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Explore Hyperelastic Invariants: I1, I2, J


Non-linear elastic materials are fully defined by how the Helmholtz free energy depend on 3 invariants (\(I_1^*, I_2^*, J\)). In this article I will explain how you can quickly and easily explore how these invariants influence the stored energy, and how you can use this information to select a suitable hyperelastic material model.

Hyperelastic Energy Functions

A non-linear elastic material model is completely defined by how its Helmholtz free energy per unit reference volume depends on the deformation gradient: \(\Psi(\mathbf{F})\). Furthermore, for an isotropic material the stored elastic energy has to only depend on deformation invariants. In other words, the stored energy cannot depend on rotations. In this case the Helmholtz free energy has the following dependence \(\Psi(I_1^*, I_2^*, J)\), and the Cauchy stress is always given by:

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

This equation is using the following definitions:

  • \(J = \det[\mathbf{F}]\)
  • \(\mathbf{b}^* = J^{-2/3} \mathbf{F} \mathbf{F}^\top\)
  • \(I_1^* = \text{tr}[\mathbf{b}^*]\)
  • \(I_2^* = \frac{1}{2} \left( I_1^{*2} – \text{tr}[\mathbf{b}^{*2}] \right) \)

Hermann von Helmholtz (1821-1894).

Experimental Data

I will use the classical Treloar data to examine the influence of the invariants \([I^*_1, I^*_2, J]\). The stress-strain data for the natural rubber that Treloar tested comes with MCalibration, and is shown in Figure 1.

Figure 1. Treloar’s stress-strain data.

The stress-strain data is what is typically plotted, and that is certainly useful for understanding how a material behaves. There are other plots, however, that are also useful to consider. Figure 2 shows how the total  energy density depends on the first invariant \(I_1^*\). Recall that the Helmholtz free energy for a Neo-Hookean hyperelastic model is given by:

\(\Psi = \displaystyle\frac{\mu}{2} ( I_1^* – 3) + \frac{\kappa}{2} (J-1)^2\).


For this equation to be true, the energy density needs to be linearly dependent on the first invariant, and the same curve should be obtained in all loading modes. Figure 2 shows that for the natural rubber that was tested, the energy is almost linearly dependent on the first invariant, but the response is not the same for the 3 loading modes. These observations indicate that the energy density also depends on the second invariant, and that a Neo-Hookean hyperelastic model will not be able to capture all aspects of the experimental data.

Figure 2. Total energy density as a function of \(I_1^*\).

Another way to plot the experimental data is to look at the energy density as a function of the second invariant, see Figure 3. In this case, different loading modes give very different responses. This indicates that the energy density is not strongly dependent on the second invariant.

Figure 3. Total energy density as a function of \(I_2^*\).

As a final example, Figure 4 shows how the second invariant depends on the first invariant for the 3 different loading modes that were tested. It can be shown (see my book), the biaxial loading results in the highest amount of \(I_2^*\), and uniaxial the lowest amount of \(I_2^*\) of the different possible loading modes. Note, I usually ignore the dependence on the second invariant since it is usually week, and it allow for material model calibration using less experimental data.

Figure 4. Invariant \(I_2^*\) as a function of \(I_1^*\).


  • If you have experimental data from different loading modes then you can use MCalibration to quickly examine how the Helmholtz free energy depends on the deformation invariants.
  • This can help you choose a suitable hyperelastic material model.

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