# Compressive pressure between interacting materials

Hello.

I'm conducting some experiments to test for the pressure exerted by a thermally expanding cylinder inside a tube. The cylinder is placed snugly inside the tube, usually with zero to a few 0.001s clearance. The cylinder has a higher coefficient of friction than the tube, therefore we expect pressure to be exerted onto the inside surface area. I've tried using
\begin{equation}
P=E\frac{\Delta R}{R}
\end{equation}
where E is the elastic modulus of the cylinder and $\Delta$ R is the constrained length (also considering the expansion of the outer tube). And R is the radius of the cylinder. Am I right in assuming the expansion is radial?

Now enter a new variable. The tube is lined with a soft fabric that will absorb some of the pressure as deflection. How can I calculate the pressure on the surface of the fabric liner caused by the thermally expanding cylinder? Is there a way to include both compressibilities (elastic modulii) into one equation? Help!

Thanks!!

anorlunda
Staff Emeritus
ping @jrmichler . Can you help with this spring cleaning question?

jrmichler
Mentor
Step 1: Given the exact clearance, calculate the temperature at which the cylinder contacts the tube.
Step 2: Search Interference fits and Shrink fits. This is a topic in advanced strength of materials. The equations can be found Roark's Formulas for Stress and Strain. You do not need the newest edition, the older editions also have the equations. There is a discussion of shrink fits in Advanced Mechanics of Materials by Seely and Smith, but that book is out of print. There are other books in print that discuss shrink fits, all with the title Advanced Mechanics of Materials.

The equations are about radial strains, stresses, and pressures. Longitudinal stresses and strains are also present, but I have never seen equations for them. Normal procedure is to calculate only radial stresses, strains, and pressure.

If the outer tube wall is thin relative to the diameter, then the calculation becomes much easier. Treat the tube as a thin wall cylinder, assume zero radial strain in the inner cylinder, and use the thin wall tube equation to calculate the radial pressure.

If the fabric liner is radially soft compared to the inner cylinder and outer tube, then the calculation is easier yet. Calculate the relative radial displacement from thermal expansion, the radial strain in the fabric, and get the stress from that.