  Creases are purposely introduced to thin structures for designing deployable
origami, artistic geometries, and functional structures with tunable nonlinear
mechanics. Modeling the mechanics of creased structures is challenging because
creases introduce geometric discontinuity and often have complex mechanical
responses due to the local material damage. In this work, we propose a
continuous description of the sharp geometry of creases and apply it to the
study of \emph{creased annuli}, made by introducing radial creases to annular
strips with the creases annealed to behave elastically. We find creased annuli
have generic bistability and can be folded into various compact shapes,
depending on the crease pattern and the overcurvature of the flat annulus. We
use a regularized Dirac delta function (RDDF) to describe the geometry of a
crease, with the finite spike of the RDDF capturing the localized curvature.
Together with anisotropic rod theory, we solve the nonlinear mechanics of
creased annuli, with its stability determined by the standard conjugate point
test. We find excellent agreement between precision tabletop models, numerical
predictions from our analytical framework, and modeling results from finite
element simulations. We further show that by varying the rest curvature of the
thin strip, dynamic switches between different states of creased annuli can be
achieved, which could inspire the design of novel deployable and morphable
structures. We believe our smooth description of discontinuous geometries will
benefit to the mechanical modeling and design of a wide spectrum of engineering
structures that embrace geometric and material discontinuities.
