Here is some experimental data for a slightly filled PTFE material that was pull in tension to 1% strain, and then held at that strain for 1,500 seconds (25 min). The figure shows that the stress relaxes from about 4.5 MPa down to 2.5 MPa during that time. The question that I will try to answer in this article is how one can extrapolate short-term data to long-term stress relaxation.
Also see our posts on: “Predicting the Long-Term Stress Relaxation Response of Rubbers“, and “The Math Behind Stress Relaxation and Creep of Polymers“. There are also some interesting ASTM standards on this topic.
Long-Term Stress Relaxation Predictions
The first step is to extract the stress relaxation portion of the experimental data, and set the time for the first data point to be zero (0). The figure to the right shows the raw experimental data.
Then, in MCalibration change the graph so that the time axis is logarithmic. I also changed the axis limits to make the data clearer. When plotting experimental stress relaxation data as a function of logarithmic time the response is typically linear, just as shown in the figure. We can now use this figure to extrapolate to longer time. For example, at a time of 1e5 seconds (28 hours) the stress will approximately be 1.5 MPa.
The key to this approach of solving this problem is to realize that the stress relaxation and creep response are energy activated processes that follow an Arrhenius rule. BTW, did you know that Svante Arrhenius was a Swedish scientist! Anyway, the exponential form of the energy activation gives a stress relaxation (and creep) response that typically changes linearly with logarithmic time.
It is also possible to predict the long-term stress relaxation response using a calibrated material model. This figure shows the predictions from the PolyUMod Three Network (TN) model. The average error in the model predictions is about 10%. Note that the green curve is volumetric compression.
Once we have an accurate calibrated material model we can predict the material response in any load history. In this figure I have plotted the predicted stress relaxation response when pulled in tension to 1% and then help at that strain for 1e5 seconds. The predictions from the Three Network model are in good agreement with both the known stress relaxation response and the estimated extrapolated stress relaxation behavior!
In other words, the calibrated TN model is able to accurately extrapolate to larger times.