If your chinchilla is fur chewing you will notice bald spots from where your chinchilla has been biting its fur. A number of things can cause this behavior, such as stress, too small of a cage, boredom, not getting enough to eat, sleep, introducing a new mate, or removing an old mate. You can try a few different things to cure this: you can add more toys, increase its intake of food,  move the mate out of the cage, or back into the cage, maybe spend more time with it, & or let it have more freedom out of the cage. Try different things to see if you can resolve the under laying issues.

Fur chewing is also a genetic chinchilla trait, so you will want to refrain from ANY breeding of this animal.