i
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
v
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
vii
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
ix
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
x
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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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
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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
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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
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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
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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.
40
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.
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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 ¯
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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]
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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
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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
ÈÎFl cos q + Fy 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
Yarn modelling
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 accura
