Thermal Expansion: Square & Hole Expansion

[Grhymn]

Homework Statement

You have a square which has a hole inside. When the temperature increases how does it expand?

Homework Equations

The Attempt at a Solution

I know that the hole will expand along with the square, but I don't see why. Can't it expand in both directions: the square getting bigger and the hole smaller?

 

[Mapes]

Here's one way to look at it: if there are no constraints on the edges of the square, then the material is stress-free. Therefore, you should be able to put a circle of the same material in the empty circle, and the interface will be stress-free. But we know that the circle will expand with increasing temperature. Therefore, the hole must expand exactly the same amount.

If the edges of the square are fixed, however, then the circular hole will shrink. But you didn't specify the edge constraints, so I'm guessing by your answer that you've assumed there are no constraints.

 

[Grhymn]

There were no constraints indeed. Thank you for the clarification.