Chemistry Reference
In-Depth Information
of the material, and the average molecular weight between cross-links. In terms
of nominal strain this equation is approximately
2
τ 5 ðρ
RT
=
M
c
Þð
3
ε1
3
ε
1 ...
(4-30)
and at low strains, Young's modulus,
Y
,is
Y
d
τ=
d
ε5
3
ρ
RT
=
M
c
(4-31)
The more tightly cross-linked the elastomer, the lower will be
M
c
and the
higher will be its modulus. That is, it will take more force to extend the polymer
a given amount at fixed temperature. Also, because the elastomer is an entropy
spring, the modulus will increase with temperature.
Equation (4-29)
is valid for small extensions only. The actual behavior of real
cross-linked elastomers in uniaxial extension is described by the Mooney
Rivlin
equation which is similar in form to
Eq. (4-29)
:
2
τ 5 ð
C
1
1
C
2
=ΛÞðΛ2
1
=Λ
Þ
(4-32)
Here
C
1
and
C
2
are empirical constants, and
C
1
is often assumed to be equal
to
ρ
RT/M
c
.
EXAMPLE 4-1
Given an SBR rubber (23.5 mol% styrene) that has an
M
n
of 100,000 before cross-linking.
/m
2
at 100% elongation of the cross-
Calculate the engineering stress in the units of
MN
linked elastomer with an
M
c
of 10,000 at 25
C. Also calculate the corresponding modulus
at very low extensions. The density of the cross-linked elastomer is 0.98 g/cm
3
.
A 100% elongation means that
Λ5
2
λ
0
/
λ5
2.
τ
and
Y
can be calculated using
Eqs. (4-
29) and (4-31)
, respectively.
5
0:98310
6
38:31433298
10;000
1
4
000
m
2
5
425
M
m
2
τ 5
2
2
425
;
0
:
330:98310
6
38:31433298
10;000
000
m
2
5
728
M
m
2
Y 5
5
728
;
0
:
4.5.2.4
Real and Ideal Rubbers
To this point, we have emphasized that the retractive force in a stretched ideal
elastomer is directly proportional to its temperature. In a cross-linked, real elasto-
mer that has been reinforced with carbon black, as is the usual practice, the force
to produce a given elongation may actually be seen to
decrease
with increased
temperature. This is because the anchor regions that hold the elastomer chains
together are not only chemical cross-links, as assumed in the ideal theory. They
also comprise physical entanglements of polymer molecules and rubber-carbon
