 Solution

Let $a$ be a characteristic crack depth and $Y$ its geometry factor. A simple fast-fracture condition isThe crack should penetrate a wall of thickness $t$ before reaching the critical size, so the simplified leak-before-break criterion is $t<a_c$, with appropriate geometry and stable-through-wall-crack assumptions.

The conventional material-based maximum thickness uses the yield stress as a conservative limiting operating stress:It ensures that the depth required for penetration is subcritical even at the stated limiting stress. Geometrical factors depend on crack shape and orientation; this is an educational screening estimate rather than a full crack-growth or leak-detection analysis. The handbook's fracture formula uses the same stress-intensity and geometry-factor convention.^[ref]

There is an important distinction if the actual pressure and radius are fixed while thickness varies. Substituting $\sigma =Pr/t$ at penetration $a=t$ givesThat is a lower, not upper, bound on thickness under the fixed-pressure model. Yield avoidance additionally requires $t\gtrsim Pr/{\sigma }_y$. It would be inconsistent to derive the fixed-pressure expression and call it a maximum. The standard maximum quoted above answers a different, yield-limited material-selection comparison.