# Stress Distribution in Compound Cylinders

## Homework Statement

A long, hollow, thick elastic cylinder (E , v) is surrounded by a thin elastic band of thickness t made of a different material ($E_2$ , $v_2$). The fit is ideal, such that neither a gap nor a pressure exists. An internal pressure p is then applied at the inside radius (r=a) of the thick cylinder. Assuming that the radial displacements is continuous at the interface (r=b), prove that the stress in the thick cylinder are given by
$\sigma_{rr} = -p\frac{\beta(1+(b/r)^2)+(1-(1-((b/r)^2)}{\beta(1+(b/a)^2)+(1-((b/a)^2)}$

where
$\beta= \frac{(1-v^2)}{(1+v) v-(bE)(1-v_b^2)/(E_b h)}$

## Homework Equations

Lame's Equations. If a long cylinder is subject to internal pressure $p_i$ at $r_i$ and external pressure $p_o$ at $r_o$ then the radial stress distribution is

$\sigma_{rr}=\frac{p_i r_i^2-p_o r_o^2}{r_o^2-r_i^2}+\frac{r_i^2 r_0^2(p_o-p_i)}{r^2(r_o^2-r_i^2)}$

and the tangential stress distribution
$\sigma_{tt}=\frac{p_i r_i^2-p_o r_o^2}{r_o^2-r_i^2}-\frac{r_i^2 r_0^2(p_o-p_i)}{r^2(r_o^2-r_i^2)}$.

and finally the displacement is

$u_r=\frac{1-v}{E}\frac{(p_i r_i^2 - p_o r_o^2)}{r_o^2-r_i^2} r + \frac{1+v}{E}\frac{r_i^2 r_o^2 (p_i-p_o)}{r_o^2-r_i^2}\frac{1}{r}$.

## The Attempt at a Solution

I attempted to match the displacements at the interface and solve for the interface pressure $P$. However, I can not get the correct solution with this approach.
The displacement of the inner cylinder at the interface is given by
$u_r(r=b)=\frac{1-v}{E}\frac{(p_i a^2 - P b^2)}{b^2-a^2} b + \frac{1+v}{E}\frac{a^2 b^2 (p_i-P)}{b^2-a^2}\frac{1}{b}$.

Canceling terms yields
$u_r(r=b)=\frac{1}{E(b^2-a^2)}(2p_i a^2 b - (1-v)P b^3 - (1+v)P a^2 b)$
The displacement at the interface of the outer cylinder is given by:

$\sigma_\theta ≈ E \epsilon_\theta→ u(r=b) = \frac{Pb^2}{E_2 t}$

because the outer elastic band is thin. Equating the displacements I get the interface pressure to

$P = \frac{2 p_i a^2}{\frac{Eb(b^2-a^2)}{E_2 t}+(1-v)b^2+(1+b)a^2}$.

I know this is the wrong interface pressure by check the solution given in the problem statement and through FEA analysis.

There is a typo in my post above because $(1+b)$ should be $(1+v)$ in the denominator of the expression for $P$. The mistake I made was using the lame's equations for plain stress, while the problem wanted to plane strain solution. The expression for $P$ matches the given result by making the plane stress to plane strain substitutions $E\rightarrow E/(1-v^2)$ and $v\rightarrow v/(1-v)$.