Problem $4(10$ points$)$

A very long pipeline is used to transport oil with an internal pressure of $\backslash ($ p$\_\{$i$\}\backslash ).$ The pipe's inner radius is $\backslash ($ r$\_\{$i$\}\backslash ),$ and outer radius $\backslash ($ r$\_\{$o$\}\backslash ).$

$(1)$ If the material yield stress is $\backslash (\backslash$sigma$\_\{$y$\}\backslash ),$ evaluate the critical internal pressure $\backslash ($ p$\_\{$i$\}\backslash )$ that will make the pipeline material starts to yield. Use the Tresca yielding criterion.

$(2)$ There is an initial crack $($detected by X$-$Ray CT$)$ at the inner surface of pipe with a radial direction depth of $\backslash ($ c $\backslash ).$ Note that $\backslash ($ c $\backslash )$ is very small compared to the pipe radius. The material fracture toughness $\backslash ($ K$\_\{$I C$\}\backslash )$ is given. Evaluate the critical internal pressure $\backslash ($ p$\_\{$i$\}\backslash )$ that will initiate a crack propagation. Hint: the geometry coefficient of stress intensity factor for an edge tension crack is assumed to be $1\mathrm{.}12.$