## Introduction

Back in 1962, Paul J. Blatz and William L. Ko wrote an article with the title “Application of Finite Elastic Theory to the Deformation of Rubbery Materials” in the journal Transactions in the Society of Rheology. They started their paper with Robert Hooke’s famous statement from 1678: “ut tensio, sic vis”. This translates from Latin to “as the extension, so the force”, which of course is the basis for Hooke’s law in solid mechanics. In this article I will try to explain the key equations for the different versions of the Blatz-Ko model that are available in modern FE software.

Original Paper by Blatz-Ko

## Blatz-Ko Model Theory - COMSOL Multiphysics

The COMSOL Multiphysics implementation of the Blatz-Ko model has the most general form that is available in commercial FE codes. COMSOL specifically uses the following equation for the strain energy density:

\( W = \phi \displaystyle\frac{\mu}{2} \left( (I_1-3) + \frac{1}{\beta} \left( I_3^{-\beta} – 1 \right) \right) + (1-\phi) \frac{\mu}{2} \left( I_2/I_3 – 3 + \frac{1}{\beta} \left( I_3^{\beta} – 1 \right) \right). \)

The parameter \(\phi\) has to be between 0 and 1, \(\mu\) is the shear modulus, and \(\beta\) controls the Poisson’s ratio. Note that if \(\beta \rightarrow \infty\) then the model becomes the Mooney-Rivlin model, and if in addition \(\phi=1\) then the model becomes the Neo-Hookean model. That is an interesting strain energy density. Here’s a MCalibration image showing the stress-strain predictions from this model.

This image shows how you specify this material models in COMSOL Multiphysics, and here’s a MCalibration image showing the stress-strain predictions from this model.

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If I try to calibrate this hyperelastic model to the classis experimental data from Treloar [The Physics of Rubber Elasticity, Oxford University Press, 1975], then I get the following results. The average error (NMAD) in the model predictions is 11%, which is OK but not great.

If I only include stress-strain data fror engineering strains that are less than 100%, then the Blatz-Ko model does significantly better. The following figure shows that the average error in this case is 2.3%. Which is great.

One interesting feature of the Blatz-Ko model is that the predicted Poisson’s ratio is constant independent of the applied strain. This is typically NOT the case for hyperelastic material models. Here are the Poisson’s ratio predictions from MCalibration.

## Blatz-Ko Model Theory - Solidworks Simulation

Solidworks Simulation supports a slightly simplified version of the Blatz-Ko material model. Specifically, the following energy function is used:

\( W = \displaystyle \frac{\mu}{2} \left(\frac{I_2}{I_3} + 2 \sqrt{I_3} – 5 \right).\)

In this model there is only one material parameter (\(\mu\)), the Poisson’s ratio is 0.25, and the Young’s modulus is 2.5 times the shear modulus. This model should only be used for soft compressible foams that have a Poisson’s ratio that is close to 0.25.

## Blatz-Ko Model Theory - Ansys

Ansys uses exactly the same simplified version of the Blatz-Ko model as is used by Solidworks Simulation. This model can be specified using the following APDL commands:

` ````
```TB, HYPER, 1,,,BLATZ
TBDATA, 1, 2.0

Here are some stress-strain predictions from MCalibration for this hyperelastic model.

## Summary

- COMSOL Multiphysics has a full implementation of the Blatz-Ko hyperelastic model. This model can accurately predict the time-independent response of rubber-like materials as long as the strains are not too large (< 100%).
- Ansys and Solidworks Simulations use a more simplified version of the Blatz-Ko model that only has one material parameter – the shear modulus. The Poisson’s ratio is always 0.25 for this model. This hyperelastic foam is less useful, and in most cases I would recommend using the Ansys Ogden Hyperfoam model instead.
- All of these models can be quickly calibrated using the MCalibration software. Get your free trial license here.