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Say you have an iron (or whatever strong material) ball filled with water, and you leave it in a frozen environment (say $-100\ \mathrm{^\circ C}$); once it gets cold enough, the water will want to freeze, and if the ball is weak enough the ball will crack open when it does. But if the ball is strong enough, as far as I know the water won't be able to freeze (assume that the ball is perfectly filled to capacity with water).

If the water can't freeze, then what happens to the energy in the water? Does the water stay relatively warm indefinitely?

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    $\begingroup$ Thermal equilibrium does not depend on the state of the water, as far as I know? Note that the freezing point of water depends on its pressure as well as its temperature. $\endgroup$ – Aaron Stevens Oct 19 at 17:58
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As you point out, if expansion of the water is prevented then the water cannot proceed through the phase transition.

It itself that is not a problem; there is the phenomenon of supercooled water. The state of supercooled water is that the temperature of the water is below its freezing point, but it hasn't crystalyzed. To some extent the crystalization needs a trigger. Under normal circumstances there is always something in the water that will act as a nucleus, and once crystalization has started it's a runaway proccess.

In this case we take it of course that the conditions for nucleation are there and that it is the confinement that prevents the expansion, preventing crystalization of all of the water.

For comparison, imagine the water is in a ball made of a material that has good elasticity; the material deforms readily.
As the water/ice expands the material stretches, and just as any other elastic material the elastic deformation acts as a repository of potential energy. That is, when the ice melts the elastic material contracts again, and that contraction releases the stored elastic energy.

A very familiar example of storing energy in an elastic material is the way a slingshot works.

If an elastic material is very stiff then a small draw stores a lot of energy. In general, the amount of stored energy is proportional to the force times the displacement.

This brings us to the material that you are asking about: a material strong enough to resist the pressure of the water inside with very little deformation.

A high tensile steel well stretch only a tiny amount. But the force required to stretch the steel that tiny amount is huge. That is how a high tensile steel sphere will store a considerable amount of elastic energy.

Well, what if you use a material with even higher tensile strength? That material will stretch even less, but the amount of energy stored will be comparable because the elastic force will be that much stronger.

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