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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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