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subchain is small compared to ncl0. Without going into full mathematical
details, let us examine how this constraint can be lifted.
The first step is to write an expression for the energy stored in a
network U as a result of the elongation a. This may be written as
where the F's are the x, y, and z components of the distorting force and
x0, y0, and z0 are the initial dimensions of the sample. Assuming that
the applied force is in the z direction, as in Fig. 3.1, we note that the
contraction in the x and ó directions is much less than the elongation in
z. The x and ó components of the force are therefore assumed to obey
Hooke's law; that is, Fx = kHx, with kH given by Eq. (3.45). A similar
expression describes F , but Fz is described differently because of the
greater elongation it experiences. For the latter the the force component
may be shown to be
in which rz is the z component of the average end-to-end distance in the
subchain. Note that Eq. (3.48) gives a force which is identical to the x
and ó components if rz < ncl0. Equation (3.48) gives the first three
terms in the series expansion of the inverse Langevin function of
rz/ncl0; this describes
óî Na azo
U = /
Fx dx + / Fy dy + / Fz dz
The Elastic and Viscoelastic States
the chain-end separation without assuming stringent limitations on
r/ncl0. Equations (3.48) and (3.45) with n = nc can be substituted into
Eq. (3.47) and the result integrated and then multiplied by v to give the
elastic energy stored per unit volume of elastomer. The derivative of
this last quantity with respect to a equals at. When the indicated
operations are carried out, the following result is obtained:
For the case of a < nc'/2, terms higher than first order in this
expression are negligible and the result becomes identical to Eq. (3.39).
However, the derivation of Eq. (3.49) does not limit its applicability to
these small values of a. Figure 3.5 shows that experimental results are
still not exactly described by Eq. (3.49), although the general shape of
the theoretical and experimental lines is similar.
ÃÒ + 7 V nc
a + 9 / a \ 297 5 / a v
fif s m ç (;/=-)
Figure 3.5 Comparison of experiment (points) and theory [Eq. (3.49)] for
the entropy elasticity of the same sample shown in Fig. 3.3. [Reprinted
with permission from H. M. James and E. Guth, J. Chem. Phys. 11:455
The Shear Modulus and the Compliances
In connection with Eq. (3.45) we noted that the deformation of
individual chains can be studied directly from random flight statistics.
Using equivalent expressions for the x, y, and z components of force and
following the procedure outlined above gives a more rigorous derivation
of Eq. (3.39) than that presented in the last section.
Even better agreement between theory and experiment has been obtained
in other theories by abandoning the notion of affine deformation and
recognizing that shorter subchains experience a greater strain than do
longer subchains for a given stress. We shall not pursue this development
any further, however, and shall turn next to a consideration of other
types of deformation.
3.6 The Shear Modulus and the Compliances
Until now we have restricted ourselves to consideration of simple tensile
deformation of the elastomer sample. This deformation is easy to
visualize and leads to a manageable mathematical description. This is by
no means the only deformation of interest, however. We shall consider
only one additional mode of deformation, namely, shear deformation.
Figure 3.6 represents an elastomer sample subject to shearing forces.
Deformation in the shear mode is the basis
Figure 3.6 Definition of variables to define the shear deformation of an
The Elastic and Viscoelastic States
for the definition of viscosity. An examination of elasticity according
to the same mode of deformation will make it easy to combine the two for
a description of viscoelasticity. We shall begin, however, by describing
the purely elastic response to a body under conditions of shear.
By analogy with Eq. (3.1), we seek a description for the
relationship between stress and strain. The former is the shearing force
per unit area, which we symbolize as as, as in Chap. 2. For shear strain
we use the symbol ys; it is the rate of change of ys that is involved in
the definition of viscosity in Eq. (2.2). As in the analysis of tensile
deformation, we write the strain AL/L, but this time AL is in the