 Elastic stress concentration in a pressurized thick cylinder

For an isotropic elastic annulus $a<r<b$ with internal excess pressure $P$ and zero outer gauge pressure, radial equilibrium gives $u=Cr+D/r$. Its stress tensor has ${\tau }_{rr}=2(\lambda +\mu )C-2\mu D/r^2$ and ${\tau }_{\theta \theta }=2(\lambda +\mu )C+2\mu D/r^2$. Enforcing ${\tau }_{rr}(a)=-P$, ${\tau }_{rr}(b)=0$ gives $2\mu D=P{a}^2{b}^2/(b^2-a^2)$, so the maximum difference occurs at the bore and equals the displayed expression. Once an appropriate yield criterion is reached, stress must be related to elastic strain while total strain includes a plastic contribution. The elastic formula alone cannot describe further irreversible deformation.