Biomedical Engineering Reference
In-Depth Information
4.2
Theories of rubber elasticity
As mentioned earlier, the decrease in conformational entropy of a stretched rubber gives
rise to an elastic restoring force; this force is, of course, the signature property of a rubber.
The classical theory of rubber elasticity (Flory,
1953
; Treloar,
1975
) is now some 70
years old (three of the most important papers were published in the same year, 1943), so it
is not surprising that it has been subsequently challenged, extended and improved
(Erman and Mark,
1997
; Mark and Erman,
1998
). That said, for elastomeric systems,
and where exact quantitative agreement between theory and experiment is not required,
its beauty and simplicity still makes it a useful starting point for any more detailed
discussion.
In the original theories (mostly pre-1975) the exact numerical factors depend upon
whether the derivation assumes
'
phantom
'
networks, in which chains have no excluded
volume and so do not
'
see
'
one another, or
'
af
ne
'
networks, so that network junction
points move af
nely following macroscopic deformation.
Pre-1975 theories give the elastic free energy as
ð
x
2
þ
y
2
þ
z
2
D
F
el
¼
ak
B
T
Þ;
ð
4
:
1
Þ
where the
Λ
is refer to macroscopic deformation ratios in the x, y and z directions, and a is a
'
-
strain behaviour of an elastomeric network can be calculated, and the stress is found to be
proportional to (
'
front factor
which depends on the particular assumption used. In this way the stress
Λ-Λ
−
2
), where
3
is simply the product (
Λ
x
Λ
y
Λ
z
) (Treloar,
1975
).
What the theory also produces is a relationship between the equilibrium elastic
modulus
Λ
in terms
of the number of cross-links (more exactly the number of junction points and chains in
the network). In one version of the theory, there is a quantitative relationship between the
modulus of the network (in this case the equilibrium elastic shear modulus G) and the
number of elastically active network chains (EANCs) (Gordon and Ross-Murphy,
1975
).
This version of the theory gives
-
the ratio of the retractive force per area, i.e. stress, per unit strain
-
G
¼
a
ν
e
k
B
T
:
ð
4
:
2
Þ
Here
ν
e
is the number density of EANCs, and is dependent upon the degree of conversion p
and the functionality f. The parameter a is again the rubber front factor, and k
B
T is the usual
Boltzmann energy term. More particularly in this formulation,
ν
e
will be zero before the gel
point, de
ned as in (
3.3
), and so, up to this point, G = 0. An EANC is de
ned as any
sequence of cross-linked units joining two units known as
'
ties
'
, and a tie is any unit which
has at least three non-extinct cross-links, i.e. is
'
tied
'
into the in
nite gel molecule by at least
three separate paths to in
ν
e
is then equal to half the
average number of non-extinct cross-links per unit present, because each EANC has two
ends. The average is calculated excluding contributions from units with only one or two
non-extinct cross-links, using methods described elsewhere. (Note that in an alternative
form of this equation, due originally to Flory (Erman and Mark,
1997
), the relevant
parameter is M
c
, the molecular mass between cross-links, rather than
nity (see
Figure 3.2
). The value of
ν
e
.)
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