantum mechanics methods can be applied to the calculation of equilibrium configurations and conformations of molecules, thermodynamic characteristics of molecules (such as heat of formation), electric multipole moments and charge distribution in a molecule, and the formation of hydrogen bonds. Energy minimisation methods (see Himmelblau, 1972; Press et al., 2002) are then used to obtain equilibrium molecular structures and their characteristics. Molecular mechanics methods take into account forces acting between individual atoms in a molecule (resulting from extension and bending of intermolecular bonds) and rotation around the bonds together with electrostatic and van der Waals interactions between non-bonded parts. These forces are considered in formulating the equations for the total potential energy of the system of atoms and molecules where, again, energy minimisation methods are used in order to find equilibrium configurations. The general disadvantage of energy minimisation methods is that they are extremely time consuming particularly for systems involving large number of particles/atoms/molecules. This makes them impossible to use in applications involving real-life size samples. An alternative approach is a computer simulation technique that deals with a small representative part of the whole system consisting of a manageable number of atoms and molecules. Computer simulations are mainly based on two approaches (Leach, 1996). The first is molecular dynamics where a large number of successive configurations of the system (some 100,000) are generated at short time intervals (typically between 10–15 and 10–14 s) by integrating differential equations of the motion of particles. All configurations generated by this process are linked in time and each configuration is characterised by the position and velocities of all particles considered. Collectively they define a trajectory along which the dynamics of the system changes with time. Statistical mechanics, developed by Boltzmann and Gibbs, is then used to calculate the average values in question where, according to ergodic hypothesis, the average over time is assumed to be equal to the average over a large number of the system configurations. The second method is the Monte Carlo method where system configurations (or states) are generated by randomly moving or rotating atoms or molecules. The set of states generates a Markov chain where, in contrast to the molecular dynamics method, each new configuration depends only on its immediate predecessor (old state) but not on any other previous state of the system. Each new configuration, which corresponds to a new value of the total potential energy of the system, is either unconditionally accepted if the new energy

52

Modelling and predicting textile behaviour

is lower than at the old state or randomly accepted if a random number x Π[0, 1] is greater than the Boltzmann energy factor

b = exp

Wnew (r N ) – Wold (r N ) kT

where Wnew and Wold are the system energy in new and old state, respectively, rN denotes the position of N particles in the system, k is Boltzmann’s constant (1.38066 ¥ 10–23 J K–1) and T is absolute temperature. There is a great deal of similarity, as we shall see, between modelling fibre properties on the basis of internal molecular structure and modelling properties of fibre assemblies on the basis of fibre arrangements.

2.5

Statistical models of fibre geometry

A very important feature of all fibres used in textile applications, including man-made and synthesised fibres, is that all fibre properties have a statistical nature. For the sake of simplicity, the distribution of properties such as fibre length, diameter and strength is often approximated by a normal distribution:

f (x ) =

1 exp Ê – (x – x )2 ˆ ÁË 2s 2 ˜¯ s 2p

[2.4]

but a more careful analysis however shows that the applicability of a normal distribution is limited and there are many properties where the distribution significantly deviates from a normal distribution.

2.5.1 Fibre length distribution Fibre length has a significant effect on the behaviour of fibres in carding and drafting, the strength of carded webs, slivers, rovings and yarns where in general longer fibres make stronger products. For this reason fibres shorter than 12 mm are not considered suitable for spinning. Fibre length distribution affects the variation of yarn linear density and strength. The distribution of cotton fibre length has been shown to be bimodal (Schneider et al., 1994; Schenek et al., 1998; Krifa, 2006, 2008). A similar feature has been noticed in acrylic fibres, Grishanov (2008). One possible approach in this case is to approximate a multi-modal distribution by a weighted sum of normal distributions: – ai f (x ) = Â e i = 1 s i 2p n



(x – xi )2 2s i2



[2.5]

Fundamental modelling of textile fibrous structures

53

where n is the number of components of function f(x), ai is the weight n

coefficient of component i, Â a i = 1 , si and xi are standard deviation i =1

and mean length of component i, respectively. Parameters in Equation [2.5] identified on the basis of measuring acrylic fibre length are given in Table 2.1; experimental and fitted fibre length distributions are given in Figures 2.2, 2.3 and 2.4. A multi-modal distribution of fibre length and fibre diameter may occur in multi-component blends as a result of choosing incompatible components. This problem has been specifically known to researchers working in the area of short flax fibre processing, Harwood et al. (2004). It is also necessary to take into account that the distribution of fibre length and diameter may change in processing. There are some interesting properties related to the distribution of the segments of the total fibre length. The distribution of the length of fibre segments on the yarn surface was found to follow a gamma distribution (Choi and Kim, 2004) the probability density for which can be expressed as follows, Korn and Korn (1968): g(x ) =



x

– 1 xa –1e b b G (a )

a

[2.6]

where a is shape parameter, b is scale parameter, x > 0; a, b > 0 and G(a) =

Ú

∞

0

t a –1 e– t dt is the gamma function.

Lengths of hairs protruding from the yarn main core can be assumed to follow an exponential distribution (Barella, 1957):

f (x) = l exp(– lx)

[2.7]

where l > 0 is the parameter of the distribution.

2.5.2 Fibre diameter distribution Fibre diameter, df, is another factor defining the behaviour of individual fibres in spinning and mechanical deformations, the minimum achievable linear density and the evenness of yarns, the dye uptake, the reflection of light and other properties. Generally, assuming that the fibre cross-section is circular, the fibre resistance to tensile deformation is proportional to the second degree of fibre diameter, whereas in bending deformation it is proportional to the moment of inertia of its cross-section which in turn is proportional to the fourth degree of fibre diameter:

54

Fibre type

Component 1

Component 2

Weight coefficient

Mean length (mm)

Standard Weight deviation coefficient (mm)

Mean length (mm)

Standard Weight deviation coefficient (mm)

Mean length (mm)

Standard deviation (mm)

Red Green Blue

0.6942 0.7782 0.6783

110.99 103.07 117.14

29.84 34.93 29.24

68.61 59.35 60.56

82.56 14.06 11.16

54.66 36.76 35.29

12.18 16.81 46.87

0.1795 0.1369 0.1456

Component 3

0.1259 0.0846 0.1757

Modelling and predicting textile behaviour

Table 2.1 Parameters of fibre length distribution

Fundamental modelling of textile fibrous structures

55

0.06 Experiment Theory

0.05

Frequency

0.04 0.03 0.02 0.01

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205

0 Fibre length (mm)

2.2 Experimental and fitted length distribution for red fibres. 0.07 Experiment

0.06

Theory

Frequency

0.05 0.04 0.03 0.02 0.01

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200

0 Fibre length (mm)

2.3 Experimental and fitted length distribution for green fibres.



I=

p df4 64

[2.8]

For this reason, a typical 12.5% variation coefficient in fibre diameter of acrylic fibre may cause a 2.12 times difference in tensile strength and a massive 4.48 times difference in fibre stiffness between the finest and the thickest fibre. Assuming that the fibre cross-section is solid and circular, it is possible to obtain a link between fibre diameter and another important characteristic of fibre fineness such as linear density as follows:

56

Modelling and predicting textile behaviour

Frequency

0.09 0.08

Experiment

0.07

Theory

0.06 0.05 0.04 0.03 0.02 0.01 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200

0 Fibre length (mm)

2.4 Experimental and fitted length distribution for blue fibres.



df = 0.0357

Tf gf

[2.9]

where Tf is the fibre linear density in tex and gf is fibre volume density in g cm–3. The fibre cross-section is often not circular, in which case a more general term for fibre fineness is applied to characterise the transversal fibre dimensions. In this respect, an effective fibre diameter is often used. The effective fibre diameter is assumed to be equal to that of a fibre with a circular cross-section and the same cross-sectional area as the original shape. If the equation for a general fibre shape in the polar coordinate system (r, q) is r = r(q) then the area of cross-section can be estimated by integrating:

A=

Ú

2p

r 2 (q ) dq 2

0

[2.10]

and then the effective diameter is:

de = 2 A p

[2.11]

An alternative approach is to assume that the effective diameter is that of a circle with the same perimeter as the fibre (Neelakantan, 1975) in which case the effective diameter is:



de = 1 p

Ú

2p

0

2

Ê dr (q )ˆ r 2 (q ) + Á dq Ë dq ˜¯

[2.12]

These approaches, of course, may help in assessing the overall fineness

Fundamental modelling of textile fibrous structures

57

of fibres which are different in their origins and geometrical characteristics but they cannot be recommended if a detailed analysis of the anisotropic mechanical behaviour of the fibre caused by a non-symmetric shape of the cross-section is to be carried out. The distribution of the diameter of single fibres such as wool is well known to follow the log-normal distribution (Linhart and Westhuyzen, 1963; Lunney and Brown, 1985; Wang and Wang, 1998; Barella, 2000) the probability density function of which is: f (x) =



Ê (ln x - x )2 ˆ expÁ for x > 0. Ë 2s 2 ˜¯ s x 2p 1

[2.13]

Sometimes a good approximation can be achieved by using a weighted sum of normal distributions which was the case in the investigation of acrylic fibre diameter conducted as a part of the EPSRC project, Grishanov (2008): ai f (d ) = S e i = 1s i 2p n



( d - di )2 2s i2

[2.14]



where n is the number of components of function f, ai is the weight coefficient of component i; si and di are standard deviation and mean diameter of component i, respectively. Parameters in Equation [2.14] identified on the basis of measuring 5000 acrylic fibre snippets by LaserScan are given in Table 2.2. It is often the case that the fibre is not a single one but a bundle of many elementary fibres or filaments glued together by interfibre cement; a typical example is the technical flax fibre. Flax fibre tends to split into elementary fibres in carding and spinning. Thus, flax fibre sample will always contain a mixture of fibre bundles and single filaments for which the estimated average diameter is much greater than a true diameter of a single fibre. A method for estimating the single flax fibre fineness based on the analysis of experimental data obtainable from LaserScan and OFDA instruments has been suggested by Grishanov et al. (2006). The probability density function for the flax fibre diameter measured by LaserScan has been obtained in the following form: gn (D, bs , s ) =

Ú



2p 0

(

È D 2 2 2 Í n n - (n - 1)cos j - bs n - (n - 1)cos j exp Í2s 2 2p 2p ns Í ÍÎ 2

2

2

) ˙˙ dj 2˘

˙ ˙˚

[2.15]

58

Fibre Component 1 Component 2 Component 3 type Weight Mean Standard Weight Mean Standard Weight Mean coefficient diameter deviation coefficient diameter deviation coefficient diameter (μm) (μm) (μm) (μm) (μm)

Standard deviation (μm)

Red Green Blue

8.02 0.86 4.95

0.8075 0.5767 0.5847

23.12 23.22 22.74

3.01 1.55 1.76

0.1657 0.3711 0.2174

23.98 22.73 26.28

7.98 3.82 1.92

0.0300 0.0521 0.1977

23.99 27.32 24.47

Modelling and predicting textile behaviour

Table 2.2 Parameters of fibre diameter distribution

Fundamental modelling of textile fibrous structures

59

where n is the number of single fibres in the fibre bundle, D is the fibre diameter measured by the LaserScan, bs is the single fibre diameter mean; s is the standard deviation of the single fibre diameter and j is the angle that determines the random position of the fibre bundle relative to the sensor in LaserScan. The distribution of the flax fibre and fibre bundle width in the mixture estimated by LaserScan measurement then will be: m



hLS = a l pl (bs , s ) + S a n gn ( D, bs, s ) n=2

[2.16]



where a1, a2, a3, …, am are unknown proportions of single fibres and fibre bundles consisting of 2, 3,…, m single fibres together, respectively, and p1 ( bs , s) is the normal distribution of the single fibre diameter. The problem of estimating the unknown characteristics of the flax fibre sample can be resolved by fitting function [2.16] to the experimental data obtained with the LaserScan. It is necessary to take into account that the average diameter of the single fibre, bs , its standard deviation, s, and the proportions of single fibre and fibre bundles, a1, a2, a3, .., and am, must satisfy obvious constraints as follows:

bs ≥ 0

[2.17]



s ≥ 0

[2.18]



0 £ aj £ 1

[2.19]

m



S aj = 1

j =1

[2.20]



where j = 1, 2, …, m. The unknown single fibre diameter, its standard deviation and the proportions of single and multiple fibres a1, a2, a3, .., am can be estimated from the experimental data by applying non-linear programming methods (Himmelblau, 1972) to minimise a penalty function with constraints:

k

m-1

j =1

j =1

S = S (hLSj - H j )2 + d 0 + S d j ( x j , L j , U j )



[2.21]

where hLSj is the jth ordinate of fibre diameter distribution function determined by equation [2.16], Hj is the jth ordinate of experimentally measured fibre diameter distribution, d0 is a penalty function for equality constraint [2.20], dj is a penalty function for inequality constraints [2.17–2.19], xj is the argument of function dj, Lj and Uj are lower and upper boundaries for the argument of function dj, respectively, m is the maximum number of single fibres in the fibre bundle.

60

Modelling and predicting textile behaviour

This method was used for classification of 83 flax accessions into groups according to the estimated fibre diameter (Grishanov et al., 2006). It was predicted theoretically and then confirmed by the experimental data that the mode of the experimental distribution can be considered as the closest estimate of the single flax fibre diameter.

2.5.3 Models of fibre cross-section One of the important factors affecting fibre behaviour in processing and the performance characteristics of the end-use products is the fibre cross-sectional shape (Nikoli and Bukoek, 1995; Roberts, 1996). Natural fibres come with a variety of cross-sectional shapes such as circular or elliptical (wool, hair), triangular (natural silk), ‘dogbone’-shaped (cotton) and (irregular) polygonal (flax). Man-made fibres can be manufactured with a specified cross-sectional shape such as circular (polyester, nylon), trilobal or multilobal (nylon) (Cook, 1984). The polymerisation process affects the cross-section of man-made fibres; this is more prominent in viscose fibres and acrylic fibres (Hearle and Peters, 1963). Fibre cross-sectional shape affects the way in which the fibre surface reflects light and thus the colour of the product made from this fibre. Fibres with a circular cross-section display a symmetric mechanical behaviour in bending because the moment of inertia of a circular cross-section is the same with respect to any axis passing through the centre. Fibres whose cross-section deviates from a circular shape behave differently depending on the direction of force with regard to the axes of symmetry of the cross-section and may display not only bending deformation but also torsion. In the case of blends containing fibres with different cross-sections, the difference in mechanical behaviour of blend components in processing can cause a significant nonuniformity in fibre distribution both over the cross-section and along the axis of fibre assembly thus creating an additional unevenness (Truevtsev et al., 1995; Su and Fang, 2006). For these reasons it is important to have a mathematical description of the fibre cross-section which can be taken into account in modelling the appearance and mechanical properties of fibres. In a series of publications Lee (2003, 2005a, 2007) proposed two models of various cross-sectional fibre shapes as follows: 1. A general cardioid-shaped cross-section described by equation in polar coordinate system (r, q):

r = a[1 + l cos nq]

[2.22]

where a is a scale factor, l is a constant, n ≥ 0 is an integer. This equation can be applied to the description of the fibre crosssections as follows: (a) a circular cross-section in which case a = df/2

Fundamental modelling of textile fibrous structures

61

and l =  0, where df is fibre diameter; this is applicable to wool fibres and man-made fibres; (b) a polygonal cross-section of flax and nettle fibre by setting parameter n equal to the number of the polygon sides and l to a suitable small value (see Fig. 2.5); (c) a star-like cross-section of man-made fibres. 2. A general elliptic cross-section: p



q

Ê yˆ Ê xˆ ÁË a ˜¯ + ÁË b ˜¯ = 1

[2.23]

or in parametric form



x = a(cos q)2/p; y = b(sin q)2/q



where a and b are semi-axes of the ellipse. This equation can be useful for the description of circular, elliptical, diamond, rectangular and lentil-like fibre cross-sections. Equation [2.22] can be generalised by a cosine Fourier series as follows: m



r = a(1 + S li cos niq ) i =1

[2.24]



where a is a scale factor, li is a constant and ni ≥ 0 is an integer. In this form it can be used for non-symmetrical and serrated cross-sections 15

¥ 0.001 mm

10

5

0 0

5

¥ 0.001 mm

10

2.5 A model of a flax fibre cross-section.

15

62

Modelling and predicting textile behaviour

as well. For example, the cross-sections of a cotton fibre, a serrated fibre and a bilobal fibre (Fig. 2.6) can be respectively approximated by equations:

r = 5(1 – 0.3 cos q – 0.3 cos 2q);

[2.25]



r = 10(1 + 0.3 cos q + 0.08 cos 10 q + 0.015 cos 100 q);

[2.26]



r = 10(1 – 0.3 cos q + 0.3 cos 2q).

[2.27]

2.5.4 Modelling fibre shape in three dimensions

30

10

20

¥ 0.001 mm

15

5

0

10

0 0

5 ¥ 0.001 mm (a)

10

0

10 20 ¥ 0.001 mm (b)

30

20

¥ 0.001 mm

¥ 0.001 mm

The spatial wavy shape of fibres is commonly described by a notion of crimp. Natural fibres are randomly shaped at their origin whereas man-made fibres initially produced as straight can then be given an artificially introduced crimp. The main reason for fibres to develop crimp is understood to be the

10

0 0

10 20 ¥ 0.001 mm (c)

30

2.6 Models of fibre cross-sections: (a) a cotton fibre, (b) a serrated fibre, (c) a bilobal fibre.

Fundamental modelling of textile fibrous structures

63

non-uniformity of the internal fibre structure and differences in mechanical properties between the structural components of the fibre (Chapman, 1969; El-Sheikh et al., 1971; Brand and Scruby, 1973; Batra, 1974a, 1974b). Owing to crimp fibre assemblies occupy a greater volume in comparison to the volume they would occupy as straight fibres. It has been generally accepted that crimp influences fibre behaviour in blending, carding, drafting and spinning. For example, a straight fibre with a length equal to the distance between the nips of the rollers in the drafting zone can be classified as ‘controlled’, whereas a fibre of the same length but crimped may behave as a ‘floating’ fibre and thus contribute to an increase in the unevenness of the product. Fibre crimp increases the cohesion between fibres by creating a number of fibre-to-fibre contact points and thus affects the tensile strength and compression behaviour of the fibre assembly (Horio and Kondo, 1953; Batra, 1974a,b; Bauer-Kurz et al., 2004; Munro and Carnaby, 1999; Munro, 2001). Fibre crimp is a complex phenomenon which has been studied by many researchers who suggested a number of parameters to describe the shape of a crimped fibre both in space and in a two-dimensional projection on a plane, see for example Brand and Scruby (1973), Xu et al. (1992) and Muraoka et al. (1995). The attributes of fibre crimp can be characterised as follows (Xu et al., 1992): (1) Natural length Cnl is the length of the fibre at rest (Fig. 2.7). (2) Extended length Cel is the length of the fully extended fibre under a certain tension without elongation. C - Cnl (3) Crimp percent C% = el [2.28] ¥ 100 . Cnl (4) Noncrimp Cnc is the length of the fibre portion that shows no change in slope.

Cw

Ca Cnc

Ch

Cnl

2.7 Fibre crimp parameters.

64

Modelling and predicting textile behaviour

(5) Crimps per centimetre Cp =

number of crimps ¥ 10 . Cnl

[2.29] (6) Spatial frequency Csf is the dominant frequency of fibre crimp waves obtained by the fast Fourier transform. (7) Crimp height Ch is the vertical distance from a peak to the next valley. (8) Crimp width Cw is the horizontal distance from a peak to the next valley. (9) Crimp angle Ca is the angle between two lines that are tangent to two shoulders of a crimp at the points of maximum slope. Cam (10) Crimp intensity Ci = , where 0 £ Cam £ (h0 + Cam), Cam is h0 + Cam the amplitude of crimp, h0 is the difference between the extrapolated rise and Cam. The value of Ci is always in the range from 0 to 1. (11) Crimp sharpness Cs is a combination of Ca and Ci: Cs = ÊÁ1 - Ca ˆ˜ Ci. Ë 180 ¯ Similar to crimp intensity, the value of Cs is always in the range from 0 to 1. The definition of crimp percent in item (3) above is essentially different from that given in Denton and Daniels (2002) which, using the same notations as above, can be expressed as follows:

C%TI =

Cel – Cnl ¥ 100 Cel

[2.30]

It can be seen that the formula for crimp percent suggested by Xu et al. (1992) in most cases will produce a greater value than the one adopted by the Textile Institute because generally Cnl £ Cel. The definition of Cp is equivalent to that of crimp frequency recommended by Denton and Daniels (2002). Modern digital image acquisition techniques and image processing methods can be employed to extract detailed information about crimp attributes (Sawyer and Chex, 1978, Xu and Ting, 1996; Koehl et al., 1998), although crimp percent and crimp frequency are the two main parameters used in practice. For example, wool fibre crimp frequency is typically in the range 2–13.5 crimps/cm (Cook and Fleischfresser, 1990). The mechanics-based characterisation of crimp considers a load–extension diagram of the crimped fibre (Bauer-Kurz et al., 2004). Owing to the complexity of the spatial shape of a crimped fibre it is difficult to develop a computational model that would accurately take into account all parameters of fibre crimp. The random spatial shape of the fibre can be modelled using a stochastic approach (Muraoka et al., 1995; Grishanov and Harwood, 1999), a

Fundamental modelling of textile fibrous structures

65

micromechanical approach (Munro, 2001), or a combination of these two methods. In the case of stochastic approach it can be assumed that fibre consists of n small straight segments of equal length Dl randomly oriented in 3D space and joined together to form a continuous body so that the fibre extended length is Cel = nDl. It follows that the same approach as in the case of molecular conformations (see Equation [2.1]) can be applied in order to predict the fibre average natural length Cnl = Dl n . Grishanov and Harwood (1999) approximated shape of hairs protruding out of the yarn main core by a 3D randomly curved line which has a dominant direction. This led to the representation of the fibre shape by a series of points in three dimensions which are the start and the end points of vectors that have constant unit length Dl and randomly deviate around the general direction:

n

n

r =1

r =1

S = S sr = S ( ixr + jyr + kzr )



[2.31]

where n is the number of vectors, i , j and k are unit vectors associated with the axes of an orthogonal Cartesian coordinate system, xr = Dl sin jr cos qr, yr = Dl sin jr sin qr, zr = Dl cos jr, jr = j + sj xr; qr = q + sq xr. Here j and q are two angles in a spherical coordinate system which give the dominant direction of fibre shape, sj and sq are standard deviations of j and q , respectively, and x r is a normally distributed random number with zero mean and unit standard deviation. An appropriate choice of Dl provides necessary accuracy for the approximation. Once fibre coordinates xr, yr and zr are calculated, the necessary smoothness of the fibre shape can be provided by a 3D cubic spline approximation of the fibre curve:



x = a3 (t – ti )3 + a2 (t – ti )2 + a1 (t – ti ) + a0 ¸ ÔÔ y = b3 (t – ti )3 + b2 (t – ti )2 + b1 (t – ti ) + b0 ˝ Ô z = c3 (t – ti )3 + c2 (t – ti )2 + c1 (t – ti ) + c0 Ô˛

[2.32]

where t Π[ti; ti+1] is a spline parameter. An approach that uses fractal dimension has been applied by Muraoka et al. (1995) for the analysis and simulation of fibre crimp. Acrylic and nylon fibres were first analysed in order to estimate the value of the box-counting dimension, DB, which, according to Falconer (1990), can be defined as follows:

DB = lim

d Æ0

log Nd – logd

[2.33]

66

Modelling and predicting textile behaviour

where Nd is the number of boxes of a regular grid that cover the image in question and d is the size of the squares constituting the grid. Modified random Koch curves, described by Falconer (1990), were then used to simulate the two-dimensional shapes of crimped fibres. The two approaches described above produced a reasonable similarity between the simulated and real fibres but questions about the degree of this similarity and the method for estimating the parameters that would provide the best fit require additional investigation. Brand and Backer (1962), El-Sheikh et al. (1971), Batra (1974a, b) and Gupta and George (1975) used the theory of elasticity for modelling fibre crimp in three dimensions; however, along the length and over the crosssection of the fibre, they did not take into account the variation in the location of regions with differing properties. The micromechanical approach applied to wool fibre suggested by Munro and Carnaby (1999) was based on modelling the geometric arrangement and the mechanics of microfibrils in paracortical and orthocortical regions of the fibre. The model takes account of the amount and position of the ortho- and paracortical regions in the fibre cross-section. The finite-element method was used to predict the variation in longitudinal strain and fibre curvature depending on ratio and position of the ortho- and paracortical cells. Munro (2001) then used a cross-sectional-area profile of a sample fibre to simulate the 3D shapes of virtual fibres using an energy minimisation method.

2.5.5 Modelling longitudinal profiles Modelling the properties of individual fibres and fibre assemblies based on statistical methods has two significantly different aspects. One is the modelling and simulation of point estimates of properties such as length, diameter, strength or any other for which the probability density distribution is known or, if not, then may be assumed to follow an experimentally obtained or one of the known distributions. In this case the generation of an instant value of a parameter is reduced to the use of one of many well-developed algorithms that implement numerical methods for generating random numbers with specified distribution (Knuth (1998); Press et al., 2002). It is, however, necessary to take into account that some parameters, for example, yarn strength and elongation at break, are not independent but display a significant correlation (Truevtsev et al., 1997). In this case it is necessary to generate two correlated random variables. According to Knuth (1998), if X1 and X2 are two independent normally distributed random variables with a mean of zero and a standard deviation of one, then Y1 = m1 + s1X1 and Y2 = m2 + s2 (rX1+ 1 – r 2 X2) are two dependent random variables normally distributed with means m1 and m2, standard deviations s1 and s2, and with a correlation coefficient r.

Fundamental modelling of textile fibrous structures

67

The second aspect is the modelling of a series of values of properties such as fibre diameter profile along the fibre length, the linear density profile along the length of a linear fibre assembly, the stress–strain diagram of a product and others. These characteristics can be considered as random functions and thus modelling in this case concerns the generation of a set of random numbers that collectively represent one realisation of a random function. This is a much more complex case than that above because it requires knowledge of correlation characteristics of the random function in question. The required values Y1, Y2 , …, Yn of this series can be expressed in the form (Knuth, 1998):

Y1 = b1 + a11X1; Y2 = b2 + a21X1 + a22X2;



Yn = bn + an1X1 + an2X2 + … + annXn

[2.34]

where Y1, Y2 , …, Yn are normally distributed dependent variables, Yi has a mean mi and YS have a given covariance matrix C(cij). Knuth (1998) suggested that coefficients of a triangular matrix A(aij) can be found from matrix equation:

AAT = C

[2.35]

The solution of this equation gives the unknown values of aij:



c1i , i > 1; a11

¸ Ô Ô j –1 Ô c ji – S a jk aik Ô k =1 , i > j;Ô aij = a jj ˝ Ô i-1 Ô aii = cii – S aik , Ô k =1 Ô Ô˛ i = 1, 2, ..., n. ai1 =

[2.36]

The necessity of incorporating information on correlation between the values of parameter being generated has not been widely recognised and may lead to production of seemingly realistic profiles but with the wrong statistical properties as might have been the case in the paper by Glass (2000). Accurate modelling of fibre diameter profiles is important for predicting the fibre breakage and ultimate tensile strength of fibre assemblies. The models briefly introduced above provide essential information about fibre geometry that is necessary for the consideration of mechanical properties of fibres which are considered in the next section.

68

Modelling and predicting textile behaviour

2.6

Modelling mechanical behaviour of single fibres

There are two main problems that are usually considered in the mechanics of fibres. One is the prediction of ultimate tensile strength and another is the characterisation of mechanical behaviour at deformations (or stresses) below the level of destruction where various degrees of non-linearity are considered. The tensile strength of single fibres, according to weakest link theory, Peirce (1926), usually is assumed to follow a Weibull distribution (Weibull, 1951): f (x, a , b ) =

bÊ xˆ a ÁË a ˜¯

b –1

Ê Ê x ˆ bˆ expÁ – Á ˜ ˜ Ë Ëa¯ ¯

[2.37]

where a > 0 is the scale parameter and b > 0 is the shape parameter of the distribution. The same approach is used for the prediction of the tensile strength of fibre assemblies (Peirce, 1926; Realff et al., 2000; Ghosh et al., 2005). However, earlier Truevtsev et al. (1997) tested the applicability of various distributions to the description of experimental data on yarn strength and extension and found that it is reasonable to use a Gaussian distribution. A simple approach to the mechanical properties of fibres can be based on the assumption of linear elastic behaviour of the fibre material when both tensile and shear stresses are proportional to the strain applied, independent of time, and fully recovered once the external forces are removed. Figure 2.8 shows that for a linear elastic behaviour a constant stress s0 instantly applied at time t = t0 causes an instant strain e0 which remains constant until the removal of stress at t = t1. Fibres in this case are usually considered as thin elastic rods with a small ratio of transversal dimensions to length and made of isotropic homogeneous material. Within this approach, s

eelastic e0

s0

0

t0

t1 (a)

t

0

t0

t1 (b)

t

2.8 Stress and strain of a linear elastic solid: (a) constant stress applied from time t0 to time t1, (b) a constant elastic response from time t0 to time t1.

Fundamental modelling of textile fibrous structures

69

however, it is necessary to distinguish two cases which are different in terms of the scale of deformation applied. If both the deformation and the deformation rate are small then a linear mechanics model can be used according to Hook’s law:

s = Ee;

(2.38)



t = Gg,

(2.39)

FT F and t = S are tensile and shear stresses, respectively, FT A A and FS are tensile and shear forces, respectively, A is the area to which the stress is applied, E and G are tensile and shear moduli, respectively, e = Dl and g = D x are tensile and shear strains, respectively, Dl and l h Dx are tensile and shear deformations and l and h are the initial sample dimensions. In volume deformation, pressure is assumed to be proportional to the bulk modulus and relative change of volume where s =



p = KeV,

[2.40]

where p is hydrostatic pressure and K is the bulk modulus; eV = DV . V The range of tensile deformations where the Hook’s law can be applied to the textile fibres can be as small as 0.03%. Similar to many materials used in engineering, textile fibres display a tendency to contract (respectively, to expand) in the direction perpendicular to the direction of extension (respectively, compression) which is measured by the Poisson’s ratio as follows:

n =–

ex ey

[2.41]

where ex and ey are strains in the direction of force F and in the direction perpendicular to the direction of extension, respectively (Fig. 2.9). All three moduli are linked by a relationship

E = 9GK G + 3K

[2.42]

but for a perfectly elastic solid only two out of four parameters characterising the mechanical properties of elastic solid are independent:

G=

E E and K = 2(1 + n ) 3(1 – 2n )

[2.43]

For engineering materials like steel, Poisson’s ratio is typically positive

70

Modelling and predicting textile behaviour

in the range from 0.23 to 0.3. Recent years have seen a growing interest in the auxetic materials which have a negative Poisson’s ratio (see Alderson et al., 2002; Ravirala et al., 2006). These materials, including polymer fibres, are characterised by unusual behaviour in mechanical deformations, that is they display lateral expansion in tensile deformation and lateral contraction in compression. The investigation of properties and developing theoretical models that can predict the structure–properties relationship of these materials is a relatively new area of research. The tensile and shear moduli can be estimated by applying a specified small force and measuring displacement or applying a specified small displacement and measuring force. In both cases only the initial modulus which is applicable if strains are small (typically less than 1%) can be estimated and considered not to depend on time, deformation rate and temperature. When considering small bending deformation it is usually assumed that the curvature of the neutral axis of the beam, k, is proportional to the second derivative of the y coordinate:

k =

d 2 y / dx 2 ª d 2 y / dx 2 [1 + (dy /dx )2 ]3/2

[2.44]

This approximation leads to the formula for the deflection of the cantilever beam as follows:

Dx =

Flf3 3EI

[2.45]

where Dx is deflection, F is bending force applied, lf is fibre length and I is the moment of inertia of the beam cross-section. Fibre diameter and mechanical properties change along the fibre length owing to the non-uniformity of the internal fibre structure. This leads to difficulties in determining the fibre moment of inertia and the tensile modulus. Therefore, quite often bending rigidity which is an integral property of a fibre is measured B = EI. Under the quasi-static conditions of deformation, the bending rigidity of the fibre, B, can be calculated from the standard formula [2.45] for bending of a cantilever at small deflections, D x, such that D x/lf < 0.1, where lf is fibre length:

B=

Fl 3f 3Dx

[2.46]

The characteristic feature of the textile fibres is that they easily develop large bending deformations under small external forces. The case when the deformation is large in comparison with the length of the fibre must be

Fundamental modelling of textile fibrous structures

71

y F

x (a) y F

x (b)

2.9 On the definition of Poisson’s ratio. y s

q

F

0

x

2.10 Large deflection of a fibre under a concentrated load.

considered by taking into account the geometric non-linearity. This, according to Popov (1986), leads to the differential equation for a cantilever (Fig. 2.10): 2



lf2 d q2 = – b 2 sin q ds

[2.47]

where q is the angle between the direction of load and the tangent to the curved axis of the fibre, s is the arc length measured along the curved axis of the fibre and b =

Flf2 is the force similarity coefficient. EI

The first integration of this equation gives:

72

Modelling and predicting textile behaviour



lf dq = 2b ds

(C – sin q2 ) 2

[2.48]

where C is an arbitrary constant defined by the initial conditions. Introducing new variables k2 = C and sin q = k sin y and substituting in 2 Equation [2.48] yields:

lf

dy = b 1 – k 2 sin 2 y ds

Integrating for the second time between the starting point (s = 0, y = y0) and an arbitrary point (s, y) finally gives:



Ú

bs = lf

y

y0

dy 1 – k 2 sin 2 y

[2.49]

The integral in the right part is known as a complete elliptic integral of the first kind which is usually denoted as

Ú

F (k ) =

y

0



dy 1 – k 2 sin 2 y

The deflections of the fibre end can be obtained in the form:

x1 =

where E (k ) =

2l f k cos y 0 ; y1 = lf {1 – 2 [ E ( k ) – E (y 0 )]} b b

Ú

y

0

[2.50]

1 – k 2 sin 2 y dy is the complete elliptic integral of the

second kind. The values of the elliptic integrals are widely available in mathematical and mechanical hand books, see for example Korn and Korn (1968, p 1033). It can be shown that the linear theory (see Equation [2.45]) gives an overestimate of the values of cantilever deflection in comparison with the results produced by the exact solution outlined above. For example, for the case when at the end of the beam angle q = 20°, linear theory gives a deflection of 0.244lf whereas the exact solution is 0.23lf (6% difference); the case of q = 60° gives respectively 1.135lf and 0.634lf, or a 79% difference. The analysis of single fibre mechanics in recent years has moved towards the application of more complex models that try taking into account a combination of factors acting in real life situations such as non-circular fibre cross-section and non-linear stress–strain relationship.

Fundamental modelling of textile fibrous structures

73

In a series of papers, Lee (2002, 2003, 2005a, 2005b, 2007) investigated torsion and bending of fibres of various cross-sectional shapes. In torsion the moment, M, and the twist angle, q, were linked by the equation M = qGD where G is shear modulus and the torsion rigidity constant, D, plays the role of the polar moment of inertia. The main difficulty was to define the constant D. This can be done by using various methods such as: ∑ ∑

conformal mapping which leads to the determination of a function that maps a unit circle into the fibre cross-section shape by a numerical method; Saint-Vernant torsion theory expressed in terms of the stress function F(x, y)



∂2 F + ∂2 F = – 2 ∂x 2 ∂y 2

which satisfies boundary conditions F = 0; the rigidity constant can be found by integration D = 2

ÚÚF(x, y) d xd y .

In bending the material was assumed to be non-linear elastic such that s = Een, where typically for polymers n < 1 and the bending moment was expressed as M = Ek n

Ú

A

y n +1 (x ) dA .

In a similar treatment of large deflections of fibres with non-linear properties, Jung and Kang (2005) suggested a modified nonlinear stress–strain relationship:

s = sign(e )[( e + e1)n – e1n ]E

[2.51]



1

Ê n ˆ 1– n where constant e1 = Á ˜ was chosen to provide ds de Ë 2¯

e =0

= 2E .

This relationship assumes that extension and compression moduli are equal, which is not always so. Four different cases of fibre loading were considered which combine distributed and concentrated loads. The differential equations obtained for the neutral curved axis were of a general form:



dk = Ê dM ˆ ds ÁË ds ˜¯

Ê ˆ Á E nt 2 [ k t + e1]n –1 d A˜ ÁË ˜¯ A

Ú

[2.52]

where k is the curvature, s is the curvilinear coordinate, M is the bending moment and t is the distance from the neutral axis.

74

Modelling and predicting textile behaviour

The numerical solution of the problem was found using a fourth-order Runge–Kutta method and showed a good agreement with the elliptic integral solution considered above for a cantilever beam (see Equations [2.47] – [2.50]). Dynamic rigidity can be estimated by measuring a resonant exciting frequency at small amplitudes and then using the formula derived by Guthrie et al. (1954):



B=

4p 2 rAlf4 f 2 mi4

[2.53]

where A is the area of fibre cross-section, r is the fibre volume density, lf is the fibre length and f is the resonant frequency. mi is a constant depending on the mode of vibration obtained as the solution of equation cos mi cosh mi = –1. For i = 0 m0 = 1.8751; i = 1 m1 = 4.6941; i = 2 m2 = 7.8548. In general, for i ≥ 2 mi = (i + 1/2)p. The dynamic value of the tensile modulus measured by the frequency method, Ea, is higher than that measured in tensile test at low speed, Ei. It can be assumed (Guthrie et al., 1954) that at low speed the conditions of deformation are isothermal whereas in the dynamic test the process is adiabatic. This leads to the formula which links two moduli measured under different conditions:

1 = 1 – a 2T Ea Ei rCJ

[2.54]

where a is the linear thermal coefficient of expansion, T is the absolute temperature, r is the specific gravity, C is the specific heat and J is Joule’s equivalent of heat. There are two general methods for testing the torsion rigidity of a single fibre. One is based on the measurement of torque required to twist the fibre by the specified angle q under quasi-static conditions (Chapman, 1971) or at a constant rate, Mitchell and Feughelmen (1960). Torsion rigidity then can be estimated as:

T =

Mlf q

[2.55]

where M is the torque applied. Another method is to apply the oscillation pendulum method. Considering the fibre as a rod of a circular cross-section made of a linear viscoelastic material, the equation describing the rotational motion of a pendulum attached at the end of the fibre under the action of external moment is:

Fundamental modelling of textile fibrous structures



2 I p d q2 + D dq + Tq = M (t ) dt dt

75

[2.56]

where Ip is the moment of inertia of the pendulum, D is damping coefficient, pr 4G T= is the torsion rigidity of the fibre, r is the fibre radius, G is the 2lf shear modulus of the fibre material, lf is the fibre length, q is angle of rotation, M(t) is the external moment and t is time. Dividing by Ip and introducing new denotations:

D/Ip = 2awn, T/Ip = w n2 and M(t)/Ip = m(t)

where a =

D is viscous damping ratio and wn is natural oscillation 2 TI p

frequency. Equation [2.56] can be rewritten to give:

d 2q + 2aw dq + w 2q = m(t ) n n dt dt 2

[2.57]

If the damping coefficient is small and there is no external moment, equation [2.57] describes simple harmonic motion. The solution of this equation with initial conditions q t =0 = q0 and dq = 0 is q = q0sin(wnt dt t =0 + d), where d is phase angle which in this particular case is equal p/2. Thus, q = q0coswnt. The natural frequency is linked to the parameters of the system



wn =

pr 4G 2 lf I p



[2.58]

This relationship, rearranged with respect to G, can be used for determining the shear modulus of a fibre of known dimensions by measuring the frequency or period of oscillation of a pendulum with a known moment of inertia. Note that in the treatment of this problem the fibre moment of inertia was assumed to be negligibly small. In the case if damping is not small and cannot be neglected, that is if the fibre displays a degree of viscosity, then in the absence of an external moment, a decaying oscillation will be observed:

q = q0 e–awnt sin(wnt + d)

[2.59]

The ratio of two successive amplitudes yields the logarithmic decrement:

76



Modelling and predicting textile behaviour

l = ln

Ê ˆ qi 2ap = ln(e– aw n (ti –ti +1 ) ) = lnÁ exp 2ap ˜ = 2 qi+1 Ë w n 1– a ¯ w n 1 – a 2



[2.60]

This approach was widely used for the estimation of shear modulus of natural and synthetic fibres (Peirce, 1923; Ray, 1947; Meredith, 1954; Guthrie et al., 1954; Karrholm et al., 1955; Owen, 1965).

2.7

Viscoelastic properties of fibres

Textile fibres do not behave as linear elastic bodies but display a combination of the behaviour of viscous fluids and elastic solids. For example, if a constant stress s0 is instantly applied at time t = t0 to a linear viscoelastic fibre (Fig. 2.11) then this will first cause an instant elastic strain e0 followed by a slowly increasing strain which approaches its ultimate value e•. This increasing component of strain is the sum of the delayed elastic strain and the strain related to viscous properties of the fibre material. It can be said that in this condition the fibre material ‘flows’ as a viscous liquid. This effect is termed creep and produced by the rearrangement of molecules under the load applied. Once the load is removed the fibre slowly returns to its original length; this is typical behaviour of the elastic material. The behaviour in tensile deformation, however, depends on the rate of the strain applied, that is the same fibre may fracture as a metal if deformed at a high rate. s s0

0

t0

t1 (a)

t

e e∞ e0 e0

0

t0

t1 (b)

t

2.11 Creep and relaxation behaviour of a linear viscoelastic solid (a) a constant stress applied from time t0 to time t1, (b) an elastic response at time t0, creep from time t0 to time t1, and recovery after time t1.

Fundamental modelling of textile fibrous structures

77

There are linear and non-linear approaches to modelling the viscoelastic properties of fibres. In the case of linear approach, the viscosity is defined by Newton’s law which states that in the liquid shear stress is proportional to the velocity gradient:

t = h ∂V ∂y

[2.61]

where h is viscosity, V is velocity and y is the direction of the velocity gradient. In application to linear viscoelastic solids it is assumed that the stresses related to strain and strain rate produce a combined effect resulting in a simple model of viscoelastic behaviour in the form:

t = Gg + h

∂g ∂t

[2.62]

This equation in fact describes the viscoelastic behaviour of the Kelvin–Voigt model discussed further in Section 2.7.1. The models presented in the sections that follow are limited to the consideration of viscoelesticity of an isotropic homogeneous material in one dimension.

2.7.1 Models of creep and stress relaxation Linear viscoelastic behaviour is usually represented by models consisting of springs which follow Hook’s law of elastic deformation and dashpots with properties of a Newtonian liquid where stress is proportional to the strain rate. The Maxwell model has a spring and a dashpot connected in series (Fig. 2.12). Because of this, the stress in each element is the same as in the whole model, whereas the strain is equal to the sum of strains of two elements:

E

h

2.12 Two-element Maxwell model.

78

Modelling and predicting textile behaviour

s 1 = Ee1



¸ Ô de s 2 = h 2 ÔÔ dt ˝ s1 = s 2 = s Ô Ô e = e1 + e 2 Ô˛

[2.63]

Combining these equations yields:

de = 1 ds + s dt E d t h

[2.64]

The Maxwell model can explain reasonably well the stress relaxation when de/dt = 0. Substituting this in Equation [2.64] and rearranging brings:

ds = – E dt s h

which after integrating under initial conditions t = 0, s = s0 gives:

s = s 0e

– t t rel

[2.65]



where trel = h/E is called relaxation time which refers to the point of a maximum slope on the stress relaxation curve plotted in logarithmic scale; at this point stress s = s0e–1. It can be seen that the Maxwell model does not adequately describe viscoelastic behaviour in creep because under constant stress, i.e. ds/dt = 0, it corresponds to the Newton’s law of viscosity (Equation [2.61]). The Voigt model shown in Fig. 2.13 has a spring and a dashpot arranged in parallel. The total stress is therefore shared between these two elements so that s = s1 + s2 but the strain of each individual element is the same e1 = e2 = e. Thus, for this model:

s = Ee + h de dt

[2.66]

Integration and rearrangement gives Equation [2.67] for creep:

e = Js (1 – e

– t t ret )



[2.67]

where J = 1/E is spring compliance and tret = h/ E is retardation time; this is the time after loading at which strain reaches the value of (1–1/e) of the equilibrium value. The Voigt model cannot describe stress relaxation at a constant strain, i.e. if de/dt = 0 it gives s = Ee which is Hook’s law.

Fundamental modelling of textile fibrous structures

E

79

h

2.13 Two-element Voigt model.

E2

E1

E2 E1

h

h

(a)

(b)

2.14 Three-element standard linear solid with a second spring in (a) parallel with the Maxwell model and (b) in series with the Voigt model.

A better model can be obtained by adding a second spring either in parallel with Maxwell model or in series with Voigt model, see Fig. 2.14a and 2.14b, respectively. Applying the same technique as above to the analysis of stresses and strains in both cases yields the model equation (Ward and Hadley, 1993):

s +

h ds h = E1e + ( E1 + E2 ) de E 2 dt E 2 dt

[2.68]

This model, commonly known as Zener model or standard linear solid, ˆ Ê has two time constants, one is for creep t ret = hÁ 1 + 1 ˜ and another for Ë E1 E2 ¯

80

Modelling and predicting textile behaviour

h , and can describe both creep and stress relaxation. E2 Here E1 and E2 are the moduli for the spring parallel to Maxwell model and the spring in series to the dashpot, respectively. The solution for creep behaviour of this model can be obtained in the form (Jaeger, 1962): stress relaxation t rel =



E1E2 t ˆ Ê – E2 e = s Á1 – e h ( E1+E2 ) ˜ E1 Ë E1 + E2 ¯

[2.69]

Differentiating with respect to time and taking logarithm yields the linear dependence of the logarithmic creep rate against time

E1E2 s E22 ln de = ln – t dt h( E1 + E2 )2 h( E1 + E2 )



[2.70]

This linear dependence is often not the case for many polymers even at small strains and non-linear models of viscoelasticity provide better results.

2.7.2 Infinitely many element models In reality the viscoelastic behaviour of polymer fibres is more complex than that described by any of the models above. A number of modifications to these models have been suggested that in general can be called ladder network models. They include arrays of Maxwell models arranged in parallel for modelling stress relaxation (Fig. 2.15) and Voigt models arranged in series for modelling creep (Fig. 2.16). These models lead to the notion of relaxation and retardation time spectra.

E1

E2

h1

E3

h2

h3

En

hn

2.15 Parallel arrangement of Maxwell models.

Fundamental modelling of textile fibrous structures h1

h2

h3

E2

E1

E3

81

hn

En

2.16 Series arrangement of Voigt models.

For a single Maxwell model at e = const the stress is s (t ) = e Ee Considering n such models in parallel gives: n



s (t ) = e S Ei e i =1

– t t rel

.

– t ti

where Ei and ti are modulus and relaxation time of element i, respectively. If there are infinitely many elements then:

s (t ) = Er e + e

Ú

•

0

F (t )e

–t t

dt

or, dividing by e, the stress relaxation modulus can be obtained:

E (t ) = Er +

•

Ú

0

–t

H (t )e t dt

[2.71]



where Er is a constant modulus corresponding to an infinite relaxation time t = • and H(t) is continuous relaxation time spectrum. In a similar way for the Voigt model with infinitely many elements the creep compliance is given by:

J (t ) = J 0 +

Ú

•

0

L (t )(1 – e

–t t

) dt



[2.72]

where J0 is the instantaneous compliance at t = 0 and L(t) is continuous retardation time spectrum. Monographs by Tobolsky (1960) and Ferry (1980) provide an overview of relationships between the spectra and methods for calculation of various viscoelastic constants. These two models are capable of describing continuous relaxation and retardation spectra over a limited range of time because their elements, although infinite in number, are characterised by a discrete set of parameters. For the model to be able to define a continuous spectrum over the entire time range it must have an infinite number of elements with continuously distributed parameters (Gross and Fuoss, 1956). These models, which combine parallel

82

Modelling and predicting textile behaviour

and series arrangements, are known as ladder networks; see for example Blizard (1951), Gross (1956) and Marin (1960).

2.7.3 Non-linear models Boltzmann made important assumptions about general viscoelastic behaviour which are used in linear viscoelastic theories; they are known as the Boltzmann superposition principles: ∑ ∑

Creep is a function of the entire past loading history of the sample. Each loading step makes an independent contribution to the final deformation.

For example, if a sample is subjected to a multistep loading test (Fig. 2.17) where incremental stresses Ds1, Ds2, …, Dsn are applied at times t1, t2, …, tn, then the total creep at time t is: n



e (t ) = S Ds i J (t – ti ) i =1

[2.73]



where J(t – ti) = 1/E(t – ti) is the creep compliance function. Equation (2.73) can be written in a general integral form for a continuous stress history as:



e (t ) = s + E

Ú

t

–•

J (t – t )

ds (t )  dt dt

[2.74]

s s(t)

Ds3 Ds2 Ds1 t1

0

t2

t3

t

t

(a) e

0

e(t)

t1

t2

t3

t

t

(b)

2.17 Creep behaviour of a linear viscoelastic solid in a multi-step loading: (a) stress applied, (b) creep behaviour.

Fundamental modelling of textile fibrous structures

83

where s is the total stress at the end of the test, E is the unrelaxed modulus, t is the current time in the stress history and t is the present time. Similar to this, the total stress at time t in response to the incremental strains De1, De2, …, Den applied at times t1, t2, …, tn is: n



s (t ) = S De i E (t – ti ) i =1



[2.75]

where E(t – ti) is the stress relaxation modulus, or in integral form:

Ú

s (t ) = E• e +

t

–•

de (t ) E (t – t )  dt dt

[2.76]

where E• is the relaxed modulus in the equilibrium state. Experimental studies of viscoelastic behaviour of textile fibres have shown that the simple assumptions of linear viscoelastic theory do not hold even in the case of small strains and stresses. It was noticed that the experimental stress–strain curves of polymer textile fibres have a yield region which is similar to that of mild steel where the strain increases under a constant stress. This leads to the fact that the fibre does not return to its original state when the external stress is removed. The total strain can be expressed as a sum of elastic (completely recoverable) and plastic (residual) strains: eS = ee + e p. Leaderman (1943) suggested a modification to the Boltzmann principle by introducing an arbitrary empirical function of stress f (s):

e (t ) = s + E

Ú

t

–•

df (s ) J (t – t )dt dt

[2.77]

The applicability of this approach, however, is limited by simple creep and relaxation. One approach to modelling the non-linear viscoelastic behaviour of polymers is to assume that the liquid in the dashpot is non-Newtonian (Halsey et al., 1945):

de = k sinh as dt

[2.78]

where k and a are two constants. Gupta and Kumar (1977) suggested using power law for the fluid:

de = As n dt

[2.79]

Both approaches give a reasonable approximation to the experimentally obtained results.

84

Modelling and predicting textile behaviour

Alternatively, Pao and Marin (1952, 1953) assumed that the total strain can be considered as a sum of three independent components:

e S = e e + e ve + e p = s + as n (1 – e– bt ) + cs n t E

[2.80]

where ee, eve and ep are elastic, viscoelastic and plastic strain, respectively and a, b, c and n are constants defined by the properties of a material. Ariyama (1994) suggested a three-element model of viscoelastic–plastic behaviour derived on the basis of the standard linear solid described in Section 2.7.1 (Fig. 2.14b). The spring element E1 was replaced by an element representing a hardening function f (ep) where ep is the plastic strain. Within this model it was assumed that the dependence of strain rate on flow stress in the plastic region is based on the effect of viscosity. The constitutive equation of this model was expressed as:

de = 1 ds + s – f (e p ) dt E 2 dt h

[2.81]

where e is the total strain, E2 is the tensile module, s is the tensile stress and h is the viscosity coefficient. This model was applied to the study of non-linear viscoelastic–plastic behaviour of polypropylene samples and showed good agreement with the experimental results. A more advanced approach to modelling the viscoelastic–plastic behaviour in comparison with Pao and Marin (1952, 1953) has been developed by Cui and Wang (1999) in application to fabrics where the creep equation contained four terms representing instantaneous elastic and plastic strains, viscoelastic strain and viscoplastic strain as follows:

e(t) = ee + ep + eve + evp

[2.82]

This model was based on Schapery’s non-linear constitutive relation developed from thermodynamic principles (Schapery, 1997). Fabric test procedures enabled the material parameters to be determined but an extended experimental verification of this model is required.

2.7.4 Temperature dependence The models of viscoelastic behaviour of polymer fibres considered in previous sections take into account the dependence of the mechanical behaviour of fibres on time and rate of deformation. However, experiments carried out over the same period of time and at the same rate of deformation but at different temperatures show a dramatic difference in relaxation modulus. The degree of this difference depends on the proportion of amorphous

Fundamental modelling of textile fibrous structures

85

and crystalline parts of the polymer. For the amorphous polymers this can be explained by the effects of temperature on rotational and translational motions of molecules and their segments. At the temperatures below the glass transition temperature the molecular segments vibrate around fixed positions. The polymer is hard and exhibits elastic behaviour characterised by a high value of the tensile modulus. As temperature increases and reaches the glass transition region, the amplitude of vibration also increases; thermal energy becomes sufficient to cause rotation and translation of molecular segments. This results in a dramatic decrease in the modulus. At temperatures higher than the glass transition the motion of molecular chains is still restricted by the local interactions; this is reflected in a nearly constant modulus and a rubber-like behaviour. Further increase in temperature produces different effects depending on whether the polymer is cross-linked or not. In the case of linear polymers, the whole molecular chain will be completely free to move, in which case the polymer starts to behave as a viscous liquid; this is accompanied by a rapid drop in the modulus. The cross-linked polymers may be considered to have a constant modulus in this temperature range owing to the cross-links that restrict the relative translation of molecules. Crystalline polymers display similar changes in the relaxation modulus to those observed in amorphous polymers but the influence of the crystalline part often makes the effect of glass transition less noticeable. The main change in the relaxation modulus of crystalline polymers happens at the melting point, after which their behaviour is very much similar to the amorphous polymers. Numerous experimental investigations of time and temperature effects have made researchers realise that the data on relaxation modulus obtained at different temperatures over the same period of time (Fig. 2.18a) can be combined by shifting individual curves horizontally along the logarithmic time axis (Fig. 2.18b). This is equivalent to dividing the time scale by a constant factor at each temperature. It is also necessary to take into account the influence of temperature on the cross-sectional area and the volume density of the sample being tested which produces a vertical shift of the data. Thus, the modulus at any time t which would be observed at a reference temperature T0 can be found through the modulus at different temperatures T as follows (Aklonis and MacKnight, 1983):

E (T0, t ) =

r(T0 )T0 E (T , t /aT ) r(T )T

[2.83]

where E is the relaxation modulus, r is the volume density of the polymer and aT is the temperature shift factor. The resultant modulus–time plot covering a wide range of time is called the master curve (Fig. 2.18b) and the procedure for its calculation is called the time reduction procedure. The polymer material for which the behaviour

86

Modelling and predicting textile behaviour

log E

log E T1 T2 T4

T3

T5 log t

log t

(a)

(b)

2.18 (a) Relaxation modulus measured at different temperatures, (b) master curve.

at transient temperatures can be predicted by using the data obtained at constant temperatures is called a thermorheologically simple material. A remarkable relationship, known as the WLF equation, was found by Williams, Landel and Ferry (see Williams et al., 1955) which is approximately true for all amorphous polymers: log aT =



C1 (T – T0 ) C2 + T – T0

[2.84]

Here C1 and C2 are constants; if T0 = Tg then for many materials it can be assumed that C1 = 17.4 and C2 = 51.6 (Ferry, 1980). The time–temperature equivalence considered above is applicable to the amorphous and single-phase polymers; more complex cases are discussed by Tobolsky (1960), Schapery (1974) and Ferry (1980).

2.8

Modelling fibre friction

Friction is a fundamental property of fibres that affects the strength of all fibre-based products such as yarns, threads, ropes, woven and knitted fabrics and nonwovens by creating forces that hold individual fibres together. Friction also plays an important role in fibre processing where, on one hand, it helps by providing cohesion between fibres in twistless products such as cotton slivers, wool tops and carded webs. Friction between fibres and air is used to transport fibre tufts via air ducts in fibre opening and cleaning and in pneumatic looms for weft insertion. On the other hand: ∑ ∑

friction makes it difficult to disentangle fibres and to remove impurities at the early stages of processing; it causes uncontrolled movement of short fibres in drafting and facilitates wool fibre felting owing to a directional friction effect;

Fundamental modelling of textile fibrous structures

∑ ∑ ∑

87

friction may create an excessive tension in yarns passing through guides in weaving, knitting and sewing; yarns and threads sliding over guides at high speeds may generate an excessive heat due to the friction; friction also affects the compression behaviour of fibres and the wrinkling behaviour of fabrics by limiting the recovery of bending deformation (Chapman, 1975) and even affecting the protective properties of singleand multi-layered body armour (Briscoe and Motamedi, 1992; Duan et al., 2005).

There are several theories that explain the relationship between the forces acting on surfaces involved in friction (Howell et al. 1959, Morton and Hearle, 1993). The surfaces of materials are not perfectly smooth but contain many asperities that generally are assumed to be spherical in shape and randomly distributed. It is usually assumed that there are three main reasons for the frictional forces as follows: 1. Union of the two surfaces at points where asperities on both sides come into actual contact; the area of these discrete contacts developed under normal load depends on the mechanical properties of the materials in question such as viscoelasticity, plasticity and anisotropy; 2. Shear deformation and breaking of these unions when relative movement of the surfaces starts; 3. for the surfaces of a different hardness, ploughing out a track in the softer material by the asperities on the harder surface. The simplest approach is to characterise friction by a friction coefficient that can be measured in static and dynamic conditions. The static friction coefficient, ms, between two dry flat solid surfaces in contact is defined as the tangential force, F, required to produce the relative movement of the surfaces divided by the normal force, N, between the surfaces:

ms = F/N

[2.85]

According to Amonton–Coulomb laws, the friction coefficient is assumed to be independent of the geometric area of contact between two bodies and the acting normal force. The kinetic friction coefficient, mk, is generally lower than the static coefficient but for the fibres there is evidence that the friction coefficient first decreases with the increase of velocity and then starts increasing. The case of fibre friction over a cylindrical surface is described by the capstan equation:

T1 = T0e mj

[2.86]

where T0 is the incoming tension, T1 is the leaving tension and j is the contact angle.

88

Modelling and predicting textile behaviour

This equation assumes that the friction does not depend on the radius of curvature. Fibres, however, do not obey simple laws indicated above. The friction coefficient for fibres and polymers increases with decreasing load, the friction force increases with decrease in surface roughness and with the increase in the contact area. For fibres, the relationship between the frictional force and normal load is usually approximated as follows:

F = aNn

[2.87]

where a and n are constants and n is typically between 2/3 and unity. In this case the fibre tension on a cylindrical surface will be described by a formula different from Equation [2.86] (Howell, 1953):

T11– n = T01– n + aj (1 – n )r1– n

[2.88]

where r is the radius of the cylinder. For the case of friction of two cylindrical fibres crossing at the right angle and thus having one area of contact, the frictional force can be expressed as:

F = kN2/(2+x)D2x/(2+x)

[2.89]

where D is the diameter of the cylinder and x is an experimentally defined constant. For more complex cross-sectional shapes discussed in Section 2.5.3 above it is possible to have several contact areas between the fibres. These cases require careful treatment based on the application of Hertz’s theory of bodies in contact (see Timoshenko and Goodier, 1970). There are fibres that show a significant difference in frictional behaviour. Cotton is known to have the kinetic friction coefficient that steadily increases over all ranges of velocities. Polytetrafluoroethylene fibres have a very low friction coefficient under all sorts of conditions (Howell et al., 1959) which can be attributed to weak intermolecular forces combined with its relatively high strength and yield limit. Wool shows a high degree of directional friction effect (DFE) which is explained by the scales on the wool surface. This leads to wool fibres displaying greater frictional force if two fibres are rubbed in a direction against the scales (from the scale tip to the scale root) than if rubbed in the opposite direction. The analysis of wool fibre friction carried out by Adams et al. (1990) resulted in a general equation for the friction force between two fibres: 2



È R R Ê 1 – n12 1 – n 22 ˆ ˘ F = pt 0 3 Í 3 1 2 Á + N + aN E2 ˜¯ ˙˚ Î 4 R1 + R2 Ë E1

[2.90]

Fundamental modelling of textile fibrous structures

89

where t0 and a are surface rheological properties which are characteristic of the surfaces of the materials, R is the fibre radius, E and n  are Young’s modulus and Poisson ratio of fibres, respectively, subscripts 1 and 2 refer to two contacting fibres and N is the normal load. It was shown that this equation can be transformed into the power form of Equation [2.87] so that:



Fw = aw N nw ¸Ô ˝ Fa = aa N na Ô˛

[2.91]

where subscripts w and a stand for friction with and against the scales, respectively. Parameters a and n for both directions of friction can then be determined from the experimentally measured forces. Many investigations confirmed that the simple approach to fibre friction expressed by equations [2.85] and [2.86] does not satisfy the available experimental data but in the majority of cases they are still used when modelling friction in fibre assemblies.

2.9

Modelling fibre assemblies

Fundamental and experimental investigations of the structure and properties of fibre assemblies have been one of the major topics in textile science for many years, yielding thousands of published papers and hundreds of books worldwide. It would be virtually impossible to mention all the scientists who contributed to the development of theoretical, experimental and technological aspects of fibre assemblies. The importance of this problem in practice is obvious because fibre-based products are among the most numerous, diverse and widely applied products which are in continuous use. All essential properties of these products depend on their structure and the physical properties of the constituting elements which are fibres. One of the difficulties in modelling performance characteristics of fibre assemblies lies in their dual nature because they are continuous products devised from a discrete set of fibres. In terms of fibre arrangement, fibre assemblies can be classified into four main types: 1. Twistless assemblies which are linear (slivers), sheet (carded webs and nonwovens) and bulk products (fibre mats), 2. Twisted assemblies which include linear products such as rovings, yarns, threads, and ropes, 3. Semi-regular structures such as needle-punched and stitch-bonded nonwovens, and 4. Regular assemblies which are knitted and woven fabrics.

90

Modelling and predicting textile behaviour

It was understood in the early stages of research in textiles that even in the simplest case of a parallel fibre bundle the properties of a fibre assembly do not equal the simple sum of the properties of all its individual fibres (Peirce, 1926). The fundamental task in the successful modelling of properties of fibre assemblies is therefore to define the minimal representative building block, a unit cell containing all essential structural features of the assembly, from which the physical properties of the unit cell and then of the whole assembly can be derived. The unit cell approach was first applied to regular structures by Peirce (1937, 1947) using high simplifications of the geometry and mechanical properties of yarn segments, and was then extended to general fibre assemblies by van Wyk (1946), Komori and Makishima (1977, 1978, 1979), Komori and Itoh (1991a, 1991b, 1994), Pan (1993a), and to yarns by Carnaby (1980), van Luijk et al. (1984a, 1984b, 1985a, 1985b), Pan (1992, 1993a, 1993b, 1996) and Pan and Postle (1995). There have been many major reviews of the mechanics of fibrous structures in monographs by Hearle et al. (1969) and Postle et al. (1988) as well as in conference proceedings (Hearle et al., 1980). There are several basic characteristics which are used to describe the geometry and structure of the fibrous unit cell such as fibre fineness and cross-sectional shape, length, orientation, 3D shape, packing density and mutual position, which are used to derive the number of contact points and then the structure–properties relationship of fibre assemblies. The geometry, structure and mechanical properties of individual fibres were considered in Sections 2.4–2.8 above. Let us now consider the structure and properties of fibre assemblies.

2.9.1 Unevenness of linear fibre assemblies The overall variability in properties of fibre assemblies, which is commonly referred to as unevenness, can be split into its components related to the variation in the geometry and properties of the fibres themselves and to the variation in their mutual position, or in other words, in the assembly’s structure:

s T = s p2 + s s2 + 2rpss ps s



[2.92]

where sT, sp and ss are total standard deviation, standard deviation of properties and standard deviation of structural parameters, respectively and rps is the correlation coefficient between the variation in properties and variation in structure. These two components of the total unevenness are not always independent. For example, it is known that in spinning there is a tendency for short fibres to group together. The exact value of the correlation coefficient rps has yet to be determined.

Fundamental modelling of textile fibrous structures

91

The most frequently used model of linear fibre assemblies is that suggested by Martindale (1945) according to which the fibres are randomly arranged along the axis of the product so that probability of a given fibre crossing a certain section follows the Poisson distribution. This assumption enabled the standard deviation, sn, and the variation coefficient, Vn, of the number of fibres per cross-section to be found:

s n = n ; Vn = 100 n

[2.93]

where n is the number of fibres in the cross-section. Then, the standard deviation, sa, and the variation coefficient, Va, of yarn cross-section area were found to be:

s a2 = s n2 A 2 + ns A2 = nA 2 (1 + 0.0001VA2 ); Va =



100 1 + 0.0001VA2 n [2.94]

where Ā is the mean area of fibre cross-section and VA is the variation coefficient of fibre cross-section. Assuming that for wool fibres VA = 2 VD, where VD is the variation coefficient of fibre diameter, the minimum possible variation coefficient of a yarn with randomly arranged fibres was obtained in the form:

Va =

100 1 + 0.0004VD2 n

[2.95]



The relationship between the number of fibres in the cross-sections of a fibre assembly, m(x), and the density of leading fibre ends per unit length of the ensemble, n(x), is important for the prediction of the unevenness of linear textile products. In the case where the ensemble consists of fibres of the same length l, the number of fibres in the cross-section is:

m(x ) =

Ú

x x –l

n(x ) dx

[2.96]



However, if the density of fibre length distribution is f(x), then the number of fibres will be:

m(x ) =

Ú

l1

l0

f (l ) dl

Ú

x x –l

n(x ) dx



[2.97]

where l0 and l1 are minimum and maximum fibre length, respectively. The relationship between the unevenness of fibre assembly and characteristics of fibre length and fibre end density distribution can be established by using Laplace transform, Sevostianov and Sevostianov (1984).

92

Modelling and predicting textile behaviour

Let N(p) = L{n(x)} and M(p) = L{m(x)} be the Laplace transforms of n(x) and m(x), respectively. Then:

Ï LÌ ÓÔ

Ú

x

0

¸ N ( p) Ï n(x ) dx˝ = and L Ì p ÓÔ ˛Ô

Ú

x –l

0

¸ N ( p )e– pl n(x ) dx˝ = p ˛Ô ,

respectively, which gives:

Ï V ( p, l ) = L Ì ÔÓ

Ú

¸ N ( p) n(x ) dx˝ = (1 – e– pl ) p Ô˛

l x-l

Substituting in [2.97] yields:

N ( p) p

M ( p) =

Ú

l1

l0

f (l )(1 – e– pl ) dl



[2.98]

and then a transfer function between the number of fibres in the cross-section and the density of fibre ends can be determined:

M ( p) 1 – F ( p) = N ( p) p

W ( p) =

[2.99]

where F( p) is the Laplace transform of f(l). For example, for an assembly consisting of fibres of a constant length, l = lc, the transfer function is W( p) = (1 – e–plc)/p. In the case of a normal fibre length distribution:

W ( p) =

where T1 =

T1 p – C1 T2 p – T3 p + C2

1

2s l2 2p

2

, T2 =

[2.100]

s l2 l , and C2 = C1/sl are , T3 = l , C1 = 2 4 2s l 2p

constants and l and sl are average and standard deviation of fibre length, respectively. In a similar way, the transfer function can be derived for other cases of fibre length distribution. Knowledge of the transfer function enables the spectral density, Sm(w), and finally the variance, Dm, and the variation coefficient of the number of fibres in the cross-section, CVm, to be determined using the relationships: 2¸ Sm (w ) = Sn (w ) W (iw ) Ô Ô • Ô Dm = Sm (w ) dw ˝ 0 Ô Ô Dm Ô CVm = 100% m ˛

Ú



[2.101]

Fundamental modelling of textile fibrous structures

Here Sm (w ) = 2 p

•

Ú

0

Rm (t )cos wt dt and Sn (w ) = 2 p

Ú

•

0

93

Rn (t )cos wt dt

are spectral densities, Rm and Rn are correlation functions for the number of fibres and the density of fibre ends in the cross-section, respectively. For example, if the occurrence of fibre ends is a random process with the average number of fibre ends m so that the spectral density is a constant Sn(w) = m /plc then the variance is:

Dm = m p lc

Ú

•

0

lc2

sin 2 (w lc /2) dw = m (w lc /2)2

[2.102]

which is the expected result. In a more realistic case, a normal fibre length distribution and a periodical change in the density of fibre ends with amplitude a and frequency wn can be assumed in the form:

n( x ) = n (1 + a cos w n x )

[2.103]

where n is the average number of fibres in the cross-section. The correlation function in this case is:

2 2 Rn (t ) = n a cos w nt 2

which after integration results in the spectral density of fibre ends:

Sn (w ) = n a 2d (w – w n ) 2

where d(w – wn) is the d-function. The spectral density of the number of fibres in the cross-section can be obtained: Sm (w ) = Sn (w ) W (iw )

2

2 w 2T12 + C12 = na d (w – w n ) 2 w 4 T22 – 2w 2 (C2T2 – T32 ) + C22

[2.104]

Substituting in Equation [2.101] and integrating produces the variance:

2 w n2T12 + C12 Dm = na 2 w n4 T22 – 2w n2 (C2T2 – T32 ) + C22

[2.105]

This approach enables the unevenness of linear textile products to be predicted from knowledge of the fibre length distribution and the distribution of the density of fibre ends.

94

Modelling and predicting textile behaviour

2.9.2 Average properties of fibre blend Industrial practice in fibre processing often uses blends of fibres where individual components have different properties; examples include blends of cotton and polyester, wool and acrylic, flax and cotton and many others. The compatibility of fibre components in the blend, the uniformity of mixing and the distribution of the properties of individual components has a significant effect on fibre blend processing and the properties of the final product. Properly selected components can complement each other’s properties thus enhancing the performance characteristics of the product whereas wrongly selected components can cause difficulty in processing and actually worsen the product’s properties. Let us consider a fibre blend of k components where each component is characterised by a set of properties X{x1, x2, …, xn} for which their distributions F{ f1(x1), f2(x2), …, fn(xn)}, average values X {x1, x2, ..., xn} and standard deviations S{s1, s2, …, sn} are known. Let a1, a2, …, ak and b1, b2, …, bk be the proportions of fibres in the blend by the number of fibres and by mass, respectively. The average properties of the blend Xb { x1b, x2 b, ..., xnb} and their standard deviations Sb{s1b, s2b, …, snb} depend on the proportions and properties of constituting fibres. The proportions ai and bi are linked as follows:



bi liTi ai = k b S j j =1 l j T j bi =



[2.106]

a i liTi

[2.107]

k

S a j l jT j



j =1

where lj and T j are average fibre length and linear density, respectively. The overall average value xib and standard deviation sib of a blend property can be calculated: k



xib = S a j xij j =1

s ib =

[2.108]



k

S a j [s ij2 + ( xij – xi )2 ]

j =1



[2.109]

The rules of mixture which were used in the derivation of Equations [2.108] and [2.109] can be applied for the prediction of tensile strength and elastic moduli (Ratnam et al., 1968; Pan and Postle, 1995; Pan, 1996).

Fundamental modelling of textile fibrous structures

95

There is, however, a limitation which must be considered when designing and predicting the blends’ properties. Pan (1996) and, much earlier, Ratnam et al. (1968) showed that in the case of a two-component mixture the strength of a blended yarn can be lower or higher than that of a yarn made entirely of the weaker fibre. Ratnam et al. (1968) in the study of a cotton–viscose mixture derived a simple semiempirical formula to predict the strength of the blended yarn:

sb = [1 – 0.7x(1 – x)][sc(1 – x) + svx]

[2.110]

where sb, sc and sv are the strengths of the blended yarn, cotton and viscose fibre, respectively and x is the proportion of viscose in the blend. It can be seen from the analysis of this formula that if fibres in the blend have the same strength then the minimum strength of the blend sb = 0.825sc = 0.825sv is achieved at x = 0.5. If, for example, cotton is stronger so that sc = 1.4sv then at x = 0.72, sb = 0.955sv. Pan (1996) conducted a more detailed analysis of the case and showed that the proportion of the stronger fibre in the blend must be greater than the critical value bcrit given as:

b crit

s2 E 2 – s E1 = 1 E2 1– E1

[2.111]

where s and E are the strength and tensile elastic moduli of the fibre, respectively and subscripts 1 and 2 refer to the weaker and stronger fibre, respectively. Pan and Postle (1995) based their analysis of the strength of a twocomponent parallel and twisted fibre bundle on a number of simplifying assumptions which neglected the main structural features of staple yarn such as fibre migration, non-uniform distribution of blend components, interaction between the fibres of different types, and effects of twist on strength and strength distribution. The distribution of fibre strength however was assumed to be of a Weibull type, Equation [2.37]. The analysis showed that owing to the interaction with fibres with different properties in tensile deformation there is a considerable difference between the strength predicted by the rules of the mixture and the results obtained on the basis of hybrid effect theory. The average value and the variance of strength of a blended yarn can be calculated:

s y = hq (V1 + V2 Q 2y =

Ef 2 )(l a b )–1/b1 e–1/b1 E f 1 c1 1 1

hq2 Ef 2 2 (V + V2 ) (lc1a1b1)–2/b1 e–1/b1 (1 – e–1/b1 ) a1 N 1 Ef1

[2.112] [2.113]

96

Modelling and predicting textile behaviour

where hq is the fibre orientation efficiency factor, V1 and V2 are volume fractions of fibre type 1 and 2, respectively, a1 is the blend ratio of fibre type 1, N is the total number of fibres in the blend, Ef1 and Ef2 are fibre tensile moduli, lc1 is the critical length of fibre type 1 and a1 and b1 are scale and shape parameters of Weibull distribution of fibre type 1; see Equation [2.37]. It was also confirmed that the strength of a twisted fibre bundle is always greater than that of a parallel bundle reaching its maximum at a certain twist level.

2.9.3 Structural mechanics of fibre assemblies In the approaches to modelling properties of fibre assemblies it is not possible to consider the properties as separate from the way in which the fibres are arranged. Therefore the studies that considered the mechanical properties of fibre assemblies either suggested their own fibre arrangement or accepted an assembly structure developed earlier by other researchers. The modes of deformation, which were the focus of interest, included compression, extension, bending, buckling and torsion of fibre assemblies. One of the first models dealing with the compression behaviour of general fibre assemblies was proposed by van Wyk (1946) where the number of the contact points between the straight cylindrical fibres in a unit mass of a random assembly was derived as:

n=

8m pr 2 D 3V

[2.114]

and the average distance l between neighbouring contact points was estimated to be

l = 2V p DL

[2.115]

where m and V are the mass and the volume of the assembly, the m/V ratio defines the fibre packing density of the assembly, r is the volume density of the fibre, D is the fibre diameter and L is the total length of fibres in the volume. These results cannot be applied to assemblies where fibres are not randomly oriented or not circular in cross-section. The relationship between the pressure, p, and volume, V, of the fibre assembly was found by considering small bending deformation of fibre segments with built-in end points under the action of force applied at the midpoint of the segment:

3Ê 1 – 1ˆ p = KEm 3 Á 3 V03 ˜¯ r ËV

[2.116]

Fundamental modelling of textile fibrous structures

97

where K is a constant and V0 is the extrapolated initial volume at p = 0. This relationship however deviates from the observed values of pressure and volume both at a very low degree of compression (including p = 0) and at high pressures. Since Wyk’s pioneering work the microstructural characteristics of fibre assemblies have received much attention owing to their fundamental importance for the analysis of mechanical behaviour of textiles. Komori and Makishima (1977) used the density function of fibre orientation specified in three-dimensional space in the form W(q, j)sinq and obtained a general formula for the number of contact points per unit length of fibre:

n = 2 DL I V

[2.117]

where I=

p

p

p

Ú Ú Ú Ú 0

dq

0

df

0

dq ¢

p 0

dj ¢ [1 – {cosq cosq ¢ + sinq siinq ¢}2 ]

¥ W(q , j )W(q ¢, j ¢ )sinq sinq ¢

where q and q¢ are polar angles and j and j¢ are azimuth angles of two fibres in contact. This approach was then applied to the cases of crimped fibres, fibres of any length distribution and fibres with an arbitrary cross-sectional shape which showed no significant influence of these factors. The last case, however, did not consider a possibility for the fibres with a multilobal cross-section to develop multiple contact points. The model suggested by Komori and Makishima (1977) was then repeatedly improved and used in the micromechanical analysis of fibre assemblies. Carnaby and Pan (1989) in effect used the same geometry of the fibre assembly as that assumed by van Wyk (1946) and Komori and Makishima (1977) but made a significant improvement by considering fibre slippage and large deformations. This enabled the theoretical prediction of compression– recovery hysteresis to be made for the first time. The predicted Poisson’s ratio exceeding 0.5 indicates that owing to the assumptions made, the volume of fibrous mass increases in compression; this cannot be considered as a reasonable result. In the work that shortly followed, Lee et al. (1990) reexamined the geometry and the mechanics of the fibrous unit cell. The main results were the introduction of multiple and randomly distributed loading points between the ends of the fibre segment and a random distribution of the acting forces at the contact points. It was shown that the number of loading points, k, follows binomial distribution f(k) = k2–k–1 with the mean value of k = 3. The most important outcome of the analysis was the understanding that the

98

Modelling and predicting textile behaviour

collection of simple bending elements accepted as a unit fibrous cell in all previous publications cannot generate the full volume of a randomly oriented fibre assembly. In the analysis of interfibre friction, however, the simple relationship [2.85] was still used and no attention was paid to the possible differential friction effects. Komori and Itoh (1991a and b) continued the analysis of micromechanics by using an energy method but neglecting the boundary condition effects. It was assumed that in compression of fibre mass, the deformation of the unit cell is proportional to that of the whole assembly; the deformation was described in terms of true or logarithmic strain de = dL/L0 or e = ln(L/L0). This led to the formulation of the relationship which links the change in fibre orientation density W (q, j) (Komori and Makishima, 1977) to the change of strain. It was shown that the Poisson’s ratio and stress–strain compression curve depend on the way in which the bending length was calculated, that is, that the length of the fibre bending element depends on its orientaton. Lee and Carnaby (1992) introduced a new energy-based micromechanical approach to the compression of a random fibre assembly. Within this approach, however, a number of model simplifications have been made, some of which are not realistic. One of these assumptions is that the assembly is considered as a collection of independent fibre segments; another is the discontinuity of fibre curvature at contact points which allows free rotation of the segments. The important aspects of this model are (i) simplification of the fibre orientation density function which was reduced to sin q, (ii) introduction of the fibre segment length density function and (iii) discrimination between bending, straightening and slipping of the fibre segments. Pan (1993a) acknowledged an important contribution made by Komori and Makishima (1977) to modelling the fibre assemblies but pointed to problems related to too high a number of contact points being predicted and too small a volume of individual unit cells which, when summed, do not give the total volume of the assembly. In their paper, Komori and Itoh (1994) showed that this was not correct. Pan (1993) also showed that the fibre contact probabilities are not constant but depend on the number of previously formed contacts; this means that the chance of forming any new contact decreases. The formula for the probability of fibre contact obtained by Komori and Makishima (1977):

p=

2 Dlf2 sin c V

[2.118]

was replaced by

pi +1 =

i 2 Dlf2 sin c i Ê 1 ˆ 1– D S Á V l sin c j ˜¯ j =1 Ë f

[2.119]

Fundamental modelling of textile fibrous structures

99

where pi+1 is the probability of the fibre creating the contact point i+1 and ci is the angle between two fibres at the contact point i. Using this, the number of contact points per unit length of fibre was estimated as: nl =



8 sVf I lf (p + 4Vf Y )

[2.120]

where s = lf/D is the fibre aspect ratio, Vf is the fibre volume fraction; Y=

Ú

p

0

dq

Ú

p

0

df J (q , f )K (q , f )W(q , f ) sin q , K(q, f) is the mean value

of 1/sin ci and J(q, f) is the average value of sin ci. These theoretical results were then applied to the cases of 3D and 2D random assemblies and a twisted assembly. However, Komori and Itoh (1994) made further improvements to the theory and refined some problems related to the derivation of Equation [2.120]. Two new concepts of forbidden length and forbidden volume were introduced that take account of the fact that fibres in contact occupy a finite volume that prevents the formation of other contacts on the same fibre and contacts between contacting fibres within this volume. The equations for the mean total forbidden length, l*(o), and mean total forbidden volume, V*(oA), were obtained under the assumption that the third fibre is oriented perpendicularly to two fibres which are already in contact as follows:

l * (o) = 2ND

p(o, o¢ )

Ú sin c (o, o') W(o¢)dw ¢ Ú

[2.121]

Ú

V * (oA ) = 1 N 2 v* (oA, o)W(o) ¥ ÈÍ p(o, o¢ )W(o, o¢ ) dw ¢˘˙ dw [2.122] 2 Î ˚

where N is the number of contacting elements, D is the fibre diameter, p(o, o¢) is the probability that two fibres oriented in directions o and o¢ make a contact, dw and dw¢ are infinitesimal solid angles around the directions o and o¢, respectively, v*(oA, o) is the forbidden volume related to two fibres in contact and subscript A denotes the third fibre in contact. These new concepts were then applied to the cases of 3D and 2D isotropic fibre assemblies. Komori and Itoh (1994) made an important comment on the applicability of fibre contact theories by warning that the formulae obtained are valid only in the sense of the statistical average taken over all possible cases of locating fibres at arbitrary allowed positions in a volume but not for the regularly oriented fibre mass. Experimental verification of the theories of fibre contact is difficult and has not yet been established.

100

Modelling and predicting textile behaviour

In the study of yarn compression, Grishanov et al. (1997) introduced the important concept of virtual locations into which the yarn cross-section, or the cross-section of any fibre assembly, can be divided. This approach enabled a realistic modelling of fibre distribution in both uncompressed and compressed yarns to be carried out. In the paper that followed (Harwood et al., 1997) a new yarn compression model was generated that removed some of van Wyk’s assumptions relating to the randomness of fibre orientation and unlimited fibre deformability. The relationship between the force applied per unit length of a yarn, Q, and relative yarn compression, h1, was obtained in the form:

Q(h1) = k

Ú

h1

1

dx = a[G (h1, h1min ) – G (1, h1min )] [2.123] x( x – h1min )3

where k and a are empirical constants, h1 = d1/d0 is the minimum relative yarn diameter in compression which corresponds to the maximum fibre packing density, d0 and d1 are initial and compressed yarn diameters, respectively and

G(h1, h1min ) =

2h1 – 3h1min h – h1min . + ln 1 h1 2h12min (h1 – h1min )2

This equation showed a better fit to the experimentally measured yarn compression than the van Wyk’s theory. The same concept of virtual locations was later used for modelling the fibre migration in staple-fibre yarn (Grishanov et al., 1999) and for 3D modelling of fibre assemblies (Sreprateep and Bohez, 2006); this concept was further developed in modelling the yarn structure by Siewe et al. (2009). An attempt to model the structure and to predict compression behaviour of a fibre assembly from the properties of individual fibres was recently undertaken by Beil and Roberts (2002a, 2002b). The model assumes a randomly oriented assembly of helical inextensible fibres with circular cross-section. The model considers the movement of fibres under the action of an external compression force and internal force in the fibre caused by bending and torsion. This leads to the fibres changing their orientation, creating new contacts and fibre slippage where the difference between the static and kinetic friction is taken into account. The model vector equations include conservation of linear momentum:   ∂F + f + mg = m ∂2 x [2.124] ∂s ∂t 2 conservation of angular momentum:   ∂M + t ¥ F + m =0 ∂s

[2.125]

Fundamental modelling of textile fibrous structures

a linear relation between the moments and the curvatures:     M = Gp + G ¢q + Ht

101

[2.126]

and the condition of inextensibility: ∂x = t [2.127] ∂s   F and M where are the internal force and the moment in the fibre, respectively,   f and m are the external applied force and the moment per unit fibre length,  respectively, m is the mass per unit length of the fibre, g is the gravitational acceleration, s is the curvilinear coordinate measured along the fibre central  line, t is time, x is the position of a point on the fibre, G, G¢ and   H are components of bending and torsion moments, respectively and p, q and t are local orthogonal unit vectors associated with the normal, bi-normal and tangent vectors, respectively. The numerical solution of these equations was found for a small representative part of the assembly about 2 mm long containing 35 and 50 fibres at initial fibre volume fractions of 0.4% and 0.8%. The results generally agreed with the linear van Wyk (1946) relationship between the number of contacts and the inverse volume but showed a considerably greater rate of increase. The model overpredicted the assembly stiffness but its decrease with the increase in the number of fibres agreed with the theoretical estimate. A significant but gradually diminishing loss of energy owing to fibre friction was noticed in modelling the compression hysteresis; this was in agreement with the experimental results by Dunlop (1974). As should be expected, the absorption of energy was greater for the assemblies where fibres were highly crimped. The effect of compression on fibre orientation was significantly lower than could be expected. Munro et al. (1997a) applied a finite-element analysis to a 2D unit cell of aligned fibres and then extended their approach to the 3D case (Munro et al., 1997b). The fibres in the unit cell were assumed to be linear elastic cylindrical bodies which may have a degree of crimp but the viscoelastic properties were neglected. The 2D square unit cell was assumed to have 12 degrees of freedom of which (i) three represented rigid body translation and rotation with no energy contribution to the system; (ii) three related to extension, three to bending and two to compression degrees each associated with the corresponding energy contribution and (iii) one was considered to be enough to describe the shear deformation. In a similar way, the 3D cubic unit cell had 48 degrees of freedom which represented the same deformations as those above and in addition eight degrees of freedom were introduced which were responsible for fibre jamming and a further eight related to torsion. The model was applied to modelling of yarn extension, compression, bending and twisting. The results showed that the numerical

102

Modelling and predicting textile behaviour

model provides a good approximation to the theoretical solution and agrees qualitatively with the expected behaviour of the real yarn. The advantage of the approach taken was that the non-linear material properties, the complex initial shape of individual fibres and fibre interactions can be included and modified independently of other model parameters.

2.10

Conclusions

The theoretical and experimental investigations of structure and properties of individual fibres and fibre assemblies employ a wide range of well-developed mechanical, mathematical and numerical methods. There is, however, a tendency to apply these methods to narrowly specified problems which often suffer from a lack of lateral thinking and thus do not provide a comprehensive consideration of the processes and properties involved. There is a general understanding among scientists working in the area of modelling fibre assemblies that their properties to a large extent depend on their structure. The structure of any fibre-based material is a direct result of a process or a sequence of processes that join fibres together in a continuous product. Nevertheless there have been very few attempts, mainly in the area of nonwoven materials, to generate a model that would be able to simulate the resultant structure from the process parameters and properties of constituent fibres. In the absolute majority of cases, the models of fibrous structures were not generic but based on experimental data concerning specific structures. This resulted in regression equations linking average structural characteristics to basic process parameters which inevitably were different even for products of similar nature, such as different types of yarns. To date there is no model that would be able to link, for example, the fibre diameter, length and strength on one hand with the yarn structure resulting from specific processing parameters and spinning methods on the other and then to explain the mechanical behaviour of this structure in various modes of deformation. In this respect it seems to be more appropriate to talk about process–structure–properties relationships rather than structure-properties relationships. It may be argued that the generation of mathematical and computational models at this level of detail is hardly needed since all the necessary information about the mechanics of textile products, such as yarns, can be obtained from simple experiments. However, lack of knowledge on the major relationships governing the process–structure–properties relationships hampers progress in fibre processing technology which is one of the oldest sections of the textile industry. Because of this it is not really possible to apply a fully engineering approach to the design of fibrous assemblies which can provide an optimal combination of fibre alignment and fibre entanglement in order to achieve maximum strength.

Fundamental modelling of textile fibrous structures

103

There are still many unresolved problems in modelling the fibrous structures and it would be appropriate to point to a few specific problems. One relates to difficulties with modelling fibrous assemblies based on blends of fibres which are only partly compatible in properties. This problem originated from the recent trend to provide new sustainable sources of fibres by using flax and other bast fibres as a partial substitute for cotton and also in blends with wool and synthetic fibres. Differences in mechanical and geometrical properties of flax in comparison with wool and cotton result in preferential outward migration of flax fibres in the yarn structure (Truevtsev et al., 1995). This creates different conditions of interfibre cohesion and friction at the yarn core and at the yarn outer layers, thus making it difficult to model and predict yarn properties. Another fundamental problem is associated with the investigation of structure–properties relationships of fibrous assemblies in general. The physical and geometrical properties of fibres and fibrous assemblies can be characterised by the numerical values of many parameters. It is therefore possible, at least in principle, to find functional dependence between these parameters. In fact nearly all research in textile modelling has been aimed at generating these types of relationship which can be called property–property relationships. In strict mathematical terms, functional dependence between two variables implies that for each value of one of them, it is possible to obtain the value of another by following the rules that define the function. From this point of view the state of the art in the study of structure–properties relationships has not yet reached the point where it would be possible to suggest a model which can provide a functional link between structure and properties. This is because up until now it has not been possible to characterise the structure of fibrous assemblies by a universal numerical parameter. This means that until such a parameter is generated it is not possible to formulate a functional relationship between structure and properties. New approaches to this problem applied to regular textile structures have been recently established in several general (Grishanov et al., 2009a, 2009b) and more specific papers (Grishanov et al., 2007; Grishanov et al., 2009c; Morton and Grishanov, 2009). These approaches are based on the application of the topological theory of knots and links to the textile structures defined by their minimum repeat or unit cell. A similar approach is being developed in application to general fibrous assemblies.

2.11

References

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mechanics’, in Mechanics of Flexible Fibre Assemblies, Hearle J W S, Thwaites J J and Amirbayat J (eds), NATO Advanced Study Institute Series: Applied Sciences. Series E; No. 38, Sijthoff & Noordhoff, Alphen aan den Rijn, 99–112. Carnaby G A and Pan N (1989), ‘Theory of compression hysteresis of fibrous assemblies’, Text. Res. J., 59(5), 275–84. Chapman B M (1969), ‘A mechanical model for wool and other keratin fibres’, Text. Res. J., 39(12), 1102–9. Chapman B M (1971), ‘An apparatus for measuring bending and torsional stress–strain–time relations of single fibres’, Text. Res. J., 41(8), 705–7. Chapman B M (1975), ‘The importance of inter-fibre friction in wrinkling’, Text. Res. J., 45(12), 825–9. Choi K F and Kim K L (2004), ‘Fiber segment length distribution on the yarn surface in relation to yarn abrasion resistance’, Text. Res. J., 74(7) 603–6. Cook G (1984), Handbook of textile fibers, II Manmade Fibres, Merrow, Shildon. Cook J R and Fleischfresser B E (1990), ‘Crimping of Wool Fibres’, Text. Res. J., 60(2), 77–85. Cui S-Z and Wang S-Y (1999), ‘Nonlinear creep characterization of textile fabrics’, Text. Res. J., 69(12), 931–4. Denton M J and Daniels P N (2002), Textile Terms and Definitions, Textile Institute, Manchester. Duan Y, Keefe M, Bogetti T A and Cheeseman B A (2005), ‘Modeling friction effects on the ballistic impact behaviour of a single-ply high-strength fabric’, Internat. J. Impact Eng., 31(8), 996–1012. Dunlop J I (1974), ‘Characterising the compression characteristics of fibre masses’, J. Text. Inst., 65, 532–6. Eichhorn S J and Davies G R (2006), ‘Modelling the crystalline deformation of native and regenerated cellulose’, Cellulose, 13(3), 291–307. Eichhorn S J, Sirichaisit J and Young R J (2001a), ‘Deformation mechanisms in cellulose fibres, paper and wood’, J. Mater. Sci., 36, 3129–35. Eichhorn S J, Young R J and Yeh W-Y (2001b), ‘Deformation processes in regenerated cellulose fibers’, Text. Res. J., 71, 121–29. El-Sheikh A, Bogdan J F and Gupta R K (1971), ‘The mechanics of bicomponent fibres. Part I: Theoretical analysis’, Text. Res. J., 41(4), 281–97. Falconer K (1990), Fractal geometry. Mathematical foundations and applications, John Wiley & Sons, Chichester. Ferry J D (1980), Viscoelastic properties of polymers, John Wiley & Sons, New York. Feughelman M. (1996), Mechanical properties and structure of alpha-keratin fibres, New South Wales University Press, Kensington, NSW. Ghosh A, Ishtiaque S M, and Rengasamy R S (2005), ‘Analysis of spun yarn failure. Part I: Tensile failure of yarns as a function of structure and testing parameters’, Text. Res. J., 75(10), 731–40. Glass M (2000), ‘A technique for generating random fibre diameter profiles using a constrained random walk’, Text. Res. J., 70(8), 744–748. Grishanov S A (2008), Modelling the Structure Dependent Colour Properties of Melange Yarns. Final report, EPSRC grants GR/S77325/01 and GR/S77318/01, Leicester, De Montfort University. Grishanov S A and Harwood R J (1999), ‘The development of 3D models for yarn CAD system’, Proceedings of World Congress: Textiles in the Millennium, University of Huddersfield, 6–7 July, 1999, UK.

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3

Yarn modelling R. Ognjanovic, Innoval Technology Limited, UK

Abstract: This chapter will explain the importance of yarn modelling, how yarns are modelled and give examples of how the models have been applied. The construction of a yarn will be described in detail, pointing out the difficulties of yarn modelling, before reviewing a number of modelling approaches that have been developed over the last 100 years. Practical examples of modelling a yarn and how it can be incorporated into a fabric model for soft body armour applications and composite lay-up will be given along with a prediction of future modelling trends and where to obtain yarn modelling advice. Key words: yarn modelling, finite element analysis (FEA), carbon fibre composites, soft body armour, textiles.

3.1

Introduction

The modelling of yarns is very important for a number of reasons. Models can be used to increase our understanding of what controls the properties of yarns and how the yarn properties can be controlled. In an industrial setting, new types of yarns can be developed more quickly using models by adjusting the design of a yarn already in production or designing a completely new yarn that would otherwise not have been thought of without the help of modelling. The risks, time and costs of designing a yarn can be reduced by modelling the yarns before the yarn is made. Equally important, modelling of yarns will enable yarns to be manufactured to satisfy the property requirements of yarns. Yarns are constituent of many products, such as threads, rope, nets, cables, textiles and composites reinforced with yarns or textiles. An effective yarn model has to be able to predict the mechanical characteristics of the yarn with accuracy. Depending on the application, some mechanical characteristics will be more important than others. They include properties like the ultimate breaking strength of the yarn, the elongation of the yarn as a function of tensile and compressive axial load applied to it, the bending force as a function of bending angle and the spread of the yarn as a lateral force is applied to the sides of the yarn. All of these will depend on the construction of the yarn and the materials it is made of. The property that has received the greatest attention is yarn tenacity as it has the greatest influence on winding, warping and weaving machines. Tenacity is the yarn breaking strength divided by its yarn count. The yarn count (dtex) is the weight in grams of 10 km of yarn. 112

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3.2

113

Yarn construction

A yarn is an assembly of fibres. The fibres come in a variety of lengths, cross-section shapes and straightness (some fibres are curled or crimped whilst others are straight). A collection of fibres may well have a distribution of lengths and cross-section area. The cross-section shape may also vary along the fibre length both in terms of magnitude and orientation. For example, a fibre with an oval shaped cross-section may have a periodic variation in thickness along its length and the orientation of the shape may change down the length of the fibre. If the cross-section area is important, it has to be modelled as a distribution function that describes the range of crosssection areas. Similarly, the other fibre parameters should be modelled as distribution functions if the parameters vary significantly. The fibre can be a long filament fibre or a short-length staple fibre. All man-made fibres are manufactured as continuous filament fibres that can be cut or stretch broken into shorter lengths to give staple fibres. The fibres can be packed into a yarn in a number of different geometries. The simplest yarn is a single yarn that has fibres that are twisted together as shown schematically in Fig. 3.1. More complicated yarn geometries include plied yarns where two or more single yarns are twisted together in Fig. 3.2. Figure 3.3 shows a cabled yarn where two or more plied yarns are twisted together. Both plied yarns and cabled yarns may have yarns that are made from different materials. The direction and amount of twist has a direct influence on the yarn properties. Generally, as the amount of twist increases in staple yarns, the yarn stiffness increases to a maximum, then falls off. The path a single fibre takes in a twisted yarn may not be perfectly helical with a constant radius from the yarn axis. Indeed, it may have a range of helical radii as it migrates through the yarn. Fibre migration is important in staple yarns as Morton (1956) explained because fibres at the yarn surface are not held if they follow perfectly helical paths. Texturizing yarns and fibres involves modifying the appearance of the fibre by adding crimp, loops, coils or crinkles to the fibre. This can increase the bulk of the yarn and change its mechanical properties. The most common texturizing process is the false twist process where a twist is imparted to a yarn then the yarn is heated. When the yarn is cooled and relaxed the fibres revert to their twisted state. The next most popular texturizing process is air-jet texturizing. Air jets are used to distort and mix the fibres as several feed stock yarns are fed into the air jets. Feed stock yarns are untexturized parallel continuous fibre yarns. Other texturizing processes include stuffer box texturizing where fibres are stuffed into a heated box to create crimped fibres, knife edge texturizing that produces a spiral crimp to the fibre as fibre is run across a hot knife edge, knit-deknit texturizing and twist-heatsetuntwist.

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3.1 Schematic of a two-fibre single yarn.

Clearly, the range of possible yarn constructions is vast because of the many possible permutations of fibre type, the different ways of placing fibres next to each other in a yarn and different methods of texturizing. This creates a problem in modelling yarns as it is difficult to have a model that works well across all yarn types. Considerable simplifications to the models have to be made to make the model usable.

3.3

Types of models to predict the structure and properties of yarns

The different modelling approaches used to predict yarn properties can be grouped into continuum models, discrete fibre models, mechanistic, statistical, neural network, fuzzy logic, expert systems and case-based reasoning

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Single yarn Fibre

Plied yarn

3.2 Schematic of a plied yarn containing two single yarns. Each single yarn has two fibres in it. Single yarn

Fibres

Plied yarn

Cable yarn

3.3 Schematic of a cable yarn with two plied yarns. The plied yarns have two single yarns in them and the single yarns contain two fibres.

models. Some models may employ more than one approach. The very first yarn model seems to be one by Gegauff (1907). Gegauff derived a simple mathematical relationship between twist angle and yarn strength. Over the years, the modelling approach has shifted from an empirical approach early in

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the 20th century, to the employment of applied mechanics of fibre assemblies in the second half of the 20th century (Hearle 2004). In the 1920s, Peirce (1930) began characterizing the properties and geometry of textiles, but there was very little applied mechanics analysis until the 1940s. The modelling challenges today are to model fibre assemblies with millions of fibres, include non-linear viscoelastic properties to describe fibres and model anisotropic structures undergoing large deformations and strains.

