Novel mechanical response of parallelogram-based origami governed by topological characteristics

Parallelogram-based origami sheets like the Miura-ori are archetypal flexible mechanical metamaterials. However, studies of their mechanical response typically either examine homogeneous properties such as the Poisson ratio or model the system as a continuum elastic sheet. In this talk, we use reciprocal-space methods to decompose the response to non-uniform loading into different periodic modes. Drawing on an analogy to quantum systems, we identify two classes of origami, governed by a topological invariant induced by a sheet's symmetries. In one class, origami response is dominated by long-wavelength modes, as in conventional systems, but in the other there are doubly degenerate low-energy modes at all wavelengths, leading to dramatically lower stiffness and to sharply nonuniform deformations. As supported by simulation, these theoretical predictions can be observed even in realistic origami sheets subject to open boundary conditions, finite-size effects and significant nonlinearities. As origami sheets continue to achieve more controlled geometries, these new phenomena represent a powerful new way of controlling origami response.