Chemistry Reference
In-Depth Information
FIGURE 4.17
Representation of a complex number z5x 1 iy by a vector on the xy plane (i 5
O
2
1).
z 5x1 iy 5 re
iθ
5 r (cos θ1 i sin θ); r 5 jzj 5 jx 1yj 5 (x
2
2
)
1/2
;1/z5e
iθ
/r 5 (1/r). (cos
1y
θ2 i sin
θ
).
and
G
0
ðωÞ 5
J
0
ðωÞ
tan
δ 5
G
vðωÞ=
J
vðωÞ=
(4-49)
The real and imaginary parts of the complex numbers used here have no phys-
ical significance. This is simply a convenient way to represent the component
vectors of stress and strain in a dynamic mechanical experiment.
Tan
measures the ratio of the work dissipated as heat to the maximum
energy stored in the specimen during one cycle of a periodic deformation. The
conversion of applied work to thermal energy in the sample is called
damping
.It
occurs because of flow of macromolecular segments past each other in the sam-
ple. The energy dissipated per cycle due to such viscoelastic losses is
δ
a
G
.
For low strains and damping the dynamic modulus
G
0
will have the same mag-
nitude as that obtained from other methods like stress relaxation or tensile tests,
provided the time scales are similar in these experiments.
Viscosity is the ratio of a stress to a strain rate [
Eq. (4-39)
]. Since the complex
modulus
G
has the units of stress, it is possible to define a complex viscosity
πγ
v
η
as the ratio of
G
to a complex rate of strain:
G
ðωÞ
i
G
0
ðωÞ 1
iG
vðωÞ
η
ðωÞ 5
5η
0
ðωÞ 2
5
i
ηvðωÞ
(4-50)
ω
i
ω
Then it follows that
η
0
ðωÞ 5GvðωÞ=ω
(4-51)
and
G
0
ðωÞ=ω
ηvðωÞ 5
(4-52)