3.3.1 The continuum model The continuum model assumes that the material making up the yarn can be treated as a continuum. A continuum is a volume that can be continually subdivided into smaller elements and those elements will have the same properties as the bulk material. This type of model cannot describe the discontinuities and variations in materials at the molecular and atomic level. However, well established mechanical equations can be applied to model the continuum in order to calculate stresses and strains, even in cases involving large and complex deformations of the yarn. The first yarn continuum model was created by Hearle et al. (1961). In their model the fibres were assumed to be elastic, following perfectly helical paths in the yarn and were evenly distributed across the yarn cross-section. The model simulated yarn extension to the point of breakage of continuous filament yarns using small strain theory. The stresses were uniformly distributed through the yarn cross-section and the stresses at any point were assumed to be constant in all directions at right angles to the fibre axis. Some of these assumptions were readily admitted by Hearle et al. not to be valid. For example, small strain theory is not valid for large extensions of the yarn and this is where many theories showed deviations from experimental results. The relative modulus (RM) is the yarn modulus divided by the fibre modulus and it was predicted by Hearle et al. to be RM =

c2 ¥ (1 + s 1 )(1 + c 2 )

Ï È 4 + 3s 1 1 + s 1 2 2s 12 + 2s 1 – 1 2 (1 + s1 ) ˘ + log e c˙ Ì(1 + s y ) Í 2(1 + s ) – 2s c – 2s (1 + s ) c 1 1 1 1 Î ˚ ÓÔ

s y È ( 3s 12 + 2s 1 – 1) 2(1 + s 1 ) 1 – 2s 12 + 2s 1 -1 c 2s1 ˘¸Ô log – c e Í ˙˝ 2 Î s1 s 12 s 12 c2 ˚Ô˛ [3.1] where c is the cosine of the surface angle of twist, s1 the axial Poisson’s

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ratio and sy the yarn lateral contraction ratio. Hearle et al. observed that the latest theories in 1961 were little improvement on the simplest expression [3.2] given by Gegauff (1907) for large yarn extensions

Yb = cos2 a Xf,b

[3.2]

where Yb is the yarn tenacity, Xf,b the fibre tenacity and a the surface angle of twist. White et al. (1976) and Huang and Funk (1975) improved the model of Hearle et al. by modelling shear forces on the yarn and by allowing the stress acting normal to the fibre surfaces to vary. Although Huang and Funk’s model applied to small extensions of the yarn, it did allow for the contraction of the yarn to be non-uniformly distributed across the yarn diameter. The theory of elastic curved rods was used in the analysis and the possibility of partial separation of fibres was considered. Huang and Funk (1975) predicted that the applied axial force ( F ) on the yarn when it is extended by a certain amount was:

F = 2p (1 – g )

Ú

R 0

ÈÎFl cos q + Fy sin q )cos q + (T cos q – Qsin q )sin q ˘˚ rdr



[3.3]

It is not intended to explain each of the terms in this expression as some of them rapidly expand into very complex expressions themselves. This equation is shown here simply to illustrate the complexity of the model of Huang and Funk. The normalized relative modulus versus the twist angle relationship is shown in Fig. 3.4 for three models, Gegauff (1907), Hearle et al. (1961) and Huang and Funk (1975), along with some experimental data for comparison. The data were chosen to represent the boundary of a larger set of data presented in Huang and Funk (1975). It can be seen that all three models lie within the range of experimental data and that the three models produce similar predictions. However, the complicated equations in two of the models contrast sharply with the simplicity of Gegauff’s. Thwaites (1980) created a continuum model to predict tensile and torsional properties of yarn. The yarn tension (T) and axial torque (Q) are given by:

T = (a11e + a12bR cot a)pR2 E 3

[3.4]

Q = (a21e + a22bR cot a)pR E cot a

[3.5]



a11 = 1 – 3f1(a) – 9f2(a)

[3.6]



a12 = a21 = f1(a) + 6f2(a)

[3.7]

where

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Modelling and predicting textile behaviour 1.2

Relative modulus

1

0.8

0.6 Gegauff (1907) Hearle et al. (1961) Huang and Funk (1975) Experiment

0.4

0.2 0

10

20 30 Twist angle

40

50

3.4 Comparison of three models and experimental measurements. The relative modulus was normalised at 10° of twist. Adapted from Huang and Funk (1975).



a22 = (m/4) tan2 a – 4f2(a)

[3.8]



f1(a) = {1 – (3m/2)}(1 – 2 cot2 a ln sec a)

[3.9]



ˆÊ 1 ˆ Ê 1 f2 (a ) = Á – m˜ Á sin 2 a – 1 + 2cot 2 a ln sec a ˜ u 1 + 2 ¯Ë ¯ Ë

[3.10]



m = 2G E

[3.11]

where e is the axial strain, b is the axial twist per unit length, R is the yarn radius, E is Young’s modulus along the fibre, a is the surface helix angle, G is the fibre shear modulus and u is the Poisson ratio. His model predictions agreed with experimental measurements of yarn tension for a high twist yarn when shear stresses were included in his model. As the yarn twist decreased, the experimental values tended towards predictions using a zero shear modulus. In all cases, the experimental values lay between the two theoretical values of shear modulus. Fibre slippage effects were found to be small as measured by hysteresis loops during the modulus tests. Singh and Sengupta (1977) found that yarn strength was reduced when fibre slippage was present. Van Luijk (1981) proposed a new type of finite element analysis (FEA) model to model fibre migration within yarns but the model was not a general one that could be easily applied to other yarns. Djaja (1989) created an element that was more conventional than the van Luijk model but it was limited in generality because it had to be used in a helically wound yarn. Carnaby

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119

and Curiskis (1987) and Lee et al. (1990) have studied the micromechanics of yarn to derive constitutive equations that are used in FEA models to describe the mechanical properties of yarn. Many models were limited to small deformations of yarn but the most interesting aspects of yarn properties occur at high deformations. Difficulties also arise in modelling yarns using FEA models because the stiffness of the yarn is generally far greater in the fibre direction than the transverse direction (Hadley et al., 1965). Jeong and Kang (2001) developed a FEA model to simulate the compressibility of yarn in the lateral direction. Yarn compression has been studied by Hadley et al. (1965), Morris (1968), Pheoenix and Skelton (1974) and Pinnock et al. (1966). The compressibility of yarn goes through a number of regimes that describe how the load is taken up by the yarn. Fibres that protrude from the yarn surface will resist the load first. Any irregularities in the yarn surface that stand out from the yarn will be deformed next inducing slippage between fibres in the yarn. As deformation increases, the yarn flattens and the fibres jam against each other. When jamming occurs the compressibility suddenly increases. The yarn surface geometry formed by the twist was shown by Kawabata et al. (1978) to influence compressibility. In Jeong and Kang’s model, a yarn was represented as a smooth solid cylinder with elastic isotropic material properties (the modulus was the same in all directions). The transverse modulus of the yarn was used since most of the compressive deformation was in the transverse direction. The transverse modulus was measured experimentally and used as an input to the material model to describe the yarn. No attempt was made to derive the transverse modulus from the fibre properties and fibre packing arrangements arising from different spinning technologies, as it was assumed this would all be captured in the experimentally determined material constants. Despite the simplifying assumptions in the yarn model, it was claimed that very good agreement with experimental data was achieved when the yarns were used in a fabric unit cell to predict the compressive deformation of four woven fabrics. Fabric compressibility is one of the most important properties that determine the way the fabric feels (Kawabata, 1980; McPhee et al., 1985; Postle et al., 1988) and is commonly described as its ‘handle’ or ‘fabric hand’. Jeong and Kang’s model has clearly shown that the yarn transverse modulus is very important in fabric compressibility predictions but it is worth noting that the transverse modulus may not be the most important parameter if, for example, fabric tension was being modelled. Van Langenhove (1997a, 1997b, 1997c) established an elegant FEA model of a yarn to predict stress–strain and torque–tensile strain curves. The model can use data that is available in the literature and the data may be in an incomplete form. In contrast to some models in the literature, the model takes into account the fibre length, and that the fibres do not have to have identical properties, they do not have to be perfectly elastic, the fibre

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path in the yarn is not limited to a helix, fibre migration is possible, large extensions can be modelled and all stresses are considered. Yarn structure was described by three parameters; twist, migration and density. The easily accessible nature of the model means that parameters can be input as distributions and functions. The density, for example, can be a constant or expressed as a function of radial coordinate or fibre strain. The fibres were allowed to slip at their ends if the tensile stress exceeded the contact forces between fibres. Figure 3.5 shows the stress in a staple fibre in a schematic form. The stress is constant over much of the fibre length but is reduced near the ends. Slippage occurs over a critical length (lc) which is dependent on stress levels in the fibre, friction coefficients, the contact surface area, fibre orientation and crimp. lc is given by:

lc =

e11EA ws 22 P

[3.12]

where e11 is the axial strain, E is the elastic modulus, A is the fibre crosssectional area, P is the fibre perimeter, w is the fibre–fibre friction coefficient and s22 is the fibre transverse stress. The value EA/(wP) is a constant for a fibre and Langenhove referred to it as a slippage constant Cs. A consequence of slippage at the fibre ends is that the average stress in the fibre length is smaller than without slippage. This will affect the fibre breakage mechanism in this model because as a fibre breaks it will not contribute to the yarn strength in the section of yarn length containing the break and in the yarn section near the break it will behave like a short fibre. To keep the model simple it was assumed the fibre broke into two equal lengths. The relative

Stress

lc

0

lc

Fibre length

3.5 Stresses in the fibre as a function of fibre length. The critical lengths, lc, are marked at both ends of the fibre.

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121

force (Fr), which is the ratio of the average force over a fibre to the force over the same fibre when there is no slippage, is given by: È s2 + l 2 Ê – t 2 ˆ È ( 2 l – l ) sl ˘ l ˘ l Fr = expÁ c ˜ Í c –l + c˙ – c +1 + f (t c )Í l ˙ 2 l l 4 l˚ l Ë ¯ Î 2p 4 lc l ˚ c Î [3.13] where l is the average fibre length, sl is the standard deviation of the fibre length, f(tc) is the cumulative normal distribution function and 2l – l tc = c . sl The stress in a yarn under tension was found by transforming the fibre stresses from a fibre coordinate system to a yarn coordinated system using a helical twist angle and a fibre migration angle. The migration angle assumes the fibre moves from the yarn core to the surface and back again. A considerable amount of work is needed to measure the yarn density function over the yarn cross-section, twist angles, migration angles, fibre length distribution, fibre strength and fibre friction. This was done by van Langenhove (1997c) and the yarn tenacity predictions agreed with the experimental measurements to within the statistical confidence limits. A limitation of the model was the breakage mechanism because the model cannot simulate sudden rupture of a yarn. The model can calculate the stress–strain curves of a yarn with excellent agreement up to the point of rupture but the stress does not go to zero after rupture. Komori (2001) proposed a constitutive theory to predict the stress–strain behaviour of a twisted continuous filament yarn from the structural parameters and moduli. The constitutive equations can be incorporated in a FEA model and are valid for large deformations of the yarn, right up to the start of breakage of fibres in the yarn. This was done by using an incremental formulation together with the differential moduli of the system. The stress increase (ds), for example, is given by:

d s = m * d e

[3.14]

where m is the differential modulus of a fibre bundle and de is the strain increment. The radial distribution of stresses and strains in the yarn can also be predicted and the model is free from a critical problem that earlier theories suffered from – an approximation that is valid only for an infinitesimally small deformation (strains less than approximately 1%) is assumed for a finite elongation where strains may be much greater than 1%. According to Komori, these models include those by Hearle (1958), Hearle at al. (1959), Hearle and Konopasek (1975), Hearle and Sakai (1979), Holdaway (1964), Kilby (1964), Onder and Baser (1996), Platt (1950), Riding and Wilson (1965), Treloar (1965), Treloar and Riding (1963) and van Luijk et al.

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(1985a, 1985b). Komori’s model assumed that the local structure of the yarn is uniform along its length but may have a radial distribution for the mass and fibre direction. The yarn behaves as an elastic material so that slippage between fibres is neglected. Although no comparison with experimental results was given by Komori, a subsequent paper of his will develop the model further and present practical examples.

3.3.2 Discrete fibre models Discrete fibre models are models that treat each fibre individually and sum up the effect of many fibres in a yarn structure to predict the yarn performance. FEA can be used to model individual fibres in a yarn. As computing power increases and FEA software improves, the size and complexity of FEA models that can be run also increases. In recent years, some very realistic simulations of yarns have been done. Morris et al. (1999) noted that one of the main reasons for the disappointing performance of most models is the large number of simplifying assumptions in the models. The most common assumption is that the yarn is treated as an idealized structure where the fibre path is not random. Morris et al. produced a model to create a yarn structure that included migration and fibre length variations. The main aim of the work was to generate a good yarn structure and then demonstrate how a fairly basic stress analysis can be used to calculate the mechanical properties of the yarn. A distribution function was used to describe the probability p(r) of a fibre occupying a small volume at a distance r from the yarn centre: b



Ï Ê r ˆ¸ ÔÔ exp(1) – expÁË rmax ˜¯ ÔÔ p(r ) = (1 – 2e ) Ì ˝ +e exp(1) – 1 Ô Ô ÓÔ ˛Ô

[3.15]

where rmax is the yarn radius and e and b are distribution parameters. This function defined the yarn cross-sections at evenly spaced positions down the length of a yarn. Fibre migration was modelled by allowing a fibre to move into unoccupied space between consecutive cross-sections. A random number generator was used to determine which vacant position would become occupied and this process was tuned to produce migration paths similar to experimental results. Periodic migration from the yarn centre to the surface and back again was modelled along with random migration. Both types of migration were superimposed onto the yarn twist. A natural consequence of this model was that fibre ends may terminate near the surface of the yarn and sometimes produce a hairy yarn. Hairiness

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123

in the model was measured by counting all the fibres where the final 3 mm of either end was near the yarn surface. The hairiness predictions were difficult to compare with experimental measurements because the model counts the number of hairs per unit length of yarn, whereas the Uster hairiness tester measures the total length of hairs per unit length of yarn. The model was able to predict the breaking strength of Tencel yarn to within 5% of the experimental results. It is possible to improve the model by including more details of the process used to make the yarn. However, the model does rely heavily on experimental results to help tune various modelling parameters and to produce realistic yarn structures. Sreprateep and Bohez (2006) have designed a sophisticated algorithm that generates a yarn with many individual fibres in a computer aided design (CAD) package (SolidWorks). The CAD model is read into a FEA package (COSMOSWorks) as a starting configuration of the yarn that can be bent, tensioned and compressed to determine its mechanical properties by computer simulation. The algorithm is capable of constructing a yarn from a distribution of fibre diameters and a radial distribution function from Morris et al. (1999) that gives the probability of a fibre being at a given position from the yarn axis. Other inputs to the model include twist angle, fibre migration parameters and material properties. The simulations were of continuous filament nylon yarns with 33 filaments but the method can be extended to staple yarns. Although the predictions were not compared with any experimental yarn data, the model was useful in showing the stress distribution in the yarn and images of the yarn after it was deformed. The model does not take into account the stressed nature of fibres in the starting configuration of the yarn. During yarn production, the fibres are twisted into a yarn and this deformation produces stresses in the fibres. The stresses can be significant in a high twist yarn. Ognjanovic has produced a yarn model that has a realistic starting structure for the yarn and includes stresses in the fibres caused by twisting in the yarn formation process. It works on three different length scales, the fibre, yarn and fabric. It uses ADINA FEA software (www.adina.com), a commercially available, general purpose FEA code, to predict the mechanical behaviour of fabric from basic material properties and geometries. The simulations are accurate, relatively fast and can be applied to a variety of fabric weaves, yarn constructions and fibre types. A single fibre which may have a non-circular cross-section is modelled first, as shown in Fig. 3.6. The mechanical response of the fibre is simulated by flexing, twisting and stretching it using the computer model and all the modelled mechanical response data are then used to develop a model of a ‘simple fibre’ with a circular cross-section shown in Fig. 3.7. This simple fibre has the same mechanical characteristics as the original fibre model but has fewer FEA elements in it to make it computationally faster.

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3.6 A single fibre with a non-circular cross-section.

3.7 A ‘simple fibre’ with a circular cross-section.

In the next stage of modelling, a yarn is modelled from a collection of simple fibres that are twisted together as shown in Fig. 3.8. The mechanical response of the yarn is predicted from FEA and a cylindrical version of the yarn, like the one in Fig. 3.9, is built with exactly the same mechanical characteristics as the collection of individual simple fibres making up the yarn. The number of elements in the ‘simple yarn’ model is considerably fewer than the full yarn model, thereby speeding up the computation of the model. In the third and final stage of the modelling, four simple yarns are used to build the 1:1 weave fabric unit cell shown in Fig. 3.10. More complicated

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125

3.8 Approximately 200 ‘simple fibres’ twisted to form a yarn.

weaves and more yarns can be modelled if needed. The mechanical characteristics of the fabric unit cell are predicted from FEA and a single element representation of the fabric unit cell in Fig. 3.11 is built which behaves in exactly the same way as the unit cell made up of simple yarns. Once a fabric model has been built, large areas of fabric can be simulated using very few elements which make the simulations run quickly. Simulations such as soft body armour simulations involving 35 layers of fabric in Fig. 3.12 have been done. The fabric models incorporate all the modelling assumptions used to develop the fabric model; these assumptions include the fibre diameter, the fibre material including plasticity, the fibre crosssection, fibre–fibre friction, yarn twist, number of fibres in the yarn, fibre length distribution and the fabric weave pattern. The models are accurate as a result of incorporating these assumptions and large deformations can be modelled. The whole modelling exercise would have to be repeated if the fibre diameter was changed because the yarn model and the fabric model will need the new fibre characteristics. However, if the fabric weave is changed,

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Modelling and predicting textile behaviour

3.9 Cylindrical version of the collection of fibres in Fig. 3.8 which represents a ‘simple yarn’.

3.10 A 1:1 weave using four ‘simple yarns’.

the simple yarns can still be used as they are simply woven into a different pattern.

3.3.3 Statistical models Regression analysis has been used to model yarn properties either as the only model or in conjunction with other models. Aggarwal (1989a, b), Deluca et al. (1990), Hafez (1978) and Neelakantan and Subramanian (1976) have

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127

3.11 A fabric unit cell representing the weave in Fig. 3.10.

3.12 FEA model of ballistic penetration of a projectile through 35 layers of fabric. Only quarter symmetry is modelled to save computational resources. The picture on the left shows the projectile just touching the first fabric layer and the projectile is just exiting the fabric layers in the picture on the right towards the end of the simulation.

used mechanistic approaches coupled with statistical tools. Others have used only regression models such as El Mogahzy (1988), El Mogahzy and Chewning (2002), Ethridge et al. (1982), Pavendham and Anbarasan (1998), Cheng and Adams (1995), Hunter (1988), Smith and Waters (1985) and El

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Modelling and predicting textile behaviour

Sourday et al. (1974). The technique involves finding statistically significant relationships between yarn processing parameters and a yarn property. Once a suitable relationship has been found, the model can be used to predict the properties of yarns if the yarns are similar to the set of yarns used to develop the model. El Mogahzy et al. (1990) used multiple regression to relate the skein break factor (Sb) to a variety of process parameters. The form of their regression model was:

Sb = B0 + B1Lf + B2Fm + B3Lu + B4Sf + B5Rd + B6Be + B7Lg [3.16]

where B0, B1...B7 are the regression coefficients, Lf is the fibre length in inches, Fm is the fibre micronaire, Lu is the length uniformity, Sf is the fibre bundle tenacity in g tex–1, Rd is the colour reflectance, Be is the percent break elongation and Lg is the leaf grade. The equation for coarser ring spun count was: Sb = – 3842.04 + 1713.5Lf – 90.3Fm + 42.5Lu + 43.0Sf    + 1.3Rd + 53.4Be + 44.2Lg

[3.17]

2

This equation had a reasonably good correlation coefficient (r ) of 0.84 and a similar equation was found for open end yarns. Swiech and Frydrych (1994) used multiple regression to predict ring spun yarn and rotor spun yarn tenacity from fibre parameters. Their equation was:

W = – 4.4409 + 0.3482Sf – 0.9759Fm + 0.2427Lf + 0.1328Ur [3.18]

where W is the predicted yarn tenacity, Sf is the bundle strength (cN tex–1), Fm is the micronaire, Lf is the 2.5% span length (mm) and Ur is the uniformity ratio. The most important factor controlling yarn tenacity was fibre strength which was found from a simple correlation for yarn strength against individual fibre parameters. Statistical models generally have two limitations. One is the data may have non-linear relationships in it and thus cannot be modelled effectively using linear regression models. The second limitation is the input parameters of the regression model have to be independent. Usually, this is not the case, as there are significant correlations between the input parameters. These limitations are not a problem for artificial neural networks.

3.3.4 Artificial neural network models During the mid-1990’s some research groups showed that artificial neural networks can be used to predict yarn properties from fibre properties. An artificial neural network is a computer program that learns the mathematical relationships that may exist between the fibre properties and the yarn properties.

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129

Figure 3.13 shows a typical neural network with three inputs, one output and two hidden layer neurons, although any number of neurons can be used in each layer and more than one hidden layer may be employed depending on the complexity of the dataset. The neural network in Fig. 3.13 could be used to learn the characteristics of a process with three variables and one property, for example, spinning fibre into yarn where the three inputs could be yarn count, twist and fibre strength and the property measured at the end of the process is yarn strength. An artificial neural network neuron is a mathematical transfer function, such as a sigmoidal function as shown in Fig. 3.14, where x is the input to the neuron. The flow of information in a neural network begins with the three normalized process variables. These are presented as inputs to the three input neurons. The input layer neurons transform the input data using the sigmoidal transfer function and pass it to the hidden layer neurons along weighted Input layer

Inputs

Hidden layer

Output layer

Output

Yarn count Yarn strength

Twist Fibre strength

Weighted connections between neurons

3.13 An artificial neural network.

1.2

y = 1/(1+exp(–x))

1 0.8 0.6 0.4 0.2 0 –10

–8

–6

–4

3.14 The sigmoidal function.

–2

0 x

2

4

6

8

10

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Modelling and predicting textile behaviour

connections. Before they reach the hidden layers, the numbers are multiplied by the corresponding weighting factor of each connection between the input and hidden layer neurons. The hidden layer neuron inputs are summed and transformed by the hidden layer sigmoidal transfer function then sent to the output layer neuron. The output layer neuron inputs are weighted according to the weights on the connections between the hidden layer and output layer. The neural network output is compared to the experimental value and the network weights are adjusted to minimize the difference during the learning phase. This is done for all the data in the learning dataset as an iterative process using the following equation to adjust the weights:

È ˘ Dw pq ( n ) = – h Í ∂E ˙ ∂ w Î pq ( n ) ˚

[3.19]

where wpq(n) is the weight connecting neurons p and q at the nth iteration, Δwpq(n) is the correction applied to wpq(n) at the nth iteration, h is the learning rate and E is the error vector. The error vector is given by:

E = 1 S (Tr – Or )2 2

[3.20]

where Tr and Or are the target output and the predicted output, respectively, at output neuron r. There are different schemes to train neural networks but the one described here is the most popular method, the back propagation algorithm. Once the network has learnt the relationships, it can be used to predict yarn properties from fibre properties so long as the fibre properties are not too dissimilar from the learning data set. The neural network is, in effect, a complex mathematical function that is fitted to the data by adjusting the connection weights, the number of hidden layers and the number of hidden layer neurons. Interactions between input variables and non-linear relationships can be modelled by the neural network and care is needed not to over fit or under fit the data by choosing a very complex or simple network respectively. The predictive performance of the network should be optimized by testing it on data not used in the learning phase. Cabeco-Silva et al. (1996) used neural networks to predict the yarn tenacity and elongation of carded cotton yarns. Ramesh et al. (1995) predicted the yarn strength and elongation of air-jet spun yarns from yarn count, the proportion of different fibres in a fibre blend and nozzle pressures. Cheng and Adams (1995) and Shanmugam et al. (2001) predicted the yarn strength of ring yarns from fibre properties and Pynckels et al. (1997) tried to predict a wide range of ring and rotor yarn properties from fibre properties and machine settings. Chattopadhyay et al. (2004) showed how the neural network performance can be improved by using principle component analysis to manipulate the data set before it is used to train the neural network. A further improvement

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131

was gained when some of the network inputs, which were highly correlated with other inputs, were removed to simplify the network. Rajamanickan et al. (1997) compared the performance of four different models–mathematical, empirical, computer simulation and neural network. Unfortunately, two different data sets were used for the models and not one data set was used for all four models. This made comparing all four models with each other difficult. Guha et al. (2001), however, have compared a mechanistic model by Frydrych (1992) with a statistical regression fitting model and a neural network based model using Frydrych’s data. The tenacity of a polyester staple fibre yarn was predicted. The performance of the mechanistic model was very similar to the statistical model and the neural network model was significantly better than the other models. This was attributed to the ability of the neural network model to capture complex interactions between fibre properties, process parameters and yarn tenacity. Among the few authors who have developed neural networks to predict single yarn tenacity from fibre properties are Majumdar et al. (2004). The network was trained on cotton properties and yarn count. The importance of each of the network inputs was found by eliminating an input from the training and gauging the effect on the quality of the network predictions. When an important input was removed, the model predictions deteriorated. Majumdar et al. showed that for ring yarn tenacity, the most important effect was fibre bundle tenacity and the least important was fibre yellowness. The ranking generally agreed with work of Ethridge and Zhu (1996) who also used neural networks. The neural networks of Majumdar et al. were able to predict the yarn tenacity with very good accuracy; 4.9% mean error for ring spinning and 1.6% mean error for rotor spinning.

3.3.5 Knowledge-based networks Case-based reasoning (CBR) is an advanced reasoning technique that models human reasoning and thinking. It is a way of solving new problems based on solutions of past similar problems. There are generally four stages in this technique: representation, indexing, retrieval and adaptation. The representation part describes a case as a problem and a solution. It is stored in a database index (the indexing part). Case retrieval means that most similar cases to the new case are retrieved from the database by searching the database. Once a matching case is retrieved, the CBR system will predict a solution by adapting the stored solution to the new case. Cheng and Cheng (2004) were the first to demonstrate a CBR system that could be used to understand the relationship between fibre properties and the tenacity of cotton yarns. In their work, eight fibre properties (strength, length, elongation, trash content, length uniformity, micronaire and reflectance) and the yarn tenacity for 25 different cotton samples were measured. Multiple regression was used to select

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Modelling and predicting textile behaviour

the most important variable as an index in the CBR system and the system correctly identified fibre strength as the most important variable controlling the yarn tenacity. When a new case is presented to the system, similar cases are retrieved according to their degree of similarity (SM):

SM =

1 1 + ad

[3.21]

where a is a normalisation factor to make the final value of SM fall between 0, 1 and d is the weighted Euclidean distance between two cases: 1

n



d = ( S w 2j ( x pj – xqj )2 ) 2 j =1

[3.22]



where xp is the query and xq the stored case, n is the number of input variables, j is an individual input variable from 1 to n and w is the importance weighting which is determined by the multiple regression method. The lowest degree of similarity is 0 and the highest is 1. If no exact match is available, the CBR system may need to adapt the closest matching solution stored in the database to the new case. If there was more than one closest match, the predicted yarn tenacity for the new case was found by taking an average of the closest solutions. The CBR steps are illustrated in Fig. 3.15. To test the predictive capabilities of their CBR system, Cheng and Cheng (2004) used five new cases and found the CBR system was able to predict the yarn tenacities with a mean error of only 2.76%. Apart from its very good predictive accuracy, one of the big advantages of CBR over other yarn Start New case Retrieve similar cases from database of previous cases

Is there an exact match?

No

Predict new solution by adapting closest matches

Yes Predicted solution

3.15 Steps used in a CBR system.

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133

models is that it is sufficiently transparent for users, who are not familiar with CBR, to understand how predictions were arrived at. In order to improve the CBR system, Cheng and Cheng suggested more work needs to be done on the similarity function and determining how the most appropriate input variables are selected.

3.3.6 Mathematical models A number of mathematical models have been developed to investigate the effect of fibre properties on yarn properties. Grosberg and Smith (1966) developed a semi-empirical expression to predict the breaking strength of low twist woollen spun yarn and showed that strength depended on fibre length and fibre strength. Yarn strength has been modelled by Zeidman and Sawhney (2002) to show that the fibre length greatly influences the yarn strength. Zeidman and Sawhney’s analytical model used the fibre length distribution from a fibrogram to calculate the yarn strength. Frictional forces between fibres that oppose slippage of one fibre along the length of another fibre were summed over all fibre lengths. If the fibres are smaller than a critical length, the fibres will slip. Fibres longer than a critical length will not slip before they break because the frictional forces are capable of creating tensile stresses in the fibre that are greater than the tensile strength of the fibre. This is shown schematically in Fig. 3.16. The following expression was derived to relate the yarn breaking strength (P) to the number of fibres in the yarn (N), the fibre breaking strength (pb), the fibre critical slipping length (lc), the area under a normalized fibrogram

Normalised proportion of fibres

1.2 Fibres shorter than lc slip 1 0.8 Fibres longer than lc can break

0.6 0.4 0.2 0 0

lc

Length

3.16 Normalized fibrogram showing the fibre length distribution. The critical fibre length lc is marked.

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Modelling and predicting textile behaviour

(Z(0)) and the area under a normalized fibrogram for fibres longer than the critical length (Z(lc)):

P = Z (0) – Z (lc ) pb N lc

[3.23]

It can be seen from this equation that the yarn strength increases if the number of fibres or fibre breaking strength increases. The yarn strength increases if the fibre critical length decreases because there are more fibres in the fibre length distribution that break before they slip when the yarn is tensioned. If there are proportionately more short fibres than long fibres, this will be reflected in the fibrogram and hence the Z(0) – Z(lc) term in Equation [3.23]. More work is needed to establish the critical length values for a variety of yarns and processing conditions. Once these have been collected, the model predictions can be tested against experimental values to determine the accuracy of the model predictions. Zeidman and Sawhney derived a new parameter called the ‘strength efficiency’ of fibres in a yarn to help understand fibre processing effectiveness. The strength efficiency of fibres in a yarn is the ratio of yarn tensile strength to the sum of all fibre strengths in a yarn cross-section. If it increases it means fewer fibres are required to produce a yarn of a particular strength and it therefore becomes a useful fibre processing measure. Aggarwal (1989a,b) developed a mathematical model to predict the breaking elongation (Ey) of ring spun yarn. It was claimed the model was highly accurate and can be applied to both carded and combed yarns. The equation describing the breaking elongation was:

E y = 0.86Ef (1 + 0.014TM 2 )(1 – K ) N

[3.24]

where Ef is the fibre elongation, TM is the twist multiplier, N is the average number of fibres in the cross-section and K is a regression constant fitted to fibre break and elongation data. Bogdan (1956) used an empirical approach to show that the strength of a yarn primarily depends on the fibre strength. His expression for skein breaking force (S) is

S=

160 Ê P [1 – 10 – 0.13( M –T )2 ] – Fˆ ˜¯ Á 1 + BM 2 Ë C

[3.25]

where B is the fibre obliquity, M is the twist multiplier, P is the intrinsic strength, T is the ineffective twist parameter, C is the cotton system yarn count and F is the drafting parameter. A skein prepared for the skein break test consists of 80 wraps, or 160 threads, to which tension is applied. There was excellent agreement between the model predictions and experimental results as shown in Fig. 3.17.

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135

400

Predicted skein break (pounds)

350 300 250 200 150 100 50 0 0

50

100

150 200 250 300 Actual skein break (pounds)

350

400

3.17 Predicted skein break versus the experimentally determined skein break strength. Adapted from Bogdan (1956).

3.3.7 Fuzzy logic models The spinnability and strength of a yarn can be predicted from rule sets that have been developed from the available data. Rule sets are ‘if-then’ rules that are generated automatically using a learning classifier system called a fuzzy efficiency-based classifier system (FECS). Each rule was rated for its efficiency, E(t), using the formula:

E (t ) =

a(t ) g(t )

[3.26]

where a(t) is the ‘accuracy’ or number of accumulated successful instances of the rule and g(t) is the ‘generalism’ or the total number of accumulated selections of the rule. E(t) varies from 0 to 1 and if a rule is successful when applied to all the data it will have an efficiency of 1. Sette et al. (2000) described how white modelling, where mathematical equations based on theoretical physical knowledge of the process, requires extensive knowledge of the process. No exact mathematical model is known to exist or is likely ever to be constructed because of the complexity of the process. The black box model, however, simply connects the inputs of the model to the outputs without being given or containing any substantial information about the process. Neural networks are black box models and they have been used by Pynckels et al. (1995, 1997) to predict yarn properties such as spinnability. The FECS approach was described by Sette et al. (2000) as ‘grey modelling’ because it establishes relationships between the inputs and outputs of the model rather like the black box approach, but also generates a rule set that

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can be interpreted to gain qualitative information about the process similar to a white model approach. The FECS has a learning classifier system that uses a genetic algorithm to generate new rules and fuzzy logic to help classify some of the input and output variables. Sette et al. (2000) used a cotton yarn database of 2160 data samples in the FECS model to predict yarn strength with good accuracy. The database had five fibre descriptors (length, uniformity, strength, elongation and micronaire) and five machine parameters (yarn count, twist, navel, breaker and rotor speed) to describe the spinning machine setup. When process conditions produced yarn they were classed as ‘spinnable’ process conditions. If yarn could not be produced the process conditions were non-spinnable. For the case where yarn was produced, the yarn strength was measured. As an example of the type of rules produced by the FECS, one of the rules that described most of the yarn strength data was: IF the yarn count, rotor speed, fibre length and fibre strength are all low AND the twist and breaker speed are low or medium, the fibre uniformity, elongation and micronaire do not matter and THEN the resulting yarn strength will be low. The FECS approach correctly found that fibre strength played an important part in predicting the yarn strength. The rule set was easily interpreted to give qualitative information and each rule had a measure assigned to it that allowed the user to identify the important, accurate rules. One of the most important findings from this work was that fuzzy logic can be used to improve the efficiency-based classifier system but if too many classes of fuzzy yarn strengths are used, the rule set becomes too general and starts to lose its accuracy. Figure 3.18 shows how yarn strength can be represented as a two, three and five class fuzzy logic output. Three fuzzy sets gave the best compromise between accuracy and generalism with a total prediction accuracy of 92% for yarn strength. Further work is needed to predict a rule set for continuous yarn strength rather than two, three or five fuzzy sets as is currently the case.

3.4

Applications and examples

A yarn model has been used by Wu et al. (1995) to predict the performance of double-braided synthetic fibre ropes that are widely used in marine applications. The braids twist in opposite directions to give a balanced structure that does not twist when tensioned. A yarn model proposed by Hearle et al. (1965) was used to model the rope. This yarn model relates the fibre axial strain to the yarn axial strain as:

es = ea cos2 Q

[3.27]

Truth value

Yarn modelling Low

1

0

0

1

0

High

Yarn strength

Low

Truth value 0

Medium

Truth value

0

High

Yarn strength

1 Very low Low Medium High

0

137

Very high

Yarn strength

3.18 Two, three and five class fuzzy classifiers. Adapted from Sette et al. (2000).

where ea is the axial strain of the yarn, es is the axial strain of the fibre and Q is the helix angle. This equation was extended to include bending of the yarn or rope and lateral contraction whilst under tension. It was used to predict the tensile and bending behaviour of the rope. Marine ropes are often bent as they are looped on a bollard, run through pulleys and terminated with eyelets. The tensile behaviour is important as the ropes are used in marine applications where rope failure can cause a large amount of damage. The effect of water immersion on the rope tensile properties was also studied. Research consultancy companies such as Tension Technology International (www.tensiontech.com) offer a range of commercial technical consultancy services for ropes, cables, nets and textiles. They have a number of proprietary computer models that are used to design ropes, nets and cables for marine and other environments. Their website lists case studies where this modelling has been applied.

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Composites, such as carbon fibre composites, are used in many applications including aircraft such as the Airbus A330 and the Boeing 787. The composite parts are made from carbon fibre fabrics impregnated with resin. Yarns in the fabric have zero twist because the fibres are so stiff. They are called tows, rather than yarns, because of the zero twist and they can be modelled using some yarn models. Ognjanovic has modelled the tows and resin impregnated woven fabrics to predict the mechanical properties of the impregnated fabric before curing. These properties are used in models of the lay-up process where a number of fabric layers are laid into a mould before curing and forming a carbon fibre laminate. A Department of Trade and Industry sponsored research project, PhysVis (www.physvis.co.uk), is developing a robot system to simulate the lay-up process using the fabric model. An industrial robot automates the lay-up process to give a consistent product. Production can be scaled up to produce more parts by using more robots and without having to rely on a limited number of highly skilled people for lay-up. The PhysVis project uses the PhysX accelerator chip by Ageia (www.ageia.com). The chip is targeted at computer games to accelerate the calculations of physics in a game in order to make the scene more realistic. In PhysVis, the chip is used to model the fabric more realistically and accurately in real time.

3.5

Future trends in yarn modelling

Yarn modelling started over 100 years ago with Gegauff proposing a simple formula to predict yarn strength. More complex analytical models began to appear in the 1940s and as new modelling techniques were invented these too were used to model yarns. FEA modelling software, for example, became commercially available in the 1970s and applications of this technique to yarn modelling continue to grow to this day. Over the last few decades, computers and FEA software have become more powerful and this has resulted in an increase in the size and complexity of FEA simulations. Individual fibres can now be modelled in a yarn with friction and contact between the fibres, viscoelastic material properties, material non-linear effects and large deformations to make the simulations even more realistic. The models can be applied to a wider range of yarn types as fewer modelling simplifications are used. New, better ways of creating the undeformed yarn structure in the FEA simulation are being developed because they have a great effect on the yarn predictions. These models will continue to evolve into more complex models as computer software and hardware increase in power. Free, open source FEA software can be used to do some yarn simulations. The software can run on powerful, inexpensive computing platforms such as Linux clusters. Computing time on a cluster can be bought from various websites so there is no need to buy the hardware outright and maintain the system. Both the free

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software and minimal computer hardware costs mean this type of modelling is much more readily accessible than it has been and this will accelerate progress in this field of research. Some recent advances in computer gaming hardware and software, such as the PhysX chip, are beginning to make an impact on modelling textiles. The PhysX accelerator chip is already used to model fabric drape in composite lay-up processes. Soft computing techniques such as neural networks, fuzzy logic and knowledge-based systems are starting to be applied to yarn modelling and much more work in these areas is to be expected. Soft computing techniques are still at a very early stage of development in industry and considerable work needs to be done to build up the industry’s confidence in these techniques. Even at this stage, before some of the techniques have become established, they are being used in conjunction with other techniques such as a fuzzy efficiency based classifier system that combines genetic algorithms, fuzzy logic and learning classifier techniques. The cross disciplinary approach to modelling has been seen before with statistical and mechanistic models. The trend is likely to continue with some of the latest techniques in artificial intelligence. There is still a considerable need for the other modelling approaches and they all have their strengths and weaknesses. There is not one modelling approach that is clearly capable of modelling yarns better than all the other modelling approaches. It is probably safe to say that, at the moment, a combination of many different types of models is needed to model yarns effectively.

3.6

Sources of further information and advice

Internet search engines, such as Google, are a good source of information on yarns. The search term ‘yarn’ returned over 22 million hits on Google, many of which may not be useful, but extra terms can be added to refine the search. A Google book search returned over 600 hits for books when ‘yarn modelling’ was searched and most hits had access to some pages in the books. These types of search quickly identify a number of articles but these do not necessarily search proprietary databases of journal publishers such as Sage (www.sagepub.co.uk). These databases require registration but yield very good results. Authors of papers can often be emailed directly to ask for advice and commercial research consultancies can be approached as well. A search of ‘textile consultancies’ in Google produced many hits. A list of textile trade and professional bodies can be found at Heriot-Watt University library website (http://www.hw.ac.uk/library/concisetextiles. html) along with other sources of useful information. There are a number of university textile departments and research groups in the UK and abroad. An internet search for ‘textile’ and ‘university’ listed a number of university textile departments.

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One organization specifically set up by the UK government Department of Trade and Industry to bring industry and the research community together is TechniTex (www.technitex.org). This organization is a useful starting point to find out more about textiles and their applications in industry. Members of the organization have access to expert advice and other government funded initiatives such as Knowledge Transfer Networks.

3.7

References

Aggarwal S K (1989a), ‘A model to estimate the breaking elongation of high twist ring spun cotton yarns: Part I: Derivation of the model for yarns from single cotton varieties’, Text Res J, 59(11), 691–5. Aggarwal S K (1989b), ‘A model to estimate the breaking elongation of high twist ring spun cotton yarns: Part II: Applicability to yarns from mixtures of cottons’, Text Res J, 59(12), 717–20. Bogdan J F (1956), ‘The characterization of spinning quality’, Text Res J, 26(9), 720–30. Cabeco-Silva M E, Cabeco-Silva A A, Samarao J L and Nasrallah B N (1996), Proceedings of the beltwide cotton conference, Nashville, TN, 2, 1481. Carnaby G A and Curiskis J I (1987), ‘The tangent compliance of staple-fibre bundles in tension’, J Text Inst, 78, 293–305. Chattopadhyay R, Guha A and Jayadeva (2004), ‘Performance of neural networks for predicting yarn properties using principal component analysis’, J Appl Poly Sci, 91, 1746–51. Cheng L and Adams D L (1995), ‘Yarn strength prediction using neural networks’, Text Res J, 65(9), 495–500. Cheng Y S J and Cheng K P S (2004), ‘Case-based reasoning system for predicting yarn tenacity’, Text Res J, 74(8), 718–22. Deluca L B, Smith B and Waters W T (1990), ‘Analysis of factors influencing ring spun yarn tenacities for a long staple cotton Part I: Determining broken fibres in yarns’, Text Res J, 60(8), 475–82. Djaja R G (1989), Finite element modelling of fibrous assemblies, Master of Engineering thesis, University of Canterbury, New Zealand. El Mogahzy Y E (1988), ‘Selecting cotton fibre properties for fitting reliable equations to HVI data’, Text Res J, 58(7), 392–7. El Mogahzy Y and Chewning C H (2002), Cotton fibre to yarn manufacturing technology – optimizing cotton production by utilizing the engineered fibre selection system, 2nd edn, Cotton Incorporated, Cary, NC. El Mogahzy Y E, Broughton Jr R and Lynch W K (1990), ‘A statistical approach for determining the technological value of cotton using HVI fibre properties’, Text Res J, Oct, 497–500. El Sourady A S, Worley S and Stith L S (1974), ‘The relative contribution of fibre properties to variations in yarn strength in upland cotton, gossypium hirsutum L’, Text Res J, 44(4), 301–6. Ethridge D and Zhu R (1996), ‘Prediction of rotor spun cotton yarn quality: a comparison of neural network and regression algorithms’, Proceedings of the Beltwide Cotton Conference, vol 2, National Cotton Council, Memphis, TN, 1314–18.

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Ethridge M D, Towery J D and Hembree J F (1982), ‘Estimating functional relationships between fibre properties and the strength of open-end spun yarns’, Text Res J, 52(1), 35–45. Frydrych I (1992), ‘A new approach for predicting strength properties of yarns’, Text Res J, 62(6), 340–8. Gegauff C (1907), ‘Strength and elasticity of cotton threads’, Bull Soc Ind Mulhouse, 77, 153–76. Grosberg P and Smith P A (1966), ‘The strength of slivers of relatively low twist’, J Text Inst, 57, T15–T23. Guha A, Chattopadhyay R and Jayadeva (2001), ‘Predicting yarn tenacity: A comparison of mechanistic, statistical and neural network models’, J Text Inst, 92(part 1, no 2), 139–45. Hadley D W, Ward I M and Ward J (1965), ‘The transverse compression of anisotropic fibre monofilaments’, J Proc Royal Society, A285, 275–86. Hafez O M A (1978), ‘Yarn-strength prediction of American cottons’, Text Res J, 48(12), 701–5. Hearle J W S (1958), ‘The mechanics of twisted yarns: The influence of transverse forces on tensile behaviour’, J Text Inst, 49, T389–T408. Hearle J W S (2004), ‘The challenge of changing from empirical craft to engineering design’, Int J Clothing Sci Technol, 16(1/2), 141–52. Hearle J W S and Konopasek M (1975), ‘On unified approaches to twisted yarn mechanics’, Appl Polym Symp, 27, 253–73. Hearle J W S and Sakai T (1979), ‘On the extended theory of mechanics of twisted yarns’, J Text Machine Soc Japan, 25, T68–T72. Hearle J W S, El-Dehery H M A E and Thakur V M (1959), ‘The mechanics of twisted yarns: Tensile properties of continuous-filament yarns’, J Text Inst, 50, T83–T111. Hearle J W S, El-Behery H and Thabur V M (1961), ‘The mechanics of twisted yarns: Theoretical development’, J Text Inst, 52, T197–T220. Hearle J W S, Grossberg P and Backer S (1965), Structural mechanics of fibres yarns and fabrics, Wiley Interscience, NY. Holdaway H W J (1964), ‘A theoretical model for predicting the strength of singles worsted yarns’, J Text Inst, 56, T121–T144. Huang N C and Funk G E (1975), ‘Theory of extension of elastic continuous filament yarns’, Text Res J, 45(1), 14–24. Hunter L (1988), ‘Prediction of cotton processing performance and yarn properties from HVI test results’, Melliand Textilberichte, 229–32 (E 123–E 124). Jeong Y J and Kang T J (2001), ‘Analysis of compressional deformation of woven fabric using finite element method’, J Text Inst, 92 (Part 1, No 1), 1–5. Kawabata S (1980), ‘The standardization and analysis of hand evaluation’, 2nd edn, The Textile Machine Society Japan, Osaka. Kawabata S, Niwa M and Kawai Y (1978), ‘Study of the compression deformation of woven fabrics. Part 1. Measurement of the compression property of yarns’, J Text Machine Soc Japan, 31, T74–T79. Kilby W F (1964), ‘The mechanical properties of twisted continuous-filament yarns’, J Text Inst, 55, T589–T632. Komori T (2001), ‘A generalized micromechanics of continuous-filament yarns – Part I: Underlying formalism’, Text Res J, 71(10), 898–904. Lee D H, Carnaby G A, Carr A J and Moss P J (1990), A review of current micromechanical models of the unit fibrous cell, WRONZ communication No C113.

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Majumdar A, Majumdar P K and Sarkar B (2004), ‘Prediction of single yarn tenacity of ring- and rotor- spun yarns from HVI results using artificial neural networks’, Indian J Fibre Text Res, 29(2), 157–62. McPhee J R, Russell K P and Shaw T (1985), ‘The role of objective measurement in the wool textile industry’, J Text Inst, 76(2), 110–21. Morris S J (1968), ‘The determination of the lateral compression modulus of fibres’, J Text Inst, 59, 536–47. Morris P J, Merkin J H and Rennell R W (1999), ‘Modelling of yarn properties from fibre properties’, J Text Inst, 90(3), 322–35. Morton W E (1956), ‘The arrangement of fibres in single yarns’, Text Res J, 26(5), 325–31. Neelakantan P and Subramanian T A (1976), ‘An attempt to quantify the translation of fibre bundle tenacity into yarn tenacity’, Text Res J, 46(11), 822–7. Onder E and Baser G (1996), ‘A comprehensive stress and breakage analysis of staple fibre yarns: Part I: Stress analysis of a staple yarn based on a yarn geometry of conical helix fibre paths’, Text Res J, 66(9), 562–75. Pavendham A and Anbarasan M (1998), ‘Fibre to yarn link’, Ind Text J, 109(2), 40–8. Peirce F T (1930), ‘The handle of cloth as a measurable quantity’, J Text Inst, 21, T377–T416. Pheoenix S L and Skelton J (1974), ‘Transverse compressive moduli and yield behaviour of some orthotropic, high modulus filaments’, Text Res J, 28, 934–40. Pinnock P R, Ward I M and Wolfe J M (1966), ‘The compression of anisotropic fibre monofilaments. II’, J Proc Royal Soc, A291, 267–78. Platt M (1950), ‘Mechanics of elastic performance of textile materials: III. Some aspects of stress analysis of textile structures – continuous-filament yarns’, Text Res J, 20(1), 1–15. Postle R, Carnaby G A and de Jong S (1988), The mechanics of wool structures, John Wiley & Sons, New York, NY, USA. Pynckels F, Sette S, van Langenhove L, Kiekens P and Impe K (1995), ‘Use of neural nets for determining the spinnability of fibres’, J Text Inst, 86(3), 425–37. Pynckels F, Sette S, van Langenhove L, Kiekens P and Impe K (1997), ‘The use of neural nets to simulate the spinning process’, J Text Inst, 88(1), 440–7. Rajamanickam R, Hansen S M and Jayaramam S (1997), ‘Analysis of modelling methodologies for predicting the strength of air-jet spun yarns’, Text Res J, 67(1), 39–44. Ramesh M C, Rajamanickam R and Jayaraman S (1995), ‘The prediction of yarn tensile properties by using artificial neural networks’, J Text Inst, 86, 459–69. Riding G and Wilson N (1965), ‘The stress–strain properties of continuous-filament yarns’, J Text Inst, 56, T205–T214. Sette S, Boullart L and van Langenhove L (2000), ‘Building a rule set for the fibre-toyarn production process by means of soft computing techniques’, Text Res J, 70(5), 375–86. Shanmugam N, Chattopadhyay S K, Vivekanandan M V and Sreenivasamurthy H V (2001), ‘Prediction of micro-spun yarn lea CSP using artificial neural networks’, Indian J Fibre Text Res, 26(4), 372–7. Singh V P and Sengupta A K (1977), ‘A new method of estimating the contribution of fibre rupture to yarn strength and its application’, Text Res J, 47, 186–7. Smith B and Waters B (1985), ‘Extending the application ranges of regression equations for yarn strength forecasting’, Text Res J, 55(12), 713–17.

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Sreprateep K and Bohez E L J (2006), ‘Computer aided modelling of fibre assemblies’, Computer-Aided Design & Applications, 3(1–4), 367–76. Swiech T and Frydrych I (1994), ‘Polish experience in the use of the 900 HVI system for predicting yarn properties’, Proceedings 22nd International Cotton Conference, Bremen, March 2–5. Harig H and Heap S A (eds), Faserinstitut Bremen, Bremer Baumwollbörse, Bremen, Germany, 119–32. Thwaites J J (1980), ‘A continuum model for yarn mechanics’, ‘Mechanics of flexible fibre assemblies’, Hearle J W S, Thwaites J I and Amirbayat J (eds), Series E, Applied Sciences, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 38, 87–97. Treloar L R G (1965), ‘A migrating-filament theory of yarn properties’, J Text Inst, 56, T359–T380. Treloar L R G and Riding G A (1963), ‘A theory of the stress–strain properties of continuous-filament yarns’, J Text Inst, 54, T156–T170. Van Langenhove L (1997a), ‘Simulating the mechanical properties of a yarn based on the properties and arrangements of its fibres. Part I: The finite element model’, Text Res J, 67(4), 263–8. Van Langenhove L (1997b), ‘Simulating the mechanical properties of a yarn based on the properties and arrangements of its fibres. Part II: Results of simulations’, Text Res J, 67(5), 342–7. Van Langenhove L (1997c), ‘Simulating the mechanical properties of a yarn based on the properties and arrangements of its fibres. Part III: Practical measurements’, Text Res J, 67(6), 406–12. Van Luijk C J (1981), Structural analysis of wool yarns, PhD thesis, University of Canterbury, New Zealand. Van Luijk C J, Carr A J and Carnaby C A (1985a), ‘The mechanics of staple-fibre yarns Part I: Modelling assumption’, J Text Inst, 76(1), 11–18. Van Luijk C J, Carr A J and Carnaby C A (1985b), ‘The mechanics of staple-fibre yarns Part II: Analysis and results’, J Text Inst, 76(1), 19–29. White J L, Cheng C C and Duckett K E (1976), ‘An approach to friction effects in twisted yarns’, Text Res J, 46, 496–501. Wu H, Seo M H, Backer S and Mandell J F (1995), ‘Structural modelling of doublebraided synthetic fibre ropes’, Text Res J, 65(11), 619–31. Zeidman M and Sawhney P S (2002), ‘Influence of fibre length distribution on strength efficiency of fibres in yarn’, Text Res J, 72(3), 216–20.

4

Modelling the structures and properties of woven fabrics

E. Vidal-Salle and P. Boisse, INSA Lyon, France

Abstract: Modelling the mechanical properties of woven fabrics has been a goal of researchers for many decades. Nowadays, numerical simulations permit different approaches at different scales. This chapter gives a nonexhaustive view of the proposed approaches and tools for modelling mechanical properties of woven fabrics. Two main aspects are dealt with: (1) macroscopic modelling that is used for textile structure deformation analyses, especially forming simulations, and (2) mesoscopic modelling, that is modelling on the scale of the unit woven cell. The specific experimental tests used for woven fabrics are described. X-ray tomography analyses are performed to validate simulations of unit cell deformations. Mesoscopic modelling is an efficient method of providing input data on the macroscopic behaviour of textile preforms and for resin flow analysis in the resin transfer moulding (RTM) processes. Key words: bending and transverse compaction tests, bias extension test, biaxial tensile test, continuous/discrete mechanical models, hypoelastic models, mechanical behaviour of textiles, mesoscopic analyses, multi-scale analysis, periodicity boundary conditions, picture frame test, unit woven cell, X-ray tomography.

4.1

Introduction: The importance and objectives of modelling woven fabrics

Modelling the mechanical properties of woven fabrics has been a goal of researchers for many decades, for instance Peirce (1930) and Kawabata et al. (1973). Until recently, however, models and numerical simulations for woven fabrics were less developed than for other materials. This is now changing, and advanced modelling and numerical simulation techniques have been developed. The two main reasons behind this are the strong need for simulation codes in fields such as textile composites for aeronautical applications (Itool European project: http://www.itool.eu) and the rapid increase in computer capabilities, which have made simulations of complex forms and local analysis of intricate textile reinforcements possible. Textiles are multiscale materials and simulation tools for multiscale materials are now available. Another important advance is progress in image processing, which gives scientists the opportunity better to understand the physics of woven behaviour. Imaging techniques not only facilitate surface observations (e.g. 144

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145

full field digital image correlation (DIC) measurement), but techniques such as X-ray tomography also allow us to look inside the fabric and to see its structure at different scales and the modifications caused by loadings. This chapter reviews the most important aspects of the mechanical properties of woven fabric reinforcements, along with the corresponding mechanical tests. We then present three dimensional (3D) simulations of the deformation of the unit woven cell at the mesoscopic scale, together with suggested imaging processes that would make it possible to validate the proposed approaches experimentally.

4.2

The mechanical behaviour of woven fabrics

The mechanical behaviour of a woven fabric is directly linked to its structure. Textiles are made up of interlacing warp and weft tows, themselves made up of thousands of fibres (Fig. 4.1). This internal structure makes relative motion possible between fibres and between yarns, and this leads to very specific mechanical behaviour. The only high stiffness is the tensile stiffness in the fibre direction, all other rigidities (shear, bending, compaction) are much weaker, even approaching zero. A woven fabric is intrinsically a multiscale material (Fig. 4.1) and, depending on the specific application of interest, one or more scales of the woven fabric have to be explored.

4.2.1 Three scales of investigation A woven fabric can be investigated on three different levels. The macroscopic level refers to the whole component level, with dimensions in the order of ten centimetres to several meters. At this level, a woven fabric can be seen as a continuous material with very specific behaviours including high anisotropy

Macroscopic scale

Mesoscopic scale Microscopic scale

4.1 The three scales of textile reinforcement (respectively macroscopic, mesoscopic and microscopic).

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and the ability to exhibit very large shearing and bending deformations. Investigation at the macroscopic level is the most popular for preforming simulations, as it allows large mechanical components to be modelled. Unfortunately, despite the large amount of work in this field, there is no widely accepted model that accurately describes all aspects of the mechanical behaviour of fabrics. Some authors have proposed different macroscopic approaches to simulate the forming of woven reinforcements, these will be described briefly in Section 4.3, below. Moreover, the macroscopic scale, even if it is suitable for simulating draping in dry or prepreg woven composite reinforcements, cannot be used for the whole process. Liquid composite moulding processes in particular require simulation of the injection phase, which necessitates a finer geometrical description of the weave. At the mesoscopic scale, the woven reinforcement is seen as an interlacing of tows, respectively the warp and the weft (or fill) yarns. Consequently, the working scale corresponds to the yarn dimension, typically one to several millimetres. For periodic materials, mesoscopic models consider the smallest elementary pattern, which can represent the whole fabric in several translations. That domain is called the representative unit cell (RUC) (see Fig. 4.2). Generally, each yarn is modelled as a continuous medium with specific behaviour which takes into account its fibrous nature via a specific constitutive law. Mesoscopic modelling is used to estimate the permeability tensor of the preformed fabric, taking into account the local geometry of the deformed reinforcement (Vandeurzen et al., 1996a; Bickerton et al., 1997; Belov et al., 2004; Loix et al., 2008). It is also widely used by scientists better to understand the physics of textile reinforcement and to propose macroscopic constitutive models using homogenization techniques. Mesoscopic analysis can also be used as a virtual test to determine the mechanical properties of a textile, in particular such analyses permit fabric behaviour to be evaluated before manufacture. Each yarn is made up of thousands of continuous fibres which interact,

(a)

(b)

4.2 Representative unit cell for: (a) an interlock fabric (G1151®), (b) a 2 ¥ 2 twill weave.

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and thus the interactions of the reinforcement can be analysed at the microscopic scale. Each fibre is considered as a 3D beam interacting with its numerous neighbours (Zhou et al., 2004; Durville, 2005; 2007; 2008; Miao et al., 2008). At the microscopic level, the characteristic dimension is about one to several micrometers. This is the only scale at which the material is actually continuous. The microscopic approach has been used to analyse the deformation of one or a few woven unit cells (see Fig. 4.3) and has interesting potential regarding fibre arrangements during deformation. Nevertheless, the number of fibres within a yarn that can be considered using the microscopic approach is currently small in comparison to the real number (3 k to 48 k in a standard composite reinforcement), which reduces the conclusions that can be obtained from such analyses.

4.2.2 Fibrous behaviour Since yarn is made up of thousands of fibres, it exhibits some behaviour particularities. The fibres which constitute the tow can move in relation to their neighbours and this makes the yarn very soft in response to certain types of loading. In particular, even though a yarn has the geometry of a beam because its transverse dimensions are small with regard to its length, its bending stiffness is small because of the relative motion that can occur between the fibres constituting the tow (Lahey et al., 2004; de Bilbao et al., 2008). Consequently, it is very easy to bend a yarn because it is like bending each fibre. The possibility of relative motion between fibres also implies a

4.3 Microscopic modelling of a sheared plain weave (Durville, 2007).

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very low in-plane shear stiffness for the yarn. Bending and in-plane shear stiffnesses have sometimes been neglected in draping simulations, in particular for the fishnet method. However, at the macroscopic level, if shear stiffness is necessary to simulate wrinkle appearance, bending stiffness allows the wrinkles to have the correct shape (Boisse et al., 2006; Hamila and Boisse 2007) (Fig. 4.4). For a single yarn, one not interlaced with other yarns in a woven fabric, there are no cohesion forces between fibres. Consequently, there is a great difference between the density of the unloaded sections and the same density when the yarn is compressed or tightened. High compressibility is another particularity of fibrous behaviour. The stiffness of a yarn in the direction of the fibre is very high and depends on the constituent material, fibre density in the section and possibly also the arrangement of fibres within the yarn (the present work considers yarns made of juxtaposed continuous fibres). As mentioned above, the evolution of section density is important when the yarn is tightened and consequently the apparent tensile stiffness of a yarn appears to be non-linear.

4.3

Different approaches for modelling the mechanical behaviour of woven fabrics at different scales

The mechanical behaviour of the fabric is modelled using different approaches depending on the level or scale of observation.

4.3.1 Geometrical approach The simplest of several approaches at the macroscopic level is the fishnet, or fisherman’s net method (Bergsma and Huisman, 1988; Van Der Ween, 1991; Cherouat et al., 2005), which does not take into account the mechanical behaviour of the fabric or the load boundary conditions (see Fig. 4.5). This approach, based on the fishnet algorithm, states that the fibres are non-

(a)

(b)

(c)

4.4 Airbag opening (a) with only tensile stiffness; (b) including inplane shear stiffness; (c) with in-plane shear and bending stiffness.

Modelling the structures and properties of woven fabrics 11.081 0

33.244 22.163

149

55.407 44.326

(a) 14.092 0

42.275 28.183

70.458 56.366

(b)

4.5 Examples of draping with the fishnet algorithm for two fabric orientations (a) 0°/90° and (b) +45°/–45° (Borouchaki and Cherovat, 2003).

stretchable and that only the shear angles between warp and weft yarns are available. Each yarn is depicted by line segments representing the in-plane fibre bundle directions, connected at their crossover points by pivots. The method has the great advantage of simplicity and numerical efficiency for simulating the formability of a woven fabric, as it is based purely on geometry. However, geometrical approaches have strong limitations. In particular, since this method does not take into account the mechanical characteristics of the fabric, it will give the same results whatever the weaving mode and whatever the nature of fibres used. Moreover, because the shear rigidity is

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Modelling and predicting textile behaviour

not taken into account, the method is unable to predict wrinkle appearance during the forming process. Some authors have tried to extend the method to estimate the mechanical characteristics of the final shapes of components (Hofstee and Van Keulen, 2001; Long, 2001). Methods that allow the mechanical behaviour of the fabric and the load boundary conditions to be taken into account are based on the finite element method and can be divided into continuous approaches and discrete approaches.

4.3.2 Continuous mechanical models The most popular models are continuous and concern the macroscopic level. They allow for the specific behaviour of the fabric, which is linked to the fibrous structure of the tows and exhibits a strong anisotropy. The first attempts were based on standard laminate theory. They used the same number of layers as the directions that had to be taken into account, and linked elastic behaviour, representing fibre effects, and viscous behaviour, representing the matrix (Rogers, 1989; Vandeurzen et al., 1996b; Spencer, 2000). The advantage of continuous models is that they can be used directly in standard finite elements and the computations are less expensive than at lower scales. The fabric is considered as an anisotropic continuum, the mechanical behaviour of which is influenced by the underlying mesoscopic structure (Fig. 4.6) (Spencer, 2000; Dong et al., 2001; King et al., 2005; Shahkarami and Vaziri, 2007). Some authors point out the importance of homogenizing the mechanical properties of the whole fabric (Carvelli and Poggi, 2001), while others give more importance to the anisotropic nature of a material exhibiting large shear angle variations. Some authors (Yu et al., 2002; 2005; Peng et al., 2005; Xue et al., 2005) have proposed specific non-orthogonal constitutive equations. Unfortunately, some of these models fail to give correct stress–strain responses for simple loadings (Badel et al., 2008a). While these models can be easily integrated in the finite element standard shell or membrane elements, the identification of homogenized material parameters is difficult, especially because these parameters change when the fabric is subjected to the large strains due to forming and when the directions of the yarns change as a consequence. This is why some authors have given particular attention to following the material directions correctly (Xiao et al., 1998; Boisse et al., 2005; Hagège et al., 2005; ten Thije et al., 2007). Another solution for avoiding the difficulty linked to strong non-linearities caused by rotations is to use hyperelastic models. Various authors have proposed anisotropic hyperelastic models for modelling textiles. Some models deal with dry fabrics (Aimène et al., 2008), while others work on composites

Modelling the structures and properties of woven fabrics Fabric structure Yarn family orientation vectors

Continuum approximation Yarn family orientation vectors

g2

151

g1

g2

g1

Unit thickness

Discrete yarn tensions Continuum stresses

4.6 Structure of a typical physically based continuous model (King et al., 2005).

during their elaboration (Wysocki et al., 2008) or in use (Holzapfel and Gasser 2001).

4.3.3 Discrete approach In discrete modelling, each fibre bundle is modelled as a simple element such as a beam or spring (Fig. 4.7) and the interaction between warp and weft directions is taken into account explicitly by considering contact behaviour (Pickett et al., 2005; Creech and Pickett 2006; Duhovic and Bhattacharyya 2006; Ben Boubaker et al., 2007; Sze and Liu, 2007). At the microscopic level, this approach is time consuming because each fibre is described as a beam. The main difficulty is the great number of contacts that have to be taken into account, especially for a woven fabric. For this reason, only very small elements of the fabric have been modelled to date (Zhou et al., 2004; Durville, 2005; 2008; Miao et al., 2008). Nevertheless, this approach is promising because it does not necessitate any assumptions regarding the continuity of the material and it provides an interesting way of taking the weaving operation into account. The computations are made for a number of fibres per yarn, significantly smaller than in reality. This implies that a compromise must be found between a precise description (which will be expensive from the computation time point of view) and a model simple enough to compute the entire forming process. This approach will continue to be interesting because a large part of the mechanical specificity of fabric behaviour is due to fibre interactions and following fibre directions is simpler than for continuous models.

4.3.4 Semi-discrete elements The semi-discrete approach is intermediate between the continuous and discrete approaches (see Fig. 4.8) (Boisse et al., 1995; 2006; Hamila and

152

Modelling and predicting textile behaviour Boundary fixation spring Cedge3 Flexional spring Ch(ij )

Cedge4 Torsional spring Ct(ij )

Stretching spring Cex(ij )

Shear spring Cs(ij )

Node mass m(ij )

Compressive load Px

Cedge1

Cedge2 Dx

Dy

Compressive load Py

Elastic foundation’s spring Cfoundation

4.7 Discrete model from ben Boubaker et al., 2007.

3

4

2 1

4.8 Semi-discrete textile finite element.

Boisse, 2007, 2008). The semi-discrete approach, as in the discrete approach, considers the components at the mesoscopic or microscopic levels (yarns, fibres, woven cells etc.) and their corresponding strain energies are calculated. But in this approach, they are considered as part of the finite elements, which sets the kinematics in these discrete components as a function of the nodal displacements of the elements. This leads to some continuity within the elements and, in particular, two initially superimposed warp and weft fibres stay superimposed during the draping. This result has been experimentally verified (Gelin et al., 1996).

Modelling the structures and properties of woven fabrics

153

The semi-discrete approach uses the principles of physics because it considers mesoscopic entities and accounts only for their significant mechanical properties. Identifying the necessary mechanical characteristics from classical tests (tensile and bias tests) is simple and straightforward. This approach is used to simulate the formation of one or several plies simultaneously. As an example, Fig. 4.9 compares the computed and experimental deformed shapes in the hemispherical formation of a very unbalanced fabric (Boisse et al., 2006). Analysing this draping process using a fishnet algorithm would lead to (1) a symmetrical shape in both warp and weft directions, (2) no wrinkles; (3) a ratio lwarp/lweft = 1 in the hemispherical zone instead of 1.8.

4.4

Structure and geometry of the unit woven cell

Finite element simulations of unit woven cell deformation need an accurate mesh. The multiscale nature of the fabric, being composed of yarns themselves made of thousands of fibres, leads to an intricate geometry. Several models and software programmes exist (Lomov et al., 2000; Robitaille et al., 2003; Verpoest and Lomov 2005; Texgen; WiseTex), but the meshes based on geometries given by these models cannot always be used to perform finite element analyses of the unit woven cell. Interpenetrations (and to a lesser extent spurious voids) are unacceptable in such analyses. A 3D geometrical model for the fabric unit cell is said to be consistent because it guarantees that there is no penetration between warp and weft

(a)

(b)

4.9 Hemispherical forming of an unbalanced composite fabric. Simulation (a) and experiment (b).

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Modelling and predicting textile behaviour

yarns (Hivet and Boisse, 2005, 2008). Experimental observations show that the assumption of a constant cross-section along the yarn is not sufficient for modelling fabric geometry. From these observations, it appears that the reorganization of the fibres in the yarn is an important phenomenon. In this model, the yarn geometry is defined by the volume generated by the sweep of a section along a trajectory. The cross-section is dissymmetric owing to the boundary conditions prescribed by other yarns and has to change along the trajectory. This aspect is not present in most of the models. Variations of section shape along the yarn are taken into account using control sections at control points (Fig. 4.10a). The trajectory is constrained by the necessary 3D consistency of the fabric model. The complete 3D model of the yarn is obtained through a smooth interpolation between the control sections which respects the trajectory imposed. The interpolation is obtained using CAD software, such as PROEngineer®, which includes a ‘swept blend’ feature that is able to build volumes using control sections and trajectories. The elemental cell of fabric is obtained by assembling m + n yarns. The standard complete 3D model of a fabric may be identified by Q

cb1

P22

M12

M11

P25 dc15 b2

M13

Z

w

M14

X

M22

b1 M16

M15

R dc12 ct1 c2

(b)

(a)

(c)

4.10 (a) Transverse cut in the direction 1 of the model for a twill m*n and (b) 3D consistent geometrical model and mesh (c) of a 4 ¥ 3 twill (Hivet and Boisse, 2008).

Modelling the structures and properties of woven fabrics

155

measuring three parameters for balanced fabrics, up to seven parameters for unbalanced fabrics. These seven parameters may include yarn width, yarn density, crimp of each direction and thickness of the fabric, which are quite easy to obtain. Figure 4.10(b) shows the consistent CAD model (i.e. without penetration) obtained for a 4 ¥ 3 twill and the corresponding finite element mesh (Fig. 4.10c).

4.5

Specific experimental tests

Because of the specific mechanical behaviour of textiles, particular experimental tests have been developed for these materials. As textiles are made of fibres, tensile stiffness in the fibre direction is much larger than in other directions. The first experimental test described is the tensile test and, more precisely, a biaxial tensile test. The other rigidities are much weaker than the tensile one, nevertheless, they can correspond to large deformations and have great importance in some loading situations. The in-plane shear strain is the principal deformation mode when a fabric is draped on a double curved surface. The bending stiffness of fabrics is very much smaller than that of continuous materials because of the fibrous constitution of the textile, but it plays an important role in the shape of wrinkles. Transverse compaction behaviour is important when a textile is compressed, in particular in some liquid moulding processes. Experiments for these four loading cases are described below. Because textiles are numerous and diverse, different systems can be used for the same loading situation. The methods described are used for testing composite reinforcement textiles. In the case of cloth fabrics, the testing apparatus is somewhat different because the fabrics are less stiff. The KES-F system is one example that is commercially available for cloth fabrics (Kawabata, 1980).

4.5.1 Biaxial tensile test Some of the different biaxial tensile devices that have been designed for fabric testing (Kawabata et al., 1973) are based on cross-shaped specimens (Buet-Gautier and Boisse, 2001; Willems et al., 2008; Carvelli et al., 2008). The tensile device (Fig. 4.11a) adds prescribed biaxial tensile strains to a cross-shaped specimen which is set directly on a standard traction/compression machine. One of the two deformable lozenges has a modifiable length to obtain different ratios (noted k) between warp and weft strains. A regulation system allows angles other than 90° to be set between the yarn directions. Only the square central part is woven and studied. During the test, transverse deformation must be freely allowed. The unwoven ends of the specimen are essential because they make complicated systems that allow transverse deformation, as in the Kawabata device, for example (Kawabata et al.,

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Modelling and predicting textile behaviour

(a)

250 k=1 k = 0.5

Load (N/yarn)

200

Yarn

150 k=2 100

Other direction free

50

0

0

0.2

0.4 Strain (%) (b)

0.6

4.11 (a) Biaxial tensile test on cross-shaped specimen and (b) load versus strain curves for carbon twill weave in biaxial tension. k = e2/ e1 (Buet-Gautier and Boisse, 2001).

1973; Kawabata, 1989). In a fabric, this operation is unnecessary because it occurs naturally. The cross shape is particularly well suited to biaxial tests on woven materials because they have weak in-plane shear stiffness. Optical strain measurements can be used to check that there is good homogeneity of the

Modelling the structures and properties of woven fabrics

157

strain field in the entire central woven part, even in the inner corners, which pose problems in cases of non-zero in-plane shear stiffness (Launay et al., 2001; Lomov et al., 2008; Willems et al., 2008). The loads versus strains measured in the warp direction are shown in Fig. 4.11(b) for a 2 ¥ 2 carbon twill fabric. The yarns are identical in the warp and weft directions. The fabric is almost balanced (that is to say symmetric regarding the warp and weft directions), so the results are only presented for one direction. Although the yarns have a linear behaviour, tensile versus strain behaviour curves of the fabric are strongly non-linear at low loads and then linear for higher loads. This non-linearity is a consequence of non-linear phenomena occurring at lower scales (undulation variations and yarn flattening). This non-linear zone is significant for biaxial textile behaviour. It depends on the imposed strain ratio k = e2/e1, which describes the biaxial aspect of fabric behaviour, each direction having an influence on the behaviour of the other. The extension of the non-linear zone is maximal for tests where the other direction is free. Indeed, in the load direction, yarns tend to reach a totally straight state, even under very low loads. When the yarn is straight, the fabric behaviour is very similar to that of the yarn alone. The value of the strain corresponding to this transition is representative of the fabric crimp in this direction. The curves obtained for different ratios k = e2/e1 can be gathered in one (or two in the case of unbalanced fabric) biaxial surface tension Taa(e1, e2) which defines the biaxial tensile behaviour of the fabric and can be used in fabric deformation simulations (Boisse et al., 2001).

4.5.2 In-plane shear tests Many studies have analysed the in-plane shear behaviour of textile materials, probably because it is the principal deformation mode of woven fabrics when they are draped on a double curved surface (Kawabata et al., 1973; Prodromou and Chen, 1997; Wang et al., 1998; Lomov et al., 2008). An international benchmark has been launched in order to compare the experiments and results obtained by different laboratories (Cao et al., 2008). Two principal devices are used: the hinged framework or ‘picture frame’, and the tensile test at 45° or ‘bias-extension test’. The picture frame is made from a hinged frame with four side bars of equal length assembled in a tensile testing machine (Fig. 4.12a). A tensile force is applied across diagonally opposing corners of the picture frame rig, causing the picture frame to move from a square configuration into a rhomboid shape. Consequently, the specimen within the picture frame is theoretically subjected to a pure shear strain field, that is, angle variation between the warp and weft directions without any stretching. The tensile test at 45° or bias-extension test is performed on a rectangular specimen such that the warp and weft directions of the tows are orientated

158

Modelling and predicting textile behaviour Numerical camera 2 Telecentric lens

Shear frame Woven specimen

Telecentric lens

Numerical camera 1 Picture frame device (a)

Bias extension test device (b)

4.12 (a) Picture frame device equipped with an optical system (Launay et al., 2008), (b) bias extension test.

initially at ± 45° to the direction of the applied tensile load (Fig. 4.12b). When the specimen is stretched from L to L + d, the fibrous and woven nature of the specimen leads to zones with different deformed states. In the central zone, the warp and weft yarns have free ends. Non-sliding at crossovers and the stretching of the specimen leads to a pure shear deformation related to d. For both tests, the load on the tensile machine can be related to the shear rigidity of the fabric (Launay et al., 2008; Cao et al., 2008). Figure 4.13 gives the shear curve obtained by a bias-extension test on an interlock G1151® fabric. Digital image correlation (DIC) analyses are performed at the mesoscopic level, that is on a zone small enough to be within a single yarn. Figure 4.13 shows the incremental displacement fields in the central zone for two shear angles. For a moderate shear, the displacement field is a pure rotation field. The yarn is subjected to a rigid body motion. The macroscopic shear strain of the specimen is due to the relative displacement of the different yarns, as is the case in an articulated four-bar mechanism. When the shear angle is large, the shear angle and the geometry of the woven fabric lead to a local transverse compression field (Lomov et al., 2008). The angle at the transition between the two deformation modes is called the locking angle. It is important because wrinkles are likely in a draped fabric if this shear angle is exceeded (Boisse et al., 2006). In practice the transition between the two zones is often progressive and this angle is not strictly defined. In the bias-extension test, the yarns of the central zone are free at their ends. This is a strong advantage for a pure shear test. In the case of the

Modelling the structures and properties of woven fabrics

Shear force Fsh (N mm–1)

0.5

159

Bias extension test

0.4

0.3

0.2

0.1

0 0

10

20

30 40 Shear angle (°)

50

60

4.13 Bias-extension test on an interlock fabric. In plane shear curve and DIC local field analysis before and after locking angle are shown (Lomov et al., 2008).

picture frame test, it is necessary to verify strictly that the test is performed without spurious tensions, which could otherwise strongly disrupt the results of the test because of the high rigidity of the fabric in the direction of the fibres (Launay et al., 2008). Nevertheless, the bias-extension test cannot be performed for very large shear angles because the theoretical behaviour of the specimen can no longer be verified (Zhu et al., 2007)

4.5.3 Bending tests The bending stiffness of a textile is very low in comparison to that of continuous materials such as metals and polymers. This is due to the fact that relative movement of the fibres within the fabric is possible. In many analyses, a membrane assumption (i.e. no bending stiffness) is made for textile materials. Nevertheless, bending stiffness may be important for some applications. In particular, the shape of the wrinkles that develop in some draping processes can depend on the bending stiffness of the fabric. To know the bending stiffness of a textile, experiments must be carried out, since there is no direct relation between tension and bending rigidity, unlike for standard continuous materials. The bending tests usually used for continuous materials, such as three-point or four-point bending tests, cannot be used because of the low bending stiffness. Specific devices have been developed, for instance the KES-FB tester, which measures torque on a bent fabric specimen (Kawabata, 1980). This device has been designed for cloth fabrics and composite reinforcement textiles are too stiff for this

160

Modelling and predicting textile behaviour

apparatus. A bending testing device based on a cantilever test is shown in Fig. 4.14 (de Bilbao et al., 2008). The textile specimen is placed upon lathes which are successively retracted during the test. The specimen is bent under its own weight. The complete test is a succession of quasi-static tests with different lengths and loadings. A digital camera records the shapes of the bent sample. From these measurements, the bending moment can be related to the bending curvature.

4.5.4 Transverse compaction The transverse compaction response is an important property, especially in the case of liquid composite moulding (LCM) processes (Comas-Cardona et al., 2007; Kelly, 2008). In these processes, the fibrous reinforcement is compacted under a load transmitted thought the mould, and then the resin is injected through the compacted material. The response of the textile reinforcement to this transverse load is critical to ensuring the quality of the process. Compaction experiments can be made on a single yarn, a thin fabric, several fabric layers or thick textiles (interlock or 3D). Figure 4.15 shows a transverse compaction device and the curves obtained for the G1151® (the unit cell of which is shown in Fig. 4.2a). As it is difficult to set the position of the start of the test, results give the transverse pressure on the fabric versus the fibre volume fraction. To determine the transverse behaviour of a single yarn, an inverse approach using a mesoscopic 3D finite element analysis is usually used because it is very difficult to obtain a constant strain state on the yarn (Gasser et al., 2000).

4.14 Flexometer (de Bilbao et al., 2008; de Bilbao 2008).

Modelling the structures and properties of woven fabrics

161

Force measurement

Displacement measurement

f 100 mm cylinder

Revolute joint (a) 250

Load (N)

200 150 100 50 0 0

10

20 30 Fibre volume ratio (%) (b)

40

50

4.15 (a) Transverse compaction device of a glass plain weave (Bigorgne, 2008); (b) typical loading curve.

4.6

3D simulation of the deformation of the unit woven cell at the mesoscopic level

In 3D mesoscopic-level simulations, the woven reinforcement is considered at the scale of the unit (representative) woven cell. The corresponding domain depends on the nature of the fabric. The yarn material is considered as a continuous medium and exhibits specific fibrous behaviour. For analytical accuracy at the mesoscopic level, the geometrical description of the elementary pattern is of great importance. Various procedures can be used to create finite element models that guarantee a good geometrical consistency, as mentioned above (see Section 4.4). It is important that the boundary conditions applied to the domain studied are representative of the whole fabric. The boundary conditions must be able to translate into both symmetry and periodicity conditions.

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Modelling and predicting textile behaviour

Mesoscopic simulations have different applications. They provide interesting tools for developing physics-based models. They give the solid skeleton that is used by flow algorithms to evaluate permeability tensors for various geometrical configurations. They can also be used as virtual tests to estimate the mechanical behaviour of the fabric at the macroscopic level without performing experimental tests. In this section, we first describe a constitutive model of the material constituting the yarns with its specificities. We then examine boundary conditions that allow the periodicity of the fabric to be taken into account and, finally, give the analyses of some loading cases that correspond to the characterization tests: biaxial tension, in-plane shear, compression and permeability.

4.6.1 Constitutive model Considering a single yarn and based on experimental observations (Potluri et al., 2006; Badel et al., 2008b), the fibre bundles constituting the yarn material can be considered to be transversly isotropic. A covariance analysis in different directions and for different deformations justifies this assumption. The main specificities of such a material lie in its strong stiffness in the fibre direction, which can differ from transverse stiffness in a ratio greater than several hundreds (Gu, 2007). In the same manner, in-plane and out-of-plane shear stiffnesses are very small (Potter, 2002; Sun and Pan, 2005). Strong anisotropy means that the mechanical model strictly follows the material directions in order to calculate stresses or stress increments in the material frame. Hypoelastic models (also called rate constitutive equations) have been proposed for material at high strain (Truesdell 1955; Xiao et al., 1997; Belytschko et al., 2000):

s— = C : D

[4.1]



where D and C are the strain rate tensor and the constitutive tensor, — respectively. s , called the objective derivative of the stress tensor, is the

derivative for an observer who is fixed with respect to the material:

s— = Q ·

( ddt (Q

T

)

· s · Q ) · QT



[4.2]

Q is the rotation from the initial orthogonal frame to the so-called rotating frame where the objective derivative is made. The most common objective derivatives are those of Green-Naghdi (Dienes, 1979) and Jaumann (Dafalias, 1983). They use the rotation of the polar decomposition of the deformation

Modelling the structures and properties of woven fabrics

163

gradient tensor F = R · U (standard in Abaqus explicit) and the corotational frame, respectively. These are routinely used for analyses of metals at high strains. It has been shown that, in the case of a material made of fibres oriented by the vector f1, the proper objective rotational derivative is based on the rotation defined by f1, that is:

F = f i ƒ ei0

[4.3]

where



f1 =

F · e10 F · e10

f2 =

F · e10 – ( F · e 02 · f 1 ) f 1

and f 3 = f 1 ¥ f 2

F · e 02 – ( F · e 02 · f 1 ) f 1



[4.4]

Equation [4.1] is integrated over a time increment Dt = tn+1 – tn using the formula of Hughes and Winget (Hughes and Winget, 1980), widely used in finite element codes at finite strains:

[s n +1 ] f n +1 = [s n ] f n + [ C n +1/2 ] f n +1/2 [De ] f n +1/2 i

i

i

i



[4.5]

where [De ] f n +1/2 = [ D] f n +1/2 D t and [ S] f n stands for the matrix of the i

i

i

components of any tensor S expressed in the basis fi ƒ fj ƒ … ƒ fm at time t n. If several rotations are used to define rotational objective derivatives, it has been shown that, in the case of a material made of fibres, only the rotating basis {fi} based on the fibre rotation is correct (Hagège et al., 2005; Boisse et al., 2005; Badel et al., 2008a; ten Thije et al., 2007). In Equation [4.5], the constitutive matrix [ C] fi is expressed in the basis

{fi} oriented by the fibres at the current time. This is a major advantage because its shape and its components are known or can be determined on this basis where longitudinal and transverse behaviour is distinguished. The first is defined by the stiffness of the fibres in direction f1 (Fig. 4.16), whereas the second characterizes the behaviour of the fibre bundle in-plane (f2, f3). Considering the transverse isotropy assumption leads to splitting the strain in the transverse section into ‘volumetric’ and ‘deviatoric’ parts. As these strains are in a plane, the term ‘surface’ will be used instead of volumetric. This assumption allows us to describe correctly the section shape changes during fabric loading. If the rigidities in the transverse section are small, the strains can be large and play a major role in the macroscopic mechanical behaviour of the woven fabric. Tomography scans show that the two modes

164

Modelling and predicting textile behaviour e02 f1 = f f2

e 01

M

4.16 Rotating frames of the hypo-elastic model.

of deformation of the cross-section can be distinguished as compaction of the network of fibres and rearrangement or change in the shape of the fibre bundle. These two modes correspond to surface and deviatoric transformations of the cross-section: the surface part of the deformation is related to fibre density changes and the deviatoric part to shape changes. In the following, these two modes are considered to be uncoupled. This can be proved in the case of isotropy in the plane (f2, f3). Let us separate longitudinal and transverse deformations within the strain tensor:



È e11 e12 Í [e] fi = Í 0 Í ÍÎ sym.

e13 ˘ È 0 0 ˙ Í e22 0 ˙+Í ˙ Í 0 ˙ Í sym. ˚ Î

0 ˘ ˙ e23 ˙ = [e L ] fi + [e T ] fi ˙ e33 ˙ ˚



[4.6]

where e , here, stands for the tensorial cumulation of DD t in the rotating frame. The following developments only concern the plane (f2, f3) and the restriction of [e T ] fi to (f2, f3):



È e22 e23 [e T ] fi = Í Í sym. e33 Î

˘ È e ˙ = Í s ˙ Í 0 ˚ Î

0 ˘ È ed ˙ + Í es ˙ Í e23 ˚ Î

e23 ˘ ˙ – ed ˙ ˚

[4.7]

Modelling the structures and properties of woven fabrics

165

e22 + e33 e – e33 is the surface strain component and ed = 22 2 2 and e23 are the deviatoric ones. This formalism, where surface and deviatoric parts are uncoupled, is widely used in plasticity models. It has also been used in fibre bundle micromechanics in Simacek and Karbhari, (1996). For the moment, the model is considered as non-linear elastic, thus the same separation is valid for stresses. Moreover, since integration is a linear operation, this separation also holds for strain and stress increments [De T ] fi and [Ds T ] fi . Thus the suggested uncoupling leads to the following where es =

relations:

Dss = ADes



Dsd = BDed



Ds23 = CDe23

[4.8]

where Dss, Dsd, Des and Ded are respectively surface and deviatoric stress and strain increments (stress components being defined in the same way as strain components in Equation [4.7]) and A, B, C are elastic coefficients. It can be shown that B = C. Eventually the transverse constitutive tensor (used in Equation [4.5]) contains two independent elastic coefficients, which correspond to an isotropic two-dimensional medium. Using Voigt notation:

 T]f [C i

È (A + B )/2 (A – B )/2 Í = Í (A – B )/2 (A + B )/2 Í 0 0 ÍÎ

0 ˘ ˙ 0 ˙ ˙ B ˙˚

[4.9]

To complete the description of this model, the form of the coefficients A and B must be specified. For this, basic physical assumptions are used. Under compaction, the material becomes stiffer in both surface and deviatoric behaviours owing to a densification of the fibre network (up to the maximum achievable density). Note also that under longitudinal tension, the surface behaviour should be stiffer. The influence of longitudinal tension on the deviatoric behaviour is not as easy to anticipate but can be assumed to be weak. For the sake of simplicity, it is considered in the present model that tension has no influence on deviatoric behaviour. From these assumptions, it is suggested to give the following form to coefficients A and B: A = A0 e– pes ene11

B = B0 e– pes



[4.10]

Finally, the transverse constitutive model requires four parameters. The

166

Modelling and predicting textile behaviour

determination of these parameters is explained in Badel et al. (2008b). Several other laws have been proposed for fabric transverse behaviour (Cai and Gutowski, 1992; Chen and Chou, 1999; Comas-Cardona et al., 2007; Kelly, 2008), but most of them concern compaction of the fabric and they are often one-dimensional models.

4.6.2 Periodicity and symmetry boundary conditions In the mesoscopic analyses, only a volume representative of the whole fabric is studied. Consequently, the choice of correct boundary conditions is important. Analysing the deformation of a periodic medium from its representative unit cell means imposing kinematic boundary conditions on the representative unit cell (RUC) in order to guarantee this periodicity (Miehe and Dettmar, 2004; Badel et al., 2007) (Fig. 4.17). If the periodicity is planar, translation vectors P can be expressed by linear combinations of two elementary vectors P1 and P2: 2



P = S ma Pa a =1

ma Œ 

[4.11]



The boundary ∂V of the RUC can be split into two pairs {∂Va– , ∂Va+ )a = 1, 2 . Let us consider two points of the boundary, images of each other by an elementary translation Pa (a = 1 or 2). Let them be called paired points. Owing to periodicity:

Xa– Œ ∂Va– ; Xa+ Œ ∂Va+

Xa+ – Xa– = Pa

[4.12]

The transformation ϕ (X) of the structure can be decomposed into a macroscopic (or average) part ϕm (X) and a periodic (or local) fluctuation

B B

A

A

4.17 Conditions to be fulfilled to guarantee correct periodicity conditions.

Modelling the structures and properties of woven fabrics

167

w(X). To ensure the periodicity of the deformed structure, we need to verify:

w(Xa– ) = w(Xa+ ) for paired points



(Xa– ) Œ ∂Va– and Xa+ Œ ∂Va+ , a = 1 or 2

[4.13]

As a consequence, in the deformed configuration:

Xa+ – Xa– = jm ( Xa+ ) – jm ( Xa– )



[4.14]

Since ϕm is known, Equation [4.14] is a kinematic condition on each point of the boundary. This periodicity condition is used in the example shown in Fig. 4.18 for plain weave and for 2 ¥ 2 twill weave. A difficulty exists when the geometrical boundary of the RUC is not made of material, because in this case it cannot be applied. In the plain weave example shown in Fig. 4.19, the first type of simulation (called RUC type 1) makes ensuring the periodicity and boundary conditions simpler because no contact change can occur on the frontier. This is particularly critical for in-plane shear analysis (Badel et al., 2007).

4.6.3 Biaxial tension Mesoscopic modelling of biaxial tension tests is useful for evaluating the effect of undulation on fabric behaviour, but it is also helpful for analysing transverse crushing (Gasser et al., 2000; Carvelli et al., 2008). Figure 4.20 shows the results of simulating the unit cell of a 2 ¥ 2 twill submitted to biaxial tension for different warp–weft strain ratios k. The results of the simulation are compared to experimental results given by a biaxial tensile test as described in Section 4.5 (Fig. 4.11). The transverse compaction

(a)

(b)

(c)

4.18 Deformed RUC for (a) plain weave at shear angle of 28°, (b) plain weave at shear angle of 54°, (c) 2 ¥ 2 twill weave.

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Modelling and predicting textile behaviour (a) RUC type 1

(b) RUC type 2

4.19 Undeformed and deformed RUC for plain weave (a) allowing consistent periodicity boundary conditions; (b) difficult to apply consistent boundary conditions.

strain can be large, especially when k = 1 (Fig. 4.20c). Because of the large compaction strain associated with tensions in the yarn direction, and because the compaction stiffness depends on the tensile state, this biaxial tensile test is used to identify the transverse law coefficients (defined in Equation [4.10] by an inverse approach.

4.6.4 In-plane shear As previously mentioned, one of the most popular in-plane shear tests is the picture frame test. Simulating a pure shear test is not easy, but it allows us to estimate the shear response of particular textile reinforcements and avoid having to perform a large number of shear tests. Moreover, it may be helpful to understand the mechanical behaviour of a fabric before manufacturing and this method can provide local mesoscopic level results such as crushing and section changes, which are difficult to obtain from experiment. Figure 4.18(a) and (b) showed the deformed representative unit cell at shear angles of 28° and 54° respectively, that is before shear locking and at the end of the transformation. It is clear that shear stiffness in the second stage is related to lateral crushing of the yarns as the square geometry becomes rhomboid. Transverse behaviour plays a major role, particularly when the locking angle is reached. Figure 4.21 compares the deformed shape obtained by tomography against that computed for a glass plain weave submitted to a 46° shear angle. The shape of the transverse section changes greatly when the shear is applied, but the experimental and computed shapes are in good agreement.

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Warp yarn Weft yarn (a) 300 k = 0.5 k=1

250 Load (N/arn)

Yarn 200 k=2

150

Free warp

100 50 0 0

0.1

0.2

0.3

0.4 0.5 Strain(%) (b)

0.6

0.7

0.8

10%

22%

30% (c)

4.20 Biaxial tension on a 2 ¥ 2 twill (a) RUC meshing; (b) comparison between tests and simulations; (c) logarithmic transverse shear strain (e33) for k = 1.

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Modelling and predicting textile behaviour Experiment

Simulation boundary

Initial section

(a)

(b)

4.21 Pure shear. Comparison of simulation with tomograph scans: (a) deformed shape of the unit cell, (b) set of yarn cross-sections along half a period of the yarn.

4.6.5 Transverse compression While the crushing of the yarns plays a major role in the global response of a woven fabric to large shear angles, experimental observations of fabric compressions are difficult to obtain (as mentioned in Section 4.5). Mesoscopiclevel simulations allow us to predict fabric response to a transverse compression (see Fig. 4.22). Such loading is very important as all the LCM processes include a compaction phase before the matrix is injected in order to ensure correct mixing between the reinforcement and the polymer matrix. A good permeability matrix calculation requires the correct compaction simulation, which we develop below.

4.6.6 Permeability computations An important application of mesoscopic simulations is to calculate the permeability tensors for computing resin flow in a preform using Darcy’s law. The position and density of the solid reinforcement at the end of the draping phase constitute the input data for the Stokes (or Stokes and Brinkman) flow calculation. While some methods have been proposed to leapfrog this step using approximate geometries (Belov et al., 2004; Verleye et al., 2006;

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Compression stress (MPa)

20 15 10 5 0 –5

15

20

25 30 35 40 45 Fibre volume fraction (%)

50

4.22 Simulations of transverse compression of plain weave fabric showing influence of relative yarn positions on the compaction (Bigorgne, 2008). pa

0.8 mm

pb e3

e2

e3

e1

e2 e1

(a)

e3

(a) p ¢a

e2

e3

e1

p¢b

e2 e1

(b)

(b)

4.23 Solid and fluid simulations for permeability simulations (a) undeformed pattern; (b) deformed pattern (Loix et al., 2008).

Demaria et al., 2007), calculations of the flow based on the geometry obtained by mesoscopic analysis is important because it takes into account the actual deformed shape of the fabric, as shown in Fig. 4.23 (Loix et al., 2008).

172

4.7

Modelling and predicting textile behaviour

Image analyses: Full field digital image correlation measurements and X-ray tomography

The two types of image processing described here are widely different. Full field digital image correlation (DIC) is a way of measuring displacement and strain fields, while X-ray tomography is a 3D tool for non-destructive observation of samples. In the first case, only the surfaces of the observed specimens are examined, whereas in the second case, the technology allows us to see what is happening inside the fabric before, during and after load application.

4.7.1 Full field digital image correlation measurements Developed in the early 1980s (Sutton et al., 1983), DIC has been widely used to measure strain fields at various scales (Touchal et al., 1997). The method allows us to obtain full strain fields quickly in the case of woven fabrics (Dumont et al., 2003; Lomov et al., 2008; Cao et al., 2008). One particular interest of the technique lies in the fact that it allows a multi-scale observation of the same sample (Fig. 4.24). Depending on the size of the observed zone, microscopic or macroscopic strain fields are obtained. The main use of full field DIC measurements is the precise identification of the shear locking angle during in-plane shear tests, but there are multiple other uses, including ensuring strain homogeneity during a test and explaining the physics of a deformation.

4.7.2 X-ray tomography X-ray tomography is analogous to a medical scanner and uses X-ray radiography to reconstruct the internal structure of an opaque material non-destructively. This is its main advantage in comparison to standard microscopic imaging, which necessitates using a matrix to freeze the actual configuration and cutting the sample. The same sample can be imaged at various stages of loading. The reconstruction involves a computed step and the final image is a 3D map of the local X-ray attenuation coefficient. Since fibres absorb far more X-rays than air, the attenuation contrast permits a straightforward subsequent thresholding of the images. X-ray tomography can be used at a mesoscopic level (the scale of the yarns) to determine the initial and deformed geometry of fibrous composite reinforcements and at a microscopic level (the scale of the fibres) to determine the density and distribution of fibres within the yarn. The information gathered from these experiments can be used to improve and justify the assumptions made during the development of the mechanical constitutive model and,

Dq1 = –0.1°

Dq1 = 1.7°

Dq2 = 1.7°

Dq2 = 2.1°

Dg = 1.8° Dg = 3.7°

(d)

(c) Dq1 = 1.8°

Dq1 = –0.2°

Dq2 = 2.0°

Dq2 = 3.2°

Dg = 3.8°

Dg = 3.4°

Mesoscale measurements

4.24 Local and global field measurement during a bias extension test.

Macroscale measurements

Modelling the structures and properties of woven fabrics

(b)

(a)

173

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Modelling and predicting textile behaviour

above all, to validate the results obtained from simulation. At the microscopic scale, X-ray tomography allows us to see the real fibre distribution within the fibre bundle before or during the load application (Badel et al., 2008b) (Figs 4.21 and 4.25).

4.8

Conclusions and future trends

Woven fabrics are multiscale materials, having very specific macroscopic mechanical behaviour owing to their mesoscopic (woven yarn) structure, with the yarns themselves made of thousands of fibres at the microscopic level. Modelling the mechanical behaviour of woven fabrics is carried out at the three different levels. Continuous models address the macroscopic scale and take the specificities of the textile material into account. They are currently the most common approach in finite element analyses of fabric deformation, although there is currently no universally accepted model. Improvements in computing allow us to perform discrete analysis at mesoscopic or microscopic levels. The main interest of these approaches is in the natural description of the meso- or macroscopic architecture and consequently of the related specificities of the mechanical behaviour of textiles. At present, these discrete analyses are limited to describing small domains such as one or a few unit cells. Future computing improvements and increasing efficiency of the numerical scheme will probably make it possible to perform draping analyses of a full woven fabric sheet. Image analysis is also an important potential source of advances in fabric behaviour analysis. Both DIC and X-ray tomography are very well suited for use on textile materials. They are the preferable, and sometimes only, method for measuring deformation at the surface (DIC) or within a textile (X-ray tomography). They can be used at different scales depending on the measurement being addressed. The use of these techniques for studying the

(a)

(b)

4.25 X-ray tomography image of a yarn (a) before and (b) during tensile loading.

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mechanics of materials is rather new. It is very probable that these techniques will permit us to understand and model the mechanical behaviour of woven fabrics better at different scales in future.

4.9

References

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Gasser A., Boisse P. and Hanklar S. (2000). Mechanical behaviour of dry fabric reinforcements. 3D simulations versus biaxial tests’, Comput Mater Sci, 17, 7–20. Gelin J.C., Cherouat A., Boisse P. and Sabhi H. (1996). ‘Manufacture of thin composite structures by the RTM process: Numerical simulation of the shaping operation’, Composites Sci Technol, 56, 711–18. Gu H. (2007). ‘Tensile behaviours of woven fabrics and laminates’, Mater Design, 28, 704–7. Hagège B., Boisse P. and Billoët J.L. (2005). ‘Finite element analyses of knitted composite reinforcement at large strain’, Eur J Comput Mech, 14, 767–76. Hamila N. and Boisse P. (2007). ‘A Meso–Macro Three Node Finite Element for Draping of Textile Composite Preforms’, Appl Composites Mater, 14, 235–50. Hamila N. and Boisse P. (2008). ‘Simulations of textile composite reinforcement draping using a new semi-discrete three node finite element’, Composites B, 39, 999–1010. Hivet G. and Boisse P. (2005). Consistent 3D geometrical model of fabric elementary cell. Application to a meshing preprocessor for 3D finite element analysis’, Finite Elements in Anal Design, 42, 25–49. Hivet G. and Boisse P. (2008). ‘Consistent mesoscopic mechanical behaviour model for woven composite reinforcements in biaxial tension’, Composites B, 39, 345–61. Hofstee J. and van Keulen F. (2001). 3-D geometric modelling of a draped woven fabric – Composite Structures, 54, 179–95. Holzapfel G.A. and Gasser T.C. (2001). ‘A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computational aspects and applications’, Comput Methods Appl Mech Eng, 190, 4379–430. Hughes T.J.R. and Winget J. (1980). ‘Finite rotation effects in numerical integration of rate constitutive equations arising in large deformation analysis’, Int J Numerical Methods Eng, 15, 1862–7. ITOOL ‘Integrated Tool for Simulation of Textile Composites’, European Specific Targeted, Research Project, Sixth framework programme, Aeronautics and Space, http://www.itool.eu. Kawabata S. (1980). The Standardization and Analysis of Hand Evaluation, The Textile Machinery Society of Japan, Osaka, Japan. Kawabata S. (1989). ‘Nonlinear mechanics of woven and knitted materials’ in: Textile Structural Composites, vol. 3, T.W. Chou and F.K. Ko (eds), Elsevier, Amsterdam, 67–116. Kawabata S., Niwa M. and Kawai H. (1973). ‘The finite deformation theory of plain weave fabrics’, J Textile Inst, 64, 21–85. Kelly P.A. (2008). ‘A compaction model for liquid composite moulding fibrous materials’, Proceedings of the 9th International Conference on Flow Processes in Composite Materials, Montréal (Québec), Canada. King M.J., Jearanaisilawong P. and Socrate S. (2005). A continuum constitutive model for the mechanical behavior of woven fabrics, Int J Solids Structures, 42, 3867–96. Lahey T.J. and Heppler G.R. (2004). ‘Mechanical modeling of fabrics in bending’, Trans ASME, 71, 32–40. Launay J., Lahmar F., Boisse P. and Vacher P. (2001). ‘Strain measurement in tests on fibre fabric by image correlation method’, Adv Composites Lett, 11–1, 7–12. Launay J., Hivet G., Duong A.V. and Boisse P. (2008). Experimental analysis of the influence of tensions on in plane shear behaviour of woven composite reinforcements’, Composites Sci Technol, 68, 506–515. Loix F., Badel P., Orgéas L., Geindreau C. and Boisse P. (2008). ‘Woven fabric permeability:

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5

Modelling of nonwoven materials

N. Mao and S. J. Russell, University of Leeds, UK

Abstract: Nonwovens are porous fibre assemblies that are engineered to meet the technical requirements of numerous industrial, medical and consumer products. In this chapter, the properties of nonwoven fabrics that govern their suitability for use in various applications are briefly summarised, and examples of analytical and empirical models that link nonwoven fabric properties and structural parameters are given. The fabric structural parameters include fibre dimensions and properties, fibre alignment, the structural properties of the bond points, the pore structure and fabric porosity, and the fabric dimensions and variation. Key words: analytical, empirical, fabric, model, nonwovens, property, structure.

