$4.(10$ Points$)$ A spherical pressure vessel made of $13$ Cr steel $(5$ mm thickness, $100$ mm diameter$)$ is used to store a yressurized saline solution $(\backslash (3\backslash \%\backslash$mathrm$\{$NaCl$\}\backslash )).$ Determine tie maximum allowable pressure for the saline solution using the leak$-$before$-$break criteria Additionally, calculate the maximum pressure if the vessel is used to store a non$-$conosive solution. For both cases, assess the maximum pressure that could cause failure due to yielding as well as the maximum pressure the vessel can sustain in the presence of a through$-$thickness crack.

Material properties of the stainless steel used to make the pressure vessel are:

Yield Strength $\backslash (=500\backslash$mathrm$\{$MPa$\}\backslash )$

Fracture toughness in non$-$oorrosive environment, $\backslash ($ K$\_\{1\backslash$mathrm$\{$C$\}\}=60\backslash$mathrm$\{$MPa$\}\backslash$mathrm$\{$ma$\}^\{0\mathrm{.}3\}\backslash )$

Fracture toughness in $\backslash (3\backslash \%\backslash$mathrm$\{$NaCl$\}\backslash )$ solution, $\backslash ($ K$\_\{13\backslash$mathrm$\{$scc$\}\}=12\backslash$mathrm$\{$MPa$\}\_\{\backslash$mathrm$\{$m$\}\}\{\}^\{0\mathrm{.}3\}\backslash )$

Assume that the constant $\backslash ($ C $\backslash )$ in equation below is equal to $1$

$\backslash [$

$\backslash$sigma$=\backslash$frac$\{$K$\_\{$t c$\}\}\{$C $\backslash$sqrt$\{\backslash$pi t $/2\}\}$

$\backslash ]$

$5.(6$ Points$)$ The standard $\backslash ($ C $\backslash$supset $\backslash )("$Jewel$"$ case$)$ cracks easily and, if broken, can scratch the CD$.$ Jewel cases are made of injection molded polystyrene, chosen because it is transparent, cheap and easy to mold. A material is sought to make CD cases that do not crack so easily. The case must still be transparent, able to be injection molded, and able to compete with polystyrene in cost$.$ Translate the requirements into material selection criteria, listing function, constraints, objective and free variable.

$6.(16$ Points$)$