Page 227 - Handbook of Properties of Textile and Technical Fibres

202 Handbook of Properties of Textile and Technical Fibres

As shown in Fig. 6.7, Eq. (6.12) does represent the stress-strain curves well. With
these functions, the slopes of the stress-strain curves can be calculated conveniently.

6.3.3.2 The history-dependent response
In Eq. (6.11) the history-dependent responses are summarized by the reduced relaxa-
tion function G(t). G(t) should be evaluated in such a way that it is consistent with
hysteresis, creep, and sinusoidal stretching experiments.
Cyclic loading experiments showed that the hysteresis loss of the fibers were some-
what insensitive to strain rates. In sinusoidal stretching experiments, the dynamic
modulus and loss modulus were quite insensitive to frequency. Such frequency insen-
sitivity has been found in a wide variety of textile fibers (Dunell and Dillon, 1951;
Halsey et al., 1945; Fujino et al., 1955), as well as in biopolymeric materials (Fung
et al., 1972; Fung, 1967). Energy dissipations characteristic of ordered polymeric
materials appear quite insensitive to frequency.

6.3.3.3 The continuous relaxation spectrum

To account for the frequency insensitive characteristics of the fibers, we need a contin-
uous distribution of relaxation times or a continuous relaxation spectrum S(s).
For a system that has a continuous relaxation spectrum S(s), we can obtain the
generalized reduced relaxation function, G(t), reduced creep function, J(t), and
complex modulus, M(u), as follows:
h t i
R N
1 þ s sðÞexp ds
0 s
GtðÞ ¼ R N (6.13)
1 þ s sðÞds
0

R N s sðÞ 1
1 exp ds
0
1 þ s sðÞ 1 þ s sðÞ
JtðÞ ¼ R N (6.14)
1 þ s sðÞds
0
" ! #
R N ut R N s sðÞds
1 þ s sðÞ ds þ i
0 1 0 1
us þ us þ
us us
M uðÞ ¼ R N (6.15)
1 þ s sðÞds
0
Our task now is to select a spectrum that represents the experimental observations
for different loading regimens.
The use of spectra to link various viscoelastic and dielectric phenomena has been
reviewed by Gross (1953), Ferry and Myers (1961),and Fung et al. (1972).