In this article I will demonstrate one common mistake that I see people do when they try to create a time-temperature-superposition (TTS) Master Curve from multiple frequency sweeps at different temperatures. It turns out that if you don’t also include how the instantaneous (or long-term) elastic response depends on the temperature, then the generated Master Curve will not be very accurate.
A common problem that engineers are faced with is determining how much stress relaxation will occur in a product over a very long time period. The challenge is that it is not easy to run a set of stress relaxation experiments over for a few months (or even years).
Figure 1. How much does the material relax over a very long time?
Common Solution Approach
One relatively common way to address this problem is to run DMA frequency sweep tests at a few different temperatures in order to get the type of data shown in Figure 2. The figure to the left shows how the storage modulus depends on the frequency, and it seems that if the curves are shifted horizontally, using the WLF equation, then one should be able to determine the master curve for the material.
Figure 2. DMA frequency sweeps at different temperatures.
Figure 3 shows the master curve that I generated by best matching the results of the individual temperature sweeps. There is not a perfect match between the data segments, but that is not unusual since experimental data often contains errors. Overall the master curve looks reasonable.
Figure 3. Master curve of the storage modulus as a function of frequency at 293 K.
Once the master curve has been established, it is easy to calibrate a linear viscoelastic material model to it. In this example I used an Abaqus Neo-Hookean hyperelastic model with a 9-term Prony series. The calibration results are shown in Figure 4. Overall the fit is pretty good. The waviness of the predicted response is due to the relatively small number of Prony series terms.
Figure 4. Viscoelastic material model calibration results.
Once the linear viscoelastic material model has been calibrated, we can use it for any purpose. Figure 5 shows the long-term stress relaxation response after a 1% strain jump. The model predicts that the stress will relax 62% over a long time period.
Figure 5. Predicted stress relaxation response from the master curve model.
So What is the Real Deal?
The problem with the approach outlined above is that it failed to consider the temperature dependence of the instantaneous (or alternatively, the long-term) response of the material. Only the viscoelastic flow rate is assumed to be temperature dependent.
One way to understand this is to consider a “really fast” tension test at different temperatures. In this case the instantaneous stress response will depend on the temperature.
In my example above I did not use real experimental data, instead I used predictions from a temperature-dependent Neo-Hookean model with a 3-term Prony series and the WLF equation. I specifically used the following Abaqus commands:
*Hyperelastic, Neo Hook, moduli=instantaneous 1.20, 0.02, 289 1.00, 0.02, 293 0.75, 0.02, 297 *Viscoelastic, time=prony ** gi, kappai, taui 0.1, 0, 0.1 0.2, 0, 1.0 0.1, 0, 10.0 *TRS, definition=WLF 293, 17.4, 51.6
In this case, since I have the real material model, I can easily predict and plot the storage modulus as a function of frequency for any temperature (see Figure 6). We can now clearly see that the original frequency curves that were horizontally shifted to create a master curve, do not actually form a master curve since the instantaneous response is also temperature dependent.
Figure 6. Real master curves for different temperatures.
The following figure shows a comparison between the stress relaxation predictions from the originally generated master curve, and relaxation behavior of the real material (model). In this case there is a very large difference between the two.
Figure 7. Correct stress relaxation response.
- Just because DMA data can be shifted does not mean that the generated “Master Curve” is correct.
- Do not forget that the instantaneous (or long-term) elastic response can also be temperature dependent.