Annealing bellows is more common in Japan than in the U.S. and Europe. In radial outward forming a small thinning occurs but this is compensated by an increase in yield strength due to cold work. When the bellows are subsequently annealed, the increase in yield is eliminated. As-formed yield strength is usually double (or more) the specification minimum annealed strength. Increased yield results in increased convolution deformation pressure and in-plane squirm pressure increase. Also, plastic strain due to deflection is reduced and this results in increased fatigue life.
The argument for annealing usually revolves about stress corrosion cracking. However, tests have shown that the threshold of stress required for stress corrosion is very low and the normal pressure stress in an annealed bellows exceeds this level.
“The work hardening of austenitic stainless steel induced during the forming of convolutions generally improves the fatigue life of an expansion joint, often to a marked degree; thus it is not normally considered beneficial to either stress relieve or anneal after forming.” (1985 EJMA Standards, for example para C-5.15)
Appendix B also contains a good article on stress corrosion cracking.
Informing high yield strength materials such as Inconel 625, an interstage anneal may be required. However, annealing is not recommended after forming except in special cases.