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# Gent Hyperelasticity

## Theory

The Gent model is a hyperelastic model that was developed by Prof. Alan Gent in 1996. It is an $$I_1$$-based hyperelastic model with a finite stretchability.  The strain energy function is given by:

$$W = \displaystyle -\frac{\mu J_m}{2}\text{ln}\left( 1 – \frac{\overline{I}_1-3}{J_m} \right) + \frac{1}{d} \left( \frac{J^2-1}{2} – \text{ln}(J)\right).$$

In this equation: $$\overline{I}_1 = J^{-2/3} \text{tr}[\mathbf{b}]$$, $$\mathbf{b} = \mathbf{F} \mathbf{F}^{\top}$$, and $$J = \det[\mathbf{F}]$$.

The material model uses 3 parameters that need to be determined from experimental data:

• The shear modulus: $$\mu$$
• The limiting value for $$\overline{I}_1-3$$ called $$J_m$$
• and the a bulk modulus value, either $$\kappa$$ or $$d$$ depending on the FE software.

The Cauchy-stress can be obtained from Equation (5.41) in my book:

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

The Gent hyperelastic model is a built-in feature of Ansys Mechanical, and COMSOL Multiphysics. The model is also available in the PolyUMod library.

## Evaluation

To evaluate the accuracy of the Gent model I compared the model predictions to experimental data for a natural rubber obtained by Treloar [“The Physics of Rubber Elasticity”, Oxford Classic Texts]. The following figure shows the calibration results from MCalibration.

## Gent Model Summary

• The Gent model is one of my favorite hyperelastic models:
• Easy to use
• Easy to calibrate
• Only need uniaxial experimental data
• Always stable
• Is similar to the Arruda-Boyce Eight-Chain model in many ways
• Available in both Ansys Mechanical and COMSOL Multiphysics
• Does not capture viscoelastic or viscoplastic effects.

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