Page 487 - Handbook of Properties of Textile and Technical Fibres

460 Handbook of Properties of Textile and Technical Fibres

(1) and (2) are very useful. These models are well suited to the macromechanical
approach to the modeling of fiber rheology. The models from group (4) need some
additional quantitative information on structural elements. The most complex models
in group (3) represent a micromechanical approach to the modeling of fiber rheology.
In this case it is necessary to have complex information about structural elements and
their mechanical behavior.

13.4.1.1 Continuum models
The continuum approach ignores the molecular nature of polymers and treats it in
terms of laws of elasticity for solids and laws of fluid dynamics and viscous flow
for liquids. Models of this type consider the fiber from the perspective of its mechan-
ical behavior as a homogeneous (non) linear viscoelastic body and the fine structure of
the fiber is neglected. The simplest (spring/dashpot) models are based on the formal
ideas of linear viscoelasticity (Christensen, 1982). They are generally expressed by
using linear differential equations with constant coefficients. The strain limit for line-
arity remains constant at about 1% throughout the glassy region, then increases very
rapidly from 1% to roughly 50% as the polymers go through the transition region and
reaches up to 100%e150% when the polymers are in the rubbery region (Yannas,
1974).
Many models use various theories of nonlinear viscoelasticity of homogeneous
bodies (Hadley and Ward, 1975). The general multiple integral constitutive
relations for a nonlinear viscoelastic material are given by the GreeneRivlin theory
(Christensen, 1982). This constitutive relation is based on an expansion of multiple in-
tegrals with multivariable relaxation functions as kernels. For nonlinear deformation of
polymeric fibers, the appropriate adaptation of the classical Boltzman integral was sug-
gested by Leadermann (1943).
A number of other approaches based on the theory of nonlinear viscoelasticity was
published by Yannas (1974). Most of the rheological models of this type cannot be
expressed analytically and are expressed by multiple integral equations with rather
complicated kernels. The nonlinear viscoelastic and viscoplastic constitutive model
based on the generalized Boltzman integral proposed by Schapery (1997) was success-
fully modified for modeling of the nonlinear viscoelastic and viscoplastic behavior of
polyester fibers (Chailleux and Davies, 2005).
Hall (1967) proposed an empirical model of stress-strain curves for fibers, where
stress is expressed as a combination of the functions of deformation and time (the prin-
ciple of separability).

sðtÞf 1 ðεÞ
¼ f 2 ðεÞþ f ðtÞ (13.9)
ε
Nonlinearity is introduced by means of functions f 1 ðεÞ and f 2 ðεÞ, which are equal to
1 and 0, respectively, if the behavior is linear. Hall found that the function f 1 ðεÞ varied
linearly with strain over the region of homogeneous deformation and was independent
of the nature of the fiber type. The function f 2 ðεÞ was found to be clearly nonzero.