Page 228 - Handbook of Properties of Textile and Technical Fibres

Engineering properties of spider silk 203

Techniques for the computation of the relaxation spectra from experimental data were
suggested by Hopkins (Hopkins and Hamming, 1957; Hopkins, 1961, 1963) and
Meluch (1974). Tobolsky (1960) used relaxation spectra to relate relaxation and
dynamic response of textile fibers.
The criteria used in this study for selection of a spectrum is that they will make G(t)
and J(t) compatible with experimental observations. In particular, the spectrum of
relaxation times should reflect the frequency insensitivity in the dynamic response,
that is to say, S(s) should be nearly constant over a wide range of frequency.
For this study, a spectrum first introduced by Becker (1925) is used:

c
SðsÞ¼ for s 1 < s < s 2 (6.16)
s
where C is a constant and (s) is the relaxation time in the interval [s 1 , s 2 ].

6.3.3.4 Computation methods

By substitution of Eq. (6.16), Eqs. (6.13)e(6.15) can be reduced to the following
computational forms:

t t
1 þ E 1 E 1
s 2 s 1
GðtÞ¼ (6.17)
s 1
1 þ C ln
s 2
t t

1 C E 1 E 2
C þ s 2 C þ s 1
JðtÞ¼ (6.18)
C þ s 2
1 Cln
C þ s 1

2 2
2 2 1 1
1 þ :5C 1 þ u s 2 ln 1 þ u s 1 þ i tan ðus 2 Þ tan ðus 1 Þ
MðuÞ¼
s 2
1 þ ln
s 1
(6.19)

where E l (z) is the exponential integral
Z N t
e
E 1 ðZÞ¼ dt (6.20)
Z t