**Theory of Structure and Mechanics of Yarns**

By Bohuslav Neckář and Dipayan Das

**Contents**

Preface xi

Introduction xv

1 Basic properties of single fibers and yarns 1

1.1 Fiber characteristics: definitions and relations 1

1.2 Characteristics of general fibrous assemblies 9

1.3 Yarn characteristics: definitions and relations 14

1.4 Yarn fineness, twist and diameter in accordance with Koechlin’s concept 24

1.5 Empirical corrections to Koechlin’s concept 28

1.6 References 31

2 Creation of yarns 33

2.1 General structure of technological process 33

2.2 Outline of principles of fiber twisting in yarn 38

2.3 Notes to methodology of studying **yarns** 40

2.4 References 42

3 General description of yarn structure 43

3.1 Description of fiber paths and fiber elementary vectors 43

3.2 A structural characteristic with respect to axial arrangement of fiber paths 62

3.3 A structural characteristic with respect to radial arrangement of fiber paths 78

3.4 A structural characteristic with respect to twisted fiber paths 89

3.5 Idea of regular yarn 94

3.6 Classification of structural models 105

3.7 References 109

4 Helical model of fibers in yarns 111

4.1 Introduction to helical model 111

4.2 An alternative introduction to helical model 118

4.3 Retraction and limit of twisting – hypothesis of neutral radius 122

4.4 Retraction and limit of twisting – hypothesis of constant fiber volume 129

4.5 Retraction and limit of twisting – hypothesis of zero axial force 134

4.6 Yarn retraction in ideal helical model – comparison of hypotheses 140

4.7 Notes to retraction of staple yarns 145

4.8 References 156

5 Models of fiber migration in yarns 157

5.1 Fundamental equation of radial fiber migration 157

5.2 An alternative way to determine the fundamental equation of radial migration 163

5.3 Treloar’s ideal radial fiber migration model 164

5.4 Treloar’s approximation 178

5.5 Equidistant radial fiber migration model 191

5.6 Approximation of equidistant radial migration model 200

5.7 Characteristics of migration proposed by Hearle and coworkers 206

5.8 Radial migration in case of non-constant packing density 220

5.9 Two examples of experimental results 229

5.10 Creation of radial migration 233

5.11 Closing notes to fiber migration in yarns 236

5.12 References 238

6 Mass unevenness of staple fiber yarns 239

6.1 General ideas of irregularity 239

6.2 Martindale’s model of sliver irregularity 240

6.3 Influences of fiber direction and fiber length on sliver irregularity 245

6.4 Some alternative ideas of yarn irregularity 261

6.5 Mass irregularity of structural units following binomial and Poisson distributions 267

6.6 References 281

7 Hairiness of staple fiber yarns 283

7.1 Introductory images and experience 283

7.2 Probabilistic model of yarn hairiness – general equations 285

7.3 Exponential model of yarn hairiness 293

7.4 Double-exponential model of yarn hairiness 303

7.5 Experimental blackening function – evaluation of hairiness 306

7.6 Experience and observed trends 317

7.7 References 326

8 Internal mechanics of twisted yarns 329

8.1 Differential equation of radial equilibrium 329

8.2 Deformations and packing density 340

8.3 Solution of differential equation of radial equilibrium 345

8.4 Some possibilities of solution of deformation law 352

8.5 A semi-empirical equation for yarn diameter 369

8.6 Suitable yarn twist 379

8.7 References 392

9 Tensile behaviour of yarns 395

9.1 Stress–strain relation 395

9.2 Notes to parallel fiber bundles – fiber blends 399

9.3 Notes to parallel fiber bundles – variable fiber properties 404

9.4 Tensile behaviour of twisted yarns – general solution 413

9.5 Tensile behaviour of twisted yarns – simplified solution 421

9.6 Tensile behaviour of generalized helical model – simplified solution 428

9.7 Experimental results 438

9.8 References 447

10 Yarn strength in relation to gauge length 449

10.1 Introduction 449

10.2 Yarn strength in relation to gauge length –Peirce’s theory 450

10.3 Yarn strength in relation to gauge length – the alternative Weibull distribution 463

10.4 Yarn strengths as SEM stochastic process 471

10.5 Yarn strengths as SEMG stochastic process 491

10.6 References 516

11 Constitutive theory of fiber-to-fiber slippage 519

11.1 Introductory remarks 519

11.2 Static equilibrium of fiber elements 520

11.3 Slippages of fiber ends 528

11.4 Slippages on the ‘middle’ part of fiber. 535

11.5 Double-sided slippage at ‘middle’ parts of fiber. 550

11.6 Simplest theoretical examples of slippage at ‘middle’ parts of fiber 560

11.7 Closing remarks on fiber-to-fiber slippage in yarns 573

11.8 Reference 578

12 Semi-empirical modelling of yarn strength 579

12.1 Introduction 579

12.2 Optimum twisting 579

12.3 Yarn strength according to Solovev 588

12.4 References 597

Appendix 1 599

Appendix 2 601

Appendix 3 607

Contents ix

Appendix 4 615

