= Radial equilibrium of a thick elastic cylinder
{title2=$\tau_\phi=(r\tau_r)',\quad r^2R''+rR'-R=0$}
An axisymmetric small radial displacement $R(r)$ gives radial <strain> $R'$ and hoop <strain> $R/r$. Radial force balance on an annular wedge gives $\tau_r'+(\tau_r-\tau_\phi)/r=0$. Under <plane stress>, isotropic <Hooke's law> gives $\tau_r=E(R'+\nu R/r)/(1-\nu^2)$ and $\tau_\phi=E(R/r+\nu R')/(1-\nu^2)$. Substitution into equilibrium gives the displayed displacement equation. Its power-law exponents are $+1$ and $-1$, so $R=c_1r+c_2/r$. The hoop-stress solution is not based on treating a thick wall as a membrane.
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