Page 504 - Handbook of Properties of Textile and Technical Fibres

Tensile failure of polyester fibers 477

where E y is the strain hardening modulus and s y is the yield stress. Haward (1993)
used a neo-Hookean relation based on the entanglement density in the amorphous
phase for the description of the strain hardening behavior of semi-crystalline polymers.
The true stress s t versus true strain ε t dependence (stress-strain curve) of semi crys-
talline fibers can be often represented by the empiric Ramberg-Osgood equation (see
Ricks (2005))

1=m

s t s t
ε t ¼ þ (13.52)
E K
where K is the strength coefficient and m is the strain hardening exponent.
The Ramberg-Osgood model was used to represent the stress-strain behavior of a
nanocomposite up to ultimate tensile strength. It was observed that there was a
good correlation between K and yield stress s y .
For the true stress/strain relationship of crystalline polymers at elevated tempera-
tures, Takayanagi and co-workers have proposed an empirical relationship (Marayama
et al., 1972)

logðs t =s Þlogðε t =ε Þ¼ c (13.53)
where s* and ε* are determined empirically by shifting the doubly logarithmic curves
along both axes. The constant c is characteristic of polymer species, being independent
of melt index, drawing temperature and deformation rate. For polyamide 6, the value
of c is 0.175. s* decreased rapidly with temperature and ε* was nearly independent of
temperature.
Exponential, parabolic, hyperbolic, and piecewise linear functions have been used
to describe the constitutive behavior of materials (Desai and Siriwardane, 1984). Of
these models, an exponential model of true stress/strain dependence in the form of
(Kurtz et al., 1996)

s t ¼ a þ b expðcε t Þ (13.54)
is widely used. Here the parameter a is the asymptotic true strain as true stress
approaches infinity (assuming c < 0), p is the rate at which the material approaches its
asymptotic strain, and c describes the curvature in the rate of approach to the
asymptote.
The simple piecewise nonlinear model for true stress/strain dependence description
was published by Hutchinson and Neale (1983).

b
s t ¼ kaε for
t ε t ε 0
(13.55)
2

s t ¼ k exp cε for ε t > ε 0
t
The variables a, b, c, and ε 0 are related by the continuity of true stress s t and its first
b
2

derivative with respect to strain at ε t ¼ ε 0 ; i.e., b ¼ 2cε and a ¼ expðb=2Þ ε . A three
0 0