Chemistry Reference
In-Depth Information
where
is called a
retardation time
in a creep experiment.
Equation (4-67)
can be made exact by using the multiplying factor
e
t/ζ
. Integration from
ζ 5G/η
τ 5τ
0
,
γ5
0at
t
5
0 gives
G
γ=τ
0
5
1
2
exp
ð2
Gt
=ηÞ 5
1
2
exp
ð2
t
=ζÞ
(4-68)
If the creep experiment is extended to infinite times, the strain in this element
does not grow indefinitely but approaches an asymptotic value equal to
τ
0
/G
.
This is almost the behavior of an ideal elastic solid as described in
Eq. (4-36) or
(4-57)
. The difference is that the strain does not assume its final value immedi-
ately on imposition of the stress but approaches its limiting value gradually. This
mechanical model exhibits delayed elasticity and is sometimes known as a
Kelvin
solid
. Similarly, in creep recovery the Maxwell body will retract instantaneously,
but not completely, whereas the Voigt model recovery is gradual but complete.
Neither simple mechanical model approximates the behavior of real polymeric
materials very well. The Kelvin element does not display stress relaxation under
constant strain conditions and the Maxwell model does not exhibit full recovery
of strain when the stress is removed. A combination of the two mechanical mod-
els can be used, however, to represent both the creep and stress relaxation beha-
viors of polymers. This is the standard linear solid, or Zener model, comprising
either a spring in series with a Kelvin element or a spring in parallel with a
Maxwell model. Details of this construction are outside the scope of this introduc-
tory text.
Limitations to the effectiveness of mechanical models occur because actual
polymers are characterized by many relaxation times instead of single values and
because use of the models mentioned assumes linear viscoelastic behavior which
is observed only at small levels of stress and strain. The linear elements are nev-
ertheless useful in constructing appropriate mathematical expressions for visco-
elastic behavior and for understanding such phenomena.
4.7.2.3
Time
Temperature Correspondence
The left-hand panel of
Fig. 4.19
contains sketches of typical stress relaxation
curves for an amorphous polymer at a fixed initial strain and a series of tempera-
tures. Such data can be obtained much more conveniently than those in the exper-
iment summarized in
Fig. 4.8
, where the modulus was measured at a given time
and a series of temperatures. It is found that the stress relaxation curves can be
caused to coincide by shifting them along the time axis. This is shown in the
right-hand panel of
Fig. 4.19
where all the curves except that for temperature
T
8
have been shifted horizontally to form a continuous “master curve” at temperature
T
8
. The glass transition temperature is shown here to be
T
5
at a time of 10
2
2
min.
The polymer behaves in a glassy manner at this temperature when a strain is
imposed within 10
2
2
min or less.
Similar curves can be constructed for creep or dynamic mechanical test data
of amorphous polymers. Because of the equivalence of time and temperature, the