5.1

Introduction

Nonwovens are porous fibre assemblies that are engineered to meet the technical requirements of numerous industrial, medical and consumer products. Nonwoven applications span single use and durable products including wipes and absorbent hygiene goods, filter media, protective clothing, wound dressings, tissue engineering scaffolds, insulation, geosynthetics, automotive interiors and floorcoverings. A nonwoven is defined as a manufactured sheet, web or batt of directionally or randomly orientated fibres, bonded by friction, and/or cohesion and/or adhesion, excluding paper and products which are woven, knitted, tufted, stitch-bonded incorporating binding yarns or filaments, or felted by wet-milling, whether or not additionally needled.1 To reflect the rapid development of the industry, an update of this standard definition has been proposed to the International Standardisation Organization (ISO) by EDANA (www.edana. org) and INDA (www.inda.org) as follows: A nonwoven is a sheet of fibres, continuous filaments, or chopped yarns of any nature or origin, that have been formed into a web by any means, and bonded together by any means, with the exception of weaving or knitting. Felts obtained by wet milling are not nonwovens. Wetlaid webs are nonwovens provided they contain a minimum of 50% of man-made fibres or other fibres of non-vegetable origin with a length to diameter ratio equal or superior to 300, or a minimum of 30% of man-made fibres with a length-to-diameter ratio equal or superior to 600, and a 180

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maximum apparent density of 0.40 g cm–3. Composite structures are considered nonwovens provided their mass is constituted of at least 50% of nonwoven as per the above definitions, or if the nonwoven component plays a prevalent role. The performance of a nonwoven fabric depends on its chemical, physical and mechanical properties, which is influenced by its composition and structure. The structure of the fabric is largely determined by the manufacturing methods used to produce the fabric and the associated process parameters. In the design and engineering of nonwoven fabrics, it is normally desirable to predict the performance and properties based on the fibre and other component properties and the structural parameters of the fabric. This involves establishing relationships between the properties of the fabric and its composition and structure. It is also necessary to understand the link between manufacturing process parameters and the resultant fabric structure. Therefore, the objective of modelling nonwoven materials is to obtain quantitative process–structure–property–performance relationships using various mathematical and computational techniques. The properties of nonwoven materials can be categorised as follows: ∑

∑

∑ ∑

∑

Mechanical properties: tensile (Young’s modulus, tenacity, elasticity, work of rupture), compression and compression recovery, bending and shear rigidity, tear resistance, burst strength, crease resistance, abrasion, frictional properties (smoothness, roughness, friction coefficient) and energy absorption. Fluid handling properties: permeability, liquid absorption (liquid absorbency, penetration time, wicking rate, rewet, bacteria/particle collection, repellency and barrier properties, run-off, strike time), water vapour transport and breathability. Physical properties: thermal and acoustic insulation and conductivity, electrostatic properties, dielectric constant and electrical conductivity, opacity and others. Chemical properties: surface wetting angle, oleophobicity and hydrophobicity, interface compatibility with binders and resins, chemical resistance and durability to wet treatments, flame resistance, dyeing capability, flammability and soiling resistance, Application specific performance: linting (particle generation), aesthetics and handle, filtration efficiency, biocompatibility, sterilisation compatibility, biodegradability and many others.

The properties of nonwoven fabrics that govern their suitability for use in various applications depend on the properties of the composition and the fabric structure. The composition in this context refers to the fibres as well as chemical binders, fillers and finishes that may be present in the fabric. The fabric structural parameters include the pore structure and porosity, fibre

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alignment, fabric dimensions and variation and the structural properties of the bond points. Some of the major parameters of a nonwoven fabric are listed below. ∑

∑ ∑ ∑

∑

Fibre dimensions and properties: fibre diameter, diameter variation, cross-sectional shape, crimp wave frequency and amplitude, fibre length and distribution, density, fibre properties (Young’s modulus, elasticity, tenacity, bending and torsional rigidity, compression, friction coefficient), fibrillation propensity, surface chemistry and wetting angle. Fabric dimensions and variation: dimensions (length, width, thickness and weight per unit area), dimensional stability, mass density and thickness uniformity. Fibre alignment: fibre orientation distribution. Structural properties of bond points: bonding type, shape, size, bonding area, bonding density, bond strength, bond point distribution, geometrical arrangement, the degree of liberty of fibre movement within the bonding points, interface properties between binder and fibre; surface properties of bond points. Porous structural parameters: fabric porosity, pore size, pore size distribution, pore shape and connectivity.

Various techniques are employed to model and simulate the structure and properties of nonwovens. The output may be physical, mathematical, visual (image), verbal or take the form of an artificial intelligence (or simulation) model. In this chapter, we concentrate on mathematical models in nonwovens, which are widely used to provide explicitly quantitative relationships between variables. Types of mathematical models available for nonwoven materials might be further classified as analytical and empirical models, continuous and numerical (discrete) models, dynamic models and artificial intelligence models. Analytical models are helpful in providing insights in to proposed mechanisms or interactions; they can show whether a mechanism is at least theoretically feasible and help to suggest experiments that might further elucidate and discriminate the influence of individual variables on fabric properties. In this chapter, we consider models of nonwoven fabric structure and introduce examples of analytical and empirical models that link nonwoven fabric structure and properties.

5.2

Constructing physical models of nonwoven structure

Nonwoven structures are so complex and diverse in terms of their geometry and uniformity that it is impossible quantitatively to describe real nonwoven structures accurately in every detail. Therefore, models of the physical

Modelling of nonwoven materials

183

structure employed in existing analytical models of nonwoven fabric properties are always simplified depending on the properties of the fabric that are targeted. Such physical models of nonwoven structure make assumptions, for example: ∑ ∑ ∑ ∑ ∑

whether the nonwoven fabric is homogeneous or heterogeneous, whether the nonwoven structure is isotropic (or random) or anisotropic, if the constituent fibres are cylindrical in cross-section or have some other shape, if the pores in the fabric structure are cylindrical, if the arrangement of the fibres (or the network of the pores) are in periodic patterns.

One of the main factors frequently considered in the simplification of nonwoven structure is the fibre orientation distribution. The fibre orientation in nonwoven fabrics refers to the fibre orientation angle as shown in Fig. 5.1. In the two-dimensional (2D) fabric plane, the fibre orientation angle is defined as the relative directional position of individual fibres in the structure relative to the machine direction (MD). The frequency distribution (or statistical function) of the fibre orientation angles in a nonwoven fabric is called the fibre orientation distribution (FOD) or ODF (orientation distribution function). The following general relationship is proposed for the fibre orientation distribution in a 2D web or fabric:

Ú

p 0

W (a ) da = 1 (W (a ) ≥ 0)

where a is the fibre orientation angle and W(a) is the fibre orientation distribution function in the examined area.

Cross direction

Centroid of fibre

Z Fibre orientation angle

Fibre

b

a

Machine direction (a)

a Y

X

(b)

5.1 Fibre orientation angle in (a) two- and (b) three-dimensional nonwoven fabrics.

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Modelling and predicting textile behaviour

Typically, simplified models of nonwoven fabric structure assume either an isotropic (or random) fibre alignment, unidirectional fibre bundles, orthotropic fibre alignment or unidirectional capillary tubes. In all these models, the nonwoven fabric is assumed to be homogeneous. The fibre orientation distribution is an important element in the modelling of fabric tensile properties,2, 3 bending rigidity,4 directional permeabilities5, 6 and directional capillary pressure,7 where the fibre orientation distribution in two and three- dimensions may be considered. The majority of fibres in nonwoven fabrics are aligned in the fabric’s x–y planes. In certain processes such as needling, a small proportion of fibres are oriented throughplane (z), which influences fabric properties. The direct measurement of the three-dimensional (3D) fibre orientation distribution in a fabric can be complex and expensive8 and, therefore, the structure of a 3D nonwoven can be simplified into a combination of 2D layers connected by fibres oriented perpendicular to the fabric plane (Fig. 5.2). The fibre orientation in such a 3D fabric can be described by 2D fibre orientation distributions in the fabric plane.9 In addition to standard microscopy techniques, methods such as microcomputed X-ray tomography and digital volumetric imaging (DVI) provide detailed information about the internal 3D structure of nonwovens that can be used directly for modelling purposes. In constructing models of nonwoven fabric structure, another factor to consider is the nature of the bond points. Assumptions are made about their type, shape, rigidity, size, density and distribution. These bond points can be grouped into two categories: rigid solid bonds and flexible elastic joints, the prevalence of which in the fabric depends on choice of manufacturing Z

X

Y

5.2 Simplified 3D nonwoven structure.

Modelling of nonwoven materials

185

process. The bond points in a mechanically bonded fabric (e.g. needled and hydroentangled) are formed by mechanical entanglement of fibres. These bonds are flexible and the component fibres are able to slip or move within the bonding points to some extent. By contrast, the bonds in thermally pointbonded and chemically bonded fabrics are formed by adhesion or cohesion between polymer surfaces, in which a small portion of the fibrous network is firmly bonded and the fibres have little freedom to move within the bond points. The bond points in thermoplastic spunbond and through-air thermally bonded fabrics are formed by melting polymer surfaces to produce bonding at fibre cross-over points and the fibres associated with these bonds cannot move individually. In meltblown fabrics, the fibres are usually not so well bonded together as in thermally bonded spunbond fabrics and in some applications, the large surface area is sufficient to give the web acceptable cohesion without the need for thermal, chemical or mechanical bonding. Stitch-bonded fabrics are stabilised by knitting fibres or yarns through the web and the bonding points are flexible but connected together by these stitched regions. The number, size and characteristics of bond points are influenced by fabric manufacturing parameters. For example, in needling, the size of the needle barb throat in relation to the fibre diameter, punch density and number of barbs in the batt on the downstroke; in hydroentangling, specific energy, jet dimensions, forming surface type; in thermal calendar bonding, the land area, pressure and temperature; and in chemical bonding, the method of binder application, for example full saturation, spray or printing and binder viscosity. The rigidity of solid bond points in most nonwoven materials can be physically characterised by the resulting tensile properties, for example, strength and elasticity, while the degree of bonding may also be determined by microscopic analysis of the fabric cross-section. In the cross-section of mechanically bonded fabrics, specifically needlepunched and hydroentangled fabrics, the depth of bent fibre loops in the bonding points can be determined, which, together with the number of bonding points, can be associated with the level of bonding in the fabric.10

5.3

Modelling of pore size and pore size distribution in nonwoven fabrics

Nonwovens are typical porous materials and the pore structure may be characterised in terms of the total pore volume (or porosity), the pore size, pore size distribution and the pore connectivity. Porosity provides information on the overall pore volume and is defined as the ratio of the non-solid volume (voids) to the total volume of nonwoven fabric. The volume fraction of solid material is defined as the ratio of solid fibre material to the total volume of

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Modelling and predicting textile behaviour

the fabric. While the fibre density is the weight of a given volume of the solid component only (i.e. not containing other materials), the porosity can be calculated as follows using the fabric bulk density and the fibre density:

f (%) =

rfabric ¥ 100% rfibre

e (%) = (1 – f) ¥ 100%

where e is the fabric porosity (%), f is the volume fraction of solid material (%), rfabric (kg m–3) is the fabric bulk density and rfibre (kg m–3) is the fibre density. Although it is arguably inappropriate to use the term ‘pore’ to describe a void in a highly connective high-loft nonwoven fabric, it is an indication of the network of void spaces in the structure and relates to the combination of fabric porosity and fibre dimensions.

5.3.1 Models of pore sizes The pore size of simplified nonwoven fabrics can be approximately estimated by Wrotnowski’s model11, 12 (Fig. 5.3) based on the assumption that fibres are circular in cross-section, straight, parallel, equidistant and arranged in a square pattern. The pore size (2r) can be obtained using the following equation:

Ê ˆ d r = Á 0.075737 Tex ˜ – f 2 rfabric ¯ Ë

d

df

2r

5.3 Wrotnowski’s model for pore size in a bundle of parallel cylindrical fibres arranged in a square pattern.

[5.1]

Modelling of nonwoven materials

187

where Tex = fibre linear density (tex), rfabric is the fabric density (g cm–3) and df is the fibre diameter (m). Several other models linking pore size, fibre diameter and fabric porosity are applicable to nonwoven materials. For example, both the largest pore size (2rmax) and the mean pore size (2r) can be predicted by Goeminne’s equation13 as follows: df 2(1 – e )



largest pore size (2rmax): rmax =



mean pore size (2r) (porosity < 0.9): r =

df 4(1 – e )

When nonwovens are modelled as assemblies of parallel and identical cylindrical capillary tubes, the pore size (2r) can be obtained from the fabric permeability based on Hagen–Poiseuille’s equation:14 [5.2] r = 8k where k is the specific permeability (m2) in Darcy’s law and e is the fabric porosity (%).

5.3.2 Models of pore size distribution Pores in nonwoven fabrics are not identical but are represented by a distribution of various sizes. If it is assumed that the fibres are randomly aligned in a nonwoven fabric following Poisson’s law, then the probability, P(r), of a circular pore of known radius, r, is distributed as follows:15

P(r) = –(2 pn¢) exp (– pr2 n¢)

[5.3]

where n¢ = 0.36/r2, and it is defined as the number of fibres per unit area. Giroud16 proposed a theoretical equation for calculating the filtration pore size of nonwoven geotextiles based on the fabric porosity, fabric thickness and fibre diameter as follows:



xe df ˘ È Of = Í 1 –1+ d (1 – e ) h ˙˚ f Î 1– e

[5.4]

where df is the fibre diameter, e is the porosity, h is the fabric thickness, x is an unknown dimensionless parameter to be obtained by calibration with test data to account for the further influence of geotextile porosity, where x = 10 for some experimental results; and Of is the filtration opening size that is usually given as the near largest constriction size of the fabric (e.g. O95).

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Modelling and predicting textile behaviour

Lambard17 and Faure18 applied Poissonian line network theory to establish a theoretical model of the ‘opening sizing’ of nonwovens. In this model, the fabric thickness is assumed to consist of randomly stacked elementary layers. Each layer has a thickness Te and is simulated by two dimensional straight lines (a Poissonian line network). Faure et al.19 and Gourc and Faure20 also presented a theoretical technique for determining constriction size based on a Poissonian polyhedral model. An epoxy-impregnated nonwoven specimen was sectioned and the fabric was modelled as a pile of elementary layers, in which fibres were randomly distributed in planar sections of the fabric. The cross-sections were obtained at a thickness of fibre diameter df and a statistical distribution of pores was calculated by inscribing a circle into each polygon defined by the fibres (Fig. 5.4). The pore size distribution, which is equivalent to the probability of passage of different spherical particles (similar to glass beads in a dry sieving test) through the layers forming the fabric, can be determined theoretically as follows:19

Ê 2 + l (d + df ) ˆ Q (d ) = (1 – f ) Á Ë 2 + l df ˜¯

2N

e– l Nd



[5.5]

(1 – f ) where l = 4 and N = T p df df Q(d) is the probability of a particle with a diameter d passing through a pore channel in the fabric, f is the fraction of solid fibre materials in the fabric, l is the total length of straight lines per unit area in a planar surface (also termed as the specific length) and N is the number of slices in a cross-sectional image. Because of the assumption in Faure’s approach that the constriction size at relatively high fabric thicknesses tends to approach zero, this model generally produces lower values18 than would be expected.

T df

5.4 Model for constriction pore size in a nonwoven fabric consisting of randomly stacked elementary layers of fibres.

Modelling of nonwoven materials

189

The use of this method is not recommended for geotextiles with a porosity of 50% or less.21

5.4

Tensile strength

Backer and Petterson2 pioneered a fibre network theory for estimating the tensile properties of a nonwoven fabric based on the fibre orientation, fibre tensile properties and the assumption that fibre segments between bonds were straight. Hearle and Stevenson3 expanded this theory by taking account of the effects of fibre curl. They indicated that the stress–strain properties of the fabric were dictated by the orientation distribution of fibre segments. Later, Hearle and Ozsanlav22 developed a further theoretical model that accounted for binder deformation. The fibre orientation distribution (FOD) is an essential parameter in the construction of these models. When the fibres in a nonwoven fabric are assumed to lie in layers parallel to the 2D fabric plane, the prediction of the stress–strain curve under uniaxial extension can be established based on three approaches: orthotropic models, the force analysis method for small strains and the energy analysis method based on the elastic energy absorption model.

5.4.1 Orthotropic models of tensile strength2 Models based on tensile properties in principal directions The model is based on the assumption that the deformation of a 2D nonwoven fabric is analogous to that of a 2D orthotropic woven fabric where stress–strain relationships are known for the two principal directions of the fabric (MD and CD). It is also assumed that the following material properties are known: ∑ ∑ ∑

elastic modulus (EX, EY) in both of the two principal directions shear modulus (GXY) between the two principal directions e ˆ Ê Poisson’s ratio Á vXY = 1 = X ˜ in the two principal directions. Ë eY ¯ vYX

s (q ) = E (q ), e (q ) the fabric modulus, E(q), and the Poisson’s ratio v(q) in the direction q are given as follows: For a unidirectional force on the fabric with a small strain,



1 = e (q ) = cos 4 q + Ê 1 – 2vXY ˆ cos 2 q sin 2 q + sin 4 q ÁË G E (q ) s (q ) EX E X ˜¯ EY XY

[5.6]

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Modelling and predicting textile behaviour

v (q ) = –

e (q ) e q + p 2

(

)

(cos 4 q + sin 4 q ) Ê 1 1 1 ˆ 2 2 ÁË E + E – G ˜¯ cos q sin q – vYX EX X Y XY =– 4 cos q + Ê 1 – 2vXY ˆ cos 2 q sin 2 q + sin 4 q ÁË G EY EX E X ˜¯ XY



[5.7]

Models based on fibre orientation distribution and fibre properties This model is based on the assumption that in nonwoven fabrics: ∑ ∑ ∑

The fibres in the fabric are straight and cylindrical with no fibre buckling, The bonding strength between fibres in the fabric is considerably higher than the fibre strength (i.e. nonwoven rupture results from fibre failure). The shear stress and shear strain are negligible.

Accordingly, the following equations for nonwoven fabric tensile properties are obtained if the fibre orientation distribution function in the fabric is W(b), where b is the fibre orientation angle:

Ú

s (q ) = Ef e x

p /2

(cos 4 b – v (q ) sin 2 b cos2 b ) W (b ) db

– p /2

p ˆ Ê s Á q + ˜ = Ef e x 2¯ Ë



Ú v(q ) =

p /2 – p /2

Ú



E (q ) =



Ú

– p /2

p /2

[5.9]

(sin b ) W (b ) db

p /2

Ú

– p /2



p /2 – p /2

(sin 2 b cos 2 b )W (b )db

– p /2 p /2

Ú

[5.10]

4

– p /2

[5.8]

(sin 2 b cos 2 b – v (q ) sin 4 b ) W (b ) db

(sin 2 b cos2 b ) W (b ) db

s (q ) = Ef e (q )

Ê Á Á cos 4 b – Á ÁË

Ú

p /2



(sin 4 b )W (b ) db

ˆ ˜ sin 2 b cos 2 b ˜ W (b )db ˜ ˜¯ [5.11]

Modelling of nonwoven materials

191

When the fabric is isotropic, i.e. W(b) = 1/p, from the above equations, we have, v(q) = 1/3.

5.4.2 Force analysis method in the small strain model The structure of nonwovens in the small strain model is assumed as follows: ∑ ∑ ∑ ∑ ∑ ∑

The fibres are assumed to be cylindrical and to lie in layers parallel to the two dimensional fabric plane. The fabric is subjected to a small strain. The fabric is a pseudo-elastic material and Hooke’s law applies. There is no lateral contraction of the fabric. There are no transverse forces between fibres. There is no fibre curl.

The stress–strain relation is established by analysis of the components of force in the fibre elements in the fabric as follows:2, 3

(1 + ej)2 = (1 + eL)2 cos2 qj

    

+ [1 + eT + (1 + eL) cot qj tan t]2 sin2 qj

[5.12]

where t is the shear undertaken by the fabric, ej is the fibre strain in the jth fibre element, eL and eT are the fabric strains in the longitudinal and transverse directions respectively and qj is the fibre orientation angle of the jth fibre element. If there is no shear in the fabric plane, we have:

(1 + ej)2 = (1 + eL)2 cos2 qj + (1 + eT)2 sin2 qj

[5.13]

5.4.3 Energy analysis method23 In the energy analysis method, the deformation geometry of the fabric is defined by minimum energy criteria and the applied stresses and strains are used for the analysis rather than the applied forces and displacements. The following assumptions are made: ∑ ∑ ∑ ∑ ∑

The fabric is a 2D planar sheet. The sheet consists of networks of fibre elements between bond points. The bond points move in a way which corresponds to the overall fabric deformation. Stored energy is derived from changes solely in the fibre length, (i.e. there is no contribution of the binder, each point is freely jointed and fibres are free to move independently between bonds). The fibres have curl (note: fibres in real nonwoven fabrics usually have various degrees of curl).

192

∑

Modelling and predicting textile behaviour

The fabric is treated as a network of energy absorbing elastic fibrous elements, where elastic energy in reversible deformation can be solely determined by changes in fibre length.

When a unidirectional force is applied, we have: 1 e ¢j = 1 [(1 + e L2 ) cos 2 q j + (1 – vXY sin 2q j )2 sin 2q j ] 2 – 1 [5.14] Cj



N

sL =

Ê

S m js j Á j =1 Ë

C 2j

[5.15]

N

S mj

j =1



ˆ (1 + e ¢j ) ˜¯

cos 2q j



where e ¢j is the strain in the jth fibre element, eL is the overall fabric strain, sj is the stress in the jth fibre element, sL is the overall fabric stress, vXY is the fabric contraction factor, which is defined as the contraction in the Y direction due to a force in the X direction, which equates to the ratio of strain in the Y direction to the strain in the X direction, qj is the orientation angle of the jth fibre element, Cj is the curl factor of the jth fibre element, mj is the mass of the jth fibre element and N is the total number of fibre elements.

5.5

Modelling the bending rigidity of nonwoven fabrics

The bending rigidity (or flexural rigidity) of chemically bonded nonwovens was evaluated by Freeston and Platt.4 In this model, the fabric is assumed to be composed of unit cells of nonwoven structure and the bending rigidity is the sum of the bending rigidities of all the unit cells in the fabric. This is defined as the bending moment times the radius of curvature of a unit cell. The following assumptions are made: ∑ ∑ ∑ ∑ ∑ ∑

The fibre cross-section is cylindrical and constant along the fibre length. The shear stresses in the fibre are negligible. The fibres are initially straight and the axes of the fibres in the bent cell follow a cylindrical helical path. The fibre diameter and fabric thickness are small compared to the radius of curvature; the neutral axis of bending is in the geometric centreline of the fibre. The fabric density is high enough that the fibre orientation distribution density function is continuous. The fabric is homogenous in the fabric plane and in the fabric thickness.

Modelling of nonwoven materials

193

The general unit cell bending rigidity, (EI)cell, is then obtained as follows: (EI )cell = N f



Ú

p /2 – p /2

[Ef I f cos 4 q + GI p sin 2 q cos 2 q ] W (q ) dq



[5.16]

where, Nf is the number of fibres in the unit cell, Ef If is the fibre bending rigidity around the fibre axis, G is the shear modulus of the fibre, Ip is the polar moment of the inertia of the fibre cross section, a torsion term, and W(q) is the the fibre orientation distribution in the direction, q. The bending rigidities of the fabric are considered for two specific cases of fibre mobility as follows: (1) ‘Complete freedom’ of relative fibre motion: If the fibres are free to twist during fabric bending as, for example, in a needled fabric, the torsion term (GIp sin2 q cos2 q) will be zero. Therefore: (EI )cell =



p df4 N f Ef 64

Ú

p /2 – p /2

W (q ) cos 4 q dq

[5.17]



where df is the fibre diameter (2) ‘No freedom’ of relative fibre motion: In chemically bonded nonwovens where an adhesive is used to stabilise the fibre network, the freedom of relative fibre motion is severely restricted. It is assumed in this case that there is no freedom of relative fibre motion and the unit cell bending rigidity, (EI)cell, is therefore given as follows: (EI )cell =



p N f Ef df2 h 48

Ú

p /2 – p /2

W (q ) cos 4 q dq



[5.18]

where h is the fabric thickness and df is the fibre diameter.

5.6

Modelling the specific permeability of nonwovens

Intrinsic permeability (also called the permeability) of a nonwoven fabric is a structure and represents the void cavity is solely determined by the nonwoven Darcy’s Law24:

specific permeability or absolute characteristic feature of the fabric through which a fluid can flow. It fabric structure and is defined in

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Modelling and predicting textile behaviour



Dp q =– k h h

[5.19]

where q is the superficial flow rate of the fluid flow through the porous structure (m s–1), h is the viscosity of the fluid (Pa.s), Dp is the pressure drop (Pa) along the conduit length of the fluid flow h (m) and k is the specific permeability of the porous material (m2). Existing theoretical models of permeability and empirical equations for fibrous structures are based on one of three groups of assumptions. In all cases the nonwoven fabric is assumed to be homogenous: (1) Isotropic – the permeability is identical in all directions throughout the entire structure). (2) Unidirectional25, 26, 35, 36, 38, 47, 48 – the two principal directions are obtained along and perpendicular to the fibre orientation. (3) Anisotropic5, 6, 9 – the directional permeabilities are different in various directions. Empirical permeability models for nonwoven fabrics have also been obtained for specific conditions.

5.6.1 Theoretical models of specific permeability Existing theoretical models of permeability applied to nonwoven fabrics can be grouped into two main categories: ∑ ∑

Capillary channel theory, e.g. Kozeny,27 Carman,28 Davies,29 Piekaar and Clarenburg30 and Dent.31 Drag force theory, e.g. Emersleben,32 Brinkman,33 Iberall,34 Happel,35 Kuwabara,36 Cox,37 and Sangani and Acrivos.38

The permeability models established using capillary channel theory are based on Hagen–Poiseuille14 and the work of Kozeny27 and Carman,28 where the flow through the fabric is treated as a conduit flow between parallel cylindrical capillary tubes. Gebart39 presented two permeability models suitable for nonwoven fabrics having a low fabric porosity (as low as 0.35). The directional permeability along the direction of fibre orientation was in the same form as the Kozeny–Carman equation, and the directional permeability perpendicular to the fibre orientation was obtained using the lubrication approximation, assuming that the narrow gaps between adjacent cylinders dominate the flow resistance. However, it has been argued that models based on capillary channel theory are unsuitable for highly porous media where the porosity is greater than 0.8, see for example, Carman.28 Permeability models based on capillary channel theory are summarised in Table 5.1.

Modelling of nonwoven materials

195

Table 5.1 Permeability models established using capillary channel theory Permeability (m2)

Theory Hagen–Poiseuille equation14

4 k = pr 8



Kozeny–Carman’s equation (structure of capillary channels) Kozeny–Carman’s equation (fibrous

materials)40



C = k 0 S02

(1 – f)3 k = 1 C f2

C =

Rushton’s equation (woven fabrics)41

C =



k // = –

d f2 16 t k 0 d f2

C = 32 x d f2

Sullivan’s equation42, 43

Gebart39(square array)

k0

2d f2 (1 – f)3 57 f f 2 5

ˆ2 4d f2 Ê p k^ = – – 1˜ Á f 4 ¯ 9p 2 Ë

Shen’s model44

(1 – f)3 2 k = 1 df 128 f2

Rollin’s model45

k = 7.376 ¥ 10–6





df

f

Note : t is the roughness factor, k0 is the Kozeny constant, x is the orientation factor, is the Si specific internal surface area and S0 is the specific surface area, where, Si S0 = and r is the radius of a cylindrical capillary tube. (1 – f)

In permeability models based on drag force theory, the fibres in the fabric, that is the walls of the pores in the structure, are treated as obstacles to an otherwise straight flow of the fluid and the fibres can not be displaced (see Scheidegger).46 The sum of all the ‘drags’ is assumed to be equal to the total resistance to flow in the porous material. Unlike capillary flow theory, drag force theory and unit cell models demonstrate the relationship between permeability and the internal structural architecture of the fabric. In drag force models, the fibres are assumed to be aligned unidirectionally in a periodic pattern such as a square, triangular or hexagonal array. The permeability of unidirectional fibrous materials can then be solved using the Navier–Stokes

196

Modelling and predicting textile behaviour

equation in the unit cell with appropriate boundary conditions. A summary of drag force models is given in Table 5.2. It is evident in Fig. 5.5 that the Kozeny equation27 and its derivations, which is based on capillary channel theory, agrees with experimental data very well when the fabric porosity is low (<0.8) but is not applicable for high porosity fabrics (>0.8). The Mao–Russell equation for isotropic fibrous structures (denoted as M_R_ISO in Fig. 5.5) is in good agreement with models from capillary theory at low porosity and is also in reasonable agreement with the results of empirical models at high porosity. Predicted results from the Mao–Russell equation are in close agreement with the empirical data from Table 5.2 Permeability models established using drag force theory Permeability (m2)

Theory

d2 Emersleben’s equation32 k = 1 f f C C // = 32 Happel’s model35, 47 S

C = 16



C ^ = – 32 T

Kuwabara36

C ^ = 64 S

Drummond and Tahir (square array) 48

C // =

32 Ê 2 0.1019426 f 4 ˆ Áf – 4f + 2 lnf + 2.952671932 + ˜ 978 f 4 ¯ 1+1.51 Ë

C^ =

32 Ê ˆ 2 f – 0.7958978 f 2 Álnf + 1.47633597 – ˜ 2 1 + 0.4891924 f – 1.60486942f ¯ Ë





Langmuir’s model49

k^ =

S d2 19.2 f f

k ^ = S d f2 9f

50

Miao



Mao–Russell_ISO6 (2D isotropic nonwoven fabric)

¸ Ï C = – 16 ÌS + T ˝ Ó ST ˛

Mao–Russell_ISO3D (3D isotropic nonwoven fabric)9

¸ Ï C = – 32 Ì 2S + T ˝ 3 Ó ST ˛

Iberall’s model34

(4 – ln Re) 1 C = 16 3 (2 – ln Re) (1 – f)



È 1 – f2 ˘ Note : (1) S = [2 ln f – 4f + 3 + f2] and T = Íln f + ˙. 1 + f 2 ˙˚ ÍÎ (2) C⁄⁄ and C^ are the coefficients of permeability in the direction parallel and perpendicular to the fibre orientation respectively in Happel’s equation.

Modelling of nonwoven materials

197

1

Specific permeability (¥10–10m2)

Iberall 0.1

Rushton C = 0.5 M_R_ISO Shen S = 8

0.01 Davies 10

Iberall Rushton Shen M_R_ISO Davies

–3

10–4 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 Porosity P (¥ 100%) (a)

Specific permeability (¥10–10m2)

10000

1000

Iberall Rushton Shen M_R_ISO Davies

100

Rushton C = 0.5 Shen S = 8

10

Iberall

1 Davies

M_R_ISO

0.1 0.90 0.91 0.92 0.93 0.94 0.95 0.96 Porosity P (¥ 100%) (b)

0.97 0.98 0.99

5.5 Comparison of existing permeability models for homogenous isotropic materials. (a) Relationship between permeability and fabric porosity when the fabric porosity P is in the range of 0.3 to 0.9, (b) shows this relationship when the fabric porosity P is in the range of 0.9 to 0.99. Note: in Rushton’s equation, the product of the roughness factor and the Kozeny constant k0 is taken as 0.5.

198

Modelling and predicting textile behaviour

both Shen’s equation at low porosity (0.5~0.8) and with Davies’ equation at higher porosity (0.85~0.99). In contrast, Iberall’s equation, also obtained from drag force theory, gives predicted permeability results that are much higher than those obtained from empirical models for low porosity fabrics (<0.8).

5.6.2 Directional permeability in three-dimensional anisotropic nonwovens9, 51 Nonwovens are 3D anisotropic structures in which fibres are oriented in preferred directions. Anisotropy in nonwoven fabric properties, including permeability, is believed to be closely related to fibre orientation.42 In Fig.  5.6, fluid flow through a 3D nonwoven fabric in thedirection t and f is the  fibre orientation. When the two direction vectors f and t are represented by angles with three principal axes in a spherical coordinate system,  Ê  Ê p p ˆ ˆ f = Á b, – b , a ˜ and t = Á b t , – b t , a t ˜ , the angle between the 2 2 Ë ¯ Ë ¯     two direction vectors f and t , y (y = – ( f , t )) is:

cos (y) = cos (a) cos (at) + sin (a) sin (at) cos (b – bt)

The Mao–Russell equations were developed to enable modelling of directional permeability in nonwovens based on measurable parameters of fabric structure. The following assumptions are made: ∑

The distance between fibres and the length of individual fibres is much larger than the fibre diameter, i.e. the structure has high porosity. The disturbance of the flow due to adjacent fibres is assumed to be negligible. Z

Æ

at a

f

tÆ

O

Y b

bt X

5.6 Liquid flows through 3D fabric structures.

Modelling of nonwoven materials

199

∑

The fibres constituting the nonwoven fabric are of the same diameter and are distributed horizontally in-plane and in two dimensions. No fibres are aligned in the Z-direction. ∑ The flow resistivity of the fibres per unit volume in the entire structure of the fabric is equal, i.e. the fabric is homogeneous. ∑ The number of fibres oriented in each direction is not the same, but obeys the function of the fibre orientation distribution, W(a) , where a is the fibre orientation angle. ∑ The inertial forces of the fluid are negligible, i.e. the fluid has a low local Reynolds number Re, and the pressure drop between planes perpendicular to the direction of the macroscopic flow is equal to the drag force on all elements between the planes. ∑ The pressure drop necessary to overcome the viscous drag is linearly additive for the various fibres, whether they are arranged parallel, perpendicular or in any other direction relative to the flow.   The directional permeability, k (t ) , in the direction t of a 3D nonwoven fabric can be written as follows:9, 6, 5, 51 d2  k (t ) = – f 32f

Ï ÔÔ Ì Ô ÔÓ

π

Ú Ú 0

π 0

¸ ÔÔ ST ˝ [T cos 2 (y ) + S sin 2 (y )] (b , a ) d b d a Ô Ô˛ [5.20]

È 1 – f2 ˘  where S = –[4f – f2 – 3 – 2 ln f] and T = Í ln f + , f is the fibre 1 + f 2 ˙˚ Î orientation, f is the volume fraction of the solid material, df is the fibre diameter, W(b, a) is the fibre orientation distribution function in direction  f , and cos(y) = cos(a) cos(at) + sin(a) sin(at) cos(b – bt) In typical nonwoven fabrics, most fibres are oriented in the fabric plane (e.g. in spunbond and meltblown nonwovens) and a small proportion of fibres are oriented in the fabric thickness (e.g. in needled and hydroentangled fabrics). In some special structured fabrics (e.g. STRUTO), a great proportion of fibres are oriented in the direction of fabric thickness. These three-dimensional fabrics can be simplified as follows (see Fig 5.2): ∑ ∑ ∑ ∑

fibres are either aligned or are perpendicular to the fabric plane. The fibre distribution in the Z-axis and in the fabric plane is homogeneous and uniform. The number of fibres perpendicular to the fabric plane represents a fraction z of the total number of fibres. The fluid flow is laminar and in-plane (i.e. the flow along the Z-axis is ignored).

200

Modelling and predicting textile behaviour

As indicated in Table 5.3, using this approach the permeability of 3D nonwovens can be obtained based on the 2D fibre orientation distribution which is easy to obtain at low cost.

5.6.3 Directional capillary pressure and anisotropic liquid wicking in nonwovens (Mao–Russell equations)7, 51 Liquid wicking in nonwovens can be studied in terms of a steady state fluid flow in porous media, although in many practical situations the liquid is in fact an unsteady state flow where the nonwoven fabric is not uniformly and completely saturated. Lucas–Washburn equation based on fabric pore sizes Traditionally, liquid wicking in textile fabrics, paper53 and nonwovens is modelled by the Lucas–Washburn equation,58, 54 in which fabrics containing capillary pores having an average diameter r are frequently modelled as an equivalent system of parallel cylindrical capillary tubes having the same diameter r. In this case, the Hagen–Poiseuille equation14 for a laminar fluid flow through a cylindrical capillary tube is employed to describe the wicking rate in nonwoven fabrics as follows:

dh = r 2 DP dt 8h h

[5.21]

where r is the radius of the capillary tube, h is the distance through which the fluid flows in time t, h is the viscosity of the fluid and DP is the capillary pressure in a cylindrical tube. This is given by Laplace’s equation:

Pcap =

2s cos g r

[5.22]

where g is the contact angle between the liquid and the capillary tube surface and s is the surface tension of the liquid. In the upwards vertical strip test, the upward driving pressure in the Hagen–Poiseuille equation is DP = Pcap – rgh (where g is the acceleration of gravity and r is the liquid density). The relationship between the rising wicking height of the liquid capillary flow in the fabric and the wicking time is given:55

{

bt = –hm log e 1 – h hm

}

–h



where hm = a/b with a = rs cosg/4h and b = r2rg/8h

[5.23]

Table 5.3 Permeability in various two-and three-dimensional nonwoven structures Fibrous structures

Fibre orientation distribution

Generalised W(a) permeability model for 3D nonwoven structures9, 51

Directional permeability k(q) Ï ¸ ST d2 Ô  Ô k(t ) = – f Ì ÈT cos2 (y ) + S sin2 (y )˘ W (b , a ) d b d a ˝ 32f Ô Ô Î ˚ Ó 0 0 ˛ cos (y) = cos (a) cos (at) + sin (a) sin (at) cos (b – bt)

ÚÚ

2 2 1 = cos (q – j ) + sin (q – j ) k (f) kX kY

Ferrandon’s equation52

cos2 (bt – j ) 1 = + k (bt , a t ) kX



Ï d2 Ô  k(t ) = – f Ì 32f Ô zT + (1 – z) S + (1 – z)(T – S) Ó



4(T + S) cos(2j ) – cos (4 j ) T –S



+ sin (4j ) 3D fabric with isotropic fibre alignment in the fabric plane9

W (a ) = 1 p

Ú



0

[W (b)cos (2b)] db

0

p

0

k X = kY = –

Ú

Ú

¸ Ô ˝ [cos2 (b – bt ) sin2 (a t )] W (b)d b Ô ˛

ST

p

[W (b)sin(2b )]db = 3

Ú

p

[W (b )cos (2b)]db

0

¸ d f2 Ï ST ¸ d f2 Ï 1 ST Ì ˝ ; kZ = – Ì ˝ 16(1 – z ) f ÓÔS + T ˛Ô 32 f ÓÔ (1 – z ) S + zT ˛Ô

x and z are the fraction of fibres aligned in the X and Z directions respectively; the Z direction is perpendicular to the fabric plane. kX, kY and xZ are the three principal permeabilities respectively.

201

È 1 – f2 ˘ S = –[4f – f2 – 3 – 2 ln f] and T = Íln f + ˙. 1 + f 2 ˙˚ ÍÎ

Modelling of nonwoven materials

W(b)

Generalised permeability model for simplified 3D nonwoven structures9, 51

Ê ˆ cos2 Á p – bt + j ˜ Ë2 ¯ cos2a t + kY kZ

202

Modelling and predicting textile behaviour

In the horizontal strip test, the gravity effect is negligible and the equation can be reduced to the form of the Lucas–Washburn equation:56–58

1

h = Ct 2

[5.24]

where C is a constant related to both the liquid properties and the fabric

{

rs cos g structure and is given as C = 2h

}

1 2

.

Whilst in capillary channel theory (Lucas–Washburn equation) wicking is assumed to be determined by the geometry of pores in a fabric, there are difficulties in quantifying the average equivalent capillary radii59 of pores in nonwovens because: ∑ ∑

The capillary channels differ in size and shape. They are interconnected as well as interdependent throughout the 3D network of fibres. The capillary channels in real nonwoven fabrics do not have circular cross-sections and are not necessarily uniform along their lengths.

Modelling of directional capillary pressure in nonwoven fabrics based on equivalent hydraulic radius theory was developed by Mao and Russell. 7,51 Directional capillary pressure and directional wicking7,51 In addition to the assumptions about nonwoven structure given in Section 5.6.2, the Mao-Russell equation for capillary pressure was established based on the following assumptions:7, 51 ∑ ∑ ∑

The capillary pressure in direction q in the fabric plane is hydraulically equivalent to a capillary tube assembly in which there are N cylindrical capillary tubes of the same hydraulic diameter. The wetted specific area in the direction q in the fabric plane, S0(q), should be identical to the capillary tube assembly. The porosity of the capillary tube assembly should be the same as the nonwoven fabric.

The overall capillary pressure in any specific direction in the fabric is represented by the capillary pressure between fibres along their longitudinal axis. This assumption is based on the fact that the capillary wicking phenomenon occurring along the direction of fibre orientation60 is much greater than the capillary wicking in the direction perpendicular to the fibre orientation. This is evidenced by theory of Cassie61 and the models by Princen.62 Therefore, it is assumed that the overall capillary pressure in a nonwoven fabric varies in different directions depending on the fibre orientation distribution. 51 Given a planar fibre orientation distribution function W(a) in the fabric,

Modelling of nonwoven materials

203

where a is the fibre orientation angle, the capillary pressure p(q) in the direction q is as follows:7 4f

p (q ) =



Ú

p 0

W (a ) | cos(q – a ) |da df (1 – f )

s cos g



[5.25]

where g is the fibre contact angle, s is the liquid surface tension, df is the fibre diameter and f is the solid fibre fraction  in the fabric. The directional capillary pressure p(t ) in the direction t of a 3D nonwoven fabric (see Fig. 5.5) can be written as:51  p(t ) =

f S0

p

Ú Ú 0

p 0

| cos (a ) cos (a t )

+ sin (a ) sin (a t ) cos (b – b t ) | W (b , a ) da d b s cos g (1 – f )



[5.26]

where W(b, a)is the fibre orientation distribution function in the fabric structure  in direction f . Combining the directional permeability in the direction t ,   both the wicking rate, V (t ) , and wicking distance, L (t ) during horizontal wicking can be determined in direction t :51  V ( t ) = V (b t , a t )



Ï ÔÔ df  =– 8x (t )(1 – f ) ÌÔ ÔÓ

 

¥

ST π

π

Ú Ú 0

0

¸ ÔÔ 0 0 ˝ [T cos2 (y ) + S sin 2 (y )] W (b , a ) da db Ô Ô˛ π

Ú Ú

π

|cos(y )|(W (b , a ) da db

s cos g h

[5.27]

1  L (t ) = L (b t , a t ) = Ct 2

[5.28]

where cos (y) = cos (a) cos (at) + sin (a) sin (at) cos (b – bt) C= Ï Ï Ô Ô 1 Ô – df Ô Ì 2 Ô (1 – f ) ÌÔ ÔÓ ÔÓ

π

Ú Ú 0

1

¸2 ¸ ST |cos(y )|(W (b , a ) da db ÔÔ s cos g ÔÔ 0 0 ˝ h ˝ π Ô [T cos 2 (y ) + S sin 2 (y )] W (b , a ) da db Ô Ô Ô˛ 0 ˛ π

Ú Ú

π



204

Modelling and predicting textile behaviour

It is therefore apparent that in addition to liquid viscosity, surface tension and the liquid contact angle the wicking rate depends on the fibre diameter, the fibre orientation distribution and the fabric porosity.

5.7

Thermal resistance and thermal conductivity

Heat is transferred in three ways:63, 64 (1) by conduction from particles (molecules, atoms and electrons) in the high-temperature region of a solid, gas and liquid, which transmit some of their energy to the adjacent lowertemperature regions through particle interaction, (2) by convection through fluid flow process and (3) by emission of electromagnetic radiation. The thermal conductivity is defined in a one-dimensional form of Fourier’s heat conduction equation for steady-state heat flow through a flat plate:

q = k DT L



[5.29]

where q (Wm–2) is the heat flow rate in the unit area, normal to the heat flow, DT (k) is the temperature difference across the plate, L(m) is the fabric thickness, k (Wm–1K–1) is the thermal conductivity. The thermal conductivity of a nonwoven fabric, k, depends on the combined effect of the thermal conductivities of the component fibres (including any other solid materials such as binders) and air trapped in the fabric. It is therefore the sum of all individual thermal conductivities resulting from the conduction, convection and radiation values of air and solid fibre (and binders if any) as follows:

k = kair conduction + kfconduction + kconvection + kradiation + kfibre–air [5.30]

where kair condution is the conduction of air, kfconduction is the conduction through fibres, kconvection is the conduction by convection, kradiation is the conduction by radiation and kfibre–air is the conduction because of the heat interaction between air and fibre. The heat transfer (i.e. the thermal conductivity) is influenced by the thermal conductivity of fibre components, the fabric structure and dimensions and the environmental temperature. The fabric bulk density, porosity and the fibre arrangement are particularly important. Conduction occurs in the solid fibre material and in the air held in the spaces between the fibres; free convection takes place in the presence of a gravity field and there is radiation from the surface of the fabric as well as internally among the fibres. Both convection and radiation can be reduced by increasing the bulk density of the fabric whereas there may be an increase in the thermal conduction by the fibres. For modelling purposes, it is assumed that (1) the fibrous structure can be approximated as a homogeneous medium of conductivity k, (2) the interaction of fibres influencing k can be averaged over a unit volume, (3)

Modelling of nonwoven materials

205

any individual fibre can be assumed to be a spheroid whose major axis is very large compared with the minor axis and (4) the fibres are perpendicular to the heat flow. By considering the fibre orientation and the thermal resistance of air based on the basic fibre–air unit cell (Fig. 5.7), Stark and Fricke65 established a model to determine the thermal conductivity, kBM:

b –1 Ê ˆ k BM = ks Á 1 + 1 + a (1 + Z (b – 1)/(b + 1)) ˜¯ Ë

[5.31]

where a = vs/va, b = ka/ks and va and vs are the fractional volumes of the medium of thermal conductivity ka, and ks, respectively, and va + vs = 1. The term Z is the fraction of fibres arranged perpendicularly to the macroscopic heat flow (Z = 1 when fibres are aligned perpendicular to the heat flux, Z = 0.66 when randomly arranged and Z = 0.83 when arranged parallel to the heat flux). The thermal resistance in the contact area between adjacent fibres (Fig. 5.7) is modelled using statistical probability and the coupling effects of the fibre and air. The thermal conductivity of the fabric based on a modified model, kMMC, may be calculated using the diagram in Fig. 5.8.65

k MMC

Ï ÔÔ x +1 1 = (m + 1) Ì m + BM 2 k Ad x k Ô ka + s f xp act ÔÓ

¸ ÔÔ ˝ Ô Ô˛

–1

[5.32]

Heat flux

Fibre length l

m¥d

Fibre-air unit cell model (basic model)

d air Contact

Act

xAct

5.7 Modified fibre–air unit cell model for the thermal conductivity of fabrics.65

206

Modelling and predicting textile behaviour

RBM

Rct

Rg

5.8 Circuit diagram of the modified model based on the thermal resistances in the basic model (RBM), fibre–fibre contact area (Rct) and air (Rg).65

where

m = l cos q 0 – 1 df 2



Ê 2l ˆ 3 x=Á ˜ Ë df ¯

1

(0.5 sin q 0 ) 3

2

p (1.5(1 – m02 ) pext /E ) 3

–1

1



1 ˆ ˆ2 Ê Ê p 1 + Á a ˜¯ ˜ Ë 1 = 1Á ˜ Á df 2 Á 0.5 sin 2 q 0 cos q 0 ˜ ÁË ˜¯

where act is the contact radius, A is the connection parameter and A = 0.611, Act is the area of one contact, df is the fibre diameter, E is Young’s modulus, l is the fibre length in the basic model of the fibre–air unit cell (diagonally across the cell), kBM is the thermal conductivity based on the basic model of the fibre–air unit cell, kMMC is the thermal conductivity based on the modified model of the fibre–air unit cell, m is the height of the fibre–air unit cell in units of the fibre diameter (df), pext is the external pressure, m0 is the Poisson number of the fibre material, x + 1 is the area of a fibre–air cell in units of the contact area (Act) and q0 is the mean fibre orientation angle. Empirical equations for the relationship between the thermal conductivity of a homogeneous and isotropic nonwoven fabric at a certain regain, k, and the proportion of its component fibres was given by Schuhmeister:66

Modelling of nonwoven materials



k1 k2 ˆ Ê k = 1 (k1 v1 + k2 v2 ) + 2 Á 3 3 Ë k1 v2 + k2 v1 ˜¯

207

[5.33]

where v1 and v2 are the fractional volumes of the medium with thermal conductivities of k1, and k2, respectively, and v1 + v2 = 1. For a nonwoven fabric containing wool fibres, Baxter67 expanded Schuhmeister’s equation:

k1 k2 Ê ˆ km = x (k1 v1 + k2 v2 ) + y Á k v Ë 1 2 + k2 v1 ˜¯

[5.34]

where x = 0.21, y = 0.79, k1 is the thermal conductivity of air (k1 = 0.0264); k2 is the thermal conductivity of wool fibre at a certain regain (k2 = 0.2226 at a regain of 0.7%; k2 = 0.1933 at a regain of 10.7%), and v1 and v2 are the fractional volumes of air and the wool fibre, respectively.

5.8

Acoustic impedance

There are three acoustical effects in nonwoven fabrics: reflection, transmission and absorption of sound waves. The present theoretical analyses of sound propagation in nonwoven fabrics are mostly focused on the reflection and absorption of small amplitude, airborne sound waves from nonwoven fabric surfaces in the audio-frequency range. They are all based on the assumption that nonwovens are two phase assemblies with a solid fibre material (rigid or flexible) and air. The theoretical models introduced in this section can provide helpful indications for the design and engineering of nonwoven sound insulation but more realistic empirical models of sound absorption in nonwovens have been developed.68–71

5.8.1 Theoretical models Three types of sound transport model are available for nonwoven fabrics:72 parallel capillary pore models, parallel fibre models and semi-empirical models based on the two preceding models. In the parallel capillary pores model, a nonwoven is assumed to be a medium containing identical parallel cylindrical capillary pores (running normal to the fabric surface). Two types of parallel fibre model are available: the first assumes the fibres are parallel to each other and to the fabric surface and the second assumes the fibres are parallel to each other but are normal to the fabric surface. Propagation of sound in an isotropic homogeneous material is determined by two complex quantities, the characteristic impedance Z0(f, df, e) = z0 (R) – jz0(I) and the propagation coefficient per metre g(f, df, e) = a + jb. The quantities zo(R) and zo(I) are the real and imaginary parts of Z0(f, df, e) respectively. The normal-incidence energy absorption coefficient is defined as follows:

208



Modelling and predicting textile behaviour

Z – r0 c0 a =1– Z + r0 c0

2

[5.35]



where f is the sound frequency, r0 and c0 are the density of free air and the speed of sound in free air, respectively. When the fabric has a small thickness, h (up to 10 cm), and is fixed on a rigid wall, the impedance Z may be calculated using the following equation:

Z0 = Z coth g (f, df, e)h

[5.36]

The absorption coefficient of the fabric given in Equation [5.35] can be rewritten:

4z0 (R )r0 c0 (z0 (R ) + r0 c0 )2 + z02 (I )

a =

[5.37]

Parallel fibre models in which the influences of the viscous and thermal effects of each cylindrical fibre are decoupled73 have been established by Zwikker and Kosten74 and by Kirby and Cummings.68 The bulk acoustic properties of a nonwoven are modelled using the combination of the effective dynamic (complex) density and bulk modulus of individual cylindrical fibres in a ‘Raleigh type’ model. A schematic of the parallel fibre model of fabric structure that is assumed68 is given in Fig. 5.9. The problem in the ‘open face’ case is defined by the following system of equations:75

Fibre

2R

r x

df

5.9 Model of sound propagation in a bundle of parallel fibres (note: the direction of the sound propagation is in the direction of fibre orientation).69

Modelling of nonwoven materials

209

The equation of motions for a fibre: ∂pfibre (x , t ) ∂ufibre (x , t ) rfabric = D (uair (x , t ) – ufibre (x , t )) [5.38] + ∂x ∂t



The equation of motions for air: ∂pair (x , t ) ∂uair (x , t ) rair = – D (uair (x , t ) – ufibre (x , t )) + ∂x ∂t



[5.39]

The continuity equations for fibre:

–

∂pfibre (x , t ) ffibre ∂pair (x , t ) ∂u (x , t ) + = K fibre fibre e ∂t ∂t ∂x

[5.40]

The continuity equations for air: –



∂pair (x , t ) ∂u (x , t ) ∂u (x , t ) = K air e air + (K air – P0 ) ffibre fibre ∂t ∂x ∂x [5.41]

The boundary conditions associated with the above equations are given as:

uair (h, t) = 0



ufibre (h, t) = 0



epfibre (0, t) = ffibre pair (0, t)



x Π[0, h]



t Π[0, T]

where r0 and rp are the densities of free air and the fibre respectively; e and ffibre are the porosity and the fibre volume fraction respectively where, 2

Êd ˆ ffibre = 1 – e; and in Fig. 5.9, e = 1 – Á f ˜ ; rair and rfabric are the bulk Ë 2R ¯ densities of the air and solid fibre components in a unit volume of nonwoven fabric respectively (i.e. rair = r0e and rfabric = rfibre ffibre); h is the thickness of the nonwoven, x is a distance, t is a time, u is the velocity, p is the pressure and u (x, t) is the sound velocity; Kf is the fibre bulk modulus, Ka is the air bulk modulus and P0 is the air pressure. The coupling parameter, D(uair (x, t) – ufibre (x, t)), representing the drag force between the solid fibre material and air is given by:74

D(uair (x, t) –ufibre(x, t)) = i wrair (m – 1) + e2 s

[5.42]

where w = 2πf is the angular frequency, m is the structure constant and s is the resistance constant. The characteristic impedance at the front face (x = 0) of a nonwoven fabric is given:

210

Modelling and predicting textile behaviour

where



1 = (ufibre + a1 uair )(uair – b 2 ufibre ) – (ufibre + a 2 uair )(uair – b1 ufibre ) Z0 (b1 – b 2 ) z f2 coth (ik2 h ) (b1 – b 2 ) z f1 coth (ik1 h )

[5.43]

aj =

bj =

k

U airj k

j U fibre

k

j Pfibre

k



2 Ê ˆ k 2 Ê j ˆ (1 – iq ) Á 1 – cf Á ˜ ˜ – iBq Ëw¯ ¯ Ë

k

Pairj

zfj =

2 Ê Ê kj ˆ ˆ uair Á 1 – cf2 Á ˜ ˜ – iBq Ëw¯ ¯ Ë = (ufibre – iq ) B

j Pfibre

k

j U fibre

=

ufibre (1– iq (1 + B )) –

= w kj

iq uair cf2

ufibre (1– iq (1 + B )) –

Ê kj ˆ ÁË w ˜¯

iq uair cf2

ufibre – iq

2

uair

Ê kj ˆ ÁË w ˜¯

2

rfabric

where cf2 = K fibre /rfabric is the square of the fibre wave velocity,

q =

D (uair (x , t ) – ufibre (x , t )) rair w

is a dimensionless coupling coefficient, B = rair/rfabric, and the subscripts air and fibre refer to the air and fibre phases, respectively. Using this approach, the sound absorption coefficient, a, can be obtained from the characteristic impedance, Z0.

5.8.2 Empirical models Given that the real part of the characteristic impedance z0(R), is the most sensitive to the influence of fibrous structure, the structural characteristic79 (Q) is usually defined as Q = z0(R) – 1. The other two dimensionless parameters, the structural factor (Ks) and the airflow resistance per unit thickness (s) in complex form are also defined as follows:76, 77

(z0 (R ) b – z0 (I )a ) Ks = e k

s = lim (z0 (R ) a – z0 (I ) b ) kÆ0

[5.44]

[5.45]

where, z0(R) and z0(I) are the real and imaginary parts of Z0 and a and b are the attenuation and phase constants.

Modelling of nonwoven materials

211

(1) Delany–Bazley equations78 When the nonwoven fabrics are thick (up to 2 m) and when the sound f waves are in the range of 10 ≤ ≤ 1000 , the Delany–Bazley equations s are as follows: –0.75



z 0 (R ) Ê fˆ = 1 + 9.08 Á ˜ r0 c0 Ës¯



z0 (I ) Ê fˆ = – 11.9 Á ˜ r0 c0 Ës¯



Ê 2p f ˆ Ê f ˆ a = 10.3 Á Ë c0 ˜¯ ÁË s ˜¯



Ê 2p f ˆ b =Á Ë c0 ˜¯

[5.46]



–0.73

[5.47]

–0.59

[5.48]



–0.70 È ˘ Ê fˆ 1 + 10.8 Í ˙ ÁË s ˜¯ ÍÎ ˙˚

[5.49]

(2) Voronina models79 Based on sound frequencies of f = 250 ~ 2000 Hz and a range of nonwovens composed of glass, silica, mineral wool and basalt fibres, respectively (df = 2 ~ 8 mm), e = 0.996 ~ 0.92), Voronina developed empirical models for sound propagation through nonwoven fibrous media:

Z(f, df, e) = (1 + Q) – jQ

g (f , df , e ) =

[5.50]

kQ (2 + Q ) + jk (1 + Q ) (1 + Q )



ks/e = 1 + 2Q



s = 2kQ

[5.51] [5.52]

To predict the sound absorption based on nonwoven fabric structure, the following empirical equations for structural characteristics (Q) and sound resistance (s) were given based on the experimental results:76, 77

Q=

(1 – e )(1 + 0.25 ¥ 10 –4 (1 – e )–2 ) e df

8h wr0 c0

s =

16 (1 – e )2 (1 + 0.25 ¥ 10 –4 (1 – e )–2 ) h e 2 df2 r0 c0



[5.53] [5.54]

212

Modelling and predicting textile behaviour

where h = 1.85 ¥ 10–5 is the dynamic viscosity of air (Pa s), r0 is the air density (kg m–3); c0 is the sound velocity in air (m s–1) and w = 2pf/c0 is the wave number (m–1).

5.9

Particle filtration in nonwoven filters

The filtration process is frequently considered in terms of dry filtration (air filtration, aerosol filtration), wet filtration (mist filtration) and liquid filtration. The prime objective of modelling filtration processes using nonwoven filter fabrics is to improve their engineering design and to obtain improved separation of targeted solid particulates from the fluid flow whilst minimising changes in pressure drop of the fluid flow across the filter thickness. The mechanism of filtration varies with the properties of the targeted particles and properties of the fluid in which the particles are suspended. Consequently, the filtration process and the performance of nonwoven filter fabrics can not be modelled by a universal model. In this section, models of air filtration for nonwovens in depth filtration are considered. The following nomenclature is adopted:



Êl ˆ CC = Cunningham slip factor CC = 1 + Á ˜ Ë dp ¯

–0.435dp ˆ Ê l 2.492 + 0 .84 e ˜¯ ÁË

Dd is the particle diffusion coefficient, (m2 s–1); this describes the degree of motion due to diffusion and is a function of the mean free path of the fluid molecules:

Dd =

CC k B T 3phdp

df is the fibre diameter dp is the particle diameter ef is the effective fibre length factor, which is the ratio of the theoretical pressure drop for a Kuwabara flow field to the experimental pressure drop86, 91 (ef = 16hui f h /(Kudf2 DPO )) E is the single fibre collection efficiency. This is the fraction of the particles that can be collected by a fibre from a normal cross-sectional area of the gas stream, which is equal to the frontal area of the fibre. Ej is the single fibre collection efficiency for a size sub-range j of a polydisperse aerosol G is the gravity parameter, G = dprpg/18hui Ku is the Kuwabara hydrodynamic factor, Ku = – 3 – 1 ln f + f – 1 f 2 4 2 4

Modelling of nonwoven materials

213

g is gravitational acceleration h is filter thickness kB is Boltzman’s constant, 1.3708 ¥ 10–23 J K–1 n is the aerosol number concentration leaving the filter, no is the aerosol number concentration entering the filter uh N¢cap is the simplified capillary number, N cap ¢ = s cos g Pe is the Peclet number, which characterises the intensity of the diffusion deposition. An increase in the Peclet number will decrease the single-fibre diffusion efficiency,

Pe =

U 0 df Dd

DP0 is the pressure drop across the dry filter R is the interception parameter, R = dp/df Ref is the Reynolds number, Ref = ruidf/h Stk is the Stokes number, which is the ratio of the particle kinetic energy to work done against the viscous drag over a distance of one fibre radius,

Stk =

rp ui dp2 9m d f

T is temperature (K) u is the superficial gas velocity (m s–1) ui is the interstitial gas velocity (m s–1) Y is filter efficiency (%) e is the porosity of filter f is the packing density of dry filter (or fibre volume fraction) h is the absolute viscosity of gas (Ns m–2); l is the mean free path of gas molecules at NTP (0.067 mm),80 which is inversely proportional to the air pressure. g is the contact angle between the liquid and fibre r is the density of the gas rp is the density of the particle s is the surface tension of the liquid (N m–1).

5.9.1 Evaluation of filter performance An ideal filter would have both a high separation capacity and a low pressure drop and these would remain constant during the service life of filter. In practice, this is difficult to achieve. The major performance criteria of a filter include the filter efficiency, pressure drop and the filter quality performance.81,82

214

Modelling and predicting textile behaviour

The filter efficiency, E, describes the ability of a filter to retain particles and it is defined as the ratio of the particle concentrations in the upstream (Pin) and downstream (Pout) fluid flows respectively:

E =1–

Pout Pin

The pressure drop is the difference of pressures in the upstream (pin) and downstream (pout) fluid flows across the filter thickness:

Dp = pin – pout

The filter quality performance Q, or the filter quality coefficient, is defined as the ratio of the filtration efficiency to the pressure drop across the filter thickness:

Q=

– ln (1 – E ) Dp

5.9.2 Filtration mechanism The capacity of a nonwoven filter to capture particles depends on the interactions between targeted particles, individual fibre surfaces and the fluid molecules. The most frequently encountered mechanisms describing the particle–fibre–fluid interactions included in generalised filtration models are straining, Brownian diffusion, direct interception and inertial impaction. The effect of electrostatic forces and gravity sedimentation are also important in specific filtration processes but are not discussed in this section. Straining is the entrapment of a particle when it is larger than the opening between fibres in the fabric. Inertial impaction (impingement) occurs when the particle inertia is so high that it breaks the air streamlines and impacts the fibre. Direct interception refers to a phenomenon where a particle can be caught by a fibre if it approaches the fibre within a small distance. It usually assumes that this distance is equal to or smaller than half the particle diameter. Brownian diffusion (Brownian motion) is the random movement of a particle in the fluid flow stream caused by the collision of other particles with the molecules of the fluid medium on a molecular scale. The diffusion mechanism is important for particles smaller than about 0.1 mm.83, 84 A nonwoven filter can be modelled as multiple layers of 2D fibre networks of high porosity. During filtration, a small proportion of the targeted particles in the fluid flow will penetrate through the total thickness; most will be captured and gradually deposited onto the fibre surfaces in the fabric. The progressive deposition of particles on the fibre surfaces leads to the formation of a filter cake and therefore, the permeability of the filter structure gradually reduces with time. Filtration is therefore not a stationary process.

Modelling of nonwoven materials

215

The formation of the filter cake makes modelling of the filtration process complex and gradually leads to an increase in the pressure drop while the filtration of small diameter particles can increase. A convenient method of modelling nonwoven filter efficiency is by means of single-fibre collection efficiency theory.

5.9.3 Filter efficiency in dry air filtration Filter efficiency based on a single fibre collection efficiency A nonwoven filter fabric is composed of many individual fibres and the filter efficiency depends in part on the particle collection efficiency of single fibres. An equation defining the overall filter efficiency of a nonwoven fabric, Y(df), for any particle size, dp, and set of conditions can be written as follows:29, 85–88

Ê – 4f Eh ˆ Y (df ) = 1 – exp Á Ë p (1 – f )df ef ˜¯

[8.55]

The particle collection efficiency of a single fibre, E, depends on the particle size, the air velocity and fibre properties based on the six primary mechanisms operating in filtration, that is impaction (EI), direct interception (ER), diffusion (ED), enhanced interception due to diffusion (EDr), gravitational settling (EG) and electrostatic attraction (Eq). Several equations have been proposed for predicting E from these different collection mechanisms including Davies:29

E = EDRI = (R + (0.25 + 0.4R)(Stk + 2Pe–1 )



– 0.0263R (Stk + 2Pe–1 )2 )(0.16 + 10.9 f – 17f 2 )

[5.56]

Friederlandler89, 90



3 Ê Ê 1 1ˆ Ê 1 1ˆ ˆ E = EDRI = 1 Á 6 Á RPe3 Re6 ˜ + 3 Á RPe3 Re6 ˜ ˜ RPe Á Ë ¯ Ë ¯ ˜¯ Ë

[5.57]

where EDRI is defined as the particle collection efficiency of a single fibre, E, depending on the mechanism of impaction (EI), direct interception (ER) and diffusion (ED). and Stenhouse98

E = ED + ER + EDr + EI + EG

[5.58]

Each of the component collection efficiencies can be determined as in the following.

216

Modelling and predicting textile behaviour

(A) Diffusion91



Ê1 – fˆ ED = 2.9 Á Ë Ku ˜¯

–1 3

–2

Pe 3 + 0.62Pe–1



[5.59]

which is valid for 0.005 < f < 0.2, 0.1 < U0 < 2 m s–1; 0.1 < df < 50 mm; Ref < 1) (B) Interception83, 85, 92



2 ˆ fˆ f (1 + R) Ê Ê 1 ˆ Ê 1 – ˜ – (1 + R) 2 ˜ 2 ln (1 + R) – 1 + f + Á Á Á ˜ 2 2 2Ku Ë 1 + R ¯ Ë ¯ Ë ¯

ER =

(1 – f )R 2

=

2 Ku (1 +

 

2 R) 3(1–f )

[5.60]

(C) Impaction93

EI =

where Stk =

(Stk ) J 2K u2

[5.61]

rd dp2 CcU 0 18hdf

J = (29.6 – 28f0.62)R2 – 27.5R2.8 for 0.01 < R < 0.4

and 0.0035 < f < 0.111

J = 2 for R > 0.4.

(D) Enhanced diffusion due to interception of diffusing particles 94 2

EDr

3 = 1.24R 1 for Pe > 100 (KuPe ) 2

[5.62]

(E) Gravitational settling95

EG @ (1 + R)G for VTS and U0 in the same direction.



EG @ –(1 + R)G, for VTS and U0 in the opposite direction.



2

EG @ – G , for VTS and U0 in the orthogonal direction.

[5.63]

Modelling of nonwoven materials

where G =

217

2 VTS rd dp Cc g = U0 18hU 0

VTS and V0 are the particle terminal setting velocity and face air velocity, respectively. (F) Electrostatic attraction:84 1



ˆ q2 Ê ' – 1ˆ 2 Ê Eq = Á ˜ Á 2 1 ' + ¯ Ë 3phdp df U 0 (2 – lnRef )˜¯ Ë

[5.64]

where ' is the dielectric constant of the particle and q is the particle charge. Filter efficiency of nonwovens with multiple fibre components When particles of different sizes are filtered by a nonwoven composed of fibres having one diameter, the filter efficiency of the fabric can be obtained from the model of collection efficiency for a single fibre, E, described in the Section above. By sub-dividing the particle size range into several sub-ranges, Ej. Ej can be obtained for each sub-range j of average particle diameter dpj from the above equations for a single fibre. The filter efficiency Y is then calculated as shown:

where

Ênˆ Y = 1 – S aj Á ˜ Ë n0 ¯ j j

[5.65]

Ênˆ Ê 4f E j h ˆ Ê nˆ ÁË n ˜¯ = exp Á p (1 – f )d e ˜ and ÁË n ˜¯ and aj are the number of 0 j Ë 0 j f f¯

particle penetrations and mass fraction of the jth size range of particles, respectively. It is observed that filters containing fibres of the same size are limited in the range of particles sizes they can effectively remove from the fluid flow. A minimum filter efficiency exists in the filtering of particles of certain sizes as indicated in Fig. 5.10. For very small particles less than dp1 in diameter, the primary filtration mechanism is diffusion. For particles between dp1 and pp2, the filter is less efficient as the particles are too large for significant diffusion to take place and are too small for a large interception effect. For particles of diameter above dp2, the filter efficiency increases because interception along with inertial impaction effects are predominant. The decrease in filter efficiency for particles with a diameter between

Modelling and predicting textile behaviour

Filter efficiency (%)

218

Diffusion regime

Diffusion and interception regime dp1

Inertia and interception regime dp2

Diameter of particles (µm)

5.10 Filter efficiency of a nonwoven fabric in relation to particle size in an air flow.9

dp1 and dp2 is unacceptable but inevitable. To design a nonwoven filter with high filter efficiency, fabrics containing blends of fibres of more than one diameter can be produced.97 If the nonwoven is composed of multiple fibre components and the fluid contains particles of multiple diameters, the filter efficiency can be written:

Where



Ênˆ Y = 1 – S aj Á ˜ Ë n0 ¯ j j Ê E j (df )h Á 4f S df Ê nˆ ÁË n ˜¯ = exp Á p (1 – f )d e 0 j f f Á Ë

ˆ ˜ ˜ ˜ ¯

[5.66]

Ej (df) = ED (df)j + ER (df)j + EDr (df)j + EI (df)j + EG (df)j + Ee (df)j

Ênˆ where Á ˜ and aj are the number of particle penetrations and the mass Ë n0 ¯ j fraction of the jth size range of particles respectively. The term Ej (df) is the collection efficiency of a single fibre having a diameter of df against a particle of diameter of dpj.

5.9.4 Pressure drop The pressure drop across a nonwoven filter in the dry air filtration, DP0, can be predicted using the expression developed by Davies:84

Modelling of nonwoven materials



DP0 =

U 0hh (64f1.5 + (1 + 56f 3)) df2

219

[5.67]

For mist filtration or filtration of liquid particles using nonwovens, the required collection efficiencies can be obtained by appropriate combinations of filter thickness, fibre diameter, packing density and gas velocity. For a specified efficiency of 90%, the required filter thickness varies according to the approximate empirical relation:98

h = 5f –1.5 df2.5

[5.68]

The corresponding pressure drop at a constant filtration efficiency is insensitive to df but varies approximately according to the relation:98

5.10

DPwet µ f0.6U0.3 where f > 0.01

[5.69]

Future trends and sources of further information and advice

Modelling of nonwoven fabric structure and properties is fundamental for developing an improved understanding of this important class of fibre assembly. This chapter has reviewed only a fraction of the available work in this area and has focused specifically on analytical and empirical models. In last decade, numerical models and techniques based on computational simulation have increasingly been used to model nonwoven materials and these techniques provide significant potential for solving and visualising complicated mathematical models. These tools apply to mathematical models of 3D fabric structure and are particularly important in providing detailed information about dynamic processes such as fluid flows through porous media using computational fluid dynamics. In practice, the reliability of solutions using numerical models and the results of computational simulation depend on two factors. The first is the availability and the accuracy of analytical (or empirical) models and the second factor is the use of realistic assumptions including the choice of appropriate boundary conditions. In addition, there is potential to explore further other modelling tools that are seldom used in modelling of nonwoven materials such as advanced mathematical and statistical tools, visual, verbal and artificial intelligence techniques. Further reading on the modelling of nonwoven materials is to be found in refereed journals including The Journal of the Textile Institute, The Textile Research Journal and The Journal of Engineering Fibers and Fabrics. General references of value are listed below. K. Vafai, Handbook of Porous Media, Taylor & Francis, Boca Raton, 2005.

220

Modelling and predicting textile behaviour

A. Kabla and L. Mahadevan, ‘Nonlinear mechanics of soft fibrous networks’, J Roy Soc Interface, 2007, 4, 99–106. R. C. Brown, Air Filtration: an integrated approach to the theory and applications of fibrous filters, Pergamon, Oxford, 1993. K. M. Entwistle, Basic Principles of the Finite Element Method, Woodhead Publishing, Cambridge, 2001. Hearle, J. W. S., Grosberg, P. and Backer, S., Structural Mechanics of Fibers, Yarns, and Fabrics, Wiley-Interscience, New York, 1969. Flow Modeling Solutions for the Nonwovens Industry, http://www.fluent. com/solutions/nonwovens/index.htm

5.11

References

1. ISO 90921988; BS EN 29092,1992 Textiles, Nonwovens, Definition. 2. Backer S and Petterson DR, ‘Some principles of nonwoven fabrics’, Textile Res J, 1960, 30(12), 704–711. 3. Hearle JWS and Stevenson PJ, ‘Studies in nonwoven fabrics: prediction of tensile properties’, Textile Res J, 1964, 34, 181–91. 4. Freeston WD and Platt MM, ‘Mechanics of elastic performance of textile materials, Part XVI: bending rigidity of nonwoven fabrics’, Textile Res J, 1965, 35(1), 48–57. 5. Mao N and Russell SJ, ‘Directional permeability of homogeneous anisotropic fibrous material, Part 1’, J Text Inst, 2000, 91, 235–43. 6. Mao N and Russell SJ, ‘Directional permeability of homogeneous anisotropic fibrous material, Part 2’, J Text Inst, 2000, 91, 244–58. 7. Mao N and Russell SJ, ‘Anisotropic liquid absorption in homogeneous two-dimensional nonwoven structures’, J Appl Phys, 2003, 94(6), 4135–38. 8. Gilmore T, Davis H and Mi Z, ‘Tomographic approaches to nonwovens structure definition’, National Textile Centre Annual Report, September, 1993, USA. 9. Mao N and Russell SJ, ‘Modelling of permeability in homogeneous three-dimensional nonwoven fabrics’, Text Res J, 2003, 91, 243–58. 10. Mao N and Russell SJ, ‘A framework for determining the bonding intensity in hydroentangled nonwoven fabrics’, Composite Sci Technol, 2006, 66(1), 66–81. 11. Wrotnowski AC, ‘Nonwoven filter media’, Chem Eng Progr, 1962, 58(12), 61–7. 12. Wrotnowski AC, ‘Felt filter media’, Filtration and Separation, September/October, 1968, 426–31. 13. Goeminne H, ‘The geometrical and filtration characteristics of metal–fiber filters – a comparative study’, Filtration and Separation, 1974, August, 350–5. 14. Poiseuille JL, CR Acad Sci Paris, 1840, 11, 961, 1041; 1841, 12, 112. 15. Rollin AL, Denis R, Estaque L and Masounave J, ‘Hydraulic behaviour of synthetic nonwoven filter fabrics’, Can J Chem Eng, 1982, 60, 226–34. 16. Giroud JP, ‘Granular filters and geotextile filters’, Proceedings Geo-filters’96, Montréal, 1996, 565–680. 17. Lambard G, Rollin A and Wolff C, ‘Theoretical and experimental opening size of heat-bonded geotextiles’, Textile Res J, 1988, April, 208–17. 18. Faure YH, Gourc JP, Milloe F and Sunjoto S, ‘Theoretical and experimental determination of the filtration opening size of geotextiles’, 3rd International Conference on Geotextiles, Vienna, Austria, 1989, 1275–80.

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19. Faure YH, Gourc JP and Gendrin P, 1990, ‘Structural study of porometry and filtration opening size of geotextiles’, Geosynthetics: microstructure and performance, ASTM STP 1076, Peggs ID (ed.), Philadelphia, 102–19. 20. Gourc JP and Faure YH, ‘Soil particle, water, and fiber – a fruitful interaction now controlled’, Proceedings, 4th International Conference on Geotextiles, Geomembranes and Related Products, The Hague, The Netherlands, 1990, 949–71. 21. Aydilek AH, Oguz SH and Edil TB, ‘Digital image analysis to determine pore opening size distribution of nonwoven geotextiles’, J Comput Civ Eng, 2002, 16(4), 280–90. 22. Hearle JWS and Ozsanlav V, ‘Nonwoven fabric studies, part 1: a theoretical model of tensile response incorporating binder deformation’, J Textile Inst, 1979, 70, 19–28. 23. Hearle JWS and Newton A, ‘Nonwoven fabric studies, Part XIV: Derivation of generalized mechanics by the energy method’, Textile Res J, 1967, 37(9), 778. 24. Darcy H, Les Fontaines Publiques de la Ville de Dijon, Victor Valmont, Paris, 1856. 25. Nogai T and Ihara M, ‘Study on air permeability of fibre assemblies oriented unidirectionary’, J Text Machine Soc Japan, 1980, 26, 10. 26. Happel J and Brenner H, Low Reynolds’s Number Hydrodynamics, Prentice Hall, 1965. 27. Kozeny J, ‘Uber Kapillare heitung des wassers in Boder’, Roy Acad Sci, Vienna, Proceedings Class 1, 1927, 136, 271. 28. Carman PC, Flow of Gases through Porous Media, Academic Press, New York, 1956. 29. Davies CN, ‘The separation of airborne dust and particles’, Proc Instn Mech Engrs, IB, 1952, 185–213. 30. Piekaar HW and Clarenburg LA, ‘Aerosol filters: Pore size distribution in fibrous filters’, Chem Eng Sci, 1967, 22, 1399. 31. Dent RW, ‘The air permeability of nonwoven fabrics’, J Textile Inst, 1976, 67, 220–23. 32. Emersleben VO, ‘Das darcysche filtergesetz’, Phsikalische Zeitschrift, 1925, 26, 601. 33. Brinkman HC, ‘On the permeability of media consisting of closely packed porous particles’, Appl Sci Res, 1948, A1, 81. 34. Iberall AS, ‘Permeability of glass wool and other highly porous media’, J Res Natl Bureau Standards, 1950, 45, 398. 35. Happel J, ‘Viscous flow relative to arrays of cylinders’, AIChE J, 1959, 5, 174–7. 36. Kuwabara SJ, ‘The forces experienced by randomly distributed parallel circular cylinder or spheres in a viscous flow at small Reynolds numbers’, J Phys Soc Japan, 1959, 14, 527. 37. Cox RG, ‘The motion of long slender bodies in a viscous fluid, Part 1’, J Fluid Mech, 1970, 44, 791–810. 38. Sangani AS and Acrivos A, ‘Slow flow past periodic arrays of cylinders with applications to heat transfer’, Int J Multiphase Flow, 1982, 8, 193–206. 39. Gebart B, ‘Permeability of unidirectional reinforcements for RTM’, J Composite Mater, 1992, 26(8), 1100. 40. Collins RE, Flow of Fluids through Porous Materials, Reinhold Publishing Corporation, New York, 1961. 41. Rushton A, ‘The analysis of textile filter media’, Separation and Filtration, November/ December, 1968, 516.

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42. Sullivan RR and Hertel KL, ‘Flow of air through porous media’, J Appl Phys, 1940 11, 761. 43. Sullivan RR, ‘Specific surface measurements on compact bundles of parallel fibres’, J. Appl Phys, 1942, 13, 725–30. 44. Shen X, An application of Needle-punched Nonwovens in the Press Casting of Concrete, PhD Thesis, University of Leeds, 1996. 45. Rollin AL, Denis R, Estaque L and Masounave J, ‘Hydraulic behaviour of synthetic nonwoven filter fabrics’, Can J Chem Eng, 1982, 60, 226–34. 46. Scheidegger AE, The Physics of Flow Through Porous Media, University of Toronto Press, Toronto, 1972. 47. Happel J and Brenner H, Low Reynolds Number Hydrodynamics, Prentice Hall, Englewood Cliffs, NJ, 1965. 48. Drummond JE and Tahir MI, ‘Laminar viscous flow through regular arrays of parallel solid cylinders’, Int J Multiphase Flow, 1983, 10, 515–40. 49. Langmuir I, Report on Smokes and Filters, Section I. US Office of Scientific Research and Development, No. 865, Part IV, 1942. 50. Miao L, The Gas Filtration Properties of Needlefelts, PhD Thesis, Department of Textile Industries, University of Leeds, 1989. 51. Mao N and Russell SJ, ‘Capillary pressure and liquid wicking in three-dimensional nonwoven materials’, J Appl Phys, 2008, 104(3), 034911–18. 52. Scheidegger AE, The Physics of Flow Through Porous Media, University of Toronto Press, Toronto, 1972. 53. Peek RL and McLean DA, Ind Eng Chem Anal Edn, 1934, 6, 85. 54. Minor FW, Schwartz AM, Buckles LC and Wulkow EA, ‘The migration of liquids in textile assemblies’, Textile Res J, 1959, 29, 931. 55. Laughlin RD and Davies JE, ‘Some aspects of capillary absorption in fibrous textile wicking’, Textile Res J, 1961, 31, 904. 56. Lucas R, ‘Ueber das Zeitgesetz des Kapillaren Aufstiegs von Flussigkeiten’, Kolloid Z, 1918, 23, 15. 57. Washburn E, ‘The dynamics of capillary flow’, Phys Rev, 1921, 17(3), 273–83. 58. Gupta BS and Wadsworth LC, ‘Differentially absorbent cotton-surfaced spunbond copoplyester and spunbond PP with wetting agent’, Proceedings of 7th Nonwovens Conference at 2004 Beltwide Cotton Conferences, San Antonio, TX, January 5–9, 2004. 59. Robinson GD, A Study of the Voids within the Interlock Structure and their Influence on Thermal Properties of Fabric, PhD Thesis, Department of Textile Industries, University of Leeds, 1982. 60. Carroll BJ, ‘Accurate measurement of contact-angle, phase contact areas, drop volume, and Laplace excess pressure in drop-on-fiber systems’, J Colloid Interface Sci, 1976, 57(3), 488–95. 61. Cassie ABD, ‘Physical properties of wool fibres and fabrics’, Wool Research (II), WIRA, Leeds, 1955. 62. Princen HM, J Colloid Interface Sci, 1969, 30, 359–71. 63. Bankvall C, ‘Heat transfer in fibrous material’, J Testing Evaluation, 1973, May, 235–43. 64. Bomberg M and Klarsfeld S, ‘Semi-empirical model of heat transfer in dry mineral fiber insulations’, J Thermal Insulation, 1983, 6(1), 157–73. 65. Stark C and Fricke J, ‘Improved heat-transfer models for fibrous insulations’, Inte J Heat Mass Transfer, 1993, 36(3), 617–25.

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66. Schuhmeister J, Ber K Akad Wien (Math-Naturw Klasse), 1877, 76, 283. 67. Baxter S, ‘The thermal conductivity of textiles’, Proc Phys Soc, 1946, 58, 105– 18. 68. Kirby R and Cummings A, ‘Prediction of the bulk acoustic properties of fibrous materials at low frequencies’, Appl Acoustics, 1999, 56(2), 101–25. 69. Burns SH, ‘Propagation constant and specific impedance of airborne sound in metal wool’, J Acoustical Soc Am, 49 (1971), 1–8. 70. Mechel FP, ‘Eine Modelltheorie zum Faserabsorber, Teil I: Regulare Faseranordnung; Teil II: Absorbermodell aus Elementarzellen und numerische Ergelnisse’, Acustica, 36 (1976/1977), 53–89. 71. Cummings A and Chang I-J, ‘Acoustic propagation in porous media with internal mean flow’, J Sound Vibration, 114 (1987), 565–81. 72. Attenborough Y, ‘Acoustical characteristics of porous materials’, Phys Rev, Phys Reports, 1982, 82(3), 179–227. 73. Tijdeman H, ‘On the propagation of sound waves in cylindrical tubes’, J Sound Vibration, 39 (1975), 1–33. 74. Zwikker C and Kosten CW, Sound Absorbing Materials, Elsevier, Amsterdam, 1949. 75. Shoshani Y and Yakubov Y, ‘Numerical assessment of maximal absorption coefficients for nonwoven fibrewebs’, Appl Acoustics, 2000, 59(1), 77–87. 76. Voronina NN, ‘Improved empirical model of sound propagation through a fibrous material’, Appl Acoustics, 1996, 48(2), 121–32. 77. Voronina NN, ‘Influence of fibrous materials structure on their acoustic properties’, Acoustic J, 29, 1983, 598–602. 78. Delany ME and Bazley EN, ‘Acoustical properties of fibrous absorbent materials’, Appl Acoustics, 1970, 3, 105–16. 79. Voronina NN, ‘Acoustic properties of fibrous materials’, Appl Acoustics, 1994, 42, 165–74. 80. Reist PC, Aerosol Science and Technology, McGraw-Hill, New York, 1993. 81. BS EN 779:2002, Particulate air filters for general ventilation — Determination of the filtration performance. 82. BS ISO 19438:2003, Diesel fuel and petrol filters for internal combustion engines – Filtration efficiency using particle counting and contaminant retention capacity. 83. Brown RC, Air Filtration—An Integrated Approach to the Theory and Applications of Fibrous Filters, Pergamon Press, Oxford, UK, 1988. 84. Davies CN (ed), Air Filtration, Academic Press, London, 1973. 85. Krish AA and Stechkina IB, ‘The theory of aerosol filtration with fibrous filters’, in Fundamentals of Aerosol Science, Shaw DT. (ed), J Wiley and Sons, New York, 1978. 86. Kirsh AA and Fuchs NA, ‘Investigation of fibrous filters: diffusional deposition of aerosols in fibrous filters’, Colloid Zh, 1968, 30(6), 836. 87. Stechkina IB, Kirsh AA and Fuchs NA, ‘Effect of inertia on the captive coefficient of aerosol particles by cylinders at low Stokes’ numbers’, Kolloid Zh, 1970, 32, 467. 88. Stechkina IB, Kirsh AA and Fuchs NA, ‘Studies on fibrous aerosol filters. IV. Calculation of aerosol deposition in model filters in the range of maximum penetration’, Ann Occup. Hyg, 1969, 12, 1–8. 89. Friedlander SK, ‘Theory of aerosol filtration’, Ind Eng Chem, 30, 1958, 1161–64.

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90. Friedlander SK, ‘Aerosol filtration by fibrous filters’, in Biochemical and Biological Engineering, Blakebrough N (ed), Vol. 1, Chap 3, Academic Press, London, 1967. 91. Steckina IB and Fuchs NA, ‘Studies on fibrous aerosol filters I: Calculation of diffusional deposition of aerosols in fibrous filters’, Ann Occup Hyg, 1966, 9, 59–64. 92. Lee KW and Gieseke JA, ‘Note on the approximation of interceptional collection efficiencies’, J Aerosol Sci, 1980, 11, 335–41. 93. Yeh, HC and Liu BYH, ‘Aerosol filtration by fibrous filters’, J Aerosol Sci, 1974, 5, 191–217. 94. Kirsch AA, Chechuev PV, ‘Diffusion deposition of aerosol in fibrous filters at intermediate Peclet numbers’, Aerosol Sci Technol, 1985, 4(1), 11–16. 95. Hinds WC, Aerosol Technology: Properties, Behaviour and Measurements of Airborne Particles, John Wiley and Sons, New York, 1999. 96. http://www.tsi.com/AppNotes/appnotes.aspx?Pid=33&lid=439&file=iti_041 97. Vaughan NP and Brown RC, ‘Observations of the microscopic structure of fibrous filters’, Filtration and Separation, 1996, 9, 741–8. 98. Stenhouse JIT, ‘Filtration of air by fibrous filters’, Filtration and Separation, 1975, 12 (May/June), 268–74.

6

Modelling and visualization of knitted fabrics

Y. Kyosev, Niederrhein University of Applied Sciences, Germany and W. Renkens, Renkens Consulting, Germany

Abstract: In this chapter an overview is presented of modelling methods for warp and weft knitted structures. The chapter begins with basic information about knitted structures, their classifications and the observation scales. After this introduction, the basic structural elements as main modelling elements at the meso-scale level are explained, followed by the basic problems and required steps during the modelling. The model generation is discussed in three main parts, checking the input data, topology generation and mechanics of the structure. For the mechanical model, two paths are presented, continuum and discrete. Some additional problems connected with the form of the yarn cross-section, yarn unevenness and contact treatment are followed. Finally, different ways of postprocessing the data are explained, including yarn volume rendering, visualization and other calculations, based on the modelled data and some application areas. Key words: knitting, warp knitted, weft knitted, structures, threedimensional models, topology, mechanics.

6.1

Aim and objectives of modelling knitted structures

The overall objective in this chapter is to provide a method of generating a geometrical model to predict the physical properties of knitted structures. By doing so virtually on a computer with all the production data as input parameters production time can be saved on the real machine. The required input parameters to generate the structures are: the type of machine and its properties, the pattern, the yarn parameters and the processes after knitting. All these parameters are known before the knitting procedure begins. The physical properties of real fabrics are evaluated by testing throughout and after the production process. In order to produce fabrics with certain properties, an iterative procedure is used: a test-sample has to be produced and measured and then the machine settings have to be changed. This cycle has to be repeated several times until the required values of the parameters are achieved. This development process requires machine time, testing devices and skilled staff and the costs of production (machinery, materials, 225

226

Modelling and predicting textile behaviour

yarns, skilled staff capable of carrying out these tasks and labour) must be assessed. If the measured values are incorrect, then changes must be made to the machine settings, to produce and measure new samples. The virtual knitting process will speed up the development process giving initial information about the possible properties of the new product, especially for technical textiles like composites, medical textiles, human implants, car seats, normal and special clothing, and so on. In all these application areas, knitted structures are widely used because of their particular properties. The prediction and optimization of these properties is the final aim of the modelling.

6.2

Classification of knitted structures

There are different classifications of knitted structures depending on the classification criteria. Because the task of this chapter is connected with the modelling of structures, only those criteria that influence the structural elements will be used. More details and other classifications can be found in Paling (1965), Spencer (2001), Wilkens (1995) and Weber and Weber (2008). There are two basic types of knitted structure that depend on the direction of creation of the loops: weft knitted (Fig. 6.1c) and warp knitted (Fig. 6.1b). The most simple way of recognizing the fabric is to follow the way of the loop feet (F) in Fig. 6.1(a). If one foot is coming down and the other goes up (F1) the structure is warp knitted, and if the feet are moving from left to right (F2) it is weft knitted. A weft knitted structure is usually created from a single or a few yarns building all the loops in one row (horizontally). A warp knitted structure uses one or more warp beams with at least as many yarns as number of needles

H

L F1

F2

F2 F1 (a)

(b)

(c)

6.1 Loop elements (a), warp (b) and weft (c) knitted structures.

Modelling and visualization of knitted fabrics

227

and all these yarns building only one loop/needle in one row (with some exceptions in some special patterns) (vertically), see Fig. 6.1. Furthermore, depending on the number of the working needle beds, both the warp and weft knitted structures can be divided into two groups: ∑ ∑

single-faced (produced usually on a single-needle bed machine) and double-faced (produced on a double-needle bed machine).

The single-faced fabrics have the faces of all the loops oriented on the same side. The double-faced fabrics have two sets of loops so that one set of loops show the faces on one side and the other set of loops show faces on the other. In Fig. 6.2(a) the face side of a loop is presented and Fig. 6.2(b) presents the reverse side. More details about the structures and classification can be found in Spencer (2001). In the weft knitting technique there are additional classes of structures: ∑ ∑

purl – where face and reverse stitches are from the same side in the same wale (using double headed needles), interlock – which could be classified as a special (sub-)class of doublefaced structures because the loops on both sets are not shifted on a half needle step like those of double face structures. But from a modelling point of view, these structures can be treated as derivatives of the double bed structures. They can be reconstructed using the same loops, but using a special arrangement of their position and orientation.

Single-faced warp knitted structures can be divided into sub-classes depending on the number and arrangement of yarns used during the knitting like:



(a)

6.2 Face (a) and reverse (b) side of a loop.

(b)

228

∑ ∑ ∑

Modelling and predicting textile behaviour

single guide bar with full threading, multiple guide bars with full threading and multiple guide bars with partial threading (general case) (Paling, 1965).

Depending on the number of the guide bars and the threading, how many yarns build loops (or other elements) on the same needle during a cycle can be calculated. This information is important for the modelling. With this data the general orientation of the yarns in the loop can be calculated. Analysing and modelling particular sub-class structures does not make sense, all the subclasses are presented in the double-faced warp knitted structures and because of this, all have to be modelled by one general method. In modelling knitted structures another criterion for classification could be very useful: classification according to the types of the different structural elements in a fabric. Fabrics can be divided into fabrics that have either only plain loops, or loops and one or more of the elements – tucks, platting loops, transferred loops, weft insertions and so on. Since each of these structural elements has to be modelled separately, they will be discussed in more detail in Section 6.4.

6.3

Scales in the structure

Knitted structures consist of a large set of structural elements described in the next section. These structural elements are built from yarns, which are presented, for simplification, as single rods or ropes. In reality, these yarns are structures of single fibres – or filaments. Three levels – structure, structural elements and single fibres present three scales known as the macro-, mesoand micro-scale levels (Verpoest and Lomov, 2005; Lomov et al., 2007). The macro-scale is important for the applications; it handles the mechanical behaviour of the structure as a continuum membrane or plate with known properties (Fig. 6.3). There are different methods for obtaining information at this macro-scale level, the most used but expensive way is to perform different mechanical tests on the samples. In order to understand macroscale behaviour or in some cases to predict it, knowledge of behaviour at the meso-scale is required. At the meso-scale, the yarns are considered as a continuum with known properties. At this level, the geometry and the mechanics of the single unit cell of the structure can be considered as known and conclusions can be drawn from the behaviour of the materials. Mesoscale models neglect the effects at the micro-scale level that are the result of the single fibre-to-fibre interactions. To model the entire knitted structure successfully, all these levels have to be considered during the modelling. The accuracy of the model depends on which of the levels are precisely modelled and from which levels only

Modelling hierarchy

er ial

M

es

l

osc ale

Yarn properties elasticity, bending and torsion rigidity, friction, cross-section

M

Geometry loop geometry and modulus, loop width and length

icr o-

Knitting process loop length, yarn tension take up speed

Structural basis

6.3 Hierarchy at the structural and material level of the knitted structures.

Fibre properties elasticity, bending and torsion rigidity, friction, cross-section Materials basis

229

Treatments and relaxations chemical treatments, temperature, moisture

ale

Knitting program chain links resp. knitting notation, threading

sc

Fibre–fibre interactions changes in crosssection

Topology elements

Modelling and visualization of knitted fabrics

at

Yarn–yarn interactions forces, contact, crosssection form changes stick–slip effect

M

ra

ale

ctu

-sc

ru

ro

St

ac

Mechanics interactions of yarns due to forces

M

Properties of the fabric stress–strain curves, cover factor, elasticity modulus

230

Modelling and predicting textile behaviour

the main information is used. For example, if after some investigation, the exact yarn cross-section form is not important, then the micro level – the interactions between the fibres inside the yarn – is not required. In other cases such as medical filters, the distance and the distribution of single fibres could be of greater importance. Then the micro-level has to be modelled and connected to the meso-level into one multi-scale model. How the information at the different levels is modelled does not play a significant role in the model. The higher the required accuracy of the model, the more that detail from the different levels has to be considered. There are two basic sources of information (Fig. 6.3), which determine most parameters of the knitted structure: the topological basis of the knitting program and the material basis of the fibre properties. With this information the typical behaviour of the knitted structure can be predicted. The knitting process parameters and the treatment of the structure present secondary information, which makes the model more accurate and gives it a more exhaustive profile in geometry and dimensions. There are different ways to build models depending on how the interactions at the different levels are considered. Including all the levels from microup to macro in one model has not been effective up until now. The high complexity of the nature and the large number of connections between the parameters make such models very clumsy and very difficult to handle. Normally only one of the levels is modelled and the information from the lower levels is given in some condensed form. In this chapter the modelling path will be shown, starting from the topological basis. This means that knitted structures will first be created from ideal yarns. No geometrical or mechanical property is applied. After this, step-by-step, the important properties at the different levels will be taken into account. This method consists of simple steps at each level. After each step it is decided if switching to the next level of complexity is useful. Depending on the final purpose of the model, the topology should already be sufficient (e.g. education). In other cases (design), geometrical models are required. Only if precise mechanical behaviour has to be predicted must the most complex and computational intensive mechanical models be used. There is another path in the simulation, following the scales in the material hierarchy: in this case fibres and yarns have to be created as single objects and after that knitted structures have to be built using these objects exactly as they are produced on the machine (Finckh, 2007) . This ‘virtual production’ has not been efficient enough until now to produce useful results, but in the next few years parallel computing could begin to be of practical interest too. The disadvantage of this method is that there are no intermediate results which can be used and the full simulation has to be done in order to obtain the results. Since the main difference between knitted structures is at the meso-scale

Modelling and visualization of knitted fabrics

231

level, modelling at this level will mainly be discussed. Micro-scale level investigation is the object of yarn mechanics and macro-scale investigation relates to the pure continuum or structure mechanics.

6.4

Structural elements of knitted structures at the meso-scale

Woven and braided fabrics can be represented as a set of single structural elements – crossing yarn pieces. Depending on which yarn (weft or warp) is in the upper position, these elements can be coded, creating the weaving pattern. This approach was extended to multilayer fabrics, where instead of warp or ‘not warp’, a number corresponding to the layer was used (Chen and Hearle, 2008; Lomov et al., 2000). The attempt to use this approach for knitted structures was implemented in software for weft knitted structures (Sobotka, 2004). The program can produce two-dimensional (2D) pictures but requires a special way of coding, since two cross points have to be given for a single loop. This approach requires the user to have special knowledge about the position of the yarns and thus has not found wide usage in modelling. Most authors use the whole loop as a single structural element during the modelling. In one loop several single cross point pairs are included but they appear at every loop, and therefore it is quite effective to make this kind of coding at the loop level. This way of coding the pattern is widely used in the commercial software like M1 of Fa. H. Stoll GmbH & Co. KG, Reutlingen, Germany, as well as in the academic community like WeftKnit (Moesen, 2002; Moesen et al., 2003; House and Breen, 2000, p 271).

6.4.1 Loops Following the definition of Spencer (2001): the interlooping consists of forming yarn(s) into loops, each of which is typically only released after a succeeding loop has been formed and intermeshed with it so that a secure ground loop structure is achieved. The loops are also held together by the yarn passing from one to the next. This rule is valid for all structures held together by loops: weft knitted, warp knitted and multilayer structures where thin loop yarn holds the heavy tows together. A loop consists of a head H (Fig. 6.1a), two side legs (limbs) L and, at each base is a foot (F). The foot meshes through the head of the loop formed during the previous knitting, cycle, usually by the same needle. The yarn passes from the foot of one loop into the foot and leg of the next loop formed by it. In weft knitting the yarn passes (normally) to the left or right

232

Modelling and predicting textile behaviour

in the same row, while in warp knitting one foot passes to the preceding and the other to the following row. If the feet are crossing, the loop is called closed (Fig. 6.4b): if not it is called open (Fig. 6.4a). The subdivision into open and closed loops is only important for warp knitted structures. In weft knitted structures the loops are normally open because the yarn continues to be supplied in one direction (with some exceptions in special cases).

