Material Specification for a Pressurized Cylindrical Tank

Consider a thin$-$walled cylindrical tank of radius $0\mathrm{.}5$m$(500$mm$)$ and wall thickness of $8\mathrm{.}0$ mm

that is to be used as a pressure vessel to contain a fluid at a pressure of $2\mathrm{.}0$ MPa $.$ Assume a crack

exists within the tank$\text{'}$s wall that is propagating from the inside to the outside as shown in Figure

$($Note: Crack propagation may occur due to cyclic loading associated with fluctuations in

pressure, or as a result of aggressive chemical attack of the wall material.$)$

Figure $1.$ Schematic diagram showing the cross section of a cylindrical pressure vessel

subjected to an internal pressure p that has a radial crack of length a located on the inside

wall.

Regarding the likelihood of failure of this pressure vessel, two scenarios are possible:

Leak$-$before$-$break. Using principles of fracture mechanics, allowance is made for the growth

of the crack through the thickness of the vessel wall prior to rapid propagation. Thus, the crack

will completely penetrate the wall without catastrophic failure, allowing for its detection by the

leaking of pressurized fluid.

Brittle fracture. When the advancing crack reaches a critical length, which is shorter than

for leak$-$before$-$break, fracture occurs by its rapid propagation through the entirety of the wall.

This event typically results in the explosive expulsion of the vessel's fluid contents.

Obviously, leak$-$before$-$break is almost always the preferred scenario.

For a cylindrical pressure vessel, the circumferential $($or hoop$)$ stress sigma$\_($n$)$ on the wall is a function

of the pressure p in the vessel and the radius r and wall thickness t according to the following

expression:

sigma$\_($h$)=($pr$)/($t$)$

Using values of p$,$r$,$ and t provided earlier, we compute the hoop stress for this vessel as follows:

$\{$:$[$sigma$\_($h$)=((2\mathrm{.}0$MPa$)(0\mathrm{.}5($m$)))/(8$xx$10^(-3)($m$))],[=125$MPa$]$:$\}$

Upon consideration of the metal alloys listed in Table B$.5$ of Appendix B $($of the print

textbook$),$ determine which satisfy the following criteria:

$($a$)$ Leak$-$before$-$break

$($b$)$ Brittle fracture

Use minimum fracture toughness values when ranges are specified in Table B$\mathrm{.}5($of the print

textbook$).$ Assume a factor of safety value of $3\mathrm{.}0$ for this problem.

Hint for solution: A propagating surface crack will assume a configuration shown

schematically in Figure $2-$having a semicircular shape in a plane perpendicular to the stress

direction and a length of $2$c $($and also a depth of a$,$ where a$=$c $).$ It can be shown as the crack

penetrates the outer wall surface, that $2$c$=2$t $($i$.$e$.,$ c$=$t$).$ Thus, the leak$-$before$-$break condition

is satisfied when a crack's length is equal to or greater than the vessel wall thickness$-$that is$,$

there is a critical crack length for leak$-$before$-$break c$\_($c$)$ defined as follows:

wall segment of a cylindrical pressure vessel; also shown is the geometry of a crack of length

$2$c and depth a that is propagating from the inside to the outside of the wall.