Chemistry Reference
In-Depth Information
η
0
(
The
) term is often called the
dynamic viscosity
. It is an energetic dissipation
term related to
G
ω
v
(
ω
) and has a value approaching that of the steady flow viscos-
ity
η
in very low frequency measurements on polymers that are not cross-linked.
4.7.2
Linear Viscoelasticity
In linear viscoelastic behavior the stress and strain both vary sinusoidally,
although they may not be in phase with each other. Also, the stress amplitude is
linearly proportional to the strain amplitude at given temperature and frequency.
Then mechanical responses observed under different test conditions can be inter-
related readily. The behavior of a material in one condition can be predicted from
measurement made under different circumstances.
Linear viscoelastic behavior is actually observed with polymers only in
very restricted circumstances involving homogeneous, isotropic, amorphous
specimens subjected to small strains at temperatures near or above
T
g
and
under test conditions that are far removed from those in which the sample
may be broken. Linear viscoelasticity theory is of limited use in predicting
the service behavior of polymeric articles, because such applications often
involve large strains, anisotropic objects, fracture phenomena, and other effects
that result in nonlinear behavior. The theory is nevertheless valuable as a ref-
erence frame for a wide range of applications,
just as the thermodynamic
equations for
ideal solutions help organize the observed behavior of
real
solutions.
The major features of linear viscoelastic behavior that will be reviewed here
are the superposition principle and time
temperature equivalence. Where
they are valid, both make it possible to calculate the mechanical response of a
material under a wide range of conditions from a limited store of experimental
information.
4.7.2.1
Boltzmann Superposition Principle
The Boltzmann principle states that the effects of mechanical history of a sample
are linearly additive. This applies when the stress depends on the strain or rate of
strain or, alternatively, where the strain is considered a function of the stress or
rate of change of stress.
In a tensile test, for example,
Eq. (4-40)
relates the strain and stress in a creep
experiment when the stress
τ
0
is applied instantaneously at time zero. If this load-
ing were followed by application of a stress
σ
1
at
time
u
1
,
then the time-
dependent strain resulting from this event alone would be
εð
t
Þ 5σ
1
D
ð
t
2
u
1
Þ
(4-53)
The total strain from the imposition of stress
σ
0
at
t
5
0 and
σ
1
at
t
5
u
1
is
εð
t
Þ 5σ
0
D
ð
t
Þ 1σ
1
D
ð
t
2
u
1
Þ
(4-54)
