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Something interesting to note is that hardening steel does not significantly change the stiffness of the material. A spring or sword that is heat treated to a soft condition (annealed) is not more floppy than the object in the normal hardened condition.

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I googled the question, and found a fascinating discussion by academics from around the world here https://www.researchgate.net/post/Will_the_Youngs_Modulus_E_...

Not only does the Young's modulus hardly change between soft and hard steel, it barely changes for any steel. This provides an interesting constraint for some types of engineering. For example, if you are making a bridge you can use better steel to make it stronger, but you can't use better steel to make the bridge have less deflection for a given load. You can only change deflection by engineering the shape and design of the bridge.

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