Table Of ContentTheory of
structure and mechanics
of fibrous assemblies
Theory of
structure and mechanics
of fibrous assemblies
Bohuslav Neckář
and
Dipayan Das
W
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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
