Chemistry Reference
In-Depth Information
So, suppose that we apply this property to our relaxation integral (eqn (4.47))
such that the relaxation spectrum is replaced by a Dirac delta function at
time t
m
:
H
¼
G
ð
N
Þ
d
ð
lnt
ln t
m
Þ
ð
4
:
129
Þ
which when substituted in the integrand for relaxation gives us
G
ð
t
Þ¼
Z
He
t
=
t
dlnt
¼
Z
þ
N
þ
N
G
ð
N
Þ
d
ð
lnt
lnt
m
Þ
e
t
=
t
dlnt
N
N
¼
G
ð
N
Þ
e
t
=
t
m
ð
4
:
130
Þ
This has now reduced the integral to a Maxwell relaxation process. In
other words, a spike or delta function in the distribution of relaxation
processes represents a Maxwell model. It has a position in time that defines
the relaxation time. So, it is not difficult to prove that two spikes equates with
two Maxwell models and so forth. However, it is not a complete description.
For a viscoelastic solid we know that we have a G(0) value. This can be
represented by a Dirac delta function but only at infinite time, which is
an esoteric concept to say the least! Nonetheless, the fact that a Maxwell
model is represented spectrally by a spike is an important concept. A spike in
the retardation spectra represents a Kelvin model. It is possible to use these
properties of the spectra and the Dirac delta function to interrelate models. It is
not straightforward, however. Throughout this chapter we have largely derived
most of the mathematical functions used. Those that have not been derived are
either repetitious examples or require the application of complicated arith-
metic. For example, a convolution integral is required to derive the relationship
between creep and relaxation and this has not been explored in detail as,
although it is interesting, it is not essential to our argument. The interrelation-
ship between spectra has been well described by Gross.
2
For a Kelvin model for
example one can derive the characteristic relaxation time and moduli of the
springs in an equivalent Maxwell and accompanying spring. As the number of
Maxwell or Kelvin models increases the calculations required to convert
between relaxation and retardation spectra increases. The mathematics is not
particularly involved but it is time consuming. As the number of models
increase polynomials arise that inevitably can only be readily solved numeri-
cally. There are useful rules that can be considered when modelling relaxation
and retardation spectra. These have been clearly stated by Gross. Put simply
these are:
When one spectra (i.e. relaxation) is represented by a Dirac delta func-
tion, the other (i.e. retardation) is also a delta function.
For a viscoelastic liquid the number of delta functions in the relaxation
spectra always exceeds by 1 the number in the retardation spectra.
For a viscoelastic solid the number of delta functions in the relaxation
spectra is the same as the number in the retardation spectra.
