Chemistry Reference
In-Depth Information
FIGURE 4.15
Effect of frequency on dynamic response of an amorphous, lightly cross-linked polymer:
(a) elastic behavior at high frequency—stress and strain are in phase, (b) liquid-like
behavior at low frequency—stress and strain are 90
out of phase, and (c) general case—
stress and strain are out of phase.
vector OB also rotates counterclockwise with frequency
ω
but it lags OA by an
angle
.
The strain vector OB can be resolved into vector OE along the direction of
OA and OF perpendicular to OA. Then the projection
OH
of OE on the vertical
axis is the magnitude of the strain, which is in phase with the stress at any time.
Similarly, projection
OI
of vector OF is the magnitude of the strain, which is 90
(one-quarter cycle) out of phase with the stress. The stress can be similarly
resolved into two components with one along the direction of OB and one leading
the strain vector by
δ
. The
loss tangent
is defined as tan
δ
/
2 rad.
When the stress is decomposed into two components the ratio of the in-phase
stress to the strain amplitude (
π
γ
a
, maximum strain) is called the
storage modulus
.
This quantity is labeled
G
0
(
) in a shear deformation experiment. The ratio of
the out-of-phase stress to the strain amplitude is the loss modulus
G
ω
).
Alternatively, if the strain vector is resolved into its components, the ratio of the
v
(
ω
