Why doesn't a metal disk expand in all directions when heated?

Question

The hole in a metal disk gets bigger when heated. This has been explained by the reason that if we take the already cut out piece of the metal and heat it separately it also gets bigger, but the metal of the cut out is a metal hence it expands , hole is the absence of metal so why doesn't the metal disc expand in all directions thus making the hole smaller?

Answer 1

Imagine you build a model of the disc out of LEGO bricks. And then you re-build the same model, brick-for -brick, out of the equivalent DUPLO bricks (they look the same but are 2x as big). Do you expect the hole in the DUPLO disc to be smaller or bigger than that in the LEGO disc?

With thermal expansion you have exactly the same principle at play. When you heat a material up, all the chemical bonds between the atoms get a bit longer. That is the reason for the overall expansion. The atoms stay exactly in their places relative to one another but all distances are (uniformly) larger. This is just like using somewhat bigger bricks to build otherwise the same model.

Answer 2

It does try to expand in all directions, but inward growth is opposed by internal pressure. Consider such atom lattice schematics :

Where blue dots crystal lattice atom positions, and red dots - possible lattice atoms positions if they would "compress inwards".

Making inside hole smaller means that lattice atoms aligned on circle perimeter should decrease it's bond lengths, as per,

$$ \tag 1 \frac lL = \frac {\theta r}{\theta R}=\frac rR \lt1 $$

Where $L,l$ are bond lengths before and after inwards expansion.

As heating metal means that bonds between ALL atoms should increase,- this compression inwards due to expansion can't be done, since it is compensated by lattice internal pressure. And as such heated metal disk can only expand outwards, making hole bigger.

BTW, for the similar reasons crystals grows outwards too, like snowflake for example. They never start from big aperture and go inwards because it's not optimal in terms of energy utilization.

Answer 3

Cut the disk up into a series of thin rings. When a ring expands, its circumference grows. This means its radius grows. This still applies if you take out the innermost rings to leave a hole.

The thickness of each ring also grows. But suppose the thickness is $1/100^{th}$ of the circumference. The thickness increase will be $1/100^{th}$ of the circumference increase. Both inner and outer edges move outward.

Answer 4

Heating a disk (of any shape or genus$^0$) causes internal stresses because every individual part of the material also seeks to expand$^1$. The material will relax into whatever shape provides mechanical equilibrium to these internal forces$^2$. What shape could that be?

Since every point on the disk is trying to expand away from every other point$^3$, the changes are equivalent to a uniform scaling transformation, which preserves shape. A shrunken hole with an expanded rim or a shrunken rim with an expanded hole represent different shapes and possess unresolved stresses (indeed compressive ones, on the hole boundary, as @AgniusVasiliauskas points out, which is different than when the hole is filled, as you point out).

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$^0$ loosely, the number of holes
$^1$ or contract; not all materials expand on heating.
$^2$ As @Karel points out, the forces are due to lengthening of the average separation between individual constituents of matter (atoms/molecules), given the increased thermal energy (but the same binding electromagnetic potential) though I avoid invoking phenomenology in this answer. Also note that equilibrium is achieved, when the increased separation leads to an increased force of attraction.
$^3$ ...by the same amount; assuming homogeneity and isotropy of the material

Answer 5

It does expand in all directions. The way to prove that is not to see the hole in the middle get smaller, but to measure the thickness of the band (as opposed to the diameter of the whole ring) and see that it gets bigger. It just doesn't get as much bigger as the entire ring, so the hole still grows.

Answer 6

This reminds me of a school physics lesson when I was 14 or so. The teacher posed this as a trick question. Some of the kids had problems understanding it...

Imagine that the disc is made of something flexible and you cut it across a radius. Straighten it out and now you have a metal bar. Heat the bar so it expands -- it gets longer. Now bend it back into a circle. Because the length of the inner circumference (the one round the hole) has increased, so has the size of the hole.

Or think of a curve in railway tracks. Let it be a perfect circle. The inner rail is shorter than the outer one because the circumference of the inner rail is less (because the radius is less). If you ever played with Lego rail tracks, you'll know what I mean. Now imagine the rails expand, i.e. get longer (don't try heating Lego rails!) Both rails are longer so the radius of both is greater. The hole gets bigger.

Answer 7

When something expands, there's exactly one spot on/in the object that does not move, and everything else moves away from that point. The farther away from the stationary point something is, the more it moves during the expansion. For example, you might nail down the top left corner; then the center of the object moves a little bit away from it and the bottom right corner moves a lot.

We can now reason in two alternative directions: either starting from this fact, and seeing that a hole must grow, or imagining that there's a hole that shrinks, and concluding that this fact couldn't be true. I'll do both, as I think both are insightful.

If stuff expands away from one stationary point, holes grow. The stationary point could be anywhere, and there's two especially interesting cases: either the stationary point is somewhere inside the hole, or somewhere outside. If it's inside, then all the points on the hole are expanding away from it, so the hole gets bigger.

Now suppose it's outside. We can always rotate our head so that the stationary point is to the left of the hole, so let's do that to make the conversation easier to follow. Now the left-hand side of the hole is expanding to the right, and so filling in part of the hole. This part matches your intuition that the hole should shrink -- parts of the metal sheet are indeed expanding into the space where the hole was. But the right-hand side of the hole is also expanding to the right, away from our stationary point; and because it's farther away from our stationary point, it's expanding away to the right faster than the left-hand side of the hole. So while the left starts filling in the hole, the right is making new hole at a faster rate, and the hole expands.

If a hole shrinks, stuff isn't expanding away from one stationary point. One of the things we can do with a part of an expanding object is track its motion to find out where a stationary point, if there is one, would be. If something is expanding to the right, the stationary point it's expanding away from must be to the left, while if something is expanding to the left, the stationary point must be off to the right.

Now we can look at the leftmost and rightmost points of a shrinking hole. (For simplicity, let's say the hole is shrinking but not moving.) If the hole is shrinking, the leftmost point must be moving to the right, which means if there's a stationary point it's expanding away from, that point must be off to the left of the hole. Similarly, the rightmost point must be moving left, indicating that any center of expansion must be off to the right. But now we have conflicting clues about where the expansion point ought to be -- so there can't be just one stationary point that everything is expanding away from!

Answer 8

It does, it expands "in all directions". The "center" of this expansion is the center of the circles.

On a PC (from a CAD to Gimp to Inkscape to Paint), just draw two circles with the same center and apply a scaling transformation.

That's what's happening to the heated ring, and you can easily see that the inner circle becomes bigger as well.