Materials for engineering, 3rd Edition - (Malestrom)

166
Materials for engineering
5.4.2 Rubber elasticity
Curve 1 of Fig. 5.4 is typical of elastomers above their T g , showing a low
modulus but, because of cross-linking between the chains, exhibiting a
reversible elasticity up to strains of several hundred per cent. In the unstressed
state, the molecules are randomly kinked between the points of cross-linkage.
On straining, the random kinking is eliminated because the polymer chains
become aligned and, at high strains, X-ray diffraction photographs of stretched
rubber show that this alignment gives rise to images typical of crystallinity.
Thermodynamically, the deformation of an elastomer may be regarded as
analogous to the compression of an ideal gas. Applying a combination of the
first and second laws of thermodynamics to tensile strains, we may write:
dU = TdS + FdL – PdV [5.1]
where F is the tensile force, L the length of the specimen, P the pressure and
V the volume. In elastomers, the Poisson ratio is found to be approximately
1 / 2 , which implies that the tensile elongation causes no change in volume.
Thus, if an ideal elastomer is extended isothermally, then dU = 0 and equation
[5.1] becomes:
⎛ ∂S
⎞
F = – T⎜
⎟ [5.2]
⎝ ∂L
⎠
T,V
The ordering of the molecules by the applied strain decreases the entropy of
the specimen. The strained state is thus energetically unfavourable and when
the specimen is released it will return to a higher-entropy state where the
chains have random conformations.
Treloar has calculated (δS/δL) T for an isolated long-chain molecule and
for a model network of randomly kinked chains cross-linked at various
points. If p is the probability of a particular configuration in an irregularly
kinked chain, then the entropy of the chain can be written:
S = k log p
where k is Boltzmann’s constant.
Treloar shows that S = constant – kb 2 L 2 , where b is a constant related to
the most probable linear distance between the ends of the chain. So from
equation [5.2], the tensile force required to extend by dL a chain whose ends
are separated by a distance L is:
F = 2kTb 2 L
The force is thus proportional to the temperature, which is in good agreement
with experiment in that elastomers differ from crystalline solids, where the
elastic moduli decrease as the temperature rises.
Elastomers do not show the Hookean behaviour of F ∝ L, however, as