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# Polymer Chemistry. The Basic Concepts - Himenz P.C.

Himenz P.C. Polymer Chemistry. The Basic Concepts - Copyright, 1984. - 736 p.
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A dynamic modulus can also be evaluated by following a procedure
similar to that used for the dynamic compliance above. The modulus is
most easily approached by considering a Maxwell element in which case the
differential equation analog to Eq. (3.77) is
ñîòî cos (cot) = - - + - a
(3.81)
G* dt ö*
Equations (3.77) and (3.81) both have the same general form dy/dt + Py =
Q, so the general solution-given in Example 3.5-is the same for both,
although the values of the constants are different. When the constants
are evaluated, the storage and loss components of the modulus are found
to be
G'(co) = co2r2 G*(l +co2t2)-1
(3.82)
and
G"(co) = gjtG*(1 + co2t2)-1
(3.83)
respectively.
The dynamic viscosity is related to the loss component of the shear
modulus through the result T?dyn = G"/co As ñî -> 0, the dynamic
viscosity approaches the zero shear viscosity of an ordinary liquid, t?n.
We commented above that the elastic and viscous effects are out of
phase with each other by some angle 5 in a viscoelastic material. Since
both vary periodically with the same frequency, stress and strain
oscillate with t, as shown in Fig. 3.14a. The phase angle 5 measures the
lag between the two waves. Another representation of this situation is
shown in Fig. 3.14b, where stress and strain are represented by arrows of
different lengths separated by an angle 5. Projections of either one onto
the other can be expressed in terms of the sine and cosine of the phase
angle. The bold arrows in Fig. 3.14b are the components of 7 parallel and
perpendicular to ñò. Thus we can say that ó cos 5 is the strain component
in phase with the stress and ó sin 6 is the component out of phase with
the stress. We have previously observed that the elastic response is in
phase with the stress and the viscous response is out of phase. Hence the
ratio of
178
The Elastic and Viscoelastic States
(a)
Çòã
Figure 3.14 General representations of stress and strain out of phase by
amount 5: (a) represented by oscillating functions and (b) represented by
vectors.
the two measures the relative contributions of these effects. These two
contributions are also measured by J'(co) and J"(co); therefore
7 sin 5
/Ýîë,
= tan 5 =
(3.84)
7 cos 5 J (ñî)
where the single parameter tan 5 is called the loss tangent. The loss
tangent is also proportional to the energy loss per cycle, so both J"(co)
and tan 5 measure viscous dissipation of mechanical energy in a sample.
Often the quantity n tan 6 is reported; it is called the logarithmic
decrement.
3.11 The Dynamic Components: Measurement and Interpretation
The wide range of variables involved makes experimental viscoelasticity
somewhat difficult to describe. In a particular study the range might be
narrowed because of the specific character of the material under
investigation. In more general terms, however, such wide ranges of
variation are involved that no single experimental design is suitable for
all purposes. Figure 3.12, for example, spans 9 orders of magnitude
variation in J(t) and 18 orders in t (no wonder log-log
The Dynamic Components: Measurement and Interpretation
179
coordinates are standard in this type of work!). In addition, samples can
vary from viscous liquids to brittle solids, and there may be
restrictions on the size or shape of available samples.
Both stress relaxation and creep experiments have their place in the
study of viscoelasticity, but measurement and interpretation of the
dynamic mechanical behavior of a polymer gives the most information on
the most convenient time scale. We shall describe the experimental
aspects of dynamic viscoelasticity only in broad conceptual terms. The
actual implementation of these concepts embraces many different kinds of
apparatus and experimental routines. In a conceptually simple apparatus
for measuring the dynamic components, the sample is sandwiched between
two surfaces, one of which is driven with a known periodic displacement.
The force at the other side of the gap is measured by a strain gauge. The
stress and strain at the two surfaces are proportional to the voltage
outputs of electromechanical transducers monitoring each plate. These can
be calibrated so that the maximum force F0 and maximum displacement x0
are read directly. The signals could be fed into a dual-beam oscilloscope
so that the sinusoidal variation of both the stress and strain appear on
the screen. The phase angle 6 can be measured from this trace, as shown
in Fig. 3.14a. Alternatively, the ratio of the two signals can be used to
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