Radial equilibrium of a thick elastic cylinder (${\tau }_{\varphi }=(r{\tau }_r{)}^{\prime },r^2{R}^{\prime \prime }+r{R}^{\prime }-R=0$)

An axisymmetric small radial displacement $R(r)$ gives radial strain $R^{\prime }$ and hoop strain $R/r$. Radial force balance on an annular wedge gives ${\tau }_r^{\prime }+({\tau }_r-{\tau }_{\varphi })/r=0$. Under plane stress, isotropic Hooke's law gives ${\tau }_r=E(R^{\prime }+\nu R/r)/(1-{\nu }^2)$ and ${\tau }_{\varphi }=E(R/r+\nu R^{\prime })/(1-{\nu }^2)$. Substitution into equilibrium gives the displayed displacement equation. Its power-law exponents are $+1$ and $-1$, so $R=c_{1}r+c_2/r$. The hoop-stress solution is not based on treating a thick wall as a membrane.

For inner and outer radii $a,b$, with internal and external pressures $p_1,p_0$, the boundary tractions are ${\tau }_r(a)=-p_1$ and ${\tau }_r(b)=-p_0$. The radial equilibrium of a thick elastic cylinder gives $P=(p_1{a}^2-p_0{b}^2)/(b^2-a^2)$ and $Q=(p_1-p_0)a^2{b}^2/(b^2-a^2)$. The radial compressive pressure is $p(r)=-P+Q/r^2$. Since hoop stress is monotone in $r$ when $Q\ne 0$, its maximum absolute value is the larger of $|P+Q/a^2|$ and $|P+Q/b^2|$. With negligible external pressure, its maximum is the tensile inner-wall value $p_1(a^2+b^2)/(b^2-a^2)$. The stress distribution itself is independent of elastic moduli in this linear homogeneous axisymmetric model; its associated displacement depends on them.