Page 505 - Handbook of Properties of Textile and Technical Fibres

478 Handbook of Properties of Textile and Technical Fibres

phase empiric piecewise model for the description of engineering stress/strain depen-
dence of PET yarns was published in the work by Serwatka et al. (2007).
The true stress/strain behavior of amorphous PET can be analyzed by using a model
composed of parallel viscoplastic s p and an orientational s or components connected in
series with Hookean springs (Vigny, 1999). The viscoplastic term is thermally acti-
vated and reflects mainly the strain rate sensitivity of the material. Its response is
defined by Eq. (13.16). The orientational term represents the internal stress of the
rubberlike network approximated by an array of Langevin springs. Its response is
defined by Eq. (13.24). The final model is modified to take into account the linear
increase of crystallinity in PET during tensile deformation.
The sigmoid stress-strain curves can be expressed by a nonlinear model composed
of a linear spring (modulus E 2 ), a Maxwell element (modulus E 1 , viscosity h) and a
2

nonlinear spring with parabolic response s ¼ bε in parallel. The linear spring in
the model is used to describe the Hookean region in the tensile curve at lower strains,
the Maxwell element is introduced to illustrate the viscoelasticity, and the nonlinear
spring to characterize the nonlinear mechanical response,

2 E 1 ε
s ¼ E 2 ε þ bε þ hv 1 exp (13.56)
hv
where v ¼ dε=dt is the rate of deformation. The asymptotic stress s as for sufficiently
2
large strains has a parabolic response s as ¼ E 2 ε þ bε þ hn. In other work (Plaseied
and Fatemi, 2008), the stress strain dependence for a standard linear viscoelastic body
model (which is in fact Eq. 13.56 for b ¼ 0) was modified by using nonlinear viscosity

h h N
0
h ¼ h N þ h i d (13.57)
2
1 þðcdε=dtÞ
where c and d are material constants. The viscosity decreases from its initial value h at
0
dε=dt ¼ 0to h N as dε=dt/N.
More complex, four element models (two springs and two dashpots with nonlinear
responses), are used in the work by Khan et al. (2006).
Drozdov and Christiansen (2003) derived the constitutive model for the mechanical
behavior of a semicrystalline polymer at small strains (below yield point, where the
stress-strain curve is monotonously increasing). A polymer is assumed to be a network
of chains bridged by permanent joints (entanglements or physical crosslinks on the
surfaces of crystallites). This network is replaced by an ensemble of mesoregions con-
nected by links (crystallites). Macroscopic deformation induces the sliding of joints
between chains and the sliding of mesoregions. The final constitutive equations
have the form

ε ε pl
s ¼ E 0 ðε ε pl Þ 1 a 1 exp (13.58a)
ε 1