Page 271 - Handbook of Properties of Textile and Technical Fibres

Tensile properties of cotton fibers: importance, research, and limitations 245

• The contribution of the variability in fiber breaking elongation between fibers has been
addressed in many studies. However, the effect of breaking elongation is also influenced
by the artificial weak-link effect imposed during testing. Furthermore, the role of fiber
elongation will largely depend on the design and the dynamics of the bundle clamps during
tensile testing (a factor, which is often ignored in such studies). Most theories dealing with
bundle strength assume a rigid clamp that moves steadily at a constant rate of elongation and
has a uniform grip on all fibers. This leads to the next assumption, which is an equally distrib-
uted loading effect on the surviving fibers. The point here is that “it may seem reasonable to
assume that the fiber bundle will hold until the fiber or fiber cluster of the highest breaking
elongation fails, but is it the fiber of the highest breaking elongation that actually fails or the
random point along the fiber of the highest breaking elongation?”
• The statistical modeling of fiber-bundle strength is typically based on the assumption that
each fiber in the bundle has a definite extension and load at which it breaks. When a given
load is applied progressively on the fiber bundle, each fiber will attempt to accommodate the
loading effect, not only via its individual extension but also via supporting adjacent fibers
(interfiber support or fiber assistance). Thus, the load applied on the bundle is initially distrib-
uted in some way between the n fibers in the bundle leading to a temporary equilibrium in
which no fiber fails. As the load exceeds a certain threshold, one of the fibers (perhaps, a fiber
cluster) breaks and the load is redistributed among the remaining fibers (n 1), each bearing a
somewhat larger share of the load than before and so being more vulnerable to fail. Because
not all fibers exhibit the same breaking elongation, some fibers will hold for a while and
others will break in a successive fashion until a point is reached when all fibers fail and
the bundle collapses. As a result, there has been a great deal of work since Daniels’s study
on developing statistical distributions of fiber strength and fiber strain using the normal
distribution or the Weibull distribution (Frydrych, 1995; Militký et al., October 3e6,
2004) for the sake of improving the relationships between bundle strength and single-fiber
strength. The point here is that “under best case scenario, the progressive failure of single
fibers during loading of the fiber bundle is far from being linear in nature; thus, a linear
empirical relationship between single-fiber strength and bundle strength would be highly
doubtful, and should be considered as a gross approximation.”
• When continuous probability distributions are used for modeling fiber failure (stress or
strain), the underlying assumption is typically that failure events are assumed to be contin-
uous and totally independent. Even if a discrete probability distribution such as the binomial
distribution is used, the failure events must be assumed as independent events. To resolve
this aspect, a different statistical approach to model the failure of a fiber bundle under tensile
stress was proposed by Dr. Elmogahzy in 2016 (unpublished) in which the failure events
were considered to be dependent, which agree with some of the physical theories of bundle
failure. Therefore, the author proposed the so-called “hypergeometric probability distribu-
tion” to simulate failure events in which each failure is likely to influence the next failure.
This work is still in progress, and the author encourages other researchers to consider the
utilization of this unique distribution.

7.11.6 The moistureestrength relationship of cotton fibers
No discussion of the tensile behavior of cotton fibers could be complete without
addressing the effect of moisture on cotton fiber strength. In the context of moisture,
the cotton fiber is categorized as a “hygroscopic fiber,” or a fiber that can absorb water
from a moist atmosphere, and conversely, lose water in a dry atmosphere. Cotton fiber
joins many other natural fibers in this important property including wool and long