Page 278 - Handbook of Properties of Textile and Technical Fibres

252 Handbook of Properties of Textile and Technical Fibres

between the transmitting and receiving transducers along the specimen is varied. A
standing wave of definite frequency is set up in the specimen and the half-wavelength
determined. The velocity V of the waves is obtained as the product of the wavelength
l and the frequency f (V ¼ l$f). Fujino et al. (1955) used this concept by exciting a
standard fiber of known modulus by longitudinal friction. The velocity of propagation
of the waves in the specimen was obtained from the length ratio when the frequencies
in both the specimen and the standard fiber were matched. Another method used
longitudinal wave resonance in which the specimen length, L, was kept constant and
the frequency of the oscillator was gradually changed. The resonance was observed at
aspecific frequency f r , and the propagation velocity V of the sound waves in the
specimen was obtained from the expression (Hamburger, 1948) V ¼ 2Lf r /n, where
n ¼ 1, 3, 5, .
In the pulse propagation method, a train of short pulses of high-frequency oscilla-
tions are transmitted through the specimen, and the transit time of the pulses along the
specimen is determined by a timing device. Hamburger (Woo, 1975) used a combina-
tion of a pulse generator and a number of frequency dividers to supply pulses at
suitable intervals. This method was only suitable for long specimens around 7 in
(175 mm) such as viscose and nylon. When a piezoelectric crystal acting as the trans-
mitter is excited by the individual pulses, the crystal vibrates at its resonant frequency
of 10 kHz. This mechanical vibration was introduced into the specimen through a
pressure contact between the crystal and the specimen. The vibration was then inter-
cepted in transit by another identical crystal at a distance along the specimen, again
the crystalespecimen contact being achieved by pressure between them. The propaga-
tion velocity V in the specimen was obtained by dividing the crystal separation L by the
transit time.
According to Woo and Postle (1974), “the pulse propagation methods only require
the measurement of the time taken for a single pulse to travel a measured or known
distance in the specimen. Therefore, the dynamic modulus obtained by this method
is independent of the dimensions of the specimen. Considering the irregular cross-
sectional and convoluted shape of cotton fibers, the errors originating from measure-
ments of fiber dimensions may easily outweigh the magnitude of variations in the
material properties.” The authors used the pulse propagation principle for measuring
the dynamic modulus of cotton fibers after a great deal of modification to accommo-
date the short lengths of cotton fibers. They produced dynamic modulusestrain
relationships, and they estimated dynamic stressestrain relationships using the
following expression:

Z
ε
s dynamic ¼ E dynamic ðεÞdε
0
Based on Woo and Postle analysis, the general relationship between the dynamic
modulus and strain and the derived relationship between the dynamic stress and strain
are as shown in Fig. 7.10. These relationships were based on testing different fiber
types such as cotton, wool, and jute. In general, the modulusestrain curve can be
divided into three main regions: the initial region, the yield region, and the postyield
(or prefracture) region. In most cases, an early peak in the modulusestrain curve was