Make Your Own Hyperelastic Model

Introduction

The Arruda-Boyce and Gent models are good general purpose models, but their predictions are not always great. Wouldn’t it be nice if we could quickly make our own hyperelastic model that is even better? You can, and in this article I will show how…

What is the problem with the Arruda-Boyce Model?

The figure to the right shows that the Arruda-Boyce hyperelastic model can be calibrated to accurately predict the uniaxial response of a natural rubber.

But since the Arruda-Boyce model has a strain energy function that is only based on the first invariant I1, the biaxial predictions will be too low. I explain this in more details in my book.

The average error of the model predictions is 8.3%. It would be nice if we could do better.

How to Fix the Problem

To get better predictions we need to modify the strain energy function like this: \( W(I_1, I_2, J) = W_{AB} + W_?, \) where the first term is from the Arruda-Boyce model, and the mystery strain energy function needs to depend on the second invariant I2.

The “trick” here is to recall that a first order (N=1) Ogden model with a negative alpha term is exactly what we need:

\( \displaystyle W = \frac{\mu_0}{\alpha} \left( \lambda_1^\alpha + \lambda_2^\alpha + \lambda_3^\alpha – 3 \right) \)

The following image shows an MCalibration image of the stress-strain predictions of a pure Ogden N=1 model in 3 different loading modes. We see a nice large spread between the uniaxial and biaxial predictions. This should allow us to predict both the large strain uniaxial response and the large strain biaxial response.

BAM Model

Combine the Arruda-Boyce hyperelastic strain energy function with a first-order Ogden model, and BAM, we get a winner. I call this the BAM model – of course.

The average error in model prediction is 2.6%. Not too bad.

Summary

  • It is easy to invent your own hyperelastic model.
  • (Arruda-Boyce) + (Oden N1) = BAM Model
  • The BAM model is significantly more accurate than the Arruda-Boyce model.
  • the BAM model require experimental data from multiple loading modes for calibration.
  • The BAM model is part of MCalibration and PolyUMod.
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