6.4.2 Platting loops Platting is the process of building loops with two or more yarns (Fig. 6.5). For weft knitted structures, platting is used as a special patterning technique, where usually one of the yarns covers a second one (with a special feeding system). For warp knitted structures, platting is a more or less natural process. Most warp knitted fabrics are produced with at least two guide bars working in opposite directions to improve the stability. For this reason, the loops consist of two yarns.

6.4.3 Held loops Held loops (Fig. 6.6, H) are typical for weft knitted structures. The needle retains a previous loop. It is not released and knocked over until the next or a later yarn feed (Spencer, 2001). Depending on the knitting technique, the length of the loop is different in the fabric and in the symbolic representation.

(a)

6.4 Loops, (a) open, (b) closed loop.

(b)

Modelling and visualization of knitted fabrics

(a)

233

(b)

6.5 Platting loops in tricot warp knitted structure with two guide bars (a) and photo of such a structure (b).

H

F

F

(a)

(b)

6.6 Held loop (H) und float loop (F) on weft knitted (a) and warp knitted (b) structures.

Normally, held loops have about the same length as all other loops in the row. Under the yarn tension it ‘borrows’ a little length from neighbour loops which become shorter.

6.4.4 The float loop Float loops (Figure 6.6, F) are freely floating pieces of yarn between other loops. The limitation that the yarn has to be between other loops is important for the modelling because being between other loops means that in the structure this yarn piece will have a special geometry. The floats from weft insertions or inline have to be treated separately in order to have distinct topological differentiations during the coding of the structures. Float loops in weft knitted

234

Modelling and predicting textile behaviour

structures are special elements which require special programming. In warp knitting, floating loops are a normal part of the knitting process produced during the underlapping of the guide bars.

6.4.5 Tucks or press loops Tucks are overlapped pieces of yarn not formed into a loop. They can be formed on a Raschel machine with a fall plate or on a tricot machine with bearded needles and a change presser (Paling, 1965; Wilkens, 1995, p 28). With weft knitting machines, tucks are formed by switching off the highest cam, so that the new yarn overlaps the needle with the old loop without knitting the old loop. As yarns move in vertical directions for the warp knitted fabrics and horizontally in weft knitting, the geometry and the effect of tucks in weft and warp knitted fabrics differ (Fig. 6.7) significantly. Depending on loop length and yarn properties, the real appearance (Fig. 6.7c) of tucks normally differs from the topological representation (Fig. 6.7b). In warp knitting, tucks are normally used to knit special patterns for decoration purposes. In weft knitting, tucks are an important structural element used regularly in the structure to change the relief of the structure and its elongation limits. For both warp or weft knitting, tucks cannot be produced independently; they are connected to a loop, changing its geometry.

6.4.6 Transferred loops The loop transfer is typical in weft knitting machines and is very seldom used in modern warp knitting (an exception is given in Paling, 1965). Transferred loops (Figure 6.8, T) are produced on one needle and afterwards are moved to another needle. The new loop on the needle, where the moved loop was made before transference, has a different form (Fig. 6.8, N). It is similar to a single tuck, without a loop. This has to be considered during modelling

(a)

(b)

6.7 Tuck in warp (a) and weft (b) and (c) knitted structures.

(c)

Modelling and visualization of knitted fabrics

235

N T

6.8 Transferred loop.

3

2 L

1

6.9 Weft inserted elements in warp knitted structure.

the geometry. There are advanced structures with higher complexity, for example braids, which require more sophisticated treatment.

6.4.7 Weft insertion Weft insertion is a normal element in warp knitted structures. It is used to stabilize the structure in the horizontal direction (Fig. 6.9) or to build net

236

Modelling and predicting textile behaviour

shaped structures (Fig. 6.17). It is the most important structural element from a mechanical point of view. It is used in light nets as well as in the structures of textile reinforced composites, where heavy tows (usually carbon or glass) are inserted as weft yarns. In warp knitted structures like these, the only task the loops have is to keep these tows together.

6.4.8 Inlay Inlays are yarn (pieces) received from a warp beam, which are normally located between the floats, loops and weft insertions. They usually have a greater thickness than the yarns building loops and for this reason they remain almost in a straight form.

6.4.9 Other or modified structural elements In some special types of structures modified elements can be found. For instance, pile samples (plush) contain loops with higher lengths. Sometimes they are cut (velvet) and sometimes they build a relief surface.

6.5

Modelling steps

The modelling process can be separated into three steps, used in computational mechanics: pre-processing, solution and post-processing. Pre-processing includes the preparation of the input data, which here includes the knitting program (chain links for the warp knitting, knitting notation for the weft knitting), the knitting process parameters and the yarn properties. Checking the consistency and the correctness of this information belongs to this step; since no model can work properly with incorrect input data. The solution step can be divided into several sub-steps: ∑ ∑ ∑

Topology generation, where the basic structural elements have to be created; Loop form calculation, where the yarn path in each structural element has to be calculated. Depending on the accuracy and the models, two sub-modules are required: – Geometrical: to adjust the geometry so that the generated yarns have the proper thickness and loop density; – Mechanical: where the yarn equilibrium state and relaxation can be calculated. The contact calculations belong to this part, from which the efficiency of the whole model depends. After the calculations are done the post-processing of the data starts. This includes three-dimensional (3D) visualization of the structure, export of the simulated structure for other programs (finite element method,

Modelling and visualization of knitted fabrics

237

FEM), or calculations of certain parameters of the structure using special algorithms (area density, mechanical parameters). All these steps will be discussed separately.

6.6

Model building

6.6.1 Preparation of input data Before setting a machine with a new pattern and material, it is normal to check if the pattern, yarn parameters and machine settings are correct. These three separate but interconnected tasks describe the knitability of this pattern with this yarn on this machine. The experienced knitters know or ‘can feel’ whether a certain pattern could be knitted with a certain material and how the machine has to be set in order to obtain the best productivity and quality. The modelling algorithms need some tools and rules in order to check the knitability of the input data before starting the modelling. How to check the knitability of the input data is still an open field for researchers. There are three main trends which are followed by researchers looking for a solution of this prob­lem (1) expert systems with formalized rules, (2) physical process simulation and (3) an engineering approach based on the definition of some limitation rules. Each of these approa­ches has its own power and its limitations. The best solution could be some combination of the advantages of these three approaches. Expert systems The expert system consists of several rules stored in a database. The rules could be defined in a clear and understandable form if the expert system has fuzzy linear systems as an engine (Peeva and Kyosev, 2004) or engine based on neural networks. The neural network requires a large number of samples for learning, the fuzzy linear systems do too, or somebody with experience who could fill in. In both cases, the expert system could produce a reasonable answer only inside the range where it is learned. If completely different input parameters from those are used during the learning process, then the error between the proper answer and that computed from the expert system could be quite high. Rule-based systems of the type ‘if A then B’ built in a logic programming language like Prolog or others have the same problems because they are mathematically equivalent to fuzzy linear systems. The advantage of the expert systems is in their ability to collect and operate with a large number of rules and because of this they are one reasonable tool for building a system for decision making.

238

Modelling and predicting textile behaviour

Physical simulation Simulation of the behaviour of the yarns during the knitting process or the calculation of the yarn tensions into a virtual structure will provide a very powerful approach with the increasing power of computers. Simulation of these structures can be based on different mechanical models like set of beams, springs and masses, 3D cylinders and so on. Building a system of equations usually works automatically inside software based on the finite element method (FEM) or particle systems. However it could be written by anyone solving mathematical equations using an appropriate numerical method. The power of the model is not limited and depends on its complexity. But more complex models require more sophisticated numerical algorithms, more calculation time and they are usually more sensitive to changes in the input data. Because of this – if an insufficiently knitable set of yarn properties, pattern and machines settings are used – it could happen that the algorithm does not converge and the process cannot be finished. This is why the physical-based simulation is not very suitable as an independent approach for checking the knitability of the system’s input data. Engineer’s approach Engineers, in contrast to physicists and mathematicians, are looking for a simplified solution to the problems based on knowledge of the process. No simulation of the knitting process is required in order to obtain maximal elongation of the yarns during the knitting process. There are some simple calculations to check if the production of the fabric is possible. The standard checks for knitted fabrics begin with the compatibility of the yarn thickness and the available space in the machine, and structure of the yarn. For both warp and weft knitting, the yarn generally may not be thicker than the one-quarter of the distance between the needles. From the well known relationships for loop geometry, developed by Doyle and Munden (Postle et al., 1988, p 228): kc l k w= w l c = kc = 1.3 w kw c=

[6.1]

and for the tightness factor of the fabric K:

K=

tex l

[6.2]

Modelling and visualization of knitted fabrics

239

could be checked if the planned fabric belongs to the ‘normal’ cases or is a very special one. In the above equations, c is the number of courses per unit length, w is the number of wales per unit lengths, kc and kw are constants, tex is the yarn linear density (g km–1), and l is the loop length (cm). According to Postle et al. (1988), the constants in normal cases are in the ranges kc = 5.0–5.5, kw = 3.8–4.1 and K = 13–15. More detailed statistical data for these coefficients for weft knitted fabrics can be found in Kurbak (1995). The next check is to verify if all loops are interlooped. If this is not the case, an open non-stable structure will be built. A possible method for the implementation of these rules is described in (Meissner and Eberhardt, 1998). At this point it has to be mentioned that there are some special knitting techniques where single yarns overlap on the needle but after that are not knitted with the next loop. In this way, free yarn length for the neighbour loops is reserved which is important for reducing the yarn tension during the knitting. For instance, the ‘braid’ pattern has several loops that have to be transferred to a needle and then moved to one or the other side in several positions. Similar cases require special attention, which is not possible in this work. But such simple rules and checks used together with physical simulation or an expert system allow a quick check of the input data. It makes the developing process of the algorithms more convenient and protects users of the modelling software from logical errors and mistakes.

6.6.2 Topology generation The graphical description of the knitting process/program is a simplified way of notating the topology of knitted structures. It provides information for understanding the topology of the structure, but it does not present the yarn interconnections exactly, as it is already available in the 2D representation (Fig. 6.10c and 6.11b). In cases where the exact 3D geometry of the knitted structure is not needed, but just the optical appearance of this structure, it is suitable to use the notations as a pattern and to map to the elements pictures of the corresponding elements (Meissner and Eberhardt, 1998; Meissner et al., 2000). This way of producing quasi 3D pictures will be discussed in Section 6.7 on post-processing. For normal engineering applications, where a real 3D representation is required, a set of points of the loops have to be created. A logical way is to start from the 2D models and then add the third dimension. Two-dimensional topology The 2D topology of loops can be defined using different key points. most used are the position of the contact points, which can be defined well over the projections of the loop.

240

Modelling and predicting textile behaviour 3–4 1–0 3–4 1–0 3–4 1–0 4

3

2

(a)

1

0

(b)

(c)

6.10 Notation of the knitting program (chain-links) for warp knitting machines (a), lapping movement and yarn threading (b), and 2D graphical representation of the fabrics (c).



(a)

(b)

6.11 Weft knitted notation (a) and 2D topological representation (b) of a Swiss rib.

Modelling and visualization of knitted fabrics

241

One of the simplest topological representations is depicted in Fig. 6.12 (Wu et al., 1994; Moesen, 2002). A similar method is used in the models for weft knitted reinforced structures by Kyosev et al. (2005) and warp knitted reinforced structures by Robitaille et al. (2000). The yarn axis of each loop is described as a curve through six points, which are the contact points between the loop at the X–Y plane. The following parameters of the loops must be known for this representation: the loop height B, the loop width L, the course A, the distance between the feet K and the height at this distance yB. With these parameters the points will be defined as follows:



P.1 : (x = ± L ; y = 0) 2 P.2 : (x = ± K ; y = yB) 2

[6.3]

The coordinates of these points for loops of the wale i and course j then could be defined as: 1i, j :

2i, j :

x = ± L + A ◊i 2 x = ± K + A ◊i 2

y = 0 + B◊ j y = yB + B ◊ j

[6.4]

Using Equation [6.3], one loop can be defined as a curve, going through the points

11,1 – 21,1 – 12,1 – 1¢2,1 – 21,1 ¢ – 11,1 ¢

[6.5] y 12,1

1¢2,1

B

K

21,1 yB

2¢1,1

11,1

1¢1,1

L A

(a)

(b)

6.12 Loop and the anchor points for building 2D loop topology: (a) main dimensions, used to determine the coordinates of the anchor points, (b) notation of anchor points.

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Modelling and predicting textile behaviour

The loop height B is defined by the take up speed of the warp knitting machine. For the weft knitting machines, where force driven take up mechanisms are used, B can be defined by the equation [6.1]. The wale is 25.4 defined as A £ , where E is the machine gauge. If the fabrics have to E be modelled in the same dimensions as they are produced on the machine, then A is equal to the space between the needles. In other cases the structure relaxes and A is usually, but not neccessary smaller. Factor K and coordinate yB present the minimal distance between the two yarn axes and they depend directly on the yarn radius r:

K ≥ 2r



yB ≥ 2r

[6.6]

At the stage of topological representation only the location of the yarn axes is important. The yarn cross-section does not play any role. If the described equations have to be used to build more precise loop geometry, then the changes in the yarn cross-section of staple and multifilament yarns have to be taken into account. The points 1, 2, 2¢ and 1¢ define the position of a loop head. All coordinates of other loops of a plain structure can be calculated by using a simple translation of these points in the X and Y directions. Based on this topology, the key points of all the structural elements can be derived using simple modifications. In Fig. 6.13(a) an example of key-point selection for weft insertions at warp knitting structures is demonstrated. To define the position for a weft insertion (of warp knitted fabrics), only the points 1¢ and 2¢ are required, as defined in Equation [6.4].

23,1

1¢3,1

13,1 2¢

13,j

1¢3,j

12,1 2¢1,1

21,1 11,1 (a)

2¢3,j

22,1 1¢



23,j

2¢3,1

1¢1,1 (b)

11,j+1

1¢1,j–1 (c)

6.13 Key points in the topological representation of the weft insertions (a), hold loops (b) and tuck loops (c).

Modelling and visualization of knitted fabrics

243

The hold loops require the points from more wales. This is why the first subscript of the point number shows to which wale (row) it belongs. A hold loop (Fig. 6.13b) then goes through the points 11,1 – 21,1 – 13,1 – 23,1 – 2¢3,1 – 1¢3,1 – 2¢1,1 – 1¢1,1. The tuck loop according to the Fig. 6.13(c) goes through the points 1¢1, j–1 – 13, j – 23, j – 2¢3, j – 1¢3, j – 1¢1, j +1. The floats are trivial at the 2D level, since they remain as a connection between the neighbour elements. For several cases like teaching, a quick check of the knitting program and so on, this simplified graphical representation of the knitted structures based on a 2D topological representation was and is still used. To make it more understandable, smoothing curves, usually splines or arcs, are drawn through the key points. Some authors draw only the visible parts of the curves on the 2D pictures to create a 3D impression. A simple picture of such a realization, from the software from Sobotka (2004) is presented in Fig. 6.14 . Three-dimensional topology The 3D topological representation of the structure allows simplified but real 3D models. It can easily be derived from the 2D representation if the coordinates of the points consider the z-axis (Fig. 6.15). To the point 1i, j are now associated two points 1i+, zj and 1i–, zj so, that 1i+, zj ( x, y, z ) ∫ (1i, j (x ), 1i, j (y), + Dz )

1i–, zj ( x, y, z ) ∫ (1i, j (x ), 1i, j (y), – Dz )

6.14 2D representation of weft knitted structure based on the topology of the cross points.

[6.7]

244

Modelling and predicting textile behaviour Dz  R 2+z 1,2

21,2

–z 21,2

1+z 1,2

11,2

–z 11,2

2+z 1,1

21,1

–z 21,1

1+z 1,1

11,1

–z 11,1

y x z

z

y

x

6.15 Key points in the topological representation of loops in 3D space.

where 1i, j (x) is the x-coordinate of the point 1i, j. Here Dz > R, where R is the yarn radius. With this notation, one-half of the plain loop can be defined –z +z +z –z as a curve going through the points: 11,1 – 21,1 – 11,2 – 21,2 .

This principle can be extended and applied to all structural elements. Usually there is not only one good choice for the position of the key points. Since the goal of this section is only the topological representation of the structural elements, the selection of the key-points position is not very important. These points have to be characteristic of the element, they have to be defined easily and the calculation of their coordinates should be not very complex. The loops of the typical warp knitting pattern consist of more then one yarn and therefore more z-positions will be required for their key points. For example, if each loop is built from two yarns, then to the point z – z + z +2z 1i, j will be associated key points 1i–2 , j , 1i , j , 1i , j , 1i , j . Examples of structures are presented in Figs 6.16 and 6.17, where loops from two yarns and weft insertions are presented, respectively. The key points of plain loops as described above were located around the local X–Y plane. Structures like rib, purl, interlock and all warp knitted structures from machines with two needle beds require two such local planes. An example for a weft knitted rib structure with the Software WeftKnit (Moesen et al., 2003) is depicted in Fig. 6.18.

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245

6.16 Example of modelled warp knitted structure with loops from two yarns.

6.17 Example of modelled warp structure with loops and two weft insertions.

In Fig. 6.19 a Spacer fabric with two yarns per loop and additional weft insertions, modelled with the same principle, is presented. Using the main key points of the loops presented in Fig. 6.15, not only plane surfaces, but structures over other surfaces can also be generated. For example, if a tubular structure has to be modelled, the key points have to be oriented along an arc (Fig. 6.20). The transformation equations for positioning the single loops around such a surface can be found in several books about computer graphics. A short description can be found in Renkens and Kyosev (2008).

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Modelling and predicting textile behaviour

6.18 Weft knitted rib structure (generated with the software WeftKnit).

6.19 Representation of a warp knitted spacer fabric.

6.6.3 Yarn path representation Once the key points of the single loops are known from the topological representation, the next step – building the geometrical model – can be started. There are two tasks for the geometrical modelling, which are performed normally together, (1) adjusting the positions of the key points according the yarn geometrical parameters and (2) calculation the yarn axis form.

Modelling and visualization of knitted fabrics

247

6.20 Part of a warp knitted tubular structure.

6.21 Warp knitted structure, consisting of yarns with different diameters.

These tasks have not been separated in major research works until now usually because only simple structures with yarns with the same cross-section have been used. In reality, the weft inserted yarns are often thicker than the yarns used to build the loops (to connect them) (Fig. 6.21). In order to model the structure more correctly from a geometrical point of view, all the distances between the key points have to be checked and adjusted according to the cross-

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Modelling and predicting textile behaviour

section information from different yarns. In this way the 3D geometry of the structures will correspond more accurately to the geometrical parameters of the yarn and will remain without interpenetration of the yarns. The key points can be adjusted using different strategies. The most simple one is based on fixing the most outer element (the loops) and then translating them in the required direction (the single key points) until the distances between them become equal to the corresponding yarn radii. The second task – generation of the form of the yarn axis – is one of the most investigated topics in knitting modelling papers, where the main task is geometrical modelling. Geometrical models Starting from the work of Dalidovich in 1933, researchers have tried to find a suitable mathematical description of the loop form. The first models in this area were based on simple geometric assumptions about the different parts of the loops, using arcs, lines and other simple geometric curves (Doyle, 1953; Munden, 1959; Postle and Munden 1967a, b). There are several more or less detailed overviews and comparisons of the most cited geometrical models in the works of Moesen (2002), Loginov et al. (2002a,b,c) and Choi and Lo (2003). Lately, user friendly 3D visualization software for fast and accurate comparison (Knit GeoModeller, 2008) has become available. The Knit GeoModeller allows application of different models (Peirce, Peirce’s Loose, Leaf & Glaskin Loose and Leaf’s Elastica) for plain, rib and purl structures and to calculate loop geometry (Fig. 6.22). Since all these models

6.22 Loop form visualization with Knit GeoModeller, of TexEng Software.

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are not applicable to more complex structures, they are more interesting from theoretical point of view. Smoothing functions An intermediate approach for building the yarn path, which could be classified as being in between geometrical and mechanical modelling, is the use of natural splines to represent the yarn axis without the need for additional points. The natural cubic spline minimizes a function, which is an approximation of the total curvature of the spline. Since the total energy of the spline is proportional to the curvature, the natural cubic spline is an approximate configuration of the minimal energy of an elastic strip constrained though several points (Sherburn, 2007).

6.6.4 Mechanical models The geometrical models discussed until now do not take into account internal and external forces. To do this, all acting forces in the already generated geometry have to be applied, where the geometry could be presented as a continuum or a discretized medium. The continuum model is more often used for theoretical and analytical investigations of simple cases. The discrete model can be treated as a discretization of the continuum model, the main advantage being the quicker and easier programming of their structure. Continuum model There are two approaches for creating a mechanical model – force and energy. These two approaches are equivalent from the point of view of the mechanics although they differ in the way they form the equilibrium equations of the loop state. The mechanical models for textile structures, introduced by Leaf and Glaskin (1955), Konopasek (1980) and Hart et al. (1985a,b), differ in the yarn properties used and the definition of the contact zone (one point, two point, contact line etc). Hart et al. (1985a,b) apply mechanical analysis for warp knitted loops too and demonstrate the influence of the bending rigidity of the yarns over the loop form. The available models differ in the assumptions made about the yarn properties, such as bending, tension, torsion, compression effects and yarn stiffness. Another point in which the models differ is in the treatment of contact between the yarns, at one, two or more points. An extensive overview of the models can be found in several papers, for example Postle et al. (1988). One of the main approaches used to calculate the loop form – taking into account the mechanical properties – is minimization of the internal energy of the loops (De Jong and Postle, 1978):

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Modelling and predicting textile behaviour



Ê min E = min Á Ë

Ú

L o

ˆ Eb + Er + Ec + Et ds˜ ¯

[6.8]

where Eb, Er, Ec, Et are the bending, torsion, lateral compression and longitudinal tension energy of the single loop. L is the loop (or one half of the loop) length and s is the curvilinear coordinate. All the energy terms can be explained as functions of the local coordinates of the parametrically defined yarn axis z = z(s). Contact between the yarns is also included through the lateral compression term:

Ec = C · g(r)

[6.9]

where C is the yarn compression rigidity and r is the distance between the reference yarn to another in the contact area:

r = z(s ) – z (s )

[6.10]

Here g(r) is defined semi-empirically and has to be able to represent the differences in the behaviour of the different yarns from (near) non-compressible yarns like monofilaments to high compressible yarns like polyacrylonitrile (PAN) bulk yarns. Continuity conditions apply as an additional constraint:

z12 + z22 + z32 = 1

[6.11]

In order to find three independent variables (since zi is connected through Equation [6.11]), new variables are introduced often defined as two rotations (one in plane, one perpendicular to the plane) and the twist rate. More detailed descriptions of the numerical approaches implementing the Lagrange multipliers and the modified Hamiltonian can be found in De Jong and Postle (1978), Postle et al. (1988) and several other works. This method demonstrates very good results if applied to single bar structures (Hart et al., 1985a, 1985b). The open problem here could be in the rapidly increasing computational complexity if large and more complex structures have to be modelled. Discrete models Discrete mechanical models are usually based on reducing the yarns to a mass–spring system. The mass–spring system (Fig. 6.23) has become more popular during the last few years because of convenience in coding. Several researchers are using these models for modelling yarns (Kyosev and Todorov, 2007), cloths in the area of computer animation and knitted structures (Kaldor et al., 2008). The yarn is divided into yarn segments with initial length Li0. All the

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251

qi

Mi ,b

mi

Ci

Ti ,i–1

mi

Kb Ti ,i+1 T = T (e)

Kb mi +1

mi +1

6.23 Model of the yarn as a mass–spring system.

forces are concentrated upon particles i between segments. Each particle has a mass mi = Li0r, where r is the linear density of the yarn and the internal and external forces are applied there as a resulting force Fi. The equation of motion of ith particle is given by the second Newton law as:

mi

d 2 ri = Fi dt 2

[6.12]

where ri = (xi, yi, zi)T is the coordinate vector of the current ith particle. The resulting force Fi is calculated as a vector sum of the forces at the particle i:

F i = T i + B i + C i + Q i

[6.13]

where Ti, Bi are the resulting nodal forces from the tension and bending of the yarns, Ci is the resulting force from the contact and lateral compression of the yarn and Qi is the resulting force for all other external influences like gravity or others. The tension force in one segment between the points i and i + 1 is defined as the linear elastic force:

Ti, i+1 = EA · e · ei, i+1

[6.14]

where EA is the initial elasticity modulus, e the elongation of the yarn and

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ei, i+1 is the unit vector of the segment between the points i and i + 1. The resulting tension force in the node is the vector sum of the tension forces from the both segments, attached to the node. The bending is modelled using linear beam theory model:

Mi = Kb · k · ni–1,i,i+1

[6.15]

where the curvature could be expressed in a simplified linear relation from the bending angle as:

qi ki = 1 = r Li, i-1 + Li, i+1

and ni–1,i,i+1 is the normal of the plane, where the links between i–1, i and i+1 are located. The bending moment is applied to the nodes through its equivalent forces. The contact forces Ci are calculated according to the model, presented by (Kaldor et al. (2008):



Ê xi – x j ˆ Ci = kcontact ◊ f Á ˜ ◊ nij Ë 2r ¯

[6.16]

where the function:



Ï 1 + d2 – 2 Ô f (d ) = Ì d 2 Ô 0 Ó

d <1 otherwise

[6.17]

allows better treatment of the contact forces. Equation [6.12] is an ordinary differential equation (ODE) system, where the forces from the right hand side are calculated by spatial derivatives. Because of this, the system is and has to be treated as a system of partial differential equations, for which for instance the Verlet algorithm (Verlet, 1967) (explicit central difference for discretization of the time derivatives) can be used. For this scheme, the coordinates at the new time step xj+1 are calculated from the coordinates of the previous time steps and the acceleration aj of the previous time step as follows:

xj+1 = 2xj – xj–1 + aj · (Dt)2

[6.18]

As an explicit integration method, the stability limit is defined in terms of maximal time step Dt depending on the highest frequency of the system (Press et al., 1992):

Dtstable < L c

[6.19]

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where L is the length of the smallest element in the system and c is the speed of sound in the current yarn. In order to avoid oscillations of the mass–spring system, different damping forces can be included. The most simple realization of the damping effect is its implementation into Equation [6.18] with damping factor fD, where 0 £ fD << 1. Equation [6.18] in this case will be:

xj+1 = (2 – fD) xj – (1 – fD) xj–1 + aj · (Dt)2

[6.20]

The system integrates until the changes in the coordinates are smaller then a given error:

| x j+1 – x j | < e

[6.21]

Figure 6.24 shows the changes in the form of a single loop after several iteration steps. This loop (a) is defined at the beginning as being simplified through eight points, where the first and last points are fixed. After several iteration steps, the edges become more smooth (b) and (c). Finally, the equilibrium state for the internal bending forces is achieved (d). Finite element method models FEM software allows different strategies for the correct mechanical modelling of knitted structures. The explicit FEM can be used to simulate the knitting process at all (Finckh, 2007) or large deformations (Kyosev, 2006). The implicit FEM is more effective for small deformations, for example calculations of small loop parts (Lomov et al., 2008). FEM software allows the use of different geometries – the quickest method is to use beams for calculation with the yarn axes. Often shell elements are used where the yarns are modelled



(a)

(b)

(c)

(d)

6.24 Simulation of a single yarn under the influence of a pure bending force, from left to right: initial configuration, intermediate states and final state. The damping was set to fD = 0.01.

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Modelling and predicting textile behaviour

as whole ropes. This way has the advantage that the yarn cross-section is better modelled, but choosing the material parameters of the shell elements of the surface so that they represent the material behaviour of the yarn volume properly, is not trivial. The most correct 3D volume representation requires volume elements with anisotropic material and is related to the longest computational time. The main disadvantage of commercial FEM software simulation of knitted structures is that at the present time it is difficult to integrate such large and complex software into a stable simulation tool for textile engineers. Because of this, its use remains mainly in academic and research areas where qualified personnel are available.

6.6.5 Contact between the yarns The identification and treatment of contact between yarns is one of the most important tasks during the mechanical modelling of knitted structures. There are different approaches to contact detection as well as to the contact treatment described in the literature for cloth simulation, computer animation, FEM and so on (Volino and Magnet-Thalmann, 2000). Which of these approaches has to be used depends on the goal, size and complexity of the modelled structure. The large number of research works in this area demonstrates that it is seldom that satisfying results combining stability of the algorithm and the calculation time are achieved. These two goals – stability and calculation time – are connected to their own algorithms and theory. The stability of the algorithm depends on the theory and the algorithms for the contact treatment (Wriggers, 2006). Time consumption depends on the contact detection algorithms. For small models, every distance checking algorithm will work. For structures with a larger size, more sophisticated techniques have to be applied like, for instance, using bounding spheres or bounding boxes, placed into a hierarchy in order to avoid checks between the yarn pieces, which are far away from each other.

6.6.6 Yarn cross-section form and the yarn path Until now it has been assumed that the yarn cross-section is constant and circular. This assumption is very close to reality only for structures like monofilaments, but not for structures like multifilament or staple yarns. In this case, the cross-section of the yarn depends on several fibre and yarn parameters (twist, friction, fibre form – textured or not, etc.) as well as the curvature of the yarn and the contact with neighbouring yarns. By bending, the single filaments change their position in space to the most effective energy point and therefore, the cross-section changes (Fig. 6.25a) (Kyosev et al.,

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y (c) x (a) (e)

(d)

y x (b)

6.25 Yarn cross-section for multifilament yarns. (a) Unstable orientation of three filaments during bending. This configuration is not stable, because (1) the outmost filament has to be elongated and the innermost filaments have to shrink in order to stay in this configuration and (2) more energy is required to bend the joint cross-section around the Y axis, because of the higher moment of inertia. (b) Usual orientation of three filaments during bending. This configuration is stable, because (1) all the filaments have the same length and there is no additional need for some of them to be elongated and (2) less energy is required (less moment of inertia) to bend the cross-section. (c) Configuration of multifilament yarn if compressed laterally from all sides. (d) and (e) Different configurations of the multifilament yarn with an almost similar yarn axis, but with different pressure and orientation at the end points.

2005). As demonstrated in the structure shown in Fig. 6.25, a multifilament yarn could have a very compact cross-section if pressed on all sides by other yarns (Fig. 6.25c). The multifilament yarn could be flat and wide at the bending points where no forces from the other sides are applied (Fig. 6.25d). Both yarn pieces shown in Fig. 6.25(d) and (e) have almost the same yarn axis, but small differences in the contact points at the upper and lower ends lead to a significant difference in their width and cross-section form. Differences in the yarn tension (e.g. as result of different lapping movements) could also lead to such differences. These changes in the yarn cross-section belong to micro-scale modelling and usually require separate models and algorithms for successful and efficient treatment. Examples can be found of how the cross-section of the multifilament yarn can be simulated by computational mechanical methods, for instance by Samadi and Robitaille (2008) and they could be used in the future to refine the meso-scale model in several local places. The other

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approach – to model all filaments as single yarns – is computationally more challenging but is possible (Durville, 2007; Mahadik and Hallett, 2008). Research in this area currently shows examples of multifilament yarns with less then 20 filaments which are fewer than in reality, but significantly more than expected by several researchers ten years ago.

6.6.7 Yarn unevenness All the geometric and energy models are usually deterministic. They describe the loop form under the assumption that the yarn has constant properties along its whole length. Actually all textiles have parameters that are subject to significant deviations, for example the cross-section area, form, elasticity modulus and so on. This means that every calculation of the loop form with constant parameters can only be used as an approximation to a mean curve of the loops in the real fabric. After some steps into the investigation of the unevenness of yarn properties, the look of the fabric is included in a quality control system (Hardt, 1996). The calculated or measured data in the yarn cross-section is mapped onto the picture generated for a plain weft knitted fabric in order to simulate the periodic defects. These investigations have usually only been until now performed in basic plain structures because the yarn axes for more complex structures were not easily calculable. Once this initial geometry is available from the geometrical models, the variations in the parameters can be implemented and investigated.

6.7

Post-processing

6.7.1 Yarn volume rendering Description or calculation of the yarn axes is the main task during the modelling of knitted structures. Building the yarn volume or the yarn surface by sweeping the yarn cross-section through the yarn axes is a simple task described in several books for computational geometry as well as in some textile-related articles like Hardt (1995), Lomov et al. (2000), Liao and Adanur (2000) and Götkepe and Harlock (2002).

6.7.2 Visualization Having the set of points defining the yarn surface, different techniques and software can be used to produce the picture. Most commonly used is the cross-language cross-platform API for writing applications for 3D graphics OpenGL. For client–server applications, the VRML language allows visualization of the structures within a standard internet browser with a suitable add-in installation. To obtain a more realistic image, it is possible to map

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the surface with a suitable texture (Meissner and Eberhardt, 1998) or using lumislice-based rendering (Chen et al., 2003). For the textile personnel who do not have enough programming experience, the program TexGen (2007) is a suitable visualization tool. It allows easy modelling and visualization of textile structures at the meso-scale level, where different cross-section forms and size at each point of the yarn are used. The program was developed by the University of Nottingham (Sherburn, 2007).

6.8

Other types of model

For different purposes, there are other types of models suitable for knitted structures, although they will not be discussed in this section. Several researchers are building meshes of beams which are mechanically equivalent to the knitted structures for small elongations (Wu et al., 1994; Araújo et al., 2003, 2004). If the material parameters are determined experimentally, another possible path is to use direct 2D membranes as a continuum. Special ‘length’ and ‘contact’ elements described by Loginov et al. (2002a,b,c) could provide a more efficient way of making mechanical calculations of the behaviour of the fabrics. None of these techniques have been discussed in this chapter because currently they still need development in order to be more useful for practical application. However, they should be considered in the selection of a modelling strategy.

6.9

Application areas of the simulated fabrics and future trends

Simulated fabrics can be used in a very wide range of areas. Most important for textile technologies is the possibility of testing the pattern and the machine settings virtually before starting production (Kyosev and Renkens, 2007; Renkens and Kyosev, 2007). Modelled fabrics are an excellent tool for explaining the complex 3D nature of knitted fabrics. Modelling software allows completely new learning methods to be used, combining theory with generation of ‘virtual structure’ which allow information about the numbers and types of structures taught in a certain time to be significantly increased. Art designers can change the colours and see the effect almost immediately, as well as observing the combined effect of the structure and colour together. Standard praxis for weft knitted structures using pattern mapping in the software is now possible for warp knitting structures, too. Most important for engineers is 3D modelling. Engineers have to predict, for example, the mechanical or thermal behaviour of a structure, reinforced composite on a basis of knitted structures, in medicine for artificial vascular prosthesis or compression stockings. Not only the pore size, but also the elasticity of the fabrics, pressure over the foot and so on have to be calculated. The simulated

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structure Fig. 6.26 demonstrates most of these application at a glance – the modelled weft knitted structure at the yarn level allows colour design with realistic rendering with clear structure mapped on the foot surface.

6.10

Conclusions

During modelling of knitted structures, most attention is required at the meso-scale level. Here the interconnections of different structural elements are described. The structural approach for modelling starts with the generation of a simple, topologically correct description of the arrangement of yarns. As a next step, the yarn axes can be constructed by a set of key points, taking into account the geometrical parameters of the yarns and the machine. This approach is general enough and allows modelling of very complex structures built from different loops, tucks, floats and so on. In cases where the geometrical model is not accurate enough to simulate the yarn axes, a mechanical model has to be built and the static or dynamic equilibrium of the system has to be calculated. In this chapter, the mass–spring model is described showing how to simulate the dynamics of the structure. Up until now a complete mechanical calculation and the realistic rendering of the yarn surfaces still takes more computational time than is realistic for the textile

6.26 Simulated weft knitted structure at the yarn level. With the helpful support and permission from J. Kaldor, D. James and S. Marschner, Cornell University.

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engineer to spend in production. However, research results demonstrate that modelling knitted structures is possible at the loop level as well as at the yarn level. It is only a question of providing suitable hardware and efficient algorithms so that these models can be used in practice. The next expected steps in modelling include combination of the models at the different levels – multi-scale modelling – where effects at the microlevel can be considered effectively in meso-scale models. These models can be implemented in the calculation of the macro-scale behaviour of knitted textile products.

6.11

References

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Spencer, D. (2001), Knitting technology, Vol. 1, Woodhead Publishing, Cambridge, UK. TexGen (2007), University of Nottingham, http://texgen.sourceforge.net/index.php/ Main_Page Verlet, L. (1967), ‘Computer “experiments” on classical fluids. (i) thermodynamical properties of lennard–jones molecules’, Physics Review, 159(1), 98–103. Verpoest, I. and Lomov, S. (2005), ‘Virtual textile composites software Wisetex: integration with micro-mechanical, permeability and structural analysis’, Composites Science and Technology, 65(15–16), 2563–74. Volino, P. and Magnet-Thalmann, N. (2000), Virtual Clothing – Theory and Practice, Springer Verlag, Berlin. Weber, K.-P. and Weber, M. O. (2008), Wirkerei und Strickerei, Deutscher Fachverlag, Heusenstamm. Wilkens, C. (1995), Warp knit fabric construction, U.Wilkens Verlag, Heusenstamm, Germany. WiseTex: http://www.mtm.kuleuven.be/Research/C2/poly/software.html Wriggers, P. (2006), Computational contact mechanics, Springer Verlag, Berlin. Wu, W., Hamada, H. and Maekawa, Z. (1994), ‘Computer simulation of the deformation of weft knitted fabtrics for composite materials’, Journal of the Textile Institute, 85(2), 198–214.

7

Modelling of fluid flow and filtration through woven fabrics

M. A. Nazarboland, X. Chen and J. W. S. Hearle, University of Manchester, UK and R. Lydon and M. Moss, Clear Edge Group, UK

Abstract: Fluid flow passing through a textile assembly is featured in numerous applications including resin impregnation for textile composites and phase separation in fluid filtration. Understanding the behaviour of fluid flow through fabrics of various geometries is essential in order to optimise the manufacturing process and to the design and engineering of intended products. Based on a recent study on textile filters for improved filtration, this chapter reports on the finite elelment (FE) simulation of liquid flow through different types of woven fabric architectures. Furthermore, this chapter reports the establishment of a robust model that incorporates the introduction of solid particles, their transport towards the filter and the capture and packing of such particles. Key words: woven fabrics, filtration, fluid flow, computational fluid dynamics, finite element simulation.

7.1

Introduction

Textile assemblies, be they woven, knitted or nonwoven, have been widely employed for many different applications and have demonstrated their advantages over other materials and structures because of the unique ways in which materials are organised together. Not only can textiles be made into the conventional sheet form, they can also be created with the required features in the third dimension. For many applications, it is of interest to understand the nature of fluid flow through textile assemblies with different structural architectures. In composite impregnation, this would help to improve the resin uniformity in the composites and therefore improve the quality and performance of the composites. Knowledge of fluid flow through fabrics would also help civil engineers in road and dam construction where geotextile materials are playing a major role in reinforcing, separation and filtration. This chapter focuses on the modelling of fabrics with applications in industrial filtration, where textile structures are the main forms of filters. Filtration is the process by which solid particles are separated from a fluid by means of a porous medium, retaining the solid particles but allowing the fluid to pass through. Filtration is widely used in industrial 265

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applications, especially in the chemical, medical, food and paper industries. Filtration operations are capable of handling suspensions with varying characteristics ranging from granular and incompressible to slimes and colloidal suspensions. The interaction between the particles in suspension and the filter medium determines to a large extent the specific mechanisms responsible for filtration. The ideal filter medium should have resistance to blinding, as well as mechanical and chemical attrition. It should also have a maximum throughput, a clear fabric and a good cake discharge at the end of the filtration cycle.1 Other principal factors for efficient filtration are the incorporation of a well-chosen material and appropriate structural design of the filter itself. A variety of different materials may be utilised to act as the separating medium. The filter structure can be either woven or nonwoven and their constituent materials could range from natural fibres, such as cotton, synthetic fibres, such as polyester, and even metallic wires. In filtration, if the medium fails to retain the solids or allow the fluid to pass through, it does not accomplish its purpose, making the choice of the filter medium and its design critical. Correct selection of the filter medium depends on its structural design to retain solids of particular dimensions, the material from which it is made and its permeability. Many filters, such as nonwoven textiles and sand, have differing pore sizes. Therefore, it is logical to see that the efficiency of the filter medium is heavily dependent on the pore size and the pore size distribution. Woven fabric filters have been favoured over other forms of filter for separating solids of particular sizes, even though they are more expensive.2 The better controlled geometrical pore sizes in woven media facilitate a more satisfactory particle cut-off, blocking all the contaminant particles larger than its pore size and letting go of particles of smaller dimensions. Simulation and modelling offer excellent methods of studying flow filtration and they lead to improved quantitative and qualitative understanding of the filtration process. The costs of experimentation, which on some occasions is a limited method of obtaining data, can be substantially reduced if these are combined with theoretical modelling and simulation. Furthermore, these investigations provide an accelerated perception of the methodology required to break down the process into its fundamental physical phenomena. Once a model has attained a significant level of confidence, it can be successfully utilised as a design tool for new processes in filtration or optimisation of existing ones. This chapter will describe the various techniques in modelling fluid flow and filtration through woven fabrics, leading to the introduction of a novel approach to simulating filtration through woven fabrics. Using the proposed methodology, analytical studies are then carried out to investigate the effect of varying fabric parameters, fluid flow or solid particle shapes on fluid flow and filtration.

Modelling of fluid flow and filtration through woven fabrics

7.2

267

Various techniques in modelling fluid flow and filtration

Textile fabrics consist of interlaced yarns or fibres and they are complex materials with a porous structure that permits the flow of fluids through their constituent elements. Analysis and modelling of fluid flow through such porous structures play an important role in the design and construction of filters. Wehner et al.3 observed through experimental analysis that the same cloth had different characteristics depending on the fluid flowing through it. However, the combination of the tortuous fabric pore path and the nature of the equations governing fluid flow preclude development of an accurate theoretical model.

7.2.1 Modelling fluid flow through textiles Fluid flow is classified into laminar and turbulent flows and is described by a set of equations based on conservation of mass [7.1] and momentum [7.2]. For an incompressible fluid, these equations are respectively:

 — ∑ V = ∂u + ∂v + ∂w = 0 ∂x ∂y ∂z

[7.1]



 È Ê ∂ui ∂u j 2d ij — ∑ V ˆ ˘ Du j ∂ ∂ P r =– + + – Ím ˜˙ Dt 3 ∂x j ∂xi Í ÁË ∂x j ∂xi ¯ ˙˚ Î

[7.2]

where i and j are x, y and z coordinate directions, m is the viscosity coefficient and d­ij is the Kronecker delta. In a review article, Dullien4 classified pressure-flow through porous media into five categories of phenomenological, geometrical and statistical models, solving the complete Navier–Stokes equations or models based on flows around submerged obstacles. These categories represent merely different approaches to the same problem and naturally there is a great deal of overlap between them. Generally, modelling fluid flow through textiles can be broken down into the two fabric types, namely woven and nonwoven fabrics. Nonwovens and deep bed filters Deep bed filtration is one of the most common filtration methods in process industries. In this category of filtration, suspended solid particles in a fluid are removed by deposition onto a packed bed of grains or fibres.5 Many models have been proposed for simulating particle transport through porous media. The first attempts in modelling such system used empirical methods, in which

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the porous medium is considered as a closed system where deposition occurs following different forms of a rate law.6 Other early investigations were generally focused at the pore level and either included very simplified models of pore system effects in an approximate way or ignored them altogether. Examples of such models are the capillary-based permeability model by Kozeny,7 the dispersion coefficient model of Saffman8 and permeability models for an array of widely separate spheres.9 However, the simplicity of such models meant that they were primarily used as a basis for extrapolating experimental data. Herzig et al.10 discussed a continuum approach, but this practice relied on the continuum conservation equations and ignored the precise specification of the morphology of the pore. Woven fabrics In the previous investigations, fluid flow through a woven fabric has been simulated using three analogies: fluid flow through an assembly of (i) orifices, (ii) randomly packed bed of fibres and (iii) creep flow around submerged obstacles. One of the earlier and most common approaches to modelling of fluid flow through woven fabrics has compared it with that of fluid flow through an assembly of orifices. Much of the work in this field was based on attempts to simplify the flow problem through multifilament filter fabrics by using wire screens as an idealised representation of woven fabrics.11 In this approach, the wire screen was considered as a flow barrier with a number of square orifices or nozzles forming flow channels. Using a similar approach, Robertson12 attained a correlation between the Reynolds number and the discharge coefficient of airflow through plain-woven metallic meshes but failed to acquire a similar degree of confidence while investigating loosely woven multifilament cloths. Peirce13 stated that the interstice of the unit cell of a fabric is too convoluted a form to permit a complete hydrodynamic analysis. However, Backer assumed that textile yarns act as flexible, inextensible circular cylinders and classified the pores occurring in woven fabrics on the basis of the mode of yarn intermesh.14 Successive slices through the pore depth were cut to obtain the cross-sectional area of the pore and it was deduced that the use of the minimum pore area greatly reduced the scatter produced when using a projected open area. An alternative approach to fluid flow modelling through woven fabrics considers the cloth as a packed bed of fibres in order to derive a quantitative relationship between the media permeability, defined by the Kozeny–Carman equation,15 and its structural properties. Davies16 used airflow through beds of fibrous materials to show the relationship between the Kozeny constant and porosity. In monofilament fabrics, the height of the bed is roughly twice the diameter of the fabric and the diameter of a pore is venturi-shaped.17 Hence this approach is not suitable for monofilament fabrics.

Modelling of fluid flow and filtration through woven fabrics

269

In the drag modelling approach, the pore walls are treated as obstacles to the flow of viscous fluids18 and simplified Navier–Stokes equations are used to estimate the drag of the fluid on each portion of the wall. This technique models the textile material as a system of cylinders.19 Hutson20 replaced the woven fabric by two perpendicular rows of equal, parallel cylinders to provide a relationship for the flow through both systems. This is a heuristic approach which is not supported by theoretical results and which uses an overtly simplified model of the woven fabric geometric structure.

7.2.2 Modelling filtration through woven fabrics Filtration as part of a more general field of solid–liquid separation has its roots in the flow of fluid in a porous medium and its quantification is based on the infamous D’Arcy equation.21 Over the past century, many scientists have extensively studied and developed equations and analyses of various filtration processes and related aspects. While some investigators have pursued experimental methods, others have developed numerical methodologies based on computer simulation. Ruth22 carried out an analysis of constant pressure cake filtration and was able to describe the constant pressure filtration behaviour of a wide range of materials. However, he was unable to describe the internal filtration mechanisms. The latter problem arises as there are major difficulties in obtaining or analysing cake samples during the cake formation process. To overcome this obstacle, Reitema23 inserted metal pins into the cake at different levels to measure the electric resistance in the cake and used mercury manometers at various levels above the cake surface to measure the pressure distribution in the cake. Baird and Perry24 precisely positioned nine pairs of metal electrode pins also to measure the electric resistance of the cake and discovered that a collapse occurred when filtration was run for some time and a critical thickness had been reached. Other researchers also used intrusive electrodes,25–27 flush mounted electrodes28 and pressure probes29, 30 to determine various aspects of cake formation such as filtrate volume, average or specific cake porosity and cake thickness. All of the analytical methods discussed use a constant pressure gradient, which is convenient in a laboratory, but not always possible in industry. Tarleton31, 32 used mechatronics technology to assess and predict pressure filtration by attaching various transducers to the experimental apparatus. Various parameters were then recorded and controlled using dedicated computer software facilities. The major drawback of the aforementioned techniques is that the cake structure is affected by the presence of these probes. Thus, alternative non-destructive methods that use X-rays,33, 34 nuclear magnetic resonance techniques35, 36 or ultrasound37 have been utilised in the investigation of porosity profiles in filter cakes. Even though these methods are quite reliable,

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they are expensive to perform, slow and also unable to measure a wide range of cake properties. Furthermore, it is questionable whether ultrasound methods can be directly applied to cakes under significant compression. Approaches used in modelling the flow of suspended particles in a porous media have classically been based on the macroscopic continuum conservation equation. These models are purely phenomenological and ignore the morphological aspect of the pore, such as its structure, connectivity and size distributions. Also, in methods using analytical solutions, even if parameters are determined accurately in the laboratory, they are often not applicable in industrial operations. The non-linearities of the mathematical models derived to solve the flow problem, together with restrictions and inaccuracies make it impossible to use analytical methods. These deficiencies led to the use of numerical analysis in order to solve the partial differential equations governing filtration. Atsumi and Akiyama38 transformed these equations into ordinary differential equations by assuming the average liquid content in the cake to be constant and the filtrate volume to be proportional to square root of time. Tosun39 solved these equations using material coordinates and a method of successive approximation and convergence, proposed by Kehoe40 as an alternative to the simplification transformation of the partial differential equations. Finite element41, 42 and finite difference43, 44 methods were used in developing numerical simulation models of one or two particles flowing in straight channels. The complexity of the pore geometry and the constant need to evolve the computational mesh make the utilisation of these methods quite difficult. Some investigators45, 46 used the lattice-Boltzmann method to overcome these problems. However, none of these models included the effects of deposition. Wakeman47 used numerical approximations to analyse the formation and growth of compressible filter cakes based on D’Arcy’s law, continuity principles and knowledge of cake porosity profiles. However, the method of treating the moving boundary of the cake made it difficult to assess the results. Wells and Dick48 combined the filtration and expression in one unified numerical model, formulating mass balance and flow equations for both solids and liquids. However, they used an absolute coordinate system that made the numerical computations complicated owing to cake structure deformation. Sorenson et al.49 overcame this stricture by using material coordinates. The deficiency of such a system is the introduction of problems associated with a moving boundary condition. Burger et al.50 imposed further simplifications to simulate the filtration as a free-boundary problem. Stamatakis and Tien51 used absolute coordinates to solve the governing equations of filtration by finite difference methods. On the other hand, design of the numerical solution requires special care, and Evje and Karlsen52 observed that numerical methods based on inexpert finite difference discretisation

Modelling of fluid flow and filtration through woven fabrics

271

may produce stable but physically incorrect solutions. Biggs et al.53 used a lattice–gas automata based numerical method to resolve the dynamics of the individual solid particles and introduced a sticking probability parameter for particles coming into contact with a pore wall or other particles. The only reservation with this work is that it only considers two-dimensional (2D) geometries.

7.3

Model design and analysis

Construction of a software tool to simulate the filtration process through woven fabric filters entails two major stages. The first stage focuses on the creation of the three-dimensional (3D) woven structure and the second on modelling of the filtration process.

7.3.1 Geometrical models for woven fabrics Over the last 100 years, various techniques have been employed to model woven fabrics, in particular, yarn path and yarn cross-section. The simplest approach was by Peirce54 in which the yarn path, in the case of plain weave, is represented by two arcs tangentially connected by a straight-line segment, whilst the yarn cross-sections were considered to be circular and elliptical, respectively. The Peirce approach is, of course, an idealised approximation for the yarn path and yarn cross-section. In more recent studies, spline curves were employed to represent the yarn path and surface. A spline curve can mathematically be described as a piecewise cubic polynomial function whose first and second derivatives are continuous across the various curve sections. In computer graphics, spline refers to any composite curve formed with polynomial sections satisfying a specified continuity condition at the boundary of the pieces. Based on the specification of yarn linear density and thread density of the fabric, a geometrical model of the fabric is created, capable of dealing with different yarn cross-sectional shapes (circular, racetrack, lenticular and irregular) and yarn paths (Peirce’s model, B-spline model and the Kochanek–Bartels Hermite spline model).55 In order for the geometrical models to be used for fluid flow analysis, graphical files of IGES format are created which are recognisable by graphicsbased computer aided design (CAD) programmes. After cleaning up and meshing, a fabric model for fluid flow analysis is established.

7.3.2 Establishing a model to simulate fluid flow Hydrodynamics and essentially significant effects arising from fluid flow can be a fundamental factor in modelling separation of solid particles suspended

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in a fluid. In filtration, the particles are supposed to be constrained by the filtering medium, whereas the fluid can flow freely through the medium. Hence fluid flow modelling becomes an essential simulation element of any filtration process.56 Only a handful of fluid flow problems can be solved analytically. In all other cases, an analytical solution is not feasible and an approximate numerical approach is sought using computational fluid dynamics (CFD). CFD is the use of computers in analysis of fluid dynamics problems. The most straightforward approach to solve the problem described by the continuum fluid dynamics is to find the solution to the Navier–Stokes equation and the equation of continuity with a given set of boundary conditions. This method entails a complex discretisation procedure of the fluid volume and implementation of all the relevant equations. Iterative methods are usually applied to solve such equations. One proposed algorithm is to import one woven fabric repeat as the representative of the entire fabric. A fabric repeat is the smallest representation of the weave pattern, which when repeated, constructs the whole fabric. In constructing the model, the fabric filter is placed in a rectangular (or circular) fluid flow chamber and the flow problem is solved by specifying the boundary conditions. The two planes of the chamber parallel to the fabric are the inlet and outlet pressure boundaries. The cross-sectional area of the chamber is related to the size of the fabric repeat. The four other containment planes of the rectangular chamber are either regarded as solid wall, when only examining the specified fabric area, or as symmetry, when the eventual results need to be expanded to consider the scaled-up fabric from the repeats. A solid wall is not permeable and thus no-slip boundary conditions apply. Figure 7.1 illustrates the schematics of a fabric repeat being placed inside the chamber and the specification of the boundary conditions needed. The choice of varying fluids and boundary condition specifications are examined in this chapter. Commercial CFD software packages such as Fluent57 can facilitate solving the fluid flow predicaments.

7.3.3 Establishing algorithms to simulate filtration To enable modelling filtration through woven fabrics, various issues relating to solid particles need to be considered. These aspects are primarily concerned with particle shapes and then the generation, transport, capture and packing of these particles. Particle generation Spherical, cylindrical, needle shaped, ellipsoid and discus shaped particle shapes, illustrated in Fig. 7.2, are considered in this section. Spherical particles

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273

Symmetry (or wall) boundary Inlet pressure

Direction of flow

Direction of flow Y

Outlet pressure/ velocity

Chamber (meshed volume) (a)

Sample fabric Z X Direction of flow

Y Z

X

(b)

7.1 Schematic diagram of a fabric inside the chamber and required boundary conditions: (a) side-view (y–z plane) and (b) front-view (x–y plane).

d1



d2 d1 (a)

t4

d3

d5

t2

t3

d4

(b)

(c)

(d)

t5 (e)

7.2 Comparing dimensions of various particle shapes: (a) spherical, (b) ellipsoid, (c) cylindrical, (d) discus, (e) needle.

are characterised by the radius of the sphere. Needle-shaped particles are constructed by joining the base of two identical cones facing in opposing directions and thus are characterised by the diameter of the circular base and the particle height. Discus and cylindrical shaped particles are defined by the diameter of their circular sides and their thickness or height. Ellipsoids are assembled using their equatorial and polar diameters. It is believed that the geometrical shapes shown in Fig. 7.2 would cover most of the particle shapes that are seen in practice. Further complicated shapes can be created easily (e.g. as a combination of the five basic shapes

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Modelling and predicting textile behaviour

specified) if necessary. The force loading details on the newly formed particle geometry may well be different, but the methods established describing the interaction of the listed particle shapes would also apply to the interaction associated with such particle geometries. The first stage prior to particle transport is consideration of the particle size and generation of their initial positions. To make this process simpler, equations tackling spherical particle shapes are first specified and then these equations are extended to incorporate the ability to consider other particle shapes. The particle size in the proposed algorithm is designed so that it will obey a log normal distribution. This gives the proposed algorithm the ability to consider a range of particle sizes according to the user specifications (i.e. particle size specified along with a standard variation value). The log normal distribution of a particle with radius r has a probability density function which can be described by:

f (r ) =

Ê (ln r – ln r )2 ˆ 1 exp Á – 2 ln 2 s ˜¯ Ë r ln s 2p

[7.3]

where s is the standard deviation and r is the mean of particle radii (for the range of particle radius dimensions specified). If the mean of the particle radii is normalised to be 1.0, then ln r approaches zero. Using this equation, the probability of variation of radius values for spherical particles is calculated. For particles other than the spherical shapes, there are two defining parameters that are employed to construct them. The probability density function specified by Equation [7.3] is used to compute the range of these defining parameters by substituting the new parameter instead of the radius value r. The initial position of a particle is generated within a cubic domain and their positioning is set to obey a standard uniform distribution so that the particle position generation algorithm has the ability to produce pseudo-random positions in the defined discrete volume space. The cubic domain corresponds to the size of the fluid flow domain calculated through the model described in the last sub-section. If there is an overlap between two particles, the position of one of the particles is rearranged to eliminate the overlaps. For each particle p, a search of other particles is conducted to find if they overlap with the particle p. If a particle o partly or wholly overlaps the spatial position of p, a new position for o is calculated using the following:     rp + ro Pp = Po + (Pp – Po ) [7.4] dpo   where Pp and Po are the vectors from the coordinate origin to the centres of particle p and o, and dpo is the centre to centre distance. Figure 7.3 illustrates

Modelling of fluid flow and filtration through woven fabrics

ppo

275

pp po

7.3 Separation of overlapped particle p from particle o.

the relation between the overlapping particles and their new positions. For non-spherical particles, the Equation [7.4] will be in a more complex form:

    lp + lo Ppo = Po + (Pp – Po ) dpo

[7.5]

where lp and lo are the perpendicular distance from particle surface to its central surface for particles p and o, respectively. For such particles, the vector from the origin to the centre of the particle is chosen recursively from one side of the particle central axis and using predefined intervals moving towards the other end. The centre to centre distance is also calculated using the identical method and instead of radius values, the perpendicular distance from the particle surface to its central axis at the points being considered is used. Subsequently, Equation [7.5] is employed to relocate the particle position so that its space will not interfere with the spaces of other particles. Particle orientation For non-spherical particles, the orientation of the rigid body also needs to be represented and then initialised. Even though particles rotate and change orientations in the fluid, it is assumed that the particles reach the filter with the orientations that they were initialised with as they were introduced to the system. The method used in the proposed algorithm is the one developed by Euler,58 which describes the orientation of rigid body in 3D Euclidean space. In the Euler approach, to give an object a specific orientation it may need to be subjected to a sequence of three rotations described by the Euler angles. This is equivalent to saying that a rotation matrix can be decomposed as a product of three elemental rotation matrices.

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Particle transport Once particle concentration, dimensions and angles have been computed and initialised, the particles are generated and the next step involves the particle transport towards the septum. A particle in a non-uniform laminar flow field experiences rotation in a lateral plane owing to the different drag forces acting upon the particle surfaces. Depending on the dimensions of the particle, other forces may also significantly affect the particle motion. For colloidal particles, the effect of diffusion is heightened, where as the particle size increases, the forces attributed to diffusion decrease and those arising from inertia and gravity are amplified. The stokes number (stk), defined as the ratio of the particle relaxation time to that of the system response time, is a useful dimensionless parameter that defines particle transport characteristics in a fluid. In principle, when stk << 1, the particle follows the fluid flow streamlines, whereas when stk is much larger, the particle follows a straight line.59 A Stokes number close to 1 could represent a scenario in which the particle motion is affected partially by the fluid dynamics and also particle inertia. Particle capture The fabric filter is chiefly responsible for capturing solid particles, especially at the early stages of filtration. Every captured particle, in turn, forms an impeding element to the filter and can act to capture other particles. Accumulation of many particles forms blocks of barriers leading to the formation of cake that becomes the main capturing entity of suspended particles. Thus, the interactions between a particle and the fabric and between particles need to be closely examined. Particle–particle interactions are assumed to be negligible before the particle reaches the proximity of the filter or the already formed cake. This assumption also implies that no particle clusters are formed in the suspension, thus the proposed methodology holds true for dilute solids concentrations. Consequently, prior to particle capture, the shape of the particle is held to be as was originally defined (as opposed to clusters of particle forming a new shape). Obviously in a solution with a higher concentration of solids, the particle geometry and, more specifically, its dimensions would change. This is a particle attribute referred to as aggregation which will not be considered in the present study because of the assumption that no particle interactions exist prior to them reaching the proximity of the fabric. Forces on a particle In general, a particle possesses translational and rotational motion, which can be described by:

Modelling of fluid flow and filtration through woven fabrics



277

Fp = mp

dvp dt

[7.6]

Tp = I p

dw p dt

[7.7]

where Fp, Tp, mp, vp, Ip and wp are the total force, torque, particle mass, translational velocity, moment of inertia and angular velocity, respectively. Whilst the particle is in motion within a fluid, the total forces (Fp¢ ) on the particle is the combination of the weight force (FW), buoyancy force (FBo), lift force (FL), the drag force (FD) and the Brownian force (FBr):       F F F F F F = + + + + [7.8] ¢ p W Bo L D Br Buoyancy force is the upward force on an object produced by the surrounding fluid in which it is fully or partially immersed, owing to the pressure difference of the fluid between the top and bottom of the object. The net upward buoyancy force is equal to the magnitude of the weight of fluid displaced by the body. This force is given by:

Êr ˆ FBo = mg Á f ˜ Ë rp ¯

[7.9]

Lift force is created as a body deflects the fluid flow. The force created by this acceleration of the fluid creates an equal and opposite force according to Newton’s third law of motion. The lift force for a particle with radius rp, and local porosity e, can be calculated using the following equation:60

FL = p rp3 (1 – e ) rf ((0.5 — ¥ uf – w p ) ¥ (uf – up )) 8

[7.10]

Finally, the most important force is the drag force which is described by the force that resists the movement of the solid particle through a fluid. This is usually made up of friction forces, which act in a direction parallel to the object’s surface (primarily along its sides, as friction forces at the front and back cancel themselves out) in addition to the pressure forces, which act in a direction prependicular the object’s surface. Thus, the drag is the sum of all the hydrodynamic forces in the direction of the external fluid flow. The drag force for a particle moving towards the filter can be computed by: 61

FD = 1 Cd Ap rf (uf – up )2 2

[7.11]

where Ap is the particle’s effective area and Cd is the drag coefficient, which can be calculated from:62

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b Re Cd = 24 (1 + b1 Reb2 ) + 3 Re b4 Re



b1 = exp(2.3288 – 6.4581f + 2.4486f2)



b2 = 0.0964 + 0.5565f

[7.12]



[7.13] [7.14]

2

3



b3 = exp (4.905 – 13.8944f + 18.4222f – 10.2599f )

[7.15]



b4 = exp (1.4681 + 12.2584f + 20.7322f2 – 15.8855f3)

[7.16]

and f = s/S is the particle sphericity with s being the surface of a sphere having the same volume as the particle and S is the actual surface area of the particle. It should be mentioned that for sub-micrometre particles, the drag force is calculated using a form of Stoke’s drag law:63

FD =

9h

rp2 rp Cc



[7.17]

where C­e is the Cunningham correction factor and in terms of molecular mean free path, l, is given by:63



C c = 1 + 2l 2rp

È Ê Ê rp ˆ ˆ ˘ Í1.257 + 0.4 exp Á –1.1 Á ˜ ˜ ˙ Ë l ¯ ¯ ˙˚ Ë ÍÎ

[7.18]

Particle–fabric interactions The simplest particle shape to consider is the spherical shape. This is because of the symmetry of the shape across every axis. Its roundness also simplifies the consideration of its first contact point with the fabric surface since this point would always signify an identical situation of contact. Nevertheless, it is possible for a spherical particle to have more than one contact point with the filter surface, most possibly on the yarn interstices. But the methodology of tackling the interactions is identical. Figure 7.4 illustrates the forces acting on a spherical particle while in contact with the fabric surface. These forces are the weight force (FW), buoyancy force (FBo), the drag force (FD), the normal contact force (FN) and the frictional force between the particle and the surface (Ff). All forces acting on the particle, except the normal contact force (FN) and the frictional force between the particle and the surface (Ff), can be resolved using already available equations (see literature review) and from other predetermined parameters, such as particle mass, fluid velocity and dimensional parameters of the particles. At the instant that the particle first comes into contact with the fabric, the particle stops momentarily. This aids calculation of the normal

Modelling of fluid flow and filtration through woven fabrics

279

FBo

FN

Particle p w Ff

a Direction of fluid velocity

Fabric surface

FD FW q

7.4 Forces acting on a spherical particle in contact with the filter surface.

force on the particle using all other forces being exerted on the particle and ultimately leading to realisation of the frictional force. The calculation of the buoyancy (FBo) and weight (FW) forces are straightforward, whereas the drag force (FD) when considering the fabric– particle interaction is calculated according to Equation [7.11]. The particle force and torque balances, which are expressed by Equations [7.19] and [7.20], are used to calculate the translational and rotational motion of the particle:        dvp Fp¢ = FW + FBo + FD + FN + Ff = mp [7.19] dt    R dw p Tp¢ = rp ¥ Ff – Tp = I p [7.20] dt where mp and Ip are the mass and moment of inertia of particle p, and vp and wp are the particle translational and rotational velocities. In calculating  the total torque on the particle, the first term in the equation (i.e. rp ¥ Ff ) is the torque on the particle due to the tangential force and TpR is the torque caused by the rolling friction to the fluid:64    [7.21] TpR = rp hr | FN | w where hr is the coefficient of rolling friction. Having obtained a method to initialise and calculate all the forces and torques being exerted on every particle coming into contact with the fabric surface, the translational and rotational velocities of the particle can be computed accordingly as time

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steps are incremented. This leads to the determination of particle direction (motion) over the fabric surface. Calculating the resultant force on other particle shapes is not so simple, owing to complexity in their dimensions (as opposed to spheres for which the radius is the single dimension) and with outer surfaces that could be flat, round and can also contain sharp edges (as opposed to spheres that have a round surface only). Thus, some levels of simplification combined with a new set of rules (that will be discussed in this section) need be applied to the calculation of forces on such particles. The simplest scenario in particle–fabric interaction modelling of other particle shapes occurs when particles make contact with the fabric surface in such way that they can roll on their principal round axis surface. Figure 7.5 illustrates the instances in which particle shapes of discus, ellipsoid and needle roll on the fabric surface using their principal round axis rather than their flat or other round or even sharp surfaces. In this case, similar techniques to those developed for spherical particles, which are described by Equations [7.17]–[7.21], can be applied. The forces on the particle are determined and Equations [7.17]–[7.19] are used to calculate the translational velocity of the particle. The rotational velocity of the particle is also determined using a modified version of Equation [7.20] which uses rp¢ instead of the sphere radius and is given as Equation [7.22] below. This new parameter equates to a vector running from the centre of the particle to the contact point with its magnitude equal to the radius of corresponding particle shape:    R dw p Tp¢ = rp¢ ¥ Ff – Tp = I p [7.22] dt  The only other difference is the need for the acquired resultant torque (Tp¢ ) to be multiplied by the transformation matrix ([R]) of the Euler angle to transform the results from inertial frame into particle frame coordinates

Round axis

w Flat axis (surface)

q (a)

w

Principal round axis

Principal round axis

q

w

q (b)

(c)

7.5 A particle rolling on the fabric surface on its principal round axis (rather than sliding on its flat axis) is simpler to consider; (a) discusshaped or cylindrical, (b) ellipsoid and (c) needle-shaped particle.

Modelling of fluid flow and filtration through woven fabrics

281

  (Tp¢¢) . This new value is then used to calculate the rotational velocity (w p ) of the particle:    dw p Tp¢¢ = [R ] · Tp¢ = I p [7.23] dt In other cases where the particle does not make first contact with the fabric surface on its principal axis, the first step involves projecting the particle to a more stable position with respect to the fabric surface as it makes contact corresponding to the direction of the fluid flow acting on the particle body. When a surface element of a particle comes into contact with a fabric surface, two methods of particle projection on the fabric surface are proposed according to the dimensions of the particle and the size of a discretised fluid flow element (mesh size). Fundamentally, for particle dimensions where more than one fluid flow vector (in cases where the particle dimension is larger than the mesh size used in the CFD model) act on the particle, a free-body diagram is employed to predict the particle projection. Figure 7.6 illustrates an example of a calculation of particle projection on to a fabric surface for a needle-shaped particle with its pointed edge touching the fabric surface first. There are three fluid velocity vectors, in the x, y and z directions, exerting force on the particle. The combination of the three fluid flow velocity vectors is used to calculate the overall drag force illustrated in Fig. 7.6. All forces acting on the particle are drawn on a free-body diagram representing the main central axis of the particle to calculate its projection and then movement. It should be noted that in such cases, another assumption in the proposed algorithm is that the particle will not slide on the point of contact and will remain stationary. This is to simplify the projection calculations. In the case where only one fluid velocity vector acts on the particle, a similar technique is used, but with an added assumption that the fluid flow vector would pass through the centre of mass of the particle. These simplification assumptions can be removed, but would lead to a reduction in computational efficiency, as the particle projection calculations would be more dynamic. Particle projection onto the fabric surface could lead to the scenario that was illustrated in Fig. 7.5, where the particle surface around the principal axis makes contact with the fabric surface. In this case, identical methodology to that previously discussed and Equations [7.16]–[7.22] can be employed to map the particle motion. In all other cases, the resultant force on the particle is calculated using the free-body diagram of Fig. 7.6 and Equation [7.18]. However, the torque caused by the tangential force between the particle and fabric surface is calculated using the shortest distance between the fabric  surface and particle axis parallel to this surface (dps ) :

282

Modelling and predicting textile behaviour V1

FBo V2

FD1

FD2

V3

First point of contact

FN

FW

FD3 q

(a)

(b)

q (c)

7.6 When a particle element comes in contact with the fabric surface and the particle dimensions are long enough for more than one of the fluid velocity vectors to act on it, the particle projection is calculated using a free-body diagram: (a) three fluid flow velocity vectors acting on the particle, (b) free-body diagram and (c) projected particle on the fabric surface.



  Tp¢ = (dps · Fp¢) – TpR

[7.24]

It should be mentioned that the above torque could cause an angular velocity with respect to the particle axis. This obviously introduces another torque caused by the resistance on the rotating body, whose contribution can be obtained by integrating along the particle length.61 Owing to the complex nature of this calculation, the new torque arising caused by resistance on the rotating body is not considered in the proposed model. Finally, the acquired

Modelling of fluid flow and filtration through woven fabrics

283

torque needs to be transformed from inertial coordinates into particle frame coordinates using a similar approach to that specified in Equation [7.22]. Particle–particle interactions A particle that is attached to the fabric surface can also act as an entity to capture other particles. Thus, modelling the particle–particle interactions gains more significance as the number of particles captured increases. In the current proposed algorithm, the effects of forces on a particle of its neighbouring particles are only considered when the particle comes in contact with one or more particles. Figure 7.7 illustrates these various forces acting on the particle. When a particle, say p, comes into contact with another particle i, the resultant force and torques on either particle are calculated to form the basis of their movement:          dvp Fp¢ = FW + FBo + FD + S (FpiT + FpiN ) + S (Fpjv + Fpje ) = mp i j dt

  Tp¢ = S (rp ¥ FpnT n

[7.25]

 N  dw p – rp hf | Fpn | w ) = I p dt

[7.26]

where FpiT and FpiN are the tangential and normal forces between particle p and the particle(s) i that it has a contact with and are calculated using Equations [7.27] and [7.28], respectively: hmin FB N F pi

e F pj

wp

v F pj

e F pj

T F pi

j

v F pi

p FD

wi

FW

i

7.7 Various forces on particle i when considering particle–particle interactions.

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Modelling and predicting textile behaviour



  FpiN = È 2 E Rxpi3/2 – g pi E R xpi (vpi · npi ) ˘ npi ˙˚ ÍÎ 3

[7.27]



È min (xT , xmax ) ˆ ˘ Ê FpiT = – sgn (xT )m | FN pi | Í1 – Á 1 – ˜¯ ˙ xmax Ë Î ˚

[7.28]

where E = Y/(1 – s2) and Y is Young’s modulus, s is the Poisson ratio, xT  and xmax are the total and maximum tangential displacement, npi is the unit vector, xpi is the overlap between particle p and i, R = rp ri /(rp + ri ) and xmax = h[(2 – s)/2(1 – s)]xpi Fpjv and Fpje are the van der Waal and electrostatic forces between the p and j particles within its range (minimum separation hmin). These forces are respectively calculated using Equations [7.29] and [7.31]:



Fpjv =

dVv dh

64rp3 rj3 (h + rp + rj )  AH · n [7.29] =– · 6 (h 2 + 2rp h + 2rj h )2 (h 2 + 2rp h + 2rj h + 4rp Rx )2 pj

where h is the separation of particle surfaces and AH is the Hamaker constant and can be found using the equation:

AH = p2Crprj

[7.30]

where C is the London–van der Waals constant. The latter constant can be evaluated by multiplying the polarisability and ionisation potential between the two particles. Finally, the electrostatic force is calculated in a similar way to the van der Waals force, but by differentiating the repulsion electric energy:69



Ê ˆ dp2 Fpje = 2p e0 ec Á exp (–k h ) y 2 2 ˜ Ë (h + dp ) ¯

[7.31]

where e0 is the absolute permittivity of free space, ec is the dielectric constant between particles, dp is the particle diameter, y is the zeta potential and k is the reciprocal of the thickness of the double layer. As before, a similar set of rules to those specified for interactions between particles of non-spherical shapes and the fabric surface are employed to describe the particle–particle interactions in cases where one or more of the particles are not spherically shaped. Essentially, if the principal round (axis) surfaces of such particles roll on each other, Equations [7.25] and [7.26] apply, otherwise a new set of rules are specified. The latter is quite similar to those rules stated in the previous section, with the only difference being that the

Modelling of fluid flow and filtration through woven fabrics

285

particles are not projected on to the surface of the other particle. In this case, only the point of contact between two particles is considered. Consequently the total forces exerted on the particle are calculated and Equation [7.24] is employed to compute the total torque on the particle. Attaining the resultant force and torque on the particle leads to the calculation of its translational and rotational velocities which in turn determine the direction of particle motion. Particle packing A particle will be in a stable position if the resultant force on the particle is zero. There are cases when the forces caused by hydrodynamics on the particle are not zero, but the direction of motion is blocked by elements of the fabric or other particles leading to impediment of particle motion. This ultimately would make the resultant force on the particle zero, stopping its motion. If other particles are preventing the motion of the particle, the forces from that particle are passed to this new particle and a new force balance for the second particle is investigated. If this force can move the blocking particle, both particles could move according to the forces acting on them. Otherwise, the particle stays stationary (idle) and the resultant force on it is stored by the algorithm temporarily. If at any other time-step the motion of other particles exerts further force or the blocking particle moves, the previous forces on the original particle are checked for validity to re-examine the possibility of its motion. This new calculation leads to the introduction of a possible phenomena of detachment. Detachment occurs when forces on the particle cause it to be separated from the fabric surface or fragmented from the cake. A detached particle could be redeposited on the cake or pass through the filter pore.

7.4

Influence of fabric parameters on flow performance

Since the geometry and porosity of the fabric filter is determined by the weave pattern and the various parameters of yarns constituting the fabric,11 it is important to optimise its structure to achieve the most efficient filtration. The past and current practice is to rely on the practical skill and experience of the fabric designers and empirical trials. Readily available computational power provides an opportunity to develop computer-aided design (CAD) procedures. CAD software would enable predictions of filter performance to be made, leading to improved filters and reduced cost of trials. In the first stages of developing CAD programs, it is necessary to produce good models of fabric structure and then to predict the flow through the fabrics.

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Modelling and predicting textile behaviour

Among the numerous outputs from the analysis, the fluid pressure, fluid velocity and shear stress on the fabric are used as performance indices.65 Fluid pressure is read on the front face and the back face of the filter fabric. Fluid velocity is taken on the planes that are one mesh size away before and after the filter fabric and on the fabric centre (middle) plane. The shear stress, on the other hand, is measured at the front side and inner side of the yarns constituting the fabric. The fluid velocity and pressure through the chamber are also simulated, giving an overall effect of the fabric filter on the fluid flow. Figure 7.8 shows the positions where the data were extracted in relation to the fabric and to the direction of the flow. In every analysis, three different fabric models (all plain) are used. Table 7.1 lists the structural parameters of these fabric models. Every fabric model Fabric ‘back’

Fabric ‘front’

ime Xd nsio

Y

n

X Z

Plane ‘after’

Plane ‘before’

Y dimension

Direction of flow

Fabric ‘front’ Fabric ‘back’ Profile ‘middle’ (a)

Z dimension/2 Plane ‘through’

Y

Z X Z dimension (b)

7.8 Fabric in relation to planes before, ‘middle’, after and through. The fabric back and front are also indicated.

Modelling of fluid flow and filtration through woven fabrics

287

Table 7.1 Different fabric model specifications used in experiments Models Changing t1 t 2 d 1 d 2 c 1 parameter (tex) (tex) (cm–1) (cm–1) (%)

Yarn WHR Porosity cross- (%) sectional shape

FS1 Weft crimp FS2 FS3 FS4 Yarn cross- sectional shape FS5 FS6

100 100 100 100

100 100 100 100

21 21 21 10

10 10 10 10

17.6 20 23.4 3

Circular Circular Circular Circular

1:1 1:1 1:1 1:1

17.59 17.59 17.59 29.09

100 100

100 100

10 10

10 10

3 3

Racetrack 1:1.6 19.81 Lenticular 1:1.6 15.19

is specified using warp yarn linear density (t1), weft yarn linear density (t2), warp density (d1), weft density (d2), warp crimp (c1), cross-sectional shape of the yarns and the width to height ratio (WHR) of the yarn cross-section. Warp crimp is then a dependent parameter. The inlet pressure for all cases is 3 bar (1 bar = 105 Pa) and the operating pressure 1 bar. Density and viscosity of the fluid were assumed to be constant, corresponding to the isothermal approach. Liquid water was used as the Newtonian fluid (with density of 998.2 kg m–3 and viscosity of 10–3 kg m–1 s–1). It should be noted that in these models, the shape of the fabric does not change under stress. Again, the CFD software tool, Fluent is used for analysis. The resulting volume of the structure was then meshed using tetrahedral elements. A denser grid cell distribution was applied to areas on the fabric surface to account for the large gradients that occur in these areas. The various results related to filtration are discussed in this section.

7.4.1 Varying weft yarn crimp The waviness of yarn in a fabric is referred to as the yarn crimp. Three fabric models FS1, FS2 and FS3 represent three levels of weft yarn crimp. These are 17.6%, 20.0% and 23.4%, respectively. For the specified fabric, the weft crimp in the model FS1 is at its smallest possible value and for FS3 is close to its potential maximum. Other conditions remaining the same, increasing the yarn crimp will increase the length of the yarns and the depth of the pore. This leads to an increase of the yarn surface area within the fabric in 3D space and increases the fluid flow resistance of the filter fabric. Figure 7.9 describes the fluid pressure on the front and back surfaces of the woven fabric and demonstrates how increasing warp yarn crimp affects the exerted pressure. The fluid pressure on the front surface of the fabric is varied according to changes in the yarn crimp. Elevating the crimp value raises the resistance of the fabric to fluid flow and increases the pressure the

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Modelling and predicting textile behaviour

fluid exerts on the front surface of the yarn. A similar trend is observed at the back surface of the filter fabrics. This may lead to increased turbulence and thus affect the fluid velocity and subsequently cake formation. This needs to be further investigated as enlarging the warp yarn crimp will also sometimes affect the weft yarn crimp value. In these cases, the weft yarn crimp is expected to reduce. The results of higher yarn crimp are an increase in pore depth and yarn surface area. This leads to a reduction in fluid velocity

4.0 FS1 3.2

2.4 FS2 1.6

0.8

FS3

X Z

0.0 Bar

Y Front

Back

(a)

3.3

Front pressure (bar)

3.1 2.9 2.7 FS1 FS2 FS3

2.5 2.3 0

0.5

1 X dimension (mm) (b)

1.5

2

7.9 Effects of increasing warp yarn crimp (from FS1 to FS3) on the fluid flow behaviour by comparing fluid pressure on the front and back surfaces of the fabric, represented in (a) contour and (b/c) graph formats.

Modelling of fluid flow and filtration through woven fabrics

289

Back pressure (bar)

2.5 FS1 FS2 FS3

2

1.5

1

0.5 0

0.5

1 X dimension (mm) (c)

1.5

2

7.9 Cont’d

before the fabric from model FS1 to FS3. The effect of changes in the weft yarn crimp on the fluid velocity is shown in Fig. 7.9 in contour format. The graphs in Fig. 7.10 show the effect of increasing weft crimp on fluid flow on the through plane across the chamber length. It can be seen that fluid flow in the proximity of the fabric has similar pressure values. The fluid pressure then reduces relative to the position of the fabric. However, model FS3 exhibits an alternative behaviour where the fluid pressure initially increases before lowering. This is probably caused by the warp yarns of the fabric being straight owing to very high crimp value. This increases the fluid resistance of the fabric to its maximum since the flow cannot slide through as freely as when the warp yarns were not straight. The increasing pore depth also has some contributing factors. After the fabric, the fluid pressure decreases for an increasing yarn crimp. The overall pressure drop in the chamber increases from model FS1 to FS3. Comparisons of fluid behaviour with increasing warp yarn crimp are better demonstrated in graph format. As the velocity graph (Fig. 7.11d) demonstrates, the fluid velocity through the fabric decreases for an increasing yarn crimp. This reduction in fluid velocity highlights the role of yarn crimp on colloidal particle capture and thus the overall filtration process. This is supported by observations carried out by Bakker et al.66 that the specific resistance of a cake formed upon the surface of a filter cloth would depend upon the initial filtration velocity. Comparing the effect that fluid flow has on the three fabrics with varying weft crimp is measured using the shear stress, the results of which are illustrated in Fig. 7.12. The variation of crimp is directly related to the yarn surface area in 3D space. The greater the yarn surface area, the lower the

Modelling and predicting textile behaviour

X Dimension

Y Dimension

Y Dimension

1.5

0.5 (c)

0.0 m s–1

Y Dimension X Dimension

X Dimension

FS2

Y Dimension

Y Dimension FS3 Y Dimension X Dimension

X Dimension

1.0

Y Dimension

X Dimension

(b)

Fluid velocity (m s–1)

2.0

FS1

Before

Fluid velocity (m s–1)

X Dimension

X Dimension

(a)

Before    ‘Middle’     After Y Dimension

Y Dimension

Fluid velocity (m s–1)

2.50

X Dimension

290

‘Middle’

After

2.1 1.4 0.7 0 0.22

0.72 1.22 Y Dimension (mm)

1.72

0.72 1.22 Y Dimension (mm)

1.72

0.73 1.23 Y Dimension (mm)

1.73

2.1 1.4 0.7 0 0.22

2.1 1.4 0.7 0 0.23

7.10 Effects of increasing warp yarn crimp (from (a) FS1 to (b) FS2 to (c) FS3) on the fluid flow behaviour by comparing fluid velocity on the planes before, ‘middle’ and after these fabrics.

shear stress on the fabric. It is observed that most of the fluid flow slides over the pore walls and it is in this region where the shear stress is at its highest. The pore walls correspond to locations in the fabric where there is no intersection of warp and weft yarns leading to a lower resistance to fluid flow. These areas correspond to the peaks observed in the trends of both graphs in Fig. 7.12 (b) and (c). This can become significant because shear stress could play an important role in the early stages of cake formation. The values obtained show the probability of particles sticking to the fabric. The usability of this parameter is further explained in the next chapter.

Modelling of fluid flow and filtration through woven fabrics Fabric position

FS1

2.84

Fabric position Z Dimension/2

Fabric position

FS2

1.92

Fabric position

1.2 0.8 0.4 0

0

1.48

FS3

1.0 Bar

Z Dimension/2

1 2 3 4 Z Dimension (mm)

FS1 FS2 FS3

(g)

Y Dimension

(c)

Y Dimension

1.6

Z Dimension Fabric position

Z Dimension

(d) Total pressure (bar)

Fluid velocity (m s–1)

0.0 m s–1

(f)

2.38

Z Dimension/2 Fabric position 0.56

Z Dimension

Y Dimension

1.04

Y Dimension

1.53

(b)

(e)

Fabric position

3.3 Y Dimension

2.01

(a)

Y Dimension

2.50

291

3.5

Fabric position

3 2.5

(h) FS1 FS2 FS3

2 1.5 1

0

1 2 3 4 Z Dimension (mm)

7.11 Effects of increasing warp yarn crimp, from FS1 to FS3 on the fluid flow behaviour by comparing fluid velocity, represented in (a), (b) and (c) vector and (d) graph formats, and fluid pressure, represented by (e), (f) and (g) profile contours and in (h) graph format, across the chamber length on the plane through.

7.4.2 Varying yarn cross-sectional shape It is important to know if changes in the yarn cross-sectional shapes can influence fluid flow. In simulating woven structures, different yarn crosssectional shapes can be assumed. Primarily, the yarn shape could change in the filter design and production stage with fabric finishing methods such as calendering. The shape could also be altered after the continuous use of the fabric filter. Varying the cross-sectional shape of the yarn alters the yarn cross-sectional width and height. It can be argued that transformations of

292

Modelling and predicting textile behaviour

0.04 FS1 0.032

0.024 FS2 0.016

0.008 X 0 Bar

FS3 Z Y

Front

Back

(a)

Front shear stress (bar)

0.025

FS1 FS2 FS3

0.02 0.015 0.01 0.005 0 0

0.5

1 Y Dimension (mm) (b)

1.5

2

7.12 Effects of increasing warp yarn crimp (from FS1 to FS3) on the fluid flow behaviour by comparing shear stress exerted on the front and back surfaces of the fabric, represented in (a) contour and (b) and (c) graph formats.

yarn cross-sectional shapes could have similar effects on fluid flow as does a combination of changing fabric density and yarn linear density. Whilst this may be true in a fabric model, the situation is different when introducing particle capture rules. In case of the latter, if for instance instead of variation in the cross-sectional shape of the yarns, the fabric density is changed, the ratio of the size of the particle to that of the pore and yarn width changes and makes the filter ineffective in stopping particles. Even in relation to fluid flow behaviour, altering the cross-sectional shape of the yarns

Modelling of fluid flow and filtration through woven fabrics

293

Back shear stress (bar)

0.025 FS1 FS2 FS3

0.02 0.015 0.01 0.005 0 0

0.5

1 Y Dimension (mm) (c)

1.5

2

7.12 Cont’d

transforms the shape of the outer surface of the yarn. This change influences the fluid flow performance in a different manner to discussed previously. In this experiment three rather ideal yarn cross-sectional shapes, namely circular, racetrack and lenticular shapes, are investigated. From circular to racetrack to lenticular, the yarn width increases as its height decreases. Figure 7.13 illustrates how variation of the yarn cross-sectional shape can affect the fluid pressure exerted on the front and back surfaces of the fabric. To change yarn cross-sectional shapes from circular to racetrack to lenticular, the fluid pressure is increased on the front face of the fabric owing to higher flow resistance and drops dramatically at the back face. The pressure that the fluid exerts on the surface of the fabric can lead to its structure being altered. In the case of yarns with a lenticular cross-sectional shape, the fluid pressure on its front surface is highest and on the back surface lowest compared to other shapes. Thus, it is more probable that the shape of the yarns with a lenticular cross-section will be altered, modifying the pore structure of the fabric filter. This alteration in the pore structure needs to be further investigated since particle capture in woven fabric filters is dependent on the pore shape (and size). Figure 7.14 illustrates the changes in the fluid velocity before, after and through (middle) the fabric and also in the chamber. The fluid velocity in proximity to the fabric is quite important as it directly affects the drag force on solid particles suspended and flowing in the fluid. The effect of varying the yarn cross-sectional shape on the pore is twofold as it changes from circular to racetrack to lenticular. The pore width decreases while the pore depth increases. The former is directly related to the fluid velocity. So, for a smaller pore width, the velocity before the fabric decreases and

294

Modelling and predicting textile behaviour 3.5

FS4 2.8

2.0 FS5 1.4

0.7 X Z 0.0 Bar

FS6 Y

Front

Back

(a)

Front pressure (bar)

3.2

2.9

2.6 FS4 FS5 FS6

2.3 0

0.5

1 Y Dimension (mm) (b)

1.5

2

7.13 Effects of varying yarn cross-sectional shape (FS4 to FS5 to FS6 are circular, racetrack and lenticular, respectively) on the fluid flow behaviour by comparing fluid pressure on the front and back surfaces of the fabric, represented in (a) contour and (b) and (c) graph formats.

for a smaller pore depth, the fluid velocity through the fabric rises. This trend continues except when comparing the fluid velocity for lenticular and racetrack through the fabric. Even though the lenticular shape yarns are

Modelling of fluid flow and filtration through woven fabrics

Back pressure (bar)

2

295

FS4 FS5 FS6

1.5

1

0.5

0 0

0.5

1 Y Dimension (mm) (c)

1.5

2

7.13 Cont’d

wider than the racetrack ones, the shape of the front surface of the yarn in the case of a racetrack shape reduces the fluid velocity. This is a flat surface and the plane is perpendicular to the direction of the flow. However, in the case of the lenticular yarn the front surface is curved, decreasing the fluid flow resistance. To obtain the overall effect on the fluid flow behaviour of varying yarn cross-sectional shape, Fig. 7.15 shows the fluid velocity vectors and pressure profiles on the through plane. In contrast to previously discussed parameters, changing yarn cross-sectional shape from circular to racetrack to lenticular reduces the pore dimensions, or to be precise, both the pore depth and width are reduced. As expected, fluid flow with similar pressure values take effect in the proximity of the fabric. Reduction in pore depth causes fluid pressure to increase slightly, before decreasing in the chamber downstream at the back of the fabric. The pressure drop increases from circular to racetrack to lenticular. The graph representations of these results provide easy-to-compare methods of fluid flow behaviour with varying yarn cross-sectional shape. The graphs in Fig. 7.15 illustrate the variations in results more objectively. The cross-sectional areas of the yarns and thus their surface areas are identical. The main change, other than increase in fluid flow resistance, is in the shape of the outer surface of the yarns. As mentioned previously, for the racetrack shape, the front surface is a flat plane perpendicular to the direction of the flow. Even though there is a higher pressure that is applied to the front surface of the yarn, the fluid cannot slide over the yarn as freely as it does on the curved surfaces. There is less tangential friction between the fluid and the surface of the yarn and hence a tiny amount of stress. In the case of lenticular shape, the curvature of the front surface is the highest

Modelling and predicting textile behaviour

FS5

0.0 m s–1

X Dimension

Y Dimension

(c)

X Dimension

0.46

X Dimension

Y Dimension

Y Dimension

Y Dimension

FS6

Fluid velocity (m s–1) Fluid velocity (m s–1)

Y Dimension

X Dimension

0.92

Y Dimension

Y Dimension

Y Dimension

(b)

Before

FS4

X Dimension

1.83

1.37

Y Dimension X Dimension

(a)

X Dimension

2.3

After

X Dimension

X Dimension

Before ‘Middle’

Fluid velocity (m s–1)

296

‘Middle’

After

2.5 2 1.5 1 0.5 0 0.2

0.7 1.2 Y Dimension (mm)

1.7

2 1.5 1 0.5 0 0.2

0.7 1.2 1.7 Y Dimension (mm)

2 1.5 1 0.5 0

0.2

0.7 1.2 Y Dimension (mm)

1.7

7.14 Effects of varying the yarn cross-sectional shape on the fluid flow behaviour by comparing fluid velocity on the planes before, ‘middle’ and after these fabrics. (a) FS4 to (b) FS5 to (c) FS6 are circular, racetrack and lenticular, respectively.

and as results show, the shear stress on the fabric is relatively higher than the other shapes. Figure 7.16 shows how changes in the yarn cross-section can vary the effects of fluid flow on the fabric. The shear stress exerted on the surface of the yarns of the back surface of a fabric with a racetrack cross-sectional shape is the lowest and that of lenticular is the highest. The more curved the surface of the yarn is, the higher the shear stress on the fabric. High shear stress could result in filtration failure in real applications. The results of shear stress could be integrated in the fabric filter design. Ultimately, the

Modelling of fluid flow and filtration through woven fabrics

(a)

Y Dimension

28.0

Fabric position 3.80

FS4 3.04

Fabric position Z Dimension/2

Fabric position 2.28

FS5 1.52 Z Dimension/2 Fabric position

FS6 0.0 Bar

Z Dimension/2

2

(d) Fabric position

1.5 1 0.5

0

1 2 3 4 Z Dimension (mm) FS4

FS5

FS6

(g)

Y Dimension

0.78

Total pressure (Bar)

Fluid velocity (m s–1)

0.0 ms–1

(f)

Z Dimension Fabric position (c)

Y Dimension

7.0

Z Dimension

Y Dimension

14.0

Y Dimension

21.0

(b)

(e)

Y Dimension

Fabric position 35

297

Z Dimension (h)

2.8

Fabric position

2.3 1.8 1.3 0.8

0

1 2 3 4 Z Dimension (mm) FS4

FS5

FS6

(d)

7.15 Effects of varying the yarn cross-sectional shape (FS4 to FS5 to FS6 are circular, racetrack and lenticular, respectively) on the fluid flow behaviour by comparing fluid velocity, represented in (a), (b) and (c) vector and (d) graph formats, and fluid pressure, represented by (e), (f) and (g) profile contours and in (h) graph format, across the chamber length on the plane through.

particle capture mechanisms along with the fluid flow behaviour govern the selection of the fabric parameters. It is therefore this combination that guides the design engineer in determining the paramount solution to a filtration dilemma. The combination of these two issues will be fully discussed in the next section.

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Modelling and predicting textile behaviour 0.05

FS4 0.04

0.03 FS5

0.02

0.01 X

FS6

Z Y 0 Bar

Front

Back

(a)

Front shear stress (bar)

0.03 FS4 FS5 FS6

0.025 0.02 0.015 0.01 0.005 0

0

0.5

1 Y Dimension (mm) (b)

1.5

2

7.16 Effects of varying yarn cross-sectional shape (FS4 to FS5 to FS6 are circular, racetrack and lenticular, respectively) on the fluid flow behaviour by comparing shear stress exerted on the front and back surfaces of the fabric, represented in (a) contour and (b) and (c) graph formats.

Modelling of fluid flow and filtration through woven fabrics

Front shear stress (bar)

0.03

299

FS4 FS5 FS6

0.025 0.02 0.015 0.01 0.005 0 0

0.5

1 Y Dimension (mm) (c)

1.5

2

7.16 Cont’d

7.5

Influence of fluid flow and fabric parameters on filtration performance

A woven filter retains particles larger than its pore size. Hence, woven fabric filters are designed in such way as to capture particles above a certain size threshold, necessitating a higher level of accuracy in predicting the pore dimensions in woven filters. This is in contrast with non-woven filters which have a range of pore shapes and dimensions. A study is carried out to investigate the effects varying the structural parameters of the woven fabric can have on the pore shapes and sizes of the filter. Subsequently, cake formation and the effectiveness of the resulting filters are investigated.

7.5.1 Fluid properties One other slurry parameter of interest is related to the properties of the fluid. A simulation-based experiment is therefore set up to examine how variation of the fluid properties affects the cake filtration when all other parameters are kept constant. Three types of fluid air, liquid water and engine oil with viscosities of 1.79 ¥ 10–5, 0.001003 and 1.06 Kg m–1 s–1, respectively, have been selected for this study. When the fluid is air with solid particles that have relatively large dimensions, the Stokes number is greater than 1 and thus particle flow paths towards the fabric are only partially affected by the fluid flow paths; the inertial forces are the major influence. Nevertheless, this experiment only compares the effects of varying fluid viscosity on the particle packing and thus it can be assumed, for the sake of this experiment only, that the solid particles follow the fluid flow paths. Figure 7.17 shows the formation of the cake at the early stages of filtration.

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(a)

(b)

(c)

7.17 The early stages of cake formation (different time scales) when using different fluids: (a) air, (b) liquid water and (c) engine oil.

The timescale used to capture the snapshot of the cake/filter is different for each of the three models since the fluid flow velocities in the three cases vary significantly. Essentially, the fluid velocity is inversely proportional to fluid viscosity and, obviously, it is the fluid velocity that determines the timescale for particles reaching the septum. Increasing the fluid viscosity, which decreases the fluid velocity, leads to different cake structures. With a lower fluid velocity, the particles are more likely to stick to the fabric surface on the point of contact as a lower fluid velocity corresponds to a lower drag force on particles. These effects are especially obvious on the warp/weft yarn interstices, with the lowest fluid viscosity model having the highest number of particles on the interstices, and vice versa. The same logic (low viscosity/high velocity) is also the underlying reason for a greater number of particles passing through the pores and the cake porosity decreasing. These comparisons are demonstrated in Fig. 7.18.

7.5.2 Driving pressure Solid–liquid separation is divided into two categories, namely sedimentation and filtration. Sedimentation is characteristically very slow in operation and therefore needs considerably more operational space. It is difficult to control the amount of bleeding in sedimentation. Filtration has proven to be a more effective approach for solid–liquid separation. Filtration requires the creation of a pressure drop across the filter medium. This can be achieved either by creating a negative pressure or by suction on the filtrate side of the

Modelling of fluid flow and filtration through woven fabrics

301

Particles bleedinig (%)

40 Air Water Engine oil

30

20

10

0 0

0.5

1 Time (s) (a)

1.5

2

1 Air Water Engine oil Porosity

0.8

0.6

0.4 0

2

Time (s) (b)

4

6

7.18 Plots of (a) % of solid particle bleeding through the filter and (b) porosity of the forming cake against time for three models with varying fluid type (air, liquid water, and engine oil) while keeping all other parameters constant.

medium, employing centrifugal force or by inducing pressure hydraulically or mechanically to force the liquid through the filter medium. The modelling approach proposed here considers the pressure inducing method. Thus, a driving pressure is defined to simulate the fluid flow characteristics. In this part of the study, three driving pressures of 2, 3, and 5 bars are used in the experiment, where all other parameters are kept constant. Figure 7.19 visually compares the cakes formed after 0.3 s on identical fabrics and particles with increasing driving pressure. The driving pressure is proportional to the fluid flow velocity and thus increasing the driving

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(a)

(b)

(c)

7.19 Comparison of cake formed on identical fabrics and under identical filtration conditions after 0.3 s but with different driving pressures of (a) 2, (b) 3 and (c) 5 bars.

pressure increases the fluid flow velocity through the chamber. This causes more particles to approach the fabric and the cake can be formed faster. This amplification of particle and fluid velocities leads to faster pore blockage and lower cake porosity. The rate of particles bleeding through the filter and the cake porosity against filtration time are illustrated in Fig. 7.20.

7.5.3 Yarn crimp In terms of woven fabric parameters, a study is carried out, simulating the cake formation on three identical fabrics but with varying warp yarn crimp. Figure 7.21 shows the cake structures forming on the three fabrics after 0.3 s of simulating filtration. As the warp yarn crimp increases, the pore depth is enlarged and the fluid velocity decreases. This leads to more particles being deposited on the intersections of warp and weft yarns making the cake less compact. These patterns are followed at the early phases of filtration. The plots of the percentage of particles passing through the pores and the dry cake mass building up as filtration time progresses is illustrated in Fig. 7.22. With increasing warp yarn crimp, the effective pore area is amplified causing fewer numbers of particles to be retained by the filter. However, the effective surface area of the fabric increases, which aids particle capture. As such, a balance between the two parameters of effective pore area and surface area should be further investigated. It should be noted that graphs of

Modelling of fluid flow and filtration through woven fabrics

303

Particles bleeding (%)

30 p = 2 bar p = 3 bar p = 5 bar 20

10

0 0

0.2

0.4

Time (s) (a)

0.6

0.8

1

1

Cake porosity

p = 2 bar p = 3 bar p = 5 bar 0.8

0.6

0.4 0

0.5

1 Time (s) (b)

1.5

2

7.20 Plots of (a) % of solid particle bleeding through the filter and (b) porosity of the forming cake against time for three models with varying driving pressure (2, 3 and 5 bars) while keeping all other parameters constant.

