Lattice mechanics of origami tessellations

Origami-based design holds promise for developing materials whose mechanical properties are tuned by crease patterns introduced to thin sheets. Although there have been heuristic developments in constructing patterns with desirable qualities, the bridge between origami and physics has yet to be fully developed. To truly consider origami structures as a class of materials, methods akin to solid mechanics need to be developed to understand their long-wavelength behavior. We introduce here a lattice theory for examining the mechanics of origami tessellations in terms of the topology of their crease pattern and the relationship between the folds at each vertex. This formulation provides a general method for associating mechanical properties with periodic folded structures and allows for a concrete connection between more conventional materials and the mechanical metamaterials constructed using origami-based design.

Figure 1

(a) Graph for a single vertex. This degree-6 vertex has its graph determined by the six sector angles ${\alpha }_i$. Each crease has a dihedral angle $f_i$ associated with it. In the flat case every $f_i=\pi$ or, equivalently, every fold angle is identically zero, since the fold angle is defined as the supplement of the dihedral angle. (b) By assigning fold angles to each crease, a three-dimensional embedding of the vertex (i.e., the folded form of the origami) is fully determined. Every face must rotate rigidly about the defined creases and the sector angles must remain constant. There is a limited set of fold angles that will solve these conditions. (c) Schematic projection of the curve of intersection between the unit sphere and the folded form origami. For an $N$-degree vertex this projection generates a spherical $N$-gon. To proceed, the $N$-gon is divided into $N-2$ spherical triangles and the interior angles (i.e., the $f_i$) follow as a result of applying the rules of spherical trigonometry. All three-dimensional origami structures are visualized using Tessellatica, a freely available online package for Mathematica [34].

Figure 2

(a) Two degree-4 vertices with labeled folds. (b) Graph for the crease pattern consisting of these two vertices contains a single crease that is shared by both vertices. In this case the constraint equation $\mathbf{D}\mathcal{F}=\mathit{0}$ simply becomes the scalar relationship $f_1^1=f_3^2$.

Figure 3

(a) While the crease pattern of a Miura-ori generally introduces only four folds per vertex, the bending of faces acts to allow two extra folds per vertex, so the crease pattern we consider is a triangulated lattice. At each vertex the dihedral angles contained in $\mathbf{f}$ are determined by specifying the state vector $\mathbf{s}$ and satisfying the geometric constraints. (b) Single-vertex origami with enclosing sphere to visualize the constraints between $\mathbf{f}$ and $\mathbf{s}$.

Figure 4

(a) Miura-ori, without the assignment of mountain and valley folds, has a simple directed graph structure with a unit cell composed of four vertices. By tessellating these four vertices, the entire pattern emerges. Note that the tessellation is rectangular, with lattice vectors ${\mathbf{a}}_1=a\widehat{\mathbf{x}}$ and ${\mathbf{a}}_2=b\widehat{\mathbf{y}}$. (b) Each vertex has six folds, labeled in the fashion shown here. (c) In Fourier space, translations associated with connecting these folds together throughout the tessellation merely amounts to a phase factor associated with the appropriate wave number and lattice vector. Shown on the left is translating in the $x$ direction. The middle shows translating in the $y$ direction. The right shows that connecting the extra folds involves a diagonal translation across the unit cell. Note that the five internal folds have a phase factor identically equal to one.

Figure 5

Shapes and energy eigenvalues for the three uniform modes for $\epsilon =\pi /2$ and $\alpha =\pi /3$. (a) The three uniform null vectors correspond to a uniform mode (I), a twisting mode (II), and a saddle mode (III). These are identical to the modes determined numerically in previous studies [6, 12]. (b) Eigenvalues associated with each of the three bulk modes as a function of face stiffness $\mathrm{\Gamma }$. Note that over a wide range the softest mode is the twisting mode (II), since it involves purely face bending.

Figure 6

Experimental observations of deformation localization in an $8\times 8$ Miura-ori tessellation. (a) An undeformed Miura-ori shows a regular periodic pattern. Under (b) small deformations, (c) large deformations, and (d) in the presence of a pop-through defect (PTD) [7], the lattice distorts to accommodate the induced strain. (e) Qualitatively, the amount of deformation localization can be easily seen by a simple image subtraction between the deformed and undeformed state. (f) Measuring strain along the horizontal axis as a function of unit cell position $n$ relative to the location of the disturbance shows a rapid decay for all three scenarios (points). For small and large amplitudes, the decays can be readily fit to an exponential function with decay length $\ell$ [upper (red) and lower (black) lines], whereas for a PTD, the decay length can be estimated to within $100\%$. Because the PTD induces an extensional distortion rather than a compression, the strain is oppositely signed. The inset is a plot of the decay length against an approximate measure of the distortion wave vector $q$ showing that the larger wave vector decays much more rapidly than the shorter wave vectors. Within error bars, this measurement is consistent with an inverse relationship between decay length and wave vector. The solid line is the theoretical prediction from Eq. (24) for $\epsilon =\pi /2$ and $\alpha =\pi /3$.

Figure 7

Eigenvalues and mode shapes as a function of wave number for a given $\mathrm{\Gamma }$. (a) Shown on the left is the mode structure for $\mathrm{\Gamma }=0.1$, 1, and 10, with $\epsilon =\pi /2$ and $\alpha =\pi /3$. At long wavelengths the saddle mode $\mathcal{I}$ is the stiffest for a wide range of $q$, since it involves both bending of the faces and deformation of the angles away from the reference state. Shown on the right is the mode structure for $\mathrm{\Gamma }=0.1$, 1, and 10, with $\epsilon =\pi /2$ and $\alpha =9\pi /20$. The labeled modes are in ascending order from largest eigenvalue (mode I) to lowest eigenvalue (mode IV). (b) Visualization of the basic modes for $q=\pi /6$.