Page 506 - Handbook of Properties of Textile and Technical Fibres

Tensile failure of polyester fibers 479

dε pl ε ε pl
¼ a 1 exp (13.58b)
dε ε 1
Parameter a [0, 1] is the rate of sliding for a fully developed flow of joints. The
strain, ε 1 , characterizes the transition to the steady viscoplastic flow and E 0 is the initial
elastic modulus. The adjustable parameters E 0 , a, and ε 1 can be found from experi-
mental stress-strain curves (Drozdov and Christiansen, 2003). A phenomenological
constitutive model to express stress dependence on strain, strain rate, and temperature
was proposed in the work (Duan et al., 2001). This constitutive model has the
capability to describe the entire range of deformation behavior of glassy and semicrys-
talline polymers, especially the intrinsic strain softening and subsequent orientation
hardening.
Engineering stress-strain dependence can be described by using models from
Section 13.4.1 usually by solving governing differential equations. For the three-
element models from Fig. 13.20, the differential equation determining the relationship
between the stress s, deformation ε, and time t has the form

ds dε h 2 i
¼ ½E 0 þ E 1 ð1 2K 1 εފ E 0 k sinh a s E 1 ðε K 1 εÞ (13.59)
dt dt
In cases when stress-strain curves are measured at a constant deformation rate
n ¼ dε=dt, the solution of this differential equation leads to the relation

1
2

s ¼ E 1 ε K 1 ε þ ln½R þ S tanh ðK 4 ε þ K 5 ފ (13.60)
2
where
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R ¼ v=k S ¼ 1 þ R 2 (13.61)
and

akE 0 S
K 4 ¼ K 5 ¼ arg tgh½ð1 RÞ=SŠ (13.62)
2v

The model parameters a; k; K 1 ; E 0 ; E 1 can be estimated from experimental data by
nonlinear regression (Meloun et al., 1994). The estimated model parameters for heat-set
1
PET fibers having a draw ratio of l ¼ 5 are a ¼ 13.9 GPa , k ¼ 7.2$10 3 s, K 1 ¼ 3,
3
E 0 ¼ 5.6 GPa, E 1 ¼ 0.52 GPa, DE s ¼ 94.2 kJ/mol, and V f ¼ 2.86$10 27 m . Details
about the application of this model for the modeling of stress-strain curves of modified
PET fibers are in the book by Militký et al. (1991).
The large tensile deformation behavior of PET fibers at high temperatures can be
described in terms of stretching springs and dashpots, approximating intermolecular
resistance acting in parallel with network resistance (see Section 13.4.1). In this rep-
resentation, the deformation of the crystalline phase is coupled with the amorphous