Griffith’s Energy Balance

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} - U_{c r a c k} + U_{s u r f a c e}

Where $U_{0}$ is the elastic energy in the uncracked plate and utilizing Inglis’s solution for an elliptical crack :

\begin{array}{r} \begin{aligned} U_{c r a c k} = \frac{π a^{2} σ_{inf}^{2} B}{E} \\ U_{s u r f a c e} = 2 (2 a B γ_{s}) \end{aligned} \end{array}

Crack growth will take place only if $\frac{d U}{d a}$ 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{array}{r} \begin{matrix} \frac{d U_{c r a c k}}{d a} = \frac{d U_{s u r f a c e}}{d a} \\ \frac{π a σ_{inf}^{2}}{E} = 2 γ_{s} \\ a_{e q} = \frac{2 γ_{s}}{π} \frac{E}{σ_{inf}^{2}} \end{matrix} \end{array}

rearranging and isolating $σ_{inf}$ we arrive at

σ_{f r a c t u r e} = {(\frac{2 E γ_{s}}{π a})}^{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 $γ_{s}$ with a more general term - $w_{f}$ the fracture energy.

$w_{f}$ can be used to account for the presence of plasticity:

w_{f} = γ_{s} + γ_{p}

Or to account for crack tortuosity : $ $w_{f} = γ_{s} * (\frac{A_{t r u e}}{A_{p r o j e c t e d}})$ $

Griffith’s Energy Balance

Contents

Griffith’s Energy Balance#

Some small modofications#