= Elastic stress concentration in a pressurized thick cylinder
{title2=$\max|\tau_{rr}-\tau_{\theta\theta}|=2Pb^2/(b^2-a^2)$}
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 $2\mu D=Pa^2b^2/(b^2-a^2)$, 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.
Back to article page
