The shape and mechanics of curved-fold origami structures

IMG_2077We develop recursion equations to describe the three-dimensional shape of a sheet upon which a series of concentric curved folds have been inscribed.In the case of no stretching outside the fold, the three-dimensional shape of a single fold prescribes the shape of the entire origami structure. To better explore these structures, we derive continuum equations, valid in the limit of vanishing spacing between folds, to describe the smooth surface intersecting all the mountain folds. We find that this surface has negative Gaussian curvature with magnitude equal to the square of the fold’s torsion. A series of open folds with constant fold angle generate a helicoid.

M. A. Dias and C. D. Santangelo EPL (2012) Arxiv and Journal.

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Geometric mechanics of curved crease origami

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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.

M. A. Dias, L. H. Dudte, L. Mahadevan, and C. D. Santangelo, PRL (2012) ArxivJournal, and Supplemental Material.

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Programmed buckling by controlled lateral swelling in a thin elastic sheet

Drum1Recent experiments have imposed controlled swelling patterns on thin polymer films, which subsequently buckle into three-dimensional shapes. We develop a solution to the design problem suggested by such systems, namely, if and how one can generate particular three-dimensional shapes from thin elastic sheets by mere imposition of a two-dimensional pattern of locally isotropic growth. Not every shape is possible. Several types of obstruction can arise, some of which depend on the sheet thickness. We provide some examples using the axisymmetric form of the problem, which is analytically tractable.

M. A. Dias, J. A. Hanna and C. D. Santangelo, PRE (2011) Arxiv and Journal.