Is there a way to weld two pieces of metal together and not have it visible from either side after it's grinded down on both sides? Will it compromise the strength of the weld? I'm not too familiar with welding but I was curious about this.

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Sure you can, one way is hammer welding. This is where you weld a short weld (usually with a gas or TIG) and then you hammer it in an on-dolly fashion till cool. You can metal finish both sides if you have the skills.

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