Page 226 - Handbook of Properties of Textile and Technical Fibres

Engineering properties of spider silk 201

For constant temperature and humidity, the stress relaxation results shown in
Fig. 6.10 suggests that K can be written approximately as

e
Kðl; tÞ ¼ T ðlÞGðtÞ (6.10)
q;h
where G(t) is a normalized function of time and is defined such as G(0) ¼ 1. T(l) is the
elastic response or the stress as a function of the stretch.
Assume the stress T(t) at any time is linearly related to the elastic response T(l) for
an arbitrary stretch. The one dimensional constitutive equation can be expressed by the
convolution integral (Fung et al., 1972):

Z t
e e _
TðtÞ¼ T lðtÞ þ T lðt sÞGðsÞds (6.11)
o
e
The first term on the right hand side of equation T ½lðtފ accounts for the elastic
response, which is reflected in the stress-strain curve while the second term accounts
for the history-dependent response, which is reflected in hysteresis, stress relaxation,
creep, and sinusoidal stretching response.
e
Since T ðlÞ is a nonlinear function of l as indicated in the experimental results
(Fig. 6.10), this model is called quasilinear despite the fact that Eq. (6.2) is a linear in-
tegral equation.

6.3.3.1 The elastic response in simple elongation
e
T ðlÞ is defined as the tensile stress generated instantly in the fiber where a step stretch
ðl 1ÞHðtÞ is imposed on the specimen. Measurements of T(l) according to this defi-
nition are impossible to obtain. However, by making use of the fact that the stress
response observed is relatively insensitive to strain rates, we may approximate
e
T ðlÞ by the tensile stress response in a rapid loading experiment.
Since the elastic responses of spider silk in this study are strongly nonlinear, quan-
titative representation of these stress-strain curves with a single function is not
possible. One way to obtain an approximate function and to smooth data is to fit the
stress-strain curve in a piecewise manner. This can be done with a cubic spline inter-
polation technique (Conte and deBoor, 1972). The generalized form of the stress-strain
relationships is shown as follows:

3
e X j
T lðÞ ¼ C ij l
i
j¼0
(6.12)
C N ¼ 0
i ¼ 1; 2; 3; .n 1

where C ij are the interpolation coefficients.