Update example_neohooke.md

docs/examples/example_neohooke.md

@@ -9,7 +9,7 @@ parent: Examples
 
 This is a very basic example on how to implement a nearly-incompressible version of the Neo-Hookean material model in a commercial FEM package (HYPELA2 for Marc). As no special two- or three-field variational principle is used in this example, it is not suitable for nearly-incompressible material behaviour. Otherwise the elements tend to show excessive volumetric locking during deformation and hence, wrong results are calculated.
 
-The strain energy density function per unit reference volume is additively splitted into an isochoric and volumetric contribution, see $$\eqref{eq:psi}$$. 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).
+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. $$\eqref{eq:psi-nh}$$.
 
 $$
 \begin{equation}
@@ -25,25 +25,53 @@ $$
 \end{equation}
 $$
 
-<a href="https://www.codecogs.com/eqnedit.php?latex=\psi(\mathbf{C})&space;=&space;\text{C}_{10}&space;(\text{I}_\mathbf{\hat&space;C}-3)&space;&plus;&space;\frac{\kappa}{2}&space;(J-1)^2" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\psi(\mathbf{C})&space;=&space;\text{C}_{10}&space;(\text{I}_\mathbf{\hat&space;C}-3)&space;&plus;&space;\frac{\kappa}{2}&space;(J-1)^2" title="\psi(\mathbf{C}) = \text{C}_{10} (\text{I}_\mathbf{\hat C}-3) + \frac{\kappa}{2} (J-1)^2" /></a>
+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. $$\eqref{eq:pk2-nh}$$.
 
-We get the second Piola-Kirchhoff stress with the derivative of the helmholtz free energy per unit reference volume with respect to one half of the right Cauchy-Green deformation tensor:
-
-<a href="https://www.codecogs.com/eqnedit.php?latex=\mathbf{S}&space;=&space;\frac{\partial&space;\psi(\mathbf{C})}{\partial&space;\frac{1}{2}\mathbf{C}}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\mathbf{S}&space;=&space;\frac{\partial&space;\psi(\mathbf{C})}{\partial&space;\frac{1}{2}\mathbf{C}}" title="\mathbf{S} = \frac{\partial \psi(\mathbf{C})}{\partial \frac{1}{2}\mathbf{C}}" /></a>
+$$
+\begin{equation}
+  \mathbf{S} = \frac{\partial \psi(\mathbf{C})}{\partial \frac{1}{2}\mathbf{C}}
+  \label{eq:pk2}
+\end{equation}
+$$
 
-<a href="https://www.codecogs.com/eqnedit.php?latex=\mathbf{S}&space;=&space;2\text{C}_{10}&space;\&space;\text{dev}(\hat{\mathbf{C}})&space;\mathbf{C}^{-1}&space;&plus;&space;\kappa&space;(J-1)&space;J&space;\mathbf{C}^{-1}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\mathbf{S}&space;=&space;2\text{C}_{10}&space;\&space;\text{dev}(\hat{\mathbf{C}})&space;\mathbf{C}^{-1}&space;&plus;&space;\kappa&space;(J-1)&space;J&space;\mathbf{C}^{-1}" title="\mathbf{S} = 2\text{C}_{10} \ \text{dev}(\hat{\mathbf{C}}) \mathbf{C}^{-1} + \kappa (J-1) J \mathbf{C}^{-1}" /></a>
+$$
+\begin{equation}
+  \mathbf{S} = 2\text{C}_{10} \ \text{dev}(\hat{\mathbf{C}}) \mathbf{C}^{-1} + \kappa (J-1) J \mathbf{C}^{-1}
+  \label{eq:pk2-nh}
+\end{equation}
+$$
 
-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:
+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. $$\eqref{eq:c4-nh}$$,
 
-<a href="https://www.codecogs.com/eqnedit.php?latex=\mathbb{C}&space;=&space;\frac{\partial&space;\mathbf{S}}{\partial\frac{1}{2}\mathbf{C}}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\mathbb{C}&space;=&space;\frac{\partial&space;\mathbf{S}}{\partial\frac{1}{2}\mathbf{C}}" title="\mathbb{C} = \frac{\partial \mathbf{S}}{\partial\frac{1}{2}\mathbf{C}}" /></a>
+$$
+\begin{equation}
+  \mathbb{C} = \frac{\partial \mathbf{S}}{\partial\frac{1}{2}\mathbf{C}}
+  \label{eq:c4}
+\end{equation}
+$$
 
