Mechanical Properties of Polymer Solids and Liquids - Polymer Science and Engineering - page 184

4.7 Polymer Viscoelasticity
An ideal elastic material is one that exhibits no time effects. When a stress

is
applied the body deforms immediately to a strain e . (These terms were defined
broadly in Section 1.8.) The sample recovers its original dimensions completely
and instantaneously when the stress is removed. Further, the strain is always pro-
portional to the stress and is independent of the rate at which the body is deformed:

σ

σ 5 Y ε (4-36)

where Y is Young's modulus if the deformation mode is a tensile stretch and
Eq. (4-36) is an expression of the familiar Hooke's law. The changes in the shape
of an isotropic, perfectly elastic material will always be proportional to the mag-
nitude of the applied stress if the body is twisted, sheared, or compressed instead
of extended, but the particular stress/strain (modulus) will differ from Y .
Figure 4.14 summarizes the concepts and symbols for the elastic constants in ten-
sile, shear, and bulk deformations.
An experiment such as that in Fig. 4.14a can produce changes in the volume
as well as the shape of the test specimen. The elastic moduli listed in this

figure are related by Poisson's ratio , which is a measure of the lateral contrac-

tion accompanying a longitudinal extension:

β

1
2

β 5 1 2 ð 1 = V Þ@ V =@ε (4-37)

where V is the volume of the sample. When there is no significant volume
change, 0 . 5. This behavior is characteristic of ideal rubbers.

Real solids dilate when extended, and values of

@V/@5 0 and β 5

down to about 0.2 are observed
for rigid, brittle materials. The moduli in the elastic behavior of isotropic solids
are related by

β

Y 5 2 G ð 1 1βÞ 5 3 K ð 1 2 2 βÞ (4-38)

At very low extensions when there is no significant amount of permanent
deformation Y/G is between about 2.5 for rigid solids and 3 for elastomeric
materials.
An ideal Newtonian fluid was described in Section 4.13 . Such a material has
no elastic character; it cannot support a strain and the instantaneous response to a
shearing stress τ is viscous flow:

τ 5ηγ (4-39)

Here γ is the shear rate or velocity gradient ( 5 d γ /dt ) and η is the viscosity

which was first defined in Section 3.3.
Polymeric (and other) solids and liquids are intermediate in behavior between
Hookean, elastic solids, and Newtonian purely viscous fluids. They often exhibit
elements of both types of responses, depending on the time scale of the experi-
ment. Application of stresses for relatively long times may cause some flow and