## 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.