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High-frequency currents induce skin-effect resistance that is used on some applications to weld metals.

Assume you have two steel rectangles of dimension $h_0, w, l$ and $h_1, w, l$ placed together along the $h$ axis. I'm wondering if placing a strong enough current through appropriate probes in the transverse faces of the rectangle, will generate enough heat to weld the touching faces uniformly, with no sticks or rods involved

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Look up ERW ( electric resistance welding ) , this is commonly used to weld steel pipe and tubes ( not necessarily round). Spot welding and projection welds have similar aspects.

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