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Update example_neohooke.md
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adtzlr committed Sep 27, 2023
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Expand Up @@ -12,17 +12,11 @@ This is a very basic example on how to implement a nearly-incompressible version
The strain energy density function per unit reference volume is additively splitted into an isochoric and volumetric contribution, see Eq. $$\eqref{eq:psi}$$. The first one is assumed to be proportional to the first invariant of the isochoric part of the right Cauchy-Green deformation tensor whereas the volumetric part is only a function of the volumetric ratio (the determinant of the deformation gradient), see Eq. $$\eqref{eq:psi-nh}$$.

$$
\begin{equation}
\psi(\mathbf{C}) = \psi(\mathbf{\hat C}) + U(J)
\label{eq:psi}
\end{equation}
$$

$$
\begin{equation}
\begin{align}
\psi(\mathbf{C}) = \psi(\mathbf{\hat C}) + U(J) \nonumber \\
\psi(\mathbf{C}) = \text{C}_{10} (\text{I}_\mathbf{\hat C}-3) + \frac{\kappa}{2} (J-1)^2
\label{eq:psi-nh}
\end{equation}
\end{align}
$$

We get the second Piola-Kirchhoff stress with the derivative of the strain energy density function per unit reference volume with respect to one half of the right Cauchy-Green deformation tensor as shown in Eq. $$\eqref{eq:pk2-nh}$$.
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