Joul-heat melting of a metallic wrench - why is it so weird? I found a video of a wrench that melts when put between two cables. 
The current is high and the wrench starts to heat-up and melt.
But WHY does it heats up quickly near the cables connection points and slower in the middle? (shown on the photo below)
I always thought that the Joul heat is because of electron movement. And that the electrons in a wrench will start all moving at once, when attached to a voltage source. Why does heating start near the ends?

 A: There is a discontinuity in the circuit (air gap, oxide layers, material difference, etc.) where the wrench is applied to the battery terminals. Since this is not a perfect connection, these areas will both experience higher resistance than the rest of the wrench-battery circuit. 
More resistance leads to more heating locally, which in turn leads to more resistance. This feedback loop continues until the wrench melts to disrupt the circuit or the battery runs flat.
A: Yes, all electrons move at the same time and the current along the length of the wrench is the same, but the heating effect of the current depends on the resistance of a conductor: the higher the resistance, the greater the heating ($I^2R$).
My best guess is that the cross-section of the handle near the ends is slightly smaller than the cross-section in the middle and therefore its resistance (per unit length) in those areas is slightly higher. The resistance there could be higher for some other reasons, that may have something to do with the way the wrench was made, but I don't have any data to support it. 
This would cause somewhat greater initial heating of those areas, which will raise their temperature and increase their resistance (as a function of temperature) faster than in the middle, which will further accelerate their heating, etc. 
In other words, due to this positive feedback effect, the areas near the sockets, presumably, with a slightly higher initial resistance, will be heated to a disproportionately higher temperatures.
A similar phenomenon could be observed in filaments of incandescent bulbs which tend to develop hot spots in the sections that are initially slightly thinner than others. 
A: Let us define three parts of the wrench: two ends and the "bar". At the sections separating the ends from the bar, the cross-section is decreasing in the directions from the ends to the bar. That means that near these sections, the current density at the surface of the bar is higher than at the surface at a section somewhere in the middle of the bar, as the current density distribution transitions from the wider distribution in the ends to the uniform distribution in the middle of the bar. Therefore, the surface of the bar heats faster near the sections separating the ends from the bar.  
