Metallurgy of Miura-ori: lattice theory for inhomogeneous deformations of origami tessellations

In nature, as well as in art, one often encounters thin materials that have been deformed by their environment or their creator into complex folded states; examples include the folds of the endoplasmic reticulum, the villi in the intestinal tract, and tessellated patterns in the ancient Japanese art of origami. One (engineering) advantage of creating a folded structure is that the geometric constraints associated with creasing imbues the construction with exotic mechanical properties, such as generating a material with a negative Poisson's ratio. Materials exhibiting novel behavior of this type, arising from the special properties of the unit cell, are generally classified as metamaterials. In this talk I consider a mechanical metamaterial known as Miura-ori, an origami tessellation pattern that displays soft modes and crystallographic defects not accounted for by a purely geometric theory of an infinitely thin material. I will discuss a method for deriving how inhomogeneous deformations arise from bending within Miura-ori, and show that this leads to a natural coherence length over which the inhomogeneity decays. Additionally, I will show how the modular nature of origami unit cells lends additional richness to the mechanical properties associated with deformation.