Curvature tuning in folded strips through hyperstatic applied rotations

Abstract

Folding a strip of paper generates extremely localized plastic strains. The relaxation of the residual stresses results in a ridge that joins two flat faces at an angle known as the dihedral angle. When constrained isostatically, the strip will be at its undeformed roof-like state. Instead, if confined hyperstatically, the flat faces will undergo bending. We demonstrate that the generated curvatures can change their sign with appropriate rotations applied at the ends. We use Euler's theory of the Elastica and a shooting method to match the applied rotations at the boundaries. We also consider a constitutive model for the discontinuous rotation that takes into account the initial dihedral angle and the rotational stiffness of the fold. We show that the curvatures on the left and the right of the fold change according to a law also confirmed by the Euler-Bernoulli beam theory for small displacements and rotations. For opposite applied rotations, the fold disappears in the limit of zero rotational stiffness; instead, for applied rotations of the same sign, there exists a theoretical non-zero critical rotational stiffness that neutralizes the fold. Below such critical value, the fold can mutate, for example, from a mountain to a valley fold.