Stress in cylinders due to thermal expansion

[ATT55]

I have two concentric thick cylinders in close tolerance (currently 50 micron gap), with a thin layer of glue between them.
Internal cylinder is made from steel and external cylinder is made from ceramic (so the thermal expansion coefficient is not the same)

The assembly is headed form inside to a given surface temperature and I'm trying to find the stress on the ceramic cylinder in order to find the optimal gap between the cylinders.

I didn't find any simple model for that, so what i did:

1. For a given surface temperature and heat flux find the the internal temperature.
2. Calculate the thermal expansion of each cylinder.
3. Calculate the radial interference due to the thermal expansion.
4. Calculate the interference pressure from the the radial interference - same as with press fit of shaft and hub analysis.
5. Calculate the stress on the cylinders according to Lame's equations.

Does it seems reasonable?

references:
https://roymech.org/Useful_Tables/Mechanics/Cylinders.html
https://courses.washington.edu/me354a/Thick Walled Cylinders.pdf

 

[Baluncore]

Yes, except for the turn on transient, when the system is not in equilibrium. The steel will be hot, the ceramic cold. That may be the critical stress on the ceramic.

You might slit the inner steel tube, like a slotted or scroll pin, so it fits inside the ceramic with a reasonably constant contact pressure.

 

[ATT55]

[Mentor Note -- Two threads on the same question have been merged.]

I have two concentric cylinders, the inner cylinder is made from stainless steel on the outer cylinder is made from ceramic. The inner cylinder is heated.

The cylinders are glued using epoxy glue. The gap between the cylinders is 50-100 micron.

Coefficient of thermal expansion:
Inner cylinder: 10.7 micron/ K*m
Outer cylinder: 10 micron/ K*m
Glue: 200 micron/ K*m, modulus: 1.7 GPa, hardness: 75 Shore D

I’m trying to understand if the glue is going extract significant stress on the other cylinder. It has high thermal expansion, but it’s thickness is quite thin and also its modulus is not high.

To reduce stress do I need glue with high or low hardness? what about the modulus?
Or because the glue layer is so thin, its thermal expansion is negligible?

 

[erobz]

I have two concentric cylinders, the inner cylinder is made from stainless steel on the outer cylinder is made from ceramic. The inner cylinder is heated.

The cylinders are glued using epoxy glue. The gap between the cylinders is 50-100 micron.

Coefficient of thermal expansion:
Inner cylinder: 10.7 micron/ K*m
Outer cylinder: 10 micron/ K*m
Glue: 200 micron/ K*m, modulus: 1.7 GPa, hardness: 75 Shore D

I’m trying to understand if the glue is going extract significant stress on the other cylinder. It has high thermal expansion, but it’s thickness is quite thin and also its modulus is not high.

To reduce stress do I need glue with high or low hardness? what about the modulus?
Or because the glue layer is so thin, its thermal expansion is negligible?

What calculations have you tried?

 

[ATT55]

What calculations have you tried?

I’m trying to understand if in principle a very thin layer, with high thermal expansion coefficient, but low modulus is placed between rigid cylinders - can it extract significant forces? I imagine it more as a soft spring.

 

[erobz]

I’m trying to understand if in principle a very thin layer, with high thermal expansion coefficient, but low modulus is placed between rigid cylinders - can it extract significant forces? I imagine it more as a soft spring.

Is the glue meant to rigidly bond the ceramic to the steel without fracturing under the thermal stress induced by the heating of dissimilar materials?

I guess its hard to say. If the inner core (steel) expands freely while glued then there wont be much stress transmitted to the ceramic. However, if the glue is meant to bond the steel and the ceramic such that they expand in unison, then there will be stress(force) transmitted to the ceramic.

 

[ATT55]

Is the glue meant to rigidly bond the ceramic to the steel without fracturing under the thermal stress induced by the heating of dissimilar materials?

Yes, it meant to bond them. Since the ceramic and steel have a similar thermal expansion coefficient the gap between them is never closing (at steady state). but the gap is filled with this glue, which expands.

A 100 micron thin flat layer of glue barley expands in absolute values. But in my case the glue is in a confined space.

 

[berkeman]

I have two concentric cylinders, the inner cylinder is made from stainless steel on the outer cylinder is made from ceramic. The inner cylinder is heated.

Is this the same question that you asked a couple weeks ago? If so, I can merge the two threads.

