Griffith's Energy Balance
============================
We will now follow Griffith and try to analyze the external stresses required to fracture a plate with an elliptical crack. 

```{image} ../LEFM/Grif.png
:alt: Grffith plate
:width: 400px
:align: center
```
We can write the potential energy on the plate as :

$$
U = U_0 - {\color{red} U_{crack}} + {\color{blue} U_{surface}}
$$

Where $U_0$ is the elastic energy in the **uncracked** plate and utilizing Inglis's solution for an elliptical crack :

$$
\begin{align*}
 &\color{red} {U_{crack}}  = \frac{\pi a^2 \sigma_{\inf} ^2 B}{E} \\
 \\ \quad \\
 &\color{blue} {U_{surface}} = 2 \left( 2aB\gamma_s\right)
 \end{align*}
 $$

 Crack growth will take place only if $\frac{dU}{d a}$ will indicate that an increase in $a$ is energetically favorable. Looking at $\frac{d U}{d a}=0$ we can thus identify the critical crack length for a given magnitude of remote loading:

 $$
 \begin{align*}
 \color{red}{ \frac{dU_{crack}}{da}}  = \color{blue}{ \frac{dU_{surface}}{da}} \\
\\ \quad \\
\color{red} {\frac{\pi a \sigma_{\inf}^2}{E}}  = \color{blue} {2\gamma_s} \\
\\ \quad \\
a_{eq} = \frac{2\gamma_s}{\pi}\frac{E}{\sigma_{\inf}^2}
\end{align*}
 $$

 rearranging and isolating $\sigma_{\inf}$ we arrive at 

$$
\sigma_{fracture} = \left( \frac{2E\gamma_s}{\pi a}\right)^{1/2}
$$

> Note that the Griffith fracture criteria is completely ignorant of the radius at the ellipse tip.

### Some small modofications

A revised expression of Griffith's criteria can be written by replacing the surface energy $\gamma_s$ with a more general term - $w_f$ the fracture energy. 

$w_f$ can be used to account for the presence of plasticity: 

$$
 w_f = \gamma_s + \gamma_p
$$
Or to account for crack tortuosity :
$$ 
w_f = \gamma_s * \left ( \frac{A_{true}}{A_{projected}} \right)
$$