@@ -206,12 +206,15 @@ $$ \Psi = \frac{c}{2} \left( \exp\left( Q(\bar{\mathbf{E}}) \right) - 1 \right)
206206
207207where $\bar{\mathbf{E}}$ is the local Green-Lagrange strain tensor, and
208208
209- $$
210- 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)
209+ <p >
211210$$
211+ Q(\bar{\mathbf{E}}) = b_{ff} \left( \bar{E}_{ff} \right)^2 + b_{ss} \left( \bar{E}_{ss}^2 + \bar{E}_{nn}^2 + \bar{E}_{sn}^2 \right) + 2b_{fs} \left( \bar{E}_{fs}^2 + \bar{E}_{fn}^2 \right)
212+ $$
213+ </p >
214+
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213216
214- In the code, $b_f = b_ {ff}$$ and $ $ b_t = b_ {ss}$.
217+ In the code, $b_f = b_ {ff}$ and $b_t = b_ {ss}$.
215218
216219<div style =" background-color : #F0F0F0 ; padding : 10px ; border : 1px solid #d0d0d0 ; border-left : 4px solid #d0d0d0 ; font-family : monospace ;" >
217220< ; <strong >Constitutive_model</strong > <i >type="Gucci"</i >> ; <br >
@@ -225,9 +228,14 @@ In the code, $b_f = b_{ff}$$ and $$b_t = b_{ss}$.
225228
226229** Holzapfel-Ogden model**
227230
228- $$
229- \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)
230- $$
231+ <p >
232+ \(
233+ \Psi_ {\text{iso}} = \frac{a}{2b} \exp\left( b (\bar{I}_ 1 - 3) \right)
234+ + \sum_ {i \in \{ f,s\} } \frac{a_i}{2b_i} \, \chi(\bar{I}_ {4i})
235+ \left( \exp\left( b_i (\bar{I}_ {4i} - 1)^2 \right) - 1 \right)
236+ + \frac{a_ {fs}}{2b_ {fs}} \left( \exp\left( b_ {fs} \bar{I}_ {8fs}^2 \right) - 1 \right)
237+ \)
238+ </p >
231239
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@@ -255,7 +263,14 @@ The heaviside function is multiplied as a switching function to turn off the fib
255263
256264This model is very similar to the Holzapfel Ogden model - the only difference is the use of full invariants instead of isochoric.
257265
258- $$ \Psi_{iso} = \frac{a}{2b} \exp \left\{ b (\bar{I}_1 - 3)\right\} + \sum_{i \in f,s} \frac{a_i}{2b_i} \chi(I_{4i}) \left\{\exp \left\{ b_i (I_{4i} - 1)^2 \right\} - 1\right\} + \frac{a_{fs}}{2b_{fs}} \left( \exp \left\{ b_{fs} I_{8fs}^2 \right\} - 1 \right) $$
266+ <p >
267+ \[
268+ \Psi_ {\text{iso}} = \frac{a}{2b} \exp\left( b (\bar{I}_ 1 - 3) \right)
269+ + \sum_ {i \in \{ f,s\} } \frac{a_i}{2b_i} \, \chi(I_ {4i})
270+ \left( \exp\left( b_i (I_ {4i} - 1)^2 \right) - 1 \right)
271+ + \frac{a_ {fs}}{2b_ {fs}} \left( \exp\left( b_ {fs} I_ {8fs}^2 \right) - 1 \right)
272+ \]
273+ </p >
259274
260275
261276where f and s are the fiber and sheet directions and the smoothed heaviside function is:
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