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.

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 :

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

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:

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

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 : $$ w_f = \gamma_s * \left ( \frac{A_{true}}{A_{projected}} \right) $$

FM036004-Notes

Griffith’s Energy Balance¶

Some small modofications¶