Consider a piece of metal of length $L$ and linear thermal expansion coefficient $\alpha$. We eat the metal $\Delta T$ degrees, causing the metal to increase to length $$ L' = L + L \alpha \Delta T$$ Now, cool the object back to the original temperature. This causes the metal to decrease in length to $$L'' = L' - L'\alpha\Delta T \\ = L + L \alpha \Delta T - (L + L \alpha \Delta T)\alpha \Delta T \\ =L (1 - \alpha^2 \Delta T^2)$$
My intuition would lead me to believe that heating up and then recooling an object would cause it to return to the same size it began at (if it did not, then bridges which repeatedly warmed and cooled would continually shrink) but this is not true according to my mathematics, which indicates that warming and recooling an object leaves it slightly smaller than it was before.
Do objects really not return to their original size when recooled? If so, then why do objects which warm and cool on a daily basis not slowly shrink?
I suspect that this may be related to $\alpha$ varying over the range of temperatures, but I didn't think that was a significant effect for solids.