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Update example_neohooke.md
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adtzlr committed Sep 27, 2023
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By evaluating the derivative of the stress with respect to one half of the right Cauchy-Green deformation tensor we get the material elasticity tensor, see Eq. $$\eqref{eq:c4-nh}$$,

$$
\begin{equation}
\mathbb{C} = \frac{\partial \mathbf{S}}{\partial\frac{1}{2}\mathbf{C}}
\label{eq:c4}
\end{equation}
$$

$$
\begin{align}
\mathbb{C} &= \frac{\partial \mathbf{S}}{\partial\frac{1}{2}\mathbf{C}} \nonumber \\
\mathbb{C} &= 2\text{C}_{10} J^{-2/3} \frac{2}{3} \ (\text{tr}(\mathbf{C}) \ \mathbb{I} - \mathbf{1} \otimes \mathbf{C}^{-1} - \mathbf{C}^{-1} \otimes \mathbf{1} + \frac{1}{3} \text{tr}(\mathbf{C}) \ \mathbf{C}^{-1} \otimes \mathbf{C}^{-1}) \nonumber \\
&+ \left(\kappa (J-1) J + \kappa J^2\right) \ \mathbf{C}^{-1} \otimes \mathbf{C}^{-1} - 2 \kappa (J-1) J \ \mathbb{I}
\label{eq:c4-nh}
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