Page 225 - Handbook of Properties of Textile and Technical Fibres

200 Handbook of Properties of Textile and Technical Fibres

anharmonic depending upon the material being tested. For viscoelastic materials, there
is a phase lag f between strain and stress response. Symbolically, for an input strain

ε ¼ ε 0 e iut (6.5)

where u ¼ frequency, t ¼ time; the stress response is

s ¼ s 0 e iðutþfÞ (6.6)

The resistance to dynamic deformation at steady state, called dynamic modulus
s 0 if
MðuÞ¼ e (6.7)
ε 0
in terms of Euler’s equation

s 0
MðuÞ¼ ðcos f þ i sin fÞ (6.8)
ε 0
For a dynamic experiment, s 0 , ε 0 , and f are the measured quantities.
In this study, the fiber was stretched to a predetermined strain on the microtensile
tester. A sinusoidal input signal from a function generator was generated and superim-
posed it on the DC bias. Thus, one end of the fiber was vibrated at a predetermined
frequency and amplitude.

6.3.3 Constitutive model
Experimental observations show that under simple elongation, the stress on the spider
dragline varies nonlinearly with the strain history and environmental conditions at all
strain levels. Hysteresis losses are not proportional to strain rates. In sinusoidal stretch-
ing experiments, dynamic stiffness and damping are rather insensitive to frequency
within the experimental range. In order to account for this complex nonlinear mechan-
ical behavior, we need a finite nonlinear viscoelastic model. Unlike traditional engi-
neering materials spider silks are oriented, anisotropic, composite structures; they do
not conform to the usual requirements or homogeneity and isotropy of continuum
mechanics. As a first approximation, the spider silk is treated as a one-dimensional
continuum, and a quasilinear viscoelastic model proposed by Fung et al. (1972) is
used to represent the experimental observations. Like many models such as those of
Leaderman (1943), Volterra (1959), and those reviewed by Ward (1971), the quasilin-
ear viscoelastic model is based on an extension or the Boltzmann superposition
principle (Boltzmann, 1874).
The history of stress response, K(t), for a suddenly applied strain history, l(t), in real
materials, is a nonlinear function of the stretch magnitude l, temperature q, humidity,
h, and the time, t:

K ¼ Kðl; q; h; .; tÞ (6.9)