Introduction - Mullins Effect
I recently wrote an article titled “All about the Mullins Effect“, and I just noticed that it is quite possible that I did not cover quite all there is to say about the Mullins Effect 😎. So I am now writing this article with 2 goals in mind: (1) visually demonstrate how the 3 parameters in the Ogden-Roxburgh Mullins Damage model influence the predicted stress-strain response; (2) demonstrate how easy it is to examine the influence of material parameters using the Parametric Study feature in MCalibration.
To start the demonstration I setup a Virtual Load Case in MCalibration. In this load case I uniaxially pull the material to an engineering strain of 0.3, then unload to a strain of 0.1, and finally reload to a strain of 0.5. I then select an Ansys Neo-Hookean hyperelastic material model with a Mullins damage model
Here’s the material model definition in Ansys APDL format:
TB, HYPER, matid, 1, 2, NEO TBDATA, 1, 1 ! mu TBDATA, 2, 0.04 ! d TB, CDM, matid, 1,3,,,PSE2 TBDATA, 1, 2 ! r TBDATA, 2, 0.1 ! m TBDATA, 3, 0.1 ! beta
To examine the influence of the different material parameters I then start a Parametric Study.
The figure to the right shows the influence of the parameter mu, which is the stiffness of the hyperelastic network. As expected, a larger mu value increases the material stiffness.
Note that r=2, m=0.1, and beta=0.1 in all 3 curves that are plotted. Only mu is changed between the curves.
A parametric study of different r values are shown to the right. The final damaged stress is (1-1/r) multiplied with the initial undamaged stress. So r should not be less than 1.
The m parameter specifies how rapidly the material gets damaged after unloading. Note that m has the dimension of stress.
Finally, the beta parameter also reduces the stress drop after a strain reversal. I recommend selecting a small value for beta (about 0.01), and instead adjusting the damage “rate” using the m parameters.
As was discussed in my original article about the Mullins damage, it is difficult to distinguish between Mullins damage and viscoelastic dissipation if all you have is a single load-unload cycle. For that reason I recommend that your experimental test program include multiple load-unload cycles to the same strain amplitude. This will allow you to quantify what part of the response is due to damage and what part is due to viscoelastic relaxation.
Here’s a movie that contain step-by-step instructions for how to perform this type of parametric study.