Case study: Safe pressure vessels that "leak before break"

You wish to design a thin$-$walled spherical pressure

vessel of known radius $R,$ that will "leak before break"

to maximize its safety. That is$,$ the critical crack length

for unstable fracture must be larger than the thickness

of the pressure vessel wall; this ensures that the

vessel will leak $($reducing the pressure$)$ before it

reaches a state where catastrophic failure $($explosion$)$

could occur.

State the function, constraint$($s$),$ objective, and free variables for this problem and derive the

materials performance index $($MPI$).$ Plot the index on the appropriate chart $($see reverse$)$ and

identify $3-4$ candidates that best meet the requirements. How would you rank these materials?

How would your selection differ if you were to apply a "yield before break" criterion, wherein the

stress required to cause the largest known flaw $($crack$)$ to grow exceeds the yield strength?

I am having trouble deriving the MPI, if someone could walk me through the steps that would be great!