Upcoming mobile devices will have flexible displays, allowing us to explore alternate forms of user authentication. On flexible displays, users can interact with the device by deforming the surface of the display through bending. In this paper, we present Bend Passwords, a new type of user authentication that uses bend gestures as its input modality. We ran three user studies to evaluate the usability and security of Bend Passwords and compared it to PINs on a mobile phone. Our first two studies evaluated the creation and memorability of user-chosen and system assigned passwords. The third study looked at the security problem of shoulder-surfing passwords on mobile devices. Our results show that bend passwords are a promising authentication mechanism for flexible display devices. We provide eight design recommendations for implementing Bend Passwords on flexible display devices.