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For shear loading, one could say that \(\tau\) and \(\gamma\) should be used in the figure. Nevertheless, \(\sigma\) and \(\epsilon\) are customarily used anyway.

\[ G^* = {\sigma_o\over \epsilon_o} \qquad \qquad J^* = {\epsilon_o \over \sigma_o} \]

Clearly \(G^* = 1 / J^*\) and vice-versa. 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 in-phase and out-of-phase 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 in-phase and out-of-phase 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 \(J'\).

\[ \tan(\delta) = {J'' \over J'} \]

\[ J' = J^* \cos(\delta) = { \cos(\delta) \over G^* } = { \cos^2(\delta) \over G' } \]

\[ J'' = J^* \sin(\delta) = { \sin(\delta) \over G^* } = { \sin^2(\delta) \over G'' } \]

It is perhaps easier to remember these as

\[ G' * J' = \cos^2(\delta) \qquad \quad \text{and} \qquad \quad G'' * J'' = \sin^2(\delta) \]

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.

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 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.)

\[ \log_{10} \left( {f_1 \over f_0} \right) = {-C_1 (\theta_1 - \theta_0) \over (C_2 + \theta_0) (C_2 + \theta_1) } \]

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.

\[ T_1 - T_g = { C_1 \theta_0 - C_2 ( C_2 + \theta_0 ) \log_{10} ( f_1 / f_0 ) \over C_1 + ( C_2 + \theta_0 ) \log_{10} ( f_1 / f_0 ) } \]

\[ \begin{eqnarray} C_1 & = & 900 \qquad & T_0 & = & 70^\circ\text{C} \\ C_2 & = & 51.6 \qquad & f_0 & = & 10 \text{ Hz} \\ T_g & = & \text{-}20^\circ\text{C} \qquad & f_1 & = & 100 \text{ Hz} \end{eqnarray} \]

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.

More extensive experimental data looks like this.