Introduction
Classical dynamic material testing involves the application of a
sinusoidal load to a sample and the recording of its displacement response.
The load and displacement data are used to calculate
stress and strain cycles. The ratio of the stress amplitude to the strain
amplitude is the
dynamic modulus. For shear loading, the usual
symbol, \(G\), is used. The phase lag, \(\delta\), between the stress
input and strain response is also recorded and usually presented as
\(\tan(\delta)\). Various combinations of these parameters are
plotted against strain amplitude, temperature, and/or frequency.
Definitions
The shear stress amplitude, \(\tau_o\), shear strain amplitude, \(\gamma_o\), and phase lag,
\(\delta\), are combined in many different ways to form different material property
definitions. The most fundamental ones are simply ratios between stress and strain.
The ratio of stress to strain gives the dynamic modulus, \(G^*\).
The ratio of strain to stress gives the dynamic compliance, \(J^*\).
\[
G^* = {\tau_o \over \gamma_o} \qquad \qquad J^* = {\gamma_o \over \tau_o}
\]
Clearly \(G^* = 1 / J^*\) and viceversa. The remaining fundamental quantity is
the tangent of the phase lag, \(\tan(\delta)\), often simply called "tan delta" and
sometimes called the "loss tangent".
The inphase and outofphase components of the dynamic modulus are known as the
storage modulus and loss modulus, respectively.
Storage Modulus
 \( \qquad G' = G^* \cos(\delta) \)

Loss Modulus
 \( \qquad G'' = G^* \sin(\delta) \)

From this, it is clear that \(\tan(\delta)\) is related to the ratio of \(G''\) to \(G'\).
\[
\tan(\delta) = {G'' \over G'}
\]
The inphase and outofphase components of the dynamic compliance are known as the
storage compliance and loss compliance, respectively.
Storage Compliance
 \( \qquad J' = J^* \cos(\delta) \)

Loss Compliance
 \( \qquad J'' = J^* \sin(\delta) \)

And it is clear that \(\tan(\delta)\) is also related to the ratio of \(J''\) to \(J'\).
\[
\tan(\delta) = {J'' \over J'}
\]
Incorrect Relationships
Do not be tempted to say that \(J' = 1 / G'\) and \(J'' = 1 / G''\), because
these are incorrect. The correct relationships are
\[
J' = J^* \cos(\delta) = { \cos(\delta) \over G^* } = { \cos^2(\delta) \over G' }
\]
\[
J'' = J^* \sin(\delta) = { \sin(\delta) \over G^* } = { \sin^2(\delta) \over G'' }
\]
It is perhaps easier to remember these as
\[
G' * J' = \cos^2(\delta) \qquad \quad \text{and} \qquad \quad
G'' * J'' = \sin^2(\delta)
\]
Strain Dependence
Here is some test data for a rubber sample. As with the uniaxial tension
test data on the previous
MooneyRivlin page,
the stiffness of the rubber decreases as the strain amplitude increases.
The curve labeled "GO" is for the portion of the test where the input load
amplitude increases with time.
The curve labeled "RETURN" is for the portion of the test where the input load
amplitude decreases with time.
The difference between the two is called the
Mullins Effect.
Below is a graph of the predicted shear strain for a sinusoidal shear stress
input signal. The predictions are based on the material stiffness from the
graph just above. Note how nonlinear the
predicted shear strain is. In other words, the ratio of stress to strain as
time passes is not constant (independent of the time delay due to the phase
lag).
However, this is not the case for actual rubber behavior during dynamic tests.
The actual strain signal is indistinguishable from the first figure at the top
of this page.
Here is test data for the phase lag of the same rubber sample.
Temperature Dependence
This is a plot of \(J'\) and \(\tan(\delta)\) versus temperature.
This is a plot of \(J''\) and \(\text{log}_{10}(G')\) versus temperature.
And here is a summary sketch. Note that temperature and frequency increase
in opposite directions.
Time Temperature Equivalence
Around the time of World War II, Williams, Landel and Ferry reported a fascinating
property of polymers: timetemperature equivalence. They observed that the dynamic
material properties of a polymer at a reference temperature and frequency could be
reproduced at a different combination of temperature and frequency that are related
to the first pair by one simple equation. That equation is now known as the
WLF Equation or
WLF Transform.
Timetemperature equivalence means that the stiffness and hysteresis of a polymer
will be the same at the proper combination of low temperature and low frequency
as at a given combination of high temperature and high frequency.
This property is very beneficial from an experimental viewpoint because it means
that the viscoelastic properties of materials at thousands of Hertz can be estimated
in the lab by testing at low frequencies and very low temperatures. (This is often
done for traction studies.)
WLF Transforms
The WLF equation is often published in many similar, though slightly different forms.
These are just specific cases of the general form given here.
\[
\log_{10} \left( {f_1 \over f_0} \right) = {C_1 (\theta_1  \theta_0) \over (C_2 + \theta_0) (C_2 + \theta_1) }
\]
where:
\(\theta_0 = T_0  T_g\)
^{ } _{ }
\(\theta_1 = T_1  T_g\)
Usual values of the constants are \(C_1\) = 900 and \(C_2\) = 51.6 .
The coefficients, \(C_1\) and \(C_2\), are obtained by curve fitting the
WLF equation to experiemental data. They change little regardless of the application.
For example, estimates from test in MFG on PL KM's gave \(C_1 = 900\) and
\(C_2 = 55\). But it must be noted that there was a great deal of uncertainty
in these estimates.
Overall, the WLF transform is considered to perform quite well, though not perfectly,
over many decades of frequency and at \(T_g\) ≤ \(T\) ≤ \(T_g\)+100°C.
The WLF transform can be rearranged to solve for \(\theta_1\), which is more useful.
\[
T_1  T_g = { C_1 \theta_0  C_2 ( C_2 + \theta_0 ) \log_{10} ( f_1 / f_0 ) \over
C_1 + ( C_2 + \theta_0 ) \log_{10} ( f_1 / f_0 ) }
\]
WLF Example
Consider the multistress test results above. The graph gives the material properties
at 10 Hz over a wide range of temperatures. Let's (arbitrarily) start with
\(\tan(\delta)\) at 10 Hz, 70°C, and at the 0.035 kg/mm
^{2} stress level, (the longdashed curve on the graph).
At these conditions, \(\tan \delta\) = 0.17 . To estimate \(\tan \delta\) at
70°C and 100 Hz, use
\[
\begin{eqnarray}
C_1 & = & 900 \qquad & T_0 & = & 70^\circ\text{C} \\
C_2 & = & 51.6 \qquad & f_0 & = & 10 \text{ Hz} \\
T_g & = & \text{}20^\circ\text{C} \qquad & f_1 & = & 100 \text{ Hz}
\end{eqnarray}
\]
to get \(T_1\) = 51°C. At 51°C and 10 Hz, \(\tan \delta\) = 0.2 .
Therefore, at \(T_1\) = 70°C and 100 Hz, \(\tan \delta\) should also
equal 0.2.
Print this file out as a transparency and it can be placed over a Multistress Test graph
to perform approximate WLF transforms.
More extensive experimental data looks like this.