In this article I will show how you can convert stress relaxation data obtained at different temperatures into master curve through time-temperature superpositioning (TTS) and a linear viscoelastic material model.
The experimental data is from an old paper by Catsiff and Tobolsky (“Stress-Relaxation of Polyisobutylene in the Transition Region”, 1955). They measured the stress relaxation response at 19 different temperature.
Method 1: First Create a Master Curve, Then Calibrate a Viscoelastic Model
The first step in analyzing the experimental data is to extract it from the image above into individual data files for each temperature. This is easy to do using the MCalibration data extraction tool. For this example, I converted the reported modulus to a stress by assuming the relaxation strain was 0.01.
I then horizontally shifted the curves using the Willims-Landel-Ferry (WLF) model. The optimal master curve was found by searching for the C1 and C2 parameters in the WLF equation, while keeping the parameter T0=-62°C. The objective function for the search was to minimize the arc-length of the shifted master curve. I performed this step using a Julia script file. We are working on implementing this feature natively in MCalibration.
All data points along the master curve was then assembled into one file. This figure shows the results from all individual tests and the resulting master relaxation curve (at T=-62°C).
The generated master curve can then be shifted to different temperatures using the WLF equation.
Now that we have a master curve, we can quickly calibrate a linear viscoelastic material model to the master curve data. In this case I used a neo-Hookean hyperelastic model with a 16-term Prony series to generate the predictions shown in the figure. The linear viscoelastic predictions are in good agreement with the experimental data.
The final temperature-dependent linear viscoelastic material model agrees well with the results from the individual stress relaxation tests from Tobolsky. The average error between the model predictions and the experimental data is 10.4%.
Method 2: Simultaneously Calibrate the Viscoelastic and WLF Models
Using MCalibration, it is also possible to calibrate the neo-Hookean hyperelastic model, the 16-term Prony series, and the WLF-parameters all at once. The approach is simply to: (1) read in the experimental data files for each temperature; (2) select a linear viscoelastic model with WLF temperature dependence; (3) Run the material model calibration. The final results are shown this figure. The average error in the model predictions is 8.6%. Which is pretty good.
The calibrated material model can also be used to predict the master curve response. As expected, also this material model matches the original Tobolsky results very well.