CONTINUUM MECHANICS for ENGINEERS

where ρ is a parameter of the distribution. Using this assumed distribution
of mean end-to-end distances, it is straightforward to find
∫
∞
2 2 3 2
r = r P() r dr = ρ
o
0
2
(8.1-6)
where the subscript o denotes that this is an intrinsic property of the chain
since it was considered alone. When the chain is placed back into a crosslinked
network of chains, the mean end-to-end distance is written as .
This latter designation takes into account the fact the chain has restrictions
placed upon it by being packed into a volume with other chains. Equating
Eqs 8.1-4 and 8.1-6 the distribution parameter ρ is found to be
r
2
i
ρ = 2n
l
3
(8.1-7)
Similar to the results of Section 5.9, the force created by stretching a uniaxial
specimen is given in terms of the Helmholtz free energy
F = ∂ψ
∂
L
θ , V
(8.1-8)
where F is the force, L is the length, and subscripts θ and V designate that
the change in length occurs at constant temperature and volume. Substitution
of Eq 5.8-9b yields the force in terms of the internal energy and entropy
u
F =
L L
∂ ∂
−θ ∂ ∂
η
(8.1-9)
where the constant temperature and volume subscripts have not been written
for convenience.
Examination of Eq 8.1-9 offers an informative comparison between metals
and ideal rubbers. In metals, the crystalline structure remains intact as the
material is deformed. Atoms are moved closer, or further, from adjacent
atoms creating a restoring force, but the relative order among the atoms
remains the same. The last term of Eq 8.1-9 has no force contribution since
the relative order of the atoms stays unchanged. For an ideal rubber, a change
in length has no effect on the internal energy. Thus, the first derivative term
of Eq 8.1-9 is zero. However, stretching of the specimen increases the mean
end-to-end distance, thus reducing the possible conformations for the chains.
This reduction in conformations gives rise to a negative change in entropy
as the length is increased.
The entropy for a single chain will be related to the conformation through
the mean end-to-end length. Noting the number of configurations is