Predicting the Long-Term Stress Relaxation Response of Thermoplastics

Problem Statement

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.


More to explore

2 thoughts on “Predicting the Long-Term Stress Relaxation Response of Thermoplastics”

  1. ivar.kjelberg

    Hei Jørgen,
    As usual nice and instructive presentation, thanks for that 🙂
    One comment: when we look at the log scale relaxation we see a slight curve and not a direct straight line, as if the relaxation saturates once, this I would say is “normal” I would expect the material once to stop relaxing.
    However, as time goes on, the displacement and delta displacements become very small, and you do not report when you hit the metrology limit, as for very small relaxations, these could also come from the apparatus used or just reach the limit of the displacement sensor used, or ?
    Do you have any values ? And you could add these to the plot such that we have some indication when we must stop optimising the material under test, as the measurements could be just be hitting some measurement noise …
    Thanks again for your interesting blogs and videos, great job 🙂

    1. Hi Ivar, Good points. I really like your comment on measurement accuracy. That is a topic that is worthy of a longer discussion. I will see if I get get some experimental data to analyze…

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