Resistance welding with copper electrodes is the standard way to spot-weld steel; for high duty cycles they are water-cooled, but low duty cycles are often just solid copper. You’d think this would totally fail because copper melts at 1084.62° while steel typically melts around 1500°, so the copper would melt long before the steel. In fact, though, at room temperature, copper has an electrical resistivity of about 16.8 nΩ m, while 1010 carbon steel is more like 143 nΩ m, and copper’s thermal conductivity of 400 W/m/K is also greater than carbon steels’ of around 30–100 W/m/K.
So the same current density running through copper and steel generates 8.5 times as much heat per unit volume in the copper, and the copper can conduct it away from the point of generation 4–12 times as fast, so the temperature rise in the copper tends to be about 30–100 times less. These numbers change at higher temperatures but I think the overall tendency remains the same. In EDM, even with the ultra-high temperatures of the arcs, copper electrodes are considered to be “free of wear”.
Of course the heat equation tells us that the copper and steel in contact immediately form a continuous temperature distribution, so what you’re really doing is melting steel under the surface, while the surface steel remains solid, chilled by the copper to under 1000°.
It occurs to me that you could use this same approach for forming the steel instead of welding pieces of it together: by locally melting it with a pulse of current, it can be formed by pressing even a soft copper ball into it. By doing this repeatedly while moving the copper ball around to precise positions in three dimensions, you can achieve arbitrarily complex surface geometry without the high side loads and noise of a mill or lathe, and regardless of how hard the steel is. If it’s a high-carbon steel, the forming process will inherently case-harden the product, as each melt is quenched by the mass of the steel. With proper planning of the toolpaths, especially the finishing toolpath, it should be possible to keep heat-induced distortion of the workpiece small by keeping the heating very localized.