Thermal Expansion: Square & Hole Expansion

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SUMMARY

The discussion focuses on the thermal expansion of a square with a hole when subjected to increased temperature. It is established that both the square and the hole expand uniformly when there are no constraints on the edges of the square. The reasoning provided indicates that the material remains stress-free, allowing the hole to expand alongside the square. In contrast, if the edges were fixed, the hole would shrink, but this scenario was clarified as not being applicable in this case.

PREREQUISITES
  • Understanding of thermal expansion principles
  • Familiarity with material stress and strain concepts
  • Knowledge of geometric constraints in physical systems
  • Basic principles of physics related to temperature effects on solids
NEXT STEPS
  • Research the principles of thermal expansion in different materials
  • Explore the effects of edge constraints on thermal expansion
  • Learn about stress-free conditions in materials science
  • Investigate real-world applications of thermal expansion in engineering
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Grhymn
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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?
 
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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.
 
There were no constraints indeed. Thank you for the clarification.
 

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