Page 32 - Handbook of Properties of Textile and Technical Fibres

Introduction to the science of fibers 13

Table 1.4 also includes continuous ceramic filaments such as the Nextel fibers from
3M and these are described in detail in Chapter 23. There are other oxide fibers that are
discontinuous and made either by projection from the melt or by sol-gel techniques,
some of the latter of which are mentioned in Chapter 23. However, there has not
been the place here to include particularly those short fibers produced by projection
that find use in high-temperature insulation such as in furnaces. There are also some
very specialized continuous monocrystalline filaments, which at one time were
thought to be going to provide the means of providing reinforcement at very high tem-
peratures in air but so far they have not found use. Finally oxide nanofibers and
ceramic whiskers are not treated in the book. In the 1960s whiskers attracted much
attention as it was realized that their small diameters, between 0.5 and 1.5 mm, would
result in very high strengths. However, their small diameters were their downfall as
they proved to be highly carcinogenic and they are now banned in many industrialized
countries. If the reader is interested in these types of fiber it is suggested that the
following reference would give a useful overview (Bunsell, 2005).

1.5 Statistical nature of fiber properties

In any fiber structure there will be thousands and often millions of fibers and the char-
acteristics of the structure depend on the sum of the fibers with which it is composed.
Such large populations of fibers require a statistical approach to understanding their
behavior not least because fibers usually show a wide scatter in their mechanical prop-
erties. Chapter 2 describes how fibers are tested in tension. The results of tensile tests
need Weibull statistics for their analysis.
A word of warning is however perhaps necessary in this type or any type of anal-
ysis. It is necessary to examine the assumptions and limits of the analysis and be aware
that not all fibers can be easily characterized in this manner. In particular, the Weibull
analysis implies that the fibers are elastic and fail without undergoing plastic deforma-
tion or creep. If they do, the values obtained cannot easily be compared to the results
for other fibers that show elastic or almost linear relationships between applied stress
and the resulting strain. For this reason the analysis is most often used for high-
performance synthetic fibers.
Materials break from their weakest point or from regions of stress concentration.
Testing a fiber in tension involves applying a load to it and determining the load at
which it breaks. If such a tensile test is conducted on many fibers, usually a large scat-
ter in breaking loads is observed within the population tested. This behavior can be
treated by Weibull statistics.
Let us consider a chain consisting of n links, as shown in Fig. 1.5. It will fail when
the weakest link breaks. The probability of failure for a link under an applied load s is
P 0 .
The probability of the chain surviving under the same stress is 1 P 0
As there are n links the survival probability of the entire chain under an applied
n
stress s is given by ð1 P 0 Þ .