Mooney-Rivlin , simple extension

[marcus]

Hi,
I am currently writing my master thesis, thus finishing my Master of Science with a major in Polymer Science. In the thesis I am using the mooney-rivlin equation sigma = 2*(lamda-1/lamda^2)*(C1 + C2/lamda) on simple extension tensile test data, where sigma is stress and lamda is elongation ratio. This equation is derived and briefly explained in literature, but does not give hints on using the equation on actual data from tests.

From papers where the equation is used, a linear regression model with sigma/(2*(lamda-1/lamda^2)) against 1/lamda is plotted for values between 1/lamda : [0,85, 0,6]. This is the "plateua" part or yield part. This gives a plausible fit when using the obtained constants C1 and C2 in the equation. In other papers a good fit is obtained but the methodology is not explained. What is correct? Or do you simply choose the interval that fits your data.

So, can anyone give me the basics on using this equation in actual experiments.

Help is much appreciated,
-Marcus

[Jorgen]

There is no right way to find the Mooney-Rivlin parameters C1 and C2. The linear regression plot method that you describe is one way, another way is to use a general minimization algorithm that gives the smallest least-squares-error between the experimental data and the model predictions.

What many people do in practice, and what I recommend, is to choose a range of strains that you are mostly interested in and then find the material parameters that best represent that range.

Note 1: The Mooney-Rivlin (MR) model is a simple hyperelastic model that can only predict a limited subset of the experimentally observed behavior. If your MR predictions are not "accurate enough", then there are other more advanced models that can be used.

Note 2: The MR model is not unconditinally stable. Therefore it is typically recommended to find the material parameters from experimental data from more than one loading mode (ie shear, biaxial, etc.)

- Jorgen

[marcus]

Thank you for your answer.
Due to time restraints I am only testing with simple extension in my thesis, but thanks for the advice on comparing MR-constants from different setups/experiments. I am starting to realize the limitations of the MR-equation, especially when comparing constants from different materials and or experiment comparison.


In your answer, second paragraph, you advised to choose the strains that one is interested in. If I understood this right: one should choose strains that limit the behaviour to hyperelastic behaviour. I am only comparing one material and modelling with the MR-equation with respect to the crosslinking induced by in my case heat. The material is surely not fully hyperelastic, but yields with increasing stress. Even though my material is far from hyperelastic, will the obtained fit fit better with increasing crosslink density and can the obtained constants be compared (as one does for hyperelastic materials) within my sample group with any relevance, or will viscous parts add errors in an unpredictable way?

- Marcus

[Jorgen]

I don't think it is easy to say with a degree of certainty that the fit always gets better with higher crosslinking density, but I suspect that that is more likely the case than not.

You should be able to compare the parameters within a given group of samples with similar properties.

- Jorgen

[marcus]

Thanks again for your replies. They have been very helpful :) .

-Marcus