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documentation/multi_physics/user-guide/material_models/material_models_list/readme.md

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@@ -206,12 +206,15 @@ $$ \Psi = \frac{c}{2} \left( \exp\left( Q(\bar{\mathbf{E}}) \right) - 1 \right)
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where $\bar{\mathbf{E}}$ is the local Green-Lagrange strain tensor, and
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$$
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Q(\mathbf{E}) = b_{ff} \bar{E}_{ff}^2 + b_{ss} \left( \bar{E}_{ss}^2 + \bar{E}_{nn}^2 + \bar{E}_{sn}^2 \right) + 2b_{fs} \left( \bar{E}_{fs}^2 + \bar{E}_{fn}^2 \right)
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<p>
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$$
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Q(\bar{\mathbf{E}}) = b_{ff} \left( \bar{E}_{ff} \right)^2 + b_{ss} \left( \bar{E}_{ss}^2 + \bar{E}_{nn}^2 + \bar{E}_{sn}^2 \right) + 2b_{fs} \left( \bar{E}_{fs}^2 + \bar{E}_{fn}^2 \right)
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$$
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</p>
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In the code, $b_f = b_{ff}$$ and $$b_t = b_{ss}$.
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In the code, $b_f = b_{ff}$ and $b_t = b_{ss}$.
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<div style="background-color: #F0F0F0; padding: 10px; border: 1px solid #d0d0d0; border-left: 4px solid #d0d0d0; font-family: monospace;">
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&lt;<strong>Constitutive_model</strong> <i>type="Gucci"</i>&gt;<br>
@@ -225,9 +228,14 @@ In the code, $b_f = b_{ff}$$ and $$b_t = b_{ss}$.
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**Holzapfel-Ogden model**
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$$
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\Psi_{iso} = \frac{a}{2b} \exp \left( b (\bar{I}_1 - 3) \right) + \sum_{i \in \{f,s\}} \frac{a_i}{2b_i} \chi (\bar{I}_{4i}) \left( \exp \left( b_i (\bar{I}_{4i} - 1)^2 \right) - 1 \right) + \frac{a_{fs}}{2b_{fs}} \left( \exp \left( b_{fs} \bar{I}_{8fs}^2 \right) - 1 \right)
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$$
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<p>
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\(
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\Psi_{\text{iso}} = \frac{a}{2b} \exp\left( b (\bar{I}_1 - 3) \right)
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+ \sum_{i \in \{f,s\}} \frac{a_i}{2b_i} \, \chi(\bar{I}_{4i})
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\left( \exp\left( b_i (\bar{I}_{4i} - 1)^2 \right) - 1 \right)
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+ \frac{a_{fs}}{2b_{fs}} \left( \exp\left( b_{fs} \bar{I}_{8fs}^2 \right) - 1 \right)
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\)
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</p>
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@@ -255,7 +263,14 @@ The heaviside function is multiplied as a switching function to turn off the fib
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This model is very similar to the Holzapfel Ogden model - the only difference is the use of full invariants instead of isochoric.
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$$ \Psi_{iso} = \frac{a}{2b} \exp \left\{ b (\bar{I}_1 - 3)\right\} + \sum_{i \in f,s} \frac{a_i}{2b_i} \chi(I_{4i}) \left\{\exp \left\{ b_i (I_{4i} - 1)^2 \right\} - 1\right\} + \frac{a_{fs}}{2b_{fs}} \left( \exp \left\{ b_{fs} I_{8fs}^2 \right\} - 1 \right) $$
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<p>
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\[
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\Psi_{\text{iso}} = \frac{a}{2b} \exp\left( b (\bar{I}_1 - 3) \right)
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+ \sum_{i \in \{f,s\}} \frac{a_i}{2b_i} \, \chi(I_{4i})
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\left( \exp\left( b_i (I_{4i} - 1)^2 \right) - 1 \right)
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+ \frac{a_{fs}}{2b_{fs}} \left( \exp\left( b_{fs} I_{8fs}^2 \right) - 1 \right)
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\]
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</p>
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where f and s are the fiber and sheet directions and the smoothed heaviside function is:

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