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\begin{document}

For a block with a hole of size R with a force/unit length of F, the
stress \( \sigma \sim \frac{F}{W} \) where W is the width. The stress near the hole is something
like: \( \sigma \sim \frac{2F}{W} \)

For a hole of length L (very ellipsoidal), with radius of curvature
at the ends, R, the energy of the distorted stress region is \( \sigma _{avg}L^{2} \). Then
\( \sigma ^{2}_{end}R\sim \sigma ^{2}_{avg}L \). The minimum R is on the level of the atomic size. This means \( \sigma _{y}\sim \sqrt{\frac{a}{L}}\mu \sim 2*10^{-4}\sqrt{\frac{L}{1cm}}\mu  \).
This is too low. creation of cracks involves surface tension. propagation
condition is \( \gamma L<L^{2}\frac{\sigma ^{2}}{\mu } \). Then \( L>\frac{\gamma \mu }{\sigma ^{2}} \)= griffith crack length. \( \Rightarrow L>\frac{a}{\epsilon ^{2}} \). This means \( \epsilon >\sqrt{\frac{a}{L}} \). Surface
is actually jagged, increasing the real crack length. \( \gamma ->W \)=work of fracture
which is typically, \( (10^{4}-10^{6})\gamma  \). Then \( \sigma _{end}>>\sigma _{y} \) sometimes, but the crack does not propagate.
propagation condition: \( \frac{W\mu }{\sigma ^{2}} \).

\begin{itemize}

\item glass: \( W=6*10^{3}erg/cm^{2} \), then \( \epsilon >.5\sqrt{\frac{a}{L}} \)

\item steel: \( W=10^{8}erg/cm^{2} \), then \( \epsilon >60\sqrt{\frac{a}{L}} \)

\end{itemize}

There should really be a kinetic term, because the cracks can propagate
at the speed of sound.

\( \frac{E}{length}\sim \frac{\sigma ^{2}}{2\mu }L^{2}+2WL+\rho L^{2}v^{2} \) for \( v\rightarrow 0 \) you get the griffith crack length.

\( v=v_{max}(1-\frac{L_{g}}{L})^{\frac{1}{2}} \). Crack can accelerate over time. Stresses tend to be perpendicular
to the crack. Instability in crack growth is onset at \( v\sim .2c_{s} \).

\end{document}
