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The question is that: they drill a hole in the middle of a metal. Then when this metal is heated, will the hole become larger or smaller? The hole will get bigger, by experiment, but I think that when material expands, the hole must get smaller. What's wrong with this?

In general, can anyone please tell me how material expands exactly?

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When the metal is heated, all inter-atomic distances increase by the same factor. This drawing may help understand why the hole also increases in size. Here, I increased all distances by a factor a two. enter image description here

Replace the atoms with galaxies, and you have a model of the expanding Universe, which may help understand why an observer in any galaxy will see herself as the center of the Universe, with all other galaxies flying away from her.

If the hole should shrink, it would mean that some distances should increase more than other. This doesn't happen:

enter image description here

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If the object is in free space, the hole shall expand proportionally as you expect.

However, a different situation is when the object is somehow restrained to expand; then the external force simultaneously causes elastic compression which limits the expansion. If the restriction of expansion is stiff enough (such as placing it in an infinitely rigid box), it may eventually lead to the material expanding mostly inwards, thus the hole getting smaller.

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Thermal expansion means that the distance between atoms increases with temperature. Picture the ring of atoms surrounding the hole: the distances between them increases, hence the circumference of the hole increases. I hope this makes it clear that the radius of the hole increases with temperature.

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