Soft elastomer foams are used in many different applications, including sports equipment, consumer electronics, aerospace and automotive parts. In this post I summarize the most common material models that are used to represent the mechanical response of soft foams. Note that although I focus only on non-linear elastic material models, many of the soft foams that are used can also be non-linear viscoelastic. I will cover material modeling techniques for non-linear viscoelastic foams in a future article.

*Photo of a soft silastic foam.*

First, can a basic hyperelastic model be used to predict the response of a foam?

## Can a Traditional Hyperelastic Model be used for Foams?

This figure shows the predicted Poisson’s ratio as a function of applied strain for a **Neo-Hookean** material model. The green line is for a case when the shear modulus mu=1.0 MPa, and the bulk modulus kappa=1.0 MPa. The figure shows that the predicted Poisson’s ratio decreases in compression, and eventually becomes negative. That is not very good!

The traditional Neo-Hookean material model is not suitable for soft polymer foams.

Switching to a **Yeoh** hyperelastic model allows us to include higher order bulk modulus terms. The green line in this figure shows that if C10=1, C20=C30=0, D1=0.8, D2=0.2, and D3=0, then the predicted Poisson’s ratio is somewhat close to 0.25 for most strains. Unfortunately, the Poisson’s ratio is still relatively strongly strain dependent.

The Yeoh hyperelastic model is therefore not suitable for soft polymer foams.

Some rubber-like materials are better modeled using an **alternative Neo-Hookean** model in which the strain energy density is not divided into deviatoric and hydrostatic components. In this case the stress is calculated from a different equation giving a modulus that is less sensitive to the applied strain. As is shown in this figure, however, the Poisson’s ratio is still relatively strongly dependent on the applied strain.

The alternative Neo-Hookean material model is not suitable for soft polymer foams.

## The Hyperfoam Class of Material Models

The “Hyperfoam” model was developed in order to create a simple hyperelastic material model that can allow for constant Poisson’s ratio predictions at large uniaxial deformations. Abaqus calls this model the *Hyperfoam* model, and provides a reference to the following paper:

As far as I can tell from reading that paper is that it refers to a previous paper by Prof. Rodney Hill, which is basically using a modified version of the Ogden hyperelastic model. This seems a bit confusing to me, and is perhaps why different FE software use different names for basically the same model: Abaqus calls it the *Hyperfoam* model, COMSOL calls it the *Storåkers* model, Ansys calls it the *Ogden Foam* model, LS-DYNA calls it the MAT_HILL_FOAM model.

Rodney Hill (1921-2011)

Bertil Storåkers (1939-2018)

Raymond Ogden

In case you are curious, Prof. Ogden’s thesis advisor was Prof. Hill. I have met both Prof. Storåkers and Prof. Ogden, but unfortunately never Prof. Hill.

## Abaqus Hyperfoam and COMSOL Storåkers Model

The Abaqus Hyperfoam and the COMSOL Storåkers model can predict constant Poisson’s ratio. The parameter beta can be calculated from the Poisson’s ratio using the equation in the figure. For this material model the parameter mu should always be positive.

If alpha > 0: the tangent modulus increases with strain in tension

If alpha < 0: the tangent modulus increases with strain in compression

## Ansys Ogden Foam

The Ansys Ogden model can predict constant Poisson’s ratio. The parameter beta can be calculated from the Poisson’s ratio using the equation in the figure.

Since alpha is not squared in the denominator of the energy function, it is typically best to select alpha to have the same sign as the mu parameter.

If mu > 0 and alpha > 0: the tangent modulus increases with strain in tension

If mu < 0 and alpha < 0: the tangent modulus increases with strain in compression

## LS-DYNA *Mat_Hill_Foam

The LS-DYNA *Mat_Hill_Foam model can predict constant Poisson’s ratio. The parameter N can be calculated from the Poisson’s ratio using the equation in the figure.

Since b is not squared in the denominator of the energy function, it is typically best to select C and b to have the same sign.

If C > 0 and b > 0: the tangent modulus increases with strain in tension.

If C < 0 and b < 0: the tangent modulus increases with strain in compression.

## Calibration to Real Foam Data

After going through all of the model theory, let’s take a look at some real experimental data for a soft closed-celled polymer foam. The foam was tested in uniaxial tension, uniaxial compression, and confined compression.

The data all look reasonable, with one surprise: the modulus in tension is not the same as the modulus in compression. In this case, it is likely that the cell walls in the foam buckle at very small compressive strains, this, of course, does not happen in tension, which leads to the asymmetry between tension and compression.

Now let’s examine how different foam material models match this data set.

Here is the best calibration that I could create using a 2-term **Ansys Ogden foam** model. The average error in the model predictions is 19%, which is not very good. As expected, the main problem is that the model cannot accurately predict both the tension and compression behavior.

If I remove the tension data, then the calibration works out much better. In this case the average error is 7%.

It is important to also consider the accuracy of the stress-strain predictions in other loading modes. This figure shows the predicted response of the calibrated Ansys Ogden Foam model in uniaxial tension+compression, confined tension+compression, and biaxial tension+compression. The figure shows that the predicted response is very different in tension and compression. This is typical of Ogden-type models. **It is very important to have enough experimental data when calibrating this type of foam model.**

There are many other material models that can be used to predict the response of soft polymer foams. In this example I’m using the PolyUMod Parallel Network (PN) model with an advanced hyperelastic component allowing for different stiffnesses in tension and compression.

This model captures the complete data set with an average error of 11%. Not too bad!

See the PolyUMod User’s manual for more info about this material model.

## Summary

- Hyperfoam models are often used to predict the response of soft polymer foams.
- Hyperfoam models requires LOTS of experimental data. Specifically, experimental data in 3 or more loading modes.
- Hyperfoam models are not always that accurate for some polymer foams. In these cases there are other more specialized material models (like the PolyUMod PN model) that can provide significantly more accurate predictions.