Destruction Stone Fragments are fragments of ore filled with Chaos Energy, that are used to upgrade your weapons in Lost Ark. These are used in the upgrade stage after Harmony Shards are used to increase the EXP to the max. Then, you combine Harmony Leap Stones with Guardian Stone Fragments or Destruction Stone Fragments to upgrade the item, giving it a huge boost.
The main source of Destruction Stone Fragments in Lost Ark is through Chaos Dungeons. They are obtainable after your first two runs as well as from Guardian Raid runs. However, are these Destruction Stone Fragments grindable in Lost Ark?
How to Farm Lost Ark Destruction Stone Fragments
In Lost Ark, you can obtain various items through trading, and this also includes the Destruction Stone Fragments.
In a Reddit post discussing the idea of farming the Fragments in Lost Ark, one user suggests farming plenty of other materials after your runs in Lost Ark and using those to trade for Destruction Stone Fragments. There is an NPC South of the Chaos Dungeon where you finish your daily runs, which you can trade with in exchange for the Destruction Stone Fragments.
This looks to be the only way to actually "farm" the Destruction Stone Fragments in Lost Ark. The stones fragments are tough to get in the game, but when you have some, they provide great boosts to your arsenal of weapons.