Methods, cleavers, and packagings for cleaving an optical fiber using an abrasive medium are disclosed. In one embodiment, the bladeless cleaver includes a body having a flexible tongue configured to receive an optical fiber. The flexible tongue is further configured to provide an arcuate surface to bend a portion of the optical fiber. The bladeless cleaver in this embodiment also includes a cleaver structure attached to the body that comprises an abrasive medium carrier configured to support an abrasive medium. The abrasive medium carrier is configured to be actuated to place the abrasive medium in contact with the portion of the optical fiber to create a flaw in the portion of the optical fiber.