Page 489 - Handbook of Properties of Textile and Technical Fibres

462 Handbook of Properties of Textile and Technical Fibres

in parallel. The response of the Eyring dashpot representing the rate and temperature-
dependent plastic flow has the form

dε
¼ k sinhðasÞ (13.10)
dt
where k and a are constants. This equation was derived from the idea of plastic
deformation as a temperature- and stress-activated process with a symmetric, simple
1
energy barrier (Morton and Hearle, 1993). The parameter a (GPa ) is related to the
minimum free volume per unit flow, V f , by the relation

V f
a ¼ (13.11)
2kT
where k ¼ 1.38 10 23 J/K is the Boltzman constant and T is the temperature. The
total activation energy, DE s , of this plastic flow (the height of the energy barrier) is
related to the parameter k according to the equation

s V f 2kT
DE ¼ RT ln lnðkÞþ ln (13.12)
V m h
Here, V m is the volume of an elementary flow unit, R is the universal gas constant, and
h ¼ 6.626 10 34 J s is Planck’s constant. For PET the value of V m is usually equal
8 3
to 2.912 10 m , which corresponds to the volume of the elementary crystalline
unit.
To obtain an analytical expression for stress-dependent viscosity, Eq. (13.10) is
substituted into relation:

s s 2s
hðsÞ¼ ¼ z (13.13)
dε=dt k sinhðasÞ k expðasÞ
The approximation sinhðxÞ z expðxÞ=2 valid for high values of x (x > 3.5) has
been also used for a description of deformation behavior near the yield stress s y
(Bauwens-Crowet et al., 1969). The Eq. (13.10) can be rewritten in terms of stress
as function of strain rate:

2kT 1 1 2kT 2
s ¼ sinh dε=dt z ln dε=dt (13.14)
V f k V f k
A plot of s y /T against lnðdε=dtÞ produces a series of straight lines (each for one
temperature), the slope of which is directly connected with activation volume V f .
The modified Eyring-like model for prediction of yield stress was published in the
work by Fotheringham and Cherry (1978). This model is based on the assumption that
yielding involves a simultaneous cooperative motion of polymer chain segments. It is
also assumed that there exists a structural parameter denoted as internal stress, s i ,