In 5e RAW, there are no generic rules for the effects of crafting armor, shields or weapons from special materials, except the following two:
- Silvering a weapon or ammunition (PHB page 148): Good to bypass the immunity or resistance of some monsters to nonmnagical weapons.
- Adamantine weapons (XGtE page 78): Unusually effective when used to break objects; each hit to objects is considered critical.
The text in XGtE also seems to preclude any benefits from applying adamantine to armor or shields:
Adamantine is an ultrahard metal found in meteorites and extraordinary mineral veins. In addition to being used to craft adamantine armor, the metal is also used for weapons.
This text is important because adamantine armor is described in the DMG as a magic item on its own:
This suit of armor is reinforced with adamantine, one of the hardest substances in existence. While you're wearing it, any critical hit against you becomes a normal hit.
So we do not get any direct benefits from building an armor from adamantine, the benefits are only present when we craft a magical armor reinforced with adamantine. And those benefits will disappear for example when we are in an anti-magic field; as you can read in twitter posts by Jeremy Crawford dating back from 8 Nov 2017:
Adamantine armor is a magic item. It follows the rules for such items, including the text in antimagic field on magic items.
The same applies for mithral armor as well.