Fig. 7.22 need to be interpreted correctly owing to the different fluid velocity patterns in the three models. Nevertheless, the relative trends observed are quite interesting and conclusive.

7.5.4 Yarn cross-sectional shape Yarn cross-sectional shape is another woven fabric parameter that can have significant effects on filtration behaviour. In the current study, three yarn cross-sections, namely, circular, lenticular and racetrack are examined while keeping all other fabric and slurry conditions constant. Figure 7.23 compares

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Modelling and predicting textile behaviour



(a)

(b)

(c)

7.21 Comparison of cake formed on identical fabrics with varying warp yarn crimps of (a) 17.6%, (b) 20% (c) and 23.4% after 0.3 s, while keeping all other parameters constant (weft yarn crimp is 3% in all cases).

the variation of cake structure formed on identical woven fabrics but with varying yarn cross-sectional shapes. The filter fabric surface of racetrack yarns demonstrates the highest retainability of particles on its surface. This is related to the flat surfaces of the yarn, which in turn lead to a very low fluid shear stress on the fabric surface. The shear stresses on the lenticular yarns are highest, but the results indicate that they capture more particles. This is attributed to the higher yarn surface for the lenticular shape in comparison to that of the circular yarns. Another feature of varying the yarn crosssectional shape, which cannot be calculated at this specific particle size, is the effect of shape and size of the orifices. In this study, the particle sizes are relatively smaller than the effective pore size and hence the pore shape influence is not examined. The contrasting effects of varying yarn cross-sectional shapes are demonstrated more clearly using graphs of the percentage of particles passing through the pores and mass of the dry cake as filtration time progresses, which are illustrated in Fig. 7.24. In comparison, the fabrics with circular yarn cross-sections have the highest ratio of solids bleeding through the filter. As mentioned previously, this is attributed to the larger effective pore areas, narrower fabric yarn thickness and relatively higher fluid shear stress on the filter. The flat surfaces of the racetrack shape, which is perpendicular to particle motion, are the most interesting parameter observed.

Modelling of fluid flow and filtration through woven fabrics

Particles bleeding (%)

30

305

MinCrimp MediCrimp MaxCrimp

20

10

0 0

Dry cake mass (mg)

0.18

0.5

Time (s) (a)

1

1.5

1

1.5

MinCrimp MediCrimp MaxCrimp

0.12

0.06

0 0

0.5

Time (s) (b)

7.22 Plots of (a) % of solid particle bleeding through the filter and (b) dry cake mass against time for three identical models but with increasing warp yarn crimps (weft yarn crimp is 3% in all cases).



(a)

(b)

(c)

7.23 Comparison of cake formed after 0.35 s on identical fabric specification but with varying yarn cross-sectional shapes (a) circular, (b) racetrack and (c) lenticular, while keeping all other parameters constant.

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Particles bleeding (%)

80 Circular Racetrack Lenticular

60

40

20

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6

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0

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2

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7.24 The plots of (a) % of solid particle bleeding through the filter and (b) dry cake mass against time for three identical models but with varying yarn cross-sectional shapes (circular, racetrack, and lenticular).

7.6

Influence of particle properties on filtration performance

To study the influence of particle properties67 on the filtration performance, an example case study is carried out to investigate the effect of varying particle properties. An identical standard filter specification is used in all simulations. The fabric filter and slurry parameters used in this study (except otherwise specified in the corresponding sub-section) are listed in Table 7.2. The smallest gap between two yarns in the above model is about 15.5 mm. This is based on the warp density (21 end per cm, epc) and weft density (10 picks per cm, ppc), making the warp spacing (1/21 = 0.04762 cm)

Modelling of fluid flow and filtration through woven fabrics

307

Table 7.2 Parameters of a woven fabric filter used in a slurry effects study

Parameters and variables

Values

Yarn properties

Warp/weft yarn linear density 100 (tex) Warp/weft yarn cross-sectional shape Circular Warp/weft yarn diameters 4.607 ¥ 10–4 (m) Yarn type Monofilament

Fabric properties

Warp crimp Weft crimp Warp density Weft density Geometrical modelling

17.6% 3% 21 (ends/cm) 10 (picks/cm) Hermite spline

Slurry properties

Solids concentration Particle density Particle shapes Fluid

1% 1602 (kg m–3) Spherical Water

Operating condition

Driving pressure

3 (bars)

0.4762 mm and weft spacing 1 mm. The yarn diameter is, according to yarn specifications, 0.4607 mm. So the projection of the pore size is 0.0155 ¥ 0.5393 (mm2). In all models, the slurry concentration is 1% and, unless specified, liquid water (density of 998 kg m–3 and viscosity of 0.00103 kg m–1 s–1) is used as the default fluid. The driving force is a pressure of 3 bars perpendicular to the plane of the fabric filter. It should be noted that the simulated fabric and forming cake in each study is illustrated at varying times according to the nature of the study. This corresponds to how long it takes for the initial cake to build up in each scenario, bearing in mind the different particle, fluid and fabric properties.

7.6.1 Particle size From values calculated for the pore size projection, that is 0.0155 ¥ 0.5393 (mm2), it is expected that the fabric filter will be able to retain particles larger than 15.5 mm. To verify this initial observation, three simulations are carried out corresponding to three different particle size specifications. Spherical particles of 8, 16 and 32 mm in diameter were used in three respective simulation studies. The structure of the forming cake in all three cases after 0.31 s is compared in Fig. 7.25. As expected, smaller particles pass through the pore and larger particles are generally captured. The first (but obvious) observation is that with increasing particle diameters, the cake thickness is also enlarged. The simulation-based study also proves wrong the initial observational assumption that particles larger than 15.5 mm should be retained, since it is observed that many particles with a diameter of 16 mm pass through the fabric filter.

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Modelling and predicting textile behaviour



(a)

(b)

(c)

7.25 Comparison of cake formed on identical fabrics and filtration conditions after .31 s but with different particle diameters of (a) 8 mm, (b) 16 mm and (c) 32 mm.

Figure 7.26 shows the results of the three simulations and their comparisons. the cake thickness and its dry mass increase with progression of time. However, plot (b) shows that for particles with the 8 and 16 mm the cake masses do not increase with a similar pattern to that of particles with 32 mm at time steps corresponding to the initial stages of cake formation. This contradicts the initial expectation based on the pore size projection that particles with diameters greater than 15.5 mm (in this case 16 mm) should be retained by the filter and hence the rate of increase of the cake mass should be higher. The inaccuracy of the original observation from pore size projection, that is the prediction that particles larger than 15.5 mm should be retained, can also be verified by the percentage of solid particles bleeding through the filter. This is illustrated in Fig. 7.27 (a). The simulation animation for a medium sized particle diameter of 16 mm was carried out again specifically to follow such particles as they move towards the fabric and through the pores. Through this observation, at every time step, the precise interactions of the particle with the fabric were noted and analysed to examine the reasons why such particles pass through the filter pores. This analysis entailed reading the data, such as the forces on the particle and its direction of motion. This closer examination indicated the importance of yarn crimps and how they can contribute to the enlargement of the pore dimensions. Consequently, this proves that 2D pore size projection has not been a correct method of measuring pore size. Yarn crimp is a parameter that leads to increasing pore depth. If yarns had been parallel and in the same plane, the effective pore size would have

Modelling of fluid flow and filtration through woven fabrics

309

Average cake thickness (mm)

0.4 Diam = 8 µm Diam = 16 µm Diam = 32 µm

0.3

0.2

0.1

0 0

0.4

Time(s) (a)

0.6

0.8

1

Diam = 8 µm Diam = 16 µm Diam = 32 µm

0.6 Cake mass (mg)

0.2

0.4

0.2

0 0

0.2

0.4

Time(s) (b)

0.6

0.8

1

7.26 The plots of cake formation against time (a) thickness and (b) dry mass

been as predicted. However, in the fabric samples, the yarns are not in the same plane owing to the yarn crimp. This implies that at one interlacing the warp is up and at the next the warp is down. Therefore, yarn crimp and the weave pattern should also be taken into account when calculating effective pore area. Subsequently the choice of particle dimensions and the threshold in size of the solids that can be retained needs to be recalculated accordingly. Figure 7.27 (b) shows the variation in the cake porosity after the pore is completely blocked. Since the particle shapes are identical for the three cases (spherical particles with varying diameters), the void space as a percentage of the total volume in their packed structures should be constant. This is

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irrespective of the fact that larger spherical particles lead to larger void volumes. To investigate the reasons for this difference, the 3D cake structure and its formation was closely inspected by noting and analysing the forces on solid particles and the resulting direction of their motion. The variation in cake porosity is attributed to the cake formation. It was observed that there is a higher probability of larger particles depositing on top of each other, whereas smaller particles are more likely to slide over one another to create a more compact structure. Larger particles have a higher mass value (solid particle densities are identical) and thus the contribution of their weight force is higher. Since all other forces on a particle are independent of its characteristics (except fluid velocity), the weight force is the only differing attribute.

7.6.2 Particle shape It is reported68 that the shape of particles plays an important role in cake formation in the filtration process. However, in most filtration simulations, particle shapes are assumed to be spherical for simplicity, although in reality particles take many different shapes. In order to quantify the effects of particle shape on filtration, this section examines the influence of the variations in particle shape for identical filtration set-ups. Spherical particles with diameter of 16 mm are used as the benchmark for this experiment. Particles of different shapes will possess identical mass and volume, but with varying dimensions. Consequently, particles in the shape of a needle, constructed by joining the base of two identical cones facing opposing directions (8.34 mm in diameter and 58.9 mm in height), discus (27.97 mm in diameter and 3.49 mm in thickness), cylinder (diameter, 8.17 mm and height, 40.85 mm) and ellipsoid (equatorial diameter, 25.4 mm and polar diameter, 12.7 mm) are all investigated. In all cases, identical woven filter and fluid properties were used. The structures of the resulting cakes after 0.31 s are illustrated in Fig. 7.28. The results of the simulation are quite interesting and are indicative of the importance of particle shape in the overall filtration design. Maintaining a constant particle volume and varying the shape essentially leads to changes in particle dimensions. Particle shapes other than the spherical shape have one dimension considerably smaller than the 16 mm size, leading to a higher number of such particles being able to pass through the pores. After 0.31 s, the percentage of solids passing through for particles of spherical, ellipsoids, discus and needle shape are 4.8%, 18.2%, 23.7% and 37.1%, respectively. The dry cake mass and local cake thickness at every time step is recorded to examine the efficiency of the filter in capturing the varying particle shapes and the cake formation structure details, respectively. The graphs of these are shown in Fig. 7.29. The dry cake mass plot illustrated in Fig. 7.29(b)

Modelling of fluid flow and filtration through woven fabrics

311

Particles bleeding (%)

100 80 60 Diam = 8 µm Diam = 16 µm Diam = 32 µm

40 20 0 0

0.2

0.4

Time (s) (a)

0.6

1

Diam = 8 µm Diam = 16 µm Diam = 32 µm

0.9

Cake porosity

0.8

0.7

0.5

0.3 0

1

2

Time (s) (b)

3

4

5

7.27 The plots of (a) % of solid particle bleeding through the filter and (b) porosity of the forming cake against time for three models with varying particle diameters of 8, 16 and 32 mm while keeping all other parameters constant

confirms the observation from Fig. 7.28. It indicates that the rate of particle capture decreases in the order of spherical, discus, ellipsoid, and needle shaped particles at the early stages of cake formation and after running the experiment for 0.31 s. The only unexpected outcome is that of the local cake thickness trend for discus shaped particles. Discus particles possess one dimension which is much smaller relative to the dimensional values of other particle shapes. Thus, it was expected to follow the expected trend and demonstrate the lowest rate of particle capture. Nevertheless, it can be argued that as its other dimension, the diameter of the discus, is quite large, this contributes to the outcome attained from the simulation. Further

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Modelling and predicting textile behaviour



(a)

(b)

(c)

(d)

7.28 Comparison of cake formed on identical fabrics and filtration conditions after 0.31 s but with different particle shapes of (a) spherical, (b) ellipsoid, (c) needle shaped and (d) discus shaped

inspection of cake formation, by analysing and noting the forces on the particles, demonstrates that in case of the discus-shaped particles, the relative motion of such particles on the fabric surface is the lowest in comparison with other particle shapes. This is because the shape of the particle is flat on its two major surfaces, causing higher friction and lacking the ability to roll over the fabric surface. The higher friction is due to the larger contact area, requiring a greater force to move the particle. The inability of two discus particles to roll over each other arise from their two flat surfaces which can only slide over each other. The local cake thickness, indicated in Fig. 7.29(a), corresponds to the thickness of cake at locations on the fabric surface where particles are attached, without taking into consideration the fabric filter surfaces where no particles are attached. This differs from the overall cake thickness in the sense that the overall value is distorted at the early stages of cake formation owing to the high proportion of the fabric surface that has not captured any particles and thus not contributed to cake thickness. Based on this definition, the needle-shaped particles have the highest local cake thickness owing to possessing the largest dimensional size (their height) in comparison with the dimensions of other particles. Needle-shaped particles have sharp points. This geometrical feature allows these particles to penetrate easily and become lodged between other particles, hence standing upright with their height perpendicular to the fabric surface. Evidently, this also can lead to a higher cake porosity as needle-shaped particles have the least capability of moving over other particles since they possess pointed edges which lodge into the fabric or between other particles at the point of impact.

Modelling of fluid flow and filtration through woven fabrics

Local cake thickness (mm)

0.25

313

Spherical Needle Ellipsoid Discus

0.2 0.15 0.1 0.05 0 0

0.2

0.4

Time (s) (a)

0.6

0.8

1

Dry cake mass (mm)

0.09 Spherical Needle Ellipsoid Discus

0.06

0.03

0 0

0.2

0.4

Time (s) (b)

0.6

0.8

1

7.29 Plots of cake (a) thickness and (b) dry mass against time of cake formation for four different models with varying particle shapes (spherical, needle shaped, ellipsoid and discus shaped) while keeping all other parameters constant.

Cake porosity and the percentage of solids bleeding through the filter for all four particle shapes against time are illustrated in Fig. 7.30. As stated previously, the cake formed by needle particles has the highest porosity. This is due to the long but thin dimensions of the particles. Essentially, the dimensional aspect ratio of the needle particles prevents them from forming a compact structure. On the other hand, cake structures composed of spherical particles with their symmetrical dimensionality are expected to have the least porosity owing to the percentage of void ratio related to their shapes. Again, the results attained from the simulation prove these initial predictions wrong. Cake formed by discus-shaped particles provides the lowest porosity

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Particles bleeding (%)

50 Spherical Needle Ellipsoid Discus

40 30 20 10 0 0

1

2 Time (s) (a)

3

4

1 Spherical Needle Ellipsoid Discus

Porosity

0.8

0.6

0.4

0.2 0

1

2 Time (s) (b)

3

4

7.30 Plots of (a) % of solid particle bleeding through the filter and (b) porosity of the forming cake against time for four different models with varying particle shapes (spherical, needle shaped, ellipsoid and discus shaped) while keeping all other parameters constant.

owing to the particles’ two flat surfaces, which allow them to form stable pile-ups.

7.6.3 Solids concentration In the previous study based on particles of varying shapes, the percentage of particles passing through the filter shown in Fig. 7.31 follows the expected trend in that particle shapes lead to various dimensions which in turn govern whether or not the particle will pass through the pores. Nevertheless, the solid

(a) Spherical

Particles bleeding (%)

After 0.31s

Spherical 40

Needle

30 20 10 0

0

1

2 Time (s) (c)

3

4

(b) Needle 30

After 0.1s

Particles bleeding (%)

Solid concentration = 8%

Spherical Needle 20

10

0 0

(e)

0.1

0.15 Time (s) (f)

0.2

0.25

0.3

315

(d)

0.05

7.31 Comparison of the effect of varying solid particle concentrations of (top) 1% and (bottom) 8%, for spherical and needleshaped particles using cake formed after 0.31s and 0.1 s, respectively and the plot of % of solid particle bleeding through the filter against time.

Modelling of fluid flow and filtration through woven fabrics

Solid concentration = 1%

50

316

Modelling and predicting textile behaviour

particle concentration should be also investigated to examine its contribution to the results of cake formation when comparing particles of different shapes. To investigate the effects of variation of solids concentration on filtration outputs, a similar experiment to that in the previous section was carried out. In this case, only needle-shaped and spherical particles were compared. Two different solids concentration values of 1% and 8% were used for both particle shapes. Figure 7.31 shows the structure of the cake being formed in the early stages of cake formation, at 0.32 and 0.1 s for the two respective solids concentration values. Closer inspection of these structures shows that pore blockage occurs faster in the case of needle-shaped particles, when comparing the two shapes with different concentration values. This is significant and is related to the higher number of particles simultaneously arriving in the filter proximity (higher solids concentration). In the test with the higher solids concentration (8%), the longer dimension of the needle-shaped particle causes faster blockage as it stops other particles freely sliding over the fabric surface. On the other hand, in the previous case in which the solids concentration was lower (1%), the particles are comparatively (compared to the 8% concentration scenario) more free (space), possessing more time to move before being blocked (retained) by other depositing particles. The two different trends observed are illustrated in the graphs of Fig. 7.31.

7.7

Application of fluid flow and filtration modelling

Developing an accurate theoretical model of particle filtration is difficult because of the nature of the filtration process, which involves particle transport in a fluid moving through complex fabric geometry. The fluid and suspended particles flowing through the maze of yarns follow a tortuous path controlled by fluid dynamics and the equations of motion for particles. Since the geometry and porosity of a fabric filter are determined by the weave pattern and the various parameters of yarns constituting the fabric, it is important to optimise its structure to achieve the most efficient filtration. Past and current practice has relied on the practical skill and experience of fabric designers and on empirical trials. Alternatively, simulation and modelling offers an excellent approach to quantitative and qualitative understanding of the filtration process. The costs of experimentation can be substantially reduced if these are combined with theoretical modelling and investigation. Furthermore, these investigations provide an accelerated perception of the methodology required to break down the process into its fundamental physical phenomena. Once a model has attained a significant level of confidence, it can be successfully utilised as a design tool for new processes of filtration or optimization of existing ones.

Modelling of fluid flow and filtration through woven fabrics

317

Results of the analysis show that fluid pressure and velocity in the proximity of the fabric and in the chamber downstream vary significantly according to the effective pore areas. The results further illustrate that fluid velocity is directly proportional to the pore width and inversely proportional to the pore depth. Results attained from the shear stress, which are influenced by the yarn surface area and yarn shape, could prove useful in design and implementation in the initial stages of cake formation. Finally, the pressure distribution on the surfaces of the fabric could facilitate calculation of deformity in the yarn shape and filter permeability. Studies undertaken signify the importance of particle shape and size. Most filtration models consider particles to be spherical with a set size. However, our results demonstrate the effects that variation of particle sizes and shapes can play in a more precise calculation of filter performance. Nevertheless, more arbitrary particle shapes such as rhomboidal shapes were not considered in this study owing to their extreme complex geometrical characteristics. The importance of accurately selecting the right woven fabric parameters constituting the fabric was also discussed. These results are especially interesting for the filter design engineer. Simulation outputs of our study have highlighted that for a range of particle sizes and shapes to be retained by the filter, careful selection of yarn linear density, fabric density and yarn crimp is essential to evaluate the effective pore area correctly and thus the particle size threshold that can be retained. Yarn cross-sectional shapes and filament types can also be quite pivotal in particle retention. Our results show that fabrics with a multifilament, racetrack yarn cross-section and low yarn crimps are the basis of the most efficient filters. However, employing multifilament yarns could lead to more complicated cake release procedures. Thus, the correct balance in selecting fabric parameters in relation to slurry conditions is paramount in designing the most efficient woven fabric filters.

7.8

Future trends

There has been rapid development and progression in computational capabilities together with an enhanced comprehension of phenomena constituting fluid flow and filtration. The ultimate objective is to implement a tool that can automatically design a filter and specify the boundary conditions needed to achieve optimum solid particle removal from particular slurries. This is different from other models discussed in the literature, which encompass a database of various models that can provide approximate solutions based on experience. Thus, the research carried out can address further improvement of current models and algorithms. More accurate and realistic methods of modelling woven fabric structures based on yarn and fabric mechanics can be instrumental in more correct prediction of filtration characteristics and efficiencies. One example is the cross-sectional shapes of yarns in a fabric

318

Modelling and predicting textile behaviour

that are found to have an irregular shape throughout their length. The yarn cross-section is narrower (width to height ratio is lower) at the cross points of warp and weft yarns. In the current algorithm, solid particles are assumed to be rigid bodies. This assumption simplifies the mechanics of the force model. Particle shapes do not change owing to pressure from other particles or the fluid flow. In reality this is not the case and hence the predicted results and structures for cakes made from compressible particles can potentially vary immensely. Furthermore, the chemical interactions between particle, fluid and the fabric filter need to be investigated, given that they could play an important role in particle capturing. There are also other forces between particles that have not been considered. Finally, cake release modelling needs to be incorporated since filters are usually designed to be re-usable. Integration of these requirements with the already established algorithm is stepping closer towards the ultimate goal set by industry to automate fabric filter selection.

7.9

References

1. M. J. Matteson and C. Orr, ‘Filtration: Principles and practices’, 2nd edition, Marcel Dekker, New York, 1986. 2. E. Hardman, ‘Some aspects of the design of filter fabrics for use in solid/liquid separation process’, Filtration & Separation, 1994, 813–18. 3. J. A. Wehner, B. Miller and L. Rebenfeld, ‘Moisture induced changes in fabric structure as evidenced by air permeability measurements’, Text. Res. J, 1987, 57, 247–56. 4. F. A. L. Dullien, ‘Single phase flow through porous media and pore structure’, Chem. Eng. J., 1975, 10, 1–34. 5. C. Tien, Granular Filtration of Aerosols and Hydrosols, Butterworths, Boston, 1989. 6. K. J. Ives, ‘Rational design of filters’, Proc. Inst. Civil Eng., 1960, 16, 189–93. 7. P. C. Carman, Flow of Gases Through Porous Media, Butterworths, London, 1956. 8. P. G. Saffman, ‘A theory of dispersion in a porous medium’, J. Fluid Mech., 1959, 6, 321–49. 9. J. Happel, ‘Viscous flow in mutliparticle systems: Slow motion of fluids relative to beds of spherical particles’, AIChE J., 1958, 4, 197–201. 10. J. P. Herzig, D. M. Leclerc and P. LeGof, ‘Flow of suspension through porous media: Application to deep bed filtration’, Ind. Eng. Chem., 1970, 62, 8–36. 11. S. Backer, ‘The relationship between the structural geometry of textile and its physical properties, I: literature review’, Text. Res. J., 1948, 18, 650–8. 12. A. F. Robertson, ‘Air porosity of open-weave fabrics. Part I: metallic meshes’, Text. Res. J., 1950, 20, 838–43. 13. F. T. Peirce, ‘Geometrical principles applicable to the design of functional fabrics’, Textile Res. J., 1947, 17, 123–47. 14. S. Backer, ‘The relationship between the structural geometry of textile and its physical properties, Part IV: interstice geometry and air permeability’, Text. Res. J., 1951, 21, 703–14.

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15. J. Bear, Dynamics of Fluids in Porous Media, Elsevier, New York, 1972. 16. C. N. Davies, ‘The separation of airborne dust and particles’, Proc. Mech. Eng. 1B, 1952, 185–98. 17. G. Bossis and J. F. Brady, ‘Self-diffusion of Brownian particles in concentrated suspensions under shear’, J. Chem. Phys., 1987, 87(9), 5437–448. 18. O. Emersleben, ‘Das Darcysche Filtergesetz’, Physik. Zeitschr, 1925, 26, 601–10. 19. T. Miyagi, ‘Viscous flow at low Reynolds number past an infinite row of equal circular cylinders’, J. Phy. Soc. Japan, 1958, 13, 493–97. 20. V. C. L. Hutson, ‘The resistance of gauze filters to slowly flowing fluid’, Chem. Eng., 1969, 232, 362–64. 21. H. D’Arcy, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris, 1856. 22. B. F. Ruth, ‘Studies in filtration: Derivation of general filtration equations’, Ind. Eng. Chem., 1935, 27, 708–23. 23. K. Rietema, ‘Stabilising effects in compressible filter cakes’, Chem. Eng. Sci., 1954, 88, 88–94. 24. R. L. Baird and M. G. Perry, ‘The distribution of porosity in filter cakes’, Filtration & Separation, 1967, 4, 471–6. 25. R. J. Wakman, ‘The formation and properties of apparently incompressible filter cakes under vacuum on downward facing surfaces’, Trans. Inst. Chem. Eng., 1981, 59, 260–70. 26. R. G. Holdich, ‘Solids concentration and pressure profiles during compressible cake filtration’, Chem. Eng. Com., 1990, 91, 255–68. 27. E. S. Tarleton, ‘The use of electrode probes in determinations of filter cake formation and batch filter scale-up’, Minerals Eng., 1999, 12, 1263–74. 28. M. Shirato, T. Aragaki, K. Ichimura and N. Ootsuti, ‘Porosity variation in filter cakes under constant pressure filtration’, J. Chem. Eng. Japan, 1971, 4, 172–7. 29. M. Fathi-Najafi and H. Theliander, ‘Determination of local filtration properties at constant pressure’, Separations Techniques, 1995, 5, 165–78. 30. M. S. Willis, R. M. Collins and W. G. Bridges, ‘Complete analysis of non-parabolic filtration behaviour’, Chem. Eng. Res. Design, 1983, 61, 96–109. 31. E. S. Tarleton, ‘Predicting the performance of pressure filters’, Filtration & Separation, 1998, 35(3), 293–8. 32. E. S. Tarleton, ‘Using mechatronics technology to assess pressure filtration’, Powder Tech., 1999, 104, 121–9. 33. B. R. Bierck, S.A. Wells and R. I. Dick, ‘Compressible cake filtration: Monitoring cake formation and shrinkage using synchrotron X-rays’, J. Water Pollution Control, 1988, 60, 645–50. 34. F. M. Tiller, N. B. Hsyung, L. Shen and W. Chen, ‘Cat scan analysis of sedimentation and constant pressure filtration’, Proceedings World Filtration Congress, Nice, France, 1990. 35. M. A. Horsefield, E. J. Fordham and L. D. Hall, ‘H NMR imaging studies of filtration in colloidal suspensions’, J. Magnetic Res., 1989, 81, 593–6. 36. E. J. La Heij, P. J. A. M. Kerkhof, K. Kopinga and L. Pel, ‘Determining porosity profiles during filtration and expression of sewage sludge by NMR imaging’, AIChE J., 1996, 42, 953–9. 37. K. Takahashi, Y. Kobayashi, T. Yokota, and K. Koyama, ‘Measurement of cake thickness on membrane for microfiltration of yeast using ultrasonic polymer concave transducer’, J. Chem. Eng. Japan, 1987, 20, 246–71.

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38. K. Atsumi and T. Akiyama, ‘A study of cake filtration: Formualtion as a Stefan problem’, J. Chem. Eng. Japan, 1975, 8, 487–92. 39. I. Tosun, ‘Formulation of cake filtration’, Chem. Eng. Sci., 1986, 41, 2563–8. 40. P. Kehoe, ‘A moving boundary problem with variable diffusivity’, Chem. Eng. Sci., 1972, 27(5), 1184–5. 41. E. Eklund and A. Jernqvist, ‘The motion of a neutrally buoyant circular cylinder in bounded shear flows’, Chem. Eng. Sci., 1994, 49, 3765–72. 42. H. H. Hu, ‘Motion of a circular cylinder in a viscous liquid between parallel plates’, Theor. Comp. Fluid Dynamics, 1995, 7, 441–55. 43. G. H. Ristow, ‘Wall correction factor for sinking cylinders in fluids’, Phys. Rev. E, 1997, 55, 2808–13. 44. A. S. Dvinsky and A. S. Popel, ‘Motion of rigid cylinder between parallel plates in stokes flow. Part 2: Poiseuille and Couette flow’, Computers and Fluids, 1987, 15, 405–19. 45. D. Qi, ‘Lattice-Boltzmann simulations of fluidization of rectangular particles’, Int. J. Multiphase Flow, 2000, 26, 421–33. 46. C. K. Adiun, Y. Lu and E. J. Ding, ‘Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation’, J. Fluid Mech, 1998, 373, 287–311. 47. R. J. Wakeman, ‘A numerical solution of the differential equations describing the formation of air flow in compressible filter cakes’, Trans. Inst. Chem. Eng., 1978, 56, 258–65. 48. S. A. Wells and R. I. Dick, ‘Mathematical modelling of compressible cake filtration’, Environmental Engineering: Proceedings of Speciality Conference, Austin, Texas, 10–12 July 1989. 49. P. B. Sorenson, P. Moldrup and J. A. Hansen, ‘Filtration and expression of compressible cakes’, Chem. Eng. Sci., 1996, 51(6), 967–79. 50. R. Burger, F. Concha and K. H. Karlsen, ‘Phenomological model of filtration processes: 1. Cake formation and expression’, Chem. Eng. Sci., 2001, 56, 4537–53. 51. K. Stamatakis and C. Tien, ‘Cake formation and growth in cake filtration’, Chem. Eng. Sci., 1991, 46(8), 1917–33. 52. S. Evje and K. H. Karlsen, ‘Monotone difference approximation of BV solutions to degenerate convection-diffusion equation’, J. Numerical Anal., 2000, 37, 1838– 60. 53. M. J. Biggs, S. J. Humby, A. Buts and U. Tuzun, ‘Explicit numerical simulation of suspension flow with deposition in porous media: influence of local flow field variation on deposition processes predicted by trajectory methods’, Chem. Eng. Sci., 2003, 58, 1271–88. 54. F. T. Peirce, ‘Geometrical principles applicable to the design of functional fabrics’, Textile Res. J., 1947, 17, 123–47. 55. M. A. Nazarboland, Computer Simulation of Filtration Through Woven Fabrics, PhD Thesis, University of Manchester, 2008. 56. L. Svarovsky, Solid–Liquid Separation, 4th edition, Butterworth Heinemann, Oxford, 2000. 57. FLUENT User’s Guide, Version 6.1, Fluent Inc., Lebanon, NH, USA, 2004 (available on www.fluent.com). 58. H. Goldstein, Classical Mechanics, 2nd edition, Reading, Addison-Wesley, 1980. 59. C. T. Crowe, R. A. Gore and T. R. Troutt, ‘Particle dispersion by coherent structures in free shear flows’, Particulate Sci. Tech. J., 1985, 3, 149–58.

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60. C. T. Crowe, M. Sommerfield and Y. Tsuji, Multiphase Flows with Droplets and Particles, CRC Press, New York, 1998. 61. C. Yin, L. Rosendahl, S. K. Kaer and H. Sorenson, ‘Modelling the motion of cylindrical particles in a non-uniform flow’, Chem. Eng. Sci., 2003, 58, 3489–98. 62. A. Haider and O. Levenspiel, ‘Drag coefficient and terminal velocity of spherical and nonspherical particles’, Powder Tech., 1989, 58, 63–70. 63. H. Ounis, G. Ahmadi and J. B. McLaughlin, ‘Brownian diffusion of submicrometer particles in the viscous sublayer’, J. Colloid Interface Sci., 1991, 143(1), 266–77. 64. N. V. Brillianatov and T. Poschel, ‘Rolling friction of a viscous sphere on a hard plane’, Europhys. Lett., 1998, 42, 511–16. 65. M. A. Nazarboland, X. Chen, J. W. S. Hearle, R. Lydon and M. Moss, ‘Modelling and simulation of filtration through woven media’, Int J. Clothing Sci. Tech., 2008, 20(3), 150–60. 66. P. J. Bakker, P. M. Heertjes and J. L. Hibou, ‘The influence of the initial filtration velocity and of vibrations on the resistance of a polystyrene filtercake’, Chem. Eng. Sci., 1959, 10, 139–49. 67. M. A. Nazarboland, X. Chen, J. W. S. Hearle, R. Lydon and M. Moss, ‘Effect of different particle shapes on the modelling of woven fabric filtration’, J. Information Computing Sci., 2007, 2(2), 111–18. 68. D. Venkateswarlu and A. P. Rao, Particle Technology, Srigam Printers, India, 1973. 69. R. J. Hunter, ‘Foundations of colloid science’, Oxford University Press, Oxford, 1967.

8

Modelling, simulation and control of textile dyeing

R. Shamey, North Carolina State University, USA

Abstract: A significant amount of work has been carried out to elucidate the dynamics of dye transfer, sorption and diffusion in a dyeing cycle. Several studies have demonstrated that a large number of factors influence the outcome of the process. The rate and direction of liquor flow affect the amount and uniformity of dye distribution within a yarn package. In order to characterise dye behaviour and the outcome of the dyeing process and specify the role of various controllable parameters such as temperature, pH, and flow on the levelness of ensuing dyeings, a comprehensive analysis of models relating this dye behaviour in a dye bath to the outcome of the properties of dyeing is given. Sorption and diffusion is discussed and the role of flow in the characterisation of the dyeing properties is explained. Fluid flow was modelled and simulated based on mathematical models that employ Darcy’s law but also take into account Navier–Stokes and Brinkman’s equations. The developed models allow for a specification of the flow rate, porous media geometry, porosity, fluid viscosity and temperature in specified domains. The results of simulation illustrate the fluid velocity profile and distribution through the spindle tube as well as the yarn assembly in a wide range of package permeabilities and are used to examine the accuracy of various models. Key words: modelling dyeing, sorption, diffusion, fluid flow, porous media.

8.1

Introduction

Dyeing is a highly complex process with a significant number of variables in which different phenomena occur simultaneously. The process requires a good understanding of different domains of science including chemistry, physical chemistry, fluid mechanics, thermodynamics, and others. Devising the most efficient dyeing process typically involves concerns about machinery design, preselection of dyes with compatible properties, use of appropriate control profiles, for example pH versus time profiles, selection of liquor ratio, flow rate and flow direction reversal times, and the type of temperature versus time profiles (Nobbs, 1991). Several groups have defined various models to describe the dyeing process and/or relate the conditions during a dyeing process to the quality of the resulting dyed material. Some models also deal with the physical chemistry of dye adsorption and the diffusion of dye molecules into the fibre; the latter is briefly discussed in the following sections. Models of 322

Modelling, simulation and control of textile dyeing

323

particular interest are those that are able to relate the conditions within the dye bath and the material being dyed to the distribution of dye within the material during and at the end of a dyeing process. These models define the relationship between dyeing parameters and the resulting quality of the dyed material and will be discussed in detail. A numerical simulation of a package dyeing process based on fluid flow within the package and taking into account mass transfer laws is explained and the simulation results are briefly discussed.

8.2

Sorption isotherms

A dyeing theory may be developed according to the behaviour of different classes of dye in relation to adsorption isotherms and the three most common adsorption isotherms therefore need to be introduced. Adsorption implies the excess concentration of solute on the surface of the solid compared to that in the surrounding liquid, while absorption refers to a more or less uniform distribution of the solute into the solid (Burdett, 1989). Dyeing is a process of adsorption and absorption in which a driving force exists which causes the transfer of dye from the solution to the fibres. The adsorption of dye into the fibres can be represented by the following mechanism (Wai, 1984):

fibre + dye ´ fibre · dye

There are two types of adsorption namely, physical adsorption and chemical adsorption. Physical adsorption refers to interactions between dye and fibre that take place through physical forces of attraction. Chemical adsorption on the other hand is a result of the formation of chemical bonds between dye and fibre. Physical adsorption normally happens rapidly and is temperature dependent (Burdett, 1989); on the other hand chemical adsorption may require a considerable amount of energy and time. Substantivity refers to the degree by which dye is transferred from a certain volume of dye liquor to the fibre, providing a uniform contact exists between dye and fibre. Exhaustion is a quantitative measure of substantivity, which is time dependent. Normally exhaustion refers to the ratio of the mass of dye being taken up by the fibre to that originally in the dye bath at equilibrium. The substantivity ratio or SR is defined as the ratio of the concentration of dye on fibre Cm to that in the solution C this is shown in Equation [8.1]:

SR =

Cm C

[8.1]

The saturation value, F is the amount of dye on fibre that cannot be increased regardless of the concentration of dye in the liquor. A dimensionless parameter f is often defined as the ratio of the concentration of dye on fibre Cf to the

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saturation value F. This dimensionless parameter is used in the definition of adsorption isotherms.

8.2.1 Nernst Isotherm In this isotherm the relationship between the concentration of dye in the liquor to the dimensionless parameter f, is shown according to Equation [8.2]:

f = Kp ¥ C for 1/Kp ≥ C ≥ 0



f=1

for

C ≥ 1/Kp

[8.2]

Here Kp is a constant at constant temperature and is referred to as the partition coefficient or the distribution coefficient. The adsorption of disperse dyes from a dispersion can be described by this isotherm. The schematic representation of this relationship is shown in Fig. 8.1.

8.2.2 Freundlich isotherm In this isotherm the saturation concentration of fibre is theoretically considered to be infinite. The Freundlich isotherm was originally derived empirically, but it can be derived theoretically for certain heterogeneous solid surfaces. The mathematical representation of this model is shown in Equation [8.3]. a for 0 < a < 1 Cf = Kp ¥ C log Cf = log Kp + a log C

[8.3]

The values of Kp and a can be found from values of Cf and C using Equation [8.3]. The Freundlich isotherm is also shown in Fig. 8.1. More details about this isotherm can be found elsewhere (Burdett, 1989, Rattee and Breuer, 1974, Sumner, 1989).

8.2.3 Langmuir isotherm In this isotherm the adsorption and desorption processes are considered to be reversible chemical reactions and their rates are equated at equilibrium. The mathematical representation of this isotherm is shown in Equation [8.4]:

f =

Kp ¥ C 1 + Kp C

[8.4]

Equation [8.4] can be rearranged to give Equation [8.5].

1 = 1+ 1 Cf F Kp ¥ C ¥ F

[8.5]

Modelling, simulation and control of textile dyeing

325

A typical representation of this isotherm can be seen in Fig. 8.1. Many dye/fibre isotherms fit the Langmuir model under laboratory conditions of dyeing; however, there are many that do not. The graph of Cf (the adsorbed dye concentration in g kg–1 or mol kg–1) versus C (the concentration of dye in the solution in g l–1 or mol 1–1) for the Langmuir isotherm in Fig. 8.1 clearly shows that the fibre becomes saturated with dye when all the available adsorption sites are filled. Langmuir-type absorption isotherms are observed when only a limited number of dye sites are available inside the fibre. In this case the concentration of dye in the polymer reaches a maximum limiting level as the concentration of dye in the dye bath is increased. This behaviour is typical of levelling acid dyes on wool and silk (Sivaraja and Srinivasan, 1984; Mitsuishi et al., 1992). The maximum number of sites in the fibre can be determined from the slope of the linear graph of C/Cf versus C according to the equation:

C = C + 1 Cf Cmax KCmax

[8.6]

where Cmax is the maximum number of adsorption sites that dye molecules can occupy in the fibre. It is assumed that any system to which the simulation model is to be applied can be represented by one of these three isotherms, or by a combination

1

Concentration of dye on fibre (Cf)

0.9 Langmuir isotherm

0.8 0.7

Freundlich isotherm

0.6 0.5 0.4 0.3 0.2 Nernst isotherm

0.1 0

0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Concentration of dye in bath (Cs)

8.1 Representation of sorption isotherms.

0.8

0.9

1

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Modelling and predicting textile behaviour

of them (Schuler, 1982). For instance, high concentrations of direct and vat dyes on cellulosic fibres commonly exhibit Freundlich-type isotherms (Aspland, 1992); this occurs apparently as a consequence of aggregation of dye molecules inside the fibres.

8.3

Dye diffusion models

Matter can be transported along a concentration gradient, that is from areas of high concentration to a low concentration as a result of random molecular motion. This mechanical diffusion process involves movement of the molecules of concern through voids and interstitial spaces between molecules in the medium through which they are moving. The movement ultimately results in an equalisation of chemical potential and concentration within the system (Johnson, 1989). In the case of more complex and larger molecules undergoing diffusion, the process takes longer until there is an equal concentration in both environments. Where there is more than one phase, the chemical potential of the diffusant is constant throughout the system only when true dynamic equilibrium has been achieved, even though the concentrations in each phase may differ considerably. In dyeing processes one phase consists initially of the fibre or filament and the other is usually a dye solution in which the undyed fibre is immersed. The mechanism of dyeing in its entirety is quite complex although it may be apparently considered a simple transfer process of dye passing from solution to the interior of the fibre. In view of the complexity of the diffusion process, real dyeing systems are studied on the basis of simpler idealised situations; however, these simpler models are based on considerable restrictions or conditions to make them amenable to mathematical treatment. It is necessary to study diffusion in terms of simple models before any advances can be made in dealing with real or experimental investigations of actual dyeing processes. Figure 8.2 shows a schematic representation of dye movement within a dyeing environment containing fibres. It shows a fast sorption process accompanied by a slower diffusion. A number of authors have investigated the diffusion of dye molecules inside fibres. Cegarra (1971) states that establishing an accurate mathematical model of diffusion is difficult. Beckman (1979) reviewed the laws of diffusion to obtain the diffusion coefficient. Vickerstaff (1950) reported that the relationship between the apparent diffusion coefficient and the structure of dye and fibre porosity for cellulosic fibres is linear. Hori et al. (1987), however, examined this relationship for acrylic fibres and stated that the results were far from linear. It ought to be pointed out that in many of these studies the influence of liquor agitation on the rate of dyeing was ignored. This is despite the fact that

Modelling, simulation and control of textile dyeing Dye (solution)

Solution transport Rapid

Possible desorption Dye (fibre)

Dye (interface) Fast

Diffusion Slow

327

Adsorption equilibrium

Dye (adsorbed)

8.2 Dye transfer from the solution into a fibre.

the kinetics of dyeing is influenced by the effects of agitation. Earlier workers have accumulated a considerable body of information concerning the relative dyeing properties of ranges of dyes, in order to enable the practical dyer to choose dyes in terms of the quantities which are adsorbed at equilibrium and the relative rates at which the dyes are taken up. In particular, it was considered that if the dyer wished to dye two or more dyes in admixture, he or she could obtain compatible mixtures by choosing dyes with the same rate of dyeing characteristics. In the earlier work, it was thought adequate to assume that, since the diffusion coefficients of the dyes in the fibres were several thousand times less than those in the liquor, the rates of dyeing could be considered to be governed by diffusion in the fibre (Vickerstaff, 1950). Peters and Ingamells (1973) summarised two main models of diffusion: the pores model and the free volume model. The first model explains diffusion by transfer of dye through channels interconnected in the fibre polymer providing that their size, shape and tortuosity is acceptable for the transfer of dye molecules into the fibre matrix. The second theory, however, refers to a part of the volume in the fibre polymer that is not occupied by constituent molecules. In the free volume model, the thermal motion of the atoms is an influential factor in the overall diffusion of the molecules since both the volume and the atomic motion increase with increasing temperature. Here motion occurs in the non-crystalline regions of the fibres. This void volume is formed by the segmental motion of polymer chains, a process that commences at the glass transition temperature of the fibre. The shift in the free volume in the non-crystalline regions promotes the diffusion of molecules that are sufficiently small (Broadbent, 2002). The pore model is mainly adopted for the diffusion of dyes into hydrophilic fibres, including cellulose, from an aqueous solution in which fibres swell markedly. Therefore it is presumed that water-filled channels of the fibres provide a transfer route for dye molecules to reach their adsorption sites (Hori et al., 1987; Peters and Ingamells, 1973). However, Peters and Ingamells (1973) concluded that the pore model in its entirety cannot be accepted. The free volume model has also been applied to diffusion processes by Fujita et al. (1960).

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Modelling and predicting textile behaviour

A number of authors have attempted to define diffusion models that could be applied to all types of fibres (Weisz, 1967; Weisz and Zollinger, 1967, 1968). These models were later applied by several authors to the dyeing of various fibres (Rys, 1973; Meyer et al., 1984). Hori and Zollinger (1986) summarised discussions on the two models in relation to dyeing wool and cellulosic fibres. The dyeing of cellulosics is claimed to be best understood in terms of the porous matrix model, while the behaviour of wool and silk most nearly approximates to the free volume model. Rohner and Zollinger (1986) however, pointed out that there was insufficient evidence for the validation of either the pore or the free volume model and in general these models represent an oversimplified version of the real situation.

8.4

Models relating dyeing parameters to the quality of dyeing

A number of authors have defined mathematical models that relate the dyeing conditions to the quality of the resulting dyeing. These models for batchwise dyeing may be divided into two types, detailed and reduced mathematical models. 1 Detailed mathematical models: In these models a comprehensive investigation into the effect of various dyeing parameters including both bulk and dispersive fluid flow is carried out. These parameters are then related to the physicochemical laws of the dyeing process. Therefore complex differential equations are derived. Major simplifications to the model would be required in order to solve these equations explicitly and therefore numerical solutions are used. 2 Reduced mathematical models: These models can be subdivided into two major groups: ∑ Models that are based on the physicochemical laws that describe the dyeing process and introduce explicit mathematical equations, but which have a restricted description of the flow of the dye liquor around the machine and within the yarn package. These models are generally used to determine dyeing conditions likely to produce a given desired exhaustion profile. ∑ Models that simulate dyeing conditions in which no dye migration occurs and fluid flow takes place only by a bulk mechanism. These models are based upon explicit mathematical equations that allow determination of levelness within a package for a ‘worst case’ dyeing scenario. Since a number of simplifying factors are used on the model, the equations defined by these models are generally simpler. Some of the most important models based on above categories are described in detail in the following sections.

Modelling, simulation and control of textile dyeing

329

8.4.1 Detailed mathematical models Boulton and Crank model Boulton and Crank (1952) noted that there are a large number of interdependent variables involved in the dyeing process. In their treatment of modelling they dealt with dye sorption using a yarn package. They considered the yarn packages to be uniform and therefore the intrinsic adsorption properties of the fibres were not considered in their model. All the factors that may cause unlevelness in a package dyeing process, except the package itself, were included in their model. The change in the concentration of dye as the dye liquor flows through the package and the exchange of dye between the fibre and the liquid at any point within the package, was defined by the two differential Equations [8.9] and [8.10], respectively. The rate of uptake of dye by the fibre from the dye bath is expressed according to Equation [8.7]:

∂F = K (Fc – F ) ∂t

[8.7]

where F is the concentration of dye on fibre at time t, and Fc is the concentration of dye on the yarn that could be taken up if a long enough time is given for the dye to reach equilibrium with the dye liquor of concentration C. K is a rate constant for the exchange of dye between liquor and yarn. The relationship between Fc and C is considered to be a Freundlich isotherm such as shown in Equation [8.8]:

Fc = K1 C0.5

[8.8]

where K1 is a constant. Assumptions made in this equation are as follows: ∑ ∑

diffusion or dispersion of dye in the liquor phase is negligibly slow compared to the rate of the bulk flow; the total amount of dye in any element of the package (yarn) is higher than that in the liquor in the same element.

The change in dye concentration is represented by a differential equation taking into account the mass balance on the element of the package. Converting Equations [8.7] and [8.8] into certain dimensionless quantities resulted in Equations [8.9] and [8.10] shown below:

∂F = C 0.5 – f ∂T



∂C = – (C 0.5 – f ) ∂X

[8.9] [8.10]

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Modelling and predicting textile behaviour

These equations, with the appropriate initial and boundary conditions, were solved and the following parameters were shown to affect the dyeing process: final percentage exhaustion of the dye bath, rate of exchange of dye (K) and a composite factor containing the rate of flow and volume of the dyebath. The main conclusions drawn from this model were summarised as follows: 1 A rapid flow rate together with a low exhaustion rate is most likely to produce level dyeing. For a given flow rate or exhaustion rate there is a critical degree of exhaustion or flow rate respectively beyond which the time of levelling increases rapidly. 2 A slow initial absorption rate followed by a steady and smooth increase of dye bath exhaustion is recommended. 3 Package thickness is advised to be reduced; however, there is a limit beyond which it would not be either technically or economically possible to do so. An overall rate constant has been used in this model to describe a complicated process of adsorption, desorption and reaction with fibre; in addition the influence of transport of dye from liquor to fibre is neglected. Later developments in these areas can be placed into two categories: (1) inclusion of the diffusion mechanism and (2) a model of the transport process of dye to the fibre surface. Some of these developments are discussed in the following sections. Hoffmann and Mueller model The first attempt to improve the Boulton and Crank model was made by Hoffmann and Mueller (1979). They considered the package as an extended plane of cylindrical fibres infinite in length and included the diffusion process in the model. They calculated the degree of unlevelness and the distribution of dye within the package for constant dyeing conditions such as temperature, pH and electrolyte concentration. This model was initially developed for a dye with a linear sorption isotherm; however, further work was carried out to derive appropriate equations for the more general case of a Freundlich equilibrium isotherm (Hoffman, 1989). Assumptions made in the development of this model were as follows: ∑ ∑ ∑

Concentration of dye outside the package is uniform; Preferential adsorption and filtration of the dye and variations in the dyeing conditions (e.g. temperature, pH) are not considered; An equilibrium state is assumed to exist throughout the dyeing.

The diffusion of dye from the surface into the interior of the fibres follows

Modelling, simulation and control of textile dyeing

331

Fick’s second equation applied to radial diffusion into a homogeneous circular cylinder according to Equation [8.11]:

È 2 ∂F ∂F ˘ = D (t ) Í ∂ F + 1 2 r ∂t ∂ r ˙˚ Î ∂r

[8.11]

where F represents F(r, X, t) which is the concentration of dye in the fibre, r is the radial coordinate of cylindrical fibre, t is time, D is the diffusion coefficient and X is the spatial coordinate in the flow direction X. The rate of uptake of dye by unit mass of fibre at time t and position X can be found from Equation [8.12]:

M (X , t ) =

2 rF2 rF

Ú

rF 0

F (r, X , T ) r dr



[8.12]

where rF is the density of the fibre, rF is the radius of fibre and M(X, t) is the mass of dye on fibre. Equations [8.11] and [8.12] were solved for constant dyeing conditions using a Laplace transformation method. The results were assessed in three terms: bath exhaustion time, levelling time and fibre penetration time (or diffusion velocity). Because of the restricted assumptions used in this model, it can only be used for a very narrow range of dyeing situations. Wai and Burley model Wai and Burley (Wai, 1984; Burley et al., 1985) considered the package as a packed bed chemical reactor. The rate of the dispersion of a dye or other solute in the liquid phase of a packed bed was described by a differential equation using a dispersion coefficient. They derived Equation [8.13] for an axial flow model, which included a dispersion term:

2 ∂C ∂C ∂F = D ∂ C2 – u –l ∂t ∂X ∂t ∂X

[8.13]

where D is the diffusion coefficient, u is the interstitial fluid velocity and l is a factor which contains the volumes of dye liquor in various parts of the dyeing machine. The net rate of absorption is calculated using Equation [8.14]: ∂F [8.14] = K a C (F * – F ) – K d F ∂t where F* is the saturation concentration of fibre, Ka is the adsorption rate constant and Kd is the desorption rate constant. They concluded that there was an optimum flow rate at which even dye

332

Modelling and predicting textile behaviour

distribution in the liquor around the individual yarns could be achieved and that further increase may result in channelling of liquor or distortion of the package. This model had a number of shortcomings which are summarised below: ∑ ∑ ∑

Adsorption, diffusion and the interaction of dye with the fibre are considered as a single stage; Temperature variations are not taken into account; The model is limited to only the Langmuir adsorption isotherm.

Vosoughi and Burley model Vosoughi (1993) further developed the earlier model of a packed bed reactor by Wai and Burley and defined two equations for axial and radial flow processes. The assumptions made in their work were as follows ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑



Packages are uniform in density and porosity; A uniform concentration of dye across the dyeing vessel exists; Fluid flow in pipes is represented as a plug flow; The same concentration of dye enters various parts of the package at any time t; The transport of dye across the package is according to a dispersed flow; The adsorption characteristics of the dyes, and their mixture if applied, onto the fibre follows a general expression shown in Equation 8.14; Packages consist of cylindrical fibres of infinite length; Diffusion of dyes into the fibre from the surface is according to Fick’s second equation applied to radial diffusion; Temperature dependence of diffusion is represented by Equation [8.15]; The temperature within the entire system including the package is uniform. 2 r ∂M ∂C + u (n ) ∂ C – DS ∂ C2 + P =0 e ∂n ∂n ∂X ∂X

[8.15]

where rP is the apparent package density, u (n) is the interstitial velocity, M represents M (X, n) which is the mass of dye absorbed in unit mass of yarn, C represents C (X, n) which is the mass of dye in unit volume of liquid, n is time, X is the axial position and Ds is the dispersion coefficient and e is the porosity of the package. Using these assumptions and defining the net amount of dye entering and exiting an element they derived Equation [8.16] for an axial flow package dyeing system:

Modelling, simulation and control of textile dyeing



2 r ∂M ∂C + u (n ) ∂ C – DS ∂ C2 + P =0 e ∂n ∂n ∂X ∂X

333

[8.16]

A similar technique has been used to develop Equation [8.17] for radial flow systems:

2 r ∂M ∂C + (u Y – DS ) 1 ∂ C – DS ∂ C2 + P =0 Y ∂Y e ∂n ∂n ∂Y



[8.17]

where Y is the radial position, u is the local velocity vector. Further equations were developed to describe the diffusion of dye into the fibre interior. These equations have been solved numerically using appropriate boundary conditions giving dimensionless relationships for both axial and radial flows. Despite the fact that a number of simplifying assumptions were made in the development of this model, the overall equations are still complicated second order differential equations. The application of this model, however, represents a sophisticated investigation of dye bath control strategy, which takes into account the diffusion process and different modes of flow throughout the dyeing vessel. However, the need for a numerical solution of the model represents a further source of error and, moreover, the predictions of the model were not experimentally verified.

8.4.2 Reduced mathematical models Medley and Holdstock model Medley and Holdstock (Medley and Holdstock, 1980; Holdstock, 1988) developed the ‘simple depletion theory’ for the direct control of exhaustion. They assumed that the rate of dye uptake by a layer in the package is proportional to the concentration of dye in the liquor at that position. They suggested that the concentration of dye in the liquor, C, decreases by passing through a radius of a package by DC as shown in Fig. 8.3. Assumptions in the development of this theory were: (1) only a single direction of flow is considered and (2) the partition coefficient between dye and fibre is independent of the concentration of dye. They stated that the fractional rate of dye uptake between the innermost and the outermost layers of the package is proportional to DC/C according to the second assumption above. In order to restrict the dye deposition error to a specified tolerance (R) throughout the dyeing, they used the condition shown in Equation [8.18] to govern the fractional rate of dye uptake:

1 ∂ C ≤ FR C C0 ∂ t L C0

[8.18]

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Modelling and predicting textile behaviour

Dq

r

C

L

C – DC

Dr

A

B

8.3 Representation of the depletion of dye across a section of the package.

with the consideration that DC/C ≤ R. Here C is the external concentration of dye at time t, C0 is the initial concentration of the dye liquor, L is the liquor ratio and F represents the flow rate. They concluded that the rate of dye uptake must progressively decrease throughout the dyeing in proportion to the concentration as shown in Equation [8.19]:

dC = FR C dt L

[8.19]

The condition is satisfied when (1/C)(dC/dt) is a constant which represents an exponential rate of exhaustion as a function of time. The condition shown in Equation [8.18] was derived from the general Equations [8.19] and [8.20] used in the theory:

1 ∂M = 1 ∂C M0 ∂t C0 ∂ t

[8.20]



1 ∂ M ≤ RF C M0 ∂t L C0

[8.21]

Modelling, simulation and control of textile dyeing

335

where M0 is the initial amount of dye in the system and M is the amount on the fibre at time t. Examining these relationships for the local difference in concentration and corresponding differences in local rate of dye uptake for fibre indicates that the slope of the dye adsorption isotherm also influences the dye deposition error. This theory is semi-empirical. Nobbs and Ren model This model was developed and extended by Nobbs and Ren (Ren, 1985; Illett, 1990) to quantify the total deposition error of dyeing. They proved both theoretically and experimentally that the quadratic exhaustion profile results in the achievement of better levelness. Illett (1990) verified this work and extended the model to include the axial flow type mechanism. This work was also further verified in a later work by Gilchrist (1995a). The model was also used in the exhaustion control of reactive dyes on cotton as well as continuous addition dyeing of cotton which is described elsewhere (Shamey and Nobbs, 1997). This model is described in additional detail here. Conventional dyeing Nobbs and co-workers’ mathematical model (Nobbs, 1991; Ren, 1985) uses a differential equation that describes the mass balance between the dye in the liquor and that on the fibre at a given time and position within the package. A second relationship is developed to relate the concentration of dye in the liquor at the entrance and exit points of the package to that in the mixing tank and the pipes leading to or from the package. For a thin layer of fibre the mass balance can be shown by Equation [8.22]:

change in dye liquor = C1 – C2 – C3

[8.22]

where C1 is the amount of dye entering the package, C2 is the amount of dye leaving the package and C3 is the amount of dye absorbed onto the package. In the schematic diagram of a section of yarn package shown in Fig. 8.3, the package is assumed to be perfectly symmetrical and so any section of it is representative of the whole. In Fig. 8.3, A is the internal radius of package (cm), B is the external radius of package (cm), q is the angle of interest (rad), r is the radius of interest (cm), and L is the height of package (cm). Assumptions made in the development of this model were: (1) Packages are uniform and no distortion or channelling occurs; (2) Flow is considered to be only bulk (dispersive flow exclusive); (3) Dye liquor is uniform throughout the tank;

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Modelling and predicting textile behaviour

(4) The rate of dye uptake at any point inside the package has a linear relationship with the liquor concentration at that point:

x ∂ M (r , t ) = K (t ) C (r, t ) ∂t (1 – x )

[8.23]

where M(r, t) is the amount of dye in the fibre at a distance r from the centre of the package (g l–1), C(r, t) is the dye concentration in the liquor at a distance r from the centre of the package (g l–1); K(t) is the dyeing rate constant within the package (l s–1), C0 is the initial dye bath concentration (g l–1), C(t) is the dye bath concentration at time t (g l–1); F is the volumetric flow rate (l s–1), V is the total volume of dye liquor present (l); Vs is the volume of dye liquor entrained in the package (l) and x is the voidage of the package which is defined as: x = 1 – (volume of yarn/volume of package). It should be noted that K(t) is considered to be independent of the local dye concentration and the amount of dye already on the yarn at that point which cannot be strictly true. Equation [8.23] was derived according to the principles highlighted here. The volume of the section in which the mass balance is calculated can be obtained using Equation [8.24]:

DV = r L Dq Dr

[8.24]

The flow rate through this section can be calculated as shown in Equation [8.25]:

flow rate = F Dq 2p

[8.25]

Having described the mass balance as shown in Equation [8.22], this equation can now be rewritten as shown in Equation [8.26]:

r L D q D r x [CA (t + Dt ) – CA (t )]= F Dq D t [C (r, t ) – C (r + Dr, t )] 2p –(1 – x )

∂ M (r, t ) Dt r L Dq Dr ∂t

[8.26]

By simplifying Equation [8.26] and taking the limit as Dt and Dr tend to zero, Equation [8.27] can be derived:

x

∂ C (r , t ) ∂ C (r , t ) ∂ M (r , t ) =– F – (1 – x ) 2p rL ∂t ∂r ∂t

[8.27]

It is then assumed that the rate of the adsorption of dye follows first order kinetics, however, it has been further assumed that the constant of proportionality may change with time which results in Equation [8.23]

Modelling, simulation and control of textile dyeing

337

described earlier. Substituting Equation [8.23] into Equation [8.27] and dividing by x results in Equation [8.28]:

∂ C (r , t ) ∂ C (r , t ) =– F – K (t ) C (r , t ) ∂t ∂r 2p rLx

[8.28]

Taking assumption (3), the dyeing rate constant K¢(t) of the liquor in the tank is defined by Equation [8.29]:

∂ C (A , t ) dC (t ) = = – K ¢ (t ) C (t ) dt ∂t

[8.29]

It is important to relate the dyeing rate constant within the package, K(t), to the rate constant of the dye bath, K¢(t), since it is K¢(t) that can be obtained by measurement during a dyeing process. Also C(r, t) is imagined to be split into two terms, a position-dependent term G(r) and a time-dependent term H(t). It follows that:

∂ C (r , t ) ∂ G (r ) = H (t ) ∂r ∂r

[8.30]

This assumption is valid if:

[G(A) – G(B)] >>> H(t) – H(t + Dt)

where Dt = (Vs/F), and is the liquor transit time through the package. Therefore ∂ C (r, t ) can be substituted in Equation [8.28] to obtain: ∂t

or



K ¢ (t ) H (t ) G (r ) = –

F H (t ) · ∂ G (r ) – K (t ) H (t ) G (r ) [8.31] ∂r 2p rLx

1 ∂ G (r ) = – 2p rLx [K (t ) – K (t )] G (r ) ¢ r ∂r F

[8.32]

The integration of Equation [8.32] leads to Equation [8.33]:

Ï p Lx ¸ C (r , t ) = C (t ) exp Ì – [K (t ) – K ¢ (t )] (r 2 –A 2 ) ˝ F Ó ˛

[8.33]

In order to apply these equations, the form of the function K¢(t) is required. By considering the mass balance in the mixing section and presuming a uniform mixing and zero time delay it can be said:

dC (t ) F [C (B , t ) – C (A , t )] = dt V – Vs

[8.34]

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Modelling and predicting textile behaviour

where the following conditions are assumed:

C(A, t) = C(t), C(•, t) = 0, C(A, 0) = C0

As a result of the uniform mixing, the concentration of dye at the inlet point A is considered to be that in the dye bath. dC (t ) F C (t ) Ï1 – C (B, t ) ¸ = Ì dt V – Vs C (A, t ) ˝˛ Ó



[8.35]

Using Equation [8.33], C(B, t) can be evaluated:



Ï p Lx ¸ C (B , t ) = C (t ) exp Ì – [K (t ) – K ¢ (t )] [B 2 – A 2 ]˝ Ó F ˛

[8.36]

Similarly substituting these values in Equation [8.35] results in Equation [8.37]: È dC (t ) Ï p Lx ¸˘ = F C (t ) Í1 – exp Ì – [K (t ) – K ¢ (t )][B 2 – A 2 ]˝ ˙ dt V – Vs F Ó ˛˚ Î



[8.37]

However, using Equation [8.29] and substituting for the rate of change of concentration results in Equation [8.38].



K ¢ (t ) =

F V – Vs

È Ï p Lx 2 2 ¸˘ Í1 – exp Ì – F [K (t ) – K ¢ (t )] (B – A ) ˝ ˙ [8.38] Ó ˛˚ Î

From this equation the relationship between K(t) and K¢(t) can be derived as shown in Equation [8.39]:

{

}

(V – VS ) K (t ) = K ¢ (t ) – F ln 1 – K ¢ (t ) Vs F

[8.39]

Using Equation [8.39], the value of C(r, t) and therefore M(r, t) can be calculated. A new parameter w(t) is defined as the ratio of K(t) and K¢(t) as shown in Equation [8.40]: Therefore:

w (t ) =

{

}

(V – VS ) K (t ) F =1– ln 1 – K ¢ (t ) K ¢ (t ) Vs K ¢ (t ) F

[8.40]

Modelling, simulation and control of textile dyeing



339

∂ M (r , t ) ∂t =

Ï V x È r 2 – A2 ˘ ¸ w (t ) K ¢ (t ) C (t ) exp Ì – S K ¢ (t ) [w (t ) – 1] Í 2 2 ˙˝ 1–x Î B – A ˚ Ô˛ ÓÔ F



[8.41]

From Equation [8.41] the distribution of dye within the dyed package may be predicted from the theoretical or measured exhaustion values and other dyeing parameters. These equations are normally solved numerically, however, the results can be compared to exact solutions for a particular idealised exhaustion profile in order to identify the errors that may be introduced by the numerical method. This has been shown elsewhere (Gilchrist, 1995a; Shamey and Nobbs, 1997). Calculation of the deposition error as an indication of the quality of dyeing The deposition error or DE, which is the difference between the amount of dye on the inner and outer layers of fibre, can be also calculated. This may provide a measure of the quality of the final dyeing, in terms of colour. The incremental deposition error can be directly calculated from the mathematical model for the in to out flow direction as shown in Equation [8.42]:

È ∂ M (A , t ) ∂ M (B , t ) ˘ DE (t ) = Í – ˙˚ ∂t ∂t Î

[8.42]

The integration of these values over the total time of dyeing gives the overall deposition error DET as shown in Equation [8.43]:

DET = M(B, td) – M(A, td)



DET = – (V – Vs ) V Vs

Ú

[8.43] 2

1 1 È dC (t ) ˘ dt F C (t ) ÍÎ dt ˙˚

[8.44]

where td is the total time of dyeing. Equation [8.44] shows that the magnitude of DE across the package is 2

dC (t ) ˘ determined by the variation of the quantity 1 1 ÈÍ with time. F C (t ) Î dt ˙˚ If this quantity is kept constant a quadratic equation results which gives the lowest DE. The value of DE calculated according to the Equation [8.43] above is the absolute deposition error. The relative deposition error, which is the deposition error relative to the total amount of dye on the fibre is, however,

340

Modelling and predicting textile behaviour

more important in terms of the dyed fibre appearance. For instance, for samples with high concentrations of dye, higher DE values may be tolerated. The relative incremental deposition error may be calculated according to Equation [8.45]:

DEREL (t ) = DE (t )

VS 1 – x DT x

[8.45]

where DEREL is the relative deposition error and DT is the total amount of dye to be absorbed by the fibre. The mathematical model described here can be used to change the conditions of the dye bath to keep DEREL below a limiting value. The limiting value is independent of dyeing conditions. Nobbs (1991) described an overall control strategy in which all the parameters of dyeing and not only one variable, such as temperature, are controlled giving final conditions.

8.4.3 Modelling stepwise addition of dye in dyeing Stepwise addition of dyes into the dye bath, when carried out isothermally with the object of achieving absorption proportional to the time, is also referred to as ‘integration dyeing’. Bayer (Hilderbrand and Haas, 1983) patented this process for dyeing cotton with reactive dyes. According to this method, dye can be added to the dye bath gradually while all the other necessary auxiliaries are already in the dye bath. It is also possible to add chemicals, including dye, simultaneously. This method of dyeing has been described by several authors (Cegarra et al., 1983, 1988, 1989, 1992; Valldeperas et al., 1990). They used the following expression to describe the constant addition of dye into the dye bath:

Cs = B t

[8.46]

where Cs is the dye concentration in g kg–1 of dye in terms of total amount of dye (g) added to the solution divided by the current amount of the solution (kg), t is the dyeing time and B is the proportionality constant or addition rate. They also defined the absorption rate of the dye by fibre as shown in Equation [8.47]:

dCf = b1 ± b2 t dt

[8.47]

where Cf is the dye concentration in g kg–1 of fibre, b1 is the rate constant in the first stages of the dyeing and b2 is the deceleration or acceleration rate. Integrating Equation [8.47] results in Equation [8.48] below:

Modelling, simulation and control of textile dyeing



Cf = b1t +

b2 t 2 2

341

[8.48]

Dividing Equation [8.47] into Equation [8.48] results in Equation [8.49]:

Cf = k1 + k2 t Cs

[8.49]

where k1 = b1/B and k2 = b2/2B:

A=

Cf = k1 + k2 t Cs

[8.50]

where A represents the dye exhaustion (concentration of dye in fibre to the total amount added to the solution) as a fraction at any time t. If k2 = 0, exhaustion is constant and so the rate of deposition onto the fibre and rate of addition of dye is constant, in this case A = k1. The temperature dependence of A is represented by a modified Arrhenius law as a quadratic term:

ln A = Z1 +

Z2 Z + 32 T T

[8.51]

where Z1 to Z3 are constants, A is the dye exhaustion after 60 min and T represents the absolute temperature. When the total amount of dye added to the dye bath as well as the influence of the concentration of dye is included, Equation [8.52] is obtained:

ln (1 – A ) = B1 +

B3 B B ˆ Ê + 42 + ln C Á B2 + 5 ˜ Ë T T ¯ T

[8.52]

where C is the total amount of added dye after 60 min dyeing. The terms B2 ln C and B5 ln C/T account for the influence of the concentration of dye and its interaction with the temperature. When the values of B1 to B5 are known, C, can be expressed by Equation [8.53]:

Ê ln (1 – A ) – B1 B3 /T – B4 /T 2 ˆ C = exp Á ˜¯ B2 + B5 /T Ë

[8.53]

Once the value of A for each temperature is known, C can be evaluated from this relationship. Cegarra and co-workers (Cegarra et al., 1988) state that the significance of Equation [8.53] is that the value of A and amount of dye absorbed under control by the fibre could be determined for each value of C and at each operating temperature with linear or slightly parabolic absorption with time. They concluded that if exhaustion is low (less than 80%), the rate constants fulfil the Arrhenius law. Rate constants are nearly independent of temperature at higher levels of exhaustion.

342

Modelling and predicting textile behaviour

Calculation of dye dosage A theoretical model has been developed by Cegarra and co-workers (Cegarra et al., 1988, 1989) to assess the amount of dye that has to be added at the end of each time interval to the solution according to the integration method. If the absorption of the fibre at the end of each interval remains constant and all the additions have to be made at the same intervals of time Dt then:

CDt = M Dt

[8.54]

The first part of the addition of dye into the dyebath is similar to that in the conventional method. Since the dye is added to a bath containing undyed fibre, the other additions are made on a fibre which contains the fraction X of the amount of dye that has to be added at each time interval. A kinetic equation including k, the rate constant and C the amount of dye on fibre at equilibrium is needed as part of the method. Figure 8.4 represents a schematic diagram of this method in which various parameters for the calculation of dye dosage have been demonstrated. Here Dt represents the time interval, C is the concentration of solution containing dye, S is the amount of dye that has to be added at the end of each interval, CDt is the amount of dye that can be absorbed at the end of each interval by fibre and M is the average rate of dye absorption in each Dt. Using the Cegarra–Puente equation the following relationship is given:

Amount of dye on material (g kg–1)



Ê C2 ˆ ln Á 1 – t2 ˜ = – kt Ë C• ¯

[8.55]

Cb(2)–CDt(2) S(2)

CDt(2)

Cb(2) Cb(1)–CDt(1)

S(1)

CDt(1)

Cb(1)

Time (min) Dt

Dt

8.4 Schematic diagram of dye dosage calculation according to the integration method.

Modelling, simulation and control of textile dyeing

343

where Ct is the amount of dye in the fibre at a time t. If A• is the exhaustion at equilibrium and Cb the quantity of dyestuff in the bath after each addition of dyestuff, then:

Ct = A• Cb 1 – e–(kt )

[8.56]

The amount of dye absorbed by the fibre at the end of the first interval is:

CDt = A• 1Cb1 1 – e–(k1 Dt )

[8.57]

Assuming Equation [8.54] occurs in all intervals:

Cb1 =

MDt A• 1 1 – e – k1 Dt

[8.58]

Cb2 =

MDt 1 – e– k2 Dt

[8.59]

A• 2

The amount of dye to be added to the bath at the beginning of the second interval of time Dt will be:

S2 = Cb2 – (Cb1 – CDt)

[8.60]

Having considered Equations [8.53]–[8.59], the general form of Equation [8.61] can be shown as:



Ê ˆ 1 1 Sn = MDt Á – + 1˜ Ë A•n 1 – e–(kn Dt ) ¯ A•n –1 1 – e –(kn –1 Dt )

[8.61]

K and A values are dependent upon temperature, initial concentration, pH, liquor ratio and so on. A also depends on X, the amount of dye already on fibre, known as predyeing. If the dyeing is carried out isothermally, the Arrhenius equation can be applied:

ln K = B0 +B11/T

[8.62]

The relationship between X, K and Cb1 was determined by experiment as shown in Equation [8.63]:

ln K = b0 + b1X + b2f(Cb)

[8.63]

where b is a coefficient and f(Cb) is a function of the concentration of dye in the bath and is different for each kind of dye. If the total exhaustion A• is measured at the end of each addition, that is from zero time until a time n ¥ Dt therefore:

344

Modelling and predicting textile behaviour

An Dt =

Cn Dt

n

S Cbi

i =1

=

Cn Dt nCb

[8.64]

where n is the number of intervals Dt until that moment. This value is different for different classes of dyes. Cegarra et al. (1989, 1992) showed by experiment that an increase of the amount of dye in the fibre, X, caused a decrease in K and further concluded that this value also decreased as the depth of dyeing increases. K is also dependent on the dye concentration in the solution and in the fibre at the moment of adding the dye, as well as on the period of time the fibre is in contact with the dyeing solution. It was found that exhaustion levels obtained for the integration method, relative to the total amount of dye used, were lower than those for conventional methods. However, the partial exhaustion of dye at each dyeing time was higher for this method than for the standard dyeing method.

8.4.4 Critical conclusion for relevant models In the previous section a number of mathematical descriptions of the dyeing process, developed by different workers, was presented. The solution of the model equations can produce results in the form of a distribution of dye throughout the system at any time during the dyeing process, which provides useful insight into the package dyeing operation as an example of a batch dyeing process. The models presented were based on convective dispersion and only describe dye transfer during dyeing. These include the works of Hoffman and Mueller (1979), Nobbs (1991) and Ren (1985), Burley et al. (1985), and Vosoughi (1993). Fluid mechanics can be used to provide a description of flow behaviour during the dyeing process. The system geometries in the majority of the models discussed here are comparatively simple, since they only model the flow in the porous media (textile assembly), while the flow behaviour in the open channel (tube) is not simultaneously considered. This can create difficulties in the definition of the boundary conditions. A simple definition of different types of flow models is therefore warranted. In a CDE (convection–dispersion equation) model, also known as the ADE (advection–dispersion equation), the centre of mass of the solute travels at the same speed as the mean water flux and the spread is normal, therefore the concentration of solute at a point changes over time. When spread is diffusion-like, the width of the solute ‘plume’ increases proportionately to the square root of time; this is a signature of a diffusive process. Conceptually the CDE is similar enough to Taylor’s tube to be thought of as just flow in a single tube.

Modelling, simulation and control of textile dyeing

345

The discussion of a fast flow in Taylor’s tube involves a stream tube model (STM). In this case there is not enough time for diffusive exchange between streamlines and therefore each streamline can be considered to act independently. If the velocity distribution is known, the dispersion can be predicted. In this case, the centre of mass of the solute still moves at the same velocity as the mean water flux, but the dispersion increases linearly with time, rather than with the square root of time. This indicates that the process is not diffusive. Conceptually it can be seen as flow in several parallel tubes of different diameters. MIM (mobile-immobile model) explains the flow taking place through a porous media, while the solute diffuses into and out of the aggregates. Such a model is commonly used to describe solute movement through textiles. This model supposes that water flows through some fraction of the total porosity, the mobile region, and not through the rest, the immobile region. The diffusive exchange between the mobile and immobile regions has the net effect of slowing down the solute movement. So solute does not move at the same velocity as the water, but it scales the same: velocity is linear with distance. If the solute moves more slowly than the water, it is retarded (slowed down) and a retardation factor is then introduced. The model of Hoffmann and Mueller (1979) related a number of dyeing parameters to dyeing quality. They assumed a uniform liquor flow through a package and Fickian diffusion of dye within the fibres. The distribution of a dye within the package and the degree of unlevelness were calculated for constant dyeing conditions of temperature, pH and electrolyte concentration. It, however, was restricted to a constant rate of liquor flow. In addition, they ignored the dispersion factor and only considered a linear adsorption relationship between the concentration of dye in the liquor and on the fibres. The work of Nobbs and Ren (Ren, 1985, Nobbs, 1991), unlike that of other work mentioned, does not produce solutions to the model equations which provide the concentration of dye at any position in the package at any time. It gives a relationship between a number of variables and the unlevelness of dyeing. The control scheme developed and described subsequently by the authors was aimed at achievement of any of the three exhaustion profiles: linear, exponential or quadratic, by online measurement of the dye bath conditions and manipulation of the temperature. The work of Burley and co-workers (Wai 1984, Vosoughi 1993) treats a number of significant operational conditions such as flow reversal, time delays and addition (and withdrawal) of dye liquor during the process, in a comprehensive analysis based on sound chemical engineering principles, which can be classified as an MIM approach. It is also supported by robust numerical techniques for treatment of complicated mathematical problems. The simulation model, however, lacks analysis of various stages of dye absorption, variable boundary conditions caused by the variable concentration

346

Modelling and predicting textile behaviour

of dye in liquor and the variable dispersion coefficient caused by the variable dyeing conditions. In both STM and MIM approaches, an exact solution of convective diffusion to a solid surface requires the solution of the hydrodynamic equations of motion of the fluid for boundary conditions appropriate to the mainstream velocity of flow as well as the geometry of the system. One major shortcoming of the dyeing models discussed is that the flow property during dyeing is not defined in a sound mathematical form. Another limitation of these models is that situations like variable boundary conditions and a variable dispersion coefficient are not considered in the numerical simulations. There are a number of other models that take into account fluid mechanics in the simulation and control of the dyeing process. These include the works of Telegin et al. (1997), Shannon et al. (2000), Scharf et al. (2002) and Karst et al. (2003). Telegin et al. (1997) emphasised that convective transfer in the package must accompany mass transfer in dyeing to result in an accurate simulation of the process. It should be noted, however, that they also ignored the dispersion component. Telegin (1998) proposed a mathematical description of convective mass transfer in the flow of solution around a single fibre and through a layer of fibres and modelled an individual cylindrical fibre with transverse liquid flow with the dissolved dye. The process of stationary diffusion near the surface of a cylinder in the case of a thin boundary layer was described in polar coordinates. He described the liquid motion near the surface of a cylinder in a simple manner using Protodiakonov’s method (Telegin, 1998). The work of Telegin and co-workers made real progress on the influence of the convective factor in the convective dispersion equation by defining the flow velocity using a sound mathematical basis. However, since the work was based on analytical methods, an exact solution of the problem at a solid surface could be obtained only where the solution flows at a steady rate past an object of simple geometrical shape. The work of Shannon et al. (2000) can be used to investigate the influence of package geometry and permeability on flow properties within the package, since it can predict pressure and velocity profiles based on user-defined package geometry, permeability profile and fluid properties. However, the model, which was obtained by combining the continuity equation for fluid flow in a porous medium and Darcy’s law, only considered the system within the package range, but did not consider the flow before approaching the surface of the package, which, if significant, can affect the flow behaviour in the system owing to the definition of the boundary conditions. The work of Scharf et al. (2002), based on the Navier–Stokes equations, simulated turbulent incompressible liquids during dyeing. They assumed the flow through bobbins to be linear and described the yarn package by a porosity model. The simulation results provided information on the static pressure

Modelling, simulation and control of textile dyeing

347

and velocity distribution at every part of the dyeing vessel. However, owing to the restricted assumptions used in this model, including direct treatment of the frictional force of the porous medium represented by the yarn bobbin as the external body forces in Navier–Stokes equations, it may only be used for a narrow range of dyeing situations. Karst et al. (2003) employed computational fluid dynamics (CFD) to model dye liquor flow in a beam dyeing and package dyeing and to determine how certain parameters affect liquor flow through the packages. The CFD approach makes it possible to accommodate complicated medium shapes, three-dimensional (3D) analysis, as well as variable parameters.

8.5

Numerical simulation of package dyeing

Zhao et al. (Zhao, 2004; Zhao et al., 2005; Shamey et al., 2003a, 2003b; Shamey and Zhao 2004a, 2004b) developed a numerical simulation of the package dyeing process using fluid flow principles. As with other models, a number of simplifying assumptions were made to enable investigation of the dyeing process quantitatively. The assumptions in this model included: 1 The packages are uniformly wound so that the package density and porosity are the same throughout a package and also from one package to another. 2 The density and porosity of the packages remain constant during the dyeing process. 3 The dye liquor is considered to be an incompressible fluid; the density of dye liquor is constant under isothermal conditions. 4 The dye liquor is assumed to be a Newtonian fluid; the viscosity is constant under isothermal conditions. 5 The dye liquor is well stirred during the dyeing process so that the concentration of dye across the entrance of the package throughout the liquor can be taken to be uniform. 6 The transport of dye liquor along the packed bed is represented by an MIM. Thus the supply of dye from the dyebath to the surface of the fibres at any position in the package is considered to consist of the additive effects of two types of transport phenomena: convective flow and dispersive flow through porous media. 7 The dye concentration at the surface of the fibre is related to the dye concentration in the liquor within the package (local dye bath concentration) according to the sorption isotherm of the dyeing equilibrium. 8 The dye adsorption and any chemical reactions with fibres occur very rapidly in comparison to the diffusion process, thus a local equilibrium can be assumed to exist between the free and immobilized components of the diffusing substance.

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Modelling and predicting textile behaviour

9 The temperature within the entire dye bath, i.e. packages and dye liquor, is uniform at all times. The basic assumptions made are more realistic than those of Hoffman, Burley, Wai and Vosoughi, although some of the assumptions may still not be strictly true. The most important assumption is the representation of transport with an MIM approach which separates convection and dispersion. These basic assumptions and some special definitions for some individual cases provide a basis for mathematical modelling of dyeing process.

8.5.1 System geometry definition To reduce the calculations, it is assumed that the mass transfer as well as flow behaviour are the same on both sides of the symmetric package axis shown in Fig. 8.5. Therefore, the system geometry can be further simplified as in Fig. 8.6. Only the left side of the symmetric axis line is considered to determine the system geometry. The derivation of the dye transfer equation within the package relies on the principle of superposition: convection and dispersion can be added together if they are linearly independent. Dispersion is a random process caused by molecular motion. Owing to dispersion, each molecule in time dt will move either one step to the left or one step to the right (i.e. ± dx). Y No-slip boundary

h

C(X1, t), u(X1, t) C(X0, t), u(X0, t)

Flow 0

X No-slip boundary

8.5 Three-dimensional representation of package geometry including the flow characteristics.

Modelling, simulation and control of textile dyeing 0.08

349

No slip boundary

R1 0.06 0.04 0.02

R2

Symmetry boundary

0 –0.02

Outflow boundary N-S/Darcy or Brinkman boundary

–0.04 –0.06 No slip boundary

–0.08 –0.1

–0.05

0

0.05

0.1

0.15

Inlet boundary

8.6 Simplified geometry of the system in two-dimensions.

Owing to convection, each molecule will also move udt in the cross-flow direction. These processes are clearly additive and independent; the presence of the cross-flow does not bias the probability that the molecule will take a diffusive step to the right or the left, it just adds something to that step. Using the mass balance principles in a multicomponent system and adding the flux law and the mass conservation law, a differential control volume was used to derive the dye transfer equation, which in vector notation is shown in Equation [8.65]:

e

∂Cp ∂C + (1 – e ) f = – — · (UCp ) + — · (D—Cp ) ∂t ∂t

[8.65]

In order to solve Equation [8.65], proper boundary conditions have to be defined. According to the system description and geometry of the package, the convective flow is very large compared with diffusion in the flow direction and dye transport by diffusion may be negligible in this direction at the inflow face. Also it is reasonable to assume that the gradients at the exit face are small and insulating conditions are assumed at all other boundaries. An overall description of the variation of fluid and solid phase concentration at a given time and position is therefore derived. Simultaneous solutions to these equations will give the actual dynamic behaviour of a packed bed dyeing machine in terms of fluid and solid phase concentration for a given time and position throughout the package. Darcy’s law and the Brinkman equation

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Modelling and predicting textile behaviour

can be used to characterise the flow property within the package. Flow in the porous domain (yarn assembly), using Darcy’s law can be described by Equations [8.66] and [8.67]:

u1 = – k —p1 h

[8.66]



— · u1 = 0

[8.67]

where k is the permeability of the porous medium, h the fluid viscosity, p1 the pressure and u1 the velocity vector in the porous medium (yarn assembly). The equations governing dye transfer, which contain factors for convection, diffusion and sorption, combined with a description of flow in both free liquor (dye liquor within the package tube) and porous media (yarn package), establish a complete mathematical model for package dyeing machinery. The solutions of the model give the dynamic behaviour of the system under given conditions of flow rate and given dyeing parameters. The results will mainly take the form of graphs showing the variation of dye on fibre concentration with time and position for given flow. It must be noted that since this is a 3D model, it is convenient to use a finite element method analysis to investigate the dynamic behaviour of the systems with any different package geometry, or any flow direction by defining the exact package geometry and appropriate boundary conditions.

8.5.2 Finite element methods The finite element method (FEM) is a numerical technique which gives approximate solutions to differential equations that model problems arising in physics and engineering (Pepper and Heinrich, 1992). FEM requires a problem defined in geometrical space (or domain) to be subdivided into a finite number of smaller regions (a mesh). In finite differences, the mesh consists of rows and columns of orthogonal lines; in finite elements, each subdivision is unique and need not be orthogonal. For example, triangles or quadrilaterals can be used in two dimensions as shown in Fig. 8.7, and tetrahedrons or hexahedrons in three dimensions as in Fig. 8.8. Over each finite element, the unknown variables (e.g. concentration, velocity, etc.) are approximated using known functions; these functions can be linear or higher order polynomial expansions that depend on the geometrical locations (nodes) used to define the finite element shape. The governing equations in the finite element method are integrated over each finite element and the solution summed (‘assembled’) over the entire problem domain. As a consequence of these operations, a set of finite linear equations is obtained in terms of a set of unknown parameters over each element. Solution of these equations is achieved using linear algebra techniques.

Modelling, simulation and control of textile dyeing

351

0.06

0.04

0.02

0

–0.02

–0.04

–0.06

–0.1

–0.08 –0.06 –0.04 –0.02

0

0.02

0.04

0.06

0.08

0.1

8.7 Mesh of yarn package for numerical solution using FEM in two dimensions.

8.5.3 Summary of the model equations As derived in the preceding section, flow properties and mass transfer during dyeing may be described by a set of equations outlined as follows. I. Flow equations The flow of free liquor in the tube can be described as:

–h—2u + r(u · —)u + —p = 0

[8.68]



— · u = 0

[8.69]

where h is the dynamic viscosity, r is the density, u the velocity field and p the pressure. The flow of liquor through the porous package can be characterised by Darcy’s law and Brinkman equation:

u1 = – k —p1 h

[8.70]



— · u1 = 0

[8.71]



u2 = – k —p2 + h— 2 u2 h

[8.72]



— · u2 = 0

[8.73]

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Modelling and predicting textile behaviour

0.14

0.12

0.1

0.08

0.06

0.04 0.02 0 0.06 0.04

0.06

0.02

0.04

0 0

–0.02 –0.04

0.02

–0.02

8.8 Mesh of yarn package for numerical solution using FEM in three dimensions.

where p1, p2, and u1, u2 are the pressure and the velocity vectors, when the flow in the porous package is described by Darcy’s law and the Brinkman equation respectively; k is the permeability of the porous package and h is the fluid viscosity. II. Mass transfer equations The differential mass balance of dye considering both convective flow and dispersive flow of dye in the liquor, as well as the absorption by the fibres can be described by Equation [8.74]:

e

∂Cp ∂C + (1 – e ) f = – — · (UCp ) + — · (D—Cp ) ∂t ∂t

[8.74]

Modelling, simulation and control of textile dyeing

353

The relationship between dye concentration at the surface of a fibre and in the liquor in contact with that point in the package at any time may generally be expressed by:

dCf = K1 C a (1 – bCf ) – K 2 Cf dt Cf =

KC a 1 + bKC a

[8.75] [8.76]

where Cp, Cf are the concentrations of dye in liquor and on the fibres at time t, respectively, U is the flow velocity vector within the package, D is the dispersion coefficient, e is the package porosity, K1 is the adsorption rate constant, K2 is the desorption rate constant, a, b are constants and K (= K1/ K2) is the adsorption coefficient.

8.5.4 Simulation results The results of a simulation of package dyeing based on a numerical modelling of dye transport in a package dyeing system are shown elsewhere (Shamey et al., 2005; Zhao et al., 2006). Figure 8.9 shows the flow velocity distribution in both tube and yarn assembly, where the flow in porous media (yarn package) is described by Darcy’s law. The shade and the length of the white arrow denotes the magnitude of the velocity and the orientation of white arrows indicate the velocity direction. It can be seen that there is a velocity direction change across the interface between the tube and yarn assembly. The velocities here are defined for x and y-directions in a two-dimensional model. However, the assumptions used in Darcy’s law simply ignore the influence of the shear stress and therefore care has to be exercised in the definition of the boundary conditions. In Fig. 8.10, the solid line and the dashed line denote the flow velocity at a flow rate of 0.1 m s–1 and 0.2 m s–1, respectively using Navier–Stokes (N-S)/Darcy’s approach. The dot–dashed line and dotted line denote the flow velocity at a flow rate of 0.1 m s–1 and 0.2 m s–1, respectively using N-S/Brinkman’s approach. This figure shows that the profile of the dyeing liquor flow within the tube for N-S/Brinkman approach fluctuates, while that when N-S/Darcy’s method is employed is stable for both flow rates. The flow velocity within the yarn assembly, however, in both flow rates, is almost the same using both N-S/Brinkman and N-S/Darcy’s approach.

8.6

Applications

The application of control strategies in dyeing machinery and within the textile industry has been reviewed elsewhere (Gilchrist, 1995b; Shamey and

354

Modelling and predicting textile behaviour max: 0.129

0.08

0.12

0.06 0.1 0.04 0.08

0.02 0

0.06

–0.02

0.04

–0.04 0.02

–0.06 –0.08 –0.1

0 –0.05

0

0.05

0.1

0.15

min: –0.00556

8.9 Distribution of flow velocity in tube and yarn bobbin, where the flow in porous media is described by Darcy’s law. 0.16 0.14 N-S/Brinkman approach for inflow rate 2, and rate 1

Velocity (m s–1)

0.12

0.1 N-S/Darcy approach for inflow rate 1 and rate 2 (solid and dashed lines)

0.08 0.06

0.04

0.02 0

0

0.01

0.02 0.03 0.04 Distance from symmetric line

0.05

0.06

8.10 Comparison of N-S/Brinkman and N-S/Darcy approaches in the characterisation of fluid flow inside a package.

Modelling, simulation and control of textile dyeing

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Nobbs, 1998a; Ferus-Comelo, 2002). Various control methodologies employed in the dyeing process both in batch as well as continuous dyeing cycles have also been highlighted and different systems have been examined. Several methodologies have been employed to implement a control strategy in the dyeing machine. Some of these methods rely on mathematical description of the physicochemical principles underlying the dyeing process while others are based on experimental or empirical approaches. Broadly speaking these methods can be classified as two distinct categories: (1) in-house proprietary or experimental methods describing the behaviour of dyes individually as well as in combination resulting in a recommendation in the form of preset, time, temperature and pH profiles and (2) the use of on-line monitoring and data processing algorithms based on models of dyeing and dye bath characteristics. Only the second category is briefly described here. A pilot scale package dyeing machine has been used by researchers to test the performance and applicability of the dye deposition error method (Gilchrist, 1998; Shamey and Nobbs 1998b; Shamey and Nobbs, 1999a,b, Shamey and Nobbs, 2000). In another study, flow injection analysis (FIA) was used as a method of standardising measuring conditions within a dye bath (Lefeber et al., 1994). The FIA approach was used for the online monitoring of indigo in a dye bath (Merritt et al., 2001). A sequential injection analsis (SIA) approach, in which a selection valve instead of an injection valve is employed, has also been advocated as a more cost effective and robust alternative to the FIA approach (Draper et al., 2001). A significant obstacle in the direct monitoring of dyebath is the wide concentration range employed throughout the dyeing process which can vary by a factor of 1000. The use of spectroscopy to obtain direct measures of dye concentration often necessitates offline sampling methods or measurement cells with variable path lengths. In addition, most methods are not suitable for direct measurement of disperse dyes as they are not completely soluble and may aggregate and scatter light and thus cause a reading error. Wilkinson (1992) used a cell in which the light simultaneously hits multiple pathlengths and employed an inverse Laplace transform to relate the absorbance of dye to the extinction coefficient. This method was successfully used in a dye bath for the online monitoring of basic dyes (Gilchrist, 1997) and reactive dyes (Shamey and Nobbs, 1998b; Shamey and Nobbs, 1999a,b). A control and monitoring system has also been implemented on a pilotscale beam dyeing machine with disperse dyes (Ferus-Comelo et al., 2005). Some of the commercial systems that have employed a control strategy include Commeureg’s Teintolab (Comeureg, 1999), Indian-based Semitronik (Anonymous, 2001), Yorkshire Chemicals (Arora and Koch, 1999) and Roaches Colortec (Smith, 1992). A promising development by HueMetrix, is based on the results of the FIA and SIA work at North Carolina State University

356

Modelling and predicting textile behaviour

(Smith, 2009). The company has developed a Dye-It-Right-Monitor™ which can monitor the exhaustion of several dyes simultaneously. In addition, the Dye-It-Right Controller™ is a system for shade control which is claimed to achieve 98+% shade accuracy without additions. Through the systematic real-time control of controllable variables such as dyestuffs, chemicals, temperature and salt, the Dye-It-Right Controller™ is claimed to compensate for variances in uncontrollable variables such as water and fibre quality.

8.7

Future trends

The need to minimize the time of dyeing while ensuring the standards are maintained necessitates the use of sophisticated control strategies. Some of the most widely accepted models used to characterize the behaviour of dye inside a dyeing vessel necessitate the monitoring and control of a number of parameters, which include temperature, pH, conductivity, liquor to goods ratio and the concentration of dye in the dyebath. A major impediment to the commercialisation of this approach has been the capital cost and the need to redesign dyeing machinery. However, with the significant reductions in the manufacturing of monitoring and control devices this approach is gaining momentum. Indeed, a number of companies utilise state of the art technology and are technologically advanced. Of interest will be machinery using automated control of conditions of dyeing and of outcome based profiles. Real time control of dyeing can result in fewer re-dyeings and attainment of a desired outcome in a shorter period. This can generally reduce shade variation, cost and have a reduced impact on the environment. It may be possible to develop systems that can respond to commands in one or less than one minute thus ensuring that faults do not occur. The shift in industry from high volume bulk productions to smaller custom made lots necessitates flexibility, accuracy and cost effectiveness. A successful strategy for the industry may be based on development of affordable monitoring and control mechanisms.

8.8

Acknowledgment

I am very grateful to Dr Xiaoming Zhao for his contribution to work and for help in generating the figures.

8.9

References

Anonymous, (2001), Int. Dyer, 186 (5) 39. Arora M. and Koch T. (2002), Rev. Progr. Color, 32, 12 (WO 9966117, 1999). Aspland J. (1992), Text. Chem. Colorist, 24, 22. Beckmann W. (1979), ‘The Dyeing of Synthetic Polymer and Acetate Fibres’, Nunn D.M. (ed.), S.D.C., Bradford, UK.

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Boulton J. and Crank J. (1952), J. Soc. Dyers, Colourists, 68, 109. Broadbent A. (2002), ‘Basic Principles of Textile Coloration’, SDC, Bradford. Burdett B. (1989), ‘The Theory of Coloration of Textiles’, Ed. Johnson A., S.D.C., Bradford, UK. Burley R., Wai P.C., McGuire G.R. (1985), ‘Appl. Math. Modelling’, 9, 33. Cegarra J. (1971), J. Soc. Dyers, Colourists, 80, 149. Cegarra J., Puente P., Valldeperas J., Pepio M., Carbonell J. (1983), IX Symposium of La Asociación Espaňola de Quimicos y Colouristas Textiles, Barcelona. Cegarra J., Puente P., Valldeperas J., Pepio M. (1988), Text Res. J., 58, 645. Cegarra J., Valldeperas J., Pepio M. (1989), J. Soc. Dyers, Colourists, 105, 349. Cegarra J., Valldeperas J., Pepio M. (1992), J. Soc. Dyers, Colourists, 108, 86. Comeureg, Teintotlab (1999), Technical Information, (www.nordet.fr/comeureg) Draper S.L., Beck K.R. and Smith B. (2001), Am. Assoc. Textile Chemist Colourists Rev, 1, (1), 24. Ferus-Comelo M. (2002), Rev. Prog. Color, 32, 1. Ferus-Comelo M., Clark M., Parker S. (2005), Color. Technolol., 121, 255. Fujita H., Kishimoto A., Matsumoto K. (1960), Trans. Faraday Soc., 56, 424. Gilchrist A. (1995a), Ph.D. Thesis, Dept. of Colour Chemistry, Leeds University, Leeds, UK. Gilchrist A. (1995b), Rev. Prog. Color, 25, 35. Gilchrist A. (1997), J. Soc. Dyers Colourists, 113, 327. Gilchrist A. (1998), J. Soc. Dyers, Colourists, 114, 327. Hoffmann F. and Mueller P.F. (1979), J. Soc. Dyers, Colourists, 95, 178. Hoffmann F. (1989), Proceedings of the International Dyeing Symposium, AATCC, Charlotte, 54. Holdstock C.R. (1988), Ph.D. Thesis, Dept. of Textile Industries, Leeds University, Leeds, UK. Hilderbrand D. and Haas F. (1983), Process for Dyeing Cellulose Materials with Reactive Dyestuffs by the Exhaustion Method, Bayer Aktiengesellschaft, US Patent 4372744, February 8, 1983. Hori T. and Zollinger H. (1986), Text Chem. Colorist, 18, 19. Hori T., Zhang H-S., Shimizu T. (1987), J. Soc. Dyers, Colourists, 103, 265. Illett S.J. (1990), Ph.D. Thesis, Dept of Colour Chemistry, Leeds University, Leeds, UK. Johnson A. (1989), ‘Theory of Coloration of Textiles’, SDC, Bradford, UK. Karst D., Yang Y., Rapp W.A. (2003), Proceeding of Industry Simulation Conference, Valencia, Spain, 457–462. Lefeber M.R., Beck K.R., Smith C.B. and McGregor R. (1994), Text. Chem. Colorist, 26 (5) 30. Medley J.A. and Holdstock C.R. (1980), J. Soc. Dyers, Colourists, 96, 286. Merritt J.T., Beck K.R., Smith C.B., Hauser P.J., Jasper W.J. (2001), AATCC Rev., 1 (4) 41. Meyer U., Rohner R.M., Zollinger H. (1984), Melliand Textilber, 65, 47. Mitsuishi M., et al., (1992), Textile Asia, 92. Nobbs J.H. (1991), J. Soc. Dyers, Colourists, 107, 430 Pepper D. and Heinrich J. (1992), ‘The Finite Element Method’, Hemisphere Publishing Corporation. Peters R.H. and Ingamells W. (1973), J. Soc. Dyers, Colourists, 89, 397. Rattee I.D. and Breuer M.M. (1974), ‘The Physical Chemistry of Dye Adsorption’, Academic Press, London and New York.

