Introduction to the Mullins Effect
The Mullins effect (also called Mullins damage) is an interesting phenomenon that only occurs in elastomers (and other crosslinked rubber-like materials). In short, the Mullins effect make the elastomer get slightly softer during the first few load cycles, the response after that is typically repeatable from cycle-to-cycle.
The following image shows the cyclic compression response of an EPDM rubber. In this case I tested the same specimen 4 times with the same applied strain rate and temperature. The figure show that the stress is significantly higher in the first load cycle compared to the following cycles.
Dr. Leonard Mullins
➡️Also see the Parametric Study of the Mullins Effect article.
The figure also shows that elastomers are typically not hyperelastic in their response. There is a large difference in the unloading response compared to the loading response. This difference is, of course, caused by the viscoelastic behavior of the material, and has nothing to do with the Mullins effect.
Here are some facts about the Mullins effect:
- Most elastomers undergo softening during the first few load cycles.
- After about 3-5 load cycles the material response becomes repeatable.
- The amount of Mullins softening increases with filler particle concentration.
- The damage is dependent on the max applied strain.
- The damage is not permanent, it recovers with time. The recovery rate depends on temperature.
Ogden-Roxburgh Model
There are many material models that have been developed for predicting the Mullins damage. The most commonly used one is the Ogden-Roxburgh model, one other model that I like is the Qi-Boyce model [J. Mech. Phys. Solids, 52, 2187, 2004]. The Ogden-Roxburgh model is a very simple model that is implemented in almost all FE solvers, so that’s the model that I’ll focus on here.
The basic equation for the Mullins damage is: $$ \eta = 1 – \frac{1}{r} \text{erf} \left( \frac{\Psi_{dev}^{max} – \Psi_{dev}}{m + \beta \Psi_{dev}^{max}} \right) $$.
A few points about this equation:
- The stress is given by: \( \boldsymbol{\sigma} = \eta \text{dev}[ \boldsymbol{\sigma} ] + \text{vol}[ \boldsymbol{\sigma}] \).
- \(\Psi_{dev}^{max}\) is a state variable for the max deviatoric strain energy that the material point has seen.
- The error function: \(\text{erf}(0)=0\) and \(\lim_{x\to \infty} \text{erf}(x) = 1\).
- During monotonic loading \(\Psi_{dev} = \Psi_{dev}^{max}\), which makes \(\eta=1\). That is, no damage is applied.
- The max damage that can be applied is given by \( \eta = 1 – 1/r \). Showing that r has to be larger than 1.
A Simple Example
The following figure shows a simple example shows the predictions from a simple example in which a rubber part is loaded in tension to an engineering strain of 0.3, then unloaded to an engineering strain of 0.1, and then loaded in tension to a strain of 0.5. The stress-strain predictions are based on an Ansys Mooney-Rivlin model with Mullins damage. The material parameters are shown on the right side of the figure, and the arrows in the figure show the path the stress-strain response takes.
The material dissipates energy during the first load cycle, and then reaches a steady-state hyperelastic response. The path 1-2-3 looks like the response of a viscoelastic material, but there is nothing viscoelastic about the Mullins effect / damage model.
A More Realistic Example
Since almost all elastomers exhibit a non-linear viscoelastic response, it is typically more accurate to represent the experimental stress-strain response using a Bergstrom-Boyce model with Mullins damage (BBM model). The following example shows how the MCalibration can quickly calibrate the BBM to experimental data for a carbon black filled chloroprene rubber. The predicted results in the figure fits the experimental data really well. The average error of the predictions is about 6.0%. Not to bad!
More Information about the Mullins Effect
For more info check out my Mechanics of Solid Polymers book, Julie Diani’s review paper on the Mullins Effect [European Polymer Journal, 45, 601, 2009], or my movie below.
