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Simulation approaches from computer graphics were some of the first to study the complex geometries and resultant elastic behaviors of curved creased origami. In this work, we present an extension of these approaches by adapting a simplified, discrete, and physics-based bar and hinge approach to simulate the mechanics of these intriguing origami structures. This model can capture stretching and shearing of the thin sheet, bending of the sheet along principle curvature directions, and bending along the prescribed curved creases. With insight from differential geometry, we use the model to explore the elastic folding sequence, the stiffness, and the global large deformation mechanics of curved crease origami.