Geometric Mechanics of Curved Crease Origami

Folding a sheet of paper along a curve can lead to structures seen in decorative art and utilitarian packing boxes. Here we present a theory for the simplest such structure: an annular circular strip that is folded along a central circular curve to form a three-dimensional buckled structure driven by geometrical frustration. We quantify this shape in terms of the radius of the circle, the dihedral angle of the fold, and the mechanical properties of the sheet of paper and the fold itself. When the sheet is isometrically deformed everywhere except along the fold itself, stiff folds result in creases with constant curvature and oscillatory torsion. However, relatively softer folds inherit the broken symmetry of the buckled shape with oscillatory curvature and torsion. Our asymptotic analysis of the isometrically deformed state is corroborated by numerical simulations that allow us to generalize our analysis to study structures with multiple curved creases.

Figure 1

A photograph of the model built by cutting a flat annulus of width $2w$ from a flat sheet of paper with central circle of radius $r$. (a) Folding along its center line buckles the structure out of plane. However, if we cut the annulus, (b), the structure collapses to an overlapping planar state, with curvature given by Eq. (1). (c) The inset shows a cross section of the fold, where the right and the left planes, ${\mathbf{X}}_{+}$ and ${\mathbf{X}}_{-}$, define the dihedral angle $\theta$.

Figure 2

(a) Perturbative fold of width $\omega =0.1$ and $\varsigma =2/\sqrt{3}$ shaded by mean curvature. The generators are indicated by the lines on the surface. The inset shows the dimensionless torsion and curvature of the crease. (b) A simulated fold of width $\omega \approx .0994$ shaded by local area change relative to the flat state.

Figure 3

(a) Angle differences $|\theta (\pi /2)-\theta (0)|$ as a function of $\omega$ with ${\theta }_0=2\pi /3$. The curve with diamonds are computed from the first-order perturbation theory with $\varsigma =2/\sqrt{3}$ and ${\theta }_0=2\pi /3$. The numerical simulations are shown in gray with $\varsigma =2\sqrt{3}$ (open circles) and $\varsigma =160/\sqrt{3}$ (open squares), and are compared with nonperturbative variational ansatz, ${\kappa }_{(2)}$ and ${\tau }_{(2)}$, described in the text with $\varsigma =\sqrt{3}/40$ (circles) and $\varsigma =\sqrt{3}/2$ (squares). Corresponding energy landscapes, as a function of $\theta (0)$ and $\theta (\pi /2)$, respectively, are shown for (b) $\omega =0.01$, (c) 0.05, and (d) 0.1, with energy minima drawn as white dots.

Figure 4

(a) A plastic model of six circular folds generated by perforating the folds at equal intervals with a laser cutter. The ratio of outer to inner boundary is 2. (b) A simulation result with the same planar geometry as the plastic model shaded by local area change. Multiple fold simulation has the same magnitude order of area change as that of single fold simulation and agrees visually with the physical model in (a).