I'm trying to calculate the pressure between a journal/shaft and a housing bore in a case of interference fit.
Considere a 2D axisymmetric (effectively 1D) journal bearing with interference fit:
Where $D_1 > D_2$ and after fitting we should have $D_1>D_4>D_2$ and I think also $D_5>D_3$. Writing the continuum equation of equilibrium for the 1D problem:
$$ \frac{d \sigma_r}{d r} +\frac{\sigma_r-\sigma_\theta}{r}=0 \tag{1}$$
and the elements of the strain tensor can be calculated from radial displacement:
$$\left\{ \begin{matrix} \epsilon_r=\frac{d u}{d r}\\ \epsilon_\theta=\frac{u}{r} \end{matrix} \right. \tag{2}$$
where $u$ is the radial displacement. And the Hooke's law including the Poisson's ratio for stress-strain tensors :
$$ \left\{ \begin{matrix} \sigma_r=\frac{\lambda}{\nu}\left( \left(1-\nu \right)\epsilon_r +\nu \epsilon_\theta \right)\\ \sigma_\theta=\frac{\lambda}{\nu}\left( \left(1-\nu \right)\epsilon_\theta +\nu \epsilon_r \right) \end{matrix} \right. \tag{3}$$
Where $\lambda=\frac{\nu E}{\left(\nu+1 \right)\left(1-2 \nu \right)}$ is the Lame’s elastic constant. Combining 1, 2 and 3 yields:
$$ u_{rr}+\frac{u_r}{r}-\frac{u}{r^2}=0 \tag{4}$$
Which is a simple second order Cauchy-Euler ODE and has an analytic solution of:
$$u=c_1 r +\frac{c_2}{r}\tag{5}$$
The issue is that this solution is singular at $r=0$, whereas based on the boundary condition we should have $u=0$ there. This doesn't make sense. I'm probably making some mistakes. I would appreciate if you could help me know what is the problem and how I can solve it.
P.S.1. The final goal is to answer my other question over here.
P.S.2. The assumption of $c_2=0$ also does not work. stress can't be constant.
P.S.3. I borrowed the equation 3 from this lecture notes of "Structural Mechanics in Nuclear Power Technology" MIT course. But it doesn't makes sense because it is exactly the same as the Cartesian one.