Blue Adamantine Shard is an in important resource in Immortals Fenyx Rising. It is used to upgrade your weapons, armor, and helmets, giving you better stats in combat, and making fights easier to manage as you get into the tougher areas of the game.
There are multiple ways to farm this resource, and as long as you are concious of grabbing it where you can, you should always have enough to any upgrades that you need.
Every common enemy that you kill in the game will drop a small amount of Blue Adamantine Shard when they die. This is good, as you will be fighting a lot of enemies, making them a constant source througout the game.
Opening chests, many of which will be well hidden, is a great way to find this resource. You can find chests as you explore the world, and using your farsight by pressing the right thumbstick and then scanning the area can reveal more locations.
Blue Shard Clusters
As you explore the world you will find large blue crystals, called Blue Shard Clusters. Attack these with a weapon will shatter them, netting you a large amount of Blue Adamantine Shard.
Chopping bushes and grass
This is an odd one, but hacking away bushes and grasses will drop very small amount of the resource. Not great for farming, but you might as well use this method to add value as you travel the world.