# Does a thermally expanding torus experience internal stress?

I'm trying to learn continuum mechanics and thermo-mechanics.

As we know, heating an object increases the mean atomic distance $a_0$ of the atoms in a rigid body. Let's assume it is a linear elastic material and we're well below the melting point. The lattice is perfect (no defects). I'm interested in whether the topology of a body plays a role when it is heated up.

I think, the cross section of a torus (topologically different to convex bodies) changes like in the sketch below (A: circumference at initial temperature, B: circumference at higher temperature).

### Question 1:

Is B still a circle or is it an ellipsoid?

### Question 2 (main question):

Is there a non-uniform internal stress field in the torus? E.g. is there more stress on the inner rim? How does this compare to the thermal expansion of a convex body (e.g. sphere)?

### Question 3:

What is the influence of the crystal structure (FCC, BCC, wurzite) to the internal stress of an thermally expanding body?

References are appreciated - especially good books on this subject.

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Q1; still circle, Q2, No , of course. Just think of that torus "sitting" in a thick slab of the same material not "carved" out of the slab. Q3 is something entirely different an should asked separately. – Georg Nov 11 '11 at 14:06
@Georg Q1 and Q2 are only trivial if the thermal coefficient of expansion is isotropic, which Q3 questions. So Q3 is vital to good answers to Q1 and Q2 and should definitely be kept as part of this question. – Rick Aug 28 at 15:56

## 1 Answer

Engineers usually treat thermal expansion as isotropic, which means the expansion occurs with the same magnitude in every direction. This means that an unconstrained object will have a constant strain and zero stress associated with thermal expansion, it's as if object just scaled up.

However, as you suspected, materials with an organized structure can be anisotropic. This means that within a single crystal one dimension might expand more than the others. So here if your torus was made of a single crystal it's possible that it would expand such that B was an ellipse. However, in this case while the strain tensor would be anisotropic, it would still be constant throughout the torus, and thus no stress would result from the thermal expansion.

If the torus was made from multiple crystals that were not aligned, then the strains from the crystals would be incompatible, but the bonds between the crystals would force them into a compatible arrangement inducing internal stresses. The shape of the torus might make it easier to create large stresses, but it not fundamentally any different from how stresses would form in a sphere.

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