Problem $1$: The hoop stress at the outer surface of a pipe of internal diameter $2r$ and external diameter $2R$ under internal pressure $P$ is given by

$=\frac{2P{r}^2}{\{R^2-r^2\}}$

When A toughened rigid PVC water pipe has $r=147\mathrm{.}5mm$ and $R=167\mathrm{.}5$ mm $.$ Tests show $K_{1C}=3\mathrm{.}1MPa{m}^{0\mathrm{.}5}$ and ${}_y=40$ MPa $.$ The largest defects are equivalent to $100m$ cracks in the outer surface.

$($a$)$ Determine whether the pipe will fail by yield or fracture in a pressurization test, and find the pressure to cause failure.

$($b$)$Find the maximum value of yield stress that will permit ductile failure in the pipe as strain rate increases, assuming that $K_{IC}$ remains constant.

Calculate maximum yield strength for poorly processed PVC pipe, for which $K_{lC}=1\mathrm{.}5MPa{m}^{0\mathrm{.}5},$ assuming that a remains unchanged at $100m.$