# 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