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256bits

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If a hole is made at the center of a metal plate and the plate is cooled to decrease its temperature, will the hole increase or decrease in size?

If you can answer my question, you should be able to answer your original question.

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mathman

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Increase: Consider what would happen if you didn't drill the hole and heated. The area where the hole would be gets bigger.

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Chestermiller

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berkeman

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AlephZero

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There are materials (e.g. many crystals, and some composite matierals) where the coefficient of thermal expansion is different in different directions. In those materials, in general the hole would not expand uniformly, and would change shape.

But apart from that:

So, imagine that you draw the circle, then change the temperature of the plate, then cut out the hole around the expanded or contracted shape of the line. Since the stress in the plate was zero everywhere before cutting, it will stay zero after cutting, and cutting the hole wll not change the shape of the plate.

Or more mathematically:

If there is no elastic stress in the plate, the strain field is ##\epsilon_{xx} = \epsilon_{yy} = \epsilon_{zz} = \alpha \Delta T## and the shear strains are all zero, where ##\alpha## is the corefficient of expansion and ##\Delta T## the temperature change.

Because the strain field is symmetrical in x y and z, this means that ANY two points that were originally a distance ##d## apart become a distance ##d(1 + \alpha \Delta T)## apart, indepedent of the shape of the object, and whether or not it contains holes.

But apart from that:

If the solid plate is at a constant temperature and the edges are free to expand, the stress in the plate will be zero everywhere, independent of the temperature.It kind of makes sense that if you draw a circle on the plate and heat it, the circle will grow with the plate.

So, imagine that you draw the circle, then change the temperature of the plate, then cut out the hole around the expanded or contracted shape of the line. Since the stress in the plate was zero everywhere before cutting, it will stay zero after cutting, and cutting the hole wll not change the shape of the plate.

Or more mathematically:

If there is no elastic stress in the plate, the strain field is ##\epsilon_{xx} = \epsilon_{yy} = \epsilon_{zz} = \alpha \Delta T## and the shear strains are all zero, where ##\alpha## is the corefficient of expansion and ##\Delta T## the temperature change.

Because the strain field is symmetrical in x y and z, this means that ANY two points that were originally a distance ##d## apart become a distance ##d(1 + \alpha \Delta T)## apart, indepedent of the shape of the object, and whether or not it contains holes.

Last edited:

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berkeman

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Ah, I didn't think of the zero stress condition. Good point. Thanks AlephZero!

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Chestermiller

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Very nicely explained!!!There are materials (e.g. many crystals, and some composite matierals) where the coefficient of thermal expansion is different in different directions. In those materials, in general the hole would not expand uniformly, and would change shape.

But apart from that:

If the solid plate is at a constant temperature and the edges are free to expand, the stress in the plate will be zero everywhere, independent of the temperature.

So, imagine that you draw the circle, then change the temperature of the plate, then cut out the hole around the expanded or contracted shape of the line. Since the stress in the plate was zero everywhere before cutting, it will stay zero after cutting, and cutting the hole wll not change the shape of the plate.

Or more mathematically:

If there is no elastic stress in the plate, the strain field is ##\epsilon_{xx} = \epsilon_{yy} = \epsilon_{zz} = \alpha \Delta T## and the shear strains are all zero, where ##\alpha## is the corefficient of expansion and ##\Delta T## the temperature change.

Because the strain field is symmetrical in x y and z, this means that ANY two points that were originally a distance ##d## apart become a distance ##d(1 + \alpha \Delta T)## apart, indepedent of the shape of the object, and whether or not it contains holes.

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