-<a href="https://www.codecogs.com/eqnedit.php?latex=\mathbb{C}&space;=&space;2\text{C}_{10}&space;J^{-2/3}&space;\frac{2}{3}&space;\&space;(\text{tr}(\mathbf{C})&space;\&space;\mathbb{I}&space;-&space;\mathbf{1}&space;\otimes&space;\mathbf{C}^{-1}&space;-&space;\mathbf{C}^{-1}&space;\otimes&space;\mathbf{1}&space;&plus;&space;\frac{1}{3}&space;\text{tr}(\mathbf{C})&space;\&space;\mathbf{C}^{-1}&space;\otimes&space;\mathbf{C}^{-1})&space;&plus;&space;\left(\kappa&space;(J-1)&space;J&space;&plus;&space;\kappa&space;J^2\right)&space;\&space;\mathbf{C}^{-1}&space;\otimes&space;\mathbf{C}^{-1}&space;-&space;2&space;\kappa&space;(J-1)&space;J&space;\&space;\mathbb{I}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\mathbb{C}&space;=&space;2\text{C}_{10}&space;J^{-2/3}&space;\frac{2}{3}&space;\&space;(\text{tr}(\mathbf{C})&space;\&space;\mathbb{I}&space;-&space;\mathbf{1}&space;\otimes&space;\mathbf{C}^{-1}&space;-&space;\mathbf{C}^{-1}&space;\otimes&space;\mathbf{1}&space;&plus;&space;\frac{1}{3}&space;\text{tr}(\mathbf{C})&space;\&space;\mathbf{C}^{-1}&space;\otimes&space;\mathbf{C}^{-1})&space;&plus;&space;\left(\kappa&space;(J-1)&space;J&space;&plus;&space;\kappa&space;J^2\right)&space;\&space;\mathbf{C}^{-1}&space;\otimes&space;\mathbf{C}^{-1}&space;-&space;2&space;\kappa&space;(J-1)&space;J&space;\&space;\mathbb{I}" title="\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}) + \left(\kappa (J-1) J + \kappa J^2\right) \ \mathbf{C}^{-1} \otimes \mathbf{C}^{-1} - 2 \kappa (J-1) J \ \mathbb{I}" /></a>
+$$
+\begin{equation}
+  \mathbb{C} = 2\text{C}_{10} J^{-2/3} \frac{2}{3} \ (\text{tr}(\mathbf{C}) \ \mathbb{I} - \mathbf{1} \otimes \mathbf{C}^{-1} - \mathbf{C}^{-1} \otimes \mathbf{1} + \frac{1}{3} \text{tr}(\mathbf{C}) \ \mathbf{C}^{-1} \otimes \mathbf{C}^{-1}) + \left(\kappa (J-1) J + \kappa J^2\right) \ \mathbf{C}^{-1} \otimes \mathbf{C}^{-1} - 2 \kappa (J-1) J \ \mathbb{I}
+  \label{eq:c4-nh}
+\end{equation}
+$$
 
-with the fourth order identity tensor
+with the fourth order identity tensor in Eq. $$\eqref{eq:i4}$$.
 
-<a href="http://www.codecogs.com/eqnedit.php?latex=\mathbb{I}=&space;\mathbf{C}^{-1}&space;\odot&space;\mathbf{C}^{-1}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\mathbb{I}=&space;\mathbf{C}^{-1}&space;\odot&space;\mathbf{C}^{-1}" title="\mathbb{I}= \mathbf{C}^{-1} \odot \mathbf{C}^{-1}" /></a>
+$$
+\begin{equation}
+  \mathbb{I}= \mathbf{C}^{-1} \odot \mathbf{C}^{-1}
+  \label{eq:i4}
+\end{equation}
+$$
 
-<a href="http://www.codecogs.com/eqnedit.php?latex=\mathbf{C}^{-1}&space;=&space;\mathbb{I}&space;:&space;\mathbf{C}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\mathbf{C}^{-1}&space;=&space;\mathbb{I}&space;:&space;\mathbf{C}" title="\mathbf{C}^{-1} = \mathbb{I} : \mathbf{C}" /></a>
+$$
+\begin{equation}
+  \mathbf{C}^{-1} = \mathbb{I} : \mathbf{C}
+  \label{eq:i4-projection}
+\end{equation}
+$$
 
 The two equations are now implemented in a Total Lagrange user subroutine  with the help of this Tensor module as follows:
 
@@ -114,4 +142,4 @@ The two equations are now implemented in a Total Lagrange user subroutine  with
       end
 ```
 
-There are also examples for a [basic understandig of the tensor toolbox](examples/script_umat.f), the implementation of the [St.Venant Kirchhoff material](example_stvenantkirchhoff.md) and a [full featured MSC.Marc Neo-Hookean material HYPELA2 user subroutine](examples/hypela2_nh_ttb.f).
+There are also examples for a [basic understandig of the tensor toolbox](examples/script_umat.f), the implementation of the [St.Venant Kirchhoff material](example_stvenantkirchhoff.md) and a [full featured Marc Neo-Hookean material HYPELA2 user subroutine](examples/hypela2_nh_ttb.f).