Dave Tweed wrote:

> Cooler material surrounding this will be subject to the radial expansion,
> and since it is also elastic, will end up being compressed radially and
> stretched circumferentially, to a degree that is related to the
> temperature gradient. =


Hm... I see. This brought me on a track. In any case, this assumes that we
stay within elastic limits. Is this plausible?

> The only way the hole can get smaller is if the material gets
> considerably softer at high temperatures, =


Check out http://midas.npl.co.uk/midas/content/mn049.html

Young's modulus of steel seems to decrease linearly up to ~600=B0C (~80% of
the modulus at 0=B0C), and then decrease sharply.

> or if something nonuniform occurs -- the hole wasn't round to begin with,
> the edge buckles, or there's an anisotropic component to the expansion.
> Any of these might apply to your blacksmiths. =


Right. If any of these apply, don't you say the hole could have become
smaller?

---------------------

Check out this:
http://physics.uwstout.edu/Statstr/statics/Stress/strs38.htm, at the
bottom. This is approximately what I was talking about (constrained thermal
expansion). =


They simplify the situation by saying let it first expand thermally, then
apply stress to bring it back into its original form. Applied to something
similar to my situation, we would have a smaller cylinder with the hole
that gets heated and a bigger constraining cylinder around it that remains
cold. (Simplified thought experiment; in reality there would a temperature
gradient, a stress gradient, heat conduction etc., probably with
qualitatively similar results.) So the smaller, heated cylinder expands in
all directions, its outer and inner diameters and its wall thickness expand
proportionally. Then the stress gets applied and the outer diameter gets
reduced to the size it was before. =


What happens to the inner diameter is determined by the elastic or plastic
forces. Maybe it just shrinks back to its original size (probably the case
if we stay within the limits of Hooke's law, but I'm not sure this would be
guaranteed). Or maybe not. But it seems we have now a different angle to
look at it: what happens to the size of a hole in a cylinder when you
compress the outer diameter? Does it shrink proportionally? If so, what are
the limits of proportionality (there definitely are some)?

This approach doesn't take into account that the elastic modulus decreases
with increasing temperature. =


Gerhard

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