One-dimensional von Karman models for elastic ribbons

By means of a variational approach we rigorously deduce three one-dimensional models for elastic ribbons from the theory of von Karman plates, passing to the limit as the width of the plate goes to zero. The one-dimensional model found starting from the “linearized” von Karman energy corresponds to that of a linearly elastic beam that can twist but can deform in just one plane; while the model found from the von Karman energy is a non-linear model that comprises stretching, bendings, and twisting. The “constrained” von Karman energy, instead, leads to a new Sadowsky type of model.