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Ren J. (1985), Ph.D. Thesis, Dept of Colour Chemistry, Leeds University, Leeds, UK. Rohner R.M. and Zollinger H. (1986), Text. Res. J., 56, 1. Rys P. (1973), Text. Res. J., 43, 24. Scharf S., Cleve E., Bach E., Schollmeyer E. and Naderwitz P. (2002), Text Res. J., 72, 783. Schuler M. (1982), Text. Chem. Colorist, 14, 13. Shamey R. and Nobbs J.H. (1997), The Use of Feed-Forward Profiles in the Control of Dyeing Machinery, AATCC International, Atlanta, USA, 279. Shamey R. and Nobbs J.H. (1998a), ‘A Review of Automation in Dyeing’, Journal of Advances in Colour Science and Technology, 2, 46. Shamey R. and Nobbs J.H. (1998b), Computer Control of Dyeing of Cotton with Reactive Dyes, AATCC International Conference, Philadelphia, USA, 251. Shamey R. and Nobbs J.H. (1999a), ‘Dyebath pH Control Under Dynamic conditions, fact or fiction?’, Text. Chem. Colorist and Am. Dyestuff Rep., 31 (3), 21. Shamey R. and Nobbs J.H. (1999b), ‘Computer Control of Batchwise Dyeing of Reactive Dyes on Cotton’, Text. Chem. Colorist, 31 (2), 21. Shamey R. and Nobbs J.H. (2000), ‘The Use of Colorimetry in the Control of Dyeing Machinery’, Text. Chem. Colorist and Am. Dyestuff Rep., 32 (2), 47. Shamey R. and Zhao X. (2004a), Numerical Simulation and Study of Fluid Flow Characteristics in Packed-bed Reactors, The 8th World Multi-Conference on Systemics, Cybernetics and Informatics (ISC), Orlando, USA., VII, 254–258. Shamey R. and Zhao X. (2004b), Numerical Study of Fluid Flow in Package Dyeing, Industrial Simulation Conference (ISC), Malaga, Spain, 453–457. Shamey R., Zhao X., Vosoughi M. (2003a), Simulation and Control of the Flow Rate and Dosing Profiles in Package Dyeing Process Based on a New Mathematical Model, International Textile Design and Engineering (INTEDEC), Edinburgh, UK. Shamey R., Zhao X., Vosoughi M. (2003b), Simulation and Control of Package Dyeing Process Based on a Mathematical Description of Dye Transfer Through the Package, Industrial Simulation Conference (ISC), Valencia, Spain, 451–457. Shamey R., Zhao X., Wardman R.H. (2005), Numerical Simulation of Dyebath and the Influence of Dispersion Factor on Dye Transport, Simulation: Mission Critical, Winter Simulation 2005 Conference, 2395–2398. Shannon B., Hendix W., Smith B., Montero G. (2000), Journal of Supercritical Fluids, 19, 87. Sivaraja S. Srinivasan D. (1984), J. Soc. Dyers, Colourists, 100, 63. Smith S.S. (1992), J. Soc. Dyers, Colourists, 108, 108. Smith B.C. (2009), Personal communication. (www.huemetrix.com) Sumner H.H. (1989), ‘The Theory of Coloration of Textiles’, Ed A Johnson, 2nd Ed., S.D.C., Bradford, UK. Telegin F. (1998), J. Soc. Dyers, Colourists, 114, 49. Telegin F., Shormanov A.V., Mel’nikov B.N. (1997), ‘Textile Chemistry – Theory, Technology and Equipment’, Ed. AP Moryganov, New York: Nova Science Publishers, 84. Valldeperas J., Cegarra J., Pepio M., Navarro J.A. (1990), Melliand Textilber, 2, 130–134. Vickerstaff T. (1950), ‘The Physical Chemistry of Dyeing’, Oliver & Boyd, London, 94. Vosoughi M. (1993), Ph.D. Thesis, Dept. of Chemical and Process Eng., Heriot-Watt University, Edinburgh, UK.

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Wai P.C. (1984), Ph.D. Thesis, Dept. of Chemical and Process Eng., Heriot-Watt University, Edinburgh, UK. Weisz P.B. (1967), Trans. Faraday Soc., 63, 1801. Weisz P.B. and Zollinger H. (1967), Trans. Faraday Soc., 63, 1815. Weisz P.B. and Zollinger H. (1968), Trans. Faraday Soc., 64, 1693. Wilkinson D. (1992), MSc Thesis, University of Leeds, UK. Zhao X. (2004), PhD Thesis, ‘Modelling of The Mass Transfer And Fluid Flow in Package Dyeing Machines’, Heriot-Watt University, UK. Zhao X., Shamey R., Wardman R.H. (2005), ‘A new approach for modelling fluid flow through a yarn package’, Research Journal of Textile and Apparel, 9 (3), 64–70. Zhao X., Wardman R.H., Shamey R. (2006), ‘Theoretical study of the influence of dispersion factor on dye transport during the dyeing process’, Color Technol, 122, 110–114.

9

Modelling colour properties for textiles

D. P. Oulton, The University of Manchester, UK

Abstract: This chapter provides an overview of how computer network systems can be used to quantify, analyse and share information about colour. A comprehensive case study of the use of virtual products in colour communication is presented and some commercially available software products in this field are identified and assessed. The relevant computer models for the creative, commercial and technical aspects of colour communication are discussed in depth and existing standardised models such as the CIE system are reviewed. Sufficient detailed guidance is given to enable the development of appropriate models for describing and visualising colour, and for calibrating the colour reproduction characteristics of computer systems. Key words: computer network, colour communication, virtual product, colour specification, colour visualisation, colour mark-up language, colorimetry.

9.1

Introduction

Colour exists in more than 1 million visually distinct shades and it subtly influences our understanding of the environment, mood and purchasing decisions. Colour is therefore studied, created and manipulated as a means of creative expression and to obtain commercial advantage in the design and production of goods or services. Artists may take a lifetime to master the subtleties of expressing their ideas in colour and the resulting creative style may then be immediately recognisable as a visually distinct and uniquely expressed message. By contrast, internationally standardised colorimetric simulation and modelling systems are used for the technical specification of colour (Berns, 2000b).

9.1.1 Key factors and principles in modelling colour Modelling is needed because colour exists only as a subjective visual sensation that is induced by light of varying wavelength. It follows that colour has no objective physically measurable existence and the only way it can be reliably specified is to create ‘psychophysical’ models of the sensation that are quantified by measuring the power of the electromagnetic light stimuli that cause it. 360

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In order to discriminate between the distinct hues of the visible spectrum at least 12 parameters are required. In practice, 30 or more are often used. A light stimulus must therefore be specified using an N‑dimensional spectral power distribution or SPD. An SPD quantifies the amount of power present in each subdivision of the spectrum and is often referred to as a ‘spectral curve’. Unlike its physical cause, the colour sensation has just three dimensions because the eye senses colour using three types of receptor cell with differing sensitivities to light at each wavelength. It follows that the receptor response can be represented numerically using only three descriptive parameters. This many-to-one relationship between the physical stimulus and the psychophysical response must be reflected in the modelling process, which in consequence has two key objectives: 1 To establish a system that uses a unique triplet of numeric ‘colour identity coordinates’ to specify each of the one million plus possible sensations of colour. Think, if you will, in terms of establishing a three-dimensional ‘colour space’ where the colours are located by their hue (blue, green red etc.), purity (neutral, dull, bright, vibrant) and lightness (light, pale, dark, full). 2 To represent the effects of additive colour mixture using a mapping by vector sum onto points in this colour space. The intent is to establish a spectrally defined model that is capable of predicting colour matches correctly, including those between physically different stimuli.

9.1.2 Conditional or metameric colour matches Visual matches between physically identical stimuli are termed ‘invariant’, but a much larger set of physically different stimuli are also ‘conditional’ or ‘metameric’ matches to any given stimulus. Such matches are only evident in a restricted range of viewing contexts, but both types of visually matching physical stimuli can in practice be identified using a vector sum model that takes the viewing context into account as in Fig. 9.1. The correctness of predictions by the vector sum mapping in Fig 9.1, is critically dependent on three factors: (1) the scaling must be precisely equated over all dimensions in which addition may occur (these are denoted in Fig. 9.1 by l), (2) The numeric scaling must be strictly constant within each dimension and (3) The ratios of proportionate value relative to the defined scaling, must be a constant property of all possible added components. These factors are all successfully realised in the CIE colorimetric model, which is described in more detail in Section 9.2. In the CIE system, the prerequisite constancy of scaling is achieved by calculating the predictive sum in terms of the units of physical stimulus cause and within the CIE

362

N fi 3 mapping by vector sum Illuminant SPD Sl

Weighted R ¢, G ¢, B ¢ colour coordinate description of effect

RGB response weightings rl l=1

SS

l

· Rl · rl fi R¢

l

· Rl · gl fi G¢

l=N

Surface colour reflectance

l=1

Rl

SS

gl

l=N l=1

SS

l

· Rl · bl fi B¢

l=N

bl

9.1 The N to 3 mapping by linear vector sum that is used to predict metameric visual matches, where N represents some arbitrary number of subdivisions of the visible spectrum.

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The cause has N defining elements or dimensions quantified by wavelength l

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model the numeric stimulus values are also normalised at the axis of visual neutrality in order to equate the psychophysical scaling interdimensionally. The scaling thus established is then reweighted over each of the long, medium and short wavelength (L, M and S) receptor-channel responses using constant proportionation ratios. Thus at a given wavelength, the response might be subdivided over the L, M and S channels in the proportions of say 0.49: 0.31: 0.20, or perhaps 0.18: 0.81: 0.01.

9.1.3 Finding the linear model Demonstrating that a descriptive vector sum makes accurate predictions clearly adds confidence to the model. It is however both possible and potentially important to quantify the statistical confidence level of those predictions and it is also possible to validate any vector sum formally by reference to its axiomatic definition (see Oulton, 2009 for more details). Historically, the use of axiomatically defined vector systems was developed in the physical domain, that is to say in the real world. The resultant vector systems are then intrinsically linear and the numeric scales, unit values and axes of spatial orientation are fundamental physical constants of Newtonian physics (Halmos, 1974). Linear vector systems can also be used to model non-linear phenomena if one or more linearisation steps are used to map the observed data values onto the descriptive spatial model. This process is described as ‘finding the linear model’ in an important paper by Roy S. Berns (Berns 1997). When establishing the linear model, it is advantageous (Oulton, 2009) to distinguish between the intradimensional scaling of the model and the essentially multidimensional vector additive effects. In Fig 9.2, the preliminary linearisation and normalisation step ‘a’ establishes a model with scales that are fully defined and uniformly equivalent within each dimension of cause and effect. The constant and linearly cross-dependent definition of additive value ‘b’ thus enabled then becomes a distinct and separately quantifiable component within the overall relationship being modelled.

9.1.4 Summary ∑ The literature on colour arises from the disciplines of vision science, colour science and practical colorimetry, and it covers many topics that are relevant to the modelling of textile colour and texture. Colour imaging and colour reproduction in such fields as computer systems, photography, television, video and the internet also have a comprehensive literature. ∑ The sensation of colour is the response of the human visual system

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Finding the linear model by three-dimensional matrix mapping of cause onto effect



(a) Two-stage mapping Redefine units A ¢ = f1(A)

The set of all possible causes by combining parameters A, B, C

B ¢ = f2(B) C ¢ = f3(C)

(b)

Define dependencies for any given effect X, Y, Z X Y  = M Z

A¢ * B¢ C¢

The set of all possible three-dimensional effects defined by X, Y, Z

where X,Y,Z and A ¢,B ¢,C ¢ now share a common unit value definition

9.2 Illustratively, a 3X3 non-linear cross-dependency can be resolved using a two-stage mapping, where the set of intradimensional scalar redefinitions, as in relationship (a), enables the linear cross dependency to be quantified by means of a constant matrix M of proportionality ratios, as in relationship (b).

∑

∑ ∑

to light stimuli, whose power content as a function of wavelength is specified by a spectral curve or SPD (spectral power distribution). Each visually distinct band of the spectrum generates a uniquely hued colour sensation. It is therefore necessary to divide the spectrum into many such bands in order to describe the colour-causing electromagnetic stimuli unambiguously. The visual response can be modelled using a three-dimensional vector sum over wavelength, because the eye discriminates colour using three distinct short, medium and long wavelength responses. In virtually all instances of colour modelling, it is advantageous to establish a linear vector sum representation of the phenomenon being modelled and it is often necessary to include a dataset linearisation step when ‘finding the linear model’ (Berns, 1997).

9.2

Types of model used

The disparate list of possible colour model types includes the standardised CIE colorimetric models, colour calibration models and models that enable colour communication and colour networking systems. It further includes the models used in photographic video and computer generated colour systems, colour appearance models and descriptive colour order systems such as colour atlases.

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9.2.1 The CIE standard observer, standard illuminant and colour coordinate models The letters CIE stand for the Commission Internationale de L’Eclairage (or in English the International Commission on Illumination) which is the main international standardisation agency for colour (CIE, 2008). The key CIE models include the 1931 and 1964 CIE standard observer models, a comprehensive set of standard illuminant definitions, and the CIELAB model for colour difference (CIE, 2008, publication 15.2004 etc.). These models are used to quantify colour identity in CIE XYZ colour space and to predict metameric colour matches by the vector sum transformation of physical SPD measurements. In the CIE system for surface colours, the physical definition of any given colour is a spectral curve that specifies the amount of light reflected at particular wavelength intervals across the visible spectrum. Fully identifiable and traceable CIE standard observer and standard illuminant data sets (see Berns, 2000c) are then used in combination with these reflectance values to weight the vector sum colour definition. Note that all three of the defining factors in Fig. 9.3 have a detailed spectral, that is to say multi element definition within the tristimulus vector sum. A given pair of surface reflectance curves may therefore be a conditional visual match that fails under some light sources, thus exhibiting ‘illuminant metamerism’. A pair of surface colours that match when viewed in daylight, can thus become a significant mismatch when viewed in artificial light of one sort or another. They may alternatively be a mismatch according to some imaging device or human observer, thereby exhibiting ‘observer metamerism’. An internationally agreed set of equations (CIE, 2008; ASTM, 2008) quantify the observed colour using the three CIE colour coordinates X, Y, Z as follows:



X =k Y =k Z =k

Ú Ú

360nm 830nm 360nm

830nm

Ú

360nm 830nm

Rl Sl xl dl Rl Sl yl dl



[9.1]

Rl Sl zl dl

where with reference to the perfect reflecting diffuser Rl is the reflectance factor (on a scale from zero, to one), Sl is the relative spectral power of a CIE standard illuminant, and xl , yl , zl is a set of CIE standard observer colour matching functions or CMFs.

366

Viewing a coloured object The eye/brain system for sensing and analysing colour

An object with a textured surface; where the incident light is diffusely reflected in all directions and some percentage of incident power Sl is absorbed at each wavelength.

9.3 The three defining factors of the object-colour sensation.

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The incident illumination which has a spectral power distribution

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The normalising factor k is given by:



k = 100

Ú

360nm 830nm

Rl Sl yl dl



[9.2]

for the perfect reflecting diffuser. In standard CIE practice (CIE, 2008), the integrals of both Equations [9.1] and [9.2] are approximated by summation over a set of N fixed-interval subdivisions of the visible spectrum:

X = k S Rl Sl xl dl l

Y = k S Rl Sl yl dl l

[9.3]



Z = k S Rl Sl zl dl l

where each element of the summation (denoted by l) is quantified as a subband of the visible spectrum l = 1 to N, and standard CMF (colour matching function) and standard illuminant tables are published for a standardised set of bandwidths and spectral range definitions (ASTM, 2008). The constant k again normalises the calculation to Y = 100 for the perfect reflecting diffuser with Rl = 1 at all wavelengths. The resulting CIE coordinates X, Y, Z specify colour identity and predict sets of metamerically equivalent visual sensations. Note however that in the spatial arrangement of point identities thus established, the scaling of visual difference is specified in terms of watts of stimulus power, not units of visible difference. Observed visual differences in colour are found to exhibit non-linear scaling relative to X, Y, Z tristimulus values and the scaling is also both wavelength and stimulus intensity dependent. Visual colour difference as distinct from colour identity is therefore represented in the standard CIE L* a* b* model using the following equations (Berns, 2000a), where the relevant X, Y, Z values are mapped by non-linear projection and axis transformation onto equivalent CIELAB values:

L* = 116 (Y/Yn)1/3 – 16



a* = 500 [(X/Xn)1/3 – (Y/Yn)1/3]



*

b = 200 [(Y/Yn)

1/3

[9.4]

1/3

– (Z/Zn) ]

where Xn, Yn and Zn are the tristimulus values for the reference white, and X/Xn, Y/Yn, Z/Zn > 0.008856. L* defines incremental steps of visual lightness, a* denotes a dimension of redness/greenness and b* denotes a dimension of yellowness/blueness.

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The perhaps more intuitive cylindrical/polar coordinates of colour, lightness, chroma and hue are then quantified as follows where chroma is a radial distance from the neutral axis. The essentially circular hue parameter loops back from blue to red via purple to complete the hue circle and hue becomes an angular measure:

* chroma Cab = (a*2 + b*2 )1/2

[9.5]



Ê *ˆ Ê *ˆ hue angle hab = tan –1Á b* ˜ = arctan Á b* ˜ Ëa ¯ Ëa ¯

[9.6]

When either X/Xn or Y/Yn or Z/Zn is less than 0.008856, the relevant cube root functions in Equation [9.4] must be replaced by a linear reweighting. A rescaling constant 7.787 that is common to all three dimensions is used and the other three rescaling constants, 116, 500 and 200 (respectively for L*, a* and b*) are retained unaltered. Thus, the value (Y/Yn)1/3 is replaced by 7.787(Y/Yn) and so on. The reverse projection from L*, a* and b* onto X, Y, Z for X/Xn, Y/Yn, Z/Zn > 0.008856 is given by: 3



* ˆ Ê L* + 16 X = Xn Á + a ˜ 500 ¯ Ë 116



ˆ Ê * Y = Yn Á L + 16 ˜ 116 ¯ Ë



* ˆ Ê L* + 16 Z = Zn Á + b ˜ 200 ¯ Ë 116

3

[9.7] 3

* and hue angle h values can also be projected back onto a*, Chroma Cab ab * b values as follows:



* a* = Cab cos (hab )

[9.8]



* b* = Cab sin (hab )

[9.9]

Under the L*, a* and b* coordinates quantified by Equations [9.4]–[9.9] the component lightness and chromatic differences are respectively designated as DL, D a* and D b*, and the Pythagorean or RMS sum distance DE, between any pair of points in colour space is then given by:

DE = ((DL*)2 + (Da*)2 + (Db*)2)1/2

[9.10]

The coordinate scaling of CIELAB colour space is used in industry (often

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with additional modelling extensions) to quantify visual match acceptability and to set production colour tolerance standards. Many more extensive accounts of the CIE system and its applications have been written; colour scientists tend to be zealous missionaries for their art. The recommended version is that given by Berns (Berns, 2000c).

9.2.2 Colour calibration models Many task-specific colour calibration models are used in chemical analysis and dye laboratory experiments. These experiments typically use the colour of clear solutions and the absorption of monochromatic light in a discipline called absorptiometry to determine coloured solute concentrations down to the parts per million level (Berns, 2000d). The light absorption property of a given solute as a function of wavelength is established by measuring fractional light transmission values for a set of known concentrations. Calibration graphs are then constructed using a well established data linearisation step that converts light transmission data into light absorbance values. The fractional light transmittance of a clear coloured glass or dissolved colorant is given by:

Tl =

It I0

[9.11]

where Tl denotes the ratio of transmitted light It to incident power I0, for some (typically monochromatic) sub-component of the visible spectrum, and Tl is a characteristically constant property of the light absorbing substance that is independent of the incident light intensity I0. Given an infinite sequence of light absorbing layers, each of which has a transmittance of say Tl = 1/2 it is readily apparent that the overall value of It relative to I0 will decrease exponentially as each successive layer absorbs half of the light that passes though it. The dependence of Tl on the light-path length and colorant concentration is thus given by:

Tl =

It = e –el l C I0

[9.12]

where l is the overall length of the light path in cm, C is the colorant concentration in Moles/litre and el is the wavelength dependent constant known as the molar coefficient of light absorption. The linear model of additive light absorbance values Al is thus given by:

Ê1ˆ Al = –log e (Tl ) = log e Á ˜ = e l l C ËT ¯

[9.13]

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where for monochromatic incident light of wavelength l, a path length l and mixtures of non-interacting colorants of concentration C1, C2 etc,

Al tot = el1 l C1 + el2 l C2 … etc.

Calculating component effects and resolving cross dependency In the above case involving coloured solutions, a true linear model of simple subtractive colour mixture may be established by the logarithmic linearisation in Equation [9.13]. The relevant constants el1,2,3 are derived from the slopes of calibration graphs of Al versus concentration C for single-colorant monochromatic light absorbance. In a three colorant mixture, a 3X3 matrix of nine coefficients is required in order to resolve the cross dependency of Al tot on the individual concentrations C1, 2, 3. A set of three calibration graphs is therefore derived for each colorant. The individual unknown concentrations of the colorants can then be calculated via optimal sets of simultaneous equations. Note that it is often assumed that the wavelengths of maximum absorption for the colorants present should be used as the analytical basis. This is, however, at best a half truth and the guiding principle must be to base the analysis on a careful inspection of the spectral curve graphs for the colorants that are present in the mixture. The objective is to differentiate their absorption properties as completely as possible. That is to say, the el values for the colorants in question should follow a ‘hi, lo, lo / lo, hi, lo / lo, lo, hi’ pattern as closely as possible at the chosen wavelengths. Calibrating digital colour reproduction systems In digital systems, colour is typically specified by a triplet of integer values (R, G and B), thereby quantifying a virtual additive mixture of red, green and blue primary-colour stimuli. The virtual mixture may then be reproduced on-screen using the visually combined output from three (R, G and B) light generating systems, which then become physical analogues of the component stimuli. The RGB triplet specification can also be reproduced by ‘subtractiveprimary’ colour mixing as a surface colour phenomenon using coloured dyes or inks. In this case and in an ideal system, each colorant subtracts exactly a third of the wavelengths from the white light being reflected by the paper or textile material. Respectively, the cyan or turquoise ink is the ‘minus red’ primary, magenta is ‘minus green’ and yellow is ‘minus blue’. Logically when all three inks are applied together at full depth, the result should be a black. Berns (Berns, 2000d) gives a more detailed and general account of the characteristics and laws of additive and subtractive colour mixing.

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On-screen colour On‑screen colour reproduction can be calibrated using the measured CIE XYZ coordinates of colour patches. The objective is to generate a reversible mapping such that on-screen colours can be specified by their CIE coordinates and colours can be reproduced accurately on demand to within a measured just visible difference relative to an input coordinate specification. It turns out (Oulton and Porat, 1992; Oulton et al., 1996) that at least for the case of a well set up CRT (cathode ray tube) screen, this objective can be met comfortably by treating the RGB/XYZ relationship as a classic 3X3 nonlinear cross dependency as indicated in Section 9.1 and Figure 9.2. Berns (Berns, 2000a) warns, however, that establishing such a scalable model of light output value, or quantifying appropriately independent models of the component primary channels may be much more difficult in some on-screen colour reproduction systems. The successful method, which involves decomposing the non-linear cross dependency into its non-linear scalar and linearly cross dependent components, was first published in the early 1990s (Oulton and Porat, 1992). The procedure was later tested more extensively as reported in (Oulton et al., 1996), where it was shown to enable an improvement in calibration accuracy by a factor of at least five. The overall non-linear cross-dependency of screen colour CIE X, Y, Z coordinates on digital gun-drive values R, G, B may be expressed as follows:

X = f1(R, G, B)



Y = f2(R, G, B)



Z = f3(R, G, B)

[9.14]

where each function f1 … f3 acts as a three-dimensionally cross-dependent component of the overall relationship that is potentially non-linear relative to R, G, B. In order to resolve relationship [9.14] into its scalar and proportionately cross-dependent components, unit value rescaling functions are first specified in each dimension R, G, B as in Fig. 9.2. Functions are specified whose products R¢, G¢, B¢ are simultaneously linearised relative to CIE Y and equated numerically. In the current three-dimensional case the relationship is constrained such that: R¢ = a1 * f4 (R), G ¢ = a2 * f5 (G ),

B¢ = a3 * f6 (B),

[9.15]

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and the equivalence R¢, G¢, B¢ ¤ X = Y = Z holds true for all R¢, G¢, B¢. In relationship [9.15], the cross-dependent elements of relationship [9.14] are held constant; f4 … f6 quantify potentially non-linear single dimension scalar transformations that map cause onto effect in each dimension and the constants a1…a3 are quantified by measuring the monitor white point. A redefinition of unit value is thus established, which maps the device dependent scaling of gun-drive R, G, B values onto the implicit trichromatic or T-unit scaling of the CIE X, Y and Z tristimulus value coordinates. This in turn enables the development of a linear matrix quantification of the cross dependency as described by W.N. Sproson (Sproson, 1983). The relevant descriptive matrix is derived from the CIE chromaticity coordinates x, y, z of the individual single gun screen colours where:

x=

X X +Y +Z

y=

Y X +Y +Z

z=

Z X +Y +Z

[9.16]

X, Y and Z denote CIE standard observer tristimulus values and the measured single gun colours and white point are represented as in the Table 9.1. The white point balance constants a1…a3 from relationship [9.15] are now quantified by simultaneous equations, such that relationship [9.17] below relates the characteristic red, green and blue gun-colour chromaticity ratios to the white point xn : yn : zn ratio.



È x1 Í Í y1 Í z Î 1

x2 y2 z2

x3 ˘ È a1 ˙ Í y3 ˙ * Í a2 z3 ˙˚ ÍÎ a3

˘ È xn /yn ˙ Í 1 ˙=Í ˙ Í zn /yn ˚ Î

˘ ˙ ˙ ˙ ˚

[9.17]

The constants a1…a3 thus established rebalance the overall relationship between the unit output values R¢, G¢, B¢ from relationship [9.15] and the CIE XYZ trichromatic or T-unit values as follows:



È a1 x1 Í Í a1 y1 Í az Î 11

a2 x2 a2 y2 a2 z2

a3 x 3 ˘ È R ¢ ˘ È X ˘ ˙ Í ˙ ˙ Í a 3 y3 ˙ * Í G ¢ ˙ = Í Y ˙ a3 z3 ˙˚ ÍÎ B¢ ˙˚ ÍÎ Z ˙˚

Table 9.1 Measured single-gun excitation and white point chromaticity coordinates denoted as they are used in relationships [9.17] and [9.18] Colour

Chromaticity coordinates

Red Green Blue White point

x1 x2 x3 xn

y1 y2 y3 yn

z1 z2 z3 zn

[9.18]

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Grey scale tracking process In principle, the definition of the unit value rescaling expressed symbolically in relationship [9.15] is both functionally distinct and potentially non-linear in each dimension R, G, B of colour generation. We therefore need to establish by practical experiment that the R¢, G¢, B¢ output values have been linearised relative to Y and that the validating constraint R¢ = G¢ = B¢ ¤ X = Y = Z holds for all R¢, G¢, B¢. Put another way, we wish to confirm that a true linear model has been generated and that the visual neutrality of the white point definition xn, yn, zn is exactly replicated at all gun-drive levels. In essence we must track a notional on-screen grey scale from black to white and establish that the R¢, G¢, B¢ values are not only equated at the white point as in relationship [9.17], they are also precisely equated at an additional representative set of points on the grey scale. The grey scale tracking process is thus in effect both a comprehensive evaluation of relationship [9.15] and a direct implementation of its validating constraint. The point is that when the grey scale tracking is exact at all levels, the scaling of R¢, G¢ and B¢ then becomes a true constant as required by the axioms of multi-dimensional addition in a vector space. Under a constant scaling that satisfies these axioms, the matrix of proportionality ratios in relationship [9.18] then also becomes a verifiably distinct and independently quantifiable constant within the overall equation system symbolised by relationship [9.14]. In effect, the proportionate and scalar components of relationship [9.14] are now fully resolved both analytically and experimentally. They can therefore be represented by independently quantifiable sub-component models derived from independently measured data sets. In the grey scale tracking process, on-screen visual neutrality is calibrated against the property of visual balance that is modelled by the CIE coordinate equivalence X = Y = Z. In the relevant practical experiment, a look-up table of R, G, B gun-drive values must be recorded and each entry must quantify a match to the grey scale. Both Bessel and Spline functions have been used successfully to model the functions f4 … f6. These three mapping functions must be simultaneously optimized by adjusting their curve fitting parameters relative to the R¢, G¢, B¢ values calculated from the look-up table and the optimisation proceeds by error minimisation until all three curves pass simultaneously through each point on the grey scale calibration axis. Note that on screen, this axis has a chromaticity xn, yn, zn that is dependent on the current white point setting; the R¢, G¢, B¢ values must therefore be appropriately balanced via relationships [9.15], [9.17] and [9.18]. The calculation via the constants a1…a3 will work for all local white point settings, but it is better practice to preset the screen chromaticity to that of a standard illuminant white such as D65 or D50.

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Primary and secondary reference bases for colour calibration The standard CIE XYZ coordinate model is referenced by axis transposition to three defining primary wavelengths Rl, Gl, Bl and scaled in terms of units of stimulus power. A proportional reweighting of unit value in which watts of stimulus power are transformed into T-units of visually equivalent effect precedes the RGB to XYZ axis transposition. The stimulus values are still scaled by reference to watts, but they are reweighted by normalisation according to their visual effect value and balanced stimuli are now represented by an axis of numeric equality R¢, G¢, B¢. The axis transposition from R¢, G¢, B¢ coordinates into X, Y, Z coordinates is then calculated in T-units and the visual equivalence model R¢ = G¢ = B¢ is preserved and projected onto the CIE X, Y, Z coordinate system such that R¢ = G¢ = B¢ ¤ X = Y = Z. The CIE model is thus validated as an exact numeric reference basis for both three-dimensional grey scale tracking and on-screen colour calibration. Thus defined, the CIE system is also validated as a predictive system of affine colour-space transformations under constant unit value that preserve proportionality and equivalence relationships. It is suggested, however (Oulton, 2009), that a primary reference basis with wider general validity should also be established. This primary reference model should be strictly context free, that is to say it should have a structure and properties that have an independent mathematical definition and are therefore free from any artefacts that might be introduced by the assumptions and quantifying datasets of the CIE system. The reference basis for the scalar value model (symbolised by relationship [9.15]) thus becomes the real number scale R+ rather than the CIE Y scale and the axiomatic constancy of additive value in a true multi-dimensional vector space is used as a comparative basis for modelling visual additive value. To avoid scalar ambiguity, relationship [9.15] must reflect the full case-specific dimensionality of cause in each colour-related phenomenon under analysis and each dimension of (typically spectral) cause must also be represented by a distinct and specific scaling equation. As a result of its potentially infinite dimensionality and fully adjustable scaling in each dimension, the general reference model can be extended into sufficient dimensions to quantify any conceivable variation in both the multi-dimensional additivity property and the intradimensional scalar property of a given quantifying data set. In practice, of course, the potential complexity of scalar and additive value modelling thus specified may not be required. The default descriptive model should therefore (as in the 1931 CIE standard observer XYZ coordinate model) have uniformly linear and interdimensionally equivalent scaling in all dimensions relative to the standard units of physical measurement. The descriptive system should only deviate from this default model if statistically significant non-linear relationships or

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equally significant super-or sub-additive proportionate ratio relationships are found in the relevant observational data sets. Applications Using the mathematically defined reference basis thus established, it is possible to re-examine the properties of the CIE system itself. The significance of chromaticity ratios in the CIE system and their status as characteristic constants of the colour response might thus be investigated. It is possible to use neutral axis matches to investigate and validate the CIE colorimetric transform definition of visual neutrality at all response levels and it is also possible to investigate any potential super-or sub-additive deviations from the standard CIE visual response model. In the latter case the neutral axis visual matches will be with spectrally specified alternative primary–triplet stimuli. It is important to remember that when investigating the human visual response or relating some colour-related phenomenon to it, only the proportionate stimulus value ratios can be measured directly in colour matching experiments. The true scaling of the visual response can only be quantified using separate magnitude estimation experiments that relate stimulus value to some scalar aspect of the visual response such as just perceptible difference. The sometimes complex colour rendering properties of colour cameras and colour reproduction systems can also be investigated and compared with the spectral response of the human visual system. This is because (as in the CRT calibration case described above) all colour capture and colour reproduction devices have physically quantifiable photometric characteristics. It is also possible to address analytical problems such as those concerned with colour appearance in context, illuminant or observer metameric conditional equivalences and the more complex modelling problems in colour capture and reproduction described by Berns (Berns, 2000d).

9.2.3 Colour profile characterisation Another colour quantifying discipline that is often used in imaging and colour reproduction is called ‘device profiling’. A device profile is an essentially fixed characterisation that is typical of and specific to a particular model and make of colour scanner, camera, computer screen or colour printer. Premeasured profiles are widely available and their main function is to enable transformations between sets of device specific colour coordinates. They also enable the resolution of inter-device compatibility problems where the chosen colour may lie outside the gamut of reproducible colours for a given device. ‘ICC (International Color Consortium) device profiling’ is a

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widely accepted and used implementation of this concept that is gradually developing into a comprehensive and accurate model (see ICC, 2008 and http://www.color.org). A typical user reaction to the device profiling approach is given at http://www.drycreekphoto.com/Learn/profiles.htm. (Dry Creek Photo, 2008). Full current-state colour calibration functions that systematically update the basic characterisation can clearly be added to the device profiling system. Calibrations of this type allow for such factors as local device adjustments or ageing or something as simple as the effects produced by a change of ink or paper supplier.

9.2.4 Colour appearance modelling Colour models are also under active research that seek to explain the subtleties of ‘colour in context’. The research objective is typically to simulate the changes in colour appearance that occur when the viewing context is changed (Fairchild, 2005). The change may be in the light source used to illuminate the scene, the effect of adjacent colours and the wider ambient context, or perhaps the changes that are related to surface texture differences. The best known model is called CIECAM 97, the most recent one to be adopted is CIECAM 2002 and both of these models are discussed in detail by Fairchild (Fairchild, 2005).

9.3

Case study in colour communication

In recent decades the opportunities for gaining advantage by modelling colour have increased significantly in parallel with the rapidly expanding use of media such as colour TV and video, on‑screen and internet computer colour and the wide availability of cheap colour printing. Take, for example, the case of the abundant and remarkably cheap clothing that we expect to be supplied in large volumes by the retailers of mass market fashion. Typically, consumers will expect the store to stock new colours that change frequently and reflect the seasons. They will look for and buy the colour that currently most appeals to them and they will be led in their choices at least in part by the ‘in colours’ of fashion predictions. They are likely to buy multiple colour coordinated accessories to these outfits if they are available. They exhibit very subtle colour preferences and it has been shown in a test marketing exercise (Marks & Spencer, 1991) that in a given garment style and quality, one specific version of the colour described as navy blue consistently outsold all of the tested alternative versions by a factor of ten to one. They will also complain about and reject retail outlets and products that have poor colour ranges or include visibly mismatching garment components.

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Models that enable optimised colour communication are thus typically required, when the need is: ∑ ∑ ∑

∑

to be in close touch with current colour fashions in the target market segment; to select, develop and deliver seasonal colour ranges for the chosen market at least four and perhaps eight or more times per year, with the shortest possible lead time; to specify the desired colour range precisely and manage its implementation in large volume production. Taking a lingerie-set as an example, this might involve coordinating the colour of 20 or more distinct components (both textile and non-textile), in successive stages of production, at multiple remotely sited production units; to establish quality control systems that are capable of monitoring colour fidelity and continuity from batch to batch in production and from the original design concept to the retail sales counter.

The simple answer is, of course, to give up any attempt to meet such requirements, other than by simply optimising the choice of suppliers and to sell what is on offer at their wholesale warehouses. However consider Fig. 9.4, where all the relevant stakeholders and participants are assumed to be in direct communication by electronic networking. Figure 9.4 illustrates the key dialogue channels and the diverse colour modelling requirements diagrammatically. The resultant creative, colorimetric and technical colour communication links can be provided by internet portals which are password and subscription protected. The first key advantage of a communication network is that communicating by electronic links is easy and instantaneous. This basic capability is delivered by the hardware of the computer network, a text description model called HTML (the ‘hypertext mark up language’) and the data mark up language XML (the ‘extended mark up language’). In combination these agreed protocol models enable universal inter-computer communication using text and images. Before useful colour communication can take place, however, appropriate colour visualisation, specification, calibration and match prediction modelling systems must also be present. It is additionally necessary to enable and standardise the physical measurement and spectral definition of all the colour stimuli involved and to standardise the calculations whereby these models predict colour matches and colour differences (see also ASTM, 2008).

9.3.1 Specification of colour identities, matches and tolerances Colorimetry is conventionally standardised by reference to the CIE system, but some users of the colour communication network will neither need to

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9.4 Some key work flow sequences and colour communication links that are present during the development of new coloured textile products are highlighted and typical data management and communication tasks are identified at each stage. © Chromashare Ltd.

know the relevant colorimetric technicalities, nor would they understand them. They may alternatively identify a given component of the colour range by a colour name, ID number, or even perhaps by a production order number. It follows that the precise colorimetric technicalities and calculations should be implemented in a hidden layer of the model, using callable subroutines whose output is only displayed on demand.

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9.3.2 Quantifying and visualising the creative input to colour range development The concerns of and methods used by creative colourists are quite distinct from those of the technical colourist, dyer, colour systems managers and the quality control team. The work of the creative colourist is essentially visual and it uses input from fashion predictions, colour libraries, sales monitoring and both physical and virtual product prototyping in order to assemble and define new colour combinations and product colour ranges. These essentially visual processes can also be aided significantly by colour modelling. Two closely interrelated and interacting colour models are needed. One of these models must implement and enable the essentially artistic work flow of the creative team and provide the users with comprehensible and appropriate tools for creating, manipulating and visualising colours. The other component must be comprehensible to the technical colourist and generate the required unambiguous spectral curve and CIE coordinate definitions for all the colours being discussed and visualised. The term comprehensible is deliberately used twice here. This is because the relevant colour models must collaborate electronically in order to bridge the often significant communication gap between the creative developers of the colour range and the technical managers and producers of coloured products. In effect, what is needed is a computer aided design or ‘CAD for colour’ system. The objectives of such creative systems include: 1. Providing the user with extensive colour-calibrated models including colour libraries and virtual product simulations, which can be shared electronically by members of the development team. 2. Allowing the users to visualise explore and develop sympathetic colour combinations, such as print colour ways and seasonal colour ranges as a whole rather than by piecemeal colour changes. 3. The ability to specify all of the visualised colours technically (using spectral curves and CIE colour coordinates), so that they can be reproduced accurately as products. The pay-back for developing and using such colour networking models is typically a dramatic reduction in both the cost and the lead times necessary for developing new colour ranges (see Oulton et al., 1996). A significantly enhanced flexibility of response to fashion changes and consumer preference is also enabled. A key factor in all such colour imaging and product visualisation systems is the use of calibrated colour imaging, colour reproduction devices and device profiling (see Section 9.4). The need for precise electronic colour visualisation and specification is clear when you understand the potential that exists for finding a subtle variant of say navy blue, raspberry or lavender, which could outsell all the other variants by ten to one.

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9.3.3 Mark up and information transmission languages for describing colour A mark up language embeds a set of computer instructions or tags in the content of a given image or text file. The name of each tag is specific and it tells the receiving computer how to treat the following section of file content. In the case of a document file, an example would be a tag that specifies the chosen font, or font size, or page layout. There are three key problems when defining a mark up language model. First, it must be universally understandable to those who wish to use it. The intent is typically to establish a standard that is accessible to all and the language must therefore have a fully standardised and published ‘schema’ (such as that for XML) that defines the semantic meaning, syntax and vocabulary of the language. Second, the language must be both comprehensive and accurately descriptive, because computers are completely unintelligent and cannot ‘talk their way around’ difficulties of semantics, syntax or vocabulary as humans might do. Third, the language must have complex data omission and error handing features, with robust default values. This allows the message-receiving interpreter to act competently on fragmentary or incomplete data. An example of an openly accessible colour oriented mark up language is the CxF (or ‘colour exchange format’) language pioneered by Gretag-Macbeth (CxF, 2002). This is a good example of a formatting language whose structure and content (which are written as an extension to the standard XML schema) are freely accessible to potential users. CxF is, however, only one attempt to solve the problems of colour communication and a number of alternative approaches are either under current research or technical discussion (see Section 9.4 for further guidance on learning about and adapting mark up languages for use in colour modelling).

9.3.4 Quantifying colorant formulations and predicting optimum colour mixtures Once the colour specifications are agreed, the task of selecting an optimum colorant recipe for a given textile product is typically delegated to the dyer or technical colourist, who will usually be supported by a proprietary computer colour-match prediction system. The relevant software will be installed with a database of spectral curves for a large range of potentially useful dyes and it will be capable of evaluating and minimising the problems associated with metameric matches. This colour recipe prediction system will use a complex non-linear search in N spectral dimensions that depends critically on the ‘Kubelka Munk’ model (see Berns, 2000a). This relates the amount of light reflected by a coloured object to the concentration of the colorants

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present. The recipe prediction model will probably also use an iteratively convergent recipe refinement algorithm originally published by Allen (see Berns, 2000a). Readers who seek to create such models are recommended to refer in the first instance to Billmeyer And Saltzman’s Principles of Color Technology (Berns, 2000a), where a wide ranging annotated bibliography of source material for building such models is made available. Colour matching to physical or numerically specified samples In traditional textile dyeing, the colour match is usually made to a physical sample of the target colour. Ideally this sample would be a piece of the same fabric to be dyed, and it should have a measurable spectral curve that can also be used as a basis for minimising metameric match problems. However, in a major industrial trial in the early 1990s, a change from physical samples to standardised numeric colour definitions was shown to contribute significantly to minimising ‘colour standard drift’, to improving product colour continuity and to improving the fidelity of the final product to the original design inspiration. The technical colourist will, in practice, also have concerns such as cost minimisation and dye fading resistance. Recent proprietary computer software systems for predicting dye recipes often now include predictions of washing and light fading properties and they also support recipe cost optimisation. On-screen computer visualisation for the predicted but as yet undyed alternative dye recipes is also gaining in popularity. In such systems (see Fig. 9.4) calibrated colour appearance models are established as ‘virtual dyeings’ and these can at least partially replace physically dyed ‘lab dip’ samples during colour range development (Oulton et al., 1996). Colour libraries Owing to the cost of colour range development, it makes good sense to seek maximum benefit from each successfully developed colour standard and maintaining a detailed inventory of previously successful colour definitions is a rewarding solution. When such a database is combined electronically with an appropriately colour oriented and preferably visual indexing system, it has been shown that as often as nine times out of ten it is possible immediately to find an existing recipe that is an acceptable match.

9.4

Future trends in colour modelling

All aspects of colour modelling are expected to continue to develop significantly and some advances will probably reflect developments in numeric analysis techniques. Axiomatically verifiable vector space models that include one or

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more data linearisation steps are also likely to be developed, particularly in order to explain and characterise the complex scalar properties of phenomena such as colour difference. Further progress in colour communication-related models is expected and new models may be developed, for example, by adding technical colour description and colorimetry extensions to hypertext languages such as HTML or XML. An expected outcome of these developments will be significant advances in imaging and colour reproduction, particularly of complex surface textures such as those of textile materials. The demand for the enabling colour models is expected to be significant and it will be driven by the available commercial advantages anticipated in Sections 9.2 and 9.3.

9.4.1 Trends in available tools and methods Colour modelling problems, particularly in vision science and colour appearance modelling, often concern phenomena in which the number of interacting variables is uncertain, or their patterns of interaction are currently unknown. It follows that the methods and tools of multivariate and numeric analysis are key elements in colour modelling. It also follows that the multidimensional nature of the sensation will be amenable to geometric description and vector space modelling. Those involved in colour modelling will typically need to be mathematically adept and may need to use affine and Riemannian geometry as well as the perhaps more familiar Euclidean geometry of Newtonian physics in order to model ‘colour space’ (Wyszecki and Stiles, 1982). The best approach in many such cases is to treat the phenomenon in question as a ‘black box’ operation that transforms inputs into outputs. The properties of such ‘black box systems’ may be analysed by altering their inputs systematically while closely studying their outputs. Four examples typify this approach and may be useful in colour modelling. They are neural network modelling (Golden, 1996), clustering analysis and simulated annealing (Arabie et al., 1996), principle component analysis or PCA (Jollife, 2002) and wavelet analysis (Torrence and Compo, 1998). Principle component analysis, also sometimes known as the hotelling transform, or orthogonal decomposition is a technique that allows the apparent dimensionality of a descriptive data set to be reduced. The relevant exploratory analysis involves the calculation of the eigenvalue decomposition of a data covariance matrix or the singular value decomposition of a data matrix (Jollife, 2002). This technique is a key tool in some recent colourrelated journal articles and it is expected to receive significant attention in future colour modelling projects. Simulated annealing is a useful approach when seeking global minima in the multidimensional data spaces typified by spectrally defined data and

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metameric equivalence. Clustering analysis is helpful when analysing the apparently noisy data sets that are generated in visual experiments. Wavelet analysis is a relatively recent development in the wider field of Fourier analysis and it allows sequentially ordered data sets with time and / or spatially dependent characteristics to be analysed. It is particularly useful when the functional relationships under analysis have discontinuities or sharp peaks. Combinations of rapidly decaying short duration wavelets each of which has a distinct wavelength and initial amplitude are used to model and predict the relevant properties (Torrence and Compo, 1998). Wavelet analysis has been applied successfully to photographic images, particularly of textile structures and a significant number of journal articles have reported interesting results using this approach. Modelling the colour sense The problem with modelling the human colour sense is threefold. First, the phenomenon is caused by an N dimensional set of spectral power inputs that all appear to cause non-linear responses. Second, the visual response appears to be a multilayer phenomenon with both parallel and sequentially acting component responses and third, the visually distinct colour sensations cannot necessarily be treated as individual responses because they often appear to exist only as complex colour appearance interdependencies. The term affine geometry refers to a multidimensional descriptive system of proportionate or ratio values with a strictly constant but not necessarily fully defined scalar metric. The term is rarely if ever used in the literature in connection with the CIE colorimetric model. However from the outset the pioneers of colorimetry appear to have intended their model to consist of a set of demonstrably valid affine rescaling and axis transformations based on ratio value datasets. The undoubted predictive success of the resultant CIE colorimetric transform appears to validate affine geometry and affine vector space transformation as potentially useful approaches to colour modelling. The key to the potential success of this approach is that the axioms of affine transformation establish a fundamental distinction between on one hand the proportionation ratios of multidimensional vector addition and on the other hand the unit value scaling that establishes the measure of spatial separation or colour difference within each dimension of colour space. Multidimensional affine constructs and sets of single dimension non-linear projections of scalar value are thus recommended as potentially useful colour modelling tools and as a basis for future extensions of the existing CIE colorimetric models. Colour measurement The general principles and practical aspects of laboratory and industrial ‘colour measurement’ are well described by Berns (Berns, 2000a). The

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term ‘colour measurement’ is placed in quotes to remind the reader that only the relevant physical stimuli can actually be measured and the visual response definition is the product of some colorimetric model, usually the CIE colorimetric transform. For those who wish to implement the CIE colorimetric model themselves, a book by Westland and Ripamonti (Westland and Ripamonti, 2004) is available. This is a particularly useful source of fully annotated and tested calculation subroutines which are recommended as a basis for implementing the CIE models. Each subroutine is denoted in MATLAB, which is a mathematical modelling language that is essentially an extension of C++. The publications by the CIE (CIE, 2008) and the ASTM (ASTM, 2008) are also obligatory source material references when implementing fully annotated and traceable CIE models.

9.4.2 Trends in colour description, definition and colour communication methods Those interested in extending mark up languages such as XML as a basis for colour modelling might perhaps start by studying a not necessarily definitive colour-related example, see (CxF, 2002). The general trends in mark up language development are best understood by visiting the authoritative and comprehensive website of the W3 consortium (W3schools, 2008). HTML concerns data presentation and visual formatting, whereas XML is a more flexible general purpose description language for describing data structure and data content (see also Section 9.3.3). The current situation regarding extensions of XML for describing colour is somewhat unclear, but an ultimate winner will no doubt emerge and it is expected to provide a solution that is both commercially and scientifically important.

9.5

Commercial vendors and their products

Konica Minolta concentrate on providing high quality instrumentation, cameras and colour reproduction hardware. The UK contact address is: Konica Minolta Sensing Europe, UK Branch, Office Suite 8, 500 Avebury Boulevard, Milton Keynes, Buckinghamshire, MK9 2BE, UK Tel: +44 (0)1908 540622. Website: http://www.konicaminolta.eu X-Rite Inc first entered the market for colour-related instrumentation and products only in 1990, but they have now become a major international vendor. The UK the address is: X-Rite Ltd, Acumen Centre, First Avenue, Poynton, Cheshire, SK12 1FJ, UK. Tel: +44 (0) 1625 871100. Datacolor Inc are well known and established as suppliers of colourrelated instrumentation and software solutions for the global textile and apparel industries. They are also pioneers of colour calibrated on-screen

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visualisation for textile products. Their contact details are: Datacolor Ltd, 6 St Georges Court, Dairyhouse Lane, Broadheath, Altrincham, Cheshire, WA14 5UA, UK. E-mail: [email protected]. Website: http://www. datacolor.com. The Sony Corporation, who are well known as Japanese global vendors of electronics and media products, are notable in the colour modelling field for their high quality Trinitron colour screen products. Their contact details are: Sony Broadcast & Professional, Jays Close, Viables, Basingstoke, Hampshire, RG22 4SB, UK. Tel: +44 (0)1256 355011. Website: http://www. sony.co.uk/ eWarna (www.ewarna.com) is a pioneering Malaysia-based company that sells software-only colour solutions. Their products are aimed specifically at the users of colour on the internet. eWarna have entered into a partnership with an industry-specific business-to-business website operator, Hangzhou Hi2000 Infotech Co Ltd in China to market its services to enterprises in that country. Hangzhou Infotech can be found on http://b2b.chemnet.com. Chromashare Ltd www.chromashare.com is also a software solutions vendor. They offer products that enable precise colour communication over the internet and are represented in the USA by ‘Precision textile color’ http:// www.precisiontex.com/. The offered overall solution is notable for its use of the ‘colour server concept’ and the use of a comprehensive proprietary colour description language to enable simple and accurate colour communication. Chromashare’s desktop application CS SmartClient delivers its functionality through ‘web services’ and collaborating users thus establish global availability at all times for their colours and their product specifications. Colour calibrated web browsing is also supported. The postal address and contact details for Chromashare Ltd are: Chromashare Ltd, 18 Lincoln Place, Hulme St, Manchester, M1 5GL, UK. Tel: +44 (0)161 237 3046.

9.6

References

Arabie, P, Hubert LJ and De Soete, G (eds) (1996). Clustering and Classification, World Scientific Press, New Jersey, USA. American Society for Testing and Materials (now known as ASTM International) (2008). (www.astm.org) 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA, 19428–2959, USA. The ASTM works to improve standardization in colorimetry and publishes widely accepted standard procedures and methods for measuring the spectral reflectance of coloured surfaces. It also certifies the required data tables and methods, for use when calculating colorimetric values such as CIE colour coordinates. Berns RS (1997). ‘A generic approach to color modeling’, Color Res & Applications, 22, 318–325. Berns RS (2000a). Billmeyer and Saltzman’s Principles of Color Technology, 3rd edition,

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Roy S. Berns (ed.), John Wiley & Sons, New York. This is an authoritative reference source for deeper study. Berns RS (2000b). Billmeyer and Saltzman’s Principles of Color Technology, 3rd edition, Roy S. Berns (ed.), John Wiley & Sons, New York. ‘Standardized illuminants’, Chapter 1, Section B, pp 3–10, ‘Standard observer systems’, Chapter 2, Section C, pp 50–54; ‘Standardized visual inspection’, Chapter 3, Section C, pp 78–82. Berns RS (2000c). ‘The CIE colorimetric system’, Billmeyer and Saltzman’s Principles of Color Technology, 3rd edition, Roy S. Berns (ed.), John Wiley & Sons, New York, Chapter 2, Section C, pp 44–63. Berns RS (2000d). ‘Absorptiometry: Subtractive colour mixing in clear solids and liquids’, Billmeyer and Saltzman’s Principles of Color Technology, 3rd edition, Roy S. Berns (ed.), John Wiley & Sons, New York, Chapter 6, Section B, pp 151–8. CIE (2008). The CIE Central Bureau, Kegelgasse 27, A–1030, Vienna, Austria, is the primary standardizing body for colour. The CIE recommends the precise way in which the basic principles of colour measurement should be applied, and CIE Publication 15:2004 Colorimetry is the latest edition of one such set of recommendations. CIE publication 15:2004 contains information on standard illuminants, standard colorimetric observers, the reference standard for reflectance, illuminating and viewing conditions for visual colour matching, the calculation of tristimulus values, chromaticity coordinates, colour spaces and colour differences and various other colorimetric practices and formulae. The CIE explicitly warrants that publication 15:2004 is consistent with the fundamental data and procedures described in the CIE Standards on colorimetry. A full list of CIE publications is available at http://cie.kee.hu/newcie/framepublications. html CxF (2002). CxF white paper 1.0 June 2002. Published by X-Rite Gretag-Macbeth. Available at http://www.color-source.net/en/Docs_Formation/CxF_Public_WhitepaperV1-01.pdf Dry Creek Photo (2008). For an informative and interesting ‘users view’ of ICC colour profiling, visit the website at http://www.drycreekphoto.com/Learn/profiles.htm Fairchild MD (2005). Colour Appearance Models, 2nd edition, Wiley – IS&T Series in Imaging Science and Technology, Wiley Chichester, UK. Golden RM (1996). Mathematical Methods for Neural Network Analysis and Design, MIT Press, Cambridge, MA, USA. Halmos PR (1974). Finite-Dimensional Vector Spaces, Springer Verlag, New York. ICC (2008). The International Color Consortium, 1899 Preston White Drive, Reston VA, 20191 USA. Website http://www.color.org/index.xalter Jollife IT (2002). Principle Component Analysis, 2nd edition, Springer Science and Business Media, New York, USA. Marks & Spencer (1991). M&S confidential customer-preference research. Private communication, C. Sargeant, 1991. Oulton DP and Porat I (1992). ‘The control of colour by using measurement and feedback’, J Textile Inst, 83(3), 454–461. Oulton DP, Boston J and Walsby R (1996). Building a Precision Colour Imaging System. Proceedings of the IS&T/SID 4th Color Imaging Conference, Scottsdale Arizona, Nov 1996. The Society for Imaging Science and Technology, Springfield, Virginia USA, 14–19. Oulton DP (2009). ‘Notes toward a verifiable vector algebraic basis for colorimetric modeling’, Color Res Applic, 34(2), 163–9. Sproson WN (1983). Colour Science in Television and Display Systems, Adam Hilger, Bristol, England, Chapter 2, 27–29.

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Torrence C and Compo GP (1998). ‘A practical guide to wavelet analysis’, Bull Am Meteorology Soc, 79(1), 61–78. W3schools (2008). This site (http://www.w3schools.com) is dedicated to explaining and defining the many distinct languages, protocols and definitions that make the internet work and all of the relevant concepts are covered by detailed tutorials. Concepts covered other than XML and HTML, include DTD (the pro-forma for ‘document type description’) and XSL (the ‘extendible style-sheet language’) which defines the relevant semantic styling templates. Westland S and Ripamonti C (2004). Computational Colour Science Using MATLAB, John Wiley and Sons, New York. Wyszecki G and WS Stiles (1982). Color Science, Concepts and Methods, 2nd edition, John Wiley and Sons, New York. This was also republished in 2000 without further significant editing in a Wiley classics library edition. This book is regarded by many in the field to be the primary reference source of colour science. It is a fundamental repository and exposition of the historical development of colour science and of the available concepts and methods of colour modelling up to the publication date. However it inevitably lacks any input on more recent topics such as colour appearance modelling and imaging science. It is also a somewhat ‘heavy read’ for non-specialists.

10

3D modelling, simulation and visualisation techniques for drape textiles and garments

F. Han and G. K. Stylios, Heriot-Watt University, UK

Abstract: The 3D modelling of fabrics and their realistic simulation into garments is an exciting field for academe and industry. An important indication of the effectiveness and success of a real-time realistic simulation model is given by a measure of the computational complexity and accuracy that such algorithmic implementations will require. In this chapter we present an overview of such techniques and their respective advantages and disadvantages and we provide a detailed description and analysis of ‘a mass– spring system’, which is commercially important owing to its simplicity and computational efficiency. Our focus is directed upon a ‘velocity and force modification algorithm’ based on a mixed implicit–explicit integration scheme, which improves both accuracy and realism without incurring additional computational complexity. In this chapter we also explore the wide-ranging benefits of a new concept of textile measurement, the FAMOUS equipment, specifically relating to its usage in textiles, clothing and retailing. Finally, in recognition of the importance of commercial application, we explore the use of this technology in virtual wearer trials and its relation to global retailing. Key words: drape simulation, drape performance, textile measurement, virtual wearer trial.

10.1

Introduction

Predicting the motion or static drape of textiles is of interest in garment design, textile engineering and computer graphics. This problem has already received considerable attention from mechanical and textile engineers, as well as computer scientists. Major advances have been made regarding the understanding of the physical behaviour of textiles and on the derivation of appropriate mathematical models and realistic simulation methods. Realistic 3D real-time garment simulation should accurately model the behaviour of a cloth during collision with any object, whilst maintaining the efficiency of the animation. If enhanced to satisfy both of these requirements, 3D cloth simulations will no doubt find uses in many aspects of daily life. For example, in fashion design and manufacturing, a useful technique would be to develop virtual try-on applications that allow consumers to see how a garment would fit/look on their individual bodies without the need physically to try on any garment. 388

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Unfortunately, the high computational complexity of the simulation makes existing systems that satisfy both requirements very difficult to implement. Meanwhile, to make matters worse, we are facing enormous and still increasing computational demands, especially for high quality cloth simulations that are based on high resolution models. Although current achievements in this area are already effective and satisfactory from the point of view of those in the movie industry (for example), to the aesthetic eye of the fashion designer there are still plenty of technical challenges waiting to be resolved, such as creating realistic simulations, achieving faster runtimes, or developing methods of constructing and simulating garments with more complex structure or texture. These ambitions have been and will continue to provide considerable motivation for extending the capability of existing cloth modelling techniques. This chapter comprises eight sections. The introductory section sets out potential future applications of cloth simulation, presents the motivation for research and briefly summarises and integrates the content of chapters that follow. After a brief review of the history of cloth simulation, we move on to provide an overview of the available approaches, including geometric methods, finite element models and a method that is one of the most likely to achieve real-time cloth performance, the mass–spring system, also known as a particle system. A new concept of fabric measurement called FAMOUS will be introduced, which can be used to measure tensile, bending, shear and compression of fabric samples simultaneously. The focus of later sections will be on the description of available techniques and their uses in textiles, clothing and retailing, as well as a discussion of a virtual-drape measure-based finite element method. A detailed description of the framework of cloth simulation will be provided, involving generic description of a mass–spring system, collision detection and reaction. Once the strengths and limitations of existing simulations have been introduced, improvements on existing models will be presented. Experiments to evaluate and tune the performance of cloth simulations are conducted and results and consequent conclusions are presented in Section 10.7. The most obvious application of our cloth-simulation system is the simulation of clothes worn by a moving avatar. Finally, through the reconstruction of a virtual human as a geometrical shape and further refinement techniques of feature cloning and animation, the fabric model is integrated into a realistic virtual wearer trial. The resultant examples serve to highlight many possibilities, not only for virtual retailing of textiles, but also potentially as tools for use in the entertainment industry.

10.2

Review of 3D textile models

Since the first geometric fabric model was introduced in 1986 by Weil (Weil, 1986), significant research into the subject of fabric simulation has been

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conducted. Initial work on the simulation of garments concentrated only on the geometrical features of the deforming cloth (Weil, 1986), as opposed to physically based models that were first proposed in the early 1990s (Carignan et al., 1992; Volino et al., 1995). Although the original approach is not really suitable for simulating cloth motion and its applications tend to be limited to specific areas, it has the advantage of being able to generate a static cloth shape quickly. On the other hand this approach is capable of generating consecutive data and can be applied to almost any cloth simulation problem. Although the generality of the second method comes at the cost of increased computational complexity, therefore requiring a considerable amount of computational time, physically based models are more realistic and easier to implement than geometrical models. Different physically based approaches have been proposed, some exploit particle systems for mechanical simulation (Breen et al., 1992; Eberhardt et al., 1996), while others employ continuous models resolved by finite elements (Eishen et al., 1996). Recent developments can be grouped into two major categories, with the focus respectively on realism and computational efficiency. The first, adopting finite element or finite difference models, aims to achieve realism, unconditionally, where the requirement for accuracy is more important than the desire for fast simulation (Bridson, 2003) and is consequently more suited to use in the textile industry. Models in the other category only seek to produce believable animations, thus sacrificing accuracy for computational efficiency. However, with advances in computer technology and ever-increasing processor speeds, even models originally designed to achieve efficiency of real-time computation are becoming increasingly realistic.

10.2.1 Geometric methods As mentioned previously, geometric methods are designed to handle only one specific situation. Weil’s approach, for example, could only simulate a hanging curtain (Weil, 1986), while in a similar way Agui et al. (1990) presented a system that modelled the sleeves on a bending arm. These and other geometric systems, unlike the physical models that superseded them, do not take into account the physical properties of the cloth but merely try to imitate the geometric appearance of the cloth under predefined conditions. Ng and Grimsdale (1996) have presented a comprehensive overview of these models.

10.2.2 Finite element models Underlying finite element models (FEMs) is the interpretation of the body (a continuum) as a set of discrete elements. The objective of such models is to find approximations to functions which satisfy the deformation equilibrium

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equations that describe the interaction between individual elements, while enforcing continuity of these functions between neighbouring elements. The finite element procedure, which is commonly used to analyse structural problems in solid continua, has several advantages over the interacting particle approach. For example, the continuum formulation is largely parameterisation independent. This independence is important for practical reasons, because cloth simulators should be able to handle arbitrarily shaped, unstructured meshes representing cloth patterns. The finite element approach is served by a large volume of literature in circulation, detailing the use of this framework in handling the non-linearity of fabrics. Various geometries have been employed to describe the elements, including plates, shells and beams. Beyond the classical application of FEMs in mechanical, civil and electrical engineering, they have also been applied to cloth simulation (Gibson and Mirtich, 1997). Collier et al. (1991) present a non-linear finite element approach using plate elements. Ascough et al. (1987) employ beam elements to improve processing time at the cost of loss of accuracy. Tan et al. (1999) introduced geometric constraints to a thin plate element model, making the assumption that lengths of threads in a fabric remain unchanged after deformation. Donald and David (2000) use a geometrically exact resultant shell theory, which includes a non-linear stress–strain relationship based on real fabric measurements. FEM models are still actively pursued, because the level of accuracy provided by these thinshell or plate-based methods cannot be matched by other models. Although the finite element procedure is technically sound and is often a versatile tool, a number of issues must be addressed if this technique is to succeed. (1) There is no straightforward or efficient way of alleviating the ‘buckling instability problem’. (2) The finite element procedure needs many more numerical operations than the particle formulation. (3) Collision resolution becomes a non-trivial task.

10.2.3 Particle systems Although finite elements have shown the greatest accuracy at high computational costs, particle systems became the preferred approach in the computer graphics community for their simplicity, flexibility and their fidelity/performance ratio. Mass–spring techniques were proposed for cloth simulation by Breen et al. (1994). Since then, others in the computer graphics community have followed suit (Provot, 1995; Desbrun et al., 1999). In a mass–spring system, the mass particles are connected by three types of springs, characterised by structure, shear and bend. Given the initial conditions, the motion of the model is calculated by applying standard Newtonian laws of motion. The main issues involved in this technique are numerical stability and accuracy. Volino and Magnenat-Thalmann (2001) compared the efficiency of

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a number of time integration methods. Eberhardt et al. (2000) and Parks and Forsyth (2002) sought more accuracy in order to obtain detailed simulations. Despite all these approaches, be they explicit or implicit, the ultimate aim of cloth simulation has yet to be achieved, namely, an accurate, generic, stable technique that can be computed in O(n) time (where n is the number of nodes, or particles, in the mesh). In addition, distance constraints, which eliminate unacceptable elongations of the cloth, are being developed in this method. Further benefits of mass–spring techniques include their ease of implementation and their ability to exhibit folding and wrinkling behaviour without the need for as fine a mesh resolution as those adopted for finite element techniques.

10.3

Automatic measurement of fabric mechanics

Academic literature regarding virtual 3D draping of fabrics is just beginning to examine fabric mechanical properties in relation to drape simulation. Textile materials are engineering structures that behave neither as solids nor as liquids, that is they are said to be ‘limp’ and they possess viscoelastic properties which enable them to take up any 3D configuration by wrapping/ hanging around solid bodies. An additional interest is that textile structures are diverse and consist of different raw materials and designs. Thus a large number of materials need to be defined as engineering structures, for reasons of design and manufacture, quality and performance, but most importantly, in our case, for realistic garment drape during modelling and animation. Measuring tensile and bending, shear and compression mechanical properties of fabric, is an important requirement to satisfy this need. After extensive scientific and industrial usage over the last 40 years, the general understanding about the current provision of equipment is that the KESF is regarded as a scientific device for research and FAST is a simplified alternative for industrial use. Both instruments offer mechanical measurements but both have shortcomings which hinder their usage, especially in modelling and animation (Kawabata, 1982; Biglia et al., 1994). Recently, under demands by academe and industry, a new concept was established which was developed into a novel device for automatically and precisely measuring textile fabrics, leather, paper and thin films, called FAMOUS (Stylios, 2000; Peirce, 1930), which stands for Fabric Automatic Measurement and Optimization Universal System. In this equipment, all tests are made using a single sample automatically and without the need of any manual handling. Therefore, structural and physical properties of the material defining the behaviour of the fabric, for example tensile, shear, bending, compression and surface roughness and friction, can now be measured quickly, precisely and automatically. Results from these measurements are then used to identify a set of suitable control parameters that completely describe both the static and dynamic material properties of the sample cloth in question.

3D modelling, simulation and visualisation techniques

393

The first aim of this instrument is to provide a five-in-one apparatus capable of conducting the various measurements using only a single fabric sample, thus reducing space and handling. The second aim is to reduce measurement time and complexity of data interpretation while increasing accuracy and reproducibility, in comparison with existing methods, keeping the measured magnitudes compatible with existing equipment, at the same time. Figure 10.1(a) is a photograph of the FAMOUS, ‘five-in-one’, portable bench-top equipment. A fabric sample is cut to 20 ¥ 20 cm and placed on the machine. All scale magnitudes and measured parameters are compatible with those of the existing equipment. All tests for each fabric are completed in 5 minutes and the measured data is interpreted into a fabric data chart automatically. A complete automatically produced fabric data chart is shown in Fig. 10.1(b). Figure 10.2 shows various typical mechanical property curves of tensile, shear hysteresis, flexural rigidity, thickness and compression, and surface roughness and friction of a wool worsted cloth produced by the equipment. These curves represent accurate measurements for the particular mode and behaviour of deformation that the fabric sample was subjected to, so that the prediction of the behaviour of the material in a 3D garment can be established.

10.4

Drape measurement and evaluation

The drape model used in the virtual measurement system (VMS) is based on a physical analogue of a deep shell system (Stylios et al., 1996). The fabric is treated as a continuous shell system initially and then is discretised by lumping its distributive mass and its mechanical properties into a large number of deformable node elements according to the particular mesh layout employed, where the size of elements can be either uniform or distinct. In order to apply this model to any flexible material, a local surface coordinate system is necessary. The coordinate system we shall adopt is defined by first selecting two orthogonal vectors tangential at a particular point on the surface as the first and second coordinates a1, a2 and then erecting a vector normal to the surface as the third coordinate a3. The deformation of a fabric element can be described by parameters u, v for displacements within the tangent plane (i.e. along a1, a2) and w for displacements along a3. The material properties of the continuum in all elements can be lumped at these deformable nodes by integrating all the energies within those elements. A suitable conversion mechanism between the local and the global systems must be established and differential equations governing fabric deformation are then derived from the discretisation of the system energies over all fabric material elements. The final global drape governing equations takes the general form:

  M x + C x + K x = F

[10.1]

394

Modelling and predicting textile behaviour

(a)

Control zone

Tensile Tensile Tensile Tensile

linearity energy resilience extension

Acceptable zone

Control zone

Ratio of warp & weft extension

Shear rigidity Shear hysteresis @ 0.5° Shear hysteresis @ 5.0° Bending rigidity Bending hysteresis Coefficient of friction Mean deviation of friction Geometrical roughness Compressional linearity Compressional energy Compressional resilience Thickness @ 0.5 g cm–2 Thickness @ 50.0 g cm–2 Compression Drape grade (b)

10.1 (a) The FAMOUS equipment which produces (b) an automatically produced fabric data chart.

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395

 , C and K are: the mass matrix, the damping matrix and the where M stiffness matrix, respectively. F is the distributed external force vector applied at each node and x is the displacement of the node, x is its velocity and  x is its accelerations. Since a large number of nodes are used for the geometrical configuration of the loop structure, it is convenient to express the equations or matrices in an implicit form. As time space is split into a series of finite time intervals t, if we know the solution at time tn, we can find the solution at time tn+1 = tn + t by using a single step algorithm called a Newmark algorithm (Zienkiewicz and Taylor, 1991). This approach has been found to be realistic, effective and accurate, and has been adopted for the prediction of 3D fabric drape deformation.

(a)

(b)

(c)

(d)

(e)

10.2 Curves describing various typical mechanical properties. (a) Fabric flexural rigidity, (b) fabric shear, (c) fabric tensile strength, (d) surface friction and roughness and (e) fabric compression.

396

Modelling and predicting textile behaviour

Here, an algorithm is presented for carrying out a number of virtual drape measurements, which were found to define fabric aesthetics more accurately and realistically. The system is the virtual sister of our true drape measurement system (M3 system) (Stylios and Zhu, 1997), with which the comparisons and the verification between real and virtual drape measurement have been made. We have tried to define aesthetic attributes using the natural psychology of consumers as well as the know-how of engineering principles relating to fabric drape (Stylios and Zhu, 1997). It has been found that the drape coefficient, although an important property for the assessment of fabric drape, does not provide an accurate or complete characterisation of drape, since two fabrics with the same drape coefficient can possess differing drape behaviour. Consequently a number of aesthetic attributes were added to define the drape behaviour, such as number of folds, variation of folds and the depth of folds. These four virtual measurements have been used to define the drapeability of a given textile material as follows. The classic technique (Cusick, 1968) for assessing drape utilises circular fabric samples draped over small cylindrical pedestals. When positioned in line with a direct light source, the draped fabric casts a shadow substantially smaller than a circle C(R0) with radius of that of the original fabric. The initial fabric configuration and the draped configuration for circular samples are shown in Fig. 10.3(a) and (b), respectively. The drape coefficient is determined by comparing the annular ring formed between C(R0) and the outline of the pedestal supporting the drape, according to the formula:

DC =

Ashadow p R02 – p r 2

[10.2]

Let the approximated boundary curve be R(a), the fold variation is determined by:

Var =

1 S (R (ai ) – 1 S R (ak )) n – 1 i =1 n k =1

[10.3]

Let the maximum and minimum values of R(a) be Rmax, and Rmin, respectively. Determine the fold depth index of a draped virtual fabric by:

De =

Rmax – Rmin R0 – r

[10.4]

The number of drape folds can be identified directly after detecting and approximating the boundary curve of the outer edge of the shaded region, cf, Fig 10.3. In order to evaluate the drape behaviour of a fabric, an experiment was carried out in which two materials were selected (one stiff and one soft). The measured mechanical properties of these materials are shown in Table 10.1.

3D modelling, simulation and visualisation techniques R (a)

397

A

R

a

r Centre

r

Centre

(a)

(b)

10.3 Vertical projections from tested fabric, (a) initial and (b) draped configurations.

Table 10.1 Fabric properties

Tensile energy (gf · cm/cm2)

Shear stiffness (gf/cm)

Bending rigidity (g cm2 cm–1)

Weight (g m–2)

Sample A Sample B

21.31 16.16

0.33 0.88

0.0218 0.1605

186.48 238.92

Table 10.2 Real drape measurements

Drape C

Fold number

Category

Sample A Sample B

0.279 0.54

6 4

Soft Stiff

Drape C is the drape coefficient.

Using the M3 system actual fabric samples were tested and their results are shown in Table 10.2. The six photographs in Fig. 10.4 provide a comparison between the real drape and the corresponding simulated drape for each of the two types of materials used. The left side of the figure shows: (a) soft material, top view, (c) soft material, side view and (e) stiff material, side view, respectively. The mechanical properties of these samples were then fed into our VMS to produce virtual measurements from the 3D drape simulations as shown in Fig. 10.4 (b), (d) and (f). Results produced from virtual 3D drape measurements are summarised in Table 10.3, which show a good agreement between actual and virtual drape measurements of both test fabric samples. The results indicate that virtual measurements are able to provide detailed information concerning aesthetic attributes and aid in higher accuracy of drape evaluation.

398

Modelling and predicting textile behaviour



(a)

(b)



(c)

(d)



(e)

(f)

10.4 Actual drape and drape simulations of two fabric samples. Table 10.3 Virtual drape measurements

Drape C

Fold number

Fold depth

Fold variation Category

Sample A Sample B

0.28 0.542

6 4

0.743 0.675

0.226 0.21

10.5

Soft Stiff

Key principles of 3D mass–spring models

10.5.1 Generic description of a mass–spring system As we mentioned earlier, one of the simplest physically based models and thus one of the most likely to achieve real-time performance, is the mass–spring system (Provot, 1995; Gavin, 1988; Chadwick et al., 1989). A deformable body in such a model is approximated by a set of masses linked by dampened springs in a fixed topology. For an illustration consider Fig. 10.5, where particle masses are represented as a rectangular grid of n ¥ n nodes (Chittaro and Corvaglia, 2003). We understand a ‘particle’ to be an entity that has mass, position, velocity and responds to both internal and external forces; the arcs between nodes represent spring elements, of which

3D modelling, simulation and visualisation techniques

399

(b)

(a)

(c)

(d)

10.5 Mass–spring topology, (a) global structure, (b) vertical and horizontal springs, (c) shearing springs and (d) bending springs (Chittaro and Corvaglia, 2003).

there are three types, but have no spatial extent. Springs which connect each mass element to its nearest horizontal and vertical neighbours are structural stretch springs. As the name suggests, these springs resist in-plane (along the surface) stretching of the material. The springs diagonal to the rectangular grid are shear springs; besides resisting in-plane shearing these inhibit stretching. The final type are longer springs that connect each mass element to those one away (represented by arcs), these resist out-of-plane bending (e.g. wrinkling, folding and waving). With regard to flexibility, the programmer is free to define several parameters such as particle mass, elastic constants for traction (in weft and warp directions), bending (in weft and warp directions), shearing, constants for damping, friction and bouncing. Thus by varying parameter settings, it is now possible to simulate many different kinds of cloth, including high performance materials. To realise mathematically the above described mass–spring system, we begin by using a series of calculations to determine the internal motions of the particles. Let pij(t), vij(t), aij(t), (where i = 1,… , m and j = 1,… , n,) be correspondingly, the position, velocity and acceleration of the masses at time t. The system is governed by Newton’s second law:

fij = maij

[10.5]

where m is the mass of each point and fij (the resultant of both internal and external forces) is the sum of forces applied at pij. Internal forces are principally due to tension in the springs. A spring, which

400

Modelling and predicting textile behaviour

deforms according to Hooke’s law, is described by the tension between its end points. The internal force experienced at pij results from the deformation of all springs linking this point to its neighbours:  ˘ È  pkl pij 0 Í fint (pij ) = – S kijkl pkl pij – lijkl  ˙ [10.6] kj Í pkl pij ˙ ˚ Î 0 where kijkl is the stiffness and lijkl the natural length, of the spring linking pij and pkl. On the other hand, depending on what type of simulation we wish to model, external forces such as gravity and wind or air friction, can be introduced into the model. The most common additions are:

∑ ∑

gravity: fgr(pij) = mg, where g is the gravity acceleration; viscous damping: f vd (p ij ) = – C vd v ij , where C vd is a damping coefficient.

The progression of the mass–spring system in time is determined by tracking the paths of individual particles making up the cloth as they move. The result is obtained by integrating numerically over the sequence of successive positions for each particle over any specific time interval. While many explicit and implicit integration schemes exist (Witkin and Baraff, 2001), the Euler methods are among the most widely used for cloth animations because of their simplicity and low computational costs. To implement such a system, the following explicit Euler integration scheme can be used:

aij (t + h ) = 1 fij (t ) m

[10.7]



vij(t + h) = vij(t) + h* aij(t + h)

[10.8]



pij(t + h) = pij(t) + h* vij(t + h)

[10.9]

where h is the chosen time step, v(t), v(t + h) (and p(t), p(t + h)) are the initial and final particle velocities (positions) at time t and at time t + h, respectively. Empirical solutions for explicit methods have been proposed by Provot (1995) and Vassilev (2000) and the main philosophy is the limiting of spring extension. Among the explicit integrators, most important are the Euler, midpoint and Runge–Kutta methods, with truncation error terms of O(h2), O(h3) and O(h5), respectively (Chadwick et al., 1989). Explicit integration methods compute the state at the next time step, through direct extrapolation from previous states and accordingly even higher order explicit methods require very small time steps in order to guarantee system stability and accuracy (Press et al., 1993). In contrast to explicit integrators, implicit Euler replaces the forces at time t with the forces at time t + h:

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401



aij (t + h ) = 1 fij (t + h ) m

[10.10]



vij(t + h) = vij(t) + h* aij(t + h)

[10.11]



pij(t + h) = pij(t) + h* vij(t + h)

[10.12]

where h denotes the time step. This simple substitution enforces stability in a distinctive way: the new positions are not blindly reached, but they correspond to a state where the force field is coherent with the displacement found, and using an approximation of the force at the next time step introduces a sort of feedback to the integration process (Baraff and Witkin, 1998; Kawabata, 1982). In this case, whatever the size of the time step, the output state of the system will have consistent forces that will not give rise to instabilities. Despite being able to use larger time steps without loss of stability or efficiency, these approaches are more complex to implement because large linear systems need to be solved at every integration step.

10.5.2 Collision detection and reaction Collision detection is the most time-consuming process in cloth simulation. Not only must collisions between distinct objects and surfaces be considered, but the delicate nature and hence ease of deformation of fabric material means that it is also necessary to consider self-collisions between different parts of the same piece of fabric. The dynamics of the associated model are considered in terms of a structure of mass points and connections that have specific physical properties. We assume the two colliding surfaces are initially separate. We take a smooth triangulation of each surface. Collisions between the surfaces are then detected by testing for penetration of vertices of these triangles through the plane of any other triangle, that is, of either material. To determine whether a particular vertex point P4 went through a particular triangle T say, with vertices initially at P1, P2, P3, the positions of the vertex are compared at the beginning and end of each time step. Suppose P4 is initially at A, and following the direction of its velocity, V4 reaches B at time t, then a collision has occurred during this time step if there is a t for which B is coplanar with T, or equivalently with the three vertices now at p1¢ , p2¢ , p3¢ , (following the directions V1, V2 and V3) which define T  (Moore and Wilhelms, 1988; Bridson et al., 2002). Defining pij = p j – pi  and vij = V j – Vi , the root(s) of the cubic equation give the time that the four points will be coplanar:       (p21 + tv21 ) ¥ (p31 + tv31 ) ∑ (p41 + tv41 ) = 0 [10.13]

402

Modelling and predicting textile behaviour

If the left hand side is evaluated to within a small tolerance, such as 10–6 m, we register a collision. To minimise the computational costs of executing the above, a preliminary step is used to check if the point P4 is closer than  some perpendicular distance h from the triangle P1P2P3 with surface normal n. We first project the point onto the plane and compute the barycentric coordinates w1, w2, w3 (w1 + w2 + w3 = 1) with respect to the triangle:



p13 p13

p13 p23

w1

p13 p23

p23 p23

w2

=

p13 p43 p23 p43

[10.14]

These are the usual equations for the least-squares problem of finding the point w1P1 + w2P2 + w3P3 in the plane closest to P4. If the barycentric coordinates are all within an interval [–d, 1 + d] where d is h divided by a characteristic length of the triangle, then the point is close. The overlap is:     [10.15] d = h – (p4 – w1 p1 – w2 p2 – w3 p3 ) ∑ nˆ If the sign of the perpendicular distance has not changed, intersection is assumed not to have occurred. If the sign has changed, then other (more expensive) tests outlined above must be done, but in practice this test is enough to eliminate most point-triangle pairs. After a collision is detected, the response of the system as a whole must be computed. In order to test a response for a particular collision, further penetration in the proximity should not be taken into account. We propose a method of avoiding such unwanted collisions which involves constructing a very thin repelling proximity field to surround the relevant region of the impact surface, similar to the methods described by in Baraff and Witkin (1998), Breen et al. (1994) and Volino et al. (1995). The region of influence of the force field is divided into small contiguous non-overlapping cells that completely surround the surface, where the displaced volume is replaced by a cell built from the points Pi of the triangle and the normals Ni at these points, as shown in Fig. 10.6. As soon as a test point enters into the cell, a feedback force is applied. The direction and the magnitude of this force depend on: N1 N3

P1

P3 N2 P2

10.6 Cell built using a triangle P1P2P3 and the normals Ni at points Pi.

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403

(a) the velocity of vertex point relative to the normal velocity of the impact surface and (b) the material composition of the colliding objects.

10.6

Clothing simulation: strengths, limitations and suggested improvements

Mass–spring models have often been adopted because of their simplicity, efficiency and ability to simulate the physical behaviour of fabrics while using coarser mesh resolutions than those used in finite element techniques. Unfortunately, all of these approaches suffer from the same problem, that to ensure stability, the time step must be inversely proportional to the square root of the stiffness, and thus is of little use in real-time applications, where very small time steps are required to ensure stability. Although in implicit integration, large time steps may be taken to reduce the amount of numerical computation while guaranteeing system stability, one drawback is the need to solve a large linear system at each time step. In order to alleviate this shortcoming, we combine Runge–Kutta explicit methods with backward Euler implicit schemes. While springs in most cloth models obey unlimited linear deformation, the mixed explicit/implicit integration scheme can be implemented by dropping the original assumption of linear spring deformation (especially with regard to stretch structure springs) in favour of nonlinear descriptions, that is, if the spring strain exceeds, for instance, a pre-defined threshold. Specifically, the velocity from the last implicit update will be used as input to the rigid body collision algorithm and constraints in the velocity update will be used to hinder motion in the direction normal to the rigid body for points experiencing a collision. Denoting position by x, velocity by v and acceleration by a, the resulting algorithm moving step n to n + 1 is as follows:

f (t n , p n , v n ) v n +1/2 = v n + Dt (explicit integration) m 2

[10.16]

Modify vn+1/2 to get v n + 1/2 in order to limit strain, etc;

p n +1 = p n + Dtv n + 1/2



f (t n +1, p n +1, v n +1 ) v n +1 = v n +1/2 + Dt (implicit integration) [10.18] 2 m

[10.17]

Modify vn+1 in order to limit strain. One interesting behaviour of fabrics is the non-linearity of its response to tension and compression, in other words, its resistance to stretching is fairly weak in the initial phase of extension where threads in the cloth are merely flattening out, but the resistance increases significantly as the threads

404

Modelling and predicting textile behaviour

themselves are extended, as verified by measurements carried out using the FAMOUS equipment. As most current cloth simulations generate motions more characteristic of rubbery material than of cloth, an obvious but nontrivial solution is therefore to adopt a non-linear model that better captures the behaviour of real fabrics. This, in effect, also models the biphasic nature of cloth; small deformations are resisted weakly until a threshold ec is reached, whereupon stiffness dramatically increases. Inspired by such ideas we present a kind of modified structural spring (MSS) which uses a non-linear function M to describe the stress–strain relationship. Suppose an arbitrary mass point pi is connected to one of its neighbours pj by a spring with natural length lij. Let pij = pj − pi, we introduce a function K(pi, pj) to modulate a spring stiffness k0: a



Ê (|pij | –lij )2 ˆ K (pi , p j ) = Á 1 + k0 lij2 e c2 ˜¯ Ë

[10.19]

and the tension generated by an MSS is:

fMSS (pi , p j ) = – K (pi , p j )

(| pij | – lij ) pij | pij |

[10.20]

where a is the ‘non-linearity parameter’ controlling modulation of the spring constant, ec is a predefined strain threshold of cloth biphasic behaviour and k0 is the basic spring stiffness. By assigning different values to a, the function K(pi, pj) can model both linear and nonlinear stress–strain relationships. Figure 10.7 illustrates (a) a superimposed graph of K(pi, pj) as a non-linear function of the spring strain and (b) the force generated by a MSS for different values of a. It can be seen that the resultant force increases dramatically as the extension strain exceeds the threshold. The stress–strain relationship of the real fabric can be approximated by tuning the parameters of a according to the current strain stage relative to a series of strain stages, with divisions made according to a predefined strain threshold. Of course, physical bodies are not perfectly elastic; energy is dissipated during deformation as a result of non-zero relative velocity between neighbouring mass points on of the spring. To account for this, viscoelastic springs are used to dampen out relative motion. Thus, each spring exerts a viscous force. It is thus more preferable to use:



Ê vijT pij ˆ fi = kd Á T ˜ pij Ë pij pij ¯

[10.21]

where kd is the spring damping constant and vij = vj – vi,. This projects the velocity difference onto the vector separating the

3D modelling, simulation and visualisation techniques

405

30 a=1 a=2 a=3

K spring constant

25

20

15

10

5

0

0

0.05

0.1

Strain (a)

0.15

0.2

0.25

0.15

0.2

0.25

6 a=1 a=2 a=3

Structure spring force

5

4

3

2

1

0 0

0.05

0.1

Strain (b)

10.7 (a) Stress–strain relationship, (b) Magnitude of the force generated by MSS for a = 0, 1, 2, and k0 = 1.0, lij = 1.0 under the condition ec = 0.1.

masses and only allows a force along the direction of the line connecting p1 and p2. Velocity regulation can also be satisfied using a relaxation iterative procedure. The velocities vi and vj of the mass point of the spring are

406

Modelling and predicting textile behaviour

resolved into the two components of vit and v tj parallel to, and vin and v nj  perpendicular to, the vector pij . At each time step and for each of the mesh edges, since the components causing the spring to stretch are vit and v tj , if v tj is different from vit , then the corrected velocities will be: 2

vit* = vit –



v t*j = v tj +

t

t

m j v tj 2 mi vit

+

2 m j v tj

mi vit

2 mi vit

+

vijt



[10.22]

2

2 m j v tj

vijt



[10.23]

t

where v ij = v j − v i and mi, mj are the particle masses of the ends i, j of the spring under consideration. The use of an explicit update for position allows us to modify the velocities in order to enforce a wide variety of constraints and, further, enables a strain limiting procedure, where springs are limited to a maximum of 10% extension (beyond their natural length) (Baraff and Witkin, 1998; Caramana et al., 1998). In order to cope with the superelasticity, while maintaining stability and accuracy, a new algorithm is presented. We first check the length of each spring, after each iteration, and if the extension exceeds this threshold, we modify the forces at both ends of the spring using a non-linear stress–strain relationship with a different value of a. Further, the velocities of these ends can now be adjusted either directly in response to these changes to end-forces; or alternatively by the method described previously (Equations [10.22] and [10.23]). Thus a constraint on the motion of the mass point can be exerted and unlimited extension prevented. The threshold usually takes a value from 1–10% depending on the type of fabric we wish to simulate. If this modification algorithm is applied to all springs, the stretching components of the velocities can be decreased, thus preventing further stretching. After several iterations of this procedure, springs that are stretched by more than this threshold are relaxed (shortened); this in turn stretches neighbouring springs, which are in turn relaxed, and so on. The iterative process stops when all springs on the edges converge to their respective natural lengths. The end result is a fast and stable way of animating arbitrarily connected, mass–spring systems in complex, dynamic environments. It should be observed that this approach is suitable for global as well as local deformations.

10.7

Experimental results and discussions

In order to demonstrate the capabilities of our animation algorithm, we present examples of simple cloth models with complex folding processes, where rectangular pieces of cloth are draped over simple objects – a sphere

3D modelling, simulation and visualisation techniques

407

and a table. Consider the piece of cloth draped over the moving sphere shown in Fig. 10.8. When the sphere moves up and away, the fabric flips back over itself resulting in a large number of contacts and collisions. Our particular model ensured that the highly complex structures of folds and wrinkles in the fabric are stable. Incorporating textures makes the simulation more realistic. Parametric studies are essential for validating numerical simulations, investigating the necessity of providing accurate experimental data for material properties and comparing different materials and their individual drape behaviours. We have constructed an interface to allow fast parameter data input for cloth description. As shown in Fig. 10.9, our interface is composed of a multi-tab window for data editing and context dialogs (on the left) and the graphics window (on the right). The appearance and texture of simulated fabric drapes can be altered by manipulating various simulation parameters, including mass, damper constant, time step and parameters characterising fabric stretching and bending properties. The fabric shapes were calculated under the same conditions except for a difference in some parameters, assuming that the fabric was hanging from a pair of fixed points in space. Of the constants that determine ‘realism’, the three most important in the simulation are the structure spring constant ks, shear spring constant kb and bend spring constant kc. The constant ks controls the stretching ability of the system. The smaller the ks is, the greater the number of particles in the mass–spring system that

10.8 Draping the cloth (texture mapping applied) over a moving sphere.

408

Modelling and predicting textile behaviour

10.9 Fabric model interface for simulating fabric drapes with varied parameters.



(a)

(b)

(c)

(d)

(e)

10.10 Effect of varying fabric stretchiness constant kc for (a) kc = 3.0, (b) kc = 5, (c) kc = 15, (d) kc = 20, (e) kc = 30 with ks and kb fixed at ks = 3.0, kb = 3.0.

can be pulled apart and any value beyond 1.5 will not produce aesthetically pleasing animations. A range of values of ks (varying from 3 to 30) were used and simulation results displayed in Fig. 10.10, with a fabric hanging from three of its fixed corners. We see that as ks increases the fabric becomes much less likely to be stretched by its own weight. The constants kb and ks control how prone the system of particles is to bending/buckling. The smaller kb and ks are the less resistance there is to buckling as shown in Fig. 10.11 and Fig. 10.12. As we decrease either constant, the fabric shows wrinkle severity. Moreover, as highly anisotropic materials, fabrics are weak in resisting bending, but show relatively high resistance to stretching. As an example of how such effects are taken into account when determining simulation

3D modelling, simulation and visualisation techniques



(a)

(b)

(c)

(d)

409

(e)

10.11 Effect of varying fabric rigidity constant kb for (a) kb = 0.3, (b) kb = 1, (c) kb = 2, (d) kb = 4, (e) kb = 8, with kc and ks fixed at kc = 20, ks = 6.



(a)

(b)

(c)

(d)

(e)

10.12 Effect of varying fabric bend kb for (a) Kb = 0.3, (b) kb = 1, (c) kb = 2, (d) kb = 4, (e) kb = 8 with kc and ks fixed at kc = 20, ks = 4.

parameter settings, two different pieces of cloth are draped over a sphere, as shown in Fig. 10.13. When comparing, the darker material is more rigid than the lighter one. As mentioned previously, excessive spring extension in a mass–spring model always produces unrealistic fabric behaviour. A simple way to minimise this is to decrease elasticity settings, but then the mass–spring topology may become stiff and lead to instability. We have already described a post-step modification algorithm of velocity employed to eliminate this excessive stretching: each time a spring is overstretched, the biphasic spring model is introduced to enable regulation of the velocities of spring-ends, eliminating undue deformation, all within the framework of a mixed explicit–implicit scheme. Figure 10.14 compares the draping of a rectangular piece of fabric over a sphere using the original elastic model with no restriction to stretching (left), with the draping of the same fabric after application of the velocity

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10.13 Drape behaviour of two different fabric types.



(a)

(b)

10.14 (a) Original elastic model, (b) model with velocity modification.

modification described above, with a 10% elongation threshold (right). The fabric produced from the original model stretches over its own weight; this is never the case with real fabric where implausible stretching does not appear, as in our improved model, where we find that the fabric postures are more accurate and realistic.

10.8

Applications and examples

In recent years through the expansion of the internet, we have witnessed a revolution in the networking of information on a global scale. The philosophy of the 21st century is based upon global retailing without frontiers. This ambition is to be realised through research that will provide the required techno-infrastructures, which will force the restructuring of industry, as well as provide new opportunities in consumer buying methods, as illustrated in Fig. 10.15. This allows customers to design and to purchase garments tailored to their specific requirements, through what are known as ‘virtual

3D modelling, simulation and visualisation techniques PIN number Barcode Password

Personal detail databank Shape measurement Bank account Structure/colour/texture

Television/computer/ telecom

Fabric development

411

Global retailer

Garment virtual wear trial Garment development

Home Concepts Textile genetic Automated retailing Global retailers engineering Home buying (genetic fingerprinting) Global manufacturing Satellite companies Intelligent Intelligent Artificial clothes textile garment Synthetic humans manufacture manufacture Virtual environments Intelligent looms Industry Intelligent sewing machines Objective measurement technologies Textile genetic engineering Textile aesthetics Despatch (72 hours?) Tailored clothes

10.15 Flowchart of the next generation of textile and garment manufacture and retailing.

wearer trials’. Customers will be able to examine how well garments fit on the bodies of their ‘virtual counterparts’, thus removing the inconvenience of physically trying on every piece of clothing. Perhaps the age when we can have a semi-tailored garment dispatched to us within two or three days using intelligent sewing environments (Stylios and Sotomi, 1994; Stylios et al., 1994, 1995) and when virtual wearer trials become as ubiquitous as internet shopping, is not that far away. A virtual human model is the basic requirement for modelling the movement of a garment as it is being worn by a virtual human, since it can act as a ‘physical’ constraint to garment draping, as well as personalising the virtual human with characteristic features in virtual wearer trials (Stylios et  al., 2000). In previous research, data captured directly from measurements and photographs of a real female model were used to construct a virtual human from a structure comprising over 50,000 polygons (Stylios and Wan 2000). Our approach has produced a set of integrated, tested and evaluated techniques and algorithms which include geometric reconstruction, digital cloning of human features, animation of virtual human bodies, cloth simulation and automatic measurement of fabric mechanics, and brings these together to form an integrated online virtual human system, as shown in Fig. 10.16. This novel system that we have created emphasises simplicity, low-cost and ease-of-use and is suitable for anyone without the need for specialist

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Modelling and predicting textile behaviour Virtual human reconstruction, digital cloning and animation

Human image capture and analysis

Synthetic cloth simulation

Dressed augmented human model

Textile material objective measurement

Interactive visualisation database network system

10.16 Online 3D virtual human system.

high cost equipment. The potential applications of such a system in many industrial end uses, including garment research and e-commerce, are widely recognised.

10.8.1 Modelling and visualisation of synthetic humans The inspiration of the reconstruction algorithm of the facial features of a virtual human arose from the identification of the human silhouette as being topologically similar to elliptical cylindrical surfaces and is thus symmetrical about its axis of rotation. The initial mesh that will approximate sufficiently closely the desired shape of the human object was constructed from a limited number of feature points measured from normal photographs. Trigonometry and bicubic spline incorporation functions (Weisstein, 2008) were used at each cross-section of the initial mesh to supply the details of the hierarchical multi-resolution mesh. The surface patches based on these 3D curves is represented by:

Bi,j(t, s) = Bi(t) Bj(s)

[10.24]

where, Pi,j is an array of control points and Bi,j (t, s) is a basis function evaluated by: n



Q (t , s ) = S

i=0

m

S Bi ,j (t , s ) Pi , j j=0



[10.25]

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where, Bi (t) and Bj (s) are defined as bicubic spline or hermite (Submissive, 1998) and trigonometric functions in the reconstruction. Figure 10.17 illustrates an example of the head reconstruction algorithm. The forehead is slightly less curved than the back of the head. Since a large transition in curvature of the geometric surfaces that describe the face always occur near the outer extremity lines at both eyes, we designed an algorithm specifically for the purpose of calculating vertex coordinates on cross-sections of the frontal half-head. Let headfront [k][m], headside [k][m] and headback [k][m] (m = 0, 1, 2) stand for the original vertices on the primary cross-sections from the outline curves in the front, side and back views, respectively. The vertex coordinates on these primary cross-sections are determined by: Ê ˆˆ Êj Ê j ˆ V [i ][ j ][0] = S [k ][0] ¥ Á 1 + a ¥ sin Á ¥ 2p ˜ ˜ ¥ cos Á ¥ p ˜ p p Ë ¯ ¯ Ë Ë ¯



[10.26]



[10.27]



V[i][j][1] = S[k][1]

Ê Ê j ˆ Êj ˆˆ V [i ][j ][2] = F [k ][2] ¥ Á 1 + b ¥ sin Á ¥ 2p ˜ ˜ ¥ cos Á ¥ p ˜ p p Ë ¯ Ë ¯ Ë ¯



[10.28]

k = 0, 1, 2, …, keyframe-1; i = 6 ¥ k; j = 0, 1, 2,…, point-1

where, a and b are feature constants and j*p/point and j*2p/point are the coincidence angles for determining, respectively, the orientation of vertices and the regulation of a and b, in a circuitry periodicity of the frontal halfhead on ith level cross-section. A similar algorithm was used to reconstruct the remainder of the head from other perspectives. Modification of the head silhouette begins by describing the geometry of a hierarchical multi-resolution control mesh in terms of the positional

10.17 Reconstruction of a human head.

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variation of a few feature control points. To enhance realism, additional feature shape details were implemented by manipulating the localised deformation near feature points, or curves on the parametric surface based on the integration algorithm of the line deformation, in combination with facial feature measurements and a curve-fitting algorithm. Thus, the parametric model with regular vertex distribution can then be deformed to approximate the real face and body, whilst only relying on a few feature points selected using photographs. A full detailed description is beyond the scope of this chapter. Virtual human cloning provides an effective means of displaying a lifelike fashion show based on the facial and body characteristics of the individual customer, with realistic complexion, posture and movement. The task of producing the digitally cloned human using client data consists of two major parts: face cloning and body cloning. The different stages of face cloning, as an example, are shown in Fig. 10.18. Our system utilises photographs taken from the front, side and back of the person (in a simple/plain background). A generic model of the human body was reconstructed to fit actual body dimensions and a set of effective plane texture mapping algorithms with

Camera

Orthogonal photographs

Generic model

Feature detection

Modification of shape

Texture fitting

10.18 Overall structure for individual head cloning.

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multiple sub-domains were then used, in conjunction with photographs of facial aspects, to clone the client’s face and body onto the pre-prepared 3D primitive virtual human model. The results are shown in Fig. 10.19. The human model described above is only effective for displaying the stationary shape/posture of a person, thus representing how a garment is actually worn; but as in a ‘catwalk’, the model must be animated to allow human walking capabilities. In order to achieve this, a skeleton model was developed. In a typical animation hierarchy, the joint at the top of the hip forms the ‘root’ of the structure while the head, hands and feet form the ‘branches’. The locations of each of the skeletal joints are controlled by a set of paths, or motion curves, over a series of time steps. After creation of a frame loop at each step of the animation cycle, the resulting transformations at each joint start from the root of the hierarchical linkage, passing down from parent to child by linking passed-down translation vectors with the local translation matrix, shown in Fig. 10.20 and Fig. 10.21. Once the skeleton model has been completed, the body surface will then be attached to the animated skeleton to form an animated cloned character. An illustration example of a walking female in a virtual environment is shown in Fig. 10.22.

10.8.2 Virtual wearer trials Garment animation is performed by handling the collision between the garment and the moving body. The constraints used at this stage are gravitational

10.19 Example of the integration of face and body cloning.

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10.20 Body skeleton for keyframe animation.

Hip

Back

Neck

Left shoulder

Right shoulder

Left hip

Right hip

Left upper arm

Right upper arm

Left upper leg

Right upper leg

Left lower arm

Right lower arm

Left lower leg

Right lower leg

Left hand

Right hand

Left foot

Right foot

Head

10.21 Hierarchical skeleton structures.

forces and forces resulting from collisions between fabric and body during movement, as well as fabric with fabric self-collisions (Carignan et al., 1992), which are applied at each deformable node of garment simulation. When the human body is moving, collisions between the garment and different parts of the body are detected and repulsive forces calculated. Before collision detection is performed, a preliminary test should be applied to reduce the number of collisions between nodes of the garment and each segment of the body. Since the reconstruction of each segment of the body is based on animated skeletal bones, the position of the skin can be integrated into a global coordinate system. Thus with each segment to be considered in a global coordinate system, an algorithm was designed to find out dynamically the list of garment–node subdivisions that apply to the collision, with the rule that nodes in the proximity bordering adjacent body segments belong to both segments.