I have two concentric thick cylinders in close tolerance (currently 50 micron gap), with a thin layer of glue between them.
Internal cylinder is made from steel and external cylinder is made from ceramic (so the thermal expansion coefficient is not the same)

The assembly is headed form inside to a given surface temperature and I'm trying to find the stress on the ceramic cylinder in order to find the optimal gap between the cylinders.

I didn't find any simple model for that, so what i did:

1. For a given surface temperature and heat flux find the the internal temperature.
2. Calculate the thermal expansion of each cylinder.
3. Calculate the radial interference due to the thermal expansion.
4. Calculate the interference pressure from the the radial interference - same as with press fit of shaft and hub analysis.
5. Calculate the stress on the cylinders according to Lame's equations.

Does it seems reasonable?

references:
https://roymech.org/Useful_Tables/Mechanics/Cylinders.html
https://courses.washington.edu/me354a/Thick Walled Cylinders.pdf

 

[erobz]

If its meant to bond them the stress is induced in the ceramic. Allow the steel to freely expand. The ceramic is a good insulator, its average temperature is going to be lower than the steel. Its also has more surface area to dissipate the heat.

For the steel expanding freely:

$$ \delta L \approx L_o \alpha \Delta T $$

Then you apply a force to the steel $P$ to "push it back" to its original length, the ( approximately) initial length of the ceramic.

$$ \delta L = \frac{PL}{A_cE_c}$$

By Newtons second ( static equilibrium ) that force is also acting on the ceramic shell.

You can use $P$ and the cross sectional area of the ceramic shell to find the normal stress acting in the ceramic shell.

This is approximation, low ball estimate. They are also going to grow radially at different rates, and combined stresses will need to be considered etc..., but it's a flag in the ground.

 

[ATT55]

Is this the same question that you asked a couple weeks ago? If so, I can merge the two threads.

Yes, you can merge them.

 

[Baluncore]

I avoid mixing materials, by looking for alternative solutions.

A coat of vitreous enamel or ceramic, could be baked onto the stainless steel tube.

If either tube, steel or ceramic, was slotted over its full length, the two could be held together by elasticity, without glue, like a roll-pin.
https://en.wikipedia.org/wiki/Spring_pin

 

[Chestermiller]

What are the axial constraints? What are the ingested and unstressed radii?

 

[Ranger Mike]

you left out the critical factor of design.

How round are each cylinder? you have a 50 micron gap that is huge. How round is each part. This must be known before any assembly, let alone stress analysis can be made and contact area.
Try fitting a hydraulic connection in a fracking application with 15,000 psi pressure and think you will not have blow by in the connection, let alone any stress analysis .
How round are they? What is the out of roundness ? I am not talking about diameter. I need to know the FORM ERROR.

 

[ATT55]

What are the axial constraints? What are the ingested and unstressed radii?

No axial constraints.
OD 130mm (unstressed diameter)
ID 100mm

 

[ATT55]

you left out the critical factor of design.

How round are each cylinder? you have a 50 micron gap that is huge. How round is each part. This must be known before any assembly, let alone stress analysis can be made and contact area.
Try fitting a hydraulic connection in a fracking application with 15,000 psi pressure and think you will not have blow by in the connection, let alone any stress analysis .
1970 chevy vega with aluminum block and cast iron cylinders...case in point. How round are they? not by diameter as this is not form error, but size.

Roundness is 50micron.
If the parts are assembled with no force required, and have similar thermal expansion coefficient, then the question, I think, what is the effect of adding glue between them, with high thermal expansion coefficient.
The parts are rigid, and the glue is polymer. Can it cause significant stress?

 

[Ranger Mike]

you can not analyze cylindricity with only a size measurement. you must know the form error of both diameters otherwise it is a waste of time.

 

[Chestermiller]

No axial constraints.
OD 130mm (unstressed diameter)
ID 100mm

If there are no axial constraints, then the axial strain in the two cylinders and the epoxy are the same and the axial force is zero. The stress strain equations in the 3 materials are

$$\epsilon_{r,j}=\frac{\sigma_{r,j}-\nu(\sigma_{\theta,j}+\sigma_{z,j})}{E}+\alpha\Delta T$$ $$\epsilon_{\theta,j}=\frac{\sigma_{\theta,j}-\nu(\sigma_{r,j}+\sigma_{z,j})}{E}+\alpha\Delta T$$ and $$\epsilon_{z,j}=\frac{\sigma_{z,j}-\nu(\sigma_{\theta,j}+\sigma_{r,j})}{E}+\alpha\Delta T$$ where j is the layer number.