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Neo Hookean Shell

Neo Hookean Shell is a constitutive model designed for simulating thin shell structures, based on a triangle element.

For the sake of simplicity, we denote the three vertices of the triangle as \(\mathbf{x}_0\), \(\mathbf{x}_1\), and \(\mathbf{x}_2\). The rest shape positions are denoted as \(\bar{\mathbf{x}}_0\), \(\bar{\mathbf{x}}_1\), and \(\bar{\mathbf{x}}_2\).

#11 Neo Hookean Shell

For a triangular shell element with vertices at positions \(\mathbf{x}_0\), \(\mathbf{x}_1\), and \(\mathbf{x}_2\), we define the edge vectors:

\[ \begin{aligned} \mathbf{e}_{01} = \mathbf{x}_1 - \mathbf{x}_0\\ \mathbf{e}_{02} = \mathbf{x}_2 - \mathbf{x}_0 \end{aligned} \]

The current metric tensor is constructed as:

\[ \mathbf{A} = \begin{bmatrix} \mathbf{e}_{01} \cdot \mathbf{e}_{01} & \mathbf{e}_{01} \cdot \mathbf{e}_{02} \\ \mathbf{e}_{02} \cdot \mathbf{e}_{01} & \mathbf{e}_{02} \cdot \mathbf{e}_{02} \end{bmatrix} \]

Similarly, for the reference configuration with positions \(\bar{\mathbf{x}}_0\), \(\bar{\mathbf{x}}_1\), and \(\bar{\mathbf{x}}_2\):

\[ \begin{aligned} \bar{\mathbf{e}}_{01} = \bar{\mathbf{x}}_1 - \bar{\mathbf{x}}_0\\ \bar{\mathbf{e}}_{02} = \bar{\mathbf{x}}_2 - \bar{\mathbf{x}}_0 \end{aligned} \]
\[ \bar{\mathbf{B}} = \begin{bmatrix} \bar{\mathbf{e}}_{01} \cdot \bar{\mathbf{e}}_{01} & \bar{\mathbf{e}}_{01} \cdot \bar{\mathbf{e}}_{02} \\ \bar{\mathbf{e}}_{02} \cdot \bar{\mathbf{e}}_{01} & \bar{\mathbf{e}}_{02} \cdot \bar{\mathbf{e}}_{02} \end{bmatrix} \]

Here we use \(\bar{\mathbf{B}}\) to denote the reference metric tensor.

Deformation Energy Density

The Neo-Hookean shell deformation energy density is given by:

\[ \Psi = \frac{\mu}{2} \left( \text{tr}(\bar{\mathbf{B}}^{-1} \mathbf{A}) - 2 - 2\ln J \right) + \frac{\lambda}{2} (\ln J)^2 \]

where:

  • \(\mu\) is the shear modulus

  • \(\lambda\) is the Lamé parameter

  • \(J = \sqrt{\det(\mathbf{A}) \det(\bar{\mathbf{B}}^{-1})}\) is the determinant of the deformation

  • \(\ln J = \frac{1}{2}\ln(\det(\mathbf{A}) \det(\bar{\mathbf{B}}^{-1}))\) is the logarithmic strain