Theory of structure and mechanics of fibrous assemblies Theory of structure and mechanics of fibrous assemblies Bohuslav Neckář and Dipayan Das W OODHEAD PUBLISHING INDIA PVT LTD New Delhi Cambridge Oxford Philadelphia Published by Woodhead Publishing India Pvt. Ltd. Woodhead Publishing India Pvt. Ltd., G-2, Vardaan House, 7/28, Ansari Road Daryaganj, New Delhi – 110002, India www.woodheadpublishingindia.com Woodhead Publishing Limited, 80 High Street, Sawston, Cambridge, CB22 3HJ UK Woodhead Publishing USA 1518 Walnut Street, Suite1100, Philadelphia www.woodheadpublishing.com First published 2012, Woodhead Publishing India Pvt. Ltd. © Woodhead Publishing India Pvt. Ltd., 2012 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission. 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 India Pvt. Ltd. The consent of Woodhead Publishing India Pvt. Ltd. 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 India Pvt. Ltd. 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. Woodhead Publishing India Pvt. Ltd. ISBN: 978-81-90800-17-4 Woodhead Publishing Ltd. U.K. ISBN: 978-1-84569-791-4 Typeset by Sunshine Graphics, New Delhi Printed and bound by Replika Press, New Delhi 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. 1. Adovasio, J. M., Hyland, D. C., and Soffer, O. (1997). Textiles and Cordage: A Preliminary Assessment, In: Svoboda, J. (ed.), Pavlov I – Northwest. The Dolní Vìstonice Studies, Volume 4, Brno, pp. 403–424. 2. “…eine rechnungsmßige Verfolgung oder Erklärung der Vorgänge beim Spinen und Weben sowohl, als auch der Eigenschaften der Endprodukte fast unmöglich ist. ” Marschik, S., Physicalish-technische Untersuchungen von Gespinsten und Geweben, Wien, 1904 (German). vi Preface (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 Preface vii out some of the research themes reported in this book and processing of 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 Introduction to fibers and fibrous assemblies A fiber is a coherent and slender entity with sufficiently high “length-to- diameter” ratio, often known as aspect ratio or slenderness ratio. The high aspect ratio is intrinsically associated with high specific surface area (ratio of surface area to mass). The high aspect ratio and the associated high specific surface area impart a unique advantage to the fibers over other materials in many critical applications. The fibers are generally procured in the form of large ensembles such as bales, tows, etc. They are mechanically processed and arranged to create fibrous assemblies of the first hierarchical level such as sliver, roving, yarn, nonwoven, etc. and, when necessary, followed by more complex hierarchical levels such as woven fabric, knitted fabric, braided fabric, etc. The fibrous assemblies of the first hierarchical level are called as simple or “primary” fibrous assemblies; they are produced directly from fibers. The fibrous assemblies of the second hierarchical level are known as composed or “secondary” fibrous assemblies; they are produced from “primary” fibrous assemblies. Sometimes, from geometrical point of view, the fibrous assemblies are divided into three categories: one- dimensional or linear fibrous assembly, two-dimensional or planar fibrous assembly and three-dimensional or spatial fibrous assembly. This book is dealt with the fibrous assemblies of the first hierarchical level, otherwise known as simple or primary fibrous assemblies. Each simple or primary fibrous assembly has its own specific characteristics as mentioned below. (1) Constituents of fibrous assembly. The simple or primary fibrous assembly consists of fibers, and the fibers are either of same type or of different types. (2) Geometrical arrangement of fibers. The geometrical arrangement of fibers in the simple or primary fibrous assembly is characterized by (a) Packing arrangement of fibers, i.e., how the fibers are packed inside the fibrous assembly (b) Directional arrangement of fibers, i.e., how the fibers are oriented in the fibrous assembly xi xii Introduction to fibers and fibrous assemblies (3) Mutual interaction of fibers. The mutual interaction among fibers in the simple or primary fibrous assembly is characterized by (a) Type of interaction, i.e., the type of force (mechanical, thermal, chemical, etc.) exerted onto the fibers to realize fiber-to-fiber contacts. (b) Characteristics of fiber-to-fiber contacts, i.e., number of fiber- to-fiber contacts and distance between neighboring contacts. Basic properties of single fibers and fibrous assemblies 1 1 Basic properties of single fibers and fibrous assemblies The basic structural element of the fibrous assemblies considered here is fiber. It is sufficiently long and thin. It is characterized by many properties such as length, fineness, diameter, aspect ratio, cross-sectional shape, surface area, specific surface area, strength, breaking elongation, etc. Numerous fibers constitute the fibrous assembly. Like fibers, the fibrous assembly is also characterized by its properties such as total length of fibers in the assembly, total surface area occupied by fibers in the assembly, etc. In this chapter, the basic characteristics of individual fibers are described and their relations to those of the fibrous assembly are derived. The fibrous assembly is considered to be made up of homogeneous or heterogeneous fibers. 1.1 Fiber characteristics: definitions and relations Starting parameters. Figure 1.1 illustrates a fiber of s p length l, mass m, volume V, and surface area A. Let f f f us assume a homogenous fibrous assembly contains N number of such identical fibers. If L, m, V, A represent length, mass, volume, and surface area of all fibers in the assembly, respectively; then we can write m L(cid:2)Nl, m(cid:2) Nm ,V (cid:2) NV , A(cid:2)N A. … (1.1) f l f f f V Fiber density ((cid:3)). Using Eq. (1.1), the fiber density f can be expressed as follows A f (cid:3)(cid:2)m V (cid:2)m V. … (1.2) f f 1.1 Scheme of a fiber. 1 2 Theory of structure and mechanics of fibrous assemblies Table 1.1 shows the density values of some commonly used fibers. Table 1.1 Fiber density values (according to Goswami et al. [1]). Fiber (cid:3)[kgm3] Cotton 1520 Linen, jute 1520 Wool 1310 Natural silk 1340 Viscose 1500 Acetate 1320 Polyester 1360 Polyamide 1140 Polyacrylonitrile 1300 Polypropylene 910 Fiber fineness (t). In practice, it is often necessary to specify the fineness characteristic of fibers. The fiber fineness is usually defined by fiber mass per unit length; in other words, it is called “linear density,” or “titre.” Using Eq. (1.1), the fiber fineness can be expressed as follows m m t(cid:2) f (cid:2) . … (1.3) l L According to the international standard unit system (“SI” system), the dimension for mass is [kg] and for length is [m]. Accordingly, the dimension for fiber fineness expressed by Eq. (1.3) is [kg m–1] (cid:4) [Mtex]“megatex” (1Mtex = 106tex), which is an impractical unit to express fiber fineness. In practice, the dimensions “tex,” [tex] = [g/km], or “decitex”, [dtex] = [g/10 km] are used as shown below t(cid:5) (cid:6) (cid:2)m(cid:5) (cid:6) L(cid:5) (cid:6) (cid:2)m(cid:5) (cid:6) L(cid:5) (cid:6), t(cid:5) (cid:6) (cid:2)10m(cid:5) (cid:6) L(cid:5) (cid:6) (cid:2)10m(cid:5) (cid:6) L(cid:5) (cid:6) . tex g km mg m dtex g km mg m In industry, fibers such as cotton, wool, manufactured, and micro fibers are used. The fiber fineness values of some commonly used fibers are given in Table 1.2. Example 1.1: Consider a cotton fiber of 1.7 dtex fineness and 28 mm length. By Eq. (1.3), the fiber mass is obtained as m = 0.00476 mg, that f Table 1.2 Fineness of different types of fibers Fibers Fineness Micro-fibers < 1 dtex Cotton and compatible manufactured fibers about 1.6 dtex Wool and compatible manufactured fibers about 3.5 dtex Carpet fibers, industrial fibers > 7 dtex