What is melting integral I^2t
What they don't teach you in school is .... that a fuse also has a thermal electrical-resistance coefficient where R rises with temperature 1:10 near 3000'K and has a thermal mass, velocity and thermal resistance coefficient that are also dynamic in times &lt; 0.1 second  so there are two power series trends for times.

In fuses  the \$I^2Rt\$ term is not accurate to determine fusing Energy because R rises with temperature but more importantly the cooling effects of longer times affects how much energy is needed

Due to the thermal time constant of the part, the self-cooling effect (Adiabatic threshold)  and non-linear thermal resistance (PTC effect) changes the power series coefficient.

Following example selected for 1A SMD 1206 fuse     


Notice the exponent of energy changes due to log-log scale linear slope.

What does this mean?   

For time > thermal time constant self-cooling demands more energy to raise resistance times current^2 to reach melting temperature of metal conductor and open fuse.

This is the steep curve for time > 0.1s.

Fuses are rated for the holding current within a certain temperature range and not the breaking current.