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10.22 Human walking animation with seamless connections.

10.23 Virtual wearer trials on a real female model.

An example simulation of a virtual wearer trial with a cloned (lifelike) female model is shown in Fig. 10.23, with different fabrics and different garment design patterns. The system is able successfully to simulate skirts; we see realistic wrinkles appearing on the dress as it flutters with body movement. A more detailed discussion on collision detection and reaction algorithms has already been given in Section 10.4.

418

10.9

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Conclusions and future trends

In this chapter, we have presented an overview of existing models and techniques in the 3D simulation of dynamic drape of textiles and garments, suggested directions for future research and development and potential applications to the textile, fashion, retailing other industries such as the entertainment industry. Through our explorations of virtual wearer trials we have provided a practical example of how these simulations may have important applications in global retailing. We began by discussing virtual drape models that were based on finite element methods and how from our own experiments a good agreement between actual and virtual drape measurements obtained using such models had been observed. But despite successes such as this, we have to also conclude that for finite element methods to be successful, issues regarding high computational costs and complexity have to be resolved. This drawback motivated the development of an alternative technique: the mass–spring system. A detailed description and analysis of such a framework, including the mass–spring system, forces acting on and within the system, the evolution of the fabric sample in time and collision detection and reaction algorithms, was provided. In order to cope with the superelasticity perceived in many existing cloth models, we proposed an effective ‘velocity and force modification algorithm’ based on a mixed explicit/implicit integration scheme, which improved both accuracy and realism without incurring additional computational complexity. The last section explored how a realistic, individually tailored virtual wearer trial should be, and can be, designed and implemented, using only twodimensional photographs and without the need of excessive amount of user data, as in the case of body scanning. The introduction of a new concept of textile measurement, the FAMOUS equipment, has been explored and its fundamental uses in textiles, clothing and retailing research and industry discussed. The difficulty of how to map intricately the material properties of real fabrics as measured using the FAMOUS equipment onto our own models has been described and discussed. We have been extending our cloth environment and collision detection algorithm to support more complex fabric shapes, structures and textures, than those presented in this chapter. Several avenues remain open for research, such as interactive clothing design systems and methods of improving simulation efficiency further. The core of all of these approaches, whether from textile researchers or from animators, is the mechanical data for the fabric. There are still many possibilities in modelling textile fabrics and processes and other related fields. Textile materials, with their viscoelastic behaviour, are the most complex of engineering structures. Textile fabrics are found in clothes, in aerospace, in automotive, in the built environment, in geotextiles,

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in filtration, in medical and other products and we will continue to seek intellectual rigour in precisely defining their dynamic, time dependent and combinatory behaviour.

10.10 References Agui T, Nagao Y and Nakajma M (1990), ‘An expression method of cylindrical cloth objects – an expression of folds of a sleeve using computer graphics’, Transactions Society Electronics, Information and Communications, J73-D –II (7), 1095–7. Ascough H E, Bez J and Bricis A M (1987), ‘A simple beam element large displacement model for the finite element simulation of cloth drape’, Journal of the Textile Institute, 87(1), 152–65. Baraff D and Witkin A (1998), ‘Large steps in cloth simulation’, Computer Graphics, 32 (Annual Conference Series), 43–54. Biglia U, Roczniok A F, Fassina C and Ly N G (1994), Textile Objective Measurement and Automation in Garment Manufacture, Stylios G (ed.), Ellis Horwood, Chichester, 139–44. Breen D E, House D H and Getto P H (1992), ‘A physical-based particle model of woven cloth’, Visual Computer, 264–77. Breen D E, House D H and Wozny M J (1994), ‘Predicting the drape of woven cloth using interacting particles’, in SIGGRAPH ‘94 Conference Proceedings, Computer Graphics and Interactive Techniques, ACM Press, 28, 365–72. Bridson R (2003), Computational Aspects of Dynamic Surfaces, PhD Thesis, Stanford University, USA. Bridson R, Fedkiw R and Anderson J (2002), ‘Robust treatment of collisions, contact and friction for cloth animation’, ACM Trans. Graph (SIGGRAPH Proceedings), 21, 594, 603. Caramana E, Burton D, Shashkov M and Whalen P (1998), ‘The construction of compatible hydrodynamics algorithms utilizing conservation of total energy’, Journal of Computational Physics, 146, 227–62. Carignan M, Yang Y, Magnenat-Thalmann and Thalmann D (1992), ‘Dressing animated synthetic actors with complex deformable clothes’, Computer Graphics Proceedings (ACM SIGGRAPH), 99–104. Chadwick J E, Haumann D R and Parent R E (1989), ‘Layered construction for deformable animated characters’, Computer Graphics, 23(3), 243–52. Chittaro L and Corvaglia D (2003), ‘3D virtual clothing: from garment design to web 3D visualization and simulation’, Proceedings of Web3D 2003: 8th International Conference on 3D Web Technology, ACM Press, New York, March, 73–84. Collier J, Collier B, O’Toole G and Sargand S (1991), ‘Drape prediction by means of finite element analysis’, Journal of the Textile Institute, 82(1), 96–107. Cusick, G E (1968), ‘The measurement of fabric drape’, Journal of the Textile Institute, 59, 253–60. Desbrun M, Schroder P and Barr A (1999), ‘Interactive animation of structured deformable objects’, Graphics Interface, 1–8. Donald H H and David E B (eds) (2000), Cloth Modelling and Animation, A K Peters, USA, Chapter 4. Eberhardt B, Weber A and Strasser W (1996), ‘A fast, flexible, particle-system for cloth draping’, in Computer Graphics in Textiles and Apparel, IEEE Computer Graphics and Applications, September, 52–9.

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Eberhardt B, Etzmuß O and Hauth M (2000), ‘Implicit explicit schemes for fast animation with particle systems’, Proc. of Eurographics Workshop on Computer Animation and Simulation in Interlaken, Switzerland, August 21–22, 2000, N Magnenat-Thalmann, D Thalmann and B. Arnaldi (eds), 137–151. Eishen J W, Deng S and Clapp T G (1996), ‘Modelling and control of flexible fabric parts’, in Computer Graphics in Textiles and Apparel, IEEE Computer Graphics and Applications, September, 71–80. Gavin M (1988), ‘The motion dynamics of snakes and worms’, August Proceedings of SIGGRAPH’88 (Atlanta, Georgia), Computer Graphics, 22(4), 169–77. Gibson S and Mirtich B (1997), A Survey of Deformable Modelling in Computer Graphics. Technical Report, Mitsubishi Electric Research Laboratory, November. Kawabata S (1982), ‘The development of the objective measurement of fabric handle’, Objective Specification of Fabric Quality, Mechanical Properties and Performance, Kawabata S, Postle R. and Niwa M., The Textile Machinery Society of Japan, 31–59. Moore M and Wilhelms J (1988), ‘Collision detection and response for computer animation’, in Proceedings. SIGGRAPH 1988, ACM Press/ACM SIGGRAPH, New York, June, Volume 22, 289–98. Ng H and Grimsdale R (1996), ‘Computer graphics techniques for modelling cloth’, IEEE Computer Graphics and Applications, 16(5), 28–41. Parks D and Forsyth D (2002), ‘Improved integration for cloth simulation’, Short Presentations in Proceedings of Eurographics, Saarbrücken, Germany, Computer Graphics Forum, Eurographics Assoc. Peirce F T (1930), ‘The handle of cloth as a measurable quantity’, Journal of the Textile Institute, 55, T377–T416. Press W H, Flannery B P, Teukolsky S A and Vetterling W T (1993), ‘Numerical recipes in C’, The Art of Scientific Computing, Cambridge University Press, York. Provot X (1995), ‘Deformation constraints in a mass–spring model to describe rigid cloth behaviour’, in Proceedings Graphics Interface, Quebec City, Canada, May17–19, 147–54. Stylios G K (2000), ‘Fabric objective measurement; FAMOUS, a new alternative to low stress measurement’ International Journal of Clothing Science and Technology, 12(1), 1–12. Stylios G and Sotomi O J (1994), A Neuro-fuzzy Control System for Intelligent Sewing Machines, Intelligent Systems Engineering Technology, IEEE Publication no. 395, 241–6. Stylios G K, Han F and Wan T R (2000), ‘A remote, on line 3D human measurement and reconstruction approach for virtual wearer trials in global retailing’, 3rd International Conference Innovation and Modelling of Clothing Engineering Proc–IMCEP 2000, Faculty of Mechanical Engineering, Maribor, Slovenia. Stylios G K and Zhu R (1997), ‘The characterisation of the static and dynamic drape of fabrics’, Journal of the Textile Institute, 88(4), 465–75. Stylios G, Sotomi O J, Zhu R, Xu Y M, Xu Y M and Deacon R (1995), ‘The mechatronic principles for intelligent sewing environment’, Mechatronics, 5(2/3), 309–19. Stylios G K, Wan T R and Powell N J (1996), ‘Modelling the dynamic drape of garments on synthetic humans in a virtual fashion show’, International Journal of Clothing Science and Technology, 8(3), 95–112. Stylios G, Fan J, Sotomi O J and Zhu R (1994), ‘An integrated sewability environment for intelligent garment manufacture’, Factory 2000; Advanced Factory Automation, IEE Proceedings no. 398, 543–51.

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Stylios G K and Wan T R (2000), ‘Artificial garments for synthetic humans in global retailing’, Digital Media: The Future, J Vince and R A Earnshaw (eds), Springer, New York, 175–18. Submissive N P A (1998), Hermite Curve Interpolation, Hamburg (Germany) http://www. cubic.org/~submissive/sourcerer/hermite.htm Tan S T, Wong T N, Zhao Y F and Chen W J (1999), ‘A constrained finite element method for modelling cloth deformation’, The Visual Computer, 15, 90–9. Vassilev T I (2000), ‘Dressing virtual people’, in SCI’2000 Conference, Orlando, July, 23–6. Volino P and Magnenat-Thalmann N (2001), ‘Comparing efficiency of integration methods for cloth simulation’, In Proceedings Computer Graphics International, July, 2001 City University of Hong Kong, 265–74. Volino P, Courchesne M and Magnenat-Thalmann N (1995), ‘Versatile and efficient techniques for simulating cloth and other deformable objects’, Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, archive, ACM Press, New York,137–44. Weil J (1986), ‘The synthesis of cloth objects’, in SIGGRAPH ’86: Proceedings of the 13th Annual International Conference on Computer Graphics and Interactive Techniques, New York, NY, USA, ACM Press, 49–54. Weisstein E (2008), Bicubic Spline, at Wolfram Mathworld, http://mathworld.wolfram. com/BicubicSpline.html Witkin A and Baraff D (2001), ‘Physically-based modelling’, ACM SIGGRAPH CourseNotes, (#25), Los Angeles, CA. Zienkiewicz D C and Taylor R L (1991), The Finite Element Method: Solid and Fluid Mechanics, Dynamics and Non-linearity, 4th edition, McGraw-Hill, London.

11

Recognition, differentiation and classification of regular repeating patterns in textiles

M. A. Hann and B. G. Thomas, University of Leeds, UK

Abstract: This chapter presents details of a well-established system by which two-dimensional repeating designs (or patterns) can be differentiated and classified by reference to their constituent symmetry characteristics. A comprehensive review is presented of the more important theoretical literature and the symmetry characteristics of motifs, border patterns and all-over patterns are identified and illustrated. In all three cases, simple notations are introduced and a systematic means by which textile and other surface patterns may be classified with respect to the symmetry characteristics of their underlying structures is explained. Key words: systematic classification, motifs, border patterns, all-over patterns, symmetry, colour counterchange.

11.1

Introduction

During the 20th century, the University of Leeds played a pivotal role in the recognition, differentiation, interpretation and classification of patterns – the three-dimensional patterns which are the basis of crystal structures and the two-dimensional patterns which are the basis of fabric design, tessellations and tilings. This role may be said to have begun with the Nobel Prizewinning work of W H Bragg, Cavendish Professor of Physics, and his son W L Bragg. Working as a team, using X-ray diffraction techniques, they solved the first crystal structures in 1913. In the 1930s, H J Woods of the Department of Textile Industries presented a comprehensive appraisal of geometry in patterns (Woods, 1935a, b, c and 1936). Drawing on concepts which have their origin in the study of crystal structures, Woods was the first to present the complete and explicit enumeration of two-colour counterchange border and all-over patterns, visionary work which was several years ahead conceptually of the theoretical developments emanating from crystallographers worldwide. Today it is acknowledged widely that Woods helped to lay the foundation for much of the current-day thinking on the geometry of regular repeating patterns (Washburn and Crowe, 1988). Between the 1930s and 1940s, W T Astbury, building on work initiated by J B Speakman, both also of the Department of Textile Industries at Leeds, pioneered the use of X-ray diffraction techniques for the elucidation of wool fibre structure, work which 422

Recognition, differentiation and classification of regular patterns

423

(it could be argued) led directly to the discovery of the structure of DNA. In fact Astbury, together with his research student Florence Bell, in 1938, took the first x-ray diffraction photographs of DNA. The Leeds tradition continues and contributions have been made to furthering the understanding of the pattern geometry of textiles produced in different cultural and historical contexts (Hann and Thomson, 1992) and to the classification, analysis and synthesis of counter-change patterns (Hann and Lin, 1995). Concepts have been developed in the field of layer symmetry, to aid advances in our understanding of the geometry of woven textiles (Scivier and Hann, 2000a, b). Tilings, tessellations and polyhedra are a research focus currently. One project at Leeds is concerned with Platonic (or regular) tilings and Archimedean (or semi-regular) tilings and their application to various forms of nonwoven, fibrous structures. Another is concerned with identifying the rules governing the patterning of Platonic, Archimedean and other polyhedra (Thomas and Hann, 2007). Consideration has been given also to fractals and scale symmetry and the use of such concepts in the design of contemporary textiles, as well as to the construction and use of aperiodic (non-repeating) Penrose-type tilings. The underlying threads throughout all the diverse work charted above are attempts to identify and to classify structures, invariably by reference to their symmetry characteristics. In fact a comprehensive means by which all possible types of repeating patterns can be grouped or classified by reference to their symmetry characteristics is now well established. The concern of this chapter therefore is to consider two-dimensional symmetry and its associated geometric operations, and to show how these geometric operations combine and are manifested in regular repeating patterns in such a way that they can act as a basis for recognition, differentiation and classification.

11.2

Study of pattern: historical precedents

Apart from a small number of exceptions (identified later in the chapter) the literature concerned with decorative patterns falls into two categories: (1) where the focus is on the aesthetic, cultural or decorative, and (2) where the emphasis is on scientific or mathematical explanation. This section considers a small number of studies from the first category and identifies the more important literature from the second category. Subsequently, reference is made to the relatively small number of publications which, although authored largely by mathematicians or scientists, are aimed at making the subject accessible to a wider audience including those with interests and expertise in the arts and humanities. Arguably the most influential 19th century European study of pattern was Owen Jones’ The Grammar of Ornament (1856), which focused on the cultural and historical principles of patterned ornament. While this work

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was not concerned with the geometrical characteristics of pattern structure, it represents the first attempt to categorise patterns with reference to their cultural and historical origins. The bulk of design literature of the late 19th and early 20th centuries focused on pattern construction rather than geometric analysis. Whilst design commentators did not adopt either the terminology or the theoretical perspectives being developed by mathematicians or crystallographers, certain observers exhibited an awareness of the fundamental geometric principles underlying the construction of regular repeating patterns; the relevant literature in this category has been identified by Hann and Thomson (1992, p 3). A few publications are of particular importance because they suggest perspectives similar to those used at a later date by scientists and mathematicians. Meyer (1894, p 3), for example, grouped designs according to spatial characteristics into enclosed spaces, ribbon-like bands, or unlimited flat patterns, corresponding to motifs, border patterns and all-over patterns, respectively. Meyer also recognised that there was a ‘…certain division, a subsidiary construction or a network’, anticipating the principle of lattices underlying the structure of all-over patterns, as explained later in the chapter. Stevenson and Suddards (1897, chapters 2–5), in their appraisal of the geometry of Jacquard woven patterns, illustrated constructions based on rectangular, rhombic, hexagonal and square lattices. Similarly, Day (1903) emphasised the geometrical construction of patterns, illustrating all-over patterns based on square, parallelogram, rhombic and hexagonal type lattices. Christie’s Pattern Design, first published in 1910, made a formal ordering of patterns according to the motifs appearing within them, rather than referencing them according to time periods or culture. This work is of significance as it represents an early stage in the categorisation of patterns in terms of their geometric properties. The 17th century astronomer Johannes Kepler (1619) conducted one of the first mathematical studies of tessellations in which he attempted to tile the plane with figures exhibiting five-fold symmetry. Subsequent to this, only limited mathematical investigation of patterns and tilings took place before the end of the 19th century (Martin, 1982, p 117). The early 20th century saw the evolution of another perspective of pattern analysis and classification: the consideration of patterns by reference to their symmetry characteristics, a perspective with origins in the scientific investigation of the structures of crystals. It is therefore from the discipline of crystallography that the first relevant investigations were initiated. These were led by Federov, the Russian crystallographer, who in the late 19th century, determined the 230 threedimensional crystallographic groups before proving that regular repeating patterns of the plane are constructed in accordance with the underlying structures of the 17 crystallographic symmetry groups (Federov, 1885 and

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1891, p 345, cited by Grünbaum and Shephard, 1987, p 55). This theorem was rediscovered in 1897 by Fricke and Klein (1897, pp 227–33), but since the focus of crystallographers was primarily towards higher dimensional phenomena it was not until the 1920s that interest in the enumeration of the two-dimensional crystallographic groups (which are of relevance to the consideration of decorative motifs and patterns) was aroused through the work of Pólya and Niggli (1924). Two years later, in 1926, Niggli identified the seven possible classes of border patterns (Niggli, 1926). Developments in the application of crystallographic theory to the understanding of two-dimensional design continued, and in 1933 Birkhoff defined and illustrated various symmetry operations and discussed their occurrence in motifs and patterns (Birkhoff, 1933). As noted in the Introduction, an important early attempt to classify regular repeating patterns, according to their underlying geometry, was made by Woods in the 1930s (Woods, 1935a, b, c and 1936). This work is of importance because it adapted a wide range of concepts from the discipline of crystallography, applied these to the understanding of the geometry of patterns, presented a comprehensive system of classification for motifs, border patterns and all-over patterns, and also considered two-colour counterchange possibilities. In terms of the concepts dealt with, it is apparent that Woods’ work anticipated work that would not be done by mathematicians and crystallographers for another 20 years (Horne, 2000, p 3). Unfortunately Woods’ efforts went largely unnoticed until nearly 40 years later when Grünbaum recognised the relevance to research on coloured tilings (observed by Crowe, 1986). Ultimately it was Crowe who brought Woods’ work to the attention of a wider international audience, initially through a paper entitled The Mosaic Patterns of H. J. Woods (Crowe, 1986). Subsequent to Woods’ work, although probably unaware of it, Buerger and Lukesh (1937) presented a series of symbols denoting lattice structures, orders of rotation and the presence of reflection and glide-reflection axes to account for the symmetries of patterns. Brainerd initiated the use of symmetry as a tool for the classification of archaeological artefacts (Brainerd, 1942). In 1952, Weyl presented a review of symmetry in art, botany and other pure sciences (Weyl, 1952). Russian crystallographers, Shubnikov and Koptsik, also provided new perspectives on symmetry (Shubnikov and Koptsik, 1974). The works of Walker and Padwick (1977), Schattschneider (1978, 1986) and Stevens (1984) made the consideration of pattern symmetry accessible to a non-mathematical audience. In 1986, Crowe and Washburn presented a flow-chart to aid the recognition of the 17 classes of all-over patterns. This was developed further to incorporate two-colour-counterchange possibilities (Washburn and Crowe, 1988). Grünbaum and Shephard (1987) charted much of what is currently known on the subject of regular tilings (which have symmetry characteristics in common with regular patterns); in fact their

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work is the key landmark to date in the mathematical investigation of the subject. Washburn and Crowe (1988) published an impressive treatise dealing with the theory and practice of pattern analysis and, in particular, with the use of symmetry concepts as an aid to the analysis of designs from different cultural settings and historical periods. This has proved to be a classic reference for anthropologists, archaeologists, art historians, mathematicians and designers. A more recent publication (Washburn and Crowe, 2004) explores how cultures use pattern to encode meaning. The extensive bibliography also updates its predecessor. Hargittai (1986, 1989) published two compendia containing over 100 papers from the sciences, arts and humanities, each paper dealing with some aspect of symmetry, its occurrence, use or application. Schattschneider’s monumental study of the work of artist M C Escher, provided an insight into the periodic drawings of the artist and also presented explanations of associated symmetry concepts in ways which were understandable to audiences without specialist knowledge of mathematics. Hargittai and Hargittai (1994) also published a profusely illustrated review of the principles of symmetry aimed at encouraging awareness amongst non-specialists. From the early 1990s, research at the University of Leeds into the areas of pattern and structure has continued, with contributions by Hann (1992) into pattern analysis within a cultural context, the analysis and construction of counterchange patterns (Hann and Lin, 1995) and into the geometry of woven fabrics (Scivier and Hann, 2000a, b). More recent undertakings have considered the potential value of the geometric concepts associated with patterns as problem-solving tools in the 21st century (Hann and Russell, 2003). A series of papers produced by Hann presents a comprehensive review of theoretical developments since the time of Woods (Hann, 2003a, b, c). A further paper, co-authored by Hann and Thomas, deals specifically with colour symmetry, a topic dealt with later in this chapter (Hann and Thomas, 2007).

11.3

Symmetry in pattern: fundamental concepts

Mathematical and scientific studies (such as Coxeter, 1948; Shubnikov and Koptsik, 1974; Schattschneider, 1978; Martin, 1982; Grünbaum and Shephard, 1987) have recognised that symmetry in plane patterns is characterised by one or more of the following geometrical actions: translation, rotation, reflection and glide-reflection (illustrated schematically in Fig. 11.1). These are known as symmetry operations. Synonymous terms include congruent transformations (Coxeter, 1948), isometries (Schattschneider, 1978), symmetry transformations (Shubnikov and Koptsik, 1974, p 13) or symmetries (Grünbaum and Shephard, 1987). The fundamental characteristic of patterns in the plane is that they exhibit systematic repetition of each constituent component of

Recognition, differentiation and classification of regular patterns



Translation

Key:

Rotation

Reflection

427

Glide-reflection

translation axis two-fold rotation reflection axis glide-reflection axis

11.1 The four symmetry operations.

the design. This systematic repetition is governed by the use of one or more of the four distinct symmetry operations which, acting on the constituent components, instigate regular repetition without change in the shape or size of these components. Depending on the particular symmetry operation or particular combination of symmetry operations, change in orientation is, however, a possibility. Each of the four symmetry operations is explained further below. Translation moves a figure without change in orientation, shape or size, across a given distance and direction (Coxeter, 1948, p 34). As observed by Schattschneider: ‘A translation of points in the plane shifts all points the same distance in the same direction.’ (Schattschneider, 1986, p 673). Translation can be represented diagrammatically through the use of an asymmetrical motif which can be shown to move a specified distance (say T) along a straight line (or translation axis, say L), without changing orientation. Displacement by a distance equal to T does not change the figure in any way and the action can be repeated as many times as desired. While border patterns admit translation in one direction only (say horizontally), all-over patterns exhibit translation in two independent directions (say horizontally and vertically) across the plane. Rotation is considered to occur through a fixed point called a centre of rotation (or rotocentre). The angle of rotation is constant and is measured in relation to this centre of rotation, with the order of rotational symmetry of a figure being the number of times (n) it repeats itself in one revolution (of 360 degrees) around this centre of rotation. Repetition therefore occurs at regular angular intervals. Schattschneider observed: ‘A rotation of points in the plane moves points by turning the plane about a fixed point (called a centre of rotation)’ (Schattschneider, 1986, p 673). A design is considered to have n-fold rotational symmetry about a fixed point when a motif (or other repeating component) in the plane is repeated by successive rotations through regular angles of 360 degrees/n about a fixed

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point (or centre of rotation), where n is an integer greater than or equal to one, which corresponds to the order of rotation. After n successive rotations of 360 degrees/n, the repeating component comes back to rest in its original position. By way of example, if a repeating component is rotated through 60 degrees, the design is considered to exhibit six-fold rotational symmetry. Thus where n is equal to one, no rotational symmetry is exhibited, as the component part (or figure) turns a full 360 degrees without interruption. As observed by Schattschneider: ‘A rotation of 360 degrees (n = 1) sends each point in the plane to its original position. This isometry has the same effect as leaving each point fixed and is called the identity isometry’ (Schattschneider, 1986, p 673). With two-fold rotational symmetry (sometimes refereed to as a half-turn), two rotations, each of 180 degrees, are required. With the repeated action of two rotations through 180 degrees, each point in the plane is returned to its original position. Three-fold rotational symmetry requires three rotations, each of 120 degrees, and four-fold rotational symmetry requires four rotations, each of 90 degrees. Although five-fold rotational symmetry (i.e. rotations through 72 degrees) can be a feature of individual motifs or figures, this order of rotation is not possible in all-over patterns which, as will be considered later in the chapter, can only exhibit one-, two-, three-, four- and six-fold rotation. A reflection produces a figure’s mirror image, equidistant, at the other side of an imaginary two-sided mirror or reflection axis (generally denoted by the letter m) positioned perpendicular to the plane of the design. Schattschneider commented: ‘A reflection of points in the plane is determined by a fixed line, called a mirror line or reflection axis; every point not on the line is sent to its mirror image with respect to the line and every point on the line is left fixed.’ (Schattschneider, 1986, p 673). Where a figure or object exhibits a single reflection through its centre, splitting it into two imaginary equal parts, this is also known as bilateral symmetry, a phenomenon which predominates in the natural and humanmade worlds (Shubnikov and Kopstik, 1974, p 11). Reflection allows the reflected figure to retain the same size and shape, but directional aspects of the figure are reversed, much the same as in a mirror. Reflection can be a characteristic of motifs, border patterns and all-over patterns. Where two or more reflection axes intersect, this has the effect of generating rotation. Thus where n is greater than two and represents the number of intersecting reflection axes in a design, then n intersecting reflection axes will generate n-fold rotation. As observed by Washburn and Crowe: ‘In two-dimensional patterns, the presence of two intersecting mirror lines implies the presence of a rotation (by an angle which is twice the angle of intersection of the two lines) about their point of intersection’ (Washburn and Crowe, 1988, p 47).

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Glide-reflection is best considered as a combination of reflection followed by translation or vice versa. This symmetry operation is often illustrated by reference to the imprint left by a person’s feet while walking on wet sand. Schattschneider observed: ‘A glide-reflection, as its name suggests, is a transformation of points in the plane which combines a translation (glide) and a reflection. It may be obtained by a reflection followed non-stop by a translation which is parallel to the mirror line or by a translation followed by a reflection in a mirror line parallel to the translation vector’ (Schattschneider, 1986, p 677). In diagrammatic representations, a glide-reflection axis is often denoted by a dashed line and by the letter G. Any figure, motif or other graphic component in the plane which exhibits one or more of the four symmetry operations described above is considered to be symmetrical. Patterns are deemed to have a particular symmetry if the operation of that symmetry, when applied to the pattern as a whole, transforms each motif or figure into another one exactly the same. Any motif or pattern may be classified by reference to its constituent symmetries or symmetry group. Regular repeating patterns are designs (classified as either border patterns or all-over patterns), which exhibit repetition of a motif (or motifs), at regular intervals across the plane. Motifs may be divided into two distinct classes dependent on their constituent symmetry characteristics. Synonymous terms include finite groups (Schattschneider, 1978), point groups (Stevens, 1984) and finite designs (Horne, 2000). There are only seven distinct classes of border patterns, each characterised by translation of a motif (or motifs) in one direction only. Synonymous terms include frieze groups (Schattschneider, 1978), line groups (Stevens, 1984), one-dimensional designs (Washburn and Crowe, 1988) and mono-translational designs (Horne, 2000). All-over patterns are patterns that exhibit translation of a motif (or motifs) in two independent non-parallel directions across the plane. Synonymous terms include wallpaper groups (Schattschneider, 1978), plane groups (Stevens, 1984), periodic patterns (Grünbaum and Shephard, 1987), two-dimensional patterns (Washburn and Crowe, 1988) and ditranslational designs (Horne, 2000). The concept of symmetry is occasionally employed to denote repetition in association with scale change. Geometric scaling involves increasing or decreasing a figure’s linear dimensions by a given factor. These transformations involving expansion (known as enlargement) or dilation (known as contraction) allow a figure to be reproduced (or repeated) in an enlarged or diminished state, respectively. The term ‘fractal’ was coined by Benoit Mandelbrot to describe complex geometrical phenomena that exhibit characteristics relating to repetition and change of scale. Fractals have two distinct properties: they tend to exhibit infinite detail and they conform to the same shape at different scales, a property known as self-similarity or scale symmetry. Fractals can be

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based on mathematical models, but are also common in real life. Examples of nature’s fractals are coastlines, lightning, clouds, trees, various vegetables (e.g. cauliflower and broccoli) and mountains. Further explanation of such concepts and their associated phenomena are outside the concern of this chapter. Interested readers are referred to Bovill (1996). The sections below provide a review of the symmetry principles by which motifs, border patterns and all-over patterns may be differentiated and classified; in each case an explanation of the most commonly used notation is given.

11.4

Classification of motifs

Motifs may be classified by reference to their constituent symmetry characteristics. The first step is to differentiate symmetrical from asymmetrical motifs; each may be used as the recurrent component of repeating patterns. Asymmetrical motifs exhibit no symmetrical properties and their constituent elements can only coincide with themselves after a full rotation of 360 degrees (Fig. 11.2). Symmetrical motifs may be categorised readily into two distinct groups: cyclic (cn) or dihedral (dn), each exhibiting particular symmetry characteristics. Motifs from class cn exhibit only rotational symmetry, with the value of n being determined by the order of rotation. Motifs from class dn exhibit reflection symmetry and, where more than one reflection axis is present, also

11.2 An asymmetrical motif.

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431

rotational symmetry. In this instance, n equals the number of reflection axes present and also the highest order of rotation. A motif of class cn is thus considered to have n-fold rotational symmetry around a fixed point. This occurs when an asymmetric unit is repeated by successive rotations through an angle of 360/n degrees. Examples of cn motifs are given in Fig. 11.3. Meanwhile the principal characteristic of class dn motifs is that they are constructed by reflections of a fundamental unit. Examples are given in Fig. 11.4.



c2

c3

c4

Key: centre of two-fold rotation (c2)

c5

c6

centre of five-fold rotation (c5)

centre of three-fold rotation (c3)

centre of six-fold rotation (c6)

centre of four-fold rotation (c4)

11.3 Examples of cn motifs.



d1

d2

Key: centre of two-fold rotation (d2)

d3

d4

d5

centre of five-fold rotation (d5)

centre of three-fold rotation (d3)

centre of six-fold rotation (d6)

centre of four-fold rotation (d4)

axis of reflection

11.4 Examples of dn motifs.

d6

432

11.5

Modelling and predicting textile behaviour

The seven classes of border patterns

Border patterns are designs characterised by the translation of a motif or other repeating unit, along a horizontal axis, known as a translation axis. Using combinations of the four symmetry operations, it is possible to produce only seven distinct classes of border patterns, considered from the viewpoint of geometric symmetry; if colour is changed systematically, further possibilities unfold. Reflection symmetry may be present in both a parallel and a perpendicular direction to the translation axis of a border pattern, but only two-fold rotation may occur in order to maintain the pattern’s orientation. Glide-reflection may occur in a direction parallel to the translation axis. Schematic illustrations of the seven classes of border pattern are given in Fig. 11.5. Border patterns can, of course, appear to be oriented vertically, horizontally or in any other direction between two parallel lines with the direction depending on the position of the viewer. To aid description below, it is assumed that the axis of orientation is from left to right and is thus of horizontal orientation. Border patterns are designs which show regular repetition of a motif or other repeating unit between two imaginary or real parallel lines. Such designs are thus of specified width and may be considered to continue to infinity. As noted above, using combinations of the four symmetry operations it is possible to produce seven (and only seven) distinct classes of border patterns. The most commonly used notation, which identifies the characteristics of each of the seven classes, is explained below.

11.5.1 Notation As indicated above, border patterns exhibit regular translation in one direction only and, from the viewpoint of underlying geometry or symmetry, there are seven distinct classes of border patterns. Various types of notation can be found in the relevant literature. The most commonly accepted notation, of the form pxyz, gives a concise and easily understandable indication of the constituent symmetry operations used in the construction of each of the seven classes. The notation is further explained below. The first letter p of the four-symbol notation prefaces all seven border patterns and is consistent with similar notation used for classifying all-over patterns. Symbols in the second, third and fourth positions (i.e. represented by x, y and z) indicate the presence (or absence) of vertical reflection, horizontal reflection or glide reflection and two-fold rotation, respectively. The letter x, at the second position in the notation, will equal m if vertical reflection (perpendicular to the longitudinal axis) is present, otherwise x will equal the number 1. The letter y, at the third position in the notation, will equal m if longitudinal reflection (parallel to the sides of the border) is

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433 p111

p1a1

pm11

p1m1

p112

pma2

pmm2

11.5 Schematic illustrations of the seven classes of border patterns.

present, will equal the letter a if glide-reflection is present and will equal the number 1 if neither is present. The letter z, at the fourth position in the notation, will equal the number 2 if two-fold rotation is present and the number 1 if no rotation is present. The seven classes of border patterns can thus be classified as follows: classes p111, p1a1, pm11, p1m1, p112, pma2 and pmm2. Descriptions of each of the seven classes are provided below.

11.5.2 The translation class p111 From the view-point of symmetry, this is the most elementary of the seven border classes since the only constituent symmetry operation is translation.

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The construction is through simple regular repetition (or translation) of a motif, or other repeating unit between two real or imaginary parallel lines. An example of this border pattern class, together with the relevant schematic illustration, is provided in Fig. 11.6.

11.5.3 The glide-reflection class p1a1 Glide reflection, as explained previously, is a combination of translation followed by a reflection in a line parallel to the translation axis of the border. Glide-reflection is characteristic of p1a1 patterns. An example of this class is given in Fig. 11.7.

11.5.4 The vertical-reflection class pm11 Class pm11 border patterns (Fig. 11.8) are characterised by the presence of translation combined with reflections across two alternating vertical reflection axes, each perpendicular to the central axis of the border.

p111

11.6 Border pattern class p111.

p1a1

11.7 Border pattern class p1a1.

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pm11

11.8 Border pattern class pm11.

p1m1

11.9 Border pattern class p1m1.

11.5.5 The horizontal-reflection class p1m1 Class p1m1 border patterns (Fig. 11.9) exhibit reflection in a longitudinal axis through the centre line of the border.

11.5.6 The two-fold-rotation class p112 Class p112 border patterns (Fig. 11.10) exhibit two-fold rotational symmetry. This class is characterised by successive translations of motifs with centres of two-fold rotation (c2 motifs or figures). A second two-fold rotation centre is thus created which alternates with the first two-fold rotation centre.

11.5.7 The four-symmetries-in-one class pma2 Class pma2 border patterns (Fig. 11.11) contain all four symmetry operations. This class of borders may be generated using one of four procedures: by successive reflection of a class c2 motif; by successive translation of two

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p112

11.10 Border pattern class p112.

pma2

11.11 Border pattern class pma2.

alternate class c2 motifs; by successive two-fold rotation of a class d1 motif; by successive glide-reflection of a class d1 motif. The translation unit consists of a generating unit which is reflected, rotated and reflected again.

11.5.8 The intersecting-reflection-axes class pmm2 Border patterns from class pmm2 (Fig. 11.12) have a horizontal reflection axis which is intersected at regular points by two alternating perpendicular reflection axes. Two-fold rotation centres are thus established in the intersection points of the axes.

11.6

The 17 classes of all-over patterns

As indicated previously, translation is the underlying symmetry feature of regularly repeating patterns. Unlike border patterns, the translational symmetry

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437

pmm2

11.12 Border pattern class pmm2.

of all-over patterns is not confined to only one direction and extends to two independent directions. The introduction of one or more of the remaining three symmetry operations to the underlying operation of translation permits a total of 17 pattern classes. Proof of the existence of only 17 all-over pattern classes was provided by Weyl (1952), Jawson (1965), Coxeter (1969), Schwarzenberger (1974) and Martin (1982). Geometric frameworks of corresponding points that form lattice structures are also present in addition to combinations of the four symmetry operations. These lattice points may be connected to produce unit cells of identical size, shape and content which, when translated in two independent non-parallel directions, produce the full repeating pattern. There are only five distinct lattice types, each with an associated unit cell: parallelogram, rectangular, rhombic, square and hexagonal, with a rhombic unit cell consisting of two equilateral triangles. Crystallographers have termed these Bravais lattices, after Bravais who initially verified that lattices could be classified into five types (Grünbaum and Shephard, 1987, p 262). The five Bravais lattice types are illustrated in Fig. 11.13. Schematic illustrations of the all-over pattern classes are provided in Fig. 11.14. Diagrams of the seventeen unit cells for all-over patterns, indicating the symmetry characteristics, unit cell and fundamental region of each pattern class, are given in Fig. 11.15. The fundamental region of each pattern class is the area taken up by the smallest individual component which when subjected to the relevant symmetry operation will produce the full repeating unit.

11.6.1 Terminology and notation There are various notations, which have been used by mathematicians and crystallographers in the classification of all-over patterns; these were reviewed

438

b

q

Modelling and predicting textile behaviour

b a

Parallelogram lattice a ≠ b, q < 90°

b q a Rhombic (centred cell) lattice a = b, q ≠ 60° or 90°

q

b a

Rectangular lattice a ≠ b, q = 90°

q

a Square lattice a = b, q = 90°

b q

a Hexagonal lattice a = b, q = 60°

11.13 The five Bravais lattice types.

by Schattschneider (1978). The most widely used notation is taken from the International Tables of X-Ray Crystallography (Henry and Lonsdale, 1952). This four-symbol notation, of the form pxyz or cxyz, indicates the type of unit cell, the highest order of rotation and the symmetry axes present in two directions. Further explanation of this notation was provided by Schattschneider (1978) and Washburn and Crowe (1988, p 58) and is summarised below. The first symbol, either the letter p or letter c, indicates whether the lattice cell is primitive or centred. Primitive cells are present in 15 of the all-over pattern classes and generate the full pattern by translation alone. The remaining two classes of all-over patterns have cells of the rhombic lattice type with the enlarged cell containing two repeating units, one contained within the centred cell and another in quarters of the enlarged cell corners. The second symbol, n, denotes the highest order of rotation present. Where rotational symmetry is present, only two-, three-, four- and six-fold orders of rotation are possible in the production of two-dimensional plane patterns, as figures cannot repeat themselves around an axis of five-fold symmetry. This is known as the crystallographic restriction and was discussed by Stevens (1984, Appendix, p 376–90). If no rotational symmetry is present, n = 1.

Recognition, differentiation and classification of regular patterns



p1

p1g1

p1m1

c1m1



p2

p2gg

p2mg

p2mm



c2mm

p3

p31m

p3m1



p4

p4mm

p4gm

p6

439

p6mm

11.14 Schematic illustrations of the 17 all-over pattern classes.

The third symbol represents a symmetry axis normal to the x-axis (i.e. perpendicular to the left side of the unit cell): m (for mirror) indicates a reflection axis, g (for glide) indicates a glide-reflection axis and 1 indicates that no reflection or glide-reflection axes are present normal to the x-axis.

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p1

p1m1

p1g1

c1m1

p2

p2mm

p2mg

p2gg

c2mm

p3

p3m1

p4

p31m

p4mm

p6

p4gm

p6mm

Key: centre of two-fold rotation

axis of reflection

centre of three-fold rotation

axis of glide-reflection

centre of four-fold rotation

translation vector

centre of six-fold rotation

outline of unit cell outline of centred cell fundamental region

11.15 The unit cells for the 17 all-over pattern classes, indicating constituent symmetry operations.

Recognition, differentiation and classification of regular patterns

441

The fourth symbol indicates a symmetry angle at angle a to the x-axis, with a dependent on n, the highest order of rotation. Angle a = 180 degrees if n = 1 or 2, a = 45 degrees if n = 4 and a = 60 degrees if n = 3 or 6. The symbols m and g denote the presence of reflection and glide reflection symmetry, as with the third symbol. The absence of symbols in the third or fourth position indicates that the pattern has neither reflection nor glidereflection. A summary of the symmetry characteristics of the 17 classes of all-over patterns is provided in Table 11.1. Descriptions of each pattern class are presented below.

11.6.2 Patterns without rotational properties There are four all-over pattern classes with no rotational symmetry, or in which the highest order of rotation is 1 (a full rotation of 360 degrees). From Table 11.1 Summary of the symmetry characteristics of the 17 classes of all over patterns

p1 3 3 3 3 3 p2 3 3 3 3 3 p1m1 3 3 p1g1 3 3 c1m1 3 3 3 p2mm 3 3 p2gg 3 3 p2mg 3 3 c2mm 3 3 3 p4 3 p4mm 3 p4gm 3 p3 3 p31m 3 p3m1 3 p6 3 p6mm 3

1 1/2 1/2 1/2 1/2 1/4 1/4 1/4 1/4 1/4 1/8 1/8 1/3 1/6 1/6 1/6 1/12

1 2 1 3 1 1 3 2 3 2 2 3 2 3 4 4 3 4 3 3 3 3 3 3 6 6 3

Glide-reflection

Reflection

Symmetry operations present Highest order of rotation

Area of fundamental region/unit cell

Hexagonal

Rhombic

Square

Rectangular

Parallelogram

Symmetry Lattice structure class

3 3 3 3 3

3 3 3 3

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the perspective of geometrical symmetry, all-over pattern class p1 (Fig. 11.16) is the simplest in terms of analysis, classification and (often) construction. Normally constructed on a parallelogram lattice, this pattern contains no reflections, glide-reflections or rotational properties and repeats simply by translation of an asymmetric figure (motif class c1) in two independent directions. The fundamental region is of an area equivalent to the unit cell, which could be based on any of the five geometric lattice structures. Class p1m1 all-over patterns (Fig. 11.17) are constructed on a rectangular or square lattice structure. They have no rotational properties and two parallel reflection axes are present on the corners of each unit cell. The repeating pattern is created through two parallel reflections and translation in the direction of the reflection axes. The fundamental region occupies half the area of the unit cell and is bounded on opposite sides by the reflection axes. Based on either a rectangular or square lattice, pattern class p1g1 (Fig. 11.18) contains two alternating and parallel glide-reflection axes falling on

p1

11.16 All-over pattern class p1.

p1m1

11.17 All-over pattern class p1m1.

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443

p1g1

11.18 All-over pattern class p1g1.

c1m1

11.19 All-over pattern class c1m1.

the corners of the unit cell, but exhibits no reflection or rotational properties. The fundamental region is equal to half the area of the unit cell. Class c1m1 all-over patterns (Fig. 11.19) are usually constructed from a rhombic lattice, although square or hexagonal lattice types may also be used. The pattern exhibits reflection at right angles to the enlarged cell of the rhombic lattice and parallel glide-reflection axes. The enlarged cell contains two repeating components, one contained within the centred cell and another in quarters at the enlarged cell corners.

11.6.3 Patterns exhibiting two-fold rotation There are a total of five all-over pattern classes in which the highest order of rotation is 2 (180 degrees rotational symmetry): p2, p2mm, p2mg, p2gg and c2mm. Each of these pattern classes appears similar when viewed upside down. Their constituent symmetry characteristics are considered below.

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Pattern class p2 (Fig. 11.20) may be based on any of the five geometrical lattice types, although the parallelogram lattice is the most frequently used. A total of nine points of two-fold rotation are associated with the unit cell: at the centre of the unit, at each corner and at the mid-points of the unit’s sides. The fundamental region occupies half the area of the unit cell. All-over pattern class p2mm (Fig. 11.21) is based upon either a rectangular or a square lattice, exhibiting two alternating axes of horizontal reflection and two alternating axes of vertical reflection. Centres of two-fold rotation are present where the reflection axes intersect. The fundamental region is one-quarter of the unit cell. Pattern class p2gg (Fig. 11.22) contains vertical glide-reflection axes that intersect with horizontal glide-reflection axes at right angles within the unit cell, the latter being based on either a rectangular or a square lattice. The fundamental region is one-quarter of the unit cell and two-fold rotational centres are positioned at the centre of the unit, at each of the unit corners and the midpoints of the unit sides.

p2

11.20 All-over pattern class p2.

p2mm

11.21 All-over pattern class p2mm.

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p2gg

11.22 All-over pattern class p2gg.

p2mg

11.23 All-over pattern class p2mg.

Constructed on either a rectangular or a square lattice, pattern class p2mg (Fig. 11.23) presents two alternating and parallel reflection axes intersecting at right angles with parallel glide-reflection axes. Centres of two-fold rotation are found on the glide-reflection axes positioned at the centre of the unit cell, at each of the unit corners and the midpoints of the unit sides. The fundamental region of pattern class p2mg is equal to one-quarter of the unit cell area. Class c2mm all-over patterns (Fig. 11.24) are based on a rhombic, a square or a hexagonal lattice structure. Parallel reflection axes and glide-reflection axes alternate with each other in both horizontal and vertical directions. Centres of two-fold rotation fall on the intersection points of both reflection axes and glide-reflection axes, with the fundamental region consisting of one-quarter of the centred cell. The rhombic-centred cell can generate the pattern by translation alone; however convention states that the larger outline cell should be the generating unit (Hann and Thomson, 1992, p 40).

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Modelling and predicting textile behaviour

c2mm

11.24 All-over pattern class c2mm.

p3

11.25 All-over pattern class p3.

11.6.4 Patterns exhibiting three-fold rotation There are three all-over pattern classes in which the highest order of rotation is 3 (rotation through 120 degrees). These pattern classes are all constructed on the hexagonal lattice with unit cells bounded by two equilateral triangles. Alternatively the pattern may be generated by translation of the hexagonal lattice unit, which is three times the area of the unit cell. The characteristics of classes p3, p3m1 and p31m are outlined briefly below. All-over pattern class p3, (Fig. 11.25) as stated above, is based on an hexagonal lattice structure. The unit cell contains three distinct centres of three-fold rotation, which are located at the unit corners and the centres of the triangular cells. The area of the fundamental region is one-third of the unit cell area. Class p3m1 all-over patterns (Fig. 11.26) are based on the hexagonal lattice and are produced through a combination of three-fold rotational centres and reflection axes. The fundamental region is one-sixth of the unit

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447

p3m1

11.26 All-over pattern class p3m1.

p31m

11.27 All-over pattern class p31m.

cell. Centres of three-fold rotational are positioned at the intersections of the reflection axes, which are present along the longest diagonal of the unit cell and alternate with glide-reflection axes in all three directions. Pattern class p31m (Fig. 11.27) is also constructed on the hexagonal lattice. Reflection axes are located on each side of the unit cell and also along the shortest diagonal of the cell. Centres of three-fold rotation occur at the centres of the two triangular units and also at the corners of the rhomboid cell where the reflection axes intersect. The fundamental region is equal to one-sixth of the area of the unit cell.

11.6.5 Patterns exhibiting four-fold rotation There are three classes of all-over patterns in which the highest order of rotation is 4 (90 degrees rotational symmetry). The three pattern classes exhibiting four-fold rotational symmetry are p4, p4mm and p4gm, all of which are based on a square lattice structure.

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All-over pattern class p4 (Fig. 11.28), constructed using the square unit cell, exhibits no reflection or glide-reflection, presenting only two- and four-fold rotation. Centres of four-fold rotation are located at the centre and corners of the unit cell, with two-fold rotational centres present at the midpoint of each side. The fundamental region is one-quarter of the unit cell area. The square lattice also provides the construction base for pattern class p4mm (Fig. 11.29). The fundamental region is equal to one-eighth of the unit-cell area. Reflection axes are present at the sides of the unit and also run diagonally across the unit cell dividing it into eight equal parts. Centres of four-fold rotation are located at the corners and centre of the unit cell. Two-fold rotational centres are positioned at the midpoint of each unit edge and are intersected at right angles by axes of glide-reflection. Pattern class p4gm (Fig. 11.30) is based on the square lattice structure and is generated through reflection and four-fold rotation of a fundamental region occupying one-eighth of the unit cell area. Centres of four-fold rotation are located at the centre and also the corners of the unit cell. Reflection

p4

11.28 All-over pattern class p4.

p4mm

11.29 All-over pattern class p4mm.

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p4gm

11.30 All-over pattern class p4gm.

p6

11.31 All-over pattern class p6.

axes intersect the unit cell at right angles on two-fold rotation centres at the midpoint of each unit side. Glide-reflection axes intersect the reflection axes at 45 degrees and 90 degrees.

11.6.6 Patterns exhibiting six-fold rotation There are two all-over pattern classes in which the highest order of rotational symmetry is 6 (60 degrees rotation). Each of the patterns is constructed using a hexagonal lattice unit bounded by two equilateral triangles, as previously seen with patterns exhibiting three-fold rotation. All-over pattern class p6 (Fig. 11.31) exhibits six-fold rotation centres at each corner of the hexagonal lattice unit cell. Centres of three-fold rotation are located at the centres of the triangular cells and centres of two-fold rotation are present at the midpoints of the triangular cell edges. All the six-fold rotational points have the same orientation; the three-fold rotational points have two different orientations; conversely, two-fold rotational centres have

450

Modelling and predicting textile behaviour

three different orientations. The generating region for this pattern class is one-sixth of the unit cell area. Pattern class p6mm (Fig. 11.32) is based on the hexagonal lattice and exhibits two-, three- and six-fold rotation combined with reflection and glide-reflection. Centres of six-fold rotation are located at each corner of the unit cell. Reflection axes connect each corner with the other three and also bisect the opposite unit side at a right angle. Centres of two- and three-fold rotational symmetry are located at the intersection points of reflection axes. The fundamental region is one-twelfth of the unit cell area and is bounded by reflection axes connecting centres of two-, three- and six-fold rotation.

11.7

Colour symmetry

The fundamental geometrical rules outlined so far place the emphasis only on symmetry operations which do not involve colour change, that is colour remains unchanged following each symmetry operation. It may, however, be the case with some patterns that colour is changed systematically, and in a continuous way, in conjunction with certain symmetry operations. Such designs are called counterchange designs, a term used by Christie (1910, chapter 10 of 1969 edition) and Gombrich (1979, p 89), as well as Woods (1936). Schwarzenberger (1984) identified over 100 research papers or other publications dealing with counterchange or colour symmetry from a mathematical perspective. Although the perfect counterchange colouring of some patterns may be ascertained by experimentation and common sense, it is group theory which offers the key to finding all possible types of perfectly coloured patterns. A readily understandable introduction to group theory was provided by Budden (1972). By introducing systematic change of colour in association with certain

p6mm

11.32 All-over pattern class p6mm.

Recognition, differentiation and classification of regular patterns

451

symmetry operations, a total of 17 two-colour counterchange border patterns is possible. Examples of the 17 possibilities based on the seven primary structures are shown schematically in Fig. 11.33. Based on the 17 primary all-over pattern structures, 46 counterchange possibilities arise where k equals two and 23 possibilities where k equals three. Ninety-six possibilities unfold where k is equal to four. Systematic colouring of this type is found in various contexts such as the Alhambra Palace (Granada, Spain) and has been applied by a number of artists (the most prominent being M C Escher). An early explanation and enumeration of two-colour counterchange pattern possibilities was given by Woods (Woods, 1935a, b, c and 1936). Further influential work was produced by Belov and Tarkhova (1964), Senechal (1979), Loeb (1971), Lockwood and Macmillan (1978), Wieting (1982). Grunbaum and Shephard (1987), Schattschneider (1986) and Washburn and Crowe (1988). A series of original two-colour counterchange illustrations was provided previously by Hann (2003b). As

11.33 Schematic illustrations of the 17 classes of two-colour counterchange border patterns.

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Modelling and predicting textile behaviour

Table 11.2 Enumeration of counterchange possibilities for each of the 17 primary pattern classes, for values of k up to 12. Source: Wieting (1982) and Grunbaum and Shephard (1987, p 407) Pattern class

Number of colours 2

3

4

5

6

7

8

9

10

11

12

p1 pg pm cm p2 pgg pmg pmm cmm p3 p31m p3m1 p4 p4g p4m p6 p6m

1 2 5 3 2 2 5 5 5 – 1 1 2 3 5 1 3

1 2 2 2 1 1 2 1 1 2 2 2 – – – 2 2

2 4 10 7 3 4 11 13 11 1 1 1 5 7 13 1 2

1 2 2 2 1 1 2 1 1 – – – 1 – – – –

1 5 11 7 2 4 11 9 8 1 5 4 2 2 2 5 11

1 2 2 2 1 1 2 1 1 1 – – – – – 1 –

2 7 16 13 4 7 19 21 21 – 1 1 9 13 28 1 3

2 3 3 3 2 2 3 2 2 3 3 3 1 1 1 3 3

1 6 12 8 2 5 12 10 9 – – – 4 3 3 – –

1 2 2 2 1 1 2 1 1 – – – – – – – –

2 11 23 17 3 9 26 25 22 4 7 7 9 10 16 8 20

Total

46

23

96

14

90

15

166

40

75

13

219

pointed out by Washburn and Crowe (1988, chapter 5), although there is no universally accepted international notation for the 46 two-colour counterchange pattern classes, the notation proposed by Belov and Tarkhova (1964, p 211) appears to be the most widely adopted. The notation is an adaptation of that used to classify the 17 primary all-over pattern classes, but is slightly more complex. Washburn and Crowe (1988) gave an easily understandable explanation, accompanied by relevant schematic illustrations. Woods (1936) himself presented a now well-known series of two-colour illustrations, but used a rather unwieldy notation. The principles governing two-colour counterchange patterns are readily applicable to three-colour and higher-colour counterchange patterns. Colours may change (or may be preserved) in association with each symmetry operation, to produce a given permutation of colours across the pattern. The prefix k may be used to denote the number of colours available. Table 11.2 summarises the possibilities for all 17 primary classes for values of k up to 12. Further discussion and illustration was given by Hann and Thomas (2007).

11.8

Conclusions

Pattern analysts are aware that the construction of regular repeating patterns is governed by the use of up to four geometric actions known as symmetry

Recognition, differentiation and classification of regular patterns

453

operations, or symmetries, which, when applied in combinations across the plane, can produce 17 distinct structures. These 17 structures act as the geometric basis for 17 possible classes of all-over patterns (or plane patterns). The series of articles authored by Woods is of great importance historically (Woods, 1935a, b, c and 1936). Woods assembled a wide range of concepts taken largely from the developing discipline of crystallography and used these to present a comprehensive and systematic means by which motifs, border patterns and all-over patterns could be identified, differentiated and classified. Although the vast majority of 20th century publications dealing with symmetry were largely understandable only to a mathematically aware audience, there were several notable exceptions. In the latter few decades of the 20th century, much was done by scholars such as Schattschneider and Crowe (both eminent mathematicians) and Washburn (a social anthropologist) to disseminate academic knowledge of symmetry beyond its parent disciplines of crystallography and mathematics. Washburn and Crowe (1988) were at the forefront internationally in the development of symmetry classification as a systematic tool to assist cultural, archaeological and historical research. Meanwhile Schattschneider, the eminent authority on the tessellating patterns of M. C. Escher, provided an academic bridge between the worlds of art and mathematics; importantly, her work proved accessible to observers on both sides of this academic bridge (Schattschneider, 1978, 1986 and 1990). By the first decade of the 21st century, symmetry had entered main stream high school education in North America and parts of Europe, and knowledge of the relevant principles and their applicability had been disseminated widely through numerous internet sites, and through academic conferences, such as Bridges (2007, online), aimed at developing school curricula. Using symmetry as the guiding force for pattern recognition and construction, several software packages and plug-ins, such as those available from Artlandia (2008, online), were developed to assist designers and analysts alike.

11.9

References

Artlandia (2008). SymmetryWorks – Powerful pattern design plug-in, From Artlandia – All patterns, all easy [online]. [Accessed 20/04/08.] Available from World Wide Web: Belov N V and Tarkhova T N (1964), ‘Dichromatic plane groups’, in Coloured Symmetry, Shubnikov A V and Belov N V (eds), Pergamon Press, New York. Birkhoff G D (1933), Aesthetic Measure, Harvard University Press, Cambridge, MA. Bovill C (1996), Fractal Geometry in Architecture and Design, Birkhauser, Boston. Brainerd G W (1942), American Antiquity, 8(2), 164. Bridges (2007). Bridges Resource Center, From The Bridges Organization – Mathematical connections in art, music, and science [online]. [Accessed 20/04/08.] Available from World Wide Web:

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Modelling and predicting textile behaviour

Budden F J (1972), The Fascination of Groups, London, Cambridge University Press. Buerger M J and Lukesh J S (1937), Technology Review, 39(8), 338. Christie A H (1910), Traditional Methods of Pattern Designing, Clarendon Press, Oxford and (1969) as Pattern design. An Introduction to the Study of Formal Ornament, Dover, New York. Coxeter H S M (1948), Regular Polytopes, Methuen, London. Coxeter H S M (1969), Introduction to Geometry, John Wiley and Sons, New York. Crowe D W (1986), ‘The mosaic patterns of H J Woods’, in Symmetry: Unifying Human Understanding, Hargittai I (ed), Pergamon, New York, 407–411. Crowe D W and Washburn D K (1986), ‘Flow charts as an aid to the symmetry classification of patterned design’, in Material Anthropology: Contemporary Approaches to Material Culture, Reynolds B and Stott M (eds), University Press of America, Lanham, Maryland, 69. Day L F (1903), Pattern Design, Batsford, London. Federov E S (1885), Elements of the Theory of Figures (in Russian), Imperial Academy of Science, St. Petersburg. Fedorov E S (1891), ‘Symmetry in the plane’ (in Russian), Zapiski Rus. Mineralog Obscestva, Ser. 2, 28, 345–90. Fricke R and Klein F (1897), Vorlesungen über die theorie der automorphen funktionen, vol.1, Teubner, Leipzig. Gombrich E H (1979), The Sense of Order. A Study of the Psychology of Decorative Arts, London, Phaidon. Grunbaum B and Shephard G C (1987), Tilings and Patterns, New York, Freeman. Hann M A (1992), ‘Symmetry in regular repeating patterns: case studies from various cultural settings’, Journal of the Textile Institute, 83(4), 579–90. Hann M A (2003a), ‘The fundamentals of pattern structure. Part I: Woods revisited’, Journal of the Textile Institute, 94 (2, nos 1 and 2), 53–65. Hann M A (2003b), ‘The fundamentals of pattern structure. Part II: the counter-change challenge’, Journal of the Textile Institute, 94 (2, nos 1 and 2), 66–80. Hann M A (2003c), ‘The fundamentals of pattern structure. Part III: the use of symmetry classification as an analytical tool’, Journal of the Textile Institute, 94 (2, nos 1 and 2), 81–8. Hann M A and Lin X (1995), Symmetry in Regular Repeating Patterns, The University Gallery, Leeds. Hann M A and Russell S J (2003), ‘Minding the gap – The principles and applications of regular tessellations’, in Fibrous Assemblies at the Design and Engineering Interface, Book of Proceedings, International Textile Design and Engineering Conference, INTEDEC 2003, 22–24 September 2003, published by Heriot-Watt University, Edinburgh. Hann M A and Thomas B G (2007). ‘Beyond black and white – A note concerning threecolour-counterchange patterns’, Journal of the Textile Institute, 98(6), 539–47. Hann M A and Thomson G M (1992), The Geometry of Regular Repeating Patterns, The Textile Institute, Manchester. Hargittai I (ed) (1986), Symmetry: Unifying Human Understanding, New York, Pergammon. Hargittai I (ed) (1989), Symmetry 2: Unifying Human Understanding, New York, Pergammon. Hargittai I and Hargittai M (1994), Symmetry: a Unifying Concept, Shelter Publications, Bolinas, California.

Recognition, differentiation and classification of regular patterns

455

Henry N F M and Lonsdale K (eds) (1952), International Tables for X-ray Crystallography: Vol 1 Symmetry Groups, Kynock Press, Birmingham, UK. Horne C E (2000), Geometric Symmetry in Patterns and Tilings, Cambridge, Woodhead Publishing. Jawson M A (1965), An Introduction to Mathematical Crystallography, London, Longman, Green and Co. Jones O (1856, reprinted 1982), The Grammar of Ornament, New York, Van Nostrand Reinhold Company. Kepler J (1619), Harmonices Mundi Libri Quinque (Five books on the harmony of the world), Joannes Plancus, Lincii, Austria, translated with commentary by Casper M (1978), Weltharmonik (Harmony of the World), R. Oldenbourg-Verlag, Munich. Lockwood E M and Macmillan R H (1978), Geometric Symmetry, London, Cambridge University Press. Loeb A L (1971). Colour and Symmetry, New York, Wiley-Interscience. Martin G E (1982), Transformation Geometry: an Introduction to Symmetry, SpringerVerlag, New York. Meyer F S (1894), Handbook of Ornament: a Grammar of Art, Industrial and Architectural, 4th edition, Hessling and Spielmayer, New York. Niggli P (1926), Zeitschrift fur Kristallographie, 1926, 63, 255. Pólya G and Niggli P (1924), ‘Über die analogie der kristallsymmetrie in der ebene’, Zeitschrift fur Kristallographie, 60, 278–98. Schattschneider D (1978), ‘The plane symmetry groups: their recognition and notation’, American Mathematical Monthly, 85(6), 439–50. Schattschneider D (1986), ‘In black and white: how to create perfectly colored symmetric patterns’, Computers and Mathsematics with Applications, 12B(3/4), 673–95. Schattschneider D (1990), Visions of Symmetry. Notebooks, Periodic Drawings and Related Works of M. C. Escher, Freeman, New York. Schwarzenberger R L E (1974), ‘The seventeen plane symmetry groups’, The Mathematical Gazette, 58, 23–131. Schwarzenberger R L E (1984), ‘Colour symmetry’, Bulletin of the London Mathematical Society, 16, 209–40. Scivier J A and Hann M A (2000a), ‘The application of symmetry principles to the classification of fundamental simple weaves’, Ars Textrina, 33, 29–50. Scivier J A and Hann M A (2000b), ‘Layer symmetry in woven textiles’, Ars Textrina, 34, 81–108. Senechal M (1979), ‘Colour groups’, Discrete Applied Mathematics, 1, 51–73. Shubnikov A V and Koptsik V A (1974), Symmetry in Science and Art, New York, Plenum Press. Stevens P S (1984), Handbook of regular patterns. An introduction to symmetry in two dimensions, Cambridge, MA, MIT Press. Stevenson C and Suddards F (1897), A Textbook Dealing with Ornamental Design for Woven Fabrics, London, Methuen. Thomas B G and Hann M A (2007), Patterns in the Plane and Beyond: Symmetry in Two and Three Dimensions, no 37 in the Ars Textrina series, University of Leeds International Textiles Archive (ULITA) in association with Leeds Philosophical and Literary Society. Walker T and Padwick R (1977), Pattern: Its Structure and Geometry, Ceolfirth Press, Sunderland Arts Centre. Washburn D K and Crowe D W (1988), Symmetries of Culture: Theory and Practice of Plane Pattern Analysis, University of Washington Press, Seattle.

456

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Washburn D K and Crowe D W (2004), Summetry Comes of Age: The Role of Pattern in Culture, University of Washington, Seattle. Wieting T W (1982), The Mathematical Theory of Chromatic Plane Ornaments, New York, Marcel Dekker. Weyl H (1952), Symmetry, Princeton, Princeton University Press. Woods H J (1935a), ‘The geometrical basis of pattern design. Part 1: point and line symmetry in simple figures and borders’, Journal of the Textile Institute, Transactions, 26, T197–T210. Woods H J (1935b), ‘The geometrical basis of pattern design. Part 2: nets and sateens’, Journal of the Textile Institute, Transactions, 26, T293–T308. Woods H J (1935c), ‘The geometrical basis of pattern design. Part 3: geometrical symmetry in plane patterns’, Journal of the Textile Institute, Transactions, 26, T341–T357. Woods H J (1936), ‘The geometrical basis of pattern design. Part 4: counterchange symmetry in plane patterns’, Journal of the Textile Institute, Transactions, 27, T305–T320.

12

Mathematical and mechanical modelling of 3D cellular textile composites for protection against trauma impact

X. Chen, The University of Manchester, UK

Abstract: This chapter describes the mathematical and mechanical modelling of the 3D cellular textile composites whose intended application is in protection against low velocity impact. Mathematical modelling of the 3D cellular textiles has led to algorithms used in computer aided design (CAD) software for such cellular structures of composites. The mechanical modelling of the cellular textile composites reveals the influence of the structural parameters on the impact performance of cellular textile composites. Key words: cellular composites, 3D textiles, energy absorption, transmitted force, simulation.

12.1

Introduction

Use of cellular structures to reduce weight and in many instances to conserve materials is abundant in nature and the various constructions of this type result in materials with high strength-to-weight ratios. Performance is optimised primarily by geometric arrangement of the material in space to form an interconnected network of cells with strut edges (Gibson and Ashby, 1997). Much effort has been made in the manufacture and use of artificial cellular materials in recent decades. They can be manufactured relatively easily from many different materials including metals, polymers, ceramics, paper and carbon, to varying densities (Bitzer, 1997). Cellular materials are characterised by their light weight and impact energy absorption. Consequently, they are widely used for shock mitigation in applications ranging from aerospace structures, where high performance is required, to general packaging for everyday transportation purposes. Much work has been carried out to study the characteristics of cellular materials, natural and manufactured alike, with irregular-shaped cells such as foams and regular-shaped cells such as honeycombs. Reid and Peng (1997) studied the general characteristics of the stress–strain curves of wood materials including oak, redwood, pine and balsa along the axial and the radial directions. Ruan et al. (2002) worked on a typical CYMAT aluminium foam material, a manufactured irregular cellular material, because of its strain–stress 457

458

Modelling and predicting textile behaviour

relationship and the localised cell deformation. Theoretical and experimental studies by Papka and Kyriakides (1998) were carried out on an aluminium honeycomb with regular cell shapes crushed in the X2 direction, leading to a good agreement between the theoretical and experimental studies. There is increasing interest in textile composites, including honeycomb textile composites, on account of their attractive properties for a variety of applications. For many composite applications, such as those in the automotive and aerospace industries, reduction in component weight is highly desirable. A wide range of fabrics available for composite reinforcement in the field of textile structural composites was reviewed by Ko (1999). Systematic work at the University of Manchester has been reported in engineering and characterising three-dimensional (3D) honeycomb composites for impact applications. Chen (2008) reported work that has been carried out on the mathematical modelling of integrated cellular woven preforms and on a Computer aided design (CAD) tool for designing cellular fabrics with various structural parameters. Tan and Chen (2005), Tan et al. (2007) and Yu and Chen (2004) carried out finite element (FE) analysis adopting quasi-static and dynamic approaches respectively, reporting on the influence of structural parameters on the mechanical behaviour of 3D honeycomb composites. This chapter explains the structural modelling, design and manufacture, experimental test and mechanical modelling of 3D cellular textile composites against trauma impact. Two general types of 3D cellular structure can be produced by weaving, one with hexagonal cells and the other with quadrilateral cells. The length of each of the cell edges (also termed cell walls) may or may not be made to be the same, therefore resulting in complicated cellular shapes. Cellular structures with regular cell shapes are easier to manufacture using the conventional textile technology. In the case of the regular hexagonal cellular shape, the cellular fabric can be opened to form a 3D cellular structure with uneven surfaces. For the quadrilateral cell shapes on the other hand, if the top and bottom edges of the cell are kept parallel to each other, the cellular fabric will open to a 3D cellular structure with flat surfaces. The 3D cellular fabric structures with uneven and flat surfaces are illustrated in Fig. 12.1, where each line segment represents a fabric session.



(a)

(b)

12.1 Cross-sectional views of cellular fabric structures, (a) with uneven surfaces, (b) with flat surfaces.

Mathematical and mechanical modelling of 3D cellular composites

459

As shown in Fig. 12.1, both types of cellular fabric are made on basis of the multilayer principle. Layers of fabric are combined with their adjacent layers and are separated from a joined layer at the pre-designed interval. For cellular fabrics with regular hexagonal cells, the diagonal cell edges at each side of a cell are the same length, whereas in the trapezoidal cellular fabrics, a type of quadrilateral, the lengths of diagonal cell edges have to be made longer than those of the horizontal ones. Figure 12.2 shows the relationships between the cellular fabrics and their constituent fabric layers.

12.2

Mathematical description of cellular textile structures

12.2.1 The hexagonal cellular structure Chen et al. (2004) has pointed out that a hexagonal cellular structure has two columns of cells containing n and (n–1) cells respectively. The whole structure is made from 2n layers of fabrics. It is assumed in this context that the tunnels formed by the cells run in the weft direction, although they can be arranged to go in the warp direction too. One repeat of the cellular structure can be divided into four regions, that is, region I, II, III and IV, as shown in Fig. 12.3. Region I corresponds to the section of the 3D cellular structure where the fabric layers are all separated from each other; region II is where the adjacent layers join together at an alternate interval; region III is the same as region I; and region IV is again Warp

Weft

(a)

(b)

12.2 Relationships between the cellular fabric and its constituent fabric layers, (a) hexagonal cells, (b) trapezoid cells.

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Modelling and predicting textile behaviour



I

II

III

IV



l f

l b

l f

lb

(a)

I

II

III

IV

(b)

12.3 Division of a cellular structure into regions, (a) cellular fabric woven before opening, (b) cellular fabric after opening.

the joining section but the joining layers are different from those in region II. Because of the nature of weaving, a cellular fabric of this type is woven with all cells flattened as indicated in Fig. 12.3(a), and the cellular structure is achieved when the fabric is opened up and consolidated as illustrated in Fig. 12.3(b). According to the definition of a cell, regions II and IV correspond to the bonded walls, with lb being the length, and regions I and III the free walls, with, lf being the length. Figure 12.3(b) shows only part of the cellular structure that would open up to from Fig. 12.3(a). A woven cellular structure can be defined by specifying a group of structural parameters. The following general coding format is used to denote a particular cellular structure:

xL(lb + lf)Pq

where x is the number of fabric layers used to form the cellular structure, lb is the length of bonded wall measured in the number of picks, lf is the length of free wall measured in the number of picks, q is the opening angle between the free cell walls and the horizontal line, L is used to denote ‘layer’ and P is used to denote ‘pick’. There are situations when the lengths of free and bonded walls are the same, that is lb = lf = y (say). In such a case, the coding format can be reduced to xLyPq. Further, when the opening angle of the cell is 60°, the coding format becomes xLyP. In all the above coding expressions, x, lb and lf are integers and x ≥ 2, lb ≥ 1, lf ≥ 1. According to the format, a 4L6P cellular structure stands for a cellular structure comprising four layers of fabric, where the length of both free and bonded walls is 6 picks and the opening angle of the cells is 60°. On the other hand, code name 8L(4+3)P refers to a cellular structure made from eight layers of fabric, where the lengths of the bonded and free walls are 4 picks and 3 picks, respectively, with a cell opening angle of 60°.

Mathematical and mechanical modelling of 3D cellular composites

461

12.2.2 The quadratic cellular structure The cross-section of a quadratic cellular structure can be created based on the shape of the cell and the arrangement of cells. A cell can be fully defined either by the coordinates of its nodes, or by lengths and angles of fabric sections (Chen and Wang, 2006). The arrangement of cells can be described by the number of cell levels and the shift between the levels. A quadratic cellular structure becomes more complicated when the cells have different geometrical features and when the cells are arranged differently in the cross-section. Figure 12.4 illustrates a quadratic cellular structure with trapezoidal cells. In general, the geometrical model of a quadratic cellular structure with a trapezoidal cross-section can be specified using the following parameters: ∑ ∑ ∑

A, B, C, D – the lengths of trapezoidal sides, measured in number of picks; Rpi – the shift between levels i and i + 1; a/A or b/B – the ratio between long and short fabric sections or a – the expansion angle.

Obviously, cos a = a/A. As can be seen from Fig. 12.4, the structures with more than one level can be classified into three groups, namely: ∑ ∑ ∑

Rpi = Rpi+1 = 0 Rpi ≠ 0 and Rpi = Rpi+1 Rpi ≠ 0 and Rpi ≠ Rpi+1

where i = 1, 2, …, n, n being the total number of levels in a trapezoidal structure.

12.2.3 A complex cellular structure For the cellular structures described above, the cellular tunnels run in either the warp or the weft direction. In a recent patent (Chen and Zhang, 2006), Rpi   A

Level (row) i + I

Level (row) i

C

B

D

a

a

b

12.4 Geometrical description of a hollow structure with trapezoidal cells.

462

Modelling and predicting textile behaviour

a woven cellular structure was described, where the cellular tunnels run in any direction and the relationship between the tunnels may be one above the other, or intersecting each other. Figure 12.5 is a photograph of such a sample.

12.3

Computer aided design/computer aided manufacturing (CAD/CAM) of 3D cellular woven fabrics

12.3.1 Mathematical modelling of weaves for two-layer of fabrics It has been a common practice to use a 2D matrix to record data for a weave (Li et al., 1988; Milasius and Reklaitis, 1988; Chen et al., 1996). In the case of a single-layer fabric, a 2D binary matrix is used to represent the weave, whose element values are either 0 or 1. ‘1’ indicates a warp-over-weft crossover, and ‘0’ means a weft-over-warp crossover. The position of each element in the matrix is located by a coordinates (x, y) where x indicates the xth column from the left and y the yth row from the bottom. This approach is adopted in generating the weaves for hexagonal cellular structures (Chen et al., 2004). For hexagonal cellular structures, if there are n cells in the thickness direction, then 2n layers of single layers of fabrics must be used. The adjacent layers of the 2n fabrics will be joined and separated from one another in a predesigned pattern. Based on such specifications, a cross-sectional view of a hexagonal cellular fabric can be defined. Weaves will be assigned to the single layers as well as the joined layers. Once all these layers have received weaves, respectively, all these weaves will be combined to form the overall weave for a hexagonal cellular structure based on the layer relationship.

12.5 Cellular structure with complex tunnel relationships.

Mathematical and mechanical modelling of 3D cellular composites

463

Consider the combination of weave of two single layer fabrics. Let M1 and M2 be the matrices for two single layer fabric sections. In order to unify the repeat size of the two layers, the lowest common multiple (lcm) is calculated based on the dimensions of these two matrices. Assume that d1 is the dimension of M1 and d2 that of M2. If lcm(d1, d2) is equal to d1 (or d2), the dimension of M2 (or M1) will be expanded to the same size as d1 (or d2). If lcm(d1, d2) is larger than both d1 and d2, then M1 and M2 will have to be expanded to the new dimension lcm(d1, d2). Let the enlarged matrix be ME. The elements of this matrix are assigned as follows:

MEk (i,j ) = M k (i%dk , j %dk ) i, j = 1, 2,..., lcm (d1 , d2 ) k = 1, 2

[12.1]

The final task in this procedure is to combine the two enlarged matrices into an overall matrix MF. Equation [12.2] describes an algorithm from which MF is enumerated:



Ï ME1(i /2, j /2 ) Ô 0 Ô MF(i, j ) = Ì 0 Ô Ô ME E2 ( i /2 , j /2 ) Ó

if i % 2 = 0

and j % 2 = 0

if i % 2 = 0

and j % 2 = 1

if i % 2 = 1

and j % 2 = 0

if i % 2 = 1

and j % 2 = 1

[12.2]

Note that i%2 and j%2 are integer divisions, returning only the value of the integer part of the result. For instance, 4%3 = 1. Figure 12.6 illustrates the weave combination from two single-layer weaves, leading to the final matrix. The same algorithm can be applied to the creation of a weave matrix from more than two single-layer weaves.

12.3.2 Generation of weaves for hexagonal cellular structures For cellular structures, two columns of cells represent one structural repeat, as indicated in Fig. 12.1. One repeat can be further divided into four areas and this is illustrated in Fig. 12.7, with two areas (I and III) identical to each other. For each of these areas, the algorithm discussed in the previous section is applied and extended to create the combined weave. As can be seen, area I contains only single layers, area II has two single layers (top and bottom) and several two-layers, area III is the same as area I, and area IV has only two-layer fabrics. A base matrix is a rectangular matrix which is prepared in order to contain the weave for an individual area. If all component weave matrices are assigned to weaves with the same weave repeat, the width and the height of the base matrix will be equal to the product of the number of layers and the

464

Modelling and predicting textile behaviour 1 0 1 0 1 0 0 1 0 1 0 1

Weave one 1

0

0

1

1 0 1 0 1 0

Enlarge

0 1 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1

Weave two 1 1 0 0 1 1

1 1 0 1 1 0 0 1 1 0 1 1

Enlarge

1 0 1 1 0 1

1 0 1

1 1 0 1 1 0 0 1 1 0 1 1

1 1 1 0 1 1 1 0 1 1 1 0

1 0 0 0 1 0 1 0 0 0 1 0

1 0 1 1 1 0 1 1 1 0 1 1

1 0 1 0 0 0 1 0 1 0 0 0

Final matrix 1 0 1 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 0 1 1 1 1 1 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 1 0 0 1 0 0

1 0 1 0 0 0 1 0 1 0 0 0

1 0 1 1 1 0 1 1 1 0 1 1

0 0 1 0 1 0 0 0 1 0 1 0

1 0 1 1 0 1

12.6 Combination of two single-layer weaves.

1 2 3 4

I

II

III

IV

12.7 One repeat of a hexagonal cellular structure.

dimension of each layer. Otherwise, the dimension of the base matrix will be the lowest common multiple of all the dimensions of the component weave matrices. One repeat in the cellular structure has three different areas (areas I and III being the same), therefore there are three base matrices involved in a weaving diagram for the cellular structure. Each base matrix is further divided along the warp direction by the dimension of weave matrix. Take Fig. 12.8, for example, where a 2/1 twill weave is used for all four layers in area I. Counting from the left-hand side, the first three columns correspond to fabric layer 1, the second three to layer 2, the third three to layer 3 and the last three to layer 4. Each section in the base matrix is constructed by superimposing the weave matrix for each layer and by insertion of ‘additional lifting points’ (Chen et al., 1996), which are added to reflect the multi-layer weaving principle that when a pick for a lower layer is to be inserted, all warp ends for layers above must be lifted. This finished base matrix will represent the overall weave for the area concerned.

Mathematical and mechanical modelling of 3D cellular composites

465

12.8 Creation of an example base matrix (2/1 twill weave, area I).



Area I

Area II

Area IV

12.9 Base matrices.

Area II includes two single layers and several two-layer fabrics and area III has only two-layer fabrics. In a similar way, the base matrices for areas II and IV can be generated. Figure 12.9 lists the base matrices for areas I, II and IV. The final weave for the whole hexagonal cellular structure is the combination of these base matrices for three areas. Figure 12.10 is the overall weave responsible for weaving one repeat of the hexagonal cellular structure described in Fig. 12.7, in the sequence of areas I, II, III and IV.

12.3.3 Generation of weaves for quadratic cellular structures Subdivision of a quadratic structure In order to generate the weave for the whole cellular structure, the repeat unit of the quadratic structure must be determined. Figure 12.11 shows an example of a three-layer single level quadratic structure with three repeat units and subdivision of each repeat unit into four different ‘areas’ (Chen and Wang, 2006). Area 1 includes three layers of fabric, which have different lengths and possibly different weaves. All layers in this area are single-layer fabrics. The length ratio of these three fabric layers in area 1 is defined as l1 : l2 : l1. However, picks of weft yarns must be distributed evenly into the three fabric layers. It is necessary to divide the number of picks for each section into the same number of groups.

466

Modelling and predicting textile behaviour

Area IV

Area III

Area II

Area I

12.10 Final weave for the hexagonal cellular structure.

One repeat I Layers

II III

a 1

2

3

4

Areas

12.11 Subdivision of a repeat unit.

Let us denote the number of such groups by k. For a quadratic structure with m layers, group i of weft yarn distribution is recorded as w1i : w2i : w3i : … : wmi, where all wqi (q = 1, 2…, m and i = 1, 2, …, k) are integers. When the weft density is taken into consideration, the following relation holds:

k

k

k

k

i =1

i =1

i =1

i =1

S w1i : S w2i : … : S wqi : … : S wmi = l1 : l2 : … : lq … : lm

Mathematical and mechanical modelling of 3D cellular composites

467

Area 2 includes two sections of fabric, each having the same length and possibly different weaves. However, the top fabric section is a combined layer made from top and middle fabric layers from area 1, and the bottom section is a single layer fabric, the same as layer III in area 1. The weave for area 3 is the same as that for area 1, and area 4 is similar to area 2 except that the top section is a single layer and the bottom section is a stitched layer in this case. Weave generation for each area The weave for each fabric layer is recorded in a 2D matrix. The 2D weave matrix representing the overall weave for each area can be generated by combining the 2D matrices of all fabric sections in the area concerned. Suppose there are n fabric layers in the area of concern. The weave for layer i is recorded in matrix Mi ( i = 1, 2, …, n), and the element of this matrix at the xth warp and the yth weft is Mi(x, y) (1 ≤ x ≤ rie, 1 ≤ y ≤ rip), where rie is the number of warp ends in Mi, and rip the number of weft picks in Mi. li is the length of layer i. The warp dimension of all constituent matrices, rlcm, can be found by calculating the lowest common multiple (LCM) of the warp repeats of all constituent weave matrices:

rlcm = LCM (r1e, r2e,…, rne)

[12.3]

Accordingly, the warp dimensions of the constituent weave matrices are expanded to rlcm. The expanded weave matrix for the constituent layer i is denoted by M i¢ and its elements can be obtained by repeating the weave of a single layer rlcm/rie times:

M i¢ (x,y) = Mi (x mod rie, y)

[12.4]

The warp repeat rie¢ of the expanded matrix M i¢ is now rlcm, while its weft repeat rip¢ remains rip. When a similar treatment is applied, the weave matrix for fabric layer i will change from M i¢ to M i¢¢ by repeating the rows of M i¢ until its weft dimension is equal to the layer length li:

M i¢¢ (x, y) = M i¢ (x, y mod rip )

[12.5]

Based on what has been achieved, all constituent weave matrices M i¢¢ in this area will be combined to generate the overall weave matrix W for this area. The elements of matrix W, W(x, y), are assigned by the following expression:

468

Modelling and predicting textile behaviour

i =m i =m –1 i =m Ï Ô 0 when x > S rie¢¢, S rip¢¢¢ < y < S rip¢¢, 0 < m < n +1 i =1 i =1 i =1 Ô W (x, y) = Ì i =m –1 i =m i =m –1 Ô 1 when S r ¢¢, < x < S r ¢¢, y > S r ¢¢, 0 < m < n +1 ip ie ie Ô i =1 i =1 i =1 Ó [12.6] i = m –1 i = m –1 Ê ˆ M i¢¢ Á x – S rie¢¢, y – S rip¢¢˜ i =1 i =1 Ë ¯

when



i =m –1

i =m

i =m –1

i =m

i =1

i =1

i =1

i =1

S rie¢¢ < x < S rie¢¢, S rip¢¢ < y < S rip¢¢, 0 < m < n + 1

Weave for the entire quadratic structure Based on the aforementioned, the weave for the entire repeat of the quadratic structure can be obtained by combining the 2D matrices for the different areas in the corresponding sequence. The procedure for combining all the areas may involve enlargement of the matrix for each area in order to give them the same warp ends. Consider a three-layer quadratic structure, where A = B = 6 picks, C = D = 4 picks, and A/a = 3:2 (a ≈ 48.2o). The weaves used for the top, middle and bottom fabric layers are 1/1, 2/1 and 1/1, respectively. The weave for one repeat unit of this quadratic structure is created and is shown in Fig. 12.12.

12.4

Experimental study of properties of 3D cellular composites

In order to investigate systematically the mechanical performance of the cellular textile composites, 14 cellular composites are designed in four groups, that is an opening angle group, a cell size group, a length ratio group and a similar thickness group. The details of these four groups of cellular composites are listed in Table 12.1. In carrying out the test, an impact head weighing 0.55 kg was raised to a typical height of 1.54 m above the top surface of the specimen, thus the typical impact was 8.3 J. Under this setup, the impact force acting on the anvil is 16.490 kN (Chen et al., 2008a, 2008b). Table 12.2 shows the structural features and the energy absorption and the transmitted force, where E is the energy absorption, K is the impact energy, Ftrans is the transmitted force and fatt is the attenuation factor.

Mathematical and mechanical modelling of 3D cellular composites

469

Area 4

Area 3

Area 2

Area 1

12.12 Weave diagram created for a three-layer quadratic structure. Table 12.1 Details of cellular composite samples Group Group Group Group Group

Composite 1: Opening angle 2: Cell size 3: Length ratio 4: Similar thickness

8L6P30, 8L6P45, 8L6P60, 8L6P75, 8L6P90 8L3P60, 8L4P60, 8L5P60, 8L6P60 8L(3 + 6)P60, 8L(4 + 6)P60, 8L6P60, 8L3P60, 8L(4 + 3)P60, 8L(6 + 3)P60 4L6P60, 6L4P60, 8L3P60

Table 12.2 Structural features and mechanical performance of cellular composites Composite

Density (g cm3)

Thickness E (J) (mm)

E/K (%)

Ftrans (kN)

fatt (%)

4L6P60 6L4P60 8L3P60 8L4P60 8L5P60 8L6P30 8L6P45 8L6P60 8L6P75 8L6P90 8L(3+6)P60 8L(4+6)P60 8L(4+3)P60 8L(6+3)P60

0.081 0.118 0.155 0.099 0.101 0.115 0.083 0.072 0.085 0.073 0.089 0.086 0.121 0.112

29.2 30.4 31.6 40.4 49.2 35.7 48.3 58.0 64.1 66.2 35.7 35.7 31.6 31.6

73.9 83.9 87.8 92.1 92.7 98.9 98.4 89.7 87.5 84.7 97.3 86.1 86.3 86.3

1.272 0.499 0.950 0.602 0.550 0.491 0.416 0.345 0.285 0.268 0.398 0.401 0.769 0.664

92.3 97.0 94.2 96.3 96.7 97.0 97.5 97.9 98.3 98.4 97.6 97.6 95.3 96.0

6.31 7.18 7.46 7.78 7.83 8.41 8.30 7.54 7.33 7.05 8.21 7.17 7.33 7.28

470

Modelling and predicting textile behaviour

12.4.1 Influence of cell opening angle The opening angle group involves composites 8L6P30, 8L6P45, 8L6P60, 8L6P75 and 8L6P90. These composites are made from the same fabric as for reinforcement but opened up to different angles, indicated by the last two digits in the codes. Figure 12.13(a) shows the relationship between the impact force and the displacement of the cellular composites during the 0.45 0.4

8L6P30 8L6P45

0.3 8L6P60

0.25

8L6P75

0.2 0.15 0.1 8L6P90

0.05 0 0

1

2

3 Displacement (cm)

4

5

(a) 9 8.5 Energy absorption (J)

Contact force (kN)

0.35

8 7.5 7 6.5 6 8L6P30

8L6P45

8L6P60 (b)

8L6P75

8L6P90

12.13 Opening angle influence on energy absorption, (a) impact force–displacement curves, (b) energy absorption.

6

Mathematical and mechanical modelling of 3D cellular composites

471

impact. No obvious trend can be followed in the curves and in the shape of the area covered by the curves to indicate that the opening angle is a major mechanism by which the impact energy was absorbed. Other structural parameters, such as the overall thickness of the cellular composites, must have played important roles too. Figure 12.13(b) depicts the influence of opening angle on the energy absorption of this group of cellular composites. The message is clear that an increase in the opening angle leads to a decrease of energy absorption. This is believed to be because the cellular composites with smaller opening angles present lower resistance to the bending of its angled cell walls and therefore create a larger deformation. All the composites made from 8L6P reinforcement contain four rows of cells, which in this case seem to be sufficiently thick not to be totally crushed by the application of the impact energy. In contrast, when the 4L6P60 composite, which has only two rows of cells, was subjected to the same impact energy, it was completely crushed, leading to lower energy absorption. Figure 12.14(a) shows the time dependency of the transmitted force associated with the five composites in the group. Whilst the peak transmitted force is reduced along with the increase in composite opening angle, it is clear that the peak forces tend to be smoothed when the composites get thicker due to the increase in the opening angle. Figure 12.14(b) displays the maximal transmitted force for the five cellular composites in the group. It shows a clear tendency for the maximal transmitted force to be reduced when the opening angle increases. The thickness of the composites is believed to have an important role to play, together with the cell geometry in the cross-section of the composites.

12.4.2 Influence of cell size All cellular composites in this group are made from eight layers of fabrics involving four regular hexagonal cells, where the six walls of each cell have the same length. The opening angle of the cells in these composites is 60∞. With the same weft density for all reinforcing fabric sections, the cell wall length changes from 3, 4, 5 to 6 picks, resulting in cells with increasing sizes. These composites are 8L3P60, 8L4P60, 8L5P60, and 8L6P60, with 8L3P60 being the thinnest and 8L6P60 the thickest. The force–displacement curves in Fig. 12.15(a) show that composites with smaller cell sizes have a higher impact modulus and those with larger cell sizes have a low impact modulus, leading to harder and softer composite materials, respectively, with similar levels of energy absorption, as indicated in Fig. 12.15(b). This experimental result suggests that for cellular composites that have the same number of cells in a column, cell sizes can be used as the key parameter for altering the softness of the composite material. Also, from the energy absorption perspective, the use of smaller cells in cellular

472

Modelling and predicting textile behaviour

0.6

Transmitted force (kN)

0.5

8L6P30

0.4

8L6P45

0.3

0.2

8L6P60

8L6P75 8L6P90

0.1

0 0

0.005

0.01

0.015 Time (s) (a)

0.02

8L6P60 (b)

8L6P75

0.025

0.03

Maximal transmitted force (kN)

0.6 0.5 0.4 0.3 0.2 0.1 0

8L6P30

8L6P45

8L6P90

12.14 Transmitted force with different opening angles, (a) time dependency of transmitted force, (b) maximal transmitted force.

composites results in less bulky materials with similar energy absorption capability. Impact force attenuation is much affected by the cell size for cellular composites as demonstrated in Fig. 12.16. It is clear that cellular composites with larger cells perform better in that the magnitude of the peak transmitted force is much reduced and the peak transmitted force gets smoothed out. It is also evident that the maximal transmitted force occurs at a later stage in

Mathematical and mechanical modelling of 3D cellular composites

473

0.9 8L3P60 8L4P60 8L5P60 8L6P60

0.8 8L3P60

Contact force (kN)

0.7 0.6

8L4P60

0.5

8L5P60

0.4 0.3

8L6P60

0.2 0.1 0 0

0.5

1

1.5

2 2.5 3 Displacement (cm)

3.5

4

4.5

5

(a) 9 8

Energy absorption (J)

7 6 5 4 3 2 1 0

8L3P60

8L4P60

8L5P60

8L6P60

(b)

12.15 Influence of cell size on energy absorption, (a) impact force– displacement curves, (b) energy absorption.

general as the cell size becomes larger. This is certainly a favourable feature for materials that are to be used for body and limb protection against trauma impact, with composites with larger cells allowing more preparation time for the human body to react to the impact, hence reducing the chances of more serious injuries.

474

Modelling and predicting textile behaviour

1 8L3P60 8L4P60 8L5P60 8L6P60

8L3P60

0.9

Transmitted force (kN)

0.8 0.7 8L4P60

0.6 0.5

8L5P60

0.4 0.3 0.2

8L6P60

0.1 0 0

0.005

0.01

Time (s)

0.015

0.02

0.025

12.16 Influence of cell size on transmitted force.

12.4.3 Influence of length ratio of the free to bonded cell walls Two subgroups of cellular composites, based on the eight-layer construction, have been created to study the influence of cell wall ratio on the impact performance. The first group involves 8L3P60, 8L(4+3)P60 and 8L(6+3)P where the free wall length is kept at 3 picks and the bonded wall length takes values of 3, 4 and 6 picks. In the second subgroup, the free length remains at 6 picks, while the length of the bonded walls change from 3, 4, to 6 picks. The cell opening angle for all the composites is 60º. Figure 12.17(a) plots the impact force–displacement curve for these composites. It is clear that the two subgroups perform distinctly differently. The composites whose length ratio of free wall to bonded wall lf/lb ≤ 1 demonstrated a higher impact modulus than the other group. With the lf/lb ≤ 1 subgroup, it is clear that 8L3P60 whose cell wall length ratio is 1 showed the highest impact modulus and 8L(6+3)P60, with a length ratio of 0.5, displayed the lowest impact modulus in the subgroup. For the subgroup with lf/lb ≥ 1, ductile behaviour was demonstrated for all three samples involved, with the 8L6P60 being the most ductile. Obviously, these two subgroups absorb impact energy in a distinctly different way. The cell wall ratio, therefore, can be used as an important parameter to influence the format in which the cellular composites absorb impact energy. Figure 12.17(b) shows, however, that the total amount of impact energy absorbed by the composites of each subgroup is about the same. The subtle differences in the absorption of

Mathematical and mechanical modelling of 3D cellular composites

475

0.9 0.8

8L(3 + 6)P60 8L(4 + 6)P60 8L6P60 8L3P60 8L(4 + 3)P60 8L(6 + 3)P60

8L3P60

Contact force (kN)

0.7 8L(4+3)P60

0.6

8L(6+3)P60

0.5 0.4

8L(3+6)P60

0.3 0.2 8L6P60

0.1

8L(4+6)P60

0 0

0.5

1

1.5

2 2.5 3 Displacement (cm) (a)

3.5

4

4.5

5

Energy absorption (J)

9 8 7 6 5 4 3 2 1 60

60

)P +3 (6 8L

+3 8L

(4

3P 8L

)P

60

60 6P 8L

)P +6 (4 8L

8L

(3

+6

)P

60

60

0

(b)

12.17 Influence of cell wall ratio on energy absorption, (a) impact force–displacement curves, (b) energy absorption.

impact energy between the composites, especially in the lf/lb ≥ 1 subgroup, require further investigation. Clear influence of the cell wall ratio on transmitted force has been demonstrated through the experiment. Figure 12.18(a) depicts the relationship between the transmitted force and impact time, where distinctive differences are observed between the two subgroups. The subgroup with lf/lb ≥ 1 clearly demonstrated a better capability for impact force attenuation than the subgroup with lf/lb ≤ 1. The maximum transmitted force of all cellular composites in this group is shown in Fig. 12.18(b). It is true for both the lf/lb ≥ 1 and lf/lb ≤ 1

476

Modelling and predicting textile behaviour

1

8L3P60

0.9

8L(4+3)P60

0.8 Transmitted force (kN)

8L(3 + 6)P60 8L(4 + 6)P60 8L6P60 8L3P60 8L(4 + 3)P60 8L(6 + 3)P60

0.7 8L(6+3)P60

0.6 0.5 0.4

8L(4+6)P60

0.3 0.2

8L(3+6)P60

0.1

8L6P60

0 0.005

0.01

1.05 Time (s) (a)

0.02

0.25

60 8L

(6

+3

)P

60 +3 8L

(4

8L

)P

60 3P

6P 8L

60 8L

(4

+6

)P

)P +6 (3 8L

60

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 60

Transmitted force (kN)

0

(b)

12.18 Influence of cell wall length ratio on transmitted force, (a) time dependency of transmitted force, (b) maximal transmitted force.

subgroups that increase in the ratio leads to a higher transmitted force and decrease in the ratio results in a lower transmitted force. It is also evident that the lf/lb ≤ 1 subgroup permits a higher transmitted force than the other subgroup.

12.5

Theoretical characterisation of 3D cellular composites

The use of 3D cellular textiles as reinforcements leads to composites with many potential properties such as light weight, bulky volume and high energy

Mathematical and mechanical modelling of 3D cellular composites

477

absorption. This section introduces modelling of 3D cellular composites, mainly of the hexagonal type, subject to quasi-static and dynamic impact loading.

12.5.1 Quasi-static compression loading In order to study the response of 3D cellular textile composites, four groups of structures are modelled, which have different opening angles, cell sizes, cell wall thickness and wall length ratios. The parameters of a cell and the definition of a cellular structure are illustrated in Fig. 12.19. In the FE modelling, quasi-static compression loads were applied to these models. The walls of the cellular composite are reinforced by woven structures and therefore the material is regarded as orthotropic. In this work, data from a glass composite were used, of which the Young’s moduli are Exx = 13.2 GPa, Eyy = 5.3 GPa and Ezz = 5.3 GPa. Poisson ratios are nxy = nyz = nzx = 0.3 and E yy the shear moduli are Gxy = Gyz = Gzx = = 2.038 GPa. 2(1 + n xy ) In this analysis, the engineering strain of a cellular composite is defined by:

e=



T0 – T1 T0

[12.7]

where T0 is the original thickness and T1 the deformed thickness of the cellular composite. The average strain energy per unit volume of a cellular composite is expressed as: e= E V



[12.8]

lf a lb

tf Free wall

Bonded wall tb lb: Bonded wall length lf: Free wall length tb: Bonded wall thickness tf: Free wall thickness a: Opening angle (a)

(b)

12.19 Definition of (a) a cell and (b) a cellular cross-section.

478

Modelling and predicting textile behaviour

where E is the total strain energy in the cellular composite and V is the total original volume of the cellular structure, V = W0L0T0, in which W0, L0 and T0 are the original width, original length and original thickness of the cellular structure, respectively. Influence of opening angle Figure 12.20 gives details for the FE modelling on quasi-static compression under specific engineering strain and specific load, respectively. When all the other parameters remain the same, a cellular composite with a 60° (isogonic) opening angle leads to the lowest volume density. When compressed by a given strain, a much smaller compression force is taken to deform the composites with small opening angles than those with large opening angles. For instance, the compression force needed to produce an engineering strain of 0.8 is about 600 times larger for the composite with a 90° opening angle than for the one with a 30° opening angle. On the other hand, when 3D cellular composites with various opening angles Structural a = 30° 45° 60° 75° 90° parameters lf = lb = 9.527 mm, tf = 0.145 mm, tb = 2tf = 0.290 mm, 12 rows ¥ 9 columns Original pattern

Strain energy per unit volume (MJ m–3)

Density

0.0284

10

0

0.2

0.0219 0.0205 Under given engineering strain 0.4 0 0.04

0.008

5

0

Applied force (kN) 1.6 0 4.2 8.4 0

0.8

0

0.8

0

0.0222

0.0276

27

54 0

129

258

0.4

0.8 0

0.4

0.8

0.3 0

0.8

0

0

0.8

0 0

0.4

0.8 0

0.4

0.8 0 0.4 0.8 0 Engineering strain Under given load of 380 N

Deformed pattern

e (kJ m–3)

6.895

1.912

0.475

0.107

12.20 Influence of opening angle on strain energy density.

0.021

Mathematical and mechanical modelling of 3D cellular composites

479

the cellular textile composites are compressed by a given force of 380 N, the composites with smaller opening angles are easily deformed and the ones with larger opening angle are hardly affected, as indicated in Fig. 12.14. The response to the two different compression modes is illustrated in Fig. 12.21(a), where it is evident that the strain energy caused by deforming the composites by a given strain of 0.8 is 1000 times higher that that caused by applying a given force of 380 N. In the former situation, the strain energy 9

Strain energy density

Strain energy density at specific strain (MJ m–3) Strain energy density at specific load (KJ m–3) 6

3

0 30

45

60 Opening angle (degree) (a)

75

90

300

9

Strain energy density at specific strain (MJ m–3)

8

Maximum load (kN)

250

7

200

6 150

5 4

100

3 2

50

1 0

0 0

20

40 60 Opening angle (degree) (b)

80

100

12.21 Strain energy density influenced by opening angle, (a) two deformation modes, (b) strain energy versus maximum load.

