➡️Also see the Extended Tube Model for COMSOL Multiphysics.

The Extended Tube (“ETube”) model is a well-known and regarded hyperelastic material model for rubber-like materials. The ETube model is available as a built-in model in Ansys Mechanical (but not any of other major FE solvers).

The strain energy density that is used by the ETube model contains two deviatoric parts, and one volumetric part. The following equation is from my Mechanics of Solid Polymers book.

As described in the original paper by Kaliske and Heinrich, the [\(G_c, \delta\)] parameters control the “cross-linking network”, and the [\(G_e, \beta\)] parameters control the confining tube parameters.

One of the most impressive features of the ETube model is that it can very accurately predict the response of the famous Treloar experimental data in tension, pure shear, and equi-biaxial tension. One of the key reasons why the model can predict this challenging data set is that it contains a first-order Ogden term in the strain energy function. This enables the model to tailor the response in biaxial loading, which cannot be done by hyperelastic models that are only based on the first invariant (I1).

To learn more about the ETube model is it useful to perform a parametric study using MCalibration. The following image shows how the parameter \(\delta\) influences the stress-strain response in uniaxial tension and uniaxial compression. The figure shows that the parameter \(\delta\) has a very strong influence. A larger \(\delta\) value causes a highly non-linear response in tension.

The figure shows shows that the modal can predict a **very** asymmetric stress-strain response between tension and compression. In the example shown in the figure, the stress magnitude in tension can be more than 6X higher than in compression (at the same strain). I have not seen that large experimental difference between tension and compression, and the predicted response is likely not physical. The asymmetry between tension and compression can, of course, be adjusted to some extent by the selection of the material parameters.

Another issue to look out for when using this model is the magnitude of the \(\delta\) parameter. If \(\delta\) is too large then the stress response in tension can become unstable, as illustrated in the following image.

## Conclusions

- The Extended Tube (ETube) model is a very capable hyperelastic material model
- The model often predicts a high non-linear stress-strain response, so it is important to use experimental data to large enough strains to ensure that the predictions are reasonable and stable for all strain levels of interest.
- The model requires experimental data in 2 or more loading modes. For example, uniaxial tension, biaxial tension, and simple shear.
- You should NOT use the Extended Tube model if you only have uniaxial tension or compression data.

Here’s a hands-on demonstration of how the Extended Tube model works.