Effective linear wave motion in periodic origami structures

We establish a dynamic homogenization framework catering for the linear elastic wave motion in periodic origami structures. The latter are modeled via “bar-and-hinge” paradigm where: (i) the folding of the structure and the bending of individual panels are modeled via elastic hinges, and (ii) the in-plane deformation of each panel is modeled with elastic bars. Using the so-formulated discrete model of an origami structure, we pursue finite wavenumber-finite frequency (FW-FF) homogenization of the wave motion in a spectral neighborhood of simple, repeated, and nearby eigenfrequencies at an arbitrary wavenumber within the first Brillouin zone. The lynchpin of the proposed approach is the “projection” of the nodal displacements over each unit cell onto a suitable Bloch eigenvector, evaluated at the “center” of the spectral region of interest. For completeness, we make an account for: (i) the source term acting at the nodes of a discrete structure, and (ii) periodic Dirichlet boundary conditions. We obtain the leading-order (system of) effective equation(s) synthesizing the wave motion in a selected spectral neighborhood, and we describe asymptotically the corresponding dispersion relationship. We illustrate the proposed framework by comparing numerically the Bloch dispersion relationship to its asymptotic approximation for (a) a 2D-periodic Miura-ori structure, and (b) a 1D-periodic Miura tube. The dispersion analysis is complemented by evaluating the effective wave motion (in terms of both “macroscopic” and “microscopic” essentials) in a 2D-periodic Miura-ori structure due to spatially-localized source term acting either inside a band gap or within a passband.