Appendix 5 623

Appendix 6 631

Appendix 7 639

Appendix 8 643

Appendix 9 651

Appendix 10 653

Appendix 11 661

Appendix 12 669

Appendix 13 675

Appendix 14 691

Index 707

**Preface**

There exist many books in the world that deal with manufacturing and properties of different kinds of yarns. Nevertheless, this book is not intended towards such type of publication. It is also not a compilation of several hundred references citing the work of other authors. Of course, the authors’ own work is cited in this book, but there is no attempt made to summarize the information and knowledge available in the literature.

Our goal was – above all – to prepare an original scientific book on a very specific but traditional fibrous assembly – single and mostly twisted yarn. Although yarn is perhaps a 27,000-year-old product1 of human civilization, the technological activities of yarn production are not much explored scientifically.

Why is it so? Since their existence, yarns were developed and used purely on an empirical basis. As a result, a high volume of experience was obtained over many years without much scientific research. (The first scientific knowledge on yarn was published as late as sometime between 18th and 19th centuries, but the major scientific studies began from the second half of 20th century.) In the last few decades, many new avenues were opened up for deeper scientific understanding of yarns in the light of newer necessities and possibilities – recent technological innovations, novel end-uses (often called ‘technical textiles’), advanced mathematical tools connecting computers, latest laboratory equipments, etc.

This book does not provide a complete recipe on how to carry out different activities in relation to yarn – it is not a ‘handbook’ of yarn production, properties, and applications. The authors of this book made attempts to scientifically explain why specific behaviours are observed in yarns with a goal to develop a better understanding of our yarns. This is considered to be a typical character of the so-called basic research work. According to the Frascati Manual 20152, ‘Basic research is experimental or theoretical work undertaken primarily to acquire new knowledge of the underlying foundations of phenomena and observable facts, without any particular application or use in view… Oriented basic research is carried out with the expectation that it will produce a broad base of knowledge likely to form the basis of the solution to recognized or expected current or future problems or possibilities’.

As in many fields, mathematics creates the basic tool for expressing our scientific understanding on yarn. (Feynman3 considered mathematics as a language and simultaneously a method of cognition, that is, language and logic together, as an instrument of thinking.) However, the textile specialists are not always having enough experience with mathematical operations. Therefore, in this book, the derivations of mathematical expressions are provided step by step so that the reader, if having less experience in formulation and manipulation of mathematical expressions, can easily follow the text. (In our first book4, we wrote ‘The authors do not like the idiom “The reader can himself easily derive…”, the so-called “easy derivation” may represent a work of one month!’.) This results in relatively large number of equations which might cause a repulsive view. Nevertheless, to keep the logical continuity of the text, some special mathematical formulations are given separately as appendixes. Let us note that the derived equations, except equations with physical dimensions stated in the subscripts, are valid in any coherent unit system (for example, international system of units – SI).

Theory of structure and mechanics of yarns represents a relatively large and non-trivial complex of relations, despite the fact that it accounts for a small segment of theory of fibrous assemblies. The simpler parts described in this book can be useful for teaching of undergraduate students in colleges or universities as well as for the technologists in textile industries. The relatively difficult parts are made for the postgraduate as well as doctoral students and also for the researchers working in the area of structure–property relationship in yarns. The most complicated theories described in this book can inspire our academic colleagues who are professionally oriented towards basic research.

We are thankful to both of our universities, Technical University of Liberec (TUL) and Indian Institute of Technology Delhi (IITD), for their support to our research work and publications. We also want to extend our thanks to the students and colleagues of our departments – Department of Textile Technologies and Structures at the Faculty of Textiles in TUL and Department of Textile Technology in IITD – for their help and support while writing this book.