Not a metallurgist, so I may not be explaining this correctly, but...brass will harden just from sitting. Unlike most metals brass gets harder just over time. There is probably some "molecular level" change or better explainable reason for it, but I never got quite that far into it other than to know that it seems like time alone will make brass hard. I am not sure why it is referred to as "air" hardening, other than it is just sitting in air??? I believe if I remember right that air hardened brass to a point can be annealed back to being as soft as the alloy can be or will allow. There is also a point over time that it cannot be recovered by annealing.
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This is interesting to me. I'm not a metallurgist either; however, I did attend a number of metallurgy classes during my formal education as an engineer. Of course, one of the metals we studied was brass. Now, many years later, I find myself enjoying the shooting sports including hand loading, so I find myself once again studying brass since I anneal all my cases.
I'm aware that there are some alloys of copper known as "precipitation hardening alloys" used mainly where particular electrical or thermodynamic properties are important. These alloys often contain metals other than zinc such as chromium, beryllium, nickel, silicon, and zirconium. While some aluminum alloys are precipitation hardened at ambient temperatures, as far as I know copper alloys are normally precipitation hardened at elevated temperatures of several hundred degrees. This is often called "age" hardening since it happens over a relatively long period of time, but not at room temperature.
But the point is moot since the brass we use is a simple copper/zinc alloy in the proportions of 70%/30%; commonly called "cartridge brass" for obvious reasons. It is not one of the precipitation hardening alloys.
Once again I must emphasize that although my formal training in metallurgy was more than most college students receive, it did not qualify me to be called a metallurgist. However, as far as I know, our cartridge brass will not become harder with age.
If I am wrong, and cartridge brass indeed becomes harder with age, I would appreciate someone pointing me to a scientific study. Surely if this were true, someone has studied this phenomenon and reported on it along with the mechanism causing it. Copper alloys have been around since the end of the stone age and I would guess have received more study than any other metal. If ordinary 70/30 brass got harder over a period of years by just sitting on the shelf, I would have thought this would have become common knowledge.
Apparently there is some anecdotal evidence that old cartridges can develop cracks. I suspect these cracked cartridge cases are the result of what is called "stress corrosion cracking". It can occur in several metals, brass being one of them, if they are simultaneously subjected to stress and chemical attack. This is not the same thing as "age" hardening, but I can see how someone might mistakenly think that cartridge case neck cracks could be caused by hardening of the brass.
As far as the term "air hardening", I am only familiar with that term as it applies to certain steels which can be hardened by cooling in air. In other words, unlike some steel alloys which are quenched in oil or water, "air hardening" steels are cooled from above their transformation temperature by simply allowing them to cool down in ordinary ambient air. But the hardening of steel is completely different from the way brass is hardened.