Problem $1.$

For a penny shaped crack embedded within

$3-$D cracks an infinite solid and loaded remotely perpendicular to the crack at a stress $,$ as sketched schematically on the right hand side, it is known that the stress intensity factor $K_{\text{I}}$ is

$K_{\text{I}}=\frac{2}{}\sqrt[\phantom{2}]{a},$

embedded pennyshaped crack in tension

$K_l=\frac{2}{}\sqrt[\phantom{2}]{a}$

with a being the radius of the penny crack.

Consider a penny crack of radius a being embedded into a circular cylinder with radius $r$ and length $L,$ suspended on a ceiling from the top $($see the schematic to the right$).$ Given that

$-ra$

$L$ is very long

the density of the material making up the cylinder is $$

the mode I fracture toughness of the material making up the cylinder is $K_{IC}$

Consider the penny crack is initially placed at $z=0$ and then moves up the cylinder in the $+z$ direction, calculate the height $z$ at which the cylinder will fracture due to the combination of the presence of the penny crack and its own weight. Please show your work process. The answer to this problem is relevant to why a $"$sky elevator" will not work.