Introduction
Using the Ogden model without thinking is a little bit like driving a car without your seat belt on – you might be OK, but it might also lead to big problems. In this article I will go through 4 features of the Ogden model that you might not know of, but will help you avoid trouble if you end up using this powerful hyperelastic model.
Note 1: My article about the hyperfoam model also covers topics that apply to the Ogden model.
Note 2: All recommendations in this article are easy to use in MCalibration.
#1: Theory
The following images show the strain energy densities that are used by some of the more common FE solvers. One of them is significantly different than the others. Can you see which one that is?
If you look carefully you will see that the strain energy function used by Abaqus for the Ogden model is using a \(\alpha^2\) in the denominator. This means that the \(\mu_i\) terms should always be positive when using Abaqus. In the other FE solvers the pre-factor is \(\mu_i/\alpha_i\), so if \(\alpha_i < 0\) then \(\mu_i\) should also be negative!
#2: Volumetric d-Parameters
There are lots of differences between how the different FE solvers have implemented the volumetric part of the strain energy density. Abaqus and Ansys use the following equation:
\( W_{vol} = \displaystyle\sum_{k=1}^N\frac{1}{d_k}(J-1)^{2k}.\)
If \(N=3\), as an example, the equation becomes:
\(W_{vol} = \displaystyle \frac{1}{d_1}(J-1)^2 + \frac{1}{d_2}(J-1)^4 + \frac{1}{d_3}(J-1)^6.\)
In almost all situations you will not have information about how the bulk modulus changes with applied pressure, and even if you had that information it would not matter very much. So my recommendation is to set \(d_1 > 0\), and \(d_i=0\) for \(i \in [2,N]\).
#3: Alpha-Parameters
The \(\alpha_i\) parameters is what make the Ogden model interesting. These parameters can be both positive and negative, and they control the shape of the predicted stress-strain response. The figures below show the stress-response for a one-term Ogden model in 2 cases: one with \(\alpha_1\) equal to 1, 2, and 4; and the other case with \(\alpha_1\) equal to -1, -2, and -4. In both cases I scaled \(\mu_1\) so that the true stress at an applied true strain of 0.7 becomes 1.0 MPa. This just makes it easier to compare the shapes of the stress-strain predictions.
The figures show that if \(\mu_1 > 0\) and \(\alpha_1 > 1\) then the stress stiffens in large strain tension, and if \(\mu_1< 0\) and \(\alpha_1 < -1\) then the stress stiffens in large strain compression. Typically you should have both \(\alpha_i > \) and \(\alpha_i < -1\) terms to make the stress-strain predictions close to symmetrical between true stress – true strain in tension and compression.
In summary, I often recommend using a 3-term Ogden model with the following structure:
- One term with \(\alpha \approx 1\)
- One term with \(\mu_i > 0\) and \(\alpha_i > 1\)
- One term with \(\mu_i<0\) and \(\alpha_i<-1\)
#4: Calibration
It is easy to calibrate a 3-term Ogden model to uniaxial tension data from Treloar (ref1, ref2). As shown below, the calibration results look good. If I then take that calibrated material model and compare the model predictions to experimental data in 2 other loading modes (pure shear, and biaxial tension) then it is clear that the model is NOT very good. In other words, one cannot calibrate an Ogden model to experimental data in one loading mode.
If I instead calibrate the 3-term Ogden hyperelastic model to both uniaxial and pure shear then the model can capture the response of both of those loading modes at once. But when applying that material model to all the experimental data the predictions do not look good. In this case, one cannot calibrate the 3-term Ogden model to experimental data in two different loading modes.
Finally, the figure below shows that I can calibrate a 3-term Ogden model to all data at once. In this case it is necessary to have experimental data from 3 different loading modes in order to calibrate the Ogden model. This is the main reason I don’t use the Ogden model very frequently – it needs so much experimental data for the calibration. Other hyperelastic models, like the Arruda-Boyce Eight-Chain model, are easier to use and almost as accurate.
4 thoughts on “4 Things You Didn’t Know About the Ogden Model”
If you want more info about this topic, then you may also find this paper interesting:
https://www.sciencedirect.com/science/article/pii/S1751616115004452
Paper title:
Control of tension–compression asymmetry in Ogden hyperelasticity with application to soft tissue modelling
Hi, Dr. Bergstrom,
Thank you for the comprehensive explanation of Ogden model. You memtioned that the alpha parameters shoud have positive and negative terms to reproduce the symmetric behavior in tension and compression. In this case, how to realized this in Abaqus which alpha only can be positive?
I just ran a simple Abaqus test case with a negative alpha parameter for one of the Ogden terms.
Why do you think that alpha cannot be negative?
/Jorgen
Oh! I found the mistake in my thought. I thought the sign of alpha in Abaqus will not lead difference in strain energy. Actually this is wrong due to alpha showing in exponental terms, which makes sign of alpha influencing the result. Thank you for the clearify!