Chemistry Reference
In-Depth Information
and
G
ð
t
Þ 5τð
t
Þ=γ
0
6¼ γð
t
Þ=τ
0
5
J
ð
t
Þ
(4-45)
Consider two experiments carried out with identical samples of a viscoelastic
material. In experiment (a) the sample is subjected to a stress
σ
1
for a time
t
. The
resulting strain at
t
is
ε
1
, and the creep compliance measured at that time is
D
1
(
t
)
σ
1
. In experiment (b) a sample is stressed to a level
s
2
such that strain
e
1
is achieved immediately. The stress is then gradually decreased so that the
strain remains at
e
1
for time
t
(i.e., the sample does not creep further). The stress
on the material at time
t
will be
5
e
1
/
σ
3
, and the corresponding relaxation modulus
will be
Y
2
(
t
)
5σ
3
/
e
1
. In measurements of this type,
it can be expected that
(
D
(
t
))
2
1
, as indicated in
Eq. (4-44)
.
G
(
t
) and
Y
(
t
) are
obtained directly only from stress relaxation measurements, while
D
(
t
) and
J
(
t
)
require creep experiments for their direct observation. These various parameters
can be related in the linear viscoelastic region described in
Section 4.7.2
.
σ
2
.σ
1
.σ
3
and
Y
(
t
)
6¼
4.7.1.1
Terminology of Dynamic Mechanical Experiments
A complete description of the viscoelastic properties of a material requires infor-
mation over very long times. Creep and stress relaxation measurements are lim-
ited by inertial and experimental limitations at short times and by the patience of
the investigator and structural changes in the test material at very long times. To
supplement these methods, the stress or the strain can be varied sinusoidally in a
dynamic mechanical experiment. The frequency of this alternation is
ν
cycles/sec
or
is qualitatively
equivalent to a creep or stress relaxation measurement at a time
t 5
ω
(
5
2
πν
) rad/sec. An alternating experiment at frequency
ω
) sec.
In a dynamic experiment, the stress will be directly proportional to the strain
if the magnitude of the strain is small enough. Then, if the stress is applied sinu-
soidally the resulting strain will also vary sinusoidally. In special cases the stress
and the strain will be in phase. A cross-linked amorphous polymer, for example,
will behave elastically at sufficiently high frequencies. This is the situation
depicted in
Fig. 4.15a
where the stress and strain are in phase and the strain is
small. At sufficiently low frequencies, the strain will be 90
out of phase with the
stress as shown in
Fig. 4.15c
. In the general case, however, stress and strain will
be out of phase (
Fig. 4.15b
).
In the last instance, the stress can be factored into two components, one of
which is in phase with the strain and the other of which leads the strain by
π
(1
/ω
/
2 rad. (Alternatively, the strain could be decomposed into a component in phase
with the stress and one which lagged behind the stress by 90
.) This is accom-
plished by use of a rotating vector scheme, as shown in
Fig. 4.16
. The magnitude
of the stress at any time is represented by the projection
OC
of the vector OA on
the vertical axis. Vector OA rotates counterclockwise in this representation with a
frequency
equal to that of the sinusoidally varying stress. The length of OA is
the stress amplitude (maximum stress) involved in the experiment. The strain is
represented by the projection
OD
of vector OB on the vertical axis. The strain
ω
