So I know that 301 stainless steel was chosen for the SpaceX Starship rocket as it can handle high temperature differences.
What about this alloy makes the steel harden when it gets colder?
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301 is the lowest alloy content of the "18-8" family. The slightly lower nickel ( compared to 304) makes the austenite less stable so when cold rolled it work hardens more than 304. It is still basically austenitic so does not become brittle at low temperatures. So it was chosen for the higher strength ( when cold worked) at all temperatures , relative to 304. Austenite is non-magnetic but as 301 is cold rolled it becomes somewhat magnetic depending on the amount of cold work. Presumable caused by formation of tiny grains of martensite ( the magnetic , strong, crystal of the iron family). The small martensite grains block the slip planes in the austenite matric making it stronger.