Page 491 - Handbook of Properties of Textile and Technical Fibres

464 Handbook of Properties of Textile and Technical Fibres

2
where s r is computed in kg/mm . For large l m Eq. (13.18) reduces to the Gaussian
equation (Treloar, 1975) and

1

s r ¼ C r l 2 (13.21)
l
where C r is the elastic network modulus. It is interesting, that the expression of Eq.
(13.21) in terms of true stress (s tr ) leads to the form

2 1
s tr ¼ C r l (13.22)
l
The strain dependence of the glassy polymers networks deformation can be also
expressed by another non-Gaussian chain statistic (Arruda and Boyce, 1993). For the
plane-strain geometry, the rubberlike stress s r generated by the entangled polymeric
chain network in the direction of loading is expressed in the form:

p ffiffiffiffi
n E h 2 1 1 l c
s r ¼ l 2 L p ffiffiffiffi (13.23)
3 l c l n

where E h is the initial strain hardening modulus of the network, n is the number of
“rigid” entanglements between crosslinks providing limiting extensibility of a
p ffiffiffi
chain (l m ¼ n), and l c is the stretch on each chain in the network
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
l c ¼ l þ 1 þ 1 l 3. The final model describing polymeric material defor-
mation behavior can be then simply expressed as the sum of s r þ s y where s y is yield
stress (Bergstroem and Boyce, 1998).
The rubberlike behavior of a polymeric network can be approximated by a model
composed from a basic cell containing eight non-Gaussian chains (Langevin springs).
This model includes the cooperative nature of the network deformation. The response
of this model to a uniaxial deformation is in the form (Arruda and Boyce, 1993; Diani,
2008)

8 19
0
0 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
p ffiffiffiffi > 2 >
G N < l þ 2=l 2 =
s r ¼ p L 1 @ p A B l 1=l C (13.24)
@q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA
ffiffiffi
ffiffiffiffiffiffi
3N 2
l 3 > >
: l þ 2=l ;
Parameter G is equal to the material shear modulus and N is a function of the stretch
at break l b , i.e.,
2
l þ 2=l b
b
N ¼ (13.25)
3