Maximum force

Strain energy density

10

480

Modelling and predicting textile behaviour

density increases as the opening angle gets bigger and the increase becomes much sharper when the opening angle opens at larger than 60°. Figure 12.21(b) shows that for a given engineering strain, the maximum load needed to deform the cellular composites has almost a linear relationship with the strain energy density with an increase of the opening angle. It needs be noted that all composites, although of different dimensions, are made of the same amount of material. Influence of cell wall length This group of models is created to reflect the change in the cell wall length and its influence on the material performance. As shown in Fig. 12.22, all models are created based on the use of regular hexagonal cells, that is the length of free wall is always the same as the bonded walls, which range from 14.29 mm down to 4.764 mm. The opening angle of all cellular composites is 60°. In order to have a similar cross-sectional area, the models involve different number of rows and columns of cells. It is obvious that the longer is the cell wall, the lower is the volume density and this is indicated in the density row. The two deformation modes are also applied to this group of models. While the given engineering strain is also 0.8, the given compression force is specified to be 15 kN owing to the change in the structures of the cellular composites. Unlike in the previous case, all models have similar dimensions, but they are made of different amounts of materials. When the cellular composite models are deformed by the given engineering strain of 0.8, the model with the smaller cell sizes exhibits much higher rigidity against the compression. In order to create the given engineering strain, a compression force 14 times greater is needed for the composites with smallest cell size than for those with the largest cell size. On the other hand, when a given compression force has been applied to all cellular composite models, the one with the largest cell size is crushed well whereas the one with the smallest cell size is hardly affected. Figure 12.23 summarises the influence of the cell size on the performance of the cellular composites under the two deformation modes for this group of composite models. Under the given engineering strain, the cellular composites’ capability for absorbing energy decreases as the cell size becomes larger. However, this trend is reversed when the cellular composites are subject to compression by a given compression load, where cellular composites with larger sizes will be more energy absorbent than those with smaller cell sizes. These phenomena are depicted in Fig. 12.23(a). Figure 12.23(b) demonstrates a linear relationship between the strain energy density and the maximum load required when compressing the cellular composites by the given engineering strain of 0.8. This, as well as

Mathematical and mechanical modelling of 3D cellular composites

3D cellular composites with various opening angles Structural parameters



lf = lb = 14.29



Row = 4 Column = 7

9.527

7.145

5.716

6 9

8 13 a = 60°, tf = 0.5 mm, tb = 0.7 mm

10 17

4.763 mm 12 19

Original pattern

Strain energy per unit volume (MJ m–3)

Density

0.0233

0

0.0378 0.0545 Under given engineering strain

12

24 0

0.4

0.8 0

23

Applied force, (kN) 46 0 55 110 0

0.0734

114

0.0947

228 0

164

328

10 5

0 0

0.4

0.8 0 0.4 0.8 0 Engineering strain

0.4

0.8 0

0.4

0.8

Under given load of 15 kN Deformed pattern

23.06

0.135

0.054

0.026

0.019

481

e (kJ m–3)

12.22 Influence of cell wall length on strain energy density.

482

Modelling and predicting textile behaviour

25 Strain energy density at specific strain (MJ m–3)

Strain energy density

20

Strain energy density at specific load (MJ m–3)

15

10

5

0 4

6

8

10 12 Cell wall length (mm) (a)

14

16

350

10 Strain energy density at specific strain (MJ m–3)

Strain energy density

8

300

Maximum load (kN)

250

7 6

200

5 4

150

3

100

2

Maximum load

9

50

1 0 4

6

8 10 12 Cell wall length (mm) (b)

14

16

0

12.23 Strain energy density influenced by cell size, (a) two deformation modes, (b) strain energy versus maximum load.

the linear relationship demonstrated between the maximum load and the cell opening angle, provides direct information about material engineering on the cellular composites. A cellular composite is always more energy absorbent for a given load when it has a smaller cell size. Influence of the cell wall length ratio Up to now, it has been assumed that the two types of cell wall lengths in the cellular composites are the same. However, the free wall length and the

Mathematical and mechanical modelling of 3D cellular composites

483

bonded wall length could well be different. In this discussion, the influence of the cell wall length ratio, lf/lb, on the performance of the cellular composites are discussed. Figure 12.24 shows the structural details of this group of models and the results of the simulated quasi-static compression under different loading modes. Figure 12.25(a) shows that increasing the cell wall length ratio, lf/lb, makes the cellular composites more rigid against compression and therefore takes more energy. This is true for both loading modes, that is compression for a given engineering strain and for a given loading force. In the case of compression for a given specific strain, while the strain energy density increases with the cell wall length ratio, the applied force in all cases seemed to be quite similar, as illustrated in Fig. 12.25(b). This, together with the results from the compression for given loading force, indicates that a cellular composite with lf/lb < 1 is a more rigid material, and the one with lf/lb > 1 leads to a softer material. This compares well with the experimental analysis described later in this chapter.

12.5.2 Dynamic impact loading with cross-sectional models In this part of the modelling, a cellular composite model made from eight fabric layers was used for analysis. The model unit is 35 mm in width and 160 mm in height, lb = lf = 5 mm, tb = tf = 1 mm and the opening angle is 60°. The same composites data were used. Although different impact velocities and impactor shapes were simulated for dynamic impact loading, this chapter will only explain the influence of impactor shapes and mass on the performance of cellular composites when impacted at 15 m s–1. Table 12.3 lists the details of the impactors used in the simulation and Table 12.4 shows the properties of the materials used for the impactors. Strain energy Figure 12.26 illustrates relationship between strain energy of the cellular composite and the impactor shape and mass. An impactor with a larger mass causes more strain in the composite, regardless of the shape. It also shows that impactors with spherical shapes cause less deformation in the cellular composite and this can attributed to the smooth curvature of the contacting surface with the composite, compared to the impact with a rectangular shaped impactor, where the strain energy is always higher. Figure 12.27 shows the different modes of deformation occurring in the composites when impacted (a) by a spherical impactor and (b) by a rectangular impactor. In the latter case, severe deformation at the corner areas is evident and the

484

3D cellular composites with various opening angles

Structural parameters

lf /lb = 2

1.333

0.8

0.667

0.5

Original pattern

Density

0.0403

0.0394

0.0387

0.0365

0.0346

Absorbed energy per unit volume (MJ m–3)

Under given engineering strain 0

21

42

0

21.2

0

0.4

0.8 0

0.4

Applied force, (kN) 42.4 0 21.6 43.2 0

21.7

43.4 0

21.2

0.8

0 0.8 0 0.4 0.8 0 Engineering strain

0.4

0.8 0

0.4

Under given load of 10 kN Deformed pattern e (kJ m–3)

42.4

1.6

90.73

75.24

56.25

12.24 Influence of cell wall length ratio on strain energy density.

50.70

43.01

0.8

Modelling and predicting textile behaviour

a = 60°, lf = 9.527 mm, tf = 0.5 mm, tb = 0.7 mm, 6 rows ¥ 9 columns

Mathematical and mechanical modelling of 3D cellular composites 1.8

100

1.6

90 80

Strain energy densigy

1.4

70

1.2

60

1

50

0.8

40

0.6 0.4

Strain energy density at specific strain (MJ m–3)

0.2

Strain energy density at specific load (kJ m–3)

0

485

0

0.5

1 1.5 Wall length ratio (lf/lb) (a)

30 20 10 0 2.5

2

1.6

40

1.4

35

1.2

30

1

25

0.8

20

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15

0.4

10

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0.2

Maximum load

Strain energy densigy

1.8

5

Maximum load (kN)

0

0 0

0.5

1 1.5 Wall length ratio (lf/lb) (b)

2

2.5

12.25 Strain energy density influenced by cell size, (a) two deformation modes, (b) strain energy versus maximum load.

Table 12.3 Impactor shapes and mass Object

Dimension (mm)

Spherical ø 80 Cylindrical (end impact) ø 70 ¥ 60 Rectangular 40 ¥ 60 ¥ 53.33 Cylindrical (side impact) ø 40 ¥ 53.33

Mass (kg) Iron

Wood

Concrete Glass

2.091 1.801 0.998 0.519

0.402 0.346 0.192 0.0999

0.67 0.577 0.32 0.167

0.697 0.6 0.333 0.173

Modelling and predicting textile behaviour Table 12.4 Mechanical properties of the impact materials Material

Young’s modulus (N mm–2)

Mass density (tonne mm–3)

Iron Wood Concrete Glass

200 000 25 000 27 000 94 000

7.8 1.5 2.5 2.6

¥ ¥ ¥ ¥

10–9 10–9 10–9 10–9

160 140 Material strain energy (J)

486

120 100 80 60 Spherical Rectangular Cylindrical end Cylindrical side

40 20 0 0

0.5

1 1.5 Mass of impactor (kg)

2

2.5

12.26 Influence of impactor shape and mass on material strain energy (v = 15 m s–1).

(a)

(b)

12.27 Graphical comparisons of composite deformation with different impactors, (a) spherical and (b) rectangular impactors.

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whole composite was deformed more obviously than in the former case about 1 ms after the impact. Transmitted force Transmitted force is an important performance index in dynamic impact, as it reflects a material’s capability to attenuate the impact force. Experiments and analysis have shown that the attenuation factor of the cellular composites can be as high as over 98%, as shown earlier in this chapter. Figure 12.28 depicts the relationship between the transmitted force and the mass of different geometrical-shaped impactors. It shows that in all cases when the mass of the impactor is below a certain value, the increase in mass does not affect the transmitted force noticeably. In another words, the transmitted force is not sensitive to change in the impacting mass. However, the cellular composite becomes very sensitive to mass when mass value gets larger. This phenomenon is a reflection of the composite material’s deformation mechanism. At the given impacting velocity, which is 15 m s–1 in this case, the low impacting mass represents low impact energy which is not sufficient to damage the cellular composite. The cellular composite will be able to absorb the impact energy through elastic and plastic deformation. On the other hand, when the impact energy is sharply increased owing to an increase in mass, the deformation of the cellular composite will be dominated by damage to the cell walls, in addition to the elastic and plastic deformation. This obviously would reduce the cellular composite’s ability to absorb impact energy and to attenuate the impact force. 2000 1800

Transmitted force (N)

1600 1400 1200 1000 800 600

Spherical Rectangular Cylindrical end Cylindrical side

400 200 0

0

0.5

1 1.5 Mass of impactor (kg)

2

12.28 Influence of mass of impactor on transmitted force.

2.5

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Modelling and predicting textile behaviour

Figure 12.28 also shows that the spherical impactor is associated with the lower transmitted force and that the flat faced impactors (rectangular and cylindrical end) would cause larger transmitted forces. Deformation In order to see the effect of the shape of the impactor on the deformation of the cellular composites, the normalised deformation (D) was defined in order to remove the influence of the mass which is different for different impactors. The normalised deformation is defined as follows: D = d (mm kg –1) m



[12.9]

where m is the impactor mass and d is the deformation of the cellular composite. Figure 12.29 illustrates the normalised deformation (in the direction of impact) caused by different shaped impactors against impact time. Basically, the normalised deformation increases with time and eventually reaches its maximum during the impact process. It can be seen that the cylindrical (end impact)-shaped impactor causes the smallest deformation per unit mass and the spherical-shaped impactor causes the largest deformation per unit mass during most of the impact time. The rectangular-shaped impactor demonstrates a quick increase of deformation but slows down in the second half of the impact process. Spherical and cylindrical (side impact) impactors 50 Normalised deformation (mm kg–1)

45 40 35 30 25 20 15

Spherical Rectangular Cylindrical end Cylindrical side

10 5 0 0

0.5

1

1.5 2 Time (ms)

2.5

3

12.29 Normalised deformation caused by different impactors.

3.5

Mathematical and mechanical modelling of 3D cellular composites

489

demonstrate the largest deformation per unit mass where the two impactors with a flat impactor surface tend to produce a smaller deformation per unit mass. Considering together the phenomena illustrated in Fig. 12.26 and 12.27, it seems possible to conclude that the impactors with a curved impact surface (spherical and cylindrical side) mainly deform the cellular composites in the impacting direction, whereas those with a flat impact surface (rectangular and cylindrical end) cause deformation in the impacting direction and also in the transverse direction. As a result, the flat surface impactors produce more strain energy than the curved surface impactors. Case study: Limb protectors from polymer foam and from cellular textile composites Limb protectors are one type of effective personal protection equipment used by the police officers when dealing with riot situations. The version that the UK riot police currently uses is basically made from a sheet of polymer foam in conjunction with a layer of hard plastic shell. A new version of limb protector made from a cellular textile composite was modelled to study its performance in comparison with the current version. Figure 12.30 shows the situation when the composite limb protector was impacted by a spherical impactor. The deformation of the cells in the cross-section is obvious and this contributes to energy absorption.

Y Z

X

12.30 Dynamic impact on a limb protector made of a cellular textile composite.

490

Modelling and predicting textile behaviour

In the case study, the performance of limb protectors made from polymer foam and from cellular textile composites was compared. The results are shown in Table 12.5. For the same impact energy 45 J, the limb protector made from cellular composite absorbs 27.7% more impact energy than the one made from polymer foam. The maximum transmitted force for the limb protector from cellular composite is almost a kiloNewton smaller than its counterpart. The maximum contact force is taken at the outside of the impacted limb protectors and it shows the contact force with the composite limb protector is only about half that for the foam limb protector, indicating that the cellular composite is easily deformed and can dissipate energy more effectively.

12.6

Discussions and conclusions

Engineered cellular structures have been manufactured in many different ways for their unique performance against impact and compression loading. However, in most cases, the cellular structures are made by joining materials together and therefore lack material continuity. The most popular type of cellular structure made from fibrous materials is the carton board which is widely used for packaging. Single sheets of fabric were also used to make cellular composites, but based on gluing. The design and manufacture of the 3D cellular fabrics with fibre continuity throughout the structure make it possible for us to exploit the advantages the cellular structures in the form of textile composites.

12.6.1 Mathematical modelling of cellular textiles The mathematical modelling of the cellular structures forms the basis of research, as the design of such complex woven structures can be sophisticated and time consuming. Based on the models described in this chapter, cellular structures have been parameterised and the algorithms developed have been used in the CAD/CAM software for cellular woven structures. For both types of the cellular structures it has been noted that the free cell walls are all made a from single layer fabric. While this could be sufficient for some applications, some other applications, such as those involving heavy duty impact, may need the cell walls to contain more material constructed in a Table 12.5 Comparison between foam and composite limb protectors Limb protector

Transmitted Contact force (N) force (N)

Total strain energy (J)

Polymer foam (18 mm) Cellular composite (18 mm)

1318.62 352.52

21.12 26.97

1759.23 932.40

Mathematical and mechanical modelling of 3D cellular composites

491

specific fashion. In addition, the cellular textiles that have been modelled all have a regular shape. The models are unable to deal with cells that are irregular, although inevitably these are of interest. When these new demands are taken into consideration, the current mathematical models seem to be insufficient to provide satisfactory solutions. It must be noted that the capacity of the weaving machines must be considered in order for CAD designed cellular structures to be manufactured. Modification of the weaving machines may be necessary in some cases.

12.6.2 Mechanical modelling Mechanical modelling of 3D cellular textile composites has been carried out based on three different approaches: (1) quasi-static compression on cross-sectional models, (2) low velocity impact on cross-sectional models and (3) low velocity impact on volume models. Modelling and simulation were able to provide data that reflect the performance of the cellular textile composites and indicate suggestions about how cellular composites should be engineered to meet the specific properties of any application. However, the weave details were not directly in the model and a change in weave structures in the cellular composites relied on physically testing the sheet composites made from particular fabric weaves. This is mainly due to the demand of computer resources which even modern computers may have difficulty providing. Better algorithms should be developed so that the weave details can be directly included in the modelling. The complexity of the cellular composite models also limits the structural detail that can be included in simulation.

12.6.3 Conclusions Cellular materials have demonstrated properties against trauma impact. In this chapter, cellular composites, as well as reinforcing 3D cellular woven structures, were modelled for their design and performance against impact. Mathematical modelling of the cellular woven structures has led to the establishment of parameterised algorithms which have been implemented in CAD software. The use of such software has increased the efficiency of the design process. Fourteen systematically designed cellular composites were created and tested for trauma impact performance, the data from which are used to guide the mechanical modelling of 3D cellular composites. Cellular textile composites were modelled and simulated through the use of FE software in three stages, quasi-static compression on cross-sectional models, low velocity impact on cross-sectional models and low velocity impact on volume models. Quasi-static compression modelling with a given load demonstrated trends which are close to experimental results. In

492

Modelling and predicting textile behaviour

addition, modelling also provided information on material response when cellular composites are subject to a given engineering strain. The structural parameters studied in this part of the modelling included the opening angle, the length of cell walls and the ratio of the free wall length to the bonded wall length. The results will be useful in guiding the design and manufacture of cellular textile composites to meet specific mechanical requirements. The second stage of modelling simulates dynamic impact on the cellular textile composites by different shapes of impactors with different masses. Strain energy and transmitted force were examined for different impactors. This showed that a limb protector made from cellular textile composites should be more effective than the limb protector which is currently used by the UK riot police.

12.7

Future trends

Natural cellular structures perform remarkably well against a harsh natural environment. The advantages of engineered cellular structures have prompted their use in many areas. The work described in this chapter demonstrates that integral cellular composite structures can be made using textile technology and that such composite materials offer excellent properties against low velocity impact. Mathematical modelling of the textile cellular structures has made it possible to design and manufacture them to various geometrical specifications and mechanical modelling has provided much information on the design of cellular composites to meet specific performance requirements. It is the author’s belief that cellular textile composites will be further exploited in the years to come. This is because of the attractive properties that they offer, including extra light weight, remarkable ability for impact energy absorption, excellent force attenuation and designable properties. Cellular textile composites can be used in any of the three principal directions and can also be used in conjunction with other materials by, for example, filling up the tunnels with different types of materials. Another direction that may attract interest is the use of the cellular textile composites that have multidirectional tunnels, which will definitely increase the choice of light weight and energy-absorbent materials.

12.8

References

Bitzer T (1997), Honeycomb Technology: Materials, design, manufacturing, applications and testing, Chapman & Hall, London. Chen X (2008), ‘CAD/CAM for 3D fabrics for conventional looms’, Proceedings 1st World Conference on 3D Fabrics and their Applications, Manchester, UK (www. texeng.co.uk) Chen X and Wang H (2006), ‘Modelling and computer aided design of 3D hollow woven fabrics’, J. Text. Inst., 97, 79–87.

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Chen X and Zhang H (2006), Woven Textile Structure, Patent GB2404669 Chen X, Knox R T, McKenna D F and Mather R R (1996), ‘Automatic generation of weaves for the CAM of 2D and 3D woven textile structures’, J. Text. Inst., 87, 356–70. Chen X, Ma Y L and Zhang H (2004), ‘CAD/CAM for cellular woven structures’, J. Text. Inst., 95, 229–41. Chen X, Sun Y and Gong X (2008a), ‘Design, manufacture, and experimental analysis of 3D honeycomb textile composites, Part I: Design and manufacture’, Text. Res. J., 78(9), 771–81. Chen X, Sun Y and Gong X (2008b), ‘Design, manufacture, and experimental analysis of 3D honeycomb textile composites, Part II: Experimental analysis’, Text. Res. J., 78(10), 1011–21. Gibson L J and Ashby M F (1997), Honeycomb Solids: structure and properties, 2nd edition, Cambridge University Press, Cambridge. Ko F K (1999), ‘3-D textile reinforcements in composite materials’, 3-D Textile Reinforcements in Composite Materials, Miravete A (ed.), Woodhead Publishing, Cambridge, 9–40. Li M, Chen X and Liu Z (1988), ‘Mathematical models for fabric weaves and their application in fabric CAD’, Text. Res. J. China, 9, 319. Milasuis V and Reklaitis V (1988), ‘The principles of weave coding’, J. Text. Inst., 79, 598. Papka S D and Kyriakides S (1998), ‘Experimental and full-scale numerical simulations of in-plane crushing of a honeycomb’, Acta Materials, 46, 2765–76. Reid S R and Peng C (1997), ‘Dynamic uniaxial crashing of wood’, Int. J. Impact Eng., 19, 431–570. Ruan D, Liu G, Chen L and Siores E (2002), ‘Compressive behaviour of aluminium foams at low and medium strain rates’, Composite Structures, 57, 331–6. Tan X and Chen X (2005), ‘Parameters affecting energy absorption and deformation in textile composite cellular structures’, Materials & Design, 26, 424–38. Tan X, Chen X, Conway P P and Yan X-T (2007), ‘Effect of plies assembling on textile cellular structures’, Materials & Design, 28, 857–70. Wierzbicki T (1983), ‘Crushing analysis of metal honeycombs’, Int. J. Impact Eng., 1, 157–174. Yu D K C and Chen X (2004), ‘Simulation of trauma impact on textile reinforced cellular composites for personal protection’, Proceedings Technical Textiles for Security and Defence (TTSD), Leeds, UK.

13

Development and application of expert systems in the textile industry

R. Shamey, W. S. Shim and J. A. Joines, North Carolina State University, USA

Abstract: The textile and color industry has experienced many technological advances, which have resulted in improvements in quality and productivity. These advances have often accompanied reductions in personnel resources and a diminishing expertise base. Conversely, the resolution of problems in the global manufacturing complex increasingly go beyond the abilities of individual experts and can be very time consuming as the process is influenced by a large number of, often, interactive variables. The application of expert systems in the textile industry can help address many of these problems more effectively and economically. In this chapter, an overview of expert system technology is given and different types of expert systems including rule-based, fault trees, model-based, machine learning and hybrid approaches are described and compared. A brief review of system principles, strengths and shortcomings is given and the development strategy is described. Finally, various applications of expert systems in different sectors of the textile industry including product components (fibre, yarn and fabric), coloration and finishing as well as supply chain and management are highlighted and future trends are briefly portrayed. Key words: expert systems, fault diagnosis, textiles, troubleshooting, dyeing.

13.1

Introduction

In our day-to-day lives, we frequently encounter situations which ultimately would reach more satisfactory or timely results if a deep and thorough knowledge of the problem or a means of obtaining such knowledge were available. In industrial environments, knowledge is often the difference between success and failure and those with foresight have long employed expert opinions to gain advantage over competitors, forecast future trends and develop a more refined and economically profitable enterprise. Experts, therefore, are people who are very good at solving specific types of problem. Their skill usually comes from extensive experience and detailed specialized knowledge of the problems they handle, where examples include consulting physicians, interpreters of oil well data and engineering experts who carry out diagnosis and repair of sophisticated technological equipment. It long has been an objective of mankind to capture and confer expertise in some manner to the next generation, be it through the word of mouth, 494

Development and application of expert systems

495

generating manuals and instructional guidelines or publishing peer-reviewed manuscripts. Expertise gained through education or professional experience in any domain, however, is often inadequately utilized and indeed in many instances wasted. For many decades, science fiction has led the way in advocating an automated form for such interactions. Recent developments in increased processing rate and memory size of personal computers have facilitated the generation of highly sophisticated domain specific interactive expertise, which in their basic form are known as expert systems. The original research in this arena started in the 1950s with the advent of the term ‘artificial intelligence’ (AI) by John McCarthy at Massachusetts Institute of Technology (McCarthy, 1960). This chapter provides a brief review of the development and application of expert systems in the textile industry.

13.1.1 What are expert systems? An important application-oriented branch of AI, which incorporates utility information as well as additional capability of providing a means of developing and utilizing pure inference in solving problems, is known as expert systems (Juang and Rabiner, 1991; Wright, 1993; Nikolopoulos, 1997). Expert systems are specific computer programs which differ from conventional programs as they solve problems by mimicking human reasoning processes which often rely on logic, beliefs, rules of thumb and experience. A computer-based expert system seeks to capture enough of the human specialist’s knowledge to give the system the ability to solve problems directly. Expert systems use specific and specialized knowledge to solve problems at a level similar to that of a human expert. The knowledge or expertise in an expert system is obtained from magazines, textbooks, scientific journals, as well as human experts. Over the past 20 years, various research groups in the AI domain have developed highly specialized expert systems. Often the terms ‘expert system’, ‘knowledge-based system’ and ‘knowledge-based expert system’ are used interchangeably. There are two general categories of expert systems, those that are used to support decisions and those employed for decision making. The former is mainly developed as a tool to remind the decision maker or human expert of alternative approaches that may have been initially overlooked as well as issues which ought to be explored prior to or during the decision making process (e.g. medical diagnostic). This type of expert system is commonly found in the area of medicine. Therasim Inc. (TheraSim, 2006; 2008) developed several knowledge-based systems to aid the training of doctors and nurses and with the diagnosis of illnesses especially in third world clinics. The latter type is developed to aid individuals in problem solving in an area that they are unfamiliar with or have inadequate inexperience in the domain. This type of expert system is typically employed in the industrial

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Modelling and predicting textile behaviour

sector, including textiles, and would potentially yield marked returns in the general domain of science and engineering.

13.1.2 Motivation for the development of an expert system Despite the fact that in many sectors of science and technology there is an abundance of expertise, the sheer volume of rapidly expanding scientific knowledge, technology, or the interactions of many parts of a complicated process is becoming increasingly difficult to grasp and process. Human experts, therefore, cannot be expected to know everything of importance even within a small sector of their specialization. In addition, in the last several decades, the highly competitive global market trends necessitated geographical shifts in certain industrial activities with subsequent shrinkage of expertise in affected domains, especially in the industrialized world. Moreover, the assessment and analysis of expertise can be highly time consuming and provision of advice, even on an occasional basis, may not be economically feasible, especially in large domains. Thus computerized expert systems can provide an alternative and practical method of providing help when needed. Expert systems enjoy widespread applications in science and are equipped to gain further deployment. Since the introduction of DENDRAL in the mid-1960s (Giarratano and Riley, 2005), expert systems have multiplied to encompass a broad range of applications and they have potential for use in virtually every field in which expert advice is needed. Currently expert systems are used in diverse fields including medical diagnosis, genetic engineering, analytical chemistry, chemical safety, textile industry and experimental design, and the range of applications continues to grow steadily.

13.1.3 Types of expert system The ability effectively to capture or encode expert knowledge into a computerized system that can be used to make decisions is critical to the success of an expert system. Currently, a variety of methods exist that can be used to create a knowledge-based system. These methods include rule-based systems, fault trees, model-based approaches, machine-learning approaches and hybrid methods of the previous four. Rule-based expert systems Rule-based diagnostic systems attempt to convert the experience of skilled diagnosticians or knowledge about the symptoms (effects) and faults (causes) into the form of IF-THEN rules and are generally expressed as ‘if some effect, then some fault(s)’. Rule-based inference involves taking information

Development and application of expert systems

497

about the problem domain and invoking rules that match this information. An example of a rule is as follows: IF the animal is green and hops THEN it is a frog. Rule-based expert systems are the most prevalent intelligent diagnostic system in the industry (Darwiche, 2000). MYCIN is one of the classic diagnostic expert systems based on IF-THEN rules, which was used to diagnose blood infections (Buchanan and Shortliffe 1984). Some other examples include a diagnostic system for thyroid cytopathology (Pandey and Bajpai, 1995) and an expert system for pulmonary tuberculosis diagnosis (Phuong et al., 1999). The system is only able to answer questions on the embedded rules. Therefore, system dependence (i.e., a new rule-base) will have to be generated for each new system type (i.e. condition, disease, etc.). Fault (decision) trees The next method of capturing the knowledge is a tree (i.e. a hierarchical data structure consisting of nodes which store information or knowledge and branches which connect the nodes) (Giarratano and Riley, 1998). A fault tree uses symptom(s) or test results as its starting point, followed by a branching decision tree, consisting of actions, decisions and, finally, repairs recommendations. The merit of a fault tree system is its simplicity and ease of use. However, for more complex systems, a full fault tree can be very large and complex. In addition, a fault tree is system dependent and even small changes in the system can require significant updates. Furthermore, this system does not offer a method of indicating the knowledge used to generate the response. Model-based approaches A model is an approximate representation of the actual system being diagnosed. Model-based diagnosis, based on model-based reasoning (MBR), involves using observations and information from the real device or system based on a model to predict faults. MBR describes a particular problem in terms of the behavior of its smaller building blocks (e.g. pumps and valves in hydraulic systems, switches in electrical systems or gates in digital circuits). A general problem-solving engine then experiments with different ways in which these subcomponents can interact to obtain solutions to problems such as ‘why doesn’t this device work?’ (i.e. diagnosis) or ‘how can I put these parts together to get a functioning system?’ (i.e. configuration and design). Some of the earliest models involved chemical plants and aerospace applications. Research on model-based diagnosis was intensified during the 1980s and 1990s. Nowadays, this is still an expansive research area with many unsolved questions. Various types of approach employed in model-based diagnosis

498

Modelling and predicting textile behaviour

include fault models (Fenton et al., 2001), causal models (Pazzani, 1989; Pardo and Sberveglieri, 2002; Grotzer and Perkins, 2000; Bhatnagar and Kanal, 1991), structural and behavioral models (Paasch and Agogino, 1993) and diagnostic inference models (Rich and Venkatasubramanian, 1992; Frank, 1990; Isermann and Balle, 1997). The main difficulty of systems based on MBR is developing the model to understand all of the interactions between the components of the model. Machine-learning approaches The next method used to create a knowledge-based system is machine-learning approaches which can be divided into three categories namely, case-based reasoning (CBR), explanation-based learning (EBL) and data-based learning (DBL). Case-based reasoning involves storing experiences of the past known cases, retrieving a suitable case to use in a new problem situation, adapting and re-using the retrieved case to suit the new problem, revising the adapted case based on its level of success or failure and retaining any useful learned experience in the case memory (Kolodner, 1993; Leake, 1996; Watson and Marir, 1994; Sharma and Sleeman, 1993; Aamodt and Plaza, 1994). CBR expert systems have been used for medical diagnosis (TheraSim 2006; 2008), and to diagnose defects in steam turbines (Georgin et al., 1995) and steel production (Schnelle and Mah, 1992). The effectiveness of CBR depends upon the availability of suitable case data and the selection of effective case representation, retrieval and adaptation methods. Explanation-based learning uses domain knowledge and a single training example to learn a new concept, while their success depends on the availability of adequate domain knowledge. Therefore, for complex domains where extensive knowledge is needed to formulate new concepts, the approach may prove to be intractable. Another approach is the extraction of knowledge base from existing databases or cause bases (Pardo and Sberveglieri, 2002). This overcomes the knowledge acquisition bottleneck and automatically generates an intelligent diagnostic system from existing resources. However, this method is only suitable where large databases of domain information are available. Thus, it is unfit for new systems where actual data is not yet available or is in a form that makes it unusable. Hybrid approaches The final method (hybrid approaches) combines techniques to produce improved diagnostic solutions by capitalizing on the advantages of the individual methods. Different diagnostic expert systems that combine ‘model-based reasoning and case-based reasoning’, ‘model-based reasoning and fuzzy logic’ and ‘case-based reasoning, artificial neural networks and fuzzy logic’ have

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been reported (Fenton et al., 2001). The diagnosis of lung diseases (Phuong et al., 2001) based on a combination of case-based reasoning with rule-based reasoning is a reported example of such approaches. Systems that combine model-based and rule-based approaches have also been reported. In order to create the best combination of approaches, many questions must be answered (Fenton et al., 2001). Three broad classes of knowledge have been applied to diagnosis: heuristic, fundamental and historical (Gallagher, 1979). Heuristic knowledge employs rules, which relate symptoms to faults, that is IF-THEN rules, often with associated certainty values or probabilities. Fundamental knowledge uses underlying principles of the working of a system. Modelbased reasoning is an example of this system. Historical knowledge employs data or experience recorded during previous diagnostic sessions, to perform new diagnoses. Case-based reasoning is an example of this technique. A comparison of the three broad classes is given in Table 13.1.

13.2

System principles

13.2.1 System development The universal steps in the development of an expert system are shown in Fig. 13.1 (Giarratano and Riley, 1998; 2005). The development is based on a dialogue between a knowledge engineer and a human expert or a group of experts as an attempt to acquire the expert’s knowledge. This step is similar to a system designer in conventional programming discussing the system requirements with a client for whom the program will be built. They then code the knowledge precisely into the knowledge base. Human experts then judge and compare the expert system’s knowledge base and provide comments for the knowledge engineer. This process is reiterated until the performance of the system is judged to be satisfactory by the human expert. A key issue in developing an expert system today involves the methodology employed to represent an expert’s knowledge. Several factors need to be Human expert Dialogue Knowledge engineer Explicit knowledge Knowledge base of expert system

13.1 Development stage of an expert system (Giarratano and Riley, 1998; 2005).

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Modelling and predicting textile behaviour

Table 13.1 Comparison of expert system diagnostic approaches (Shamey and Hussain, 2003) Approach

Pros

Cons

Rule-based ∑ ∑ ∑ ∑ ∑ ∑

Easy to understand owing ∑ to their simplicity Inference sequence can be easily traced ∑ Shells are widely available and this eases development ∑ The technique is well proven ∑ Many rule-based systems have been deployed in real applications ∑ Rules are inherently modular, facilitating maintenance and updating of the system

Development can be long and time consuming if large number of rules are involved For systems with short life cycles it may not be worth the development effort Knowledge acquisition bottleneck Generally only faults anticipated during the design phase can be diagnosed System dependence, i.e. a new rule-base has to be designed for each new system type

Model-based ∑ ∑

Models based on structure ∑ It is hard to develop a complete and behavior would seem and consistent model to offer an ideal solution ∑ Development times can be long for many diagnostic applications Theoretically, the models should be easily generated with help of appropriate software and all defects can be diagnosed

Case-based ∑ Reduces the knowledge ∑ acquisition effort ∑ Makes use of existing data, e.g. in databases ∑ Improves over time and ∑ adapts to changes in the environment ∑ Higher user acceptance owing to existence of precedent ∑ ∑ A fairly intuitive and easy to understand process ∑

Effectiveness depends upon the availability of suitable case data, number of cases and correctness of cases A case-base lacks deeper knowledge of the system under observation, hence there is no way of proving the diagnosis of the system The retrieval and matching algorithms must be carefully developed Less common faults will be more difficult to diagnose owing to lack of presence in the initial case base

considered when selecting the knowledge representation scheme (Duda and Shortliffe, 1983; Bench-Capon, 1993; Giarratano and Riley, 2005) which include the following:

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∑ ∑

Specialized knowledge of the expert should be represented faithfully; The knowledge must be available in a form that can be interpreted by the programmer; ∑ Domain knowledge should be separated from the interpretation component of the program to allow editing of the knowledge-base without having to reprogram the interpreter; and ∑ A line of reasoning which can be understood and critiqued by a human should be supported.

13.2.2 Language, tool and shell It is important to distinguish between the language and development tools used in the construction of an expert system. Language is a translator of commands written in a specific syntax. An expert system language such as LISP, PROLOG and CLIPS provides an inference engine to execute the statements of the code. To provide the reader with a context it may be useful to describe the historical development of some of these languages. LISP was invented by John McCarthy in 1958 at MIT. McCarthy published its design in a paper in Communications of the ACM (McCarthy, 1960). PROLOG was created in 1972 by Alain Colmerauer and Philippe Roussel, based on Robert Kowalski’s procedural interpretation of Horn clauses. It was motivated in part by the desire to reconcile the use of logic as a declarative knowledge representation language with the procedural representation of knowledge that was popular in North America in the late 1960s and early 1970s. It is a general purpose language, which is especially associated with artificial intelligence and computational linguistics (Bratko, 2001). The development of CLIPS started in 1984 at NASA-Johnson Space Center (as an alternative to their existing ART inference system) and continued until the mid-1990s when the development group’s responsibilities ceased to focus on expert system technology. CLIPS is an acronym for ‘C Language Integrated Production System’. Like other expert system languages, CLIPS deals with rules and facts, where various facts can make a rule applicable. CLIPS is a productive development and delivery expert system tool which provides a complete environment for the construction of rule and/or objectbased expert systems. CLIPS is probably the most widely used expert system tool, with application in several sectors including the government, industry and academia, because it is fast, efficient and can be downloaded from the world wide web free of charge. The key features of CLIPS are its knowledge representation capability, portability, extensibility and validation ability (Ford and Rager, 1995; Aikins, 1983; Behera et al., 2004; Giarratano and Riley, 2002; Dlodlo et al., 2007a). From a knowledge representation perspective, the CLIPS environment provides a cohesive tool for handling a wide variety of knowledge with support for three different programming

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paradigms: rule-based, object-oriented and procedural. It is written in C for portability and speed and has been installed on many different operating systems without code changes. Operating systems on which CLIPS has been tested include Windows 95/98/XP/VISTA, MacOS X and Unix. CLIPS can be ported to any system which has an ANSI compliant C or C++ compiler. CLIPS can be embedded within a procedural code, called as a subroutine, and integrated with languages such as C, Java, FORTRAN and ADA providing for complete integration into other systems and/or the ability to be extended to perform some new future task. The standard version of CLIPS provides an interactive, text-oriented development environment, including debugging aids, on-line help and an integrated editor. The interface provides features such as pull-down menus, integrated editors and multiple windows which have also been developed for the MacOS, Windows XP and VISTA environments. One of the key issues in developing an expert system is the verification and validation of the system and CLIPS includes a number of features to support the verification and validation of expert systems including support for modular design and partitioning of a knowledge base. Depending on the implementation, the inference engine may provide forward chaining, backward chaining or both (i.e. bidirectional chaining). Expert system languages focus on providing flexible and robust ways to represent knowledge. Forward chaining starts with all the data available and uses the inference rules in the knowledge system to match the known data to extract more data. The system then uses the new data to try to satisfy other inference rules until the final goal has been inferred (i.e. data is chained to gather new data which is chained to other data). For example, consider a system with the following four rules: 1. If 2. If 3. If 4. If

someone is a fourth year student, then they need a job. someone is a fourth year student, then they live off campus. someone needs a job, they will become a chemist. a person is a chemist, they will work for DuPont.

Suppose we put the following new fact into our system: Woo Sub is a fourth year student. As soon as this data is presented to the system, it searches all the rules for any whose conditions were not true before but are true now. It then adds their conclusions to the known data. In this case, rules one and two will fire and have conditions which match this new fact. The system will immediately create and add the two facts below: Woo Sub needs a job and Woo Sub lives off campus. These facts in turn can trigger other rules. As each arrives, the system would look for yet more rules that are made true. In this case, the fact Woo Sub needs a job would trigger/fire rule three, resulting in the addition of another fact: Woo Sub will become a chemist which in turn fires the final rule Woo Sub will work for DuPont. It

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is therefore inferred that Woo Sub will work for DuPont. On the other hand, backward chaining works in the opposite direction by requiring a piece of data. It searches rules to answer the question. Suppose we put the same following fact into our system: Woo Sub is a fourth year student. The system will not do anything until a goal is presented to the system: Is anyone going to work for DuPont? The system will try to answer the question by searching for either a fact that gives the answer directly or for a rule by which the answer could be inferred. It searches for all rules in the knowledge system whose conclusions, if made true will answer the question. There are no direct rules, but rule four if fired, would answer the question. Next the system would check to see if anybody is a chemist which in turn would fire rule three since there are no facts that answer this particular question. Now the system will check to see if anybody needs a job which in turn fires rule one and finally there is a fact (Woo Sub) that answers there is a student who is a fourth year. So we have proved rule 1 that proved rule three which in turn proved rule four, and that has answered the question. The expert system paradigm allows two levels of abstraction: data abstraction and knowledge abstraction. An example of this separation is that of facts (data abstraction) and rules (knowledge abstraction) in a rule-based expert system language. Compared to the wide variation in domain knowledge, only a small number of artificial intelligence methods are known that are useful in expert systems. That is, currently there are a few methods that represent knowledge, or to make inferences, or to generate explanations. Systems therefore, can be built that contain these useful methods without any domainspecific knowledge. Such systems are known as tools, or simply shells. A tool is composed of a language plus any associated utility programs to facilitate the development, debugging and delivery of application programs. Utility programs may include text and graphics editors, debuggers, file management and even code generators. A tool may be integrated with all its utility programs in one environment to present a common interface to the user. This approach minimizes the need for the user to leave the environment to perform a task. For example, a simple tool may not provide facilities for file management and so the user would have to exit the tool to give conventional commands in a C host language to manage the generated file (Bertram et al., 2000; Bourbakis, 1998; Giarratano and Riley, 2005). A shell on the other hand is a special purpose tool designed for certain types of applications in which the user must supply only knowledge base. The first shell, called MYCIN, was created at Stanford University in the mid-1970s when the knowledge base was stripped from an expert system to diagnose blood infections. This shell was made by removing the medical knowledge base of the MYCIN expert system. MYCIN was designed as a backward chaining system to diagnose disease. By simply removing the

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medical knowledge, EMYCIN was created as a shell containing knowledge about other kinds of consultative systems that use backward chaining (Buchanan and Shortliffe, 1984; David and Krivine, 1993; Giarratano and Riley, 2005). There are advantages as well as disadvantages in using an expert system development tool. The greatest advantage is the amount of time saved by using a tool contrasted by starting with no rule base or inference engine. Other advantages are relatively obvious and include the following, as identified by Withers (Withers, 1986): a person skilled in specialized AI languages is not needed, the shell can be used many times for many different applications and prototypes can be easily and rapidly constructed to monitor the development of the expert system. The main disadvantage of using a shell is the imposed inference structure and the knowledge representation schemes which must be employed. These imposed structures limit the system designers’ flexibility in providing the best method to reason with data and the best manner by which knowledge can be represented for a particular problem. However, the best tools remedy these problems by providing different control structures and knowledge representation designs which can be selected by the user for optimum application (Walker, 1986). The following are some of the additional shortcomings which may be experienced when using a development tool (Antonelli, 1985; Duda and Shortliffe, 1983): ∑ ∑ ∑ ∑ ∑

The old framework may be inappropriate for the new task. The old rule language may be inappropriate for the new task. The tool may be appropriate for one problem domain, but inappropriate for others because of various characteristics of the domain. The control structure embodied in the inference engine may not sufficiently match the new expert’s way of solving problems. There may be task specific knowledge hidden in the old system in unrecognized ways.

13.2.3 Components The basic components of a general expert system are shown in Fig. 13.2 and consist of the rule creator, the knowledge base, the problem application, interface, inference engine, result and finally the user or decision maker (Engelmore and Feigenbaum, 1993; Hunt, 1986; John, 1994): ∑ ∑

Rule creator: This portion allows the creation of the program and the ability of the human expert to enter information into the knowledge base. The information entered here forms the basis of the knowledge base. Knowledge base: This component actually stores the factual and heuristic knowledge. An expert system provides one or more knowledge representation schemes to express knowledge about the application domain. Some tools use both frames (objects) and IF-THEN rules.

Development and application of expert systems Rule creator

User

User interface

Solution/ comment

505

Faced problem

Knowledge base

Inference engine

13.2 Basic components of a expert system. Reproduced from Engelmore and Feigenbaum (1993), Hunt (1986) and John (1994).

∑

Faced problem: This is the area that gathers the data or changing conditions of the problem. ∑ User interface: One of the critical components which provides the means of communication with the user/decision maker. The user interface is generally not a part of the expert system technology and was not given much attention in the past. However, it is now widely accepted that the user interface can make a critical difference in the perceived utility of a system regardless of the system’s performance. ∑ Inference engine: This section contains the logical programming to examine the information provided by the user, as well as facts and rules specified within the knowledge base. It evaluates the current problem situation and then seeks rules that will provide advice about the situation. This is not usually accessible to the expert system designer, but is instead built into the system that the designer uses. ∑ Solution/comment: The result which has been generated from the knowledge base using the problem information gathered from the user through the user interface is presented in this section of the system. ∑ User: Finally and perhaps the most important component of the expert system, the user is the individual who requires/needs the expert system to help them solve the problem in the real world. Moreover, expert system development usually proceeds through several phases including problem selection, knowledge acquisition, knowledge representation, programming, testing and evaluation.

13.2.4 Validation, verification and evaluation The verification, validation and evaluation of expert systems is claimed to be an integral part of the design and development process. As already described,

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an expert system is a computer program that includes a representation of the experience, knowledge and reasoning processes of an expert. Verification of an expert system, or any computer system, is the task of determining that the system is built according to a certain set of specifications. Validation is the process of determining that the system actually fulfills the purpose for which it was intended. Evaluation reflects the acceptance of the system by the end users and its performance in the field. Several researchers conclude that one verifies to show that the system is built correctly, validates to show that the right system has been developed and evaluates to show the availability of the system (Adrion et al., 1982; Chen et al., 2000; Smith, 1993, Verdaguer et al., 1992). The purpose of the verification is to check whether the system has been built correctly (i.e. Does the code exactly reflect the detailed design? Does the design reflect the requirements? Is the code accurate with respect to the syntax of language? and Does the detailed design show the design objectives?) (Adrion et al., 1982; Chen et al., 2000; Smith, 1993, Verdaguer et al., 1992). When the program has been verified, it is assured that there are no bugs or technical errors. The purpose of the validation is to make sure the system does the job that it was intended to do. Thus, validation is the determination that the completed expert system performs the functions in the requirements specification and is usable for the intended purpose. The scope of the specification is rarely precise and it is practically impossible to test a system under all the rare events that are possible. Therefore, it is impossible to have an absolute guarantee that a program satisfies its specification; only a degree of confidence that a program is valid can be obtained. Evaluation addresses the issue ‘Is the system valuable?’ This is reflected by the acceptance of the system by its end users and the performance of the system in its application (i.e. Does the expert system provide an improvement over the practices it is intended to supplement? Is the system valuable as a training tool? Does the detailed design show the design aims? Is the system user friendly? and Is the system maintainable by people?) (Adrion et al., 1982; Chen et al., 2000; Smith, 1993, Verdaguer et al., 1992). To illustrate the difference, the task might be to build a system that computes the serviceability coefficient of a pavement. The specifications for the system are contained in textbooks defining the coefficient. Verification involves completeness and consistency checks, as well as examining for technical correctness using techniques such as are described in this handbook. To validate the system, the serviceability of the program must be tested on examples in the texts and other test cases and the results of the program compared with independently computed coefficients for the same examples. It is important to use a test set covering all the important cases which contains enough examples to ensure that correct results are not just anomalies. The final step is evaluation. For the serviceability program, this means giving the

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system to engineers to use in computing the coefficient. Although the system is known to produce the correct result, it could fail the evaluation because it is too cumbersome to use, requires data that are not readily available, does not really save any effort, does something that can be estimated accurately enough without a computer, solves a problem rarely needed in practice or produces a result not universally accepted because different people define the coefficient in different ways.

13.3

Strengths and limitations of expert systems

Expert systems are beginning to gain wide acceptance owing to their success in various fields. However, certain difficulties in developing an expert system for a particular problem exist that need to be addressed.

13.3.1 Challenges in constructing expert systems The construction of an expert system can be very complex and challenging because obtaining a vast amount of information from several sources is highly time consuming and once this task is completed, the knowledge base has to be coded using an appropriate tool. The development of an expert system has to overcome several challenges (Kaewert and frost, 1990; Badiru, 1992; Cowan, 2001; Liao, 2005; Shamey and Hussain, 2007). The first challenge is that the person collecting the information has to have an extensive background knowledge in order to effectively collect, study and classify several sources such as peer reviewed papers, textbooks, World Wide Web and human expert’s interviews. The feasibility of constructing an expert system that contains hundreds or thousands of codes/rules is not straightforward and undoubtedly time consuming irrespective of the computer language tool employed. In order to represent knowledge in an expert system, it must at least be articulable. Thus, there are two explanations for the non-codification of a piece of knowledge: either it really is inarticulable and therefore cannot be represented or relative to the benefits, the cost of representing it in the knowledge base is too high. Often the method used to construct such systems (i.e. knowledge engineering piece) requires one to extract a set of rules and data from experts through extensive questioning. The evidence collected in the knowledge base of the expert system with uncertainty is usually internally conflicted owing to different sources of information from the reported literature or experienced knowledge. This has to be resolved in some manner.

13.3.2 Comparison with human experts Expert systems have several advantages, if created successfully, compared with their human expert counterpart (Walker, 1986; Bench-Capon, 1993;

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Engelmore and Feigenbaum, 1993; Jin, 2005). Expert systems have the ability to imitate human thought and reasoning and make expert level recommendations understandable to basic users as well as to offer the ability to generate solutions when information is uncertain. For example, dyeing machinery operators do not need a graduate degree to understand how dyes interact in order to control the dyeing process as they can employ a suitable expert system to diagnose potential problems quickly. Computerized knowledge-based systems reduce employee training costs, danger caused by human error, and loss of the expert owing to retirement and relocation (i.e. they provide permanence for the expert knowledge). They combine multiple human experts’ knowledge and other sources into one system which facilitates the reduction of missing data and provides solutions that may have been overlooked by only one person and make it easy to modify/ update the knowledge stored.

13.3.3 Other advantages of expert systems The previous section demonstrated the advantages compared to human experts. Knowledge-based expert systems differ from conventional computer programs in four main aspects (Braspenning et al., 1995; David and Krivine, 1993; Marik et al., 1993; Medsker and Liebowitz, 1994; Mital and Anand, 1993). Unlike conventional computer programs, expert systems are partitioned in a manner that separates the knowledge base required to solve the problem from the reasoning mechanism (Reich et al., 1999; Steels, 1994; Tzafestas, 1993). Moreover, conventional programs rely upon sequential execution of program steps, while expert systems execute rules and procedures in any order. Unlike conventional computer programs, expert systems can operate with uncertainty in data and knowledge and more importantly can offer multiple solutions ranked in the most likely order. In addition, expert systems have the ability to partition an area of rules with the ability to explain itself. This is known as transparency and enables the expert system to be very user friendly.

13.3.4 Disadvantages of expert systems Although expert systems offer significant advantages compared to their human counterparts or conventional programs, knowledge-based expert systems also have limitations (Braspenning et al., 1995; David and Krivine, 1993; Marik et al., 1993; Medsker and Liebowitz, 1994; Mital and Anand, 1993; Reich et al., 1999; Steels, 1994; Tzafestas, 1993). Several examples could be listed to elucidate this point and some important cases are given. Undoubtedly many human experts are highly creative and can respond to unusual circumstances that may arise. It is difficult to embed human common sense which may be needed in some decision making into the expert system.

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Sometimes domain experts cannot explain their logic and the reasons why they make certain decisions. Also some processes may be very complicated and their automation can be very challenging. Even though expert systems can respond to uncertain data they may occasionally lack flexibility and the ability to adapt to changing environments owing to a static set of rules that defines their boundaries. In addition, when no apparent answer is available for a given query, the expert system may demonstrate difficulty in recognizing such situations. Also, as was stated previously, expert systems are often expensive and time consuming to develop and difficult to verify. However, despite these limitations, expert systems have been widely successful and proved their value in a number of important applications that require analysis of any practical problem that lies within a set of defined parameters. The following section reviews some of the important applications of expert systems in the textile industry.

13.4

Applications of expert systems in the textile industry

The textile and color industry, like most industries that have existed for a long time, has experienced many technological advances, which have brought about improvements in quality and productivity. Control technology is concerned with regulating the state of control of productivity and quality of a process. Usually, this state of control is assessed by computer-based monitors, which measure production and/or quality characteristics directly at the machine. The development of an electronic decision support system requires the combined efforts of experts from many fields of textiles and must be developed with the cooperation of the manufacturers who use them. Experts tend to be trained in rather narrow domains and are best at solving problems within those domains. However, there is a growing realization that the complex problems faced by manufacturers go beyond the abilities of individual experts. Interdisciplinary teams of experts must work in unison to formulate solutions to textile-related problems. The textile complex must be viewed as a system of interacting parts where the perturbation of one part affects many others. With limited personnel resources for problem solving and a diminishing expertise base as younger generations tend to move around in their careers or not be interested in becoming domain experts, interest in the development and use of expert systems in the textile industry has been growing. In textiles, expert systems are capable of integrating the perspectives of individual disciplines (e.g. pretreatment, dyeing and finishing) into a framework that best addresses the type of decision making required for modern textiles. Expert systems can be one of the most useful tools for accomplishing the task of providing the textile industry with the day-to-day, integrated decision

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support needed to produce/sell their textile products. Expert systems have been applied to many sectors of the entire textile complex from fiber identification, to dyeing and finishing, all the way to the retail sector. The next sections describe the application of expert systems in textile and color complexes. These are divided into three broad categories including initial components which include fiber, yarn and fabric, processes including production, dyeing and finishing and finally decision making which encompasses the supply chain, and the retail sector as shown in Table 13.2.

13.4.1 Product components (fiber/yarn/fabric) Several systems have been built that deal with potential issues at the fiber level. FIBRE was built using an inexpensive expert system shell called MICROEXPERT (Thomson and Thomson, 1985). Another expert system utilizing a machine-based learning approach dealing with fibers was developed for color grading of cotton. This system was trained using a statistic method based on Bayes’ theorem, which relates the probability of particular events occurring to the probability that events conditional upon them have taken place, as well as a genetic algorithm (Cheng et al., 1999; Doraiswamy et al., Table 13.2 Applications of expert systems in textile sections Category

Sub-section

Author(s)

Product Fiber components Yarn Fabric

Thomson and Thomson (1985), Cheng et al. (1999), Doraiswamy et al. (2005) McDermott (1981), Vasconcelos (1996), Kondler (1996), Dlodlo et al. (2007a) Lin et al. (1995), Standridge (1985), Fan and Hunter (1998a), Fan and Hunter (1998b), Ollenhauser-Ries (1998), Anagnostopoulos et al. (2004), Hu and Tsai (2000), Dlodlo et al. (2007b)

Coloration Dyeing and finishing Color Finishing

Frei and Walliser (1991), Lange et al. (1992), Ruttiger (1988), Frei and Poppenwinner (1992), Lin et al. (2001), Shamey and Hussain (2003), Hussain et al. (2005a, 2005b), Shamey and Hussain (2007) Karamantscheva (1994, 1995), Convert et al. (2000) Aspland et al. (1991)

Supply chain Supply chain management Retail

Karacapilidis and Pappis (1996), Turban (2001), Chiplunkar et al. (2001), Kritchanchai and Wasusri (2007), Wong et al. (2008a), Wong et al. (2008b) Ford and Rager (1995), Nwankwo et al. (2002), Rao and Miller (2004), Thomassey et al. (2005), Jung et al. (2007)

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2005). McDermott (1981) developed an expert system for the selection of all the sizing parameters. By selecting the most suitable sizing agents (i.e. setting the required size add-on level as well as a variety of data for the sizing machine), environmental charges and energy consumption were reduced and the weaving process was optimized. This reduction and subsequently optimized weaving process led to reduced costs and a higher quality of the produced warp fabric (McDermott, 1981). Lin et al. described an intelligent diagnosis system utilizing an expert system capable of tracing possible breakdown causes of fabric defects which was applied to the fabric inspection processing (Lin et al., 1995). TESS (Terren Expert System Shell), an expert system for diagnosing defects in woven textiles, was also developed by the Swiss Federal Laboratories for the Materials Science and Technology Research Institute (EMPA) (Standridge, 1985). WOFAX, a Worsted Fabric Expert System, was designed to provide guidelines or assistance during each stage of the fabric forming process and to predict fabric properties directly (Fan and Hunter, 1998a; 1998b). Behera et al. (2004) used an artificial neural network (ANN) embedded expert system as a tool to advise designers in the prediction of the technical properties of textiles. Several other expert systems have been developed to assist the textile manufacturer in the design of fabrics. For example, a system developed by Ford and Rager (1995), known as Clothing Design Expert System (CDES), contains two separate tools: the alteration definition tool (ADT) and the pattern requirement language (PRL). These two tools provided a flexible way to change and expand the system’s expertise. The ADT, similar to a computer aided design (CAD) environment, enabled the user to generate and store sets of alteration sequences to modify a pattern geometrically. The PRL executed the alteration sequences generated by the ADT. Data describing the customer’s order was processed and the PRL would generate an altered pattern. CDES captured the expertise of a decreasing number of knowledgeable experts in the pattern alteration industry (Ford and Rager, 1995). The next set of expert systems deal with machine control throughout the stages of fabric formation. The MODEX expert system was designed to enable users to perform classical steps in product development in the usual manner and to profit from the application of CAD technology, from drafting to knitting machine control (Ollenhauser-Ries, 1998). Expert systems were also used in the design of industrial fabrics and custom clothing. The Institute of Textile Technology (ITT) and a textile machinery vendor developed an expert system called Texpert to help technicians with machines exhibiting excessive warp and weft stops, quality problems, as well as mechanical and electrical malfunctions. Texpert incorporated graphic images of the machine parts in question to illustrate the corrective action suggested to solve the problem (Turban, 2001).

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13.4.2 Dyeing/color/finishing Once the fabrics or garments are produced, the products need to be dyed, printed and have finishing applied to them. Wooly, developed by Sandoz Products, was developed to enable the dyer to read off important chemical and physical data about the products to be used in wool finishing, instead of having to consult reference works such as shade cards or technical information bulletins. Product data and application procedures could thus be much more easily identified and adapted (Frei and Walliser, 1991). BAFAREX, developed by BASF, was designed to determine dyeing recipes for cotton and polyester/cotton articles, using vat and disperse dyes (Lange et al., 1992). OPTIMIST, also developed by BASF, was used to optimize the dyeing processes (Ruttiger, 1988). TEXPERTO, developed by SANDOZ, was designed to optimize textile finishing recipes and their performance (Frei and Poppenwinner, 1992). Another expert system for use in the dyeing and finishing sector was designed to deal with the selection of suitable fluorescent whiteners (Aspland et al., 1991). A number of expert systems have also been developed to assist in dyeing, For instance, SMARTMATCH was developed by Datacolor International for color matching (Karamantscheva, 1994; 1995) and another expert system was designed for the determination of dye recipes (Convert et al., 2000). An expert system for exhaust dyeing of polyester was developed by the China Textile Institute. The system was designed to integrate practical experience and academic theory to help manufacturers to accomplish their essential technological objectives and to provide technicians with valuable technical information (Lin et al., 1995, 2001). More recently, FuzzyCLIPS, a fuzzy version of CLIPS, was used to develop a diagnostic expert system DEXPERT (Shamey and Hussain, 2003; Hussain et al., 2005a, 2005b, Shamey and Hussain, 2007), primarily to troubleshoot problems in dyeing cotton material. This system comprised a total number of 4786 rules and was capable of diagnosing about 132 faults in the pretreatment and dyeing of cotton in a woven, knitted and yarn package form with direct, reactive, vat, sulphur and azoic dyes and also provided suggestions for corrective measures.

13.4.3 Supply chain/retail A series of expert systems have been used to streamline the decision-making process in the textile supply chain. Knowledge-based management system (KBMS), a production scheduling system, for instance, utilizes an icon-based user interface to enhance the user’s productivity. Once a purchase order is entered into the system, KBMS breaks down the order items into the raw materials required for processing. The system schedules production for one week and gives priority to the orders most easily processed. KBMS then

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informs the materials handling department when orders can or cannot be produced within the following week. It has been stated that KBMS greatly improved customer satisfaction and the company’s ability to schedule orders (Turban, 2001). Expert systems can be used in all facets of the textile supply chain complex. Fashion designers often create collections of products composed of several items which could also include hats, shawls and other accessories. Such items are often meant to be exhibited collectively to create a predetermined coordinated look. However, in practice limited store and exhibition space, among other limitations, does not always provide the managers with such possibilities. It is therefore up to the sales associates to suggest to the customer the products which might be suitable/complementary additions to their item of choice. This task requires a very good knowledge of the store’s layout and a sharp memory to identify quickly the location of various items within the store in order to minimize customer’s waiting period. In addition, in the current competitive market, retailers continually strive to differentiate themselves and increase sales by providing high levels of customer service. This conventionally requires associates to be trained to be knowledgeable about fashion collections and to have a well mannered sales force disposed within easy reach of the customer. However, training associates to be fashion designers can be very challenging and, moreover, many customers often may not wish to be continually disturbed by salespersons. To address some of these challenges, researchers have developed a fashion mix-and-match expert system to expedite and enhance customers’ shopping experience (Wong et al., 2008a; 2008b). This expert system is based on the fuzzy screening technique which utilizes radiofrequency identification (RFID) of embedded tags to determine the type and location of accessories or coordinated products based on the designer’s predetermined selections. The combination of expert system technology with the non-contact radiofrequency identification method within a smart dressing room accomplishes the above task in minimal time and without error. This method allows product combinations to be exhibited to a prospective customer on a computer monitor. Items selected by the customer can be quickly located by the sales associates and brought directly to the dressing rooms, thus increasing customer satisfaction as well as the sales volume of the collections. Ford and Rager (1995) developed an expert system to assist in a vertical production planning process for a variety of end products. The system allows the decision maker to select a particular end product (e.g. denim jeans) and determine a series of steps for the selection of appropriate fibre type, yarn count group, spinning process as well as the processes required to prepare the fibres and yarns. In the end, the results are summarized for each of the stages providing the production manager with a plan to create the fabric for the product.

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Modelling and predicting textile behaviour

Future trends

As was briefly shown, expert systems in the textile industry have been and are being employed in a variety of fields, from fiber identification to fashion merchandizing. Some of these systems are simple and others are designed to deal with complex multistage processes. Hybrid expert systems combining the effectiveness and simplicity of fuzzy rules and artificial neural networks, together with conventional expert systems are being increasingly deployed as new tools to tackle the large array of problems in the textile industry. These tools will be used to complement experts’ knowledge, train novice employees and quickly respond to challenges. New systems will be more powerful, technically competent and will be adaptive to various challenges that require instant, accurate and reliable responses. However, the full integration of expert systems within various sectors of the industry will be greatly influenced by the implementation methods. To this end, flexible and adaptable features that enable the users to personalize the system and create a seamless and intelligent interface will be among the challenges of expert system development in the future.

13.6

Sources of further information and advice

∑

CLIPS expert system tool: A candidate for the Diagnostic System engine: http://rd13doc.cern.ch/Atlas/Notes/108/Note108-1.html ∑ A tool for building expert systems: http://clipsrules.sourceforge.net/ ∑ Expert Systems: http://www.cs.cofc.edu/~manaris/ai-education-repository/ expert-systems-tools.html ∑ NASA CLIPS rule-based language: http://www.siliconvalleyone.com/ clips.htm ∑ FuzzyCLIPS: http://www.nrc-cnrc.gc.ca/eng/projects/iit/fuzzy-reasoning. html http://www.nrc-cnrc.gc.ca/eng/licensing/iit/non-commercial9.html CLIPS question web search: http://www.isphouston.com/clipsearch/

13.7

References

Aamodt A and Plaza E (1994), ‘Case-based reasoning: foundational issues, methodological variations, and system approaches’, AICOM, 1, 39–59. Adrion W, Branstad M and Cherniavsky J (1982), ‘Validation, verification and testing of computer software’, ACM Comp Survey, 14(2), 159–92. Aikins J S (1983), ‘Prototypical knowledge for expert systems’, Artif Intell, 20, 25–6. Anagnostopoulos C, Anagnostopoulos I, Vergados D, Kayafas E and Loumos V (2004), ‘Sliding windows: a software method suitable for real-time inspection of textile surfaces’, Text Res J, 74, 646–51. Antonelli D R (1985), Artificial Intelligence in Maintenance, Noyes publications, NJ.

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Index

ABACUS, 36 additive light absorbance, 369 ADINA FEA software, 123 advection–dispersion equation. see convection–dispersion equation affine geometry, 383 Ageia, 138 air bulk modulus, 208 air-jet spinning, 14 air-jet texturising, 113 air pressure, 208 Airbus A330, 138 algorithms filtration simulation, 272–85 forces on a particle, 276–8 particle capture, 276 particle generation, 272–5 shape dimensions, 273 particle orientation, 275 particle packing, 285 particle-particle interactions, 283–5 forces acting on the particle, 283 particle transport, 276 particle–fabric interactions, 278–83 discus, ellipsoid and needle-shaped particle, 280 forces on particle in contact with filter surface, 279 free-body diagram, 282 all-over patterns, 436–9, 441–50 five Bravais lattice types, 438 four-fold rotation, 447–9 All-over pattern class p4, 448 All-over pattern class p4gm, 449 All-over pattern class p4mm, 448 schematic illustrations, 439 six-fold rotation, 449–50

All-over pattern class p6, 449 All-over pattern class p6mm, 450 symmetry characteristics of the 19 classes of all over patterns, 441 terminology and notation, 437–9, 441 three-fold rotation, 446–7 All-over pattern class p3, 446 All-over pattern class p3m1, 447 All-over pattern class p31m, 447 two-fold rotation, 443–5 All-over pattern class c2mm, 446 All-over pattern class p2, 444 All-over pattern class p2gg, 445 All-over pattern class p2mg, 445 All-over pattern class p2mm, 444 unit cells, 440 without rotational properties, 441–3 All-over pattern class c1m1, 443 All-over pattern class p1, 442 All-over pattern class p1g1, 443 All-over pattern class p1m1, 442 Amonton–Coulomb laws, 87 anisotropy, 198 Arrhenius law, 341 artificial neural network models, 128–31 an artificial neural network, 129 sigmoidal function, 129 atactic, 49 back propagation algorithm, 130 BAFAREX, 512 bending modulus, 35 bending rigidity, 192–3 complete freedom, 193 no freedom, 193 Bessel functions, 373 bias-extension test, 157

521

522

Index



on interlock fabric, 159 local and global field measurement, 173 BIOSYM Technologies, 5 black box model, 135 Boeing 787, 138 Boltzmann energy factor, 52 Boltzmann principle, 83 Boltzmann superposition principles, 82 Boltzmann’s constant, 52, 213 border patterns, 432–6 four-symmetries-in-one class pma2, 435–6 Border pattern class pma2, 436 glide-reflection class p1a1, 434 Border pattern class p1a1, 434 horizontal-reflection class p1m1, 435 Border pattern class p1m1, 435 intersecting-reflection-axes class pmm2, 436 Border pattern class pmm2, 437 notation, 432–3 schematic illustration, 433 translation class p111, 433–4 Border pattern class p111, 434 two-fold-rotation class p112, 435 Border pattern class p112, 436 vertical-reflection class pm11, 434 Border pattern class pm11, 435 Boulton and Crank model, 329–30 breaking elongation, 134 Brinkman equation, 351 Brownian movement, 214 bulked continuous filament yarns, 14 capillary-based permeability model, 268 capillary channel theory, 194 capstan equation, 87 carbon fibre composites, 138 carbon fibres, 11 3D Cartesian coordinate system, 50 case-based reasoning, 131–3, 498 four stages, 131 steps used, 132 CDES, 511 cell opening angle, 470–1 impact force-displacement curves and energy absorption, 470 transmitted force, 472 cell size, 471–3



impact force-displacement and energy absorption, 473 transmitted force, 474 cell wall ratio, 474–5 impact force-displacement and energy absorption, 475 transmitted force and impact time, 476 cellular composites mathematical and mechanical modelling for protection against trauma impact, 457–92 CAD/CAM of 3D cellular woven fabrics, 462–8 discussions and conclusions, 490–2 experimental study of properties, 468, 470–6 future trends, 492 textile structures, 459–62 theoretical characterisation, 476–80, 482–3, 487–90 cellulose fibres, 11 centre of rotation. see rotocentre ceramic fibres, 11 CHARMM force field, 5 chromaticity, 373 CIE. see Commission Internationale de L’Eclairage CIECAM 97, 376 CIECAM 2002, 376 CIELAB model, 365 CLIPS, 501–2 clothing simulation, 403–6 stress-strain relationship, 405 clustering analysis, 383 collision detection, 401–3 colour modelling properties for textiles, 360–85 case study in colour communication, 376–81 commercial vendors and their products, 384–5 conditional or metameric colour matches, 361, 363 future trends, 381–4 key factors and principles, 360–1 linear model, 363 summary, 363–4 types of model used, 364–5, 367–76 colour communication, 376–81

Index

colorant formulations and optimum colour mixtures, 380–1 colour identities, matches and tolerances, 377–8 creative input in colour range development, 379 mark up and information transmission languages, 380 work flow sequences and colour communication links, 378 colour measurement, 383–4 colour symmetry, 450–2 17 classes of two-colour counterchange border patterns, 451 counterchange possibilities of the 17 primary pattern classes, 452 colourimetry, 383 Commission Internationale de L’Eclairage, 361, 365 complete elliptic integral, 72 computational fluid dynamics, 272, 347 computer-aided design, 285 colour system, 379 computer-aided design/computer-aided manufacture (CAD/CAM), 462–8 generation for hexagonal cellular structures, 463–5 base matrices, 465 base matrix example, 465 final weave, 466 one repeat, 464 generation for quadratic cellular structures, 465–8 repeat unit subdivision, 466 subdivision of quadratic structure, 465–7 weave diagram for a three-layer quadratic structure, 469 weave generation for the entire quadratic structure, 468 weave generation per area, 467 mathematical modelling of weaves, 462–3 two single-layer weaves, 464 computer aided design package, 123 conduction, 204 conservation of mass, 349 constitutive theory, 121 continuous filament yarns, 14

523

continuum model, 114–22, 249–50 convection–dispersion equation, 344 conventional dyeing, 335–9 cosine Fourier series, 60 COSMOSWorks, 123 cotton, 6, 14, 47–8, 88 fibre structure, 7 structural features, 8 Coulomb equation, 50 Coulomb forces, 50 creep, 76, 82 behaviour of linear viscoelastic solid in multi-step loading, 82 and relaxation behaviour of linear viscoelastic solid, 76 and stress relaxation models, 77–80 creep compliance function, 82 creep equation, 84 crimp fibre, 63 crystallographic theory, 425 Cunningham correction factor, 278 Cunningham slip factor, 212 CxF, 380, 384 damping coefficient, 75 D’Arcy equation, 269 Darcy’s law, 170, 187, 193, 350, 353 data-based learning, 498 Davies’ equation, 198 deep bed filtration, 267–8 deformation, 488–9 Delany–Bazley equations, 211 DENDRAL, 496 deposition error, 335, 339–40 device profile, 375–6 DEXPERT, 512 diffusion, 216 diffusion coefficient, 326, 332 digital colour reproduction systems, 370 digital image correlation, 158 digital volumetric imaging, 184 direct interception, 214 directional friction effect, 88 discrete models, 122–6, 250–3 1:1 weave using four simple yarns, 126 cylindrical version of collection of fibres, 126 fabric unit cell representing the weave, 127

524

Index



FEA model of ballistic penetration of projectile through 35 layers of fabric, 127 simple fibre with circular cross-section, 124 simple fibres twisted to form a yarn, 125 simulation using a bending force, 253 single fibre with non-circular cross-section, 124 yarn as mass-spring system, 251 dispersion coefficient model, 268 drag force theory, 194, 195, 198 driving pressure, 300–2 cake formation on identical fabrics and filtration conditions, 302 solid particle through the filter and porosity of cake formed, 301, 303 dye diffusion, 14 Dye-It-Right Controller, 356 Dye-It-Right-Monitor, 356 dyeing modelling, simulation and control, 322–56 applications, 353, 355–6 dye diffusion models, 326–8 future trends, 356 numerical simulation and package dyeing, 347–53 parameters and dyeing quality, 328–47 sorption isotherms, 323–6 dynamic impact loading, 483, 487–90 deformation, 488–9 normalised deformation from different impactors, 488 impactor shapes and mass, 485 mechanical properties of impact materials, 486 strain energy, 483, 487 composite deformation with different impactors, 486 impactor shape and mass, 486 transmitted force, 487–8 impactor mass, 487 Dyneema, 11 effective diameter, 56 elastic modulus, 120, 189

electrostatic attraction, 217 electrostatic force, 284 electrostatic interactions, 51 EMYCIN, 503 engineer’s approach, 238–9 enhanced diffusion due to interception of diffusing particles, 216 entanglement, 18, 37 environment–garment–body system, 45 EPSRC project, 57 equivalent hydraulic radius theory, 202 ergodic hypothesis, 51 error vector, 130 Euclidean distance, 132 Euler methods, 275, 400, 403 expert systems, 495–6 applications, 509–13 dyeing/color/finishing, 512 product components, 510–11 supply chain/retail, 512–13 development and application in textile industry, 494–514 future trends, 514 motivation for development, 496 strengths and limitations, 507–9 challenges in construction, 507 comparison with human experts, 507–8 disadvantages, 508–9 other advantages, 508 system principles, 499–507 basic components of an expert system, 505 components, 504–5 development stage, 501 language, tool and shell, 500–4 system development, 499 validation, verification and evaluation, 505–7 types, 496–9 comparison of diagnostic approaches, 500 fault (decision) trees, 497 hybrid approaches, 498–9 machine-learning approaches, 498 model-based approaches, 497–8 rule-based expert systems, 496–7 explanation-based learning, 498

Index Eyring’s three element model, 13 fabric geometrical modelling, 24–34 weft knitted fabrics, 30–4 woven fabrics, 25–30 mechanical modelling, 34–7 modelling, 18–37 comparison of different models, 35 parameters influencing flow performance, 285–97 fabric model specifications, 287 fabric planes, 286 varying weft yarn crimp, 287–90 structure, 19–24 knitted fabrics, 23–4 woven fabrics, 19–23 fabric hand, 119 fabric modulus, 189 fabric repeat, 272 placement inside the chamber and required boundary conditions, 273 false twist process, 113 FAMOUS, 392–3, 404 portable bench-top equipment, 394 FAST, 392 FIBRE, 510 fibre, xxi characterising attributes of fibre crimp, 63–4 classification table of textile fibres, 44 crimp parameters, 63 definition, 43 experimental and fitted length distribution blue fibres, 56 green fibres, 55 red fibres, 55 fibre classification, 43 functions in textile materials and composites, 43–7 fundamental modelling of textile fibrous structures, 41–103 migration, 122 model fibre cross-section cotton fibre, serrated fibre and bilobal fibre, 62 flax fibre, 61 modelling fibre assemblies, 89–102



525

average properties of fibre blend, 94–6 classification of fibre assemblies, 89 structural mechanics of fibre assemblies, 96–102 unevenness of linear fibre assemblies, 90–3 modelling fibre behaviour, 11–14 mechanical responses, 11–13 other properties, 14 spring and dashpot models, 12 modelling fibre friction, 86–9 modelling from molecular level, 4–11 formation, structure and properties, 4–5 inorganic fibres, 11 manufactured polymer fibres, 10–11 mechanical modelling sequence for modelling, 9 natural polymer fibres, 6–10 nylon fibres structure, 10 polymeric form, 5–6 wool fibre, 9 parameters of fibre diameter distribution, 58 parameters of fibre length distribution, 54 properties classification chemical and biomedical properties, 46 general mechanical properties, 46 heat/mass transfer, 46 optical/electromagnetic properties, 46 single fibres mechanical behaviour modelling, 68–76 on the definition of Poisson’s ratio, 71 large deflection of fibre under concentrated load, 71 stress and strain of linear elastic solid, 68 statistical models of fibre geometry, 52–67 fibre cross-section models, 60–2 fibre diameter distribution, 53, 55–7, 59–60 fibre length distribution, 52–3 modelling fibre shape in three dimensions, 62–6

526

Index

modelling longitudinal profiles, 66–7 structure modelling, 47–52 natural fibres, 47–9 synthetic fibres, 49–52 viscoelastic properties of fibres, 76–86 infinitely many element models, 80–2 models of creep and stress relaxation, 77–80 non-linear models, 82–4 relaxation modulus and master curve, 86 temperature dependence, 84–6 three-element standard linear solid with second spring, 79 fibre-air unit cell, 206 fibre breakage mechanism, 120 fibre bulk modulus, 208 fibre crimp, 63 fibre density, 186 fibre friction, 13 fibre linear density, 56 fibre orientation angle, 183 fibre orientation distribution, 183, 189 fibre orientation efficiency factor, 96 fibre volume density, 56 fibre–fibre friction coefficient, 120 filter efficiency, 213–14, 217 filter quality coefficient, 214 filter quality performance, 214 filtration, 306–14, 316 particle shape, 310–14 cake formation and dry mass against time, 313 comparison of cake formed on identical fabric and filtration condition, 312 varying particle shapes of solid particle through the filter and porosity of cake formed, 314 particle size, 307–10 cake formation vs time, 309 comparison of cake formed on identical fabric and filtration condition, 308 varying diameters of solid particle through the filter and porosity of cake formed, 311 solid concentration, 314, 316



cake formation in spherical and needle-shaped particles, 315 finite element methods, 25, 118, 238, 253–4, 350 numerical solution in three dimension, 352 numerical solution in two dimension, 351 fishnet method, 148, 153 flexometer, 160 float loops, 233–4 weft and warp knitted structures, 233 flow injection analysis, 355 Fluent, 272, 287 fluid flow application and filtration modelling, 316–17 fabric parameters on filtration performance, 299–304 driving pressure, 300–2 fluid properties, 299–300 yarn crimp, 302 yarn cross-sectional shape, 303–4 fabric parameters on flow performance, 285–97 weft yarn crimp, 287–90 yarn cross-sectional shape, 291–7 future trends, 317–18 model design and analysis, 271–85 algorithms for filtration simulation, 272–85 flow simulation, 271–2 geometrical models for woven fabrics, 271 modelling through woven fabrics, 265–318 particle properties on filtration performance, 306–14, 316 parameters of woven fabric filter, 307 particle shape, 310–14 particle size, 307–10 solids concentration, 314, 316 various modelling techniques, 267–71 textiles, 267–9 woven fabrics, 269–71 flux law, 349 Fourier’s heat conduction through fibres, 204 fractals, 429–30

Index fractional light transmittance, 369 free convection, 204 free volume model, 327 full field digital image correlation measurement, 145, 172 fuzzy efficiency-based classifier system, 135 fuzzy logic models, 135–6

integration dyeing, 340 interception, 216 interlacing, 18 interloping, 18 International Color Consortium, 375 intrinsic permeability, 193 isotactic, 49 ITOOL European project, 144

G1151 fabric, 158, 160 Gaussian distribution, 68 glass composite, 477 glass fibres, 11 glass transition temperature, 47, 85 glide-reflection, 429 Goeminne’s equation, 187 gravitational settling, 217 grey modelling, 135 grey scale tracking process, 373

Jacquard woven patterns, 424 Jeong and Kang’s model, 119 Joule’s equivalent of heat, 74

Hagen–Poiseuille equation, 187, 200 hairiness, 122–3 Hamaker constant, 284 hand spinning, 14 head reconstruction algorithm, 413 heat conduction equation, 204 heat transfer, 204 held loops, 232–3 weft and warp knitted structures, 233 Hertz’s theory of bodies in contact, 88 hinged framework, 157 Hoffmann and Mueller model, 330–1, 345 Hooke’s law, 69, 77, 78, 191, 400 Horn clause, 501 hotelling transform. see principle component analysis HTML, 377, 384 hybrid effect theory, 95 hyperelastic model, 151 hypoelastic model, 162 Iberall’s equation, 198 impaction, 216 inertial impaction, 214 inlay, 236 inorganic fibres, 11 input data, 237–9 engineer’s approach, 238–9 expert systems, 237 physical simulation, 238

527

Kawabata device, 157 Kelvin–Voigt model, 77 KES-F system, 155 KES-FB tester, 160 KESF, 392 Kevlar, 11 kinetic friction coefficient, 87 knife edge texturising, 113 knit-deknit texturising, 113 Knit GeoModeller, 248 knitted fabrics, 36 geometrical modelling, 30–4 Leaf and Glaskin’s plain stitch model, 33 Leaf’s elastica plain stitch model, 34 modelling and visualisation, 225–59 aims and objectives, 225–6 application areas of simulated fabrics and future trends, 257–8 classification of knitted structures, 226–8 model building, 237–9, 241–56 other types of models, 257 post-processing, 256–7 process, 236–7 scales in the structure, 228, 230–1 structural elements at mesoscale, 231–6 Peirce loose plain stitch model, 31 structure, 23–4 basic knitted structures, 24 plain stitches, 24 knitted structures classification, 226–8 face and reverse sides of a loop, 227 loop elements, 226 scales, 228, 230–1

528

Index



structural and material level hierarchy, 229 structural elements as the meso-scale, 231–6 float loop, 233–4 held loops, 232–3 inlay, 236 loops, 231–2 other modified structural elements, 236 platting loops, 232 transferred loops, 234–5 tucks or press loops, 234 weft insertion, 235–6 knowledge-based management system, 512–13 knowledge-based networks, 131–3 Komori’s model, 122 Kozeny equation, 196 Kozeny–Carman equation, 194, 268 Kronecker delta, 267 Kubelka Munk model, 380–1 Kuwabara flow field, 212 Kuwabara hydrodynamic factor, 212

Lucas–Washburn equation, 200, 202

Laplace transform, 91–2, 200, 331, 355 LaserScan, 57, 59 lattice-Boltzmann method, 270 Leaf and Glaskin’s plain stitch model, 32–3 Leaf’s elastica plain stitch model, 33–4 limb protectors, 489–90 foam vs composite limb protectors, 490 made of cellular textile composite, 489 linear model, 363 linearisation and normalisation step, 364 linear theory, 72 linear van Wyk relationship, 101 linear viscoelastic theory, 83 Linux clusters, 138 liquid composite moulding, 160 LISP, 501 locking angle, 159 loops, 231–2. see also float loops; held loops; platting loops; transferred loops open and closed, 232

M3 system, 396–7 Mao–Russell equation, 196, 198 Markov chain, 51 mass balance, 335, 349 mass transfer equations, 352 mass–spring system, 250, 398–403 collision detection and reaction, 401–3 region of influence of the force field, 402 generic description, 398–401 topology, 399 yarn modelling, 251 master curve, 85 mathematical modelling, 490–1 MATLAB, 384 Maxwell model, 78, 79, 81 parallel arrangement, 80 series model, 12 two-element, 77 mechanical modelling, 491 Medley and Holdstock model, 333–5 dye depletion across a section of the package, 334 microcomputed X-ray tomography, 184 MICROEXPERT, 510 mobile-immobile model, 345 model-based reasoning, 497–8 modelling, 236–7 applications and examples, 410–17 flow chart of next generation textile and garment manufacture and retail, 411 online 3D virtual human system, 412 synthetic human, 412–15 virtual wearer trials, 415–17 automatic measurement of fabric mechanics, 392–3 FAMOUS equipment, 394 various typical mechanical properties, 395 3D textile models, 389–92 finite element models, 390–1 geometric methods, 390 particle systems, 391–2 drape measurement and evaluation, 393–7 actual drape and drape simulations, 398

Index fabric properties, 397 real drape measurements, 397 vertical projections, 397 virtual drape measurements, 398 dye diffusion models, 326–8 dye transfer from solution to fibre, 327 experimental results and discussion, 406–10 drape behaviour of two different fabric types, 410 fabric bend, 409 fabric model interface for drape simulation, 408 fabric rigidity constant, 409 fabric stretchiness constant, 408 original elastic and with velocity modification model, 410 texture mapping application, 407 fundamental modelling of textile fibrous structures, 41–103 model building, 237–9, 241–56 input data preparation, 237–9 mechanical models, 249–54 topology generation, 239, 241–5 yarn contacts, 254 yarn cross-section form and path, 254–6 yarn path representation, 246–9 yarn unevenness, 256 nonwovens materials, 180–219 parameters and dyeing quality, 328–47 critical conclusion for relevant models, 344–7 dye dosage calculation, 342 mathematical models, 329–33 reduced mathematical models, 333–44 simulation and visualisation techniques for drape textiles and garments, 388–419 textile colour properties, 360–85 types of model used, 364–5, 367–76 CIE standard observer, illuminant and colour coordinate models, 365, 367–9 colour appearance modelling, 376 colour calibration models, 369–75 colour profile characterisation, 375–6

529

yarn modelling, 112–39 MODEX, 511 modified random Koch curves, 66 modified structural spring, 404 Monte Carlo method, 5, 51 M_R_ISO, 196 mule spinning, 14 MYCIN, 497, 503 natural polymer fibres, 6–10 Navier–Stokes equations, 195, 267, 269, 346 Newmark algorithm, 395 Newton’s law of viscosity, 77, 78 Nobbs and Ren model, 335, 345 non-linear phenomena, 157 nonwovens acoustic impedance, 207–12 empirical models, 210–12 theoretical models, 207–10 bending rigidity modelling, 192–3 categories of properties application specific performance, 181 chemical properties, 181 fluid handling properties, 181 mechanical properties, 181 circuit diagram based on the thermal resistance in basic model, 206 constructing physical models, 182–5 fibre orientation angle in two- and three-dimensional fabrics, 183 simplified 3D structure, 184 definition, 180 description, 180 fabrics, 36–7 filter efficiency in dry air filtration, 215–18 based on single fibre collection efficiency, 215–17 filter efficiency in relation to particle size, 218 multiple fibre components, 217–18 single fibre collection efficiency, 215–17 future trends, 219 major parameters fabric dimensions and variation, 182 fibre alignment, 182 fibre dimensions and properties, 182

530

Index

porous structural parameters, 182 structural properties of bond points, 182 modelling of materials, 180–219 modelling of pore size and pore size distribution, 185–9 model for constriction pore size, 188 pore sizes distribution model, 187–9 pore sizes model, 186–7 Wrotnowski’s model for pore size, 186 modified fibre–air unit cell model for thermal conductivity of fabrics, 205 parallel fibre model of fabric structure, 208 particle filtration, 212–19 evaluation of filter performance, 213–14 filtration mechanism, 214–15 pressure drop, 218–19 permeability in various two- and threedimensional structures, 201 specific permeability modelling, 193–204 comparison of permeability models, 197 directional capillary pressure and anisotropic liquid wicking, 200, 201–3 directional permeability, 198–200 liquid flows through 3D fabric structures, 198 permeability models established using capillary channel theory, 195 permeability models established using drag force theory, 196 theoretical models, 194–8 tensile strength, 189–92 energy analysis method, 191–2 force analysis method in small strain model, 191 orthotropic models, 189–91 thermal resistance and thermal conductivity, 204–7 numerical simulation, 347–53 finite element methods, 350



yarn package for numerical solution in three dimension, 352 yarn package for numerical solution in two dimension, 351 model equations, 351–3 simulation results, 353 flow velocity distribution, 354 N-S/Brinkman vs N-S/Darcy approaches, 354 system geometry, 348–50 system in two dimension, 349 three dimensional representation & flow characteristics, 348 nylon 6, 10 nylon 66, 10 object-colour sensation, 366 OFDA instrument, 57 on-screen colour, 371–2 single-gun excitation and white point chromaticity coordinates, 372 OpenGL, 256 ordinary differential equation, 252 orientation distribution function, 183 orthogonal Cartesian coordinate system, 65 orthogonal decomposition. see principle component analysis oscillation pendulum method, 74 parallel fibre models, 208 particle diffusion coefficient, 212 Peclet number, 213 Peirce derivative models, 26 Peirce loose plain stitch model, 30–2 Peirce’s circular cross-section model, 27 periodic migration, 122 PhysVis, 138 PhysX chip, 138, 139 picture frame, 157, 168 Planck’s constant, 50 platting loops, 232 in tricot warp knitted structure, 233 Poisson distribution, 16, 69–70, 89, 91, 97, 98, 118, 187, 189, 206 Poissonian line network theory, 188 Poissonian polyhedral model, 188 polar coordinate system, 60 polybutylene terephthalate, 10 polyethylene, 49 polyethylene 2,6-naphthalate, 10

Index polyethylene terephthalate, 5, 10 data on molecular modelling, 6 poly(methyl methacrylate), 49 polypropylene, 49 polystyrene, 49 polytetrafluoroethylene fibres, 88 poly(vinyl)chloride, 49 pore model, 327 porosity, 185 pressure drop, 214 principle component analysis, 130, 382 PROEngineer, 155 PROLOG, 237, 501 quantum mechanics method, 51 Quantum theory, 5 quasi-static compression loading, 477–80, 482–3, 487–90 cell and cellular cross-section, 477 influence of cell wall length, 480, 482 strain energy density, 481 strain energy vs maximum load, 482 influence of cell wall length ratio, 482–3 strain energy density, 484 strain energy vs maximum load, 485 influence of opening angle, 478–80 strain energy density, 478 strain energy vs maximum load, 479 radiofrequency identification, 513 Raleigh type model, 208 Raschel machine, 234 rate constitutive equations, 162 reflection, 428 relaxation modulus, 85 relaxation time, 78 relaxed modulus, 83, 116 repeating patterns classification of motifs, 430–1 asymmetrical motif, 430 cyclic (cn) motif, 431 dihedral (dn) motif, 431 fundamental concepts of pattern symmetry, 426–30 four symmetry operations, 427 historical precedents in the study of patterns, 423–6 recognition, differentiation and classification, 422–53

531

seven classes of border patterns, 432–6 four-symmetries-in-one class pma2, 435–6 glide-reflection class p1a1, 434 horizontal-reflection class p1m1, 435 intersecting-reflection-axes class pmm2, 436 notation, 432–3 schematic illustration, 433 translation class p111, 433–4 two-fold-rotation class p112, 435 vertical-reflection class pm11, 434 seventeen classes of all-over patterns, 436–9, 441–50 five Bravais lattice types, 438 four-fold rotation, 447–9 schematic illustrations, 439 six-fold rotation, 449–50 symmetry characteristics of the 19 classes of all over patterns, 441 terminology and notation, 437–9, 441 three-fold rotation, 446–7 two-fold rotation, 443–5 unit cells, 440 without rotational properties, 441–3 Reynolds number, 199, 213, 268 ring twisting, 14 rotation, 427 rotocentre, 427 rotor spinning, 14 RUC type 1, 167 Runge–Kutta methods, 74, 400, 403 Saint-Vernant torsion theory, 73 3D scanning electron microscopy, 49 Schapery’s non-linear constitutive relation, 84 Schuhmeister’s equation, 207 sequential injection analysis, 355 Shanahan and Hearle’s plain woven fabric model, 27 shear modulus, 35, 75, 118, 189 shell, 503–4 Shen’s equation, 198 short flax fibre processing, 53 Shrodinger equation, 50, 51 simple yarn model, 124 simulated annealing, 382–3

532

Index

sinusoidal cyclic deformation, 13 skein break, 128, 134 slippage constant, 120 slippage factor, 17 small strain theory, 116 SMARTMATCH, 512 soft computing techniques, 139 solid–liquid separation, 300 SolidWorks, 123 sorption isotherms, 323–6 Freundlich isotherm, 324 representations, 325 Langmuir isotherm, 324–5 representations, 325 Nernst Isotherm, 324 representations, 325 Spectra, 11 spectral power distribution, 361 spinnable process conditions, 136 Spline functions, 373 standard beam-bending theory, 13 standard laminate theory, 150 static friction coefficient, 87 Stoke’s drag law, 278 Stokes flow calculation, 170 Stokes number, 213, 276 strain energy, 483, 487 straining, 214 stream tube model, 345 strength efficiency, 134 stress relaxation modulus, 83 STRUTO, 199 stuffer box texturising, 113 subroutine, 502 superposition principle, 348–9 syndiotactic, 49 synthetic human, 412–15 body skeleton for keyframe animation, 416 hierarchical skeleton structures, 416 human head reconstruction, 413 human walking animation, 417 integration of face and body cloning, 415 structure for head cloning, 414 technical flax fibre, 57 tenacity, 112 Tencel yarn, 123 tensile deformation, 76

tensile modulus, 35 tensile strength, 189–92 energy analysis method, 191–2 force analysis method in small strain model, 191 orthotropic models, 189–91 models based on fibre orientation distribution and fibre properties, 190–1 models based on tensile properties in principal directions, 189–90 Tension Technology International, 137 TESS, 511 tessellations, 424 TexGen, 257 TEXPERTO, 512 textile hierarchy, 3–4, xxi–xxii textile industry development and application of expert systems, 494–514 applications, 509–13 future trends, 514 strengths and limitations, 507–9 system principles, 499–507 textile patterns recognition, differentiation and classification, 422–53 colour symmetry, 450–2 conclusions, 452–3 fundamental concepts of pattern symmetry, 426–30 historical study of patterns, 423–6 motif classification, 430–1 seven classes of border patterns, 432–6 seventeen classes of all-over patterns, 436–9, 441–50 Textile Terms and Definitions, 43, 64 textiles, xxi cellular textile structures, 459–62 complex cellular structure, 461–2 complex tunnel relationships, 462 division into regions, 460 hexagonal structure, 459–60 hollow structure with trapezoidal cells, 461 quadratic structure, 461 3D modelling, simulation and visualisation techniques for draping, 388–419

Index

applications and examples, 410–17 automatic measurement of fabric mechanics, 392–3 clothing simulation, 403–6 conclusions and future trends, 418–19 drape measurement and evaluation, 393–7 experimental results and discussion, 406–10 key principles of 3D mass–spring models, 398–403 review of 3D textile models, 389–92 experimental studies of 3D cellular composites, 468, 470–6 cell opening angle, 470–1 cell size, 471–3 length ratio of free to bonded cell walls, 474–6 sample details, 469 structural features and mechanical performance, 469 mathematical and mechanical modelling of 3D cellular composites for protection against trauma, 457–92 cellular fabric structures crosssection, 458 relationship of cellular fabric structures and its constituent fabric layers, 459 structural hierarchy in materials, 3–37 fabrics modelling, 18–37 fibre behaviour modelling, 11–14 fibres from molecular level modelling, 4–11 textile hierarchy, 3–4 yarns and cords modelling, 14–18 textile structural hierarchy, 4 theory of elastic curved rods, 117 theory of linear viscoelasticity, 13 TheraSim, 495–6 thermal resistance, 205 time reduction procedure, 85 tool, 503 topological theory of knots and links, 103 topology, 239, 241–5 3D representation, 244 key point selection for weft insertions, 242



533

loop and anchor points for 2D loop topology, 241 warp knitted spacer fabric, 246 warp knitted structure from 2 yarns, 245 warp knitted structure with two weft insertions, 245 warp knitted tubular structure, 247 warp knitting machine notation, 240 weft knitted notation, 240 weft knitted rib structure, 246 weft knitted structures based on topology of cross points, 243 torsion rigidity, 74 tows, 138 transferred loops, 234–5 translation, 427 transmitted force, 487–8 transverse law coefficients, 168 transverse modulus, 119 tucks, 234 warp and weft structures, 234 Twaron, 11 twist-heatset-untwist, 113 Uster hairiness tester, 123 validation, 506 van der Waals interactions, 51 van Luijk model, 118 van Wyk’s theory, 100 vector sum mapping, 362 verification, 506 virtual measurement system, 393 virtual wearer trials, 415–17 using a real female model, 417 viscose rayon, 11 viscosity, 77 Voigt model, 78 parallel, 12 series arrangement, 81 two-element, 79 Voigt notation, 165 volume fraction, 185 Voronina models, 211 Vosoughi and Burley model, 332–3 VRML, 256 W3 consortium, 384 Wai and Burley model, 331–2

534

Index

warp knitted, 226–8 loop elements, 226 wavelet analysis, 383 weakest link theory, 68 weft insertion, 235–6 inserted elements in warp knitted structure, 235 weft knitted, 226–8 loop elements, 226 simulated structure at the yarn level, 258 WeftKnit, 244 Weibull distribution, 68, 95, 96 wide angle X-ray scattering technique, 48 WLF equation, 86 WOFAX, 511 wool, 13, 14, 47, 88 fibre, 9 woven fabrics, 268–9 bias-extension test on interlock fabric, 159 biaxial tensile test on cross-shaped specimen, 156 biaxial tension on 2 x 2 twill, 169 computer-generated geometrical model, 29 conditions to be fulfilled to guarantee correct periodicity conditions, 166 3D simulation of deformation of unit woven cell at mesoscopic level, 161–71 biaxial tension, 167–8 constitutive model, 162–6 in-plane shear, 168 periodicity and symmetry boundary conditions, 166–7 permeability computations, 170–1 transverse compression, 170 deformed RUC for plain weave and twill weave, 167 different approaches for modelling mechanical behaviour at different scales, 148–53 continuous mechanical models, 150–1 discrete approach, 151–2 discrete model, 152 geometrical approach, 148–50 hemispherical forming of

unbalanced composite fabric, 153 semi-discrete elements, 152–3 semi discrete textile finite element, 152 structure of typical physically based continuous model, 151 draping with the fishnet algorithm for two fabric orientations, 149 flexometer, 160 geometrical modelling, 25–30 image analyses, 172–4 full field digital image correlation measurements, 172 x-ray tomography, 172, 174 Kemp’s racetrack model, 27 local and global field measurement during bias-extension test, 173 mechanical behaviour, 145–8 airbag opening, 148 fibrous behaviour, 147–8 microscopic modelling of sheared plain weave, 147 representative unit cell, 146 three scales of investigation, 145–7 modelling of fluid flow and filtration, 265–318 application, 316–17 design and analysis, 271–85 fabric parameters on flow performance, 285–97 fluid flow and fabric parameters on filtration performance, 299–304 future trends, 317–18 particle properties on filtration performance, 306–14, 316 various modelling techniques, 267–71 Peirce’s model, 25 picture frame device equipped with optical system, 158 plain weave fabric transverse compression simulations, 171 pure shear, 170 rotating frames of hypo-elastic model, 164 Shanahan and Hearle’s lenticular model, 28 solid and fluid simulation for permeability simulations, 171

Index specific experimental tests, 155–61 bending tests, 159–60 biaxial tensile test, 155–7 in-plane shear tests, 157–9 transverse compaction, 160–1 structure, 19–23 3D model and weave for orthogonal woven fabric, 22 elementary weaves, 19 plain derivative weaves, 20 twill derivatives achieved by inverse mirroring, 21 twill derivatives achieved by mirroring, 20 twill derivatives achieved by using other principles, 21 structures and properties modelling, 144–74 conclusions and future trends, 174 importance and objectives, 144–5 structure and geometry of unit woven cell, 153–5 three scales of textile reinforcement, 145 transverse compaction device of glass plain weave, 161 undeformed and deformed RUC for plain weave, 168 variations of section shape along the yarn, 154 x-ray tomography image of yarn, 174 wrapping, 18 Wrotnowski’s model, 186 X-ray tomography, 145, 172, 174 XML, 377, 384 yarn, xxi cable yarn with two plied yarns, 115 comparison of three models and experimental measurements, 118 contact, 254 crimp, 302–3 comparison of cake formation on identical fabric with varying crimps, 304 fluid pressure on the front vs back fabric surface, 288–9 fluid velocity graph, 291 fluid velocity on fabric planes, 290



535

shear stress on front vs back fabric surface, 292–3 solid particle through the filter and dry cake mass against time, 305 cross-section form and path, 254–6 multifilament yarns, 255 cross-sectional shape, 291–7, 303–4 comparison of cake formed on identical fabric, 305 fluid pressure on the front and back surfaces, 294–5 fluid velocity on fabric planes, 296 pressure profiles and fluid velocity vectors, 297 shear stress on front and back surfaces, 298–9 solid particle through the filter and dry cake mass against time, 306 idealised helical yarn geometry, 15 migration, 16 modelling, 14–18, 112–39 applications and examples, 136–8 future trends, 138–9 types of yarn, 14 yarn construction, 113–14 yarn geometry, 15–16 yarn mechanics, 16–18 normalised fibrogram showing fibre length distribution, 133 path representation, 246–9 Knit GeoModeller, 248 warp knitted structure with different diameters, 247 plied yarn containing two single yarns, 115 post-processing, 256–7 visualisation, 256–7 volume rendering, 256 predicted skein break vs experimentally determined skein break strength, 135 slip at fibre ends, 17 stresses in fibre as function of fibre length, 120 two, three and five class fuzzy classifiers, 137 two-fibre single yarn, 114 types of models to predict structure and properties, 114–36

536

Index

artificial neural network models, 128–31 continuum model, 114–22 discrete fibre models, 122–6 fuzzy logic models, 135–6 knowledge-based networks, 131–3 mathematical models, 133–4 statistical models, 126–8 unevenness, 256

woollen spun yarns, 18 worsted yarns, 18 yarn count, 112 yarn geometry, 154 Young’s modulus, 89, 118, 206 Zeidman and Sawhney’s analytical model, 133 Zener model, 79