Physical Meaning of Joule Integral

The Joule integral is actually used to characterize fuses. There are two extremes for specifying the current-carrying capability of a conductor.

a) Over a long time, all heat is lost to the environment. The current is given for some permitted temperature rise. Typically this is the temperature rating of the insulation.

b) Over a very short time, no heat is lost to the environment. This is the so-called adiabatic case.

As the power generated in a conductor is $I^2R$, the heat energy deposited is $I^2Rt$. For any given conductor, R is constant, so the energy (per unit resistance) to reach some temperature for the short-time case is usually given as $I^2t$.

For a fuse, typical temperatures are the 'guaranteed to still work' temperature and the 'guaranteed to break' temperature or melting temperature.

When you're protecting something else that has a quoted survival of $I^2t$, for instance a rectifier diode, you would want the rating of the diode to exceed the 'guaranteed to break' rating of the fuse, also known as the 'let-through' energy.

That's the reason we define $I^2t$ to express the amount of energy (per resistance) required to actuate the fuse.