Q$4(20$ Points$)$

A cylindrical pressure vessel made of Type $304$ stainless steel is used to store a corrosive chemical solution at a temperature of $\backslash (50^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The vessel has an internal diameter of $1\mathrm{.}5$ meters and a wall thickness of $15$ mm $.$ During a routine inspection, a small crack of $2$ mm length is detected on the inner surface of the vessel. Given that

$-$ The yield strength of Type $304$ stainless steel at $\backslash (50^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ is $205$ MPa $.$

$-$ The fracture toughness $\backslash (\backslash$left$(\backslash$mathrm$\{$K$\}\_\{\backslash$mathrm$\{$IC$\}\}\backslash$right$)\backslash )$ of the material under these conditions is $\backslash (85\backslash$mathrm$\{$MPa$\}\backslash$sqrt$\{\backslash$mathrm$\{$m$\}\}\backslash ).$

$-$ The threshold stress intensity factor for stress corrosion cracking $(\backslash (\backslash$mathrm$\{$K$\}\_\{\backslash$mathrm$\{$IScc$\}\}\backslash ))$ for this material$-$environment combination is $\backslash (22\backslash$mathrm$\{$MPa$\}\backslash$sqrt$\{\}\backslash$mathrm$\{$m$\}\backslash ).$

$-$ The vessel operates at an internal pressure of $2$ MPa $.$

a$)(4$ Points$)$ Calculate the hoop stress in the vessel wall under normal operating conditions.

b$)(4$ Points$)$ Determine the stress intensity factor $\backslash (\backslash$left$(\backslash$mathrm$\{$K$\}\_\{\backslash$mathrm$\{$I$\}\}\backslash$right$)\backslash )$ for the detected crack, assuming it is a surface crack in a thick$-$walled cylinder. Use the approximation in $\backslash (\backslash$mathrm$\{$K$\}\_\{\backslash$mathrm$\{$I$\}\}=1\mathrm{.}12\backslash$sigma $\backslash$sqrt$\{\}(\backslash$mathrm$\{\backslash$pi a$\})\backslash ),$ where $\backslash (\backslash$sigma $\backslash )$ is the hoop stress and a is half the crack length.

c$)(6$ Points$)$ Assess whether the current operating conditions could lead to stress corrosion cracking. Explain your reasoning.

d$)(6$ Points$)$ Calculate the critical crack length that would lead to fast fracture under the current operating conditions. Recommend a safe operating pressure to prevent both stress corrosion cracking and fast fracture, assuming no change in crack length. Justify your recommendation.