**Theory of Structure and Mechanics of Fibrous Assemblies**

**Contents **

Preface v

Introduction to fibers and fibrous assemblies xi

1 Basic properties of single fibers and fibrous assemblies 1

1.1 Fiber characteristics: definitions and relations 1

1.2 Characteristics of fibrous assemblies 10

1.3 Characteristics of fiber blend 19

References 28

2 Pores in fibrous assemblies 29

2.1 Pores and their general characteristics 29

2.2 Some special variants of pores 34

2.3 Some possible applications 38

2.4 References 44

3 Arrangement of fibers – fiber orientation 46

3.1 Fiber orientation and its general description 46

3.2 Fiber orientation in a sectional plane 51

3.3 Mean sectional area of fiber, coefficient kn and number of fibers in a section 54

3.4 Model of fiber orientation in plane 57

3.5 Section of planar fiber assembly 69

3.6 Model of fiber orientation in a linear fibrous assembly – sliver 76

3.8 References 95

4 Fiber-to-fiber contacts 97

4.1 Contacts according to van Wyk 97

4.2 Density and distances of contacts 103

4.3 Distribution of contact places 107

4.4 Modified probability of fiber-to-fiber contacts 116

4.5 Modified distribution of fiber contacts 124

4.6 Modified parameters of fiber contacts 130

4.7 References 134

5 Compression behavior of fibrous assemblies 135

5.1 General relations and modeling strategy 135

5.2 Model of compression according to van Wyk 142

5.3 Empirical modification of van Wyk’s model 147

5.4 An alternative model of compression 152

5.5 Experimental results of Baljasov 169

5.6 Model of biaxial deformation 172

5.7 References 186

6 Mechanics of parallel fiber bundles 187

6.1 Mechanics of simplest ideal parallel fiber bundle 187

6.2 Mechanics of bi-component parallel fiber bundle 189

6.3 Mechanics of multi-component parallel fiber bundle 201

6.4 Mechanics of a complex parallel fiber bundle with variable fiber properties 219

6.5 Notes to bundle from staple fibers 240

6.6 References 247

7 Mechanics of non-parallel fiber bundles 249

7.1 Multi-axial textiles 249

7.2 Regular multi-axial textiles 273

7.3 Planar bundle with a continuous distribution of directions 284

7.4 References 293

Appendices

1 Evaluation of the integral /2 ( ) 0 cos d ∫ð ϑ u ϑ ϑ for fiber orientation 295

2 Function (ex – 1)n 299

3 Useful terms and quantities in mechanics 307

4 Derivative of deformation energy dE(ä,î) dì 310

5 Angular utilization coefficient of special alternative to simple planar bundle 313

Index 321

**Preface**

“The imagination is more important than the knowledge.” —Albert Einstein

It is known that the textile fibrous assembly has accompanied human civilization since its inception. It is claimed by the archeological discoveries in the Czech Republic that textiles were existing before the 27th century BC1 . It is obvious that during this extremely long period, people have gained a wide range of empirical knowledge and experience about the manufacturing of textile materials and their behaviors.

One can say that along this extremely long period, empiricism has been the main and might be the only source of development of textile materials and manufacturing technologies. In fact, during the industrial revolution in the 19th century the handwork was replaced by the machines, but textiles – material, structure, and end-use characteristics – have not been changed significantly. The other engineering branches began to develop exact concepts on the basis of knowledge of natural science, but the textile fibrous assemblies did not deeply follow this way of thinking as it was found to be very difficult at that time. (Before 100 years ago, Marschik, in one of his earlier work, tried to establish a mathematical model for yarn and fabric and wrote “….theoretical investigation and clarifying of phenomenon occur during spinning and weaving processes and determinations of end product properties are almost impossible”2 ). Firstly, during the second half of the 20th century, the textile researchers applied exact methods such as mathematical modeling of textile fibrous assemblies. Gradually, it was recognized that the routine application of results gained from the other technical branches would not be a too much successful way; namely, textiles have their specific structure, which is manifested in a unique way.

(Therefore, the textile fibrous assemblies are used also as specific technical materials). To understand the structure of textile fibrous assemblies, it needs specific methods such as mathematical modeling. At the present time, it appears that the exact knowledge about textile structure is widely spreaded, thanks to the computer application and computer-aided design.

A lot of textbooks have dealt with the methods of manufacturing of textile materials, but there are only a few books available on the exact formulation of the internal structure of textile materials and the mathematical modeling on the behavior of textiles. The main aim of this book is to introduce the theory of structure and mechanics of fibrous assemblies in order to partially fulfill the shortage of literature in this field. It includes mainly the original results of the theoretical researches carried out on the structure and mechanics of fibrous assemblies. We hope that this book will be used as a textbook in universities and as a special study material for scientific researchers. Each topic is therefore started with very basic and simple discussions and gradually continued to the more sophisticated formulations for specialists only. We have tried to keep the continuity of logical way of thinking in deriving the relationships needed for explanation without any discontinuation. The derivation of the mathematical expressions is provided relatively in a detailed manner so that the reader, who has less experience in formulation and manipulation of mathematical expressions, can easily follow the text. (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 may cause a “repulsive” view. The dimensionless equations are valid in any coherent unit system (for example, international SI system). To keep the logical continuity of the text, some special mathematical formulations are given separately in appendices.

This book should be useful for the university students as well as the experienced researches. A few topics mentioned in this book can be used for teaching of the undergraduate and postgraduate students and the other special and sophisticated topics can be studied by the doctoral students and the scientific researchers. We would like to remind our dear readers that any topic of this book cannot be automatically studied and mechanically processed. The topics of this book should be understood as a “road map” only, which would guide us to create our own ideas and own understanding of the structure and mechanics of fibrous assemblies using our own mind. We hope that this book will prove to be very useful by the readers.

We gratefully acknowledge the support received from the Grant Agency of Czech Republic GAÈR under project number 106/09/1916 for carrying the manuscript of this book. We are also thankful to our universities Technical University of Liberec and Indian Institute of Technology Delhi for supporting our research work.

Bohuslav Neckáø

Dipayan Das