# Predicting the Long-Term Stress Relaxation Response of Rubbers

## Problem Statement

##### The question I will try to answer:

How can one extract the relaxation data to longer relaxation times (for a rubber).  ## How to Predict the Long-Term Stress Relaxation

The first step is to extract the stress relaxation segments from the experimental time-strain-stress data. This is easy to do using MCalibration (see the YouTube video below). The figure to the right shows the results. The amount of stress relaxation is large and appears to happen rapidly, but when plotted this way it does not seem like the two curves will ever converge to the same value (even at infinite time). By plotting the stress as a function of logarithmic time we see a different trend. For uncrosslinked polymers, plots of this type tend to be basically linear for many, many, decades of time (see, for example, my article about stress relaxation of thermoplastics). For thermoplastics, the deviation from a linear response often does not occur until long enough times that the material starts to degrade by chain scission or other mechanisms.

For rubbers, there rate of relaxation typically reaches a final equilibrium value due to the chemical crosslinks in the material. We can see some of that effect in this experimental data set already after 10 minutes. If I had held the strain constant even longer, for example for 24 hours (about 1e5 sec), the two relaxation curves would likely have converged to almost the same value.  Let’s now try to understand what is causing this behavior. For rubbers it is often useful to think of the microstructure as if it consisted of two entangled networks, each contributing to the overall deformation response. One network (A) has a “perfect” crosslinking structure, and the second network  (B) has built-in defects (see figure to the left). This physical idealization of the microstructure is the foundation of the Bergstrom-Boyce model, which I developed in my M.I.T. basement lab, and can be used to understand why the stress may either increase or decrease during a stress relaxation segment. It also explains why the stress cannot relax to zero even in the limit of infinite times.

One common way to predict the long-term stress relaxation (or creep) response of a rubber is to calibrate a suitable viscoelastic material model to experimental data, and then use the calibrated model to perform the extrapolation. The two figures below show the calibration results using the Bergstrom-Boyce-Mullins model, and the predicted stress relaxation response. The figures show that the non-linear viscoelastic model captures the essence of the experimental data.  