Residual Stresses 1

[MetalsInc]

Here's one I can't wrap my head around...

I have two tubes of differing material, that are concentric to each other and bonded. At 2000F they are both solid and intimately bonded throughout. I now cool them to room temp...

The outer material has a COTE of 1 and then inner material is 2 (relative to each other). They are the same thickness.

What will the stress patter in the outer material look like? Radial stresses and circumfrential?

 

[RPstress]

For 'long'* or 'short'** tubes basic principle is simple enough, quantification less so. Inner tries to shrink more than outer: outer pulls radially outward on inner; inner pulls radially inward on outer. Bond is in radial tension, no shear. Radial stresses are tensile, peaking at interface and dropping to zero at outer surface of outer and inner surface of inner. Actual value of stress depends on modulus of tubes and CTE x delta_T. Circumferential stress in tubes is tensile inner, compressive outer. Again value depends on E and CTE x delta_T. Just knowing the ratio of CTE's isn't quite enough. Also, peak radial and average circumferential are intimately related: radial_stress_in_bond x radius = circumferential_stress x tube_thickness (for tubes thin compared with their radius).

For tubes with 'big'*** radius cf. thickness, I think (will need to check) (CTE_inner - CTE_outer) x delta_T x modulus = circumferential_stress (tension inner, compression outer). It's the difference in CTE's that matters, not the ratio. Also this ignores small stuff to do with stiffness and thickness of any bond material present.

Any help?

-R.

* 'long' approx. >= 4 x combined thickness of tubes.
** 'short' approx. <= 1/4 of combined thickness.
*** 'big' means radius approx. >= 5 x combined thickness.

 

[MetalsInc]

Yes, lots of help. I am intersted in qualitative results so you're answer is perfect for my needs...

Now let me compicate it. Add a third tube on the outside of the (previously) outer material. This tube has COTE 2. So the sandwich goes 2-1-2.

Same process, and the third tube (far outside) is removed at the end.

Can you walk me through that?

 

[MetalsInc]

I should have added the option of removing the inner most tube as well.

So the set up would be reversed.

 

[Lcubed]

RPstress-
No shear? Aren't the tubes changing in relative length as well as diameter?
Regards.