New or vintage shotgun barrels, made with the appropriate steel that is free of defects, do not "fatigue" unless stressed above the yield strength, by loads for which they were not designed, obstruction, or corrosion/pitting.

We have published data from the gun makers regarding yield and tensile strength, and a good collection of actual testing for vintage pattern welded, decarbonized and fluid steel barrels
https://docs.google.com/document/d/1cvqRzkg0wEjhAAcFWr8gFi7aPFRsSIJ_hahfDxmrNAU/edit
https://docs.google.com/document/d/1dnRLZgcuHfx7uFOHvHCUGnGFiLiset-DTTEK8OtPYVA/edit