The Holzapfel-Gasser-Ogden material model description is confusing

[ktbolt]

Problem

The description of the Holzapfel-Gasser-Ogden material model looks like this

However, in the literature the model is given by

It seems that the a4 parameter is k1 / 2 but that is not clear.

Solution

Include in the description of the Holzapfel-Gasser-Ogden material model how the input parameters relate to those used in the literature.

Additional context

No response

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[ktbolt]

@divya-adil Please have a look at this; we have a user that is slightly confused and rightly so !

[divya-adil]

@ktbolt There is a mistake in the strain energy function (SEF) in the documentation - there should be a factor of 2 in there to match the equation in the literature. The correct SEF is:
$\Psi_{aniso} = \frac{a_4}{2b_4} \left( \exp\left( b_4\left( \kappa \bar{I}_1 + (1-3\kappa)\bar{I}_4 - 1\right)^2 \right) - 1 \right) + \frac{a_6}{2b_6} \left( \exp\left( b_6\left( \kappa \bar{I}_1 + (1-3\kappa)\bar{I}_6 - 1 \right)^2 \right) - 1 \right)$.

I double-checked the stress calculation from the SEF against the code. See details below:

Define the two fiber-strain scalars(Eff, Ess) - this definition is used throughout the code:

$$E_1 \;=\; \kappa \bar I_1+(1-3\kappa)\bar I_4-1 \qquad(\texttt{Eff})$$ $$E_2 \;=\; \kappa \bar I_1+(1-3\kappa)\bar I_6-1 \qquad(\texttt{Ess})$$

The second Piola–Kirchhoff stress is calculated as

$$\bar{\mathbf S} = 2\frac{\partial \Psi_{\text{aniso}}}{\partial \mathbf C} = 2\frac{\partial\Psi_{\text{aniso}}}{\partial E_1}\frac{\partial E_1}{\partial \mathbf C} + 2\frac{\partial\Psi_{\text{aniso}}}{\partial E_2}\frac{\partial E_2}{\partial \mathbf C}$$

For a single fiber family, differentiate the scalar energy with respect to its strain measure $E$:

$$\frac{\partial\Psi_{\text{aniso}}}{\partial E_1} = a_4\,E_1\,e^{b_4E_1^2}, \qquad \frac{\partial\Psi_{\text{aniso}}}{\partial E_2} = a_6\,E_2\,e^{b_6E_2^2}$$

This is matches what the code computes:

double g2 = stM.aff * Eff * exp(stM.bff*Eff*Eff);   //  = a_4 * E_1 * exp(b_4 * E_1^2)
double g3 = stM.ass * Ess * exp(stM.bss*Ess*Ess);   //  = a_6 * E_2 * exp(b_6 * E_2^2)

and the outer factor of 2 in

Matrix<nsd> S_bar = 2.0*(g1*Idm + g2*Hff_disp + g3*Hss_disp);

So, the parameters $a_4$ and $b_4$ correspond to $k_1$ and $k_2$ from the original paper (applied to the $\bar I_4$ fiber family), and $a_6$ and $b_6$ correspond to $k_1$ and $k_2$ (applied to the $\bar I_6$ fiber family). In the original Gasser–Ogden–Holzapfel formulation both fiber families share identical $k_1,k_2$ values (i.e. $a_4=a_6=k_1, b_4=b_6=k_2$); this implementation generalizes the model by allowing each family its own pair of stiffness parameters.

I will update the documentation and submit a pull request.

[ktbolt]

@divya-adil Good work checking the consistency of the code !