## Introduction

In this article I will discuss the theory for the Abaqus Marlow and Ansys Response Function Hyperelastic Models. I will show that these models are easy to use, but not necessarily better than any of the other I_{1}-based hyperelastic models.

## Response Function Hyperelastic Model Theory

The stress for the most general invariant-based hyperelastic model can be obtained from:

\[\boldsymbol{\sigma} = \displaystyle \frac{2}{J} \left[ \frac{\partial\Psi}{\partial I_1^*} + \frac{\partial\Psi}{\partial I_2^*} I_1^* \right] \mathbf{b}^* – \frac{2}{J} \frac{\partial\Psi}{\partial I_2^*} (\mathbf{b}^*)^2 + \left[ \frac{\partial\Psi}{\partial J} – \frac{2I_1^*}{3J} \frac{\partial\Psi}{\partial I_1^*} – \frac{4I_2^*}{3J} \frac{\partial\Psi}{\partial I_2^*} \right] \mathbf{I}.\]

Note that the stress is completely defined by the energy function \(\Psi(I_1^*, I_2^*)\). For most industrially important polymers the dependence of the second invariant (\(I_2^*\)) is very small, and can often be ignored. In this case the stress can be obtained from the simpler equation:

\[\boldsymbol{\sigma} = \displaystyle \frac{2}{J} \frac{\partial\Psi}{\partial I_1^*} \text{dev}[\mathbf{b}^*] + \frac{\partial \Psi}{J} \mathbf{I}.\]

This shows that for an almost incompressible hyperelastic model, the stress only depends on the partial derivative of the energy function with respect to the first invariant. **In other words, if the stress strain response is known, then the term \({\partial\Psi}/{\partial I_1^*}\) can be obtained from the experimental data.** That is the whole concept of the Response Function type hyperelastic material models. In other words, the energy function is directly obtained from one experimental test, and no material parameters need to be found.

## Examine the Accuracy of Response Function Hyperelasticity

MCalibration comes with the classic Treloar data for natural rubber (uniaxial tension, biaxial tension, pure shear). MCalibration can also initialize the Abaqus Marlow and Ansys Response Function models to the experimental data. The figure below shows that this class of hyperelastic models matches the uniaxial tension data very accurately, but does not predict the biaxial response very well.

As a comparison I also calibrated the Ansys Yeoh model (3rd order) to the same experimental data. The results are shown in the figure below. The average error in this case is 6.3%, which is slightly better than the Response function (and Marlow) model.

The most accurate hyperelastic model for this data set is the Ansys Extended Tube model (see Figure below). The average error of this model predictions is 2.6%. Note that the Extended Tube model is based on an energy function that depends on both the first and second invariants, and therefore needs experimental data from multiple loading modes for calibration.